Group #30638775 — subgroups

Subgroups — #30638775

Order 16256.

260 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 81 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×#16861217	C₂	1	1	yes	maximal, characteristic
2	#22549912	C₂	1	2	yes	maximal, complemented
3	#22549849	C₂	1	2	yes	maximal
4	#22549633	C₂	1	1	yes	maximal, characteristic, complemented
5	H(3,ℤ/4)×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14523248	C₂²	1	1	yes	characteristic
7	#16861217	—	2	2	no	—
8	#16861217	—	2	2	no	—
9	C₂×#14523231	C₄	1	2	yes	complemented
10	C₂²×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₈×D₁₂₇	C₄	1	2	yes	complemented
12	#16861168	C₂²	1	2	yes	—
13	#16861142	—	2	2	no	—
14	#16861140	—	2	2	no	—
15	#16861123	—	2	4	no	—
16	C₂×M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
17	#117×C₁₂₇	C₂²	1	2	yes	—
18	#14523248	—	2	2	no	—
19	#14523248	—	2	2	no	—
20	C₂×Dic₂₅₄	Q₈	1	1	yes	characteristic
21	C₂×D₅₀₈	Q₈	1	1	yes	characteristic
22	C₂×C₂²⋊D₁₂₇	D₄	1	2	yes	—
23	C₂×C₄×D₁₂₇	C₂×C₄	1	2	yes	—
24	#14523231	—	2	4	no	—
25	C₂×C₁₂₇⋊C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₂₇⋊C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₂₇⋊C₈	D₄	1	2	yes	—
28	C₂×C₁₂₇⋊C₈	D₄	1	2	yes	—
29	C₈×D₁₂₇	—	2	2	no	—
30	C₈×D₁₂₇	—	2	2	no	—
31	C₄⋊Dic₁₂₇	—	2	2	no	—
32	C₂²⋊Dic₁₂₇	—	2	2	no	—
33	C₄×Dic₁₂₇	—	2	2	no	—
34	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
35	M₁₆×C₁₂₇	—	2	2	no	—
36	C₂×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
37	C₄⋊C₄×C₁₂₇	—	2	2	no	—
38	C₂²⋊C₄×C₁₂₇	—	2	2	no	—
39	C₄²×C₁₂₇	—	2	2	no	—
40	C₂²×D₁₂₇	C₄⋊C₄	1	2	yes	—
41	Dic₂₅₄	—	2	2	no	—
42	D₅₀₈	—	2	2	no	—
43	C₂²⋊D₁₂₇	—	2	4	no	—
44	C₂×Dic₁₂₇	—	4	2	no	—
45	C₂×Dic₁₂₇	C₄⋊C₄	1	2	yes	—
46	C₄×D₁₂₇	—	2	2	no	—
47	C₄×D₁₂₇	—	2	2	no	—
48	C₁₂₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
49	C₁₂₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
50	C₁₂₇⋊C₈	C₂²⋊C₄	1	2	yes	—
51	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₁₂₇	—	4	2	no	—
53	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₄×C₁₂₇	#41	1	2	yes	—
57	C₈×C₁₂₇	—	2	2	no	—
58	C₂²×C₁₂₇	—	4	1	no	—
59	C₂²×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₁₂₇	#121	1	2	yes	—
61	C₂×D₁₂₇	—	2	2	no	—
62	C₂×D₁₂₇	—	4	2	no	—
63	C₄×C₁₂₇	—	8	2	no	—
64	C₄×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₄×C₁₂₇	#117	1	1	yes	characteristic
66	C₄×C₁₂₇	C₂²⋊Q₈	1	2	yes	—
67	Dic₁₂₇	—	2	2	no	—
68	Dic₁₂₇	—	8	2	no	—
69	C₂×C₁₂₇	—	2	2	no	—
70	C₂×C₁₂₇	—	2	1	no	—
71	C₂×C₁₂₇	#380	1	1	yes	characteristic
72	D₁₂₇	—	4	2	no	—
73	#1812	—	127	1	no	maximal, Sylow
74	C₁₂₇	#1812	1	1	yes	characteristic, complemented, Sylow
75	C₂×#131	—	127	1	no	—
76	#427	—	127	2	no	—
77	#410	—	127	2	no	—
78	H(3,ℤ/4)	—	127	1	no	—
79	H(3,ℤ/4)	D₁₂₇	1	1	yes	characteristic, complemented
80	C₂×#41	—	127	1	no	—
81	#131	—	254	2	no	—
82	#131	—	254	2	no	—
83	C₂×M₁₆	—	127	2	no	—
84	C₂×M₁₆	C₂×D₁₂₇	1	1	yes	characteristic
85	C₂²×C₈	—	127	1	no	—
86	C₂²×C₈	—	127	2	no	—
87	#117	—	127	2	no	—
88	#117	C₂×D₁₂₇	1	2	yes	—
89	#107	—	254	2	no	—
90	#106	—	254	2	no	—
91	#99	—	254	4	no	—
92	#41	—	254	2	no	—
93	#41	—	254	2	no	—
94	C₂×Q₈	—	127	1	no	—
95	C₂×D₄	—	127	2	no	—
96	C₂×D₄	—	127	1	no	—
97	C₂²×C₄	—	127	2	no	—
98	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
99	M₁₆	—	2	2	no	—
100	M₁₆	—	254	4	no	—
101	C₂×C₈	—	127	1	no	—
102	C₂×C₈	—	127	1	no	—
103	C₂×C₈	—	127	2	no	—
104	C₂×C₈	—	127	2	no	—
105	C₂×C₈	—	254	2	no	—
106	C₂×C₈	—	254	2	no	—
107	C₂×C₈	C₄×D₁₂₇	1	2	yes	—
108	C₄⋊C₄	—	2	2	no	—
109	C₄⋊C₄	—	254	2	no	—
110	C₂²⋊C₄	—	2	2	no	—
111	C₂²⋊C₄	—	254	2	no	—
112	C₄²	—	2	2	no	—
113	C₄²	—	254	2	no	—
114	C₂³	—	127	2	no	—
115	C₂³	D₄×D₁₂₇	1	1	yes	characteristic
116	Q₈	—	254	2	no	—
117	D₄	—	254	4	no	—
118	D₄	—	254	2	no	—
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	—	127	2	no	—
121	C₂×C₄	—	508	2	no	—
122	C₂×C₄	—	254	2	no	—
123	C₂×C₄	—	254	2	no	—
124	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
125	C₂×C₄	D₄×D₁₂₇	1	2	yes	—
126	C₂×C₄	Q₈×D₁₂₇	1	1	yes	characteristic
127	C₂×C₄	#14523248	1	2	yes	—
128	C₈	—	2	2	no	—
129	C₈	—	127	1	no	—
130	C₈	—	127	1	no	—
131	C₈	—	127	2	no	—
132	C₂²	—	4	1	no	—
133	C₂²	—	508	2	no	—
134	C₂²	—	254	2	no	—
135	C₂²	C₄⋊C₄×D₁₂₇	1	1	yes	characteristic
136	C₂²	#16861182	1	2	yes	—
137	C₄	—	8	2	no	—
138	C₄	—	1016	2	no	—
139	C₄	—	254	2	no	—
140	C₄	C₂²⋊C₄×D₁₂₇	1	1	yes	characteristic
141	C₄	#16861166	1	1	yes	characteristic
142	C₄	#16861187	1	2	yes	—
143	C₂	—	2	2	no	—
144	C₂	—	2	1	no	—
145	C₂	—	508	2	no	—
146	C₂	#22549735	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16861217

