Subgroups — #27136165

Order 12352.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20259958	C₂	1	2	yes	maximal
2	#20259888	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊#11715871	—	2	2	no	—
4	C₂×D₇₇₂	C₂²	1	1	yes	characteristic
5	#15690532	—	2	2	no	—
6	C₄×Dic₁₉₃	C₄	1	1	yes	characteristic, complemented
7	C₂×Q₈×C₁₉₃	C₄	1	1	yes	characteristic, complemented
8	C₂×#11715871	—	4	2	no	—
9	C₂²×D₁₉₃	D₄	1	2	yes	—
10	D₇₇₂	—	4	1	no	—
11	C₂×Dic₁₉₃	—	2	1	no	—
12	Q₈×C₁₉₃	—	4	1	no	—
13	C₂×C₄×C₁₉₃	—	2	1	no	—
14	C₂×C₄×C₁₉₃	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₁₉₃	C₂²⋊C₄	1	1	yes	characteristic
16	C₂×D₁₉₃	—	2	2	no	—
17	C₂×D₁₉₃	—	4	2	no	—
18	#11715871	—	8	2	no	—
19	C₄×C₁₉₃	—	2	1	no	—
20	C₄×C₁₉₃	—	4	1	no	—
21	Dic₁₉₃	—	4	1	no	—
22	C₂×C₁₉₃	—	2	1	no	—
23	C₂×C₁₉₃	#99	1	1	yes	characteristic
24	D₁₉₃	—	4	2	no	—
25	C₁₉₃	#354	1	1	yes	characteristic, complemented, Sylow
26	#354	—	193	1	no	maximal, Sylow
27	#124	—	193	1	no	—
28	#99	—	193	2	no	—
29	C₂×Q₈	#11715871	1	1	yes	characteristic, complemented
30	C₂×D₄	—	193	1	no	—
31	C₂²⋊C₄	—	386	2	no	—
32	C₂²⋊C₄	—	386	2	no	—
33	C₄²	—	193	1	no	—
34	C₂³	—	193	2	no	—
35	Q₈	—	4	1	no	—
36	D₄	—	772	1	no	—
37	C₂×C₄	—	2	1	no	—
38	C₂×C₄	—	772	2	no	—
39	C₂×C₄	—	386	1	no	—
40	C₂×C₄	C₂×#11715871	1	1	yes	characteristic
41	C₂²	—	772	2	no	—
42	C₂²	—	386	2	no	—
43	C₂²	C₂²⋊#11715871	1	1	yes	characteristic
44	C₄	—	2	1	no	—
45	C₄	—	4	1	no	—
46	C₄	—	772	1	no	—
47	C₄	—	1544	2	no	—
48	C₂	—	2	1	no	—
49	C₂	—	772	2	no	—
50	C₂	#20259957	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20259957
commutator	C₂×C₄×C₁₉₃	C₂×C₄
Frattini	C₂×C₄	C₂×#11715871
Fitting	C₂×Q₈×C₁₉₃	C₄
socle	C₂×C₁₉₃	#99
prime radical, cosocle	C₂×D₇₇₂	C₂²
hypercenter	C₂×Q₈	#11715871
hypocenter	C₁₉₃	#354
LCS	G ▷ C₂×C₄×C₁₉₃ ▷ C₂²×C₁₉₃ ▷ C₂×C₁₉₃ ▷ C₁₉₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
6. C₄×Dic₁₉₃	C₄	semidirect
7. C₂×Q₈×C₁₉₃	C₄	semidirect
25. C₁₉₃	#354	semidirect
29. C₂×Q₈	#11715871	semidirect

Maximal subgroups

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

subgroup	index
#20259958	2
#20259888	2
#354	193