Subgroups — C₂×#23776805 #35195113

Order 20224.

7022 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23776805	C₂	1	32	yes	maximal, direct product factor
2	C₂×#19099194	C₂	1	4	yes	maximal, complemented
3	C₂×#19099191	C₂	1	8	yes	maximal, complemented
4	C₂×#19099189	C₂	1	4	yes	maximal
5	C₂×#19099120	C₂	1	4	yes	maximal, complemented
6	C₂×#19099116	C₂	1	8	yes	maximal, complemented
7	C₂²×#14579120	C₂	1	2	yes	maximal
8	C₂²×#117×C₇₉	C₂	1	1	yes	maximal, characteristic
9	#19099194	C₂²	1	64	yes	complemented
10	#19099191	C₂²	1	128	yes	complemented
11	#19099189	C₂²	1	64	yes	—
12	#19099120	C₂²	1	64	yes	complemented
13	#19099116	C₂²	1	128	yes	complemented
14	C₂×#14579161	C₂²	1	8	yes	complemented
15	C₂×#14579160	C₂²	1	16	yes	complemented
16	C₂×#14579153	C₂²	1	8	yes	complemented
17	C₂×#14579152	C₂²	1	16	yes	complemented
18	C₂×#14579140	C₂²	1	16	yes	—
19	C₂×#14579138	C₂²	1	32	yes	—
20	C₂×C₄×Dic₁₅₈	C₂²	1	16	yes	complemented
21	C₂×#14579120	C₂²	1	4	yes	—
22	C₂×#14579120	C₂²	1	4	yes	—
23	C₂×#14579120	C₂²	1	32	yes	—
24	C₂×#14579120	C₂²	1	4	yes	—
25	C₂×#14579120	C₂²	1	4	yes	—
26	C₂²×C₄⋊Dic₇₉	C₂²	1	4	yes	—
27	C₂²×C₂²⋊Dic₇₉	C₂²	1	4	yes	—
28	C₂²×C₄×Dic₇₉	C₂²	1	4	yes	—
29	C₂⁴×C₄×C₇₉	C₂²	1	1	yes	characteristic
30	C₂×#117×C₇₉	C₂²	1	4	yes	—
31	C₂×#117×C₇₉	C₂²	1	16	yes	—
32	C₂×#117×C₇₉	C₂²	1	4	yes	—
33	C₂²×C₄⋊C₄×C₇₉	C₂²	1	2	yes	—
34	C₂²×C₂²⋊C₄×C₇₉	C₂²	1	2	yes	—
35	C₂²×C₄²×C₇₉	C₂²	1	2	yes	—
36	C₂²×Dic₁₅₈	C₂³	1	16	yes	—
37	C₂³×Dic₇₉	C₂³	1	8	yes	—
38	#14579161	C₂³	1	64	yes	complemented
39	#14579160	C₂³	1	128	yes	complemented
40	#14579153	C₂³	1	64	yes	complemented
41	#14579152	C₂³	1	128	yes	complemented
42	#14579140	C₂³	1	128	yes	—
43	#14579138	C₂³	1	256	yes	—
44	C₄×Dic₁₅₈	C₂³	1	128	yes	complemented
45	#14579120	C₂³	1	32	yes	—
46	#14579120	C₂³	1	32	yes	—
47	#14579120	C₂³	1	32	yes	—
48	#14579120	C₂³	1	32	yes	—
49	C₂×C₄⋊Dic₇₉	C₂³	1	32	yes	—
