Subgroups — #27135636

Order 12352.

162 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₅₄₄	C₂	1	4	yes	maximal, complemented
2	#20259896	C₂	1	2	yes	maximal, complemented
3	C₄×C₈×C₁₉₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₇₇₂	—	2	4	no	—
5	C₂×D₇₇₂	C₂²	1	4	yes	—
6	D₁₅₄₄	—	2	8	no	—
7	C₂×C₈×C₁₉₃	C₂²	1	2	yes	—
8	C₄²×C₁₉₃	C₂²	1	1	yes	characteristic
9	C₂²×D₁₉₃	—	2	4	no	—
10	D₇₇₂	—	2	8	no	—
11	D₇₇₂	—	4	8	no	—
12	C₂×C₄×C₁₉₃	D₄	1	2	yes	—
13	C₂×C₄×C₁₉₃	C₂³	1	1	yes	characteristic
14	C₈×C₁₉₃	D₄	1	4	yes	complemented
15	C₂²×C₁₉₃	C₂×D₄	1	1	yes	characteristic
16	C₂×D₁₉₃	—	4	4	no	—
17	C₂×D₁₉₃	—	4	8	no	—
18	C₄×C₁₉₃	D₈	1	4	yes	complemented
19	C₄×C₁₉₃	C₂×D₄	1	2	yes	—
20	C₂×C₁₉₃	#127	1	1	yes	characteristic
21	C₂×C₁₉₃	C₂×D₈	1	2	yes	—
22	D₁₉₃	—	8	4	no	—
23	C₁₉₃	#493	1	1	yes	characteristic, complemented, Sylow
24	#493	—	193	1	no	maximal, Sylow
25	C₂×D₈	—	193	4	no	—
26	#127	—	193	2	no	—
27	C₄×C₈	D₁₉₃	1	1	yes	characteristic, complemented
28	C₂×D₄	—	386	4	no	—
29	C₂×D₄	—	193	4	no	—
30	D₈	—	386	8	no	—
31	C₂×C₈	C₂×D₁₉₃	1	2	yes	—
32	C₄²	C₂×D₁₉₃	1	1	yes	characteristic
33	C₂³	—	386	4	no	—
34	D₄	—	772	8	no	—
35	D₄	—	386	8	no	—
36	C₂×C₄	D₇₇₂	1	2	yes	—
37	C₂×C₄	C₂²×D₁₉₃	1	1	yes	characteristic
38	C₈	D₇₇₂	1	4	yes	complemented
39	C₂²	—	772	4	no	—
40	C₂²	—	772	8	no	—
41	C₂²	C₂×D₇₇₂	1	1	yes	characteristic
42	C₄	D₁₅₄₄	1	4	yes	complemented
43	C₄	C₂×D₇₇₂	1	2	yes	—
44	C₂	—	1544	4	no	—
45	C₂	#20259896	1	1	yes	characteristic
46	C₂	C₂×D₁₅₄₄	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₇₇₂
commutator, cosocle	C₂×C₄×C₁₉₃	C₂³
Frattini	C₂×C₄	C₂²×D₁₉₃
Fitting	C₄×C₈×C₁₉₃	C₂
socle	C₂²×C₁₉₃	C₂×D₄
hypercenter	C₄×C₈	D₁₉₃
hypocenter	C₁₉₃	#493
LCS	G ▷ C₂×C₄×C₁₉₃ ▷ C₂×C₁₉₃ ▷ C₁₉₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
1. C₂×D₁₅₄₄	C₂	semidirect
2. #20259896	C₂	semidirect
3. C₄×C₈×C₁₉₃	C₂	semidirect
14. C₈×C₁₉₃	D₄	semidirect
18. C₄×C₁₉₃	D₈	semidirect
23. C₁₉₃	#493	semidirect
27. C₄×C₈	D₁₉₃	semidirect
38. C₈	D₇₇₂	semidirect
42. C₄	D₁₅₄₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₁₅₄₄	2
#20259896	2
C₄×C₈×C₁₉₃	2
#493	193