Subgroups — #34276228

Order 20224.

710 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23774349	C₂	1	2	yes	maximal
2	#23773984	C₂	1	1	yes	maximal, characteristic
3	#23773941	C₂	1	2	yes	maximal
4	#500×D₇₉	C₂	1	1	yes	maximal, characteristic
5	#23763079	C₂	1	2	yes	maximal, complemented
6	#23763074	C₂	1	2	yes	maximal, complemented
7	#23763018	C₂	1	1	yes	maximal, characteristic, complemented
8	#23763017	C₂	1	1	yes	maximal, characteristic, complemented
9	#23762994	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×#11929607	C₂	1	1	yes	maximal, characteristic
11	C₇₉×#1707	C₂	1	1	yes	maximal, characteristic, complemented
12	#19099118	C₂²	1	2	yes	—
13	C₄⋊Q₈×D₇₉	C₂²	1	2	yes	—
14	#19098970	—	2	4	no	—
15	#19098968	C₂²	1	1	yes	characteristic
16	#19098968	C₂²	1	1	yes	characteristic
17	#19098968	C₂²	1	2	yes	—
18	#19098967	—	2	2	no	—
19	#19098966	C₂²	1	2	yes	—
20	#19098966	C₂²	1	2	yes	—
21	#19098959	C₂²	1	2	yes	—
22	#19098943	C₂²	1	2	yes	—
23	#19098942	C₂²	1	2	yes	—
24	#19098915	C₂²	1	2	yes	—
25	#19098913	C₂²	1	2	yes	—
26	#19098801	—	2	2	no	—
27	#19098772	—	2	4	no	—
28	#19098720	—	2	4	no	—
29	Q₁₆×Dic₇₉	—	2	2	no	—
30	#19098614	C₂²	1	1	yes	characteristic
31	#19098602	—	2	4	no	—
32	#107×D₇₉	—	2	2	no	—
33	C₄×#11929607	C₂²	1	1	yes	characteristic
34	C₄×C₈×D₇₉	C₂²	1	1	yes	characteristic
35	#19098285	C₂²	1	1	yes	characteristic
36	#19098284	C₂²	1	1	yes	characteristic
37	C₈×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
38	#500×C₇₉	C₂²	1	1	yes	characteristic
39	#498×C₇₉	C₂²	1	2	yes	—
40	#494×C₇₉	C₂²	1	1	yes	characteristic
41	#487×C₇₉	C₂²	1	2	yes	—
42	C₈²×C₇₉	C₂²	1	1	yes	characteristic
43	C₂×Q₈×D₇₉	—	2	2	no	—
44	C₂×Dic₃₁₆	—	2	4	no	—
45	C₂×#11929614	—	2	4	no	—
46	C₂×#11929607	D₄	1	2	yes	—
47	C₂×C₈×D₇₉	D₄	1	2	yes	—
48	#14579161	C₂³	1	2	yes	—
49	#14579161	C₂³	1	2	yes	—
50	#14579160	—	2	2	no	—
51	#14579160	—	2	2	no	—
52	#14579159	—	2	2	no	—
53	#14579152	—	2	2	no	—
54	#14579141	—	2	2	no	—
55	#14579137	—	2	2	no	—
56	Q₈×Dic₇₉	—	2	2	no	—
57	#14579126	—	2	2	no	—
58	#14579119	—	2	2	no	—
59	C₄⋊C₄×D₇₉	—	2	2	no	—
60	C₄⋊C₄×D₇₉	—	2	2	no	—
61	C₄²×D₇₉	C₂³	1	1	yes	characteristic
62	#14579095	—	2	2	no	—
63	#14579095	—	2	2	no	—
64	#14579095	—	2	4	no	—
65	#14579094	—	2	4	no	—
66	#14579094	—	2	4	no	—
67	#14579092	—	2	4	no	—
68	Dic₇₉⋊C₈	D₄	1	2	yes	—
69	#14579085	—	2	4	no	—
70	#14579084	—	2	4	no	—
71	(C₂×D₇₉)⋊C₈	D₄	1	2	yes	—
72	#14579071	D₄	1	2	yes	—
73	C₄×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
74	C₄×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
75	C₄×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
76	C₈×Dic₇₉	D₄	1	2	yes	—
