Subgroups — #27161471

Order 12608.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15691348	C₂	1	2	yes	maximal, complemented
2	C₂×#15691345	C₂	1	2	yes	maximal
3	#20263314	C₂	1	1	yes	maximal, characteristic
4	#20263237	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄⋊Q₈×C₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₃₉₄	C₂²	1	1	yes	characteristic
7	C₂×D₇₈₈	C₂²	1	1	yes	characteristic
8	#15691348	—	2	4	no	—
9	#15691345	—	2	4	no	—
10	C₂×#13096802	C₂²	1	2	yes	—
11	#15691335	—	2	2	no	—
12	C₂×Q₈×C₁₉₇	C₂²	1	2	yes	—
13	C₄⋊C₄×C₁₉₇	—	2	2	no	—
14	C₄²×C₁₉₇	C₂²	1	1	yes	characteristic
15	C₂²×D₁₉₇	—	2	1	no	—
16	Dic₃₉₄	—	2	2	no	—
17	D₇₈₈	—	2	2	no	—
18	C₂×Dic₁₉₇	—	2	1	no	—
19	#13096802	D₄	1	4	yes	—
20	Q₈×C₁₉₇	—	2	2	no	—
21	Q₈×C₁₉₇	—	2	2	no	—
22	C₂×C₄×C₁₉₇	—	2	2	no	—
23	C₂×C₄×C₁₉₇	D₄	1	2	yes	—
24	C₂×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
25	C₂²×C₁₉₇	C₂×D₄	1	1	yes	characteristic
26	C₂×D₁₉₇	—	4	1	no	—
27	C₂×D₁₉₇	—	4	2	no	—
28	C₄×C₁₉₇	—	2	2	no	—
29	C₄×C₁₉₇	—	4	2	no	—
30	C₄×C₁₉₇	C₂×D₄	1	2	yes	—
31	Dic₁₉₇	—	4	1	no	—
32	C₂×C₁₉₇	#127	1	1	yes	characteristic
33	C₂×C₁₉₇	#137	1	2	yes	—
34	D₁₉₇	—	8	1	no	—
35	C₁₉₇	#497	1	1	yes	characteristic, complemented, Sylow
36	#497	—	197	1	no	maximal, Sylow
37	C₂×Q₁₆	—	197	2	no	—
38	C₂×SD₁₆	—	197	2	no	—
39	C₄⋊Q₈	D₁₉₇	1	1	yes	characteristic, complemented
40	#124	—	197	1	no	—
41	#97	—	197	1	no	—
42	C₂×Q₈	—	197	1	no	—
43	C₂×Q₈	C₂×D₁₉₇	1	2	yes	—
44	C₂×D₄	—	197	1	no	—
45	Q₁₆	—	394	4	no	—
46	SD₁₆	—	394	4	no	—
47	C₂×C₈	—	197	2	no	—
48	C₄⋊C₄	—	2	2	no	—
49	C₂²⋊C₄	—	394	2	no	—
50	C₄²	C₂×D₁₉₇	1	1	yes	characteristic
51	C₂³	—	394	1	no	—
52	Q₈	—	2	2	no	—
53	Q₈	—	2	2	no	—
54	Q₈	—	394	2	no	—
55	D₄	—	394	2	no	—
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	—	394	1	no	—
58	C₂×C₄	C₂²⋊D₁₉₇	1	2	yes	—
59	C₂×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
60	C₈	—	197	4	no	—
61	C₂²	—	788	1	no	—
62	C₂²	—	788	2	no	—
63	C₂²	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
64	C₄	—	2	2	no	—
65	C₄	—	4	2	no	—
66	C₄	—	788	1	no	—
67	C₄	D₄×D₁₉₇	1	2	yes	—
68	C₂	—	1576	1	no	—
69	C₂	#20263322	1	1	yes	characteristic
70	C₂	#20263357	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₉₇
commutator, cosocle	C₂×C₄×C₁₉₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₉₇
Fitting	C₄⋊Q₈×C₁₉₇	C₂
socle	C₂²×C₁₉₇	C₂×D₄
prime radical	C₂×D₇₈₈	C₂²
hypercenter	C₄⋊Q₈	D₁₉₇
hypocenter	C₁₉₇	#497
LCS	G ▷ C₂×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×#15691348	C₂	semidirect
4. #20263237	C₂	semidirect
5. C₄⋊Q₈×C₁₉₇	C₂	semidirect
35. C₁₉₇	#497	semidirect
39. C₄⋊Q₈	D₁₉₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15691348	2
C₂×#15691345	2
#20263314	2
#20263237	2
C₄⋊Q₈×C₁₉₇	2
#497	197