Group #38743678 — subgroups

Subgroups — #38743678

Order 25216.

252 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 113 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	C₂×C₄×#13096802	C₂	1	1	yes	maximal, characteristic
2	#27160945	C₂	1	1	yes	maximal, characteristic
3	#27160841	C₂	1	4	yes	maximal
4	C₂×#125×C₁₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#13096802	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₁₉₇	C₂²	1	2	yes	—
7	C₄×#13096802	—	2	2	no	—
8	#20263233	—	2	2	no	—
9	#125×C₁₉₇	—	2	2	no	—
10	#125×C₁₉₇	C₄	1	4	yes	complemented
11	C₂×C₄⋊C₄×C₁₉₇	—	2	2	no	—
12	C₂×C₄⋊C₄×C₁₉₇	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₉₇	C₂²	1	1	yes	characteristic
14	C₂²×Dic₁₉₇	—	2	2	no	—
15	C₂×#13096802	—	2	2	no	—
16	C₂×#13096802	—	2	2	no	—
17	C₂×#13096802	Q₈	1	4	yes	—
18	C₄⋊Dic₁₉₇	—	2	4	no	—
19	C₂²×C₄×C₁₉₇	—	2	2	no	—
20	C₂²×C₄×C₁₉₇	D₄	1	2	yes	—
21	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
22	C₄⋊C₄×C₁₉₇	—	2	8	no	—
23	C₄⋊C₄×C₁₉₇	—	2	2	no	—
24	C₄⋊C₄×C₁₉₇	C₂×C₄	1	4	yes	—
25	C₄²×C₁₉₇	C₂×C₄	1	2	yes	—
26	C₄²×C₁₉₇	D₄	1	2	yes	—
27	C₂×Dic₁₉₇	—	4	2	no	—
28	C₂×Dic₁₉₇	—	4	4	no	—
29	#13096802	—	2	4	no	—
30	C₂³×C₁₉₇	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₁₉₇	—	2	4	no	—
32	C₂×C₄×C₁₉₇	—	2	4	no	—
33	C₂×C₄×C₁₉₇	—	4	4	no	—
34	C₂×C₄×C₁₉₇	C₂²⋊C₄	1	4	yes	—
35	C₂×C₄×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₁₉₇	C₂×Q₈	1	2	yes	—
38	C₂×C₄×C₁₉₇	#41	1	2	yes	—
39	C₂²×C₁₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
40	C₂²×C₁₉₇	#124	1	1	yes	characteristic
41	C₂²×C₁₉₇	C₄⋊Q₈	1	1	yes	characteristic
42	C₂²×C₁₉₇	#135	1	4	yes	—
43	C₄×C₁₉₇	—	2	4	no	—
44	C₄×C₁₉₇	—	4	4	no	—
45	C₄×C₁₉₇	C₄×Q₈	1	2	yes	—
46	C₄×C₁₉₇	C₂²⋊Q₈	1	2	yes	—
47	Dic₁₉₇	—	8	2	no	—
