Subgroups — #27135222

Order 12352.

190 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×D₁₉₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#15690539	C₂	1	1	yes	maximal, characteristic, complemented
3	#20259811	C₂	1	4	yes	maximal, complemented
4	C₂×M₁₆×C₁₉₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×D₁₉₃	C₄	1	1	yes	characteristic
6	C₂²×Dic₁₉₃	C₄	1	1	yes	characteristic
7	C₂×C₄×D₁₉₃	—	2	2	no	—
8	C₂×C₄×D₁₉₃	—	2	2	no	—
9	C₂×C₄×D₁₉₃	C₄	1	2	yes	—
10	C₂×C₄×D₁₉₃	C₂²	1	2	yes	—
11	#15690539	—	2	2	no	—
12	C₂×#13095393	C₂²	1	2	yes	complemented
13	C₂²×C₄×C₁₉₃	C₂²	1	1	yes	characteristic
14	M₁₆×C₁₉₃	—	2	2	no	—
15	C₂×C₈×C₁₉₃	C₂²	1	2	yes	complemented
16	C₂²×D₁₉₃	—	2	2	no	—
17	C₂²×D₁₉₃	—	2	4	no	—
18	C₂²×D₁₉₃	C₂×C₄	1	2	yes	—
19	C₂×Dic₁₉₃	—	2	2	no	—
20	C₂×Dic₁₉₃	C₂×C₄	1	2	yes	—
21	C₄×D₁₉₃	—	2	4	no	—
22	C₄×D₁₉₃	—	2	2	no	—
23	C₄×D₁₉₃	—	2	2	no	—
24	#13095393	—	2	2	no	—
25	C₂³×C₁₉₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×C₁₉₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×C₁₉₃	D₄	1	2	yes	—
28	C₂×C₄×C₁₉₃	D₄	1	2	yes	—
29	C₂×C₄×C₁₉₃	C₂³	1	1	yes	characteristic
30	C₈×C₁₉₃	—	2	2	no	—
31	C₂²×C₁₉₃	—	2	2	no	—
32	C₂²×C₁₉₃	C₂²⋊C₄	1	2	yes	—
33	C₂²×C₁₉₃	C₂²×C₄	1	1	yes	characteristic
34	C₂×D₁₉₃	—	2	8	no	—
35	C₂×D₁₉₃	—	2	2	no	—
36	C₂×D₁₉₃	—	2	2	no	—
37	C₂×D₁₉₃	M₁₆	1	4	yes	complemented
38	C₄×C₁₉₃	C₂²⋊C₄	1	2	yes	—
39	C₄×C₁₉₃	C₂×D₄	1	1	yes	characteristic
40	C₄×C₁₉₃	C₂×D₄	1	1	yes	characteristic
41	Dic₁₉₃	—	2	2	no	—
42	C₂×C₁₉₃	—	2	2	no	—
43	C₂×C₁₉₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₁₉₃	C₂×M₁₆	1	2	yes	—
45	D₁₉₃	—	2	4	no	—
46	C₁₉₃	#407	1	1	yes	characteristic, complemented, Sylow
47	#407	—	193	1	no	maximal, Sylow
48	C₂³×C₄	—	193	1	no	—
49	C₂×M₁₆	—	193	1	no	—
50	C₂×M₁₆	D₁₉₃	1	1	yes	characteristic, complemented
51	C₂²⋊C₈	—	193	4	no	—
52	C₂⁴	—	193	1	no	—
53	C₂²×C₄	—	386	2	no	—
54	C₂²×C₄	—	386	2	no	—
55	C₂²×C₄	—	193	1	no	—
56	C₂²×C₄	—	193	2	no	—
57	C₂²×C₄	—	193	2	no	—
58	C₂²×C₄	C₂×D₁₉₃	1	1	yes	characteristic
59	M₁₆	—	2	2	no	—
60	M₁₆	—	386	2	no	—
61	C₂×C₈	—	193	2	no	—
62	C₂×C₈	C₂×D₁₉₃	1	2	yes	complemented
63	C₂³	—	386	4	no	—
64	C₂³	—	386	2	no	—
65	C₂³	—	193	2	no	—
66	C₂³	C₄×D₁₉₃	1	1	yes	characteristic
67	C₂×C₄	—	386	2	no	—
68	C₂×C₄	—	386	4	no	—
69	C₂×C₄	—	386	2	no	—
70	C₂×C₄	—	386	2	no	—
71	C₂×C₄	—	193	2	no	—
72	C₂×C₄	C₄×D₁₉₃	1	1	yes	characteristic
73	C₂×C₄	C₂²⋊D₁₉₃	1	2	yes	—
74	C₂×C₄	D₇₇₂	1	2	yes	—
75	C₂×C₄	C₂²×D₁₉₃	1	1	yes	characteristic
76	C₈	—	2	2	no	—
77	C₈	—	386	2	no	—
78	C₂²	—	2	2	no	—
79	C₂²	—	386	8	no	—
80	C₂²	—	386	2	no	—
81	C₂²	—	386	2	no	—
82	C₂²	—	193	4	no	—
83	C₂²	#15690532	1	2	yes	—
84	C₂²	C₂×C₄×D₁₉₃	1	1	yes	characteristic
85	C₄	—	386	2	no	—
86	C₄	#15690532	1	2	yes	—
87	C₄	C₂×C₂²⋊D₁₉₃	1	1	yes	characteristic
88	C₄	C₂×D₇₇₂	1	1	yes	characteristic
89	C₂	—	2	2	no	—
90	C₂	—	386	4	no	—
91	C₂	C₂×#15690532	1	1	yes	characteristic
92	C₂	M₁₆×D₁₉₃	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₁₉₃
commutator, socle	C₂²×C₁₉₃	C₂²×C₄
Fitting	C₂×M₁₆×C₁₉₃	C₂
prime radical	C₂³×D₁₉₃	C₄
cosocle	C₂×C₄×C₁₉₃	C₂³
hypercenter	C₂×M₁₆	D₁₉₃
hypocenter	C₁₉₃	#407
LCS	G ▷ C₂²×C₁₉₃ ▷ C₁₉₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×#15690539	C₂	semidirect
3. #20259811	C₂	semidirect
4. C₂×M₁₆×C₁₉₃	C₂	semidirect
12. C₂×#13095393	C₂²	semidirect
15. C₂×C₈×C₁₉₃	C₂²	semidirect
37. C₂×D₁₉₃	M₁₆	semidirect
46. C₁₉₃	#407	semidirect
50. C₂×M₁₆	D₁₉₃	semidirect
62. C₂×C₈	C₂×D₁₉₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×D₁₉₃	2
C₂×#15690539	2
#20259811	2
C₂×M₁₆×C₁₉₃	2
#407	193