Subgroups — #27135061

Order 12352.

66 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
2	#20259833	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₆×C₁₉₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₇₇₂	—	2	1	no	—
5	D₁₅₄₄	—	2	1	no	—
6	D₁₅₄₄	C₄	1	2	yes	complemented
7	C₄⋊Dic₁₉₃	—	2	1	no	—
8	C₂×C₈×C₁₉₃	C₂²	1	1	yes	characteristic
9	C₁₆×C₁₉₃	—	2	1	no	—
10	C₂²×D₁₉₃	—	4	1	no	—
11	D₇₇₂	—	2	2	no	—
12	D₇₇₂	—	4	1	no	—
13	C₂×Dic₁₉₃	—	4	1	no	—
14	C₂×C₄×C₁₉₃	D₄	1	1	yes	characteristic
15	C₈×C₁₉₃	C₂×C₄	1	1	yes	characteristic
16	C₈×C₁₉₃	D₄	1	1	yes	characteristic
17	C₂²×C₁₉₃	D₈	1	1	yes	characteristic
18	C₂×D₁₉₃	—	4	2	no	—
19	C₂×D₁₉₃	—	8	1	no	—
20	C₂×D₁₉₃	—	8	1	no	—
21	C₄×C₁₉₃	C₂²⋊C₄	1	1	yes	characteristic
22	C₄×C₁₉₃	SD₁₆	1	1	yes	characteristic
23	Dic₁₉₃	—	8	1	no	—
24	C₂×C₁₉₃	#102	1	1	yes	characteristic
25	C₂×C₁₉₃	D₁₆	1	1	yes	characteristic
26	C₂×C₁₉₃	SD₃₂	1	1	yes	characteristic
27	D₁₉₃	—	8	2	no	—
28	C₁₉₃	#357	1	1	yes	characteristic, complemented, Sylow
29	#357	—	193	1	no	maximal, Sylow
30	C₂×D₈	—	193	1	no	—
31	C₂×C₁₆	D₁₉₃	1	1	yes	characteristic, complemented
32	#107	—	193	1	no	—
33	C₂×D₄	—	386	1	no	—
34	D₈	—	386	1	no	—
35	D₈	—	193	2	no	—
36	C₂×C₈	C₂×D₁₉₃	1	1	yes	characteristic
37	C₄⋊C₄	—	386	1	no	—
38	C₁₆	—	2	1	no	—
39	C₂³	—	772	1	no	—
40	D₄	—	772	1	no	—
41	D₄	—	386	2	no	—
42	C₂×C₄	—	772	1	no	—
43	C₂×C₄	D₇₇₂	1	1	yes	characteristic
44	C₈	C₄×D₁₉₃	1	1	yes	characteristic
45	C₈	C₂²⋊D₁₉₃	1	1	yes	characteristic
46	C₂²	—	772	2	no	—
47	C₂²	—	1544	1	no	—
48	C₂²	—	1544	1	no	—
49	C₂²	D₁₅₄₄	1	1	yes	characteristic
50	C₄	—	1544	1	no	—
51	C₄	#15690532	1	1	yes	characteristic
52	C₄	#15690544	1	1	yes	characteristic
53	C₂	—	1544	2	no	—
54	C₂	#20259821	1	1	yes	characteristic
55	C₂	D₃₀₈₈	1	1	yes	characteristic
56	C₂	#20259844	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₁₅₄₄
commutator	C₈×C₁₉₃	C₂×C₄
Frattini	C₂×C₈	C₂×D₁₉₃
Fitting	C₂×C₁₆×C₁₉₃	C₂
socle	C₂²×C₁₉₃	D₈
prime radical	C₂×D₁₅₄₄	C₂
cosocle	C₂×C₈×C₁₉₃	C₂²
hypercenter	C₂×C₁₆	D₁₉₃
hypocenter	C₁₉₃	#357
LCS	G ▷ C₈×C₁₉₃ ▷ C₄×C₁₉₃ ▷ C₂×C₁₉₃ ▷ C₁₉₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
2. #20259833	C₂	semidirect
3. C₂×C₁₆×C₁₉₃	C₂	semidirect
6. D₁₅₄₄	C₄	semidirect
28. C₁₉₃	#357	semidirect
31. C₂×C₁₆	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
#20259833	2
C₂×C₁₆×C₁₉₃	2
#357	193