Subgroups — #38742328

Order 25216.

192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27161482	C₂	1	4	yes	maximal
2	#27161459	C₂	1	1	yes	maximal, characteristic
3	C₈×#13096802	C₂	1	1	yes	maximal, characteristic
4	#494×C₁₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₇₈₈	—	2	2	no	—
6	#20263326	C₂²	1	2	yes	—
7	#20263259	—	2	8	no	—
8	C₄×#13096802	C₂²	1	2	yes	—
9	C₂×Q₁₆×C₁₉₇	—	2	2	no	—
10	C₄⋊Q₈×C₁₉₇	C₂²	1	2	yes	—
11	C₄×C₈×C₁₉₇	C₂²	1	1	yes	characteristic
12	C₂×Dic₃₉₄	—	2	2	no	—
13	Dic₇₈₈	—	4	4	no	—
14	C₂×#13096802	D₄	1	4	yes	—
15	C₄⋊Dic₁₉₇	—	2	4	no	—
16	C₂×Q₈×C₁₉₇	—	2	2	no	—
17	Q₁₆×C₁₉₇	—	4	4	no	—
18	C₂×C₈×C₁₉₇	D₄	1	2	yes	—
19	C₄⋊C₄×C₁₉₇	—	2	4	no	—
20	C₄²×C₁₉₇	C₂³	1	1	yes	characteristic
21	Dic₃₉₄	—	4	4	no	—
22	C₂×Dic₁₉₇	—	4	2	no	—
23	#13096802	Q₁₆	1	8	yes	complemented
24	Q₈×C₁₉₇	—	4	4	no	—
25	C₂×C₄×C₁₉₇	—	4	2	no	—
26	C₂×C₄×C₁₉₇	C₂×D₄	1	2	yes	—
27	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
28	C₈×C₁₉₇	D₈	1	4	yes	—
29	C₂²×C₁₉₇	#127	1	1	yes	characteristic
30	C₄×C₁₉₇	—	8	2	no	—
31	C₄×C₁₉₇	C₂×D₈	1	2	yes	—
32	C₄×C₁₉₇	C₂×Q₁₆	1	4	yes	—
33	Dic₁₉₇	—	8	2	no	—
34	C₂×C₁₉₇	#493	1	1	yes	characteristic
35	C₂×C₁₉₇	#494	1	2	yes	—
36	C₁₉₇	#1713	1	1	yes	characteristic, complemented, Sylow
37	#1713	—	197	1	no	maximal, Sylow
38	#500	—	197	4	no	—
39	#494	—	197	1	no	—
40	#494	D₁₉₇	1	1	yes	characteristic
41	C₈²	—	197	1	no	—
42	C₂×Q₁₆	—	2	2	no	—
43	C₂×Q₁₆	—	394	2	no	—
44	C₄⋊Q₈	—	197	2	no	—
45	C₄⋊Q₈	C₂×D₁₉₇	1	2	yes	—
46	#107	—	394	8	no	—
47	C₄×C₈	—	197	2	no	—
48	C₄×C₈	C₂×D₁₉₇	1	1	yes	characteristic
49	C₂×Q₈	—	2	2	no	—
50	C₂×Q₈	—	394	2	no	—
51	Q₁₆	—	4	4	no	—
52	Q₁₆	—	788	4	no	—
53	C₂×C₈	—	197	4	no	—
54	C₂×C₈	C₂²⋊D₁₉₇	1	2	yes	—
55	C₄⋊C₄	—	2	4	no	—
56	C₄⋊C₄	—	394	4	no	—
57	C₄²	C₂²×D₁₉₇	1	1	yes	characteristic
58	Q₈	—	4	4	no	—
59	Q₈	—	788	4	no	—
60	C₂×C₄	—	4	2	no	—
61	C₂×C₄	—	788	2	no	—
62	C₂×C₄	D₄×D₁₉₇	1	2	yes	—
63	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
64	C₈	—	197	8	no	—
65	C₈	#15691343	1	4	yes	—
66	C₂²	#20263322	1	1	yes	characteristic
67	C₄	—	8	2	no	—
68	C₄	—	1576	2	no	—
69	C₄	C₂×#15691343	1	2	yes	—
70	C₄	Q₁₆×D₁₉₇	1	4	yes	—
71	C₂	#27161455	1	1	yes	characteristic
72	C₂	#27161457	1	2	yes	—

	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	#1713
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
23. #13096802	Q₁₆	semidirect
36. C₁₉₇	#1713	semidirect

Maximal subgroups

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

subgroup	index
#27161482	2
#27161459	2
C₈×#13096802	2
#494×C₁₉₇	2
#1713	197