50	C₂×C₄⋊Dic₇₉	C₂³	1	8	yes	—
51	C₂×C₄⋊Dic₇₉	C₂³	1	8	yes	—
52	C₂×C₄⋊Dic₇₉	C₂³	1	16	yes	—
53	C₂×Dic₇₉⋊C₄	C₂³	1	32	yes	—
54	C₂×C₂²⋊Dic₇₉	C₂³	1	32	yes	—
55	C₂×C₂²⋊Dic₇₉	C₂³	1	8	yes	—
56	C₂×C₂²⋊Dic₇₉	C₂³	1	8	yes	—
57	C₂×C₄×Dic₇₉	C₂³	1	8	yes	—
58	C₂×C₄×Dic₇₉	C₂³	1	8	yes	—
59	C₂×C₄×Dic₇₉	C₂³	1	32	yes	—
60	C₂⁵×C₇₉	C₂³	1	1	yes	characteristic
61	C₂³×C₄×C₇₉	Q₈	1	16	yes	complemented
62	C₂³×C₄×C₇₉	C₂³	1	4	yes	—
63	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
64	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
65	C₂³×C₄×C₇₉	C₂³	1	2	yes	—
66	C₂³×C₄×C₇₉	C₂³	1	2	yes	—
67	C₂³×C₄×C₇₉	C₂³	1	8	yes	—
68	#117×C₇₉	C₂³	1	32	yes	—
69	#117×C₇₉	C₂³	1	32	yes	—
70	C₂×C₄⋊C₄×C₇₉	C₂³	1	16	yes	—
71	C₂×C₄⋊C₄×C₇₉	C₂³	1	8	yes	—
72	C₂×C₂²⋊C₄×C₇₉	C₂³	1	16	yes	—
73	C₂×C₂²⋊C₄×C₇₉	C₂³	1	8	yes	—
74	C₂×C₄²×C₇₉	C₂³	1	4	yes	—
75	C₂×C₄²×C₇₉	C₂³	1	4	yes	—
76	C₂×C₄²×C₇₉	C₂³	1	16	yes	—
77	C₂×Dic₁₅₈	—	4	32	no	—
78	C₂×Dic₁₅₈	C₂⁴	1	64	yes	—
79	C₂²×Dic₇₉	—	4	16	no	—
80	C₂²×Dic₇₉	C₂⁴	1	16	yes	—
81	C₂²×Dic₇₉	C₂⁴	1	32	yes	—
82	C₄⋊Dic₇₉	C₂⁴	1	32	yes	—
83	C₄⋊Dic₇₉	C₂⁴	1	32	yes	—
84	C₄⋊Dic₇₉	C₂⁴	1	64	yes	—
85	Dic₇₉⋊C₄	C₂⁴	1	128	yes	—
86	C₂²⋊Dic₇₉	C₂⁴	1	32	yes	—
87	C₂²⋊Dic₇₉	C₂⁴	1	32	yes	—
88	C₄×Dic₇₉	C₂⁴	1	32	yes	—
89	C₄×Dic₇₉	C₂⁴	1	32	yes	—
90	C₂⁴×C₇₉	—	4	4	no	—
91	C₂⁴×C₇₉	C₂×Q₈	1	8	yes	—
92	C₂⁴×C₇₉	C₂⁴	1	2	yes	—
93	C₂⁴×C₇₉	C₂⁴	1	1	yes	characteristic
94	C₂⁴×C₇₉	C₂⁴	1	4	yes	—
95	C₂²×C₄×C₇₉	—	4	8	no	—
96	C₂²×C₄×C₇₉	C₂×Q₈	1	8	yes	—
97	C₂²×C₄×C₇₉	C₂×Q₈	1	8	yes	—
98	C₂²×C₄×C₇₉	C₂×Q₈	1	16	yes	—
99	C₂²×C₄×C₇₉	C₂×Q₈	1	16	yes	—
100	C₂²×C₄×C₇₉	C₂×Q₈	1	64	yes	complemented
101	C₂²×C₄×C₇₉	C₂⁴	1	8	yes	—
102	C₂²×C₄×C₇₉	C₂⁴	1	16	yes	—
103	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
104	C₂²×C₄×C₇₉	C₂⁴	1	4	yes	—
105	C₂²×C₄×C₇₉	C₂⁴	1	2	yes	—
106	C₂²×C₄×C₇₉	C₂⁴	1	4	yes	—
107	C₂²×C₄×C₇₉	C₂⁴	1	8	yes	—
108	C₂²×C₄×C₇₉	C₂⁴	1	8	yes	—
109	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
