Subgroups — #34442954

Order 20224.

556 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23773936	C₂	1	1	yes	maximal, characteristic, complemented
2	#23772252	C₂	1	1	yes	maximal, characteristic
3	#23765559	C₂	1	2	yes	maximal, complemented
4	#23765555	C₂	1	2	yes	maximal, complemented
5	#23765545	C₂	1	2	yes	maximal, complemented
6	#23765535	C₂	1	2	yes	maximal, complemented
7	#23765186	C₂	1	1	yes	maximal, characteristic, complemented
8	#23765184	C₂	1	1	yes	maximal, characteristic, complemented
9	#23765158	C₂	1	1	yes	maximal, characteristic, complemented
10	#23765116	C₂	1	1	yes	maximal, characteristic
11	C₂×#357×C₇₉	C₂	1	1	yes	maximal, characteristic
12	#19099194	—	2	1	no	—
13	#19099038	—	2	1	no	—
14	C₂×#14579101	C₂²	1	1	yes	characteristic
15	#19098971	—	2	2	no	—
16	#19098967	C₂²	1	2	yes	complemented
17	#19098964	C₂²	1	2	yes	complemented
18	#19098675	—	2	2	no	—
19	D₈×Dic₇₉	C₂²	1	2	yes	—
20	D₈×Dic₇₉	C₂²	1	2	yes	—
21	#19098603	—	2	1	no	—
22	#19098601	C₂²	1	1	yes	characteristic
23	#19098601	C₂²	1	1	yes	characteristic
24	#19098568	—	2	1	no	—
25	#19098518	C₂²	1	1	yes	characteristic
26	#19098391	C₂²	1	1	yes	characteristic, complemented
27	#19098391	C₂²	1	1	yes	characteristic, complemented
28	#19098390	C₂²	1	2	yes	complemented
29	#19098389	C₂²	1	1	yes	characteristic, complemented
30	#19098389	C₂²	1	1	yes	characteristic, complemented
31	#19098388	C₂²	1	2	yes	complemented
32	Dic₇₉⋊C₁₆	C₂²	1	2	yes	complemented
33	Dic₇₉⋊C₁₆	C₂²	1	2	yes	complemented
34	#19098364	C₂²	1	2	yes	—
35	#19098364	C₂²	1	2	yes	—
36	C₂²×D₈×C₇₉	C₂²	1	1	yes	characteristic
37	C₂²×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
38	C₂×#107×C₇₉	C₂²	1	1	yes	characteristic
39	#357×C₇₉	C₂²	1	2	yes	—
40	#357×C₇₉	C₂²	1	2	yes	—
41	C₂×#11929608	D₄	1	1	yes	characteristic
42	#14579160	—	2	2	no	—
43	#14579160	—	4	1	no	—
44	#14579152	—	2	2	no	—
45	#14579152	—	4	1	no	—
46	#14579140	—	4	2	no	—
47	D₄×Dic₇₉	—	2	2	no	—
48	D₄×Dic₇₉	—	2	2	no	—
49	D₄×Dic₇₉	—	4	1	no	—
50	D₄×Dic₇₉	—	4	1	no	—
51	#14579120	—	2	1	no	—
52	#14579120	—	2	1	no	—
53	#14579120	D₄	1	1	yes	characteristic
54	C₂×C₂²⋊Dic₇₉	—	4	2	no	—
55	#14579101	—	2	2	no	—
56	C₂×C₇₉⋊C₁₆	C₂³	1	2	yes	—
57	#14579095	C₂³	1	1	yes	characteristic
58	#14579095	C₂³	1	1	yes	characteristic
