Group #38744266 — subgroups

Subgroups — #38744266

Order 25216.

210 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 67 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	#27161128	C₂	1	1	yes	maximal, characteristic, complemented
2	#27161106	C₂	1	2	yes	maximal, complemented
3	#27161097	C₂	1	2	yes	maximal
4	#27161047	C₂	1	1	yes	maximal, characteristic
5	H(3,ℤ/4)×C₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15691358	C₂²	1	1	yes	characteristic
7	C₂×#15691342	C₂²	1	2	yes	—
8	C₂²×#13096802	C₂²	1	1	yes	characteristic
9	#20263262	—	2	2	no	—
10	C₄⋊#13096802	—	2	4	no	—
11	#20263254	—	2	4	no	—
12	#20263238	—	2	2	no	—
13	C₂×M₁₆×C₁₉₇	C₂²	1	1	yes	characteristic
14	#117×C₁₉₇	C₂²	1	2	yes	—
15	#15691358	—	2	2	no	—
16	#15691358	—	2	2	no	—
17	C₂×Dic₃₉₄	Q₈	1	1	yes	characteristic
18	C₂×D₇₈₈	Q₈	1	1	yes	characteristic
19	C₂×C₂²⋊D₁₉₇	—	2	1	no	—
20	C₂×C₄×D₁₉₇	—	2	1	no	—
21	#15691342	—	2	2	no	—
22	#15691342	—	2	2	no	—
23	C₂×#13096802	—	2	1	no	—
24	C₂×#13096802	—	2	2	no	—
25	C₂×#13096802	—	2	2	no	—
26	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
27	M₁₆×C₁₉₇	—	2	2	no	—
28	C₂×C₈×C₁₉₇	D₄	1	2	yes	—
29	C₄⋊C₄×C₁₉₇	—	2	2	no	—
30	C₂²⋊C₄×C₁₉₇	—	2	2	no	—
31	C₄²×C₁₉₇	D₄	1	4	yes	—
32	C₂²×D₁₉₇	—	2	1	no	—
33	Dic₃₉₄	—	2	2	no	—
34	D₇₈₈	—	2	2	no	—
35	C₂²⋊D₁₉₇	—	4	2	no	—
36	C₂×Dic₁₉₇	—	2	1	no	—
37	C₄×D₁₉₇	—	4	1	no	—
38	C₄×D₁₉₇	—	4	1	no	—
39	#13096802	—	4	1	no	—
40	#13096802	—	4	2	no	—
41	C₂³×C₁₉₇	#41	1	1	yes	characteristic
42	C₂×C₄×C₁₉₇	—	2	4	no	—
43	C₂×C₄×C₁₉₇	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₉₇	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₁₉₇	#41	1	2	yes	—
47	C₈×C₁₉₇	—	2	2	no	—
48	C₂²×C₁₉₇	—	4	1	no	—
49	C₂²×C₁₉₇	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂²×C₁₉₇	#123	1	2	yes	—
51	C₂×D₁₉₇	—	4	2	no	—
52	C₂×D₁₉₇	—	4	1	no	—
53	C₄×C₁₉₇	—	4	4	no	—
54	C₄×C₁₉₇	C₂²⋊Q₈	1	2	yes	—
55	C₄×C₁₉₇	#123	1	1	yes	characteristic
56	C₄×C₁₉₇	#127	1	1	yes	characteristic
57	Dic₁₉₇	—	4	1	no	—
58	C₂×C₁₉₇	—	2	2	no	—
59	C₂×C₁₉₇	—	2	1	no	—
60	C₂×C₁₉₇	#399	1	1	yes	characteristic
61	D₁₉₇	—	8	1	no	—
62	C₁₉₇	#2098	1	1	yes	characteristic, complemented, Sylow
63	#2098	—	197	1	no	maximal, Sylow
64	#430	—	197	1	no	—
65	#424	—	197	2	no	—
66	#421	—	197	2	no	—
67	#408	—	197	1	no	—
68	H(3,ℤ/4)	D₁₉₇	1	1	yes	characteristic, complemented
69	C₂×#41	—	197	1	no	—
70	C₂×M₁₆	—	197	2	no	—
71	C₂×M₁₆	C₂×D₁₉₇	1	1	yes	characteristic
72	C₂²×C₈	—	197	1	no	—
73	#117	C₂×D₁₉₇	1	2	yes	—
74	#108	—	394	2	no	—
75	C₄⋊C₈	—	394	4	no	—
76	#104	—	394	4	no	—
77	C₂²⋊C₈	—	394	2	no	—
78	#41	—	394	2	no	—
79	#41	—	394	2	no	—
80	C₂×Q₈	—	197	1	no	—
81	C₂×D₄	—	394	1	no	—
82	C₂×D₄	—	197	1	no	—
83	C₂²×C₄	—	394	1	no	—
84	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
85	M₁₆	—	2	2	no	—
86	M₁₆	—	394	2	no	—
87	M₁₆	—	394	2	no	—
88	C₂×C₈	—	394	1	no	—
89	C₂×C₈	—	394	2	no	—
90	C₂×C₈	—	394	2	no	—
91	C₂×C₈	D₇₈₈	1	2	yes	—
92	C₄⋊C₄	—	2	2	no	—
93	C₂²⋊C₄	—	2	2	no	—
94	C₄²	D₇₈₈	1	4	yes	—
95	C₂³	—	394	1	no	—
96	C₂³	#15691357	1	1	yes	characteristic
97	Q₈	—	394	2	no	—
98	D₄	—	788	2	no	—
99	D₄	—	394	2	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	788	1	no	—
102	C₂×C₄	—	788	1	no	—
103	C₂×C₄	—	394	1	no	—
104	C₂×C₄	C₂×D₇₈₈	1	2	yes	—
105	C₂×C₄	C₂×D₇₈₈	1	1	yes	characteristic
106	C₂×C₄	Q₈×D₁₉₇	1	1	yes	characteristic
107	C₂×C₄	#15691357	1	2	yes	—
108	C₈	—	2	2	no	—
109	C₈	—	788	1	no	—
110	C₈	—	788	2	no	—
111	C₂²	—	4	1	no	—
112	C₂²	—	788	2	no	—
113	C₂²	—	788	1	no	—
114	C₂²	#20263306	1	1	yes	characteristic
115	C₂²	#20263310	1	2	yes	—
116	C₄	—	4	4	no	—
117	C₄	—	788	1	no	—
118	C₄	#20263306	1	2	yes	—
119	C₄	#20263310	1	1	yes	characteristic
120	C₄	#20263323	1	1	yes	characteristic
121	C₂	—	2	2	no	—
122	C₂	—	2	1	no	—
123	C₂	—	1576	1	no	—
124	C₂	#27161016	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27161128

