Group #38738539 — subgroups

Subgroups — #1605×C₁₉₇ #38738539

Order 25216.

238 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 68 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	#544×C₁₉₇	C₂	1	1	yes	maximal, characteristic
2	#451×C₁₉₇	C₂	1	2	yes	maximal
3	#395×C₁₉₇	C₂	1	1	yes	maximal, characteristic
4	#328×C₁₉₇	C₂	1	2	yes	maximal
5	#324×C₁₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₈×C₁₉₇	C₂²	1	1	yes	characteristic
7	C₂×Q₁₆×C₁₉₇	—	2	2	no	—
8	C₄⋊Q₈×C₁₉₇	—	2	2	no	—
9	#125×C₁₉₇	—	2	2	no	—
10	C₂²⋊Q₈×C₁₉₇	—	2	1	no	—
11	C₂²⋊Q₈×C₁₉₇	C₂²	1	2	yes	—
12	#117×C₁₉₇	—	2	1	no	—
13	C₂×C₄⋊C₄×C₁₉₇	—	2	1	no	—
14	C₂×C₄⋊C₄×C₁₉₇	C₂²	1	1	yes	characteristic
15	#103×C₁₉₇	—	2	2	no	—
16	C₂²⋊C₈×C₁₉₇	C₂²	1	2	yes	—
17	#95×C₁₉₇	—	2	1	no	—
18	#95×C₁₉₇	C₂²	1	1	yes	characteristic
19	C₂×Q₈×C₁₉₇	—	2	1	no	—
20	C₂×Q₈×C₁₉₇	—	2	2	no	—
21	C₂×Q₈×C₁₉₇	—	4	2	no	—
22	C₂×Q₈×C₁₉₇	D₄	1	2	yes	—
23	C₂²×C₄×C₁₉₇	—	2	1	no	—
24	C₂²×C₄×C₁₉₇	—	2	1	no	—
25	C₂²×C₄×C₁₉₇	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₁₉₇	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
28	Q₁₆×C₁₉₇	—	4	4	no	—
29	C₂×C₈×C₁₉₇	—	2	2	no	—
30	C₄⋊C₄×C₁₉₇	—	2	1	no	—
31	C₄⋊C₄×C₁₉₇	—	2	1	no	—
32	C₄⋊C₄×C₁₉₇	—	2	1	no	—
33	C₄⋊C₄×C₁₉₇	—	2	2	no	—
34	C₄⋊C₄×C₁₉₇	—	2	2	no	—
35	C₄⋊C₄×C₁₉₇	—	2	2	no	—
36	C₄⋊C₄×C₁₉₇	—	4	2	no	—
37	C₄⋊C₄×C₁₉₇	D₄	1	2	yes	—
38	C₂²⋊C₄×C₁₉₇	—	2	2	no	—
39	C₄²×C₁₉₇	—	2	2	no	—
40	C₂³×C₁₉₇	C₂×D₄	1	1	yes	characteristic
41	Q₈×C₁₉₇	—	2	4	no	—
42	Q₈×C₁₉₇	—	4	2	no	—
43	Q₈×C₁₉₇	—	4	2	no	—
44	Q₈×C₁₉₇	—	8	1	no	—
45	C₂×C₄×C₁₉₇	—	2	1	no	—
46	C₂×C₄×C₁₉₇	—	2	1	no	—
47	C₂×C₄×C₁₉₇	—	2	2	no	—
48	C₂×C₄×C₁₉₇	—	2	2	no	—
49	C₂×C₄×C₁₉₇	—	2	2	no	—
50	C₂×C₄×C₁₉₇	—	2	2	no	—
51	C₂×C₄×C₁₉₇	—	2	1	no	—
52	C₂×C₄×C₁₉₇	—	4	1	no	—
53	C₂×C₄×C₁₉₇	—	4	1	no	—
54	C₂×C₄×C₁₉₇	—	4	1	no	—
55	C₂×C₄×C₁₉₇	—	4	1	no	—
56	C₂×C₄×C₁₉₇	—	4	1	no	—
57	C₂×C₄×C₁₉₇	—	4	1	no	—
58	C₂×C₄×C₁₉₇	Q₁₆	1	2	yes	complemented
59	C₂×C₄×C₁₉₇	C₂×D₄	1	2	yes	—
60	C₈×C₁₉₇	—	4	2	no	—
61	C₂²×C₁₉₇	—	2	1	no	—
62	C₂²×C₁₉₇	—	2	1	no	—
63	C₂²×C₁₉₇	#120	1	1	yes	characteristic
64	C₂²×C₁₉₇	C₂×Q₁₆	1	1	yes	characteristic
65	C₂²×C₁₉₇	#137	1	1	yes	characteristic
66	C₄×C₁₉₇	—	2	2	no	—
67	C₄×C₁₉₇	—	2	2	no	—
68	C₄×C₁₉₇	—	4	2	no	—
69	C₄×C₁₉₇	—	4	2	no	—
70	C₄×C₁₉₇	—	4	2	no	—
71	C₄×C₁₉₇	—	4	1	no	—
72	C₄×C₁₉₇	—	4	2	no	—
