Group #30639872 — subgroups

Subgroups — #30639872

Order 16256.

290 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 91 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×#16861209	C₂	1	1	yes	maximal, characteristic
2	C₂×#16861134	C₂	1	2	yes	maximal
3	#22549782	C₂	1	1	yes	maximal, characteristic
4	#22549639	C₂	1	2	yes	maximal
5	#336×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₅₄	—	2	1	no	—
7	C₂²×Dic₂₅₄	C₂²	1	1	yes	characteristic
8	#16861209	—	2	4	no	—
9	C₂×C₄⋊Dic₁₂₇	—	2	1	no	—
10	C₂×C₄⋊Dic₁₂₇	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
12	#16861134	—	2	4	no	—
13	C₁₂₇⋊#95	—	2	1	no	—
14	C₁₂₇⋊#95	—	2	1	no	—
15	C₂²×C₈×C₁₂₇	C₂²	1	2	yes	—
16	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
17	C₂×Dic₂₅₄	—	2	4	no	—
18	C₂×Dic₂₅₄	—	2	2	no	—
19	C₂×Dic₂₅₄	—	2	1	no	—
20	C₂×Dic₂₅₄	—	2	2	no	—
21	C₂×Dic₂₅₄	—	4	2	no	—
22	C₂×Dic₂₅₄	—	4	2	no	—
23	C₂×Dic₂₅₄	Q₈	1	2	yes	complemented
24	C₂²×Dic₁₂₇	—	2	2	no	—
25	C₂²×Dic₁₂₇	—	2	1	no	—
26	C₂²×Dic₁₂₇	—	2	1	no	—
27	C₄⋊Dic₁₂₇	—	2	4	no	—
28	C₄⋊Dic₁₂₇	—	2	2	no	—
29	C₄⋊Dic₁₂₇	—	2	2	no	—
30	C₄⋊Dic₁₂₇	D₄	1	4	yes	—
31	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
34	C₂×C₈×C₁₂₇	—	2	2	no	—
35	C₂×C₈×C₁₂₇	—	2	2	no	—
36	C₂×C₈×C₁₂₇	—	2	2	no	—
37	C₄²×C₁₂₇	—	2	2	no	—
38	Dic₂₅₄	—	2	4	no	—
39	Dic₂₅₄	—	4	8	no	—
40	Dic₂₅₄	—	4	2	no	—
41	C₂×Dic₁₂₇	—	2	4	no	—
42	C₂×Dic₁₂₇	—	2	2	no	—
43	C₂×Dic₁₂₇	—	4	1	no	—
44	C₂×Dic₁₂₇	—	4	1	no	—
45	C₂×Dic₁₂₇	—	4	2	no	—
46	C₂×Dic₁₂₇	—	4	2	no	—
47	C₂×Dic₁₂₇	—	4	1	no	—
48	C₂×Dic₁₂₇	—	4	1	no	—
49	C₂×Dic₁₂₇	—	4	1	no	—
50	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₁₂₇	—	2	1	no	—
52	C₂×C₄×C₁₂₇	—	2	1	no	—
53	C₂×C₄×C₁₂₇	—	2	1	no	—
54	C₂×C₄×C₁₂₇	—	2	1	no	—
55	C₂×C₄×C₁₂₇	SD₁₆	1	2	yes	—
56	C₂×C₄×C₁₂₇	Q₁₆	1	2	yes	—
57	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
58	C₂×C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
59	C₂×C₄×C₁₂₇	#41	1	1	yes	characteristic
60	C₂×C₄×C₁₂₇	#41	1	2	yes	—
61	C₈×C₁₂₇	—	4	2	no	—
62	C₂²×C₁₂₇	#120	1	1	yes	characteristic
63	C₂²×C₁₂₇	C₂²⋊Q₈	1	1	yes	characteristic
64	C₂²×C₁₂₇	#124	1	1	yes	characteristic
65	C₂²×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
66	C₂²×C₁₂₇	C₂×Q₁₆	1	1	yes	characteristic
67	C₂²×C₁₂₇	AGL(1,ℤ/8)	1	1	yes	characteristic
68	C₂²×C₁₂₇	#137	1	1	yes	characteristic
69	C₄×C₁₂₇	—	2	2	no	—
70	C₄×C₁₂₇	—	2	2	no	—
71	C₄×C₁₂₇	C₂²⋊Q₈	1	2	yes	—
72	C₄×C₁₂₇	#124	1	2	yes	—
73	Dic₁₂₇	—	4	2	no	—
74	Dic₁₂₇	—	4	4	no	—
75	Dic₁₂₇	—	8	1	no	—
76	C₂×C₁₂₇	#393	1	1	yes	characteristic
77	C₂×C₁₂₇	#450	1	1	yes	characteristic
