Group #28332701 — subgroups

Subgroups — C₁₀₇×#2018 #28332701

Order 13696.

382 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 92 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×AGL(1,ℤ/8)×C₁₀₇	C₂	1	2	yes	maximal, complemented
2	C₂²×D₈×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#100×C₁₀₇	C₂	1	2	yes	maximal, complemented
4	#408×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#341×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#41×C₁₀₇	C₂²	1	1	yes	characteristic
7	C₂²×D₄×C₁₀₇	C₂²	1	2	yes	—
8	AGL(1,ℤ/8)×C₁₀₇	—	2	4	no	—
9	AGL(1,ℤ/8)×C₁₀₇	—	2	4	no	—
10	C₂×SD₁₆×C₁₀₇	—	2	2	no	—
11	C₂×D₈×C₁₀₇	—	2	4	no	—
12	C₂×D₈×C₁₀₇	—	2	2	no	—
13	C₂×D₈×C₁₀₇	—	2	2	no	—
14	C₂×M₁₆×C₁₀₇	C₂²	1	1	yes	characteristic
15	C₂×M₁₆×C₁₀₇	C₂²	1	2	yes	—
16	C₂²×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
17	#100×C₁₀₇	—	2	4	no	—
18	C₂²⋊C₈×C₁₀₇	—	2	2	no	—
19	C₂⁴×C₁₀₇	—	2	2	no	—
20	#41×C₁₀₇	—	2	2	no	—
21	#41×C₁₀₇	—	2	2	no	—
22	C₂×Q₈×C₁₀₇	D₄	1	1	yes	characteristic
23	C₂×D₄×C₁₀₇	—	2	1	no	—
24	C₂×D₄×C₁₀₇	—	2	2	no	—
25	C₂×D₄×C₁₀₇	—	2	2	no	—
26	C₂×D₄×C₁₀₇	—	2	2	no	—
27	C₂×D₄×C₁₀₇	—	2	2	no	—
28	C₂×D₄×C₁₀₇	—	2	2	no	—
29	C₂×D₄×C₁₀₇	D₄	1	1	yes	characteristic
30	C₂×D₄×C₁₀₇	D₄	1	4	yes	—
31	C₂²×C₄×C₁₀₇	—	2	1	no	—
32	C₂²×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
33	SD₁₆×C₁₀₇	—	2	8	no	—
34	D₈×C₁₀₇	—	2	8	no	—
35	D₈×C₁₀₇	—	2	8	no	—
36	M₁₆×C₁₀₇	—	2	2	no	—
37	M₁₆×C₁₀₇	—	2	2	no	—
38	M₁₆×C₁₀₇	D₄	1	4	yes	—
39	C₂×C₈×C₁₀₇	—	2	2	no	—
40	C₂×C₈×C₁₀₇	—	2	2	no	—
41	C₂×C₈×C₁₀₇	—	2	1	no	—
42	C₂×C₈×C₁₀₇	D₄	1	2	yes	—
43	C₂³×C₁₀₇	—	2	1	no	—
44	C₂³×C₁₀₇	—	2	4	no	—
45	C₂³×C₁₀₇	—	4	2	no	—
46	C₂³×C₁₀₇	—	4	2	no	—
47	C₂³×C₁₀₇	—	8	4	no	—
