Group #41033080 — subgroups

Subgroups — C₁₀₇×#7116 #41033080

Order 27392.

282 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 74 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	C₁₀₇×#1961	C₂	1	1	yes	maximal, characteristic
2	C₁₀₇×#1375	C₂	1	2	yes	maximal, complemented
3	C₂×#127×C₁₀₇	C₄	1	1	yes	characteristic
4	C₂×M₃₂×C₁₀₇	—	2	2	no	—
5	C₂×#102×C₁₀₇	—	2	2	no	—
6	C₂×#97×C₁₀₇	C₂²	1	1	yes	characteristic
7	#382×C₁₀₇	C₄	1	1	yes	characteristic
8	C₂²×D₄×C₁₀₇	—	2	1	no	—
9	C₂²×D₄×C₁₀₇	—	4	1	no	—
10	C₂²×C₈×C₁₀₇	D₄	1	2	yes	—
11	#127×C₁₀₇	—	2	1	no	—
12	#127×C₁₀₇	—	4	1	no	—
13	#127×C₁₀₇	C₈	1	2	yes	—
14	C₂×C₄⋊C₄×C₁₀₇	—	2	1	no	—
15	C₂×C₄²×C₁₀₇	C₂×C₄	1	1	yes	characteristic
16	M₃₂×C₁₀₇	—	4	2	no	—
17	M₃₂×C₁₀₇	—	4	2	no	—
18	C₂×C₁₆×C₁₀₇	—	2	4	no	—
19	#102×C₁₀₇	—	4	2	no	—
20	#102×C₁₀₇	—	4	2	no	—
21	#97×C₁₀₇	—	4	1	no	—
22	#95×C₁₀₇	—	4	1	no	—
23	C₂⁴×C₁₀₇	—	4	1	no	—
24	C₂×D₄×C₁₀₇	—	2	2	no	—
25	C₂×D₄×C₁₀₇	—	4	2	no	—
26	C₂×D₄×C₁₀₇	—	4	2	no	—
27	C₂×D₄×C₁₀₇	—	4	1	no	—
28	C₂×D₄×C₁₀₇	—	4	2	no	—
29	C₂×D₄×C₁₀₇	—	4	2	no	—
30	C₂×D₄×C₁₀₇	—	8	4	no	—
31	C₂²×C₄×C₁₀₇	—	2	1	no	—
32	C₂²×C₄×C₁₀₇	—	4	1	no	—
33	C₂²×C₄×C₁₀₇	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₈×C₁₀₇	—	2	2	no	—
35	C₂×C₈×C₁₀₇	—	2	2	no	—
36	C₂×C₈×C₁₀₇	D₈	1	2	yes	—
37	C₂×C₈×C₁₀₇	SD₁₆	1	2	yes	—
38	C₄⋊C₄×C₁₀₇	—	4	2	no	—
39	C₄²×C₁₀₇	—	2	1	no	—
40	C₄²×C₁₀₇	C₂×C₈	1	1	yes	characteristic
41	C₄²×C₁₀₇	M₁₆	1	1	yes	characteristic
42	C₁₆×C₁₀₇	—	4	4	no	—
43	C₂³×C₁₀₇	—	4	2	no	—
44	C₂³×C₁₀₇	—	8	4	no	—
45	C₂³×C₁₀₇	—	8	1	no	—
46	C₂³×C₁₀₇	—	8	1	no	—
47	C₂³×C₁₀₇	#100	1	1	yes	characteristic
48	D₄×C₁₀₇	—	4	4	no	—
49	D₄×C₁₀₇	—	8	4	no	—
50	D₄×C₁₀₇	—	8	4	no	—
51	D₄×C₁₀₇	—	8	2	no	—
52	C₂×C₄×C₁₀₇	—	2	1	no	—
53	C₂×C₄×C₁₀₇	—	2	1	no	—
54	C₂×C₄×C₁₀₇	—	4	2	no	—
55	C₂×C₄×C₁₀₇	—	8	2	no	—
56	C₂×C₄×C₁₀₇	—	8	1	no	—
57	C₂×C₄×C₁₀₇	C₂²⋊C₈	1	1	yes	characteristic
58	C₂×C₄×C₁₀₇	#99	1	1	yes	characteristic
59	C₂×C₄×C₁₀₇	#102	1	2	yes	—
60	C₂×C₄×C₁₀₇	#104	1	2	yes	—
61	C₈×C₁₀₇	—	2	2	no	—
62	C₈×C₁₀₇	—	2	2	no	—
63	C₂²×C₁₀₇	—	2	2	no	—
64	C₂²×C₁₀₇	—	8	4	no	—
65	C₂²×C₁₀₇	—	8	2	no	—
66	C₂²×C₁₀₇	—	16	2	no	—
67	C₂²×C₁₀₇	—	16	2	no	—
68	C₂²×C₁₀₇	#323	1	1	yes	characteristic
69	C₂²×C₁₀₇	#329	1	1	yes	characteristic
70	C₂²×C₁₀₇	#331	1	1	yes	characteristic
71	C₄×C₁₀₇	—	4	1	no	—
72	C₄×C₁₀₇	—	4	1	no	—
73	C₄×C₁₀₇	—	16	1	no	—
74	C₄×C₁₀₇	D₄⋊C₈	1	2	yes	—
75	C₄×C₁₀₇	#327	1	2	yes	—