78	C₂×C₁₂₇	#451	1	1	yes	characteristic
79	C₂×C₁₂₇	#475	1	1	yes	characteristic
80	C₂×C₁₂₇	#477	1	1	yes	characteristic
81	C₂×C₁₂₇	#487	1	1	yes	characteristic
82	C₂×C₁₂₇	#489	1	1	yes	characteristic
83	#2024	—	127	1	no	maximal, Sylow
84	C₁₂₇	#2024	1	1	yes	characteristic, complemented, Sylow
85	C₂×C₄⋊Q₈	—	127	1	no	—
86	C₂×#103	—	127	2	no	—
87	#391	—	127	1	no	—
88	#340	—	127	2	no	—
89	#336	D₁₂₇	1	1	yes	characteristic
90	C₂²×Q₈	—	127	1	no	—
91	C₂²×Q₈	—	254	1	no	—
92	C₂²×C₈	C₂×D₁₂₇	1	2	yes	—
93	C₄⋊Q₈	—	254	4	no	—
94	C₂×C₄⋊C₄	—	127	2	no	—
95	C₂×C₄⋊C₄	—	127	1	no	—
96	C₂×C₄⋊C₄	—	254	1	no	—
97	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
98	#103	—	254	4	no	—
99	#95	—	254	1	no	—
100	#95	—	254	1	no	—
101	C₂×Q₈	—	127	2	no	—
102	C₂×Q₈	—	508	2	no	—
103	C₂×Q₈	—	508	2	no	—
104	C₂×Q₈	—	254	4	no	—
105	C₂×Q₈	—	254	2	no	—
106	C₂×Q₈	—	254	2	no	—
107	C₂×Q₈	—	254	1	no	—
108	C₂²×C₄	—	254	1	no	—
109	C₂²×C₄	—	254	1	no	—
110	C₂²×C₄	—	254	2	no	—
111	C₂²×C₄	D₅₀₈	1	1	yes	characteristic
112	C₂²×C₄	D₅₀₈	1	1	yes	characteristic
113	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	2	2	no	—
117	C₄⋊C₄	—	127	4	no	—
118	C₄⋊C₄	—	254	4	no	—
119	C₄⋊C₄	—	254	2	no	—
120	C₄⋊C₄	—	254	2	no	—
121	C₄²	—	2	2	no	—
122	C₂³	C₂×D₅₀₈	1	1	yes	characteristic
123	Q₈	—	508	8	no	—
124	Q₈	—	508	2	no	—
125	Q₈	—	254	4	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	508	1	no	—
131	C₂×C₄	—	508	1	no	—
132	C₂×C₄	—	508	1	no	—
133	C₂×C₄	—	508	1	no	—
134	C₂×C₄	—	508	2	no	—
135	C₂×C₄	—	508	2	no	—
136	C₂×C₄	—	508	1	no	—
137	C₂×C₄	—	254	2	no	—
138	C₂×C₄	—	254	4	no	—
139	C₂×C₄	#14523237	1	2	yes	—
140	C₂×C₄	Dic₅₀₈	1	2	yes	—
141	C₂×C₄	D₄×D₁₂₇	1	2	yes	—
142	C₂×C₄	Q₈×D₁₂₇	1	1	yes	characteristic
143	C₂×C₄	#14523247	1	1	yes	characteristic
144	C₂×C₄	#14523248	1	2	yes	—
145	C₈	—	4	2	no	—
146	C₂²	#16861179	1	1	yes	characteristic
147	C₂²	#16861189	1	1	yes	characteristic
148	C₂²	#16861197	1	1	yes	characteristic
149	C₂²	C₂×#14523237	1	1	yes	characteristic
150	C₂²	C₂×Dic₅₀₈	1	1	yes	characteristic
151	C₂²	#16861237	1	1	yes	characteristic
152	C₂²	#16861242	1	1	yes	characteristic
153	C₄	—	2	2	no	—
154	C₄	—	2	2	no	—
155	C₄	—	1016	1	no	—
156	C₄	—	508	4	no	—
157	C₄	—	508	2	no	—
158	C₄	#16861187	1	2	yes	—
159	C₄	#16861195	1	2	yes	—
160	C₂	#22549789	1	1	yes	characteristic
161	C₂	#22550029	1	1	yes	characteristic
162	C₂	#22550034	1	1	yes	characteristic
163	C₂	#22550162	1	1	yes	characteristic
164	C₂	#22550172	1	1	yes	characteristic
165	C₂	#22550227	1	1	yes	characteristic
166	C₂	#22550236	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16861209