48	C₂³×C₁₀₇	#41	1	1	yes	characteristic
49	Q₈×C₁₀₇	—	2	2	no	—
50	D₄×C₁₀₇	—	2	4	no	—
51	D₄×C₁₀₇	—	2	2	no	—
52	D₄×C₁₀₇	—	2	4	no	—
53	D₄×C₁₀₇	—	4	2	no	—
54	D₄×C₁₀₇	—	4	2	no	—
55	D₄×C₁₀₇	—	4	2	no	—
56	C₂×C₄×C₁₀₇	—	2	1	no	—
57	C₂×C₄×C₁₀₇	—	4	1	no	—
58	C₂×C₄×C₁₀₇	—	4	1	no	—
59	C₂×C₄×C₁₀₇	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₁₀₇	C₂×D₄	1	2	yes	—
62	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
63	C₈×C₁₀₇	—	2	2	no	—
64	C₈×C₁₀₇	—	2	4	no	—
65	C₈×C₁₀₇	—	4	1	no	—
66	C₂²×C₁₀₇	—	2	2	no	—
67	C₂²×C₁₀₇	—	4	1	no	—
68	C₂²×C₁₀₇	—	4	4	no	—
69	C₂²×C₁₀₇	—	4	2	no	—
70	C₂²×C₁₀₇	—	8	4	no	—
71	C₂²×C₁₀₇	—	8	4	no	—
72	C₂²×C₁₀₇	—	8	4	no	—
73	C₂²×C₁₀₇	#121	1	2	yes	—
74	C₂²×C₁₀₇	#121	1	1	yes	characteristic
75	C₄×C₁₀₇	—	4	1	no	—
76	C₄×C₁₀₇	#120	1	1	yes	characteristic
77	C₄×C₁₀₇	#120	1	2	yes	—
78	C₄×C₁₀₇	#127	1	1	yes	characteristic
79	C₂×C₁₀₇	—	2	2	no	—
80	C₂×C₁₀₇	—	8	1	no	—
81	C₂×C₁₀₇	—	8	4	no	—
82	C₂×C₁₀₇	#392	1	1	yes	characteristic
83	C₂×C₁₀₇	#472	1	2	yes	—
84	#2018	C₁₀₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
85	C₁₀₇	#2018	1	1	yes	characteristic, direct product factor, Sylow
86	C₂×AGL(1,ℤ/8)	C₂×C₁₀₇	1	2	yes	complemented
87	C₂²×D₈	C₂×C₁₀₇	1	1	yes	characteristic, complemented
88	C₂×#100	C₂×C₁₀₇	1	2	yes	complemented
89	#408	C₂×C₁₀₇	1	1	yes	characteristic, complemented
90	#341	C₂×C₁₀₇	1	1	yes	characteristic, complemented
91	C₂×#41	C₂²×C₁₀₇	1	1	yes	characteristic
92	C₂²×D₄	C₂²×C₁₀₇	1	2	yes	—
93	AGL(1,ℤ/8)	—	2	4	no	—
94	AGL(1,ℤ/8)	—	2	4	no	—
95	C₂×SD₁₆	—	2	2	no	—
96	C₂×D₈	—	2	4	no	—
97	C₂×D₈	—	2	2	no	—
98	C₂×D₈	—	2	2	no	—
99	C₂×M₁₆	C₂²×C₁₀₇	1	1	yes	characteristic
100	C₂×M₁₆	C₂²×C₁₀₇	1	2	yes	—
101	C₂²×C₈	C₂²×C₁₀₇	1	1	yes	characteristic