110	C₄⋊C₄×C₇₉	C₂⁴	1	32	yes	—
111	C₂²⋊C₄×C₇₉	C₂⁴	1	32	yes	—
112	C₄²×C₇₉	C₂⁴	1	16	yes	—
113	C₄²×C₇₉	C₂⁴	1	16	yes	—
114	C₂³×C₇₉	—	2	8	no	—
115	C₂³×C₇₉	—	2	4	no	—
116	C₂³×C₇₉	—	4	16	no	—
117	C₂³×C₇₉	—	4	8	no	—
118	C₂³×C₇₉	C₂²×Q₈	1	8	yes	—
119	C₂³×C₇₉	C₂²×Q₈	1	4	yes	—
120	C₂³×C₇₉	C₂²×Q₈	1	16	yes	—
121	C₂³×C₇₉	C₂⁵	1	2	yes	—
122	C₂³×C₇₉	C₂⁵	1	4	yes	—
123	C₂³×C₇₉	C₂⁵	1	1	yes	characteristic
124	C₂×C₄×C₇₉	—	2	16	no	—
125	C₂×C₄×C₇₉	—	4	16	no	—
126	C₂×C₄×C₇₉	C₂²×Q₈	1	4	yes	—
127	C₂×C₄×C₇₉	C₂²×Q₈	1	8	yes	—
128	C₂×C₄×C₇₉	C₂²×Q₈	1	4	yes	—
129	C₂×C₄×C₇₉	C₂²×Q₈	1	16	yes	—
130	C₂×C₄×C₇₉	C₂²×Q₈	1	16	yes	—
131	C₂×C₄×C₇₉	C₂²×Q₈	1	32	yes	—
132	C₂×C₄×C₇₉	C₂²×Q₈	1	32	yes	—
133	C₂×C₄×C₇₉	C₂⁵	1	16	yes	—
134	C₂×C₄×C₇₉	C₂⁵	1	2	yes	—
135	C₂×C₄×C₇₉	C₂⁵	1	4	yes	—
136	C₂×C₄×C₇₉	C₂⁵	1	2	yes	—
137	C₂×Dic₇₉	—	2	32	no	—
138	C₂×Dic₇₉	—	4	32	no	—
139	C₂×Dic₇₉	C₂⁵	1	32	yes	—
140	Dic₁₅₈	—	4	64	no	—
141	C₂²×C₇₉	—	2	4	no	—
142	C₂²×C₇₉	—	2	8	no	—
143	C₂²×C₇₉	—	2	16	no	—
144	C₂²×C₇₉	—	2	8	no	—
145	C₂²×C₇₉	—	4	16	no	—
146	C₂²×C₇₉	C₂³×Q₈	1	4	yes	—
147	C₂²×C₇₉	C₂³×Q₈	1	8	yes	—
148	C₂²×C₇₉	C₂³×Q₈	1	2	yes	—
149	C₂²×C₇₉	#585	1	4	yes	—
150	C₂²×C₇₉	C₂⁶	1	1	yes	characteristic
151	C₄×C₇₉	—	2	16	no	—
152	C₄×C₇₉	C₂³×Q₈	1	4	yes	—
153	C₄×C₇₉	C₂³×Q₈	1	8	yes	—
154	C₄×C₇₉	C₂³×Q₈	1	4	yes	—
155	Dic₇₉	—	2	32	no	—
156	C₂×#3458	—	79	1	no	maximal, Sylow
157	C₂×C₇₉	—	2	4	no	—
158	C₂×C₇₉	—	2	8	no	—
159	C₂×C₇₉	#3458	1	4	yes	complemented
160	C₂×C₇₉	C₂⁴×Q₈	1	1	yes	characteristic
161	C₂×C₇₉	C₂×#585	1	2	yes	—
162	#3458	—	79	32	no	—
163	C₂×#544	—	79	8	no	—
164	C₂×#544	—	79	4	no	—
165	C₂×#543	—	79	4	no	—
166	C₂×#533	—	79	4	no	—
167	C₂×#533	—	79	8	no	—
168	C₂²×#117	—	79	2	no	—
169	C₂²×#117	D₇₉	1	1	yes	characteristic
170	C₇₉	C₂×#3458	1	1	yes	characteristic, complemented, Sylow
171	C₂⁴×C₄	C₂×D₇₉	1	1	yes	characteristic
172	#544	—	79	128	no	—
173	#544	—	79	64	no	—
174	#543	—	79	64	no	—
