Subgroups — #41011571

Order 27008.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27212945	C₂	1	2	yes	maximal
2	#491×C₂₁₁	C₂	1	1	yes	maximal, characteristic
3	#20269896	—	2	2	no	—
4	C₄⋊Q₈×C₂₁₁	C₂²	1	1	yes	characteristic
5	#125×C₂₁₁	C₄	1	1	yes	characteristic
6	#103×C₂₁₁	—	2	2	no	—
7	#97×C₂₁₁	C₄	1	1	yes	characteristic
8	#16809946	—	4	2	no	—
9	C₂×Q₈×C₂₁₁	D₄	1	2	yes	—
10	C₂×C₈×C₂₁₁	—	2	1	no	—
11	C₄⋊C₄×C₂₁₁	—	2	1	no	—
12	C₄⋊C₄×C₂₁₁	—	4	1	no	—
13	C₄⋊C₄×C₂₁₁	—	4	1	no	—
14	C₄²×C₂₁₁	C₂×C₄	1	1	yes	characteristic
15	C₂₁₁⋊C₈	—	8	2	no	—
16	Q₈×C₂₁₁	—	4	2	no	—
17	C₂×C₄×C₂₁₁	—	2	1	no	—
18	C₂×C₄×C₂₁₁	—	2	2	no	—
19	C₂×C₄×C₂₁₁	—	4	1	no	—
20	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
21	C₈×C₂₁₁	—	4	1	no	—
22	C₂²×C₂₁₁	#99	1	1	yes	characteristic
23	C₄×C₂₁₁	—	2	1	no	—
24	C₄×C₂₁₁	—	4	1	no	—
25	C₄×C₂₁₁	—	4	2	no	—
26	C₄×C₂₁₁	—	8	1	no	—
27	C₂×C₂₁₁	—	2	1	no	—
28	C₂×C₂₁₁	#354	1	1	yes	characteristic
29	C₂₁₁	#1413	1	1	yes	characteristic, complemented, Sylow
30	#1413	—	211	1	no	maximal, Sylow
31	#491	D₂₁₁	1	1	yes	characteristic
32	#356	—	211	2	no	—
33	C₄⋊Q₈	C₂×D₂₁₁	1	1	yes	characteristic
34	#125	Dic₂₁₁	1	1	yes	characteristic
35	#103	—	2	2	no	—
36	#101	—	422	2	no	—
37	#97	Dic₂₁₁	1	1	yes	characteristic
38	C₂×Q₈	C₂²⋊D₂₁₁	1	2	yes	—
39	M₁₆	—	844	2	no	—
40	C₂×C₈	—	2	1	no	—
41	C₄⋊C₄	—	2	1	no	—
42	C₄⋊C₄	—	4	1	no	—
43	C₄⋊C₄	—	4	1	no	—
44	C₄²	C₂×Dic₂₁₁	1	1	yes	characteristic
45	Q₈	—	4	2	no	—
46	C₂×C₄	—	2	1	no	—
47	C₂×C₄	—	2	2	no	—
48	C₂×C₄	—	4	1	no	—
49	C₂×C₄	C₂²⋊Dic₂₁₁	1	1	yes	characteristic
50	C₈	—	4	1	no	—
51	C₈	—	1688	2	no	—
52	C₂²	#20269892	1	1	yes	characteristic
53	C₄	—	2	1	no	—
54	C₄	—	4	1	no	—
55	C₄	—	4	2	no	—
56	C₄	—	8	1	no	—
57	C₂	—	2	1	no	—
58	C₂	#27212941	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#27212941
commutator	C₄²×C₂₁₁	C₂×C₄
Frattini	C₄⋊Q₈	C₂×D₂₁₁
Fitting	#491×C₂₁₁	C₂
socle	C₂×C₂₁₁	#354
prime radical	C₂²×C₂₁₁	#99
cosocle	C₄⋊Q₈×C₂₁₁	C₂²
hypercenter	#491	D₂₁₁
hypocenter	C₂₁₁	#1413
LCS	G ▷ C₄²×C₂₁₁ ▷ C₂×C₄×C₂₁₁ ▷ C₂²×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊Q₈ ◁ #491

Complements

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

N	H ≅ G/N	type
29. C₂₁₁	#1413	semidirect

Maximal subgroups

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

subgroup	index
#27212945	2
#491×C₂₁₁	2
#1413	211