Subgroups — #34175007

Order 20224.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23761464	C₂	1	2	yes	maximal
2	C₇₉×#1311	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₇₉⋊C₁₆)	—	2	2	no	—
4	C₄×C₁₆×C₇₉	C₄	1	1	yes	characteristic
5	C₄×C₁₆×C₇₉	C₄	1	1	yes	characteristic
6	C₈²×C₇₉	C₂²	1	1	yes	characteristic
7	C₂×C₇₉⋊C₁₆	—	4	2	no	—
8	C₂×C₁₆×C₇₉	—	2	1	no	—
9	C₂×C₁₆×C₇₉	C₈	1	1	yes	characteristic
10	C₂×C₁₆×C₇₉	C₈	1	1	yes	characteristic
11	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
12	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
13	C₄×C₈×C₇₉	Q₈	1	1	yes	characteristic
14	C₇₉⋊C₁₆	—	8	2	no	—
15	C₂×C₈×C₇₉	—	2	1	no	—
16	C₂×C₈×C₇₉	C₂×C₈	1	1	yes	characteristic
17	C₂×C₈×C₇₉	M₁₆	1	1	yes	characteristic
18	C₂×C₈×C₇₉	SD₁₆	1	2	yes	—
19	C₄²×C₇₉	C₄⋊C₄	1	1	yes	characteristic
20	C₁₆×C₇₉	—	2	2	no	—
21	C₁₆×C₇₉	—	2	1	no	—
22	C₁₆×C₇₉	C₁₆	1	2	yes	complemented
23	C₂×C₄×C₇₉	C₄⋊C₈	1	1	yes	characteristic
24	C₂×C₄×C₇₉	#106	1	1	yes	characteristic
25	C₂×C₄×C₇₉	#108	1	1	yes	characteristic
26	C₈×C₇₉	—	2	1	no	—
27	C₈×C₇₉	—	2	2	no	—
28	C₈×C₇₉	—	4	1	no	—
29	C₈×C₇₉	C₂×C₁₆	1	1	yes	characteristic
30	C₈×C₇₉	M₃₂	1	1	yes	characteristic
31	C₂²×C₇₉	#334	1	1	yes	characteristic
32	C₄×C₇₉	—	2	1	no	—
33	C₄×C₇₉	C₄⋊C₁₆	1	1	yes	characteristic
34	C₄×C₇₉	#364	1	1	yes	characteristic
35	C₄×C₇₉	#365	1	2	yes	—
36	#7099	—	79	1	no	maximal, Sylow
37	C₂×C₇₉	#1367	1	1	yes	characteristic
38	C₂×C₇₉	#1368	1	1	yes	characteristic
39	C₂×C₇₉	#1369	1	1	yes	characteristic
40	#1367	—	79	2	no	—
41	#1311	D₇₉	1	1	yes	characteristic
42	C₇₉	#7099	1	1	yes	characteristic, complemented, Sylow
43	C₄⋊C₁₆	—	158	2	no	—
44	C₄×C₁₆	Dic₇₉	1	1	yes	characteristic
45	C₄×C₁₆	Dic₇₉	1	1	yes	characteristic
46	C₈²	C₂×D₇₉	1	1	yes	characteristic
47	C₂×C₁₆	—	2	1	no	—
48	C₂×C₁₆	—	316	2	no	—
49	C₂×C₁₆	C₇₉⋊C₈	1	1	yes	characteristic
50	C₂×C₁₆	C₇₉⋊C₈	1	1	yes	characteristic
51	C₄×C₈	Dic₁₅₈	1	1	yes	characteristic
52	C₄×C₈	D₃₁₆	1	1	yes	characteristic
53	C₄×C₈	C₂×Dic₇₉	1	1	yes	characteristic
54	C₂×C₈	—	2	1	no	—
55	C₂×C₈	C₂×C₇₉⋊C₈	1	1	yes	characteristic
56	C₂×C₈	#11929608	1	1	yes	characteristic
57	C₂×C₈	#11929613	1	2	yes	—
58	C₄²	C₄⋊Dic₇₉	1	1	yes	characteristic
59	C₁₆	—	2	2	no	—
60	C₁₆	—	2	1	no	—
61	C₁₆	—	632	2	no	—
62	C₁₆	C₇₉⋊C₁₆	1	2	yes	complemented
63	C₂×C₄	C₄⋊(C₇₉⋊C₈)	1	1	yes	characteristic
64	C₂×C₄	#14579093	1	1	yes	characteristic
65	C₂×C₄	#14579097	1	1	yes	characteristic
66	C₈	—	2	1	no	—
67	C₈	—	2	2	no	—
68	C₈	—	4	1	no	—
69	C₈	C₂×C₇₉⋊C₁₆	1	1	yes	characteristic
70	C₈	#14579101	1	1	yes	characteristic
71	C₂²	#19098314	1	1	yes	characteristic
72	C₄	—	2	1	no	—
73	C₄	C₄⋊(C₇₉⋊C₁₆)	1	1	yes	characteristic
74	C₄	#19098384	1	1	yes	characteristic
75	C₄	#19098386	1	2	yes	—
76	C₂	#23761464	1	1	yes	characteristic
77	C₂	#23761466	1	1	yes	characteristic
78	C₂	#23761468	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14579093
commutator	C₈×C₇₉	C₂×C₁₆
Frattini	C₈²	C₂×D₇₉
Fitting	C₇₉×#1311	C₂
socle, prime radical	C₂²×C₇₉	#334
cosocle	C₈²×C₇₉	C₂²
hypercenter	#1311	D₇₉
hypocenter	C₇₉	#7099
LCS	G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈² ◁ #1311

Complements

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

N	H ≅ G/N	type
22. C₁₆×C₇₉	C₁₆	semidirect
42. C₇₉	#7099	semidirect
62. C₁₆	C₇₉⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
#23761464	2
C₇₉×#1311	2
#7099	79