Group #30662255 — subgroups

Subgroups — #2072×C₁₃₁ #30662255

Order 16768.

238 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 100 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×#103×C₁₃₁	C₂	1	2	yes	maximal
2	#391×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	#384×C₁₃₁	C₂	1	1	yes	maximal, characteristic
4	#340×C₁₃₁	C₂	1	2	yes	maximal
5	#336×C₁₃₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
7	C₂²×C₈×C₁₃₁	C₂²	1	2	yes	—
8	C₂×C₄⋊C₄×C₁₃₁	—	2	1	no	—
9	C₂×C₄⋊C₄×C₁₃₁	—	2	1	no	—
10	C₂×C₄⋊C₄×C₁₃₁	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊C₄×C₁₃₁	C₂²	1	2	yes	—
12	C₂×C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
13	#103×C₁₃₁	—	2	4	no	—
14	#95×C₁₃₁	—	2	1	no	—
15	#95×C₁₃₁	—	2	1	no	—
16	C₂×Q₈×C₁₃₁	—	2	1	no	—
17	C₂×Q₈×C₁₃₁	—	2	2	no	—
18	C₂×Q₈×C₁₃₁	—	2	2	no	—
19	C₂×Q₈×C₁₃₁	D₄	1	2	yes	—
20	C₂²×C₄×C₁₃₁	—	2	1	no	—
21	C₂²×C₄×C₁₃₁	—	2	1	no	—
22	C₂²×C₄×C₁₃₁	—	2	2	no	—
23	C₂²×C₄×C₁₃₁	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₁₃₁	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
26	C₂×C₈×C₁₃₁	—	2	2	no	—
27	C₂×C₈×C₁₃₁	—	2	2	no	—
28	C₂×C₈×C₁₃₁	—	2	2	no	—
29	C₄⋊C₄×C₁₃₁	—	2	2	no	—
30	C₄⋊C₄×C₁₃₁	—	2	2	no	—
31	C₄⋊C₄×C₁₃₁	—	2	2	no	—
32	C₄⋊C₄×C₁₃₁	—	4	2	no	—
33	C₄⋊C₄×C₁₃₁	—	4	2	no	—
34	C₄²×C₁₃₁	—	2	2	no	—
35	C₂³×C₁₃₁	C₂×D₄	1	1	yes	characteristic
36	Q₈×C₁₃₁	—	2	4	no	—
37	Q₈×C₁₃₁	—	4	2	no	—
38	C₂×C₄×C₁₃₁	—	2	1	no	—
39	C₂×C₄×C₁₃₁	—	2	1	no	—
40	C₂×C₄×C₁₃₁	—	2	1	no	—
41	C₂×C₄×C₁₃₁	—	2	1	no	—
42	C₂×C₄×C₁₃₁	—	2	2	no	—
43	C₂×C₄×C₁₃₁	—	4	1	no	—
44	C₂×C₄×C₁₃₁	—	4	1	no	—
45	C₂×C₄×C₁₃₁	—	4	2	no	—
46	C₂×C₄×C₁₃₁	—	4	1	no	—
47	C₂×C₄×C₁₃₁	—	4	2	no	—
48	C₂×C₄×C₁₃₁	—	4	2	no	—
49	C₂×C₄×C₁₃₁	—	4	1	no	—
50	C₂×C₄×C₁₃₁	—	4	1	no	—
51	C₂×C₄×C₁₃₁	SD₁₆	1	2	yes	—
52	C₂×C₄×C₁₃₁	Q₁₆	1	2	yes	—
53	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₁₃₁	#41	1	2	yes	—
55	C₂×C₄×C₁₃₁	#41	1	2	yes	—
56	C₂×C₄×C₁₃₁	#41	1	1	yes	characteristic
57	C₈×C₁₃₁	—	4	2	no	—
58	C₂²×C₁₃₁	#121	1	1	yes	characteristic
59	C₂²×C₁₃₁	#123	1	1	yes	characteristic
60	C₂²×C₁₃₁	#124	1	1	yes	characteristic
61	C₂²×C₁₃₁	C₂×SD₁₆	1	1	yes	characteristic
62	C₂²×C₁₃₁	C₂×Q₁₆	1	1	yes	characteristic
63	C₂²×C₁₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
64	C₂²×C₁₃₁	#137	1	1	yes	characteristic
65	C₄×C₁₃₁	—	2	2	no	—
66	C₄×C₁₃₁	—	2	2	no	—
67	C₄×C₁₃₁	—	4	2	no	—
68	C₄×C₁₃₁	—	8	2	no	—
69	C₄×C₁₃₁	—	8	1	no	—
70	C₄×C₁₃₁	#123	1	2	yes	—
71	C₄×C₁₃₁	#126	1	2	yes	—
72	C₂×C₁₃₁	#397	1	1	yes	characteristic
73	C₂×C₁₃₁	#460	1	1	yes	characteristic
74	C₂×C₁₃₁	#462	1	1	yes	characteristic
75	C₂×C₁₃₁	#483	1	1	yes	characteristic
