Group #30640784 — subgroups

Subgroups — #30640784

Order 16256.

170 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 47 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×#16861154	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16861150	C₂	1	1	yes	maximal, characteristic
3	#22550259	C₂	1	1	yes	maximal, characteristic, complemented
4	#22549958	C₂	1	1	yes	maximal, characteristic, complemented
5	Dic₁₂₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	#22549679	C₂	1	1	yes	maximal, characteristic, complemented
7	#357×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₅₀₈	C₂²	1	1	yes	characteristic
9	C₂×#14523236	—	2	1	no	—
10	C₂×D₁₀₁₆	C₂²	1	1	yes	characteristic
11	#16861196	—	2	1	no	—
12	#16861171	—	2	1	no	—
13	#16861154	—	2	2	no	—
14	#16861150	—	2	2	no	—
15	C₂×C₁₂₇⋊C₁₆	C₂²	1	1	yes	characteristic
16	#16861129	—	2	1	no	—
17	C₈×Dic₁₂₇	C₂²	1	1	yes	characteristic
18	C₂×D₈×C₁₂₇	C₂²	1	1	yes	characteristic
19	C₂×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
20	#107×C₁₂₇	C₂²	1	1	yes	characteristic
21	C₂×Dic₂₅₄	—	2	1	no	—
22	C₂×D₅₀₈	—	2	1	no	—
23	C₂×C₄×D₁₂₇	—	4	1	no	—
24	Dic₅₀₈	—	2	1	no	—
25	Dic₅₀₈	—	2	1	no	—
26	#14523236	—	4	2	no	—
27	D₁₀₁₆	—	2	1	no	—
28	D₁₀₁₆	D₄	1	2	yes	—
29	C₂×C₁₂₇⋊C₈	D₄	1	1	yes	characteristic
30	C₂²⋊Dic₁₂₇	—	4	1	no	—
31	C₂²⋊Dic₁₂₇	—	4	1	no	—
32	(C₂×D₁₂₇)⋊C₄	—	4	1	no	—
33	C₄×Dic₁₂₇	D₄	1	1	yes	characteristic
34	C₁₂₇⋊C₁₆	—	2	1	no	—
35	C₂×D₄×C₁₂₇	—	2	1	no	—
36	D₈×C₁₂₇	—	2	1	no	—
37	D₈×C₁₂₇	—	2	1	no	—
38	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
39	C₄⋊C₄×C₁₂₇	—	2	1	no	—
40	C₁₆×C₁₂₇	—	2	1	no	—
41	C₂²×D₁₂₇	—	4	1	no	—
42	Dic₂₅₄	—	4	1	no	—
43	Dic₂₅₄	—	4	1	no	—
44	D₅₀₈	—	2	2	no	—
45	D₅₀₈	—	4	1	no	—
46	C₂×Dic₁₂₇	—	4	1	no	—
47	C₂×Dic₁₂₇	D₈	1	1	yes	characteristic
48	C₂×Dic₁₂₇	D₈	1	1	yes	characteristic
49	C₄×D₁₂₇	—	8	2	no	—
50	C₁₂₇⋊C₈	—	2	1	no	—
51	C₂³×C₁₂₇	—	4	1	no	—
52	D₄×C₁₂₇	—	4	1	no	—
53	D₄×C₁₂₇	—	4	1	no	—
54	C₂×C₄×C₁₂₇	—	4	1	no	—
55	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
56	C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
57	C₈×C₁₂₇	#41	1	1	yes	characteristic
58	C₂²×C₁₂₇	—	8	1	no	—
59	C₂²×C₁₂₇	—	8	1	no	—
60	C₂²×C₁₂₇	—	8	1	no	—
61	C₂²×C₁₂₇	C₂×D₈	1	1	yes	characteristic
62	C₂×D₁₂₇	—	4	2	no	—
63	C₂×D₁₂₇	—	8	1	no	—
64	C₂×D₁₂₇	—	8	1	no	—
65	C₄×C₁₂₇	—	8	1	no	—
66	C₄×C₁₂₇	#121	1	1	yes	characteristic
