Group #38742793 — subgroups

Subgroups — #38742793

Order 25216.

272 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 85 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×#20263285	C₂	1	1	yes	maximal, characteristic
2	#27161043	C₂	1	1	yes	maximal, characteristic
3	#27160847	C₂	1	4	yes	maximal, complemented
4	#407×C₁₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#15691342	C₂²	1	2	yes	—
6	#20263285	—	2	2	no	—
7	#20263285	—	2	2	no	—
8	C₂×C₄⋊Dic₁₉₇	C₂²	1	2	yes	—
9	C₂×C₂²⋊Dic₁₉₇	C₄	1	2	yes	—
10	C₂×C₄×Dic₁₉₇	C₄	1	2	yes	—
11	C₂²⋊#13096802	—	2	2	no	—
12	C₂³×C₄×C₁₉₇	C₂²	1	1	yes	characteristic
13	C₂×M₁₆×C₁₉₇	C₂²	1	2	yes	—
14	C₂²⋊C₈×C₁₉₇	—	2	2	no	—
15	C₂²×Dic₁₉₇	C₂×C₄	1	4	yes	—
16	#15691342	—	2	4	no	—
17	C₂×#13096802	—	2	2	no	—
18	C₄⋊Dic₁₉₇	—	2	2	no	—
19	C₄⋊Dic₁₉₇	—	2	2	no	—
20	C₂²⋊Dic₁₉₇	—	2	4	no	—
21	C₄×Dic₁₉₇	—	2	4	no	—
22	C₂⁴×C₁₉₇	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₁₉₇	—	2	4	no	—
24	C₂²×C₄×C₁₉₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₁₉₇	D₄	1	2	yes	—
26	C₂²×C₄×C₁₉₇	D₄	1	2	yes	—
27	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
28	M₁₆×C₁₉₇	—	2	4	no	—
29	C₂×C₈×C₁₉₇	—	2	2	no	—
30	C₂×Dic₁₉₇	—	2	4	no	—
31	C₂×Dic₁₉₇	—	4	4	no	—
32	#13096802	—	4	2	no	—
33	C₂³×C₁₉₇	—	2	2	no	—
34	C₂³×C₁₉₇	—	4	2	no	—
35	C₂³×C₁₉₇	C₂²⋊C₄	1	2	yes	—
36	C₂³×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₁₉₇	—	2	2	no	—
38	C₂×C₄×C₁₉₇	—	2	4	no	—
39	C₂×C₄×C₁₉₇	—	2	4	no	—
40	C₂×C₄×C₁₉₇	C₂²⋊C₄	1	2	yes	—
41	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₁₉₇	#41	1	4	yes	—
44	C₈×C₁₉₇	—	4	2	no	—
45	C₂²×C₁₉₇	—	2	2	no	—
46	C₂²×C₁₉₇	—	2	2	no	—
47	C₂²×C₁₉₇	—	2	4	no	—
48	C₂²×C₁₉₇	—	4	4	no	—
49	C₂²×C₁₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₁₉₇	#117	1	2	yes	—
51	C₄×C₁₉₇	—	2	2	no	—
52	C₄×C₁₉₇	#123	1	2	yes	—
53	C₄×C₁₉₇	#123	1	2	yes	—
