Subgroups — C₇₉×#3274 #23760896

Order 10112.

556 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×AGL(1,ℤ/8)×C₇₉	C₂	1	2	yes	maximal, complemented
2	#550×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄²×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#493×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
5	#469×C₇₉	C₂	1	2	yes	maximal, complemented
6	#460×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
7	#459×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#449×C₇₉	C₂	1	2	yes	maximal, complemented
9	#447×C₇₉	C₂	1	2	yes	maximal, complemented
10	C₄×SD₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
11	#436×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₇₉	C₂²	1	2	yes	complemented
13	C₂²×D₄×C₇₉	—	2	1	no	—
14	C₂²×D₄×C₇₉	C₂²	1	2	yes	—
15	AGL(1,ℤ/8)×C₇₉	—	2	8	no	—
16	C₂×SD₁₆×C₇₉	C₂²	1	1	yes	characteristic
17	C₂×SD₁₆×C₇₉	C₂²	1	2	yes	complemented
18	C₂×D₈×C₇₉	C₂²	1	2	yes	—
19	C₂×D₈×C₇₉	C₂²	1	2	yes	complemented
20	C₂×D₈×C₇₉	C₂²	1	2	yes	complemented
21	C₂×M₁₆×C₇₉	C₂²	1	2	yes	complemented
22	#127×C₇₉	—	2	1	no	—
23	#127×C₇₉	C₂²	1	1	yes	characteristic
24	#127×C₇₉	C₂²	1	1	yes	characteristic, complemented
25	#121×C₇₉	—	2	1	no	—
26	#121×C₇₉	—	2	2	no	—
27	#121×C₇₉	C₂²	1	2	yes	—
28	#121×C₇₉	C₂²	1	2	yes	complemented
29	#120×C₇₉	—	2	2	no	—
30	C₄×Q₈×C₇₉	C₂²	1	1	yes	characteristic, complemented
31	C₄×D₄×C₇₉	—	2	1	no	—
32	C₄×D₄×C₇₉	C₂²	1	1	yes	characteristic
33	C₄×D₄×C₇₉	C₂²	1	1	yes	characteristic
34	#106×C₇₉	C₂²	1	1	yes	characteristic, complemented
35	C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic, complemented
36	#103×C₇₉	C₂²	1	1	yes	characteristic, complemented
37	#102×C₇₉	C₂²	1	1	yes	characteristic
38	#102×C₇₉	C₂²	1	2	yes	complemented
39	#102×C₇₉	C₂²	1	2	yes	complemented
40	C₂²⋊C₈×C₇₉	C₂²	1	2	yes	complemented
41	C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic, complemented
42	C₂⁴×C₇₉	—	2	2	no	—
43	#41×C₇₉	—	2	4	no	—
44	#41×C₇₉	—	4	2	no	—
45	C₂×Q₈×C₇₉	C₂³	1	1	yes	characteristic
46	C₂×D₄×C₇₉	—	2	4	no	—
47	C₂×D₄×C₇₉	—	2	2	no	—
48	C₂×D₄×C₇₉	—	2	2	no	—
49	C₂×D₄×C₇₉	—	2	2	no	—
50	C₂×D₄×C₇₉	—	2	1	no	—
51	C₂×D₄×C₇₉	—	2	2	no	—
52	C₂×D₄×C₇₉	—	2	1	no	—
53	C₂×D₄×C₇₉	—	2	1	no	—
54	C₂×D₄×C₇₉	—	2	1	no	—
55	C₂×D₄×C₇₉	—	4	2	no	—
56	C₂×D₄×C₇₉	—	4	4	no	—
57	C₂×D₄×C₇₉	D₄	1	1	yes	characteristic
