Group #30647597 — subgroups

Subgroups — #30647597

Order 16256.

456 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 181 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	#22550353	C₂	1	2	yes	maximal
3	#22549982	C₂	1	4	yes	maximal, complemented
4	#22549970	C₂	1	4	yes	maximal, complemented
5	#22549884	C₂	1	2	yes	maximal
6	#22549831	C₂	1	1	yes	maximal, characteristic
7	#427×C₁₂₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₂₅₄	C₂²	1	2	yes	—
9	C₂×Dic₅₀₈	C₄	1	8	yes	complemented
10	C₂×Dic₅₀₈	C₂²	1	2	yes	—
11	C₂×Dic₅₀₈	C₂²	1	2	yes	—
12	C₂×#14523232	C₂²	1	1	yes	characteristic
13	C₄×Dic₂₅₄	—	2	4	no	—
14	#16861174	C₂²	1	8	yes	—
15	#16861168	C₂²	1	1	yes	characteristic
16	#16861165	—	2	4	no	—
17	#16861132	C₂²	1	8	yes	—
18	#16861119	C₂²	1	2	yes	—
19	C₈×Dic₁₂₇	C₂²	1	2	yes	—
20	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
21	#117×C₁₂₇	C₂²	1	2	yes	—
22	#107×C₁₂₇	C₂²	1	2	yes	—
23	#106×C₁₂₇	C₂²	1	2	yes	—
24	C₂×Dic₂₅₄	—	2	2	no	—
25	C₂×Dic₂₅₄	—	2	4	no	—
26	C₂×Dic₂₅₄	C₂×C₄	1	8	yes	—
27	C₂×Dic₂₅₄	C₂³	1	4	yes	—
28	C₂²×Dic₁₂₇	—	2	2	no	—
29	Dic₅₀₈	C₂×C₄	1	8	yes	—
30	Dic₅₀₈	C₂×C₄	1	8	yes	—
31	#14523232	—	2	2	no	—
32	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
33	C₄⋊Dic₁₂₇	D₄	1	2	yes	—
34	Dic₁₂₇⋊C₄	—	2	8	no	—
35	C₂²⋊Dic₁₂₇	D₄	1	2	yes	—
36	C₄×Dic₁₂₇	—	2	8	no	—
37	C₄×Dic₁₂₇	C₂³	1	2	yes	—
38	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
39	C₂×C₈×C₁₂₇	C₂×C₄	1	4	yes	—
40	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
41	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
42	C₄⋊C₄×C₁₂₇	C₂³	1	4	yes	—
43	C₂²⋊C₄×C₁₂₇	—	2	2	no	—
44	C₄²×C₁₂₇	—	2	2	no	—
45	Dic₂₅₄	—	2	8	no	—
46	Dic₂₅₄	—	2	4	no	—
47	Dic₂₅₄	C₂²×C₄	1	8	yes	—
48	C₂×Dic₁₂₇	—	2	8	no	—
49	C₂×Dic₁₂₇	—	2	4	no	—
50	C₂×Dic₁₂₇	C₂×D₄	1	4	yes	—
51	C₁₂₇⋊C₈	—	2	2	no	—
52	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₁₂₇	—	2	4	no	—
54	C₂×C₄×C₁₂₇	C₂²×C₄	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	C₂⁴	1	1	yes	characteristic
58	C₈×C₁₂₇	C₂²×C₄	1	2	yes	—
59	C₈×C₁₂₇	C₂²×C₄	1	2	yes	—
60	C₂²×C₁₂₇	—	2	2	no	—
61	C₂²×C₁₂₇	C₄×D₄	1	2	yes	—
62	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
63	C₄×C₁₂₇	—	4	4	no	—
64	C₄×C₁₂₇	C₄×D₄	1	2	yes	—
65	C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
66	C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
