Subgroups — #35233137

Order 21184.

64 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24902608	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₃₃₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₈×C₃₃₁	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₆₆₂	—	2	1	no	—
5	C₂×C₃₃₁⋊C₈	—	2	1	no	—
6	C₄⋊Dic₃₃₁	—	2	1	no	—
7	C₄⋊Dic₃₃₁	C₄	1	2	yes	—
8	C₂×C₈×C₃₃₁	—	2	1	no	—
9	C₄²×C₃₃₁	C₂²	1	1	yes	characteristic
10	Dic₆₆₂	—	4	1	no	—
11	Dic₆₆₂	—	4	1	no	—
12	C₂×Dic₃₃₁	—	2	2	no	—
13	C₃₃₁⋊C₈	—	4	1	no	—
14	C₂×C₄×C₃₃₁	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄×C₃₃₁	D₄	1	1	yes	characteristic
16	C₂×C₄×C₃₃₁	D₄	1	1	yes	characteristic
17	C₈×C₃₃₁	—	4	1	no	—
18	C₂²×C₃₃₁	C₂²⋊C₄	1	1	yes	characteristic
19	C₄×C₃₃₁	—	2	1	no	—
20	C₄×C₃₃₁	D₈	1	1	yes	characteristic
21	C₄×C₃₃₁	SD₁₆	1	1	yes	characteristic
22	C₄×C₃₃₁	SD₁₆	1	1	yes	characteristic
23	C₄×C₃₃₁	Q₁₆	1	1	yes	characteristic
24	Dic₃₃₁	—	4	2	no	—
25	C₂×C₃₃₁	#101	1	1	yes	characteristic
26	C₂×C₃₃₁	#102	1	1	yes	characteristic
27	C₂×C₃₃₁	#103	1	1	yes	characteristic
28	C₃₃₁	#332	1	1	yes	characteristic, complemented, Sylow
29	#332	—	331	1	no	maximal, Sylow
30	C₄⋊Q₈	—	331	1	no	—
31	C₄⋊C₈	—	331	1	no	—
32	C₄⋊C₈	D₃₃₁	1	1	yes	characteristic
33	C₂×Q₈	—	662	1	no	—
34	C₂×C₈	—	2	1	no	—
35	C₂×C₈	—	662	1	no	—
36	C₄⋊C₄	—	331	2	no	—
37	C₄⋊C₄	—	662	1	no	—
38	C₄²	C₂×D₃₃₁	1	1	yes	characteristic
39	Q₈	—	1324	1	no	—
40	Q₈	—	1324	1	no	—
41	C₂×C₄	—	662	2	no	—
42	C₂×C₄	C₄×D₃₃₁	1	1	yes	characteristic
43	C₂×C₄	C₂²⋊D₃₃₁	1	1	yes	characteristic
44	C₂×C₄	D₁₃₂₄	1	1	yes	characteristic
45	C₈	—	4	1	no	—
46	C₈	—	1324	1	no	—
47	C₂²	(C₂×D₃₃₁)⋊C₄	1	1	yes	characteristic
48	C₄	—	2	1	no	—
49	C₄	—	1324	2	no	—
50	C₄	D₂₆₄₈	1	1	yes	characteristic
51	C₄	#19127024	1	1	yes	characteristic
52	C₄	#19127025	1	1	yes	characteristic
53	C₄	#19127026	1	1	yes	characteristic
54	C₂	#24902526	1	1	yes	characteristic
55	C₂	#24902530	1	1	yes	characteristic
56	C₂	#24902531	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	(C₂×D₃₃₁)⋊C₄
commutator	C₂×C₄×C₃₃₁	C₂×C₄
Frattini	C₄²	C₂×D₃₃₁
Fitting	C₄⋊C₈×C₃₃₁	C₂
socle, prime radical	C₂²×C₃₃₁	C₂²⋊C₄
cosocle	C₄²×C₃₃₁	C₂²
hypercenter	C₄⋊C₈	D₃₃₁
hypocenter	C₃₃₁	#332
LCS	G ▷ C₂×C₄×C₃₃₁ ▷ C₂²×C₃₃₁ ▷ C₃₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
28. C₃₃₁	#332	semidirect

Maximal subgroups

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

subgroup	index
#24902608	2
C₄⋊(C₃₃₁⋊C₈)	2
C₄⋊C₈×C₃₃₁	2
#332	331