76	C₂×C₁₃₁	#484	1	1	yes	characteristic
77	C₂×C₁₃₁	#487	1	1	yes	characteristic
78	C₂×C₁₃₁	#489	1	1	yes	characteristic
79	C₁₃₁	#2072	1	1	yes	characteristic, direct product factor, Sylow
80	#2072	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
81	C₂×#103	C₂×C₁₃₁	1	2	yes	—
82	#391	C₂×C₁₃₁	1	1	yes	characteristic
83	#384	C₂×C₁₃₁	1	1	yes	characteristic
84	#340	C₂×C₁₃₁	1	2	yes	—
85	#336	C₂×C₁₃₁	1	1	yes	characteristic
86	C₂²×Q₈	C₂²×C₁₃₁	1	1	yes	characteristic
87	C₂²×C₈	C₂²×C₁₃₁	1	2	yes	—
88	C₂×C₄⋊C₄	—	2	1	no	—
89	C₂×C₄⋊C₄	—	2	1	no	—
90	C₂×C₄⋊C₄	C₂²×C₁₃₁	1	1	yes	characteristic
91	C₂×C₄⋊C₄	C₂²×C₁₃₁	1	2	yes	—
92	C₂×C₄²	C₂²×C₁₃₁	1	1	yes	characteristic
93	#103	—	2	4	no	—
94	#95	—	2	1	no	—
95	#95	—	2	1	no	—
96	C₂×Q₈	—	2	1	no	—
97	C₂×Q₈	—	2	2	no	—
98	C₂×Q₈	—	2	2	no	—
99	C₂×Q₈	D₄×C₁₃₁	1	2	yes	—
100	C₂²×C₄	—	2	1	no	—
101	C₂²×C₄	—	2	1	no	—
102	C₂²×C₄	—	2	2	no	—
103	C₂²×C₄	D₄×C₁₃₁	1	1	yes	characteristic
104	C₂²×C₄	D₄×C₁₃₁	1	1	yes	characteristic
105	C₂²×C₄	C₂³×C₁₃₁	1	1	yes	characteristic
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	2	2	no	—
109	C₄⋊C₄	—	2	2	no	—
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	—	2	2	no	—
112	C₄⋊C₄	—	4	2	no	—
113	C₄⋊C₄	—	4	2	no	—
114	C₄²	—	2	2	no	—
115	C₂³	C₂×D₄×C₁₃₁	1	1	yes	characteristic
116	Q₈	—	2	4	no	—
117	Q₈	—	4	2	no	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	4	1	no	—
124	C₂×C₄	—	4	1	no	—
125	C₂×C₄	—	4	2	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	4	2	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	—	4	1	no	—
131	C₂×C₄	SD₁₆×C₁₃₁	1	2	yes	—
132	C₂×C₄	Q₁₆×C₁₃₁	1	2	yes	—
133	C₂×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
134	C₂×C₄	#41×C₁₃₁	1	2	yes	—
135	C₂×C₄	#41×C₁₃₁	1	2	yes	—
136	C₂×C₄	#41×C₁₃₁	1	1	yes	characteristic
137	C₈	—	4	2	no	—
138	C₂²	#121×C₁₃₁	1	1	yes	characteristic
139	C₂²	#123×C₁₃₁	1	1	yes	characteristic
140	C₂²	#124×C₁₃₁	1	1	yes	characteristic
141	C₂²	C₂×SD₁₆×C₁₃₁	1	1	yes	characteristic
142	C₂²	C₂×Q₁₆×C₁₃₁	1	1	yes	characteristic
143	C₂²	AGL(1,ℤ/8)×C₁₃₁	1	1	yes	characteristic
144	C₂²	#137×C₁₃₁	1	1	yes	characteristic
145	C₄	—	2	2	no	—
146	C₄	—	2	2	no	—
147	C₄	—	4	2	no	—
148	C₄	—	8	2	no	—
149	C₄	—	8	1	no	—
150	C₄	#123×C₁₃₁	1	2	yes	—
151	C₄	#126×C₁₃₁	1	2	yes	—
152	C₂	#397×C₁₃₁	1	1	yes	characteristic
153	C₂	#460×C₁₃₁	1	1	yes	characteristic
154	C₂	#462×C₁₃₁	1	1	yes	characteristic
155	C₂	#483×C₁₃₁	1	1	yes	characteristic
156	C₂	#484×C₁₃₁	1	1	yes	characteristic
157	C₂	#487×C₁₃₁	1	1	yes	characteristic
158	C₂	#489×C₁₃₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#103×C₁₃₁

