Subgroups — #27161333

Order 12608.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₅₇₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15691341	C₂	1	1	yes	maximal, characteristic, complemented
3	#20263301	C₂	1	2	yes	maximal, complemented
4	#20263246	C₂	1	2	yes	maximal, complemented
5	#106×C₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₇₈₈	—	2	2	no	—
7	C₂×D₇₈₈	C₂²	1	2	yes	—
8	C₂×C₄×D₁₉₇	C₂²	1	1	yes	characteristic
9	D₁₅₇₆	—	2	2	no	—
10	#15691341	—	2	2	no	—
11	C₂×#13096802	C₂²	1	1	yes	characteristic
12	#15691335	—	2	2	no	—
13	C₂×C₈×C₁₉₇	C₂²	1	1	yes	characteristic
14	C₄⋊C₄×C₁₉₇	C₂²	1	2	yes	—
15	C₂²×D₁₉₇	—	2	2	no	—
16	C₂²×D₁₉₇	D₄	1	1	yes	characteristic
17	D₇₈₈	—	2	2	no	—
18	D₇₈₈	—	2	2	no	—
19	D₇₈₈	—	4	4	no	—
20	C₂×Dic₁₉₇	D₄	1	1	yes	characteristic
21	C₄×D₁₉₇	—	2	1	no	—
22	C₄×D₁₉₇	—	2	1	no	—
23	#13096802	—	2	1	no	—
24	C₂×C₄×C₁₉₇	—	2	2	no	—
25	C₂×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
26	C₈×C₁₉₇	D₄	1	2	yes	complemented
27	C₂²×C₁₉₇	C₂×D₄	1	1	yes	characteristic
28	C₂×D₁₉₇	—	2	1	no	—
29	C₂×D₁₉₇	—	2	2	no	—
30	C₂×D₁₉₇	—	4	2	no	—
31	C₂×D₁₉₇	—	4	2	no	—
32	C₂×D₁₉₇	—	4	2	no	—
33	C₄×C₁₉₇	—	4	2	no	—
34	C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
35	C₄×C₁₉₇	#41	1	1	yes	characteristic
36	Dic₁₉₇	—	2	1	no	—
37	C₂×C₁₉₇	#121	1	1	yes	characteristic
38	C₂×C₁₉₇	AGL(1,ℤ/8)	1	2	yes	—
39	D₁₉₇	—	4	1	no	—
40	D₁₉₇	—	8	2	no	—
41	C₁₉₇	#469	1	1	yes	characteristic, complemented, Sylow
42	#469	—	197	1	no	maximal, Sylow
43	C₂×D₈	—	197	1	no	—
44	C₂×M₁₆	—	197	1	no	—
45	#121	—	197	2	no	—
46	#106	D₁₉₇	1	1	yes	characteristic, complemented
47	#102	—	197	2	no	—
48	C₂×D₄	—	394	2	no	—
49	C₂×D₄	—	197	2	no	—
50	C₂²×C₄	—	197	1	no	—
51	D₈	—	394	2	no	—
52	M₁₆	—	394	2	no	—
53	C₂×C₈	—	197	1	no	—
54	C₂×C₈	C₂×D₁₉₇	1	1	yes	characteristic
55	C₄⋊C₄	C₂×D₁₉₇	1	2	yes	—
56	C₂²⋊C₄	—	394	2	no	—
57	C₂³	—	394	2	no	—
58	C₂³	—	197	1	no	—
59	D₄	—	788	4	no	—
60	D₄	—	394	2	no	—
61	D₄	—	394	2	no	—
62	C₂×C₄	—	2	2	no	—
63	C₂×C₄	—	394	1	no	—
64	C₂×C₄	—	394	1	no	—
65	C₂×C₄	—	197	1	no	—
66	C₂×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
67	C₈	—	394	1	no	—
68	C₈	D₇₈₈	1	2	yes	complemented
69	C₂²	—	788	2	no	—
70	C₂²	—	788	2	no	—
71	C₂²	—	788	2	no	—
72	C₂²	—	394	1	no	—
73	C₂²	—	394	2	no	—
74	C₂²	D₄×D₁₉₇	1	1	yes	characteristic
75	C₄	—	4	2	no	—
76	C₄	—	394	1	no	—
77	C₄	C₂×D₇₈₈	1	1	yes	characteristic
78	C₄	#15691359	1	1	yes	characteristic
79	C₂	—	788	1	no	—
80	C₂	—	1576	2	no	—
81	C₂	#20263301	1	1	yes	characteristic
82	C₂	#20263353	1	2	yes	—

	subgroup	G/H
center	C₂²	D₄×D₁₉₇
commutator, cosocle	C₂×C₄×C₁₉₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₉₇
Fitting	#106×C₁₉₇	C₂
socle	C₂²×C₁₉₇	C₂×D₄
hypercenter	#106	D₁₉₇
hypocenter	C₁₉₇	#469
LCS	G ▷ C₂×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

Complements

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

N	H ≅ G/N	type
1. C₂×D₁₅₇₆	C₂	semidirect
2. C₂×#15691341	C₂	semidirect
3. #20263301	C₂	semidirect
4. #20263246	C₂	semidirect
5. #106×C₁₉₇	C₂	semidirect
26. C₈×C₁₉₇	D₄	semidirect
41. C₁₉₇	#469	semidirect
46. #106	D₁₉₇	semidirect
68. C₈	D₇₈₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₁₅₇₆	2
C₂×#15691341	2
#20263301	2
#20263246	2
#106×C₁₉₇	2
#469	197