Subgroups — #38740772

Order 25216.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27161481	C₂	1	1	yes	maximal, characteristic
2	#27160900	C₂	1	1	yes	maximal, characteristic
3	#364×C₁₉₇	C₂	1	1	yes	maximal, characteristic
4	#20263318	C₄	1	2	yes	—
5	#20263266	—	2	1	no	—
6	#20263260	—	2	1	no	—
7	#20263258	—	2	1	no	—
8	M₃₂×C₁₉₇	—	2	1	no	—
9	C₄×C₈×C₁₉₇	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₁₉₇	—	2	2	no	—
11	#15691337	—	4	2	no	—
12	#15691333	—	4	1	no	—
13	C₂×C₈×C₁₉₇	D₄	1	1	yes	characteristic
14	C₂×C₈×C₁₉₇	D₄	1	1	yes	characteristic
15	C₄²×C₁₉₇	C₂×C₄	1	1	yes	characteristic
16	C₁₆×C₁₉₇	—	4	1	no	—
17	C₂×Dic₁₉₇	—	4	2	no	—
18	C₂×C₄×C₁₉₇	—	2	1	no	—
19	C₂×C₄×C₁₉₇	C₂²⋊C₄	1	1	yes	characteristic
20	C₈×C₁₉₇	SD₁₆	1	1	yes	characteristic
21	C₈×C₁₉₇	SD₁₆	1	1	yes	characteristic
22	C₈×C₁₉₇	Q₁₆	1	1	yes	characteristic
23	C₈×C₁₉₇	Q₁₆	1	1	yes	characteristic
24	C₂²×C₁₉₇	#100	1	1	yes	characteristic
25	C₄×C₁₉₇	—	4	1	no	—
26	C₄×C₁₉₇	#103	1	1	yes	characteristic
27	C₄×C₁₉₇	#103	1	1	yes	characteristic
28	Dic₁₉₇	—	8	2	no	—
29	C₂×C₁₉₇	—	2	1	no	—
30	C₂×C₁₉₇	#333	1	1	yes	characteristic
31	C₁₉₇	#1366	1	1	yes	characteristic, complemented, Sylow
32	#1366	—	197	1	no	maximal, Sylow
33	#499	—	197	1	no	—
34	#364	—	197	1	no	—
35	#364	D₁₉₇	1	1	yes	characteristic
36	#125	—	197	2	no	—
37	M₃₂	—	2	1	no	—
38	M₃₂	—	394	1	no	—
39	#107	—	394	1	no	—
40	#106	—	394	1	no	—
41	C₄×C₈	C₂×D₁₉₇	1	1	yes	characteristic
42	C₂×C₈	C₂²⋊D₁₉₇	1	1	yes	characteristic
43	C₂×C₈	D₇₈₈	1	1	yes	characteristic
44	C₄⋊C₄	—	788	2	no	—
45	C₄⋊C₄	—	394	2	no	—
46	C₄²	C₄×D₁₉₇	1	1	yes	characteristic
47	C₁₆	—	4	1	no	—
48	C₁₆	—	788	1	no	—
49	C₂×C₄	—	2	1	no	—
50	C₂×C₄	—	788	2	no	—
51	C₂×C₄	#15691335	1	1	yes	characteristic
52	C₈	#15691346	1	1	yes	characteristic
53	C₈	#15691347	1	1	yes	characteristic
54	C₈	#15691348	1	1	yes	characteristic
55	C₈	Dic₇₈₈	1	1	yes	characteristic
56	C₂²	#20263242	1	1	yes	characteristic
57	C₄	—	4	1	no	—
58	C₄	—	1576	2	no	—
59	C₄	#20263249	1	1	yes	characteristic
60	C₄	#20263251	1	1	yes	characteristic
61	C₂	—	2	1	no	—
62	C₂	#27160827	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#27160827
commutator	C₄²×C₁₉₇	C₂×C₄
Frattini	C₄×C₈	C₂×D₁₉₇
Fitting	#364×C₁₉₇	C₂
socle	C₂×C₁₉₇	#333
prime radical	C₂²×C₁₉₇	#100
cosocle	C₄×C₈×C₁₉₇	C₂²
hypercenter	#364	D₁₉₇
hypocenter	C₁₉₇	#1366
LCS	G ▷ C₄²×C₁₉₇ ▷ C₂×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

Complements

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

N	H ≅ G/N	type
31. C₁₉₇	#1366	semidirect

Maximal subgroups

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

subgroup	index
#27161481	2
#27160900	2
#364×C₁₉₇	2
#1366	197