Group #30638561 — subgroups

Subgroups — #30638561

Order 16256.

324 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 129 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	C₂×#16861134	C₂	1	2	yes	maximal
3	C₄×C₄⋊Dic₁₂₇	C₂	1	1	yes	maximal, characteristic
4	#22549639	C₂	1	2	yes	maximal
5	C₂×#97×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₅₄	C₂²	1	1	yes	characteristic
7	C₄×Dic₂₅₄	—	2	2	no	—
8	C₄×Dic₂₅₄	C₄	1	4	yes	complemented
9	C₂×C₄⋊Dic₁₂₇	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
11	C₂×Dic₁₂₇⋊C₄	—	2	1	no	—
12	C₂×C₄×Dic₁₂₇	—	2	1	no	—
13	C₂×C₄×Dic₁₂₇	—	2	1	no	—
14	#16861134	C₄	1	8	yes	complemented
15	C₁₂₇⋊#95	—	2	1	no	—
16	C₂²×C₈×C₁₂₇	C₂²	1	2	yes	—
17	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
18	#97×C₁₂₇	—	2	2	no	—
19	C₂×Dic₂₅₄	—	2	1	no	—
20	C₂×Dic₂₅₄	—	2	2	no	—
21	C₂×Dic₂₅₄	C₂×C₄	1	4	yes	—
22	C₂×Dic₂₅₄	C₂×C₄	1	2	yes	—
23	C₂²×Dic₁₂₇	—	2	1	no	—
24	C₂²×Dic₁₂₇	—	2	2	no	—
25	C₂²×Dic₁₂₇	—	2	1	no	—
26	C₄⋊Dic₁₂₇	—	2	1	no	—
27	C₄⋊Dic₁₂₇	—	2	2	no	—
28	C₄⋊Dic₁₂₇	C₂×C₄	1	4	yes	—
29	C₄⋊Dic₁₂₇	C₂×C₄	1	2	yes	—
30	Dic₁₂₇⋊C₄	—	2	4	no	—
31	C₄×Dic₁₂₇	—	2	4	no	—
32	C₄×Dic₁₂₇	—	4	2	no	—
33	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
36	C₂×C₈×C₁₂₇	—	2	2	no	—
37	C₂×C₈×C₁₂₇	—	2	2	no	—
38	C₂×C₈×C₁₂₇	C₂×C₄	1	4	yes	—
39	C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
40	C₄²×C₁₂₇	D₄	1	2	yes	—
41	Dic₂₅₄	—	2	2	no	—
42	Dic₂₅₄	—	2	4	no	—
43	Dic₂₅₄	C₄²	1	4	yes	complemented
44	C₂×Dic₁₂₇	—	2	2	no	—
45	C₂×Dic₁₂₇	—	2	4	no	—
46	C₂×Dic₁₂₇	—	2	2	no	—
47	C₂×Dic₁₂₇	—	2	4	no	—
48	C₂×Dic₁₂₇	—	4	2	no	—
49	C₂×Dic₁₂₇	—	4	1	no	—
50	C₂×Dic₁₂₇	—	4	1	no	—
51	C₂×Dic₁₂₇	—	4	2	no	—
52	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₁₂₇	—	2	1	no	—
54	C₂×C₄×C₁₂₇	—	2	1	no	—
55	C₂×C₄×C₁₂₇	—	2	2	no	—
56	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	2	yes	—
57	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	2	yes	—
58	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
60	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₁₂₇	#41	1	2	yes	—
62	C₈×C₁₂₇	—	2	4	no	—
63	C₂²×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₁₂₇	C₄×D₄	1	2	yes	—
65	C₂²×C₁₂₇	AGL(1,ℤ/8)	1	1	yes	characteristic
66	C₂²×C₁₂₇	#137	1	1	yes	characteristic
67	C₂²×C₁₂₇	#137	1	2	yes	—
68	C₄×C₁₂₇	—	2	2	no	—
69	C₄×C₁₂₇	—	2	2	no	—
70	C₄×C₁₂₇	C₂×C₄²	1	1	yes	characteristic
71	C₄×C₁₂₇	#117	1	1	yes	characteristic
72	C₄×C₁₂₇	C₄×D₄	1	2	yes	—
73	Dic₁₂₇	—	2	4	no	—
74	Dic₁₂₇	—	4	2	no	—
75	Dic₁₂₇	—	4	4	no	—
76	C₂×C₁₂₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₁₂₇	#417	1	1	yes	characteristic