76	#7116	C₁₀₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₂×C₁₀₇	—	2	1	no	—
78	C₂×C₁₀₇	—	2	1	no	—
79	C₂×C₁₀₇	—	16	2	no	—
80	C₂×C₁₀₇	#1270	1	1	yes	characteristic
81	C₂×C₁₀₇	#1355	1	2	yes	—
82	#1961	C₂×C₁₀₇	1	1	yes	characteristic
83	#1375	C₂×C₁₀₇	1	2	yes	complemented
84	C₁₀₇	#7116	1	1	yes	characteristic, direct product factor, Sylow
85	C₂×#127	C₄×C₁₀₇	1	1	yes	characteristic
86	C₂×M₃₂	—	2	2	no	—
87	C₂×#102	—	2	2	no	—
88	C₂×#97	C₂²×C₁₀₇	1	1	yes	characteristic
89	#382	C₄×C₁₀₇	1	1	yes	characteristic
90	C₂²×D₄	—	2	1	no	—
91	C₂²×D₄	—	4	1	no	—
92	C₂²×C₈	D₄×C₁₀₇	1	2	yes	—
93	#127	—	2	1	no	—
94	#127	—	4	1	no	—
95	#127	C₈×C₁₀₇	1	2	yes	—
96	C₂×C₄⋊C₄	—	2	1	no	—
97	C₂×C₄²	C₂×C₄×C₁₀₇	1	1	yes	characteristic
98	M₃₂	—	4	2	no	—
99	M₃₂	—	4	2	no	—
100	C₂×C₁₆	—	2	4	no	—
101	#102	—	4	2	no	—
102	#102	—	4	2	no	—
103	#97	—	4	1	no	—
104	#95	—	4	1	no	—
105	C₂⁴	—	4	1	no	—
106	C₂×D₄	—	2	2	no	—
107	C₂×D₄	—	4	2	no	—
108	C₂×D₄	—	4	2	no	—
109	C₂×D₄	—	4	1	no	—
110	C₂×D₄	—	4	2	no	—
111	C₂×D₄	—	4	2	no	—
112	C₂×D₄	—	8	4	no	—
113	C₂²×C₄	—	2	1	no	—
114	C₂²×C₄	—	4	1	no	—
115	C₂²×C₄	C₂²⋊C₄×C₁₀₇	1	1	yes	characteristic
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	D₈×C₁₀₇	1	2	yes	—
119	C₂×C₈	SD₁₆×C₁₀₇	1	2	yes	—
120	C₄⋊C₄	—	4	2	no	—
121	C₄²	—	2	1	no	—
122	C₄²	C₂×C₈×C₁₀₇	1	1	yes	characteristic
123	C₄²	M₁₆×C₁₀₇	1	1	yes	characteristic
124	C₁₆	—	4	4	no	—
125	C₂³	—	4	2	no	—
126	C₂³	—	8	4	no	—
127	C₂³	—	8	1	no	—
128	C₂³	—	8	1	no	—
129	C₂³	#100×C₁₀₇	1	1	yes	characteristic
130	D₄	—	4	4	no	—
131	D₄	—	8	4	no	—
132	D₄	—	8	4	no	—
133	D₄	—	8	2	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	4	2	no	—
137	C₂×C₄	—	8	2	no	—
138	C₂×C₄	—	8	1	no	—
139	C₂×C₄	C₂²⋊C₈×C₁₀₇	1	1	yes	characteristic
140	C₂×C₄	#99×C₁₀₇	1	1	yes	characteristic
141	C₂×C₄	#102×C₁₀₇	1	2	yes	—
142	C₂×C₄	#104×C₁₀₇	1	2	yes	—
143	C₈	—	2	2	no	—
144	C₈	—	2	2	no	—
145	C₂²	—	2	2	no	—
146	C₂²	—	8	4	no	—
147	C₂²	—	8	2	no	—
148	C₂²	—	16	2	no	—
149	C₂²	—	16	2	no	—
150	C₂²	#323×C₁₀₇	1	1	yes	characteristic
151	C₂²	#329×C₁₀₇	1	1	yes	characteristic
152	C₂²	#331×C₁₀₇	1	1	yes	characteristic
153	C₄	—	4	1	no	—
154	C₄	—	4	1	no	—
155	C₄	—	16	1	no	—
156	C₄	D₄⋊C₈×C₁₀₇	1	2	yes	—
157	C₄	#327×C₁₀₇	1	2	yes	—
158	C₂	—	2	1	no	—
159	C₂	—	2	1	no	—
160	C₂	—	16	2	no	—
161	C₂	C₁₀₇×#1270	1	1	yes	characteristic
162	C₂	C₁₀₇×#1355	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₀₇×#1961

