Group #30640037 — subgroups

Subgroups — #30640037

Order 16256.

264 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 63 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	#22550676	C₂	1	1	yes	maximal, characteristic
2	#22549904	C₂	1	1	yes	maximal, characteristic, complemented
3	#22549870	C₂	1	2	yes	maximal, complemented
4	#22549849	C₂	1	2	yes	maximal, complemented
5	H(3,ℤ/4)×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14523248	C₂²	1	2	yes	—
7	#16861242	—	2	2	no	—
8	#16861237	—	2	2	no	—
9	#16861231	—	2	2	no	—
10	C₂²×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
11	#16861168	C₂²	1	1	yes	characteristic
12	Dic₁₂₇⋊C₈	—	2	2	no	—
13	#16861132	—	2	2	no	—
14	#16861129	—	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	#14523248	—	2	2	no	—
21	#14523248	—	2	2	no	—
22	C₂×Dic₂₅₄	D₄	1	2	yes	—
23	C₂×D₅₀₈	D₄	1	2	yes	—
24	C₂×C₂²⋊D₁₂₇	—	2	2	no	—
25	C₂×C₄×D₁₂₇	—	2	2	no	—
26	Dic₅₀₈	—	2	2	no	—
27	#14523237	—	2	4	no	—
28	D₁₀₁₆	—	2	2	no	—
29	C₂×C₁₂₇⋊C₈	—	2	2	no	—
30	C₂×C₁₂₇⋊C₈	—	2	1	no	—
31	C₄⋊Dic₁₂₇	—	2	1	no	—
32	C₂²⋊Dic₁₂₇	D₄	1	2	yes	—
33	C₄×Dic₁₂₇	—	2	1	no	—
34	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
35	M₁₆×C₁₂₇	—	2	2	no	—
36	C₂×C₈×C₁₂₇	D₄	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	—	2	2	no	—
41	Dic₂₅₄	—	2	2	no	—
42	Dic₂₅₄	—	2	2	no	—
43	D₅₀₈	—	2	2	no	—
44	D₅₀₈	—	2	2	no	—
45	C₂²⋊D₁₂₇	—	4	2	no	—
46	C₂²⋊D₁₂₇	—	4	2	no	—
47	C₂×Dic₁₂₇	—	2	2	no	—
48	C₂×Dic₁₂₇	—	2	2	no	—
49	C₄×D₁₂₇	—	4	2	no	—
50	C₄×D₁₂₇	—	4	2	no	—
51	C₁₂₇⋊C₈	—	4	1	no	—
52	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
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	—	4	1	no	—
60	C₂²×C₁₂₇	#121	1	1	yes	characteristic
61	C₂²×C₁₂₇	#121	1	2	yes	—
62	C₂×D₁₂₇	—	4	2	no	—
63	C₂×D₁₂₇	—	8	2	no	—
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	—	4	2	no	—
69	Dic₁₂₇	—	4	2	no	—
70	C₂×C₁₂₇	—	2	1	no	—
71	C₂×C₁₂₇	—	2	2	no	—
72	C₂×C₁₂₇	#394	1	1	yes	characteristic
73	D₁₂₇	—	8	2	no	—
74	#2053	—	127	1	no	maximal, Sylow
75	C₁₂₇	#2053	1	1	yes	characteristic, complemented, Sylow
76	#575	—	127	1	no	—
77	#424	—	127	1	no	—
78	#416	—	127	2	no	—
79	#410	—	127	2	no	—
80	H(3,ℤ/4)	D₁₂₇	1	1	yes	characteristic, complemented
81	C₂×#41	—	127	2	no	—
82	#137	—	254	2	no	—
83	AGL(1,ℤ/8)	—	254	2	no	—
84	#135	—	254	2	no	—
85	C₂×M₁₆	C₂×D₁₂₇	1	1	yes	characteristic
86	C₂²×C₈	—	127	1	no	—
87	#117	—	127	1	no	—
88	#117	C₂×D₁₂₇	1	2	yes	—
89	C₄⋊C₈	—	254	2	no	—
90	#103	—	254	2	no	—
91	#102	—	254	2	no	—
92	#99	—	254	4	no	—
93	#41	—	254	2	no	—
94	#41	—	254	2	no	—
95	#41	—	254	2	no	—
96	#41	—	254	2	no	—
97	C₂×Q₈	—	127	2	no	—
98	C₂×D₄	—	127	2	no	—
99	C₂×D₄	—	254	2	no	—
100	C₂²×C₄	—	254	2	no	—
101	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
102	Q₁₆	—	254	2	no	—
103	SD₁₆	—	254	4	no	—
104	D₈	—	254	2	no	—
105	M₁₆	—	2	2	no	—
106	C₂×C₈	—	254	2	no	—
107	C₂×C₈	—	254	1	no	—
108	C₂×C₈	D₅₀₈	1	2	yes	—
109	C₄⋊C₄	—	2	2	no	—
110	C₄⋊C₄	—	254	1	no	—
111	C₂²⋊C₄	—	2	2	no	—
112	C₂²⋊C₄	—	127	2	no	—
113	C₄²	—	2	2	no	—
114	C₄²	—	254	1	no	—
115	C₂³	—	254	2	no	—
116	C₂³	D₄×D₁₂₇	1	1	yes	characteristic
117	Q₈	—	254	2	no	—
118	Q₈	—	254	2	no	—
119	D₄	—	508	2	no	—
120	D₄	—	508	2	no	—
121	D₄	—	254	2	no	—
122	D₄	—	254	2	no	—
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	508	2	no	—
125	C₂×C₄	—	508	2	no	—
126	C₂×C₄	—	254	2	no	—
127	C₂×C₄	—	254	2	no	—
128	C₂×C₄	D₄×D₁₂₇	1	2	yes	—
129	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
130	C₂×C₄	#14523248	1	2	yes	—
131	C₂×C₄	#14523249	1	1	yes	characteristic
132	C₈	—	2	2	no	—
133	C₈	—	508	1	no	—
134	C₂²	—	4	1	no	—
135	C₂²	—	1016	2	no	—
136	C₂²	—	508	2	no	—
137	C₂²	#16861182	1	2	yes	—
138	C₂²	#16861184	1	1	yes	characteristic
139	C₄	—	8	2	no	—
140	C₄	—	508	2	no	—
141	C₄	—	508	2	no	—
142	C₄	#16861179	1	1	yes	characteristic
143	C₄	#16861194	1	2	yes	—
144	C₄	#16861197	1	1	yes	characteristic
145	C₂	—	2	2	no	—
146	C₂	—	2	1	no	—
147	C₂	—	1016	2	no	—
148	C₂	#22549794	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22550676

