Group #30651372 — subgroups

Subgroups — #30651372

Order 16256.

370 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 141 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	#22550605	C₂	1	1	yes	maximal, characteristic, complemented
2	#22550228	C₂	1	3	yes	maximal, complemented
3	#22550158	C₂	1	3	yes	maximal, complemented
4	#22550081	C₂	1	3	yes	maximal
5	Q₈×C₁₂₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#14523236	C₂	1	3	yes	maximal
7	D₄×Q₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#14523236	C₂²	1	3	yes	—
9	#16861209	C₂²	1	3	yes	—
10	#16861200	—	2	3	no	—
11	Q₈×Dic₁₂₇	—	2	1	no	—
12	C₄×Dic₂₅₄	C₂²	1	3	yes	—
13	#16861140	C₂²	1	3	yes	—
14	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	3	yes	—
15	#16861132	C₂²	1	6	yes	—
16	#16861132	C₂²	1	3	yes	—
17	#16861130	C₂²	1	1	yes	characteristic
18	C₄×C₁₂₇⋊C₈	C₂²	1	3	yes	—
19	C₂²×Q₈×C₁₂₇	—	2	1	no	—
20	C₄⋊Q₈×C₁₂₇	C₂²	1	3	yes	—
21	C₂²⋊Q₈×C₁₂₇	—	2	3	no	—
22	C₄×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
23	C₄×D₄×C₁₂₇	C₂²	1	3	yes	—
24	C₂×Dic₂₅₄	C₂³	1	3	yes	—
25	#14523236	—	2	6	no	—
26	C₂×C₁₂₇⋊C₈	C₂³	1	3	yes	—
27	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
28	C₄⋊Dic₁₂₇	—	2	3	no	—
29	C₄⋊Dic₁₂₇	—	2	3	no	—
30	C₄⋊Dic₁₂₇	C₂³	1	1	yes	characteristic
31	Dic₁₂₇⋊C₄	—	2	3	no	—
32	C₄×Dic₁₂₇	—	2	3	no	—
33	C₂×Q₈×C₁₂₇	—	2	3	no	—
34	C₂×Q₈×C₁₂₇	—	4	4	no	—
35	C₂×Q₈×C₁₂₇	D₄	1	1	yes	characteristic
36	C₂×D₄×C₁₂₇	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₁₂₇	—	2	3	no	—
38	C₄⋊C₄×C₁₂₇	—	2	3	no	—
39	C₄⋊C₄×C₁₂₇	D₄	1	3	yes	—
40	C₄⋊C₄×C₁₂₇	C₂³	1	3	yes	—
41	C₂²⋊C₄×C₁₂₇	—	2	3	no	—
42	C₄²×C₁₂₇	C₂³	1	3	yes	—
43	Dic₂₅₄	—	2	3	no	—
44	Dic₂₅₄	—	2	3	no	—
45	C₂×Dic₁₂₇	—	2	1	no	—
46	C₂×Dic₁₂₇	—	2	3	no	—
47	C₁₂₇⋊C₈	—	2	3	no	—
48	C₁₂₇⋊C₈	—	2	1	no	—
49	C₂³×C₁₂₇	—	2	1	no	—
50	Q₈×C₁₂₇	—	4	6	no	—
51	Q₈×C₁₂₇	SD₁₆	1	4	yes	complemented
52	D₄×C₁₂₇	—	2	1	no	—
53	D₄×C₁₂₇	#41	1	2	yes	—
54	C₂×C₄×C₁₂₇	—	2	3	no	—
55	C₂×C₄×C₁₂₇	—	4	6	no	—
56	C₂×C₄×C₁₂₇	C₂×D₄	1	3	yes	—
57	C₂×C₄×C₁₂₇	C₂×D₄	1	3	yes	—
58	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
59	C₂²×C₁₂₇	—	2	2	no	—
60	C₂²×C₁₂₇	—	4	1	no	—
61	C₂²×C₁₂₇	—	4	1	no	—
62	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
63	C₄×C₁₂₇	—	2	3	no	—
64	C₄×C₁₂₇	—	4	3	no	—
65	C₄×C₁₂₇	C₂×SD₁₆	1	6	yes	—
66	C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