C₂

yes

maximal, characteristic

C₁₀₇×#1375

C₂

yes

maximal, complemented

C₂×#127×C₁₀₇

C₄

yes

characteristic

C₂×M₃₂×C₁₀₇

—

C₂×#102×C₁₀₇

—

C₂×#97×C₁₀₇

C₂²

yes

characteristic

#382×C₁₀₇

C₄

yes

characteristic

C₂²×D₄×C₁₀₇

—

C₂²×D₄×C₁₀₇

—

C₂²×C₈×C₁₀₇

D₄

yes

—

#127×C₁₀₇

—

#127×C₁₀₇

—

#127×C₁₀₇

C₈

yes

—

C₂×C₄⋊C₄×C₁₀₇

—

C₂×C₄²×C₁₀₇

C₂×C₄

yes

characteristic

M₃₂×C₁₀₇

—

M₃₂×C₁₀₇

—

C₂×C₁₆×C₁₀₇

—

#102×C₁₀₇

—

#102×C₁₀₇

—

#97×C₁₀₇

—

#95×C₁₀₇

—

C₂⁴×C₁₀₇

—

C₂×D₄×C₁₀₇

—

C₂×D₄×C₁₀₇

—

C₂×D₄×C₁₀₇

—

C₂×D₄×C₁₀₇

—

C₂×D₄×C₁₀₇

—

C₂×D₄×C₁₀₇

—

C₂×D₄×C₁₀₇

—

C₂²×C₄×C₁₀₇

—

C₂²×C₄×C₁₀₇

—

C₂²×C₄×C₁₀₇

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₁₀₇

—

C₂×C₈×C₁₀₇

—

C₂×C₈×C₁₀₇

D₈

yes

—

C₂×C₈×C₁₀₇

SD₁₆

yes

—

C₄⋊C₄×C₁₀₇

—

C₄²×C₁₀₇

—

C₄²×C₁₀₇

C₂×C₈

yes

characteristic

C₄²×C₁₀₇

M₁₆

yes

characteristic

C₁₆×C₁₀₇

—

C₂³×C₁₀₇

—

C₂³×C₁₀₇

—

C₂³×C₁₀₇

—

C₂³×C₁₀₇

—

C₂³×C₁₀₇

#100

yes

characteristic

D₄×C₁₀₇

—

D₄×C₁₀₇

—

D₄×C₁₀₇

—

D₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

C₂²⋊C₈

yes

C₂×C₄×C₁₀₇

yes

C₂×C₄×C₁₀₇

yes

—

C₂×C₄×C₁₀₇

#104

yes

—

C₈×C₁₀₇

—

C₈×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

yes

C₂²×C₁₀₇

yes

C₂²×C₁₀₇

yes

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

D₄⋊C₈

yes

—

C₄×C₁₀₇

#327

yes

—

#7116

C₁₀₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₀₇

—

C₂×C₁₀₇

—

C₂×C₁₀₇

—

C₂×C₁₀₇

#1270

yes

characteristic

C₂×C₁₀₇

#1355

yes

—

#1961

C₂×C₁₀₇

yes

characteristic

#1375

C₂×C₁₀₇

yes

complemented

C₁₀₇

#7116

yes

characteristic, direct product factor, Sylow

C₂×#127

C₄×C₁₀₇

yes

characteristic

C₂×M₃₂

—

C₂×#102

—

C₂×#97

C₂²×C₁₀₇

yes

characteristic

#382

C₄×C₁₀₇

yes

characteristic

C₂²×D₄

—

C₂²×D₄

—

C₂²×C₈

D₄×C₁₀₇

yes

—

#127

—

#127

—

#127

C₈×C₁₀₇

yes

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×C₄×C₁₀₇

yes

characteristic

M₃₂

—

M₃₂

—

100

C₂×C₁₆

—

101

#102

—

102

#102

—

103

#97

—

104

#95

—

105

C₂⁴

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂×D₄

—

110

C₂×D₄

—

111

C₂×D₄

—

112

