Subgroups — #34175000

Order 20224.

186 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23761470	C₂	1	2	yes	maximal, complemented
2	C₇₉×#1712	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₇₉⋊C₁₆)	—	2	2	no	—
4	#493×C₇₉	—	2	2	no	—
5	#493×C₇₉	C₄	1	2	yes	—
6	C₈²×C₇₉	C₂²	1	1	yes	characteristic
7	C₂×C₇₉⋊C₁₆	—	4	2	no	—
8	C₂×D₈×C₇₉	—	4	2	no	—
9	C₂×D₈×C₇₉	—	4	2	no	—
10	C₂×D₈×C₇₉	—	4	2	no	—
11	#127×C₇₉	—	2	2	no	—
12	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
13	C₄×C₈×C₇₉	D₄	1	2	yes	—
14	C₇₉⋊C₁₆	—	8	2	no	—
15	C₂×D₄×C₇₉	—	4	2	no	—
16	C₂×D₄×C₇₉	—	4	4	no	—
17	D₈×C₇₉	—	8	4	no	—
18	D₈×C₇₉	—	8	4	no	—
19	D₈×C₇₉	—	8	2	no	—
20	D₈×C₇₉	—	8	2	no	—
21	C₂×C₈×C₇₉	—	2	1	no	—
22	C₂×C₈×C₇₉	D₈	1	2	yes	—
23	C₂×C₈×C₇₉	SD₁₆	1	2	yes	—
24	C₄²×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
25	C₂³×C₇₉	—	8	2	no	—
26	D₄×C₇₉	—	8	4	no	—
27	D₄×C₇₉	—	8	4	no	—
28	D₄×C₇₉	—	8	4	no	—
29	C₂×C₄×C₇₉	#100	1	1	yes	characteristic
30	C₂×C₄×C₇₉	#102	1	2	yes	—
31	C₈×C₇₉	—	2	2	no	—
32	C₈×C₇₉	—	2	2	no	—
33	C₈×C₇₉	D₁₆	1	2	yes	—
34	C₈×C₇₉	SD₃₂	1	2	yes	—
35	C₂²×C₇₉	—	16	2	no	—
36	C₂²×C₇₉	—	16	4	no	—
37	C₂²×C₇₉	#331	1	1	yes	characteristic
38	C₄×C₇₉	—	2	1	no	—
39	C₄×C₇₉	#357	1	2	yes	—
40	C₄×C₇₉	#361	1	2	yes	—
41	#7096	—	79	1	no	maximal, Sylow
42	C₂×C₇₉	—	32	2	no	—
43	C₂×C₇₉	#1357	1	1	yes	characteristic
44	C₂×C₇₉	#1361	1	2	yes	—
45	#1712	D₇₉	1	1	yes	characteristic
46	#1370	—	79	2	no	—
47	C₇₉	#7096	1	1	yes	characteristic, complemented, Sylow
48	#493	—	2	2	no	—
49	#493	Dic₇₉	1	2	yes	—
50	C₄⋊C₁₆	—	158	2	no	—
51	C₈²	C₂×D₇₉	1	1	yes	characteristic
52	C₂×D₈	—	4	2	no	—
53	C₂×D₈	—	4	2	no	—
54	C₂×D₈	—	4	2	no	—
55	#127	—	2	2	no	—
56	C₂×C₁₆	—	316	2	no	—
57	C₄×C₈	C₂×Dic₇₉	1	1	yes	characteristic
58	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
59	C₂×D₄	—	4	2	no	—
60	C₂×D₄	—	4	4	no	—
61	D₈	—	8	4	no	—
62	D₈	—	8	4	no	—
63	D₈	—	8	2	no	—
64	D₈	—	8	2	no	—
65	C₂×C₈	—	2	1	no	—
66	C₂×C₈	#11929609	1	2	yes	—
67	C₂×C₈	#11929612	1	2	yes	—
68	C₄²	C₂²⋊Dic₇₉	1	1	yes	characteristic
69	C₁₆	—	632	2	no	—
70	C₂³	—	8	2	no	—
71	D₄	—	8	4	no	—
72	D₄	—	8	4	no	—
73	D₄	—	8	4	no	—
74	C₂×C₄	#14579078	1	1	yes	characteristic
75	C₂×C₄	#14579082	1	2	yes	—
76	C₈	—	2	2	no	—
77	C₈	—	2	2	no	—
78	C₈	#14579102	1	2	yes	—
79	C₈	#14579105	1	2	yes	—
80	C₂²	—	16	2	no	—
81	C₂²	—	16	4	no	—
82	C₂²	#19098307	1	1	yes	characteristic
83	C₄	—	2	1	no	—
84	C₄	#19098364	1	2	yes	—
85	C₄	#19098376	1	2	yes	—
86	C₂	—	32	2	no	—
87	C₂	#23761437	1	1	yes	characteristic
88	C₂	#23761446	1	2	yes	—

	subgroup	G/H
center	C₂²	#19098307
commutator	C₄×C₈×C₇₉	C₂×C₄
Frattini	C₈²	C₂×D₇₉
Fitting, prime radical	C₇₉×#1712	C₂
socle	C₂²×C₇₉	#331
cosocle	C₈²×C₇₉	C₂²
hypercenter	#1712	D₇₉
hypocenter	C₇₉	#7096
LCS	G ▷ C₄×C₈×C₇₉ ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1712

Complements

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

N	H ≅ G/N	type
1. #23761470	C₂	semidirect
47. C₇₉	#7096	semidirect

Maximal subgroups

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

subgroup	index
#23761470	2
C₇₉×#1712	2
#7096	79