Group #30640774 — subgroups

Subgroups — #30640774

Order 16256.

166 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 49 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×#16861154	C₂	1	1	yes	maximal, characteristic
2	C₂×#16861152	C₂	1	1	yes	maximal, characteristic
3	#22550254	C₂	1	1	yes	maximal, characteristic, complemented
4	#22549958	C₂	1	1	yes	maximal, characteristic
5	Dic₁₂₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	#22549679	C₂	1	1	yes	maximal, characteristic, complemented
7	#358×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₅₀₈	C₂²	1	1	yes	characteristic
9	C₂×#14523238	—	2	1	no	—
10	C₂×D₁₀₁₆	C₂²	1	1	yes	characteristic
11	#16861207	—	2	1	no	—
12	#16861171	—	2	1	no	—
13	#16861154	—	2	2	no	—
14	#16861152	—	2	2	no	—
15	C₂×C₁₂₇⋊C₁₆	C₂²	1	1	yes	characteristic
16	#16861129	—	2	1	no	—
17	C₈×Dic₁₂₇	C₂²	1	1	yes	characteristic
18	C₂×Q₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
19	C₂×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
20	#107×C₁₂₇	C₂²	1	1	yes	characteristic
21	C₂×Dic₂₅₄	—	2	1	no	—
22	C₂×D₅₀₈	—	2	1	no	—
23	C₂×C₄×D₁₂₇	—	4	1	no	—
24	Dic₅₀₈	—	2	1	no	—
25	Dic₅₀₈	—	2	1	no	—
26	#14523238	—	4	2	no	—
27	D₁₀₁₆	—	2	1	no	—
28	D₁₀₁₆	D₄	1	2	yes	—
29	C₂×C₁₂₇⋊C₈	D₄	1	1	yes	characteristic
30	Dic₁₂₇⋊C₄	—	4	1	no	—
31	Dic₁₂₇⋊C₄	—	4	1	no	—
32	(C₂×D₁₂₇)⋊C₄	—	4	1	no	—
33	C₄×Dic₁₂₇	D₄	1	1	yes	characteristic
34	C₁₂₇⋊C₁₆	—	2	1	no	—
35	C₂×Q₈×C₁₂₇	—	2	1	no	—
36	Q₁₆×C₁₂₇	—	2	1	no	—
37	Q₁₆×C₁₂₇	—	2	1	no	—
38	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
39	C₄⋊C₄×C₁₂₇	—	2	1	no	—
40	C₁₆×C₁₂₇	—	2	1	no	—
41	C₂²×D₁₂₇	—	4	1	no	—
42	Dic₂₅₄	—	4	1	no	—
43	Dic₂₅₄	—	4	1	no	—
44	D₅₀₈	—	2	2	no	—
45	D₅₀₈	—	4	1	no	—
46	C₂×Dic₁₂₇	—	4	1	no	—
47	C₂×Dic₁₂₇	D₈	1	1	yes	characteristic
48	C₂×Dic₁₂₇	D₈	1	1	yes	characteristic
49	C₄×D₁₂₇	—	8	2	no	—
50	C₁₂₇⋊C₈	—	2	1	no	—
51	Q₈×C₁₂₇	—	4	1	no	—
52	Q₈×C₁₂₇	—	4	1	no	—
53	C₂×C₄×C₁₂₇	—	4	1	no	—
54	C₂×C₄×C₁₂₇	—	4	1	no	—
55	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
56	C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
57	C₈×C₁₂₇	#41	1	1	yes	characteristic
58	C₂²×C₁₂₇	C₂×D₈	1	1	yes	characteristic
59	C₂×D₁₂₇	—	4	2	no	—
60	C₂×D₁₂₇	—	8	1	no	—
61	C₂×D₁₂₇	—	8	1	no	—
62	C₄×C₁₂₇	—	8	1	no	—
63	C₄×C₁₂₇	—	8	1	no	—
64	C₄×C₁₂₇	#121	1	1	yes	characteristic