58	C₂×D₄×C₇₉	C₂³	1	2	yes	—
59	C₂×D₄×C₇₉	C₂³	1	2	yes	—
60	C₂×D₄×C₇₉	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₇₉	—	2	2	no	—
62	C₂²×C₄×C₇₉	—	2	1	no	—
63	C₂²×C₄×C₇₉	C₂³	1	2	yes	—
64	SD₁₆×C₇₉	—	2	4	no	—
65	SD₁₆×C₇₉	D₄	1	4	yes	complemented
66	D₈×C₇₉	—	2	4	no	—
67	D₈×C₇₉	—	2	4	no	—
68	D₈×C₇₉	—	4	2	no	—
69	M₁₆×C₇₉	—	2	4	no	—
70	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
71	C₂×C₈×C₇₉	C₂³	1	2	yes	—
72	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₇₉	—	2	1	no	—
74	C₄⋊C₄×C₇₉	D₄	1	1	yes	characteristic
75	C₄⋊C₄×C₇₉	C₂³	1	1	yes	characteristic
76	C₄⋊C₄×C₇₉	C₂³	1	1	yes	characteristic
77	C₂²⋊C₄×C₇₉	—	2	2	no	—
78	C₂²⋊C₄×C₇₉	—	2	1	no	—
79	C₂²⋊C₄×C₇₉	—	2	2	no	—
80	C₂²⋊C₄×C₇₉	D₄	1	2	yes	—
81	C₄²×C₇₉	—	2	1	no	—
82	C₄²×C₇₉	C₂³	1	1	yes	characteristic
83	C₂³×C₇₉	—	2	1	no	—
84	C₂³×C₇₉	—	2	2	no	—
85	C₂³×C₇₉	—	2	2	no	—
86	C₂³×C₇₉	—	4	4	no	—
87	C₂³×C₇₉	—	4	4	no	—
88	C₂³×C₇₉	—	8	2	no	—
89	C₂³×C₇₉	C₂×D₄	1	2	yes	—
90	Q₈×C₇₉	—	2	1	no	—
91	Q₈×C₇₉	C₂×D₄	1	2	yes	complemented
92	D₄×C₇₉	—	2	2	no	—
93	D₄×C₇₉	—	2	2	no	—
94	D₄×C₇₉	—	2	2	no	—
95	D₄×C₇₉	—	2	1	no	—
96	D₄×C₇₉	—	2	2	no	—
97	D₄×C₇₉	—	2	2	no	—
98	D₄×C₇₉	—	4	4	no	—
99	D₄×C₇₉	—	4	4	no	—
100	D₄×C₇₉	—	4	4	no	—
101	D₄×C₇₉	—	4	2	no	—
102	D₄×C₇₉	—	4	2	no	—
103	D₄×C₇₉	—	8	1	no	—
104	D₄×C₇₉	—	8	1	no	—
105	D₄×C₇₉	—	8	2	no	—
106	D₄×C₇₉	C₂×D₄	1	2	yes	—
107	C₂×C₄×C₇₉	—	2	1	no	—
108	C₂×C₄×C₇₉	—	2	2	no	—
109	C₂×C₄×C₇₉	—	2	2	no	—
110	C₂×C₄×C₇₉	—	2	1	no	—
111	C₂×C₄×C₇₉	—	2	1	no	—
112	C₂×C₄×C₇₉	—	4	4	no	—
113	C₂×C₄×C₇₉	—	4	2	no	—
114	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
116	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
118	C₈×C₇₉	—	2	2	no	—
119	C₈×C₇₉	—	2	1	no	—
120	C₈×C₇₉	C₂×D₄	1	2	yes	complemented
121	C₂²×C₇₉	—	2	2	no	—
122	C₂²×C₇₉	—	2	2	no	—
123	C₂²×C₇₉	—	2	2	no	—
124	C₂²×C₇₉	—	2	2	no	—
125	C₂²×C₇₉	—	4	1	no	—
126	C₂²×C₇₉	—	4	1	no	—
127	C₂²×C₇₉	—	4	2	no	—
128	C₂²×C₇₉	—	4	2	no	—
129	C₂²×C₇₉	—	4	2	no	—
130	C₂²×C₇₉	—	4	2	no	—
131	C₂²×C₇₉	—	4	2	no	—
132	C₂²×C₇₉	—	4	2	no	—
133	C₂²×C₇₉	—	8	8	no	—
134	C₂²×C₇₉	C₂²×D₄	1	1	yes	characteristic
135	C₄×C₇₉	—	2	2	no	—
136	C₄×C₇₉	—	2	2	no	—
137	C₄×C₇₉	—	2	1	no	—
138	C₄×C₇₉	—	4	1	no	—
139	C₄×C₇₉	—	4	1	no	—
140	C₄×C₇₉	AGL(1,ℤ/8)	1	2	yes	complemented
141	C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
