Group #30639529 — subgroups

Subgroups — #30639529

Order 16256.

384 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 121 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⋊(C₁₂₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂²⋊(C₁₂₇⋊C₈)	C₂	1	4	yes	maximal, complemented
3	C₂×#16861120	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄×D₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₂₇⋊C₈	C₂²	1	2	yes	—
6	C₂²×C₁₂₇⋊C₈	C₂²	1	2	yes	—
7	C₄⋊(C₁₂₇⋊C₈)	—	2	2	no	—
8	C₂²⋊(C₁₂₇⋊C₈)	—	2	8	no	—
9	#16861120	—	2	2	no	—
10	C₂²×D₄×C₁₂₇	C₄	1	1	yes	characteristic
11	C₂³×C₄×C₁₂₇	C₂²	1	2	yes	—
12	C₄×D₄×C₁₂₇	—	2	2	no	—
13	C₄×D₄×C₁₂₇	—	2	2	no	—
14	C₂×C₄⋊C₄×C₁₂₇	C₄	1	1	yes	characteristic
15	C₂×C₂²⋊C₄×C₁₂₇	C₄	1	2	yes	—
16	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
17	C₂×C₁₂₇⋊C₈	—	2	2	no	—
18	C₂×C₁₂₇⋊C₈	—	2	4	no	—
19	C₂×C₁₂₇⋊C₈	—	2	2	no	—
20	C₂×C₁₂₇⋊C₈	—	2	2	no	—
21	C₂×C₁₂₇⋊C₈	D₄	1	4	yes	complemented
22	C₂⁴×C₁₂₇	C₂×C₄	1	2	yes	—
23	C₂×D₄×C₁₂₇	—	2	4	no	—
24	C₂×D₄×C₁₂₇	—	2	2	no	—
25	C₂²×C₄×C₁₂₇	—	2	2	no	—
26	C₂²×C₄×C₁₂₇	—	2	2	no	—
27	C₂²×C₄×C₁₂₇	—	2	4	no	—
28	C₂²×C₄×C₁₂₇	—	2	4	no	—
29	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₁₂₇	—	2	2	no	—
34	C₂²⋊C₄×C₁₂₇	—	2	4	no	—
35	C₄²×C₁₂₇	D₄	1	2	yes	—
36	C₄²×C₁₂₇	D₄	1	2	yes	—
37	C₁₂₇⋊C₈	—	2	4	no	—
38	C₁₂₇⋊C₈	—	4	2	no	—
39	C₂³×C₁₂₇	—	2	2	no	—
40	C₂³×C₁₂₇	—	2	4	no	—
41	C₂³×C₁₂₇	—	2	2	no	—
42	C₂³×C₁₂₇	—	2	4	no	—
43	C₂³×C₁₂₇	M₁₆	1	4	yes	—
44	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
45	D₄×C₁₂₇	—	2	8	no	—
46	C₂×C₄×C₁₂₇	—	2	2	no	—
47	C₂×C₄×C₁₂₇	—	2	2	no	—
48	C₂×C₄×C₁₂₇	—	2	2	no	—
49	C₂×C₄×C₁₂₇	—	2	4	no	—
50	C₂×C₄×C₁₂₇	—	4	4	no	—
51	C₂×C₄×C₁₂₇	—	4	2	no	—
52	C₂×C₄×C₁₂₇	—	4	2	no	—
53	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	2	yes	—
54	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	2	yes	—
55	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
56	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₁₂₇	#41	1	2	yes	—
59	C₂²×C₁₂₇	—	2	4	no	—
60	C₂²×C₁₂₇	—	2	8	no	—
61	C₂²×C₁₂₇	—	4	2	no	—
62	C₂²×C₁₂₇	—	4	2	no	—
63	C₂²×C₁₂₇	—	4	4	no	—
64	C₂²×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₁₂₇	C₄×D₄	1	2	yes	—
66	C₂²×C₁₂₇	C₂×M₁₆	1	2	yes	—
67	C₂²×C₁₂₇	#131	1	2	yes	—
68	C₄×C₁₂₇	—	2	2	no	—
69	C₄×C₁₂₇	—	2	2	no	—
70	C₄×C₁₂₇	—	4	2	no	—
71	C₄×C₁₂₇	#121	1	2	yes	—
72	C₄×C₁₂₇	#124	1	1	yes	characteristic
73	C₄×C₁₂₇	#127	1	1	yes	characteristic
74	C₂×C₁₂₇	—	4	4	no	—
75	C₂×C₁₂₇	#390	1	1	yes	characteristic
