Subgroups — #34275954

Order 20224.

750 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23773986	C₂	1	1	yes	maximal, characteristic, complemented
2	#23773979	C₂	1	1	yes	maximal, characteristic
3	#23773959	C₂	1	2	yes	maximal, complemented
4	#23773723	C₂	1	2	yes	maximal, complemented
5	#23763012	C₂	1	2	yes	maximal, complemented
6	#23763010	C₂	1	1	yes	maximal, characteristic, complemented
7	#23763006	C₂	1	1	yes	maximal, characteristic, complemented
8	#23763001	C₂	1	2	yes	maximal, complemented
9	#23762997	C₂	1	1	yes	maximal, characteristic, complemented
10	#23762992	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₈²×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#19099120	C₂²	1	2	yes	—
13	#19099113	C₂²	1	2	yes	—
14	#19098968	C₂²	1	1	yes	characteristic
15	#19098968	C₂²	1	1	yes	characteristic
16	#19098961	C₂²	1	2	yes	—
17	#19098943	C₂²	1	1	yes	characteristic
18	#19098940	C₂²	1	1	yes	characteristic
19	#19098936	C₂²	1	2	yes	—
20	#19098918	—	2	4	no	—
21	#19098914	C₂²	1	2	yes	—
22	#19098914	C₂²	1	2	yes	—
23	#19098914	C₂²	1	2	yes	—
24	#19098914	C₂²	1	2	yes	—
25	#19098910	C₂²	1	2	yes	—
26	#19098910	C₂²	1	2	yes	—
27	#19098910	C₂²	1	2	yes	—
28	#19098910	C₂²	1	2	yes	—
29	#19098804	—	2	2	no	—
30	#19098799	—	2	2	no	—
31	#19098792	—	2	4	no	—
32	C₄×Dic₃₁₆	—	2	2	no	—
33	C₄×#11929613	—	2	4	no	—
34	C₄×D₆₃₂	—	2	2	no	—
35	#19098560	—	2	4	no	—
36	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
37	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
39	C₈²×C₇₉	C₂²	1	1	yes	characteristic
40	C₈²×C₇₉	C₂²	1	1	yes	characteristic
41	C₈²×C₇₉	C₂²	1	1	yes	characteristic
42	C₈²×C₇₉	C₂²	1	1	yes	characteristic
43	C₂×#11929624	—	2	2	no	—
44	C₂×Dic₃₁₆	—	2	2	no	—
45	C₂×#11929613	—	2	4	no	—
46	C₂×D₆₃₂	—	2	2	no	—
47	#14579161	C₂³	1	2	yes	—
48	#14579161	C₂³	1	2	yes	—
49	#14579161	C₂³	1	2	yes	—
50	#14579158	C₂³	1	2	yes	—
51	#14579153	—	2	2	no	—
52	#14579149	—	2	2	no	—
53	#14579140	—	2	2	no	—
54	#14579140	—	2	2	no	—
55	#14579135	—	2	2	no	—
56	#14579135	—	2	2	no	—
57	C₄×Dic₁₅₈	—	2	2	no	—
58	C₄×Dic₁₅₈	—	2	2	no	—
59	C₄×D₃₁₆	—	2	2	no	—
60	C₄×D₃₁₆	—	2	2	no	—
61	#14579120	—	2	2	no	—
62	#14579095	—	2	2	no	—
63	#14579095	—	2	2	no	—
64	#14579093	—	2	4	no	—
65	#14579086	—	2	4	no	—
66	#14579086	—	2	4	no	—
67	#14579086	—	2	4	no	—
68	#14579086	—	2	4	no	—
69	#14579083	—	2	4	no	—
70	#14579083	—	2	4	no	—
71	#14579083	—	2	4	no	—
72	#14579083	—	2	4	no	—
73	C₂²×C₈×C₇₉	D₄	1	2	yes	—
74	C₂²×C₈×C₇₉	D₄	1	2	yes	—
75	C₂²×C₈×C₇₉	D₄	1	2	yes	—
76	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
77	C₄×C₈×C₇₉	D₄	1	2	yes	—
78	C₄×C₈×C₇₉	D₄	1	2	yes	—
79	C₄×C₈×C₇₉	D₄	1	2	yes	—
80	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
81	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
82	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