C₂

yes

maximal, characteristic, complemented

#27161106

C₂

yes

maximal, complemented

#27161097

C₂

yes

maximal

#27161047

C₂

yes

maximal, characteristic

H(3,ℤ/4)×C₁₉₇

C₂

yes

maximal, characteristic, complemented

C₂×#15691358

C₂²

yes

characteristic

C₂×#15691342

C₂²

yes

—

C₂²×#13096802

C₂²

yes

—

—

—

—

C₂×M₁₆×C₁₉₇

C₂²

yes

characteristic

#117×C₁₉₇

C₂²

yes

—

#15691358

—

#15691358

—

C₂×Dic₃₉₄

Q₈

yes

characteristic

C₂×D₇₈₈

Q₈

yes

characteristic

C₂×C₂²⋊D₁₉₇

—

C₂×C₄×D₁₉₇

—

—

—

—

—

—

C₂²×C₄×C₁₉₇

C₂³

yes

characteristic

M₁₆×C₁₉₇

—

C₂×C₈×C₁₉₇

D₄

yes

—

C₄⋊C₄×C₁₉₇

—

C₂²⋊C₄×C₁₉₇

—

C₄²×C₁₉₇

D₄

yes

—

C₂²×D₁₉₇

—

Dic₃₉₄

—

D₇₈₈

—

C₂²⋊D₁₉₇

—

C₂×Dic₁₉₇

—

C₄×D₁₉₇

—

C₄×D₁₉₇

—

#13096802

—

#13096802

—

C₂³×C₁₉₇

#41

yes

characteristic

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

C₂×D₄

yes

—

C₂×C₄×C₁₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₉₇

#41

yes

—

C₈×C₁₉₇

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₉₇

#123

yes

—

C₂×D₁₉₇

—

C₂×D₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

C₂²⋊Q₈

yes

—

C₄×C₁₉₇

yes

C₄×C₁₉₇

yes

Dic₁₉₇

—

C₂×C₁₉₇

—

C₂×C₁₉₇

—

C₂×C₁₉₇

#399

yes

characteristic

D₁₉₇

—

C₁₉₇

#2098

yes

characteristic, complemented, Sylow

#2098

—

197

maximal, Sylow

#430

—

197

—

#424

—

197

—

#421

—

197

—

#408

—

197

—

H(3,ℤ/4)