77	C₂×Q₁₆×C₇₉	—	2	2	no	—
78	C₄⋊Q₈×C₇₉	C₂³	1	2	yes	—
79	C₄⋊Q₈×C₇₉	C₂³	1	2	yes	—
80	#107×C₇₉	—	2	2	no	—
81	#106×C₇₉	—	2	4	no	—
82	#103×C₇₉	—	2	4	no	—
83	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
84	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
85	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
86	C₂×Dic₁₅₈	—	2	2	no	—
87	C₂×Dic₁₅₈	—	2	2	no	—
88	C₂×Dic₁₅₈	—	4	2	no	—
89	Q₈×D₇₉	—	8	4	no	—
90	C₂×C₄×D₇₉	—	4	2	no	—
91	C₂×C₄×D₇₉	C₂×D₄	1	1	yes	characteristic
92	C₂×C₄×D₇₉	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄×D₇₉	C₂×D₄	1	1	yes	characteristic
94	Dic₃₁₆	—	8	4	no	—
95	#11929614	—	8	4	no	—
96	#11929607	Q₁₆	1	8	yes	complemented
97	C₂×C₇₉⋊C₈	C₂×D₄	1	2	yes	—
98	C₂×C₇₉⋊C₈	C₂×D₄	1	2	yes	—
99	C₂×C₇₉⋊C₈	C₂×D₄	1	2	yes	—
100	C₈×D₇₉	—	2	4	no	—
101	C₄⋊Dic₇₉	—	2	2	no	—
102	C₄⋊Dic₇₉	—	2	2	no	—
103	C₄⋊Dic₇₉	—	2	2	no	—
104	C₄⋊Dic₇₉	—	2	2	no	—
105	C₄⋊Dic₇₉	—	4	2	no	—
106	Dic₇₉⋊C₄	—	4	2	no	—
107	Dic₇₉⋊C₄	—	4	2	no	—
108	Dic₇₉⋊C₄	—	4	2	no	—
109	(C₂×D₇₉)⋊C₄	—	4	2	no	—
110	C₄×Dic₇₉	—	4	2	no	—
111	C₄×Dic₇₉	C₂×D₄	1	1	yes	characteristic
112	C₄×Dic₇₉	C₂×D₄	1	1	yes	characteristic
113	C₄×Dic₇₉	C₂×D₄	1	1	yes	characteristic
114	C₂×Q₈×C₇₉	—	2	2	no	—
115	C₂×Q₈×C₇₉	—	2	2	no	—
116	Q₁₆×C₇₉	—	4	4	no	—
117	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
118	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
119	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
120	C₄⋊C₄×C₇₉	—	2	2	no	—
121	C₄⋊C₄×C₇₉	—	2	2	no	—
122	C₄⋊C₄×C₇₉	—	2	2	no	—
123	C₄⋊C₄×C₇₉	—	2	2	no	—
124	C₄²×C₇₉	C₂⁴	1	1	yes	characteristic
125	C₂²×D₇₉	#127	1	1	yes	characteristic
126	Q₈×C₇₉	—	4	4	no	—
127	Q₈×C₇₉	—	4	4	no	—
128	C₂×C₄×C₇₉	—	4	2	no	—
129	C₂×C₄×C₇₉	—	4	2	no	—
130	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
133	C₂×Dic₇₉	—	4	2	no	—
134	C₂×Dic₇₉	—	4	2	no	—
135	C₂×Dic₇₉	#127	1	1	yes	characteristic
136	C₂×Dic₇₉	#127	1	1	yes	characteristic
137	C₂×Dic₇₉	#127	1	1	yes	characteristic
138	C₄×D₇₉	—	2	2	no	—
139	C₄×D₇₉	—	2	2	no	—
140	C₄×D₇₉	—	8	4	no	—
141	C₄×D₇₉	C₂×Q₁₆	1	4	yes	—
142	Dic₁₅₈	—	4	4	no	—
143	Dic₁₅₈	—	4	4	no	—
144	Dic₁₅₈	—	8	4	no	—
145	C₈×C₇₉	—	2	2	no	—
146	C₈×C₇₉	C₂×Q₁₆	1	4	yes	—
147	C₈×C₇₉	#137	1	4	yes	—
148	C₇₉⋊C₈	—	2	2	no	—
149	C₇₉⋊C₈	—	2	2	no	—
150	C₇₉⋊C₈	C₂×Q₁₆	1	4	yes	—
151	C₂²×C₇₉	C₂×#127	1	1	yes	characteristic
152	C₂×D₇₉	—	2	1	no	—
153	C₂×D₇₉	—	2	1	no	—
154	C₂×D₇₉	#494	1	2	yes	—
155	C₄×C₇₉	—	8	2	no	—
156	C₄×C₇₉	—	8	2	no	—
157	C₄×C₇₉	C₂²×Q₁₆	1	2	yes	—
158	C₄×C₇₉	C₂×#137	1	2	yes	—
159	C₄×C₇₉	#576	1	2	yes	—
160	Dic₇₉	—	2	1	no	—
161	Dic₇₉	—	2	1	no	—
162	Dic₇₉	—	8	2	no	—
163	Dic₇₉	—	8	2	no	—
164	Dic₇₉	#494	1	2	yes	—
165	#19633	—	79	1	no	maximal, Sylow