C₂

yes

maximal, characteristic

#22549912

C₂

yes

maximal, complemented

#22549849

C₂

yes

maximal

#22549633

C₂

yes

maximal, characteristic, complemented

H(3,ℤ/4)×C₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂×#14523248

C₂²

yes

—

—

C₄

yes

complemented

C₂²×C₁₂₇⋊C₈

C₂²

yes

characteristic

C₂×C₈×D₁₂₇

C₄

yes

complemented

#16861168

C₂²

yes

—

#16861142

—

#16861140

—

#16861123

—

C₂×M₁₆×C₁₂₇

C₂²

yes

characteristic

#117×C₁₂₇

C₂²

yes

—

#14523248

—

#14523248

—

C₂×Dic₂₅₄

Q₈

yes

characteristic

C₂×D₅₀₈

Q₈

yes

characteristic

C₂×C₂²⋊D₁₂₇

D₄

yes

—

C₂×C₄×D₁₂₇

C₂×C₄

yes

—

#14523231

—

C₂×C₁₂₇⋊C₈

C₂×C₄

yes

characteristic

C₂×C₁₂₇⋊C₈

C₂×C₄

yes

characteristic

C₂×C₁₂₇⋊C₈

D₄

yes

—

C₂×C₁₂₇⋊C₈

D₄

yes

—

C₈×D₁₂₇

—

C₈×D₁₂₇

—

C₄⋊Dic₁₂₇

—

C₂²⋊Dic₁₂₇

—

C₄×Dic₁₂₇

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

M₁₆×C₁₂₇

—

C₂×C₈×C₁₂₇

C₂×C₄

yes

—

C₄⋊C₄×C₁₂₇

—

C₂²⋊C₄×C₁₂₇

—

C₄²×C₁₂₇

—

C₂²×D₁₂₇

C₄⋊C₄

yes

—

Dic₂₅₄

—

D₅₀₈

—

C₂²⋊D₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

C₄⋊C₄

yes

—

C₄×D₁₂₇

—

C₄×D₁₂₇

—

C₁₂₇⋊C₈

C₂²⋊C₄

yes

characteristic

C₁₂₇⋊C₈

C₂²⋊C₄

yes

characteristic

C₁₂₇⋊C₈

C₂²⋊C₄

yes

—

C₂³×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×D₄

yes

—

C₂×C₄×C₁₂₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₂₇

#41

yes

—

C₈×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₂₇

#121

yes

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₂₇

#117

yes

characteristic

C₄×C₁₂₇

C₂²⋊Q₈

yes

—

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

#380

yes

characteristic

D₁₂₇

—

#1812

—

127

maximal, Sylow

C₁₂₇

#1812

yes

characteristic, complemented, Sylow

C₂×#131

—

127

—

#427

—

127

—

#410

—

127

—

H(3,ℤ/4)

