Subgroups — C₂×#23777249 #35196217

Order 20224.

7994 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23777249	C₂	1	32	yes	maximal, direct product factor
2	C₂×#19099295	C₂	1	4	yes	maximal, complemented
3	C₂×#19099287	C₂	1	2	yes	maximal, complemented
4	C₂×#19099278	C₂	1	2	yes	maximal, complemented
5	C₂×#19099259	C₂	1	4	yes	maximal, complemented
6	C₂×#19099233	C₂	1	2	yes	maximal, complemented
7	C₂×#19099217	C₂	1	2	yes	maximal, complemented
8	C₂×#19099150	C₂	1	4	yes	maximal, complemented
9	C₂×#19099127	C₂	1	2	yes	maximal, complemented
10	C₂×#19099120	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#19099073	C₂	1	4	yes	maximal, complemented
12	C₂×#19099046	C₂	1	2	yes	maximal, complemented
13	C₂²×C₄×D₃₁₆	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×#532×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂²×#11929623	C₂²	1	4	yes	complemented
16	C₂³×D₃₁₆	C₂²	1	1	yes	characteristic
17	C₂³×C₄×D₇₉	C₂²	1	2	yes	—
18	#19099295	C₂²	1	64	yes	complemented
19	#19099287	C₂²	1	32	yes	complemented
20	#19099278	C₂²	1	32	yes	complemented
21	#19099259	C₂²	1	64	yes	complemented
22	#19099233	C₂²	1	32	yes	complemented
23	#19099217	C₂²	1	32	yes	complemented
24	#19099150	C₂²	1	64	yes	complemented
25	#19099127	C₂²	1	32	yes	complemented
26	#19099120	C₂²	1	16	yes	complemented
27	C₂×#14579161	C₂²	1	1	yes	characteristic, complemented
28	C₂×#14579161	C₂²	1	1	yes	characteristic, complemented
29	C₂×#14579157	C₂²	1	4	yes	—
30	C₂×#14579156	C₂²	1	8	yes	complemented
31	C₂×#14579153	C₂²	1	2	yes	complemented
32	C₂×#14579150	C₂²	1	4	yes	—
33	C₂×#14579147	C₂²	1	8	yes	complemented
34	#19099073	C₂²	1	64	yes	complemented
35	C₂×#14579145	C₂²	1	8	yes	complemented
36	C₂×#14579143	C₂²	1	8	yes	complemented
37	C₂×#14579141	C₂²	1	4	yes	complemented
38	C₂×#14579140	C₂²	1	2	yes	—
39	C₂×#14579140	C₂²	1	2	yes	—
40	C₂×#14579135	C₂²	1	4	yes	—
41	C₂×#14579135	C₂²	1	4	yes	—
42	C₂×#14579133	C₂²	1	8	yes	complemented
43	C₂×#14579132	C₂²	1	4	yes	complemented
44	C₂×#14579129	C₂²	1	8	yes	—
45	#19099046	C₂²	1	32	yes	complemented
46	C₂×Q₈×Dic₇₉	C₂²	1	4	yes	complemented
47	C₂×C₄×Dic₁₅₈	C₂²	1	2	yes	complemented
48	C₂×C₄×Dic₁₅₈	C₂²	1	2	yes	complemented
49	C₂×C₄×D₃₁₆	C₂²	1	2	yes	—
50	C₂×C₄×D₃₁₆	C₂²	1	2	yes	—
51	C₂×C₄×D₃₁₆	C₂²	1	2	yes	—
52	C₂×C₄×D₃₁₆	C₂²	1	2	yes	—
53	C₂×C₄×D₃₁₆	C₂²	1	16	yes	complemented
54	C₂×D₄×Dic₇₉	C₂²	1	8	yes	—
55	C₂×D₄×Dic₇₉	C₂²	1	4	yes	complemented
56	C₂×(C₂²⋊D₇₉)⋊C₄	C₂²	1	8	yes	complemented
57	C₂×#14579120	C₂²	1	2	yes	—
58	C₂×#14579120	C₂²	1	4	yes	complemented
59	C₂×#14579119	C₂²	1	4	yes	complemented
60	C₂²×C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
61	C₂²×C₄⋊Dic₇₉	C₂²	1	4	yes	complemented
62	C₂²×(C₂×D₇₉)⋊C₄	C₂²	1	2	yes	—
63	C₂²×#41×C₇₉	C₂²	1	2	yes	—
64	#532×C₇₉	C₂²	1	16	yes	complemented
65	C₂×C₄⋊Q₈×C₇₉	C₂²	1	1	yes	characteristic, complemented
66	C₂×#127×C₇₉	C₂²	1	1	yes	characteristic, complemented
67	C₂×#124×C₇₉	C₂²	1	2	yes	complemented
68	C₂×#121×C₇₉	C₂²	1	2	yes	—
69	C₂×#121×C₇₉	C₂²	1	2	yes	—
70	C₂×C₄×D₄×C₇₉	C₂²	1	2	yes	complemented