102	#100	—	2	4	no	—
103	C₂²⋊C₈	—	2	2	no	—
104	C₂⁴	—	2	2	no	—
105	#41	—	2	2	no	—
106	#41	—	2	2	no	—
107	C₂×Q₈	D₄×C₁₀₇	1	1	yes	characteristic
108	C₂×D₄	—	2	1	no	—
109	C₂×D₄	—	2	2	no	—
110	C₂×D₄	—	2	2	no	—
111	C₂×D₄	—	2	2	no	—
112	C₂×D₄	—	2	2	no	—
113	C₂×D₄	—	2	2	no	—
114	C₂×D₄	D₄×C₁₀₇	1	1	yes	characteristic
115	C₂×D₄	D₄×C₁₀₇	1	4	yes	—
116	C₂²×C₄	—	2	1	no	—
117	C₂²×C₄	C₂³×C₁₀₇	1	1	yes	characteristic
118	SD₁₆	—	2	8	no	—
119	D₈	—	2	8	no	—
120	D₈	—	2	8	no	—
121	M₁₆	—	2	2	no	—
122	M₁₆	—	2	2	no	—
123	M₁₆	D₄×C₁₀₇	1	4	yes	—
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	D₄×C₁₀₇	1	2	yes	—
128	C₂³	—	2	1	no	—
129	C₂³	—	2	4	no	—
130	C₂³	—	4	2	no	—
131	C₂³	—	4	2	no	—
132	C₂³	—	8	4	no	—
133	C₂³	#41×C₁₀₇	1	1	yes	characteristic
134	Q₈	—	2	2	no	—
135	D₄	—	2	4	no	—
136	D₄	—	2	2	no	—
137	D₄	—	2	4	no	—
138	D₄	—	4	2	no	—
139	D₄	—	4	2	no	—
140	D₄	—	4	2	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	4	1	no	—
143	C₂×C₄	—	4	1	no	—
144	C₂×C₄	C₂×D₄×C₁₀₇	1	1	yes	characteristic
145	C₂×C₄	C₂×D₄×C₁₀₇	1	2	yes	—
146	C₂×C₄	C₂×D₄×C₁₀₇	1	1	yes	characteristic
147	C₂×C₄	C₂×D₄×C₁₀₇	1	2	yes	—
148	C₈	—	2	2	no	—
149	C₈	—	2	4	no	—
150	C₈	—	4	1	no	—
151	C₂²	—	2	2	no	—
152	C₂²	—	4	1	no	—
153	C₂²	—	4	4	no	—
154	C₂²	—	4	2	no	—
155	C₂²	—	8	4	no	—
156	C₂²	—	8	4	no	—
157	C₂²	—	8	4	no	—
158	C₂²	#121×C₁₀₇	1	2	yes	—
159	C₂²	#121×C₁₀₇	1	1	yes	characteristic
160	C₄	—	4	1	no	—
161	C₄	#120×C₁₀₇	1	1	yes	characteristic
162	C₄	#120×C₁₀₇	1	2	yes	—
163	C₄	#127×C₁₀₇	1	1	yes	characteristic
164	C₂	—	2	2	no	—
165	C₂	—	8	1	no	—
166	C₂	—	8	4	no	—
167	C₂	#392×C₁₀₇	1	1	yes	characteristic
168	C₂	#472×C₁₀₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×AGL(1,ℤ/8)×C₁₀₇

