Group #30649133 — subgroups

Subgroups — #30649133

Order 16256.

392 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 133 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	#22550426	C₂	1	1	yes	maximal, characteristic
2	#22550246	C₂	1	2	yes	maximal, complemented
3	#22550242	C₂	1	2	yes	maximal, complemented
4	#22550233	C₂	1	4	yes	maximal, complemented
5	#22549875	C₂	1	4	yes	maximal, complemented
6	C₄×#14523232	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#125×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14523248	C₂²	1	2	yes	—
9	C₂×#14523232	C₂²	1	2	yes	—
10	#16861209	C₂²	1	1	yes	characteristic
11	#16861206	C₂²	1	1	yes	characteristic
12	#16861197	C₂²	1	2	yes	—
13	#16861183	—	2	2	no	—
14	C₄×D₅₀₈	—	2	2	no	—
15	#16861132	C₂²	1	8	yes	—
16	#16861129	C₂²	1	8	yes	—
17	#16861120	C₂²	1	2	yes	—
18	C₄×C₁₂₇⋊C₈	C₂²	1	2	yes	—
19	#125×C₁₂₇	—	2	2	no	—
20	#125×C₁₂₇	C₂²	1	2	yes	—
21	#125×C₁₂₇	C₂²	1	2	yes	—
22	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
23	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
24	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
25	#14523248	—	2	8	no	—
26	C₂×Dic₂₅₄	C₂³	1	2	yes	—
27	C₂×D₅₀₈	—	2	2	no	—
28	C₂×D₅₀₈	C₂³	1	2	yes	—
29	C₂×C₂²⋊D₁₂₇	—	2	2	no	—
30	C₂×C₄×D₁₂₇	—	2	2	no	—
31	#14523232	—	2	4	no	—
32	C₂×C₁₂₇⋊C₈	C₂³	1	4	yes	—
33	C₄⋊Dic₁₂₇	—	2	2	no	—
34	(C₂×D₁₂₇)⋊C₄	—	2	4	no	—
35	C₂²×C₄×C₁₂₇	—	2	2	no	—
36	C₂²×C₄×C₁₂₇	D₄	1	2	yes	—
37	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
38	C₄⋊C₄×C₁₂₇	—	2	4	no	—
39	C₄⋊C₄×C₁₂₇	—	2	4	no	—
40	C₄⋊C₄×C₁₂₇	—	2	2	no	—
41	C₄⋊C₄×C₁₂₇	C₂³	1	4	yes	—
42	C₄²×C₁₂₇	D₄	1	2	yes	—
43	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
44	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
45	C₂²×D₁₂₇	—	2	2	no	—
46	Dic₂₅₄	—	2	4	no	—
47	D₅₀₈	—	2	4	no	—
48	D₅₀₈	—	4	4	no	—
49	C₂²⋊D₁₂₇	—	4	4	no	—
50	C₂×Dic₁₂₇	—	2	2	no	—
51	C₄×D₁₂₇	—	4	4	no	—
52	C₁₂₇⋊C₈	—	2	4	no	—
53	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₁₂₇	—	2	4	no	—
55	C₂×C₄×C₁₂₇	—	4	2	no	—
56	C₂×C₄×C₁₂₇	—	4	4	no	—
57	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
58	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₁₂₇	#41	1	4	yes	—
61	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
62	C₂²×C₁₂₇	—	2	2	no	—
63	C₂²×C₁₂₇	#124	1	2	yes	—
64	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
65	C₂×D₁₂₇	—	4	2	no	—
66	C₂×D₁₂₇	—	4	4	no	—
67	C₄×C₁₂₇	—	2	2	no	—
68	C₄×C₁₂₇	—	2	2	no	—
69	C₄×C₁₂₇	—	4	4	no	—
70	C₄×C₁₂₇	#124	1	2	yes	—
