Group #22549769 — subgroups

Subgroups — #22549769

Order 8128.

190 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 75 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⋊Dic₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂²⋊Dic₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂²⋊Dic₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×Dic₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₂₇⋊#95	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₂₇⋊#95	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₂²⋊C₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₂₇	C₂²	1	1	yes	characteristic
9	C₂²×Dic₁₂₇	C₂²	1	1	yes	characteristic
10	C₂²×Dic₁₂₇	C₂²	1	1	yes	characteristic
11	C₂²×Dic₁₂₇	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₁₂₇	—	2	2	no	—
13	C₂²⋊Dic₁₂₇	—	2	2	no	—
14	C₂²⋊Dic₁₂₇	—	2	2	no	—
15	C₄×Dic₁₂₇	—	2	2	no	—
16	C₂⁴×C₁₂₇	C₂²	1	1	yes	characteristic
17	C₂²×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
18	C₂²×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
19	C₂²⋊C₄×C₁₂₇	C₄	1	4	yes	complemented
20	C₂×Dic₁₂₇	—	2	1	no	—
21	C₂×Dic₁₂₇	—	2	1	no	—
22	C₂×Dic₁₂₇	—	2	1	no	—
23	C₂×Dic₁₂₇	—	2	1	no	—
24	C₂×Dic₁₂₇	—	2	1	no	—
25	C₂×Dic₁₂₇	—	2	1	no	—
26	C₂×Dic₁₂₇	—	2	1	no	—
27	C₂×Dic₁₂₇	—	2	1	no	—
28	C₂×Dic₁₂₇	—	2	1	no	—
29	C₂×Dic₁₂₇	—	2	1	no	—
30	C₂×Dic₁₂₇	D₄	1	2	yes	—
31	C₂×Dic₁₂₇	D₄	1	2	yes	—
32	C₂³×C₁₂₇	—	2	1	no	—
33	C₂³×C₁₂₇	—	2	1	no	—
34	C₂³×C₁₂₇	—	2	1	no	—
35	C₂³×C₁₂₇	—	2	1	no	—
36	C₂³×C₁₂₇	—	2	1	no	—
37	C₂³×C₁₂₇	—	2	1	no	—
38	C₂³×C₁₂₇	C₂×C₄	1	2	yes	—
39	C₂³×C₁₂₇	C₂³	1	1	yes	characteristic
40	C₂×C₄×C₁₂₇	—	2	1	no	—
41	C₂×C₄×C₁₂₇	—	2	1	no	—
42	C₂×C₄×C₁₂₇	—	2	1	no	—
43	C₂×C₄×C₁₂₇	—	2	1	no	—
44	C₂×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
45	C₂×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
46	C₂²×C₁₂₇	—	2	2	no	—
47	C₂²×C₁₂₇	—	2	2	no	—
48	C₂²×C₁₂₇	—	2	2	no	—
49	C₂²×C₁₂₇	—	4	1	no	—
50	C₂²×C₁₂₇	—	4	1	no	—
51	C₂²×C₁₂₇	—	4	1	no	—
52	C₂²×C₁₂₇	—	4	1	no	—
53	C₂²×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₁₂₇	#41	1	1	yes	characteristic
57	C₂²×C₁₂₇	#41	1	1	yes	characteristic
58	C₂²×C₁₂₇	#41	1	1	yes	characteristic
59	C₂²×C₁₂₇	#41	1	1	yes	characteristic
60	C₄×C₁₂₇	—	2	2	no	—
61	C₄×C₁₂₇	—	2	2	no	—
62	Dic₁₂₇	—	2	2	no	—
63	Dic₁₂₇	—	2	2	no	—
64	Dic₁₂₇	—	4	1	no	—
65	Dic₁₂₇	—	4	1	no	—
66	C₂×C₁₂₇	—	4	2	no	—
67	C₂×C₁₂₇	#117	1	1	yes	characteristic
68	C₂×C₁₂₇	C₄×D₄	1	1	yes	characteristic