71	C₂×C₄×D₄×C₇₉	C₂²	1	2	yes	complemented
72	C₂²×C₄²×C₇₉	C₂²	1	1	yes	characteristic
73	C₂⁴×D₇₉	C₂³	1	2	yes	—
74	C₂×#11929623	—	4	16	no	—
75	C₂×#11929623	C₂³	1	32	yes	complemented
76	C₂²×Dic₁₅₈	C₂³	1	2	yes	—
77	C₂²×Dic₁₅₈	C₂³	1	2	yes	—
78	C₂²×D₃₁₆	—	2	8	no	—
79	C₂²×D₃₁₆	C₂³	1	2	yes	—
80	C₂²×D₃₁₆	C₂³	1	2	yes	—
81	C₂²×D₃₁₆	C₂³	1	8	yes	—
82	C₂²×C₂²⋊D₇₉	C₂³	1	8	yes	—
83	C₂²×C₂²⋊D₇₉	C₂³	1	8	yes	—
84	C₂³×Dic₇₉	C₂³	1	2	yes	—
85	C₂³×Dic₇₉	C₂³	1	8	yes	—
86	C₂²×C₄×D₇₉	—	4	8	no	—
87	C₂²×C₄×D₇₉	C₂³	1	4	yes	—
88	C₂²×C₄×D₇₉	C₂³	1	4	yes	—
89	C₂²×C₄×D₇₉	C₂³	1	16	yes	—
90	#14579161	C₂³	1	8	yes	complemented
91	#14579161	C₂³	1	8	yes	complemented
92	#14579157	C₂³	1	32	yes	—
93	#14579156	C₂³	1	64	yes	complemented
94	#14579153	C₂³	1	16	yes	complemented
95	#14579150	C₂³	1	32	yes	—
96	#14579147	C₂³	1	64	yes	complemented
97	#14579145	C₂³	1	64	yes	complemented
98	#14579143	C₂³	1	64	yes	complemented
99	#14579141	C₂³	1	32	yes	complemented
100	#14579140	C₂³	1	16	yes	—
101	#14579140	C₂³	1	16	yes	—
102	#14579135	C₂³	1	32	yes	—
103	#14579135	C₂³	1	32	yes	—
104	#14579133	C₂³	1	64	yes	complemented
105	#14579132	C₂³	1	32	yes	complemented
106	#14579129	C₂³	1	64	yes	—
107	Q₈×Dic₇₉	C₂³	1	32	yes	complemented
108	C₄×Dic₁₅₈	C₂³	1	16	yes	complemented
109	C₄×Dic₁₅₈	C₂³	1	16	yes	complemented
110	C₄×D₃₁₆	C₂³	1	16	yes	—
111	C₄×D₃₁₆	C₂³	1	16	yes	—
112	C₄×D₃₁₆	C₂³	1	16	yes	—
113	C₄×D₃₁₆	C₂³	1	16	yes	—
114	D₄×Dic₇₉	C₂³	1	64	yes	—
115	D₄×Dic₇₉	C₂³	1	32	yes	complemented
116	(C₂²⋊D₇₉)⋊C₄	C₂³	1	64	yes	complemented
117	#14579120	C₂³	1	16	yes	—
118	#14579120	C₂³	1	32	yes	complemented
119	#14579119	C₂³	1	32	yes	complemented
120	C₂×C₄⋊Dic₇₉	C₂³	1	2	yes	—
121	C₂×C₄⋊Dic₇₉	C₂³	1	2	yes	—
122	C₂×C₄⋊Dic₇₉	C₂³	1	8	yes	—
123	C₂×C₄⋊Dic₇₉	C₂³	1	2	yes	—
124	C₂×C₄⋊Dic₇₉	C₂³	1	8	yes	—
125	C₂×C₄⋊Dic₇₉	C₂³	1	2	yes	—
126	C₂×C₄⋊Dic₇₉	C₂³	1	2	yes	—
127	C₂×C₄⋊Dic₇₉	C₂³	1	2	yes	—
128	C₂×C₄⋊Dic₇₉	C₂³	1	32	yes	complemented
129	C₂×Dic₇₉⋊C₄	C₂³	1	4	yes	—
130	C₂×Dic₇₉⋊C₄	C₂³	1	4	yes	—
131	C₂×C₂²⋊Dic₇₉	C₂³	1	4	yes	—
132	C₂×C₂²⋊Dic₇₉	C₂³	1	8	yes	—
133	C₂×(C₂×D₇₉)⋊C₄	C₂³	1	16	yes	—
134	C₂×(C₂×D₇₉)⋊C₄	C₂³	1	4	yes	—
135	C₂×(C₂×D₇₉)⋊C₄	C₂³	1	4	yes	—
136	C₂×(C₂×D₇₉)⋊C₄	C₂³	1	8	yes	—
137	C₂×(C₂×D₇₉)⋊C₄	C₂³	1	8	yes	—
138	C₂×C₄×Dic₇₉	C₂³	1	8	yes	—
139	C₂×C₄×Dic₇₉	C₂³	1	4	yes	—
140	C₂×#41×C₇₉	—	2	16	no	—
141	C₂×#41×C₇₉	C₂³	1	16	yes	—
142	C₂²×Q₈×C₇₉	C₂³	1	2	yes	—
143	C₂²×D₄×C₇₉	C₂³	1	2	yes	—
144	C₂²×D₄×C₇₉	C₂³	1	4	yes	—
145	C₂²×D₄×C₇₉	C₂³	1	2	yes	—
146	C₂²×D₄×C₇₉	C₂³	1	2	yes	—
147	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
148	C₂³×C₄×C₇₉	C₂³	1	4	yes	—
149	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
150	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
151	C₄⋊Q₈×C₇₉	C₂³	1	8	yes	complemented
152	#127×C₇₉	C₂³	1	8	yes	complemented
153	#124×C₇₉	C₂³	1	16	yes	complemented
154	#121×C₇₉	C₂³	1	16	yes	—
155	#121×C₇₉	C₂³	1	16	yes	—
156	C₄×D₄×C₇₉	C₂³	1	16	yes	complemented
157	C₄×D₄×C₇₉	C₂³	1	16	yes	complemented
158	C₂×C₄⋊C₄×C₇₉	C₂³	1	2	yes	—
159	C₂×C₄⋊C₄×C₇₉	C₂³	1	2	yes	—