48	C₂×C₁₉₇	#391	1	1	yes	characteristic
49	C₂×C₁₉₇	#416	1	2	yes	—
50	C₂×C₁₉₇	#488	1	2	yes	—
51	C₂×C₁₉₇	#499	1	2	yes	—
52	C₁₉₇	#1991	1	1	yes	characteristic, complemented, Sylow
53	#1991	—	197	1	no	maximal, Sylow
54	C₂×#125	D₁₉₇	1	1	yes	characteristic
55	C₂×C₄×C₈	—	197	1	no	—
56	#382	—	197	1	no	—
57	#340	—	197	4	no	—
58	C₂²×C₈	—	197	2	no	—
59	#125	—	2	2	no	—
60	#125	Dic₁₉₇	1	4	yes	complemented
61	C₂×C₄⋊C₄	—	2	2	no	—
62	C₂×C₄⋊C₄	—	197	2	no	—
63	C₂×C₄⋊C₄	C₂×D₁₉₇	1	2	yes	—
64	C₂×C₄²	C₂×D₁₉₇	1	1	yes	characteristic
65	C₄×C₈	—	394	2	no	—
66	#95	—	394	2	no	—
67	C₂²×C₄	—	2	2	no	—
68	C₂²×C₄	—	394	2	no	—
69	C₂²×C₄	C₂²⋊D₁₉₇	1	2	yes	—
70	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
71	C₂×C₈	—	394	2	no	—
72	C₂×C₈	—	394	2	no	—
73	C₂×C₈	—	197	4	no	—
74	C₄⋊C₄	—	2	8	no	—
75	C₄⋊C₄	—	2	2	no	—
76	C₄⋊C₄	—	394	4	no	—
77	C₄⋊C₄	C₂×Dic₁₉₇	1	4	yes	—
78	C₄²	C₂×Dic₁₉₇	1	2	yes	—
79	C₄²	C₂²⋊D₁₉₇	1	2	yes	—
80	C₂³	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
81	C₂×C₄	—	2	4	no	—
82	C₂×C₄	—	2	4	no	—
83	C₂×C₄	—	4	4	no	—
84	C₂×C₄	—	788	2	no	—
85	C₂×C₄	—	788	4	no	—
86	C₂×C₄	C₂²⋊Dic₁₉₇	1	4	yes	—
87	C₂×C₄	C₂²×Dic₁₉₇	1	1	yes	characteristic
88	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
89	C₂×C₄	Q₈×D₁₉₇	1	2	yes	—
90	C₂×C₄	#15691359	1	2	yes	—
91	C₈	—	394	4	no	—
92	C₂²	C₂×C₂²⋊Dic₁₉₇	1	1	yes	characteristic
93	C₂²	#20263315	1	1	yes	characteristic
94	C₂²	#20263324	1	1	yes	characteristic
95	C₂²	#20263346	1	4	yes	—
96	C₄	—	2	4	no	—
97	C₄	—	4	4	no	—
98	C₄	—	1576	2	no	—
99	C₄	Q₈×Dic₁₉₇	1	2	yes	—
100	C₄	#20263302	1	2	yes	—
101	C₂	#27160986	1	1	yes	characteristic
102	C₂	#27161075	1	2	yes	—
103	C₂	#27161436	1	2	yes	—
104	C₂	#27161478	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×#13096802

