Group #30640138 — subgroups

Subgroups — #30640138

Order 16256.

238 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 89 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×#16861133	C₂	1	2	yes	maximal
2	#22549746	C₂	1	1	yes	maximal, characteristic
3	#22549638	C₂	1	2	yes	maximal
4	#22549631	C₂	1	1	yes	maximal, characteristic
5	#391×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₂₇⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₁₂₇	—	2	1	no	—
8	C₂×C₄⋊Dic₁₂₇	—	2	1	no	—
9	C₂×C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
10	#16861133	—	2	4	no	—
11	C₂²×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
14	#95×C₁₂₇	—	2	1	no	—
15	#95×C₁₂₇	—	2	1	no	—
16	C₂²×Dic₁₂₇	—	2	2	no	—
17	C₂×C₁₂₇⋊C₈	—	2	2	no	—
18	C₂×C₁₂₇⋊C₈	—	2	2	no	—
19	C₂×C₁₂₇⋊C₈	—	2	2	no	—
20	C₄⋊Dic₁₂₇	—	2	2	no	—
21	C₄⋊Dic₁₂₇	—	2	2	no	—
22	C₄⋊Dic₁₂₇	—	4	2	no	—
23	C₄⋊Dic₁₂₇	—	4	2	no	—
24	C₂×Q₈×C₁₂₇	—	2	1	no	—
25	C₂×Q₈×C₁₂₇	—	2	2	no	—
26	C₂×Q₈×C₁₂₇	—	2	2	no	—
27	C₂×Q₈×C₁₂₇	D₄	1	2	yes	—
28	C₂²×C₄×C₁₂₇	—	2	1	no	—
29	C₂²×C₄×C₁₂₇	—	2	1	no	—
30	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₁₂₇	—	2	2	no	—
34	C₄²×C₁₂₇	—	2	2	no	—
35	C₂×Dic₁₂₇	—	4	2	no	—
36	C₂×Dic₁₂₇	—	4	2	no	—
37	C₂×Dic₁₂₇	—	4	2	no	—
38	C₁₂₇⋊C₈	—	4	2	no	—
39	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
40	Q₈×C₁₂₇	—	2	4	no	—
41	Q₈×C₁₂₇	—	4	2	no	—
42	C₂×C₄×C₁₂₇	—	2	1	no	—
43	C₂×C₄×C₁₂₇	—	2	1	no	—
44	C₂×C₄×C₁₂₇	—	2	1	no	—
45	C₂×C₄×C₁₂₇	—	2	1	no	—
46	C₂×C₄×C₁₂₇	—	2	2	no	—
47	C₂×C₄×C₁₂₇	—	4	1	no	—
48	C₂×C₄×C₁₂₇	—	4	1	no	—
49	C₂×C₄×C₁₂₇	—	4	1	no	—
50	C₂×C₄×C₁₂₇	—	4	1	no	—
51	C₂×C₄×C₁₂₇	—	4	1	no	—
52	C₂×C₄×C₁₂₇	SD₁₆	1	2	yes	—
53	C₂×C₄×C₁₂₇	Q₁₆	1	2	yes	—
54	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₁₂₇	#41	1	2	yes	—
56	C₂×C₄×C₁₂₇	#41	1	2	yes	—
57	C₂×C₄×C₁₂₇	#41	1	1	yes	characteristic
58	C₂²×C₁₂₇	#121	1	1	yes	characteristic
59	C₂²×C₁₂₇	#123	1	1	yes	characteristic
60	C₂²×C₁₂₇	#124	1	1	yes	characteristic
61	C₂²×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
62	C₂²×C₁₂₇	C₂×Q₁₆	1	1	yes	characteristic
63	C₂²×C₁₂₇	AGL(1,ℤ/8)	1	1	yes	characteristic
64	C₂²×C₁₂₇	#137	1	1	yes	characteristic
65	C₄×C₁₂₇	—	2	2	no	—
66	C₄×C₁₂₇	—	2	2	no	—
67	C₄×C₁₂₇	—	4	2	no	—
68	C₄×C₁₂₇	—	8	1	no	—
69	C₄×C₁₂₇	#123	1	2	yes	—
70	C₄×C₁₂₇	#126	1	2	yes	—
71	Dic₁₂₇	—	8	2	no	—
72	C₂×C₁₂₇	#397	1	1	yes	characteristic
73	C₂×C₁₂₇	#460	1	1	yes	characteristic
74	C₂×C₁₂₇	#462	1	1	yes	characteristic
75	C₂×C₁₂₇	#483	1	1	yes	characteristic
76	C₂×C₁₂₇	#484	1	1	yes	characteristic
