Subgroups — #34275930

Order 20224.

814 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23773958	C₂	1	2	yes	maximal, complemented
2	#23773957	C₂	1	2	yes	maximal, complemented
3	#23773911	C₂	1	1	yes	maximal, characteristic, complemented
4	#23773827	C₂	1	1	yes	maximal, characteristic
5	#23763000	C₂	1	2	yes	maximal, complemented
6	#23762998	C₂	1	1	yes	maximal, characteristic, complemented
7	#23762997	C₂	1	1	yes	maximal, characteristic, complemented
8	#23762989	C₂	1	1	yes	maximal, characteristic, complemented
9	#23762986	C₂	1	2	yes	maximal, complemented
10	C₈²×D₇₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₉×#1706	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14579180	—	2	2	no	—
13	#19099118	C₂²	1	2	yes	—
14	#19099111	C₂²	1	2	yes	—
15	#19098968	C₂²	1	1	yes	characteristic
16	#19098964	—	2	2	no	—
17	#19098961	C₂²	1	2	yes	—
18	#19098959	C₂²	1	2	yes	—
19	#19098959	C₂²	1	2	yes	—
20	#19098946	—	2	2	no	—
21	#19098943	C₂²	1	1	yes	characteristic
22	#19098935	C₂²	1	2	yes	—
23	#19098935	C₂²	1	2	yes	—
24	#19098914	C₂²	1	2	yes	—
25	#19098913	C₂²	1	2	yes	—
26	#19098909	C₂²	1	2	yes	—
27	#19098791	—	2	4	no	—
28	#19098789	—	2	4	no	—
29	#19098654	—	2	4	no	—
30	SD₁₆×Dic₇₉	—	2	4	no	—
31	#19098600	—	2	2	no	—
32	C₄×C₈×D₇₉	C₂²	1	1	yes	characteristic
33	C₄×C₈×D₇₉	C₂²	1	1	yes	characteristic
34	C₄×C₈×D₇₉	C₂²	1	1	yes	characteristic
35	C₈×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
36	C₈×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
37	C₈×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
38	#500×C₇₉	C₂²	1	1	yes	characteristic
39	#493×C₇₉	C₂²	1	1	yes	characteristic
40	#492×C₇₉	C₂²	1	2	yes	—
41	#486×C₇₉	C₂²	1	2	yes	—
42	C₈²×C₇₉	C₂²	1	1	yes	characteristic
43	C₂×#11929623	—	2	2	no	—
44	#14579180	—	4	8	no	—
45	C₂×Dic₃₁₆	—	2	2	no	—
46	C₂×#11929612	—	2	4	no	—
47	C₂×#11929612	—	2	4	no	—
48	C₂×C₈×D₇₉	D₄	1	2	yes	—
49	C₂×C₈×D₇₉	D₄	1	2	yes	—
50	C₂×C₈×D₇₉	D₄	1	2	yes	—
51	#14579161	C₂³	1	2	yes	—
52	#14579161	C₂³	1	2	yes	—
53	#14579152	—	2	2	no	—
54	#14579148	—	2	2	no	—
55	#14579141	—	2	2	no	—
56	#14579137	—	2	2	no	—
57	#14579132	—	2	2	no	—
58	#14579132	—	2	2	no	—
59	Q₈×Dic₇₉	—	2	2	no	—
60	#14579126	—	2	2	no	—
61	D₄×Dic₇₉	—	2	2	no	—
62	D₄×Dic₇₉	—	2	2	no	—
63	#14579119	—	2	2	no	—
