Group #30639591 — subgroups

Subgroups — #30639591

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	C₂²×Dic₅₀₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#16861143	C₂	1	1	yes	maximal, characteristic
3	C₂×#16861134	C₂	1	2	yes	maximal
4	#22549782	C₂	1	2	yes	maximal
5	C₂×C₄×C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₅₄	C₂²	1	2	yes	—
7	C₂×Dic₅₀₈	—	2	2	no	—
8	C₂×Dic₅₀₈	—	2	2	no	—
9	C₂×Dic₅₀₈	C₄	1	4	yes	complemented
10	C₂×C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
11	#16861143	—	2	2	no	—
12	#16861134	—	2	4	no	—
13	C₁₂₇⋊#95	—	2	4	no	—
14	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
16	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
17	C₄×C₈×C₁₂₇	—	2	2	no	—
18	C₂×Dic₂₅₄	—	2	2	no	—
19	C₂×Dic₂₅₄	—	2	2	no	—
20	C₂×Dic₂₅₄	—	2	2	no	—
21	C₂×Dic₂₅₄	—	2	2	no	—
22	C₂×Dic₂₅₄	—	2	2	no	—
23	C₂×Dic₂₅₄	C₂×C₄	1	4	yes	—
24	C₂²×Dic₁₂₇	—	2	2	no	—
25	C₂²×Dic₁₂₇	—	2	2	no	—
26	Dic₅₀₈	—	2	8	no	—
27	C₄⋊Dic₁₂₇	—	2	2	no	—
28	C₄⋊Dic₁₂₇	—	2	2	no	—
29	C₂²×C₄×C₁₂₇	D₄	1	2	yes	—
30	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
31	C₂×C₈×C₁₂₇	—	2	1	no	—
32	C₂×C₈×C₁₂₇	—	2	1	no	—
33	C₂×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
34	C₂×C₈×C₁₂₇	D₄	1	2	yes	—
35	C₂×C₈×C₁₂₇	D₄	1	2	yes	—
36	C₂×C₈×C₁₂₇	D₄	1	2	yes	—
37	C₄²×C₁₂₇	—	2	1	no	—
38	C₄²×C₁₂₇	—	2	1	no	—
39	Dic₂₅₄	—	2	4	no	—
40	Dic₂₅₄	—	2	4	no	—
41	Dic₂₅₄	—	4	2	no	—
42	Dic₂₅₄	—	4	2	no	—
43	C₂×Dic₁₂₇	—	2	4	no	—
44	C₂×Dic₁₂₇	—	4	2	no	—
45	C₂×Dic₁₂₇	—	4	2	no	—
46	C₂×Dic₁₂₇	—	4	2	no	—
47	C₂×Dic₁₂₇	—	4	4	no	—
48	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₂₇	—	2	2	no	—
50	C₂×C₄×C₁₂₇	—	2	2	no	—
51	C₂×C₄×C₁₂₇	Q₁₆	1	4	yes	complemented
52	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₁₂₇	C₂×D₄	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	#41	1	1	yes	characteristic
57	C₂×C₄×C₁₂₇	#41	1	1	yes	characteristic
58	C₈×C₁₂₇	—	2	2	no	—
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	—	2	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	—	4	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	—	127	1	no	—
80	C₂×#107	—	127	1	no	—
81	C₂×#103	—	127	2	no	—
82	C₂×C₄×C₈	D₁₂₇	1	1	yes	characteristic
83	#391	—	127	2	no	—
84	C₂²×Q₈	—	127	2	no	—
85	C₂×Q₁₆	—	127	4	no	—
86	C₂×Q₁₆	—	254	2	no	—
87	C₂×Q₁₆	—	254	2	no	—
88	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
89	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
90	C₂×C₄⋊C₄	—	127	2	no	—
91	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
92	#107	—	254	2	no	—
93	#103	—	254	4	no	—
94	C₄×C₈	—	2	2	no	—
95	#95	—	254	4	no	—
96	C₂×Q₈	—	127	4	no	—
97	C₂×Q₈	—	254	2	no	—
98	C₂×Q₈	—	254	2	no	—
99	C₂×Q₈	—	254	2	no	—
100	C₂×Q₈	—	254	2	no	—
101	C₂×Q₈	—	254	2	no	—
102	C₂²×C₄	—	254	2	no	—
103	C₂²×C₄	—	254	2	no	—
104	C₂²×C₄	D₅₀₈	1	2	yes	—
105	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
106	Q₁₆	—	254	8	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	C₄×D₁₂₇	1	2	yes	—
110	C₂×C₈	C₂²⋊D₁₂₇	1	2	yes	—
111	C₂×C₈	D₅₀₈	1	2	yes	—
112	C₂×C₈	D₅₀₈	1	2	yes	—
113	C₄⋊C₄	—	254	2	no	—
114	C₄⋊C₄	—	254	2	no	—
115	C₄²	—	2	1	no	—
116	C₄²	—	2	1	no	—
117	C₂³	C₂×D₅₀₈	1	1	yes	characteristic
118	Q₈	—	508	2	no	—
119	Q₈	—	508	2	no	—
120	Q₈	—	254	4	no	—
121	Q₈	—	254	4	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	508	2	no	—
125	C₂×C₄	—	508	4	no	—
126	C₂×C₄	—	508	2	no	—
127	C₂×C₄	—	508	2	no	—
128	C₂×C₄	—	254	4	no	—
129	C₂×C₄	Dic₅₀₈	1	4	yes	complemented
130	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
131	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
132	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
133	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
134	C₂×C₄	#14523248	1	1	yes	characteristic
135	C₂×C₄	#14523248	1	1	yes	characteristic
136	C₈	—	2	2	no	—
137	C₈	(C₂×D₁₂₇)⋊C₄	1	4	yes	—
138	C₂²	C₄×D₅₀₈	1	1	yes	characteristic
139	C₂²	#16861183	1	1	yes	characteristic
140	C₂²	#16861206	1	1	yes	characteristic
141	C₂²	C₂×Dic₅₀₈	1	2	yes	—
142	C₂²	#16861231	1	2	yes	—
143	C₄	—	2	4	no	—
144	C₄	—	1016	2	no	—
145	C₄	—	508	4	no	—
146	C₄	C₂×(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
147	C₄	C₄×D₅₀₈	1	1	yes	characteristic
148	C₄	#16861183	1	1	yes	characteristic
149	C₄	#16861197	1	1	yes	characteristic
150	C₂	#22549778	1	1	yes	characteristic
151	C₂	C₄×Dic₅₀₈	1	2	yes	—
152	C₂	#22550117	1	2	yes	—
153	C₂	#22550256	1	1	yes	characteristic
154	C₂	#22550260	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₅₀₈

