Group #30639592 — subgroups

Subgroups — #30639592

Order 16256.

312 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 117 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×#16861143	C₂	1	1	yes	maximal, characteristic
2	C₂×#16861133	C₂	1	2	yes	maximal
3	C₂×C₈×Dic₁₂₇	C₂	1	1	yes	maximal, characteristic
4	#22549783	C₂	1	2	yes	maximal
5	C₂²×Q₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
8	C₂×C₄×Dic₁₂₇	C₂²	1	1	yes	characteristic
9	#16861143	—	2	2	no	—
10	#16861133	—	2	4	no	—
11	C₈×Dic₁₂₇	—	2	2	no	—
12	C₁₂₇⋊#95	—	2	4	no	—
13	C₂²×Q₈×C₁₂₇	C₂²	1	2	yes	—
14	C₂×Q₁₆×C₁₂₇	—	2	2	no	—
15	C₂×Q₁₆×C₁₂₇	—	2	2	no	—
16	C₂×Q₁₆×C₁₂₇	C₄	1	4	yes	complemented
17	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
18	C₂²×Dic₁₂₇	—	2	2	no	—
19	C₂²×Dic₁₂₇	D₄	1	2	yes	—
20	C₂×C₁₂₇⋊C₈	—	2	1	no	—
21	C₂×C₁₂₇⋊C₈	—	2	1	no	—
22	C₂×C₁₂₇⋊C₈	D₄	1	2	yes	—
23	C₄⋊Dic₁₂₇	—	2	2	no	—
24	C₄⋊Dic₁₂₇	—	2	2	no	—
25	C₄×Dic₁₂₇	—	2	1	no	—
26	C₄×Dic₁₂₇	—	2	1	no	—
27	C₂×Q₈×C₁₂₇	—	2	2	no	—
28	C₂×Q₈×C₁₂₇	—	2	2	no	—
29	C₂×Q₈×C₁₂₇	—	2	2	no	—
30	C₂×Q₈×C₁₂₇	—	2	2	no	—
31	C₂×Q₈×C₁₂₇	—	2	2	no	—
32	C₂×Q₈×C₁₂₇	C₂×C₄	1	4	yes	—
33	C₂²×C₄×C₁₂₇	—	2	2	no	—
34	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
35	Q₁₆×C₁₂₇	—	2	8	no	—
36	C₂×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
37	C₂×C₈×C₁₂₇	D₄	1	2	yes	—
38	C₂×C₈×C₁₂₇	D₄	1	2	yes	—
39	C₂×Dic₁₂₇	—	2	2	no	—
40	C₂×Dic₁₂₇	—	2	2	no	—
41	C₂×Dic₁₂₇	—	4	2	no	—
42	C₂×Dic₁₂₇	—	4	4	no	—
43	C₂×Dic₁₂₇	Q₁₆	1	4	yes	complemented
44	C₁₂₇⋊C₈	—	2	2	no	—
45	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
46	Q₈×C₁₂₇	—	2	4	no	—
47	Q₈×C₁₂₇	—	2	4	no	—
48	Q₈×C₁₂₇	—	4	2	no	—
49	Q₈×C₁₂₇	—	4	2	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	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₁₂₇	#41	1	1	yes	characteristic
58	C₂×C₄×C₁₂₇	#41	1	1	yes	characteristic
59	C₈×C₁₂₇	C₂²⋊C₄	1	4	yes	—
60	C₂²×C₁₂₇	C₄×D₄	1	1	yes	characteristic
61	C₂²×C₁₂₇	#121	1	1	yes	characteristic
62	C₂²×C₁₂₇	#127	1	1	yes	characteristic
63	C₂²×C₁₂₇	C₂×Q₁₆	1	2	yes	—
64	C₂²×C₁₂₇	#135	1	2	yes	—
65	C₄×C₁₂₇	—	4	4	no	—
66	C₄×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₄×C₁₂₇	C₄×D₄	1	1	yes	characteristic
68	C₄×C₁₂₇	#121	1	1	yes	characteristic
69	C₄×C₁₂₇	#124	1	1	yes	characteristic
70	Dic₁₂₇	—	2	4	no	—
71	Dic₁₂₇	—	8	2	no	—
72	C₂×C₁₂₇	#390	1	1	yes	characteristic
73	C₂×C₁₂₇	C₄×Q₁₆	1	2	yes	—
74	C₂×C₁₂₇	#467	1	2	yes	—
75	C₂×C₁₂₇	#494	1	1	yes	characteristic
