Subgroups — #27136139

Order 12352.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15690564	C₂	1	2	yes	maximal
2	C₂×#15690534	C₂	1	1	yes	maximal, characteristic
3	#15690564	—	2	2	no	—
4	#15690564	C₄	1	4	yes	complemented
5	C₂×#13095400	—	2	2	no	—
6	C₂²×Dic₁₉₃	C₄	1	1	yes	characteristic
7	C₂²×Dic₁₉₃	C₂²	1	1	yes	characteristic
8	#15690534	—	2	2	no	—
9	C₂²×C₄×C₁₉₃	C₄	1	1	yes	characteristic
10	#13095400	—	2	4	no	—
11	C₂×Dic₁₉₃	—	2	1	no	—
12	C₂×Dic₁₉₃	—	4	1	no	—
13	C₂×Dic₁₉₃	C₂×C₄	1	2	yes	—
14	C₂×Dic₁₉₃	D₄	1	1	yes	characteristic
15	C₂×Dic₁₉₃	D₄	1	2	yes	—
16	C₂×Dic₁₉₃	Q₈	1	1	yes	characteristic
17	C₂³×C₁₉₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₁₉₃	—	2	1	no	—
19	C₂×C₄×C₁₉₃	—	4	1	no	—
20	C₂²×C₁₉₃	—	2	1	no	—
21	C₂²×C₁₉₃	—	2	1	no	—
22	C₂²×C₁₉₃	C₄²	1	1	yes	characteristic
23	C₂²×C₁₉₃	C₂²⋊C₄	1	1	yes	characteristic
24	C₂²×C₁₉₃	C₄⋊C₄	1	1	yes	characteristic
25	C₄×C₁₉₃	—	4	1	no	—
26	Dic₁₉₃	—	4	1	no	—
27	Dic₁₉₃	C₂²⋊C₄	1	2	yes	—
28	Dic₁₉₃	C₄⋊C₄	1	2	yes	—
29	C₂×C₁₉₃	—	2	1	no	—
30	C₂×C₁₉₃	—	2	1	no	—
31	C₂×C₁₉₃	#95	1	1	yes	characteristic
32	C₂×C₁₉₃	#100	1	1	yes	characteristic
33	C₂×C₁₉₃	#101	1	1	yes	characteristic
34	C₁₉₃	#343	1	1	yes	characteristic, complemented, Sylow
35	#343	—	193	1	no	maximal, Sylow
36	C₂×M₁₆	—	193	2	no	—
37	C₂×C₄⋊C₄	—	193	1	no	—
38	C₂²×C₄	—	193	1	no	—
39	C₂²×C₄	—	193	1	no	—
40	C₂²×C₄	#11715871	1	1	yes	characteristic
41	M₁₆	—	386	2	no	—
42	M₁₆	—	193	4	no	—
43	C₂×C₈	—	386	2	no	—
44	C₄⋊C₄	—	386	2	no	—
45	C₂³	C₂×#11715871	1	1	yes	characteristic
46	C₂×C₄	—	2	1	no	—
47	C₂×C₄	—	4	1	no	—
48	C₂×C₄	—	772	1	no	—
49	C₂×C₄	—	386	1	no	—
50	C₂×C₄	—	193	2	no	—
51	C₂×C₄	—	193	2	no	—
52	C₂×C₄	—	193	1	no	—
53	C₂×C₄	—	193	1	no	—
54	C₈	—	386	4	no	—
55	C₂²	—	2	1	no	—
56	C₂²	—	2	1	no	—
57	C₂²	C₄×#11715871	1	1	yes	characteristic
58	C₂²	C₂²⋊#11715871	1	1	yes	characteristic
59	C₂²	C₄⋊#11715871	1	1	yes	characteristic
60	C₄	—	4	1	no	—
61	C₄	—	772	1	no	—
62	C₄	—	193	2	no	—
63	C₄	—	193	2	no	—
64	C₂	—	2	1	no	—
65	C₂	—	2	1	no	—
66	C₂	(C₂×#11715871)⋊C₄	1	1	yes	characteristic
67	C₂	#20259959	1	1	yes	characteristic
68	C₂	#20259960	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊#11715871
commutator	C₂²×C₁₉₃	C₄²
Frattini	C₂³	C₂×#11715871
Fitting	C₂²×C₄×C₁₉₃	C₄
socle	C₂²×C₁₉₃	C₂²⋊C₄
prime radical	C₂³×C₁₉₃	C₂×C₄
cosocle	C₂²×Dic₁₉₃	C₂²
hypercenter	C₂²×C₄	#11715871
hypocenter	C₁₉₃	#343
LCS	G ▷ C₂²×C₁₉₃ ▷ C₂×C₁₉₃ ▷ C₁₉₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
4. #15690564	C₄	semidirect
34. C₁₉₃	#343	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15690564	2
C₂×#15690534	2
#343	193