Subgroups — #38740768

Order 25216.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27161484	C₂	1	1	yes	maximal, characteristic
2	C₄⋊#15691333	C₂	1	1	yes	maximal, characteristic
3	#335×C₁₉₇	C₂	1	1	yes	maximal, characteristic
4	#20263326	—	2	1	no	—
5	C₂×#15691333	—	2	1	no	—
6	#20263260	—	2	1	no	—
7	#20263260	C₄	1	2	yes	—
8	C₄⋊C₈×C₁₉₇	—	2	1	no	—
9	C₄×C₈×C₁₉₇	C₂²	1	1	yes	characteristic
10	C₂×Dic₃₉₄	—	4	1	no	—
11	C₄⋊Dic₁₉₇	—	2	2	no	—
12	C₄⋊Dic₁₉₇	—	4	1	no	—
13	#15691333	—	4	1	no	—
14	C₂×C₈×C₁₉₇	—	4	1	no	—
15	C₂×C₈×C₁₉₇	C₂×C₄	1	1	yes	characteristic
16	C₂×C₈×C₁₉₇	D₄	1	1	yes	characteristic
17	C₄²×C₁₉₇	D₄	1	1	yes	characteristic
18	Dic₃₉₄	—	8	1	no	—
19	Dic₃₉₄	—	8	1	no	—
20	C₂×Dic₁₉₇	—	4	2	no	—
21	C₂×C₄×C₁₉₇	C₂²⋊C₄	1	1	yes	characteristic
22	C₂×C₄×C₁₉₇	D₈	1	1	yes	characteristic
23	C₂×C₄×C₁₉₇	SD₁₆	1	1	yes	characteristic
24	C₈×C₁₉₇	—	2	1	no	—
25	C₈×C₁₉₇	—	8	1	no	—
26	C₈×C₁₉₇	SD₁₆	1	1	yes	characteristic
27	C₈×C₁₉₇	Q₁₆	1	1	yes	characteristic
28	C₂²×C₁₉₇	#102	1	1	yes	characteristic
29	C₄×C₁₉₇	—	2	1	no	—
30	C₄×C₁₉₇	#101	1	1	yes	characteristic
31	C₄×C₁₉₇	#103	1	1	yes	characteristic
32	C₄×C₁₉₇	D₁₆	1	1	yes	characteristic
33	C₄×C₁₉₇	SD₃₂	1	1	yes	characteristic
34	Dic₁₉₇	—	8	2	no	—
35	C₂×C₁₉₇	#332	1	1	yes	characteristic
36	C₂×C₁₉₇	#357	1	1	yes	characteristic
37	C₂×C₁₉₇	#362	1	1	yes	characteristic
38	C₁₉₇	#1364	1	1	yes	characteristic, complemented, Sylow
39	#1364	—	197	1	no	maximal, Sylow
40	#500	—	197	1	no	—
41	C₄⋊C₁₆	—	197	1	no	—
42	#335	D₁₉₇	1	1	yes	characteristic
43	C₄⋊Q₈	—	394	1	no	—
44	C₂×C₁₆	—	394	1	no	—
45	#107	—	394	1	no	—
46	#107	—	197	2	no	—
47	C₄⋊C₈	—	2	1	no	—
48	C₄×C₈	C₂×D₁₉₇	1	1	yes	characteristic
49	C₂×Q₈	—	788	1	no	—
50	C₂×C₈	—	4	1	no	—
51	C₂×C₈	C₄×D₁₉₇	1	1	yes	characteristic
52	C₂×C₈	D₇₈₈	1	1	yes	characteristic
53	C₄⋊C₄	—	788	1	no	—
54	C₄⋊C₄	—	394	2	no	—
55	C₄²	C₂²⋊D₁₉₇	1	1	yes	characteristic
56	C₁₆	—	788	1	no	—
57	Q₈	—	1576	1	no	—
58	Q₈	—	1576	1	no	—
59	C₂×C₄	—	788	2	no	—
60	C₂×C₄	#15691335	1	1	yes	characteristic
61	C₂×C₄	#15691343	1	1	yes	characteristic
62	C₂×C₄	#15691345	1	1	yes	characteristic
63	C₈	—	2	1	no	—
64	C₈	—	8	1	no	—
65	C₈	#15691347	1	1	yes	characteristic
66	C₈	Dic₇₈₈	1	1	yes	characteristic
67	C₂²	#20263246	1	1	yes	characteristic
68	C₄	—	2	1	no	—
69	C₄	—	1576	2	no	—
70	C₄	#20263244	1	1	yes	characteristic
71	C₄	#20263251	1	1	yes	characteristic
72	C₄	#20263267	1	1	yes	characteristic
73	C₄	#20263269	1	1	yes	characteristic
74	C₂	#27160826	1	1	yes	characteristic
75	C₂	#27160879	1	1	yes	characteristic
76	C₂	#27160894	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#20263246
commutator	C₂×C₈×C₁₉₇	C₂×C₄
Frattini	C₄×C₈	C₂×D₁₉₇
Fitting	#335×C₁₉₇	C₂
socle, prime radical	C₂²×C₁₉₇	#102
cosocle	C₄×C₈×C₁₉₇	C₂²
hypercenter	#335	D₁₉₇
hypocenter	C₁₉₇	#1364
LCS	G ▷ C₂×C₈×C₁₉₇ ▷ C₂×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #335

Complements

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

N	H ≅ G/N	type
38. C₁₉₇	#1364	semidirect

Maximal subgroups

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

subgroup	index
#27161484	2
C₄⋊#15691333	2
#335×C₁₉₇	2
#1364	197