Group #30640769 — subgroups

Subgroups — #30640769

Order 16256.

174 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 174 conjugacy classes of subgroups — too many to draw. Only its 49 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×#16861153	C₂	1	1	yes	maximal, characteristic, complemented
3	#22550251	C₂	1	1	yes	maximal, characteristic
4	#22549970	C₂	1	1	yes	maximal, characteristic, complemented
5	Dic₁₂₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	#22549676	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×D₁₀₁₆	C₂²	1	1	yes	characteristic
10	C₂×#14523233	—	2	1	no	—
11	#16861205	—	2	1	no	—
12	#16861174	—	2	1	no	—
13	#16861154	—	2	2	no	—
14	#16861153	—	2	2	no	—
15	C₂×C₁₂₇⋊C₁₆	C₂²	1	1	yes	characteristic
16	#16861132	—	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×C₂²⋊D₁₂₇	—	4	1	no	—
24	C₂×C₂²⋊D₁₂₇	—	4	1	no	—
25	Dic₅₀₈	—	2	1	no	—
26	Dic₅₀₈	D₄	1	2	yes	complemented
27	D₁₀₁₆	—	2	1	no	—
28	D₁₀₁₆	—	2	1	no	—
29	#14523233	—	4	2	no	—
30	C₂×C₁₂₇⋊C₈	D₄	1	1	yes	characteristic
31	Dic₁₂₇⋊C₄	—	4	1	no	—
32	C₄×Dic₁₂₇	—	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	—	2	2	no	—
43	Dic₂₅₄	—	4	1	no	—
44	D₅₀₈	—	4	1	no	—
45	D₅₀₈	—	4	1	no	—
46	C₂²⋊D₁₂₇	—	8	2	no	—
47	C₂²⋊D₁₂₇	—	8	2	no	—
48	C₂×Dic₁₂₇	—	4	1	no	—
49	C₂×Dic₁₂₇	D₈	1	1	yes	characteristic
50	C₂×Dic₁₂₇	D₈	1	1	yes	characteristic
51	C₁₂₇⋊C₈	—	2	1	no	—
52	C₂³×C₁₂₇	—	4	1	no	—
53	D₄×C₁₂₇	—	4	1	no	—
54	D₄×C₁₂₇	—	4	1	no	—
55	C₂×C₄×C₁₂₇	—	4	1	no	—
56	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
57	C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
58	C₈×C₁₂₇	#41	1	1	yes	characteristic
59	C₂²×C₁₂₇	—	8	1	no	—
60	C₂²×C₁₂₇	—	8	1	no	—
61	C₂²×C₁₂₇	—	8	1	no	—
62	C₂²×C₁₂₇	C₂×D₈	1	1	yes	characteristic
63	C₂×D₁₂₇	—	8	1	no	—
64	C₂×D₁₂₇	—	8	1	no	—
65	C₂×D₁₂₇	—	8	1	no	—
66	C₄×C₁₂₇	—	8	1	no	—
67	C₄×C₁₂₇	#121	1	1	yes	characteristic
68	C₄×C₁₂₇	AGL(1,ℤ/8)	1	1	yes	characteristic
69	Dic₁₂₇	—	2	1	no	—
70	Dic₁₂₇	—	4	2	no	—
71	Dic₁₂₇	SD₃₂	1	2	yes	complemented
72	C₂×C₁₂₇	—	16	1	no	—
73	C₂×C₁₂₇	#459	1	1	yes	characteristic
74	C₂×C₁₂₇	C₂×SD₃₂	1	1	yes	characteristic
75	C₂×C₁₂₇	#509	1	1	yes	characteristic