69	C₂×C₁₂₇	C₄×D₄	1	1	yes	characteristic
70	C₂×C₁₂₇	#121	1	1	yes	characteristic
71	C₂×C₁₂₇	#123	1	1	yes	characteristic
72	C₂×C₁₂₇	#124	1	1	yes	characteristic
73	C₂×C₁₂₇	#126	1	1	yes	characteristic
74	C₁₂₇	#388	1	1	yes	characteristic, complemented, Sylow
75	#388	—	127	1	no	maximal, Sylow
76	C₂×C₄⋊C₄	—	127	1	no	—
77	C₂×C₂²⋊C₄	—	127	1	no	—
78	C₂×C₂²⋊C₄	—	127	1	no	—
79	C₂×C₂²⋊C₄	D₁₂₇	1	1	yes	characteristic
80	C₂×C₄²	—	127	1	no	—
81	#95	—	127	1	no	—
82	#95	—	127	1	no	—
83	C₂⁴	C₂×D₁₂₇	1	1	yes	characteristic
84	C₂²×C₄	—	127	1	no	—
85	C₂²×C₄	—	127	1	no	—
86	C₂²×C₄	—	127	1	no	—
87	C₂²×C₄	—	127	1	no	—
88	C₂²×C₄	C₂×D₁₂₇	1	1	yes	characteristic
89	C₂²×C₄	C₂×D₁₂₇	1	1	yes	characteristic
90	C₄⋊C₄	—	254	2	no	—
91	C₂²⋊C₄	—	254	2	no	—
92	C₂²⋊C₄	—	254	2	no	—
93	C₂²⋊C₄	Dic₁₂₇	1	4	yes	complemented
94	C₄²	—	254	2	no	—
95	C₂³	—	2	1	no	—
96	C₂³	—	2	1	no	—
97	C₂³	—	2	1	no	—
98	C₂³	—	2	1	no	—
99	C₂³	—	2	1	no	—
100	C₂³	—	2	1	no	—
101	C₂³	C₂×Dic₁₂₇	1	2	yes	—
102	C₂³	C₂²×D₁₂₇	1	1	yes	characteristic
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	2	1	no	—
105	C₂×C₄	—	2	1	no	—
106	C₂×C₄	—	2	1	no	—
107	C₂×C₄	—	127	2	no	—
108	C₂×C₄	—	127	2	no	—
109	C₂×C₄	—	254	1	no	—
110	C₂×C₄	—	254	1	no	—
111	C₂×C₄	—	254	1	no	—
112	C₂×C₄	—	254	1	no	—
113	C₂×C₄	—	254	1	no	—
114	C₂×C₄	—	254	1	no	—
115	C₂×C₄	—	254	1	no	—
116	C₂×C₄	—	254	1	no	—
117	C₂×C₄	—	254	1	no	—
118	C₂×C₄	—	254	1	no	—
119	C₂×C₄	C₂×Dic₁₂₇	1	2	yes	—
120	C₂×C₄	C₂×Dic₁₂₇	1	2	yes	—
121	C₂²	—	2	2	no	—
122	C₂²	—	2	2	no	—
123	C₂²	—	2	2	no	—
124	C₂²	—	4	1	no	—
125	C₂²	—	4	1	no	—
126	C₂²	—	4	1	no	—
127	C₂²	—	4	1	no	—
128	C₂²	C₂²×Dic₁₂₇	1	1	yes	characteristic
129	C₂²	D₄×D₁₂₇	1	1	yes	characteristic
130	C₂²	D₄×D₁₂₇	1	1	yes	characteristic
131	C₂²	#14523247	1	1	yes	characteristic
132	C₂²	#14523247	1	1	yes	characteristic
133	C₂²	#14523248	1	1	yes	characteristic
134	C₂²	#14523248	1	1	yes	characteristic
135	C₄	—	2	2	no	—
136	C₄	—	2	2	no	—
137	C₄	—	508	1	no	—
138	C₄	—	508	1	no	—
139	C₄	—	254	2	no	—
140	C₄	—	254	2	no	—
141	C₂	—	4	2	no	—
142	C₂	#16861168	1	1	yes	characteristic
143	C₂	D₄×Dic₁₂₇	1	1	yes	characteristic
144	C₂	D₄×Dic₁₂₇	1	1	yes	characteristic
145	C₂	#16861182	1	1	yes	characteristic
146	C₂	#16861191	1	1	yes	characteristic
147	C₂	#16861195	1	1	yes	characteristic
148	C₂	#16861202	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊Dic₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂×C₂²⋊Dic₁₂₇

