Subgroups — #34176082

Order 20224.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23761611	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₇₉⋊C₆₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇₉×#2258	C₂	1	1	yes	maximal, characteristic
4	C₇₉⋊C₆₄	—	2	1	no	—
5	C₇₉⋊C₆₄	—	2	1	no	—
6	#504×C₇₉	C₄	1	1	yes	characteristic
7	M₆₄×C₇₉	—	2	1	no	—
8	M₆₄×C₇₉	C₄	1	1	yes	characteristic
9	C₂×C₃₂×C₇₉	—	2	1	no	—
10	C₂×C₃₂×C₇₉	C₂²	1	1	yes	characteristic
11	#131×C₇₉	C₈	1	1	yes	characteristic
12	M₃₂×C₇₉	—	2	1	no	—
13	M₃₂×C₇₉	C₈	1	1	yes	characteristic
14	C₂×C₁₆×C₇₉	—	2	1	no	—
15	C₂×C₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
16	C₃₂×C₇₉	—	2	1	no	—
17	C₃₂×C₇₉	D₄	1	1	yes	characteristic
18	C₃₂×C₇₉	D₄	1	1	yes	characteristic
19	#41×C₇₉	C₁₆	1	1	yes	characteristic
20	M₁₆×C₇₉	—	2	1	no	—
21	M₁₆×C₇₉	C₁₆	1	1	yes	characteristic
22	C₂×C₈×C₇₉	—	2	1	no	—
23	C₂×C₈×C₇₉	C₂×C₈	1	1	yes	characteristic
24	C₁₆×C₇₉	—	2	1	no	—
25	C₁₆×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
26	C₁₆×C₇₉	M₁₆	1	1	yes	characteristic
27	Q₈×C₇₉	C₃₂	1	1	yes	characteristic
28	D₄×C₇₉	—	2	1	no	—
29	D₄×C₇₉	C₃₂	1	1	yes	characteristic
30	C₂×C₄×C₇₉	—	2	1	no	—
31	C₂×C₄×C₇₉	C₂×C₁₆	1	1	yes	characteristic
32	C₈×C₇₉	—	2	1	no	—
33	C₈×C₇₉	C₂²⋊C₈	1	1	yes	characteristic
34	C₈×C₇₉	M₃₂	1	1	yes	characteristic
35	C₂²×C₇₉	—	2	1	no	—
36	C₂²×C₇₉	M₆₄	1	1	yes	characteristic
37	C₄×C₇₉	—	2	1	no	—
38	C₄×C₇₉	C₂²⋊C₁₆	1	1	yes	characteristic
39	C₄×C₇₉	C₂×C₃₂	1	1	yes	characteristic
40	#7514	—	79	1	no	maximal, Sylow
41	C₂×C₇₉	—	2	1	no	—
42	C₂×C₇₉	—	4	1	no	—
43	C₂×C₇₉	C₂²⋊C₃₂	1	1	yes	characteristic
44	#2258	D₇₉	1	1	yes	characteristic
45	M₁₂₈	—	79	1	no	—
46	C₂×C₆₄	—	79	1	no	—
47	C₇₉	#7514	1	1	yes	characteristic, complemented, Sylow
48	#504	Dic₇₉	1	1	yes	characteristic
49	M₆₄	—	2	1	no	—
50	M₆₄	Dic₇₉	1	1	yes	characteristic
51	C₂×C₃₂	—	2	1	no	—
52	C₂×C₃₂	C₂×D₇₉	1	1	yes	characteristic
53	C₆₄	—	158	1	no	—
54	C₆₄	—	158	1	no	—
55	#131	C₇₉⋊C₈	1	1	yes	characteristic
56	M₃₂	—	2	1	no	—
57	M₃₂	C₇₉⋊C₈	1	1	yes	characteristic
58	C₂×C₁₆	—	2	1	no	—
59	C₂×C₁₆	C₂×Dic₇₉	1	1	yes	characteristic
60	C₃₂	—	2	1	no	—
61	C₃₂	C₂²⋊D₇₉	1	1	yes	characteristic
62	C₃₂	C₂²⋊D₇₉	1	1	yes	characteristic
63	#41	C₇₉⋊C₁₆	1	1	yes	characteristic
64	M₁₆	—	2	1	no	—
65	M₁₆	C₇₉⋊C₁₆	1	1	yes	characteristic
66	C₂×C₈	—	2	1	no	—
67	C₂×C₈	C₂×C₇₉⋊C₈	1	1	yes	characteristic
68	C₁₆	—	2	1	no	—
69	C₁₆	C₂²⋊Dic₇₉	1	1	yes	characteristic
70	C₁₆	#11929608	1	1	yes	characteristic
71	Q₈	C₇₉⋊C₃₂	1	1	yes	characteristic
72	D₄	—	2	1	no	—
73	D₄	C₇₉⋊C₃₂	1	1	yes	characteristic
74	C₂×C₄	—	2	1	no	—
75	C₂×C₄	C₂×C₇₉⋊C₁₆	1	1	yes	characteristic
76	C₈	—	2	1	no	—
77	C₈	C₂²⋊(C₇₉⋊C₈)	1	1	yes	characteristic
78	C₈	#14579101	1	1	yes	characteristic
79	C₂²	—	2	1	no	—
80	C₂²	#19098397	1	1	yes	characteristic
81	C₄	—	2	1	no	—
82	C₄	C₂²⋊(C₇₉⋊C₁₆)	1	1	yes	characteristic
83	C₄	C₂×C₇₉⋊C₃₂	1	1	yes	characteristic
84	C₂	—	2	1	no	—
85	C₂	—	4	1	no	—
86	C₂	C₂²⋊(C₇₉⋊C₃₂)	1	1	yes	characteristic

	subgroup	G/H
center	C₃₂	C₂²⋊D₇₉
commutator	C₄×C₇₉	C₂×C₃₂
Frattini	C₂×C₃₂	C₂×D₇₉
Fitting	C₇₉×#2258	C₂
socle	C₂×C₇₉	C₂²⋊C₃₂
prime radical	#41×C₇₉	C₁₆
cosocle	C₂×C₃₂×C₇₉	C₂²
hypercenter	#2258	D₇₉
hypocenter	C₇₉	#7514
LCS	G ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₃₂ ◁ C₂×C₃₂ ◁ #2258

Complements

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

N	H ≅ G/N	type
1. #23761611	C₂	semidirect
2. C₂×C₇₉⋊C₆₄	C₂	semidirect
47. C₇₉	#7514	semidirect

Maximal subgroups

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

subgroup	index
#23761611	2
C₂×C₇₉⋊C₆₄	2
C₇₉×#2258	2
#7514	79