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	#11929624	—	4	8	no	—
87	C₂×Dic₁₅₈	—	2	2	no	—
88	C₂×Dic₁₅₈	—	2	2	no	—
89	C₂×Dic₁₅₈	—	2	2	no	—
90	C₂×D₃₁₆	—	2	2	no	—
91	C₂×D₃₁₆	—	2	2	no	—
92	C₂×D₃₁₆	—	2	2	no	—
93	C₂×C₂²⋊D₇₉	—	4	2	no	—
94	C₂×C₄×D₇₉	—	4	2	no	—
95	Dic₃₁₆	—	4	4	no	—
96	#11929613	—	4	8	no	—
97	D₆₃₂	—	4	4	no	—
98	C₄⋊Dic₇₉	—	2	2	no	—
99	C₄⋊Dic₇₉	—	2	2	no	—
100	C₄⋊Dic₇₉	—	2	2	no	—
101	C₄⋊Dic₇₉	—	2	2	no	—
102	C₄⋊Dic₇₉	—	2	2	no	—
103	C₄⋊Dic₇₉	—	2	2	no	—
104	Dic₇₉⋊C₄	—	4	2	no	—
105	Dic₇₉⋊C₄	—	4	2	no	—
106	C₂²⋊Dic₇₉	—	4	2	no	—
107	(C₂×D₇₉)⋊C₄	—	4	2	no	—
108	(C₂×D₇₉)⋊C₄	—	4	2	no	—
109	C₄×Dic₇₉	—	4	2	no	—
110	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
113	C₂×C₈×C₇₉	—	2	4	no	—
114	C₂×C₈×C₇₉	—	2	4	no	—
115	C₂×C₈×C₇₉	—	2	4	no	—
116	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
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	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
121	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
122	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
123	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
124	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
125	C₄²×C₇₉	C₂⁴	1	1	yes	characteristic
126	C₂³×C₇₉	#127	1	1	yes	characteristic
127	C₂²×D₇₉	—	4	2	no	—
128	C₂×C₄×C₇₉	—	2	2	no	—
129	C₂×C₄×C₇₉	—	2	2	no	—
130	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
131	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
132	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
133	C₂×C₄×C₇₉	#135	1	4	yes	—
134	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
137	C₂²⋊D₇₉	—	8	4	no	—
138	C₂×Dic₇₉	—	4	2	no	—
139	C₂×Dic₇₉	—	4	2	no	—
140	C₂×Dic₇₉	—	4	2	no	—
141	D₃₁₆	—	4	4	no	—
142	D₃₁₆	—	4	4	no	—
143	D₃₁₆	—	4	4	no	—
144	C₄×D₇₉	—	8	4	no	—
145	Dic₁₅₈	—	4	4	no	—
146	Dic₁₅₈	—	4	4	no	—
147	Dic₁₅₈	—	4	4	no	—
148	C₈×C₇₉	—	2	2	no	—
149	C₈×C₇₉	—	2	2	no	—
150	C₈×C₇₉	—	2	2	no	—
151	C₈×C₇₉	—	2	2	no	—
152	C₈×C₇₉	#135	1	4	yes	—
153	C₈×C₇₉	#135	1	4	yes	—
154	C₂²×C₇₉	—	2	1	no	—
155	C₂²×C₇₉	—	2	1	no	—
156	C₂²×C₇₉	#495	1	2	yes	—
157	C₂²×C₇₉	C₂×#127	1	1	yes	characteristic
158	C₂×D₇₉	—	8	2	no	—
159	C₂×D₇₉	—	8	2	no	—
160	C₂×D₇₉	—	8	2	no	—
161	C₄×C₇₉	—	2	1	no	—
162	C₄×C₇₉	—	2	1	no	—
163	C₄×C₇₉	#495	1	2	yes	—
164	C₄×C₇₉	C₂×#135	1	2	yes	—
165	C₄×C₇₉	C₂×#135	1	2	yes	—
166	C₄×C₇₉	C₂×#135	1	2	yes	—
167	Dic₇₉	—	8	2	no	—
168	Dic₇₉	—	8	2	no	—
169	Dic₇₉	—	8	2	no	—
170	#19600	—	79	1	no	maximal, Sylow
171	C₂×C₇₉	—	2	2	no	—
172	C₂×C₇₉	C₂×#495	1	1	yes	characteristic
173	C₂×C₇₉	#3147	1	1	yes	characteristic
174	C₂×C₇₉	#3147	1	1	yes	characteristic
175	D₇₉	—	16	2	no	—
176	#3170	—	79	1	no	—
177	#3169	—	79	1	no	—
178	#3167	—	79	2	no	—
179	#3131	—	79	2	no	—
180	#1711	—	79	2	no	—
181	#1710	—	79	1	no	—
182	#1709	—	79	1	no	—
183	#1708	—	79	2	no	—
184	#1707	—	79	1	no	—
