Group #30640058 — subgroups

Subgroups — #30640058

Order 16256.

282 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 97 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×#16861142	C₂	1	2	yes	maximal
3	#22549638	C₂	1	2	yes	maximal
4	#22549631	C₂	1	1	yes	maximal, characteristic
5	#389×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₅₄	—	2	1	no	—
7	C₂²×C₁₂₇⋊C₈	C₂²	1	2	yes	—
8	#16861209	—	2	4	no	—
9	C₂×C₄⋊Dic₁₂₇	—	2	1	no	—
10	C₂×C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
11	#16861142	—	2	4	no	—
12	C₂×C₄⋊C₄×C₁₂₇	—	2	1	no	—
13	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
16	#95×C₁₂₇	—	2	1	no	—
17	C₂×Dic₂₅₄	—	2	4	no	—
18	C₂×Dic₂₅₄	—	4	2	no	—
19	C₂×Dic₂₅₄	—	4	2	no	—
20	C₂²×Dic₁₂₇	—	2	2	no	—
21	C₂×C₁₂₇⋊C₈	—	2	2	no	—
22	C₂×C₁₂₇⋊C₈	—	2	2	no	—
23	C₂×C₁₂₇⋊C₈	Q₈	1	4	yes	complemented
24	C₄⋊Dic₁₂₇	—	2	4	no	—
25	C₄⋊Dic₁₂₇	—	2	2	no	—
26	C₄⋊Dic₁₂₇	D₄	1	4	yes	—
27	C₂²×C₄×C₁₂₇	—	2	1	no	—
28	C₂²×C₄×C₁₂₇	—	2	1	no	—
29	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
30	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
32	C₄⋊C₄×C₁₂₇	—	2	1	no	—
33	C₄⋊C₄×C₁₂₇	—	2	2	no	—
34	C₄⋊C₄×C₁₂₇	—	4	2	no	—
35	C₄⋊C₄×C₁₂₇	Q₈	1	2	yes	complemented
36	C₄²×C₁₂₇	—	2	2	no	—
37	Dic₂₅₄	—	4	8	no	—
38	C₂×Dic₁₂₇	—	2	4	no	—
39	C₂×Dic₁₂₇	—	4	2	no	—
40	C₂×Dic₁₂₇	—	4	2	no	—
41	C₁₂₇⋊C₈	—	2	4	no	—
42	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
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	1	no	—
47	C₂×C₄×C₁₂₇	—	2	2	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	—	4	1	no	—
53	C₂×C₄×C₁₂₇	D₈	1	2	yes	—
54	C₂×C₄×C₁₂₇	Q₁₆	1	2	yes	—
55	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
56	C₂×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
57	C₂×C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₄×C₁₂₇	#41	1	1	yes	characteristic
59	C₂²×C₁₂₇	#120	1	1	yes	characteristic
60	C₂²×C₁₂₇	C₂²⋊Q₈	1	1	yes	characteristic
61	C₂²×C₁₂₇	C₄⋊Q₈	1	1	yes	characteristic
62	C₂²×C₁₂₇	C₂×D₈	1	1	yes	characteristic
63	C₂²×C₁₂₇	C₂×Q₁₆	1	1	yes	characteristic
64	C₂²×C₁₂₇	AGL(1,ℤ/8)	1	1	yes	characteristic
65	C₂²×C₁₂₇	#137	1	1	yes	characteristic
66	C₄×C₁₂₇	—	2	2	no	—
67	C₄×C₁₂₇	—	2	2	no	—
68	C₄×C₁₂₇	—	4	2	no	—
69	C₄×C₁₂₇	—	8	1	no	—
70	C₄×C₁₂₇	C₂²⋊Q₈	1	2	yes	—
71	C₄×C₁₂₇	C₄⋊Q₈	1	2	yes	—
72	Dic₁₂₇	—	4	4	no	—
73	C₂×C₁₂₇	#395	1	1	yes	characteristic
74	C₂×C₁₂₇	#447	1	1	yes	characteristic
75	C₂×C₁₂₇	#451	1	1	yes	characteristic
76	C₂×C₁₂₇	#474	1	1	yes	characteristic
77	C₂×C₁₂₇	#477	1	1	yes	characteristic