76	C₂×C₁₂₇	#495	1	1	yes	characteristic
77	#1969	—	127	1	no	maximal, Sylow
78	C₁₂₇	#1969	1	1	yes	characteristic, complemented, Sylow
79	C₂²×Q₁₆	D₁₂₇	1	1	yes	characteristic
80	C₂×#107	—	127	1	no	—
81	C₂×#103	—	127	2	no	—
82	C₂×C₄×C₈	—	127	1	no	—
83	#391	—	127	2	no	—
84	C₂²×Q₈	C₂×D₁₂₇	1	2	yes	—
85	C₂×Q₁₆	—	2	2	no	—
86	C₂×Q₁₆	—	2	2	no	—
87	C₂×Q₁₆	Dic₁₂₇	1	4	yes	complemented
88	C₂²×C₈	—	127	1	no	—
89	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
90	C₂×C₄⋊C₄	—	127	2	no	—
91	C₂×C₄²	—	127	1	no	—
92	#107	—	254	2	no	—
93	#103	—	254	4	no	—
94	C₄×C₈	—	254	2	no	—
95	#95	—	254	4	no	—
96	C₂×Q₈	—	2	2	no	—
97	C₂×Q₈	—	2	2	no	—
98	C₂×Q₈	—	2	2	no	—
99	C₂×Q₈	—	2	2	no	—
100	C₂×Q₈	—	2	2	no	—
101	C₂×Q₈	C₂×Dic₁₂₇	1	4	yes	—
102	C₂²×C₄	—	2	2	no	—
103	C₂²×C₄	—	127	2	no	—
104	C₂²×C₄	—	254	2	no	—
105	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
106	Q₁₆	—	2	8	no	—
107	C₂×C₈	—	127	2	no	—
108	C₂×C₈	—	254	1	no	—
109	C₂×C₈	—	254	1	no	—
110	C₂×C₈	C₂×Dic₁₂₇	1	2	yes	—
111	C₂×C₈	C₂²⋊D₁₂₇	1	2	yes	—
112	C₂×C₈	C₂²⋊D₁₂₇	1	2	yes	—
113	C₄⋊C₄	—	254	2	no	—
114	C₄⋊C₄	—	254	2	no	—
115	C₄²	—	254	1	no	—
116	C₄²	—	254	1	no	—
117	C₂³	D₄×D₁₂₇	1	1	yes	characteristic
118	Q₈	—	2	4	no	—
119	Q₈	—	2	4	no	—
120	Q₈	—	4	2	no	—
121	Q₈	—	4	2	no	—
122	C₂×C₄	—	2	4	no	—
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	—	127	4	no	—
126	C₂×C₄	—	508	2	no	—
127	C₂×C₄	—	508	4	no	—
128	C₂×C₄	—	254	2	no	—
129	C₂×C₄	—	254	2	no	—
130	C₂×C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
131	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
132	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
133	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
134	C₂×C₄	#14523247	1	1	yes	characteristic
135	C₂×C₄	#14523247	1	1	yes	characteristic
136	C₈	—	254	2	no	—
137	C₈	C₂²⋊Dic₁₂₇	1	4	yes	—
138	C₂²	D₄×Dic₁₂₇	1	1	yes	characteristic
139	C₂²	#16861180	1	1	yes	characteristic
140	C₂²	#16861205	1	1	yes	characteristic
141	C₂²	Q₁₆×D₁₂₇	1	2	yes	—
142	C₂²	#16861232	1	2	yes	—
143	C₄	—	4	4	no	—
144	C₄	—	1016	2	no	—
145	C₄	—	254	4	no	—
146	C₄	C₂×C₂²⋊Dic₁₂₇	1	1	yes	characteristic
147	C₄	D₄×Dic₁₂₇	1	1	yes	characteristic
148	C₄	#16861180	1	1	yes	characteristic
149	C₄	#16861196	1	1	yes	characteristic
150	C₂	#22549779	1	1	yes	characteristic
151	C₂	Q₁₆×Dic₁₂₇	1	2	yes	—
152	C₂	#22550114	1	2	yes	—
153	C₂	#22550254	1	1	yes	characteristic
154	C₂	#22550261	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16861143

