Subgroups — #27135213

Order 12352.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15690538	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×D₁₉₃	C₂	1	1	yes	maximal, characteristic
3	#20259856	C₂	1	1	yes	maximal, characteristic
4	#20259810	C₂	1	1	yes	maximal, characteristic, complemented
5	#20259809	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×Dic₁₉₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₈×C₁₉₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₁₉₃	C₂²	1	1	yes	characteristic
9	#15690538	C₄	1	4	yes	complemented
10	C₂×#13095393	C₂²	1	1	yes	characteristic
11	C₂×#13095393	C₂²	1	1	yes	characteristic
12	C₈×D₁₉₃	C₄	1	4	yes	complemented
13	#15690534	C₄	1	2	yes	—
14	#15690532	C₄	1	2	yes	—
15	C₄×Dic₁₉₃	C₂²	1	1	yes	characteristic
16	C₂×C₈×C₁₉₃	C₂²	1	1	yes	characteristic
17	C₂×C₈×C₁₉₃	C₂²	1	1	yes	characteristic
18	C₄²×C₁₉₃	C₂²	1	1	yes	characteristic
19	C₂²×D₁₉₃	C₂×C₄	1	1	yes	characteristic
20	C₂×Dic₁₉₃	C₂×C₄	1	2	yes	—
21	C₂×Dic₁₉₃	C₂×C₄	1	1	yes	characteristic
22	C₄×D₁₉₃	C₂×C₄	1	2	yes	—
23	C₄×D₁₉₃	C₂×C₄	1	2	yes	—
24	#13095393	C₂×C₄	1	2	yes	—
25	#13095393	C₂×C₄	1	2	yes	—
26	C₂×C₄×C₁₉₃	C₂×C₄	1	2	yes	—
27	C₂×C₄×C₁₉₃	C₂³	1	1	yes	characteristic
28	C₈×C₁₉₃	C₂×C₄	1	2	yes	—
29	C₈×C₁₉₃	C₂×C₄	1	2	yes	—
30	C₂²×C₁₉₃	C₂²×C₄	1	1	yes	characteristic
31	C₂×D₁₉₃	—	2	2	no	—
32	C₂×D₁₉₃	C₄²	1	2	yes	—
33	C₄×C₁₉₃	—	2	2	no	—
34	C₄×C₁₉₃	C₂²×C₄	1	1	yes	characteristic
35	C₄×C₁₉₃	C₂²×C₄	1	1	yes	characteristic
36	Dic₁₉₃	—	2	2	no	—
37	Dic₁₉₃	C₄²	1	2	yes	—
38	C₂×C₁₉₃	C₂×C₄²	1	1	yes	characteristic
39	C₂×C₁₉₃	#131	1	2	yes	—
40	D₁₉₃	—	2	2	no	—
41	C₁₉₃	#405	1	1	yes	characteristic, complemented, Sylow
42	#405	—	193	1	no	maximal, Sylow
43	C₂×M₁₆	—	193	1	no	—
44	C₂²×C₈	—	193	1	no	—
45	#117	—	193	1	no	—
46	#97	—	193	1	no	—
47	#97	—	193	1	no	—
48	C₄×C₈	—	193	1	no	—
49	C₄×C₈	D₁₉₃	1	1	yes	characteristic, complemented
50	C₂²×C₄	—	193	1	no	—
51	M₁₆	—	193	4	no	—
52	C₂×C₈	—	193	1	no	—
53	C₂×C₈	—	193	1	no	—
54	C₂×C₈	—	193	4	no	—
55	C₂×C₈	C₂×D₁₉₃	1	1	yes	characteristic
56	C₂×C₈	C₂×D₁₉₃	1	1	yes	characteristic
57	C₄⋊C₄	—	193	2	no	—
58	C₂²⋊C₄	—	193	2	no	—
59	C₄²	—	193	1	no	—
60	C₄²	C₂×D₁₉₃	1	1	yes	characteristic
61	C₂³	—	193	1	no	—
62	C₂×C₄	—	193	2	no	—
63	C₂×C₄	—	193	1	no	—
64	C₂×C₄	—	193	2	no	—
65	C₂×C₄	—	193	2	no	—
66	C₂×C₄	C₄×D₁₉₃	1	2	yes	—
67	C₂×C₄	C₂²×D₁₉₃	1	1	yes	characteristic
68	C₈	—	193	2	no	—
69	C₈	—	193	2	no	—
70	C₈	C₄×D₁₉₃	1	2	yes	—
71	C₈	C₄×D₁₉₃	1	2	yes	—
72	C₂²	—	386	2	no	—
73	C₂²	—	193	2	no	—
74	C₂²	C₂×C₄×D₁₉₃	1	1	yes	characteristic
75	C₄	—	2	2	no	—
76	C₄	—	386	2	no	—
77	C₄	—	193	2	no	—
78	C₄	C₂×C₄×D₁₉₃	1	1	yes	characteristic
79	C₄	C₂×C₄×D₁₉₃	1	1	yes	characteristic
80	C₂	—	386	2	no	—
81	C₂	C₄²×D₁₉₃	1	1	yes	characteristic
82	C₂	#20259905	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂×D₁₉₃
commutator	C₂×C₁₉₃	C₂×C₄²
Frattini	C₂×C₄	C₂²×D₁₉₃
Fitting	C₄×C₈×C₁₉₃	C₂
socle	C₂²×C₁₉₃	C₂²×C₄
prime radical	C₂²×D₁₉₃	C₂×C₄
cosocle	C₂×C₄×C₁₉₃	C₂³
hypercenter	C₄×C₈	D₁₉₃
hypocenter	C₁₉₃	#405
LCS	G ▷ C₂×C₁₉₃ ▷ C₁₉₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
4. #20259810	C₂	semidirect
5. #20259809	C₂	semidirect
6. C₈×Dic₁₉₃	C₂	semidirect
7. C₄×C₈×C₁₉₃	C₂	semidirect
9. #15690538	C₄	semidirect
12. C₈×D₁₉₃	C₄	semidirect
41. C₁₉₃	#405	semidirect
49. C₄×C₈	D₁₉₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15690538	2
C₂×C₈×D₁₉₃	2
#20259856	2
#20259810	2
#20259809	2
C₈×Dic₁₉₃	2
C₄×C₈×C₁₉₃	2
#405	193