Subgroups — #34203142

Order 20224.

586 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14579101	C₂	1	1	yes	maximal, characteristic
2	C₂²×#14579101	C₂	1	1	yes	maximal, characteristic
3	#23761489	C₂	1	4	yes	maximal, complemented
4	C₂×#405×C₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#14579101	—	2	1	no	—
6	C₂×#14579101	—	2	1	no	—
7	C₂×#14579101	—	2	1	no	—
8	C₂×#14579101	—	2	2	no	—
9	C₂×#14579101	—	2	2	no	—
10	C₂×#14579101	C₄	1	4	yes	complemented
11	C₂×#14579101	C₄	1	4	yes	complemented
12	C₂×#14579101	C₂²	1	2	yes	—
13	C₂²×C₇₉⋊C₁₆	—	2	1	no	—
14	C₂²×C₇₉⋊C₁₆	C₂²	1	2	yes	—
15	C₂²×M₁₆×C₇₉	C₄	1	1	yes	characteristic
16	C₂³×C₈×C₇₉	C₂²	1	1	yes	characteristic
17	C₂×#117×C₇₉	C₄	1	1	yes	characteristic
18	#405×C₇₉	—	2	1	no	—
19	#405×C₇₉	—	2	1	no	—
20	#405×C₇₉	C₄	1	2	yes	—
21	#405×C₇₉	C₂²	1	2	yes	—
22	C₂×#97×C₇₉	—	2	1	no	—
23	C₂×C₄×C₈×C₇₉	—	2	1	no	—
24	#14579101	—	2	4	no	—
25	#14579101	—	2	4	no	—
26	#14579101	—	2	4	no	—
27	#14579101	—	2	2	no	—
28	#14579101	C₂×C₄	1	4	yes	complemented
29	C₂×C₇₉⋊C₁₆	—	2	4	no	—
30	C₂×C₇₉⋊C₁₆	—	2	2	no	—
31	C₂×C₇₉⋊C₁₆	—	2	2	no	—
32	C₂×C₇₉⋊C₁₆	—	2	2	no	—
33	C₂×C₇₉⋊C₁₆	C₂×C₄	1	4	yes	complemented
34	C₂³×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
35	C₂×M₁₆×C₇₉	—	2	1	no	—
36	C₂×M₁₆×C₇₉	—	2	2	no	—
37	C₂×M₁₆×C₇₉	C₈	1	4	yes	—
38	C₂×M₁₆×C₇₉	C₂×C₄	1	2	yes	—
39	C₂²×C₈×C₇₉	—	2	1	no	—
40	C₂²×C₈×C₇₉	C₂×C₄	1	2	yes	—
41	C₂²×C₈×C₇₉	C₂×C₄	1	2	yes	—
42	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₇₉	D₄	1	2	yes	—
48	C₂²×C₈×C₇₉	Q₈	1	2	yes	—
49	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
50	#117×C₇₉	—	2	1	no	—
51	#117×C₇₉	—	2	1	no	—
52	#117×C₇₉	C₈	1	2	yes	—
53	#117×C₇₉	C₂×C₄	1	2	yes	—
54	C₂×C₄⋊C₄×C₇₉	C₈	1	2	yes	—
55	C₂×C₂²⋊C₄×C₇₉	—	2	1	no	—
56	C₂×C₄²×C₇₉	—	2	1	no	—
57	#97×C₇₉	—	2	2	no	—
58	#97×C₇₉	—	2	2	no	—
59	C₄×C₈×C₇₉	—	2	2	no	—
60	C₄×C₈×C₇₉	—	2	2	no	—
61	C₇₉⋊C₁₆	—	2	4	no	—
62	C₇₉⋊C₁₆	—	2	4	no	—
63	C₂⁴×C₇₉	M₁₆	1	1	yes	characteristic
64	C₂²×C₄×C₇₉	—	2	2	no	—
65	C₂²×C₄×C₇₉	C₄²	1	2	yes	—
66	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₇₉	C₄⋊C₄	1	2	yes	—
69	C₂²×C₄×C₇₉	C₂×C₈	1	2	yes	—
70	C₂²×C₄×C₇₉	C₂×C₈	1	1	yes	characteristic
71	C₂²×C₄×C₇₉	C₂×C₈	1	1	yes	characteristic
