Group #30640054 — subgroups

Subgroups — #30640054

Order 16256.

288 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 85 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×Dic₁₂₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×#16861132	C₂	1	2	yes	maximal
4	#22549781	C₂	1	2	yes	maximal
5	#340×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₅₄	C₂²	1	2	yes	—
7	C₂×Dic₅₀₈	—	2	4	no	—
8	C₂×Dic₅₀₈	—	2	2	no	—
9	C₂²×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₁₂₇	C₂²	1	1	yes	characteristic
11	Dic₁₂₇⋊C₈	—	2	2	no	—
12	#16861132	—	2	4	no	—
13	C₁₂₇⋊#95	—	2	2	no	—
14	C₁₂₇⋊#95	—	2	2	no	—
15	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
17	C₂×Dic₂₅₄	—	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	D₄	1	4	yes	—
23	C₂²×Dic₁₂₇	—	2	2	no	—
24	C₂²×Dic₁₂₇	D₄	1	1	yes	characteristic
25	C₂²×Dic₁₂₇	D₄	1	1	yes	characteristic
26	Dic₅₀₈	—	2	8	no	—
27	C₂×C₁₂₇⋊C₈	—	2	2	no	—
28	C₂×C₁₂₇⋊C₈	—	2	1	no	—
29	C₄×Dic₁₂₇	—	2	1	no	—
30	C₄×Dic₁₂₇	—	2	1	no	—
31	C₂²×C₄×C₁₂₇	—	2	2	no	—
32	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
33	C₂×C₈×C₁₂₇	—	2	2	no	—
34	C₂×C₈×C₁₂₇	D₄	1	2	yes	—
35	C₄⋊C₄×C₁₂₇	—	2	2	no	—
36	C₄⋊C₄×C₁₂₇	—	2	2	no	—
37	Dic₂₅₄	—	2	4	no	—
38	Dic₂₅₄	—	2	4	no	—
39	Dic₂₅₄	—	4	2	no	—
40	Dic₂₅₄	—	4	2	no	—
41	C₂×Dic₁₂₇	—	2	1	no	—
42	C₂×Dic₁₂₇	—	2	1	no	—
43	C₂×Dic₁₂₇	—	2	1	no	—
44	C₂×Dic₁₂₇	—	2	4	no	—
45	C₂×Dic₁₂₇	—	2	2	no	—
46	C₂×Dic₁₂₇	—	4	2	no	—
47	C₂×Dic₁₂₇	—	4	2	no	—
48	C₂×Dic₁₂₇	Q₁₆	1	2	yes	complemented
49	C₁₂₇⋊C₈	—	4	1	no	—
50	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₁₂₇	—	4	2	no	—
52	C₂×C₄×C₁₂₇	—	4	2	no	—
53	C₂×C₄×C₁₂₇	—	4	2	no	—
54	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₁₂₇	#41	1	2	yes	—
57	C₂×C₄×C₁₂₇	#41	1	1	yes	characteristic
58	C₈×C₁₂₇	—	2	2	no	—
59	C₂²×C₁₂₇	#121	1	1	yes	characteristic
60	C₂²×C₁₂₇	#121	1	2	yes	—
61	C₂²×C₁₂₇	C₂×Q₁₆	1	1	yes	characteristic
62	C₂²×C₁₂₇	#135	1	1	yes	characteristic
63	C₂²×C₁₂₇	#137	1	2	yes	—
64	C₄×C₁₂₇	—	8	2	no	—
65	C₄×C₁₂₇	#120	1	1	yes	characteristic
66	C₄×C₁₂₇	#123	1	2	yes	—
67	C₄×C₁₂₇	#124	1	1	yes	characteristic
68	Dic₁₂₇	—	2	2	no	—
69	Dic₁₂₇	—	4	4	no	—
70	Dic₁₂₇	—	4	1	no	—
71	C₂×C₁₂₇	#394	1	1	yes	characteristic
72	C₂×C₁₂₇	#462	1	2	yes	—
73	C₂×C₁₂₇	#464	1	2	yes	—
74	C₂×C₁₂₇	#467	1	1	yes	characteristic
