Subgroups — #34288635

Order 20224.

616 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23763549	C₂	1	2	yes	maximal, complemented
2	#23763524	C₂	1	2	yes	maximal, complemented
3	#23763504	C₂	1	2	yes	maximal
4	#23763450	C₂	1	1	yes	maximal, characteristic
5	#23763446	C₂	1	1	yes	maximal, characteristic
6	#23763279	C₂	1	2	yes	maximal, complemented
7	#23763200	C₂	1	2	yes	maximal
8	#23763155	C₂	1	2	yes	maximal, complemented
9	C₄×#131×C₇₉	C₂	1	1	yes	maximal, characteristic
10	C₂×#14579097	C₂²	1	2	yes	—
11	#19098601	C₂²	1	2	yes	—
12	#19098547	C₂²	1	2	yes	—
13	#19098537	C₂²	1	2	yes	—
14	#19098518	C₂²	1	4	yes	—
15	#19098334	C₂²	1	4	yes	—
16	#19098323	C₂²	1	4	yes	—
17	C₇₉⋊#338	C₂²	1	2	yes	—
18	#19098320	C₂²	1	2	yes	—
19	#19098319	C₂²	1	4	yes	—
20	C₂×#131×C₇₉	C₂²	1	1	yes	characteristic
21	C₂×#131×C₇₉	C₂²	1	1	yes	characteristic
22	C₄×#41×C₇₉	C₂²	1	1	yes	characteristic
23	#405×C₇₉	C₄	1	2	yes	complemented
24	#405×C₇₉	C₄	1	2	yes	complemented
25	#405×C₇₉	C₂²	1	1	yes	characteristic
26	#405×C₇₉	C₂²	1	1	yes	characteristic
27	C₄×M₁₆×C₇₉	C₄	1	2	yes	complemented
28	C₄×M₁₆×C₇₉	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₈×C₇₉	C₄	1	2	yes	complemented
30	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
31	C₂×#11929608	C₂³	1	4	yes	—
32	#14579120	C₂³	1	4	yes	—
33	#14579097	—	2	2	no	—
34	#14579097	—	2	2	no	—
35	#14579095	—	2	2	no	—
36	#14579093	—	2	2	no	—
37	#14579080	—	2	2	no	—
38	#14579078	—	2	2	no	—
39	#14579076	—	2	4	no	—
40	#14579071	—	2	4	no	—
41	C₈×Dic₇₉	—	2	4	no	—
42	C₂×#41×C₇₉	C₂³	1	1	yes	characteristic
43	#131×C₇₉	—	2	2	no	—
44	#131×C₇₉	—	2	2	no	—
45	#131×C₇₉	—	2	2	no	—
46	#131×C₇₉	—	2	2	no	—
47	C₂×M₁₆×C₇₉	C₂×C₄	1	2	yes	—
48	C₂×M₁₆×C₇₉	C₂×C₄	1	2	yes	—
49	C₂×M₁₆×C₇₉	C₂³	1	1	yes	characteristic
50	C₂×M₁₆×C₇₉	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₇₉	C₂×C₄	1	2	yes	—
52	C₂²×C₈×C₇₉	C₂×C₄	1	2	yes	—
53	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
55	C₄×Q₈×C₇₉	—	2	1	no	—
56	C₄×D₄×C₇₉	—	2	2	no	—
57	C₄×D₄×C₇₉	—	2	1	no	—
58	#117×C₇₉	C₂×C₄	1	2	yes	—
59	#117×C₇₉	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₇₉	C₂×C₄	1	2	yes	—
61	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
62	#97×C₇₉	C₂×C₄	1	2	yes	—
63	#97×C₇₉	C₂×C₄	1	2	yes	—
64	#97×C₇₉	C₂×C₄	1	2	yes	—
65	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
66	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
67	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
68	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
69	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
70	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
71	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
72	#11929608	—	2	4	no	—