72	C₂²×C₄×C₇₉	M₁₆	1	1	yes	characteristic
73	C₂²×C₄×C₇₉	M₁₆	1	2	yes	—
74	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
75	M₁₆×C₇₉	—	2	4	no	—
76	M₁₆×C₇₉	—	2	2	no	—
77	M₁₆×C₇₉	C₂×C₈	1	4	yes	—
78	C₂×C₈×C₇₉	—	2	4	no	—
79	C₂×C₈×C₇₉	—	2	2	no	—
80	C₂×C₈×C₇₉	C₄²	1	2	yes	—
81	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
82	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
83	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
84	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
85	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₈×C₇₉	C₄⋊C₄	1	2	yes	—
88	C₂×C₈×C₇₉	C₄⋊C₄	1	2	yes	—
89	C₂×C₈×C₇₉	C₄⋊C₄	1	2	yes	—
90	C₂×C₈×C₇₉	C₄⋊C₄	1	2	yes	—
91	C₂×C₈×C₇₉	C₄⋊C₄	1	2	yes	—
92	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
98	C₄⋊C₄×C₇₉	—	2	2	no	—
99	C₄⋊C₄×C₇₉	C₂×C₈	1	4	yes	—
100	C₂²⋊C₄×C₇₉	—	2	2	no	—
101	C₂²⋊C₄×C₇₉	—	2	2	no	—
102	C₄²×C₇₉	—	2	2	no	—
103	C₄²×C₇₉	—	2	2	no	—
104	C₂³×C₇₉	—	4	2	no	—
105	C₂³×C₇₉	#97	1	2	yes	—
106	C₂³×C₇₉	C₂²⋊C₈	1	1	yes	characteristic
107	C₂³×C₇₉	C₂²⋊C₈	1	1	yes	characteristic
108	C₂³×C₇₉	C₄⋊C₈	1	2	yes	—
109	C₂³×C₇₉	C₂×M₁₆	1	1	yes	characteristic
110	C₂×C₄×C₇₉	—	2	4	no	—
111	C₂×C₄×C₇₉	—	4	4	no	—
112	C₂×C₄×C₇₉	#95	1	2	yes	—
113	C₂×C₄×C₇₉	#95	1	2	yes	—
114	C₂×C₄×C₇₉	C₄×C₈	1	2	yes	—
115	C₂×C₄×C₇₉	C₄×C₈	1	2	yes	—
116	C₂×C₄×C₇₉	#97	1	2	yes	—
117	C₂×C₄×C₇₉	C₂²⋊C₈	1	1	yes	characteristic
118	C₂×C₄×C₇₉	C₂²⋊C₈	1	1	yes	characteristic
119	C₂×C₄×C₇₉	C₂²⋊C₈	1	2	yes	—
120	C₂×C₄×C₇₉	C₂²⋊C₈	1	2	yes	—
121	C₂×C₄×C₇₉	C₄⋊C₈	1	2	yes	—
122	C₂×C₄×C₇₉	C₄⋊C₈	1	2	yes	—
123	C₂×C₄×C₇₉	C₄⋊C₈	1	2	yes	—
124	C₂×C₄×C₇₉	C₂×C₄²	1	1	yes	characteristic
125	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₇₉	C₂²×C₈	1	1	yes	characteristic
128	C₂×C₄×C₇₉	C₂²×C₈	1	1	yes	characteristic
129	C₂×C₄×C₇₉	C₂×M₁₆	1	1	yes	characteristic
130	C₈×C₇₉	—	2	4	no	—
131	C₈×C₇₉	#95	1	2	yes	—
132	C₈×C₇₉	#95	1	2	yes	—
133	C₈×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₈×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₈×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
136	C₈×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₂²×C₇₉	—	2	2	no	—
138	C₂²×C₇₉	—	2	2	no	—
139	C₂²×C₇₉	—	2	2	no	—
140	C₂²×C₇₉	—	4	2	no	—
