Subgroups — #36348823

Order 22112.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26010865	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×D₆₉₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#26010855	C₂	1	1	yes	maximal, characteristic, complemented
4	#26010854	C₂	1	1	yes	maximal, characteristic, complemented
5	#26010851	C₂	1	1	yes	maximal, characteristic, complemented
6	#26010849	C₂	1	1	yes	maximal, characteristic, complemented
7	D₈×C₆₉₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₆₉₁	—	2	1	no	—
9	Dic₁₃₈₂	C₂²	1	1	yes	characteristic, complemented
10	D₂₇₆₄	C₂²	1	1	yes	characteristic
11	C₂²⋊D₆₉₁	—	2	1	no	—
12	C₂²⋊D₆₉₁	—	2	1	no	—
13	C₂×Dic₆₉₁	—	2	1	no	—
14	C₄×D₆₉₁	C₂²	1	1	yes	characteristic
15	C₆₉₁⋊C₈	C₂²	1	1	yes	characteristic, complemented
16	D₄×C₆₉₁	C₂²	1	1	yes	characteristic
17	D₄×C₆₉₁	C₂²	1	1	yes	characteristic, complemented
18	C₈×C₆₉₁	C₂²	1	1	yes	characteristic, complemented
19	C₂²×C₆₉₁	—	2	1	no	—
20	C₂²×C₆₉₁	—	2	1	no	—
21	C₂×D₆₉₁	—	2	1	no	—
22	C₂×D₆₉₁	—	4	2	no	—
23	C₂×D₆₉₁	D₄	1	1	yes	characteristic
24	C₄×C₆₉₁	C₂³	1	1	yes	characteristic
25	Dic₆₉₁	—	2	1	no	—
26	Dic₆₉₁	D₄	1	1	yes	characteristic
27	C₂×C₆₉₁	—	4	1	no	—
28	C₂×C₆₉₁	—	4	1	no	—
29	C₂×C₆₉₁	C₂×D₄	1	1	yes	characteristic
30	D₆₉₁	—	2	1	no	—
31	D₆₉₁	—	4	1	no	—
32	C₆₉₁	AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
33	AGL(1,ℤ/8)	—	691	1	no	maximal, Sylow
34	#41	—	691	1	no	—
35	C₂×D₄	—	691	1	no	—
36	SD₁₆	—	691	1	no	—
37	SD₁₆	—	691	1	no	—
38	D₈	—	691	1	no	—
39	D₈	D₆₉₁	1	1	yes	characteristic, complemented
40	M₁₆	—	691	1	no	—
41	C₂³	—	1382	1	no	—
42	Q₈	—	691	1	no	—
43	D₄	—	1382	1	no	—
44	D₄	—	1382	1	no	—
45	D₄	—	691	1	no	—
46	D₄	C₂×D₆₉₁	1	1	yes	characteristic
47	D₄	C₂×D₆₉₁	1	1	yes	characteristic, complemented
48	C₂×C₄	—	1382	1	no	—
49	C₂×C₄	—	691	1	no	—
50	C₈	—	691	1	no	—
51	C₈	C₂×D₆₉₁	1	1	yes	characteristic, complemented
52	C₂²	—	2	1	no	—
53	C₂²	—	2	1	no	—
54	C₂²	—	1382	1	no	—
55	C₂²	—	691	1	no	—
56	C₂²	—	2764	2	no	—
57	C₄	—	1382	1	no	—
58	C₄	—	691	1	no	—
59	C₄	C₂²×D₆₉₁	1	1	yes	characteristic
60	C₂	—	4	1	no	—
61	C₂	—	4	1	no	—
62	C₂	—	1382	1	no	—
63	C₂	—	2764	1	no	—
64	C₂	D₄×D₆₉₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×D₆₉₁
commutator, cosocle	C₄×C₆₉₁	C₂³
Frattini	C₄	C₂²×D₆₉₁
Fitting	D₈×C₆₉₁	C₂
socle	C₂×C₆₉₁	C₂×D₄
hypercenter	D₈	D₆₉₁
hypocenter	C₆₉₁	AGL(1,ℤ/8)
LCS	G ▷ C₄×C₆₉₁ ▷ C₂×C₆₉₁ ▷ C₆₉₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ D₈

Complements

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

N	H ≅ G/N	type
1. #26010865	C₂	semidirect
2. D₄×D₆₉₁	C₂	semidirect
3. #26010855	C₂	semidirect
4. #26010854	C₂	semidirect
5. #26010851	C₂	semidirect
6. #26010849	C₂	semidirect
7. D₈×C₆₉₁	C₂	semidirect
9. Dic₁₃₈₂	C₂²	semidirect
15. C₆₉₁⋊C₈	C₂²	semidirect
17. D₄×C₆₉₁	C₂²	semidirect
18. C₈×C₆₉₁	C₂²	semidirect
32. C₆₉₁	AGL(1,ℤ/8)	semidirect
39. D₈	D₆₉₁	semidirect
47. D₄	C₂×D₆₉₁	semidirect
51. C₈	C₂×D₆₉₁	semidirect

Maximal subgroups

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

subgroup	index
#26010865	2
D₄×D₆₉₁	2
#26010855	2
#26010854	2
#26010851	2
#26010849	2
D₈×C₆₉₁	2
AGL(1,ℤ/8)	691