C₂

yes

maximal, characteristic

#22549904

C₂

yes

maximal, characteristic, complemented

#22549870

C₂

yes

maximal, complemented

#22549849

C₂

yes

maximal, complemented

H(3,ℤ/4)×C₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂×#14523248

C₂²

yes

—

#16861242

—

#16861237

—

#16861231

—

C₂²×C₁₂₇⋊C₈

C₂²

yes

characteristic

#16861168

C₂²

yes

characteristic

Dic₁₂₇⋊C₈

—

#16861132

—

#16861129

—

#16861123

—

C₂×M₁₆×C₁₂₇

C₂²

yes

characteristic

#117×C₁₂₇

C₂²

yes

—

—

—

—

—

C₂×Dic₂₅₄

D₄

yes

—

C₂×D₅₀₈

D₄

yes

—

C₂×C₂²⋊D₁₂₇

—

C₂×C₄×D₁₂₇

—

Dic₅₀₈

—

#14523237

—

D₁₀₁₆

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

—

C₄⋊Dic₁₂₇

—

C₂²⋊Dic₁₂₇

D₄

yes

—

C₄×Dic₁₂₇

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

M₁₆×C₁₂₇

—

C₂×C₈×C₁₂₇

D₄

yes

—

C₄⋊C₄×C₁₂₇

—

C₂²⋊C₄×C₁₂₇

—

C₄²×C₁₂₇

—

C₂²×D₁₂₇

—

Dic₂₅₄

—

Dic₂₅₄

—

D₅₀₈

—

D₅₀₈

—

C₂²⋊D₁₂₇

—

C₂²⋊D₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₄×D₁₂₇

—

C₄×D₁₂₇

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

C₂×D₄

yes

characteristic

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

—

C₂²×C₁₂₇

#121

yes

characteristic

C₂²×C₁₂₇

#121

yes

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

#120

yes

characteristic

C₄×C₁₂₇

#123

yes

—

C₄×C₁₂₇

#124

yes

characteristic

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

#394

yes

characteristic

D₁₂₇

—

#2053

—

127

maximal, Sylow

C₁₂₇

#2053

yes

characteristic, complemented, Sylow

#575

—

127

—

#424

—

127

—

#416

—

127

—

#410

—

127

—

H(3,ℤ/4)

