Subgroups — #34181578

Order 20224.

384 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14579095	C₂	1	1	yes	maximal, characteristic, complemented
2	#23762163	C₂	1	1	yes	maximal, characteristic, complemented
3	#23761925	C₂	1	2	yes	maximal
4	#23761257	C₂	1	2	yes	maximal, complemented
5	C₇₉×#2966	C₂	1	1	yes	maximal, characteristic
6	#19098620	C₂²	1	2	yes	—
7	C₂×#14579095	—	2	1	no	—
8	C₄×C₄⋊Dic₇₉	C₂²	1	2	yes	—
9	#19098326	—	2	1	no	—
10	#19098316	—	2	1	no	—
11	#19098314	—	2	1	no	—
12	#19098300	—	2	2	no	—
13	#19098297	—	2	2	no	—
14	#529×C₇₉	C₂²	1	1	yes	characteristic
15	#446×C₇₉	C₄	1	1	yes	characteristic, complemented
16	C₈×Q₈×C₇₉	C₄	1	1	yes	characteristic, complemented
17	#436×C₇₉	C₄	1	1	yes	characteristic, complemented
18	#435×C₇₉	—	2	1	no	—
19	C₈×D₄×C₇₉	—	2	1	no	—
20	C₈×D₄×C₇₉	C₄	1	1	yes	characteristic, complemented
21	#433×C₇₉	—	2	1	no	—
22	#432×C₇₉	C₂²	1	1	yes	characteristic
23	#405×C₇₉	—	2	1	no	—
24	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊Dic₇₉	—	2	2	no	—
26	C₂×C₄×Dic₇₉	—	2	2	no	—
27	#14579095	—	2	4	no	—
28	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
29	C₂²⋊(C₇₉⋊C₈)	—	2	2	no	—
30	#14579072	—	2	2	no	—
31	C₇₉⋊#95	—	2	2	no	—
32	C₂×M₁₆×C₇₉	—	2	1	no	—
33	C₂²×C₈×C₇₉	—	2	1	no	—
34	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
36	#126×C₇₉	—	2	1	no	—
37	#125×C₇₉	Q₈	1	1	yes	characteristic
38	#124×C₇₉	Q₈	1	1	yes	characteristic
39	#123×C₇₉	—	2	1	no	—
40	C₂²⋊Q₈×C₇₉	D₄	1	1	yes	characteristic
41	#121×C₇₉	D₄	1	1	yes	characteristic
42	C₄×Q₈×C₇₉	C₂×C₄	1	1	yes	characteristic
43	C₄×D₄×C₇₉	—	2	1	no	—
44	C₄×D₄×C₇₉	C₂×C₄	1	1	yes	characteristic
45	#117×C₇₉	—	2	1	no	—
46	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
47	C₄⋊C₈×C₇₉	—	2	1	no	—
48	C₄⋊C₈×C₇₉	—	2	1	no	—
49	C₄⋊C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
50	C₄⋊C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
51	C₂²⋊C₈×C₇₉	—	2	1	no	—
52	C₂²⋊C₈×C₇₉	—	2	1	no	—
53	C₂²⋊C₈×C₇₉	—	2	1	no	—
54	#97×C₇₉	—	2	1	no	—
55	#97×C₇₉	—	2	1	no	—
56	C₄×C₈×C₇₉	—	2	1	no	—
57	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
58	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
59	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
60	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
61	C₂²×Dic₇₉	—	4	2	no	—
62	C₂×C₇₉⋊C₈	—	4	2	no	—
63	C₄⋊Dic₇₉	—	2	4	no	—
64	C₄⋊Dic₇₉	—	4	2	no	—
65	C₄×Dic₇₉	—	8	2	no	—
66	C₂×Q₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
67	C₂×D₄×C₇₉	—	2	1	no	—
68	C₂×D₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₇₉	—	2	1	no	—
70	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
