Subgroups — #27160837

Order 12608.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15691342	C₂	1	2	yes	maximal
2	C₂×M₁₆×C₁₉₇	C₂	1	1	yes	maximal, characteristic
3	#15691342	—	2	2	no	—
4	#15691342	—	2	2	no	—
5	C₂×#13096802	C₄	1	4	yes	—
6	C₂²×C₄×C₁₉₇	C₂²	1	1	yes	characteristic
7	M₁₆×C₁₉₇	—	2	2	no	—
8	C₂×C₈×C₁₉₇	C₄	1	2	yes	—
9	#13096802	—	2	4	no	—
10	C₂³×C₁₉₇	Q₈	1	1	yes	characteristic
11	C₂×C₄×C₁₉₇	C₂×C₄	1	2	yes	—
12	C₂×C₄×C₁₉₇	C₂×C₄	1	1	yes	characteristic
13	C₂×C₄×C₁₉₇	D₄	1	2	yes	—
14	C₂×C₄×C₁₉₇	D₄	1	1	yes	characteristic
15	C₈×C₁₉₇	—	2	2	no	—
16	C₂²×C₁₉₇	—	4	1	no	—
17	C₂²×C₁₉₇	C₄⋊C₄	1	2	yes	—
18	C₂²×C₁₉₇	C₄⋊C₄	1	1	yes	characteristic
19	C₄×C₁₉₇	C₄²	1	1	yes	characteristic
20	C₄×C₁₉₇	C₂²⋊C₄	1	2	yes	—
21	C₄×C₁₉₇	C₂²⋊C₄	1	1	yes	characteristic
22	C₂×C₁₉₇	—	2	2	no	—
23	C₂×C₁₉₇	—	2	1	no	—
24	C₂×C₁₉₇	#95	1	1	yes	characteristic
25	C₁₉₇	#338	1	1	yes	characteristic, complemented, Sylow
26	#338	—	197	1	no	maximal, Sylow
27	C₂×M₁₆	—	197	2	no	—
28	C₂×M₁₆	D₁₉₇	1	1	yes	characteristic
29	C₂²×C₄	C₂×D₁₉₇	1	1	yes	characteristic
30	M₁₆	—	2	2	no	—
31	M₁₆	—	394	2	no	—
32	M₁₆	—	394	2	no	—
33	C₂×C₈	—	197	4	no	—
34	C₂×C₈	Dic₁₉₇	1	2	yes	—
35	C₂³	Dic₃₉₄	1	1	yes	characteristic
36	C₂×C₄	C₄×D₁₉₇	1	2	yes	—
37	C₂×C₄	C₂×Dic₁₉₇	1	1	yes	characteristic
38	C₂×C₄	C₂²⋊D₁₉₇	1	2	yes	—
39	C₂×C₄	D₇₈₈	1	1	yes	characteristic
40	C₈	—	2	2	no	—
41	C₈	—	394	4	no	—
42	C₂²	—	4	1	no	—
43	C₂²	#15691337	1	2	yes	—
44	C₂²	C₄⋊Dic₁₉₇	1	1	yes	characteristic
45	C₄	C₄×Dic₁₉₇	1	1	yes	characteristic
46	C₄	#15691335	1	2	yes	—
47	C₄	C₂²⋊Dic₁₉₇	1	1	yes	characteristic
48	C₂	—	2	2	no	—
49	C₂	—	2	1	no	—
50	C₂	#20263233	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₄×Dic₁₉₇
commutator	C₄×C₁₉₇	C₄²
Frattini	C₂²×C₄	C₂×D₁₉₇
Fitting	C₂×M₁₆×C₁₉₇	C₂
socle	C₂×C₁₉₇	#95
prime radical	C₂³×C₁₉₇	Q₈
cosocle	C₂²×C₄×C₁₉₇	C₂²
hypercenter	C₂×M₁₆	D₁₉₇
hypocenter	C₁₉₇	#338
LCS	G ▷ C₄×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
25. C₁₉₇	#338	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15691342	2
C₂×M₁₆×C₁₉₇	2
#338	197