160	C₂×C₂²⋊C₄×C₇₉	C₂³	1	4	yes	—
161	C₂×C₂²⋊C₄×C₇₉	C₂³	1	4	yes	—
162	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
163	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
164	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
165	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
166	C₂×C₄²×C₇₉	C₂³	1	8	yes	—
167	C₂³×D₇₉	—	2	8	no	—
168	C₂³×D₇₉	—	4	4	no	—
169	C₂³×D₇₉	—	4	4	no	—
170	C₂³×D₇₉	C₂⁴	1	4	yes	—
171	C₂³×D₇₉	C₂⁴	1	8	yes	—
172	#11929623	—	4	64	no	—
173	C₂×Dic₁₅₈	—	2	8	no	—
174	C₂×Dic₁₅₈	—	4	4	no	—
175	C₂×Dic₁₅₈	C₂⁴	1	8	yes	—
176	C₂×Dic₁₅₈	C₂⁴	1	8	yes	—
177	C₂×D₃₁₆	—	2	8	no	—
178	C₂×D₃₁₆	—	2	8	no	—
179	C₂×D₃₁₆	—	2	32	no	—
180	C₂×D₃₁₆	C₂⁴	1	8	yes	—
181	C₂×D₃₁₆	C₂⁴	1	8	yes	—
182	C₂×C₂²⋊D₇₉	—	4	16	no	—
183	C₂×C₂²⋊D₇₉	—	4	16	no	—
184	C₂×C₂²⋊D₇₉	C₂⁴	1	32	yes	—
185	C₂×C₂²⋊D₇₉	C₂⁴	1	32	yes	—
186	C₂²×Dic₇₉	—	4	16	no	—
187	C₂²×Dic₇₉	—	4	4	no	—
188	C₂²×Dic₇₉	C₂⁴	1	4	yes	—
189	C₂²×Dic₇₉	C₂⁴	1	4	yes	—
190	C₂²×Dic₇₉	C₂⁴	1	4	yes	—
191	C₂²×Dic₇₉	C₂⁴	1	32	yes	—
192	C₂²×Dic₇₉	C₂⁴	1	8	yes	—
193	C₂×C₄×D₇₉	—	2	16	no	—
194	C₂×C₄×D₇₉	—	4	8	no	—
195	C₂×C₄×D₇₉	—	4	32	no	—
196	C₂×C₄×D₇₉	C₂⁴	1	16	yes	—
197	C₂×C₄×D₇₉	C₂⁴	1	16	yes	—
198	C₄⋊Dic₇₉	C₂⁴	1	8	yes	—
199	C₄⋊Dic₇₉	C₂⁴	1	8	yes	—
200	C₄⋊Dic₇₉	C₂⁴	1	8	yes	—
201	C₄⋊Dic₇₉	C₂⁴	1	32	yes	—
202	C₄⋊Dic₇₉	C₂⁴	1	8	yes	—
203	C₄⋊Dic₇₉	C₂⁴	1	8	yes	—
204	C₄⋊Dic₇₉	C₂⁴	1	8	yes	—
205	Dic₇₉⋊C₄	C₂⁴	1	16	yes	—
206	Dic₇₉⋊C₄	C₂⁴	1	16	yes	—
207	C₂²⋊Dic₇₉	C₂⁴	1	16	yes	—
208	C₂²⋊Dic₇₉	C₂⁴	1	32	yes	—
209	(C₂×D₇₉)⋊C₄	C₂⁴	1	16	yes	—
210	(C₂×D₇₉)⋊C₄	C₂⁴	1	16	yes	—
211	(C₂×D₇₉)⋊C₄	C₂⁴	1	32	yes	—
212	(C₂×D₇₉)⋊C₄	C₂⁴	1	32	yes	—
213	C₄×Dic₇₉	C₂⁴	1	32	yes	—
214	C₄×Dic₇₉	C₂⁴	1	16	yes	—
215	C₂⁴×C₇₉	C₂⁴	1	4	yes	—
216	C₂⁴×C₇₉	C₂⁴	1	1	yes	characteristic
217	#41×C₇₉	—	2	64	no	—
218	C₂×Q₈×C₇₉	—	2	8	no	—
219	C₂×Q₈×C₇₉	C₂⁴	1	8	yes	—
220	C₂×D₄×C₇₉	—	2	16	no	—
221	C₂×D₄×C₇₉	—	2	8	no	—
222	C₂×D₄×C₇₉	—	2	8	no	—
223	C₂×D₄×C₇₉	—	4	4	no	—
224	C₂×D₄×C₇₉	C₂⁴	1	8	yes	—
225	C₂×D₄×C₇₉	C₂⁴	1	16	yes	—
226	C₂×D₄×C₇₉	C₂⁴	1	8	yes	—
227	C₂×D₄×C₇₉	C₂⁴	1	8	yes	—
228	C₂²×C₄×C₇₉	—	2	16	no	—
229	C₂²×C₄×C₇₉	—	2	4	no	—
230	C₂²×C₄×C₇₉	—	4	2	no	—
231	C₂²×C₄×C₇₉	#41	1	8	yes	complemented
232	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
233	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
234	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
235	C₂²×C₄×C₇₉	C₂⁴	1	4	yes	—
236	C₂²×C₄×C₇₉	C₂⁴	1	4	yes	—
237	C₂²×C₄×C₇₉	C₂⁴	1	16	yes	—
238	C₂²×C₄×C₇₉	C₂⁴	1	4	yes	—
239	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
240	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
241	C₂²×C₄×C₇₉	C₂⁴	1	4	yes	—
242	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
243	C₄⋊C₄×C₇₉	C₂⁴	1	8	yes	—
244	C₄⋊C₄×C₇₉	C₂⁴	1	8	yes	—
245	C₂²⋊C₄×C₇₉	C₂⁴	1	16	yes	—
246	C₂²⋊C₄×C₇₉	C₂⁴	1	16	yes	—
247	C₄²×C₇₉	C₂⁴	1	4	yes	—
248	C₄²×C₇₉	C₂⁴	1	4	yes	—
249	C₄²×C₇₉	C₂⁴	1	4	yes	—
250	C₄²×C₇₉	C₂⁴	1	4	yes	—
251	C₂³×C₇₉	—	2	2	no	—
252	C₂³×C₇₉	—	2	2	no	—
253	C₂³×C₇₉	—	2	8	no	—
254	C₂³×C₇₉	—	2	8	no	—
255	C₂³×C₇₉	—	2	8	no	—
256	C₂³×C₇₉	C₂×#41	1	4	yes	—
