Subgroups — #27136178

Order 12352.

54 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20259974	C₂	1	1	yes	maximal, characteristic
2	#20259969	C₂	1	1	yes	maximal, characteristic
3	Q₁₆×D₁₉₃	C₂	1	1	yes	maximal, characteristic
4	#15690565	—	2	1	no	—
5	#15690558	—	2	1	no	—
6	Q₈×D₁₉₃	—	2	1	no	—
7	Dic₇₇₂	C₄	1	1	yes	characteristic
8	#15690545	—	2	1	no	—
9	C₈×D₁₉₃	C₂²	1	1	yes	characteristic
10	Q₁₆×C₁₉₃	C₄	1	1	yes	characteristic
11	#13095400	—	4	1	no	—
12	Dic₃₈₆	—	2	1	no	—
13	Dic₃₈₆	—	4	1	no	—
14	C₄×D₁₉₃	—	4	1	no	—
15	C₄×D₁₉₃	D₄	1	1	yes	characteristic
16	#13095393	D₄	1	1	yes	characteristic
17	Q₈×C₁₉₃	—	2	1	no	—
18	C₈×C₁₉₃	C₂×C₄	1	1	yes	characteristic
19	C₂×D₁₉₃	SD₁₆	1	1	yes	characteristic
20	C₄×C₁₉₃	—	4	1	no	—
21	C₄×C₁₉₃	C₂²⋊C₄	1	1	yes	characteristic
22	Dic₁₉₃	—	4	1	no	—
23	Dic₁₉₃	D₈	1	1	yes	characteristic
24	C₂×C₁₉₃	#102	1	1	yes	characteristic
25	D₁₉₃	—	2	1	no	—
26	C₁₉₃	#362	1	1	yes	characteristic, complemented, Sylow
27	#362	—	193	1	no	maximal, Sylow
28	C₂×Q₁₆	—	193	1	no	—
29	M₃₂	—	193	1	no	—
30	#108	—	193	1	no	—
31	C₂×Q₈	—	386	1	no	—
32	Q₁₆	—	386	1	no	—
33	Q₁₆	—	193	1	no	—
34	Q₁₆	#11715871	1	1	yes	characteristic
35	M₁₆	—	386	1	no	—
36	C₂×C₈	—	193	1	no	—
37	C₁₆	—	386	1	no	—
38	Q₈	—	2	1	no	—
39	Q₈	—	772	1	no	—
40	Q₈	—	386	1	no	—
41	C₂×C₄	—	772	1	no	—
42	C₂×C₄	—	193	1	no	—
43	C₈	—	772	1	no	—
44	C₈	—	193	1	no	—
45	C₈	C₂×#11715871	1	1	yes	characteristic
46	C₂²	—	193	1	no	—
47	C₄	—	4	1	no	—
48	C₄	—	772	1	no	—
49	C₄	—	193	1	no	—
50	C₄	C₂²⋊#11715871	1	1	yes	characteristic
51	C₂	—	386	1	no	—
52	C₂	#20259961	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20259961
commutator	C₈×C₁₉₃	C₂×C₄
Frattini	C₈	C₂×#11715871
Fitting	Q₁₆×C₁₉₃	C₄
socle	C₂×C₁₉₃	#102
prime radical	C₂×D₁₉₃	SD₁₆
cosocle	C₈×D₁₉₃	C₂²
hypercenter	Q₁₆	#11715871
hypocenter	C₁₉₃	#362
LCS	G ▷ C₈×C₁₉₃ ▷ C₄×C₁₉₃ ▷ C₂×C₁₉₃ ▷ C₁₉₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₁₆

Complements

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

N	H ≅ G/N	type
26. C₁₉₃	#362	semidirect

Maximal subgroups

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

subgroup	index
#20259974	2
#20259969	2
Q₁₆×D₁₉₃	2
#362	193