166	C₂×C₇₉	C₂×#494	1	1	yes	characteristic
167	C₂×C₇₉	#3152	1	1	yes	characteristic
168	C₂×C₇₉	#3155	1	1	yes	characteristic
169	D₇₉	—	2	2	no	—
170	#3209	—	79	2	no	—
171	#3170	—	79	1	no	—
172	#3165	—	79	2	no	—
173	C₂×#500	—	79	1	no	—
174	#1723	—	79	2	no	—
175	#1723	—	79	2	no	—
176	#1713	—	79	1	no	—
177	#1713	—	79	1	no	—
178	#1707	—	79	1	no	—
179	#1707	D₇₉	1	1	yes	characteristic, complemented
180	C₈×M₁₆	—	79	1	no	—
181	C₇₉	#19633	1	1	yes	characteristic, complemented, Sylow
182	#533	—	79	2	no	—
183	C₂×C₄⋊Q₈	—	79	2	no	—
184	#501	—	158	4	no	—
185	#500	—	79	2	no	—
186	#500	—	79	2	no	—
187	#500	—	79	1	no	—
188	#500	—	79	2	no	—
189	#500	—	79	1	no	—
190	#500	—	158	2	no	—
191	#500	C₂×D₇₉	1	1	yes	characteristic
192	#498	—	79	2	no	—
193	#498	C₂×D₇₉	1	2	yes	—
194	#494	—	79	2	no	—
195	#494	—	79	2	no	—
196	#494	C₂×D₇₉	1	1	yes	characteristic
197	#487	—	79	2	no	—
198	#487	—	79	2	no	—
199	#487	C₂×D₇₉	1	2	yes	—
200	#467	—	158	2	no	—
201	#462	—	158	4	no	—
202	#451	—	158	4	no	—
203	C₄×Q₁₆	—	158	2	no	—
204	#431	—	79	1	no	—
205	#428	—	158	4	no	—
206	C₂×#107	—	158	2	no	—
207	C₄×M₁₆	—	79	1	no	—
208	C₂×C₄×C₈	—	79	1	no	—
209	#322	—	79	1	no	—
210	#322	—	79	1	no	—
211	C₈²	—	79	1	no	—
212	C₈²	C₂×D₇₉	1	1	yes	characteristic
213	C₂²×Q₈	—	158	2	no	—
214	C₂×Q₁₆	—	2	2	no	—
215	C₂×Q₁₆	—	158	4	no	—
216	C₂×Q₁₆	—	158	4	no	—
217	C₂×M₁₆	—	79	2	no	—
218	C₂²×C₈	—	79	2	no	—
219	C₄⋊Q₈	—	79	2	no	—
220	C₄⋊Q₈	—	79	2	no	—
221	C₄⋊Q₈	—	158	2	no	—
222	C₄⋊Q₈	—	158	2	no	—
223	C₄⋊Q₈	—	158	2	no	—
224	C₄⋊Q₈	C₂²×D₇₉	1	2	yes	—
225	C₄⋊Q₈	C₂²×D₇₉	1	2	yes	—
226	#125	—	158	2	no	—
227	C₂²⋊Q₈	—	158	2	no	—
228	C₂²⋊Q₈	—	158	2	no	—
229	C₄×Q₈	—	158	2	no	—
230	C₄×Q₈	—	158	2	no	—
231	#117	—	158	2	no	—
232	C₂×C₄⋊C₄	—	158	2	no	—
233	C₂×C₄⋊C₄	—	158	2	no	—
234	C₂×C₄²	—	79	1	no	—
235	#107	—	2	2	no	—
236	#107	—	158	4	no	—
237	#107	—	158	4	no	—
238	#107	—	158	2	no	—
239	#107	—	158	4	no	—
240	#107	—	158	2	no	—
241	#106	—	2	4	no	—
242	#106	—	158	4	no	—
243	C₄⋊C₈	—	79	2	no	—
244	#103	—	2	4	no	—
245	#103	—	158	4	no	—
246	#103	—	158	4	no	—
247	C₂²⋊C₈	—	79	2	no	—
248	#97	—	79	2	no	—
249	C₄×C₈	—	79	2	no	—
250	C₄×C₈	—	79	1	no	—
251	C₄×C₈	—	79	1	no	—
252	C₄×C₈	—	79	1	no	—
253	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
254	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
255	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
256	C₂×Q₈	—	2	2	no	—
257	C₂×Q₈	—	2	2	no	—
258	C₂×Q₈	—	316	2	no	—
259	C₂×Q₈	—	632	4	no	—
260	C₂×Q₈	—	158	2	no	—
261	C₂×Q₈	—	158	2	no	—
262	C₂²×C₄	—	316	2	no	—
263	C₂²×C₄	—	79	1	no	—
264	C₂²×C₄	—	79	1	no	—
265	C₂²×C₄	—	79	1	no	—
266	Q₁₆	—	4	4	no	—
267	Q₁₆	—	632	4	no	—
268	Q₁₆	—	632	4	no	—
