Group #41033713 — subgroups

Subgroups — C₁₀₇×#7749 #41033713

Order 27392.

588 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 106 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×#2811	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₀₇×#2014	C₂	1	3	yes	maximal
3	C₁₀₇×#1272	C₂	1	3	yes	maximal, complemented
4	C₂²×SD₁₆×C₁₀₇	—	2	3	no	—
5	C₂×#127×C₁₀₇	C₂²	1	1	yes	characteristic
6	C₂×#102×C₁₀₇	—	2	3	no	—
7	#395×C₁₀₇	—	2	3	no	—
8	#391×C₁₀₇	—	2	3	no	—
9	#391×C₁₀₇	C₂²	1	3	yes	—
10	#336×C₁₀₇	C₂²	1	3	yes	—
11	C₂²×Q₈×C₁₀₇	—	2	1	no	—
12	C₂²×Q₈×C₁₀₇	D₄	1	3	yes	—
13	C₂²×D₄×C₁₀₇	—	2	3	no	—
14	C₂×SD₁₆×C₁₀₇	—	4	6	no	—
15	C₂×SD₁₆×C₁₀₇	—	4	6	no	—
16	C₂×SD₁₆×C₁₀₇	—	4	6	no	—
17	C₂²×C₈×C₁₀₇	—	2	3	no	—
18	#127×C₁₀₇	—	4	2	no	—
19	C₂×C₄⋊C₄×C₁₀₇	—	2	3	no	—
20	C₂×C₄⋊C₄×C₁₀₇	D₄	1	3	yes	—
21	C₂×C₄²×C₁₀₇	—	2	1	no	—
22	C₂×C₄²×C₁₀₇	C₂³	1	1	yes	characteristic
23	#102×C₁₀₇	—	4	6	no	—
24	#95×C₁₀₇	—	2	3	no	—
25	#95×C₁₀₇	—	2	6	no	—
26	C₂⁴×C₁₀₇	—	4	1	no	—
27	C₂×Q₈×C₁₀₇	—	2	3	no	—
28	C₂×Q₈×C₁₀₇	—	4	3	no	—
29	C₂×Q₈×C₁₀₇	—	4	3	no	—
30	C₂×Q₈×C₁₀₇	—	8	3	no	—
31	C₂×Q₈×C₁₀₇	SD₁₆	1	6	yes	complemented
32	C₂×D₄×C₁₀₇	—	2	6	no	—
33	C₂×D₄×C₁₀₇	—	4	3	no	—
34	C₂×D₄×C₁₀₇	—	4	3	no	—
35	C₂×D₄×C₁₀₇	—	4	3	no	—
36	C₂×D₄×C₁₀₇	—	4	3	no	—
37	C₂×D₄×C₁₀₇	—	4	3	no	—
38	C₂²×C₄×C₁₀₇	—	2	3	no	—
39	C₂²×C₄×C₁₀₇	—	2	3	no	—
40	C₂²×C₄×C₁₀₇	C₂×D₄	1	3	yes	—
41	SD₁₆×C₁₀₇	—	4	24	no	—
42	C₂×C₈×C₁₀₇	—	2	6	no	—
43	C₂×C₈×C₁₀₇	—	4	3	no	—
44	C₂×C₈×C₁₀₇	—	4	3	no	—
45	C₄⋊C₄×C₁₀₇	—	4	3	no	—
46	C₄⋊C₄×C₁₀₇	—	8	3	no	—
47	C₄²×C₁₀₇	—	4	1	no	—
48	C₄²×C₁₀₇	—	8	1	no	—
49	C₂³×C₁₀₇	—	4	2	no	—
50	C₂³×C₁₀₇	—	8	3	no	—
51	C₂³×C₁₀₇	—	8	3	no	—
52	C₂³×C₁₀₇	#120	1	1	yes	characteristic
53	Q₈×C₁₀₇	—	4	6	no	—
54	Q₈×C₁₀₇	—	8	3	no	—
55	Q₈×C₁₀₇	—	8	4	no	—
56	D₄×C₁₀₇	—	4	6	no	—
57	D₄×C₁₀₇	—	4	6	no	—
58	D₄×C₁₀₇	—	8	3	no	—
59	D₄×C₁₀₇	—	8	3	no	—
60	C₂×C₄×C₁₀₇	—	2	3	no	—
61	C₂×C₄×C₁₀₇	—	2	3	no	—
62	C₂×C₄×C₁₀₇	—	2	6	no	—
63	C₂×C₄×C₁₀₇	—	4	3	no	—
64	C₂×C₄×C₁₀₇	—	4	3	no	—
65	C₂×C₄×C₁₀₇	—	4	3	no	—
66	C₂×C₄×C₁₀₇	—	4	6	no	—
67	C₂×C₄×C₁₀₇	C₂×SD₁₆	1	3	yes	—
68	C₂×C₄×C₁₀₇	#137	1	3	yes	—
69	C₈×C₁₀₇	—	4	6	no	—
70	C₂²×C₁₀₇	—	8	6	no	—
71	C₂²×C₁₀₇	—	16	3	no	—
72	C₂²×C₁₀₇	—	16	1	no	—
73	C₂²×C₁₀₇	#448	1	3	yes	—
74	C₂²×C₁₀₇	#453	1	3	yes	—
75	C₂²×C₁₀₇	B(4,𝔽₂)	1	1	yes	characteristic
76	C₄×C₁₀₇	—	2	3	no	—
77	C₄×C₁₀₇	—	2	3	no	—
78	C₄×C₁₀₇	—	4	6	no	—
