Subgroups — #10604367

Order 640.

160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#3998	C₂	1	1	yes	maximal, characteristic
2	C₂²×#674	C₂	1	1	yes	maximal, characteristic
3	#65286	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#674	C₂	1	4	yes	maximal, complemented
5	C₅×#117	C₄	1	1	yes	characteristic
6	C₂×#681	C₄	1	1	yes	characteristic
7	C₂²×#168	C₂²	1	1	yes	characteristic
8	#3998	—	2	2	no	—
9	C₂×#674	—	2	2	no	—
10	C₂×#674	C₂²	1	2	yes	—
11	C₂×#674	C₂²	1	2	yes	—
12	#3935	C₄	1	2	yes	—
13	C₄×#168	C₂²	1	2	yes	—
14	#2151	—	5	1	no	maximal, Sylow
15	C₂²×C₄×C₅	C₂×C₄	1	1	yes	characteristic
16	C₅×C₄⋊C₄	C₈	1	2	yes	—
17	C₅×C₂²⋊C₄	C₈	1	2	yes	—
18	C₄²×C₅	C₂×C₄	1	2	yes	—
19	#681	—	2	2	no	—
20	C₂×#168	C₂×C₄	1	2	yes	—
21	C₂×#168	C₂×C₄	1	1	yes	characteristic
22	C₂×#168	D₄	1	2	yes	—
23	C₂×#168	Q₈	1	2	yes	—
24	C₂×#168	C₂³	1	1	yes	characteristic
25	#674	—	2	2	no	—
26	#674	—	2	2	no	—
27	C₂×M₃₂	—	5	1	no	—
28	C₂²×C₁₆	—	5	1	no	—
29	#405	—	5	1	no	—
30	C₄⋊C₁₆	—	5	4	no	—
31	C₂³×C₅	C₂×C₈	1	1	yes	characteristic
32	C₂×C₄×C₅	C₄⋊C₄	1	2	yes	—
33	C₂×C₄×C₅	C₂×C₈	1	4	yes	—
34	C₂×C₄×C₅	C₂×C₈	1	1	yes	characteristic
35	C₂×C₄×C₅	M₁₆	1	2	yes	—
36	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
37	#168	—	2	2	no	—
38	#168	C₄⋊C₄	1	2	yes	—
39	#168	C₂×D₄	1	1	yes	characteristic
40	#168	C₂×Q₈	1	1	yes	characteristic
41	C₂×M₁₆	—	5	1	no	—
42	C₂²×C₈	—	5	1	no	—
43	#117	AGL(1,𝔽₅)	1	1	yes	characteristic
44	M₃₂	—	10	2	no	—
45	C₂×C₁₆	—	5	2	no	—
46	C₂×C₁₆	—	5	2	no	—
47	C₂×C₁₆	—	10	2	no	—
48	#97	—	5	2	no	—
49	C₄×C₈	—	5	2	no	—
50	C₂²×C₅	—	2	2	no	—
51	C₂²×C₅	C₄⋊C₈	1	2	yes	—
52	C₂²×C₅	C₂²×C₈	1	1	yes	characteristic
53	C₄×C₅	—	2	4	no	—
54	C₄×C₅	C₄⋊C₈	1	2	yes	—
55	C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
57	C₂²×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
58	M₁₆	—	10	2	no	—
59	C₂×C₈	—	5	2	no	—
60	C₂×C₈	—	5	2	no	—
61	C₂×C₈	—	5	1	no	—
62	C₂×C₈	—	5	1	no	—
63	C₂×C₈	—	5	2	no	—
64	C₄⋊C₄	#170	1	2	yes	—
65	C₂²⋊C₄	#170	1	2	yes	—
66	C₄²	C₂×AGL(1,𝔽₅)	1	2	yes	—
67	C₁₆	—	10	2	no	—
68	C₁₆	—	10	2	no	—
69	C₂×C₅	—	2	2	no	—
70	C₂×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
71	C₂×C₅	#504	1	2	yes	—
72	C₂³	C₂×#170	1	1	yes	characteristic
73	C₂×C₄	#700	1	2	yes	—
74	C₂×C₄	C₄⋊AGL(1,𝔽₅)	1	2	yes	—
75	C₂×C₄	C₂×#170	1	4	yes	—
76	C₂×C₄	C₂×#170	1	1	yes	characteristic
77	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
78	C₈	—	5	1	no	—
79	C₈	—	5	2	no	—
80	C₈	—	5	1	no	—
81	C₈	—	10	2	no	—
82	C₅	#2151	1	1	yes	characteristic, complemented, Sylow
83	C₂²	—	2	2	no	—
84	C₂²	C₄⋊#170	1	2	yes	—
85	C₂²	C₂²×#170	1	1	yes	characteristic
86	C₄	—	2	4	no	—
87	C₄	C₄⋊#170	1	2	yes	—
88	C₄	C₂×#700	1	1	yes	characteristic
89	C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
90	C₂	—	2	2	no	—
91	C₂	C₂×C₄⋊#170	1	1	yes	characteristic
92	C₂	#65791	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×AGL(1,𝔽₅)
commutator, socle	C₂²×C₅	C₂²×C₈
Fitting	C₅×#117	C₄
prime radical	C₂³×C₅	C₂×C₈
cosocle	C₂×#168	C₂³
hypercenter	#117	AGL(1,𝔽₅)
hypocenter	C₅	#2151
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
4. C₄⋊#674	C₂	semidirect
82. C₅	#2151	semidirect

Maximal subgroups

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

subgroup	index
C₂×#3998	2
C₂²×#674	2
#65286	2
C₄⋊#674	2
#2151	5