Subgroups — #38744985

Order 25216.

128 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	#27160904	C₂	1	4	yes	maximal
3	C₄×#15691333	C₂	1	1	yes	maximal, characteristic
4	#500×C₁₉₇	C₂	1	1	yes	maximal, characteristic
5	#20263326	—	2	1	no	—
6	C₂×#15691333	C₂²	1	2	yes	—
7	#20263260	—	2	1	no	—
8	#20263260	C₂²	1	2	yes	—
9	C₄⋊Q₈×C₁₉₇	—	2	1	no	—
10	#107×C₁₉₇	—	2	1	no	—
11	#107×C₁₉₇	C₂²	1	2	yes	—
12	C₄×C₈×C₁₉₇	C₂²	1	1	yes	characteristic
13	C₂×Dic₃₉₄	—	4	1	no	—
14	C₄⋊Dic₁₉₇	—	2	2	no	—
15	C₄⋊Dic₁₉₇	—	4	1	no	—
16	#15691333	Q₈	1	4	yes	complemented
17	C₂×Q₈×C₁₉₇	—	4	1	no	—
18	C₂×C₈×C₁₉₇	D₄	1	1	yes	characteristic
19	C₂×C₈×C₁₉₇	C₂³	1	1	yes	characteristic
20	C₄⋊C₄×C₁₉₇	—	2	2	no	—
21	C₄⋊C₄×C₁₉₇	—	4	1	no	—
22	C₄²×C₁₉₇	D₄	1	1	yes	characteristic
23	Dic₃₉₄	—	8	2	no	—
24	C₂×Dic₁₉₇	—	4	2	no	—
25	Q₈×C₁₉₇	—	8	2	no	—
26	C₂×C₄×C₁₉₇	—	4	2	no	—
27	C₂×C₄×C₁₉₇	D₈	1	1	yes	characteristic
28	C₂×C₄×C₁₉₇	D₈	1	1	yes	characteristic
29	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
30	C₈×C₁₉₇	Q₁₆	1	2	yes	—
31	C₈×C₁₉₇	C₂×Q₈	1	2	yes	—
32	C₂²×C₁₉₇	C₂×D₈	1	1	yes	characteristic
33	C₄×C₁₉₇	—	8	2	no	—
34	C₄×C₁₉₇	D₁₆	1	2	yes	—
35	C₄×C₁₉₇	Q₃₂	1	2	yes	—
36	C₄×C₁₉₇	C₄⋊Q₈	1	1	yes	characteristic
37	C₄×C₁₉₇	C₂×Q₁₆	1	1	yes	characteristic
38	Dic₁₉₇	—	8	2	no	—
39	C₂×C₁₉₇	#500	1	1	yes	characteristic
40	C₂×C₁₉₇	C₂×D₁₆	1	1	yes	characteristic
41	C₂×C₁₉₇	C₂×Q₃₂	1	1	yes	characteristic
42	C₁₉₇	#2252	1	1	yes	characteristic, complemented, Sylow
43	#2252	—	197	1	no	maximal, Sylow
44	#500	—	197	1	no	—
45	#500	D₁₉₇	1	1	yes	characteristic
46	#366	—	197	4	no	—
47	C₄×C₁₆	—	197	1	no	—
48	C₄⋊Q₈	—	2	1	no	—
49	C₄⋊Q₈	—	394	1	no	—
50	C₂×C₁₆	—	197	2	no	—
51	#107	—	2	1	no	—
52	#107	—	394	1	no	—
53	#107	—	197	2	no	—
54	#107	C₂×D₁₉₇	1	2	yes	—
55	C₄×C₈	C₂×D₁₉₇	1	1	yes	characteristic
56	C₂×Q₈	—	4	1	no	—
57	C₂×Q₈	—	788	1	no	—
58	C₂×C₈	C₂²⋊D₁₉₇	1	1	yes	characteristic
59	C₂×C₈	C₂²×D₁₉₇	1	1	yes	characteristic
60	C₄⋊C₄	—	2	2	no	—
61	C₄⋊C₄	—	4	1	no	—
62	C₄⋊C₄	—	788	1	no	—
63	C₄⋊C₄	—	394	2	no	—
64	C₄²	C₂²⋊D₁₉₇	1	1	yes	characteristic
65	C₁₆	—	197	4	no	—
66	Q₈	—	8	2	no	—
67	Q₈	—	1576	2	no	—
68	C₂×C₄	—	4	2	no	—
69	C₂×C₄	—	788	2	no	—
70	C₂×C₄	#15691343	1	1	yes	characteristic
71	C₂×C₄	#15691343	1	1	yes	characteristic
72	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
73	C₈	#15691348	1	2	yes	—
74	C₈	Q₈×D₁₉₇	1	2	yes	—
75	C₂²	C₂×#15691343	1	1	yes	characteristic
76	C₄	—	8	2	no	—
77	C₄	—	1576	2	no	—
78	C₄	#20263267	1	2	yes	—
79	C₄	#20263272	1	2	yes	—
80	C₄	#20263324	1	1	yes	characteristic
81	C₄	C₂×#15691348	1	1	yes	characteristic
82	C₂	#27161482	1	1	yes	characteristic
83	C₂	C₂×#20263267	1	1	yes	characteristic
84	C₂	C₂×#20263272	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#15691343
commutator, cosocle	C₂×C₈×C₁₉₇	C₂³
Frattini	C₂×C₈	C₂²×D₁₉₇
Fitting	#500×C₁₉₇	C₂
socle, prime radical	C₂²×C₁₉₇	C₂×D₈
hypercenter	#500	D₁₉₇
hypocenter	C₁₉₇	#2252
LCS	G ▷ C₂×C₈×C₁₉₇ ▷ C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #500

Complements

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

N	H ≅ G/N	type
16. #15691333	Q₈	semidirect
42. C₁₉₇	#2252	semidirect

Maximal subgroups

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

subgroup	index
#27161484	2
#27160904	2
C₄×#15691333	2
#500×C₁₉₇	2
#2252	197