C₂

yes

maximal, characteristic

#27160945

C₂

yes

maximal, characteristic

#27160841

C₂

yes

maximal

C₂×#125×C₁₉₇

C₂

yes

maximal, characteristic

C₂²×#13096802

C₂²

yes

—

C₂×C₄⋊Dic₁₉₇

C₂²

yes

—

C₄×#13096802

—

#20263233

—

#125×C₁₉₇

—

#125×C₁₉₇

C₄

yes

complemented

C₂×C₄⋊C₄×C₁₉₇

—

C₂×C₄⋊C₄×C₁₉₇

C₂²

yes

—

C₂×C₄²×C₁₉₇

C₂²

yes

characteristic

C₂²×Dic₁₉₇

—

C₂×#13096802

—

C₂×#13096802

—

C₂×#13096802

Q₈

yes

—

C₄⋊Dic₁₉₇

—

C₂²×C₄×C₁₉₇

—

C₂²×C₄×C₁₉₇

D₄

yes

—

C₂²×C₄×C₁₉₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₉₇

—

C₄⋊C₄×C₁₉₇

—

C₄⋊C₄×C₁₉₇

C₂×C₄

yes

—

C₄²×C₁₉₇

C₂×C₄

yes

—

C₄²×C₁₉₇

D₄

yes

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

#13096802

—

C₂³×C₁₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₇

C₂×Q₈

yes

—

C₂×C₄×C₁₉₇

#41

yes

—

C₂²×C₁₉₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₉₇

#124

yes

characteristic

C₂²×C₁₉₇

C₄⋊Q₈

yes

characteristic

C₂²×C₁₉₇

#135

yes

—

C₄×C₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

C₄×Q₈

yes

—

C₄×C₁₉₇

C₂²⋊Q₈

yes

—

Dic₁₉₇

—

C₂×C₁₉₇

#391

yes

characteristic

C₂×C₁₉₇

#416

yes

—

C₂×C₁₉₇

#488

yes

—

C₂×C₁₉₇

#499

yes

—

C₁₉₇

#1991

yes

characteristic, complemented, Sylow

#1991

—

197

maximal, Sylow

C₂×#125

D₁₉₇

yes

characteristic

C₂×C₄×C₈

—

197

—

#382

—

197

—

#340

—

197

—

C₂²×C₈

—

197

—

#125

—

#125

Dic₁₉₇

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

197

—

C₂×C₄⋊C₄

C₂×D₁₉₇

yes

—

C₂×C₄²

C₂×D₁₉₇

yes

characteristic

C₄×C₈

—

394

—

#95

—

394

—

C₂²×C₄

—

C₂²×C₄

—

394

—

C₂²×C₄

C₂²⋊D₁₉₇

yes

—

C₂²×C₄

C₂²×D₁₉₇

yes

characteristic

C₂×C₈

—

394

—

C₂×C₈

—

394

—

C₂×C₈

—

197

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

394

—

C₄⋊C₄

C₂×Dic₁₉₇

yes

—

C₄²

C₂×Dic₁₉₇

yes

—

C₄²

C₂²⋊D₁₉₇

yes

—

C₂³

C₂×C₂²⋊D₁₉₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

788

—

C₂×C₄

—

788

—

C₂×C₄

C₂²⋊Dic₁₉₇

yes

—

C₂×C₄

C₂²×Dic₁₉₇

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₁₉₇

yes

characteristic

C₂×C₄

Q₈×D₁₉₇

yes

—

C₂×C₄

#15691359

yes

—

C₈

—

394

—

C₂²

C₂×C₂²⋊Dic₁₉₇

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₄

—

C₄

—

C₄

—

1576

—

C₄

Q₈×Dic₁₉₇

yes

—

100

C₄

#20263302

yes

—

101

C₂

#27160986

yes

characteristic

102

C₂

#27161075

yes

—

103

C₂

#27161436

yes

—

104

C₂

#27161478

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₉₇
commutator	C₂×C₄×C₁₉₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₉₇
Fitting	C₂×#125×C₁₉₇	C₂
socle, prime radical	C₂³×C₁₉₇	C₂×D₄
cosocle	C₂²×C₄×C₁₉₇	C₂³
hypercenter	C₂×#125	D₁₉₇
hypocenter	C₁₉₇	#1991
LCS	G ▷ C₂×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₉₇

commutator

C₂×C₄×C₁₉₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₉₇

Fitting

C₂×#125×C₁₉₇

C₂

socle, prime radical

C₂³×C₁₉₇

C₂×D₄

cosocle

C₂²×C₄×C₁₉₇

C₂³

D₁₉₇

C₁₉₇

G ▷ C₂×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
10. #125×C₁₉₇	C₄	semidirect
52. C₁₉₇	#1991	semidirect
60. #125	Dic₁₉₇	semidirect

H ≅ G/N

type

10. #125×C₁₉₇

C₄

semidirect

52. C₁₉₇

#1991

semidirect

60. #125

Dic₁₉₇

semidirect

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

subgroup	index
C₂×C₄×#13096802	2
#27160945	2
#27160841	2
C₂×#125×C₁₉₇	2
#1991	197

subgroup

index

C₂×#125×C₁₉₇

197

Subgroups — #38743678

Conjugacy classes of subgroups

Complements

Maximal subgroups