59	#14579095	C₂³	1	1	yes	characteristic
60	#14579095	C₂³	1	1	yes	characteristic
61	#14579094	—	2	2	no	—
62	#14579093	—	2	1	no	—
63	#14579093	—	2	1	no	—
64	#14579092	—	2	2	no	—
65	#14579082	—	2	2	no	—
66	#14579082	—	2	2	no	—
67	#14579071	D₄	1	2	yes	—
68	C₈×Dic₇₉	C₂³	1	2	yes	—
69	C₂²×D₄×C₇₉	—	2	1	no	—
70	C₂×D₈×C₇₉	—	2	1	no	—
71	C₂×D₈×C₇₉	—	2	2	no	—
72	C₂×D₈×C₇₉	Q₈	1	4	yes	complemented
73	C₂×D₈×C₇₉	C₂³	1	2	yes	—
74	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
75	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
76	C₂×C₁₆×C₇₉	—	2	2	no	—
77	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
78	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
79	#107×C₇₉	—	2	1	no	—
80	#107×C₇₉	C₂³	1	2	yes	—
81	C₂×Dic₁₅₈	—	4	2	no	—
82	C₂²×Dic₇₉	—	4	2	no	—
83	C₂²×Dic₇₉	—	4	2	no	—
84	#11929608	—	2	2	no	—
85	C₂×C₇₉⋊C₈	C₂×D₄	1	2	yes	—
86	C₄⋊Dic₇₉	—	2	1	no	—
87	C₄⋊Dic₇₉	—	2	1	no	—
88	C₄⋊Dic₇₉	—	2	1	no	—
89	C₄⋊Dic₇₉	—	2	1	no	—
90	C₄⋊Dic₇₉	—	4	2	no	—
91	C₄⋊Dic₇₉	D₈	1	1	yes	characteristic
92	C₄⋊Dic₇₉	D₈	1	1	yes	characteristic
93	Dic₇₉⋊C₄	—	4	2	no	—
94	Dic₇₉⋊C₄	—	4	2	no	—
95	C₂²⋊Dic₇₉	—	4	2	no	—
96	C₂²⋊Dic₇₉	—	4	2	no	—
97	C₂²⋊Dic₇₉	—	4	1	no	—
98	C₂²⋊Dic₇₉	—	4	1	no	—
99	C₂²⋊Dic₇₉	D₈	1	2	yes	—
100	C₄×Dic₇₉	—	4	1	no	—
101	C₄×Dic₇₉	—	4	1	no	—
102	C₄×Dic₇₉	C₂×D₄	1	2	yes	—
103	C₇₉⋊C₁₆	—	2	2	no	—
104	C₂⁴×C₇₉	—	4	1	no	—
105	C₂×D₄×C₇₉	—	2	4	no	—
106	C₂×D₄×C₇₉	—	2	2	no	—
107	C₂×D₄×C₇₉	—	4	1	no	—
108	C₂×D₄×C₇₉	—	4	2	no	—
109	C₂²×C₄×C₇₉	—	4	1	no	—
110	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
111	D₈×C₇₉	—	2	4	no	—
112	D₈×C₇₉	—	2	2	no	—
113	D₈×C₇₉	C₂×Q₈	1	4	yes	—
114	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
115	C₂×C₈×C₇₉	C₂×Q₈	1	2	yes	—
116	C₂×C₈×C₇₉	#41	1	2	yes	—
117	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
118	C₄⋊C₄×C₇₉	—	2	2	no	—
119	C₄⋊C₄×C₇₉	—	4	1	no	—
120	C₁₆×C₇₉	—	2	1	no	—
121	C₁₆×C₇₉	—	2	1	no	—
122	C₂³×C₇₉	—	4	2	no	—
123	C₂³×C₇₉	—	4	4	no	—
124	C₂³×C₇₉	—	16	1	no	—
125	C₂³×C₇₉	—	16	1	no	—
126	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
127	D₄×C₇₉	—	2	4	no	—
128	D₄×C₇₉	—	4	2	no	—
129	D₄×C₇₉	—	4	4	no	—
130	C₂×C₄×C₇₉	—	4	2	no	—
131	C₂×C₄×C₇₉	—	16	1	no	—
132	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
133	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₇₉	#137	1	2	yes	—