76	C₂×C₁₂₇	#407	1	1	yes	characteristic
77	C₂×C₁₂₇	#408	1	1	yes	characteristic
78	C₂×C₁₂₇	#435	1	4	yes	—
79	#1955	—	127	1	no	maximal, Sylow
80	C₁₂₇	#1955	1	1	yes	characteristic, complemented, Sylow
81	C₂×C₄×D₄	D₁₂₇	1	1	yes	characteristic
82	C₂×C₄⋊C₈	—	127	1	no	—
83	C₂×C₂²⋊C₈	—	127	4	no	—
84	C₂×#97	—	127	1	no	—
85	C₂²×D₄	Dic₁₂₇	1	1	yes	characteristic
86	C₂³×C₄	C₂×D₁₂₇	1	2	yes	—
87	C₂²×C₈	—	127	2	no	—
88	C₂²×C₈	—	127	2	no	—
89	C₄×D₄	—	2	2	no	—
90	C₄×D₄	—	2	2	no	—
91	C₂×C₄⋊C₄	Dic₁₂₇	1	1	yes	characteristic
92	C₂×C₂²⋊C₄	Dic₁₂₇	1	2	yes	—
93	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
94	C₄⋊C₈	—	254	2	no	—
95	C₂²⋊C₈	—	254	8	no	—
96	#97	—	254	2	no	—
97	C₂⁴	C₂×Dic₁₂₇	1	2	yes	—
98	C₂×D₄	—	2	4	no	—
99	C₂×D₄	—	2	2	no	—
100	C₂²×C₄	—	2	4	no	—
101	C₂²×C₄	—	2	2	no	—
102	C₂²×C₄	—	2	2	no	—
103	C₂²×C₄	—	2	4	no	—
104	C₂²×C₄	C₂×Dic₁₂₇	1	2	yes	—
105	C₂²×C₄	C₂×Dic₁₂₇	1	1	yes	characteristic
106	C₂²×C₄	C₂×Dic₁₂₇	1	1	yes	characteristic
107	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
108	C₂×C₈	—	127	4	no	—
109	C₂×C₈	—	254	4	no	—
110	C₂×C₈	—	254	2	no	—
111	C₂×C₈	—	254	2	no	—
112	C₂×C₈	—	254	2	no	—
113	C₄⋊C₄	—	2	2	no	—
114	C₂²⋊C₄	—	2	4	no	—
115	C₄²	C₂²⋊D₁₂₇	1	2	yes	—
116	C₄²	C₂²⋊D₁₂₇	1	2	yes	—
117	C₂³	—	2	2	no	—
118	C₂³	—	2	4	no	—
119	C₂³	—	2	2	no	—
120	C₂³	—	2	4	no	—
121	C₂³	#14523232	1	4	yes	—
122	C₂³	C₂²×Dic₁₂₇	1	1	yes	characteristic
123	D₄	—	2	8	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	4	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	—	4	2	no	—
130	C₂×C₄	—	4	4	no	—
131	C₂×C₄	C₂²⋊Dic₁₂₇	1	2	yes	—
132	C₂×C₄	C₂²⋊Dic₁₂₇	1	2	yes	—
133	C₂×C₄	D₄×D₁₂₇	1	2	yes	—
134	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
135	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
136	C₂×C₄	#14523247	1	2	yes	—
137	C₈	—	508	2	no	—
138	C₈	—	254	4	no	—
139	C₂²	—	2	4	no	—
140	C₂²	—	2	8	no	—
141	C₂²	—	4	2	no	—
142	C₂²	—	4	2	no	—
143	C₂²	—	4	4	no	—
144	C₂²	C₂×C₂²⋊Dic₁₂₇	1	1	yes	characteristic
145	C₂²	D₄×Dic₁₂₇	1	2	yes	—
146	C₂²	C₂×#14523232	1	2	yes	—
147	C₂²	#16861216	1	2	yes	—
148	C₄	—	2	2	no	—
149	C₄	—	2	2	no	—
150	C₄	—	4	2	no	—
151	C₄	#16861180	1	2	yes	—
152	C₄	#16861196	1	1	yes	characteristic
153	C₄	#16861205	1	1	yes	characteristic
154	C₂	—	4	4	no	—
155	C₂	#22549779	1	1	yes	characteristic
156	C₂	#22549840	1	1	yes	characteristic
157	C₂	#22549843	1	1	yes	characteristic
158	C₂	#22549948	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₁₂₇⋊C₈)

