Group #44576513 — subgroups

Subgroups — C₁₂₇×#12840 #44576513

Order 32512.

354 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 154 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×#504×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#368×C₁₂₇	C₂	1	2	yes	maximal, complemented
3	C₁₂₇×#1813	C₂	1	2	yes	maximal
4	C₁₂₇×#1385	C₂	1	2	yes	maximal, complemented
5	C₂×#131×C₁₂₇	C₂²	1	1	yes	characteristic
6	#504×C₁₂₇	—	2	4	no	—
7	C₂×M₃₂×C₁₂₇	C₄	1	2	yes	—
8	C₂×M₃₂×C₁₂₇	C₂²	1	1	yes	characteristic
9	C₂²×C₁₆×C₁₂₇	C₄	1	2	yes	—
10	C₂²×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
11	C₂×#108×C₁₂₇	C₂²	1	4	yes	—
12	#413×C₁₂₇	—	2	2	no	—
13	#368×C₁₂₇	—	2	4	no	—
14	#338×C₁₂₇	—	2	2	no	—
15	C₂×#41×C₁₂₇	D₄	1	1	yes	characteristic
16	#131×C₁₂₇	—	2	2	no	—
17	#131×C₁₂₇	—	2	2	no	—
18	C₂×M₁₆×C₁₂₇	—	2	4	no	—
19	C₂×M₁₆×C₁₂₇	D₄	1	1	yes	characteristic
20	C₂×M₁₆×C₁₂₇	Q₈	1	2	yes	—
21	C₂²×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
23	M₃₂×C₁₂₇	—	2	2	no	—
24	M₃₂×C₁₂₇	—	2	4	no	—
25	C₂×C₁₆×C₁₂₇	—	2	4	no	—
26	C₂×C₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
27	C₂×C₁₆×C₁₂₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₆×C₁₂₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₁₆×C₁₂₇	D₄	1	4	yes	—
30	#108×C₁₂₇	—	2	4	no	—
31	#108×C₁₂₇	—	2	4	no	—
32	#101×C₁₂₇	—	4	2	no	—
33	#100×C₁₂₇	—	4	2	no	—
34	#41×C₁₂₇	—	2	2	no	—
35	#41×C₁₂₇	—	2	2	no	—
36	C₂×Q₈×C₁₂₇	D₈	1	1	yes	characteristic
37	C₂×D₄×C₁₂₇	D₈	1	1	yes	characteristic
38	C₂×D₄×C₁₂₇	Q₁₆	1	2	yes	—
39	C₂²×C₄×C₁₂₇	C₄⋊C₄	1	2	yes	—
40	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
41	M₁₆×C₁₂₇	—	2	2	no	—
42	M₁₆×C₁₂₇	—	2	4	no	—
43	M₁₆×C₁₂₇	—	4	4	no	—
44	M₁₆×C₁₂₇	—	4	4	no	—
45	C₂×C₈×C₁₂₇	—	2	2	no	—
46	C₂×C₈×C₁₂₇	—	2	2	no	—
47	C₂×C₈×C₁₂₇	—	4	4	no	—
48	C₂×C₈×C₁₂₇	C₄⋊C₄	1	2	yes	—
49	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
51	C₂×C₈×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₈×C₁₂₇	#41	1	2	yes	—