79	C₄×C₁₀₇	—	8	3	no	—
80	#7749	C₁₀₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
81	C₂×C₁₀₇	—	16	2	no	—
82	C₂×C₁₀₇	#1596	1	3	yes	—
83	C₂×C₁₀₇	#1623	1	3	yes	—
84	C₂×C₁₀₇	#1655	1	1	yes	characteristic
85	#2811	C₂×C₁₀₇	1	1	yes	characteristic, complemented
86	#2014	C₂×C₁₀₇	1	3	yes	—
87	#1272	C₂×C₁₀₇	1	3	yes	complemented
88	C₁₀₇	#7749	1	1	yes	characteristic, direct product factor, Sylow
89	C₂²×SD₁₆	—	2	3	no	—
90	C₂×#127	C₂²×C₁₀₇	1	1	yes	characteristic
91	C₂×#102	—	2	3	no	—
92	#395	—	2	3	no	—
93	#391	—	2	3	no	—
94	#391	C₂²×C₁₀₇	1	3	yes	—
95	#336	C₂²×C₁₀₇	1	3	yes	—
96	C₂²×Q₈	—	2	1	no	—
97	C₂²×Q₈	D₄×C₁₀₇	1	3	yes	—
98	C₂²×D₄	—	2	3	no	—
99	C₂×SD₁₆	—	4	6	no	—
100	C₂×SD₁₆	—	4	6	no	—
101	C₂×SD₁₆	—	4	6	no	—
102	C₂²×C₈	—	2	3	no	—
103	#127	—	4	2	no	—
104	C₂×C₄⋊C₄	—	2	3	no	—
105	C₂×C₄⋊C₄	D₄×C₁₀₇	1	3	yes	—
106	C₂×C₄²	—	2	1	no	—
107	C₂×C₄²	C₂³×C₁₀₇	1	1	yes	characteristic
108	#102	—	4	6	no	—
109	#95	—	2	3	no	—
110	#95	—	2	6	no	—
111	C₂⁴	—	4	1	no	—
112	C₂×Q₈	—	2	3	no	—
113	C₂×Q₈	—	4	3	no	—
114	C₂×Q₈	—	4	3	no	—
115	C₂×Q₈	—	8	3	no	—
116	C₂×Q₈	SD₁₆×C₁₀₇	1	6	yes	complemented
117	C₂×D₄	—	2	6	no	—
118	C₂×D₄	—	4	3	no	—
119	C₂×D₄	—	4	3	no	—
120	C₂×D₄	—	4	3	no	—
121	C₂×D₄	—	4	3	no	—
122	C₂×D₄	—	4	3	no	—
123	C₂²×C₄	—	2	3	no	—
124	C₂²×C₄	—	2	3	no	—
125	C₂²×C₄	C₂×D₄×C₁₀₇	1	3	yes	—
126	SD₁₆	—	4	24	no	—
127	C₂×C₈	—	2	6	no	—
128	C₂×C₈	—	4	3	no	—
129	C₂×C₈	—	4	3	no	—
130	C₄⋊C₄	—	4	3	no	—
131	C₄⋊C₄	—	8	3	no	—
132	C₄²	—	4	1	no	—
133	C₄²	—	8	1	no	—
134	C₂³	—	4	2	no	—
135	C₂³	—	8	3	no	—
136	C₂³	—	8	3	no	—
137	C₂³	#120×C₁₀₇	1	1	yes	characteristic
138	Q₈	—	4	6	no	—
139	Q₈	—	8	3	no	—
140	Q₈	—	8	4	no	—
141	D₄	—	4	6	no	—
142	D₄	—	4	6	no	—
143	D₄	—	8	3	no	—
144	D₄	—	8	3	no	—
145	C₂×C₄	—	2	3	no	—
146	C₂×C₄	—	2	3	no	—
147	C₂×C₄	—	2	6	no	—
148	C₂×C₄	—	4	3	no	—
149	C₂×C₄	—	4	3	no	—
150	C₂×C₄	—	4	3	no	—
151	C₂×C₄	—	4	6	no	—
152	C₂×C₄	C₂×SD₁₆×C₁₀₇	1	3	yes	—
153	C₂×C₄	#137×C₁₀₇	1	3	yes	—
154	C₈	—	4	6	no	—
155	C₂²	—	8	6	no	—
156	C₂²	—	16	3	no	—
157	C₂²	—	16	1	no	—
158	C₂²	#448×C₁₀₇	1	3	yes	—
159	C₂²	#453×C₁₀₇	1	3	yes	—
160	C₂²	B(4,𝔽₂)×C₁₀₇	1	1	yes	characteristic
161	C₄	—	2	3	no	—
162	C₄	—	2	3	no	—
163	C₄	—	4	6	no	—
164	C₄	—	8	3	no	—
165	C₂	—	16	2	no	—
166	C₂	C₁₀₇×#1596	1	3	yes	—
167	C₂	C₁₀₇×#1623	1	3	yes	—
168	C₂	C₁₀₇×#1655	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₀₇×#2811