65	C₄×C₁₂₇	AGL(1,ℤ/8)	1	1	yes	characteristic
66	Dic₁₂₇	—	2	1	no	—
67	Dic₁₂₇	—	8	1	no	—
68	Dic₁₂₇	SD₃₂	1	2	yes	complemented
69	C₂×C₁₂₇	#459	1	1	yes	characteristic
70	C₂×C₁₂₇	C₂×SD₃₂	1	1	yes	characteristic
71	C₂×C₁₂₇	#510	1	1	yes	characteristic
72	D₁₂₇	—	8	2	no	—
73	#2208	—	127	1	no	maximal, Sylow
74	C₁₂₇	#2208	1	1	yes	characteristic, complemented, Sylow
75	C₂×SD₃₂	—	127	1	no	—
76	C₂×SD₃₂	—	127	1	no	—
77	#494	—	127	1	no	—
78	C₄×D₈	—	127	1	no	—
79	C₄⋊C₁₆	—	127	1	no	—
80	#358	—	127	1	no	—
81	#358	D₁₂₇	1	1	yes	characteristic, complemented
82	C₂×Q₁₆	—	127	1	no	—
83	C₂×Q₁₆	—	254	1	no	—
84	C₂×Q₁₆	C₂×D₁₂₇	1	1	yes	characteristic
85	C₂×D₈	—	127	1	no	—
86	C₄⋊Q₈	—	254	1	no	—
87	C₄×D₄	—	254	1	no	—
88	SD₃₂	—	254	2	no	—
89	SD₃₂	—	254	2	no	—
90	C₂×C₁₆	—	127	1	no	—
91	C₂×C₁₆	C₂×D₁₂₇	1	1	yes	characteristic
92	#107	C₂×D₁₂₇	1	1	yes	characteristic
93	#102	—	254	1	no	—
94	C₄×C₈	—	127	1	no	—
95	C₂×Q₈	—	2	1	no	—
96	C₂×Q₈	—	254	1	no	—
97	C₂×D₄	—	254	1	no	—
98	C₂²×C₄	—	508	1	no	—
99	Q₁₆	—	2	1	no	—
100	Q₁₆	—	2	1	no	—
101	Q₁₆	—	508	2	no	—
102	Q₁₆	—	254	1	no	—
103	Q₁₆	—	254	1	no	—
104	D₈	—	127	2	no	—
105	D₈	—	254	1	no	—
106	C₂×C₈	—	127	1	no	—
107	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
108	C₄⋊C₄	—	2	1	no	—
109	C₄⋊C₄	—	508	1	no	—
110	C₄⋊C₄	—	508	1	no	—
111	C₂²⋊C₄	—	508	1	no	—
112	C₄²	—	127	1	no	—
113	C₁₆	—	2	1	no	—
114	C₁₆	—	254	1	no	—
115	C₂³	—	508	1	no	—
116	Q₈	—	4	1	no	—
117	Q₈	—	4	1	no	—
118	Q₈	—	508	1	no	—
119	Q₈	—	508	1	no	—
120	D₄	—	508	1	no	—
121	D₄	—	254	2	no	—
122	C₂×C₄	—	4	1	no	—
123	C₂×C₄	—	4	1	no	—
124	C₂×C₄	—	127	1	no	—
125	C₂×C₄	—	127	1	no	—
126	C₂×C₄	—	1016	2	no	—
127	C₂×C₄	—	508	1	no	—
128	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
129	C₈	—	254	1	no	—
130	C₈	D₄×D₁₂₇	1	1	yes	characteristic
131	C₈	#14523248	1	1	yes	characteristic
132	C₂²	—	1016	1	no	—
133	C₂²	—	1016	1	no	—
134	C₂²	—	508	2	no	—
135	C₂²	D₈×D₁₂₇	1	1	yes	characteristic
136	C₄	—	8	1	no	—
137	C₄	—	8	1	no	—
138	C₄	—	127	2	no	—
139	C₄	—	1016	1	no	—
140	C₄	—	254	1	no	—
141	C₄	#16861182	1	1	yes	characteristic
142	C₄	#16861236	1	1	yes	characteristic
143	C₂	—	1016	2	no	—
144	C₂	#22550070	1	1	yes	characteristic
145	C₂	SD₃₂×D₁₂₇	1	1	yes	characteristic
146	C₂	#22550318	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16861154

