Group #38738394 — subgroups

Subgroups — #1460×C₁₉₇ #38738394

Order 25216.

312 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 124 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×D₄×C₁₉₇	C₂	1	1	yes	maximal, characteristic
2	#431×C₁₉₇	C₂	1	2	yes	maximal, complemented
3	#323×C₁₉₇	C₂	1	4	yes	maximal, complemented
4	C₂²×D₄×C₁₉₇	C₄	1	1	yes	characteristic
5	C₂³×C₄×C₁₉₇	C₄	1	2	yes	—
6	C₄×D₄×C₁₉₇	—	2	2	no	—
7	C₄×D₄×C₁₉₇	—	2	2	no	—
8	C₂×C₄⋊C₄×C₁₉₇	C₄	1	1	yes	characteristic
9	C₂×C₂²⋊C₄×C₁₉₇	C₂²	1	2	yes	—
10	C₂×C₄²×C₁₉₇	C₂²	1	1	yes	characteristic
11	C₄⋊C₈×C₁₉₇	—	2	2	no	—
12	C₂²⋊C₈×C₁₉₇	C₂²	1	4	yes	—
13	C₄×C₈×C₁₉₇	—	2	2	no	—
14	C₂⁴×C₁₉₇	C₂×C₄	1	2	yes	—
15	C₂×D₄×C₁₉₇	—	2	2	no	—
16	C₂×D₄×C₁₉₇	—	2	2	no	—
17	C₂×D₄×C₁₉₇	C₈	1	4	yes	complemented
18	C₂²×C₄×C₁₉₇	—	2	2	no	—
19	C₂²×C₄×C₁₉₇	—	2	2	no	—
20	C₂²×C₄×C₁₉₇	—	4	4	no	—
21	C₂²×C₄×C₁₉₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₁₉₇	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₁₉₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
25	C₂×C₈×C₁₉₇	—	2	4	no	—
26	C₄⋊C₄×C₁₉₇	—	2	2	no	—
27	C₂²⋊C₄×C₁₉₇	—	2	4	no	—
28	C₄²×C₁₉₇	D₄	1	2	yes	—
29	C₄²×C₁₉₇	D₄	1	2	yes	—
30	C₂³×C₁₉₇	—	2	2	no	—
31	C₂³×C₁₉₇	—	2	2	no	—
32	C₂³×C₁₉₇	—	4	2	no	—
33	C₂³×C₁₉₇	—	4	2	no	—
34	C₂³×C₁₉₇	C₂×C₈	1	4	yes	—
35	C₂³×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
36	D₄×C₁₉₇	—	2	8	no	—
37	C₂×C₄×C₁₉₇	—	2	2	no	—
38	C₂×C₄×C₁₉₇	—	2	2	no	—
39	C₂×C₄×C₁₉₇	—	4	1	no	—
40	C₂×C₄×C₁₉₇	—	4	4	no	—
41	C₂×C₄×C₁₉₇	—	4	4	no	—
42	C₂×C₄×C₁₉₇	—	4	2	no	—
43	C₂×C₄×C₁₉₇	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₁₉₇	C₂²⋊C₄	1	2	yes	—
45	C₂×C₄×C₁₉₇	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₁₉₇	C₂×C₈	1	2	yes	—
47	C₂×C₄×C₁₉₇	M₁₆	1	2	yes	—
48	C₂×C₄×C₁₉₇	C₂×D₄	1	2	yes	—
49	C₈×C₁₉₇	—	4	4	no	—
50	C₂²×C₁₉₇	—	2	1	no	—
51	C₂²×C₁₉₇	—	2	1	no	—
52	C₂²×C₁₉₇	—	2	4	no	—
53	C₂²×C₁₉₇	—	4	4	no	—
54	C₂²×C₁₉₇	—	4	4	no	—
55	C₂²×C₁₉₇	—	4	2	no	—
56	C₂²×C₁₉₇	—	4	2	no	—
57	C₂²×C₁₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₁₉₇	C₂²×C₈	1	1	yes	characteristic
59	C₂²×C₁₉₇	C₂×M₁₆	1	1	yes	characteristic
60	C₄×C₁₉₇	—	2	2	no	—
61	C₄×C₁₉₇	—	2	1	no	—
62	C₄×C₁₉₇	—	4	2	no	—
63	C₄×C₁₉₇	C₂²⋊C₈	1	4	yes	—
64	C₄×C₁₉₇	#100	1	2	yes	—
65	C₂×C₁₉₇	—	2	1	no	—
66	C₂×C₁₉₇	—	2	1	no	—
67	C₂×C₁₉₇	—	4	4	no	—