C₂

yes

maximal, characteristic

C₂×#16861143

C₂

yes

maximal, characteristic

C₂×#16861134

C₂

yes

maximal

#22549782

C₂

yes

maximal

C₂×C₄×C₈×C₁₂₇

C₂

yes

maximal, characteristic

C₂²×Dic₂₅₄

C₂²

yes

—

C₂×Dic₅₀₈

—

C₂×Dic₅₀₈

—

C₂×Dic₅₀₈

C₄

yes

complemented

C₂×C₄⋊Dic₁₂₇

C₂²

yes

—

#16861143

—

#16861134

—

C₁₂₇⋊#95

—

C₂²×C₈×C₁₂₇

C₂²

yes

characteristic

C₂²×C₈×C₁₂₇

C₂²

yes

characteristic

C₂×C₄²×C₁₂₇

C₂²

yes

characteristic

C₄×C₈×C₁₂₇

—

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

C₂×C₄

yes

—

C₂²×Dic₁₂₇

—

C₂²×Dic₁₂₇

—

Dic₅₀₈

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

C₂²×C₄×C₁₂₇

D₄

yes

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

C₂×C₄

yes

—

C₂×C₈×C₁₂₇

D₄

yes

—

C₂×C₈×C₁₂₇

D₄

yes

—

C₂×C₈×C₁₂₇

D₄

yes

—

C₄²×C₁₂₇

—

C₄²×C₁₂₇

—

Dic₂₅₄

—

Dic₂₅₄

—

Dic₂₅₄

—

Dic₂₅₄

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂³×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