135	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
136	C₂×Dic₇₉	—	4	1	no	—
137	C₂×Dic₇₉	—	4	1	no	—
138	C₂×Dic₇₉	—	4	1	no	—
139	C₂×Dic₇₉	—	4	1	no	—
140	C₂×Dic₇₉	—	16	2	no	—
141	C₂×Dic₇₉	—	16	2	no	—
142	C₂×Dic₇₉	C₂×D₈	1	2	yes	—
143	C₂×Dic₇₉	C₂×D₈	1	2	yes	—
144	Dic₁₅₈	—	16	2	no	—
145	C₈×C₇₉	C₂²⋊Q₈	1	2	yes	—
146	C₈×C₇₉	C₂²×Q₈	1	1	yes	characteristic
147	C₈×C₇₉	C₂×#41	1	1	yes	characteristic
148	C₇₉⋊C₈	—	2	2	no	—
149	C₂²×C₇₉	—	2	1	no	—
150	C₂²×C₇₉	—	2	1	no	—
151	C₂²×C₇₉	—	4	4	no	—
152	C₂²×C₇₉	—	8	2	no	—
153	C₂²×C₇₉	—	8	2	no	—
154	C₂²×C₇₉	—	16	4	no	—
155	C₂²×C₇₉	#474	1	2	yes	—
156	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
157	C₄×C₇₉	—	8	2	no	—
158	C₄×C₇₉	#474	1	2	yes	—
159	C₄×C₇₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
160	C₄×C₇₉	C₂×#137	1	1	yes	characteristic
161	Dic₇₉	—	2	2	no	—
162	Dic₇₉	—	2	2	no	—
163	Dic₇₉	—	8	1	no	—
164	Dic₇₉	—	8	1	no	—
165	Dic₇₉	—	8	1	no	—
166	Dic₇₉	—	8	1	no	—
167	#33631	—	79	1	no	maximal, Sylow
168	C₂×C₇₉	—	2	2	no	—
169	C₂×C₇₉	—	8	4	no	—
170	C₂×C₇₉	C₂×#474	1	1	yes	characteristic
171	C₂×C₇₉	#3415	1	1	yes	characteristic
172	C₂×C₇₉	#3416	1	1	yes	characteristic
173	#3164	—	79	1	no	—
174	#2948	—	79	1	no	—
175	#2228	—	79	2	no	—
176	#2228	—	79	2	no	—
177	#2226	—	79	2	no	—
178	#2224	—	79	2	no	—
179	#2159	—	79	1	no	—
180	#2158	—	79	1	no	—
181	#2151	—	79	1	no	—
182	#2140	—	79	1	no	—
183	C₂×#357	D₇₉	1	1	yes	characteristic
184	C₇₉	#33631	1	1	yes	characteristic, complemented, Sylow
185	C₂²×D₈	C₂×D₇₉	1	1	yes	characteristic
186	#544	—	158	1	no	—
187	#518	—	158	1	no	—
188	C₂×M₃₂	—	79	1	no	—
189	C₂²×C₁₆	C₂×D₇₉	1	1	yes	characteristic
190	#501	—	158	2	no	—
191	#500	—	79	2	no	—
192	#499	—	79	2	no	—
193	#442	—	158	2	no	—
194	C₄×D₈	—	79	2	no	—
195	C₄×D₈	—	79	2	no	—
196	#428	—	158	1	no	—
197	#427	—	79	1	no	—
198	#427	—	79	1	no	—
199	C₂×#107	C₂×D₇₉	1	1	yes	characteristic
200	#418	—	158	1	no	—
201	#405	—	79	1	no	—
202	#367	—	79	1	no	—
203	#367	—	79	1	no	—
204	#367	—	79	2	no	—
205	#366	—	79	1	no	—
206	#366	—	79	1	no	—
207	#366	—	79	2	no	—
208	C₄⋊C₁₆	—	79	2	no	—
209	C₄⋊C₁₆	—	79	2	no	—
210	#357	—	79	2	no	—
211	#357	—	79	2	no	—
212	#357	C₂×D₇₉	1	2	yes	—
213	#357	C₂×D₇₉	1	2	yes	—
214	C₂²×D₄	—	2	1	no	—
215	C₂×D₈	—	2	1	no	—
216	C₂×D₈	—	2	2	no	—