185	#1706	—	79	1	no	—
186	C₂×C₈²	D₇₉	1	1	yes	characteristic, complemented
187	C₇₉	#19600	1	1	yes	characteristic, complemented, Sylow
188	#533	—	79	2	no	—
189	#532	—	79	2	no	—
190	#500	—	79	1	no	—
191	#500	—	79	1	no	—
192	#498	—	79	2	no	—
193	#494	—	79	1	no	—
194	#493	—	79	1	no	—
195	#492	—	79	2	no	—
196	#488	—	158	4	no	—
197	#487	—	79	2	no	—
198	#487	—	79	2	no	—
199	#487	—	79	2	no	—
200	#487	—	79	2	no	—
201	#486	—	79	2	no	—
202	#486	—	79	2	no	—
203	#486	—	79	2	no	—
204	#486	—	79	2	no	—
205	#467	—	158	2	no	—
206	#466	—	158	2	no	—
207	#465	—	158	4	no	—
208	C₄×Q₁₆	—	158	2	no	—
209	C₄×SD₁₆	—	158	4	no	—
210	C₄×D₈	—	158	2	no	—
211	#416	—	158	4	no	—
212	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
213	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
214	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
215	C₈²	C₂×D₇₉	1	1	yes	characteristic
216	C₈²	C₂×D₇₉	1	1	yes	characteristic
217	C₈²	C₂×D₇₉	1	1	yes	characteristic
218	C₈²	C₂×D₇₉	1	1	yes	characteristic
219	C₂×#41	—	158	2	no	—
220	C₂×Q₁₆	—	158	2	no	—
221	C₂×SD₁₆	—	158	4	no	—
222	C₂×D₈	—	158	2	no	—
223	C₂²×C₈	D₃₁₆	1	2	yes	—
224	C₂²×C₈	D₃₁₆	1	2	yes	—
225	C₂²×C₈	D₃₁₆	1	2	yes	—
226	C₄⋊Q₈	—	79	2	no	—
227	C₄⋊Q₈	—	79	2	no	—
228	C₄⋊Q₈	—	79	2	no	—
229	#127	—	79	2	no	—
230	#125	—	158	2	no	—
231	#124	—	158	2	no	—
232	C₂²⋊Q₈	—	158	2	no	—
233	C₂²⋊Q₈	—	158	2	no	—
234	#121	—	158	2	no	—
235	#121	—	158	2	no	—
236	C₄×Q₈	—	158	2	no	—
237	C₄×Q₈	—	158	2	no	—
238	C₄×D₄	—	158	2	no	—
239	C₄×D₄	—	158	2	no	—
240	#117	—	158	2	no	—
241	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
242	#107	—	158	2	no	—
243	#107	—	158	2	no	—
244	#106	—	158	4	no	—
245	#103	—	158	4	no	—
246	#103	—	158	4	no	—
247	#103	—	158	4	no	—
248	#103	—	158	4	no	—
249	#102	—	158	4	no	—
250	#102	—	158	4	no	—
251	#102	—	158	4	no	—
252	#102	—	158	4	no	—
253	C₄×C₈	D₃₁₆	1	2	yes	—
254	C₄×C₈	D₃₁₆	1	2	yes	—
255	C₄×C₈	D₃₁₆	1	2	yes	—
256	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
257	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
258	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
259	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
260	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
261	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
262	#41	—	316	8	no	—
263	C₂×Q₈	—	158	2	no	—
264	C₂×Q₈	—	158	2	no	—
265	C₂×Q₈	—	158	2	no	—
266	C₂×D₄	—	316	2	no	—
267	C₂×D₄	—	158	2	no	—
268	C₂×D₄	—	158	2	no	—
269	C₂×D₄	—	158	2	no	—
270	C₂²×C₄	—	316	2	no	—
271	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
272	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
273	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
274	Q₁₆	—	316	4	no	—
275	SD₁₆	—	316	8	no	—
276	D₈	—	316	4	no	—
277	C₂×C₈	—	2	4	no	—
278	C₂×C₈	—	2	4	no	—
279	C₂×C₈	—	2	4	no	—
280	C₂×C₈	C₂×D₃₁₆	1	2	yes	—