C₂

yes

maximal, complemented

C₂²×D₈×C₁₀₇

C₂

yes

maximal, characteristic, complemented

C₂×#100×C₁₀₇

C₂

yes

maximal, complemented

#408×C₁₀₇

C₂

yes

maximal, characteristic, complemented

#341×C₁₀₇

C₂

yes

maximal, characteristic, complemented

C₂×#41×C₁₀₇

C₂²

yes

characteristic

C₂²×D₄×C₁₀₇

C₂²

yes

—

AGL(1,ℤ/8)×C₁₀₇

—

AGL(1,ℤ/8)×C₁₀₇

—

C₂×SD₁₆×C₁₀₇

—

C₂×D₈×C₁₀₇

—

C₂×D₈×C₁₀₇

—

C₂×D₈×C₁₀₇

—

C₂×M₁₆×C₁₀₇

C₂²

yes

characteristic

C₂×M₁₆×C₁₀₇

C₂²

yes

—

C₂²×C₈×C₁₀₇

C₂²

yes

characteristic

#100×C₁₀₇

—

C₂²⋊C₈×C₁₀₇

—

C₂⁴×C₁₀₇

—

#41×C₁₀₇

—

#41×C₁₀₇

—

C₂×Q₈×C₁₀₇

D₄

yes

characteristic

C₂×D₄×C₁₀₇

—

C₂×D₄×C₁₀₇

—

C₂×D₄×C₁₀₇

—

C₂×D₄×C₁₀₇

—

C₂×D₄×C₁₀₇

—

C₂×D₄×C₁₀₇

—

C₂×D₄×C₁₀₇

D₄

yes

characteristic

C₂×D₄×C₁₀₇

D₄

yes

—

C₂²×C₄×C₁₀₇

—

C₂²×C₄×C₁₀₇

C₂³

yes

characteristic

SD₁₆×C₁₀₇

—

D₈×C₁₀₇

—

D₈×C₁₀₇

—

M₁₆×C₁₀₇

—

M₁₆×C₁₀₇

—

M₁₆×C₁₀₇

D₄

yes

—

C₂×C₈×C₁₀₇

—

C₂×C₈×C₁₀₇

—

C₂×C₈×C₁₀₇

—

C₂×C₈×C₁₀₇

D₄

yes

—

C₂³×C₁₀₇

—

C₂³×C₁₀₇

—

C₂³×C₁₀₇

—

C₂³×C₁₀₇

—

C₂³×C₁₀₇

—

C₂³×C₁₀₇

#41

yes

characteristic

Q₈×C₁₀₇

—

D₄×C₁₀₇

—

D₄×C₁₀₇

—

D₄×C₁₀₇

—

D₄×C₁₀₇

—

D₄×C₁₀₇

—

D₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

C₂×D₄

yes

—

C₂×C₄×C₁₀₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₇

C₂×D₄

yes

—

C₂×C₄×C₁₀₇

C₂×D₄

yes

characteristic

C₈×C₁₀₇

—

C₈×C₁₀₇

—

C₈×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

#121

yes

—

C₂²×C₁₀₇

#121

yes

characteristic

C₄×C₁₀₇

—

C₄×C₁₀₇

#120

yes

characteristic

C₄×C₁₀₇

#120

yes

—

C₄×C₁₀₇

#127

yes

characteristic

C₂×C₁₀₇

—

C₂×C₁₀₇

—

C₂×C₁₀₇

—

C₂×C₁₀₇

#392

yes

characteristic

C₂×C₁₀₇

#472

yes

—

#2018

C₁₀₇

yes

maximal, characteristic, direct product factor, Sylow

C₁₀₇

#2018

yes

characteristic, direct product factor, Sylow

C₂×AGL(1,ℤ/8)

C₂×C₁₀₇

yes

complemented

C₂²×D₈

C₂×C₁₀₇

yes

characteristic, complemented

C₂×#100

C₂×C₁₀₇

yes

complemented

#408

C₂×C₁₀₇

yes

characteristic, complemented

#341

C₂×C₁₀₇

yes

characteristic, complemented

C₂×#41

C₂²×C₁₀₇

yes

characteristic

C₂²×D₄

C₂²×C₁₀₇

yes

—

AGL(1,ℤ/8)

—

AGL(1,ℤ/8)

