Group #28332699 — subgroups

Subgroups — C₁₀₇×#2016 #28332699

Order 13696.

370 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 116 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	C₂²×Q₁₆×C₁₀₇	C₂	1	2	yes	maximal
2	C₂×C₄⋊Q₈×C₁₀₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#103×C₁₀₇	C₂	1	2	yes	maximal
4	#391×C₁₀₇	C₂	1	1	yes	maximal, characteristic
5	#336×C₁₀₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₈×C₁₀₇	C₂²	1	1	yes	characteristic
7	C₂²×Q₈×C₁₀₇	C₂²	1	2	yes	—
8	C₂×Q₁₆×C₁₀₇	—	2	4	no	—
9	C₂×Q₁₆×C₁₀₇	—	2	4	no	—
10	C₂×Q₁₆×C₁₀₇	—	2	4	no	—
11	C₂²×C₈×C₁₀₇	C₂²	1	2	yes	—
12	C₄⋊Q₈×C₁₀₇	—	2	4	no	—
13	C₂×C₄⋊C₄×C₁₀₇	—	2	2	no	—
14	C₂×C₄⋊C₄×C₁₀₇	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₁₀₇	C₂²	1	1	yes	characteristic
16	#103×C₁₀₇	—	2	4	no	—
17	#95×C₁₀₇	—	2	2	no	—
18	C₂×Q₈×C₁₀₇	—	2	2	no	—
19	C₂×Q₈×C₁₀₇	—	2	1	no	—
20	C₂×Q₈×C₁₀₇	—	2	2	no	—
21	C₂×Q₈×C₁₀₇	—	2	2	no	—
22	C₂×Q₈×C₁₀₇	—	2	2	no	—
23	C₂×Q₈×C₁₀₇	—	2	2	no	—
24	C₂×Q₈×C₁₀₇	—	2	2	no	—
25	C₂×Q₈×C₁₀₇	—	2	2	no	—
26	C₂×Q₈×C₁₀₇	D₄	1	4	yes	—
27	C₂×Q₈×C₁₀₇	D₄	1	2	yes	—
28	C₂²×C₄×C₁₀₇	—	2	1	no	—
29	C₂²×C₄×C₁₀₇	—	2	2	no	—
30	C₂²×C₄×C₁₀₇	—	2	1	no	—
31	C₂²×C₄×C₁₀₇	D₄	1	2	yes	—
32	C₂²×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
33	Q₁₆×C₁₀₇	—	2	16	no	—
34	C₂×C₈×C₁₀₇	—	2	2	no	—
35	C₂×C₈×C₁₀₇	—	2	2	no	—
36	C₂×C₈×C₁₀₇	D₄	1	4	yes	—
37	C₄⋊C₄×C₁₀₇	—	2	8	no	—
38	C₄⋊C₄×C₁₀₇	—	2	2	no	—
39	C₄²×C₁₀₇	—	2	2	no	—
40	C₂³×C₁₀₇	C₂×D₄	1	1	yes	characteristic
41	Q₈×C₁₀₇	—	2	4	no	—
42	Q₈×C₁₀₇	—	2	4	no	—
43	Q₈×C₁₀₇	—	2	4	no	—
44	Q₈×C₁₀₇	—	4	2	no	—
45	Q₈×C₁₀₇	—	4	2	no	—
46	Q₈×C₁₀₇	—	4	2	no	—
47	C₂×C₄×C₁₀₇	—	2	2	no	—
48	C₂×C₄×C₁₀₇	—	2	2	no	—
49	C₂×C₄×C₁₀₇	—	2	4	no	—
50	C₂×C₄×C₁₀₇	—	2	2	no	—
51	C₂×C₄×C₁₀₇	—	4	1	no	—
52	C₂×C₄×C₁₀₇	—	4	2	no	—
53	C₂×C₄×C₁₀₇	—	4	1	no	—
54	C₂×C₄×C₁₀₇	—	4	2	no	—
55	C₂×C₄×C₁₀₇	—	4	1	no	—
56	C₂×C₄×C₁₀₇	—	4	2	no	—
57	C₂×C₄×C₁₀₇	Q₁₆	1	4	yes	complemented
58	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₁₀₇	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₁₀₇	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₁₀₇	#41	1	1	yes	characteristic
62	C₈×C₁₀₇	—	2	4	no	—
63	C₂²×C₁₀₇	#120	1	1	yes	characteristic
64	C₂²×C₁₀₇	#121	1	1	yes	characteristic
65	C₂²×C₁₀₇	#127	1	1	yes	characteristic
66	C₂²×C₁₀₇	C₂×Q₁₆	1	2	yes	—
67	C₂²×C₁₀₇	#137	1	2	yes	—
68	C₄×C₁₀₇	—	2	4	no	—
69	C₄×C₁₀₇	—	4	4	no	—
70	C₄×C₁₀₇	—	4	2	no	—
71	C₄×C₁₀₇	—	8	1	no	—
72	C₄×C₁₀₇	#120	1	2	yes	—
73	C₄×C₁₀₇	#121	1	2	yes	—