68	C₂×C₁₉₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
69	C₂×C₁₉₇	#410	1	1	yes	characteristic
70	C₂×C₁₉₇	C₂×#100	1	1	yes	characteristic
71	C₁₉₇	#1460	1	1	yes	characteristic, direct product factor, Sylow
72	#1460	C₁₉₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
73	C₂×C₄×D₄	C₂×C₁₉₇	1	1	yes	characteristic
74	#431	C₂×C₁₉₇	1	2	yes	complemented
75	#323	C₂×C₁₉₇	1	4	yes	complemented
76	C₂²×D₄	C₄×C₁₉₇	1	1	yes	characteristic
77	C₂³×C₄	C₄×C₁₉₇	1	2	yes	—
78	C₄×D₄	—	2	2	no	—
79	C₄×D₄	—	2	2	no	—
80	C₂×C₄⋊C₄	C₄×C₁₉₇	1	1	yes	characteristic
81	C₂×C₂²⋊C₄	C₂²×C₁₉₇	1	2	yes	—
82	C₂×C₄²	C₂²×C₁₉₇	1	1	yes	characteristic
83	C₄⋊C₈	—	2	2	no	—
84	C₂²⋊C₈	C₂²×C₁₉₇	1	4	yes	—
85	C₄×C₈	—	2	2	no	—
86	C₂⁴	C₂×C₄×C₁₉₇	1	2	yes	—
87	C₂×D₄	—	2	2	no	—
88	C₂×D₄	—	2	2	no	—
89	C₂×D₄	C₈×C₁₉₇	1	4	yes	complemented
90	C₂²×C₄	—	2	2	no	—
91	C₂²×C₄	—	2	2	no	—
92	C₂²×C₄	—	4	4	no	—
93	C₂²×C₄	C₂×C₄×C₁₉₇	1	1	yes	characteristic
94	C₂²×C₄	C₂×C₄×C₁₉₇	1	2	yes	—
95	C₂²×C₄	C₂×C₄×C₁₉₇	1	1	yes	characteristic
96	C₂²×C₄	C₂³×C₁₉₇	1	1	yes	characteristic
97	C₂×C₈	—	2	4	no	—
98	C₄⋊C₄	—	2	2	no	—
99	C₂²⋊C₄	—	2	4	no	—
100	C₄²	D₄×C₁₉₇	1	2	yes	—
101	C₄²	D₄×C₁₉₇	1	2	yes	—
102	C₂³	—	2	2	no	—
103	C₂³	—	2	2	no	—
104	C₂³	—	4	2	no	—
105	C₂³	—	4	2	no	—
106	C₂³	C₂×C₈×C₁₉₇	1	4	yes	—
107	C₂³	C₂²×C₄×C₁₉₇	1	1	yes	characteristic
108	D₄	—	2	8	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	4	1	no	—
112	C₂×C₄	—	4	4	no	—
113	C₂×C₄	—	4	4	no	—
114	C₂×C₄	—	4	2	no	—
115	C₂×C₄	C₂²⋊C₄×C₁₉₇	1	1	yes	characteristic
116	C₂×C₄	C₂²⋊C₄×C₁₉₇	1	2	yes	—
117	C₂×C₄	C₂²⋊C₄×C₁₉₇	1	1	yes	characteristic
118	C₂×C₄	C₂×C₈×C₁₉₇	1	2	yes	—
119	C₂×C₄	M₁₆×C₁₉₇	1	2	yes	—
120	C₂×C₄	C₂×D₄×C₁₉₇	1	2	yes	—
121	C₈	—	4	4	no	—
122	C₂²	—	2	1	no	—
123	C₂²	—	2	1	no	—
124	C₂²	—	2	4	no	—
125	C₂²	—	4	4	no	—
126	C₂²	—	4	4	no	—
127	C₂²	—	4	2	no	—
128	C₂²	—	4	2	no	—
129	C₂²	C₂×C₂²⋊C₄×C₁₉₇	1	1	yes	characteristic
130	C₂²	C₂²×C₈×C₁₉₇	1	1	yes	characteristic
131	C₂²	C₂×M₁₆×C₁₉₇	1	1	yes	characteristic
132	C₄	—	2	2	no	—
133	C₄	—	2	1	no	—
134	C₄	—	4	2	no	—
135	C₄	C₂²⋊C₈×C₁₉₇	1	4	yes	—
136	C₄	#100×C₁₉₇	1	2	yes	—
137	C₂	—	2	1	no	—
138	C₂	—	2	1	no	—
139	C₂	—	4	4	no	—
140	C₂	C₂×C₂²⋊C₈×C₁₉₇	1	1	yes	characteristic
141	C₂	#410×C₁₉₇	1	1	yes	characteristic
142	C₂	C₂×#100×C₁₉₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×D₄×C₁₉₇

