Subgroups — C₇₉×#3242 #23760864

Order 10112.

308 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#544×C₇₉	C₂	1	1	yes	maximal, characteristic
2	#541×C₇₉	C₂	1	1	yes	maximal, characteristic
3	#501×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#499×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
5	#491×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#487×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
7	#485×C₇₉	C₂	1	1	yes	maximal, characteristic
8	#485×C₇₉	C₂	1	1	yes	maximal, characteristic
9	#484×C₇₉	C₂	1	1	yes	maximal, characteristic
10	#483×C₇₉	C₂	1	1	yes	maximal, characteristic
11	#479×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#479×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
13	#477×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
14	#475×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
15	#433×C₇₉	C₂	1	1	yes	maximal, characteristic
16	C₄⋊Q₈×C₇₉	—	2	1	no	—
17	C₄⋊Q₈×C₇₉	C₂²	1	1	yes	characteristic
18	C₄⋊Q₈×C₇₉	C₂²	1	1	yes	characteristic
19	C₄⋊Q₈×C₇₉	C₂²	1	1	yes	characteristic
20	#126×C₇₉	—	2	1	no	—
21	#126×C₇₉	—	2	1	no	—
22	#125×C₇₉	—	2	1	no	—
23	#125×C₇₉	—	2	1	no	—
24	#125×C₇₉	—	2	1	no	—
25	#125×C₇₉	C₂²	1	1	yes	characteristic
26	#125×C₇₉	C₂²	1	1	yes	characteristic
27	#125×C₇₉	C₂²	1	1	yes	characteristic
28	C₂²⋊Q₈×C₇₉	—	2	1	no	—
29	C₂²⋊Q₈×C₇₉	—	2	1	no	—
30	C₂²⋊Q₈×C₇₉	C₂²	1	1	yes	characteristic
31	C₂²⋊Q₈×C₇₉	C₂²	1	1	yes	characteristic
32	C₄×Q₈×C₇₉	C₂²	1	1	yes	characteristic
33	C₄×Q₈×C₇₉	C₂²	1	1	yes	characteristic
34	#117×C₇₉	C₂²	1	1	yes	characteristic
35	#117×C₇₉	C₂²	1	1	yes	characteristic
36	C₂×C₄⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
37	#107×C₇₉	C₂²	1	1	yes	characteristic
38	#107×C₇₉	C₂²	1	1	yes	characteristic
39	#107×C₇₉	C₂²	1	1	yes	characteristic
40	#107×C₇₉	C₂²	1	1	yes	characteristic
41	#106×C₇₉	C₂²	1	1	yes	characteristic
42	#106×C₇₉	C₂²	1	1	yes	characteristic
43	#106×C₇₉	C₂²	1	1	yes	characteristic
44	#106×C₇₉	C₂²	1	1	yes	characteristic
45	C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
46	C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
47	#103×C₇₉	C₂²	1	1	yes	characteristic
48	#103×C₇₉	C₂²	1	1	yes	characteristic
49	#103×C₇₉	C₂²	1	1	yes	characteristic
50	#103×C₇₉	C₂²	1	1	yes	characteristic