74	C₂×C₁₀₇	#392	1	1	yes	characteristic
75	C₂×C₁₀₇	#451	1	2	yes	—
76	C₂×C₁₀₇	#462	1	2	yes	—
77	C₂×C₁₀₇	#494	1	1	yes	characteristic
78	C₂×C₁₀₇	#497	1	1	yes	characteristic
79	#2016	C₁₀₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
80	C₁₀₇	#2016	1	1	yes	characteristic, direct product factor, Sylow
81	C₂²×Q₁₆	C₂×C₁₀₇	1	2	yes	—
82	C₂×C₄⋊Q₈	C₂×C₁₀₇	1	1	yes	characteristic
83	C₂×#103	C₂×C₁₀₇	1	2	yes	—
84	#391	C₂×C₁₀₇	1	1	yes	characteristic
85	#336	C₂×C₁₀₇	1	1	yes	characteristic
86	C₂²×Q₈	C₂²×C₁₀₇	1	2	yes	—
87	C₂²×Q₈	C₂²×C₁₀₇	1	1	yes	characteristic
88	C₂×Q₁₆	—	2	4	no	—
89	C₂×Q₁₆	—	2	4	no	—
90	C₂×Q₁₆	—	2	4	no	—
91	C₂²×C₈	C₂²×C₁₀₇	1	2	yes	—
92	C₄⋊Q₈	—	2	4	no	—
93	C₂×C₄⋊C₄	—	2	2	no	—
94	C₂×C₄⋊C₄	C₂²×C₁₀₇	1	1	yes	characteristic
95	C₂×C₄²	C₂²×C₁₀₇	1	1	yes	characteristic
96	#103	—	2	4	no	—
97	#95	—	2	2	no	—
98	C₂×Q₈	—	2	2	no	—
99	C₂×Q₈	—	2	1	no	—
100	C₂×Q₈	—	2	2	no	—
101	C₂×Q₈	—	2	2	no	—
102	C₂×Q₈	—	2	2	no	—
103	C₂×Q₈	—	2	2	no	—
104	C₂×Q₈	—	2	2	no	—
105	C₂×Q₈	—	2	2	no	—
106	C₂×Q₈	D₄×C₁₀₇	1	4	yes	—
107	C₂×Q₈	D₄×C₁₀₇	1	2	yes	—
108	C₂²×C₄	—	2	1	no	—
109	C₂²×C₄	—	2	2	no	—
110	C₂²×C₄	—	2	1	no	—
111	C₂²×C₄	D₄×C₁₀₇	1	2	yes	—
112	C₂²×C₄	C₂³×C₁₀₇	1	1	yes	characteristic
113	Q₁₆	—	2	16	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	D₄×C₁₀₇	1	4	yes	—
117	C₄⋊C₄	—	2	8	no	—
118	C₄⋊C₄	—	2	2	no	—
119	C₄²	—	2	2	no	—
120	C₂³	C₂×D₄×C₁₀₇	1	1	yes	characteristic
121	Q₈	—	2	4	no	—
122	Q₈	—	2	4	no	—
123	Q₈	—	2	4	no	—
124	Q₈	—	4	2	no	—
125	Q₈	—	4	2	no	—
126	Q₈	—	4	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	4	1	no	—
132	C₂×C₄	—	4	2	no	—
133	C₂×C₄	—	4	1	no	—
134	C₂×C₄	—	4	2	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	—	4	2	no	—
137	C₂×C₄	Q₁₆×C₁₀₇	1	4	yes	complemented
138	C₂×C₄	C₂×D₄×C₁₀₇	1	1	yes	characteristic
139	C₂×C₄	C₂×D₄×C₁₀₇	1	2	yes	—
140	C₂×C₄	C₂×D₄×C₁₀₇	1	2	yes	—
141	C₂×C₄	#41×C₁₀₇	1	1	yes	characteristic
142	C₈	—	2	4	no	—
143	C₂²	#120×C₁₀₇	1	1	yes	characteristic
144	C₂²	#121×C₁₀₇	1	1	yes	characteristic
145	C₂²	#127×C₁₀₇	1	1	yes	characteristic
146	C₂²	C₂×Q₁₆×C₁₀₇	1	2	yes	—
147	C₂²	#137×C₁₀₇	1	2	yes	—
148	C₄	—	2	4	no	—
149	C₄	—	4	4	no	—
150	C₄	—	4	2	no	—
151	C₄	—	8	1	no	—
152	C₄	#120×C₁₀₇	1	2	yes	—
153	C₄	#121×C₁₀₇	1	2	yes	—
154	C₂	#392×C₁₀₇	1	1	yes	characteristic
155	C₂	#451×C₁₀₇	1	2	yes	—
156	C₂	#462×C₁₀₇	1	2	yes	—
157	C₂	#494×C₁₀₇	1	1	yes	characteristic
158	C₂	#497×C₁₀₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Q₁₆×C₁₀₇