175	#533	—	79	128	no	—
176	#533	—	79	64	no	—
177	C₂×C₄⋊Q₈	—	79	16	no	—
178	C₂×C₄⋊Q₈	—	79	8	no	—
179	C₂×#125	—	79	16	no	—
180	C₂×#125	—	79	8	no	—
181	C₂×C₂²⋊Q₈	—	79	16	no	—
182	C₂×C₂²⋊Q₈	—	79	32	no	—
183	C₂×C₄×Q₈	—	79	16	no	—
184	C₂×#117	—	79	4	no	—
185	C₂×#117	—	79	4	no	—
186	C₂×#117	—	79	4	no	—
187	C₂×#117	—	79	4	no	—
188	C₂×#117	—	79	32	no	—
189	C₂×#117	C₂×D₇₉	1	4	yes	—
190	C₂×#117	C₂×D₇₉	1	4	yes	—
191	C₂×#117	C₂×D₇₉	1	16	yes	—
192	C₂²×C₄⋊C₄	—	79	4	no	—
193	C₂²×C₄⋊C₄	C₂×D₇₉	1	2	yes	—
194	C₂²×C₂²⋊C₄	—	79	4	no	—
195	C₂²×C₂²⋊C₄	C₂×D₇₉	1	2	yes	—
196	C₂²×C₄²	—	79	4	no	—
197	C₂²×C₄²	C₂×D₇₉	1	2	yes	—
198	C₂⁵	C₂²×D₇₉	1	1	yes	characteristic
199	C₂²×Q₈	—	79	16	no	—
200	C₂³×C₄	—	79	8	no	—
201	C₂³×C₄	Dic₁₅₈	1	16	yes	complemented
202	C₂³×C₄	C₂²×D₇₉	1	4	yes	—
203	C₂³×C₄	C₂²×D₇₉	1	2	yes	—
204	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
205	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
206	C₂³×C₄	C₂²×D₇₉	1	2	yes	—
207	C₂³×C₄	C₂²×D₇₉	1	8	yes	—
208	C₄⋊Q₈	—	79	128	no	—
209	C₄⋊Q₈	—	79	64	no	—
210	#125	—	79	128	no	—
211	#125	—	79	64	no	—
212	C₂²⋊Q₈	—	79	128	no	—
213	C₂²⋊Q₈	—	79	256	no	—
214	C₄×Q₈	—	79	128	no	—
215	#117	—	79	32	no	—
216	#117	—	79	32	no	—
217	#117	—	79	32	no	—
218	#117	—	79	32	no	—
219	#117	C₂²×D₇₉	1	32	yes	—
220	#117	C₂²×D₇₉	1	32	yes	—
221	C₂×C₄⋊C₄	—	79	8	no	—
222	C₂×C₄⋊C₄	—	79	8	no	—
223	C₂×C₄⋊C₄	—	79	32	no	—
224	C₂×C₄⋊C₄	—	79	32	no	—
225	C₂×C₄⋊C₄	—	79	16	no	—
226	C₂×C₄⋊C₄	C₂²×D₇₉	1	8	yes	—
227	C₂×C₄⋊C₄	C₂²×D₇₉	1	16	yes	—
228	C₂×C₂²⋊C₄	—	79	8	no	—
229	C₂×C₂²⋊C₄	—	79	8	no	—
230	C₂×C₂²⋊C₄	—	79	32	no	—
231	C₂×C₂²⋊C₄	C₂²×D₇₉	1	8	yes	—
232	C₂×C₂²⋊C₄	C₂²×D₇₉	1	16	yes	—
233	C₂×C₄²	—	79	8	no	—
234	C₂×C₄²	—	79	32	no	—
235	C₂×C₄²	—	79	8	no	—
236	C₂×C₄²	C₂²×D₇₉	1	4	yes	—
237	C₂×C₄²	C₂²×D₇₉	1	16	yes	—
238	C₂×C₄²	C₂²×D₇₉	1	4	yes	—
239	C₂⁴	—	4	4	no	—
240	C₂⁴	C₂×Dic₁₅₈	1	8	yes	—
241	C₂⁴	C₂³×D₇₉	1	1	yes	characteristic
