Subgroups — #27135367

Order 12352.

108 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	C₄×Dic₃₈₆	C₂	1	2	yes	maximal
3	#20259831	C₂	1	1	yes	maximal, characteristic
4	#20259824	C₂	1	2	yes	maximal
5	#97×C₁₉₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₈₆	C₂²	1	2	yes	—
7	Dic₇₇₂	C₄	1	4	yes	complemented
8	C₄⋊Dic₁₉₃	C₂²	1	2	yes	—
9	#15690534	—	2	2	no	—
10	C₄×Dic₁₉₃	—	2	2	no	—
11	C₂×C₈×C₁₉₃	C₂²	1	1	yes	characteristic
12	C₂×C₈×C₁₉₃	C₂²	1	1	yes	characteristic
13	C₄²×C₁₉₃	C₂²	1	1	yes	characteristic
14	Dic₃₈₆	—	2	2	no	—
15	Dic₃₈₆	C₂×C₄	1	4	yes	—
16	C₂×Dic₁₉₃	—	2	2	no	—
17	C₂×Dic₁₉₃	—	2	2	no	—
18	C₂×C₄×C₁₉₃	D₄	1	2	yes	—
19	C₂×C₄×C₁₉₃	C₂³	1	1	yes	characteristic
20	C₈×C₁₉₃	—	2	1	no	—
21	C₈×C₁₉₃	C₂×C₄	1	2	yes	—
22	C₂²×C₁₉₃	C₂×D₄	1	1	yes	characteristic
23	C₄×C₁₉₃	—	2	2	no	—
24	C₄×C₁₉₃	C₂²×C₄	1	1	yes	characteristic
25	C₄×C₁₉₃	#41	1	1	yes	characteristic
26	Dic₁₉₃	—	2	4	no	—
27	Dic₁₉₃	—	4	2	no	—
28	C₂×C₁₉₃	C₄×D₄	1	1	yes	characteristic
29	C₂×C₁₉₃	#137	1	2	yes	—
30	C₁₉₃	#441	1	1	yes	characteristic, complemented, Sylow
31	#441	—	193	1	no	maximal, Sylow
32	C₂×Q₁₆	—	193	1	no	—
33	C₄×Q₈	—	193	2	no	—
34	#106	—	193	1	no	—
35	#103	—	193	2	no	—
36	#97	D₁₉₃	1	1	yes	characteristic
37	C₂×Q₈	—	193	2	no	—
38	Q₁₆	—	193	4	no	—
39	C₂×C₈	C₂×D₁₉₃	1	1	yes	characteristic
40	C₂×C₈	C₂×D₁₉₃	1	1	yes	characteristic
41	C₄⋊C₄	—	386	2	no	—
42	C₄⋊C₄	—	193	2	no	—
43	C₄²	—	386	2	no	—
44	C₄²	C₂×D₁₉₃	1	1	yes	characteristic
45	Q₈	—	386	2	no	—
46	Q₈	—	193	4	no	—
47	C₂×C₄	—	386	2	no	—
48	C₂×C₄	—	386	2	no	—
49	C₂×C₄	D₇₇₂	1	2	yes	—
50	C₂×C₄	C₂²×D₁₉₃	1	1	yes	characteristic
51	C₈	—	2	1	no	—
52	C₈	C₄×D₁₉₃	1	2	yes	—
53	C₂²	C₂×D₇₇₂	1	1	yes	characteristic
54	C₄	—	2	2	no	—
55	C₄	—	772	2	no	—
56	C₄	—	386	4	no	—
57	C₄	C₂×C₄×D₁₉₃	1	1	yes	characteristic
58	C₄	#15690555	1	1	yes	characteristic
59	C₂	C₄×D₇₇₂	1	1	yes	characteristic
60	C₂	#20259932	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₇₇₂
commutator	C₄×C₁₉₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₉₃
Fitting	#97×C₁₉₃	C₂
socle, prime radical	C₂²×C₁₉₃	C₂×D₄
cosocle	C₂×C₄×C₁₉₃	C₂³
hypercenter	#97	D₁₉₃
hypocenter	C₁₉₃	#441
LCS	G ▷ C₄×C₁₉₃ ▷ C₂×C₁₉₃ ▷ C₁₉₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

Complements

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

N	H ≅ G/N	type
7. Dic₇₇₂	C₄	semidirect
30. C₁₉₃	#441	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
C₄×Dic₃₈₆	2
#20259831	2
#20259824	2
#97×C₁₉₃	2
#441	193