67	Dic₁₂₇	—	2	8	no	—
68	Dic₁₂₇	—	2	4	no	—
69	C₂×C₁₂₇	—	2	2	no	—
70	C₂×C₁₂₇	C₂×C₄×D₄	1	1	yes	characteristic
71	C₂×C₁₂₇	#578	1	2	yes	—
72	#2950	—	127	1	no	maximal, Sylow
73	C₁₂₇	#2950	1	1	yes	characteristic, complemented, Sylow
74	C₂²×Q₁₆	—	127	1	no	—
75	#519	—	127	2	no	—
76	#441	—	127	4	no	—
77	C₄×Q₁₆	—	127	4	no	—
78	#427	D₁₂₇	1	1	yes	characteristic
79	#419	—	127	2	no	—
80	#405	—	127	1	no	—
81	C₂²×Q₈	—	127	2	no	—
82	C₂×Q₁₆	—	127	8	no	—
83	C₂×Q₁₆	—	127	2	no	—
84	C₂×Q₁₆	—	127	2	no	—
85	C₂×M₁₆	—	127	1	no	—
86	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
87	C₄×Q₈	—	127	8	no	—
88	C₄×Q₈	—	254	4	no	—
89	#117	—	127	1	no	—
90	#117	—	254	4	no	—
91	#117	C₂×D₁₂₇	1	2	yes	—
92	#107	C₂×D₁₂₇	1	2	yes	—
93	#106	C₂×D₁₂₇	1	2	yes	—
94	#103	—	127	8	no	—
95	#97	—	127	2	no	—
96	C₄×C₈	—	127	2	no	—
97	C₂×Q₈	—	127	8	no	—
98	C₂×Q₈	—	127	4	no	—
99	C₂×Q₈	—	254	2	no	—
100	C₂×Q₈	—	254	4	no	—
101	C₂²×C₄	—	254	2	no	—
102	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
103	Q₁₆	—	127	8	no	—
104	Q₁₆	—	127	8	no	—
105	M₁₆	—	254	2	no	—
106	C₂×C₈	—	127	2	no	—
107	C₂×C₈	C₄×D₁₂₇	1	4	yes	—
108	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
109	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
110	C₄⋊C₄	—	127	2	no	—
111	C₄⋊C₄	—	254	8	no	—
112	C₄⋊C₄	C₂²×D₁₂₇	1	4	yes	—
113	C₂²⋊C₄	—	2	2	no	—
114	C₂²⋊C₄	—	127	2	no	—
115	C₄²	—	2	2	no	—
116	C₄²	—	127	2	no	—
117	C₄²	—	254	8	no	—
118	C₂³	D₄×D₁₂₇	1	1	yes	characteristic
119	Q₈	—	127	8	no	—
120	Q₈	—	254	4	no	—
121	Q₈	—	254	8	no	—
122	C₂×C₄	—	2	4	no	—
123	C₂×C₄	—	127	4	no	—
124	C₂×C₄	—	254	4	no	—
125	C₂×C₄	—	254	8	no	—
126	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
127	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
128	C₂×C₄	#14523249	1	2	yes	—
129	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
130	C₈	—	254	2	no	—
131	C₈	C₂×C₄×D₁₂₇	1	2	yes	—
132	C₈	C₂×C₄×D₁₂₇	1	2	yes	—
133	C₂²	—	2	2	no	—
134	C₂²	#16861171	1	2	yes	—
135	C₂²	C₂×D₄×D₁₂₇	1	1	yes	characteristic
136	C₄	—	4	4	no	—
137	C₄	—	254	8	no	—
138	C₄	—	254	4	no	—
139	C₄	#16861171	1	2	yes	—
140	C₄	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
141	C₄	C₂×#14523249	1	1	yes	characteristic
142	C₂	—	2	2	no	—
143	C₂	C₂×#16861171	1	1	yes	characteristic
144	C₂	#22550690	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₅₀₈