73	M₁₆×C₇₉	—	4	2	no	—
74	C₂×C₈×C₇₉	—	2	2	no	—
75	C₂×C₈×C₇₉	—	2	1	no	—
76	C₂×C₈×C₇₉	—	2	1	no	—
77	C₂×C₈×C₇₉	—	4	1	no	—
78	C₂×C₈×C₇₉	—	4	2	no	—
79	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
83	C₄⋊C₄×C₇₉	—	2	1	no	—
84	C₄⋊C₄×C₇₉	—	2	1	no	—
85	C₄⋊C₄×C₇₉	—	2	1	no	—
86	C₄⋊C₄×C₇₉	—	2	1	no	—
87	C₄⋊C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
88	C₄⋊C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
89	C₂²⋊C₄×C₇₉	—	2	1	no	—
90	C₂²⋊C₄×C₇₉	—	2	1	no	—
91	C₂²⋊C₄×C₇₉	—	2	1	no	—
92	C₂²⋊C₄×C₇₉	—	2	1	no	—
93	C₂²⋊C₄×C₇₉	—	2	1	no	—
94	C₄²×C₇₉	—	2	1	no	—
95	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₇₉	C₂×Q₈	1	1	yes	characteristic
98	C₄²×C₇₉	#41	1	1	yes	characteristic
99	C₂³×C₇₉	—	2	1	no	—
100	C₂³×C₇₉	#121	1	1	yes	characteristic
101	Q₈×C₇₉	—	4	1	no	—
102	D₄×C₇₉	—	4	1	no	—
103	D₄×C₇₉	—	4	1	no	—
104	D₄×C₇₉	—	4	1	no	—
105	C₂×C₄×C₇₉	—	2	1	no	—
106	C₂×C₄×C₇₉	—	2	1	no	—
107	C₂×C₄×C₇₉	—	2	1	no	—
108	C₂×C₄×C₇₉	—	2	1	no	—
109	C₂×C₄×C₇₉	—	2	1	no	—
110	C₂×C₄×C₇₉	—	2	1	no	—
111	C₂×C₄×C₇₉	—	2	1	no	—
112	C₂×C₄×C₇₉	—	4	1	no	—
113	C₂×C₄×C₇₉	—	4	1	no	—
114	C₂×C₄×C₇₉	—	4	1	no	—
115	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₇₉	#117	1	1	yes	characteristic
118	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
119	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
120	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
121	C₂×Dic₇₉	—	4	4	no	—
122	C₂×Dic₇₉	—	8	2	no	—
123	C₂×Dic₇₉	—	8	2	no	—
124	C₈×C₇₉	—	2	1	no	—
125	C₈×C₇₉	—	2	1	no	—
126	C₈×C₇₉	—	2	1	no	—
127	C₈×C₇₉	—	4	1	no	—
128	C₈×C₇₉	—	4	1	no	—
129	C₈×C₇₉	#104	1	2	yes	—
130	C₇₉⋊C₈	—	8	2	no	—
131	C₂²×C₇₉	—	2	1	no	—
132	C₂²×C₇₉	—	2	1	no	—
133	C₂²×C₇₉	—	4	1	no	—
134	C₂²×C₇₉	—	4	1	no	—
135	C₂²×C₇₉	—	4	1	no	—
136	C₂²×C₇₉	#380	1	1	yes	characteristic
137	C₂²×C₇₉	#472	1	1	yes	characteristic
138	C₂²×C₇₉	#473	1	1	yes	characteristic
139	C₄×C₇₉	—	2	1	no	—
140	C₄×C₇₉	—	2	1	no	—
141	C₄×C₇₉	—	2	1	no	—
142	C₄×C₇₉	—	2	1	no	—
143	C₄×C₇₉	—	2	1	no	—
144	C₄×C₇₉	—	4	1	no	—
145	C₄×C₇₉	—	4	1	no	—
146	C₄×C₇₉	—	4	1	no	—
147	C₄×C₇₉	C₂×#104	1	1	yes	characteristic
148	C₄×C₇₉	#421	1	1	yes	characteristic
149	Dic₇₉	—	8	4	no	—
150	#8695	—	79	1	no	maximal, Sylow
151	C₂×C₇₉	—	2	1	no	—
152	C₂×C₇₉	—	2	1	no	—
153	C₂×C₇₉	—	8	1	no	—
154	C₂×C₇₉	#1806	1	1	yes	characteristic
155	C₂×C₇₉	#1812	1	1	yes	characteristic
156	C₂×C₇₉	#1816	1	1	yes	characteristic
157	#2966	D₇₉	1	1	yes	characteristic
158	C₄×#107	—	79	1	no	—
159	#1568	—	79	1	no	—
160	#1509	—	79	2	no	—
161	#1285	—	79	2	no	—
162	C₇₉	#8695	1	1	yes	characteristic, complemented, Sylow
163	#529	C₂×D₇₉	1	1	yes	characteristic