73	#11929608	—	2	4	no	—
74	C₂×C₇₉⋊C₈	—	2	4	no	—
75	C₄⋊Dic₇₉	—	2	4	no	—
76	C₂²⋊Dic₇₉	—	2	4	no	—
77	C₄×Dic₇₉	—	2	4	no	—
78	#41×C₇₉	—	2	2	no	—
79	#41×C₇₉	—	2	2	no	—
80	C₂×Q₈×C₇₉	#41	1	1	yes	characteristic
81	C₂×D₄×C₇₉	#41	1	2	yes	—
82	C₂×D₄×C₇₉	#41	1	1	yes	characteristic
83	C₂²×C₄×C₇₉	—	2	2	no	—
84	C₂²×C₄×C₇₉	—	2	1	no	—
85	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
86	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
87	M₁₆×C₇₉	—	2	4	no	—
88	M₁₆×C₇₉	—	2	4	no	—
89	M₁₆×C₇₉	—	2	2	no	—
90	M₁₆×C₇₉	—	2	2	no	—
91	C₂×C₈×C₇₉	—	2	2	no	—
92	C₂×C₈×C₇₉	—	2	2	no	—
93	C₂×C₈×C₇₉	—	2	2	no	—
94	C₂×C₈×C₇₉	—	2	2	no	—
95	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
96	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
97	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₈×C₇₉	#41	1	2	yes	—
102	C₂×C₈×C₇₉	#41	1	2	yes	—
103	C₂×C₈×C₇₉	#41	1	2	yes	—
104	C₂×C₈×C₇₉	#41	1	2	yes	—
105	C₄⋊C₄×C₇₉	—	2	2	no	—
106	C₄⋊C₄×C₇₉	—	2	1	no	—
107	C₂²⋊C₄×C₇₉	—	2	2	no	—
108	C₂²⋊C₄×C₇₉	—	2	1	no	—
109	C₄²×C₇₉	—	2	2	no	—
110	C₄²×C₇₉	—	2	1	no	—
111	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
112	C₄²×C₇₉	C₂²×C₄	1	2	yes	—
113	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
114	C₂³×C₇₉	#117	1	2	yes	—
115	C₂³×C₇₉	C₂×#41	1	1	yes	characteristic
116	Q₈×C₇₉	—	2	2	no	—
117	D₄×C₇₉	—	2	4	no	—
118	D₄×C₇₉	—	2	2	no	—
119	C₂×C₄×C₇₉	—	2	1	no	—
120	C₂×C₄×C₇₉	—	2	2	no	—
121	C₂×C₄×C₇₉	—	2	2	no	—
122	C₂×C₄×C₇₉	—	2	2	no	—
123	C₂×C₄×C₇₉	—	2	1	no	—
124	C₂×C₄×C₇₉	—	4	2	no	—
125	C₂×C₄×C₇₉	—	4	4	no	—
126	C₂×C₄×C₇₉	#117	1	2	yes	—
127	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
128	C₂×C₄×C₇₉	C₂×#41	1	2	yes	—
129	C₂×C₄×C₇₉	C₂×#41	1	2	yes	—
130	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
131	C₂×Dic₇₉	—	4	4	no	—
132	C₈×C₇₉	—	2	2	no	—
133	C₈×C₇₉	—	2	2	no	—
134	C₈×C₇₉	#117	1	1	yes	characteristic
135	C₈×C₇₉	#117	1	1	yes	characteristic
136	C₈×C₇₉	#117	1	1	yes	characteristic
137	C₈×C₇₉	#117	1	2	yes	—
138	C₈×C₇₉	#117	1	2	yes	—
139	C₈×C₇₉	#117	1	1	yes	characteristic
140	C₇₉⋊C₈	—	4	4	no	—
141	C₂²×C₇₉	—	2	2	no	—
142	C₂²×C₇₉	—	4	1	no	—
143	C₂²×C₇₉	—	4	2	no	—
144	C₂²×C₇₉	C₂×#117	1	1	yes	characteristic
145	C₂²×C₇₉	#529	1	2	yes	—
146	C₄×C₇₉	—	2	2	no	—
147	C₄×C₇₉	—	2	2	no	—
148	C₄×C₇₉	—	4	2	no	—
149	C₄×C₇₉	—	4	1	no	—
150	C₄×C₇₉	C₂×#117	1	1	yes	characteristic
151	C₄×C₇₉	C₂×#117	1	1	yes	characteristic
152	C₄×C₇₉	#529	1	2	yes	—
153	Dic₇₉	—	8	4	no	—
154	#20787	—	79	1	no	maximal, Sylow
155	C₂×C₇₉	—	2	2	no	—
156	C₂×C₇₉	—	2	1	no	—
157	C₂×C₇₉	—	4	2	no	—
158	C₂×C₇₉	#2283	1	1	yes	characteristic
159	C₄×#131	D₇₉	1	1	yes	characteristic
160	#1841	—	79	2	no	—
161	#1833	—	79	2	no	—
162	#1827	—	79	2	no	—
163	#1813	—	79	1	no	—