76	D₁₂₇	—	16	1	no	—
77	#2207	—	127	1	no	maximal, Sylow
78	C₁₂₇	#2207	1	1	yes	characteristic, complemented, Sylow
79	C₂×SD₃₂	—	127	1	no	—
80	C₂×SD₃₂	—	127	1	no	—
81	#493	—	127	1	no	—
82	C₄×Q₁₆	—	127	1	no	—
83	C₄⋊C₁₆	—	127	1	no	—
84	#357	—	127	1	no	—
85	#357	D₁₂₇	1	1	yes	characteristic, complemented
86	C₂×Q₁₆	—	127	1	no	—
87	C₂×D₈	—	127	1	no	—
88	C₂×D₈	—	254	1	no	—
89	C₂×D₈	C₂×D₁₂₇	1	1	yes	characteristic
90	#127	—	254	1	no	—
91	C₄×Q₈	—	254	1	no	—
92	SD₃₂	—	254	2	no	—
93	SD₃₂	—	254	2	no	—
94	C₂×C₁₆	—	127	1	no	—
95	C₂×C₁₆	C₂×D₁₂₇	1	1	yes	characteristic
96	#107	C₂×D₁₂₇	1	1	yes	characteristic
97	#103	—	254	1	no	—
98	C₄×C₈	—	127	1	no	—
99	C₂×Q₈	—	254	1	no	—
100	C₂×D₄	—	2	1	no	—
101	C₂×D₄	—	508	1	no	—
102	C₂×D₄	—	508	1	no	—
103	C₂×D₄	—	254	1	no	—
104	Q₁₆	—	127	2	no	—
105	Q₁₆	—	254	1	no	—
106	D₈	—	2	1	no	—
107	D₈	—	2	1	no	—
108	D₈	—	508	2	no	—
109	D₈	—	254	1	no	—
110	D₈	—	254	1	no	—
111	C₂×C₈	—	127	1	no	—
112	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
113	C₄⋊C₄	—	2	1	no	—
114	C₄⋊C₄	—	508	1	no	—
115	C₄²	—	127	1	no	—
116	C₄²	—	508	1	no	—
117	C₁₆	—	2	1	no	—
118	C₁₆	—	254	1	no	—
119	C₂³	—	4	1	no	—
120	C₂³	—	508	1	no	—
121	Q₈	—	508	1	no	—
122	Q₈	—	254	2	no	—
123	D₄	—	4	1	no	—
124	D₄	—	4	1	no	—
125	D₄	—	1016	2	no	—
126	D₄	—	1016	2	no	—
127	D₄	—	508	1	no	—
128	D₄	—	508	1	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	—	127	1	no	—
131	C₂×C₄	—	127	1	no	—
132	C₂×C₄	—	508	1	no	—
133	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
134	C₈	—	254	1	no	—
135	C₈	D₄×D₁₂₇	1	1	yes	characteristic
136	C₈	#14523248	1	1	yes	characteristic
137	C₂²	—	8	1	no	—
138	C₂²	—	8	1	no	—
139	C₂²	—	8	1	no	—
140	C₂²	—	1016	1	no	—
141	C₂²	—	1016	1	no	—
142	C₂²	—	1016	1	no	—
143	C₂²	D₈×D₁₂₇	1	1	yes	characteristic
144	C₄	—	8	1	no	—
145	C₄	—	127	2	no	—
146	C₄	—	508	2	no	—
147	C₄	—	254	1	no	—
148	C₄	#16861182	1	1	yes	characteristic
149	C₄	#16861236	1	1	yes	characteristic
150	C₂	—	16	1	no	—
151	C₂	—	2032	1	no	—
152	C₂	#22550070	1	1	yes	characteristic
153	C₂	SD₃₂×D₁₂₇	1	1	yes	characteristic
154	C₂	#22550310	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16861154