67	C₄×C₁₂₇	2₋¹⁺⁴	1	1	yes	characteristic
68	Dic₁₂₇	—	4	3	no	—
69	Dic₁₂₇	—	4	1	no	—
70	C₂×C₁₂₇	—	4	2	no	—
71	C₂×C₁₂₇	#548	1	1	yes	characteristic
72	C₂×C₁₂₇	C₂²×SD₁₆	1	1	yes	characteristic
73	C₂×C₁₂₇	#578	1	1	yes	characteristic
74	#3359	—	127	1	no	maximal, Sylow
75	C₁₂₇	#3359	1	1	yes	characteristic, complemented, Sylow
76	#557	—	127	1	no	—
77	D₄×Q₈	D₁₂₇	1	1	yes	characteristic
78	#487	—	127	3	no	—
79	#475	—	127	3	no	—
80	#461	—	127	3	no	—
81	C₈×Q₈	—	127	1	no	—
82	C₄×SD₁₆	—	127	3	no	—
83	C₂²×Q₈	—	2	1	no	—
84	C₂×SD₁₆	—	127	3	no	—
85	C₄⋊Q₈	—	127	3	no	—
86	C₄⋊Q₈	C₂×D₁₂₇	1	3	yes	—
87	#125	—	254	3	no	—
88	C₂²⋊Q₈	—	2	3	no	—
89	C₄×Q₈	—	127	3	no	—
90	C₄×Q₈	—	254	1	no	—
91	C₄×Q₈	C₂×D₁₂₇	1	1	yes	characteristic
92	C₄×D₄	C₂×D₁₂₇	1	3	yes	—
93	#106	—	127	3	no	—
94	C₄⋊C₈	—	127	3	no	—
95	#103	—	127	3	no	—
96	#103	—	127	6	no	—
97	#102	—	127	1	no	—
98	C₄×C₈	—	127	3	no	—
99	C₂×Q₈	—	2	3	no	—
100	C₂×Q₈	—	4	4	no	—
101	C₂×Q₈	—	127	3	no	—
102	C₂×Q₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
103	C₂×D₄	C₂²×D₁₂₇	1	1	yes	characteristic
104	C₂²×C₄	—	2	3	no	—
105	SD₁₆	—	254	6	no	—
106	C₂×C₈	—	127	1	no	—
107	C₂×C₈	—	127	3	no	—
108	C₄⋊C₄	—	2	3	no	—
109	C₄⋊C₄	—	127	1	no	—
110	C₄⋊C₄	—	254	3	no	—
111	C₄⋊C₄	—	254	3	no	—
112	C₄⋊C₄	—	254	3	no	—
113	C₄⋊C₄	C₂²⋊D₁₂₇	1	3	yes	—
114	C₄⋊C₄	C₂²×D₁₂₇	1	3	yes	—
115	C₂²⋊C₄	—	2	3	no	—
116	C₄²	—	254	3	no	—
117	C₄²	C₂²×D₁₂₇	1	3	yes	—
118	C₂³	—	2	1	no	—
119	Q₈	—	4	6	no	—
120	Q₈	—	254	3	no	—
121	Q₈	—	254	3	no	—
122	Q₈	#14523236	1	4	yes	complemented
123	D₄	—	2	1	no	—
124	D₄	#14523249	1	2	yes	—
125	C₂×C₄	—	2	3	no	—
126	C₂×C₄	—	4	6	no	—
127	C₂×C₄	—	254	1	no	—
128	C₂×C₄	—	254	3	no	—
129	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	3	yes	—
130	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	3	yes	—
131	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
132	C₈	—	254	3	no	—
133	C₈	—	254	1	no	—
134	C₂²	—	2	2	no	—
135	C₂²	—	4	1	no	—
136	C₂²	—	4	1	no	—
137	C₂²	C₂²×C₂²⋊D₁₂₇	1	1	yes	characteristic
138	C₄	—	2	3	no	—
139	C₄	—	4	3	no	—
140	C₄	—	508	3	no	—
141	C₄	—	508	1	no	—
142	C₄	C₂×#14523236	1	6	yes	—
143	C₄	C₂×#14523249	1	1	yes	characteristic
144	C₄	#16861257	1	1	yes	characteristic
145	C₂	—	4	2	no	—
146	C₂	#22550537	1	1	yes	characteristic
147	C₂	C₂²×#14523236	1	1	yes	characteristic
148	C₂	#22550691	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22550605