C₂

yes

maximal

#391×C₁₃₁

C₂

yes

maximal, characteristic

#384×C₁₃₁

C₂

yes

maximal, characteristic

#340×C₁₃₁

C₂

yes

maximal

#336×C₁₃₁

C₂

yes

maximal, characteristic

C₂²×Q₈×C₁₃₁

C₂²

yes

characteristic

C₂²×C₈×C₁₃₁

C₂²

yes

—

C₂×C₄⋊C₄×C₁₃₁

—

C₂×C₄⋊C₄×C₁₃₁

—

C₂×C₄⋊C₄×C₁₃₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₃₁

C₂²

yes

—

C₂×C₄²×C₁₃₁

C₂²

yes

characteristic

#103×C₁₃₁

—

#95×C₁₃₁

—

#95×C₁₃₁

—

C₂×Q₈×C₁₃₁

—

C₂×Q₈×C₁₃₁

—

C₂×Q₈×C₁₃₁

—

C₂×Q₈×C₁₃₁

D₄

yes

—

C₂²×C₄×C₁₃₁

—

C₂²×C₄×C₁₃₁

—

C₂²×C₄×C₁₃₁

—

C₂²×C₄×C₁₃₁

D₄

yes

characteristic

C₂²×C₄×C₁₃₁

D₄

yes

characteristic

C₂²×C₄×C₁₃₁

C₂³

yes

characteristic

C₂×C₈×C₁₃₁

—

C₂×C₈×C₁₃₁

—

C₂×C₈×C₁₃₁

—

C₄⋊C₄×C₁₃₁

—

C₄⋊C₄×C₁₃₁

—

C₄⋊C₄×C₁₃₁

—

C₄⋊C₄×C₁₃₁

—

C₄⋊C₄×C₁₃₁

—

C₄²×C₁₃₁

—

C₂³×C₁₃₁

C₂×D₄

yes

characteristic

Q₈×C₁₃₁

—

Q₈×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

SD₁₆

yes

—

C₂×C₄×C₁₃₁

Q₁₆

yes

—

C₂×C₄×C₁₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₁

#41

yes

—

C₂×C₄×C₁₃₁

#41

yes

—

C₂×C₄×C₁₃₁

#41

yes

characteristic

C₈×C₁₃₁

—

C₂²×C₁₃₁

yes

C₂²×C₁₃₁

yes

C₂²×C₁₃₁

yes

C₂²×C₁₃₁

C₂×SD₁₆

yes

characteristic

C₂²×C₁₃₁

C₂×Q₁₆

yes

C₂²×C₁₃₁

yes

C₂²×C₁₃₁

yes

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

#123

yes

—

C₄×C₁₃₁

#126

yes

—

C₂×C₁₃₁

yes

C₂×C₁₃₁

yes

C₂×C₁₃₁

yes

C₂×C₁₃₁

yes

C₂×C₁₃₁

yes

C₂×C₁₃₁

yes

C₂×C₁₃₁

yes

C₁₃₁

yes

characteristic, direct product factor, Sylow

#2072

C₁₃₁

yes

maximal, characteristic, direct product factor, Sylow

C₂×#103

C₂×C₁₃₁

yes

—

#391

C₂×C₁₃₁

yes

characteristic

#384

C₂×C₁₃₁

yes

characteristic

#340

C₂×C₁₃₁

yes

—

#336

C₂×C₁₃₁

yes

characteristic

C₂²×Q₈

C₂²×C₁₃₁

yes

characteristic

C₂²×C₈

C₂²×C₁₃₁

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²×C₁₃₁

yes

characteristic

C₂×C₄⋊C₄

C₂²×C₁₃₁

yes

—

C₂×C₄²

C₂²×C₁₃₁

yes

—

—

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

D₄×C₁₃₁

yes

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

D₄×C₁₃₁

yes

characteristic