C₂

yes

maximal, characteristic

C₂×#16861134

C₂

yes

maximal

#22549782

C₂

yes

maximal, characteristic

#22549639

C₂

yes

maximal

#336×C₁₂₇

C₂

yes

maximal, characteristic

C₂²×Dic₂₅₄

—

C₂²×Dic₂₅₄

C₂²

yes

characteristic

#16861209

—

C₂×C₄⋊Dic₁₂₇

—

C₂×C₄⋊Dic₁₂₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₂₇

C₂²

yes

—

#16861134

—

C₁₂₇⋊#95

—

C₁₂₇⋊#95

—

C₂²×C₈×C₁₂₇

C₂²

yes

—

C₂×C₄²×C₁₂₇

C₂²

yes

characteristic

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

Q₈

yes

complemented

C₂²×Dic₁₂₇

—

C₂²×Dic₁₂₇

—

C₂²×Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

D₄

yes

—

C₂²×C₄×C₁₂₇

D₄

yes

characteristic

C₂²×C₄×C₁₂₇

D₄

yes

characteristic

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₄²×C₁₂₇

—

Dic₂₅₄

—

Dic₂₅₄

—

Dic₂₅₄

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂³×C₁₂₇

C₂×D₄

yes

characteristic

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

—

C₂×C₄×C₁₂₇

C₂×Q₈

yes

C₂×C₄×C₁₂₇

yes

C₂×C₄×C₁₂₇

yes

—

C₈×C₁₂₇

—

C₂²×C₁₂₇

#120

yes

characteristic

C₂²×C₁₂₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₂₇

#124

yes

characteristic

C₂²×C₁₂₇

C₂×SD₁₆

yes

characteristic

C₂²×C₁₂₇

C₂×Q₁₆

yes

C₂²×C₁₂₇

yes

C₂²×C₁₂₇

yes

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂²⋊Q₈

yes

—

C₄×C₁₂₇

#124

yes

—

Dic₁₂₇

—

Dic₁₂₇

—

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

#2024

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

127

—

C₂×#103

—

127

—

#391

—

127

—

#340

—

127

—

#336

D₁₂₇

yes

characteristic

C₂²×Q₈

—

127

—

C₂²×Q₈

—

254

—

C₂²×C₈

C₂×D₁₂₇

yes

—

C₄⋊Q₈

—

254

—

C₂×C₄⋊C₄

—

127

—

C₂×C₄⋊C₄

—

127

—

C₂×C₄⋊C₄

—

254

—

C₂×C₄²

C₂×D₁₂₇

yes

characteristic

#103

—

254

—

#95

—

254

—

100

#95

—

254

—

101

C₂×Q₈

—

127

—

102

C₂×Q₈

—

508

—

103

C₂×Q₈

—

508

—

104

C₂×Q₈

—

254

—

105

C₂×Q₈

—

254

—

106

C₂×Q₈

—

254

—

107

C₂×Q₈

—

254

—

108

C₂²×C₄

—

254

—

109

C₂²×C₄

—

254

—

110

C₂²×C₄

—

254

—

111

C₂²×C₄

D₅₀₈

yes

characteristic

112

C₂²×C₄

D₅₀₈

yes

characteristic

113

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₄⋊C₄

—

127

—

118

C₄⋊C₄

—

254

—

119