141	C₂²×C₇₉	—	4	2	no	—
142	C₂²×C₇₉	#336	1	2	yes	—
143	C₂²×C₇₉	#336	1	2	yes	—
144	C₂²×C₇₉	C₂×#97	1	1	yes	characteristic
145	C₂²×C₇₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
146	C₂²×C₇₉	C₂×C₄⋊C₈	1	1	yes	characteristic
147	C₄×C₇₉	—	4	4	no	—
148	C₄×C₇₉	#336	1	2	yes	—
149	C₄×C₇₉	#336	1	2	yes	—
150	C₄×C₇₉	C₂×#95	1	1	yes	characteristic
151	C₄×C₇₉	C₂×C₄×C₈	1	1	yes	characteristic
152	C₄×C₇₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
153	C₄×C₇₉	C₂×C₄⋊C₈	1	1	yes	characteristic
154	#12554	—	79	1	no	maximal, Sylow
155	C₂×C₇₉	—	2	2	no	—
156	C₂×C₇₉	—	2	2	no	—
157	C₂×C₇₉	—	2	1	no	—
158	C₂×C₇₉	—	2	1	no	—
159	C₂×C₇₉	C₂×#336	1	1	yes	characteristic
160	C₂×C₇₉	#2117	1	2	yes	—
161	C₂²×M₃₂	—	79	1	no	—
162	C₂²×M₃₂	—	79	1	no	—
163	C₂×#405	D₇₉	1	1	yes	characteristic
164	#1379	—	79	4	no	—
165	C₇₉	#12554	1	1	yes	characteristic, complemented, Sylow
166	C₂²×M₁₆	Dic₇₉	1	1	yes	characteristic
167	C₂³×C₈	C₂×D₇₉	1	1	yes	characteristic
168	C₂×#117	Dic₇₉	1	1	yes	characteristic
169	C₂×M₃₂	—	79	4	no	—
170	C₂×M₃₂	—	79	2	no	—
171	C₂×M₃₂	—	79	4	no	—
172	C₂×M₃₂	—	158	1	no	—
173	C₂×M₃₂	—	158	1	no	—
174	C₂×M₃₂	—	158	1	no	—
175	C₂×M₃₂	—	158	2	no	—
176	C₂×M₃₂	—	158	2	no	—
177	C₂²×C₁₆	—	79	2	no	—
178	C₂²×C₁₆	—	158	1	no	—
179	#405	—	2	1	no	—
180	#405	—	2	1	no	—
181	#405	Dic₇₉	1	2	yes	—
182	#405	C₂×D₇₉	1	2	yes	—
183	C₂×#97	—	2	1	no	—
184	C₂×C₄×C₈	—	2	1	no	—
185	C₂³×C₄	C₂×Dic₇₉	1	1	yes	characteristic
186	C₂×M₁₆	—	2	1	no	—
187	C₂×M₁₆	—	2	2	no	—
188	C₂×M₁₆	C₇₉⋊C₈	1	4	yes	—
189	C₂×M₁₆	C₂×Dic₇₉	1	2	yes	—
190	C₂²×C₈	—	2	1	no	—
191	C₂²×C₈	Dic₁₅₈	1	2	yes	—
192	C₂²×C₈	C₄×D₇₉	1	2	yes	—
193	C₂²×C₈	C₄×D₇₉	1	2	yes	—
194	C₂²×C₈	D₃₁₆	1	2	yes	—
195	C₂²×C₈	C₂×Dic₇₉	1	1	yes	characteristic
196	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
197	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
198	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
199	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
200	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
201	#117	—	2	1	no	—
202	#117	—	2	1	no	—
203	#117	C₇₉⋊C₈	1	2	yes	—
204	#117	C₂×Dic₇₉	1	2	yes	—
205	C₂×C₄⋊C₄	C₇₉⋊C₈	1	2	yes	—
206	C₂×C₂²⋊C₄	—	2	1	no	—
207	C₂×C₄²	—	2	1	no	—
208	M₃₂	—	79	4	no	—