77	C₂×C₁₂₇	#487	1	1	yes	characteristic
78	C₂×C₁₂₇	#489	1	1	yes	characteristic
79	#2072	—	127	1	no	maximal, Sylow
80	C₁₂₇	#2072	1	1	yes	characteristic, complemented, Sylow
81	C₂×#103	—	127	2	no	—
82	#391	D₁₂₇	1	1	yes	characteristic
83	#384	—	127	1	no	—
84	#340	—	127	2	no	—
85	#336	—	127	1	no	—
86	C₂²×Q₈	C₂×D₁₂₇	1	1	yes	characteristic
87	C₂²×C₈	—	127	2	no	—
88	C₂×C₄⋊C₄	—	127	2	no	—
89	C₂×C₄⋊C₄	—	254	1	no	—
90	C₂×C₄⋊C₄	—	254	1	no	—
91	C₂×C₄⋊C₄	C₂×D₁₂₇	1	1	yes	characteristic
92	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
93	#103	—	254	4	no	—
94	#95	—	2	1	no	—
95	#95	—	2	1	no	—
96	C₂×Q₈	—	2	1	no	—
97	C₂×Q₈	—	2	2	no	—
98	C₂×Q₈	—	2	2	no	—
99	C₂×Q₈	D₅₀₈	1	2	yes	—
100	C₂²×C₄	—	2	1	no	—
101	C₂²×C₄	—	2	1	no	—
102	C₂²×C₄	—	254	2	no	—
103	C₂²×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
104	C₂²×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
105	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
106	C₂×C₈	—	254	2	no	—
107	C₂×C₈	—	254	2	no	—
108	C₂×C₈	—	254	2	no	—
109	C₄⋊C₄	—	2	2	no	—
110	C₄⋊C₄	—	508	2	no	—
111	C₄⋊C₄	—	508	2	no	—
112	C₄⋊C₄	—	254	2	no	—
113	C₄⋊C₄	—	254	2	no	—
114	C₄²	—	2	2	no	—
115	C₂³	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
116	Q₈	—	2	4	no	—
117	Q₈	—	4	2	no	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	4	1	no	—
124	C₂×C₄	—	4	1	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	508	2	no	—
129	C₂×C₄	—	508	2	no	—
130	C₂×C₄	—	508	2	no	—
131	C₂×C₄	#14523235	1	2	yes	—
132	C₂×C₄	#14523238	1	2	yes	—
133	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
134	C₂×C₄	#14523247	1	2	yes	—
135	C₂×C₄	#14523247	1	2	yes	—
136	C₂×C₄	#14523248	1	1	yes	characteristic
137	C₈	—	508	2	no	—
138	C₂²	#16861183	1	1	yes	characteristic
139	C₂²	#16861190	1	1	yes	characteristic
140	C₂²	#16861196	1	1	yes	characteristic
141	C₂²	C₂×#14523235	1	1	yes	characteristic
142	C₂²	C₂×#14523238	1	1	yes	characteristic
143	C₂²	#16861234	1	1	yes	characteristic
144	C₂²	#16861239	1	1	yes	characteristic
145	C₄	—	2	2	no	—
146	C₄	—	2	2	no	—
147	C₄	—	4	2	no	—
148	C₄	—	8	1	no	—
149	C₄	—	1016	2	no	—
150	C₄	#16861193	1	2	yes	—
151	C₄	#16861202	1	2	yes	—
152	C₂	#22549801	1	1	yes	characteristic
153	C₂	#22550076	1	1	yes	characteristic
154	C₂	#22550086	1	1	yes	characteristic
155	C₂	#22550204	1	1	yes	characteristic
156	C₂	#22550209	1	1	yes	characteristic
157	C₂	#22550226	1	1	yes	characteristic
158	C₂	#22550235	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16861133

