Subgroups — #27135556

Order 12352.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×D₇₇₂	C₂	1	1	yes	maximal, characteristic
2	#20259832	C₂	1	1	yes	maximal, characteristic, complemented
3	#20259830	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄⋊#13095393	C₂	1	1	yes	maximal, characteristic, complemented
5	#20259819	C₂	1	1	yes	maximal, characteristic
6	#20259819	C₂	1	1	yes	maximal, characteristic
7	#125×C₁₉₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₇₇₂	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₁₉₃	—	2	1	no	—
10	C₂×#13095393	C₂²	1	1	yes	characteristic
11	C₂×#13095393	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₁₉₃	C₂²	1	1	yes	characteristic
13	#15690532	—	2	1	no	—
14	C₄⋊C₄×C₁₉₃	—	2	1	no	—
15	C₄⋊C₄×C₁₉₃	—	2	1	no	—
16	C₄⋊C₄×C₁₉₃	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×C₁₉₃	C₂²	1	1	yes	characteristic
18	C₄²×C₁₉₃	C₂²	1	1	yes	characteristic
19	C₂²×D₁₉₃	—	2	1	no	—
20	D₇₇₂	—	2	1	no	—
21	D₇₇₂	Q₈	1	2	yes	—
22	C₂×Dic₁₉₃	—	2	1	no	—
23	C₄×D₁₉₃	—	4	2	no	—
24	#13095393	—	2	1	no	—
25	#13095393	—	2	1	no	—
26	C₂×C₄×C₁₉₃	—	2	1	no	—
27	C₂×C₄×C₁₉₃	—	2	1	no	—
28	C₂×C₄×C₁₉₃	D₄	1	1	yes	characteristic
29	C₂×C₄×C₁₉₃	D₄	1	1	yes	characteristic
30	C₂×C₄×C₁₉₃	C₂³	1	1	yes	characteristic
31	C₂²×C₁₉₃	C₂×D₄	1	1	yes	characteristic
32	C₂×D₁₉₃	—	2	2	no	—
33	C₂×D₁₉₃	—	4	1	no	—
34	C₂×D₁₉₃	—	4	1	no	—
35	C₄×C₁₉₃	—	2	1	no	—
36	C₄×C₁₉₃	—	2	1	no	—
37	C₄×C₁₉₃	—	4	1	no	—
38	C₄×C₁₉₃	—	4	1	no	—
39	C₄×C₁₉₃	C₂×Q₈	1	1	yes	characteristic
40	C₄×C₁₉₃	#41	1	1	yes	characteristic
41	Dic₁₉₃	—	4	1	no	—
42	C₂×C₁₉₃	C₂²⋊Q₈	1	1	yes	characteristic
43	C₂×C₁₉₃	#135	1	1	yes	characteristic
44	C₂×C₁₉₃	AGL(1,ℤ/8)	1	1	yes	characteristic
45	D₁₉₃	—	4	2	no	—
46	C₁₉₃	#478	1	1	yes	characteristic, complemented, Sylow
47	#478	—	193	1	no	maximal, Sylow
48	#125	D₁₉₃	1	1	yes	characteristic, complemented
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	—	193	1	no	—
55	C₂×D₄	—	193	1	no	—
56	C₂²×C₄	—	386	1	no	—
57	C₂×C₈	—	193	1	no	—
58	C₂×C₈	—	193	1	no	—
59	C₄⋊C₄	—	2	1	no	—
60	C₄⋊C₄	—	2	1	no	—
61	C₄⋊C₄	—	193	1	no	—
62	C₄⋊C₄	C₂×D₁₉₃	1	1	yes	characteristic
63	C₄⋊C₄	C₂×D₁₉₃	1	1	yes	characteristic
64	C₂²⋊C₄	—	386	1	no	—
65	C₄²	C₂×D₁₉₃	1	1	yes	characteristic
66	C₂³	—	386	1	no	—
67	D₄	—	386	1	no	—
68	D₄	—	193	2	no	—
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	772	2	no	—
72	C₂×C₄	—	386	1	no	—
73	C₂×C₄	C₂²⋊D₁₉₃	1	1	yes	characteristic
74	C₂×C₄	C₂²⋊D₁₉₃	1	1	yes	characteristic
75	C₂×C₄	C₂²×D₁₉₃	1	1	yes	characteristic
76	C₈	—	386	1	no	—
77	C₈	—	386	1	no	—
78	C₂²	—	772	1	no	—
79	C₂²	—	772	1	no	—
80	C₂²	—	386	2	no	—
81	C₂²	C₂×C₂²⋊D₁₉₃	1	1	yes	characteristic
82	C₄	—	2	1	no	—
83	C₄	—	2	1	no	—
84	C₄	—	4	1	no	—
85	C₄	—	4	1	no	—
86	C₄	—	772	1	no	—
87	C₄	Q₈×D₁₉₃	1	1	yes	characteristic
88	C₄	#15690556	1	1	yes	characteristic
89	C₂	—	772	2	no	—
90	C₂	#20259875	1	1	yes	characteristic
91	C₂	#20259919	1	1	yes	characteristic
92	C₂	#20259924	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₉₃
commutator, cosocle	C₂×C₄×C₁₉₃	C₂³
Frattini	C₂×C₄	C₂²×D₁₉₃
Fitting	#125×C₁₉₃	C₂
socle	C₂²×C₁₉₃	C₂×D₄
prime radical	C₂×D₇₇₂	C₂²
hypercenter	#125	D₁₉₃
hypocenter	C₁₉₃	#478
LCS	G ▷ C₂×C₄×C₁₉₃ ▷ C₂×C₁₉₃ ▷ C₁₉₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

Complements

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

N	H ≅ G/N	type
2. #20259832	C₂	semidirect
3. #20259830	C₂	semidirect
4. C₄⋊#13095393	C₂	semidirect
7. #125×C₁₉₃	C₂	semidirect
46. C₁₉₃	#478	semidirect
48. #125	D₁₉₃	semidirect

Maximal subgroups

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

subgroup	index
C₄×D₇₇₂	2
#20259832	2
#20259830	2
C₄⋊#13095393	2
#20259819	2
#125×C₁₉₃	2
#478	193