Subgroups — #38744229

Order 25216.

222 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27161128	C₂	1	3	yes	maximal, complemented
2	#27161042	C₂	1	3	yes	maximal
3	#338×C₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₄×D₁₉₇	C₂²	1	1	yes	characteristic
5	C₂×#15691342	C₂²	1	3	yes	—
6	#20263262	—	2	6	no	—
7	#20263238	—	2	6	no	—
8	C₂×M₁₆×C₁₉₇	C₂²	1	3	yes	—
9	C₂³×D₁₉₇	Q₈	1	1	yes	characteristic
10	C₂²×Dic₁₉₇	Q₈	1	1	yes	characteristic
11	C₂×C₄×D₁₉₇	—	2	3	no	—
12	C₂×C₄×D₁₉₇	—	2	3	no	—
13	#15691342	—	2	6	no	—
14	C₂×#13096802	—	2	3	no	—
15	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
16	M₁₆×C₁₉₇	—	2	6	no	—
17	C₂×C₈×C₁₉₇	D₄	1	6	yes	—
18	C₂²×D₁₉₇	—	2	3	no	—
19	C₂²×D₁₉₇	—	4	2	no	—
20	C₂×Dic₁₉₇	—	2	3	no	—
21	C₄×D₁₉₇	—	4	3	no	—
22	C₄×D₁₉₇	—	4	1	no	—
23	#13096802	—	4	3	no	—
24	C₂³×C₁₉₇	C₂×Q₈	1	1	yes	characteristic
25	C₂×C₄×C₁₉₇	C₂×D₄	1	3	yes	—
26	C₂×C₄×C₁₉₇	#41	1	3	yes	—
27	C₈×C₁₉₇	—	2	6	no	—
28	C₂²×C₁₉₇	—	4	1	no	—
29	C₂²×C₁₉₇	C₂²⋊Q₈	1	3	yes	—
30	C₂×D₁₉₇	—	2	6	no	—
31	C₂×D₁₉₇	—	4	3	no	—
32	C₂×D₁₉₇	—	4	1	no	—
33	C₄×C₁₉₇	#123	1	3	yes	—
34	C₄×C₁₉₇	#127	1	1	yes	characteristic
35	Dic₁₉₇	—	4	1	no	—
36	C₂×C₁₉₇	—	2	3	no	—
37	C₂×C₁₉₇	#399	1	1	yes	characteristic
38	D₁₉₇	—	4	2	no	—
39	C₁₉₇	#2091	1	1	yes	characteristic, complemented, Sylow
40	#2091	—	197	1	no	maximal, Sylow
41	#430	—	197	3	no	—
42	#407	—	197	3	no	—
43	#338	D₁₉₇	1	1	yes	characteristic, complemented
44	C₂³×C₄	—	197	1	no	—
45	C₂×M₁₆	—	197	3	no	—
46	C₂×M₁₆	C₂×D₁₉₇	1	3	yes	—
47	#108	—	394	6	no	—
48	C₂²⋊C₈	—	394	6	no	—
49	C₂⁴	—	197	1	no	—
50	C₂²×C₄	—	394	3	no	—
51	C₂²×C₄	—	394	3	no	—
52	C₂²×C₄	—	197	1	no	—
53	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
54	M₁₆	—	2	6	no	—
55	M₁₆	—	394	6	no	—
56	C₂×C₈	—	394	3	no	—
57	C₂×C₈	D₇₈₈	1	6	yes	—
58	C₂³	—	788	2	no	—
59	C₂³	—	394	3	no	—
60	C₂³	Q₈×D₁₉₇	1	1	yes	characteristic
61	C₂×C₄	—	788	3	no	—
62	C₂×C₄	—	788	1	no	—
63	C₂×C₄	—	394	3	no	—
64	C₂×C₄	C₂×D₇₈₈	1	3	yes	—
65	C₂×C₄	#15691357	1	3	yes	—
66	C₈	—	2	6	no	—
67	C₈	—	788	3	no	—
68	C₂²	—	4	1	no	—
69	C₂²	—	788	3	no	—
70	C₂²	—	788	1	no	—
71	C₂²	—	394	6	no	—
72	C₂²	#20263306	1	3	yes	—
73	C₄	—	788	1	no	—
74	C₄	#20263310	1	3	yes	—
75	C₄	#20263323	1	1	yes	characteristic
76	C₂	—	2	3	no	—
77	C₂	—	788	2	no	—
78	C₂	#27161016	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#20263323
commutator, cosocle	C₂²×C₄×C₁₉₇	C₂³
Frattini	C₂²×C₄	C₂²×D₁₉₇
Fitting	#338×C₁₉₇	C₂
socle	C₂×C₁₉₇	#399
prime radical	C₂³×D₁₉₇	Q₈
hypercenter	#338	D₁₉₇
hypocenter	C₁₉₇	#2091
LCS	G ▷ C₂²×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #338

Complements

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

N	H ≅ G/N	type
1. #27161128	C₂	semidirect
3. #338×C₁₉₇	C₂	semidirect
39. C₁₉₇	#2091	semidirect
43. #338	D₁₉₇	semidirect

Maximal subgroups

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

subgroup	index
#27161128	2
#27161042	2
#338×C₁₉₇	2
#2091	197