Subgroups — #27161362

Order 12608.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20263325	C₂	1	1	yes	maximal, characteristic
2	Q₈×Dic₁₉₇	C₂	1	1	yes	maximal, characteristic
3	#20263258	C₂	1	1	yes	maximal, characteristic
4	#20263257	C₂	1	1	yes	maximal, characteristic
5	#20263255	C₂	1	1	yes	maximal, characteristic
6	#20263250	C₂	1	1	yes	maximal, characteristic
7	#103×C₁₉₇	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₃₉₄	—	2	1	no	—
9	C₂×#13096802	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₁₉₇	—	2	1	no	—
11	C₄⋊Dic₁₉₇	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₁₉₇	C₂²	1	1	yes	characteristic
13	#15691337	—	2	1	no	—
14	C₄×Dic₁₉₇	—	2	1	no	—
15	C₄×Dic₁₉₇	C₂²	1	1	yes	characteristic
16	C₂×Q₈×C₁₉₇	C₂²	1	1	yes	characteristic
17	C₂×C₈×C₁₉₇	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₁₉₇	C₂²	1	1	yes	characteristic
19	Dic₃₉₄	—	4	2	no	—
20	C₂×Dic₁₉₇	—	2	1	no	—
21	C₂×Dic₁₉₇	—	2	1	no	—
22	C₂×Dic₁₉₇	D₄	1	1	yes	characteristic
23	C₂×Dic₁₉₇	D₄	1	1	yes	characteristic
24	#13096802	—	2	1	no	—
25	Q₈×C₁₉₇	—	2	1	no	—
26	Q₈×C₁₉₇	Q₈	1	2	yes	complemented
27	C₂×C₄×C₁₉₇	—	2	1	no	—
28	C₂×C₄×C₁₉₇	—	2	1	no	—
29	C₂×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
30	C₈×C₁₉₇	—	2	1	no	—
31	C₂²×C₁₉₇	C₂×D₄	1	1	yes	characteristic
32	C₄×C₁₉₇	—	2	2	no	—
33	C₄×C₁₉₇	—	4	1	no	—
34	C₄×C₁₉₇	C₂×Q₈	1	1	yes	characteristic
35	C₄×C₁₉₇	#41	1	1	yes	characteristic
36	Dic₁₉₇	—	2	1	no	—
37	Dic₁₉₇	—	4	1	no	—
38	Dic₁₉₇	—	4	1	no	—
39	Dic₁₉₇	SD₁₆	1	2	yes	—
40	C₂×C₁₉₇	C₂²⋊Q₈	1	1	yes	characteristic
41	C₂×C₁₉₇	C₂×SD₁₆	1	1	yes	characteristic
42	C₂×C₁₉₇	#137	1	1	yes	characteristic
43	C₁₉₇	#475	1	1	yes	characteristic, complemented, Sylow
44	#475	—	197	1	no	maximal, Sylow
45	C₄⋊Q₈	—	197	1	no	—
46	C₄×Q₈	—	197	1	no	—
47	#106	—	197	1	no	—
48	#106	—	197	1	no	—
49	C₄⋊C₈	—	197	1	no	—
50	#103	—	197	1	no	—
51	#103	D₁₉₇	1	1	yes	characteristic
52	C₂×Q₈	—	394	1	no	—
53	C₂×Q₈	C₂×D₁₉₇	1	1	yes	characteristic
54	C₂×C₈	—	197	1	no	—
55	C₂×C₈	C₂×D₁₉₇	1	1	yes	characteristic
56	C₄⋊C₄	—	394	1	no	—
57	C₄⋊C₄	—	394	1	no	—
58	C₄⋊C₄	—	197	1	no	—
59	C₄⋊C₄	—	197	1	no	—
60	C₄⋊C₄	C₂×D₁₉₇	1	1	yes	characteristic
61	C₄²	—	394	1	no	—
62	C₄²	—	197	1	no	—
63	Q₈	—	2	1	no	—
64	Q₈	—	788	2	no	—
65	Q₈	Dic₃₉₄	1	2	yes	complemented
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	—	2	1	no	—
68	C₂×C₄	—	394	1	no	—
69	C₂×C₄	—	394	1	no	—
70	C₂×C₄	—	197	1	no	—
71	C₂×C₄	—	197	1	no	—
72	C₂×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
73	C₈	—	2	1	no	—
74	C₈	—	394	1	no	—
75	C₂²	D₄×D₁₉₇	1	1	yes	characteristic
76	C₄	—	2	2	no	—
77	C₄	—	4	1	no	—
78	C₄	—	788	1	no	—
79	C₄	—	788	1	no	—
80	C₄	—	394	1	no	—
81	C₄	—	197	2	no	—
82	C₄	C₂×Dic₃₉₄	1	1	yes	characteristic
83	C₄	#15691357	1	1	yes	characteristic
84	C₂	#20263303	1	1	yes	characteristic
85	C₂	SD₁₆×D₁₉₇	1	1	yes	characteristic
86	C₂	#20263358	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₉₇
commutator, cosocle	C₂×C₄×C₁₉₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₉₇
Fitting	#103×C₁₉₇	C₂
socle, prime radical	C₂²×C₁₉₇	C₂×D₄
hypercenter	#103	D₁₉₇
hypocenter	C₁₉₇	#475
LCS	G ▷ C₂×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

Complements

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

N	H ≅ G/N	type
26. Q₈×C₁₉₇	Q₈	semidirect
43. C₁₉₇	#475	semidirect
65. Q₈	Dic₃₉₄	semidirect

Maximal subgroups

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

subgroup	index
#20263325	2
Q₈×Dic₁₉₇	2
#20263258	2
#20263257	2
#20263255	2
#20263250	2
#103×C₁₉₇	2
#475	197