71	C₄×C₁₂₇	C₂×#41	1	2	yes	—
72	Dic₁₂₇	—	4	2	no	—
73	C₂×C₁₂₇	—	2	2	no	—
74	C₂×C₁₂₇	C₂×#124	1	1	yes	characteristic
75	C₂×C₁₂₇	#575	1	2	yes	—
76	D₁₂₇	—	8	2	no	—
77	#3142	—	127	1	no	maximal, Sylow
78	C₁₂₇	#3142	1	1	yes	characteristic, complemented, Sylow
79	#532	—	127	1	no	—
80	C₂×#125	D₁₂₇	1	1	yes	characteristic, complemented
81	#491	—	127	2	no	—
82	#490	—	127	2	no	—
83	#488	—	127	4	no	—
84	#417	—	127	4	no	—
85	C₄×M₁₆	—	127	1	no	—
86	C₂×#41	—	127	2	no	—
87	C₂×M₁₆	—	127	2	no	—
88	C₄⋊Q₈	—	127	1	no	—
89	#127	—	127	1	no	—
90	#125	—	2	2	no	—
91	#125	C₂×D₁₂₇	1	2	yes	—
92	#125	C₂×D₁₂₇	1	2	yes	—
93	#124	—	127	2	no	—
94	#121	—	254	2	no	—
95	C₄×D₄	—	254	2	no	—
96	C₂×C₄⋊C₄	—	2	2	no	—
97	C₂×C₄⋊C₄	C₂×D₁₂₇	1	2	yes	—
98	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
99	#103	—	127	8	no	—
100	#102	—	127	8	no	—
101	#97	—	127	2	no	—
102	C₄×C₈	—	127	2	no	—
103	#41	—	254	8	no	—
104	C₂×Q₈	—	127	2	no	—
105	C₂×D₄	—	127	2	no	—
106	C₂×D₄	—	254	2	no	—
107	C₂×D₄	—	254	2	no	—
108	C₂²×C₄	—	2	2	no	—
109	C₂²×C₄	—	254	2	no	—
110	C₂²×C₄	C₂²⋊D₁₂₇	1	2	yes	—
111	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
112	M₁₆	—	254	4	no	—
113	C₂×C₈	—	127	4	no	—
114	C₄⋊C₄	—	2	4	no	—
115	C₄⋊C₄	—	2	4	no	—
116	C₄⋊C₄	—	2	2	no	—
117	C₄⋊C₄	—	254	2	no	—
118	C₄⋊C₄	C₂²×D₁₂₇	1	4	yes	—
119	C₂²⋊C₄	—	254	4	no	—
120	C₄²	C₂²⋊D₁₂₇	1	2	yes	—
121	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
122	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
123	C₂³	—	254	2	no	—
124	C₂³	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
125	Q₈	—	254	4	no	—
126	D₄	—	508	4	no	—
127	D₄	—	508	4	no	—
128	D₄	—	254	4	no	—
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	—	4	2	no	—
131	C₂×C₄	—	4	4	no	—
132	C₂×C₄	—	508	4	no	—
133	C₂×C₄	—	254	2	no	—
134	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	2	yes	—
135	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	2	yes	—
136	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
137	C₂×C₄	#14523249	1	4	yes	—
138	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
139	C₈	—	254	4	no	—
140	C₂²	—	2	2	no	—
141	C₂²	—	508	2	no	—
142	C₂²	—	508	4	no	—
143	C₂²	#16861198	1	2	yes	—
144	C₂²	C₂²×C₂²⋊D₁₂₇	1	1	yes	characteristic
145	C₄	—	2	2	no	—
146	C₄	—	2	2	no	—
147	C₄	—	4	4	no	—
148	C₄	—	508	2	no	—
149	C₄	#16861198	1	2	yes	—
150	C₄	C₂×#14523249	1	2	yes	—
151	C₂	—	2	2	no	—
152	C₂	—	1016	2	no	—
153	C₂	C₂×#16861198	1	1	yes	characteristic
154	C₂	#22550674	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22550426

