Subgroups — #36348807

Order 22112.

62 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26010866	C₂	1	1	yes	maximal, characteristic
2	#26010849	C₂	1	2	yes	maximal, complemented
3	C₂×C₆₉₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈×D₆₉₁	C₂	1	2	yes	maximal, complemented
5	M₁₆×C₆₉₁	C₂	1	1	yes	maximal, characteristic, complemented
6	Dic₁₃₈₂	C₄	1	1	yes	characteristic
7	D₂₇₆₄	C₄	1	1	yes	characteristic
8	C₂²⋊D₆₉₁	C₄	1	2	yes	—
9	C₄×D₆₉₁	C₂²	1	2	yes	—
10	C₆₉₁⋊C₈	C₂²	1	1	yes	characteristic
11	C₆₉₁⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₆₉₁	C₂²	1	1	yes	characteristic
13	C₈×C₆₉₁	C₂²	1	2	yes	—
14	C₂²×C₆₉₁	C₂×C₄	1	1	yes	characteristic
15	C₂×D₆₉₁	C₂×C₄	1	2	yes	—
16	C₄×C₆₉₁	C₂×C₄	1	1	yes	characteristic
17	C₄×C₆₉₁	C₂³	1	1	yes	characteristic
18	Dic₆₉₁	C₂×C₄	1	2	yes	—
19	C₂×C₆₉₁	—	2	1	no	—
20	C₂×C₆₉₁	C₂²×C₄	1	1	yes	characteristic
21	D₆₉₁	—	2	2	no	—
22	C₆₉₁	#131	1	1	yes	characteristic, complemented, Sylow
23	#131	—	691	1	no	maximal, Sylow
24	#41	—	691	1	no	—
25	M₁₆	—	691	2	no	—
26	M₁₆	D₆₉₁	1	1	yes	characteristic, complemented
27	C₂×C₈	—	691	1	no	—
28	C₂×C₈	—	691	2	no	—
29	Q₈	—	691	1	no	—
30	D₄	—	691	2	no	—
31	D₄	—	691	1	no	—
32	C₂×C₄	—	691	2	no	—
33	C₂×C₄	C₂×D₆₉₁	1	1	yes	characteristic
34	C₈	—	691	1	no	—
35	C₈	—	691	1	no	—
36	C₈	C₂×D₆₉₁	1	2	yes	—
37	C₂²	—	691	2	no	—
38	C₂²	C₄×D₆₉₁	1	1	yes	characteristic
39	C₄	—	691	2	no	—
40	C₄	C₄×D₆₉₁	1	1	yes	characteristic
41	C₄	C₂²×D₆₉₁	1	1	yes	characteristic
42	C₂	—	2	1	no	—
43	C₂	—	1382	2	no	—
44	C₂	C₂×C₄×D₆₉₁	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂²×D₆₉₁
commutator, socle	C₂×C₆₉₁	C₂²×C₄
Fitting	M₁₆×C₆₉₁	C₂
prime radical	#26010866	C₂
cosocle	C₄×C₆₉₁	C₂³
hypercenter	M₁₆	D₆₉₁
hypocenter	C₆₉₁	#131
LCS	G ▷ C₂×C₆₉₁ ▷ C₆₉₁
UCS	C₁ ◁ C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
2. #26010849	C₂	semidirect
3. C₂×C₆₉₁⋊C₈	C₂	semidirect
4. C₈×D₆₉₁	C₂	semidirect
5. M₁₆×C₆₉₁	C₂	semidirect
22. C₆₉₁	#131	semidirect
26. M₁₆	D₆₉₁	semidirect

Maximal subgroups

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

subgroup	index
#26010866	2
#26010849	2
C₂×C₆₉₁⋊C₈	2
C₈×D₆₉₁	2
M₁₆×C₆₉₁	2
#131	691