C₂

yes

maximal, characteristic, complemented

C₁₀₇×#2014

C₂

yes

maximal

C₁₀₇×#1272

C₂

yes

maximal, complemented

C₂²×SD₁₆×C₁₀₇

—

C₂×#127×C₁₀₇

C₂²

yes

characteristic

C₂×#102×C₁₀₇

—

#395×C₁₀₇

—

#391×C₁₀₇

—

#391×C₁₀₇

C₂²

yes

—

#336×C₁₀₇

C₂²

yes

—

C₂²×Q₈×C₁₀₇

—

C₂²×Q₈×C₁₀₇

D₄

yes

—

C₂²×D₄×C₁₀₇

—

C₂×SD₁₆×C₁₀₇

—

C₂×SD₁₆×C₁₀₇

—

C₂×SD₁₆×C₁₀₇

—

C₂²×C₈×C₁₀₇

—

#127×C₁₀₇

—

C₂×C₄⋊C₄×C₁₀₇

—

C₂×C₄⋊C₄×C₁₀₇

D₄

yes

—

C₂×C₄²×C₁₀₇

—

C₂×C₄²×C₁₀₇

C₂³

yes

characteristic

#102×C₁₀₇

—

#95×C₁₀₇

—

#95×C₁₀₇

—

C₂⁴×C₁₀₇

—

C₂×Q₈×C₁₀₇

—

C₂×Q₈×C₁₀₇

—

C₂×Q₈×C₁₀₇

—

C₂×Q₈×C₁₀₇

—

C₂×Q₈×C₁₀₇

SD₁₆

yes

complemented

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

C₂×D₄

yes

—

SD₁₆×C₁₀₇

—

C₂×C₈×C₁₀₇

—

C₂×C₈×C₁₀₇

—

C₂×C₈×C₁₀₇

—

C₄⋊C₄×C₁₀₇

—

C₄⋊C₄×C₁₀₇

—

C₄²×C₁₀₇

—

C₄²×C₁₀₇

—

C₂³×C₁₀₇

—

C₂³×C₁₀₇

—

C₂³×C₁₀₇

—

C₂³×C₁₀₇

#120

yes

characteristic

Q₈×C₁₀₇

—

Q₈×C₁₀₇

—

Q₈×C₁₀₇

—

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

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

C₂×SD₁₆

yes

—

C₂×C₄×C₁₀₇

#137

yes

—

C₈×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

#448

yes

—

C₂²×C₁₀₇

#453

yes

—

C₂²×C₁₀₇

B(4,𝔽₂)