C₂

yes

maximal, characteristic

C₂×#16861152

C₂

yes

maximal, characteristic

#22550254

C₂

yes

maximal, characteristic, complemented

#22549958

C₂

yes

maximal, characteristic

Dic₁₂₇⋊C₁₆

C₂

yes

maximal, characteristic, complemented

#22549679

C₂

yes

maximal, characteristic, complemented

#358×C₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂×Dic₅₀₈

C₂²

yes

characteristic

C₂×#14523238

—

C₂×D₁₀₁₆

C₂²

yes

—

—

—

—

C₂×C₁₂₇⋊C₁₆

C₂²

yes

characteristic

#16861129

—

C₈×Dic₁₂₇

C₂²

yes

characteristic

C₂×Q₁₆×C₁₂₇

C₂²

yes

characteristic

C₂×C₁₆×C₁₂₇

C₂²

yes

characteristic

#107×C₁₂₇

C₂²

yes

characteristic

C₂×Dic₂₅₄

—

C₂×D₅₀₈

—

C₂×C₄×D₁₂₇

—

Dic₅₀₈

—

Dic₅₀₈

—

#14523238

—

D₁₀₁₆

—

D₁₀₁₆

D₄

yes

—

C₂×C₁₂₇⋊C₈

D₄

yes

characteristic

Dic₁₂₇⋊C₄

—

Dic₁₂₇⋊C₄

—

(C₂×D₁₂₇)⋊C₄

—

C₄×Dic₁₂₇

D₄

yes

characteristic

C₁₂₇⋊C₁₆

—

C₂×Q₈×C₁₂₇

—

Q₁₆×C₁₂₇

—

Q₁₆×C₁₂₇

—

C₂×C₈×C₁₂₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₂₇

—

C₁₆×C₁₂₇

—

C₂²×D₁₂₇

—

Dic₂₅₄

—

Dic₂₅₄

—

D₅₀₈

—

D₅₀₈

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

D₈

yes

characteristic

C₂×Dic₁₂₇

D₈

yes

characteristic

C₄×D₁₂₇

—

C₁₂₇⋊C₈

—

Q₈×C₁₂₇

—

Q₈×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₈×C₁₂₇

C₂×D₄

yes

characteristic

C₈×C₁₂₇

#41

yes

characteristic

C₂²×C₁₂₇

C₂×D₈

yes

characteristic

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

yes

C₄×C₁₂₇

yes

Dic₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

SD₃₂

yes

complemented

C₂×C₁₂₇

#459

yes

characteristic

C₂×C₁₂₇

C₂×SD₃₂

yes

characteristic

C₂×C₁₂₇

#510

yes

characteristic

D₁₂₇

—

#2208

—

127

maximal, Sylow

C₁₂₇

#2208

yes

characteristic, complemented, Sylow

C₂×SD₃₂

—

127

—

C₂×SD₃₂

—

127

—

#494

—

127

—

C₄×D₈

—

127

—

C₄⋊C₁₆

—

127

—

#358

—

127

—

#358

D₁₂₇

yes

characteristic, complemented

C₂×Q₁₆

—

127

—

C₂×Q₁₆

—

254

—

C₂×Q₁₆

C₂×D₁₂₇

yes

characteristic

C₂×D₈

—

127

—

C₄⋊Q₈

—

254

—

C₄×D₄

—

254

—

SD₃₂

—

254

—

SD₃₂

—

254

—

C₂×C₁₆

—

127

—

C₂×C₁₆

C₂×D₁₂₇

yes

characteristic

#107

C₂×D₁₂₇

yes

characteristic

#102

—

254

—

C₄×C₈

—

127

—

C₂×Q₈

—

C₂×Q₈

—

254

—

C₂×D₄

—

254

—

C₂²×C₄

—

508

—

Q₁₆

—

100

Q₁₆

—

101

Q₁₆

—

508

—

102

Q₁₆

—

254

—

103

Q₁₆

—

254

—