C₂

yes

maximal, characteristic, complemented

C₂×C₂²⋊(C₁₂₇⋊C₈)

C₂

yes

maximal, complemented

C₂×#16861120

C₂

yes

maximal, characteristic, complemented

C₂×C₄×D₄×C₁₂₇

C₂

yes

maximal, characteristic

C₂²×C₁₂₇⋊C₈

C₂²

yes

—

C₂²×C₁₂₇⋊C₈

C₂²

yes

—

C₄⋊(C₁₂₇⋊C₈)

—

C₂²⋊(C₁₂₇⋊C₈)

—

#16861120

—

C₂²×D₄×C₁₂₇

C₄

yes

characteristic

C₂³×C₄×C₁₂₇

C₂²

yes

—

C₄×D₄×C₁₂₇

—

C₄×D₄×C₁₂₇

—

C₂×C₄⋊C₄×C₁₂₇

C₄

yes

characteristic

C₂×C₂²⋊C₄×C₁₂₇

C₄

yes

—

C₂×C₄²×C₁₂₇

C₂²

yes

characteristic

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

D₄

yes

complemented

C₂⁴×C₁₂₇

C₂×C₄

yes

—

C₂×D₄×C₁₂₇

—

C₂×D₄×C₁₂₇

—

C₂²×C₄×C₁₂₇

—

C₂²×C₄×C₁₂₇

—

C₂²×C₄×C₁₂₇

—

C₂²×C₄×C₁₂₇

—

C₂²×C₄×C₁₂₇

C₂×C₄

yes

—

C₂²×C₄×C₁₂₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₂₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₂₇

—

C₂²⋊C₄×C₁₂₇

—

C₄²×C₁₂₇

D₄

yes

—

C₄²×C₁₂₇

D₄

yes

—

C₁₂₇⋊C₈

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

—

C₂³×C₁₂₇

—

C₂³×C₁₂₇

—

C₂³×C₁₂₇

—

C₂³×C₁₂₇

M₁₆

yes

—

C₂³×C₁₂₇

C₂²×C₄

yes

characteristic

D₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₂₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