51	#103×C₇₉	C₂²	1	1	yes	characteristic
52	#103×C₇₉	C₂²	1	1	yes	characteristic
53	#103×C₇₉	C₂²	1	1	yes	characteristic
54	#103×C₇₉	C₂²	1	1	yes	characteristic
55	C₂²⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
56	C₂²⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
57	#97×C₇₉	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₇₉	—	2	1	no	—
60	C₂×Q₈×C₇₉	—	2	1	no	—
61	C₂×Q₈×C₇₉	C₂³	1	1	yes	characteristic
62	C₂×Q₈×C₇₉	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₇₉	—	2	1	no	—
64	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
65	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
66	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
67	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
68	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
69	C₄⋊C₄×C₇₉	—	2	1	no	—
70	C₄⋊C₄×C₇₉	—	2	1	no	—
71	C₄⋊C₄×C₇₉	—	2	1	no	—
72	C₄⋊C₄×C₇₉	—	2	1	no	—
73	C₄⋊C₄×C₇₉	—	2	1	no	—
74	C₄⋊C₄×C₇₉	—	2	1	no	—
75	C₄⋊C₄×C₇₉	—	2	1	no	—
76	C₄⋊C₄×C₇₉	—	2	1	no	—
77	C₄⋊C₄×C₇₉	—	2	1	no	—
78	C₄⋊C₄×C₇₉	—	2	1	no	—
79	C₄⋊C₄×C₇₉	—	2	1	no	—
80	C₄⋊C₄×C₇₉	—	2	1	no	—
81	C₄⋊C₄×C₇₉	—	2	1	no	—
82	C₄⋊C₄×C₇₉	—	2	1	no	—
83	C₄⋊C₄×C₇₉	—	2	1	no	—
84	C₄⋊C₄×C₇₉	D₄	1	1	yes	characteristic
85	C₄⋊C₄×C₇₉	D₄	1	1	yes	characteristic
86	C₄⋊C₄×C₇₉	C₂³	1	1	yes	characteristic
87	C₄⋊C₄×C₇₉	C₂³	1	1	yes	characteristic
88	C₄⋊C₄×C₇₉	C₂³	1	1	yes	characteristic
89	C₄⋊C₄×C₇₉	C₂³	1	1	yes	characteristic
90	C₄⋊C₄×C₇₉	C₂³	1	1	yes	characteristic
91	C₄⋊C₄×C₇₉	C₂³	1	1	yes	characteristic
92	C₂²⋊C₄×C₇₉	—	2	1	no	—
93	C₂²⋊C₄×C₇₉	—	2	1	no	—
94	C₂²⋊C₄×C₇₉	—	2	1	no	—
95	C₂²⋊C₄×C₇₉	D₄	1	1	yes	characteristic
96	C₂²⋊C₄×C₇₉	D₄	1	1	yes	characteristic
97	C₄²×C₇₉	—	2	1	no	—
98	C₄²×C₇₉	—	2	1	no	—
99	C₄²×C₇₉	—	2	1	no	—
100	C₄²×C₇₉	C₂³	1	1	yes	characteristic
101	C₄²×C₇₉	C₂³	1	1	yes	characteristic
102	C₂³×C₇₉	C₂×D₄	1	1	yes	characteristic
103	Q₈×C₇₉	—	2	1	no	—
104	Q₈×C₇₉	—	2	1	no	—
105	Q₈×C₇₉	—	2	1	no	—
106	Q₈×C₇₉	—	2	1	no	—
107	Q₈×C₇₉	—	8	1	no	—
108	Q₈×C₇₉	—	8	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	—	2	1	no	—
113	C₂×C₄×C₇₉	—	2	1	no	—
114	C₂×C₄×C₇₉	—	2	1	no	—
115	C₂×C₄×C₇₉	—	2	1	no	—
116	C₂×C₄×C₇₉	—	2	1	no	—
117	C₂×C₄×C₇₉	—	2	1	no	—
118	C₂×C₄×C₇₉	—	2	1	no	—
119	C₂×C₄×C₇₉	—	8	1	no	—
120	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