75	C₂×C₁₂₇	#470	1	1	yes	characteristic
76	#2056	—	127	1	no	maximal, Sylow
77	C₁₂₇	#2056	1	1	yes	characteristic, complemented, Sylow
78	C₂²×Q₁₆	—	127	1	no	—
79	C₂×C₄⋊C₈	—	127	1	no	—
80	C₂×#103	—	127	2	no	—
81	#391	—	127	2	no	—
82	#340	D₁₂₇	1	1	yes	characteristic
83	C₂²×Q₈	—	127	2	no	—
84	C₂×Q₁₆	—	254	4	no	—
85	C₂×Q₁₆	—	254	2	no	—
86	C₂²×C₈	—	127	1	no	—
87	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
88	C₂×C₄⋊C₄	C₂×D₁₂₇	1	2	yes	—
89	C₂×C₄²	—	127	1	no	—
90	C₄⋊C₈	—	254	2	no	—
91	#103	—	254	4	no	—
92	#95	—	254	2	no	—
93	#95	—	254	2	no	—
94	C₂×Q₈	—	127	4	no	—
95	C₂×Q₈	—	254	2	no	—
96	C₂×Q₈	—	254	2	no	—
97	C₂×Q₈	—	254	2	no	—
98	C₂×Q₈	—	254	2	no	—
99	C₂×Q₈	—	254	2	no	—
100	C₂²×C₄	—	2	2	no	—
101	C₂²×C₄	—	127	1	no	—
102	C₂²×C₄	—	127	1	no	—
103	C₂²×C₄	—	254	2	no	—
104	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
105	Q₁₆	—	254	8	no	—
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	254	2	no	—
108	C₂×C₈	—	254	1	no	—
109	C₂×C₈	D₅₀₈	1	2	yes	—
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	—	2	2	no	—
112	C₄²	—	254	1	no	—
113	C₄²	—	254	1	no	—
114	C₂³	D₄×D₁₂₇	1	1	yes	characteristic
115	Q₈	—	508	2	no	—
116	Q₈	—	508	2	no	—
117	Q₈	—	254	4	no	—
118	Q₈	—	254	4	no	—
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	—	4	2	no	—
121	C₂×C₄	—	4	2	no	—
122	C₂×C₄	—	127	2	no	—
123	C₂×C₄	—	508	2	no	—
124	C₂×C₄	—	508	2	no	—
125	C₂×C₄	—	254	1	no	—
126	C₂×C₄	—	254	2	no	—
127	C₂×C₄	—	254	4	no	—
128	C₂×C₄	—	254	1	no	—
129	C₂×C₄	—	254	1	no	—
130	C₂×C₄	D₄×D₁₂₇	1	2	yes	—
131	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
132	C₂×C₄	#14523248	1	2	yes	—
133	C₂×C₄	#14523249	1	1	yes	characteristic
134	C₈	—	2	2	no	—
135	C₈	—	508	1	no	—
136	C₂²	#16861182	1	2	yes	—
137	C₂²	#16861184	1	1	yes	characteristic
138	C₂²	Q₁₆×D₁₂₇	1	1	yes	characteristic
139	C₂²	#16861228	1	1	yes	characteristic
140	C₂²	#16861238	1	2	yes	—
141	C₄	—	8	2	no	—
142	C₄	—	508	4	no	—
143	C₄	—	508	1	no	—
144	C₄	—	254	2	no	—
145	C₄	#16861179	1	1	yes	characteristic
146	C₄	#16861194	1	2	yes	—
147	C₄	#16861197	1	1	yes	characteristic
148	C₂	#22549794	1	1	yes	characteristic
149	C₂	#22550085	1	2	yes	—
150	C₂	#22550097	1	2	yes	—
151	C₂	#22550118	1	1	yes	characteristic
152	C₂	#22550135	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₅₀₈