C₂

yes

maximal

C₂×C₄⋊Q₈×C₁₀₇

C₂

yes

maximal, characteristic

C₂×#103×C₁₀₇

C₂

yes

maximal

#391×C₁₀₇

C₂

yes

maximal, characteristic

#336×C₁₀₇

C₂

yes

maximal, characteristic

C₂²×Q₈×C₁₀₇

C₂²

yes

characteristic

C₂²×Q₈×C₁₀₇

C₂²

yes

—

C₂×Q₁₆×C₁₀₇

—

C₂×Q₁₆×C₁₀₇

—

C₂×Q₁₆×C₁₀₇

—

C₂²×C₈×C₁₀₇

C₂²

yes

—

C₄⋊Q₈×C₁₀₇

—

C₂×C₄⋊C₄×C₁₀₇

—

C₂×C₄⋊C₄×C₁₀₇

C₂²

yes

characteristic

C₂×C₄²×C₁₀₇

C₂²

yes

characteristic

#103×C₁₀₇

—

#95×C₁₀₇

—

C₂×Q₈×C₁₀₇

—

C₂×Q₈×C₁₀₇

—

C₂×Q₈×C₁₀₇

—

C₂×Q₈×C₁₀₇

—

C₂×Q₈×C₁₀₇

—

C₂×Q₈×C₁₀₇

—

C₂×Q₈×C₁₀₇

—

C₂×Q₈×C₁₀₇

—

C₂×Q₈×C₁₀₇

D₄

yes

—

C₂×Q₈×C₁₀₇

D₄

yes

—

C₂²×C₄×C₁₀₇

—

C₂²×C₄×C₁₀₇

—

C₂²×C₄×C₁₀₇

—

C₂²×C₄×C₁₀₇

D₄

yes

—

C₂²×C₄×C₁₀₇

C₂³

yes

characteristic

Q₁₆×C₁₀₇

—

C₂×C₈×C₁₀₇

—

C₂×C₈×C₁₀₇

—

C₂×C₈×C₁₀₇

D₄

yes

—

C₄⋊C₄×C₁₀₇

—

C₄⋊C₄×C₁₀₇

—

C₄²×C₁₀₇

—

C₂³×C₁₀₇

C₂×D₄

yes

characteristic

Q₈×C₁₀₇

—

Q₈×C₁₀₇

—

Q₈×C₁₀₇

—

Q₈×C₁₀₇

—

Q₈×C₁₀₇

—

Q₈×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

Q₁₆

yes

complemented

C₂×C₄×C₁₀₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₇

C₂×D₄

yes

—

C₂×C₄×C₁₀₇

C₂×D₄

yes

—

C₂×C₄×C₁₀₇

#41

yes

characteristic

C₈×C₁₀₇

—

C₂²×C₁₀₇

yes

C₂²×C₁₀₇

yes

C₂²×C₁₀₇

yes

C₂²×C₁₀₇

C₂×Q₁₆

yes

—

C₂²×C₁₀₇

#137

yes

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

#120

yes

—

C₄×C₁₀₇

#121

yes

—

C₂×C₁₀₇

#392

yes

characteristic

C₂×C₁₀₇

#451

yes

—

C₂×C₁₀₇

#462

yes

—

C₂×C₁₀₇

yes

C₂×C₁₀₇

yes

C₁₀₇

yes

maximal, characteristic, direct product factor, Sylow

C₁₀₇

#2016

yes

characteristic, direct product factor, Sylow

C₂²×Q₁₆

C₂×C₁₀₇

yes

—

C₂×C₄⋊Q₈

C₂×C₁₀₇

yes

characteristic

C₂×#103

C₂×C₁₀₇

yes

—

#391

C₂×C₁₀₇

yes

characteristic

#336

C₂×C₁₀₇

yes

characteristic

C₂²×Q₈

C₂²×C₁₀₇

yes

—

C₂²×Q₈

C₂²×C₁₀₇

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂²×C₈

C₂²×C₁₀₇

yes

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²×C₁₀₇

yes

characteristic

C₂×C₄²

C₂²×C₁₀₇

yes

characteristic

#103

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×Q₈

D₄×C₁₀₇

yes

—

107

C₂×Q₈

D₄×C₁₀₇

yes

