Subgroups — C₂×#26010850 #36348804

Order 22112.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26010850	C₂	1	4	yes	maximal, direct product factor
2	C₂×C₆₉₁⋊C₈	C₂	1	2	yes	maximal, complemented
3	C₂²×C₄×C₆₉₁	C₂	1	1	yes	maximal, characteristic
4	C₆₉₁⋊C₈	C₂²	1	4	yes	complemented
5	C₂³×C₆₉₁	C₄	1	1	yes	characteristic
6	C₂×C₄×C₆₉₁	C₄	1	2	yes	—
7	C₂×C₄×C₆₉₁	C₄	1	1	yes	characteristic
8	C₂×C₄×C₆₉₁	C₂²	1	2	yes	—
9	C₂×C₄×C₆₉₁	C₂²	1	1	yes	characteristic
10	C₂²×C₆₉₁	—	2	2	no	—
11	C₂²×C₆₉₁	C₂×C₄	1	2	yes	—
12	C₂²×C₆₉₁	C₂×C₄	1	1	yes	characteristic
13	C₄×C₆₉₁	C₂×C₄	1	2	yes	—
14	C₄×C₆₉₁	C₂×C₄	1	1	yes	characteristic
15	C₄×C₆₉₁	C₂³	1	1	yes	characteristic
16	C₂×C₆₉₁	—	2	2	no	—
17	C₂×C₆₉₁	M₁₆	1	2	yes	complemented
18	C₂×C₆₉₁	C₂²×C₄	1	1	yes	characteristic
19	C₆₉₁	C₂×M₁₆	1	1	yes	characteristic, complemented, Sylow
20	C₂×M₁₆	—	691	1	no	maximal, Sylow
21	C₂²×C₄	D₆₉₁	1	1	yes	characteristic
22	M₁₆	—	691	4	no	—
23	C₂×C₈	—	691	2	no	—
24	C₂³	Dic₆₉₁	1	1	yes	characteristic
25	C₂×C₄	Dic₆₉₁	1	1	yes	characteristic
26	C₂×C₄	Dic₆₉₁	1	2	yes	—
27	C₂×C₄	C₂×D₆₉₁	1	2	yes	—
28	C₂×C₄	C₂×D₆₉₁	1	1	yes	characteristic
29	C₈	—	691	4	no	—
30	C₂²	—	2	2	no	—
31	C₂²	C₂×Dic₆₉₁	1	2	yes	—
32	C₂²	C₂×Dic₆₉₁	1	1	yes	characteristic
33	C₄	C₂×Dic₆₉₁	1	2	yes	—
34	C₄	C₂×Dic₆₉₁	1	1	yes	characteristic
35	C₄	C₂²×D₆₉₁	1	1	yes	characteristic
36	C₂	—	2	2	no	—
37	C₂	#26010850	1	2	yes	direct product factor
38	C₂	C₂²×Dic₆₉₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₆₉₁
commutator	C₂×C₆₉₁	C₂²×C₄
Frattini	C₄	C₂²×D₆₉₁
Fitting	C₂²×C₄×C₆₉₁	C₂
socle	C₂²×C₆₉₁	C₂×C₄
prime radical	C₂³×C₆₉₁	C₄
cosocle	C₄×C₆₉₁	C₂³
hypercenter	C₂²×C₄	D₆₉₁
hypocenter	C₆₉₁	C₂×M₁₆
LCS	G ▷ C₂×C₆₉₁ ▷ C₆₉₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
1. #26010850	C₂	direct (N × H)
2. C₂×C₆₉₁⋊C₈	C₂	semidirect
4. C₆₉₁⋊C₈	C₂²	semidirect
17. C₂×C₆₉₁	M₁₆	semidirect
19. C₆₉₁	C₂×M₁₆	semidirect
37. C₂	#26010850	direct (N × H)

Maximal subgroups

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

subgroup	index
#26010850	2
C₂×C₆₉₁⋊C₈	2
C₂²×C₄×C₆₉₁	2
C₂×M₁₆	691

Subgroups — C2×#26010850 #36348804

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26010850 #36348804