257	C₂³×C₇₉	C₂⁵	1	8	yes	—
258	C₂³×C₇₉	C₂⁵	1	2	yes	—
259	C₂³×C₇₉	C₂⁵	1	1	yes	characteristic
260	C₂²×D₇₉	—	2	16	no	—
261	C₂²×D₇₉	—	2	8	no	—
262	C₂²×D₇₉	—	2	8	no	—
263	C₂²×D₇₉	—	2	8	no	—
264	C₂²×D₇₉	—	4	8	no	—
265	C₂²×D₇₉	—	4	8	no	—
266	C₂²×D₇₉	—	4	32	no	—
267	C₂²×D₇₉	C₂⁵	1	8	yes	—
268	Q₈×C₇₉	—	2	16	no	—
269	D₄×C₇₉	—	2	32	no	—
270	D₄×C₇₉	—	2	16	no	—
271	D₄×C₇₉	—	2	16	no	—
272	D₄×C₇₉	—	4	8	no	—
273	C₂×C₄×C₇₉	—	2	8	no	—
274	C₂×C₄×C₇₉	—	2	2	no	—
275	C₂×C₄×C₇₉	—	2	2	no	—
276	C₂×C₄×C₇₉	—	2	2	no	—
277	C₂×C₄×C₇₉	—	2	32	no	—
278	C₂×C₄×C₇₉	—	2	8	no	—
279	C₂×C₄×C₇₉	—	4	4	no	—
280	C₂×C₄×C₇₉	C₂×#41	1	4	yes	—
281	C₂×C₄×C₇₉	C₂×#41	1	4	yes	—
282	C₂×C₄×C₇₉	C₂×#41	1	16	yes	complemented
283	C₂×C₄×C₇₉	H(5,𝔽₂)	1	4	yes	—
284	C₂×C₄×C₇₉	C₂⁵	1	8	yes	—
285	C₂×C₄×C₇₉	C₂⁵	1	2	yes	—
286	C₂×C₄×C₇₉	C₂⁵	1	2	yes	—
287	C₂×C₄×C₇₉	C₂⁵	1	2	yes	—
288	C₂×C₄×C₇₉	C₂⁵	1	2	yes	—
289	C₂×C₄×C₇₉	C₂⁵	1	2	yes	—
290	C₂×C₄×C₇₉	C₂⁵	1	2	yes	—
291	C₂²⋊D₇₉	—	4	32	no	—
292	C₂²⋊D₇₉	—	4	32	no	—
293	C₂×Dic₇₉	—	2	8	no	—
294	C₂×Dic₇₉	—	2	8	no	—
295	C₂×Dic₇₉	—	2	8	no	—
296	C₂×Dic₇₉	—	4	8	no	—
297	C₂×Dic₇₉	—	4	32	no	—
298	C₂×Dic₇₉	C₂⁵	1	8	yes	—
299	C₂×Dic₇₉	C₂⁵	1	8	yes	—
300	C₂×Dic₇₉	C₂⁵	1	8	yes	—
301	D₃₁₆	—	2	16	no	—
302	D₃₁₆	—	2	16	no	—
303	C₄×D₇₉	—	2	32	no	—
304	C₄×D₇₉	—	4	16	no	—
305	Dic₁₅₈	—	2	16	no	—
306	Dic₁₅₈	—	4	8	no	—
307	C₂²×C₇₉	—	2	2	no	—
308	C₂²×C₇₉	—	2	2	no	—
309	C₂²×C₇₉	—	2	8	no	—
310	C₂²×C₇₉	—	2	8	no	—
311	C₂²×C₇₉	—	2	8	no	—
312	C₂²×C₇₉	—	2	8	no	—
313	C₂²×C₇₉	—	4	16	no	—
314	C₂²×C₇₉	C₂²×#41	1	4	yes	—
315	C₂²×C₇₉	C₂²×#41	1	2	yes	—
316	C₂²×C₇₉	C₂×H(5,𝔽₂)	1	2	yes	—
317	C₂²×C₇₉	#585	1	2	yes	—
318	C₂²×C₇₉	C₂⁶	1	1	yes	characteristic
319	C₂×D₇₉	—	2	8	no	—
320	C₂×D₇₉	—	2	8	no	—
321	C₂×D₇₉	—	2	8	no	—
322	C₂×D₇₉	—	4	32	no	—
323	C₂×D₇₉	—	4	16	no	—
324	C₄×C₇₉	—	2	8	no	—
325	C₄×C₇₉	—	2	2	no	—
326	C₄×C₇₉	—	2	2	no	—
327	C₄×C₇₉	—	2	2	no	—
328	C₄×C₇₉	C₂²×#41	1	4	yes	—
329	C₄×C₇₉	C₂²×#41	1	4	yes	—
330	C₄×C₇₉	C₂×H(5,𝔽₂)	1	4	yes	—
331	Dic₇₉	—	2	8	no	—
332	Dic₇₉	—	2	8	no	—
333	Dic₇₉	—	2	8	no	—
334	C₂×#3506	—	79	1	no	maximal, Sylow
335	C₂×C₇₉	—	2	4	no	—
336	C₂×C₇₉	—	4	8	no	—
337	C₂×C₇₉	#3506	1	4	yes	complemented
338	C₂×C₇₉	C₂³×#41	1	1	yes	characteristic
339	C₂×C₇₉	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
340	C₂×C₇₉	C₂×#585	1	1	yes	characteristic
341	D₇₉	—	4	8	no	—
342	#3506	—	79	32	no	—
343	C₂×#559	—	79	4	no	—
344	C₂×#556	—	79	2	no	—
345	C₂×#555	—	79	2	no	—
346	C₂×#553	—	79	4	no	—
347	C₂×#550	—	79	2	no	—
348	C₂×#547	—	79	2	no	—
349	C₂×#538	—	79	4	no	—
350	C₂×#535	—	79	2	no	—
351	C₂×#533	—	79	1	no	—
352	C₂×#532	D₇₉	1	1	yes	characteristic, complemented
353	C₂×#525	—	79	4	no	—
354	C₂×#520	—	79	2	no	—
355	C₂²×C₄×D₄	—	79	1	no	—
356	C₇₉	C₂×#3506	1	1	yes	characteristic, complemented, Sylow
357	C₂²×#41	—	79	4	no	—
358	C₂²×#41	C₂×D₇₉	1	2	yes	—
359	C₂³×D₄	—	79	1	no	—
360	C₂⁴×C₄	—	79	2	no	—
361	#559	—	79	64	no	—
362	#556	—	79	32	no	—
363	#555	—	79	32	no	—
364	#553	—	79	64	no	—
365	#550	—	79	32	no	—
366	#547	—	79	32	no	—
367	#538	—	79	64	no	—
368	#535	—	79	32	no	—
369	#533	—	79	16	no	—