67	C₄×C₁₂₇	AGL(1,ℤ/8)	1	1	yes	characteristic
68	Dic₁₂₇	—	2	1	no	—
69	Dic₁₂₇	—	2	1	no	—
70	Dic₁₂₇	—	8	1	no	—
71	C₂×C₁₂₇	—	16	1	no	—
72	C₂×C₁₂₇	#459	1	1	yes	characteristic
73	C₂×C₁₂₇	#508	1	1	yes	characteristic
74	C₂×C₁₂₇	#509	1	1	yes	characteristic
75	D₁₂₇	—	8	2	no	—
76	#2210	—	127	1	no	maximal, Sylow
77	C₁₂₇	#2210	1	1	yes	characteristic, complemented, Sylow
78	C₂×SD₃₂	—	127	1	no	—
79	C₂×D₁₆	—	127	1	no	—
80	#495	—	127	1	no	—
81	C₄×D₈	—	127	1	no	—
82	C₄⋊C₁₆	—	127	1	no	—
83	#358	—	127	1	no	—
84	#357	D₁₂₇	1	1	yes	characteristic, complemented
85	C₂×Q₁₆	—	127	1	no	—
86	C₂×SD₁₆	—	254	1	no	—
87	C₂×D₈	—	127	1	no	—
88	C₂×D₈	C₂×D₁₂₇	1	1	yes	characteristic
89	#124	—	254	1	no	—
90	C₄×D₄	—	254	1	no	—
91	SD₃₂	—	254	2	no	—
92	D₁₆	—	254	2	no	—
93	C₂×C₁₆	—	127	1	no	—
94	C₂×C₁₆	C₂×D₁₂₇	1	1	yes	characteristic
95	#107	C₂×D₁₂₇	1	1	yes	characteristic
96	#102	—	254	1	no	—
97	C₄×C₈	—	127	1	no	—
98	C₂×Q₈	—	254	1	no	—
99	C₂×D₄	—	2	1	no	—
100	C₂×D₄	—	254	1	no	—
101	C₂²×C₄	—	508	1	no	—
102	Q₁₆	—	254	1	no	—
103	Q₁₆	—	254	1	no	—
104	SD₁₆	—	508	2	no	—
105	D₈	—	2	1	no	—
106	D₈	—	2	1	no	—
107	D₈	—	127	2	no	—
108	D₈	—	254	1	no	—
109	C₂×C₈	—	127	1	no	—
110	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
111	C₄⋊C₄	—	2	1	no	—
112	C₂²⋊C₄	—	508	1	no	—
113	C₂²⋊C₄	—	508	1	no	—
114	C₂²⋊C₄	—	508	1	no	—
115	C₄²	—	127	1	no	—
116	C₁₆	—	2	1	no	—
117	C₁₆	—	254	1	no	—
118	C₂³	—	4	1	no	—
119	C₂³	—	508	1	no	—
120	Q₈	—	508	1	no	—
121	Q₈	—	508	1	no	—
122	D₄	—	4	1	no	—
123	D₄	—	4	1	no	—
124	D₄	—	508	1	no	—
125	D₄	—	254	2	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	127	1	no	—
128	C₂×C₄	—	127	1	no	—
129	C₂×C₄	—	1016	2	no	—
130	C₂×C₄	—	508	1	no	—
131	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
132	C₈	—	254	1	no	—
133	C₈	D₄×D₁₂₇	1	1	yes	characteristic
134	C₈	#14523248	1	1	yes	characteristic
135	C₂²	—	8	1	no	—
136	C₂²	—	8	1	no	—
137	C₂²	—	8	1	no	—
138	C₂²	—	1016	1	no	—
139	C₂²	—	1016	1	no	—
140	C₂²	—	508	2	no	—
141	C₂²	D₈×D₁₂₇	1	1	yes	characteristic
142	C₄	—	8	1	no	—
143	C₄	—	1016	1	no	—
144	C₄	—	254	1	no	—
145	C₄	—	254	1	no	—
146	C₄	#16861182	1	1	yes	characteristic
147	C₄	#16861236	1	1	yes	characteristic
148	C₂	—	16	1	no	—
149	C₂	—	1016	2	no	—
150	C₂	#22550070	1	1	yes	characteristic
151	C₂	#22550307	1	1	yes	characteristic
152	C₂	#22550310	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16861154