122	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
126	C₈×C₇₉	—	2	1	no	—
127	C₈×C₇₉	—	2	1	no	—
128	C₈×C₇₉	—	2	1	no	—
129	C₈×C₇₉	—	2	1	no	—
130	C₂²×C₇₉	—	2	1	no	—
131	C₂²×C₇₉	—	2	1	no	—
132	C₂²×C₇₉	—	2	1	no	—
133	C₂²×C₇₉	C₂²×D₄	1	1	yes	characteristic
134	C₄×C₇₉	—	2	1	no	—
135	C₄×C₇₉	—	2	1	no	—
136	C₄×C₇₉	—	2	1	no	—
137	C₄×C₇₉	—	2	1	no	—
138	C₄×C₇₉	—	2	1	no	—
139	C₄×C₇₉	—	4	1	no	—
140	C₄×C₇₉	—	4	1	no	—
141	C₄×C₇₉	—	4	1	no	—
142	C₄×C₇₉	—	4	1	no	—
143	C₄×C₇₉	—	4	1	no	—
144	C₄×C₇₉	—	4	1	no	—
145	C₄×C₇₉	—	4	1	no	—
146	C₄×C₇₉	—	4	1	no	—
147	C₄×C₇₉	2₋¹⁺⁴	1	1	yes	characteristic
148	C₄×C₇₉	2₋¹⁺⁴	1	1	yes	characteristic
149	C₂×C₇₉	—	4	1	no	—
150	C₂×C₇₉	#536	1	1	yes	characteristic
151	C₂×C₇₉	#577	1	1	yes	characteristic
152	C₂×C₇₉	#578	1	1	yes	characteristic
153	#3242	C₇₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
154	C₇₉	#3242	1	1	yes	characteristic, direct product factor, Sylow
155	#544	C₂×C₇₉	1	1	yes	characteristic
156	#541	C₂×C₇₉	1	1	yes	characteristic
157	#501	C₂×C₇₉	1	1	yes	characteristic, complemented
158	#499	C₂×C₇₉	1	1	yes	characteristic, complemented
159	#491	C₂×C₇₉	1	1	yes	characteristic, complemented
160	#487	C₂×C₇₉	1	1	yes	characteristic, complemented
161	#485	C₂×C₇₉	1	1	yes	characteristic
162	#485	C₂×C₇₉	1	1	yes	characteristic
163	#484	C₂×C₇₉	1	1	yes	characteristic
164	#483	C₂×C₇₉	1	1	yes	characteristic
165	#479	C₂×C₇₉	1	1	yes	characteristic, complemented
166	#479	C₂×C₇₉	1	1	yes	characteristic, complemented
167	#477	C₂×C₇₉	1	1	yes	characteristic, complemented
168	#475	C₂×C₇₉	1	1	yes	characteristic, complemented
169	#433	C₂×C₇₉	1	1	yes	characteristic
170	C₄⋊Q₈	—	2	1	no	—
171	C₄⋊Q₈	C₂²×C₇₉	1	1	yes	characteristic
172	C₄⋊Q₈	C₂²×C₇₉	1	1	yes	characteristic
173	C₄⋊Q₈	C₂²×C₇₉	1	1	yes	characteristic
174	#126	—	2	1	no	—
175	#126	—	2	1	no	—
176	#125	—	2	1	no	—
177	#125	—	2	1	no	—
178	#125	—	2	1	no	—
179	#125	C₂²×C₇₉	1	1	yes	characteristic
180	#125	C₂²×C₇₉	1	1	yes	characteristic
181	#125	C₂²×C₇₉	1	1	yes	characteristic
182	C₂²⋊Q₈	—	2	1	no	—
183	C₂²⋊Q₈	—	2	1	no	—
184	C₂²⋊Q₈	C₂²×C₇₉	1	1	yes	characteristic
185	C₂²⋊Q₈	C₂²×C₇₉	1	1	yes	characteristic
186	C₄×Q₈	C₂²×C₇₉	1	1	yes	characteristic
187	C₄×Q₈	C₂²×C₇₉	1	1	yes	characteristic
188	#117	C₂²×C₇₉	1	1	yes	characteristic