53	C₁₆×C₁₂₇	—	2	2	no	—
54	C₁₆×C₁₂₇	C₂²⋊C₄	1	2	yes	—
55	C₁₆×C₁₂₇	C₂²⋊C₄	1	2	yes	—
56	C₂³×C₁₂₇	#107	1	2	yes	—
57	C₂³×C₁₂₇	C₂×Q₁₆	1	1	yes	characteristic
58	Q₈×C₁₂₇	—	2	2	no	—
59	D₄×C₁₂₇	—	2	4	no	—
60	D₄×C₁₂₇	—	2	2	no	—
61	C₂×C₄×C₁₂₇	—	2	2	no	—
62	C₂×C₄×C₁₂₇	—	2	2	no	—
63	C₂×C₄×C₁₂₇	#107	1	2	yes	—
64	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₁₂₇	#121	1	2	yes	—
66	C₂×C₄×C₁₂₇	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₁₂₇	#135	1	2	yes	—
68	C₈×C₁₂₇	—	2	2	no	—
69	C₈×C₁₂₇	—	8	4	no	—
70	C₈×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₈×C₁₂₇	#117	1	1	yes	characteristic
72	C₈×C₁₂₇	C₂²⋊Q₈	1	2	yes	—
73	C₂²×C₁₂₇	—	2	2	no	—
74	C₂²×C₁₂₇	—	4	1	no	—
75	C₂²×C₁₂₇	—	4	2	no	—
76	C₂²×C₁₂₇	C₂×#107	1	1	yes	characteristic
77	C₂²×C₁₂₇	#467	1	2	yes	—
78	C₄×C₁₂₇	—	2	2	no	—
79	C₄×C₁₂₇	#380	1	1	yes	characteristic
80	C₄×C₁₂₇	#427	1	1	yes	characteristic
81	C₄×C₁₂₇	#466	1	2	yes	—
82	#12840	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
83	C₂×C₁₂₇	—	2	1	no	—
84	C₂×C₁₂₇	—	2	2	no	—
85	C₂×C₁₂₇	—	4	2	no	—
86	C₂×C₁₂₇	#1808	1	1	yes	characteristic
87	C₂×#504	C₂×C₁₂₇	1	1	yes	characteristic
88	C₂×#368	C₂×C₁₂₇	1	2	yes	complemented
89	#1813	C₂×C₁₂₇	1	2	yes	—
90	#1385	C₂×C₁₂₇	1	2	yes	complemented
91	C₁₂₇	#12840	1	1	yes	characteristic, direct product factor, Sylow
92	C₂×#131	C₂²×C₁₂₇	1	1	yes	characteristic
93	#504	—	2	4	no	—
94	C₂×M₃₂	C₄×C₁₂₇	1	2	yes	—
95	C₂×M₃₂	C₂²×C₁₂₇	1	1	yes	characteristic
96	C₂²×C₁₆	C₄×C₁₂₇	1	2	yes	—
97	C₂²×C₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
98	C₂×#108	C₂²×C₁₂₇	1	4	yes	—
99	#413	—	2	2	no	—
100	#368	—	2	4	no	—
101	#338	—	2	2	no	—
102	C₂×#41	D₄×C₁₂₇	1	1	yes	characteristic
103	#131	—	2	2	no	—
104	#131	—	2	2	no	—
105	C₂×M₁₆	—	2	4	no	—
106	C₂×M₁₆	D₄×C₁₂₇	1	1	yes	characteristic
107	C₂×M₁₆	Q₈×C₁₂₇	1	2	yes	—
108	C₂²×C₈	C₂×C₄×C₁₂₇	1	2	yes	—
109	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
110	M₃₂	—	2	2	no	—