142	C₄×C₇₉	H(5,𝔽₂)	1	1	yes	characteristic
143	C₂×C₇₉	—	4	2	no	—
144	C₂×C₇₉	—	4	2	no	—
145	C₂×C₇₉	—	8	2	no	—
146	C₂×C₇₉	—	8	2	no	—
147	C₂×C₇₉	D₄²	1	1	yes	characteristic
148	C₂×C₇₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
149	C₂×C₇₉	#576	1	1	yes	characteristic
150	#3274	C₇₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
151	C₇₉	#3274	1	1	yes	characteristic, direct product factor, Sylow
152	C₂×AGL(1,ℤ/8)	C₂×C₇₉	1	2	yes	complemented
153	#550	C₂×C₇₉	1	1	yes	characteristic, complemented
154	D₄²	C₂×C₇₉	1	1	yes	characteristic, complemented
155	#493	C₂×C₇₉	1	1	yes	characteristic, complemented
156	#469	C₂×C₇₉	1	2	yes	complemented
157	#460	C₂×C₇₉	1	1	yes	characteristic, complemented
158	#459	C₂×C₇₉	1	1	yes	characteristic, complemented
159	#449	C₂×C₇₉	1	2	yes	complemented
160	#447	C₂×C₇₉	1	2	yes	complemented
161	C₄×SD₁₆	C₂×C₇₉	1	1	yes	characteristic, complemented
162	#436	C₂×C₇₉	1	1	yes	characteristic, complemented
163	C₂×#41	C₂²×C₇₉	1	2	yes	complemented
164	C₂²×D₄	—	2	1	no	—
165	C₂²×D₄	C₂²×C₇₉	1	2	yes	—
166	AGL(1,ℤ/8)	—	2	8	no	—
167	C₂×SD₁₆	C₂²×C₇₉	1	1	yes	characteristic
168	C₂×SD₁₆	C₂²×C₇₉	1	2	yes	complemented
169	C₂×D₈	C₂²×C₇₉	1	2	yes	—
170	C₂×D₈	C₂²×C₇₉	1	2	yes	complemented
171	C₂×D₈	C₂²×C₇₉	1	2	yes	complemented
172	C₂×M₁₆	C₂²×C₇₉	1	2	yes	complemented
173	#127	—	2	1	no	—
174	#127	C₂²×C₇₉	1	1	yes	characteristic
175	#127	C₂²×C₇₉	1	1	yes	characteristic, complemented
176	#121	—	2	1	no	—
177	#121	—	2	2	no	—
178	#121	C₂²×C₇₉	1	2	yes	—
179	#121	C₂²×C₇₉	1	2	yes	complemented
180	#120	—	2	2	no	—
181	C₄×Q₈	C₂²×C₇₉	1	1	yes	characteristic, complemented
182	C₄×D₄	—	2	1	no	—
183	C₄×D₄	C₂²×C₇₉	1	1	yes	characteristic
184	C₄×D₄	C₂²×C₇₉	1	1	yes	characteristic
185	#106	C₂²×C₇₉	1	1	yes	characteristic, complemented
186	C₄⋊C₈	C₂²×C₇₉	1	1	yes	characteristic, complemented
187	#103	C₂²×C₇₉	1	1	yes	characteristic, complemented
188	#102	C₂²×C₇₉	1	1	yes	characteristic
189	#102	C₂²×C₇₉	1	2	yes	complemented
190	#102	C₂²×C₇₉	1	2	yes	complemented
191	C₂²⋊C₈	C₂²×C₇₉	1	2	yes	complemented
192	C₄×C₈	C₂²×C₇₉	1	1	yes	characteristic, complemented
193	C₂⁴	—	2	2	no	—
194	#41	—	2	4	no	—
195	#41	—	4	2	no	—
196	C₂×Q₈	C₂³×C₇₉	1	1	yes	characteristic
197	C₂×D₄	—	2	4	no	—
198	C₂×D₄	—	2	2	no	—
199	C₂×D₄	—	2	2	no	—
200	C₂×D₄	—	2	2	no	—