209	M₃₂	—	158	4	no	—
210	M₃₂	—	158	4	no	—
211	M₃₂	—	158	4	no	—
212	M₃₂	—	158	2	no	—
213	C₂×C₁₆	—	79	4	no	—
214	C₂×C₁₆	—	158	4	no	—
215	C₂×C₁₆	—	158	2	no	—
216	C₂×C₁₆	—	158	2	no	—
217	C₂×C₁₆	—	158	2	no	—
218	#97	—	2	2	no	—
219	#97	—	2	2	no	—
220	C₄×C₈	—	2	2	no	—
221	C₄×C₈	—	2	2	no	—
222	C₂⁴	#11929608	1	1	yes	characteristic
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	C₄×Dic₇₉	1	2	yes	—
225	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
226	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
227	C₂²×C₄	C₄⋊Dic₇₉	1	2	yes	—
228	C₂²×C₄	C₂×C₇₉⋊C₈	1	2	yes	—
229	C₂²×C₄	C₂×C₇₉⋊C₈	1	1	yes	characteristic
230	C₂²×C₄	C₂×C₇₉⋊C₈	1	1	yes	characteristic
231	C₂²×C₄	#11929608	1	1	yes	characteristic
232	C₂²×C₄	#11929608	1	2	yes	—
233	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
234	M₁₆	—	2	4	no	—
235	M₁₆	—	2	2	no	—
236	M₁₆	C₂×C₇₉⋊C₈	1	4	yes	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	C₄×Dic₇₉	1	2	yes	—
240	C₂×C₈	(C₂×D₇₉)⋊C₄	1	2	yes	—
241	C₂×C₈	(C₂×D₇₉)⋊C₄	1	2	yes	—
242	C₂×C₈	(C₂×D₇₉)⋊C₄	1	2	yes	—
243	C₂×C₈	(C₂×D₇₉)⋊C₄	1	2	yes	—
244	C₂×C₈	C₂²⋊Dic₇₉	1	1	yes	characteristic
245	C₂×C₈	C₂²⋊Dic₇₉	1	1	yes	characteristic
246	C₂×C₈	Dic₇₉⋊C₄	1	2	yes	—
247	C₂×C₈	Dic₇₉⋊C₄	1	2	yes	—
248	C₂×C₈	Dic₇₉⋊C₄	1	2	yes	—
249	C₂×C₈	Dic₇₉⋊C₄	1	2	yes	—
250	C₂×C₈	C₄⋊Dic₇₉	1	2	yes	—
251	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
252	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
253	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
254	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
255	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
256	C₂×C₈	C₂×Dic₁₅₈	1	1	yes	characteristic
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	C₂×C₇₉⋊C₈	1	4	yes	—
259	C₂²⋊C₄	—	2	2	no	—
260	C₂²⋊C₄	—	2	2	no	—
261	C₄²	—	2	2	no	—
262	C₄²	—	2	2	no	—
263	C₁₆	—	158	4	no	—
264	C₁₆	—	158	4	no	—
265	C₂³	—	4	2	no	—
266	C₂³	#14579072	1	2	yes	—
267	C₂³	C₂²⋊(C₇₉⋊C₈)	1	1	yes	characteristic
268	C₂³	C₂²⋊(C₇₉⋊C₈)	1	1	yes	characteristic
269	C₂³	C₄⋊(C₇₉⋊C₈)	1	2	yes	—
270	C₂³	C₂×#11929608	1	1	yes	characteristic
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	4	4	no	—
273	C₂×C₄	C₇₉⋊#95	1	2	yes	—
274	C₂×C₄	C₇₉⋊#95	1	2	yes	—