54	Dic₁₉₇	—	4	4	no	—
55	C₂×C₁₉₇	—	2	2	no	—
56	C₂×C₁₉₇	—	4	2	no	—
57	C₂×C₁₉₇	#381	1	1	yes	characteristic
58	C₂×C₁₉₇	#413	1	2	yes	—
59	C₁₉₇	#1822	1	1	yes	characteristic, complemented, Sylow
60	#1822	—	197	1	no	maximal, Sylow
61	C₂×#117	—	197	1	no	—
62	#407	—	197	1	no	—
63	#407	D₁₉₇	1	1	yes	characteristic
64	#343	—	197	4	no	—
65	C₂³×C₄	C₂×D₁₉₇	1	1	yes	characteristic
66	C₂×M₁₆	—	197	2	no	—
67	C₂×M₁₆	C₂×D₁₉₇	1	2	yes	—
68	#117	—	394	2	no	—
69	#117	—	394	2	no	—
70	C₂×C₄⋊C₄	—	197	2	no	—
71	C₂×C₂²⋊C₄	—	197	2	no	—
72	C₂×C₄²	—	197	2	no	—
73	C₂²⋊C₈	—	2	2	no	—
74	C₂²⋊C₈	—	394	2	no	—
75	C₂⁴	C₄×D₁₉₇	1	1	yes	characteristic
76	C₂²×C₄	—	2	4	no	—
77	C₂²×C₄	—	197	4	no	—
78	C₂²×C₄	C₄×D₁₉₇	1	1	yes	characteristic
79	C₂²×C₄	C₂²⋊D₁₉₇	1	2	yes	—
80	C₂²×C₄	D₇₈₈	1	2	yes	—
81	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
82	M₁₆	—	2	4	no	—
83	M₁₆	—	394	4	no	—
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	394	2	no	—
86	C₄⋊C₄	—	394	2	no	—
87	C₄⋊C₄	—	394	2	no	—
88	C₂²⋊C₄	—	394	4	no	—
89	C₄²	—	394	4	no	—
90	C₂³	—	2	2	no	—
91	C₂³	—	4	2	no	—
92	C₂³	#15691335	1	2	yes	—
93	C₂³	C₂×C₄×D₁₉₇	1	1	yes	characteristic
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	—	2	4	no	—
97	C₂×C₄	—	788	4	no	—
98	C₂×C₄	—	394	4	no	—
99	C₂×C₄	#15691335	1	2	yes	—
100	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
101	C₂×C₄	C₂×D₇₈₈	1	1	yes	characteristic
102	C₂×C₄	#15691357	1	4	yes	—
103	C₈	—	4	2	no	—
104	C₈	—	788	2	no	—
105	C₂²	—	2	2	no	—
106	C₂²	—	2	2	no	—
107	C₂²	—	2	4	no	—
108	C₂²	—	4	4	no	—
109	C₂²	C₂×#15691335	1	1	yes	characteristic
110	C₂²	#20263282	1	2	yes	—
111	C₄	—	2	2	no	—
112	C₄	—	788	4	no	—
113	C₄	#20263307	1	2	yes	—
114	C₄	#20263310	1	2	yes	—
115	C₂	—	2	2	no	—
116	C₂	—	4	2	no	—
117	C₂	#27160940	1	1	yes	characteristic
118	C₂	#27161061	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20263285

