Subgroups — #38744958

Order 25216.

132 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	#27161459	C₂	1	1	yes	maximal, characteristic
3	#27160884	C₂	1	4	yes	maximal
4	#346×C₁₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₇₈₈	—	2	1	no	—
6	C₂×Dic₇₈₈	C₂²	1	2	yes	—
7	#20263326	—	2	1	no	—
8	#20263318	—	2	1	no	—
9	#20263260	C₂²	1	2	yes	—
10	#20263258	—	2	1	no	—
11	C₂×C₁₆×C₁₉₇	C₂²	1	2	yes	—
12	C₄×C₈×C₁₉₇	C₂²	1	1	yes	characteristic
13	C₂×Dic₃₉₄	—	2	2	no	—
14	Dic₇₈₈	—	2	4	no	—
15	Dic₇₈₈	—	4	2	no	—
16	C₄⋊Dic₁₉₇	—	2	2	no	—
17	C₄⋊Dic₁₉₇	—	4	2	no	—
18	#15691337	—	4	2	no	—
19	C₂×C₈×C₁₉₇	D₄	1	1	yes	characteristic
20	C₂×C₈×C₁₉₇	C₂³	1	1	yes	characteristic
21	C₄²×C₁₉₇	D₄	1	1	yes	characteristic
22	C₁₆×C₁₉₇	—	2	2	no	—
23	Dic₃₉₄	—	4	2	no	—
24	Dic₃₉₄	—	4	2	no	—
25	C₂×Dic₁₉₇	—	4	2	no	—
26	C₂×Dic₁₉₇	—	4	2	no	—
27	C₂×C₄×C₁₉₇	D₈	1	2	yes	—
28	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
29	C₈×C₁₉₇	SD₁₆	1	2	yes	—
30	C₈×C₁₉₇	#41	1	2	yes	—
31	C₂²×C₁₉₇	C₂×D₈	1	1	yes	characteristic
32	C₄×C₁₉₇	—	2	2	no	—
33	C₄×C₁₉₇	#124	1	1	yes	characteristic
34	C₄×C₁₉₇	C₂×SD₁₆	1	1	yes	characteristic
35	Dic₁₉₇	—	8	2	no	—
36	Dic₁₉₇	—	8	2	no	—
37	C₂×C₁₉₇	#486	1	1	yes	characteristic
38	C₂×C₁₉₇	#510	1	2	yes	—
39	C₁₉₇	#2245	1	1	yes	characteristic, complemented, Sylow
40	#2245	—	197	1	no	maximal, Sylow
41	#499	—	197	1	no	—
42	#494	—	197	1	no	—
43	#358	—	197	4	no	—
44	#346	D₁₉₇	1	1	yes	characteristic
45	C₂×Q₁₆	—	394	1	no	—
46	C₂×Q₁₆	—	197	2	no	—
47	C₄⋊Q₈	—	394	1	no	—
48	#125	—	394	1	no	—
49	C₂×C₁₆	C₂×D₁₉₇	1	2	yes	—
50	#107	—	197	2	no	—
51	#106	—	394	1	no	—
52	C₄×C₈	C₂×D₁₉₇	1	1	yes	characteristic
53	C₂×Q₈	—	394	2	no	—
54	Q₁₆	—	788	2	no	—
55	Q₁₆	—	394	4	no	—
56	C₂×C₈	D₇₈₈	1	1	yes	characteristic
57	C₂×C₈	C₂²×D₁₉₇	1	1	yes	characteristic
58	C₄⋊C₄	—	788	2	no	—
59	C₄⋊C₄	—	788	2	no	—
60	C₄⋊C₄	—	394	2	no	—
61	C₄²	D₇₈₈	1	1	yes	characteristic
62	C₁₆	—	2	2	no	—
63	Q₈	—	788	2	no	—
64	Q₈	—	788	2	no	—
65	C₂×C₄	—	788	2	no	—
66	C₂×C₄	—	788	2	no	—
67	C₂×C₄	D₁₅₇₆	1	2	yes	—
68	C₂×C₄	C₂×D₇₈₈	1	1	yes	characteristic
69	C₈	#15691347	1	2	yes	—
70	C₈	#15691358	1	2	yes	—
71	C₂²	C₂×D₁₅₇₆	1	1	yes	characteristic
72	C₄	—	2	2	no	—
73	C₄	—	1576	2	no	—
74	C₄	—	1576	2	no	—
75	C₄	#20263314	1	1	yes	characteristic
76	C₄	C₂×#15691347	1	1	yes	characteristic
77	C₂	#27161426	1	1	yes	characteristic
78	C₂	#27161522	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₁₅₇₆
commutator, cosocle	C₂×C₈×C₁₉₇	C₂³
Frattini	C₂×C₈	C₂²×D₁₉₇
Fitting	#346×C₁₉₇	C₂
socle, prime radical	C₂²×C₁₉₇	C₂×D₈
hypercenter	#346	D₁₉₇
hypocenter	C₁₉₇	#2245
LCS	G ▷ C₂×C₈×C₁₉₇ ▷ C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
39. C₁₉₇	#2245	semidirect

Maximal subgroups

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

subgroup	index
#27161481	2
#27161459	2
#27160884	2
#346×C₁₉₇	2
#2245	197