C₂

yes

maximal, characteristic

C₂×C₂²⋊Dic₁₂₇

C₂

yes

maximal, characteristic

C₂×C₄×Dic₁₂₇

C₂

yes

maximal, characteristic, complemented

C₁₂₇⋊#95

C₂

yes

maximal, characteristic, complemented

C₁₂₇⋊#95

C₂

yes

maximal, characteristic, complemented

C₂×C₂²⋊C₄×C₁₂₇

C₂

yes

maximal, characteristic

C₂²×Dic₁₂₇

C₂²

yes

characteristic

C₂²×Dic₁₂₇

C₂²

yes

characteristic

C₂²×Dic₁₂₇

C₂²

yes

characteristic

C₂²×Dic₁₂₇

C₂²

yes

characteristic

C₄⋊Dic₁₂₇

—

C₂²⋊Dic₁₂₇

—

C₂²⋊Dic₁₂₇

—

C₄×Dic₁₂₇

—

C₂⁴×C₁₂₇

C₂²

yes

characteristic

C₂²×C₄×C₁₂₇

C₂²

yes

characteristic

C₂²×C₄×C₁₂₇

C₂²

yes

characteristic

C₂²⋊C₄×C₁₂₇

C₄

yes

complemented

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

D₄

yes

—

C₂×Dic₁₂₇

D₄

yes

—

C₂³×C₁₂₇

—

C₂³×C₁₂₇

—

C₂³×C₁₂₇

—

C₂³×C₁₂₇

—

C₂³×C₁₂₇

—

C₂³×C₁₂₇

—

C₂³×C₁₂₇

C₂×C₄

yes

—

C₂³×C₁₂₇

C₂³

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂×C₄

yes

—

C₂×C₄×C₁₂₇

C₂×C₄

yes

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂²×C₄

yes

characteristic

C₂²×C₁₂₇

C₂×D₄

yes

characteristic

C₂²×C₁₂₇

C₂×D₄

yes

C₂²×C₁₂₇

yes

C₂²×C₁₂₇

yes

C₂²×C₁₂₇

yes

C₂²×C₁₂₇

yes

C₄×C₁₂₇

—

C₄×C₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

#117

yes

characteristic

C₂×C₁₂₇

C₄×D₄

yes

characteristic

C₂×C₁₂₇

C₄×D₄

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₁₂₇

yes

characteristic, complemented, Sylow

#388

—

127

maximal, Sylow

C₂×C₄⋊C₄

—

127

—

C₂×C₂²⋊C₄

—

127

—

C₂×C₂²⋊C₄

—

127

—

C₂×C₂²⋊C₄

D₁₂₇

yes

characteristic

C₂×C₄²

—

127

—

#95

—

127

—

#95

—

127

—

C₂⁴

C₂×D₁₂₇

yes

characteristic

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

C₂×D₁₂₇

yes

characteristic

C₂²×C₄

C₂×D₁₂₇

yes

characteristic

C₄⋊C₄

—

254

—

C₂²⋊C₄

—

254

—

C₂²⋊C₄

—

254

—

C₂²⋊C₄

Dic₁₂₇

yes

complemented

C₄²

—

254

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

100

C₂³

—

101

C₂³

C₂×Dic₁₂₇

yes

—

102

C₂³

C₂²×D₁₂₇

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

127

—

108

C₂×C₄

—

127

—

109

C₂×C₄

—

254

—

110

C₂×C₄

—

254

—

111

C₂×C₄

—

254

—

112

C₂×C₄

—

254

—

113

C₂×C₄

—

254

—

114

C₂×C₄

—

254

—

115

C₂×C₄

—

254

—

116

C₂×C₄

—

254

—

117

C₂×C₄

—

254

—

118

C₂×C₄

—

254

—

119

C₂×C₄

C₂×Dic₁₂₇

yes

—

120

C₂×C₄

C₂×Dic₁₂₇

yes

—

121

C₂²

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

C₂²×Dic₁₂₇

yes

characteristic

129

C₂²

D₄×D₁₂₇

yes

characteristic

130

C₂²

D₄×D₁₂₇

yes

characteristic

131

C₂²

#14523247

yes

characteristic

132

C₂²

#14523247

yes

characteristic

133

C₂²

#14523248

yes

characteristic

134

C₂²

#14523248

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

508

—

138

C₄

—

508

—

139

C₄

—

254

—

140

C₄

—

254

—

141

C₂

—

142

C₂

#16861168

yes

characteristic

143

C₂

D₄×Dic₁₂₇

yes

characteristic

144

C₂

D₄×Dic₁₂₇

yes

characteristic

145

C₂

#16861182

yes

characteristic

146

C₂

#16861191

yes

characteristic

147

C₂

#16861195

yes

characteristic

148

C₂

#16861202

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₁₂₇
commutator	C₂²×C₁₂₇	C₂²×C₄
Fitting	C₂×C₂²⋊C₄×C₁₂₇	C₂
socle	C₂³×C₁₂₇	C₂³
hypercenter	C₂×C₂²⋊C₄	D₁₂₇
hypocenter	C₁₂₇	#388
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₂²⋊C₄

subgroup

G/H

center, Frattini

C₂³

C₂²×D₁₂₇

commutator

C₂²×C₁₂₇

C₂²×C₄

Fitting

C₂×C₂²⋊C₄×C₁₂₇

C₂

socle

C₂³×C₁₂₇

C₂³

hypercenter

C₂×C₂²⋊C₄

D₁₂₇

C₁₂₇

G ▷ C₂²×C₁₂₇ ▷ C₁₂₇

C₁ ◁ C₂³ ◁ C₂×C₂²⋊C₄

N	H ≅ G/N	type
1. C₂×C₄⋊Dic₁₂₇	C₂	semidirect
4. C₂×C₄×Dic₁₂₇	C₂	semidirect
5. C₁₂₇⋊#95	C₂	semidirect
6. C₁₂₇⋊#95	C₂	semidirect
19. C₂²⋊C₄×C₁₂₇	C₄	semidirect
74. C₁₂₇	#388	semidirect
93. C₂²⋊C₄	Dic₁₂₇	semidirect

H ≅ G/N

type

1. C₂×C₄⋊Dic₁₂₇

C₂

semidirect

4. C₂×C₄×Dic₁₂₇

C₂

semidirect

5. C₁₂₇⋊#95

C₂

semidirect

6. C₁₂₇⋊#95

C₂

semidirect

19. C₂²⋊C₄×C₁₂₇

C₄

semidirect

74. C₁₂₇

#388

semidirect

93. C₂²⋊C₄

Dic₁₂₇

semidirect

Subgroups — #22549769

Conjugacy classes of subgroups

Complements