78	C₂×C₁₂₇	#500	1	1	yes	characteristic
79	C₂×C₁₂₇	#501	1	1	yes	characteristic
80	#2057	—	127	1	no	maximal, Sylow
81	C₁₂₇	#2057	1	1	yes	characteristic, complemented, Sylow
82	C₂×C₄⋊Q₈	—	127	1	no	—
83	C₂×#107	—	127	2	no	—
84	#389	D₁₂₇	1	1	yes	characteristic
85	#340	—	127	2	no	—
86	#336	—	127	1	no	—
87	C₂²×Q₈	—	254	1	no	—
88	C₂²×C₈	—	127	2	no	—
89	C₄⋊Q₈	—	254	4	no	—
90	C₂×C₄⋊C₄	—	2	1	no	—
91	C₂×C₄⋊C₄	—	127	2	no	—
92	C₂×C₄⋊C₄	—	254	1	no	—
93	C₂×C₄⋊C₄	C₂×D₁₂₇	1	1	yes	characteristic
94	C₂×C₄⋊C₄	C₂×D₁₂₇	1	1	yes	characteristic
95	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
96	#107	—	254	4	no	—
97	#95	—	2	1	no	—
98	C₂×Q₈	—	508	2	no	—
99	C₂×Q₈	—	508	2	no	—
100	C₂×Q₈	—	254	4	no	—
101	C₂²×C₄	—	2	1	no	—
102	C₂²×C₄	—	2	1	no	—
103	C₂²×C₄	—	254	2	no	—
104	C₂²×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
105	C₂²×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
106	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
107	C₂×C₈	—	127	4	no	—
108	C₂×C₈	—	254	2	no	—
109	C₂×C₈	—	254	2	no	—
110	C₄⋊C₄	—	2	1	no	—
111	C₄⋊C₄	—	2	2	no	—
112	C₄⋊C₄	—	4	2	no	—
113	C₄⋊C₄	—	127	4	no	—
114	C₄⋊C₄	—	254	4	no	—
115	C₄⋊C₄	—	254	2	no	—
116	C₄⋊C₄	Dic₂₅₄	1	2	yes	complemented
117	C₄²	—	2	2	no	—
118	C₂³	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
119	Q₈	—	508	8	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	4	1	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	—	508	2	no	—
131	C₂×C₄	—	508	2	no	—
132	C₂×C₄	—	254	4	no	—
133	C₂×C₄	#14523233	1	2	yes	—
134	C₂×C₄	#14523238	1	2	yes	—
135	C₂×C₄	D₄×D₁₂₇	1	2	yes	—
136	C₂×C₄	Q₈×D₁₂₇	1	2	yes	—
137	C₂×C₄	C₂×Dic₂₅₄	1	1	yes	characteristic
138	C₂×C₄	#14523248	1	1	yes	characteristic
139	C₈	—	254	4	no	—
140	C₂²	#16861177	1	1	yes	characteristic
141	C₂²	#16861188	1	1	yes	characteristic
142	C₂²	#16861207	1	1	yes	characteristic
143	C₂²	C₂×#14523233	1	1	yes	characteristic
144	C₂²	C₂×#14523238	1	1	yes	characteristic
145	C₂²	#16861234	1	1	yes	characteristic
146	C₂²	#16861239	1	1	yes	characteristic
147	C₄	—	2	2	no	—
148	C₄	—	2	2	no	—
149	C₄	—	4	2	no	—
150	C₄	—	8	1	no	—
151	C₄	—	508	4	no	—
152	C₄	#16861187	1	2	yes	—
153	C₄	#16861208	1	2	yes	—
154	C₂	#22549796	1	1	yes	characteristic
155	C₂	#22550009	1	1	yes	characteristic
156	C₂	#22550031	1	1	yes	characteristic
157	C₂	#22550156	1	1	yes	characteristic
158	C₂	#22550171	1	1	yes	characteristic
159	C₂	#22550279	1	1	yes	characteristic
160	C₂	#22550286	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16861209