C₂

yes

maximal, characteristic, complemented

C₂×#16861153

C₂

yes

maximal, characteristic, complemented

#22550251

C₂

yes

maximal, characteristic

#22549970

C₂

yes

maximal, characteristic, complemented

Dic₁₂₇⋊C₁₆

C₂

yes

maximal, characteristic, complemented

#22549676

C₂

yes

maximal, characteristic, complemented

#357×C₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂×Dic₅₀₈

C₂²

yes

characteristic

C₂×D₁₀₁₆

C₂²

yes

—

—

—

—

—

C₂×C₁₂₇⋊C₁₆

C₂²

yes

characteristic

#16861132

—

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×C₂²⋊D₁₂₇

—

C₂×C₂²⋊D₁₂₇

—

Dic₅₀₈

—

Dic₅₀₈

D₄

yes

complemented

D₁₀₁₆

—

D₁₀₁₆

—

#14523233

—

C₂×C₁₂₇⋊C₈

D₄

yes

characteristic

Dic₁₂₇⋊C₄

—

C₄×Dic₁₂₇

—

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

—

C₂²⋊D₁₂₇

—

C₂²⋊D₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

D₈

yes

characteristic

C₂×Dic₁₂₇

D₈

yes

characteristic

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

SD₃₂

yes

complemented

C₂×C₁₂₇

—

C₂×C₁₂₇

#459

yes

characteristic

C₂×C₁₂₇

C₂×SD₃₂

yes

characteristic

C₂×C₁₂₇

#509

yes

characteristic

D₁₂₇

—

#2207

—

127

maximal, Sylow

C₁₂₇

#2207

yes

characteristic, complemented, Sylow

C₂×SD₃₂

—

127

—

C₂×SD₃₂

—

127

—

#493

—

127

—

C₄×Q₁₆

—

127

—

C₄⋊C₁₆

—

127

—

#357

—

127

—

#357

D₁₂₇

yes

characteristic, complemented

C₂×Q₁₆

—

127

—

C₂×D₈

—

127

—

C₂×D₈

—

254

—

C₂×D₈

C₂×D₁₂₇

yes

characteristic

#127

—

254

—

C₄×Q₈

—

254

—

SD₃₂

—

254

—

SD₃₂

—

254

—

C₂×C₁₆

—

127

—

C₂×C₁₆

C₂×D₁₂₇

yes

characteristic

#107

C₂×D₁₂₇

yes

characteristic

#103

—

254

—

C₄×C₈

—

127

—

C₂×Q₈

—

254

—

100

C₂×D₄

—

101

C₂×D₄

—

508

—

102

C₂×D₄

—

508

—

103

C₂×D₄

—

254

—

104

Q₁₆

—

127

—

105

Q₁₆

—

254

—

106

D₈

—

107

D₈

—

108

D₈

—

508

—

109

D₈

—

254

—

110

D₈

—

254

—

111

C₂×C₈

—

127

—

112

C₂×C₈

C₂²×D₁₂₇

yes

characteristic

113

C₄⋊C₄

—

114

C₄⋊C₄

—

508

—

115

C₄²

—

127

—

116

C₄²

—

508

—

117

C₁₆

—

118

C₁₆

—

254

—

119

C₂³

—

120

C₂³

—

508

—

121

Q₈

—

508

—

122

Q₈

—

254

—

123

D₄

—

124

D₄

—

125

D₄

—

1016

—

126

D₄

—

1016

—

127

D₄

—

508

—

128

D₄

—

508

—

129

C₂×C₄

—

130

C₂×C₄

—

127

—

131

C₂×C₄

—

127

—

132

C₂×C₄

—

508

—

133

C₂×C₄

D₄×D₁₂₇

yes

characteristic

134

C₈

—

254

—

135

C₈

D₄×D₁₂₇

yes

characteristic

136

C₈

#14523248

yes

characteristic

137

C₂²

—

138

C₂²

—

139

C₂²

—

140

C₂²

—

1016

—

141

C₂²

—

1016

—

142

C₂²

—

1016

—

143

C₂²

D₈×D₁₂₇

yes

characteristic

144

C₄

—

145

C₄

—

127

—

146

C₄

—

508

—

147

C₄

—

254

—

148

C₄

#16861182

yes

characteristic

149

C₄

#16861236

yes

characteristic

150

C₂

—

151

C₂

—

2032

—

152

C₂

#22550070

yes

characteristic

153

C₂

SD₃₂×D₁₂₇

yes

characteristic

154

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	#22550251	C₂
hypercenter	#357	D₁₂₇
hypocenter	C₁₂₇	#2207
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
2. C₂×#16861153	C₂	semidirect
4. #22549970	C₂	semidirect
5. Dic₁₂₇⋊C₁₆	C₂	semidirect
6. #22549676	C₂	semidirect
7. #357×C₁₂₇	C₂	semidirect
26. Dic₅₀₈	D₄	semidirect
71. Dic₁₂₇	SD₃₂	semidirect
78. C₁₂₇	#2207	semidirect
85. #357	D₁₂₇	semidirect

H ≅ G/N

type

1. C₂×#16861154

C₂

semidirect

2. C₂×#16861153

C₂

semidirect

4. #22549970

C₂

semidirect

5. Dic₁₂₇⋊C₁₆

C₂

semidirect

6. #22549676

C₂

semidirect

7. #357×C₁₂₇

C₂

semidirect

26. Dic₅₀₈

D₄

semidirect

71. Dic₁₂₇

SD₃₂

semidirect

78. C₁₂₇

#2207

semidirect

85. #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×#16861153	2
#22550251	2
#22549970	2
Dic₁₂₇⋊C₁₆	2
#22549676	2
#357×C₁₂₇	2
#2207	127

subgroup

index

Dic₁₂₇⋊C₁₆

#357×C₁₂₇

127

Subgroups — #30640769

Conjugacy classes of subgroups

Complements

Maximal subgroups