Group #30650169 — subgroups

Subgroups — #30650169

Order 16256.

370 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 117 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	#22550481	C₂	1	2	yes	maximal, complemented
2	#22550177	C₂	1	4	yes	maximal, complemented
3	#22550129	C₂	1	2	yes	maximal, complemented
4	#22550112	C₂	1	2	yes	maximal, complemented
5	#22550072	C₂	1	4	yes	maximal, complemented
6	#424×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×D₁₀₁₆	C₂²	1	2	yes	—
8	C₂×D₁₀₁₆	C₂²	1	2	yes	—
9	#16861206	C₂²	1	2	yes	—
10	#16861201	C₂²	1	2	yes	—
11	#16861194	—	2	4	no	—
12	#16861183	C₂²	1	4	yes	—
13	#16861182	—	2	4	no	—
14	C₄×D₅₀₈	C₂²	1	4	yes	—
15	#16861143	C₂²	1	2	yes	—
16	#16861141	C₂²	1	2	yes	—
17	#16861131	C₂²	1	4	yes	—
18	#16861131	C₂²	1	4	yes	—
19	C₂×M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
20	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
21	#117×C₁₂₇	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₁₂₇	C₂²	1	4	yes	—
23	C₂×D₅₀₈	—	2	4	no	—
24	C₂×D₅₀₈	C₂³	1	4	yes	—
25	C₂×C₂²⋊D₁₂₇	—	2	4	no	—
26	C₂×C₄×D₁₂₇	—	2	4	no	—
27	D₁₀₁₆	—	4	2	no	—
28	D₁₀₁₆	—	4	2	no	—
29	C₄⋊Dic₁₂₇	C₂³	1	4	yes	—
30	Dic₁₂₇⋊C₄	—	2	4	no	—
31	(C₂×D₁₂₇)⋊C₄	—	2	4	no	—
32	(C₂×D₁₂₇)⋊C₄	—	2	4	no	—
33	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
34	M₁₆×C₁₂₇	—	4	1	no	—
35	C₂×C₈×C₁₂₇	—	4	1	no	—
36	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
37	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
38	C₄⋊C₄×C₁₂₇	D₄	1	2	yes	—
39	C₂²⋊C₄×C₁₂₇	D₄	1	2	yes	—
40	C₄²×C₁₂₇	C₂³	1	2	yes	—
41	C₂²×D₁₂₇	—	2	4	no	—
42	D₅₀₈	—	2	4	no	—
43	D₅₀₈	—	2	4	no	—
44	D₅₀₈	—	8	4	no	—
45	C₂²⋊D₁₂₇	—	8	4	no	—
46	C₂×Dic₁₂₇	—	2	4	no	—
47	C₄×D₁₂₇	—	8	4	no	—
48	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₂₇	—	2	1	no	—
50	C₂×C₄×C₁₂₇	—	2	1	no	—
51	C₂×C₄×C₁₂₇	C₂×D₄	1	4	yes	—
52	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
54	C₈×C₁₂₇	—	2	2	no	—
55	C₈×C₁₂₇	—	2	2	no	—
56	C₂²×C₁₂₇	—	2	1	no	—
57	C₂²×C₁₂₇	—	2	2	no	—
58	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
59	C₂×D₁₂₇	—	4	4	no	—
60	C₂×D₁₂₇	—	4	4	no	—
61	C₂×D₁₂₇	—	4	4	no	—
62	C₄×C₁₂₇	—	2	1	no	—
63	C₄×C₁₂₇	—	2	4	no	—
64	C₄×C₁₂₇	H(5,𝔽₂)	1	1	yes	characteristic
65	C₄×C₁₂₇	2₋¹⁺⁴	1	1	yes	characteristic
66	Dic₁₂₇	—	4	4	no	—
67	C₂×C₁₂₇	—	4	1	no	—
68	C₂×C₁₂₇	#537	1	1	yes	characteristic
69	C₂×C₁₂₇	#576	1	2	yes	—
70	D₁₂₇	—	8	4	no	—
71	#3259	—	127	1	no	maximal, Sylow
72	C₁₂₇	#3259	1	1	yes	characteristic, complemented, Sylow
73	#540	—	127	2	no	—
74	#478	—	127	4	no	—
75	#469	—	127	2	no	—
76	#466	—	127	2	no	—
77	#459	—	127	4	no	—
78	#424	D₁₂₇	1	1	yes	characteristic, complemented
79	C₂×D₈	—	127	2	no	—
80	C₂×D₈	—	127	2	no	—
81	C₂×M₁₆	C₂×D₁₂₇	1	1	yes	characteristic
82	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
83	#127	—	127	2	no	—
84	#125	—	127	2	no	—
85	#123	—	254	4	no	—
86	#121	—	127	4	no	—
87	#121	—	254	4	no	—
88	C₄×D₄	—	127	4	no	—
89	#117	C₂×D₁₂₇	1	1	yes	characteristic
90	#107	—	127	2	no	—
91	#106	—	127	2	no	—
92	C₄⋊C₈	C₂×D₁₂₇	1	4	yes	—
93	#102	—	127	4	no	—
94	#102	—	127	4	no	—
95	C₂×D₄	—	127	4	no	—
96	C₂×D₄	—	254	4	no	—
97	C₂×D₄	—	254	4	no	—
98	C₂²×C₄	—	254	4	no	—
99	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
100	D₈	—	508	2	no	—
101	D₈	—	508	2	no	—
102	M₁₆	—	4	1	no	—
103	C₂×C₈	—	4	1	no	—
104	C₂×C₈	C₂²×D₁₂₇	1	2	yes	—
105	C₂×C₈	C₂²×D₁₂₇	1	2	yes	—
106	C₄⋊C₄	—	127	4	no	—
107	C₄⋊C₄	—	254	4	no	—
108	C₄⋊C₄	D₅₀₈	1	2	yes	—
109	C₂²⋊C₄	—	254	4	no	—
110	C₂²⋊C₄	—	254	4	no	—
111	C₂²⋊C₄	D₅₀₈	1	2	yes	—
112	C₄²	C₂²×D₁₂₇	1	2	yes	—
113	C₂³	—	254	4	no	—
114	C₂³	C₂×D₅₀₈	1	1	yes	characteristic
115	D₄	—	1016	4	no	—
116	D₄	—	1016	4	no	—
117	D₄	—	254	4	no	—
118	D₄	—	254	4	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	1016	4	no	—
122	C₂×C₄	—	254	4	no	—
123	C₂×C₄	C₂×D₅₀₈	1	4	yes	—
124	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
125	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
126	C₈	—	2	2	no	—
127	C₈	—	2	2	no	—
128	C₂²	—	2	2	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	508	4	no	—
131	C₂²	—	508	4	no	—
132	C₂²	—	508	4	no	—
133	C₂²	C₂²×D₅₀₈	1	1	yes	characteristic
134	C₄	—	2	4	no	—
135	C₄	—	2	1	no	—
136	C₄	—	508	4	no	—
137	C₄	#16861254	1	1	yes	characteristic
138	C₄	#16861257	1	1	yes	characteristic
139	C₂	—	4	1	no	—
140	C₂	—	1016	4	no	—
141	C₂	#22550457	1	1	yes	characteristic
142	C₂	#22550681	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22550481