217	C₂×D₈	Dic₁₅₈	1	4	yes	complemented
218	C₂×D₈	C₂²×D₇₉	1	2	yes	—
219	C₂×M₁₆	—	79	1	no	—
220	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
221	C₄⋊Q₈	—	316	1	no	—
222	C₄⋊Q₈	—	158	2	no	—
223	#125	—	316	1	no	—
224	#125	—	158	2	no	—
225	C₂²⋊Q₈	—	316	2	no	—
226	C₄×D₄	—	316	1	no	—
227	C₄×D₄	—	316	1	no	—
228	C₄×D₄	—	158	2	no	—
229	C₄×D₄	—	158	2	no	—
230	#117	—	79	1	no	—
231	#117	—	158	1	no	—
232	#117	—	158	1	no	—
233	C₂×C₄⋊C₄	—	2	1	no	—
234	C₂×C₂²⋊C₄	—	316	2	no	—
235	M₃₂	—	158	2	no	—
236	C₂×C₁₆	—	2	2	no	—
237	C₂×C₁₆	—	79	2	no	—
238	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
239	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
240	#107	—	2	1	no	—
241	#107	—	79	1	no	—
242	#107	—	79	1	no	—
243	#107	—	79	1	no	—
244	#107	—	79	1	no	—
245	#107	—	158	2	no	—
246	#107	C₂²×D₇₉	1	2	yes	—
247	#106	—	158	2	no	—
248	#106	—	158	1	no	—
249	#106	—	158	1	no	—
250	#102	—	158	2	no	—
251	#102	—	158	2	no	—
252	#97	—	79	2	no	—
253	C₄×C₈	—	79	2	no	—
254	C₂⁴	—	4	1	no	—
255	C₂×Q₈	—	316	2	no	—
256	C₂×D₄	—	2	4	no	—
257	C₂×D₄	—	2	2	no	—
258	C₂×D₄	—	4	1	no	—
259	C₂×D₄	—	4	2	no	—
260	C₂²×C₄	—	4	1	no	—
261	C₂²×C₄	—	316	2	no	—
262	C₂²×C₄	—	316	2	no	—
263	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
264	D₈	—	2	2	no	—
265	D₈	—	2	4	no	—
266	D₈	C₂×Dic₁₅₈	1	4	yes	—
267	M₁₆	—	158	2	no	—
268	C₂×C₈	—	79	2	no	—
269	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
270	C₂×C₈	C₂×Dic₁₅₈	1	2	yes	—
271	C₂×C₈	#11929623	1	2	yes	—
272	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	4	1	no	—
275	C₄⋊C₄	—	316	2	no	—
276	C₄⋊C₄	—	316	2	no	—
277	C₄⋊C₄	—	316	2	no	—
278	C₄⋊C₄	—	79	1	no	—
279	C₄⋊C₄	—	79	1	no	—
280	C₄⋊C₄	—	158	1	no	—
281	C₄⋊C₄	—	158	1	no	—
282	C₄⋊C₄	—	158	1	no	—
283	C₄⋊C₄	—	158	1	no	—
284	C₂²⋊C₄	—	316	1	no	—
285	C₂²⋊C₄	—	316	1	no	—
286	C₂²⋊C₄	—	316	2	no	—
287	C₂²⋊C₄	—	316	2	no	—
288	C₂²⋊C₄	—	79	2	no	—
289	C₄²	—	316	1	no	—
290	C₄²	—	316	1	no	—
291	C₄²	—	79	2	no	—
292	C₁₆	—	2	1	no	—
293	C₁₆	—	2	1	no	—
294	C₁₆	—	158	2	no	—
295	C₂³	—	4	2	no	—
296	C₂³	—	4	4	no	—
297	C₂³	—	16	1	no	—
298	C₂³	—	16	1	no	—
299	C₂³	D₈×D₇₉	1	1	yes	characteristic
300	Q₈	—	1264	2	no	—
301	D₄	—	2	4	no	—
302	D₄	—	4	4	no	—
303	D₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	16	1	no	—
306	C₂×C₄	—	316	1	no	—
307	C₂×C₄	—	316	1	no	—