78	C₂×C₁₂₇	#419	1	1	yes	characteristic
79	C₂×C₁₂₇	#440	1	2	yes	—
80	C₂×C₁₂₇	#441	1	2	yes	—
81	#1763	—	127	1	no	maximal, Sylow
82	C₁₂₇	#1763	1	1	yes	characteristic, complemented, Sylow
83	C₂×C₄×Q₈	—	127	1	no	—
84	C₂×#103	—	127	2	no	—
85	C₂×#97	D₁₂₇	1	1	yes	characteristic
86	C₄×C₄⋊C₄	—	127	1	no	—
87	#340	—	127	2	no	—
88	C₂²×Q₈	—	127	1	no	—
89	C₂²×C₈	C₂×D₁₂₇	1	2	yes	—
90	C₄×Q₈	—	127	4	no	—
91	C₄×Q₈	—	254	2	no	—
92	C₂×C₄⋊C₄	—	127	1	no	—
93	C₂×C₄⋊C₄	—	127	2	no	—
94	C₂×C₄⋊C₄	—	254	1	no	—
95	C₂×C₄²	—	254	1	no	—
96	C₂×C₄²	—	254	1	no	—
97	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
98	#103	—	127	8	no	—
99	#97	—	2	2	no	—
100	#95	—	254	1	no	—
101	C₂×Q₈	—	127	4	no	—
102	C₂×Q₈	—	127	2	no	—
103	C₂×Q₈	—	254	1	no	—
104	C₂×Q₈	—	254	2	no	—
105	C₂²×C₄	—	254	2	no	—
106	C₂²×C₄	—	254	1	no	—
107	C₂²×C₄	—	254	1	no	—
108	C₂²×C₄	D₅₀₈	1	1	yes	characteristic
109	C₂²×C₄	D₅₀₈	1	1	yes	characteristic
110	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	—	2	2	no	—
113	C₂×C₈	C₄×D₁₂₇	1	4	yes	—
114	C₄⋊C₄	—	127	4	no	—
115	C₄⋊C₄	—	127	2	no	—
116	C₄⋊C₄	—	254	4	no	—
117	C₄⋊C₄	—	254	1	no	—
118	C₄⋊C₄	—	254	2	no	—
119	C₄²	—	508	2	no	—
120	C₄²	—	254	4	no	—
121	C₄²	C₄×D₁₂₇	1	2	yes	—
122	C₄²	C₂²⋊D₁₂₇	1	2	yes	—
123	C₂³	C₂×D₅₀₈	1	1	yes	characteristic
124	Q₈	—	127	4	no	—
125	Q₈	—	254	2	no	—
126	Q₈	—	254	4	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	508	2	no	—
131	C₂×C₄	—	508	1	no	—
132	C₂×C₄	—	508	1	no	—
133	C₂×C₄	—	508	2	no	—
134	C₂×C₄	—	254	2	no	—
135	C₂×C₄	—	254	4	no	—
136	C₂×C₄	—	254	2	no	—
137	C₂×C₄	—	254	4	no	—
138	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	2	yes	—
139	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	2	yes	—
140	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
141	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
142	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
143	C₂×C₄	#14523248	1	2	yes	—
144	C₈	—	2	4	no	—
145	C₂²	C₂×(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
146	C₂²	C₄×D₅₀₈	1	2	yes	—
147	C₂²	#16861237	1	1	yes	characteristic
148	C₂²	#16861242	1	1	yes	characteristic
149	C₂²	#16861242	1	2	yes	—
150	C₄	—	2	2	no	—
151	C₄	—	2	2	no	—
152	C₄	—	508	2	no	—
153	C₄	—	508	4	no	—
154	C₄	—	254	4	no	—
155	C₄	C₄²×D₁₂₇	1	1	yes	characteristic
156	C₄	#16861166	1	1	yes	characteristic
157	C₄	C₄×C₂²⋊D₁₂₇	1	2	yes	—
158	C₂	C₄×(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
159	C₂	#22549878	1	1	yes	characteristic
160	C₂	#22549886	1	1	yes	characteristic
161	C₂	#22549978	1	2	yes	—
162	C₂	#22549984	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₂₅₄

