Subgroups — #27135694

Order 12352.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₅₄₄	C₂	1	1	yes	maximal, characteristic, complemented
2	#20259842	C₂	1	2	yes	maximal, complemented
3	#20259840	C₂	1	2	yes	maximal, complemented
4	#20259839	C₂	1	1	yes	maximal, characteristic, complemented
5	#135×C₁₉₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₇₇₂	—	2	1	no	—
7	D₁₅₄₄	—	2	1	no	—
8	D₁₅₄₄	C₂²	1	2	yes	—
9	#15690530	C₂²	1	2	yes	complemented
10	#41×C₁₉₃	—	2	1	no	—
11	Q₁₆×C₁₉₃	C₂²	1	1	yes	characteristic, complemented
12	SD₁₆×C₁₉₃	—	2	1	no	—
13	D₈×C₁₉₃	C₂²	1	1	yes	characteristic, complemented
14	C₂×C₈×C₁₉₃	C₂²	1	1	yes	characteristic
15	C₂²×D₁₉₃	—	4	1	no	—
16	D₇₇₂	—	2	2	no	—
17	D₇₇₂	—	4	1	no	—
18	Q₈×C₁₉₃	—	2	1	no	—
19	D₄×C₁₉₃	—	2	1	no	—
20	D₄×C₁₉₃	—	4	1	no	—
21	C₂×C₄×C₁₉₃	—	4	1	no	—
22	C₂×C₄×C₁₉₃	D₄	1	1	yes	characteristic
23	C₈×C₁₉₃	D₄	1	1	yes	characteristic
24	C₈×C₁₉₃	C₂³	1	1	yes	characteristic
25	C₂²×C₁₉₃	—	4	1	no	—
26	C₂²×C₁₉₃	D₈	1	1	yes	characteristic
27	C₂×D₁₉₃	—	4	2	no	—
28	C₂×D₁₉₃	—	8	2	no	—
29	C₄×C₁₉₃	—	4	1	no	—
30	C₄×C₁₉₃	D₈	1	1	yes	characteristic
31	C₄×C₁₉₃	C₂×D₄	1	1	yes	characteristic
32	C₂×C₁₉₃	—	2	1	no	—
33	C₂×C₁₉₃	—	8	1	no	—
34	C₂×C₁₉₃	C₂×D₈	1	1	yes	characteristic
35	D₁₉₃	—	8	2	no	—
36	C₁₉₃	#509	1	1	yes	characteristic, complemented, Sylow
37	#509	—	193	1	no	maximal, Sylow
38	#135	D₁₉₃	1	1	yes	characteristic, complemented
39	C₂×D₈	—	193	1	no	—
40	SD₃₂	—	193	2	no	—
41	D₁₆	—	193	2	no	—
42	M₃₂	—	193	1	no	—
43	#41	—	2	1	no	—
44	C₂×D₄	—	386	1	no	—
45	Q₁₆	C₂×D₁₉₃	1	1	yes	characteristic, complemented
46	SD₁₆	—	2	1	no	—
47	D₈	—	386	1	no	—
48	D₈	—	193	2	no	—
49	D₈	C₂×D₁₉₃	1	1	yes	characteristic, complemented
50	C₂×C₈	C₂×D₁₉₃	1	1	yes	characteristic
51	C₁₆	—	193	2	no	—
52	C₂³	—	772	1	no	—
53	Q₈	—	2	1	no	—
54	D₄	—	2	1	no	—
55	D₄	—	4	1	no	—
56	D₄	—	772	1	no	—
57	D₄	—	386	2	no	—
58	C₂×C₄	—	4	1	no	—
59	C₂×C₄	C₂²⋊D₁₉₃	1	1	yes	characteristic
60	C₈	C₂²⋊D₁₉₃	1	1	yes	characteristic
61	C₈	C₂²×D₁₉₃	1	1	yes	characteristic
62	C₂²	—	4	1	no	—
63	C₂²	—	772	2	no	—
64	C₂²	—	1544	2	no	—
65	C₂²	#15690540	1	1	yes	characteristic
66	C₄	—	4	1	no	—
67	C₄	#15690540	1	1	yes	characteristic
68	C₄	C₂×C₂²⋊D₁₉₃	1	1	yes	characteristic
69	C₂	—	2	1	no	—
70	C₂	—	8	1	no	—
71	C₂	—	1544	2	no	—
72	C₂	C₂×#15690540	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#15690540
commutator, cosocle	C₈×C₁₉₃	C₂³
Frattini	C₈	C₂²×D₁₉₃
Fitting	#135×C₁₉₃	C₂
socle	C₂×C₁₉₃	C₂×D₈
hypercenter	#135	D₁₉₃
hypocenter	C₁₉₃	#509
LCS	G ▷ C₈×C₁₉₃ ▷ C₄×C₁₉₃ ▷ C₂×C₁₉₃ ▷ C₁₉₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #135

Complements

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

N	H ≅ G/N	type
1. C₂×D₁₅₄₄	C₂	semidirect
2. #20259842	C₂	semidirect
3. #20259840	C₂	semidirect
4. #20259839	C₂	semidirect
5. #135×C₁₉₃	C₂	semidirect
9. #15690530	C₂²	semidirect
11. Q₁₆×C₁₉₃	C₂²	semidirect
13. D₈×C₁₉₃	C₂²	semidirect
36. C₁₉₃	#509	semidirect
38. #135	D₁₉₃	semidirect
45. Q₁₆	C₂×D₁₉₃	semidirect
49. D₈	C₂×D₁₉₃	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
#20259842	2
#20259840	2
#20259839	2
#135×C₁₉₃	2
#509	193