370	#532	C₂×D₇₉	1	16	yes	complemented
371	C₂×C₄⋊Q₈	—	79	1	no	—
372	C₂×C₄⋊Q₈	—	79	1	no	—
373	C₂×C₄⋊Q₈	C₂×D₇₉	1	1	yes	characteristic, complemented
374	C₂×#127	—	79	4	no	—
375	C₂×#127	C₂×D₇₉	1	1	yes	characteristic, complemented
376	C₂×#126	—	79	8	no	—
377	C₂×#125	—	79	2	no	—
378	C₂×#124	—	79	4	no	—
379	C₂×#124	—	79	8	no	—
380	C₂×#124	C₂×D₇₉	1	2	yes	complemented
381	#525	—	79	64	no	—
382	C₂×#123	—	79	8	no	—
383	C₂×#123	—	79	8	no	—
384	C₂×C₂²⋊Q₈	—	79	2	no	—
385	C₂×C₂²⋊Q₈	—	79	2	no	—
386	C₂×C₂²⋊Q₈	—	79	4	no	—
387	C₂×#121	—	79	4	no	—
388	C₂×#121	—	79	8	no	—
389	C₂×#121	—	79	4	no	—
390	C₂×#121	—	79	4	no	—
391	C₂×#121	C₂×D₇₉	1	2	yes	—
392	C₂×#121	C₂×D₇₉	1	2	yes	—
393	C₂×#120	—	79	8	no	—
394	#520	—	79	32	no	—
395	C₂×C₄×Q₈	—	79	4	no	—
396	C₂×C₄×Q₈	—	79	2	no	—
397	C₂×C₄×Q₈	—	79	2	no	—
398	C₂×C₄×D₄	—	79	8	no	—
399	C₂×C₄×D₄	—	79	4	no	—
400	C₂×C₄×D₄	—	79	8	no	—
401	C₂×C₄×D₄	—	79	2	no	—
402	C₂×C₄×D₄	—	79	2	no	—
403	C₂×C₄×D₄	—	79	2	no	—
404	C₂×C₄×D₄	—	79	2	no	—
405	C₂×C₄×D₄	—	79	16	no	—
406	C₂×C₄×D₄	C₂×D₇₉	1	2	yes	complemented
407	C₂×C₄×D₄	C₂×D₇₉	1	2	yes	complemented
408	C₂×#117	—	79	2	no	—
409	C₂×#117	—	79	4	no	—
410	C₂×#117	—	79	4	no	—
411	C₂²×C₄⋊C₄	—	79	1	no	—
412	C₂²×C₄⋊C₄	—	79	4	no	—
413	C₂²×C₂²⋊C₄	—	79	2	no	—
414	C₂²×C₄²	C₂×D₇₉	1	1	yes	characteristic
415	C₂⁵	—	79	2	no	—
416	C₂×#41	—	2	16	no	—
417	C₂×#41	—	316	16	no	—
418	C₂×#41	—	79	32	no	—
419	C₂×#41	C₂²×D₇₉	1	16	yes	—
420	C₂²×Q₈	—	79	2	no	—
421	C₂²×Q₈	—	79	2	no	—
422	C₂²×Q₈	C₂²×D₇₉	1	2	yes	—
423	C₂²×D₄	—	79	8	no	—
424	C₂²×D₄	—	79	8	no	—
425	C₂²×D₄	—	79	2	no	—
426	C₂²×D₄	—	79	2	no	—
427	C₂²×D₄	—	79	8	no	—
428	C₂²×D₄	—	158	8	no	—
429	C₂²×D₄	C₂²×D₇₉	1	4	yes	—
430	C₂²×D₄	C₂²×D₇₉	1	2	yes	—
431	C₂²×D₄	C₂²×D₇₉	1	2	yes	—
432	C₂²×D₄	C₂²×D₇₉	1	2	yes	—
433	C₂³×C₄	—	316	8	no	—
434	C₂³×C₄	—	79	2	no	—
435	C₂³×C₄	—	79	8	no	—
436	C₂³×C₄	—	79	4	no	—
437	C₂³×C₄	—	79	16	no	—
438	C₂³×C₄	—	79	4	no	—
439	C₂³×C₄	C₂²×D₇₉	1	4	yes	—
440	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
441	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
442	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
443	C₄⋊Q₈	—	79	8	no	—
444	C₄⋊Q₈	—	79	8	no	—
445	C₄⋊Q₈	C₂²×D₇₉	1	8	yes	complemented
446	#127	—	79	32	no	—
447	#127	C₂²×D₇₉	1	8	yes	complemented
448	#126	—	79	64	no	—
449	#125	—	79	16	no	—
450	#124	—	79	32	no	—
451	#124	—	79	64	no	—
452	#124	C₂²×D₇₉	1	16	yes	complemented
453	#123	—	79	64	no	—
454	#123	—	79	64	no	—
455	C₂²⋊Q₈	—	79	16	no	—
456	C₂²⋊Q₈	—	79	16	no	—
457	C₂²⋊Q₈	—	79	32	no	—
458	#121	—	79	32	no	—
459	#121	—	79	64	no	—
460	#121	—	79	32	no	—
461	#121	—	79	32	no	—
462	#121	C₂²×D₇₉	1	16	yes	—
463	#121	C₂²×D₇₉	1	16	yes	—
464	#120	—	79	64	no	—
465	C₄×Q₈	—	79	32	no	—
466	C₄×Q₈	—	79	16	no	—
467	C₄×Q₈	—	79	16	no	—
468	C₄×D₄	—	79	64	no	—
469	C₄×D₄	—	79	32	no	—
470	C₄×D₄	—	79	64	no	—
471	C₄×D₄	—	79	16	no	—
472	C₄×D₄	—	79	16	no	—
473	C₄×D₄	—	79	16	no	—
474	C₄×D₄	—	79	16	no	—
475	C₄×D₄	C₂²×D₇₉	1	16	yes	complemented
476	C₄×D₄	C₂²×D₇₉	1	16	yes	complemented
477	#117	—	79	16	no	—
478	#117	—	79	32	no	—
