Subgroups — #27134998

Order 12352.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20259887	C₂	1	1	yes	maximal, characteristic
2	#20259810	C₂	1	1	yes	maximal, characteristic, complemented
3	#97×C₁₉₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×Dic₃₈₆	C₄	1	1	yes	characteristic
5	C₂×D₇₇₂	C₄	1	1	yes	characteristic
6	C₂×#13095393	—	2	1	no	—
7	#15690532	—	2	1	no	—
8	#15690532	—	2	1	no	—
9	C₂×C₈×C₁₉₃	—	2	1	no	—
10	C₄²×C₁₉₃	C₂²	1	1	yes	characteristic
11	C₂²×D₁₉₃	—	2	1	no	—
12	Dic₃₈₆	—	4	1	no	—
13	D₇₇₂	—	4	1	no	—
14	C₂×Dic₁₉₃	—	2	1	no	—
15	#13095393	—	2	2	no	—
16	C₂×C₄×C₁₉₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₁₉₃	D₄	1	1	yes	characteristic
18	C₂×C₄×C₁₉₃	D₄	1	1	yes	characteristic
19	C₈×C₁₉₃	—	2	2	no	—
20	C₂²×C₁₉₃	C₂²⋊C₄	1	1	yes	characteristic
21	C₂×D₁₉₃	—	4	1	no	—
22	C₂×D₁₉₃	—	4	1	no	—
23	C₂×D₁₉₃	—	4	1	no	—
24	C₄×C₁₉₃	—	2	1	no	—
25	C₄×C₁₉₃	—	2	1	no	—
26	C₄×C₁₉₃	—	2	1	no	—
27	Dic₁₉₃	—	4	1	no	—
28	C₂×C₁₉₃	#100	1	1	yes	characteristic
29	C₂×C₁₉₃	#104	1	1	yes	characteristic
30	C₂×C₁₉₃	#104	1	1	yes	characteristic
31	D₁₉₃	—	8	1	no	—
32	C₁₉₃	#329	1	1	yes	characteristic, complemented, Sylow
33	#329	—	193	1	no	maximal, Sylow
34	#124	—	193	1	no	—
35	#97	—	193	1	no	—
36	#97	D₁₉₃	1	1	yes	characteristic, complemented
37	C₂×Q₈	—	193	1	no	—
38	C₂×D₄	—	193	1	no	—
39	C₂×C₈	—	2	1	no	—
40	C₂×C₈	—	386	1	no	—
41	C₂²⋊C₄	—	386	1	no	—
42	C₂²⋊C₄	—	386	1	no	—
43	C₄²	C₂×D₁₉₃	1	1	yes	characteristic
44	C₂³	—	386	1	no	—
45	Q₈	—	772	1	no	—
46	D₄	—	772	1	no	—
47	C₂×C₄	—	386	1	no	—
48	C₂×C₄	C₄×D₁₉₃	1	1	yes	characteristic
49	C₂×C₄	C₂²⋊D₁₉₃	1	1	yes	characteristic
50	C₂×C₄	D₇₇₂	1	1	yes	characteristic
51	C₈	—	2	2	no	—
52	C₈	—	386	2	no	—
53	C₂²	—	772	1	no	—
54	C₂²	—	772	1	no	—
55	C₂²	—	772	1	no	—
56	C₂²	#15690532	1	1	yes	characteristic
57	C₄	—	2	1	no	—
58	C₄	—	2	1	no	—
59	C₄	—	2	1	no	—
60	C₄	—	772	1	no	—
61	C₂	—	1544	1	no	—
62	C₂	#20259815	1	1	yes	characteristic
63	C₂	#20259825	1	1	yes	characteristic
64	C₂	#20259827	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15690532
commutator	C₂×C₄×C₁₉₃	C₂×C₄
Frattini	C₄²	C₂×D₁₉₃
Fitting	#97×C₁₉₃	C₂
socle	C₂²×C₁₉₃	C₂²⋊C₄
prime radical	C₂×D₇₇₂	C₄
cosocle	C₄²×C₁₉₃	C₂²
hypercenter	#97	D₁₉₃
hypocenter	C₁₉₃	#329
LCS	G ▷ C₂×C₄×C₁₉₃ ▷ C₂²×C₁₉₃ ▷ C₁₉₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

Complements

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

N	H ≅ G/N	type
2. #20259810	C₂	semidirect
3. #97×C₁₉₃	C₂	semidirect
32. C₁₉₃	#329	semidirect
36. #97	D₁₉₃	semidirect

Maximal subgroups

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

subgroup	index
#20259887	2
#20259810	2
#97×C₁₉₃	2
#329	193