Subgroups — #10590022

Order 576.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#157	C₂	1	1	yes	maximal, characteristic
2	#63542	C₂	1	2	yes	maximal
3	C₄×#157	C₄	1	1	yes	characteristic
4	#3787	C₄	1	1	yes	characteristic
5	C₈×C₃⋊D₃	C₂²	1	1	yes	characteristic
6	C₃×C₃⋊C₁₆	—	2	2	no	—
7	D₃×C₁₆	—	6	2	no	—
8	C₂×#157	C₈	1	1	yes	characteristic
9	C₂×#157	C₈	1	1	yes	characteristic
10	C₄×C₃⋊D₃	C₂×C₄	1	1	yes	characteristic
11	#622	—	2	1	no	—
12	C₃²×C₈	D₄	1	1	yes	characteristic
13	C₃⋊(C₃⋊C₈)	Q₈	1	1	yes	characteristic
14	C₄⋊C₁₆	—	9	1	no	maximal, Sylow
15	D₃×C₈	—	6	2	no	—
16	C₃×C₁₆	—	6	2	no	—
17	C₃⋊C₁₆	—	2	2	no	—
18	C₂×C₃⋊D₃	C₂×C₈	1	1	yes	characteristic
19	#157	—	2	1	no	—
20	#157	C₁₆	1	2	yes	complemented
21	C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
22	C₃⋊Dic₃	M₁₆	1	1	yes	characteristic
23	C₂×C₁₆	—	9	2	no	—
24	C₄×C₈	—	9	1	no	—
25	C₄×D₃	—	6	2	no	—
26	C₃×C₈	—	2	2	no	—
27	C₃⋊C₈	—	6	2	no	—
28	C₂×C₃²	C₄⋊C₈	1	1	yes	characteristic
29	C₃⋊D₃	C₂×C₁₆	1	1	yes	characteristic
30	C₃⋊D₃	M₃₂	1	1	yes	characteristic
31	C₂×C₈	—	9	1	no	—
32	C₂×C₈	—	9	1	no	—
33	C₄²	—	9	1	no	—
34	C₁₆	—	6	2	no	—
35	C₂×D₃	—	6	2	no	—
36	C₃×C₄	—	2	2	no	—
37	Dic₃	—	6	2	no	—
38	C₃²	C₄⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
39	C₂×C₄	—	9	1	no	—
40	C₂×C₄	—	9	1	no	—
41	C₂×C₄	—	9	1	no	—
42	C₈	—	9	1	no	—
43	C₈	—	18	1	no	—
44	C₈	GO⁺(4,𝔽₂)	1	1	yes	characteristic
45	C₂×C₃	—	2	2	no	—
46	D₃	—	6	2	no	—
47	D₃	—	6	2	no	—
48	C₂²	—	9	1	no	—
49	C₄	—	9	2	no	—
50	C₄	—	9	1	no	—
51	C₄	—	18	1	no	—
52	C₄	#3773	1	1	yes	characteristic
53	C₃	—	2	2	no	—
54	C₂	—	9	1	no	—
55	C₂	—	9	1	no	—
56	C₂	#63732	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₈	GO⁺(4,𝔽₂)
commutator	C₃⋊D₃	C₂×C₁₆
Fitting	C₃²×C₈	D₄
socle	C₂×C₃²	C₄⋊C₈
prime radical	C₂×C₃⋊D₃	C₂×C₈
cosocle	C₈×C₃⋊D₃	C₂²
hypocenter	C₃²	C₄⋊C₁₆
LCS	G ▷ C₃⋊D₃ ▷ C₃²
UCS	C₁ ◁ C₈

Complements

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

N	H ≅ G/N	type
20. #157	C₁₆	semidirect
38. C₃²	C₄⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₈×#157	2
#63542	2
C₄⋊C₁₆	9