D₁₂₇

yes

characteristic, complemented

C₂×#41

—

127

—

#137

—

254

—

AGL(1,ℤ/8)

—

254

—

#135

—

254

—

C₂×M₁₆

C₂×D₁₂₇

yes

characteristic

C₂²×C₈

—

127

—

#117

—

127

—

#117

C₂×D₁₂₇

yes

—

C₄⋊C₈

—

254

—

#103

—

254

—

#102

—

254

—

#99

—

254

—

#41

—

254

—

#41

—

254

—

#41

—

254

—

#41

—

254

—

C₂×Q₈

—

127

—

C₂×D₄

—

127

—

C₂×D₄

—

254

—

100

C₂²×C₄

—

254

—

101

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

102

Q₁₆

—

254

—

103

SD₁₆

—

254

—

104

D₈

—

254

—

105

M₁₆

—

106

C₂×C₈

—

254

—

107

C₂×C₈

—

254

—

108

C₂×C₈

D₅₀₈

yes

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

254

—

111

C₂²⋊C₄

—

112

C₂²⋊C₄

—

127

—

113

C₄²

—

114

C₄²

—

254

—

115

C₂³

—

254

—

116

C₂³

D₄×D₁₂₇

yes

characteristic

117

Q₈

—

254

—

118

Q₈

—

254

—

119

D₄

—

508

—

120

D₄

—

508

—

121

D₄

—

254

—

122

D₄

—

254

—

123

C₂×C₄

—

124

C₂×C₄

—

508

—

125

C₂×C₄

—

508

—

126

C₂×C₄

—

254

—

127

C₂×C₄

—

254

—

128

C₂×C₄

D₄×D₁₂₇

yes

—

129

C₂×C₄

C₂×D₅₀₈

yes

characteristic

130

C₂×C₄

#14523248

yes

—

131

C₂×C₄

#14523249

yes

characteristic

132

C₈

—

133

C₈

—

508

—

134

C₂²

—

135

C₂²

—

1016

—

136

C₂²

—

508

—

137

C₂²

#16861182

yes

—

138

C₂²

#16861184

yes

characteristic

139

C₄

—

140

C₄

—

508

—

141

C₄

—

508

—

142

C₄

#16861179

yes

characteristic

143

C₄

#16861194

yes

—

144

C₄

#16861197

yes

characteristic

145

C₂

—

146

C₂

—

147

C₂

—

1016

—

148

C₂

#22549794

yes

characteristic

	subgroup	G/H
center	C₄	#16861197
commutator, cosocle	C₂²×C₄×C₁₂₇	C₂³
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	H(3,ℤ/4)×C₁₂₇	C₂
socle	C₂×C₁₂₇	#394
prime radical	#22550676	C₂
hypercenter	H(3,ℤ/4)	D₁₂₇
hypocenter	C₁₂₇	#2053
LCS	G ▷ C₂²×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

#16861197

commutator, cosocle

C₂²×C₄×C₁₂₇

C₂³

Frattini

C₂²×C₄

C₂²×D₁₂₇

Fitting

H(3,ℤ/4)×C₁₂₇

C₂

C₂×C₁₂₇

C₂

D₁₂₇

C₁₂₇

G ▷ C₂²×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
2. #22549904	C₂	semidirect
3. #22549870	C₂	semidirect
4. #22549849	C₂	semidirect
5. H(3,ℤ/4)×C₁₂₇	C₂	semidirect
75. C₁₂₇	#2053	semidirect
80. H(3,ℤ/4)	D₁₂₇	semidirect

H ≅ G/N

type

2. #22549904

C₂

semidirect

3. #22549870

C₂

semidirect

4. #22549849

C₂

semidirect

5. H(3,ℤ/4)×C₁₂₇

C₂

semidirect

75. C₁₂₇

#2053

semidirect

80. 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
#22550676	2
#22549904	2
#22549870	2
#22549849	2
H(3,ℤ/4)×C₁₂₇	2
#2053	127

subgroup

index

H(3,ℤ/4)×C₁₂₇

127

Subgroups — #30640037

Conjugacy classes of subgroups

Complements

Maximal subgroups