C₂

yes

maximal, characteristic

#22550246

C₂

yes

maximal, complemented

#22550242

C₂

yes

maximal, complemented

#22550233

C₂

yes

maximal, complemented

#22549875

C₂

yes

maximal, complemented

C₄×#14523232

C₂

yes

maximal, characteristic, complemented

C₂×#125×C₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂×#14523248

C₂²

yes

—

C₂×#14523232

C₂²

yes

—

#16861209

C₂²

yes

characteristic

#16861206

C₂²

yes

characteristic

#16861197

C₂²

yes

—

#16861183

—

C₄×D₅₀₈

—

#16861132

C₂²

yes

—

#16861129

C₂²

yes

—

#16861120

C₂²

yes

—

C₄×C₁₂₇⋊C₈

C₂²

yes

—

#125×C₁₂₇

—

#125×C₁₂₇

C₂²

yes

—

#125×C₁₂₇

C₂²

yes

—

C₂×C₄⋊C₄×C₁₂₇

—

C₂×C₄⋊C₄×C₁₂₇

C₂²

yes

—

C₂×C₄²×C₁₂₇

C₂²

yes

characteristic

#14523248

—

C₂×Dic₂₅₄

C₂³

yes

—

C₂×D₅₀₈

—

C₂×D₅₀₈

C₂³

yes

—

C₂×C₂²⋊D₁₂₇

—

C₂×C₄×D₁₂₇

—

#14523232

—

C₂×C₁₂₇⋊C₈

C₂³

yes

—

C₄⋊Dic₁₂₇

—

(C₂×D₁₂₇)⋊C₄

—

C₂²×C₄×C₁₂₇

—

C₂²×C₄×C₁₂₇

D₄

yes

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

C₂³

yes

—

C₄²×C₁₂₇

D₄

yes

—

C₄²×C₁₂₇

C₂³

yes

characteristic

C₄²×C₁₂₇

C₂³

yes

characteristic

C₂²×D₁₂₇

—

Dic₂₅₄

—

D₅₀₈

—

D₅₀₈

—

C₂²⋊D₁₂₇

—

C₂×Dic₁₂₇

—

C₄×D₁₂₇

—

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

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

#41

yes

—

C₂×C₄×C₁₂₇

C₂⁴

yes

characteristic

C₂²×C₁₂₇

—

C₂²×C₁₂₇

#124

yes

—

C₂²×C₁₂₇

C₂²×D₄

yes

characteristic

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

#124

yes

—

C₄×C₁₂₇

C₂×#41

yes

—

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

C₂×#124

yes

characteristic

C₂×C₁₂₇

#575

yes

—

D₁₂₇

—

#3142

—

127

maximal, Sylow

C₁₂₇

#3142

yes

characteristic, complemented, Sylow

#532

—

127

—

C₂×#125

D₁₂₇

yes

characteristic, complemented

#491

—

127

—

#490

—

127

—

#488

—

127

—

#417

—

127

—

C₄×M₁₆

—

127

—

C₂×#41

—

127

—

C₂×M₁₆

—

127

—

C₄⋊Q₈

—

127

—

#127

—

127

—

#125

—

#125

C₂×D₁₂₇

yes

—

#125

C₂×D₁₂₇

yes

—

#124

—

127

—

#121

—

254

—

C₄×D₄

—

254

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₂₇

yes

—

C₂×C₄²

C₂×D₁₂₇

yes

characteristic

#103

—

127

—

100

#102

—

127

—

101

#97

—

127

—

102

C₄×C₈

—

127

—

103

#41

—

254

—

104

C₂×Q₈

—

127

—

105

C₂×D₄

—

127

—

106

C₂×D₄

—

254

—

107

C₂×D₄

—

254

—

108

C₂²×C₄

—

109

C₂²×C₄

—

254

—

110

C₂²×C₄

C₂²⋊D₁₂₇

yes

—

111

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

112

M₁₆

—

254

—

113

C₂×C₈

—

127

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

254

—

118

C₄⋊C₄

C₂²×D₁₂₇

yes

—

119

C₂²⋊C₄

—

254

—

120

C₄²

C₂²⋊D₁₂₇

yes

—

121

C₄²

C₂²×D₁₂₇

yes

characteristic

122

C₄²

C₂²×D₁₂₇

yes

characteristic

123

C₂³

—

254

—

124

C₂³

C₂×C₂²⋊D₁₂₇

yes

characteristic

125

Q₈

—

254

—

126

D₄

—

508

—

127

D₄

—

508

—

128

D₄

—

254

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

508

—

133

C₂×C₄

—

254

—

134

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

—

135

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

—

136

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

characteristic

137

C₂×C₄

#14523249

yes

—

138

C₂×C₄

C₂³×D₁₂₇

yes

characteristic

139

C₈

—

254

—

140

C₂²

—

141

C₂²

—

508

—

142

C₂²

—

508

—

143

C₂²

#16861198

yes

—

144

C₂²

C₂²×C₂²⋊D₁₂₇

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

508

—

149

C₄

#16861198

yes

—

150

C₄

C₂×#14523249

yes

—

151

C₂

—

152

C₂

—

1016

—

153

C₂

C₂×#16861198

yes

characteristic

154

C₂

#22550674

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₂₇
commutator, cosocle	C₂×C₄×C₁₂₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	C₂×#125×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂²×D₄
prime radical	#22550426	C₂
hypercenter	C₂×#125	D₁₂₇
hypocenter	C₁₂₇	#3142
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₁₂₇

commutator, cosocle

C₂×C₄×C₁₂₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₂₇

Fitting

C₂×#125×C₁₂₇

C₂

socle

C₂²×C₁₂₇

C₂²×D₄

C₂

D₁₂₇

C₁₂₇

G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
2. #22550246	C₂	semidirect
3. #22550242	C₂	semidirect
4. #22550233	C₂	semidirect
5. #22549875	C₂	semidirect
6. C₄×#14523232	C₂	semidirect
7. C₂×#125×C₁₂₇	C₂	semidirect
78. C₁₂₇	#3142	semidirect
80. C₂×#125	D₁₂₇	semidirect

H ≅ G/N

type

2. #22550246

C₂

semidirect

3. #22550242

C₂

semidirect

4. #22550233

C₂

semidirect

5. #22549875

C₂

semidirect

6. C₄×#14523232

C₂

semidirect

7. C₂×#125×C₁₂₇

C₂

semidirect

78. C₁₂₇

#3142

semidirect

80. C₂×#125

D₁₂₇

semidirect

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

subgroup	index
#22550426	2
#22550246	2
#22550242	2
#22550233	2
#22549875	2
C₄×#14523232	2
C₂×#125×C₁₂₇	2
#3142	127

subgroup

index

C₂×#125×C₁₂₇

127

Subgroups — #30649133

Conjugacy classes of subgroups

Complements

Maximal subgroups