C₂

yes

maximal, complemented

#22550177

C₂

yes

maximal, complemented

#22550129

C₂

yes

maximal, complemented

#22550112

C₂

yes

maximal, complemented

#22550072

C₂

yes

maximal, complemented

#424×C₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂×D₁₀₁₆

C₂²

yes

—

C₂×D₁₀₁₆

C₂²

yes

—

#16861206

C₂²

yes

—

#16861201

C₂²

yes

—

#16861194

—

#16861183

C₂²

yes

—

#16861182

—

C₄×D₅₀₈

C₂²

yes

—

#16861143

C₂²

yes

—

#16861141

C₂²

yes

—

#16861131

C₂²

yes

—

#16861131

C₂²

yes

—

C₂×M₁₆×C₁₂₇

C₂²

yes

characteristic

C₂²×C₈×C₁₂₇

C₂²

yes

characteristic

#117×C₁₂₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₂₇

C₂²

yes

—

C₂×D₅₀₈

—

C₂×D₅₀₈

C₂³

yes

—

C₂×C₂²⋊D₁₂₇

—

C₂×C₄×D₁₂₇

—

D₁₀₁₆

—

D₁₀₁₆

—

C₄⋊Dic₁₂₇

C₂³

yes

—

Dic₁₂₇⋊C₄

—

(C₂×D₁₂₇)⋊C₄

—

(C₂×D₁₂₇)⋊C₄

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

M₁₆×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

C₂³

yes

—

C₂×C₈×C₁₂₇

C₂³

yes

—

C₄⋊C₄×C₁₂₇

D₄

yes

—

C₂²⋊C₄×C₁₂₇

D₄

yes

—

C₄²×C₁₂₇

C₂³

yes

—

C₂²×D₁₂₇

—

D₅₀₈

—

D₅₀₈

—

D₅₀₈

—

C₂²⋊D₁₂₇

—

C₂×Dic₁₂₇

—

C₄×D₁₂₇

—

C₂³×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂⁴

yes

characteristic

C₈×C₁₂₇

—

C₈×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂²×D₄

yes

characteristic

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

H(5,𝔽₂)

