Subgroups — #27161414

Order 12608.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20263305	C₂	1	1	yes	maximal, characteristic
2	#20263259	C₂	1	2	yes	maximal, complemented
3	#20263249	C₂	1	2	yes	maximal, complemented
4	C₂²⋊#13096802	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊C₄×C₁₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₉₄	C₂²	1	1	yes	characteristic
7	C₂×#13096802	C₂²	1	2	yes	—
8	C₄⋊Dic₁₉₇	C₂²	1	1	yes	characteristic
9	#15691337	—	2	1	no	—
10	C₂²⋊Dic₁₉₇	—	2	1	no	—
11	C₂²×C₄×C₁₉₇	—	2	1	no	—
12	C₂²×C₄×C₁₉₇	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₁₉₇	—	2	1	no	—
14	C₄⋊C₄×C₁₉₇	C₂²	1	2	yes	—
15	Dic₃₉₄	—	2	2	no	—
16	C₂×Dic₁₉₇	—	2	1	no	—
17	C₂×Dic₁₉₇	—	2	1	no	—
18	#13096802	—	2	2	no	—
19	C₂³×C₁₉₇	D₄	1	1	yes	characteristic
20	C₂×C₄×C₁₉₇	—	2	2	no	—
21	C₂×C₄×C₁₉₇	—	2	2	no	—
22	C₂×C₄×C₁₉₇	—	4	2	no	—
23	C₂×C₄×C₁₉₇	D₄	1	1	yes	characteristic
24	C₂×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
25	C₂²×C₁₉₇	—	2	1	no	—
26	C₂²×C₁₉₇	—	2	1	no	—
27	C₂²×C₁₉₇	Q₁₆	1	2	yes	—
28	C₂²×C₁₉₇	C₂×D₄	1	1	yes	characteristic
29	C₄×C₁₉₇	—	2	1	no	—
30	C₄×C₁₉₇	—	4	2	no	—
31	C₄×C₁₉₇	#41	1	2	yes	—
32	Dic₁₉₇	—	4	1	no	—
33	Dic₁₉₇	—	4	1	no	—
34	C₂×C₁₉₇	—	2	2	no	—
35	C₂×C₁₉₇	#123	1	1	yes	characteristic
36	C₂×C₁₉₇	C₂×Q₁₆	1	1	yes	characteristic
37	C₂×C₁₉₇	AGL(1,ℤ/8)	1	1	yes	characteristic
38	C₁₉₇	#484	1	1	yes	characteristic, complemented, Sylow
39	#484	—	197	1	no	maximal, Sylow
40	C₂²⋊Q₈	—	197	1	no	—
41	C₂×C₄⋊C₄	D₁₉₇	1	1	yes	characteristic
42	#107	—	197	2	no	—
43	#103	—	197	2	no	—
44	C₂²⋊C₈	—	197	1	no	—
45	C₂×Q₈	—	197	1	no	—
46	C₂²×C₄	—	2	1	no	—
47	C₂²×C₄	C₂×D₁₉₇	1	1	yes	characteristic
48	C₂×C₈	—	197	2	no	—
49	C₄⋊C₄	—	2	1	no	—
50	C₄⋊C₄	—	394	1	no	—
51	C₄⋊C₄	—	197	1	no	—
52	C₄⋊C₄	C₂×D₁₉₇	1	2	yes	—
53	C₂²⋊C₄	—	394	1	no	—
54	C₂³	C₂²⋊D₁₉₇	1	1	yes	characteristic
55	Q₈	—	394	2	no	—
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	—	2	2	no	—
58	C₂×C₄	—	4	2	no	—
59	C₂×C₄	—	394	1	no	—
60	C₂×C₄	—	394	1	no	—
61	C₂×C₄	C₂²⋊D₁₉₇	1	1	yes	characteristic
62	C₂×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
63	C₈	—	394	2	no	—
64	C₂²	—	2	1	no	—
65	C₂²	—	2	1	no	—
66	C₂²	#15691348	1	2	yes	—
67	C₂²	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
68	C₄	—	2	1	no	—
69	C₄	—	4	2	no	—
70	C₄	—	788	1	no	—
71	C₄	—	788	1	no	—
72	C₄	#15691358	1	2	yes	—
73	C₂	—	2	2	no	—
74	C₂	#20263309	1	1	yes	characteristic
75	C₂	C₂×#15691348	1	1	yes	characteristic
76	C₂	#20263352	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₉₇
commutator, cosocle	C₂×C₄×C₁₉₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₉₇
Fitting	C₂×C₄⋊C₄×C₁₉₇	C₂
socle	C₂²×C₁₉₇	C₂×D₄
prime radical	C₂³×C₁₉₇	D₄
hypercenter	C₂×C₄⋊C₄	D₁₉₇
hypocenter	C₁₉₇	#484
LCS	G ▷ C₂×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. #20263259	C₂	semidirect
3. #20263249	C₂	semidirect
38. C₁₉₇	#484	semidirect

Maximal subgroups

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

subgroup	index
#20263305	2
#20263259	2
#20263249	2
C₂²⋊#13096802	2
C₂×C₄⋊C₄×C₁₉₇	2
#484	197