C₂

yes

maximal

#22549746

C₂

yes

maximal, characteristic

#22549638

C₂

yes

maximal

#22549631

C₂

yes

maximal, characteristic

#391×C₁₂₇

C₂

yes

maximal, characteristic

C₂²×C₁₂₇⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₁₂₇

—

C₂×C₄⋊Dic₁₂₇

—

C₂×C₄⋊Dic₁₂₇

C₂²

yes

—

#16861133

—

C₂²×Q₈×C₁₂₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₂₇

C₂²

yes

characteristic

C₂×C₄²×C₁₂₇

C₂²

yes

characteristic

#95×C₁₂₇

—

#95×C₁₂₇

—

C₂²×Dic₁₂₇

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

C₂×Q₈×C₁₂₇

—

C₂×Q₈×C₁₂₇

—

C₂×Q₈×C₁₂₇

—

C₂×Q₈×C₁₂₇

D₄

yes

—

C₂²×C₄×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

—

C₄²×C₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

C₂×D₄

yes

characteristic

Q₈×C₁₂₇

—

Q₈×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

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

SD₁₆

yes

—

C₂×C₄×C₁₂₇

Q₁₆

yes

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

#41

yes

—

C₂×C₄×C₁₂₇

#41

yes

—

C₂×C₄×C₁₂₇

yes

C₂²×C₁₂₇

yes

C₂²×C₁₂₇

yes

C₂²×C₁₂₇

yes

C₂²×C₁₂₇

C₂×SD₁₆

yes

characteristic

C₂²×C₁₂₇

C₂×Q₁₆

yes

C₂²×C₁₂₇

yes

C₂²×C₁₂₇

yes

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

#123

yes

—

C₄×C₁₂₇

#126

yes

—

Dic₁₂₇

—

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

—

127

maximal, Sylow

C₁₂₇

#2072

yes

characteristic, complemented, Sylow

C₂×#103

—

127

—

#391

D₁₂₇

yes

characteristic

#384

—

127

—

#340

—

127

—

#336

—

127

—

C₂²×Q₈

C₂×D₁₂₇

yes

characteristic

C₂²×C₈

—

127

—

C₂×C₄⋊C₄

—

127

—

C₂×C₄⋊C₄

—

254

—

C₂×C₄⋊C₄

—

254

—

C₂×C₄⋊C₄

C₂×D₁₂₇

yes

characteristic

C₂×C₄²

C₂×D₁₂₇

yes

characteristic

#103

—

254

—

#95

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

D₅₀₈

yes

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

—

254

—

103

C₂²×C₄

C₂²⋊D₁₂₇

yes

characteristic

104

C₂²×C₄

C₂²⋊D₁₂₇

yes

characteristic

105

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

106

C₂×C₈

—

254

—

107

C₂×C₈

—

254

—

108

C₂×C₈

—

254

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

508

—

111

C₄⋊C₄

—

508

—

112

C₄⋊C₄

—

254

—

113

C₄⋊C₄

—

254

—

114

C₄²

—

115

C₂³

C₂×C₂²⋊D₁₂₇

yes

characteristic

116

Q₈

—

117

Q₈

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

508

—

129

C₂×C₄

—

508

—

130

C₂×C₄

—

508

—

131

C₂×C₄

#14523235

yes

—

132

C₂×C₄

#14523238

yes

—

133

C₂×C₄

C₂×D₅₀₈

yes

characteristic

134

C₂×C₄

#14523247

yes

—

135

C₂×C₄

#14523247

yes

—

136

C₂×C₄

#14523248

yes

characteristic

137

C₈

—

508

—

138

C₂²

#16861183

yes

characteristic

139

C₂²

#16861190

yes

characteristic

140

C₂²

#16861196

yes

characteristic

141

C₂²

C₂×#14523235

yes

characteristic

142

C₂²

C₂×#14523238

yes

characteristic

143

C₂²

#16861234

yes

characteristic

144

C₂²

#16861239

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

1016

—

150

C₄

#16861193

yes

—

151

C₄

#16861202

yes

—

152

C₂

#22549801

yes

characteristic

153

C₂

#22550076

yes

characteristic

154

C₂

#22550086

yes

characteristic

155

C₂

#22550204

yes

characteristic

156

C₂

#22550209

yes

characteristic

157

C₂

#22550226

yes

characteristic

158

C₂

#22550235

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₂₇
commutator, cosocle	C₂²×C₄×C₁₂₇	C₂³
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	#391×C₁₂₇	C₂
socle, prime radical	C₂³×C₁₂₇	C₂×D₄
hypercenter	#391	D₁₂₇
hypocenter	C₁₂₇	#2072
LCS	G ▷ C₂²×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₂₇

commutator, cosocle

C₂²×C₄×C₁₂₇

C₂³

Frattini

C₂²×C₄

C₂²×D₁₂₇

Fitting

#391×C₁₂₇

C₂

socle, prime radical

C₂³×C₁₂₇

C₂×D₄

D₁₂₇

C₁₂₇

G ▷ C₂²×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

N	H ≅ G/N	type
80. C₁₂₇	#2072	semidirect

H ≅ G/N

type

80. C₁₂₇

#2072

semidirect

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

subgroup	index
C₂×#16861133	2
#22549746	2
#22549638	2
#22549631	2
#391×C₁₂₇	2
#2072	127

subgroup

index

#391×C₁₂₇

127

Subgroups — #30640138

Conjugacy classes of subgroups

Complements

Maximal subgroups