479	#117	—	79	32	no	—
480	C₂×C₄⋊C₄	—	79	2	no	—
481	C₂×C₄⋊C₄	—	79	2	no	—
482	C₂×C₄⋊C₄	—	79	2	no	—
483	C₂×C₄⋊C₄	—	79	8	no	—
484	C₂×C₄⋊C₄	—	79	8	no	—
485	C₂×C₄⋊C₄	—	79	32	no	—
486	C₂×C₄⋊C₄	—	79	2	no	—
487	C₂×C₄⋊C₄	—	79	2	no	—
488	C₂×C₄⋊C₄	—	79	4	no	—
489	C₂×C₄⋊C₄	—	79	4	no	—
490	C₂×C₄⋊C₄	—	79	2	no	—
491	C₂×C₄⋊C₄	C₂²×D₇₉	1	2	yes	—
492	C₂×C₄⋊C₄	C₂²×D₇₉	1	2	yes	—
493	C₂×C₂²⋊C₄	—	79	4	no	—
494	C₂×C₂²⋊C₄	—	79	8	no	—
495	C₂×C₂²⋊C₄	—	79	4	no	—
496	C₂×C₂²⋊C₄	—	79	4	no	—
497	C₂×C₂²⋊C₄	—	79	16	no	—
498	C₂×C₂²⋊C₄	—	79	8	no	—
499	C₂×C₂²⋊C₄	—	79	8	no	—
500	C₂×C₂²⋊C₄	C₂²×D₇₉	1	4	yes	—
501	C₂×C₂²⋊C₄	C₂²×D₇₉	1	4	yes	—
502	C₂×C₄²	—	79	8	no	—
503	C₂×C₄²	—	79	4	no	—
504	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
505	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
506	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
507	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
508	C₂×C₄²	C₂²×D₇₉	1	8	yes	—
509	C₂⁴	—	316	4	no	—
510	C₂⁴	—	316	4	no	—
511	C₂⁴	—	79	8	no	—
512	C₂⁴	—	79	4	no	—
513	C₂⁴	—	158	8	no	—
514	C₂⁴	C₂³×D₇₉	1	4	yes	—
515	C₂⁴	C₂³×D₇₉	1	1	yes	characteristic
516	#41	—	2	64	no	—
517	#41	—	316	64	no	—
518	C₂×Q₈	—	2	8	no	—
519	C₂×Q₈	—	316	4	no	—
520	C₂×Q₈	—	79	8	no	—
521	C₂×Q₈	—	79	8	no	—
522	C₂×Q₈	—	158	8	no	—
523	C₂×Q₈	C₂³×D₇₉	1	8	yes	—
524	C₂×D₄	—	2	16	no	—
525	C₂×D₄	—	2	8	no	—
526	C₂×D₄	—	2	8	no	—
527	C₂×D₄	—	4	4	no	—
528	C₂×D₄	—	316	16	no	—
529	C₂×D₄	—	316	16	no	—
530	C₂×D₄	—	79	32	no	—
531	C₂×D₄	—	79	32	no	—
532	C₂×D₄	—	79	8	no	—
533	C₂×D₄	—	79	8	no	—
534	C₂×D₄	—	158	8	no	—
535	C₂×D₄	—	158	8	no	—
536	C₂×D₄	—	158	32	no	—
537	C₂×D₄	C₂³×D₇₉	1	16	yes	—
538	C₂×D₄	C₂³×D₇₉	1	8	yes	—
539	C₂×D₄	C₂³×D₇₉	1	8	yes	—
540	C₂×D₄	C₂³×D₇₉	1	8	yes	—
541	C₂²×C₄	—	2	16	no	—
542	C₂²×C₄	—	2	4	no	—
543	C₂²×C₄	—	4	2	no	—
544	C₂²×C₄	—	316	8	no	—
545	C₂²×C₄	—	316	32	no	—
546	C₂²×C₄	—	316	4	no	—
547	C₂²×C₄	—	316	16	no	—
548	C₂²×C₄	—	79	8	no	—
549	C₂²×C₄	—	79	32	no	—
550	C₂²×C₄	—	79	16	no	—
551	C₂²×C₄	—	79	4	no	—
552	C₂²×C₄	—	79	4	no	—
553	C₂²×C₄	—	79	4	no	—
554	C₂²×C₄	—	79	16	no	—
555	C₂²×C₄	—	158	16	no	—
556	C₂²×C₄	#11929623	1	8	yes	complemented
557	C₂²×C₄	C₂³×D₇₉	1	4	yes	—
558	C₂²×C₄	C₂³×D₇₉	1	16	yes	—
559	C₂²×C₄	C₂³×D₇₉	1	4	yes	—
560	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
561	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
562	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
563	C₂²×C₄	C₂³×D₇₉	1	4	yes	—
564	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
565	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
566	C₂²×C₄	C₂³×D₇₉	1	4	yes	—
567	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
568	C₄⋊C₄	—	79	16	no	—
569	C₄⋊C₄	—	79	16	no	—
570	C₄⋊C₄	—	79	8	no	—
571	C₄⋊C₄	—	79	8	no	—
572	C₄⋊C₄	—	79	8	no	—
573	C₄⋊C₄	—	79	32	no	—
574	C₄⋊C₄	—	79	8	no	—
575	C₄⋊C₄	—	79	8	no	—
576	C₄⋊C₄	—	79	8	no	—
577	C₄⋊C₄	C₂³×D₇₉	1	8	yes	—
578	C₄⋊C₄	C₂³×D₇₉	1	8	yes	—
579	C₂²⋊C₄	—	79	16	no	—
580	C₂²⋊C₄	—	79	32	no	—
581	C₂²⋊C₄	—	79	16	no	—
582	C₂²⋊C₄	—	79	16	no	—
583	C₂²⋊C₄	—	79	32	no	—
584	C₂²⋊C₄	—	79	32	no	—