64	C₄²×D₇₉	C₂³	1	1	yes	characteristic
65	#14579095	—	2	2	no	—
66	#14579093	—	2	4	no	—
67	#14579092	—	2	4	no	—
68	#14579092	—	2	4	no	—
69	#14579086	—	2	4	no	—
70	#14579085	—	2	4	no	—
71	#14579082	—	2	4	no	—
72	C₄×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
73	C₄×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
74	C₄×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
75	C₈×Dic₇₉	D₄	1	2	yes	—
76	C₈×Dic₇₉	D₄	1	2	yes	—
77	C₈×Dic₇₉	D₄	1	2	yes	—
78	C₂×SD₁₆×C₇₉	—	2	4	no	—
79	C₂×D₈×C₇₉	—	2	2	no	—
80	C₄⋊Q₈×C₇₉	C₂³	1	2	yes	—
81	#127×C₇₉	C₂³	1	2	yes	—
82	#107×C₇₉	—	2	2	no	—
83	#102×C₇₉	—	2	4	no	—
84	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
85	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
86	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
87	#11929623	—	4	8	no	—
88	C₂×Dic₁₅₈	—	2	2	no	—
89	C₂×Dic₁₅₈	—	2	2	no	—
90	C₂×C₂²⋊D₇₉	—	4	2	no	—
91	C₂²×Dic₇₉	—	4	2	no	—
92	C₂×C₄×D₇₉	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄×D₇₉	C₂×D₄	1	1	yes	characteristic
94	C₂×C₄×D₇₉	C₂×D₄	1	1	yes	characteristic
95	Dic₃₁₆	—	4	4	no	—
96	#11929612	—	4	8	no	—
97	#11929612	—	4	8	no	—
98	C₂×C₇₉⋊C₈	C₂×D₄	1	2	yes	—
99	C₂×C₇₉⋊C₈	C₂×D₄	1	2	yes	—
100	C₂×C₇₉⋊C₈	C₂×D₄	1	2	yes	—
101	C₈×D₇₉	—	2	4	no	—
102	C₈×D₇₉	—	2	4	no	—
103	C₈×D₇₉	SD₁₆	1	8	yes	complemented
104	C₄⋊Dic₇₉	—	2	2	no	—
105	C₄⋊Dic₇₉	—	2	2	no	—
106	C₄⋊Dic₇₉	—	2	2	no	—
107	C₄⋊Dic₇₉	—	2	2	no	—
108	C₄⋊Dic₇₉	—	4	2	no	—
109	Dic₇₉⋊C₄	—	4	2	no	—
110	C₂²⋊Dic₇₉	—	4	2	no	—
111	C₂²⋊Dic₇₉	—	4	2	no	—
112	(C₂×D₇₉)⋊C₄	—	4	2	no	—
113	C₄×Dic₇₉	—	4	2	no	—
114	C₄×Dic₇₉	C₂×D₄	1	1	yes	characteristic
115	C₄×Dic₇₉	C₂×D₄	1	1	yes	characteristic
116	C₄×Dic₇₉	C₂×D₄	1	1	yes	characteristic
117	C₂×Q₈×C₇₉	—	2	2	no	—
118	C₂×D₄×C₇₉	—	2	2	no	—
119	C₂×D₄×C₇₉	—	2	2	no	—
120	C₂×D₄×C₇₉	—	2	2	no	—
121	SD₁₆×C₇₉	—	4	8	no	—
122	D₈×C₇₉	—	4	4	no	—
123	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
124	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
125	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
126	C₄⋊C₄×C₇₉	—	2	2	no	—
127	C₄⋊C₄×C₇₉	—	2	2	no	—
128	C₄²×C₇₉	C₂⁴	1	1	yes	characteristic
129	C₂³×C₇₉	—	4	2	no	—
130	C₂²×D₇₉	#127	1	1	yes	characteristic
131	Q₈×C₇₉	—	4	4	no	—
132	D₄×C₇₉	—	4	4	no	—
133	D₄×C₇₉	—	4	4	no	—