Q₁₆

yes

complemented

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

—

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

—

127

—

C₂×#107

—

127

—

C₂×#103

—

127

—

C₂×C₄×C₈

D₁₂₇

yes

characteristic

#391

—

127

—

C₂²×Q₈

—

127

—

C₂×Q₁₆

—

127

—

C₂×Q₁₆

—

254

—

C₂×Q₁₆

—

254

—

C₂²×C₈

C₂×D₁₂₇

yes

characteristic

C₂²×C₈

C₂×D₁₂₇

yes

characteristic

C₂×C₄⋊C₄

—

127

—

C₂×C₄²

C₂×D₁₂₇

yes

characteristic

#107

—

254

—

#103

—

254

—

C₄×C₈

—

#95

—

254

—

C₂×Q₈

—

127

—

C₂×Q₈

—

254

—

C₂×Q₈

—

254

—

C₂×Q₈

—

254

—

100

C₂×Q₈

—

254

—

101

C₂×Q₈

—

254

—

102

C₂²×C₄

—

254

—

103

C₂²×C₄

—

254

—

104

C₂²×C₄

D₅₀₈

yes

—

105

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

106

Q₁₆

—

254

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

C₄×D₁₂₇

yes

—

110

C₂×C₈

C₂²⋊D₁₂₇

yes

—

111

C₂×C₈

D₅₀₈

yes

—

112

C₂×C₈

D₅₀₈

yes

—

113

C₄⋊C₄

—

254

—

114

C₄⋊C₄

—

254

—

115

C₄²

—

116

C₄²

—

117

C₂³

C₂×D₅₀₈

yes

characteristic

118

Q₈

—

508

—

119

Q₈

—

508

—

120

Q₈

—

254

—

121

Q₈

—

254

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

508

—

125

C₂×C₄

—

508

—

126

C₂×C₄

—

508

—

127

C₂×C₄

—

508

—

128

C₂×C₄

—

254

—

129

C₂×C₄

Dic₅₀₈

yes

complemented

130

C₂×C₄

C₂×C₄×D₁₂₇

yes

characteristic

131

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

characteristic

132

C₂×C₄

C₂×D₅₀₈

yes

characteristic

133

C₂×C₄

C₂×D₅₀₈

yes

characteristic

134

C₂×C₄

#14523248

yes

characteristic

135

C₂×C₄

#14523248

yes

characteristic

136

C₈

—

137

C₈

(C₂×D₁₂₇)⋊C₄

yes

—

138

C₂²

C₄×D₅₀₈

yes

characteristic

139

C₂²

#16861183

yes

characteristic

140

C₂²

#16861206

yes

characteristic

141

C₂²

C₂×Dic₅₀₈

yes

—

142

C₂²

#16861231

yes

—

143

C₄

—

144

C₄

—

1016

—

145

C₄

—

508

—

146

C₄

C₂×(C₂×D₁₂₇)⋊C₄

yes

characteristic

147

C₄

C₄×D₅₀₈

yes

characteristic

148

C₄

#16861183

yes

characteristic

149

C₄

#16861197

yes

characteristic

150

C₂

#22549778

yes

characteristic

151

C₂

C₄×Dic₅₀₈

yes

—

152

C₂

#22550117

yes

—

153

C₂

#22550256

yes

characteristic

154

C₂

#22550260

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₅₀₈
commutator	C₂×C₄×C₁₂₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	C₂×C₄×C₈×C₁₂₇	C₂
socle, prime radical	C₂³×C₁₂₇	C₂×D₄
cosocle	C₂²×C₄×C₁₂₇	C₂³
hypercenter	C₂×C₄×C₈	D₁₂₇
hypocenter	C₁₂₇	#1969
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂³

C₂×D₅₀₈

commutator

C₂×C₄×C₁₂₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₂₇

Fitting

C₂×C₄×C₈×C₁₂₇

C₂

socle, prime radical

C₂³×C₁₂₇

C₂×D₄

cosocle

C₂²×C₄×C₁₂₇

C₂³

hypercenter

C₂×C₄×C₈

D₁₂₇

hypocenter

C₁₂₇

#1969

LCS

G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
9. C₂×Dic₅₀₈	C₄	semidirect
51. C₂×C₄×C₁₂₇	Q₁₆	semidirect
78. C₁₂₇	#1969	semidirect
129. C₂×C₄	Dic₅₀₈	semidirect

H ≅ G/N

type

9. C₂×Dic₅₀₈

C₄

semidirect

51. C₂×C₄×C₁₂₇

Q₁₆

semidirect

78. C₁₂₇

#1969

semidirect

129. C₂×C₄

Dic₅₀₈

semidirect

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

subgroup	index
C₂²×Dic₅₀₈	2
C₂×#16861143	2
C₂×#16861134	2
#22549782	2
C₂×C₄×C₈×C₁₂₇	2
#1969	127

subgroup

index

C₂²×Dic₅₀₈

C₂×C₄×C₈×C₁₂₇

127

Subgroups — #30639591

Conjugacy classes of subgroups

Complements

Maximal subgroups