Subgroups — #27135557

Order 12352.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20259890	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×Dic₁₉₃	C₂	1	1	yes	maximal, characteristic
3	#20259833	C₂	1	1	yes	maximal, characteristic, complemented
4	#20259830	C₂	1	1	yes	maximal, characteristic, complemented
5	#20259828	C₂	1	1	yes	maximal, characteristic, complemented
6	#20259820	C₂	1	1	yes	maximal, characteristic
7	#102×C₁₉₃	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₉₃	—	2	1	no	—
9	C₂×#13095393	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₁₉₃	—	2	1	no	—
11	C₄⋊Dic₁₉₃	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₁₉₃	C₂²	1	1	yes	characteristic
13	#15690534	—	2	1	no	—
14	C₂²⋊Dic₁₉₃	—	2	1	no	—
15	C₄×Dic₁₉₃	C₂²	1	1	yes	characteristic
16	C₂×D₄×C₁₉₃	C₂²	1	1	yes	characteristic
17	C₂×C₈×C₁₉₃	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₁₉₃	C₂²	1	1	yes	characteristic
19	C₂×Dic₁₉₃	—	2	1	no	—
20	C₂×Dic₁₉₃	—	2	1	no	—
21	C₂×Dic₁₉₃	—	4	2	no	—
22	C₂×Dic₁₉₃	D₄	1	1	yes	characteristic
23	C₂×Dic₁₉₃	D₄	1	1	yes	characteristic
24	#13095393	—	2	1	no	—
25	C₂³×C₁₉₃	—	2	1	no	—
26	D₄×C₁₉₃	—	2	1	no	—
27	D₄×C₁₉₃	Q₈	1	2	yes	—
28	C₂×C₄×C₁₉₃	—	2	1	no	—
29	C₂×C₄×C₁₉₃	C₂³	1	1	yes	characteristic
30	C₈×C₁₉₃	—	2	1	no	—
31	C₂²×C₁₉₃	—	2	2	no	—
32	C₂²×C₁₉₃	—	4	1	no	—
33	C₂²×C₁₉₃	—	4	1	no	—
34	C₂²×C₁₉₃	C₂×D₄	1	1	yes	characteristic
35	C₄×C₁₉₃	—	4	1	no	—
36	C₄×C₁₉₃	C₂×Q₈	1	1	yes	characteristic
37	C₄×C₁₉₃	#41	1	1	yes	characteristic
38	Dic₁₉₃	—	2	1	no	—
39	Dic₁₉₃	—	2	1	no	—
40	Dic₁₉₃	—	4	1	no	—
41	Dic₁₉₃	—	4	1	no	—
42	C₂×C₁₉₃	—	4	2	no	—
43	C₂×C₁₉₃	C₂²⋊Q₈	1	1	yes	characteristic
44	C₂×C₁₉₃	#135	1	1	yes	characteristic
45	C₂×C₁₉₃	AGL(1,ℤ/8)	1	1	yes	characteristic
46	C₁₉₃	#478	1	1	yes	characteristic, complemented, Sylow
47	#478	—	193	1	no	maximal, Sylow
48	#125	—	193	1	no	—
49	C₄×D₄	—	193	1	no	—
50	#107	—	193	1	no	—
51	#106	—	193	1	no	—
52	C₄⋊C₈	—	193	1	no	—
53	#102	—	193	1	no	—
54	#102	D₁₉₃	1	1	yes	characteristic
55	C₂×D₄	C₂×D₁₉₃	1	1	yes	characteristic
56	C₂²×C₄	—	386	1	no	—
57	C₂×C₈	—	193	1	no	—
58	C₂×C₈	C₂×D₁₉₃	1	1	yes	characteristic
59	C₄⋊C₄	—	386	1	no	—
60	C₄⋊C₄	—	386	1	no	—
61	C₄⋊C₄	—	193	1	no	—
62	C₄⋊C₄	—	193	1	no	—
63	C₄⋊C₄	C₂×D₁₉₃	1	1	yes	characteristic
64	C₂²⋊C₄	—	386	1	no	—
65	C₄²	—	193	1	no	—
66	C₂³	—	2	1	no	—
67	D₄	—	2	1	no	—
68	D₄	Dic₃₈₆	1	2	yes	—
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	772	2	no	—
71	C₂×C₄	—	386	1	no	—
72	C₂×C₄	—	386	1	no	—
73	C₂×C₄	—	193	1	no	—
74	C₂×C₄	—	193	1	no	—
75	C₂×C₄	C₂²×D₁₉₃	1	1	yes	characteristic
76	C₈	—	2	1	no	—
77	C₈	—	386	1	no	—
78	C₂²	—	2	2	no	—
79	C₂²	—	4	1	no	—
80	C₂²	—	4	1	no	—
81	C₂²	D₄×D₁₉₃	1	1	yes	characteristic
82	C₄	—	4	1	no	—
83	C₄	—	772	1	no	—
84	C₄	—	772	1	no	—
85	C₄	—	386	1	no	—
86	C₄	—	386	1	no	—
87	C₄	C₂×Dic₃₈₆	1	1	yes	characteristic
88	C₄	#15690554	1	1	yes	characteristic
89	C₂	—	4	2	no	—
90	C₂	#20259876	1	1	yes	characteristic
91	C₂	#20259918	1	1	yes	characteristic
92	C₂	#20259926	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₉₃
commutator, cosocle	C₂×C₄×C₁₉₃	C₂³
Frattini	C₂×C₄	C₂²×D₁₉₃
Fitting	#102×C₁₉₃	C₂
socle	C₂²×C₁₉₃	C₂×D₄
prime radical	C₂×D₄×C₁₉₃	C₂²
hypercenter	#102	D₁₉₃
hypocenter	C₁₉₃	#478
LCS	G ▷ C₂×C₄×C₁₉₃ ▷ C₂×C₁₉₃ ▷ C₁₉₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #102

Complements

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

N	H ≅ G/N	type
1. #20259890	C₂	semidirect
3. #20259833	C₂	semidirect
4. #20259830	C₂	semidirect
5. #20259828	C₂	semidirect
46. C₁₉₃	#478	semidirect

Maximal subgroups

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

subgroup	index
#20259890	2
D₄×Dic₁₉₃	2
#20259833	2
#20259830	2
#20259828	2
#20259820	2
#102×C₁₉₃	2
#478	193