#41

yes

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₂₇

C₄×D₄

yes

—

C₂²×C₁₂₇

C₂×M₁₆

yes

—

C₂²×C₁₂₇

#131

yes

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

#121

yes

—

C₄×C₁₂₇

yes

C₄×C₁₂₇

yes

C₂×C₁₂₇

—

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

—

#1955

—

127

maximal, Sylow

C₁₂₇

#1955

yes

characteristic, complemented, Sylow

C₂×C₄×D₄

D₁₂₇

yes

characteristic

C₂×C₄⋊C₈

—

127

—

C₂×C₂²⋊C₈

—

127

—

C₂×#97

—

127

—

C₂²×D₄

Dic₁₂₇

yes

characteristic

C₂³×C₄

C₂×D₁₂₇

yes

—

C₂²×C₈

—

127

—

C₂²×C₈

—

127

—

C₄×D₄

—

C₄×D₄

—

C₂×C₄⋊C₄

Dic₁₂₇

yes

characteristic

C₂×C₂²⋊C₄

Dic₁₂₇

yes

—

C₂×C₄²

C₂×D₁₂₇

yes

characteristic

C₄⋊C₈

—

254

—

C₂²⋊C₈

—

254

—

#97

—

254

—

C₂⁴

C₂×Dic₁₂₇

yes

—

C₂×D₄

—

C₂×D₄

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

C₂×Dic₁₂₇

yes

—

105

C₂²×C₄

C₂×Dic₁₂₇

yes

characteristic

106

C₂²×C₄

C₂×Dic₁₂₇

yes

characteristic

107

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

108

C₂×C₈

—

127

—

109

C₂×C₈

—

254

—

110

C₂×C₈

—

254

—

111

C₂×C₈

—

254

—

112

C₂×C₈

—

254

—

113

C₄⋊C₄

—

114

C₂²⋊C₄

—

115

C₄²

C₂²⋊D₁₂₇

yes

—

116

C₄²

C₂²⋊D₁₂₇

yes

—

117

C₂³

—

118

C₂³

—

119

C₂³

—

120

C₂³

—

121

C₂³

#14523232

yes

—

122

C₂³

C₂²×Dic₁₂₇

yes

characteristic

123

D₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

C₂²⋊Dic₁₂₇

yes

—

132

C₂×C₄

C₂²⋊Dic₁₂₇

yes

—

133

C₂×C₄

D₄×D₁₂₇

yes

—

134

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

characteristic

135

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

characteristic

136

C₂×C₄

#14523247

yes

—

137

C₈

—

508

—

138

C₈

—

254

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

C₂×C₂²⋊Dic₁₂₇

yes

characteristic

145

C₂²

D₄×Dic₁₂₇

yes

—

146

C₂²

C₂×#14523232

yes

—

147

C₂²

#16861216

yes

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

#16861180

yes

—

152

C₄

#16861196

yes

characteristic

153

C₄

#16861205

yes

characteristic

154

C₂

—

155

C₂

#22549779

yes

characteristic

156

C₂

#22549840

yes

characteristic

157

C₂

#22549843

yes

characteristic

158

C₂

#22549948

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₂₇
commutator, socle	C₂³×C₁₂₇	C₂²×C₄
Fitting	C₂×C₄×D₄×C₁₂₇	C₂
prime radical	C₂²×D₄×C₁₂₇	C₄
cosocle	C₂²×C₄×C₁₂₇	C₂³
hypercenter	C₂×C₄×D₄	D₁₂₇
hypocenter	C₁₂₇	#1955
LCS	G ▷ C₂³×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×D₄

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₂₇

commutator, socle

C₂³×C₁₂₇

C₂²×C₄

Fitting

C₂×C₄×D₄×C₁₂₇

C₂

prime radical

C₂²×D₄×C₁₂₇

C₄

cosocle

C₂²×C₄×C₁₂₇

C₂³

hypercenter

C₂×C₄×D₄

D₁₂₇

C₁₂₇

G ▷ C₂³×C₁₂₇ ▷ C₁₂₇

C₁ ◁ C₂²×C₄ ◁ C₂×C₄×D₄

N	H ≅ G/N	type
1. C₂×C₄⋊(C₁₂₇⋊C₈)	C₂	semidirect
2. C₂×C₂²⋊(C₁₂₇⋊C₈)	C₂	semidirect
3. C₂×#16861120	C₂	semidirect
21. C₂×C₁₂₇⋊C₈	D₄	semidirect
80. C₁₂₇	#1955	semidirect

H ≅ G/N

type

1. C₂×C₄⋊(C₁₂₇⋊C₈)

C₂

semidirect

2. C₂×C₂²⋊(C₁₂₇⋊C₈)

C₂

semidirect

3. C₂×#16861120

C₂

semidirect

21. C₂×C₁₂₇⋊C₈

D₄

semidirect

80. C₁₂₇

#1955

semidirect

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

subgroup	index
C₂×C₄⋊(C₁₂₇⋊C₈)	2
C₂×C₂²⋊(C₁₂₇⋊C₈)	2
C₂×#16861120	2
C₂×C₄×D₄×C₁₂₇	2
#1955	127

subgroup

index

C₂×C₄⋊(C₁₂₇⋊C₈)

C₂×C₂²⋊(C₁₂₇⋊C₈)

C₂×#16861120

C₂×C₄×D₄×C₁₂₇

#1955

127

Subgroups — #30639529

Conjugacy classes of subgroups

Complements

Maximal subgroups