—

C₂×SD₁₆

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×M₁₆

C₂²×C₁₀₇

yes

characteristic

100

C₂×M₁₆

C₂²×C₁₀₇

yes

—

101

C₂²×C₈

C₂²×C₁₀₇

yes

characteristic

102

#100

—

103

C₂²⋊C₈

—

104

C₂⁴

—

105

#41

—

106

#41

—

107

C₂×Q₈

D₄×C₁₀₇

yes

characteristic

108

C₂×D₄

—

109

C₂×D₄

—

110

C₂×D₄

—

111

C₂×D₄

—

112

C₂×D₄

—

113

C₂×D₄

—

114

C₂×D₄

D₄×C₁₀₇

yes

characteristic

115

C₂×D₄

D₄×C₁₀₇

yes

—

116

C₂²×C₄

—

117

C₂²×C₄

C₂³×C₁₀₇

yes

characteristic

118

SD₁₆

—

119

D₈

—

120

D₈

—

121

M₁₆

—

122

M₁₆

—

123

M₁₆

D₄×C₁₀₇

yes

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

D₄×C₁₀₇

yes

—

128

C₂³

—

129

C₂³

—

130

C₂³

—

131

C₂³

—

132

C₂³

—

133

C₂³

#41×C₁₀₇

yes

characteristic

134

Q₈

—

135

D₄

—

136

D₄

—

137

D₄

—

138

D₄

—

139

D₄

—

140

D₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

C₂×D₄×C₁₀₇

yes

characteristic

145

C₂×C₄

C₂×D₄×C₁₀₇

yes

—

146

C₂×C₄

C₂×D₄×C₁₀₇

yes

characteristic

147

C₂×C₄

C₂×D₄×C₁₀₇

yes

—

148

C₈

—

149

C₈

—

150

C₈

—

151

C₂²

—

152

C₂²

—

153

C₂²

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

—

158

C₂²

#121×C₁₀₇

yes

—

159

C₂²

#121×C₁₀₇

yes

characteristic

160

C₄

—

161

C₄

#120×C₁₀₇

yes

characteristic

162

C₄

#120×C₁₀₇

yes

—

163

C₄

#127×C₁₀₇

yes

characteristic

164

C₂

—

165

C₂

—

166

C₂

—

167

C₂

#392×C₁₀₇

yes

characteristic

168

C₂

#472×C₁₀₇

yes

—

	subgroup	G/H
center, socle	C₂²×C₁₀₇	#121
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₁₀₇
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₇ ◁ C₂²×C₄×C₁₀₇ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₀₇

#121

commutator, Frattini, cosocle

C₂²×C₄

C₂³×C₁₀₇

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₀₇ ◁ C₂²×C₄×C₁₀₇ ◁ G

N	H ≅ G/N	type
1. C₂×AGL(1,ℤ/8)×C₁₀₇	C₂	semidirect
2. C₂²×D₈×C₁₀₇	C₂	semidirect
3. C₂×#100×C₁₀₇	C₂	semidirect
4. #408×C₁₀₇	C₂	semidirect
5. #341×C₁₀₇	C₂	semidirect
84. #2018	C₁₀₇	direct (N × H)
85. C₁₀₇	#2018	direct (N × H)
86. C₂×AGL(1,ℤ/8)	C₂×C₁₀₇	semidirect
87. C₂²×D₈	C₂×C₁₀₇	semidirect
88. C₂×#100	C₂×C₁₀₇	semidirect
89. #408	C₂×C₁₀₇	semidirect
90. #341	C₂×C₁₀₇	semidirect

H ≅ G/N

type

1. C₂×AGL(1,ℤ/8)×C₁₀₇

C₂

semidirect

2. C₂²×D₈×C₁₀₇

C₂

semidirect

3. C₂×#100×C₁₀₇

C₂

semidirect

4. #408×C₁₀₇

C₂

semidirect

5. #341×C₁₀₇

C₂

semidirect

84. #2018

C₁₀₇

direct (N × H)

85. C₁₀₇

#2018

direct (N × H)

86. C₂×AGL(1,ℤ/8)

C₂×C₁₀₇

semidirect

87. C₂²×D₈

C₂×C₁₀₇

semidirect

88. C₂×#100

C₂×C₁₀₇

semidirect

89. #408

C₂×C₁₀₇

semidirect

90. #341

C₂×C₁₀₇

semidirect

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

subgroup	index
C₂×AGL(1,ℤ/8)×C₁₀₇	2
C₂²×D₈×C₁₀₇	2
C₂×#100×C₁₀₇	2
#408×C₁₀₇	2
#341×C₁₀₇	2
#2018	107

subgroup

index

C₂×AGL(1,ℤ/8)×C₁₀₇

C₂²×D₈×C₁₀₇

C₂×#100×C₁₀₇

#408×C₁₀₇

#341×C₁₀₇

#2018

107

Subgroups — C₁₀₇×#2018 #28332701

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C107×#2018 #28332701

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₇×#2018 #28332701