C₂

yes

maximal, characteristic, complemented

C₂×#16861150

C₂

yes

maximal, characteristic

#22550259

C₂

yes

maximal, characteristic, complemented

#22549958

C₂

yes

maximal, characteristic, complemented

Dic₁₂₇⋊C₁₆

C₂

yes

maximal, characteristic, complemented

#22549679

C₂

yes

maximal, characteristic, complemented

#357×C₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂×Dic₅₀₈

C₂²

yes

characteristic

C₂×#14523236

—

C₂×D₁₀₁₆

C₂²

yes

—

—

—

—

C₂×C₁₂₇⋊C₁₆

C₂²

yes

characteristic

#16861129

—

C₈×Dic₁₂₇

C₂²

yes

characteristic

C₂×D₈×C₁₂₇

C₂²

yes

characteristic

C₂×C₁₆×C₁₂₇

C₂²

yes

characteristic

#107×C₁₂₇

C₂²

yes

characteristic

C₂×Dic₂₅₄

—

C₂×D₅₀₈

—

C₂×C₄×D₁₂₇

—

Dic₅₀₈

—

Dic₅₀₈

—

#14523236

—

D₁₀₁₆

—

D₁₀₁₆

D₄

yes

—

C₂×C₁₂₇⋊C₈

D₄

yes

characteristic

C₂²⋊Dic₁₂₇

—

C₂²⋊Dic₁₂₇

—

(C₂×D₁₂₇)⋊C₄

—

C₄×Dic₁₂₇

D₄

yes

characteristic

C₁₂₇⋊C₁₆

—

C₂×D₄×C₁₂₇

—

D₈×C₁₂₇

—

D₈×C₁₂₇

—

C₂×C₈×C₁₂₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₂₇

—

C₁₆×C₁₂₇

—

C₂²×D₁₂₇

—

Dic₂₅₄

—

Dic₂₅₄

—

D₅₀₈

—

D₅₀₈

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

D₈

yes

characteristic

C₂×Dic₁₂₇

D₈

yes

characteristic

C₄×D₁₂₇

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

—

D₄×C₁₂₇

—

D₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₈×C₁₂₇

C₂×D₄

yes

characteristic

C₈×C₁₂₇

#41

yes

characteristic

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂×D₈

yes

characteristic

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

yes

C₄×C₁₂₇

yes

Dic₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

D₁₂₇

—

#2210

—

127

maximal, Sylow

C₁₂₇

#2210

yes

characteristic, complemented, Sylow

C₂×SD₃₂

—

127

—

C₂×D₁₆

—

127

—

#495

—

127

—

C₄×D₈

—

127

—

C₄⋊C₁₆

—

127

—

#358

—

127

—

#357

D₁₂₇

yes

characteristic, complemented

C₂×Q₁₆

—

127

—

C₂×SD₁₆

—

254

—

C₂×D₈

—

127

—

C₂×D₈

C₂×D₁₂₇

yes

characteristic

#124

—

254

—

C₄×D₄

—

254

—

SD₃₂

—

254

—

D₁₆

—

254

—

C₂×C₁₆

—

127

—

C₂×C₁₆

C₂×D₁₂₇

yes

characteristic

#107

C₂×D₁₂₇

yes

characteristic

#102

—

254

—

C₄×C₈

—

127

—

C₂×Q₈

—

254

—

C₂×D₄

—

100

C₂×D₄

—

254

—

101

C₂²×C₄

—

508

—

102

Q₁₆

—

254

—

103

Q₁₆

—

254

—

104

SD₁₆

—

508

—

105

D₈

—

106

D₈

—

107

D₈

—

127

—

108

D₈

—

254

—

109

C₂×C₈