73	C₂×C₁₉₇	—	2	1	no	—
74	C₂×C₁₉₇	—	2	1	no	—
75	C₂×C₁₉₇	#451	1	1	yes	characteristic
76	C₂×C₁₉₇	#456	1	1	yes	characteristic
77	C₂×C₁₉₇	B(4,𝔽₂)	1	1	yes	characteristic
78	C₁₉₇	#1605	1	1	yes	characteristic, direct product factor, Sylow
79	#1605	C₁₉₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
80	#544	C₂×C₁₉₇	1	1	yes	characteristic
81	#451	C₂×C₁₉₇	1	2	yes	—
82	#395	C₂×C₁₉₇	1	1	yes	characteristic
83	#328	C₂×C₁₉₇	1	2	yes	—
84	#324	C₂×C₁₉₇	1	1	yes	characteristic
85	C₂²×Q₈	C₂²×C₁₉₇	1	1	yes	characteristic
86	C₂×Q₁₆	—	2	2	no	—
87	C₄⋊Q₈	—	2	2	no	—
88	#125	—	2	2	no	—
89	C₂²⋊Q₈	—	2	1	no	—
90	C₂²⋊Q₈	C₂²×C₁₉₇	1	2	yes	—
91	#117	—	2	1	no	—
92	C₂×C₄⋊C₄	—	2	1	no	—
93	C₂×C₄⋊C₄	C₂²×C₁₉₇	1	1	yes	characteristic
94	#103	—	2	2	no	—
95	C₂²⋊C₈	C₂²×C₁₉₇	1	2	yes	—
96	#95	—	2	1	no	—
97	#95	C₂²×C₁₉₇	1	1	yes	characteristic
98	C₂×Q₈	—	2	1	no	—
99	C₂×Q₈	—	2	2	no	—
100	C₂×Q₈	—	4	2	no	—
101	C₂×Q₈	D₄×C₁₉₇	1	2	yes	—
102	C₂²×C₄	—	2	1	no	—
103	C₂²×C₄	—	2	1	no	—
104	C₂²×C₄	D₄×C₁₉₇	1	1	yes	characteristic
105	C₂²×C₄	D₄×C₁₉₇	1	1	yes	characteristic
106	C₂²×C₄	C₂³×C₁₉₇	1	1	yes	characteristic
107	Q₁₆	—	4	4	no	—
108	C₂×C₈	—	2	2	no	—
109	C₄⋊C₄	—	2	1	no	—
110	C₄⋊C₄	—	2	1	no	—
111	C₄⋊C₄	—	2	1	no	—
112	C₄⋊C₄	—	2	2	no	—
113	C₄⋊C₄	—	2	2	no	—
114	C₄⋊C₄	—	2	2	no	—
115	C₄⋊C₄	—	4	2	no	—
116	C₄⋊C₄	D₄×C₁₉₇	1	2	yes	—
117	C₂²⋊C₄	—	2	2	no	—
118	C₄²	—	2	2	no	—
119	C₂³	C₂×D₄×C₁₉₇	1	1	yes	characteristic
120	Q₈	—	2	4	no	—
121	Q₈	—	4	2	no	—
122	Q₈	—	4	2	no	—
123	Q₈	—	8	1	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	4	1	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	—	4	1	no	—
134	C₂×C₄	—	4	1	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	—	4	1	no	—
137	C₂×C₄	Q₁₆×C₁₉₇	1	2	yes	complemented
138	C₂×C₄	C₂×D₄×C₁₉₇	1	2	yes	—
139	C₈	—	4	2	no	—
140	C₂²	—	2	1	no	—
141	C₂²	—	2	1	no	—
142	C₂²	#120×C₁₉₇	1	1	yes	characteristic
143	C₂²	C₂×Q₁₆×C₁₉₇	1	1	yes	characteristic
144	C₂²	#137×C₁₉₇	1	1	yes	characteristic
145	C₄	—	2	2	no	—
146	C₄	—	2	2	no	—
147	C₄	—	4	2	no	—
148	C₄	—	4	2	no	—
149	C₄	—	4	2	no	—
150	C₄	—	4	1	no	—
151	C₄	—	4	2	no	—
152	C₂	—	2	1	no	—
153	C₂	—	2	1	no	—
154	C₂	#451×C₁₉₇	1	1	yes	characteristic
155	C₂	#456×C₁₉₇	1	1	yes	characteristic
156	C₂	B(4,𝔽₂)×C₁₉₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#544×C₁₉₇