242	C₂⁴	C₂³×D₇₉	1	2	yes	—
243	C₂⁴	C₂³×D₇₉	1	4	yes	—
244	C₂×Q₈	—	316	32	no	—
245	C₂×Q₈	—	79	64	no	—
246	C₂²×C₄	—	4	8	no	—
247	C₂²×C₄	—	316	16	no	—
248	C₂²×C₄	—	79	32	no	—
249	C₂²×C₄	—	79	16	no	—
250	C₂²×C₄	C₂×Dic₁₅₈	1	16	yes	—
251	C₂²×C₄	C₂×Dic₁₅₈	1	8	yes	—
252	C₂²×C₄	C₂×Dic₁₅₈	1	8	yes	—
253	C₂²×C₄	C₂×Dic₁₅₈	1	16	yes	—
254	C₂²×C₄	C₂×Dic₁₅₈	1	64	yes	complemented
255	C₂²×C₄	C₂³×D₇₉	1	8	yes	—
256	C₂²×C₄	C₂³×D₇₉	1	16	yes	—
257	C₂²×C₄	C₂³×D₇₉	1	2	yes	—
258	C₂²×C₄	C₂³×D₇₉	1	8	yes	—
259	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
260	C₂²×C₄	C₂³×D₇₉	1	4	yes	—
261	C₂²×C₄	C₂³×D₇₉	1	8	yes	—
262	C₂²×C₄	C₂³×D₇₉	1	4	yes	—
263	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
264	C₄⋊C₄	—	79	128	no	—
265	C₄⋊C₄	—	79	32	no	—
266	C₄⋊C₄	—	79	32	no	—
267	C₄⋊C₄	—	79	64	no	—
268	C₄⋊C₄	C₂³×D₇₉	1	32	yes	—
269	C₂²⋊C₄	—	79	32	no	—
270	C₂²⋊C₄	—	79	32	no	—
271	C₂²⋊C₄	C₂³×D₇₉	1	32	yes	—
272	C₄²	—	79	32	no	—
273	C₄²	—	79	32	no	—
274	C₄²	C₂³×D₇₉	1	16	yes	—
275	C₄²	C₂³×D₇₉	1	16	yes	—
276	C₂³	—	2	4	no	—
277	C₂³	—	2	8	no	—
278	C₂³	—	4	16	no	—
279	C₂³	—	4	8	no	—
280	C₂³	C₂²×Dic₁₅₈	1	4	yes	—
281	C₂³	C₂²×Dic₁₅₈	1	8	yes	—
282	C₂³	C₂²×Dic₁₅₈	1	16	yes	—
283	C₂³	C₂⁴×D₇₉	1	2	yes	—
284	C₂³	C₂⁴×D₇₉	1	4	yes	—
285	C₂³	C₂⁴×D₇₉	1	1	yes	characteristic
286	Q₈	—	316	64	no	—
287	C₂×C₄	—	2	16	no	—
288	C₂×C₄	—	4	16	no	—
289	C₂×C₄	—	316	32	no	—
290	C₂×C₄	—	79	32	no	—
291	C₂×C₄	—	158	32	no	—
292	C₂×C₄	C₂²×Dic₁₅₈	1	8	yes	—
293	C₂×C₄	C₂²×Dic₁₅₈	1	4	yes	—
294	C₂×C₄	C₂²×Dic₁₅₈	1	4	yes	—
295	C₂×C₄	C₂²×Dic₁₅₈	1	32	yes	—
296	C₂×C₄	C₂²×Dic₁₅₈	1	16	yes	—
297	C₂×C₄	C₂²×Dic₁₅₈	1	32	yes	—
298	C₂×C₄	C₂²×Dic₁₅₈	1	16	yes	—
299	C₂×C₄	C₂⁴×D₇₉	1	16	yes	—
300	C₂×C₄	C₂⁴×D₇₉	1	2	yes	—
301	C₂×C₄	C₂⁴×D₇₉	1	4	yes	—
302	C₂×C₄	C₂⁴×D₇₉	1	2	yes	—
303	C₂²	—	2	8	no	—
304	C₂²	—	2	16	no	—
305	C₂²	—	2	4	no	—
306	C₂²	—	2	8	no	—
307	C₂²	—	4	16	no	—
308	C₂²	C₂³×Dic₁₅₈	1	4	yes	—
309	C₂²	C₂³×Dic₁₅₈	1	8	yes	—
