Subgroups — #38742306

Order 25216.

180 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	#27161475	C₂	1	2	yes	maximal
3	#27161431	C₂	1	2	yes	maximal
4	C₈×#13096802	C₂	1	1	yes	maximal, characteristic
5	#494×C₁₉₇	C₂	1	1	yes	maximal, characteristic
6	#20263326	C₂²	1	2	yes	—
7	#20263260	—	2	2	no	—
8	#20263257	—	2	4	no	—
9	#20263249	—	2	4	no	—
10	C₄×#13096802	C₂²	1	1	yes	characteristic
11	C₄×#13096802	C₂²	1	1	yes	characteristic
12	C₂×Q₁₆×C₁₉₇	—	2	2	no	—
13	C₄⋊Q₈×C₁₉₇	C₂²	1	2	yes	—
14	C₄×C₈×C₁₉₇	C₂²	1	1	yes	characteristic
15	C₂×Dic₃₉₄	—	2	2	no	—
16	C₂×#13096802	D₄	1	2	yes	—
17	C₂×#13096802	D₄	1	2	yes	—
18	C₄⋊Dic₁₉₇	—	2	2	no	—
19	C₄⋊Dic₁₉₇	—	2	2	no	—
20	C₂×Q₈×C₁₉₇	—	2	2	no	—
21	Q₁₆×C₁₉₇	—	4	4	no	—
22	C₂×C₈×C₁₉₇	D₄	1	2	yes	—
23	C₄⋊C₄×C₁₉₇	—	2	2	no	—
24	C₄⋊C₄×C₁₉₇	—	2	2	no	—
25	C₄²×C₁₉₇	C₂³	1	1	yes	characteristic
26	Dic₃₉₄	—	4	4	no	—
27	C₂×Dic₁₉₇	—	4	2	no	—
28	#13096802	—	2	2	no	—
29	#13096802	Q₁₆	1	4	yes	complemented
30	Q₈×C₁₉₇	—	4	4	no	—
31	C₂×C₄×C₁₉₇	—	4	2	no	—
32	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
35	C₈×C₁₉₇	SD₁₆	1	4	yes	—
36	C₂²×C₁₉₇	#127	1	1	yes	characteristic
37	C₄×C₁₉₇	—	8	2	no	—
38	C₄×C₁₉₇	C₂×SD₁₆	1	2	yes	—
39	C₄×C₁₉₇	C₂×Q₁₆	1	2	yes	—
40	C₄×C₁₉₇	#135	1	2	yes	—
41	Dic₁₉₇	—	8	2	no	—
42	C₂×C₁₉₇	#492	1	1	yes	characteristic
43	C₂×C₁₉₇	#494	1	1	yes	characteristic
44	C₂×C₁₉₇	#495	1	1	yes	characteristic
45	C₁₉₇	#1707	1	1	yes	characteristic, complemented, Sylow
46	#1707	—	197	1	no	maximal, Sylow
47	#500	—	197	1	no	—
48	#498	—	197	2	no	—
49	#494	D₁₉₇	1	1	yes	characteristic
50	#487	—	197	2	no	—
51	C₈²	—	197	1	no	—
52	C₂×Q₁₆	—	2	2	no	—
53	C₄⋊Q₈	—	197	2	no	—
54	C₄⋊Q₈	C₂×D₁₉₇	1	2	yes	—
55	#107	—	394	2	no	—
56	#106	—	394	4	no	—
57	#103	—	394	4	no	—
58	C₄×C₈	—	197	1	no	—
59	C₄×C₈	—	197	1	no	—
60	C₄×C₈	C₂×D₁₉₇	1	1	yes	characteristic
61	C₂×Q₈	—	2	2	no	—
62	C₂×Q₈	—	394	2	no	—
63	Q₁₆	—	4	4	no	—
64	C₂×C₈	—	197	2	no	—
65	C₂×C₈	—	197	2	no	—
66	C₂×C₈	C₂²⋊D₁₉₇	1	2	yes	—
67	C₄⋊C₄	—	2	2	no	—
68	C₄⋊C₄	—	2	2	no	—
69	C₄⋊C₄	—	394	2	no	—
70	C₄⋊C₄	—	394	2	no	—
71	C₄²	C₂²×D₁₉₇	1	1	yes	characteristic
72	Q₈	—	4	4	no	—
73	Q₈	—	788	4	no	—
74	C₂×C₄	—	4	2	no	—
75	C₂×C₄	—	788	2	no	—
76	C₂×C₄	D₄×D₁₉₇	1	1	yes	characteristic
77	C₂×C₄	D₄×D₁₉₇	1	1	yes	characteristic
78	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
79	C₈	—	394	2	no	—
80	C₈	—	197	4	no	—
81	C₈	#15691346	1	4	yes	—
82	C₂²	#20263322	1	1	yes	characteristic
83	C₄	—	8	2	no	—
84	C₄	—	1576	2	no	—
85	C₄	C₂×#15691346	1	2	yes	—
86	C₄	Q₁₆×D₁₉₇	1	2	yes	—
87	C₄	#20263349	1	2	yes	—
88	C₂	#27161451	1	1	yes	characteristic
89	C₂	#27161457	1	1	yes	characteristic
90	C₂	#27161464	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#20263322
commutator, cosocle	C₄²×C₁₉₇	C₂³
Frattini	C₄²	C₂²×D₁₉₇
Fitting	#494×C₁₉₇	C₂
socle, prime radical	C₂²×C₁₉₇	#127
hypercenter	#494	D₁₉₇
hypocenter	C₁₉₇	#1707
LCS	G ▷ C₄²×C₁₉₇ ▷ C₂²×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

Complements

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

N	H ≅ G/N	type
29. #13096802	Q₁₆	semidirect
45. C₁₉₇	#1707	semidirect

Maximal subgroups

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

subgroup	index
#27161484	2
#27161475	2
#27161431	2
C₈×#13096802	2
#494×C₁₉₇	2
#1707	197