C₂

yes

maximal, characteristic

C₂×#16861134

C₂

yes

maximal

C₄×C₄⋊Dic₁₂₇

C₂

yes

maximal, characteristic

#22549639

C₂

yes

maximal

C₂×#97×C₁₂₇

C₂

yes

maximal, characteristic

C₂²×Dic₂₅₄

C₂²

yes

characteristic

C₄×Dic₂₅₄

—

C₄×Dic₂₅₄

C₄

yes

complemented

C₂×C₄⋊Dic₁₂₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₂₇

C₂²

yes

—

C₂×Dic₁₂₇⋊C₄

—

C₂×C₄×Dic₁₂₇

—

C₂×C₄×Dic₁₂₇

—

#16861134

C₄

yes

complemented

C₁₂₇⋊#95

—

C₂²×C₈×C₁₂₇

C₂²

yes

—

C₂×C₄²×C₁₂₇

C₂²

yes

characteristic

#97×C₁₂₇

—

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

C₂×C₄

yes

—

C₂×Dic₂₅₄

C₂×C₄

yes

—

C₂²×Dic₁₂₇

—

C₂²×Dic₁₂₇

—

C₂²×Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

C₂×C₄

yes

—

C₄⋊Dic₁₂₇

C₂×C₄

yes

—

Dic₁₂₇⋊C₄

—

C₄×Dic₁₂₇

—

C₄×Dic₁₂₇

—

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

C₂×C₄

yes

—

C₄²×C₁₂₇

C₂×C₄

yes

—

C₄²×C₁₂₇

D₄

yes

—

Dic₂₅₄

—

Dic₂₅₄

—

Dic₂₅₄

C₄²

yes

complemented

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

C₂²⋊C₄

yes

—

C₂×C₄×C₁₂₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂²×C₄

yes

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

#41

yes

—

C₈×C₁₂₇

—

C₂²×C₁₂₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₂₇

C₄×D₄

yes

—

C₂²×C₁₂₇

yes

C₂²×C₁₂₇

yes

C₂²×C₁₂₇

yes

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂×C₄²

yes

characteristic

C₄×C₁₂₇

#117

yes

characteristic

C₄×C₁₂₇

C₄×D₄

yes

—

Dic₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

C₄×C₂²⋊C₄

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

—

C₂×C₁₂₇

#441

yes

—

#1763

—

127

maximal, Sylow

C₁₂₇

#1763

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

—

127

—

C₂×#103

—

127

—

C₂×#97

D₁₂₇

yes

characteristic

C₄×C₄⋊C₄

—

127

—

#340

—

127

—

C₂²×Q₈

—

127

—

C₂²×C₈

C₂×D₁₂₇

yes

—

C₄×Q₈

—

127

—

C₄×Q₈

—

254

—

C₂×C₄⋊C₄

—

127

—

C₂×C₄⋊C₄

—

127

—

C₂×C₄⋊C₄

—

254

—

C₂×C₄²

—

254

—

C₂×C₄²

—

254

—

C₂×C₄²

C₂×D₁₂₇

yes

characteristic

#103

—

127

—

#97

—

100

#95

—

254

—

101

C₂×Q₈

—

127

—

102

C₂×Q₈

—

127

—

103

C₂×Q₈

—

254

—

104

C₂×Q₈

—

254

—

105

C₂²×C₄

—

254

—

106

C₂²×C₄

—

254

—

107

C₂²×C₄

—

254

—

108

C₂²×C₄

D₅₀₈

yes

characteristic

109

C₂²×C₄

D₅₀₈

yes

characteristic

110

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

C₄×D₁₂₇

yes

—

114

C₄⋊C₄