yes

characteristic

C₄×C₁₂₇

2₋¹⁺⁴

yes

characteristic

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

#537

yes

characteristic

C₂×C₁₂₇

#576

yes

—

D₁₂₇

—

#3259

—

127

maximal, Sylow

C₁₂₇

#3259

yes

characteristic, complemented, Sylow

#540

—

127

—

#478

—

127

—

#469

—

127

—

#466

—

127

—

#459

—

127

—

#424

D₁₂₇

yes

characteristic, complemented

C₂×D₈

—

127

—

C₂×D₈

—

127

—

C₂×M₁₆

C₂×D₁₂₇

yes

characteristic

C₂²×C₈

C₂×D₁₂₇

yes

characteristic

#127

—

127

—

#125

—

127

—

#123

—

254

—

#121

—

127

—

#121

—

254

—

C₄×D₄

—

127

—

#117

C₂×D₁₂₇

yes

characteristic

#107

—

127

—

#106

—

127

—

C₄⋊C₈

C₂×D₁₂₇

yes

—

#102

—

127

—

#102

—

127

—

C₂×D₄

—

127

—

C₂×D₄

—

254

—

C₂×D₄

—

254

—

C₂²×C₄

—

254

—

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

100

D₈

—

508

—

101

D₈

—

508

—

102

M₁₆

—

103

C₂×C₈

—

104

C₂×C₈

C₂²×D₁₂₇

yes

—

105

C₂×C₈

C₂²×D₁₂₇

yes

—

106

C₄⋊C₄

—

127

—

107

C₄⋊C₄

—

254

—

108

C₄⋊C₄

D₅₀₈

yes

—

109

C₂²⋊C₄

—

254

—

110

C₂²⋊C₄

—

254

—

111

C₂²⋊C₄

D₅₀₈

yes

—

112

C₄²

C₂²×D₁₂₇

yes

—

113

C₂³

—

254

—

114

C₂³

C₂×D₅₀₈

yes

characteristic

115

D₄

—

1016

—

116

D₄

—

1016

—

117

D₄

—

254

—

118

D₄

—

254

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

1016

—

122

C₂×C₄

—

254

—

123

C₂×C₄

C₂×D₅₀₈

yes

—

124

C₂×C₄

C₂×D₅₀₈

yes

characteristic

125

C₂×C₄

C₂³×D₁₂₇

yes

characteristic

126

C₈

—

127

C₈

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

508

—

131

C₂²

—

508

—

132

C₂²

—

508

—

133

C₂²

C₂²×D₅₀₈

yes

characteristic

134

C₄

—

135

C₄

—

136

C₄

—

508

—

137

C₄

#16861254

yes

characteristic

138

C₄

#16861257

yes

characteristic

139

C₂

—

140

C₂

—

1016

—

141

C₂

#22550457

yes

characteristic

142

C₂

#22550681

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₅₀₈
commutator, cosocle	C₂×C₄×C₁₂₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	#424×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂²×D₄
hypercenter	#424	D₁₂₇
hypocenter	C₁₂₇	#3259
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #424

subgroup

G/H

center

C₂²

C₂²×D₅₀₈

commutator, cosocle

C₂×C₄×C₁₂₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₂₇

Fitting

#424×C₁₂₇

C₂

socle

C₂²×C₁₂₇

C₂²×D₄

D₁₂₇

C₁₂₇

G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂² ◁ #117 ◁ #424

N	H ≅ G/N	type
1. #22550481	C₂	semidirect
2. #22550177	C₂	semidirect
3. #22550129	C₂	semidirect
4. #22550112	C₂	semidirect
5. #22550072	C₂	semidirect
6. #424×C₁₂₇	C₂	semidirect
72. C₁₂₇	#3259	semidirect
78. #424	D₁₂₇	semidirect

H ≅ G/N

type

1. #22550481

C₂

semidirect

2. #22550177

C₂

semidirect

3. #22550129

C₂

semidirect

4. #22550112

C₂

semidirect

5. #22550072

C₂

semidirect

6. #424×C₁₂₇

C₂

semidirect

72. C₁₂₇

#3259

semidirect

78. #424

D₁₂₇

semidirect

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

subgroup	index
#22550481	2
#22550177	2
#22550129	2
#22550112	2
#22550072	2
#424×C₁₂₇	2
#3259	127

subgroup

index

#424×C₁₂₇

127

Subgroups — #30650169

Conjugacy classes of subgroups

Complements

Maximal subgroups