Group #38743875 — subgroups

Subgroups — #38743875

Order 25216.

466 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 71 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27160934	C₂	1	3	yes	maximal
2	#27160846	C₂	1	3	yes	maximal, complemented
3	#543×C₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂⁴×D₁₉₇	C₂²	1	1	yes	characteristic
5	C₂×C₂²⋊Dic₁₉₇	C₂²	1	3	yes	—
6	C₂×#15691335	—	2	6	no	—
7	C₂²⋊Q₈×C₁₉₇	—	2	6	no	—
8	C₂×C₂²⋊C₄×C₁₉₇	C₂²	1	3	yes	—
9	C₂³×D₁₉₇	—	2	3	no	—
10	C₂³×D₁₉₇	—	2	3	no	—
11	C₂³×D₁₉₇	—	4	4	no	—
12	C₂³×D₁₉₇	Q₈	1	2	yes	complemented
13	C₂²×Dic₁₉₇	—	2	3	no	—
14	C₂²⋊Dic₁₉₇	—	2	6	no	—
15	#15691335	—	4	12	no	—
16	C₂⁴×C₁₉₇	C₂³	1	1	yes	characteristic
17	C₂×Q₈×C₁₉₇	—	2	2	no	—
18	C₂²×C₄×C₁₉₇	D₄	1	6	yes	—
19	C₄⋊C₄×C₁₉₇	—	2	6	no	—
20	C₂²⋊C₄×C₁₉₇	—	2	6	no	—
21	C₂²×D₁₉₇	—	2	12	no	—
22	C₂²×D₁₉₇	—	2	6	no	—
23	C₂²×D₁₉₇	—	4	4	no	—
24	C₂²×D₁₉₇	—	4	12	no	—
25	C₂²×D₁₉₇	—	4	6	no	—
26	C₂²×D₁₉₇	—	4	1	no	—
27	C₂²×D₁₉₇	—	4	3	no	—
28	C₂×Dic₁₉₇	—	4	6	no	—
29	C₂×Dic₁₉₇	—	4	3	no	—
30	C₂³×C₁₉₇	—	4	2	no	—
31	C₂³×C₁₉₇	C₂×D₄	1	3	yes	—
32	C₂³×C₁₉₇	C₂×Q₈	1	1	yes	characteristic
33	C₂³×C₁₉₇	#41	1	3	yes	—
34	Q₈×C₁₉₇	—	8	2	no	—
35	C₂×C₄×C₁₉₇	—	2	6	no	—
36	C₂×C₄×C₁₉₇	—	4	6	no	—
37	C₂²×C₁₉₇	—	2	6	no	—
38	C₂²×C₁₉₇	—	4	1	no	—
39	C₂²×C₁₉₇	—	4	3	no	—
40	C₂²×C₁₉₇	#120	1	1	yes	characteristic
41	C₂²×C₁₉₇	C₂²⋊Q₈	1	3	yes	—
42	C₂²×C₁₉₇	#124	1	3	yes	—
43	C₂×D₁₉₇	—	2	12	no	—
44	C₂×D₁₉₇	—	4	12	no	—
45	C₂×D₁₉₇	—	4	6	no	—
46	C₂×D₁₉₇	—	4	2	no	—
47	C₂×D₁₉₇	—	4	4	no	—
48	C₄×C₁₉₇	—	4	6	no	—
49	Dic₁₉₇	—	8	3	no	—
50	C₂×C₁₉₇	—	2	3	no	—
51	C₂×C₁₉₇	—	2	3	no	—
52	C₂×C₁₉₇	#393	1	1	yes	characteristic
53	C₂×C₁₉₇	B(4,𝔽₂)	1	2	yes	—
54	D₁₉₇	—	4	4	no	—
55	C₁₉₇	#2029	1	1	yes	characteristic, complemented, Sylow
56	#2029	—	197	1	no	maximal, Sylow
57	#543	D₁₉₇	1	1	yes	characteristic, complemented
58	#379	—	197	3	no	—
59	#342	—	197	3	no	—
60	C₂⁵	—	197	1	no	—
61	C₂²⋊Q₈	—	2	6	no	—
62	C₂×C₂²⋊C₄	—	394	6	no	—
63	C₂×C₂²⋊C₄	—	197	3	no	—
64	C₂×C₂²⋊C₄	C₂×D₁₉₇	1	3	yes	—
65	C₂⁴	—	788	4	no	—
66	C₂⁴	—	394	3	no	—
67	C₂⁴	—	394	3	no	—
68	C₂⁴	—	197	2	no	—
69	C₂⁴	C₂²×D₁₉₇	1	1	yes	characteristic
70	C₂×Q₈	—	2	2	no	—
71	C₂²×C₄	—	394	3	no	—
72	C₂²×C₄	C₂²⋊D₁₉₇	1	6	yes	—
73	C₄⋊C₄	—	2	6	no	—
74	C₂²⋊C₄	—	2	6	no	—
75	C₂²⋊C₄	—	788	12	no	—
76	C₂²⋊C₄	—	394	6	no	—
77	C₂³	—	4	2	no	—
78	C₂³	—	788	6	no	—
79	C₂³	—	788	12	no	—
80	C₂³	—	788	4	no	—
81	C₂³	—	788	1	no	—
82	C₂³	—	788	3	no	—
83	C₂³	—	394	12	no	—
84	C₂³	—	394	6	no	—
85	C₂³	C₂×C₂²⋊D₁₉₇	1	3	yes	—
86	C₂³	Q₈×D₁₉₇	1	1	yes	characteristic
87	C₂³	#15691359	1	3	yes	—
88	Q₈	—	8	2	no	—
89	C₂×C₄	—	2	6	no	—
90	C₂×C₄	—	4	6	no	—
91	C₂×C₄	—	788	6	no	—
92	C₂×C₄	—	788	3	no	—
93	C₂²	—	2	6	no	—
94	C₂²	—	4	1	no	—
95	C₂²	—	4	3	no	—
96	C₂²	—	788	12	no	—
97	C₂²	—	788	6	no	—
98	C₂²	—	788	2	no	—
99	C₂²	—	788	4	no	—
100	C₂²	—	394	12	no	—
101	C₂²	#20263295	1	1	yes	characteristic
102	C₂²	#20263302	1	3	yes	—
103	C₂²	#20263315	1	3	yes	—
104	C₄	—	4	6	no	—
105	C₄	—	1576	3	no	—
106	C₂	—	2	3	no	—
107	C₂	—	2	3	no	—
108	C₂	—	788	4	no	—
109	C₂	#27160990	1	1	yes	characteristic
110	C₂	#27161266	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#27160934