201	C₂×D₄	—	2	1	no	—
202	C₂×D₄	—	2	2	no	—
203	C₂×D₄	—	2	1	no	—
204	C₂×D₄	—	2	1	no	—
205	C₂×D₄	—	2	1	no	—
206	C₂×D₄	—	4	2	no	—
207	C₂×D₄	—	4	4	no	—
208	C₂×D₄	D₄×C₇₉	1	1	yes	characteristic
209	C₂×D₄	C₂³×C₇₉	1	2	yes	—
210	C₂×D₄	C₂³×C₇₉	1	2	yes	—
211	C₂×D₄	C₂³×C₇₉	1	1	yes	characteristic
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	1	no	—
214	C₂²×C₄	C₂³×C₇₉	1	2	yes	—
215	SD₁₆	—	2	4	no	—
216	SD₁₆	D₄×C₇₉	1	4	yes	complemented
217	D₈	—	2	4	no	—
218	D₈	—	2	4	no	—
219	D₈	—	4	2	no	—
220	M₁₆	—	2	4	no	—
221	C₂×C₈	C₂³×C₇₉	1	1	yes	characteristic
222	C₂×C₈	C₂³×C₇₉	1	2	yes	—
223	C₂×C₈	C₂³×C₇₉	1	1	yes	characteristic
224	C₄⋊C₄	—	2	1	no	—
225	C₄⋊C₄	D₄×C₇₉	1	1	yes	characteristic
226	C₄⋊C₄	C₂³×C₇₉	1	1	yes	characteristic
227	C₄⋊C₄	C₂³×C₇₉	1	1	yes	characteristic
228	C₂²⋊C₄	—	2	2	no	—
229	C₂²⋊C₄	—	2	1	no	—
230	C₂²⋊C₄	—	2	2	no	—
231	C₂²⋊C₄	D₄×C₇₉	1	2	yes	—
232	C₄²	—	2	1	no	—
233	C₄²	C₂³×C₇₉	1	1	yes	characteristic
234	C₂³	—	2	1	no	—
235	C₂³	—	2	2	no	—
236	C₂³	—	2	2	no	—
237	C₂³	—	4	4	no	—
238	C₂³	—	4	4	no	—
239	C₂³	—	8	2	no	—
240	C₂³	C₂×D₄×C₇₉	1	2	yes	—
241	Q₈	—	2	1	no	—
242	Q₈	C₂×D₄×C₇₉	1	2	yes	complemented
243	D₄	—	2	2	no	—
244	D₄	—	2	2	no	—
245	D₄	—	2	2	no	—
246	D₄	—	2	1	no	—
247	D₄	—	2	2	no	—
248	D₄	—	2	2	no	—
249	D₄	—	4	4	no	—
250	D₄	—	4	4	no	—
251	D₄	—	4	4	no	—
252	D₄	—	4	2	no	—
253	D₄	—	4	2	no	—
254	D₄	—	8	1	no	—
255	D₄	—	8	1	no	—
256	D₄	—	8	2	no	—
257	D₄	C₂×D₄×C₇₉	1	2	yes	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	C₂×D₄×C₇₉	1	1	yes	characteristic
266	C₂×C₄	C₂×D₄×C₇₉	1	2	yes	—
267	C₂×C₄	C₂×D₄×C₇₉	1	1	yes	characteristic
268	C₂×C₄	C₂⁴×C₇₉	1	1	yes	characteristic
269	C₈	—	2	2	no	—
270	C₈	—	2	1	no	—
271	C₈	C₂×D₄×C₇₉	1	2	yes	complemented
272	C₂²	—	2	2	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	2	2	no	—
276	C₂²	—	4	1	no	—
277	C₂²	—	4	1	no	—
278	C₂²	—	4	2	no	—
279	C₂²	—	4	2	no	—
280	C₂²	—	4	2	no	—
281	C₂²	—	4	2	no	—
282	C₂²	—	4	2	no	—
283	C₂²	—	4	2	no	—
284	C₂²	—	8	8	no	—
285	C₂²	C₂²×D₄×C₇₉	1	1	yes	characteristic
286	C₄	—	2	2	no	—
287	C₄	—	2	2	no	—
288	C₄	—	2	1	no	—
289	C₄	—	4	1	no	—
290	C₄	—	4	1	no	—
291	C₄	AGL(1,ℤ/8)×C₇₉	1	2	yes	complemented
292	C₄	C₂²×D₄×C₇₉	1	1	yes	characteristic
293	C₄	H(5,𝔽₂)×C₇₉	1	1	yes	characteristic
294	C₂	—	4	2	no	—
295	C₂	—	4	2	no	—
296	C₂	—	8	2	no	—
297	C₂	—	8	2	no	—