—

127

—

110

C₂×C₈

C₂²×D₁₂₇

yes

characteristic

111

C₄⋊C₄

—

112

C₂²⋊C₄

—

508

—

113

C₂²⋊C₄

—

508

—

114

C₂²⋊C₄

—

508

—

115

C₄²

—

127

—

116

C₁₆

—

117

C₁₆

—

254

—

118

C₂³

—

119

C₂³

—

508

—

120

Q₈

—

508

—

121

Q₈

—

508

—

122

D₄

—

123

D₄

—

124

D₄

—

508

—

125

D₄

—

254

—

126

C₂×C₄

—

127

C₂×C₄

—

127

—

128

C₂×C₄

—

127

—

129

C₂×C₄

—

1016

—

130

C₂×C₄

—

508

—

131

C₂×C₄

D₄×D₁₂₇

yes

characteristic

132

C₈

—

254

—

133

C₈

D₄×D₁₂₇

yes

characteristic

134

C₈

#14523248

yes

characteristic

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

—

1016

—

139

C₂²

—

1016

—

140

C₂²

—

508

—

141

C₂²

D₈×D₁₂₇

yes

characteristic

142

C₄

—

143

C₄

—

1016

—

144

C₄

—

254

—

145

C₄

—

254

—

146

C₄

#16861182

yes

characteristic

147

C₄

#16861236

yes

characteristic

148

C₂

—

149

C₂

—

1016

—

150

C₂

#22550070

yes

characteristic

151

C₂

#22550307

yes

characteristic

152

C₂

#22550310

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₁₂₇
commutator, cosocle	C₂×C₈×C₁₂₇	C₂³
Frattini	C₂×C₈	C₂²×D₁₂₇
Fitting	#357×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂×D₈
prime radical	C₂×#16861150	C₂
hypercenter	#357	D₁₂₇
hypocenter	C₁₂₇	#2210
LCS	G ▷ C₂×C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #357

subgroup

G/H

center

C₂²

D₈×D₁₂₇

commutator, cosocle

C₂×C₈×C₁₂₇

C₂³

Frattini

C₂×C₈

C₂²×D₁₂₇

Fitting

#357×C₁₂₇

C₂

socle

C₂²×C₁₂₇

C₂×D₈

C₂

D₁₂₇

C₁₂₇

G ▷ C₂×C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #357

N	H ≅ G/N	type
1. C₂×#16861154	C₂	semidirect
3. #22550259	C₂	semidirect
4. #22549958	C₂	semidirect
5. Dic₁₂₇⋊C₁₆	C₂	semidirect
6. #22549679	C₂	semidirect
7. #357×C₁₂₇	C₂	semidirect
77. C₁₂₇	#2210	semidirect
84. #357	D₁₂₇	semidirect

H ≅ G/N

type

1. C₂×#16861154

C₂

semidirect

3. #22550259

C₂

semidirect

4. #22549958

C₂

semidirect

5. Dic₁₂₇⋊C₁₆

C₂

semidirect

6. #22549679

C₂

semidirect

7. #357×C₁₂₇

C₂

semidirect

77. C₁₂₇

#2210

semidirect

84. #357

D₁₂₇

semidirect

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

subgroup	index
C₂×#16861154	2
C₂×#16861150	2
#22550259	2
#22549958	2
Dic₁₂₇⋊C₁₆	2
#22549679	2
#357×C₁₂₇	2
#2210	127

subgroup

index

Dic₁₂₇⋊C₁₆

#357×C₁₂₇

127

Subgroups — #30640784

Conjugacy classes of subgroups

Complements

Maximal subgroups