C₂

yes

maximal, characteristic

C₂×#16861133

C₂

yes

maximal

C₂×C₈×Dic₁₂₇

C₂

yes

maximal, characteristic

#22549783

C₂

yes

maximal

C₂²×Q₁₆×C₁₂₇

C₂

yes

maximal, characteristic

C₂²×C₁₂₇⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₂₇

C₂²

yes

—

C₂×C₄×Dic₁₂₇

C₂²

yes

characteristic

#16861143

—

#16861133

—

C₈×Dic₁₂₇

—

C₁₂₇⋊#95

—

C₂²×Q₈×C₁₂₇

C₂²

yes

—

C₂×Q₁₆×C₁₂₇

—

C₂×Q₁₆×C₁₂₇

—

C₂×Q₁₆×C₁₂₇

C₄

yes

complemented

C₂²×C₈×C₁₂₇

C₂²

yes

characteristic

C₂²×Dic₁₂₇

—

C₂²×Dic₁₂₇

D₄

yes

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

D₄

yes

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

C₄×Dic₁₂₇

—

C₄×Dic₁₂₇

—

C₂×Q₈×C₁₂₇

—

C₂×Q₈×C₁₂₇

—

C₂×Q₈×C₁₂₇

—

C₂×Q₈×C₁₂₇

—

C₂×Q₈×C₁₂₇

—

C₂×Q₈×C₁₂₇

C₂×C₄

yes

—

C₂²×C₄×C₁₂₇

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

Q₁₆×C₁₂₇

—

C₂×C₈×C₁₂₇

C₂×C₄

yes

—

C₂×C₈×C₁₂₇

D₄

yes

—

C₂×C₈×C₁₂₇

D₄

yes

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

Q₁₆

yes

complemented

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

C₂×D₄

yes

characteristic

Q₈×C₁₂₇

—

Q₈×C₁₂₇

—

Q₈×C₁₂₇

—

Q₈×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×D₄

yes

C₂×C₄×C₁₂₇

yes

C₂×C₄×C₁₂₇

yes

C₈×C₁₂₇

C₂²⋊C₄

yes

—

C₂²×C₁₂₇

C₄×D₄

yes

C₂²×C₁₂₇

yes

C₂²×C₁₂₇

yes

C₂²×C₁₂₇

C₂×Q₁₆

yes

—

C₂²×C₁₂₇

#135

yes

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₂₇

C₄×D₄

yes

C₄×C₁₂₇

yes

C₄×C₁₂₇

yes

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

#390

yes

characteristic

C₂×C₁₂₇

C₄×Q₁₆

yes

—

C₂×C₁₂₇

#467

yes

—

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

—

127

maximal, Sylow

C₁₂₇

#1969

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

D₁₂₇

yes

characteristic

C₂×#107

—

127

—

C₂×#103

—

127

—

C₂×C₄×C₈

—

127

—

#391

—

127

—

C₂²×Q₈

C₂×D₁₂₇

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

Dic₁₂₇

yes

complemented

C₂²×C₈

—

127

—

C₂²×C₈

C₂×D₁₂₇

yes

characteristic

C₂×C₄⋊C₄

—

127

—

C₂×C₄²

—

127

—

#107

—

254

—

#103

—

254

—

C₄×C₈

—

254

—

#95

—

254

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

C₂×Dic₁₂₇

yes

—

102

C₂²×C₄

—

103

C₂²×C₄

—

127

—

104

C₂²×C₄

—

254

—

105

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

106

Q₁₆

—

107

C₂×C₈

—

127

—

108

C₂×C₈

—

254

—

109

C₂×C₈

—

254

—

110

C₂×C₈

C₂×Dic₁₂₇

yes

—

111

C₂×C₈

C₂²⋊D₁₂₇

yes

—

112

C₂×C₈

C₂²⋊D₁₂₇

yes