585	C₂²⋊C₄	C₂³×D₇₉	1	16	yes	—
586	C₂²⋊C₄	C₂³×D₇₉	1	16	yes	—
587	C₄²	—	79	32	no	—
588	C₄²	—	79	16	no	—
589	C₄²	C₂³×D₇₉	1	4	yes	—
590	C₄²	C₂³×D₇₉	1	4	yes	—
591	C₄²	C₂³×D₇₉	1	4	yes	—
592	C₄²	C₂³×D₇₉	1	4	yes	—
593	C₂³	—	2	2	no	—
594	C₂³	—	2	2	no	—
595	C₂³	—	2	8	no	—
596	C₂³	—	2	8	no	—
597	C₂³	—	2	8	no	—
598	C₂³	—	316	32	no	—
599	C₂³	—	316	8	no	—
600	C₂³	—	316	8	no	—
601	C₂³	—	79	8	no	—
602	C₂³	—	158	8	no	—
603	C₂³	—	158	8	no	—
604	C₂³	—	158	16	no	—
605	C₂³	—	158	8	no	—
606	C₂³	C₂×#11929623	1	4	yes	—
607	C₂³	C₂⁴×D₇₉	1	8	yes	—
608	C₂³	C₂⁴×D₇₉	1	2	yes	—
609	C₂³	C₂⁴×D₇₉	1	1	yes	characteristic
610	Q₈	—	2	16	no	—
611	Q₈	—	316	8	no	—
612	Q₈	—	158	16	no	—
613	D₄	—	2	32	no	—
614	D₄	—	2	16	no	—
615	D₄	—	2	16	no	—
616	D₄	—	4	8	no	—
617	D₄	—	316	32	no	—
618	D₄	—	316	32	no	—
619	D₄	—	158	16	no	—
620	D₄	—	158	16	no	—
621	C₂×C₄	—	2	2	no	—
622	C₂×C₄	—	2	2	no	—
623	C₂×C₄	—	2	2	no	—
624	C₂×C₄	—	2	8	no	—
625	C₂×C₄	—	2	8	no	—
626	C₂×C₄	—	2	32	no	—
627	C₂×C₄	—	4	4	no	—
628	C₂×C₄	—	316	16	no	—
629	C₂×C₄	—	316	8	no	—
630	C₂×C₄	—	316	32	no	—
631	C₂×C₄	—	79	8	no	—
632	C₂×C₄	—	79	8	no	—
633	C₂×C₄	—	79	8	no	—
634	C₂×C₄	—	158	8	no	—
635	C₂×C₄	—	158	8	no	—
636	C₂×C₄	—	158	8	no	—
637	C₂×C₄	—	158	32	no	—
638	C₂×C₄	C₂×#11929623	1	4	yes	—
639	C₂×C₄	C₂×#11929623	1	4	yes	—
640	C₂×C₄	C₂×#11929623	1	16	yes	complemented
641	C₂×C₄	#14579207	1	4	yes	—
642	C₂×C₄	C₂⁴×D₇₉	1	8	yes	—
643	C₂×C₄	C₂⁴×D₇₉	1	2	yes	—
644	C₂×C₄	C₂⁴×D₇₉	1	2	yes	—
645	C₂×C₄	C₂⁴×D₇₉	1	2	yes	—
646	C₂×C₄	C₂⁴×D₇₉	1	2	yes	—
647	C₂×C₄	C₂⁴×D₇₉	1	2	yes	—
648	C₂×C₄	C₂⁴×D₇₉	1	2	yes	—
649	C₂²	—	2	2	no	—
650	C₂²	—	2	2	no	—
651	C₂²	—	2	8	no	—
652	C₂²	—	2	8	no	—
653	C₂²	—	2	8	no	—
654	C₂²	—	2	8	no	—
655	C₂²	—	4	16	no	—
656	C₂²	—	316	32	no	—
657	C₂²	—	316	16	no	—
658	C₂²	—	158	8	no	—
659	C₂²	—	158	8	no	—
660	C₂²	—	158	8	no	—
661	C₂²	C₂²×#11929623	1	4	yes	—
662	C₂²	C₂²×#11929623	1	2	yes	—
663	C₂²	C₂×#14579207	1	2	yes	—
664	C₂²	#19099399	1	2	yes	—
665	C₂²	C₂⁵×D₇₉	1	1	yes	characteristic
666	C₄	—	2	8	no	—
667	C₄	—	2	2	no	—
668	C₄	—	2	2	no	—
669	C₄	—	2	2	no	—
670	C₄	—	158	8	no	—
671	C₄	—	158	8	no	—
672	C₄	—	158	8	no	—
673	C₄	C₂²×#11929623	1	4	yes	—
674	C₄	C₂²×#11929623	1	4	yes	—
675	C₄	C₂×#14579207	1	4	yes	—
676	C₂	—	2	4	no	—
677	C₂	—	4	8	no	—
678	C₂	—	316	8	no	—
679	C₂	#23777249	1	4	yes	direct product factor
680	C₂	C₂³×#11929623	1	1	yes	characteristic
681	C₂	C₂²×#14579207	1	1	yes	characteristic
682	C₂	C₂×#19099399	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂⁴×D₇₉
commutator, cosocle	C₂²×C₇₉	C₂⁶
Frattini	C₂²	C₂⁵×D₇₉
Fitting	C₂×#532×C₇₉	C₂
socle	C₂³×C₇₉	C₂⁵
hypercenter	C₂×#532	D₇₉
hypocenter	C₇₉	C₂×#3506
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×#532

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #23777249	C₂	direct (N × H)
2. C₂×#19099295	C₂	semidirect
3. C₂×#19099287	C₂	semidirect
4. C₂×#19099278	C₂	semidirect
5. C₂×#19099259	C₂	semidirect
6. C₂×#19099233	C₂	semidirect
7. C₂×#19099217	C₂	semidirect
8. C₂×#19099150	C₂	semidirect
9. C₂×#19099127	C₂	semidirect