164	#446	Dic₇₉	1	1	yes	characteristic, complemented
165	C₈×Q₈	Dic₇₉	1	1	yes	characteristic, complemented
166	#436	Dic₇₉	1	1	yes	characteristic, complemented
167	#435	—	2	1	no	—
168	C₈×D₄	—	2	1	no	—
169	C₈×D₄	Dic₇₉	1	1	yes	characteristic, complemented
170	#433	—	2	1	no	—
171	#432	—	79	2	no	—
172	#432	C₂×D₇₉	1	1	yes	characteristic
173	C₂×#107	—	158	1	no	—
174	#405	—	2	1	no	—
175	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
176	C₄×C₄⋊C₄	—	79	2	no	—
177	#340	—	158	1	no	—
178	#335	—	158	1	no	—
179	#334	—	158	1	no	—
180	#328	—	158	2	no	—
181	#327	—	158	2	no	—
182	C₂×M₁₆	—	2	1	no	—
183	C₂²×C₈	—	2	1	no	—
184	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
185	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
186	#126	—	2	1	no	—
187	#125	Dic₁₅₈	1	1	yes	characteristic
188	#124	Dic₁₅₈	1	1	yes	characteristic
189	#123	—	2	1	no	—
190	C₂²⋊Q₈	D₃₁₆	1	1	yes	characteristic
191	#121	D₃₁₆	1	1	yes	characteristic
192	C₄×Q₈	C₂×Dic₇₉	1	1	yes	characteristic
193	C₄×D₄	—	2	1	no	—
194	C₄×D₄	C₂×Dic₇₉	1	1	yes	characteristic
195	#117	—	2	1	no	—
196	C₂×C₄⋊C₄	—	158	2	no	—
197	C₂×C₄²	—	158	2	no	—
198	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
199	#107	—	158	4	no	—
200	C₄⋊C₈	—	2	1	no	—
201	C₄⋊C₈	—	2	1	no	—
202	C₄⋊C₈	—	158	2	no	—
203	C₄⋊C₈	C₂×Dic₇₉	1	1	yes	characteristic
204	C₄⋊C₈	C₂×Dic₇₉	1	1	yes	characteristic
205	C₂²⋊C₈	—	2	1	no	—
206	C₂²⋊C₈	—	2	1	no	—
207	C₂²⋊C₈	—	2	1	no	—
208	C₂²⋊C₈	—	158	2	no	—
209	#97	—	2	1	no	—
210	#97	—	2	1	no	—
211	#97	—	158	2	no	—
212	C₄×C₈	—	2	1	no	—
213	C₄×C₈	C₂×Dic₇₉	1	1	yes	characteristic
214	C₄×C₈	C₂×Dic₇₉	1	1	yes	characteristic
215	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
216	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
217	#95	—	158	2	no	—
218	C₂×Q₈	C₄⋊Dic₇₉	1	1	yes	characteristic
219	C₂×D₄	—	2	1	no	—
220	C₂×D₄	C₄⋊Dic₇₉	1	1	yes	characteristic
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	316	2	no	—
223	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
224	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
225	C₂²×C₄	#11929624	1	1	yes	characteristic
226	M₁₆	—	4	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	1	no	—
229	C₂×C₈	—	2	1	no	—
230	C₂×C₈	—	4	1	no	—
231	C₂×C₈	—	4	2	no	—
232	C₂×C₈	—	316	2	no	—
233	C₂×C₈	C₂²⋊Dic₇₉	1	1	yes	characteristic
234	C₂×C₈	C₂²⋊Dic₇₉	1	1	yes	characteristic
235	C₂×C₈	C₂²⋊Dic₇₉	1	1	yes	characteristic
236	C₂×C₈	C₂²⋊Dic₇₉	1	1	yes	characteristic
237	C₄⋊C₄	—	2	1	no	—
238	C₄⋊C₄	—	2	1	no	—
239	C₄⋊C₄	—	2	1	no	—
240	C₄⋊C₄	—	2	1	no	—
241	C₄⋊C₄	—	316	2	no	—
242	C₄⋊C₄	—	158	4	no	—
243	C₄⋊C₄	C₄⋊Dic₇₉	1	1	yes	characteristic