308	C₂×C₄	—	316	1	no	—
309	C₂×C₄	—	316	1	no	—
310	C₂×C₄	—	79	2	no	—
311	C₂×C₄	—	79	2	no	—
312	C₂×C₄	—	1264	2	no	—
313	C₂×C₄	—	1264	2	no	—
314	C₂×C₄	#14579138	1	2	yes	—
315	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
316	C₂×C₄	#14579190	1	2	yes	—
317	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
318	C₈	—	158	2	no	—
319	C₈	#14579138	1	2	yes	—
320	C₈	C₂²×Dic₁₅₈	1	1	yes	characteristic
321	C₈	C₂×#11929623	1	1	yes	characteristic
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	4	4	no	—
325	C₂²	—	8	2	no	—
326	C₂²	—	8	2	no	—
327	C₂²	—	16	4	no	—
328	C₂²	#19098841	1	2	yes	—
329	C₂²	C₂×D₈×D₇₉	1	1	yes	characteristic
330	C₄	—	8	2	no	—
331	C₄	—	632	1	no	—
332	C₄	—	632	1	no	—
333	C₄	—	632	1	no	—
334	C₄	—	632	1	no	—
335	C₄	—	158	2	no	—
336	C₄	—	158	2	no	—
337	C₄	#19098841	1	2	yes	—
338	C₄	C₂×#14579138	1	1	yes	characteristic
339	C₄	C₂×#14579190	1	1	yes	characteristic
340	C₂	—	2	2	no	—
341	C₂	—	8	4	no	—
342	C₂	C₂×#19098841	1	1	yes	characteristic
343	C₂	#23776537	1	1	yes	characteristic
344	C₂	#23776542	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₇₉
commutator, cosocle	C₂×C₈×C₇₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₂×#357×C₇₉	C₂
socle	C₂²×C₇₉	C₂²×D₈
prime radical	C₂²×D₈×C₇₉	C₂²
hypercenter	C₂×#357	D₇₉
hypocenter	C₇₉	#33631
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#357

Complements

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

N	H ≅ G/N	type
1. #23773936	C₂	semidirect
3. #23765559	C₂	semidirect
4. #23765555	C₂	semidirect
5. #23765545	C₂	semidirect
6. #23765535	C₂	semidirect
7. #23765186	C₂	semidirect
8. #23765184	C₂	semidirect
9. #23765158	C₂	semidirect
16. #19098967	C₂²	semidirect
17. #19098964	C₂²	semidirect
26. #19098391	C₂²	semidirect
27. #19098391	C₂²	semidirect
28. #19098390	C₂²	semidirect
29. #19098389	C₂²	semidirect
30. #19098389	C₂²	semidirect
31. #19098388	C₂²	semidirect
32. Dic₇₉⋊C₁₆	C₂²	semidirect
33. Dic₇₉⋊C₁₆	C₂²	semidirect
72. C₂×D₈×C₇₉	Q₈	semidirect
184. C₇₉	#33631	semidirect
217. C₂×D₈	Dic₁₅₈	semidirect

Maximal subgroups

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

subgroup	index
#23773936	2
#23772252	2
#23765559	2
#23765555	2
#23765545	2
#23765535	2
#23765186	2
#23765184	2
#23765158	2
#23765116	2
C₂×#357×C₇₉	2
#33631	79