134	D₄×C₇₉	—	4	4	no	—
135	C₂×C₄×C₇₉	—	4	2	no	—
136	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
139	C₂²⋊D₇₉	—	8	4	no	—
140	C₂×Dic₇₉	—	4	2	no	—
141	C₂×Dic₇₉	—	4	2	no	—
142	C₂×Dic₇₉	—	8	4	no	—
143	C₂×Dic₇₉	#127	1	1	yes	characteristic
144	C₂×Dic₇₉	#127	1	1	yes	characteristic
145	C₂×Dic₇₉	#127	1	1	yes	characteristic
146	C₄×D₇₉	—	2	2	no	—
147	C₄×D₇₉	—	2	2	no	—
148	C₄×D₇₉	C₂×SD₁₆	1	4	yes	—
149	Dic₁₅₈	—	4	4	no	—
150	Dic₁₅₈	—	4	4	no	—
151	C₈×C₇₉	—	2	2	no	—
152	C₈×C₇₉	C₂×SD₁₆	1	4	yes	—
153	C₈×C₇₉	#135	1	4	yes	complemented
154	C₇₉⋊C₈	—	2	2	no	—
155	C₇₉⋊C₈	C₂×SD₁₆	1	4	yes	—
156	C₇₉⋊C₈	#135	1	4	yes	complemented
157	C₂²×C₇₉	—	8	2	no	—
158	C₂²×C₇₉	—	8	2	no	—
159	C₂²×C₇₉	—	8	2	no	—
160	C₂²×C₇₉	C₂×#127	1	1	yes	characteristic
161	C₂×D₇₉	—	2	1	no	—
162	C₂×D₇₉	—	2	1	no	—
163	C₂×D₇₉	#492	1	2	yes	—
164	C₄×C₇₉	—	8	2	no	—
165	C₄×C₇₉	C₂²×SD₁₆	1	2	yes	—
166	C₄×C₇₉	C₂×#135	1	2	yes	—
167	C₄×C₇₉	C₂×#135	1	2	yes	—
168	Dic₇₉	—	2	1	no	—
169	Dic₇₉	—	2	1	no	—
170	Dic₇₉	—	8	2	no	—
171	Dic₇₉	—	8	2	no	—
172	Dic₇₉	#492	1	2	yes	—
173	#19596	—	79	1	no	maximal, Sylow
174	C₂×C₇₉	—	16	2	no	—
175	C₂×C₇₉	C₂×#492	1	1	yes	characteristic
176	C₂×C₇₉	#3147	1	1	yes	characteristic
177	C₂×C₇₉	#3147	1	1	yes	characteristic
178	D₇₉	—	2	2	no	—
179	#3167	—	79	2	no	—
180	#3167	—	79	2	no	—
181	#3160	—	79	1	no	—
182	#3147	—	79	1	no	—
183	#1708	—	79	2	no	—
184	#1707	—	79	1	no	—
185	#1707	—	79	1	no	—
186	#1706	—	79	1	no	—
187	#1706	D₇₉	1	1	yes	characteristic, complemented
188	#1705	—	79	2	no	—
189	C₂×C₈²	—	79	1	no	—
190	C₇₉	#19596	1	1	yes	characteristic, complemented, Sylow
191	C₂×#135	—	158	2	no	—
192	#533	—	79	2	no	—
193	#532	—	79	2	no	—
194	#500	—	79	1	no	—
195	#500	C₂×D₇₉	1	1	yes	characteristic
196	#499	—	158	2	no	—
197	#498	—	79	2	no	—
198	#498	—	79	2	no	—
199	#498	—	79	2	no	—
200	#495	—	158	2	no	—
201	#494	—	79	1	no	—
202	#493	C₂×D₇₉	1	1	yes	characteristic
203	#492	—	79	2	no	—
204	#492	—	79	2	no	—
205	#492	C₂×D₇₉	1	2	yes	—
206	#487	—	79	2	no	—
207	#487	—	79	2	no	—
208	#486	—	79	2	no	—
209	#486	C₂×D₇₉	1	2	yes	—
210	#465	—	158	4	no	—
211	#465	—	158	4	no	—
212	C₄×SD₁₆	—	158	4	no	—
213	C₄×SD₁₆	—	158	4	no	—
214	#427	—	158	2	no	—
215	C₂×C₄×C₈	—	79	1	no	—
216	C₂×C₄×C₈	—	79	1	no	—
217	C₂×C₄×C₈	—	79	1	no	—
218	C₈²	—	79	1	no	—
