Subgroups — #36348845

Order 22112.

166 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26010867	C₂	1	2	yes	maximal, complemented
2	#26010866	C₂	1	3	yes	maximal, complemented
3	C₂×D₂₇₆₄	C₂	1	3	yes	maximal, complemented
4	D₄×D₆₉₁	C₂	1	6	yes	maximal, complemented
5	#41×C₆₉₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₆₉₁	C₂²	1	6	yes	complemented
7	Dic₁₃₈₂	C₂²	1	1	yes	characteristic, complemented
8	D₂₇₆₄	C₂²	1	6	yes	complemented
9	D₂₇₆₄	C₂²	1	3	yes	complemented
10	C₂²⋊D₆₉₁	C₂²	1	6	yes	complemented
11	C₄×D₆₉₁	C₂²	1	6	yes	complemented
12	Q₈×C₆₉₁	C₂²	1	1	yes	characteristic, complemented
13	D₄×C₆₉₁	C₂²	1	3	yes	complemented
14	C₂×C₄×C₆₉₁	C₂²	1	3	yes	complemented
15	C₂²×C₆₉₁	C₂³	1	3	yes	—
16	C₂×D₆₉₁	—	4	6	no	—
17	C₂×D₆₉₁	C₂³	1	6	yes	—
18	C₄×C₆₉₁	C₂³	1	1	yes	characteristic
19	C₄×C₆₉₁	C₂³	1	3	yes	—
20	Dic₆₉₁	C₂³	1	2	yes	—
21	C₂×C₆₉₁	—	2	3	no	—
22	C₂×C₆₉₁	C₂⁴	1	1	yes	characteristic
23	D₆₉₁	—	2	6	no	—
24	C₆₉₁	H(5,𝔽₂)	1	1	yes	characteristic, complemented, Sylow
25	H(5,𝔽₂)	—	691	1	no	maximal, Sylow
26	#41	—	691	2	no	—
27	#41	—	691	3	no	—
28	#41	D₆₉₁	1	1	yes	characteristic, complemented
29	C₂×D₄	—	691	6	no	—
30	C₂×D₄	—	691	3	no	—
31	C₂³	—	691	6	no	—
32	Q₈	—	691	1	no	—
33	Q₈	C₂×D₆₉₁	1	1	yes	characteristic, complemented
34	D₄	—	691	6	no	—
35	D₄	—	691	6	no	—
36	D₄	—	691	3	no	—
37	D₄	C₂×D₆₉₁	1	3	yes	complemented
38	C₂×C₄	—	691	6	no	—
39	C₂×C₄	C₂×D₆₉₁	1	3	yes	complemented
40	C₂²	—	691	6	no	—
41	C₂²	—	2764	6	no	—
42	C₂²	C₂²×D₆₉₁	1	3	yes	—
43	C₄	—	691	2	no	—
44	C₄	C₂²×D₆₉₁	1	3	yes	—
45	C₄	C₂²×D₆₉₁	1	1	yes	characteristic
46	C₂	—	2	3	no	—
47	C₂	—	1382	6	no	—
48	C₂	C₂³×D₆₉₁	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂³×D₆₉₁
commutator, socle, cosocle	C₂×C₆₉₁	C₂⁴
Fitting	#41×C₆₉₁	C₂
hypercenter	#41	D₆₉₁
hypocenter	C₆₉₁	H(5,𝔽₂)
LCS	G ▷ C₂×C₆₉₁ ▷ C₆₉₁
UCS	C₁ ◁ C₂ ◁ #41

Complements

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

N	H ≅ G/N	type
1. #26010867	C₂	semidirect
2. #26010866	C₂	semidirect
3. C₂×D₂₇₆₄	C₂	semidirect
4. D₄×D₆₉₁	C₂	semidirect
5. #41×C₆₉₁	C₂	semidirect
6. C₂²×D₆₉₁	C₂²	semidirect
7. Dic₁₃₈₂	C₂²	semidirect
8. D₂₇₆₄	C₂²	semidirect
9. D₂₇₆₄	C₂²	semidirect
10. C₂²⋊D₆₉₁	C₂²	semidirect
11. C₄×D₆₉₁	C₂²	semidirect
12. Q₈×C₆₉₁	C₂²	semidirect
13. D₄×C₆₉₁	C₂²	semidirect
14. C₂×C₄×C₆₉₁	C₂²	semidirect
24. C₆₉₁	H(5,𝔽₂)	semidirect
28. #41	D₆₉₁	semidirect
33. Q₈	C₂×D₆₉₁	semidirect
37. D₄	C₂×D₆₉₁	semidirect
39. C₂×C₄	C₂×D₆₉₁	semidirect

Maximal subgroups

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

subgroup	index
#26010867	2
#26010866	2
C₂×D₂₇₆₄	2
D₄×D₆₉₁	2
#41×C₆₉₁	2
H(5,𝔽₂)	691