269	M₁₆	—	79	8	no	—
270	C₂×C₈	—	79	2	no	—
271	C₂×C₈	—	79	2	no	—
272	C₂×C₈	—	79	2	no	—
273	C₂×C₈	—	158	4	no	—
274	C₂×C₈	D₄×D₇₉	1	2	yes	—
275	C₂×C₈	D₄×D₇₉	1	2	yes	—
276	C₂×C₈	D₄×D₇₉	1	2	yes	—
277	C₄⋊C₄	—	2	2	no	—
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	—	2	2	no	—
280	C₄⋊C₄	—	2	2	no	—
281	C₄⋊C₄	—	316	2	no	—
282	C₄⋊C₄	—	316	2	no	—
283	C₄⋊C₄	—	316	2	no	—
284	C₄⋊C₄	—	316	2	no	—
285	C₄⋊C₄	—	158	2	no	—
286	C₄⋊C₄	—	158	2	no	—
287	C₄⋊C₄	—	158	2	no	—
288	C₄⋊C₄	—	158	2	no	—
289	C₂²⋊C₄	—	316	2	no	—
290	C₄²	—	316	2	no	—
291	C₄²	—	79	1	no	—
292	C₄²	—	79	1	no	—
293	C₄²	—	79	1	no	—
294	C₄²	C₂³×D₇₉	1	1	yes	characteristic
295	C₂³	—	79	1	no	—
296	Q₈	—	4	4	no	—
297	Q₈	—	4	4	no	—
298	Q₈	—	316	4	no	—
299	Q₈	—	316	4	no	—
300	Q₈	—	632	4	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	316	2	no	—
304	C₂×C₄	—	316	2	no	—
305	C₂×C₄	—	79	1	no	—
306	C₂×C₄	—	79	4	no	—
307	C₂×C₄	—	79	1	no	—
308	C₂×C₄	—	79	1	no	—
309	C₂×C₄	—	632	4	no	—
310	C₂×C₄	—	158	2	no	—
311	C₂×C₄	—	158	2	no	—
312	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
313	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
314	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
315	C₈	—	2	2	no	—
316	C₈	—	79	4	no	—
317	C₈	—	158	2	no	—
318	C₈	—	158	2	no	—
319	C₈	Q₁₆×D₇₉	1	4	yes	—
320	C₈	#14579190	1	4	yes	—
321	C₂²	—	79	2	no	—
322	C₂²	—	158	1	no	—
323	C₂²	—	158	1	no	—
324	C₂²	#127×D₇₉	1	1	yes	characteristic
325	C₄	—	8	2	no	—
326	C₄	—	8	2	no	—
327	C₄	—	79	2	no	—
328	C₄	—	632	2	no	—
329	C₄	—	632	2	no	—
330	C₄	—	158	1	no	—
331	C₄	—	158	1	no	—
332	C₄	C₂×Q₁₆×D₇₉	1	2	yes	—
333	C₄	C₂×#14579190	1	2	yes	—
334	C₄	#19099365	1	2	yes	—
335	C₂	—	158	2	no	—
336	C₂	#494×D₇₉	1	1	yes	characteristic
337	C₂	#23773854	1	1	yes	characteristic
338	C₂	#23773882	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#127×D₇₉
commutator, cosocle	C₄²×C₇₉	C₂⁴
Frattini	C₄²	C₂³×D₇₉
Fitting	C₇₉×#1707	C₂
socle	C₂²×C₇₉	C₂×#127
prime radical	C₂²×D₇₉	#127
hypercenter	#1707	D₇₉
hypocenter	C₇₉	#19633
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1707

Complements

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

N	H ≅ G/N	type
5. #23763079	C₂	semidirect
6. #23763074	C₂	semidirect
7. #23763018	C₂	semidirect
8. #23763017	C₂	semidirect
9. #23762994	C₂	semidirect
11. C₇₉×#1707	C₂	semidirect
96. #11929607	Q₁₆	semidirect
179. #1707	D₇₉	semidirect
181. C₇₉	#19633	semidirect

Maximal subgroups

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

subgroup	index
#23774349	2
#23773984	2
#23773941	2
#500×D₇₉	2
#23763079	2
#23763074	2
#23763018	2
#23763017	2
#23762994	2
C₈×#11929607	2
C₇₉×#1707	2
#19633	79