219	C₈²	—	79	1	no	—
220	C₈²	—	79	1	no	—
221	C₈²	C₂×D₇₉	1	1	yes	characteristic
222	C₂×#41	—	158	2	no	—
223	#135	—	316	8	no	—
224	C₂×Q₁₆	—	158	2	no	—
225	C₂×SD₁₆	—	2	4	no	—
226	C₂×SD₁₆	—	158	4	no	—
227	C₂×SD₁₆	—	158	4	no	—
228	C₂×D₈	—	2	2	no	—
229	C₂²×C₈	—	79	2	no	—
230	C₂²×C₈	—	79	2	no	—
231	C₂²×C₈	—	79	2	no	—
232	C₄⋊Q₈	—	79	2	no	—
233	C₄⋊Q₈	—	79	2	no	—
234	C₄⋊Q₈	C₂²×D₇₉	1	2	yes	—
235	#127	C₂²×D₇₉	1	2	yes	—
236	#125	—	158	2	no	—
237	#124	—	158	2	no	—
238	C₂²⋊Q₈	—	158	2	no	—
239	C₂²⋊Q₈	—	158	2	no	—
240	#121	—	158	2	no	—
241	#121	—	158	2	no	—
242	C₄×Q₈	—	158	2	no	—
243	C₄×Q₈	—	158	2	no	—
244	C₄×D₄	—	158	2	no	—
245	C₄×D₄	—	158	2	no	—
246	#117	—	158	2	no	—
247	C₂×C₄²	—	79	1	no	—
248	#107	—	2	2	no	—
249	#107	—	158	2	no	—
250	#106	—	158	4	no	—
251	#106	—	158	4	no	—
252	#106	—	158	4	no	—
253	#103	—	158	4	no	—
254	#103	—	158	4	no	—
255	#102	—	2	4	no	—
256	#102	—	158	4	no	—
257	C₄×C₈	—	79	2	no	—
258	C₄×C₈	—	79	2	no	—
259	C₄×C₈	—	79	2	no	—
260	C₄×C₈	—	79	1	no	—
261	C₄×C₈	—	79	1	no	—
262	C₄×C₈	—	79	1	no	—
263	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
264	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
265	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
266	#41	—	316	8	no	—
267	C₂×Q₈	—	2	2	no	—
268	C₂×Q₈	—	158	2	no	—
269	C₂×Q₈	—	158	2	no	—
270	C₂×D₄	—	2	2	no	—
271	C₂×D₄	—	2	2	no	—
272	C₂×D₄	—	2	2	no	—
273	C₂×D₄	—	316	2	no	—
274	C₂²×C₄	—	316	2	no	—
275	C₂²×C₄	—	79	1	no	—
276	C₂²×C₄	—	79	1	no	—
277	C₂²×C₄	—	79	1	no	—
278	Q₁₆	—	316	4	no	—
279	SD₁₆	—	4	8	no	—
280	SD₁₆	—	316	8	no	—
281	SD₁₆	—	316	8	no	—
282	D₈	—	4	4	no	—
283	C₂×C₈	—	79	2	no	—
284	C₂×C₈	—	79	2	no	—
285	C₂×C₈	—	79	8	no	—
286	C₂×C₈	—	79	2	no	—
287	C₂×C₈	—	158	4	no	—
288	C₂×C₈	—	158	4	no	—
289	C₂×C₈	D₄×D₇₉	1	2	yes	—
290	C₂×C₈	D₄×D₇₉	1	2	yes	—
291	C₂×C₈	D₄×D₇₉	1	2	yes	—
292	C₄⋊C₄	—	2	2	no	—
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	—	316	2	no	—
295	C₄⋊C₄	—	316	2	no	—
296	C₄⋊C₄	—	158	2	no	—
297	C₄⋊C₄	—	158	2	no	—
298	C₄⋊C₄	—	158	2	no	—
299	C₄⋊C₄	—	158	2	no	—
300	C₂²⋊C₄	—	316	2	no	—
301	C₂²⋊C₄	—	316	2	no	—
302	C₂²⋊C₄	—	316	2	no	—
303	C₄²	—	316	2	no	—
304	C₄²	—	79	1	no	—
305	C₄²	—	79	1	no	—
306	C₄²	—	79	1	no	—
307	C₄²	C₂³×D₇₉	1	1	yes	characteristic
308	C₂³	—	4	2	no	—
309	C₂³	—	79	1	no	—
310	Q₈	—	4	4	no	—