yes

characteristic

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

#7749

C₁₀₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₀₇

—

C₂×C₁₀₇

#1596

yes

—

C₂×C₁₀₇

#1623

yes

—

C₂×C₁₀₇

#1655

yes

characteristic

#2811

C₂×C₁₀₇

yes

characteristic, complemented

#2014

C₂×C₁₀₇

yes

—

#1272

C₂×C₁₀₇

yes

complemented

C₁₀₇

#7749

yes

characteristic, direct product factor, Sylow

C₂²×SD₁₆

—

C₂×#127

C₂²×C₁₀₇

yes

—

—

—

C₂²×C₁₀₇

yes

—

#336

C₂²×C₁₀₇

yes

—

C₂²×Q₈

—

C₂²×Q₈

D₄×C₁₀₇

yes

—

C₂²×D₄

—

C₂×SD₁₆

—

100

C₂×SD₁₆

—

101

C₂×SD₁₆

—

102

C₂²×C₈

—

103

#127

—

104

C₂×C₄⋊C₄

—

105

C₂×C₄⋊C₄

D₄×C₁₀₇

yes

—

106

C₂×C₄²

—

107

C₂×C₄²

C₂³×C₁₀₇

yes

characteristic

108

#102

—

109

#95

—

110

#95

—

111

C₂⁴

—

112

C₂×Q₈

—

113

C₂×Q₈

—

114

C₂×Q₈

—

115

C₂×Q₈

—

116

C₂×Q₈

SD₁₆×C₁₀₇

yes

complemented

117

C₂×D₄

—

118

C₂×D₄

—

119

C₂×D₄

—

120

C₂×D₄

—

121

C₂×D₄

—

122

C₂×D₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

C₂×D₄×C₁₀₇

yes

—

126

SD₁₆

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄²

—

133

C₄²

—

134

C₂³

—

135

C₂³

—

136

C₂³

—

137

C₂³

#120×C₁₀₇

yes

characteristic

138

Q₈

—

139

Q₈

—

140

Q₈

—

141

D₄

—

142

D₄

—

143

D₄

—

144

D₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

C₂×SD₁₆×C₁₀₇

yes

—

153

C₂×C₄

#137×C₁₀₇

yes

—

154

C₈

—

155

C₂²

—

156

C₂²

—

157

C₂²

—

158

C₂²

#448×C₁₀₇

yes

—

159

C₂²

#453×C₁₀₇

yes

—

160

C₂²

B(4,𝔽₂)×C₁₀₇

yes

characteristic

161

C₄

—

162

C₄

—

163

C₄

—

164

C₄

—

165

C₂

—

166

C₂

C₁₀₇×#1596

yes

—

167

C₂

C₁₀₇×#1623

yes

—

168

C₂

C₁₀₇×#1655

yes

characteristic

	subgroup	G/H
center, socle	C₂³×C₁₀₇	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₇
prime radical	C₂×#127×C₁₀₇	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₇ ◁ C₂×C₄²×C₁₀₇ ◁ G

subgroup

G/H

center, socle

C₂³×C₁₀₇

#120

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₁₀₇

prime radical

C₂×#127×C₁₀₇

C₂²

LCS

G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂³×C₁₀₇ ◁ C₂×C₄²×C₁₀₇ ◁ G

N	H ≅ G/N	type
1. C₁₀₇×#2811	C₂	semidirect
3. C₁₀₇×#1272	C₂	semidirect
31. C₂×Q₈×C₁₀₇	SD₁₆	semidirect
80. #7749	C₁₀₇	direct (N × H)
85. #2811	C₂×C₁₀₇	semidirect
87. #1272	C₂×C₁₀₇	semidirect
88. C₁₀₇	#7749	direct (N × H)
116. C₂×Q₈	SD₁₆×C₁₀₇	semidirect

H ≅ G/N

type

1. C₁₀₇×#2811

C₂

semidirect

3. C₁₀₇×#1272

C₂

semidirect

31. C₂×Q₈×C₁₀₇

SD₁₆

semidirect

80. #7749

C₁₀₇

direct (N × H)

85. #2811

C₂×C₁₀₇

semidirect

87. #1272

C₂×C₁₀₇

semidirect

88. C₁₀₇

#7749

direct (N × H)

116. C₂×Q₈

SD₁₆×C₁₀₇

semidirect

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

subgroup	index
C₁₀₇×#2811	2
C₁₀₇×#2014	2
C₁₀₇×#1272	2
#7749	107

subgroup

index

C₁₀₇×#2811

C₁₀₇×#2014

C₁₀₇×#1272

#7749

107

Subgroups — C₁₀₇×#7749 #41033713

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C107×#7749 #41033713

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₇×#7749 #41033713