C₂

yes

maximal

#27160846

C₂

yes

maximal, complemented

#543×C₁₉₇

C₂

yes

maximal, characteristic, complemented

C₂⁴×D₁₉₇

C₂²

yes

characteristic

C₂×C₂²⋊Dic₁₉₇

C₂²

yes

—

C₂×#15691335

—

C₂²⋊Q₈×C₁₉₇

—

C₂×C₂²⋊C₄×C₁₉₇

C₂²

yes

—

C₂³×D₁₉₇

—

C₂³×D₁₉₇

—

C₂³×D₁₉₇

—

C₂³×D₁₉₇

Q₈

yes

complemented

C₂²×Dic₁₉₇

—

C₂²⋊Dic₁₉₇

—

#15691335

—

C₂⁴×C₁₉₇

C₂³

yes

characteristic

C₂×Q₈×C₁₉₇

—

C₂²×C₄×C₁₉₇

D₄

yes

—

C₄⋊C₄×C₁₉₇

—

C₂²⋊C₄×C₁₉₇

—

C₂²×D₁₉₇

—

C₂²×D₁₉₇

—

C₂²×D₁₉₇

—

C₂²×D₁₉₇

—

C₂²×D₁₉₇

—

C₂²×D₁₉₇

—

C₂²×D₁₉₇

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

C₂³×C₁₉₇

—

C₂³×C₁₉₇

C₂×D₄

yes

—

C₂³×C₁₉₇

C₂×Q₈

yes

characteristic

C₂³×C₁₉₇

#41

yes

—

Q₈×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

#120

yes

characteristic

C₂²×C₁₉₇

C₂²⋊Q₈

yes

—

C₂²×C₁₉₇

#124

yes

—

C₂×D₁₉₇

—

C₂×D₁₉₇

—

C₂×D₁₉₇

—

C₂×D₁₉₇

—

C₂×D₁₉₇

—

C₄×C₁₉₇

—

Dic₁₉₇

—

C₂×C₁₉₇

—

C₂×C₁₉₇

—

C₂×C₁₉₇

#393

yes

characteristic

C₂×C₁₉₇

B(4,𝔽₂)