C₂

yes

maximal, characteristic

#451×C₁₉₇

C₂

yes

maximal

#395×C₁₉₇

C₂

yes

maximal, characteristic

#328×C₁₉₇

C₂

yes

maximal

#324×C₁₉₇

C₂

yes

maximal, characteristic

C₂²×Q₈×C₁₉₇

C₂²

yes

characteristic

C₂×Q₁₆×C₁₉₇

—

C₄⋊Q₈×C₁₉₇

—

#125×C₁₉₇

—

C₂²⋊Q₈×C₁₉₇

—

C₂²⋊Q₈×C₁₉₇

C₂²

yes

—

#117×C₁₉₇

—

C₂×C₄⋊C₄×C₁₉₇

—

C₂×C₄⋊C₄×C₁₉₇

C₂²

yes

characteristic

#103×C₁₉₇

—

C₂²⋊C₈×C₁₉₇

C₂²

yes

—

#95×C₁₉₇

—

#95×C₁₉₇

C₂²

yes

characteristic

C₂×Q₈×C₁₉₇

—

C₂×Q₈×C₁₉₇

—

C₂×Q₈×C₁₉₇

—

C₂×Q₈×C₁₉₇

D₄

yes

—

C₂²×C₄×C₁₉₇

—

C₂²×C₄×C₁₉₇

—

C₂²×C₄×C₁₉₇

D₄

yes

characteristic

C₂²×C₄×C₁₉₇

D₄

yes

characteristic

C₂²×C₄×C₁₉₇

C₂³

yes

characteristic

Q₁₆×C₁₉₇

—

C₂×C₈×C₁₉₇

—

C₄⋊C₄×C₁₉₇

—

C₄⋊C₄×C₁₉₇

—

C₄⋊C₄×C₁₉₇

—

C₄⋊C₄×C₁₉₇

—

C₄⋊C₄×C₁₉₇

—

C₄⋊C₄×C₁₉₇

—

C₄⋊C₄×C₁₉₇

—

C₄⋊C₄×C₁₉₇

D₄

yes

—

C₂²⋊C₄×C₁₉₇

—

C₄²×C₁₉₇

—

C₂³×C₁₉₇

C₂×D₄

yes

characteristic

Q₈×C₁₉₇

—

Q₈×C₁₉₇

—

Q₈×C₁₉₇

—

Q₈×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

Q₁₆

yes

complemented

C₂×C₄×C₁₉₇

C₂×D₄

yes

—

C₈×C₁₉₇

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

#120

yes

characteristic

C₂²×C₁₉₇

C₂×Q₁₆

yes

characteristic

C₂²×C₁₉₇

#137

yes

characteristic

C₄×C₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

—

C₂×C₁₉₇

—

C₂×C₁₉₇

—

C₂×C₁₉₇

yes

C₂×C₁₉₇

yes

C₂×C₁₉₇

yes

C₁₉₇

yes

characteristic, direct product factor, Sylow

#1605

C₁₉₇

yes

maximal, characteristic, direct product factor, Sylow

#544

C₂×C₁₉₇

yes

characteristic

#451

C₂×C₁₉₇

yes

—

#395

C₂×C₁₉₇

yes

characteristic

#328

C₂×C₁₉₇

yes

—

#324

C₂×C₁₉₇

yes

characteristic

C₂²×Q₈

C₂²×C₁₉₇

yes

characteristic

C₂×Q₁₆

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

C₂²×C₁₉₇

yes

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²×C₁₉₇

yes

characteristic

#103

—

C₂²⋊C₈

C₂²×C₁₉₇

yes

—

#95

—

#95

C₂²×C₁₉₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

D₄×C₁₉₇