164	#1812	—	79	1	no	—
165	#1772	—	79	2	no	—
166	#1757	—	79	2	no	—
167	#1747	—	79	2	no	—
168	C₇₉	#20787	1	1	yes	characteristic, complemented, Sylow
169	C₂×#131	C₂×D₇₉	1	1	yes	characteristic
170	C₂×#131	C₂×D₇₉	1	1	yes	characteristic
171	C₄×#41	C₂×D₇₉	1	1	yes	characteristic
172	C₂×#108	—	79	2	no	—
173	#427	—	79	2	no	—
174	#413	—	79	2	no	—
175	#410	—	79	2	no	—
176	#405	—	79	4	no	—
177	#405	Dic₇₉	1	2	yes	complemented
178	#405	Dic₇₉	1	2	yes	complemented
179	#405	C₂×D₇₉	1	1	yes	characteristic
180	#405	C₂×D₇₉	1	1	yes	characteristic
181	C₄×M₁₆	Dic₇₉	1	2	yes	complemented
182	C₄×M₁₆	C₂×D₇₉	1	1	yes	characteristic
183	C₂×C₄×C₈	Dic₇₉	1	2	yes	complemented
184	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
185	#344	—	79	4	no	—
186	#339	—	79	4	no	—
187	#338	—	79	2	no	—
188	H(3,ℤ/4)	—	79	2	no	—
189	H(3,ℤ/4)	—	79	4	no	—
190	C₂×#41	C₂²×D₇₉	1	1	yes	characteristic
191	#131	—	2	2	no	—
192	#131	—	2	2	no	—
193	#131	—	2	2	no	—
194	#131	—	2	2	no	—
195	C₂×M₁₆	—	79	4	no	—
196	C₂×M₁₆	C₂×Dic₇₉	1	2	yes	—
197	C₂×M₁₆	C₂×Dic₇₉	1	2	yes	—
198	C₂×M₁₆	C₂²×D₇₉	1	1	yes	characteristic
199	C₂×M₁₆	C₂²×D₇₉	1	1	yes	characteristic
200	C₂²×C₈	C₂×Dic₇₉	1	2	yes	—
201	C₂²×C₈	C₂×Dic₇₉	1	2	yes	—
202	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
203	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
204	C₄×Q₈	—	2	1	no	—
205	C₄×D₄	—	2	1	no	—
206	C₄×D₄	—	2	2	no	—
207	#117	—	79	4	no	—
208	#117	C₂×Dic₇₉	1	2	yes	—
209	#117	C₂²×D₇₉	1	1	yes	characteristic
210	C₂×C₄²	C₂×Dic₇₉	1	2	yes	—
211	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
212	#108	—	158	2	no	—
213	#108	—	158	2	no	—
214	#107	—	158	2	no	—
215	#106	—	158	2	no	—
216	#101	—	158	2	no	—
217	#100	—	158	2	no	—
218	#99	—	158	4	no	—
219	#97	—	158	4	no	—
220	#97	C₂×Dic₇₉	1	2	yes	—
221	#97	C₂×Dic₇₉	1	2	yes	—
222	#97	C₂×Dic₇₉	1	2	yes	—
223	C₄×C₈	—	158	4	no	—
224	C₄×C₈	C₂×Dic₇₉	1	1	yes	characteristic
225	C₄×C₈	C₂×Dic₇₉	1	2	yes	—
226	C₄×C₈	C₂×Dic₇₉	1	2	yes	—
227	C₄×C₈	C₂×Dic₇₉	1	2	yes	—
228	C₄×C₈	C₂×Dic₇₉	1	1	yes	characteristic
229	C₄×C₈	C₂×Dic₇₉	1	1	yes	characteristic
230	C₄×C₈	C₂×Dic₇₉	1	1	yes	characteristic
231	#41	—	2	2	no	—
232	#41	—	2	2	no	—
233	C₂×Q₈	#11929624	1	1	yes	characteristic
234	C₂×D₄	#11929624	1	1	yes	characteristic
235	C₂×D₄	#11929624	1	2	yes	—
236	C₂²×C₄	—	2	1	no	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	C₂²×Dic₇₉	1	2	yes	—
239	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
240	M₁₆	—	2	2	no	—
241	M₁₆	—	2	2	no	—
242	M₁₆	—	2	4	no	—
243	M₁₆	—	2	4	no	—
244	M₁₆	—	158	4	no	—
245	M₁₆	—	158	4	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	158	4	no	—
251	C₂×C₈	C₂²×Dic₇₉	1	2	yes	—
252	C₂×C₈	C₂²×Dic₇₉	1	2	yes	—
253	C₂×C₈	C₂²×Dic₇₉	1	1	yes	characteristic