C₂

yes

maximal, characteristic, complemented

#22550228

C₂

yes

maximal, complemented

#22550158

C₂

yes

maximal, complemented

#22550081

C₂

yes

maximal

Q₈×C₁₂₇⋊C₈

C₂

yes

maximal, characteristic, complemented

C₄×#14523236

C₂

yes

maximal

D₄×Q₈×C₁₂₇

C₂

yes

maximal, characteristic

C₂×#14523236

C₂²

yes

—

#16861209

C₂²

yes

—

#16861200

—

Q₈×Dic₁₂₇

—

C₄×Dic₂₅₄

C₂²

yes

—

#16861140

C₂²

yes

—

C₄⋊(C₁₂₇⋊C₈)

C₂²

yes

—

#16861132

C₂²

yes

—

#16861132

C₂²

yes

—

#16861130

C₂²

yes

characteristic

C₄×C₁₂₇⋊C₈

C₂²

yes

—

C₂²×Q₈×C₁₂₇

—

C₄⋊Q₈×C₁₂₇

C₂²

yes

—

C₂²⋊Q₈×C₁₂₇

—

C₄×Q₈×C₁₂₇

C₂²

yes

characteristic

C₄×D₄×C₁₂₇

C₂²

yes

—

C₂×Dic₂₅₄

C₂³

yes

—

#14523236

—

C₂×C₁₂₇⋊C₈

C₂³

yes

—

C₂×C₁₂₇⋊C₈

C₂³

yes

characteristic

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

C₂³

yes

characteristic

Dic₁₂₇⋊C₄

—

C₄×Dic₁₂₇

—

C₂×Q₈×C₁₂₇

—

C₂×Q₈×C₁₂₇

—

C₂×Q₈×C₁₂₇

D₄

yes

characteristic

C₂×D₄×C₁₂₇

C₂³

yes

characteristic

C₂²×C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

D₄

yes

—

C₄⋊C₄×C₁₂₇

C₂³

yes

—

C₂²⋊C₄×C₁₂₇

—

C₄²×C₁₂₇

C₂³

yes

—

Dic₂₅₄

—

Dic₂₅₄

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₁₂₇⋊C₈

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

—

Q₈×C₁₂₇

—

Q₈×C₁₂₇

SD₁₆

yes

complemented

D₄×C₁₂₇

—

D₄×C₁₂₇

#41

yes

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

—

C₂×C₄×C₁₂₇

C₂⁴

yes

characteristic

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂²×D₄

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂×SD₁₆

yes

—

C₄×C₁₂₇

C₂×#41

yes

characteristic

C₄×C₁₂₇

2₋¹⁺⁴

yes

characteristic

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

#548

yes

characteristic

C₂×C₁₂₇

C₂²×SD₁₆

yes

C₂×C₁₂₇

yes

—

127

maximal, Sylow

C₁₂₇

#3359

yes

characteristic, complemented, Sylow

#557

—

127

—

D₄×Q₈

D₁₂₇

yes

characteristic

#487

—

127

—

#475

—

127

—

#461

—

127

—

C₈×Q₈

—

127

—

C₄×SD₁₆

—

127

—

C₂²×Q₈

—

C₂×SD₁₆

—

127

—

C₄⋊Q₈

—

127

—

C₄⋊Q₈

C₂×D₁₂₇

yes

—

#125

—

254

—

C₂²⋊Q₈

—

C₄×Q₈

—

127

—

C₄×Q₈

—

254

—

C₄×Q₈

C₂×D₁₂₇

yes

characteristic

C₄×D₄

C₂×D₁₂₇

yes

—

#106

—

127

—

C₄⋊C₈

—

127

—

#103

—

127

—

#103

—

127

—

#102

—

127

—

C₄×C₈

—

127

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

—

127

—

102

C₂×Q₈

C₂²⋊D₁₂₇

yes

characteristic

103

C₂×D₄

C₂²×D₁₂₇

yes

characteristic

104

C₂²×C₄