—

127

—

H(3,ℤ/4)

D₁₂₇

yes

characteristic, complemented

C₂×#41

—

127

—

#131

—

254

—

#131

—

254

—

C₂×M₁₆

—

127

—

C₂×M₁₆

C₂×D₁₂₇

yes

characteristic

C₂²×C₈

—

127

—

C₂²×C₈

—

127

—

#117

—

127

—

#117

C₂×D₁₂₇

yes

—

#107

—

254

—

#106

—

254

—

#99

—

254

—

#41

—

254

—

#41

—

254

—

C₂×Q₈

—

127

—

C₂×D₄

—

127

—

C₂×D₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

M₁₆

—

100

M₁₆

—

254

—

101

C₂×C₈

—

127

—

102

C₂×C₈

—

127

—

103

C₂×C₈

—

127

—

104

C₂×C₈

—

127

—

105

C₂×C₈

—

254

—

106

C₂×C₈

—

254

—

107

C₂×C₈

C₄×D₁₂₇

yes

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

254

—

110

C₂²⋊C₄

—

111

C₂²⋊C₄

—

254

—

112

C₄²

—

113

C₄²

—

254

—

114

C₂³

—

127

—

115

C₂³

D₄×D₁₂₇

yes

characteristic

116

Q₈

—

254

—

117

D₄

—

254

—

118

D₄

—

254

—

119

C₂×C₄

—

120

C₂×C₄

—

127

—

121

C₂×C₄

—

508

—

122

C₂×C₄

—

254

—

123

C₂×C₄

—

254

—

124

C₂×C₄

C₂×C₄×D₁₂₇

yes

characteristic

125

C₂×C₄

D₄×D₁₂₇

yes

—

126

C₂×C₄

Q₈×D₁₂₇

yes

characteristic

127

C₂×C₄

#14523248

yes

—

128

C₈

—

129

C₈

—

127

—

130

C₈

—

127

—

131

C₈

—

127

—

132

C₂²

—

133

C₂²

—

508

—

134

C₂²

—

254

—

135

C₂²

C₄⋊C₄×D₁₂₇

yes

characteristic

136

C₂²

#16861182

yes

—

137

C₄

—

138

C₄

—

1016

—

139

C₄

—

254

—

140

C₄

C₂²⋊C₄×D₁₂₇

yes

characteristic

141

C₄

#16861166

yes

characteristic

142

C₄

#16861187

yes

—

143

C₂

—

144

C₂

—

145

C₂

—

508

—

146

C₂

#22549735

yes

characteristic

	subgroup	G/H
center	C₄	#16861166
commutator	C₂×C₄×C₁₂₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	H(3,ℤ/4)×C₁₂₇	C₂
socle	C₂×C₁₂₇	#380
prime radical	C₂×#14523248	C₂²
cosocle	C₂²×C₄×C₁₂₇	C₂³
hypercenter	H(3,ℤ/4)	D₁₂₇
hypocenter	C₁₂₇	#1812
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

#16861166

commutator

C₂×C₄×C₁₂₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₂₇

Fitting

H(3,ℤ/4)×C₁₂₇

C₂

C₂×C₁₂₇

C₂²

C₂²×C₄×C₁₂₇

C₂³

D₁₂₇

C₁₂₇

G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
2. #22549912	C₂	semidirect
4. #22549633	C₂	semidirect
5. H(3,ℤ/4)×C₁₂₇	C₂	semidirect
9. C₂×#14523231	C₄	semidirect
11. C₂×C₈×D₁₂₇	C₄	semidirect
74. C₁₂₇	#1812	semidirect
79. H(3,ℤ/4)	D₁₂₇	semidirect

H ≅ G/N

type

2. #22549912

C₂

semidirect

4. #22549633

C₂

semidirect

5. H(3,ℤ/4)×C₁₂₇

C₂

semidirect

9. C₂×#14523231

C₄

semidirect

11. C₂×C₈×D₁₂₇

C₄

semidirect

74. C₁₂₇

#1812

semidirect

79. H(3,ℤ/4)

D₁₂₇

semidirect

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

subgroup	index
C₂×#16861217	2
#22549912	2
#22549849	2
#22549633	2
H(3,ℤ/4)×C₁₂₇	2
#1812	127

subgroup

index

H(3,ℤ/4)×C₁₂₇

127

Subgroups — #30638775

Conjugacy classes of subgroups

Complements

Maximal subgroups