C₂

yes

maximal, characteristic

#27161043

C₂

yes

maximal, characteristic

#27160847

C₂

yes

maximal, complemented

#407×C₁₉₇

C₂

yes

maximal, characteristic

C₂×#15691342

C₂²

yes

—

#20263285

—

#20263285

—

C₂×C₄⋊Dic₁₉₇

C₂²

yes

—

C₂×C₂²⋊Dic₁₉₇

C₄

yes

—

C₂×C₄×Dic₁₉₇

C₄

yes

—

C₂²⋊#13096802

—

C₂³×C₄×C₁₉₇

C₂²

yes

characteristic

C₂×M₁₆×C₁₉₇

C₂²

yes

—

C₂²⋊C₈×C₁₉₇

—

C₂²×Dic₁₉₇

C₂×C₄

yes

—

#15691342

—

C₂×#13096802

—

C₄⋊Dic₁₉₇

—

C₄⋊Dic₁₉₇

—

C₂²⋊Dic₁₉₇

—

C₄×Dic₁₉₇

—

C₂⁴×C₁₉₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₉₇

—

C₂²×C₄×C₁₉₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₉₇

D₄

yes

—

C₂²×C₄×C₁₉₇

D₄

yes

—

C₂²×C₄×C₁₉₇

C₂³

yes

characteristic

M₁₆×C₁₉₇

—

C₂×C₈×C₁₉₇

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

#13096802

—

C₂³×C₁₉₇

—

C₂³×C₁₉₇

—

C₂³×C₁₉₇

C₂²⋊C₄

yes

—

C₂³×C₁₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₇

#41

yes

—

C₈×C₁₉₇

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₉₇

#117

yes

—

C₄×C₁₉₇

—

C₄×C₁₉₇

#123

yes

—

C₄×C₁₉₇

#123

yes

—

Dic₁₉₇

—

C₂×C₁₉₇

—

C₂×C₁₉₇

—

C₂×C₁₉₇

#381

yes

characteristic

C₂×C₁₉₇

#413

yes

—

C₁₉₇

#1822

yes

characteristic, complemented, Sylow

#1822

—

197

maximal, Sylow

C₂×#117

—

197

—

#407

—

197

—

#407

D₁₉₇

yes

characteristic

#343

—

197

—

C₂³×C₄

C₂×D₁₉₇

yes

characteristic

C₂×M₁₆

—

197

—

C₂×M₁₆

C₂×D₁₉₇

yes

—

#117

—

394

—

#117

—

394

—

C₂×C₄⋊C₄

—

197

—

C₂×C₂²⋊C₄

—

197

—

C₂×C₄²

—

197

—

C₂²⋊C₈

—

C₂²⋊C₈

—

394

—

C₂⁴

C₄×D₁₉₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

197

—

C₂²×C₄

C₄×D₁₉₇

yes

characteristic

C₂²×C₄

C₂²⋊D₁₉₇

yes

—

C₂²×C₄

D₇₈₈

yes

—

C₂²×C₄

C₂²×D₁₉₇

yes

characteristic

M₁₆

—

M₁₆

—

394

—

C₂×C₈

—

C₂×C₈

—

394

—

C₄⋊C₄

—

394

—

C₄⋊C₄

—

394

—

C₂²⋊C₄

—

394

—

C₄²

—

394

—

C₂³

—

C₂³

—

C₂³

#15691335

yes

—

C₂³

C₂×C₄×D₁₉₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

788

—

C₂×C₄

—

394

—

C₂×C₄

#15691335

yes

—

100

C₂×C₄

C₂×C₂²⋊D₁₉₇

yes

characteristic

101

C₂×C₄

C₂×D₇₈₈

yes

characteristic

102

C₂×C₄

#15691357

yes

—

103

C₈

—

104

C₈

—

788

—

105

C₂²

—

106

C₂²

—

107

C₂²

—

108

C₂²

—

109

C₂²

C₂×#15691335

yes

characteristic

110

C₂²

#20263282

yes

—

111

C₄

—

112

C₄

—

788

—

113

C₄

#20263307

yes

—

114

C₄

#20263310

yes

—

115

C₂

—

116

C₂

—

117

C₂

#27160940

yes

characteristic

118

C₂

#27161061

yes

—

	subgroup	G/H
center	C₂²	C₂×#15691335
commutator	C₂³×C₁₉₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₉₇
Fitting	#407×C₁₉₇	C₂
socle	C₂²×C₁₉₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₁₉₇	C₂×C₄
cosocle	C₂²×C₄×C₁₉₇	C₂³
hypercenter	#407	D₁₉₇
hypocenter	C₁₉₇	#1822
LCS	G ▷ C₂³×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #407

subgroup

G/H

center

C₂²

C₂×#15691335

commutator

C₂³×C₁₉₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₉₇

Fitting

#407×C₁₉₇

C₂

socle

C₂²×C₁₉₇

C₂×C₂²⋊C₄

prime radical

C₂⁴×C₁₉₇

C₂×C₄

cosocle

C₂²×C₄×C₁₉₇

C₂³

D₁₉₇

C₁₉₇

G ▷ C₂³×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #407

N	H ≅ G/N	type
3. #27160847	C₂	semidirect
59. C₁₉₇	#1822	semidirect

H ≅ G/N

type

3. #27160847

C₂

semidirect

59. C₁₉₇

#1822

semidirect

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

subgroup	index
C₂×#20263285	2
#27161043	2
#27160847	2
#407×C₁₉₇	2
#1822	197

subgroup

index

#407×C₁₉₇

197

Subgroups — #38742793

Conjugacy classes of subgroups

Complements

Maximal subgroups