189	#117	C₂²×C₇₉	1	1	yes	characteristic
190	C₂×C₄⋊C₄	C₂²×C₇₉	1	1	yes	characteristic
191	#107	C₂²×C₇₉	1	1	yes	characteristic
192	#107	C₂²×C₇₉	1	1	yes	characteristic
193	#107	C₂²×C₇₉	1	1	yes	characteristic
194	#107	C₂²×C₇₉	1	1	yes	characteristic
195	#106	C₂²×C₇₉	1	1	yes	characteristic
196	#106	C₂²×C₇₉	1	1	yes	characteristic
197	#106	C₂²×C₇₉	1	1	yes	characteristic
198	#106	C₂²×C₇₉	1	1	yes	characteristic
199	C₄⋊C₈	C₂²×C₇₉	1	1	yes	characteristic
200	C₄⋊C₈	C₂²×C₇₉	1	1	yes	characteristic
201	#103	C₂²×C₇₉	1	1	yes	characteristic
202	#103	C₂²×C₇₉	1	1	yes	characteristic
203	#103	C₂²×C₇₉	1	1	yes	characteristic
204	#103	C₂²×C₇₉	1	1	yes	characteristic
205	#103	C₂²×C₇₉	1	1	yes	characteristic
206	#103	C₂²×C₇₉	1	1	yes	characteristic
207	#103	C₂²×C₇₉	1	1	yes	characteristic
208	#103	C₂²×C₇₉	1	1	yes	characteristic
209	C₂²⋊C₈	C₂²×C₇₉	1	1	yes	characteristic
210	C₂²⋊C₈	C₂²×C₇₉	1	1	yes	characteristic
211	#97	C₂²×C₇₉	1	1	yes	characteristic
212	C₄×C₈	C₂²×C₇₉	1	1	yes	characteristic
213	C₂×Q₈	—	2	1	no	—
214	C₂×Q₈	—	2	1	no	—
215	C₂×Q₈	C₂³×C₇₉	1	1	yes	characteristic
216	C₂×Q₈	C₂³×C₇₉	1	1	yes	characteristic
217	C₂²×C₄	—	2	1	no	—
218	C₂²×C₄	C₂³×C₇₉	1	1	yes	characteristic
219	C₂×C₈	C₂³×C₇₉	1	1	yes	characteristic
220	C₂×C₈	C₂³×C₇₉	1	1	yes	characteristic
221	C₂×C₈	C₂³×C₇₉	1	1	yes	characteristic
222	C₂×C₈	C₂³×C₇₉	1	1	yes	characteristic
223	C₄⋊C₄	—	2	1	no	—
224	C₄⋊C₄	—	2	1	no	—
225	C₄⋊C₄	—	2	1	no	—
226	C₄⋊C₄	—	2	1	no	—
227	C₄⋊C₄	—	2	1	no	—
228	C₄⋊C₄	—	2	1	no	—
229	C₄⋊C₄	—	2	1	no	—
230	C₄⋊C₄	—	2	1	no	—
231	C₄⋊C₄	—	2	1	no	—
232	C₄⋊C₄	—	2	1	no	—
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	—	2	1	no	—
235	C₄⋊C₄	—	2	1	no	—
236	C₄⋊C₄	—	2	1	no	—
237	C₄⋊C₄	—	2	1	no	—
238	C₄⋊C₄	D₄×C₇₉	1	1	yes	characteristic
239	C₄⋊C₄	D₄×C₇₉	1	1	yes	characteristic
240	C₄⋊C₄	C₂³×C₇₉	1	1	yes	characteristic
241	C₄⋊C₄	C₂³×C₇₉	1	1	yes	characteristic
242	C₄⋊C₄	C₂³×C₇₉	1	1	yes	characteristic
243	C₄⋊C₄	C₂³×C₇₉	1	1	yes	characteristic
244	C₄⋊C₄	C₂³×C₇₉	1	1	yes	characteristic
245	C₄⋊C₄	C₂³×C₇₉	1	1	yes	characteristic
246	C₂²⋊C₄	—	2	1	no	—
247	C₂²⋊C₄	—	2	1	no	—
248	C₂²⋊C₄	—	2	1	no	—
249	C₂²⋊C₄	D₄×C₇₉	1	1	yes	characteristic
250	C₂²⋊C₄	D₄×C₇₉	1	1	yes	characteristic
251	C₄²	—	2	1	no	—
252	C₄²	—	2	1	no	—
253	C₄²	—	2	1	no	—