—

127

—

115

C₄⋊C₄

—

127

—

116

C₄⋊C₄

—

254

—

117

C₄⋊C₄

—

254

—

118

C₄⋊C₄

—

254

—

119

C₄²

—

508

—

120

C₄²

—

254

—

121

C₄²

C₄×D₁₂₇

yes

—

122

C₄²

C₂²⋊D₁₂₇

yes

—

123

C₂³

C₂×D₅₀₈

yes

characteristic

124

Q₈

—

127

—

125

Q₈

—

254

—

126

Q₈

—

254

—

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

—

254

—

135

C₂×C₄

—

254

—

136

C₂×C₄

—

254

—

137

C₂×C₄

—

254

—

138

C₂×C₄

(C₂×D₁₂₇)⋊C₄

yes

—

139

C₂×C₄

(C₂×D₁₂₇)⋊C₄

yes

—

140

C₂×C₄

C₂×C₄×D₁₂₇

yes

characteristic

141

C₂×C₄

C₂×C₄×D₁₂₇

yes

—

142

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

characteristic

143

C₂×C₄

#14523248

yes

—

144

C₈

—

145

C₂²

C₂×(C₂×D₁₂₇)⋊C₄

yes

characteristic

146

C₂²

C₄×D₅₀₈

yes

—

147

C₂²

#16861237

yes

characteristic

148

C₂²

#16861242

yes

characteristic

149

C₂²

#16861242

yes

—

150

C₄

—

151

C₄

—

152

C₄

—

508

—

153

C₄

—

508

—

154

C₄

—

254

—

155

C₄

C₄²×D₁₂₇

yes

characteristic

156

C₄

#16861166

yes

characteristic

157

C₄

C₄×C₂²⋊D₁₂₇

yes

—

158

C₂

C₄×(C₂×D₁₂₇)⋊C₄

yes

characteristic

159

C₂

#22549878

yes

characteristic

160

C₂

#22549886

yes

characteristic

161

C₂

#22549978

yes

—

162

C₂

#22549984

yes

—

	subgroup	G/H
center	C₂³	C₂×D₅₀₈
commutator	C₄×C₁₂₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	C₂×#97×C₁₂₇	C₂
socle, prime radical	C₂³×C₁₂₇	C₂×D₄
cosocle	C₂²×C₄×C₁₂₇	C₂³
hypercenter	C₂×#97	D₁₂₇
hypocenter	C₁₂₇	#1763
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

C₂×D₅₀₈

commutator

C₄×C₁₂₇

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₁₂₇

Fitting

C₂×#97×C₁₂₇

C₂

socle, prime radical

C₂³×C₁₂₇

C₂×D₄

cosocle

C₂²×C₄×C₁₂₇

C₂³

hypercenter

C₂×#97

D₁₂₇

hypocenter

C₁₂₇

#1763

LCS

G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
8. C₄×Dic₂₅₄	C₄	semidirect
14. #16861134	C₄	semidirect
43. Dic₂₅₄	C₄²	semidirect
82. C₁₂₇	#1763	semidirect

H ≅ G/N

type

8. C₄×Dic₂₅₄

C₄

semidirect

14. #16861134

C₄

semidirect

43. Dic₂₅₄

C₄²

semidirect

82. C₁₂₇

#1763

semidirect

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

subgroup	index
C₂×C₄×Dic₂₅₄	2
C₂×#16861134	2
C₄×C₄⋊Dic₁₂₇	2
#22549639	2
C₂×#97×C₁₂₇	2
#1763	127

subgroup

index

C₂×C₄×Dic₂₅₄

C₂×#16861134

C₄×C₄⋊Dic₁₂₇

#22549639

C₂×#97×C₁₂₇

#1763

127

Subgroups — #30638561

Conjugacy classes of subgroups

Complements

Maximal subgroups