Group #30648492 — subgroups

Subgroups — C₂×#22550158 #30648492

Order 16256.

416 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 173 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×C₄×Dic₂₅₄	C₂	1	1	yes	maximal, characteristic
2	#22550158	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16861140	C₂	1	2	yes	maximal
4	C₂×C₄⋊(C₁₂₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
5	C₂×#16861132	C₂	1	2	yes	maximal
6	C₂×C₄⋊Q₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₂₅₄	C₂²	1	1	yes	characteristic
8	C₂²×C₁₂₇⋊C₈	C₂²	1	2	yes	—
9	C₄×Dic₂₅₄	—	2	2	no	—
10	C₄×Dic₂₅₄	C₂²	1	4	yes	—
11	C₂×C₄⋊Dic₁₂₇	C₂²	1	1	yes	characteristic
12	C₂×Dic₁₂₇⋊C₄	—	2	1	no	—
13	C₂×C₄×Dic₁₂₇	—	2	1	no	—
14	#16861140	C₂²	1	8	yes	—
15	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	4	yes	—
16	#16861132	C₂²	1	8	yes	—
17	C₂²×Q₈×C₁₂₇	—	2	1	no	—
18	C₄⋊Q₈×C₁₂₇	—	2	2	no	—
19	C₄⋊Q₈×C₁₂₇	C₂²	1	4	yes	—
20	C₂×C₄⋊C₄×C₁₂₇	—	2	1	no	—
21	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
22	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
23	C₂×Dic₂₅₄	—	2	1	no	—
24	C₂×Dic₂₅₄	—	2	2	no	—
25	C₂×Dic₂₅₄	Q₈	1	4	yes	complemented
26	C₂×Dic₂₅₄	C₂³	1	2	yes	—
27	C₂²×Dic₁₂₇	—	2	1	no	—
28	C₂²×Dic₁₂₇	—	2	1	no	—
29	C₂×C₁₂₇⋊C₈	—	2	4	no	—
30	C₂×C₁₂₇⋊C₈	C₂³	1	4	yes	—
31	C₄⋊Dic₁₂₇	—	2	1	no	—
32	C₄⋊Dic₁₂₇	C₂³	1	2	yes	—
33	Dic₁₂₇⋊C₄	—	2	4	no	—
34	C₄×Dic₁₂₇	—	2	4	no	—
35	C₂×Q₈×C₁₂₇	—	2	4	no	—
36	C₂×Q₈×C₁₂₇	—	4	4	no	—
37	C₂²×C₄×C₁₂₇	—	2	2	no	—
38	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
41	C₄⋊C₄×C₁₂₇	—	2	4	no	—
42	C₄⋊C₄×C₁₂₇	—	2	2	no	—
43	C₄⋊C₄×C₁₂₇	C₂³	1	4	yes	—
44	C₄²×C₁₂₇	D₄	1	2	yes	—
45	C₄²×C₁₂₇	C₂³	1	2	yes	—
46	Dic₂₅₄	—	2	4	no	—
47	Dic₂₅₄	—	2	2	no	—
48	Dic₂₅₄	C₂×Q₈	1	4	yes	complemented
49	C₂×Dic₁₂₇	—	2	4	no	—
50	C₂×Dic₁₂₇	—	2	2	no	—
51	C₂×Dic₁₂₇	—	2	2	no	—
52	C₂×Dic₁₂₇	—	4	2	no	—
53	C₁₂₇⋊C₈	—	2	4	no	—
54	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
55	Q₈×C₁₂₇	—	4	8	no	—
56	C₂×C₄×C₁₂₇	—	2	2	no	—
57	C₂×C₄×C₁₂₇	—	2	4	no	—
58	C₂×C₄×C₁₂₇	—	4	4	no	—
59	C₂×C₄×C₁₂₇	SD₁₆	1	4	yes	—
60	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
63	C₂×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
64	C₂×C₄×C₁₂₇	#41	1	2	yes	—
65	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
66	C₂²×C₁₂₇	C₂²⋊Q₈	1	2	yes	—
67	C₂²×C₁₂₇	C₂×SD₁₆	1	2	yes	—
68	C₂²×C₁₂₇	#137	1	2	yes	—
69	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
70	C₄×C₁₂₇	—	2	2	no	—
71	C₄×C₁₂₇	—	4	4	no	—
72	C₄×C₁₂₇	C₂²⋊Q₈	1	2	yes	—
73	C₄×C₁₂₇	C₂×SD₁₆	1	4	yes	—
74	C₄×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
75	C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
76	Dic₁₂₇	—	2	4	no	—
77	Dic₁₂₇	—	4	2	no	—
78	C₂×C₁₂₇	#475	1	4	yes	complemented