111	M₃₂	—	2	4	no	—
112	C₂×C₁₆	—	2	4	no	—
113	C₂×C₁₆	C₂×C₄×C₁₂₇	1	2	yes	—
114	C₂×C₁₆	C₂×C₄×C₁₂₇	1	1	yes	characteristic
115	C₂×C₁₆	C₂×C₄×C₁₂₇	1	1	yes	characteristic
116	C₂×C₁₆	D₄×C₁₂₇	1	4	yes	—
117	#108	—	2	4	no	—
118	#108	—	2	4	no	—
119	#101	—	4	2	no	—
120	#100	—	4	2	no	—
121	#41	—	2	2	no	—
122	#41	—	2	2	no	—
123	C₂×Q₈	D₈×C₁₂₇	1	1	yes	characteristic
124	C₂×D₄	D₈×C₁₂₇	1	1	yes	characteristic
125	C₂×D₄	Q₁₆×C₁₂₇	1	2	yes	—
126	C₂²×C₄	C₄⋊C₄×C₁₂₇	1	2	yes	—
127	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
128	M₁₆	—	2	2	no	—
129	M₁₆	—	2	4	no	—
130	M₁₆	—	4	4	no	—
131	M₁₆	—	4	4	no	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	4	4	no	—
135	C₂×C₈	C₄⋊C₄×C₁₂₇	1	2	yes	—
136	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
137	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
138	C₂×C₈	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
139	C₂×C₈	#41×C₁₂₇	1	2	yes	—
140	C₁₆	—	2	2	no	—
141	C₁₆	C₂²⋊C₄×C₁₂₇	1	2	yes	—
142	C₁₆	C₂²⋊C₄×C₁₂₇	1	2	yes	—
143	C₂³	#107×C₁₂₇	1	2	yes	—
144	C₂³	C₂×Q₁₆×C₁₂₇	1	1	yes	characteristic
145	Q₈	—	2	2	no	—
146	D₄	—	2	4	no	—
147	D₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	#107×C₁₂₇	1	2	yes	—
151	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
152	C₂×C₄	#121×C₁₂₇	1	2	yes	—
153	C₂×C₄	C₂×D₈×C₁₂₇	1	1	yes	characteristic
154	C₂×C₄	#135×C₁₂₇	1	2	yes	—
155	C₈	—	2	2	no	—
156	C₈	—	8	4	no	—
157	C₈	C₂×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
158	C₈	#117×C₁₂₇	1	1	yes	characteristic
159	C₈	C₂²⋊Q₈×C₁₂₇	1	2	yes	—
160	C₂²	—	2	2	no	—
161	C₂²	—	4	1	no	—
162	C₂²	—	4	2	no	—
163	C₂²	C₂×#107×C₁₂₇	1	1	yes	characteristic
164	C₂²	#467×C₁₂₇	1	2	yes	—
165	C₄	—	2	2	no	—
166	C₄	#380×C₁₂₇	1	1	yes	characteristic
167	C₄	#427×C₁₂₇	1	1	yes	characteristic
168	C₄	#466×C₁₂₇	1	2	yes	—
169	C₂	—	2	1	no	—
170	C₂	—	2	2	no	—
171	C₂	—	4	2	no	—
172	C₂	C₁₂₇×#1808	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#504×C₁₂₇

