Subgroups — #65707

Order 320.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#4033	C₂	1	1	yes	maximal, characteristic
2	#4004	C₂	1	1	yes	maximal, characteristic
3	#4004	C₂	1	1	yes	maximal, characteristic
4	#4002	C₂	1	1	yes	maximal, characteristic
5	#4002	C₂	1	1	yes	maximal, characteristic
6	C₄⋊#170	C₂	1	1	yes	maximal, characteristic
7	C₄×#170	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₁₀	C₄	1	1	yes	characteristic
9	C₂×#170	C₂²	1	1	yes	characteristic
10	C₂×#170	C₂²	1	1	yes	characteristic
11	C₂×#170	C₂²	1	1	yes	characteristic
12	C₂×#170	C₂²	1	1	yes	characteristic
13	C₅×C₄⋊C₄	C₄	1	1	yes	characteristic
14	#683	C₄	1	1	yes	characteristic
15	#683	C₄	1	1	yes	characteristic
16	C₄×Dic₅	C₂²	1	1	yes	characteristic
17	C₄×Dic₅	C₂²	1	1	yes	characteristic
18	C₄×Dic₅	C₂²	1	1	yes	characteristic
19	#446	—	5	1	no	maximal, Sylow
20	C₂×C₄×C₅	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₅	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×C₅	C₂×C₄	1	1	yes	characteristic
23	C₂×Dic₅	C₂×C₄	1	1	yes	characteristic
24	C₂×Dic₅	C₂×C₄	1	1	yes	characteristic
25	C₂×Dic₅	C₂×C₄	1	1	yes	characteristic
26	C₂×Dic₅	C₂³	1	1	yes	characteristic
27	Dic₁₀	—	2	2	no	—
28	#170	—	2	1	no	—
29	#170	—	2	1	no	—
30	#170	—	2	1	no	—
31	#170	Q₈	1	2	yes	complemented
32	C₄×Q₈	—	5	1	no	—
33	C₄⋊C₈	—	5	1	no	—
34	C₄⋊C₈	—	5	1	no	—
35	C₄⋊C₈	—	5	1	no	—
36	#97	—	5	1	no	—
37	#97	—	5	1	no	—
38	C₄×C₈	—	5	1	no	—
39	C₂²×C₅	C₂²×C₄	1	1	yes	characteristic
40	C₄×C₅	—	2	1	no	—
41	C₄×C₅	—	2	1	no	—
42	C₄×C₅	—	2	1	no	—
43	Dic₅	—	2	1	no	—
44	Dic₅	—	2	1	no	—
45	Dic₅	M₁₆	1	2	yes	—
46	Dic₅	C₂×Q₈	1	1	yes	characteristic
47	Dic₅	#41	1	1	yes	characteristic
48	C₂×Q₈	—	5	1	no	—
49	C₂×C₈	—	5	1	no	—
50	C₂×C₈	—	5	1	no	—
51	C₂×C₈	—	5	1	no	—
52	C₂×C₈	—	5	1	no	—
53	C₄⋊C₄	—	5	1	no	—
54	C₄⋊C₄	—	5	1	no	—
55	C₄⋊C₄	AGL(1,𝔽₅)	1	1	yes	characteristic
56	C₄²	—	5	1	no	—
57	C₄²	—	5	1	no	—
58	C₄²	—	5	1	no	—
59	C₂×C₅	C₄×Q₈	1	1	yes	characteristic
60	C₂×C₅	C₂×M₁₆	1	1	yes	characteristic
61	C₂×C₅	#131	1	1	yes	characteristic
62	Q₈	—	10	2	no	—
63	C₂×C₄	—	5	1	no	—
64	C₂×C₄	—	5	1	no	—
65	C₂×C₄	—	5	1	no	—
66	C₂×C₄	—	5	1	no	—
67	C₂×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
68	C₂×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
69	C₂×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
70	C₈	—	5	2	no	—
71	C₈	—	10	1	no	—
72	C₈	—	10	1	no	—
73	C₈	—	10	1	no	—
74	C₅	#446	1	1	yes	characteristic, complemented, Sylow
75	C₂²	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
76	C₄	—	2	1	no	—
77	C₄	—	2	1	no	—
78	C₄	—	2	1	no	—
79	C₄	—	5	2	no	—
80	C₄	—	5	1	no	—
81	C₄	—	5	1	no	—
82	C₄	—	10	1	no	—
83	C₄	—	10	1	no	—
84	C₂	D₅⋊M₁₆	1	1	yes	characteristic
85	C₂	#4131	1	1	yes	characteristic
86	C₂	Q₈×AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×AGL(1,𝔽₅)
commutator, socle, prime radical	C₂²×C₅	C₂²×C₄
Fitting	C₅×C₄⋊C₄	C₄
cosocle	C₂×Dic₅	C₂³
hypercenter	C₄⋊C₄	AGL(1,𝔽₅)
hypocenter	C₅	#446
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
31. #170	Q₈	semidirect
74. C₅	#446	semidirect

Maximal subgroups

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

subgroup	index
#4033	2
#4004	2
#4002	2
C₄⋊#170	2
C₄×#170	2
#446	5