D₁₉₇

yes

characteristic, complemented

C₂×#41

—

197

—

C₂×M₁₆

—

197

—

C₂×M₁₆

C₂×D₁₉₇

yes

characteristic

C₂²×C₈

—

197

—

#117

C₂×D₁₉₇

yes

—

#108

—

394

—

C₄⋊C₈

—

394

—

#104

—

394

—

C₂²⋊C₈

—

394

—

#41

—

394

—

#41

—

394

—

C₂×Q₈

—

197

—

C₂×D₄

—

394

—

C₂×D₄

—

197

—

C₂²×C₄

—

394

—

C₂²×C₄

C₂²×D₁₉₇

yes

characteristic

M₁₆

—

M₁₆

—

394

—

M₁₆

—

394

—

C₂×C₈

—

394

—

C₂×C₈

—

394

—

C₂×C₈

—

394

—

C₂×C₈

D₇₈₈

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

D₇₈₈

yes

—

C₂³

—

394

—

C₂³

#15691357

yes

characteristic

Q₈

—

394

—

D₄

—

788

—

D₄

—

394

—

100

C₂×C₄

—

101

C₂×C₄

—

788

—

102

C₂×C₄

—

788

—

103

C₂×C₄

—

394

—

104

C₂×C₄

C₂×D₇₈₈

yes

—

105

C₂×C₄

C₂×D₇₈₈

yes

characteristic

106

C₂×C₄

Q₈×D₁₉₇

yes

characteristic

107

C₂×C₄

#15691357

yes

—

108

C₈

—

109

C₈

—

788

—

110

C₈

—

788

—

111

C₂²

—

112

C₂²

—

788

—

113

C₂²

—

788

—

114

C₂²

#20263306

yes

characteristic

115

C₂²

#20263310

yes

—

116

C₄

—

117

C₄

—

788

—

118

C₄

#20263306

yes

—

119

C₄

#20263310

yes

characteristic

120

C₄

#20263323

yes

characteristic

121

C₂

—

122

C₂

—

123

C₂

—

1576

—

124

C₂

#27161016

yes

characteristic

	subgroup	G/H
center	C₄	#20263323
commutator, cosocle	C₂²×C₄×C₁₉₇	C₂³
Frattini	C₂²×C₄	C₂²×D₁₉₇
Fitting	H(3,ℤ/4)×C₁₉₇	C₂
socle	C₂×C₁₉₇	#399
prime radical	C₂×#15691358	C₂²
hypercenter	H(3,ℤ/4)	D₁₉₇
hypocenter	C₁₉₇	#2098
LCS	G ▷ C₂²×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

#20263323

commutator, cosocle

C₂²×C₄×C₁₉₇

C₂³

Frattini

C₂²×C₄

C₂²×D₁₉₇

Fitting

H(3,ℤ/4)×C₁₉₇

C₂

C₂×C₁₉₇

C₂²

D₁₉₇

C₁₉₇

G ▷ C₂²×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
1. #27161128	C₂	semidirect
2. #27161106	C₂	semidirect
5. H(3,ℤ/4)×C₁₉₇	C₂	semidirect
62. C₁₉₇	#2098	semidirect
68. H(3,ℤ/4)	D₁₉₇	semidirect

H ≅ G/N

type

1. #27161128

C₂

semidirect

2. #27161106

C₂

semidirect

5. H(3,ℤ/4)×C₁₉₇

C₂

semidirect

62. C₁₉₇

#2098

semidirect

68. H(3,ℤ/4)

D₁₉₇

semidirect

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

subgroup	index
#27161128	2
#27161106	2
#27161097	2
#27161047	2
H(3,ℤ/4)×C₁₉₇	2
#2098	197

subgroup

index

H(3,ℤ/4)×C₁₉₇

197

Subgroups — #38744266

Conjugacy classes of subgroups

Complements

Maximal subgroups