275	C₂×C₄	C₄×C₇₉⋊C₈	1	2	yes	—
276	C₂×C₄	C₄×C₇₉⋊C₈	1	2	yes	—
277	C₂×C₄	#14579072	1	2	yes	—
278	C₂×C₄	C₂²⋊(C₇₉⋊C₈)	1	2	yes	—
279	C₂×C₄	C₂²⋊(C₇₉⋊C₈)	1	2	yes	—
280	C₂×C₄	C₂²⋊(C₇₉⋊C₈)	1	1	yes	characteristic
281	C₂×C₄	C₂²⋊(C₇₉⋊C₈)	1	1	yes	characteristic
282	C₂×C₄	C₄⋊(C₇₉⋊C₈)	1	2	yes	—
283	C₂×C₄	C₄⋊(C₇₉⋊C₈)	1	2	yes	—
284	C₂×C₄	C₄⋊(C₇₉⋊C₈)	1	2	yes	—
285	C₂×C₄	C₂×C₄×Dic₇₉	1	1	yes	characteristic
286	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
287	C₂×C₄	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
288	C₂×C₄	C₂²×C₇₉⋊C₈	1	1	yes	characteristic
289	C₂×C₄	C₂²×C₇₉⋊C₈	1	1	yes	characteristic
290	C₂×C₄	C₂×#11929608	1	1	yes	characteristic
291	C₈	—	2	4	no	—
292	C₈	C₇₉⋊#95	1	2	yes	—
293	C₈	C₇₉⋊#95	1	2	yes	—
294	C₈	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
295	C₈	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
296	C₈	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
297	C₈	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	4	2	no	—
302	C₂²	—	4	2	no	—
303	C₂²	#19098318	1	2	yes	—
304	C₂²	#19098318	1	2	yes	—
305	C₂²	C₂×#14579072	1	1	yes	characteristic
306	C₂²	C₂×C₂²⋊(C₇₉⋊C₈)	1	1	yes	characteristic
307	C₂²	C₂×C₄⋊(C₇₉⋊C₈)	1	1	yes	characteristic
308	C₄	—	4	4	no	—
309	C₄	#19098318	1	2	yes	—
310	C₄	#19098318	1	2	yes	—
311	C₄	C₂×C₇₉⋊#95	1	1	yes	characteristic
312	C₄	C₂×C₄×C₇₉⋊C₈	1	1	yes	characteristic
313	C₄	C₂×C₂²⋊(C₇₉⋊C₈)	1	1	yes	characteristic
314	C₄	C₂×C₄⋊(C₇₉⋊C₈)	1	1	yes	characteristic
315	C₂	—	2	1	no	—
316	C₂	—	2	2	no	—
317	C₂	—	2	2	no	—
318	C₂	—	2	1	no	—
319	C₂	C₂×#19098318	1	1	yes	characteristic
320	C₂	#23765028	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₇₉
commutator	C₄×C₇₉	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₂×#405×C₇₉	C₂
socle	C₂²×C₇₉	C₂×C₂²⋊C₈
prime radical	C₂⁴×C₇₉	M₁₆
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	C₂×#405	D₇₉
hypocenter	C₇₉	#12554
LCS	G ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
3. #23761489	C₂	semidirect
10. C₂×#14579101	C₄	semidirect
11. C₂×#14579101	C₄	semidirect
28. #14579101	C₂×C₄	semidirect
33. C₂×C₇₉⋊C₁₆	C₂×C₄	semidirect
165. C₇₉	#12554	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14579101	2
#23761489	2
C₂×#405×C₇₉	2
#12554	79