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

D₄×C₁₀₇

yes

—

112

C₂²×C₄

C₂³×C₁₀₇

yes

characteristic

113

Q₁₆

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

D₄×C₁₀₇

yes

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄²

—

120

C₂³

C₂×D₄×C₁₀₇

yes

characteristic

121

Q₈

—

122

Q₈

—

123

Q₈

—

124

Q₈

—

125

Q₈

—

126

Q₈

—

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×C₁₀₇

yes

complemented

138

C₂×C₄

C₂×D₄×C₁₀₇

yes

characteristic

139

C₂×C₄

C₂×D₄×C₁₀₇

yes

—

140

C₂×C₄

C₂×D₄×C₁₀₇

yes

—

141

C₂×C₄

#41×C₁₀₇

yes

characteristic

142

C₈

—

143

C₂²

#120×C₁₀₇

yes

characteristic

144

C₂²

#121×C₁₀₇

yes

characteristic

145

C₂²

#127×C₁₀₇

yes

characteristic

146

C₂²

C₂×Q₁₆×C₁₀₇

yes

—

147

C₂²

#137×C₁₀₇

yes

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

#120×C₁₀₇

yes

—

153

C₄

#121×C₁₀₇

yes

—

154

C₂

#392×C₁₀₇

yes

characteristic

155

C₂

#451×C₁₀₇

yes

—

156

C₂

#462×C₁₀₇

yes

—

157

C₂

#494×C₁₀₇

yes

characteristic

158

C₂

#497×C₁₀₇

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₀₇	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₁₀₇
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₇ ◁ C₂×C₄²×C₁₀₇ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₁₀₇

C₂×D₄

commutator, Frattini, cosocle

C₂²×C₄

C₂³×C₁₀₇

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₁₀₇ ◁ C₂×C₄²×C₁₀₇ ◁ G

N	H ≅ G/N	type
57. C₂×C₄×C₁₀₇	Q₁₆	semidirect
79. #2016	C₁₀₇	direct (N × H)
80. C₁₀₇	#2016	direct (N × H)
137. C₂×C₄	Q₁₆×C₁₀₇	semidirect

H ≅ G/N

type

57. C₂×C₄×C₁₀₇

Q₁₆

semidirect

79. #2016

C₁₀₇

direct (N × H)

80. C₁₀₇

#2016

direct (N × H)

137. C₂×C₄

Q₁₆×C₁₀₇

semidirect

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

subgroup	index
C₂²×Q₁₆×C₁₀₇	2
C₂×C₄⋊Q₈×C₁₀₇	2
C₂×#103×C₁₀₇	2
#391×C₁₀₇	2
#336×C₁₀₇	2
#2016	107

subgroup

index

C₂²×Q₁₆×C₁₀₇

C₂×C₄⋊Q₈×C₁₀₇

C₂×#103×C₁₀₇

#391×C₁₀₇

#336×C₁₀₇

#2016

107

Subgroups — C₁₀₇×#2016 #28332699

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C107×#2016 #28332699

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₇×#2016 #28332699