104

D₈

—

127

—

105

D₈

—

254

—

106

C₂×C₈

—

127

—

107

C₂×C₈

C₂²×D₁₂₇

yes

characteristic

108

C₄⋊C₄

—

109

C₄⋊C₄

—

508

—

110

C₄⋊C₄

—

508

—

111

C₂²⋊C₄

—

508

—

112

C₄²

—

127

—

113

C₁₆

—

114

C₁₆

—

254

—

115

C₂³

—

508

—

116

Q₈

—

117

Q₈

—

118

Q₈

—

508

—

119

Q₈

—

508

—

120

D₄

—

508

—

121

D₄

—

254

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

127

—

125

C₂×C₄

—

127

—

126

C₂×C₄

—

1016

—

127

C₂×C₄

—

508

—

128

C₂×C₄

D₄×D₁₂₇

yes

characteristic

129

C₈

—

254

—

130

C₈

D₄×D₁₂₇

yes

characteristic

131

C₈

#14523248

yes

characteristic

132

C₂²

—

1016

—

133

C₂²

—

1016

—

134

C₂²

—

508

—

135

C₂²

D₈×D₁₂₇

yes

characteristic

136

C₄

—

137

C₄

—

138

C₄

—

127

—

139

C₄

—

1016

—

140

C₄

—

254

—

141

C₄

#16861182

yes

characteristic

142

C₄

#16861236

yes

characteristic

143

C₂

—

1016

—

144

C₂

#22550070

yes

characteristic

145

C₂

SD₃₂×D₁₂₇

yes

characteristic

146

C₂

#22550318

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₁₂₇
commutator, cosocle	C₂×C₈×C₁₂₇	C₂³
Frattini	C₂×C₈	C₂²×D₁₂₇
Fitting	#358×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂×D₈
prime radical	C₂×D₁₀₁₆	C₂²
hypercenter	#358	D₁₂₇
hypocenter	C₁₂₇	#2208
LCS	G ▷ C₂×C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

subgroup

G/H

center

C₂²

D₈×D₁₂₇

commutator, cosocle

C₂×C₈×C₁₂₇

C₂³

Frattini

C₂×C₈

C₂²×D₁₂₇

Fitting

#358×C₁₂₇

C₂

socle

C₂²×C₁₂₇

C₂×D₈

prime radical

C₂×D₁₀₁₆

C₂²

D₁₂₇

C₁₂₇

G ▷ C₂×C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

N	H ≅ G/N	type
3. #22550254	C₂	semidirect
5. Dic₁₂₇⋊C₁₆	C₂	semidirect
6. #22549679	C₂	semidirect
7. #358×C₁₂₇	C₂	semidirect
68. Dic₁₂₇	SD₃₂	semidirect
74. C₁₂₇	#2208	semidirect
81. #358	D₁₂₇	semidirect

H ≅ G/N

type

3. #22550254

C₂

semidirect

5. Dic₁₂₇⋊C₁₆

C₂

semidirect

6. #22549679

C₂

semidirect

7. #358×C₁₂₇

C₂

semidirect

68. Dic₁₂₇

SD₃₂

semidirect

74. C₁₂₇

#2208

semidirect

81. #358

D₁₂₇

semidirect

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

subgroup	index
C₂×#16861154	2
C₂×#16861152	2
#22550254	2
#22549958	2
Dic₁₂₇⋊C₁₆	2
#22549679	2
#358×C₁₂₇	2
#2208	127

subgroup

index

Dic₁₂₇⋊C₁₆

#358×C₁₂₇

127

Subgroups — #30640774

Conjugacy classes of subgroups

Complements

Maximal subgroups