C₂

yes

maximal, characteristic

C₂×#16861142

C₂

yes

maximal

#22549638

C₂

yes

maximal

#22549631

C₂

yes

maximal, characteristic

#389×C₁₂₇

C₂

yes

maximal, characteristic

C₂²×Dic₂₅₄

—

C₂²×C₁₂₇⋊C₈

C₂²

yes

—

#16861209

—

C₂×C₄⋊Dic₁₂₇

—

C₂×C₄⋊Dic₁₂₇

C₂²

yes

—

#16861142

—

C₂×C₄⋊C₄×C₁₂₇

—

C₂×C₄⋊C₄×C₁₂₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₂₇

C₂²

yes

characteristic

C₂×C₄²×C₁₂₇

C₂²

yes

characteristic

#95×C₁₂₇

—

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

—

C₂²×Dic₁₂₇

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

Q₈

yes

complemented

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

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

—

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

Q₈

yes

complemented

C₄²×C₁₂₇

—

Dic₂₅₄

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

C₂×D₄

yes

characteristic

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

D₈

yes

—

C₂×C₄×C₁₂₇

Q₁₆

yes

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

—

C₂×C₄×C₁₂₇

C₂×Q₈

yes

—

C₂×C₄×C₁₂₇

C₂×Q₈

yes

C₂×C₄×C₁₂₇

yes

C₂²×C₁₂₇

yes

C₂²×C₁₂₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₂₇

C₄⋊Q₈

yes

characteristic

C₂²×C₁₂₇

C₂×D₈

yes

characteristic

C₂²×C₁₂₇

C₂×Q₁₆

yes

C₂²×C₁₂₇

yes

C₂²×C₁₂₇

yes

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂²⋊Q₈

yes

—

C₄×C₁₂₇

C₄⋊Q₈

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

#2057

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

127

—

C₂×#107

—

127

—

#389

D₁₂₇

yes

characteristic

#340

—

127

—

#336

—

127

—

C₂²×Q₈

—

254

—

C₂²×C₈

—

127

—

C₄⋊Q₈

—

254

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

127

—

C₂×C₄⋊C₄

—

254

—

C₂×C₄⋊C₄

C₂×D₁₂₇

yes

characteristic

C₂×C₄⋊C₄

C₂×D₁₂₇

yes

characteristic

C₂×C₄²

C₂×D₁₂₇

yes

characteristic

#107

—

254

—

#95

—

C₂×Q₈

—

508

—

C₂×Q₈

—

508

—

100

C₂×Q₈

—

254

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

254

—

104

C₂²×C₄

C₂²⋊D₁₂₇

yes

characteristic

105

C₂²×C₄

C₂²⋊D₁₂₇

yes

characteristic

106

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

107

C₂×C₈

—

127

—

108

C₂×C₈

—

254

—

109

C₂×C₈

—

254

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

127

—

114

C₄⋊C₄

—

254

—

115

C₄⋊C₄

—

254

—

116

C₄⋊C₄

Dic₂₅₄

yes

complemented

117

C₄²

—

118

C₂³

C₂×C₂²⋊D₁₂₇

yes

characteristic

119

Q₈

—

508

—

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

—

129

C₂×C₄

—

130

C₂×C₄

—

508

—

131

C₂×C₄

—

508

—

132

C₂×C₄

—

254

—

133

C₂×C₄

#14523233

yes

—

134

C₂×C₄

#14523238

yes

—

135

C₂×C₄

D₄×D₁₂₇

yes

—

136

C₂×C₄

Q₈×D₁₂₇

yes

—

137

C₂×C₄

C₂×Dic₂₅₄

yes

characteristic

138

C₂×C₄

#14523248

yes

characteristic

139

C₈

—

254

—

140

C₂²

#16861177

yes

characteristic

141

C₂²

#16861188

yes

characteristic

142

C₂²

#16861207

yes

characteristic

143

C₂²

C₂×#14523233

yes

characteristic

144

C₂²

C₂×#14523238

yes

characteristic

145

C₂²

#16861234

yes

characteristic

146

C₂²

#16861239

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

508

—

152

C₄

#16861187

yes

—

153

C₄

#16861208

yes

—

154

C₂

#22549796

yes

characteristic

155

C₂

#22550009

yes

characteristic

156

C₂

#22550031

yes

characteristic

157

C₂

#22550156

yes

characteristic

158

C₂

#22550171

yes

characteristic

159

C₂

#22550279

yes

characteristic

160

C₂

#22550286

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₂₇
commutator, cosocle	C₂²×C₄×C₁₂₇	C₂³
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	#389×C₁₂₇	C₂
socle, prime radical	C₂³×C₁₂₇	C₂×D₄
hypercenter	#389	D₁₂₇
hypocenter	C₁₂₇	#2057
LCS	G ▷ C₂²×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #389

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₂₇

commutator, cosocle

C₂²×C₄×C₁₂₇

C₂³

Frattini

C₂²×C₄

C₂²×D₁₂₇

Fitting

#389×C₁₂₇

C₂

socle, prime radical

C₂³×C₁₂₇

C₂×D₄

D₁₂₇

C₁₂₇

G ▷ C₂²×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #389

N	H ≅ G/N	type
23. C₂×C₁₂₇⋊C₈	Q₈	semidirect
35. C₄⋊C₄×C₁₂₇	Q₈	semidirect
81. C₁₂₇	#2057	semidirect
116. C₄⋊C₄	Dic₂₅₄	semidirect

H ≅ G/N

type

23. C₂×C₁₂₇⋊C₈

Q₈

semidirect

35. C₄⋊C₄×C₁₂₇

Q₈

semidirect

81. C₁₂₇

#2057

semidirect

116. C₄⋊C₄

Dic₂₅₄

semidirect

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

subgroup	index
C₂×#16861209	2
C₂×#16861142	2
#22549638	2
#22549631	2
#389×C₁₂₇	2
#2057	127

subgroup

index

#389×C₁₂₇

127

Subgroups — #30640058

Conjugacy classes of subgroups

Complements

Maximal subgroups