Group #30640055 — subgroups

Subgroups — #30640055

Order 16256.

282 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 99 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×#16861142	C₂	1	2	yes	maximal
2	#22549774	C₂	1	1	yes	maximal, characteristic
3	#22549638	C₂	1	2	yes	maximal
4	#22549631	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Q₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₂₇⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₁₂₇	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₁₂₇	C₂²	1	1	yes	characteristic
9	C₂×Dic₁₂₇⋊C₄	—	2	1	no	—
10	#16861142	—	2	4	no	—
11	C₁₂₇⋊#95	—	2	1	no	—
12	C₂²×Q₈×C₁₂₇	—	2	1	no	—
13	C₄⋊Q₈×C₁₂₇	—	2	4	no	—
14	C₂×C₄⋊C₄×C₁₂₇	—	2	1	no	—
15	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
16	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
17	C₂²×Dic₁₂₇	—	2	1	no	—
18	C₂²×Dic₁₂₇	—	2	1	no	—
19	C₂×C₁₂₇⋊C₈	—	2	2	no	—
20	C₂×C₁₂₇⋊C₈	—	2	2	no	—
21	C₂×C₁₂₇⋊C₈	Q₈	1	4	yes	complemented
22	C₄⋊Dic₁₂₇	—	2	2	no	—
23	C₄⋊Dic₁₂₇	—	2	1	no	—
24	C₄⋊Dic₁₂₇	Q₈	1	2	yes	complemented
25	Dic₁₂₇⋊C₄	—	4	2	no	—
26	C₂×Q₈×C₁₂₇	—	2	4	no	—
27	C₂×Q₈×C₁₂₇	—	4	2	no	—
28	C₂×Q₈×C₁₂₇	—	4	2	no	—
29	C₂²×C₄×C₁₂₇	—	2	2	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	4	no	—
34	C₄⋊C₄×C₁₂₇	—	2	2	no	—
35	C₄⋊C₄×C₁₂₇	D₄	1	4	yes	—
36	C₄²×C₁₂₇	—	2	2	no	—
37	C₂×Dic₁₂₇	—	2	2	no	—
38	C₂×Dic₁₂₇	—	4	1	no	—
39	C₂×Dic₁₂₇	—	4	1	no	—
40	C₂×Dic₁₂₇	—	4	1	no	—
41	C₂×Dic₁₂₇	—	4	1	no	—
42	C₂×Dic₁₂₇	—	4	1	no	—
43	C₁₂₇⋊C₈	—	2	4	no	—
44	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
45	Q₈×C₁₂₇	—	4	8	no	—
46	C₂×C₄×C₁₂₇	—	2	1	no	—
47	C₂×C₄×C₁₂₇	—	2	1	no	—
48	C₂×C₄×C₁₂₇	—	2	1	no	—
49	C₂×C₄×C₁₂₇	—	2	1	no	—
50	C₂×C₄×C₁₂₇	—	2	4	no	—
51	C₂×C₄×C₁₂₇	—	4	2	no	—
52	C₂×C₄×C₁₂₇	—	4	2	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	4	no	—
69	C₄×C₁₂₇	C₂²⋊Q₈	1	2	yes	—
70	C₄×C₁₂₇	C₄⋊Q₈	1	2	yes	—
71	Dic₁₂₇	—	4	2	no	—
72	Dic₁₂₇	—	8	1	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	D₁₂₇	1	1	yes	characteristic
83	C₂×#107	—	127	2	no	—
84	#389	—	127	1	no	—
85	#340	—	127	2	no	—
86	#336	—	127	1	no	—
87	C₂²×Q₈	—	2	1	no	—
88	C₂²×C₈	—	127	2	no	—
89	C₄⋊Q₈	—	2	4	no	—
90	C₂×C₄⋊C₄	—	2	1	no	—
91	C₂×C₄⋊C₄	—	127	1	no	—
92	C₂×C₄⋊C₄	—	127	1	no	—
93	C₂×C₄⋊C₄	—	254	1	no	—
94	C₂×C₄⋊C₄	C₂×D₁₂₇	1	2	yes	—
95	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
96	#107	—	254	4	no	—
97	#95	—	254	1	no	—
98	C₂×Q₈	—	2	4	no	—
99	C₂×Q₈	—	4	2	no	—
100	C₂×Q₈	—	4	2	no	—
101	C₂²×C₄	—	2	2	no	—
102	C₂²×C₄	—	254	1	no	—
103	C₂²×C₄	—	254	1	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	4	no	—
111	C₄⋊C₄	—	2	2	no	—
112	C₄⋊C₄	—	127	2	no	—
113	C₄⋊C₄	—	508	2	no	—
114	C₄⋊C₄	—	254	2	no	—
115	C₄⋊C₄	—	254	1	no	—
116	C₄⋊C₄	C₂²⋊D₁₂₇	1	4	yes	—
117	C₄²	—	2	2	no	—
118	C₂³	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
119	Q₈	—	4	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	4	no	—
125	C₂×C₄	—	4	2	no	—
126	C₂×C₄	—	4	2	no	—
127	C₂×C₄	—	508	1	no	—
128	C₂×C₄	—	508	1	no	—
129	C₂×C₄	—	508	1	no	—
130	C₂×C₄	—	508	1	no	—
131	C₂×C₄	—	508	1	no	—
132	C₂×C₄	—	254	2	no	—
133	C₂×C₄	#14523233	1	2	yes	—
134	C₂×C₄	#14523238	1	2	yes	—
135	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	2	yes	—
136	C₂×C₄	Q₈×D₁₂₇	1	2	yes	—
137	C₂×C₄	Q₈×D₁₂₇	1	1	yes	characteristic
138	C₂×C₄	#14523249	1	1	yes	characteristic
139	C₈	—	254	4	no	—
140	C₂²	#16861178	1	1	yes	characteristic
141	C₂²	#16861185	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	#16861235	1	1	yes	characteristic
146	C₂²	#16861240	1	1	yes	characteristic
147	C₄	—	2	2	no	—
148	C₄	—	2	2	no	—
149	C₄	—	4	4	no	—
150	C₄	—	1016	1	no	—
151	C₄	—	508	2	no	—
152	C₄	#16861185	1	2	yes	—
153	C₄	#16861207	1	2	yes	—
154	C₂	#22549795	1	1	yes	characteristic
155	C₂	#22550010	1	1	yes	characteristic
156	C₂	#22550032	1	1	yes	characteristic
157	C₂	#22550153	1	1	yes	characteristic
158	C₂	#22550168	1	1	yes	characteristic
159	C₂	#22550279	1	1	yes	characteristic
160	C₂	#22550282	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16861142