310	C₂²	C₂³×Dic₁₅₈	1	2	yes	—
311	C₂²	#19099399	1	4	yes	—
312	C₂²	C₂⁵×D₇₉	1	1	yes	characteristic
313	C₄	—	2	16	no	—
314	C₄	—	158	32	no	—
315	C₄	C₂³×Dic₁₅₈	1	8	yes	—
316	C₄	C₂³×Dic₁₅₈	1	4	yes	—
317	C₄	C₂³×Dic₁₅₈	1	4	yes	—
318	C₂	—	2	4	no	—
319	C₂	—	2	8	no	—
320	C₂	#23776805	1	4	yes	direct product factor
321	C₂	C₂⁴×Dic₁₅₈	1	1	yes	characteristic
322	C₂	C₂×#19099399	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂³×D₇₉
commutator, cosocle	C₂²×C₇₉	C₂⁶
Frattini	C₂²	C₂⁵×D₇₉
Fitting	C₂²×#117×C₇₉	C₂
socle	C₂³×C₇₉	C₂⁵
prime radical	C₂⁵×C₇₉	C₂³
hypercenter	C₂²×#117	D₇₉
hypocenter	C₇₉	C₂×#3458
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#117

Complements

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

N	H ≅ G/N	type
1. #23776805	C₂	direct (N × H)
2. C₂×#19099194	C₂	semidirect
3. C₂×#19099191	C₂	semidirect
5. C₂×#19099120	C₂	semidirect
6. C₂×#19099116	C₂	semidirect
9. #19099194	C₂²	semidirect
10. #19099191	C₂²	semidirect
12. #19099120	C₂²	semidirect
13. #19099116	C₂²	semidirect
14. C₂×#14579161	C₂²	semidirect
15. C₂×#14579160	C₂²	semidirect
16. C₂×#14579153	C₂²	semidirect
17. C₂×#14579152	C₂²	semidirect
20. C₂×C₄×Dic₁₅₈	C₂²	semidirect
38. #14579161	C₂³	semidirect
39. #14579160	C₂³	semidirect
40. #14579153	C₂³	semidirect
41. #14579152	C₂³	semidirect
44. C₄×Dic₁₅₈	C₂³	semidirect
61. C₂³×C₄×C₇₉	Q₈	semidirect
100. C₂²×C₄×C₇₉	C₂×Q₈	semidirect
159. C₂×C₇₉	#3458	semidirect
170. C₇₉	C₂×#3458	semidirect
201. C₂³×C₄	Dic₁₅₈	semidirect
254. C₂²×C₄	C₂×Dic₁₅₈	semidirect
320. C₂	#23776805	direct (N × H)

Maximal subgroups

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

subgroup	index
#23776805	2
C₂×#19099194	2
C₂×#19099191	2
C₂×#19099189	2
C₂×#19099120	2
C₂×#19099116	2
C₂²×#14579120	2
C₂²×#117×C₇₉	2
C₂×#3458	79

Subgroups — C2×#23776805 #35195113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23776805 #35195113