104

C₂²×C₄

D₄×C₁₃₁

yes

characteristic

105

C₂²×C₄

C₂³×C₁₃₁

yes

characteristic

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄²

—

115

C₂³

C₂×D₄×C₁₃₁

yes

characteristic

116

Q₈

—

117

Q₈

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

SD₁₆×C₁₃₁

yes

—

132

C₂×C₄

Q₁₆×C₁₃₁

yes

—

133

C₂×C₄

C₂×D₄×C₁₃₁

yes

characteristic

134

C₂×C₄

#41×C₁₃₁

yes

—

135

C₂×C₄

#41×C₁₃₁

yes

—

136

C₂×C₄

#41×C₁₃₁

yes

characteristic

137

C₈

—

138

C₂²

#121×C₁₃₁

yes

characteristic

139

C₂²

#123×C₁₃₁

yes

characteristic

140

C₂²

#124×C₁₃₁

yes

characteristic

141

C₂²

C₂×SD₁₆×C₁₃₁

yes

characteristic

142

C₂²

C₂×Q₁₆×C₁₃₁

yes

characteristic

143

C₂²

AGL(1,ℤ/8)×C₁₃₁

yes

characteristic

144

C₂²

#137×C₁₃₁

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

#123×C₁₃₁

yes

—

151

C₄

#126×C₁₃₁

yes

—

152

C₂

#397×C₁₃₁

yes

characteristic

153

C₂

#460×C₁₃₁

yes

characteristic

154

C₂

#462×C₁₃₁

yes

characteristic

155

C₂

#483×C₁₃₁

yes

characteristic

156

C₂

#484×C₁₃₁

yes

characteristic

157

C₂

#487×C₁₃₁

yes

characteristic

158

C₂

#489×C₁₃₁

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃₁	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₁₃₁
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₁ ◁ C₂×C₄²×C₁₃₁ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₁₃₁

C₂×D₄

commutator, Frattini, cosocle

C₂²×C₄

C₂³×C₁₃₁

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₁₃₁ ◁ C₂×C₄²×C₁₃₁ ◁ G

N	H ≅ G/N	type
79. C₁₃₁	#2072	direct (N × H)
80. #2072	C₁₃₁	direct (N × H)

H ≅ G/N

type

79. C₁₃₁

#2072

direct (N × H)

80. #2072

C₁₃₁

direct (N × H)

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

subgroup	index
C₂×#103×C₁₃₁	2
#391×C₁₃₁	2
#384×C₁₃₁	2
#340×C₁₃₁	2
#336×C₁₃₁	2
#2072	131

subgroup

index

C₂×#103×C₁₃₁

#391×C₁₃₁

#384×C₁₃₁

#340×C₁₃₁

#336×C₁₃₁

#2072

131

Subgroups — #2072×C₁₃₁ #30662255

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2072×C131 #30662255

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2072×C₁₃₁ #30662255