—

113

C₄⋊C₄

—

254

—

114

C₄⋊C₄

—

254

—

115

C₄²

—

254

—

116

C₄²

—

254

—

117

C₂³

D₄×D₁₂₇

yes

characteristic

118

Q₈

—

119

Q₈

—

120

Q₈

—

121

Q₈

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

127

—

126

C₂×C₄

—

508

—

127

C₂×C₄

—

508

—

128

C₂×C₄

—

254

—

129

C₂×C₄

—

254

—

130

C₂×C₄

C₂²×Dic₁₂₇

yes

characteristic

131

C₂×C₄

D₄×D₁₂₇

yes

characteristic

132

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

characteristic

133

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

characteristic

134

C₂×C₄

#14523247

yes

characteristic

135

C₂×C₄

#14523247

yes

characteristic

136

C₈

—

254

—

137

C₈

C₂²⋊Dic₁₂₇

yes

—

138

C₂²

D₄×Dic₁₂₇

yes

characteristic

139

C₂²

#16861180

yes

characteristic

140

C₂²

#16861205

yes

characteristic

141

C₂²

Q₁₆×D₁₂₇

yes

—

142

C₂²

#16861232

yes

—

143

C₄

—

144

C₄

—

1016

—

145

C₄

—

254

—

146

C₄

C₂×C₂²⋊Dic₁₂₇

yes

characteristic

147

C₄

D₄×Dic₁₂₇

yes

characteristic

148

C₄

#16861180

yes

characteristic

149

C₄

#16861196

yes

characteristic

150

C₂

#22549779

yes

characteristic

151

C₂

Q₁₆×Dic₁₂₇

yes

—

152

C₂

#22550114

yes

—

153

C₂

#22550254

yes

characteristic

154

C₂

#22550261

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₁₂₇
commutator	C₂×C₄×C₁₂₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	C₂²×Q₁₆×C₁₂₇	C₂
socle, prime radical	C₂³×C₁₂₇	C₂×D₄
cosocle	C₂²×C₄×C₁₂₇	C₂³
hypercenter	C₂²×Q₁₆	D₁₂₇
hypocenter	C₁₂₇	#1969
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₁₆

subgroup

G/H

center

C₂³

D₄×D₁₂₇

commutator

C₂×C₄×C₁₂₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₂₇

Fitting

C₂²×Q₁₆×C₁₂₇

C₂

socle, prime radical

C₂³×C₁₂₇

C₂×D₄

cosocle

C₂²×C₄×C₁₂₇

C₂³

hypercenter

C₂²×Q₁₆

D₁₂₇

hypocenter

C₁₂₇

#1969

LCS

G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₁₆

N	H ≅ G/N	type
16. C₂×Q₁₆×C₁₂₇	C₄	semidirect
43. C₂×Dic₁₂₇	Q₁₆	semidirect
78. C₁₂₇	#1969	semidirect
87. C₂×Q₁₆	Dic₁₂₇	semidirect

H ≅ G/N

type

16. C₂×Q₁₆×C₁₂₇

C₄

semidirect

43. C₂×Dic₁₂₇

Q₁₆

semidirect

78. C₁₂₇

#1969

semidirect

87. C₂×Q₁₆

Dic₁₂₇

semidirect

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

subgroup	index
C₂×#16861143	2
C₂×#16861133	2
C₂×C₈×Dic₁₂₇	2
#22549783	2
C₂²×Q₁₆×C₁₂₇	2
#1969	127

subgroup

index

C₂×C₈×Dic₁₂₇

C₂²×Q₁₆×C₁₂₇

127

Subgroups — #30639592

Conjugacy classes of subgroups

Complements

Maximal subgroups