yes

—

D₁₉₇

—

C₁₉₇

#2029

yes

characteristic, complemented, Sylow

#2029

—

197

maximal, Sylow

#543

D₁₉₇

yes

characteristic, complemented

#379

—

197

—

#342

—

197

—

C₂⁵

—

197

—

C₂²⋊Q₈

—

C₂×C₂²⋊C₄

—

394

—

C₂×C₂²⋊C₄

—

197

—

C₂×C₂²⋊C₄

C₂×D₁₉₇

yes

—

C₂⁴

—

788

—

C₂⁴

—

394

—

C₂⁴

—

394

—

C₂⁴

—

197

—

C₂⁴

C₂²×D₁₉₇

yes

characteristic

C₂×Q₈

—

C₂²×C₄

—

394

—

C₂²×C₄

C₂²⋊D₁₉₇

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

788

—

C₂²⋊C₄

—

394

—

C₂³

—

C₂³

—

788

—

C₂³

—

788

—

C₂³

—

788

—

C₂³

—

788

—

C₂³

—

788

—

C₂³

—

394

—

C₂³

—

394

—

C₂³

C₂×C₂²⋊D₁₉₇

yes

—

C₂³

Q₈×D₁₉₇

yes

characteristic

C₂³

#15691359

yes

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

788

—

C₂×C₄

—

788

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

788

—

C₂²

—

788

—

C₂²

—

788

—

C₂²

—

788

—

100

C₂²

—

394

—

101

C₂²

#20263295

yes

characteristic

102

C₂²

#20263302

yes

—

103

C₂²

#20263315

yes

—

104

C₄

—

105

C₄

—

1576

—

106

C₂

—

107

C₂

—

108

C₂

—

788

—

109

C₂

#27160990

yes

characteristic

110

C₂

#27161266

yes

—

	subgroup	G/H
center	C₂²	#20263295
commutator, cosocle	C₂⁴×C₁₉₇	C₂³
Frattini	C₂⁴	C₂²×D₁₉₇
Fitting	#543×C₁₉₇	C₂
socle	C₂²×C₁₉₇	#120
prime radical	C₂⁴×D₁₉₇	C₂²
hypercenter	#543	D₁₉₇
hypocenter	C₁₉₇	#2029
LCS	G ▷ C₂⁴×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ #543

subgroup

G/H

center

C₂²

#20263295

commutator, cosocle

C₂⁴×C₁₉₇

C₂³

Frattini

C₂⁴

C₂²×D₁₉₇

Fitting

#543×C₁₉₇

C₂

socle

C₂²×C₁₉₇

#120

prime radical

C₂⁴×D₁₉₇

C₂²

D₁₉₇

C₁₉₇

G ▷ C₂⁴×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇

UCS

C₁ ◁ C₂² ◁ C₂⁴ ◁ #543

N	H ≅ G/N	type
2. #27160846	C₂	semidirect
3. #543×C₁₉₇	C₂	semidirect
12. C₂³×D₁₉₇	Q₈	semidirect
55. C₁₉₇	#2029	semidirect
57. #543	D₁₉₇	semidirect

H ≅ G/N

type

2. #27160846

C₂

semidirect

3. #543×C₁₉₇

C₂

semidirect

12. C₂³×D₁₉₇

Q₈

semidirect

55. C₁₉₇

#2029

semidirect

57. #543

D₁₉₇

semidirect

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

subgroup	index
#27160934	2
#27160846	2
#543×C₁₉₇	2
#2029	197

subgroup

index

#27160934

#27160846

#543×C₁₉₇

#2029

197

Subgroups — #38743875

Conjugacy classes of subgroups

Complements

Maximal subgroups