Subgroups — #27135283

Order 12352.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15690539	C₂	1	2	yes	maximal
2	C₄⋊#13095393	C₂	1	4	yes	maximal, complemented
3	C₂×C₄²×C₁₉₃	C₂	1	1	yes	maximal, characteristic
4	#15690539	—	2	4	no	—
5	C₂×#13095393	C₂²	1	4	yes	—
6	C₂²×C₄×C₁₉₃	C₄	1	2	yes	—
7	C₂²×C₄×C₁₉₃	C₂²	1	1	yes	characteristic
8	C₄²×C₁₉₃	C₄	1	1	yes	characteristic
9	C₄²×C₁₉₃	C₄	1	1	yes	characteristic
10	C₄²×C₁₉₃	C₂²	1	2	yes	—
11	#13095393	—	2	4	no	—
12	C₂³×C₁₉₃	C₂×C₄	1	1	yes	characteristic
13	C₂×C₄×C₁₉₃	—	2	4	no	—
14	C₂×C₄×C₁₉₃	C₂×C₄	1	2	yes	—
15	C₂×C₄×C₁₉₃	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×C₁₉₃	C₂×C₄	1	2	yes	—
17	C₂×C₄×C₁₉₃	D₄	1	2	yes	—
18	C₂×C₄×C₁₉₃	Q₈	1	2	yes	—
19	C₂×C₄×C₁₉₃	C₂³	1	1	yes	characteristic
20	C₂²×C₁₉₃	—	2	2	no	—
21	C₂²×C₁₉₃	C₄⋊C₄	1	2	yes	—
22	C₂²×C₁₉₃	C₂²×C₄	1	1	yes	characteristic
23	C₄×C₁₉₃	—	2	2	no	—
24	C₄×C₁₉₃	C₄⋊C₄	1	2	yes	—
25	C₄×C₁₉₃	M₁₆	1	4	yes	complemented
26	C₄×C₁₉₃	C₂×D₄	1	1	yes	characteristic
27	C₄×C₁₉₃	C₂×Q₈	1	1	yes	characteristic
28	C₂×C₁₉₃	—	2	2	no	—
29	C₂×C₁₉₃	C₂×C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₁₉₃	C₂×M₁₆	1	2	yes	—
31	C₁₉₃	#423	1	1	yes	characteristic, complemented, Sylow
32	#423	—	193	1	no	maximal, Sylow
33	C₂×M₁₆	—	193	2	no	—
34	C₂×C₄²	D₁₉₃	1	1	yes	characteristic
35	C₄⋊C₈	—	193	4	no	—
36	C₂²×C₄	Dic₁₉₃	1	2	yes	—
37	C₂²×C₄	C₂×D₁₉₃	1	1	yes	characteristic
38	M₁₆	—	386	4	no	—
39	C₂×C₈	—	193	4	no	—
40	C₄²	Dic₁₉₃	1	1	yes	characteristic
41	C₄²	Dic₁₉₃	1	1	yes	characteristic
42	C₄²	C₂×D₁₉₃	1	2	yes	—
43	C₂³	C₂×Dic₁₉₃	1	1	yes	characteristic
44	C₂×C₄	—	2	4	no	—
45	C₂×C₄	C₂×Dic₁₉₃	1	2	yes	—
46	C₂×C₄	C₂×Dic₁₉₃	1	1	yes	characteristic
47	C₂×C₄	C₂×Dic₁₉₃	1	2	yes	—
48	C₂×C₄	D₇₇₂	1	2	yes	—
49	C₂×C₄	Dic₃₈₆	1	2	yes	—
50	C₂×C₄	C₂²×D₁₉₃	1	1	yes	characteristic
51	C₈	—	386	4	no	—
52	C₂²	—	2	2	no	—
53	C₂²	C₄⋊Dic₁₉₃	1	2	yes	—
54	C₂²	C₂²×Dic₁₉₃	1	1	yes	characteristic
55	C₄	—	2	2	no	—
56	C₄	C₄⋊Dic₁₉₃	1	2	yes	—
57	C₄	#15690539	1	4	yes	complemented
58	C₄	C₂×D₇₇₂	1	1	yes	characteristic
59	C₄	C₂×Dic₃₈₆	1	1	yes	characteristic
60	C₂	—	2	2	no	—
61	C₂	C₂×C₄⋊Dic₁₉₃	1	1	yes	characteristic
62	C₂	C₂×#15690539	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₁₉₃
commutator, socle	C₂²×C₁₉₃	C₂²×C₄
Fitting	C₂×C₄²×C₁₉₃	C₂
prime radical	C₂³×C₁₉₃	C₂×C₄
cosocle	C₂×C₄×C₁₉₃	C₂³
hypercenter	C₂×C₄²	D₁₉₃
hypocenter	C₁₉₃	#423
LCS	G ▷ C₂²×C₁₉₃ ▷ C₁₉₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
2. C₄⋊#13095393	C₂	semidirect
25. C₄×C₁₉₃	M₁₆	semidirect
31. C₁₉₃	#423	semidirect
57. C₄	#15690539	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15690539	2
C₄⋊#13095393	2
C₂×C₄²×C₁₉₃	2
#423	193