yes

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

D₄×C₁₉₇

yes

characteristic

105

C₂²×C₄

D₄×C₁₉₇

yes

characteristic

106

C₂²×C₄

C₂³×C₁₉₇

yes

characteristic

107

Q₁₆

—

108

C₂×C₈

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

D₄×C₁₉₇

yes

—

117

C₂²⋊C₄

—

118

C₄²

—

119

C₂³

C₂×D₄×C₁₉₇

yes

characteristic

120

Q₈

—

121

Q₈

—

122

Q₈

—

123

Q₈

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

Q₁₆×C₁₉₇

yes

complemented

138

C₂×C₄

C₂×D₄×C₁₉₇

yes

—

139

C₈

—

140

C₂²

—

141

C₂²

—

142

C₂²

#120×C₁₉₇

yes

characteristic

143

C₂²

C₂×Q₁₆×C₁₉₇

yes

characteristic

144

C₂²

#137×C₁₉₇

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₂

—

153

C₂

—

154

C₂

#451×C₁₉₇

yes

characteristic

155

C₂

#456×C₁₉₇

yes

characteristic

156

C₂

B(4,𝔽₂)×C₁₉₇

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉₇	#120
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₁₉₇
prime radical	C₂³×C₁₉₇	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉₇ ◁ C₂²×C₄×C₁₉₇ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₉₇

#120

commutator, Frattini, cosocle

C₂²×C₄

C₂³×C₁₉₇

prime radical

C₂³×C₁₉₇

C₂×D₄

LCS

G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₁₉₇ ◁ C₂²×C₄×C₁₉₇ ◁ G

N	H ≅ G/N	type
58. C₂×C₄×C₁₉₇	Q₁₆	semidirect
78. C₁₉₇	#1605	direct (N × H)
79. #1605	C₁₉₇	direct (N × H)
137. C₂×C₄	Q₁₆×C₁₉₇	semidirect

H ≅ G/N

type

58. C₂×C₄×C₁₉₇

Q₁₆

semidirect

78. C₁₉₇

#1605

direct (N × H)

79. #1605

C₁₉₇

direct (N × H)

137. C₂×C₄

Q₁₆×C₁₉₇

semidirect

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

subgroup	index
#544×C₁₉₇	2
#451×C₁₉₇	2
#395×C₁₉₇	2
#328×C₁₉₇	2
#324×C₁₉₇	2
#1605	197

subgroup

index

#544×C₁₉₇

#451×C₁₉₇

#395×C₁₉₇

#328×C₁₉₇

#324×C₁₉₇

#1605

197

Subgroups — #1605×C₁₉₇ #38738539

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1605×C197 #38738539

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1605×C₁₉₇ #38738539