281	C₂×C₈	C₂×D₃₁₆	1	2	yes	—
282	C₂×C₈	C₂×D₃₁₆	1	2	yes	—
283	C₂×C₈	C₂×D₃₁₆	1	2	yes	—
284	C₂×C₈	C₂×D₃₁₆	1	2	yes	—
285	C₂×C₈	C₂×D₃₁₆	1	2	yes	—
286	C₄⋊C₄	—	316	2	no	—
287	C₄⋊C₄	—	316	2	no	—
288	C₄⋊C₄	—	158	2	no	—
289	C₄⋊C₄	—	158	2	no	—
290	C₄⋊C₄	—	158	2	no	—
291	C₄⋊C₄	—	158	2	no	—
292	C₄⋊C₄	—	158	2	no	—
293	C₄⋊C₄	—	158	2	no	—
294	C₂²⋊C₄	—	316	2	no	—
295	C₂²⋊C₄	—	316	2	no	—
296	C₂²⋊C₄	—	316	2	no	—
297	C₄²	—	316	2	no	—
298	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
299	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
300	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
301	C₄²	C₂³×D₇₉	1	1	yes	characteristic
302	C₂³	—	316	2	no	—
303	C₂³	#14579158	1	1	yes	characteristic
304	Q₈	—	316	4	no	—
305	Q₈	—	316	4	no	—
306	Q₈	—	316	4	no	—
307	D₄	—	316	4	no	—
308	D₄	—	316	4	no	—
309	D₄	—	316	4	no	—
310	D₄	—	632	4	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	316	2	no	—
314	C₂×C₄	—	316	2	no	—
315	C₂×C₄	—	316	2	no	—
316	C₂×C₄	—	632	4	no	—
317	C₂×C₄	#14579158	1	1	yes	characteristic
318	C₂×C₄	#14579158	1	1	yes	characteristic
319	C₂×C₄	#14579158	1	1	yes	characteristic
320	C₂×C₄	#14579183	1	4	yes	—
321	C₂×C₄	C₂²×D₃₁₆	1	1	yes	characteristic
322	C₂×C₄	C₂²×D₃₁₆	1	1	yes	characteristic
323	C₂×C₄	C₂²×D₃₁₆	1	1	yes	characteristic
324	C₈	—	2	2	no	—
325	C₈	—	2	2	no	—
326	C₈	—	2	2	no	—
327	C₈	—	2	2	no	—
328	C₈	#14579183	1	4	yes	—
329	C₈	#14579183	1	4	yes	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	632	2	no	—
333	C₂²	—	632	2	no	—
334	C₂²	—	632	2	no	—
335	C₂²	#19098947	1	2	yes	—
336	C₂²	C₂×#14579158	1	1	yes	characteristic
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	632	2	no	—
340	C₄	—	632	2	no	—
341	C₄	—	632	2	no	—
342	C₄	#19098947	1	2	yes	—
343	C₄	C₂×#14579183	1	2	yes	—
344	C₄	C₂×#14579183	1	2	yes	—
345	C₄	C₂×#14579183	1	2	yes	—
346	C₂	—	2	2	no	—
347	C₂	—	1264	2	no	—
348	C₂	C₂×#19098947	1	1	yes	characteristic
349	C₂	#23773828	1	1	yes	characteristic
350	C₂	#23773828	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14579158
commutator, cosocle	C₄²×C₇₉	C₂⁴
Frattini	C₄²	C₂³×D₇₉
Fitting	C₂×C₈²×C₇₉	C₂
socle	C₂²×C₇₉	C₂×#127
prime radical	#23773979	C₂
hypercenter	C₂×C₈²	D₇₉
hypocenter	C₇₉	#19600
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
1. #23773986	C₂	semidirect
3. #23773959	C₂	semidirect
4. #23773723	C₂	semidirect
5. #23763012	C₂	semidirect
6. #23763010	C₂	semidirect
7. #23763006	C₂	semidirect
8. #23763001	C₂	semidirect
9. #23762997	C₂	semidirect
10. #23762992	C₂	semidirect
11. C₂×C₈²×C₇₉	C₂	semidirect
186. C₂×C₈²	D₇₉	semidirect
187. C₇₉	#19600	semidirect

Maximal subgroups

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

subgroup	index
#23773986	2
#23773979	2
#23773959	2
#23773723	2
#23763012	2
#23763010	2
#23763006	2
#23763001	2
#23762997	2
#23762992	2
C₂×C₈²×C₇₉	2
#19600	79