298	C₂	D₄²×C₇₉	1	1	yes	characteristic
299	C₂	C₂×AGL(1,ℤ/8)×C₇₉	1	1	yes	characteristic
300	C₂	#576×C₇₉	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂×AGL(1,ℤ/8)×C₇₉	C₂	semidirect
2. #550×C₇₉	C₂	semidirect
3. D₄²×C₇₉	C₂	semidirect
4. #493×C₇₉	C₂	semidirect
5. #469×C₇₉	C₂	semidirect
6. #460×C₇₉	C₂	semidirect
7. #459×C₇₉	C₂	semidirect
8. #449×C₇₉	C₂	semidirect
9. #447×C₇₉	C₂	semidirect
10. C₄×SD₁₆×C₇₉	C₂	semidirect
11. #436×C₇₉	C₂	semidirect
12. C₂×#41×C₇₉	C₂²	semidirect
17. C₂×SD₁₆×C₇₉	C₂²	semidirect
19. C₂×D₈×C₇₉	C₂²	semidirect
20. C₂×D₈×C₇₉	C₂²	semidirect
21. C₂×M₁₆×C₇₉	C₂²	semidirect
24. #127×C₇₉	C₂²	semidirect
28. #121×C₇₉	C₂²	semidirect
30. C₄×Q₈×C₇₉	C₂²	semidirect
34. #106×C₇₉	C₂²	semidirect
35. C₄⋊C₈×C₇₉	C₂²	semidirect
36. #103×C₇₉	C₂²	semidirect
38. #102×C₇₉	C₂²	semidirect
39. #102×C₇₉	C₂²	semidirect
40. C₂²⋊C₈×C₇₉	C₂²	semidirect
41. C₄×C₈×C₇₉	C₂²	semidirect
65. SD₁₆×C₇₉	D₄	semidirect
91. Q₈×C₇₉	C₂×D₄	semidirect
120. C₈×C₇₉	C₂×D₄	semidirect
140. C₄×C₇₉	AGL(1,ℤ/8)	semidirect
150. #3274	C₇₉	direct (N × H)
151. C₇₉	#3274	direct (N × H)
152. C₂×AGL(1,ℤ/8)	C₂×C₇₉	semidirect
153. #550	C₂×C₇₉	semidirect
154. D₄²	C₂×C₇₉	semidirect
155. #493	C₂×C₇₉	semidirect
156. #469	C₂×C₇₉	semidirect
157. #460	C₂×C₇₉	semidirect
158. #459	C₂×C₇₉	semidirect
159. #449	C₂×C₇₉	semidirect
160. #447	C₂×C₇₉	semidirect
161. C₄×SD₁₆	C₂×C₇₉	semidirect
162. #436	C₂×C₇₉	semidirect
163. C₂×#41	C₂²×C₇₉	semidirect
168. C₂×SD₁₆	C₂²×C₇₉	semidirect
170. C₂×D₈	C₂²×C₇₉	semidirect
171. C₂×D₈	C₂²×C₇₉	semidirect
172. C₂×M₁₆	C₂²×C₇₉	semidirect
175. #127	C₂²×C₇₉	semidirect
179. #121	C₂²×C₇₉	semidirect
181. C₄×Q₈	C₂²×C₇₉	semidirect
185. #106	C₂²×C₇₉	semidirect
186. C₄⋊C₈	C₂²×C₇₉	semidirect
187. #103	C₂²×C₇₉	semidirect
189. #102	C₂²×C₇₉	semidirect
190. #102	C₂²×C₇₉	semidirect
191. C₂²⋊C₈	C₂²×C₇₉	semidirect
192. C₄×C₈	C₂²×C₇₉	semidirect
216. SD₁₆	D₄×C₇₉	semidirect
242. Q₈	C₂×D₄×C₇₉	semidirect
271. C₈	C₂×D₄×C₇₉	semidirect
291. C₄	AGL(1,ℤ/8)×C₇₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×AGL(1,ℤ/8)×C₇₉	2
#550×C₇₉	2
D₄²×C₇₉	2
#493×C₇₉	2
#469×C₇₉	2
#460×C₇₉	2
#459×C₇₉	2
#449×C₇₉	2
#447×C₇₉	2
C₄×SD₁₆×C₇₉	2
#436×C₇₉	2
#3274	79

Subgroups — C79×#3274 #23760896

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₉×#3274 #23760896