C₂×D₄

—

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂²×C₄

C₂²⋊C₄×C₁₀₇

yes

characteristic

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

D₈×C₁₀₇

yes

—

119

C₂×C₈

SD₁₆×C₁₀₇

yes

—

120

C₄⋊C₄

—

121

C₄²

—

122

C₄²

C₂×C₈×C₁₀₇

yes

characteristic

123

C₄²

M₁₆×C₁₀₇

yes

characteristic

124

C₁₆

—

125

C₂³

—

126

C₂³

—

127

C₂³

—

128

C₂³

—

129

C₂³

#100×C₁₀₇

yes

characteristic

130

D₄

—

131

D₄

—

132

D₄

—

133

D₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

C₂²⋊C₈×C₁₀₇

yes

characteristic

140

C₂×C₄

#99×C₁₀₇

yes

characteristic

141

C₂×C₄

#102×C₁₀₇

yes

—

142

C₂×C₄

#104×C₁₀₇

yes

—

143

C₈

—

144

C₈

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

150

C₂²

#323×C₁₀₇

yes

characteristic

151

C₂²

#329×C₁₀₇

yes

characteristic

152

C₂²

#331×C₁₀₇

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

D₄⋊C₈×C₁₀₇

yes

—

157

C₄

#327×C₁₀₇

yes

—

158

C₂

—

159

C₂

—

160

C₂

—

161

C₂

C₁₀₇×#1270

yes

characteristic

162

C₂

C₁₀₇×#1355

yes

—

	subgroup	G/H
center, socle	C₂²×C₁₀₇	#329
commutator	C₄²	C₂×C₈×C₁₀₇
Frattini, cosocle	C₂×#97	C₂²×C₁₀₇
prime radical	C₂×#127×C₁₀₇	C₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₇ ◁ C₂²×C₄×C₁₀₇ ◁ C₂×#97×C₁₀₇ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₀₇

#329

commutator

C₄²

C₂×C₈×C₁₀₇

Frattini, cosocle

C₂×#97

C₂²×C₁₀₇

prime radical

C₂×#127×C₁₀₇

C₄

LCS

G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₀₇ ◁ C₂²×C₄×C₁₀₇ ◁ C₂×#97×C₁₀₇ ◁ G

N	H ≅ G/N	type
2. C₁₀₇×#1375	C₂	semidirect
76. #7116	C₁₀₇	direct (N × H)
83. #1375	C₂×C₁₀₇	semidirect
84. C₁₀₇	#7116	direct (N × H)

H ≅ G/N

type

2. C₁₀₇×#1375

C₂

semidirect

76. #7116

C₁₀₇

direct (N × H)

83. #1375

C₂×C₁₀₇

semidirect

84. C₁₀₇

#7116

direct (N × H)

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

subgroup	index
C₁₀₇×#1961	2
C₁₀₇×#1375	2
#7116	107

subgroup

index

C₁₀₇×#1961

C₁₀₇×#1375

#7116

107

Subgroups — C₁₀₇×#7116 #41033080

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C107×#7116 #41033080

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₇×#7116 #41033080