C₂

yes

maximal, characteristic

C₂×#368×C₁₂₇

C₂

yes

maximal, complemented

C₁₂₇×#1813

C₂

yes

maximal

C₁₂₇×#1385

C₂

yes

maximal, complemented

C₂×#131×C₁₂₇

C₂²

yes

characteristic

#504×C₁₂₇

—

C₂×M₃₂×C₁₂₇

C₄

yes

—

C₂×M₃₂×C₁₂₇

C₂²

yes

characteristic

C₂²×C₁₆×C₁₂₇

C₄

yes

—

C₂²×C₁₆×C₁₂₇

C₂²

yes

characteristic

C₂×#108×C₁₂₇

C₂²

yes

—

#413×C₁₂₇

—

#368×C₁₂₇

—

#338×C₁₂₇

—

C₂×#41×C₁₂₇

D₄

yes

characteristic

#131×C₁₂₇

—

#131×C₁₂₇

—

C₂×M₁₆×C₁₂₇

—

C₂×M₁₆×C₁₂₇

D₄

yes

characteristic

C₂×M₁₆×C₁₂₇

Q₈

yes

—

C₂²×C₈×C₁₂₇

C₂×C₄

yes

—

C₂²×C₈×C₁₂₇

C₂³

yes

characteristic

M₃₂×C₁₂₇

—

M₃₂×C₁₂₇

—

C₂×C₁₆×C₁₂₇

—

C₂×C₁₆×C₁₂₇

C₂×C₄

yes

—

C₂×C₁₆×C₁₂₇

C₂×C₄

yes

characteristic

C₂×C₁₆×C₁₂₇

C₂×C₄

yes

characteristic

C₂×C₁₆×C₁₂₇

D₄

yes

—

#108×C₁₂₇

—

#108×C₁₂₇

—

#101×C₁₂₇

—

#100×C₁₂₇

—

#41×C₁₂₇

—

#41×C₁₂₇

—

C₂×Q₈×C₁₂₇

D₈

yes

characteristic

C₂×D₄×C₁₂₇

D₈

yes

characteristic

C₂×D₄×C₁₂₇

Q₁₆

yes

—

C₂²×C₄×C₁₂₇

C₄⋊C₄

yes

—

C₂²×C₄×C₁₂₇

C₂×D₄

yes

characteristic

M₁₆×C₁₂₇

—

M₁₆×C₁₂₇

—

M₁₆×C₁₂₇

—

M₁₆×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

C₄⋊C₄

yes

—

C₂×C₈×C₁₂₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₁₂₇

C₂×D₄

yes

—

C₂×C₈×C₁₂₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₁₂₇

#41

yes

—

C₁₆×C₁₂₇

—

C₁₆×C₁₂₇

C₂²⋊C₄

yes

—

C₁₆×C₁₂₇

C₂²⋊C₄

yes

—

C₂³×C₁₂₇

#107

yes

—

C₂³×C₁₂₇

C₂×Q₁₆

yes

characteristic

Q₈×C₁₂₇

—

D₄×C₁₂₇

—

D₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

#107

yes

—

C₂×C₄×C₁₂₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₂₇

#121

yes

—

C₂×C₄×C₁₂₇

C₂×D₈

yes

characteristic

C₂×C₄×C₁₂₇

#135

yes

—

C₈×C₁₂₇

—

C₈×C₁₂₇

—

C₈×C₁₂₇

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₁₂₇

#117

yes

characteristic

C₈×C₁₂₇

C₂²⋊Q₈

yes

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂×#107

yes

characteristic

C₂²×C₁₂₇

#467

yes

—

C₄×C₁₂₇

—

C₄×C₁₂₇

yes

C₄×C₁₂₇

yes

C₄×C₁₂₇

yes

—

#12840

C₁₂₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

#1808

yes

characteristic

C₂×#504

C₂×C₁₂₇

yes

characteristic

C₂×#368

C₂×C₁₂₇

yes

complemented

#1813

C₂×C₁₂₇

yes

—

#1385

C₂×C₁₂₇

yes

complemented

C₁₂₇

#12840

yes

characteristic, direct product factor, Sylow

C₂×#131

C₂²×C₁₂₇

yes

characteristic

#504

—

C₂×M₃₂

C₄×C₁₂₇

yes

—

C₂×M₃₂

C₂²×C₁₂₇

yes

characteristic

C₂²×C₁₆

C₄×C₁₂₇

yes

—

C₂²×C₁₆

C₂²×C₁₂₇

yes

characteristic

C₂×#108

C₂²×C₁₂₇

yes

—

#413

—

100

#368

—

101

#338

—

102

C₂×#41

D₄×C₁₂₇

yes

characteristic

103

#131

—

104

#131

—

105

C₂×M₁₆

—

106

C₂×M₁₆

D₄×C₁₂₇

yes

characteristic

107

C₂×M₁₆

Q₈×C₁₂₇

yes

—

108

C₂²×C₈

C₂×C₄×C₁₂₇

yes

—

109

C₂²×C₈

C₂³×C₁₂₇

yes

characteristic

110

M₃₂

—

111

M₃₂

—

112

C₂×C₁₆

—

113

C₂×C₁₆

C₂×C₄×C₁₂₇