254	C₂×C₈	C₂²×Dic₇₉	1	1	yes	characteristic
255	C₂×C₈	C₂²×Dic₇₉	1	1	yes	characteristic
256	C₂×C₈	C₂²×Dic₇₉	1	1	yes	characteristic
257	C₂×C₈	#11929624	1	2	yes	—
258	C₂×C₈	#11929624	1	2	yes	—
259	C₂×C₈	#11929624	1	2	yes	—
260	C₂×C₈	#11929624	1	2	yes	—
261	C₄⋊C₄	—	2	1	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	158	4	no	—
264	C₂²⋊C₄	—	2	1	no	—
265	C₂²⋊C₄	—	2	2	no	—
266	C₂²⋊C₄	—	158	4	no	—
267	C₄²	—	2	1	no	—
268	C₄²	—	2	2	no	—
269	C₄²	—	158	4	no	—
270	C₄²	C₂²×Dic₇₉	1	1	yes	characteristic
271	C₄²	C₂²×Dic₇₉	1	1	yes	characteristic
272	C₄²	C₂²×Dic₇₉	1	2	yes	—
273	C₂³	#14579120	1	2	yes	—
274	C₂³	C₂×#11929624	1	1	yes	characteristic
275	Q₈	—	2	2	no	—
276	D₄	—	2	4	no	—
277	D₄	—	2	2	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	4	no	—
285	C₂×C₄	—	316	4	no	—
286	C₂×C₄	#14579120	1	2	yes	—
287	C₂×C₄	C₂³×Dic₇₉	1	1	yes	characteristic
288	C₂×C₄	C₂×#11929624	1	2	yes	—
289	C₂×C₄	C₂×#11929624	1	2	yes	—
290	C₂×C₄	C₂×#11929624	1	1	yes	characteristic
291	C₈	—	2	2	no	—
292	C₈	—	2	2	no	—
293	C₈	—	316	4	no	—
294	C₈	#14579120	1	1	yes	characteristic
295	C₈	#14579120	1	1	yes	characteristic
296	C₈	#14579120	1	1	yes	characteristic
297	C₈	#14579120	1	2	yes	—
298	C₈	#14579120	1	2	yes	—
299	C₈	#14579120	1	1	yes	characteristic
300	C₂²	—	2	2	no	—
301	C₂²	—	4	1	no	—
302	C₂²	—	4	2	no	—
303	C₂²	C₂×#14579120	1	1	yes	characteristic
304	C₂²	#19099097	1	2	yes	—
305	C₄	—	2	2	no	—
306	C₄	—	2	2	no	—
307	C₄	—	4	1	no	—
308	C₄	—	4	2	no	—
309	C₄	—	632	4	no	—
310	C₄	C₂×#14579120	1	1	yes	characteristic
311	C₄	C₂×#14579120	1	1	yes	characteristic
312	C₄	#19099097	1	2	yes	—
313	C₂	—	2	1	no	—
314	C₂	—	2	2	no	—
315	C₂	—	4	2	no	—
316	C₂	#23765798	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#14579120
commutator	C₂×C₄×C₇₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₉
Fitting	C₄×#131×C₇₉	C₂
socle	C₂×C₇₉	#2283
prime radical	C₂×#41×C₇₉	C₂³
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	C₄×#131	D₇₉
hypocenter	C₇₉	#20787
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₈ ◁ #405 ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
1. #23763549	C₂	semidirect
2. #23763524	C₂	semidirect
6. #23763279	C₂	semidirect
8. #23763155	C₂	semidirect
23. #405×C₇₉	C₄	semidirect
24. #405×C₇₉	C₄	semidirect
27. C₄×M₁₆×C₇₉	C₄	semidirect
29. C₂×C₄×C₈×C₇₉	C₄	semidirect
168. C₇₉	#20787	semidirect
177. #405	Dic₇₉	semidirect
178. #405	Dic₇₉	semidirect
181. C₄×M₁₆	Dic₇₉	semidirect
183. C₂×C₄×C₈	Dic₇₉	semidirect

Maximal subgroups

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

subgroup	index
#23763549	2
#23763524	2
#23763504	2
#23763450	2
#23763446	2
#23763279	2
#23763200	2
#23763155	2
C₄×#131×C₇₉	2
#20787	79