Subgroups — #27161010

Order 12608.

150 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15691337	C₂	1	2	yes	maximal
2	C₂×#15691337	C₂	1	1	yes	maximal, characteristic
3	#20263233	C₂	1	2	yes	maximal
4	#20263233	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊C₄×C₁₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₉₇	C₂²	1	1	yes	characteristic
7	C₂²×Dic₁₉₇	C₂²	1	1	yes	characteristic
8	C₂²×Dic₁₉₇	C₂²	1	2	yes	—
9	#15691337	—	2	4	no	—
10	#15691337	—	2	2	no	—
11	C₂²×C₄×C₁₉₇	C₂²	1	2	yes	—
12	C₂²×C₄×C₁₉₇	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₁₉₇	—	2	2	no	—
14	C₂×Dic₁₉₇	—	2	2	no	—
15	C₂×Dic₁₉₇	—	2	1	no	—
16	C₂×Dic₁₉₇	—	2	1	no	—
17	C₂×Dic₁₉₇	—	2	2	no	—
18	C₂×Dic₁₉₇	—	2	1	no	—
19	C₂×Dic₁₉₇	—	2	2	no	—
20	C₂×Dic₁₉₇	D₄	1	2	yes	—
21	C₂×Dic₁₉₇	Q₈	1	4	yes	—
22	C₂³×C₁₉₇	C₂³	1	1	yes	characteristic
23	C₂×C₄×C₁₉₇	—	2	2	no	—
24	C₂×C₄×C₁₉₇	—	2	2	no	—
25	C₂×C₄×C₁₉₇	—	2	2	no	—
26	C₂×C₄×C₁₉₇	—	2	2	no	—
27	C₂×C₄×C₁₉₇	D₄	1	2	yes	—
28	C₂²×C₁₉₇	C₂×D₄	1	1	yes	characteristic
29	C₂²×C₁₉₇	C₂×D₄	1	1	yes	characteristic
30	C₂²×C₁₉₇	C₂×Q₈	1	2	yes	—
31	C₂²×C₁₉₇	#41	1	1	yes	characteristic
32	C₂²×C₁₉₇	#41	1	2	yes	—
33	C₄×C₁₉₇	—	2	2	no	—
34	C₄×C₁₉₇	—	4	2	no	—
35	Dic₁₉₇	—	2	2	no	—
36	Dic₁₉₇	—	2	4	no	—
37	Dic₁₉₇	—	4	1	no	—
38	C₂×C₁₉₇	#121	1	1	yes	characteristic
39	C₂×C₁₉₇	C₂²⋊Q₈	1	2	yes	—
40	C₂×C₁₉₇	#123	1	1	yes	characteristic
41	C₂×C₁₉₇	#125	1	2	yes	—
42	C₂×C₁₉₇	C₄⋊Q₈	1	1	yes	characteristic
43	C₁₉₇	#398	1	1	yes	characteristic, complemented, Sylow
44	#398	—	197	1	no	maximal, Sylow
45	C₂×C₄⋊C₄	—	197	2	no	—
46	C₂×C₄⋊C₄	—	197	1	no	—
47	C₂×C₄⋊C₄	D₁₉₇	1	1	yes	characteristic
48	#95	—	197	2	no	—
49	#95	—	197	1	no	—
50	C₂²×C₄	—	197	1	no	—
51	C₂²×C₄	—	197	2	no	—
52	C₂²×C₄	—	197	1	no	—
53	C₂²×C₄	C₂×D₁₉₇	1	2	yes	—
54	C₂²×C₄	C₂×D₁₉₇	1	1	yes	characteristic
55	C₄⋊C₄	—	2	2	no	—
56	C₄⋊C₄	—	394	4	no	—
57	C₄⋊C₄	—	394	2	no	—
58	C₂³	C₂²×D₁₉₇	1	1	yes	characteristic
59	C₂×C₄	—	2	2	no	—
60	C₂×C₄	—	2	2	no	—
61	C₂×C₄	—	2	2	no	—
62	C₂×C₄	—	2	2	no	—
63	C₂×C₄	—	394	1	no	—
64	C₂×C₄	—	394	2	no	—
65	C₂×C₄	—	394	2	no	—
66	C₂×C₄	—	394	1	no	—
67	C₂×C₄	—	394	1	no	—
68	C₂×C₄	—	394	2	no	—
69	C₂×C₄	—	197	2	no	—
70	C₂×C₄	—	197	4	no	—
71	C₂×C₄	C₂²⋊D₁₉₇	1	2	yes	—
72	C₂²	D₄×D₁₉₇	1	1	yes	characteristic
73	C₂²	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
74	C₂²	Q₈×D₁₉₇	1	2	yes	—
75	C₂²	#15691357	1	1	yes	characteristic
76	C₂²	#15691358	1	2	yes	—
77	C₄	—	2	2	no	—
78	C₄	—	4	2	no	—
79	C₄	—	788	1	no	—
80	C₄	—	394	2	no	—
81	C₄	—	394	4	no	—
82	C₂	#20263297	1	1	yes	characteristic
83	C₂	#20263304	1	2	yes	—
84	C₂	#20263309	1	1	yes	characteristic
85	C₂	#20263316	1	2	yes	—
86	C₂	#20263324	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₁₉₇
commutator, socle, prime radical, cosocle	C₂³×C₁₉₇	C₂³
Fitting	C₂×C₄⋊C₄×C₁₉₇	C₂
hypercenter	C₂×C₄⋊C₄	D₁₉₇
hypocenter	C₁₉₇	#398
LCS	G ▷ C₂³×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
43. C₁₉₇	#398	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15691337	2
#20263233	2
C₂×C₄⋊C₄×C₁₉₇	2
#398	197