Subgroups — #41011462

Order 27008.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊(C₂₁₁⋊C₁₆)	C₂	1	2	yes	maximal
2	C₈²×C₂₁₁	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₁₁⋊C₁₆	—	2	2	no	—
4	C₄×C₈×C₂₁₁	C₄	1	1	yes	characteristic
5	C₄×C₈×C₂₁₁	C₄	1	1	yes	characteristic
6	C₄×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
7	C₂₁₁⋊C₁₆	—	4	2	no	—
8	C₂×C₈×C₂₁₁	—	2	1	no	—
9	C₂×C₈×C₂₁₁	C₈	1	1	yes	characteristic
10	C₂×C₈×C₂₁₁	C₈	1	1	yes	characteristic
11	C₂×C₈×C₂₁₁	D₄	1	1	yes	characteristic
12	C₂×C₈×C₂₁₁	Q₈	1	1	yes	characteristic
13	C₄²×C₂₁₁	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄×C₂₁₁	C₄⋊C₄	1	1	yes	characteristic
15	C₂×C₄×C₂₁₁	C₂×C₈	1	1	yes	characteristic
16	C₂×C₄×C₂₁₁	M₁₆	1	1	yes	characteristic
17	C₈×C₂₁₁	—	2	2	no	—
18	C₈×C₂₁₁	—	2	1	no	—
19	C₈×C₂₁₁	—	2	1	no	—
20	C₈×C₂₁₁	C₁₆	1	2	yes	complemented
21	C₈×C₂₁₁	SD₁₆	1	2	yes	—
22	C₂²×C₂₁₁	C₄⋊C₈	1	1	yes	characteristic
23	C₄×C₂₁₁	—	2	1	no	—
24	C₄×C₂₁₁	#106	1	1	yes	characteristic
25	C₄×C₂₁₁	#108	1	1	yes	characteristic
26	C₄×C₂₁₁	C₂×C₁₆	1	1	yes	characteristic
27	C₄×C₂₁₁	M₃₂	1	1	yes	characteristic
28	C₂×C₂₁₁	#334	1	1	yes	characteristic
29	C₂×C₂₁₁	C₄⋊C₁₆	1	1	yes	characteristic
30	C₂×C₂₁₁	#364	1	1	yes	characteristic
31	C₂₁₁	#1367	1	1	yes	characteristic, complemented, Sylow
32	#1367	—	211	1	no	maximal, Sylow
33	C₄⋊C₁₆	—	211	2	no	—
34	C₈²	D₂₁₁	1	1	yes	characteristic
35	C₂×C₁₆	—	422	2	no	—
36	C₄×C₈	Dic₂₁₁	1	1	yes	characteristic
37	C₄×C₈	Dic₂₁₁	1	1	yes	characteristic
38	C₄×C₈	C₂×D₂₁₁	1	1	yes	characteristic
39	C₂×C₈	—	2	1	no	—
40	C₂×C₈	C₂₁₁⋊C₈	1	1	yes	characteristic
41	C₂×C₈	C₂₁₁⋊C₈	1	1	yes	characteristic
42	C₂×C₈	D₈₄₄	1	1	yes	characteristic
43	C₂×C₈	Dic₄₂₂	1	1	yes	characteristic
44	C₄²	C₂×Dic₂₁₁	1	1	yes	characteristic
45	C₁₆	—	844	2	no	—
46	C₂×C₄	C₄⋊Dic₂₁₁	1	1	yes	characteristic
47	C₂×C₄	C₂×C₂₁₁⋊C₈	1	1	yes	characteristic
48	C₂×C₄	#16809946	1	1	yes	characteristic
49	C₈	—	2	2	no	—
50	C₈	—	2	1	no	—
51	C₈	—	2	1	no	—
52	C₈	C₂₁₁⋊C₁₆	1	2	yes	complemented
53	C₈	#16809951	1	2	yes	—
54	C₂²	C₄⋊(C₂₁₁⋊C₈)	1	1	yes	characteristic
55	C₄	—	2	1	no	—
56	C₄	#20269909	1	1	yes	characteristic
57	C₄	#20269913	1	1	yes	characteristic
58	C₄	C₂×C₂₁₁⋊C₁₆	1	1	yes	characteristic
59	C₄	#20269917	1	1	yes	characteristic
60	C₂	#27212897	1	1	yes	characteristic
61	C₂	C₄⋊(C₂₁₁⋊C₁₆)	1	1	yes	characteristic
62	C₂	#27212967	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₈₄₄
commutator	C₄×C₂₁₁	C₂×C₁₆
Frattini	C₄×C₈	C₂×D₂₁₁
Fitting	C₈²×C₂₁₁	C₂
socle, prime radical	C₂²×C₂₁₁	C₄⋊C₈
cosocle	C₄×C₈×C₂₁₁	C₂²
hypercenter	C₈²	D₂₁₁
hypocenter	C₂₁₁	#1367
LCS	G ▷ C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈²

Complements

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

N	H ≅ G/N	type
20. C₈×C₂₁₁	C₁₆	semidirect
31. C₂₁₁	#1367	semidirect
52. C₈	C₂₁₁⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊(C₂₁₁⋊C₁₆)	2
C₈²×C₂₁₁	2
#1367	211