Subgroups — #27136280

Order 12352.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#13095400	C₂	1	2	yes	maximal, complemented
2	C₂²⋊#13095400	C₂	1	2	yes	maximal
3	#20259954	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×Dic₁₉₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#13095400	C₂²	1	4	yes	—
6	C₂²×Dic₁₉₃	C₄	1	1	yes	characteristic
7	C₂²×Dic₁₉₃	C₂²	1	1	yes	characteristic
8	C₄×Dic₁₉₃	C₄	1	1	yes	characteristic
9	C₄×Dic₁₉₃	C₄	1	1	yes	characteristic
10	C₄×Dic₁₉₃	C₂²	1	1	yes	characteristic
11	C₄×Dic₁₉₃	C₂²	1	1	yes	characteristic
12	C₂²×C₄×C₁₉₃	C₄	1	1	yes	characteristic
13	#13095400	—	2	4	no	—
14	C₂×Dic₁₉₃	—	2	2	no	—
15	C₂×Dic₁₉₃	—	2	2	no	—
16	C₂×Dic₁₉₃	C₂×C₄	1	1	yes	characteristic
17	C₂×Dic₁₉₃	C₂×C₄	1	1	yes	characteristic
18	C₂×Dic₁₉₃	C₂×C₄	1	1	yes	characteristic
19	C₂×Dic₁₉₃	C₂³	1	1	yes	characteristic
20	C₂³×C₁₉₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₁₉₃	—	2	2	no	—
22	C₂×C₄×C₁₉₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×C₁₉₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₁₉₃	—	2	1	no	—
25	C₂²×C₁₉₃	—	2	1	no	—
26	C₂²×C₁₉₃	M₁₆	1	2	yes	—
27	C₂²×C₁₉₃	C₂²×C₄	1	1	yes	characteristic
28	C₄×C₁₉₃	—	2	1	no	—
29	C₄×C₁₉₃	M₁₆	1	2	yes	—
30	Dic₁₉₃	—	2	1	no	—
31	Dic₁₉₃	—	2	1	no	—
32	Dic₁₉₃	—	2	1	no	—
33	Dic₁₉₃	#41	1	2	yes	—
34	C₂×C₁₉₃	—	2	2	no	—
35	C₂×C₁₉₃	#117	1	1	yes	characteristic
36	C₂×C₁₉₃	C₂×M₁₆	1	1	yes	characteristic
37	C₂×C₁₉₃	C₂×M₁₆	1	1	yes	characteristic
38	C₁₉₃	#432	1	1	yes	characteristic, complemented, Sylow
39	#432	—	193	1	no	maximal, Sylow
40	C₂×C₄²	—	193	1	no	—
41	C₄⋊C₈	—	193	2	no	—
42	C₂²⋊C₈	—	193	2	no	—
43	#97	—	193	2	no	—
44	C₂²×C₄	—	193	1	no	—
45	C₂²×C₄	—	193	1	no	—
46	C₂²×C₄	#11715871	1	1	yes	characteristic
47	C₂×C₈	—	193	4	no	—
48	C₄²	—	193	1	no	—
49	C₄²	—	193	1	no	—
50	C₄²	—	193	1	no	—
51	C₄²	—	193	1	no	—
52	C₂³	C₂×#11715871	1	1	yes	characteristic
53	C₂×C₄	—	2	2	no	—
54	C₂×C₄	—	386	2	no	—
55	C₂×C₄	—	386	2	no	—
56	C₂×C₄	—	193	1	no	—
57	C₂×C₄	—	193	1	no	—
58	C₂×C₄	—	193	1	no	—
59	C₂×C₄	—	193	1	no	—
60	C₂×C₄	C₂×#11715871	1	1	yes	characteristic
61	C₂×C₄	C₂×#11715871	1	1	yes	characteristic
62	C₈	—	386	4	no	—
63	C₂²	—	2	1	no	—
64	C₂²	—	2	1	no	—
65	C₂²	#15690565	1	2	yes	—
66	C₂²	C₂²×#11715871	1	1	yes	characteristic
67	C₄	—	2	1	no	—
68	C₄	—	386	1	no	—
69	C₄	—	386	1	no	—
70	C₄	—	386	1	no	—
71	C₄	—	193	2	no	—
72	C₄	#15690564	1	2	yes	—
73	C₂	—	2	2	no	—
74	C₂	#20259978	1	1	yes	characteristic
75	C₂	C₂×#15690564	1	1	yes	characteristic
76	C₂	C₂×#15690565	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×#11715871
commutator, socle	C₂²×C₁₉₃	C₂²×C₄
Fitting	C₂²×C₄×C₁₉₃	C₄
prime radical	C₂³×C₁₉₃	C₂×C₄
cosocle	C₂×Dic₁₉₃	C₂³
hypercenter	C₂²×C₄	#11715871
hypocenter	C₁₉₃	#432
LCS	G ▷ C₂²×C₁₉₃ ▷ C₁₉₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
1. C₄⋊#13095400	C₂	semidirect
3. #20259954	C₂	semidirect
38. C₁₉₃	#432	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊#13095400	2
C₂²⋊#13095400	2
#20259954	2
C₂×C₄×Dic₁₉₃	2
#432	193