79	C₂×C₁₂₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
80	C₂×C₁₂₇	C₂²×SD₁₆	1	1	yes	characteristic
81	C₂×C₁₂₇	C₂×#137	1	1	yes	characteristic
82	C₂×#475	—	127	1	no	maximal, Sylow
83	C₁₂₇	C₂×#475	1	1	yes	characteristic, complemented, Sylow
84	C₂×C₄⋊Q₈	D₁₂₇	1	1	yes	characteristic
85	C₂×C₄×Q₈	—	127	1	no	—
86	#475	—	127	8	no	—
87	C₂×#106	—	127	2	no	—
88	C₂×C₄⋊C₈	—	127	1	no	—
89	C₂×#103	—	127	2	no	—
90	C₂²×Q₈	—	2	1	no	—
91	C₂²×Q₈	—	127	1	no	—
92	C₂²×C₈	—	127	2	no	—
93	C₄⋊Q₈	—	2	2	no	—
94	C₄⋊Q₈	C₂×D₁₂₇	1	4	yes	—
95	C₄×Q₈	—	127	4	no	—
96	C₄×Q₈	—	254	2	no	—
97	C₂×C₄⋊C₄	—	2	1	no	—
98	C₂×C₄⋊C₄	—	127	1	no	—
99	C₂×C₄⋊C₄	—	254	1	no	—
100	C₂×C₄⋊C₄	C₂×D₁₂₇	1	2	yes	—
101	C₂×C₄²	—	254	1	no	—
102	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
103	#106	—	127	8	no	—
104	C₄⋊C₈	—	127	4	no	—
105	#103	—	127	8	no	—
106	C₂×Q₈	—	2	4	no	—
107	C₂×Q₈	—	4	4	no	—
108	C₂×Q₈	—	127	4	no	—
109	C₂×Q₈	—	127	2	no	—
110	C₂×Q₈	—	254	1	no	—
111	C₂×Q₈	—	254	2	no	—
112	C₂²×C₄	—	2	2	no	—
113	C₂²×C₄	—	254	1	no	—
114	C₂²×C₄	—	254	1	no	—
115	C₂²×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
116	C₂²×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
117	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
118	C₂×C₈	—	127	4	no	—
119	C₂×C₈	—	254	4	no	—
120	C₄⋊C₄	—	2	4	no	—
121	C₄⋊C₄	—	2	2	no	—
122	C₄⋊C₄	—	127	2	no	—
123	C₄⋊C₄	—	254	4	no	—
124	C₄⋊C₄	—	254	1	no	—
125	C₄⋊C₄	C₂²×D₁₂₇	1	4	yes	—
126	C₄²	—	254	4	no	—
127	C₄²	C₂²⋊D₁₂₇	1	2	yes	—
128	C₄²	C₂²×D₁₂₇	1	2	yes	—
129	C₂³	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
130	Q₈	—	4	8	no	—
131	Q₈	—	127	4	no	—
132	Q₈	—	254	4	no	—
133	Q₈	—	254	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	4	4	no	—
137	C₂×C₄	—	508	2	no	—
138	C₂×C₄	—	254	2	no	—
139	C₂×C₄	—	254	4	no	—
140	C₂×C₄	—	254	2	no	—
141	C₂×C₄	#14523236	1	4	yes	—
142	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	2	yes	—
143	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
144	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	2	yes	—
145	C₂×C₄	Q₈×D₁₂₇	1	2	yes	—
146	C₂×C₄	#14523249	1	2	yes	—
147	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
148	C₈	—	254	4	no	—
149	C₂²	#16861185	1	2	yes	—
150	C₂²	C₂×#14523236	1	2	yes	—
151	C₂²	#16861240	1	2	yes	—
152	C₂²	C₂²×C₂²⋊D₁₂₇	1	1	yes	characteristic
153	C₄	—	2	2	no	—
154	C₄	—	4	4	no	—
155	C₄	—	508	2	no	—
156	C₄	—	254	4	no	—
157	C₄	#16861185	1	2	yes	—
158	C₄	C₂×#14523236	1	4	yes	—
159	C₄	C₂×Q₈×D₁₂₇	1	1	yes	characteristic
160	C₄	C₂×#14523249	1	1	yes	characteristic
161	C₂	#22550158	1	4	yes	direct product factor
162	C₂	C₂×#16861185	1	1	yes	characteristic
163	C₂	C₂²×#14523236	1	1	yes	characteristic
164	C₂	C₂×#16861240	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₂₅₄

