Subgroups — #27135062

Order 12352.

58 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₇₇₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#15690530	C₂	1	1	yes	maximal, characteristic
3	#107×C₁₉₃	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₃₈₆	—	2	1	no	—
5	Dic₇₇₂	—	2	1	no	—
6	Dic₇₇₂	C₄	1	2	yes	complemented
7	#15690530	—	2	1	no	—
8	C₂×C₈×C₁₉₃	C₂²	1	1	yes	characteristic
9	C₄⋊C₄×C₁₉₃	—	2	1	no	—
10	Dic₃₈₆	—	2	2	no	—
11	Dic₃₈₆	—	4	1	no	—
12	C₂×Dic₁₉₃	—	4	1	no	—
13	C₂×C₄×C₁₉₃	—	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	C₄×C₁₉₃	—	8	1	no	—
19	C₄×C₁₉₃	C₂²⋊C₄	1	1	yes	characteristic
20	C₄×C₁₉₃	SD₁₆	1	1	yes	characteristic
21	Dic₁₉₃	—	4	2	no	—
22	C₂×C₁₉₃	#102	1	1	yes	characteristic
23	C₂×C₁₉₃	SD₃₂	1	1	yes	characteristic
24	C₂×C₁₉₃	Q₃₂	1	1	yes	characteristic
25	C₁₉₃	#358	1	1	yes	characteristic, complemented, Sylow
26	#358	—	193	1	no	maximal, Sylow
27	C₂×Q₁₆	—	193	1	no	—
28	C₂×C₁₆	—	193	1	no	—
29	#107	D₁₉₃	1	1	yes	characteristic
30	C₂×Q₈	—	386	1	no	—
31	Q₁₆	—	386	1	no	—
32	Q₁₆	—	193	2	no	—
33	C₂×C₈	C₂×D₁₉₃	1	1	yes	characteristic
34	C₄⋊C₄	—	2	1	no	—
35	C₁₆	—	386	1	no	—
36	Q₈	—	772	1	no	—
37	Q₈	—	386	2	no	—
38	C₂×C₄	—	4	1	no	—
39	C₂×C₄	—	772	1	no	—
40	C₂×C₄	C₂²⋊D₁₉₃	1	1	yes	characteristic
41	C₈	C₄×D₁₉₃	1	1	yes	characteristic
42	C₈	D₇₇₂	1	1	yes	characteristic
43	C₂²	#15690540	1	1	yes	characteristic
44	C₄	—	772	2	no	—
45	C₄	—	8	1	no	—
46	C₄	#15690532	1	1	yes	characteristic
47	C₄	#15690542	1	1	yes	characteristic
48	C₂	#20259819	1	1	yes	characteristic
49	C₂	#20259843	1	1	yes	characteristic
50	C₂	#20259845	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15690540
commutator	C₈×C₁₉₃	C₂×C₄
Frattini	C₂×C₈	C₂×D₁₉₃
Fitting	#107×C₁₉₃	C₂
socle, prime radical	C₂²×C₁₉₃	D₈
cosocle	C₂×C₈×C₁₉₃	C₂²
hypercenter	#107	D₁₉₃
hypocenter	C₁₉₃	#358
LCS	G ▷ C₈×C₁₉₃ ▷ C₄×C₁₉₃ ▷ C₂×C₁₉₃ ▷ C₁₉₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #107

Complements

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

N	H ≅ G/N	type
6. Dic₇₇₂	C₄	semidirect
25. C₁₉₃	#358	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₇₇₂	2
C₂×#15690530	2
#107×C₁₉₃	2
#358	193