C₂

yes

maximal, characteristic

#22550353

C₂

yes

maximal

#22549982

C₂

yes

maximal, complemented

#22549970

C₂

yes

maximal, complemented

#22549884

C₂

yes

maximal

#22549831

C₂

yes

maximal, characteristic

#427×C₁₂₇

C₂

yes

maximal, characteristic

C₂²×Dic₂₅₄

C₂²

yes

—

C₂×Dic₅₀₈

C₄

yes

complemented

C₂×Dic₅₀₈

C₂²

yes

—

C₂×Dic₅₀₈

C₂²

yes

—

C₂×#14523232

C₂²

yes

characteristic

C₄×Dic₂₅₄

—

#16861174

C₂²

yes

—

#16861168

C₂²

yes

characteristic

#16861165

—

#16861132

C₂²

yes

—

#16861119

C₂²

yes

—

C₈×Dic₁₂₇

C₂²

yes

—

C₂²×C₈×C₁₂₇

C₂²

yes

characteristic

#117×C₁₂₇

C₂²

yes

—

#107×C₁₂₇

C₂²

yes

—

#106×C₁₂₇

C₂²

yes

—

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

C₂×C₄

yes

—

C₂×Dic₂₅₄

C₂³

yes

—

C₂²×Dic₁₂₇

—

Dic₅₀₈

C₂×C₄

yes

—

Dic₅₀₈

C₂×C₄

yes

—

#14523232

—

C₂×C₁₂₇⋊C₈

C₂³

yes

—

C₄⋊Dic₁₂₇

D₄

yes

—

Dic₁₂₇⋊C₄

—

C₂²⋊Dic₁₂₇

D₄

yes

—

C₄×Dic₁₂₇

—

C₄×Dic₁₂₇

C₂³

yes

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

C₂×C₈×C₁₂₇

C₂×C₄

yes

—

C₂×C₈×C₁₂₇

C₂³

yes

characteristic

C₂×C₈×C₁₂₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₂₇

C₂³

yes

—

C₂²⋊C₄×C₁₂₇

—

C₄²×C₁₂₇

—

Dic₂₅₄

—

Dic₂₅₄

—

Dic₂₅₄

C₂²×C₄

yes

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

C₂×D₄

yes

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

#41

yes

—

C₂×C₄×C₁₂₇

C₂⁴

yes

characteristic

C₈×C₁₂₇

C₂²×C₄

yes

—

C₈×C₁₂₇

C₂²×C₄

yes

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₄×D₄

yes

—

C₂²×C₁₂₇

C₂²×D₄

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

C₄×D₄

yes

—

C₄×C₁₂₇

C₂³×C₄

yes

characteristic

C₄×C₁₂₇

C₂×#41

yes

characteristic

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

C₂×C₄×D₄

yes

characteristic

C₂×C₁₂₇

#578

yes

—

#2950

—

127

maximal, Sylow

C₁₂₇

#2950

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

127

—

#519

—

127

—

#441

—

127

—

C₄×Q₁₆

—

127

—

#427

D₁₂₇

yes

characteristic

#419

—

127

—

#405

—

127

—

C₂²×Q₈

—

127

—

C₂×Q₁₆

—

127

—

C₂×Q₁₆

—

127

—

C₂×Q₁₆

—

127

—

C₂×M₁₆

—

127

—

C₂²×C₈

C₂×D₁₂₇

yes

characteristic

C₄×Q₈

—

127

—

C₄×Q₈

—

254

—

#117

—

127

—

#117

—

254

—

#117

C₂×D₁₂₇

yes

—

#107

C₂×D₁₂₇

yes

—

#106

C₂×D₁₂₇

yes

—

#103

—

127

—

#97

—

127

—

C₄×C₈

—

127

—

C₂×Q₈

—

127

—

C₂×Q₈

—

127

—

C₂×Q₈

—

254

—

100

C₂×Q₈

—

254

—

101

C₂²×C₄

—

254

—

102

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

103

Q₁₆

—

127

—

104

Q₁₆

—

127

—

105

M₁₆

—

254

—

106

C₂×C₈

—

127

—

107

C₂×C₈

C₄×D₁₂₇

yes

—

108

C₂×C₈

C₂²×D₁₂₇

yes

characteristic

109

C₂×C₈

C₂²×D₁₂₇

yes

characteristic

110

C₄⋊C₄

—

127

—

111

C₄⋊C₄

—

254

—

112

C₄⋊C₄

C₂²×D₁₂₇

yes

—

113

C₂²⋊C₄

—

114

C₂²⋊C₄

—

127

—

115

C₄²

—

116

C₄²

—

127

—

117

C₄²

—

254

—

118

C₂³

D₄×D₁₂₇

yes

characteristic

119

Q₈

—

127

—

120

Q₈

—

254

—

121

Q₈

—

254

—

122

C₂×C₄

—

123

C₂×C₄

—

127

—

124

C₂×C₄

—

254

—

125

C₂×C₄

—

254

—

126

C₂×C₄

C₂×C₄×D₁₂₇

yes

—

127

C₂×C₄

D₄×D₁₂₇

yes

characteristic

128

C₂×C₄

#14523249

yes

—

129

C₂×C₄

C₂³×D₁₂₇

yes

characteristic

130

C₈

—

254

—

131

C₈

C₂×C₄×D₁₂₇

yes

—

132

C₈

C₂×C₄×D₁₂₇

yes

—

133

C₂²

—

134

C₂²

#16861171

yes

—

135

C₂²

C₂×D₄×D₁₂₇

yes

characteristic

136

C₄

—

137

C₄

—

254

—

138

C₄

—

254

—

139

C₄

#16861171

yes

—

140

C₄

C₂²×C₄×D₁₂₇

yes

characteristic

141

C₄

C₂×#14523249

yes

characteristic

142

C₂

—

143

C₂

C₂×#16861171

yes

characteristic

144

C₂

#22550690

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₂₇
commutator	C₄×C₁₂₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	#427×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂²×D₄
prime radical	C₂³×C₁₂₇	C₂×D₄
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	#427	D₁₂₇
hypocenter	C₁₂₇	#2950
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂²

C₂×D₄×D₁₂₇

commutator

C₄×C₁₂₇

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₁₂₇

Fitting

#427×C₁₂₇

C₂

socle

C₂²×C₁₂₇

C₂²×D₄

prime radical

C₂³×C₁₂₇

C₂×D₄

cosocle

C₂×C₄×C₁₂₇

C₂⁴

D₁₂₇

C₁₂₇

G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
3. #22549982	C₂	semidirect
4. #22549970	C₂	semidirect
9. C₂×Dic₅₀₈	C₄	semidirect
73. C₁₂₇	#2950	semidirect

H ≅ G/N

type

3. #22549982

C₂

semidirect

4. #22549970

C₂

semidirect

9. C₂×Dic₅₀₈

C₄

semidirect

73. C₁₂₇

#2950

semidirect

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

subgroup	index
C₂²×Dic₅₀₈	2
#22550353	2
#22549982	2
#22549970	2
#22549884	2
#22549831	2
#427×C₁₂₇	2
#2950	127

subgroup

index

C₂²×Dic₅₀₈

#427×C₁₂₇

127

Subgroups — #30647597

Conjugacy classes of subgroups

Complements

Maximal subgroups