Subgroups — #38740900

Order 25216.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27161511	C₂	1	1	yes	maximal, characteristic
2	#27160907	C₂	1	1	yes	maximal, characteristic, complemented
3	M₆₄×C₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#20263348	—	2	1	no	—
5	Dic₁₅₇₆	C₄	1	1	yes	characteristic, complemented
6	#20263271	—	2	1	no	—
7	D₃₁₅₂	C₄	1	1	yes	characteristic, complemented
8	#20263260	—	2	1	no	—
9	C₂×C₁₆×C₁₉₇	C₂²	1	1	yes	characteristic
10	C₃₂×C₁₉₇	—	2	1	no	—
11	#15691358	—	4	1	no	—
12	Dic₇₈₈	—	2	1	no	—
13	#15691347	—	4	1	no	—
14	D₁₅₇₆	—	2	1	no	—
15	C₄⋊Dic₁₉₇	—	4	1	no	—
16	C₂×C₈×C₁₉₇	D₄	1	1	yes	characteristic
17	C₁₆×C₁₉₇	C₂×C₄	1	1	yes	characteristic
18	C₁₆×C₁₉₇	D₄	1	1	yes	characteristic
19	Dic₃₉₄	—	4	1	no	—
20	D₇₈₈	—	4	1	no	—
21	C₂²⋊D₁₉₇	—	8	1	no	—
22	C₂×Dic₁₉₇	—	8	1	no	—
23	C₄×D₁₉₇	—	8	1	no	—
24	C₂×C₄×C₁₉₇	D₈	1	1	yes	characteristic
25	C₈×C₁₉₇	C₂²⋊C₄	1	1	yes	characteristic
26	C₈×C₁₉₇	SD₁₆	1	1	yes	characteristic
27	C₂²×C₁₉₇	D₁₆	1	1	yes	characteristic
28	C₂×D₁₉₇	—	8	1	no	—
29	C₄×C₁₉₇	#102	1	1	yes	characteristic
30	C₄×C₁₉₇	SD₃₂	1	1	yes	characteristic
31	Dic₁₉₇	—	8	1	no	—
32	Dic₁₉₇	—	16	1	no	—
33	C₂×C₁₉₇	—	2	1	no	—
34	C₂×C₁₉₇	#357	1	1	yes	characteristic
35	D₁₉₇	—	16	1	no	—
36	C₁₉₇	#1418	1	1	yes	characteristic, complemented, Sylow
37	#1418	—	197	1	no	maximal, Sylow
38	#508	—	197	1	no	—
39	M₆₄	D₁₉₇	1	1	yes	characteristic, complemented
40	#367	—	197	1	no	—
41	#135	—	394	1	no	—
42	Q₃₂	—	197	1	no	—
43	SD₃₂	—	394	1	no	—
44	D₁₆	—	197	1	no	—
45	C₂×C₁₆	C₂×D₁₉₇	1	1	yes	characteristic
46	#107	—	394	1	no	—
47	C₃₂	—	2	1	no	—
48	#41	—	788	1	no	—
49	Q₁₆	—	394	1	no	—
50	SD₁₆	—	788	1	no	—
51	D₈	—	394	1	no	—
52	C₂×C₈	D₇₈₈	1	1	yes	characteristic
53	C₄⋊C₄	—	788	1	no	—
54	C₁₆	C₄×D₁₉₇	1	1	yes	characteristic
55	C₁₆	C₂²⋊D₁₉₇	1	1	yes	characteristic
56	Q₈	—	788	1	no	—
57	D₄	—	788	1	no	—
58	D₄	—	1576	1	no	—
59	C₂×C₄	—	1576	1	no	—
60	C₂×C₄	—	1576	1	no	—
61	C₂×C₄	D₁₅₇₆	1	1	yes	characteristic
62	C₈	#15691335	1	1	yes	characteristic
63	C₈	#15691347	1	1	yes	characteristic
64	C₂²	—	1576	1	no	—
65	C₂²	D₃₁₅₂	1	1	yes	characteristic
66	C₄	—	1576	1	no	—
67	C₄	—	3152	1	no	—
68	C₄	#20263248	1	1	yes	characteristic
69	C₄	#20263271	1	1	yes	characteristic
70	C₂	—	2	1	no	—
71	C₂	—	3152	1	no	—
72	C₂	#27160881	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#27160881
commutator	C₁₆×C₁₉₇	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₁₉₇
Fitting	M₆₄×C₁₉₇	C₂
socle	C₂×C₁₉₇	#357
prime radical	#27161511	C₂
cosocle	C₂×C₁₆×C₁₉₇	C₂²
hypercenter	M₆₄	D₁₉₇
hypocenter	C₁₉₇	#1418
LCS	G ▷ C₁₆×C₁₉₇ ▷ C₈×C₁₉₇ ▷ C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
2. #27160907	C₂	semidirect
3. M₆₄×C₁₉₇	C₂	semidirect
5. Dic₁₅₇₆	C₄	semidirect
7. D₃₁₅₂	C₄	semidirect
36. C₁₉₇	#1418	semidirect
39. M₆₄	D₁₉₇	semidirect

Maximal subgroups

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

subgroup	index
#27161511	2
#27160907	2
M₆₄×C₁₉₇	2
#1418	197