C₂

yes

maximal

#22549774

C₂

yes

maximal, characteristic

#22549638

C₂

yes

maximal

#22549631

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₁₂₇

C₂

yes

maximal, characteristic

C₂²×C₁₂₇⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₁₂₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₂₇

C₂²

yes

characteristic

C₂×Dic₁₂₇⋊C₄

—

#16861142

—

C₁₂₇⋊#95

—

C₂²×Q₈×C₁₂₇

—

C₄⋊Q₈×C₁₂₇

—

C₂×C₄⋊C₄×C₁₂₇

—

C₂×C₄⋊C₄×C₁₂₇

C₂²

yes

—

C₂×C₄²×C₁₂₇

C₂²

yes

characteristic

C₂²×Dic₁₂₇

—

C₂²×Dic₁₂₇

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

Q₈

yes

complemented

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

Q₈

yes

complemented

Dic₁₂₇⋊C₄

—

C₂×Q₈×C₁₂₇

—

C₂×Q₈×C₁₂₇

—

C₂×Q₈×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

D₄

yes

—

C₄²×C₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

C₂×D₄

yes

characteristic

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

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

C₂²⋊Q₈

yes

—

C₄×C₁₂₇

C₄⋊Q₈

yes

—

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

#2057

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

D₁₂₇

yes

characteristic

C₂×#107

—

127

—

#389

—

127

—

#340

—

127

—

#336

—

127

—

C₂²×Q₈

—

C₂²×C₈

—

127

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

127

—

C₂×C₄⋊C₄

—

127

—

C₂×C₄⋊C₄

—

254

—

C₂×C₄⋊C₄

C₂×D₁₂₇

yes

—

C₂×C₄²

C₂×D₁₂₇

yes

characteristic

#107

—

254

—

#95

—

254

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂²×C₄

—

102

C₂²×C₄

—

254

—

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

—

127

—

113

C₄⋊C₄

—

508

—

114

C₄⋊C₄

—

254

—

115

C₄⋊C₄

—

254

—

116

C₄⋊C₄

C₂²⋊D₁₂₇

yes

—

117

C₄²

—

118

C₂³

C₂×C₂²⋊D₁₂₇

yes

characteristic

119

Q₈

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

508

—

128

C₂×C₄

—

508

—

129

C₂×C₄

—

508

—

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

C₂×C₂²⋊D₁₂₇

yes

—

136

C₂×C₄

Q₈×D₁₂₇

yes

—

137

C₂×C₄

Q₈×D₁₂₇

yes

characteristic

138

C₂×C₄

#14523249

yes

characteristic

139

C₈

—

254

—

140

C₂²

#16861178

yes

characteristic

141

C₂²

#16861185

yes

characteristic

142

C₂²

#16861207

yes

characteristic

143

C₂²

C₂×#14523233

yes

characteristic

144

C₂²

C₂×#14523238

yes

characteristic

145

C₂²

#16861235

yes

characteristic

146

C₂²

#16861240

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

1016

—

151

C₄

—

508

—

152

C₄

#16861185

yes

—

153

C₄

#16861207

yes

—

154

C₂

#22549795

yes

characteristic

155

C₂

#22550010

yes

characteristic

156

C₂

#22550032

yes

characteristic

157

C₂

#22550153

yes

characteristic

158

C₂

#22550168

yes

characteristic

159

C₂

#22550279

yes

characteristic

160

C₂

#22550282

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₂₇
commutator, cosocle	C₂²×C₄×C₁₂₇	C₂³
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	C₂×C₄⋊Q₈×C₁₂₇	C₂
socle, prime radical	C₂³×C₁₂₇	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₁₂₇
hypocenter	C₁₂₇	#2057
LCS	G ▷ C₂²×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₂₇

commutator, cosocle

C₂²×C₄×C₁₂₇

C₂³

Frattini

C₂²×C₄

C₂²×D₁₂₇

Fitting

C₂×C₄⋊Q₈×C₁₂₇

C₂

socle, prime radical

C₂³×C₁₂₇

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₁₂₇

hypocenter

C₁₂₇

#2057

LCS

G ▷ C₂²×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
21. C₂×C₁₂₇⋊C₈	Q₈	semidirect
24. C₄⋊Dic₁₂₇	Q₈	semidirect
81. C₁₂₇	#2057	semidirect

H ≅ G/N

type

21. C₂×C₁₂₇⋊C₈

Q₈

semidirect

24. C₄⋊Dic₁₂₇

Q₈

semidirect

81. C₁₂₇

#2057

semidirect

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

subgroup	index
C₂×#16861142	2
#22549774	2
#22549638	2
#22549631	2
C₂×C₄⋊Q₈×C₁₂₇	2
#2057	127

subgroup

index

C₂×C₄⋊Q₈×C₁₂₇

127

Subgroups — #30640055

Conjugacy classes of subgroups

Complements

Maximal subgroups