244	C₄⋊C₄	C₄⋊Dic₇₉	1	1	yes	characteristic
245	C₂²⋊C₄	—	2	1	no	—
246	C₂²⋊C₄	—	2	1	no	—
247	C₂²⋊C₄	—	2	1	no	—
248	C₂²⋊C₄	—	2	1	no	—
249	C₂²⋊C₄	—	2	1	no	—
250	C₄²	—	2	1	no	—
251	C₄²	—	632	2	no	—
252	C₄²	C₂²×Dic₇₉	1	1	yes	characteristic
253	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
254	C₄²	C₂×Dic₁₅₈	1	1	yes	characteristic
255	C₄²	#11929624	1	1	yes	characteristic
256	C₂³	—	2	1	no	—
257	C₂³	#14579135	1	1	yes	characteristic
258	Q₈	—	4	1	no	—
259	D₄	—	4	1	no	—
260	D₄	—	4	1	no	—
261	D₄	—	4	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	4	1	no	—
270	C₂×C₄	—	4	1	no	—
271	C₂×C₄	—	4	1	no	—
272	C₂×C₄	—	316	4	no	—
273	C₂×C₄	—	632	2	no	—
274	C₂×C₄	—	632	2	no	—
275	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
276	C₂×C₄	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
277	C₂×C₄	#14579120	1	1	yes	characteristic
278	C₂×C₄	#14579135	1	1	yes	characteristic
279	C₂×C₄	#14579140	1	1	yes	characteristic
280	C₂×C₄	#14579140	1	1	yes	characteristic
281	C₈	—	2	1	no	—
282	C₈	—	2	1	no	—
283	C₈	—	2	1	no	—
284	C₈	—	4	1	no	—
285	C₈	—	4	1	no	—
286	C₈	—	632	2	no	—
287	C₈	#14579088	1	2	yes	—
288	C₂²	—	2	1	no	—
289	C₂²	—	2	1	no	—
290	C₂²	—	4	1	no	—
291	C₂²	—	4	1	no	—
292	C₂²	—	4	1	no	—
293	C₂²	#19098423	1	1	yes	characteristic
294	C₂²	#19098833	1	1	yes	characteristic
295	C₂²	#19098838	1	1	yes	characteristic
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	2	1	no	—
301	C₄	—	4	1	no	—
302	C₄	—	4	1	no	—
303	C₄	—	4	1	no	—
304	C₄	—	632	4	no	—
305	C₄	C₂×#14579088	1	1	yes	characteristic
306	C₄	#19098580	1	1	yes	characteristic
307	C₂	—	2	1	no	—
308	C₂	—	2	1	no	—
309	C₂	—	8	1	no	—
310	C₂	#23763420	1	1	yes	characteristic
311	C₂	#23763446	1	1	yes	characteristic
312	C₂	#23763464	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19098423
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#2966	C₂
socle	C₂²×C₇₉	#380
prime radical	#121×C₇₉	D₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#2966	D₇₉
hypocenter	C₇₉	#8695
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966

Complements

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

N	H ≅ G/N	type
1. C₄×#14579095	C₂	semidirect
2. #23762163	C₂	semidirect
4. #23761257	C₂	semidirect
15. #446×C₇₉	C₄	semidirect
16. C₈×Q₈×C₇₉	C₄	semidirect
17. #436×C₇₉	C₄	semidirect
20. C₈×D₄×C₇₉	C₄	semidirect
162. C₇₉	#8695	semidirect
164. #446	Dic₇₉	semidirect
165. C₈×Q₈	Dic₇₉	semidirect
166. #436	Dic₇₉	semidirect
169. C₈×D₄	Dic₇₉	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14579095	2
#23762163	2
#23761925	2
#23761257	2
C₇₉×#2966	2
#8695	79