C₄⋊C₄

—

254

—

120

C₄⋊C₄

—

254

—

121

C₄²

—

122

C₂³

C₂×D₅₀₈

yes

characteristic

123

Q₈

—

508

—

124

Q₈

—

508

—

125

Q₈

—

254

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

508

—

131

C₂×C₄

—

508

—

132

C₂×C₄

—

508

—

133

C₂×C₄

—

508

—

134

C₂×C₄

—

508

—

135

C₂×C₄

—

508

—

136

C₂×C₄

—

508

—

137

C₂×C₄

—

254

—

138

C₂×C₄

—

254

—

139

C₂×C₄

#14523237

yes

—

140

C₂×C₄

Dic₅₀₈

yes

—

141

C₂×C₄

D₄×D₁₂₇

yes

—

142

C₂×C₄

Q₈×D₁₂₇

yes

characteristic

143

C₂×C₄

#14523247

yes

characteristic

144

C₂×C₄

#14523248

yes

—

145

C₈

—

146

C₂²

#16861179

yes

characteristic

147

C₂²

#16861189

yes

characteristic

148

C₂²

#16861197

yes

characteristic

149

C₂²

C₂×#14523237

yes

characteristic

150

C₂²

C₂×Dic₅₀₈

yes

characteristic

151

C₂²

#16861237

yes

characteristic

152

C₂²

#16861242

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

1016

—

156

C₄

—

508

—

157

C₄

—

508

—

158

C₄

#16861187

yes

—

159

C₄

#16861195

yes

—

160

C₂

#22549789

yes

characteristic

161

C₂

#22550029

yes

characteristic

162

C₂

#22550034

yes

characteristic

163

C₂

#22550162

yes

characteristic

164

C₂

#22550172

yes

characteristic

165

C₂

#22550227

yes

characteristic

166

C₂

#22550236

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₅₀₈
commutator, cosocle	C₂²×C₄×C₁₂₇	C₂³
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	#336×C₁₂₇	C₂
socle, prime radical	C₂³×C₁₂₇	C₂×D₄
hypercenter	#336	D₁₂₇
hypocenter	C₁₂₇	#2024
LCS	G ▷ C₂²×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×D₅₀₈

commutator, cosocle

C₂²×C₄×C₁₂₇

C₂³

Frattini

C₂²×C₄

C₂²×D₁₂₇

Fitting

#336×C₁₂₇

C₂

socle, prime radical

C₂³×C₁₂₇

C₂×D₄

D₁₂₇

C₁₂₇

G ▷ C₂²×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
23. C₂×Dic₂₅₄	Q₈	semidirect
84. C₁₂₇	#2024	semidirect

H ≅ G/N

type

23. C₂×Dic₂₅₄

Q₈

semidirect

84. C₁₂₇

#2024

semidirect

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

subgroup	index
C₂×#16861209	2
C₂×#16861134	2
#22549782	2
#22549639	2
#336×C₁₂₇	2
#2024	127

subgroup

index

#336×C₁₂₇

127

Subgroups — #30639872

Conjugacy classes of subgroups

Complements

Maximal subgroups