C₂

yes

maximal, characteristic

#22550158

C₂

yes

maximal, direct product factor

C₂×#16861140

C₂

yes

maximal

C₂×C₄⋊(C₁₂₇⋊C₈)

C₂

yes

maximal, characteristic

C₂×#16861132

C₂

yes

maximal

C₂×C₄⋊Q₈×C₁₂₇

C₂

yes

maximal, characteristic

C₂²×Dic₂₅₄

C₂²

yes

characteristic

C₂²×C₁₂₇⋊C₈

C₂²

yes

—

C₄×Dic₂₅₄

—

C₄×Dic₂₅₄

C₂²

yes

—

C₂×C₄⋊Dic₁₂₇

C₂²

yes

characteristic

C₂×Dic₁₂₇⋊C₄

—

C₂×C₄×Dic₁₂₇

—

#16861140

C₂²

yes

—

C₄⋊(C₁₂₇⋊C₈)

C₂²

yes

—

#16861132

C₂²

yes

—

C₂²×Q₈×C₁₂₇

—

C₄⋊Q₈×C₁₂₇

—

C₄⋊Q₈×C₁₂₇

C₂²

yes

—

C₂×C₄⋊C₄×C₁₂₇

—

C₂×C₄⋊C₄×C₁₂₇

C₂²

yes

—

C₂×C₄²×C₁₂₇

C₂²

yes

characteristic

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

Q₈

yes

complemented

C₂×Dic₂₅₄

C₂³

yes

—

C₂²×Dic₁₂₇

—

C₂²×Dic₁₂₇

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

C₂³

yes

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

C₂³

yes

—

Dic₁₂₇⋊C₄

—

C₄×Dic₁₂₇

—

C₂×Q₈×C₁₂₇

—

C₂×Q₈×C₁₂₇

—

C₂²×C₄×C₁₂₇

—

C₂²×C₄×C₁₂₇

D₄

yes

characteristic

C₂²×C₄×C₁₂₇

D₄

yes

characteristic

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

C₂³

yes

—

C₄²×C₁₂₇

D₄

yes

—

C₄²×C₁₂₇

C₂³

yes

—

Dic₂₅₄

—

Dic₂₅₄

—

Dic₂₅₄

C₂×Q₈

yes

complemented

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

C₂×D₄

yes

characteristic

Q₈×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

SD₁₆

yes

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×D₄

yes

—

C₂×C₄×C₁₂₇

C₂×Q₈

yes

—

C₂×C₄×C₁₂₇

#41

yes

—

C₂×C₄×C₁₂₇

C₂⁴

yes

characteristic

C₂²×C₁₂₇

C₂²⋊Q₈

yes

—

C₂²×C₁₂₇

C₂×SD₁₆

yes

—

C₂²×C₁₂₇

#137

yes

—

C₂²×C₁₂₇

C₂²×D₄

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂²⋊Q₈

yes

—

C₄×C₁₂₇

C₂×SD₁₆

yes

—

C₄×C₁₂₇

C₂²×Q₈

yes

characteristic

C₄×C₁₂₇

C₂×#41

yes

characteristic

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

#475

yes

complemented

C₂×C₁₂₇

C₂×C₂²⋊Q₈

yes

characteristic

C₂×C₁₂₇

C₂²×SD₁₆

yes

C₂×C₁₂₇

yes

—

127

maximal, Sylow

C₁₂₇

C₂×#475

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

D₁₂₇

yes

characteristic

C₂×C₄×Q₈

—

127

—

#475

—

127

—

C₂×#106

—

127

—

C₂×C₄⋊C₈

—

127

—

C₂×#103

—

127

—

C₂²×Q₈

—

C₂²×Q₈

—

127

—

C₂²×C₈

—

127

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₁₂₇

yes

—

C₄×Q₈

—

127

—

C₄×Q₈

—

254

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

127

—

C₂×C₄⋊C₄

—

254

—

100

C₂×C₄⋊C₄

C₂×D₁₂₇

yes

—

101

C₂×C₄²

—

254

—

102

C₂×C₄²

C₂×D₁₂₇

yes

characteristic

103

#106

—

127

—

104

C₄⋊C₈

—

127

—

105

#103

—

127

—

106

C₂×Q₈

—

107

C₂×Q₈

—

108

C₂×Q₈

—

127

—

109

C₂×Q₈

—

127

—

110

C₂×Q₈

—

254

—

111

C₂×Q₈

—

254

—

112

C₂²×C₄

—

113

C₂²×C₄

—

254

—

114

C₂²×C₄

—

254

—

115

C₂²×C₄

C₂²⋊D₁₂₇

yes

characteristic

116

C₂²×C₄

C₂²⋊D₁₂₇

yes

characteristic