C₂

yes

maximal, characteristic

#431×C₁₉₇

C₂

yes

maximal, complemented

#323×C₁₉₇

C₂

yes

maximal, complemented

C₂²×D₄×C₁₉₇

C₄

yes

characteristic

C₂³×C₄×C₁₉₇

C₄

yes

—

C₄×D₄×C₁₉₇

—

C₄×D₄×C₁₉₇

—

C₂×C₄⋊C₄×C₁₉₇

C₄

yes

characteristic

C₂×C₂²⋊C₄×C₁₉₇

C₂²

yes

—

C₂×C₄²×C₁₉₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₉₇

—

C₂²⋊C₈×C₁₉₇

C₂²

yes

—

C₄×C₈×C₁₉₇

—

C₂⁴×C₁₉₇

C₂×C₄

yes

—

C₂×D₄×C₁₉₇

—

C₂×D₄×C₁₉₇

—

C₂×D₄×C₁₉₇

C₈

yes

complemented

C₂²×C₄×C₁₉₇

—

C₂²×C₄×C₁₉₇

—

C₂²×C₄×C₁₉₇

—

C₂²×C₄×C₁₉₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₉₇

C₂×C₄

yes

—

C₂²×C₄×C₁₉₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₉₇

C₂³

yes

characteristic

C₂×C₈×C₁₉₇

—

C₄⋊C₄×C₁₉₇

—

C₂²⋊C₄×C₁₉₇

—

C₄²×C₁₉₇

D₄

yes

—

C₄²×C₁₉₇

D₄

yes

—

C₂³×C₁₉₇

—

C₂³×C₁₉₇

—

C₂³×C₁₉₇

—

C₂³×C₁₉₇

—

C₂³×C₁₉₇

C₂×C₈

yes

—

C₂³×C₁₉₇

C₂²×C₄

yes

characteristic

D₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₉₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₉₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₉₇

