Subgroups — #28357242

Order 13856.

62 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20294097	C₂	1	2	yes	maximal, complemented
2	Dic₁₇₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#20294086	C₂	1	2	yes	maximal, complemented
4	D₃₄₆₄	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₈×C₄₃₃	C₂	1	1	yes	maximal, characteristic, complemented
6	Dic₈₆₆	C₂²	1	2	yes	—
7	D₁₇₃₂	C₂²	1	2	yes	—
8	C₂²⋊D₄₃₃	—	2	2	no	—
9	C₄×D₄₃₃	—	2	2	no	—
10	C₂×C₄×C₄₃₃	C₂²	1	1	yes	characteristic
11	C₈×C₄₃₃	C₂²	1	1	yes	characteristic
12	C₈×C₄₃₃	C₂²	1	1	yes	characteristic
13	C₂²×C₄₃₃	D₄	1	1	yes	characteristic
14	C₂×D₄₃₃	—	2	2	no	—
15	C₄×C₄₃₃	D₄	1	1	yes	characteristic
16	C₄×C₄₃₃	C₂³	1	1	yes	characteristic
17	Dic₄₃₃	—	2	2	no	—
18	C₂×C₄₃₃	—	2	1	no	—
19	C₂×C₄₃₃	C₂×D₄	1	1	yes	characteristic
20	D₄₃₃	—	4	2	no	—
21	C₄₃₃	#135	1	1	yes	characteristic, complemented, Sylow
22	#135	—	433	1	no	maximal, Sylow
23	#41	—	433	2	no	—
24	Q₁₆	—	433	1	no	—
25	SD₁₆	—	433	2	no	—
26	D₈	—	433	1	no	—
27	C₂×C₈	D₄₃₃	1	1	yes	characteristic, complemented
28	Q₈	—	433	2	no	—
29	D₄	—	866	2	no	—
30	D₄	—	433	2	no	—
31	C₂×C₄	—	866	2	no	—
32	C₂×C₄	C₂×D₄₃₃	1	1	yes	characteristic
33	C₈	C₂×D₄₃₃	1	1	yes	characteristic
34	C₈	C₂×D₄₃₃	1	1	yes	characteristic
35	C₂²	—	866	2	no	—
36	C₂²	D₁₇₃₂	1	1	yes	characteristic
37	C₄	—	866	2	no	—
38	C₄	D₁₇₃₂	1	1	yes	characteristic
39	C₄	C₂²×D₄₃₃	1	1	yes	characteristic
40	C₂	—	2	1	no	—
41	C₂	—	1732	2	no	—
42	C₂	C₂×D₁₇₃₂	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₁₇₃₂
commutator, cosocle	C₄×C₄₃₃	C₂³
Frattini	C₄	C₂²×D₄₃₃
Fitting	C₂×C₈×C₄₃₃	C₂
socle	C₂×C₄₃₃	C₂×D₄
hypercenter	C₂×C₈	D₄₃₃
hypocenter	C₄₃₃	#135
LCS	G ▷ C₄×C₄₃₃ ▷ C₂×C₄₃₃ ▷ C₄₃₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
1. #20294097	C₂	semidirect
2. Dic₁₇₃₂	C₂	semidirect
3. #20294086	C₂	semidirect
4. D₃₄₆₄	C₂	semidirect
5. C₂×C₈×C₄₃₃	C₂	semidirect
21. C₄₃₃	#135	semidirect
27. C₂×C₈	D₄₃₃	semidirect

Maximal subgroups

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

subgroup	index
#20294097	2
Dic₁₇₃₂	2
#20294086	2
D₃₄₆₄	2
C₂×C₈×C₄₃₃	2
#135	433