C₂

yes

maximal, characteristic

C₂×Dic₁₂₇⋊C₈

C₂

yes

maximal, characteristic

C₂×#16861132

C₂

yes

maximal

#22549781

C₂

yes

maximal

#340×C₁₂₇

C₂

yes

maximal, characteristic

C₂²×Dic₂₅₄

C₂²

yes

—

C₂×Dic₅₀₈

—

C₂×Dic₅₀₈

—

C₂²×C₁₂₇⋊C₈

C₂²

yes

characteristic

C₂×C₄×Dic₁₂₇

C₂²

yes

characteristic

Dic₁₂₇⋊C₈

—

#16861132

—

C₁₂₇⋊#95

—

C₁₂₇⋊#95

—

C₂²×C₈×C₁₂₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₂₇

C₂²

yes

—

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

D₄

yes

—

C₂²×Dic₁₂₇

—

C₂²×Dic₁₂₇

D₄

yes

characteristic

C₂²×Dic₁₂₇

D₄

yes

characteristic

Dic₅₀₈

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

—

C₄×Dic₁₂₇

—

C₄×Dic₁₂₇

—

C₂²×C₄×C₁₂₇

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

D₄

yes

—

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

—

Dic₂₅₄

—

Dic₂₅₄

—

Dic₂₅₄

—

Dic₂₅₄

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

Q₁₆

yes

complemented

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

#41

yes

—

C₂×C₄×C₁₂₇

#41

yes

characteristic

C₈×C₁₂₇

—

C₂²×C₁₂₇

#121

yes

characteristic

C₂²×C₁₂₇

#121

yes

—

C₂²×C₁₂₇

C₂×Q₁₆

yes

C₂²×C₁₂₇

yes

C₂²×C₁₂₇

yes

—

C₄×C₁₂₇

—

C₄×C₁₂₇

#120

yes

characteristic

C₄×C₁₂₇

#123

yes

—

C₄×C₁₂₇

#124

yes

characteristic

Dic₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

#394

yes

characteristic

C₂×C₁₂₇

#462

yes

—

C₂×C₁₂₇

#464

yes

—

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

—

127

maximal, Sylow

C₁₂₇

#2056

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

127

—

C₂×C₄⋊C₈

—

127

—

C₂×#103

—

127

—

#391

—

127

—

#340

D₁₂₇

yes

characteristic

C₂²×Q₈

—

127

—

C₂×Q₁₆

—

254

—

C₂×Q₁₆

—

254

—

C₂²×C₈

—

127

—

C₂²×C₈

C₂×D₁₂₇

yes

characteristic

C₂×C₄⋊C₄

C₂×D₁₂₇

yes

—

C₂×C₄²

—

127

—

C₄⋊C₈

—

254

—

#103

—

254

—

#95

—

254

—

#95

—

254

—

C₂×Q₈

—

127

—

C₂×Q₈

—

254

—

C₂×Q₈

—

254

—

C₂×Q₈

—

254

—

C₂×Q₈

—

254

—

C₂×Q₈

—

254

—

100

C₂²×C₄

—

101

C₂²×C₄

—

127

—

102

C₂²×C₄

—

127

—

103

C₂²×C₄

—

254

—

104

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

105

Q₁₆

—

254

—

106

C₂×C₈

—

107

C₂×C₈

—

254

—

108

C₂×C₈

—

254

—

109

C₂×C₈

D₅₀₈

yes

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄²

—

254

—

113

C₄²

—

254

—

114

C₂³

D₄×D₁₂₇

yes

characteristic

115

Q₈

—

508

—

116

Q₈

—

508

—

117

Q₈

—

254

—

118

Q₈

—

254

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

127

—

123

C₂×C₄

—

508

—

124

C₂×C₄

—

508

—

125

C₂×C₄

—

254

—

126

C₂×C₄

—

254

—

127

C₂×C₄

—

254

—

128

C₂×C₄

—

254

—

129

C₂×C₄

—

254

—

130

C₂×C₄

D₄×D₁₂₇

yes

—

131

C₂×C₄

C₂×D₅₀₈

yes

characteristic

132

C₂×C₄

#14523248

yes

—

133

C₂×C₄

#14523249

yes

characteristic

134

C₈

—

135

C₈

—

508

—

136

C₂²

#16861182

yes

—

137

C₂²

#16861184

yes

characteristic

138

C₂²

Q₁₆×D₁₂₇

yes

characteristic

139

C₂²

#16861228

yes

characteristic

140

C₂²

#16861238

yes

—

141

C₄

—

142

C₄

—

508

—

143

C₄

—

508

—

144

C₄

—

254

—

145

C₄

#16861179

yes

characteristic

146

C₄

#16861194

yes

—

147

C₄

#16861197

yes

characteristic

148

C₂

#22549794

yes

characteristic

149

C₂

#22550085

yes

—

150

C₂

#22550097

yes

—

151

C₂

#22550118

yes

characteristic

152

C₂

#22550135

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₁₂₇
commutator, cosocle	C₂²×C₄×C₁₂₇	C₂³
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	#340×C₁₂₇	C₂
socle, prime radical	C₂³×C₁₂₇	C₂×D₄
hypercenter	#340	D₁₂₇
hypocenter	C₁₂₇	#2056
LCS	G ▷ C₂²×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

subgroup

G/H

center

C₂³

D₄×D₁₂₇

commutator, cosocle

C₂²×C₄×C₁₂₇

C₂³

Frattini

C₂²×C₄

C₂²×D₁₂₇

Fitting

#340×C₁₂₇

C₂

socle, prime radical

C₂³×C₁₂₇

C₂×D₄

D₁₂₇

C₁₂₇

G ▷ C₂²×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

N	H ≅ G/N	type
48. C₂×Dic₁₂₇	Q₁₆	semidirect
77. C₁₂₇	#2056	semidirect

H ≅ G/N

type

48. C₂×Dic₁₂₇

Q₁₆

semidirect

77. C₁₂₇

#2056

semidirect

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

subgroup	index
C₂²×Dic₅₀₈	2
C₂×Dic₁₂₇⋊C₈	2
C₂×#16861132	2
#22549781	2
#340×C₁₂₇	2
#2056	127

subgroup

index

C₂²×Dic₅₀₈

C₂×Dic₁₂₇⋊C₈

C₂×#16861132

#22549781

#340×C₁₂₇

#2056

127

Subgroups — #30640054

Conjugacy classes of subgroups

Complements

Maximal subgroups