Subgroups — #27136305

Order 12352.

104 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	#20259969	C₂	1	1	yes	maximal, characteristic, complemented
3	#20259964	C₂	1	2	yes	maximal, complemented
4	#20259953	C₂	1	1	yes	maximal, characteristic, complemented
5	#20259922	C₂	1	1	yes	maximal, characteristic
6	#15690565	—	2	2	no	—
7	#15690565	C₂²	1	2	yes	—
8	#15690563	—	2	2	no	—
9	#15690563	C₂²	1	1	yes	characteristic
10	C₄×#11715871	C₂²	1	1	yes	characteristic
11	#15690556	C₂²	1	2	yes	—
12	#15690542	C₄	1	2	yes	complemented
13	D₁₅₄₄	C₄	1	1	yes	characteristic, complemented
14	C₈×D₁₉₃	C₂²	1	1	yes	characteristic
15	Q₁₆×C₁₉₃	C₄	1	1	yes	characteristic, complemented
16	C₂×#11715871	—	2	1	no	—
17	#13095400	—	2	2	no	—
18	#13095400	D₄	1	2	yes	—
19	D₇₇₂	—	2	2	no	—
20	D₇₇₂	C₂×C₄	1	2	yes	—
21	C₄×D₁₉₃	—	2	2	no	—
22	C₄×D₁₉₃	C₂³	1	1	yes	characteristic
23	#13095393	C₂×C₄	1	1	yes	characteristic
24	Q₈×C₁₉₃	C₂×C₄	1	2	yes	—
25	C₈×C₁₉₃	C₂×C₄	1	1	yes	characteristic
26	C₂×D₁₉₃	—	2	2	no	—
27	C₂×D₁₉₃	#41	1	1	yes	characteristic
28	#11715871	—	4	1	no	—
29	C₄×C₁₉₃	—	2	2	no	—
30	C₄×C₁₉₃	C₂²×C₄	1	1	yes	characteristic
31	Dic₁₉₃	C₂×D₄	1	1	yes	characteristic
32	C₂×C₁₉₃	C₄×D₄	1	1	yes	characteristic
33	D₁₉₃	—	2	1	no	—
34	D₁₉₃	—	4	2	no	—
35	C₁₉₃	#444	1	1	yes	characteristic, complemented, Sylow
36	#444	—	193	1	no	maximal, Sylow
37	#135	—	193	1	no	—
38	#131	—	193	2	no	—
39	#108	—	193	1	no	—
40	#104	—	193	2	no	—
41	#97	—	193	1	no	—
42	#41	—	193	2	no	—
43	Q₁₆	#11715871	1	1	yes	characteristic, complemented
44	SD₁₆	—	193	2	no	—
45	D₈	—	193	1	no	—
46	M₁₆	—	386	2	no	—
47	M₁₆	—	193	2	no	—
48	C₂×C₈	—	386	2	no	—
49	C₂×C₈	—	193	1	no	—
50	C₂×C₈	—	193	1	no	—
51	C₄²	—	193	1	no	—
52	Q₈	C₂×#11715871	1	2	yes	—
53	D₄	—	386	2	no	—
54	D₄	—	193	2	no	—
55	C₂×C₄	—	386	2	no	—
56	C₂×C₄	—	386	1	no	—
57	C₂×C₄	—	193	1	no	—
58	C₈	—	386	2	no	—
59	C₈	—	193	2	no	—
60	C₈	—	193	1	no	—
61	C₈	C₂×#11715871	1	1	yes	characteristic
62	C₂²	—	386	2	no	—
63	C₂²	—	193	1	no	—
64	C₄	—	2	2	no	—
65	C₄	—	772	1	no	—
66	C₄	—	193	1	no	—
67	C₄	C₂²×#11715871	1	1	yes	characteristic
68	C₂	—	772	2	no	—
69	C₂	—	386	1	no	—
70	C₂	D₄×#11715871	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×#11715871
commutator	C₄×C₁₉₃	C₂²×C₄
Frattini	C₄	C₂²×#11715871
Fitting	Q₁₆×C₁₉₃	C₄
socle	C₂×C₁₉₃	C₄×D₄
prime radical	#20259922	C₂
cosocle	C₄×D₁₉₃	C₂³
hypercenter	Q₁₆	#11715871
hypocenter	C₁₉₃	#444
LCS	G ▷ C₄×C₁₉₃ ▷ C₂×C₁₉₃ ▷ C₁₉₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₁₆

Complements

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

N	H ≅ G/N	type
1. #20259987	C₂	semidirect
2. #20259969	C₂	semidirect
3. #20259964	C₂	semidirect
4. #20259953	C₂	semidirect
12. #15690542	C₄	semidirect
13. D₁₅₄₄	C₄	semidirect
15. Q₁₆×C₁₉₃	C₄	semidirect
35. C₁₉₃	#444	semidirect
43. Q₁₆	#11715871	semidirect

Maximal subgroups

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

subgroup	index
#20259987	2
#20259969	2
#20259964	2
#20259953	2
#20259922	2
#444	193