254	C₄²	C₂³×C₇₉	1	1	yes	characteristic
255	C₄²	C₂³×C₇₉	1	1	yes	characteristic
256	C₂³	C₂×D₄×C₇₉	1	1	yes	characteristic
257	Q₈	—	2	1	no	—
258	Q₈	—	2	1	no	—
259	Q₈	—	2	1	no	—
260	Q₈	—	2	1	no	—
261	Q₈	—	8	1	no	—
262	Q₈	—	8	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	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	8	1	no	—
274	C₂×C₄	C₂×D₄×C₇₉	1	1	yes	characteristic
275	C₂×C₄	C₂×D₄×C₇₉	1	1	yes	characteristic
276	C₂×C₄	C₂×D₄×C₇₉	1	1	yes	characteristic
277	C₂×C₄	C₂×D₄×C₇₉	1	1	yes	characteristic
278	C₂×C₄	C₂×D₄×C₇₉	1	1	yes	characteristic
279	C₂×C₄	C₂⁴×C₇₉	1	1	yes	characteristic
280	C₈	—	2	1	no	—
281	C₈	—	2	1	no	—
282	C₈	—	2	1	no	—
283	C₈	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	—	2	1	no	—
286	C₂²	—	2	1	no	—
287	C₂²	C₂²×D₄×C₇₉	1	1	yes	characteristic
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	1	no	—
293	C₄	—	4	1	no	—
294	C₄	—	4	1	no	—
295	C₄	—	4	1	no	—
296	C₄	—	4	1	no	—
297	C₄	—	4	1	no	—
298	C₄	—	4	1	no	—
299	C₄	—	4	1	no	—
300	C₄	—	4	1	no	—
301	C₄	2₋¹⁺⁴×C₇₉	1	1	yes	characteristic
302	C₄	2₋¹⁺⁴×C₇₉	1	1	yes	characteristic
303	C₂	—	4	1	no	—
304	C₂	#536×C₇₉	1	1	yes	characteristic
305	C₂	#577×C₇₉	1	1	yes	characteristic
306	C₂	#578×C₇₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₇₉
prime radical	C₂³×C₇₉	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₉ ◁ #117×C₇₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. #501×C₇₉	C₂	semidirect
4. #499×C₇₉	C₂	semidirect
5. #491×C₇₉	C₂	semidirect
6. #487×C₇₉	C₂	semidirect
11. #479×C₇₉	C₂	semidirect
12. #479×C₇₉	C₂	semidirect
13. #477×C₇₉	C₂	semidirect
14. #475×C₇₉	C₂	semidirect
153. #3242	C₇₉	direct (N × H)
154. C₇₉	#3242	direct (N × H)
157. #501	C₂×C₇₉	semidirect
158. #499	C₂×C₇₉	semidirect
159. #491	C₂×C₇₉	semidirect
160. #487	C₂×C₇₉	semidirect
165. #479	C₂×C₇₉	semidirect
166. #479	C₂×C₇₉	semidirect
167. #477	C₂×C₇₉	semidirect
168. #475	C₂×C₇₉	semidirect

Maximal subgroups

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

subgroup	index
#544×C₇₉	2
#541×C₇₉	2
#501×C₇₉	2
#499×C₇₉	2
#491×C₇₉	2
#487×C₇₉	2
#485×C₇₉	2
#484×C₇₉	2
#483×C₇₉	2
#479×C₇₉	2
#477×C₇₉	2
#475×C₇₉	2
#433×C₇₉	2
#3242	79

Subgroups — C79×#3242 #23760864

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₉×#3242 #23760864