Subgroups — #10604448

Order 640.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#168	C₂	1	1	yes	maximal, characteristic
2	#64859	C₂	1	2	yes	maximal
3	C₄⋊#674	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#674	C₂	1	2	yes	maximal
5	C₄×#674	C₂	1	1	yes	maximal, characteristic
6	C₄×C₅×Q₈	C₄	1	1	yes	characteristic
7	C₂×#674	C₂²	1	1	yes	characteristic
8	C₂×#674	C₂²	1	2	yes	—
9	C₂×#674	C₂²	1	1	yes	characteristic
10	C₄⋊#168	C₄	1	2	yes	—
11	C₄⋊#168	C₄	1	1	yes	characteristic
12	C₄×#168	C₂²	1	1	yes	characteristic
13	C₄×#168	C₂²	1	2	yes	—
14	#2183	—	5	1	no	maximal, Sylow
15	C₂×C₅×Q₈	C₈	1	1	yes	characteristic
16	C₅×C₄⋊C₄	C₈	1	2	yes	—
17	C₅×C₄⋊C₄	C₈	1	1	yes	characteristic
18	C₄²×C₅	C₂×C₄	1	2	yes	—
19	C₄²×C₅	C₂×C₄	1	1	yes	characteristic
20	C₂×#168	C₂×C₄	1	1	yes	characteristic
21	C₂×#168	C₂×C₄	1	2	yes	—
22	C₂×#168	C₂³	1	1	yes	characteristic
23	#674	—	2	1	no	—
24	#674	—	2	2	no	—
25	#674	Q₈	1	2	yes	complemented
26	C₈×Q₈	—	5	1	no	—
27	C₄⋊C₁₆	—	5	1	no	—
28	C₄⋊C₁₆	—	5	2	no	—
29	#346	—	5	2	no	—
30	C₄×C₁₆	—	5	1	no	—
31	C₅×Q₈	—	2	2	no	—
32	C₂×C₄×C₅	C₂×C₈	1	2	yes	—
33	C₂×C₄×C₅	C₂×C₈	1	1	yes	characteristic
34	C₂×C₄×C₅	C₂×C₈	1	2	yes	—
35	C₂×C₄×C₅	C₂×C₈	1	1	yes	characteristic
36	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
37	#168	—	2	1	no	—
38	#168	—	2	2	no	—
39	#168	C₂×Q₈	1	1	yes	characteristic
40	#168	#41	1	1	yes	characteristic
41	C₄×Q₈	AGL(1,𝔽₅)	1	1	yes	characteristic
42	C₂×C₁₆	—	5	1	no	—
43	C₂×C₁₆	—	5	1	no	—
44	C₂×C₁₆	—	5	2	no	—
45	C₄⋊C₈	—	5	2	no	—
46	C₄⋊C₈	—	5	1	no	—
47	C₄×C₈	—	5	1	no	—
48	C₄×C₈	—	5	2	no	—
49	C₂²×C₅	C₂²×C₈	1	1	yes	characteristic
50	C₄×C₅	—	2	2	no	—
51	C₄×C₅	—	2	2	no	—
52	C₄×C₅	—	2	1	no	—
53	C₄×C₅	M₃₂	1	2	yes	—
54	C₄×C₅	C₄×Q₈	1	1	yes	characteristic
55	C₄×C₅	#131	1	1	yes	characteristic
56	C₂×Q₈	#170	1	1	yes	characteristic
57	C₂×C₈	—	5	1	no	—
58	C₂×C₈	—	5	1	no	—
59	C₂×C₈	—	5	2	no	—
60	C₄⋊C₄	#170	1	2	yes	—
61	C₄⋊C₄	#170	1	1	yes	characteristic
62	C₄²	C₂×AGL(1,𝔽₅)	1	2	yes	—
63	C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
64	C₁₆	—	5	2	no	—
65	C₁₆	—	10	1	no	—
66	C₁₆	—	10	2	no	—
67	C₂×C₅	C₈×Q₈	1	1	yes	characteristic
68	C₂×C₅	C₂×M₃₂	1	1	yes	characteristic
69	C₂×C₅	#504	1	1	yes	characteristic
70	Q₈	—	2	2	no	—
71	C₂×C₄	C₂×#170	1	2	yes	—
72	C₂×C₄	C₂×#170	1	1	yes	characteristic
73	C₂×C₄	C₂×#170	1	2	yes	—
74	C₂×C₄	C₂×#170	1	1	yes	characteristic
75	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
76	C₈	—	5	1	no	—
77	C₈	—	5	1	no	—
78	C₈	—	10	1	no	—
79	C₈	—	10	2	no	—
80	C₅	#2183	1	1	yes	characteristic, complemented, Sylow
81	C₂²	C₂²×#170	1	1	yes	characteristic
82	C₄	—	2	2	no	—
83	C₄	—	2	2	no	—
84	C₄	—	2	1	no	—
85	C₄	#3998	1	2	yes	—
86	C₄	#4133	1	1	yes	characteristic
87	C₄	Q₈×AGL(1,𝔽₅)	1	1	yes	characteristic
88	C₂	C₂×#3998	1	1	yes	characteristic
89	C₂	Q₈×#170	1	1	yes	characteristic
90	C₂	#65791	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×AGL(1,𝔽₅)
commutator, socle, prime radical	C₂²×C₅	C₂²×C₈
Fitting	C₄×C₅×Q₈	C₄
cosocle	C₂×#168	C₂³
hypercenter	C₄×Q₈	AGL(1,𝔽₅)
hypocenter	C₅	#2183
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
25. #674	Q₈	semidirect
80. C₅	#2183	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#168	2
#64859	2
C₄⋊#674	2
C₄×#674	2
#2183	5