yes

—

114

C₂×C₁₆

C₂×C₄×C₁₂₇

yes

characteristic

115

C₂×C₁₆

C₂×C₄×C₁₂₇

yes

characteristic

116

C₂×C₁₆

D₄×C₁₂₇

yes

—

117

#108

—

118

#108

—

119

#101

—

120

#100

—

121

#41

—

122

#41

—

123

C₂×Q₈

D₈×C₁₂₇

yes

characteristic

124

C₂×D₄

D₈×C₁₂₇

yes

characteristic

125

C₂×D₄

Q₁₆×C₁₂₇

yes

—

126

C₂²×C₄

C₄⋊C₄×C₁₂₇

yes

—

127

C₂²×C₄

C₂×D₄×C₁₂₇

yes

characteristic

128

M₁₆

—

129

M₁₆

—

130

M₁₆

—

131

M₁₆

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

C₄⋊C₄×C₁₂₇

yes

—

136

C₂×C₈

C₂²×C₄×C₁₂₇

yes

characteristic

137

C₂×C₈

C₂×D₄×C₁₂₇

yes

—

138

C₂×C₈

C₂×Q₈×C₁₂₇

yes

characteristic

139

C₂×C₈

#41×C₁₂₇

yes

—

140

C₁₆

—

141

C₁₆

C₂²⋊C₄×C₁₂₇

yes

—

142

C₁₆

C₂²⋊C₄×C₁₂₇

yes

—

143

C₂³

#107×C₁₂₇

yes

—

144

C₂³

C₂×Q₁₆×C₁₂₇

yes

characteristic

145

Q₈

—

146

D₄

—

147

D₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

#107×C₁₂₇

yes

—

151

C₂×C₄

C₂×C₄⋊C₄×C₁₂₇

yes

characteristic

152

C₂×C₄

#121×C₁₂₇

yes

—

153

C₂×C₄

C₂×D₈×C₁₂₇

yes

characteristic

154

C₂×C₄

#135×C₁₂₇

yes

—

155

C₈

—

156

C₈

—

157

C₈

C₂×C₂²⋊C₄×C₁₂₇

yes

characteristic

158

C₈

#117×C₁₂₇

yes

characteristic

159

C₈

C₂²⋊Q₈×C₁₂₇

yes

—

160

C₂²

—

161

C₂²

—

162

C₂²

—

163

C₂²

C₂×#107×C₁₂₇

yes

characteristic

164

C₂²

#467×C₁₂₇

yes

—

165

C₄

—

166

C₄

#380×C₁₂₇

yes

characteristic

167

C₄

#427×C₁₂₇

yes

characteristic

168

C₄

#466×C₁₂₇

yes

—

169

C₂

—

170

C₂

—

171

C₂

—

172

C₂

C₁₂₇×#1808

yes

characteristic

	subgroup	G/H
center	C₄×C₁₂₇	#427
commutator	C₂×C₈	C₂²×C₄×C₁₂₇
Frattini, cosocle	C₂²×C₈	C₂³×C₁₂₇
socle	C₂×C₁₂₇	#1808
prime radical	C₂×#41×C₁₂₇	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₂₇ ◁ C₂×M₁₆×C₁₂₇ ◁ C₂×#131×C₁₂₇ ◁ G

subgroup

G/H

center

C₄×C₁₂₇

#427

commutator

C₂×C₈

C₂²×C₄×C₁₂₇

Frattini, cosocle

C₂²×C₈

C₂³×C₁₂₇

socle

C₂×C₁₂₇

#1808

prime radical

C₂×#41×C₁₂₇

D₄

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₄×C₁₂₇ ◁ C₂×M₁₆×C₁₂₇ ◁ C₂×#131×C₁₂₇ ◁ G

N	H ≅ G/N	type
2. C₂×#368×C₁₂₇	C₂	semidirect
4. C₁₂₇×#1385	C₂	semidirect
82. #12840	C₁₂₇	direct (N × H)
88. C₂×#368	C₂×C₁₂₇	semidirect
90. #1385	C₂×C₁₂₇	semidirect
91. C₁₂₇	#12840	direct (N × H)

H ≅ G/N

type

2. C₂×#368×C₁₂₇

C₂

semidirect

4. C₁₂₇×#1385

C₂

semidirect

82. #12840

C₁₂₇

direct (N × H)

88. C₂×#368

C₂×C₁₂₇

semidirect

90. #1385

C₂×C₁₂₇

semidirect

91. C₁₂₇

#12840

direct (N × H)

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

subgroup	index
C₂×#504×C₁₂₇	2
C₂×#368×C₁₂₇	2
C₁₂₇×#1813	2
C₁₂₇×#1385	2
#12840	127

subgroup

index

C₂×#504×C₁₂₇

C₂×#368×C₁₂₇

C₁₂₇×#1813

C₁₂₇×#1385

#12840

127

Subgroups — C₁₂₇×#12840 #44576513

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C127×#12840 #44576513

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#12840 #44576513