117

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

118

C₂×C₈

—

127

—

119

C₂×C₈

—

254

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

127

—

123

C₄⋊C₄

—

254

—

124

C₄⋊C₄

—

254

—

125

C₄⋊C₄

C₂²×D₁₂₇

yes

—

126

C₄²

—

254

—

127

C₄²

C₂²⋊D₁₂₇

yes

—

128

C₄²

C₂²×D₁₂₇

yes

—

129

C₂³

C₂×C₂²⋊D₁₂₇

yes

characteristic

130

Q₈

—

131

Q₈

—

127

—

132

Q₈

—

254

—

133

Q₈

—

254

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

508

—

138

C₂×C₄

—

254

—

139

C₂×C₄

—

254

—

140

C₂×C₄

—

254

—

141

C₂×C₄

#14523236

yes

—

142

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

—

143

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

characteristic

144

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

—

145

C₂×C₄

Q₈×D₁₂₇

yes

—

146

C₂×C₄

#14523249

yes

—

147

C₂×C₄

C₂³×D₁₂₇

yes

characteristic

148

C₈

—

254

—

149

C₂²

#16861185

yes

—

150

C₂²

C₂×#14523236

yes

—

151

C₂²

#16861240

yes

—

152

C₂²

C₂²×C₂²⋊D₁₂₇

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

508

—

156

C₄

—

254

—

157

C₄

#16861185

yes

—

158

C₄

C₂×#14523236

yes

—

159

C₄

C₂×Q₈×D₁₂₇

yes

characteristic

160

C₄

C₂×#14523249

yes

characteristic

161

C₂

#22550158

yes

direct product factor

162

C₂

C₂×#16861185

yes

characteristic

163

C₂

C₂²×#14523236

yes

characteristic

164

C₂

C₂×#16861240

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₂₇
commutator, cosocle	C₂×C₄×C₁₂₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	C₂×C₄⋊Q₈×C₁₂₇	C₂
socle, prime radical	C₂³×C₁₂₇	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₁₂₇
hypocenter	C₁₂₇	C₂×#475
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₂₇

commutator, cosocle

C₂×C₄×C₁₂₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₂₇

Fitting

C₂×C₄⋊Q₈×C₁₂₇

C₂

socle, prime radical

C₂³×C₁₂₇

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₁₂₇

hypocenter

C₁₂₇

C₂×#475

LCS

G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
2. #22550158	C₂	direct (N × H)
25. C₂×Dic₂₅₄	Q₈	semidirect
48. Dic₂₅₄	C₂×Q₈	semidirect
78. C₂×C₁₂₇	#475	semidirect
83. C₁₂₇	C₂×#475	semidirect
161. C₂	#22550158	direct (N × H)

H ≅ G/N

type

2. #22550158

C₂

direct (N × H)

25. C₂×Dic₂₅₄

Q₈

semidirect

48. Dic₂₅₄

C₂×Q₈

semidirect

78. C₂×C₁₂₇

#475

semidirect

83. C₁₂₇

C₂×#475

semidirect

161. C₂

#22550158

direct (N × H)

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

subgroup	index
C₂×C₄×Dic₂₅₄	2
#22550158	2
C₂×#16861140	2
C₂×C₄⋊(C₁₂₇⋊C₈)	2
C₂×#16861132	2
C₂×C₄⋊Q₈×C₁₂₇	2
C₂×#475	127

subgroup

index

C₂×C₄×Dic₂₅₄

C₂×C₄⋊(C₁₂₇⋊C₈)

C₂×C₄⋊Q₈×C₁₂₇

127

Subgroups — C₂×#22550158 #30648492

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#22550158 #30648492

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22550158 #30648492