C₂×C₈

yes

—

C₂×C₄×C₁₉₇

M₁₆

yes

—

C₂×C₄×C₁₉₇

C₂×D₄

yes

—

C₈×C₁₉₇

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₉₇

C₂²×C₈

yes

characteristic

C₂²×C₁₉₇

C₂×M₁₆

yes

characteristic

C₄×C₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

C₂²⋊C₈

yes

—

C₄×C₁₉₇

#100

yes

—

C₂×C₁₉₇

—

C₂×C₁₉₇

—

C₂×C₁₉₇

—

C₂×C₁₉₇

C₂×C₂²⋊C₈

yes

C₂×C₁₉₇

yes

C₂×C₁₉₇

yes

C₁₉₇

yes

characteristic, direct product factor, Sylow

#1460

C₁₉₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₄×D₄

C₂×C₁₉₇

yes

characteristic

#431

C₂×C₁₉₇

yes

complemented

#323

C₂×C₁₉₇

yes

complemented

C₂²×D₄

C₄×C₁₉₇

yes

characteristic

C₂³×C₄

C₄×C₁₉₇

yes

—

C₄×D₄

—

C₄×D₄

—

C₂×C₄⋊C₄

C₄×C₁₉₇

yes

characteristic

C₂×C₂²⋊C₄

C₂²×C₁₉₇

yes

—

C₂×C₄²

C₂²×C₁₉₇

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

C₂²×C₁₉₇

yes

—

C₄×C₈

—

C₂⁴

C₂×C₄×C₁₉₇

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₈×C₁₉₇

yes

complemented

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×C₄×C₁₉₇

yes

characteristic

C₂²×C₄

C₂×C₄×C₁₉₇

yes

—

C₂²×C₄

C₂×C₄×C₁₉₇

yes

characteristic

C₂²×C₄

C₂³×C₁₉₇

yes

characteristic

C₂×C₈

—

C₄⋊C₄

—

C₂²⋊C₄

—

100

C₄²

D₄×C₁₉₇

yes

—

101

C₄²

D₄×C₁₉₇

yes

—

102

C₂³

—

103

C₂³

—

104

C₂³

—

105

C₂³

—

106

C₂³

C₂×C₈×C₁₉₇

yes

—

107

C₂³

C₂²×C₄×C₁₉₇

yes

characteristic

108

D₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

C₂²⋊C₄×C₁₉₇

yes

characteristic

116

C₂×C₄

C₂²⋊C₄×C₁₉₇

yes

—

117

C₂×C₄

C₂²⋊C₄×C₁₉₇

yes

characteristic

118

C₂×C₄

C₂×C₈×C₁₉₇

yes

—

119

C₂×C₄

M₁₆×C₁₉₇

yes

—

120

C₂×C₄

C₂×D₄×C₁₉₇

yes

—

121

C₈

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

C₂×C₂²⋊C₄×C₁₉₇

yes

characteristic

130

C₂²

C₂²×C₈×C₁₉₇

yes

characteristic

131

C₂²

C₂×M₁₆×C₁₉₇

yes

characteristic

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

C₂²⋊C₈×C₁₉₇

yes

—

136

C₄

#100×C₁₉₇

yes

—

137

C₂

—

138

C₂

—

139

C₂

—

140

C₂

C₂×C₂²⋊C₈×C₁₉₇

yes

characteristic

141

C₂

#410×C₁₉₇

yes

characteristic

142

C₂

C₂×#100×C₁₉₇

yes

characteristic

	subgroup	G/H
center	C₂×C₄×C₁₉₇	C₂²⋊C₄
commutator	C₂²	C₂²×C₈×C₁₉₇
Frattini, cosocle	C₂²×C₄	C₂³×C₁₉₇
socle	C₂²×C₁₉₇	C₂×C₂²⋊C₄
prime radical	C₂²×D₄×C₁₉₇	C₄
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₉₇ ◁ C₂×C₄²×C₁₉₇ ◁ G

subgroup

G/H

center

C₂×C₄×C₁₉₇

C₂²⋊C₄

commutator

C₂²

C₂²×C₈×C₁₉₇

Frattini, cosocle

C₂²×C₄

C₂³×C₁₉₇

socle

C₂²×C₁₉₇

C₂×C₂²⋊C₄

prime radical

C₂²×D₄×C₁₉₇

C₄

LCS

G ▷ C₂² ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₁₉₇ ◁ C₂×C₄²×C₁₉₇ ◁ G

N	H ≅ G/N	type
2. #431×C₁₉₇	C₂	semidirect
3. #323×C₁₉₇	C₂	semidirect
17. C₂×D₄×C₁₉₇	C₈	semidirect
71. C₁₉₇	#1460	direct (N × H)
72. #1460	C₁₉₇	direct (N × H)
74. #431	C₂×C₁₉₇	semidirect
75. #323	C₂×C₁₉₇	semidirect
89. C₂×D₄	C₈×C₁₉₇	semidirect

H ≅ G/N

type

2. #431×C₁₉₇

C₂

semidirect

3. #323×C₁₉₇

C₂

semidirect

17. C₂×D₄×C₁₉₇

C₈

semidirect

71. C₁₉₇

#1460

direct (N × H)

72. #1460

C₁₉₇

direct (N × H)

74. #431

C₂×C₁₉₇

semidirect

75. #323

C₂×C₁₉₇

semidirect

89. C₂×D₄

C₈×C₁₉₇

semidirect

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

subgroup	index
C₂×C₄×D₄×C₁₉₇	2
#431×C₁₉₇	2
#323×C₁₉₇	2
#1460	197

subgroup

index

C₂×C₄×D₄×C₁₉₇

#431×C₁₉₇

#323×C₁₉₇

#1460

197

Subgroups — #1460×C₁₉₇ #38738394

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1460×C197 #38738394

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1460×C₁₉₇ #38738394