Subgroups — #27160862

Order 12608.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20263332	C₂	1	1	yes	maximal, characteristic
2	#20263266	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₆×C₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#15691358	C₄	1	1	yes	characteristic
5	#15691342	C₄	1	1	yes	characteristic
6	#15691341	—	2	1	no	—
7	C₈×D₁₉₇	—	2	1	no	—
8	#15691333	—	2	1	no	—
9	C₂×C₈×C₁₉₇	C₂²	1	1	yes	characteristic
10	C₁₆×C₁₉₇	—	2	1	no	—
11	Dic₃₉₄	C₈	1	1	yes	characteristic
12	D₇₈₈	C₈	1	1	yes	characteristic
13	C₂²⋊D₁₉₇	—	2	1	no	—
14	C₄×D₁₉₇	—	2	1	no	—
15	#13096802	—	2	1	no	—
16	C₂×C₄×C₁₉₇	C₂×C₄	1	1	yes	characteristic
17	C₈×C₁₉₇	D₄	1	1	yes	characteristic
18	C₈×C₁₉₇	D₄	1	1	yes	characteristic
19	C₂²×C₁₉₇	M₁₆	1	1	yes	characteristic
20	C₂×D₁₉₇	—	2	1	no	—
21	C₄×C₁₉₇	C₂²⋊C₄	1	1	yes	characteristic
22	C₄×C₁₉₇	C₂×C₈	1	1	yes	characteristic
23	Dic₁₉₇	—	2	1	no	—
24	C₂×C₁₉₇	—	2	1	no	—
25	C₂×C₁₉₇	C₂²⋊C₈	1	1	yes	characteristic
26	D₁₉₇	—	4	1	no	—
27	C₁₉₇	#350	1	1	yes	characteristic, complemented, Sylow
28	#350	—	197	1	no	maximal, Sylow
29	#131	—	197	1	no	—
30	M₃₂	—	197	1	no	—
31	C₂×C₁₆	D₁₉₇	1	1	yes	characteristic, complemented
32	#41	—	197	1	no	—
33	M₁₆	—	394	1	no	—
34	M₁₆	—	197	1	no	—
35	C₂×C₈	—	394	1	no	—
36	C₂×C₈	C₂×D₁₉₇	1	1	yes	characteristic
37	C₁₆	—	2	1	no	—
38	C₁₆	—	394	1	no	—
39	Q₈	—	197	1	no	—
40	D₄	—	394	1	no	—
41	D₄	—	197	1	no	—
42	C₂×C₄	—	394	1	no	—
43	C₂×C₄	C₄×D₁₉₇	1	1	yes	characteristic
44	C₈	—	394	1	no	—
45	C₈	C₂²⋊D₁₉₇	1	1	yes	characteristic
46	C₈	D₇₈₈	1	1	yes	characteristic
47	C₂²	—	394	1	no	—
48	C₂²	#15691341	1	1	yes	characteristic
49	C₄	—	394	1	no	—
50	C₄	#15691335	1	1	yes	characteristic
51	C₄	C₈×D₁₉₇	1	1	yes	characteristic
52	C₂	—	2	1	no	—
53	C₂	—	788	1	no	—
54	C₂	#20263238	1	1	yes	characteristic

	subgroup	G/H
center	C₈	D₇₈₈
commutator	C₄×C₁₉₇	C₂×C₈
Frattini	C₂×C₈	C₂×D₁₉₇
Fitting	C₂×C₁₆×C₁₉₇	C₂
socle	C₂×C₁₉₇	C₂²⋊C₈
prime radical	#15691358	C₄
cosocle	C₂×C₈×C₁₉₇	C₂²
hypercenter	C₂×C₁₆	D₁₉₇
hypocenter	C₁₉₇	#350
LCS	G ▷ C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
2. #20263266	C₂	semidirect
3. C₂×C₁₆×C₁₉₇	C₂	semidirect
27. C₁₉₇	#350	semidirect
31. C₂×C₁₆	D₁₉₇	semidirect

Maximal subgroups

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

subgroup	index
#20263332	2
#20263266	2
C₂×C₁₆×C₁₉₇	2
#350	197