311	Q₈	—	316	4	no	—
312	Q₈	—	316	4	no	—
313	D₄	—	4	4	no	—
314	D₄	—	4	4	no	—
315	D₄	—	4	4	no	—
316	D₄	—	632	4	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	316	2	no	—
319	C₂×C₄	—	316	2	no	—
320	C₂×C₄	—	79	1	no	—
321	C₂×C₄	—	79	4	no	—
322	C₂×C₄	—	79	1	no	—
323	C₂×C₄	—	79	1	no	—
324	C₂×C₄	—	632	4	no	—
325	C₂×C₄	—	158	2	no	—
326	C₂×C₄	—	158	2	no	—
327	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
328	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
329	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
330	C₈	—	2	2	no	—
331	C₈	—	79	4	no	—
332	C₈	—	79	4	no	—
333	C₈	—	158	2	no	—
334	C₈	SD₁₆×D₇₉	1	4	yes	—
335	C₈	#14579180	1	4	yes	complemented
336	C₂²	—	8	2	no	—
337	C₂²	—	8	2	no	—
338	C₂²	—	8	2	no	—
339	C₂²	—	79	2	no	—
340	C₂²	—	158	1	no	—
341	C₂²	—	158	1	no	—
342	C₂²	#127×D₇₉	1	1	yes	characteristic
343	C₄	—	8	2	no	—
344	C₄	—	79	2	no	—
345	C₄	—	632	2	no	—
346	C₄	—	632	2	no	—
347	C₄	—	158	1	no	—
348	C₄	—	158	1	no	—
349	C₄	C₂×SD₁₆×D₇₉	1	2	yes	—
350	C₄	#135×D₇₉	1	2	yes	—
351	C₄	C₂×#14579180	1	2	yes	—
352	C₂	—	16	2	no	—
353	C₂	—	158	2	no	—
354	C₂	#492×D₇₉	1	1	yes	characteristic
355	C₂	#23773825	1	1	yes	characteristic
356	C₂	#23773827	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#127×D₇₉
commutator, cosocle	C₄²×C₇₉	C₂⁴
Frattini	C₄²	C₂³×D₇₉
Fitting	C₇₉×#1706	C₂
socle	C₂²×C₇₉	C₂×#127
prime radical	#23773827	C₂
hypercenter	#1706	D₇₉
hypocenter	C₇₉	#19596
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1706

Complements

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

N	H ≅ G/N	type
1. #23773958	C₂	semidirect
2. #23773957	C₂	semidirect
3. #23773911	C₂	semidirect
5. #23763000	C₂	semidirect
6. #23762998	C₂	semidirect
7. #23762997	C₂	semidirect
8. #23762989	C₂	semidirect
9. #23762986	C₂	semidirect
10. C₈²×D₇₉	C₂	semidirect
11. C₇₉×#1706	C₂	semidirect
103. C₈×D₇₉	SD₁₆	semidirect
153. C₈×C₇₉	#135	semidirect
156. C₇₉⋊C₈	#135	semidirect
187. #1706	D₇₉	semidirect
190. C₇₉	#19596	semidirect
335. C₈	#14579180	semidirect

Maximal subgroups

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

subgroup	index
#23773958	2
#23773957	2
#23773911	2
#23773827	2
#23763000	2
#23762998	2
#23762997	2
#23762989	2
#23762986	2
C₈²×D₇₉	2
C₇₉×#1706	2
#19596	79