Subgroups — #27212949

Order 13504.

58 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₈₄₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₁₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#107×C₂₁₁	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₄₂₂	—	2	1	no	—
5	Dic₈₄₄	—	2	1	no	—
6	Dic₈₄₄	C₄	1	2	yes	complemented
7	C₂₁₁⋊C₁₆	—	2	1	no	—
8	C₂×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
9	C₄⋊C₄×C₂₁₁	—	2	1	no	—
10	Dic₄₂₂	—	2	2	no	—
11	Dic₄₂₂	—	4	1	no	—
12	C₂×Dic₂₁₁	—	4	1	no	—
13	C₂×C₄×C₂₁₁	—	4	1	no	—
14	C₂×C₄×C₂₁₁	D₄	1	1	yes	characteristic
15	C₈×C₂₁₁	C₂×C₄	1	1	yes	characteristic
16	C₈×C₂₁₁	D₄	1	1	yes	characteristic
17	C₂²×C₂₁₁	D₈	1	1	yes	characteristic
18	C₄×C₂₁₁	—	8	1	no	—
19	C₄×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
20	C₄×C₂₁₁	SD₁₆	1	1	yes	characteristic
21	Dic₂₁₁	—	4	2	no	—
22	C₂×C₂₁₁	#102	1	1	yes	characteristic
23	C₂×C₂₁₁	SD₃₂	1	1	yes	characteristic
24	C₂×C₂₁₁	Q₃₂	1	1	yes	characteristic
25	C₂₁₁	#358	1	1	yes	characteristic, complemented, Sylow
26	#358	—	211	1	no	maximal, Sylow
27	C₂×Q₁₆	—	211	1	no	—
28	C₂×C₁₆	—	211	1	no	—
29	#107	D₂₁₁	1	1	yes	characteristic
30	C₂×Q₈	—	422	1	no	—
31	Q₁₆	—	422	1	no	—
32	Q₁₆	—	211	2	no	—
33	C₂×C₈	C₂×D₂₁₁	1	1	yes	characteristic
34	C₄⋊C₄	—	2	1	no	—
35	C₁₆	—	422	1	no	—
36	Q₈	—	844	1	no	—
37	Q₈	—	422	2	no	—
38	C₂×C₄	—	4	1	no	—
39	C₂×C₄	—	844	1	no	—
40	C₂×C₄	C₂²⋊D₂₁₁	1	1	yes	characteristic
41	C₈	C₄×D₂₁₁	1	1	yes	characteristic
42	C₈	D₈₄₄	1	1	yes	characteristic
43	C₂²	#16809947	1	1	yes	characteristic
44	C₄	—	8	1	no	—
45	C₄	—	844	2	no	—
46	C₄	(C₂×D₂₁₁)⋊C₄	1	1	yes	characteristic
47	C₄	#16809949	1	1	yes	characteristic
48	C₂	#20269897	1	1	yes	characteristic
49	C₂	#20269921	1	1	yes	characteristic
50	C₂	#20269923	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16809947
commutator	C₈×C₂₁₁	C₂×C₄
Frattini	C₂×C₈	C₂×D₂₁₁
Fitting	#107×C₂₁₁	C₂
socle, prime radical	C₂²×C₂₁₁	D₈
cosocle	C₂×C₈×C₂₁₁	C₂²
hypercenter	#107	D₂₁₁
hypocenter	C₂₁₁	#358
LCS	G ▷ C₈×C₂₁₁ ▷ C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #107

Complements

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

N	H ≅ G/N	type
6. Dic₈₄₄	C₄	semidirect
25. C₂₁₁	#358	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₈₄₄	2
C₂×C₂₁₁⋊C₁₆	2
#107×C₂₁₁	2
#358	211