Subgroups — #27136338

Order 12352.

258 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20259987	C₂	1	2	yes	maximal, complemented
2	#20259986	C₂	1	6	yes	maximal, complemented
3	C₂×#15690565	C₂	1	3	yes	maximal, complemented
4	D₁₉₃⋊M₁₆	C₂	1	3	yes	maximal, complemented
5	#41×D₁₉₃	C₂	1	1	yes	maximal, characteristic
6	#15690565	C₂²	1	1	yes	characteristic
7	#15690565	C₂²	1	3	yes	—
8	#15690565	C₂²	1	6	yes	complemented
9	#15690564	C₂²	1	6	yes	complemented
10	#15690563	C₂²	1	6	yes	complemented
11	C₂×#13095400	C₂²	1	6	yes	complemented
12	#15690556	C₂²	1	1	yes	characteristic
13	#15690555	C₄	1	3	yes	—
14	#15690554	C₂²	1	3	yes	—
15	Q₈×D₁₉₃	C₄	1	1	yes	characteristic
16	D₄×D₁₉₃	C₄	1	3	yes	—
17	C₂×C₄×D₁₉₃	C₂²	1	3	yes	—
18	#41×C₁₉₃	C₄	1	1	yes	characteristic
19	#13095400	C₂³	1	6	yes	—
20	#13095400	C₂³	1	2	yes	—
21	C₂²×D₁₉₃	C₂×C₄	1	3	yes	—
22	Dic₃₈₆	C₂×C₄	1	3	yes	—
23	D₇₇₂	C₂×C₄	1	3	yes	—
24	C₂²⋊D₁₉₃	C₂×C₄	1	6	yes	—
25	C₂×Dic₁₉₃	C₂³	1	3	yes	—
26	C₄×D₁₉₃	C₂×C₄	1	3	yes	—
27	C₄×D₁₉₃	C₂×C₄	1	3	yes	—
28	C₄×D₁₉₃	C₂³	1	1	yes	characteristic
29	C₄×D₁₉₃	C₂³	1	3	yes	—
30	Q₈×C₁₉₃	C₂×C₄	1	1	yes	characteristic
31	D₄×C₁₉₃	C₂×C₄	1	3	yes	—
32	C₂×C₄×C₁₉₃	C₂×C₄	1	3	yes	—
33	C₂²×C₁₉₃	C₂²×C₄	1	3	yes	—
34	C₂×D₁₉₃	—	4	3	no	—
35	C₂×D₁₉₃	C₂²×C₄	1	1	yes	characteristic
36	C₂×D₁₉₃	C₂²×C₄	1	3	yes	—
37	C₄×C₁₉₃	C₂²×C₄	1	1	yes	characteristic
38	C₄×C₁₉₃	C₂²×C₄	1	3	yes	—
39	Dic₁₉₃	C₂²×C₄	1	3	yes	—
40	Dic₁₉₃	C₂⁴	1	1	yes	characteristic
41	C₂×C₁₉₃	—	2	3	no	—
42	C₂×C₁₉₃	C₂³×C₄	1	1	yes	characteristic
43	D₁₉₃	—	2	1	no	—
44	D₁₉₃	—	2	3	no	—
45	C₁₉₃	#568	1	1	yes	characteristic, complemented, Sylow
46	#568	—	193	1	no	maximal, Sylow
47	C₂×#41	—	193	1	no	—
48	#131	—	193	6	no	—
49	#131	—	193	2	no	—
50	C₂×M₁₆	—	193	3	no	—
51	C₂×M₁₆	—	193	3	no	—
52	#41	—	193	1	no	—
53	#41	—	193	3	no	—
54	#41	—	193	3	no	—
55	#41	#11715871	1	1	yes	characteristic
56	C₂×Q₈	—	193	1	no	—
57	C₂×D₄	—	193	3	no	—
58	C₂²×C₄	—	193	3	no	—
59	M₁₆	—	193	6	no	—
60	M₁₆	—	193	1	no	—
61	M₁₆	—	193	3	no	—
62	M₁₆	—	193	6	no	—
63	C₂×C₈	—	193	6	no	—
64	C₂×C₈	—	193	6	no	—
65	C₂³	—	193	3	no	—
66	Q₈	—	193	3	no	—
67	Q₈	C₂×#11715871	1	1	yes	characteristic
68	D₄	—	193	6	no	—
69	D₄	—	193	3	no	—
70	D₄	C₂×#11715871	1	3	yes	—
71	C₂×C₄	—	193	3	no	—
72	C₂×C₄	—	193	3	no	—
73	C₂×C₄	—	193	1	no	—
74	C₂×C₄	—	193	3	no	—
75	C₂×C₄	—	193	3	no	—
76	C₂×C₄	C₂×#11715871	1	3	yes	—
77	C₈	—	193	6	no	—
78	C₈	—	193	2	no	—
79	C₂²	—	772	3	no	—
80	C₂²	—	193	1	no	—
81	C₂²	—	193	3	no	—
82	C₂²	C₂²×#11715871	1	3	yes	—
83	C₄	—	193	1	no	—
84	C₄	—	193	3	no	—
85	C₄	C₂²×#11715871	1	1	yes	characteristic
86	C₄	C₂²×#11715871	1	3	yes	—
87	C₂	—	2	3	no	—
88	C₂	—	386	1	no	—
89	C₂	—	386	3	no	—
90	C₂	C₂³×#11715871	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂³×#11715871
commutator, socle	C₂×C₁₉₃	C₂³×C₄
Fitting	#41×C₁₉₃	C₄
prime radical	#41×D₁₉₃	C₂
cosocle	Dic₁₉₃	C₂⁴
hypercenter	#41	#11715871
hypocenter	C₁₉₃	#568
LCS	G ▷ C₂×C₁₉₃ ▷ C₁₉₃
UCS	C₁ ◁ C₂ ◁ #41

Complements

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

N	H ≅ G/N	type
1. #20259987	C₂	semidirect
2. #20259986	C₂	semidirect
3. C₂×#15690565	C₂	semidirect
4. D₁₉₃⋊M₁₆	C₂	semidirect
8. #15690565	C₂²	semidirect
9. #15690564	C₂²	semidirect
10. #15690563	C₂²	semidirect
11. C₂×#13095400	C₂²	semidirect
45. C₁₉₃	#568	semidirect

Maximal subgroups

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

subgroup	index
#20259987	2
#20259986	2
C₂×#15690565	2
D₁₉₃⋊M₁₆	2
#41×D₁₉₃	2
#568	193