10. C₂×#19099120	C₂	semidirect
11. C₂×#19099073	C₂	semidirect
12. C₂×#19099046	C₂	semidirect
13. C₂²×C₄×D₃₁₆	C₂	semidirect
14. C₂×#532×C₇₉	C₂	semidirect
15. C₂²×#11929623	C₂²	semidirect
18. #19099295	C₂²	semidirect
19. #19099287	C₂²	semidirect
20. #19099278	C₂²	semidirect
21. #19099259	C₂²	semidirect
22. #19099233	C₂²	semidirect
23. #19099217	C₂²	semidirect
24. #19099150	C₂²	semidirect
25. #19099127	C₂²	semidirect
26. #19099120	C₂²	semidirect
27. C₂×#14579161	C₂²	semidirect
28. C₂×#14579161	C₂²	semidirect
30. C₂×#14579156	C₂²	semidirect
31. C₂×#14579153	C₂²	semidirect
33. C₂×#14579147	C₂²	semidirect
34. #19099073	C₂²	semidirect
35. C₂×#14579145	C₂²	semidirect
36. C₂×#14579143	C₂²	semidirect
37. C₂×#14579141	C₂²	semidirect
42. C₂×#14579133	C₂²	semidirect
43. C₂×#14579132	C₂²	semidirect
45. #19099046	C₂²	semidirect
46. C₂×Q₈×Dic₇₉	C₂²	semidirect
47. C₂×C₄×Dic₁₅₈	C₂²	semidirect
48. C₂×C₄×Dic₁₅₈	C₂²	semidirect
53. C₂×C₄×D₃₁₆	C₂²	semidirect
55. C₂×D₄×Dic₇₉	C₂²	semidirect
56. C₂×(C₂²⋊D₇₉)⋊C₄	C₂²	semidirect
58. C₂×#14579120	C₂²	semidirect
59. C₂×#14579119	C₂²	semidirect
61. C₂²×C₄⋊Dic₇₉	C₂²	semidirect
64. #532×C₇₉	C₂²	semidirect
65. C₂×C₄⋊Q₈×C₇₉	C₂²	semidirect
66. C₂×#127×C₇₉	C₂²	semidirect
67. C₂×#124×C₇₉	C₂²	semidirect
70. C₂×C₄×D₄×C₇₉	C₂²	semidirect
71. C₂×C₄×D₄×C₇₉	C₂²	semidirect
75. C₂×#11929623	C₂³	semidirect
90. #14579161	C₂³	semidirect
91. #14579161	C₂³	semidirect
93. #14579156	C₂³	semidirect
94. #14579153	C₂³	semidirect
96. #14579147	C₂³	semidirect
97. #14579145	C₂³	semidirect
98. #14579143	C₂³	semidirect
99. #14579141	C₂³	semidirect
104. #14579133	C₂³	semidirect
105. #14579132	C₂³	semidirect
107. Q₈×Dic₇₉	C₂³	semidirect
108. C₄×Dic₁₅₈	C₂³	semidirect
109. C₄×Dic₁₅₈	C₂³	semidirect
115. D₄×Dic₇₉	C₂³	semidirect
116. (C₂²⋊D₇₉)⋊C₄	C₂³	semidirect
118. #14579120	C₂³	semidirect
119. #14579119	C₂³	semidirect
128. C₂×C₄⋊Dic₇₉	C₂³	semidirect
151. C₄⋊Q₈×C₇₉	C₂³	semidirect
152. #127×C₇₉	C₂³	semidirect
153. #124×C₇₉	C₂³	semidirect
156. C₄×D₄×C₇₉	C₂³	semidirect
157. C₄×D₄×C₇₉	C₂³	semidirect
231. C₂²×C₄×C₇₉	#41	semidirect
282. C₂×C₄×C₇₉	C₂×#41	semidirect
337. C₂×C₇₉	#3506	semidirect
352. C₂×#532	D₇₉	semidirect
356. C₇₉	C₂×#3506	semidirect
370. #532	C₂×D₇₉	semidirect
373. C₂×C₄⋊Q₈	C₂×D₇₉	semidirect
375. C₂×#127	C₂×D₇₉	semidirect
380. C₂×#124	C₂×D₇₉	semidirect
406. C₂×C₄×D₄	C₂×D₇₉	semidirect
407. C₂×C₄×D₄	C₂×D₇₉	semidirect
445. C₄⋊Q₈	C₂²×D₇₉	semidirect
447. #127	C₂²×D₇₉	semidirect
452. #124	C₂²×D₇₉	semidirect
475. C₄×D₄	C₂²×D₇₉	semidirect
476. C₄×D₄	C₂²×D₇₉	semidirect
556. C₂²×C₄	#11929623	semidirect
640. C₂×C₄	C₂×#11929623	semidirect
679. C₂	#23777249	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#23777249	2
C₂×#19099295	2
C₂×#19099287	2
C₂×#19099278	2
C₂×#19099259	2
C₂×#19099233	2
C₂×#19099217	2
C₂×#19099150	2
C₂×#19099127	2
C₂×#19099120	2
C₂×#19099073	2
C₂×#19099046	2
C₂²×C₄×D₃₁₆	2
C₂×#532×C₇₉	2
C₂×#3506	79

Subgroups — C2×#23777249 #35196217

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23777249 #35196217