—

105

SD₁₆

—

254

—

106

C₂×C₈

—

127

—

107

C₂×C₈

—

127

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

127

—

110

C₄⋊C₄

—

254

—

111

C₄⋊C₄

—

254

—

112

C₄⋊C₄

—

254

—

113

C₄⋊C₄

C₂²⋊D₁₂₇

yes

—

114

C₄⋊C₄

C₂²×D₁₂₇

yes

—

115

C₂²⋊C₄

—

116

C₄²

—

254

—

117

C₄²

C₂²×D₁₂₇

yes

—

118

C₂³

—

119

Q₈

—

120

Q₈

—

254

—

121

Q₈

—

254

—

122

Q₈

#14523236

yes

complemented

123

D₄

—

124

D₄

#14523249

yes

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

254

—

128

C₂×C₄

—

254

—

129

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

—

130

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

—

131

C₂×C₄

C₂³×D₁₂₇

yes

characteristic

132

C₈

—

254

—

133

C₈

—

254

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

C₂²×C₂²⋊D₁₂₇

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

508

—

141

C₄

—

508

—

142

C₄

C₂×#14523236

yes

—

143

C₄

C₂×#14523249

yes

characteristic

144

C₄

#16861257

yes

characteristic

145

C₂

—

146

C₂

#22550537

yes

characteristic

147

C₂

C₂²×#14523236

yes

characteristic

148

C₂

#22550691

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₂₇
commutator, cosocle	C₂×C₄×C₁₂₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	D₄×Q₈×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂²×D₄
prime radical	C₂×D₄×C₁₂₇	C₂³
hypercenter	D₄×Q₈	D₁₂₇
hypocenter	C₁₂₇	#3359
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ D₄×Q₈

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₁₂₇

commutator, cosocle

C₂×C₄×C₁₂₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₂₇

Fitting

D₄×Q₈×C₁₂₇

C₂

socle

C₂²×C₁₂₇

C₂²×D₄

prime radical

C₂×D₄×C₁₂₇

C₂³

hypercenter

D₄×Q₈

D₁₂₇

hypocenter

C₁₂₇

#3359

LCS

G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ D₄×Q₈

N	H ≅ G/N	type
1. #22550605	C₂	semidirect
2. #22550228	C₂	semidirect
3. #22550158	C₂	semidirect
5. Q₈×C₁₂₇⋊C₈	C₂	semidirect
51. Q₈×C₁₂₇	SD₁₆	semidirect
75. C₁₂₇	#3359	semidirect
122. Q₈	#14523236	semidirect

H ≅ G/N

type

1. #22550605

C₂

semidirect

2. #22550228

C₂

semidirect

3. #22550158

C₂

semidirect

5. Q₈×C₁₂₇⋊C₈

C₂

semidirect

51. Q₈×C₁₂₇

SD₁₆

semidirect

75. C₁₂₇

#3359

semidirect

122. Q₈

#14523236

semidirect

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

subgroup	index
#22550605	2
#22550228	2
#22550158	2
#22550081	2
Q₈×C₁₂₇⋊C₈	2
C₄×#14523236	2
D₄×Q₈×C₁₂₇	2
#3359	127

subgroup

index

Q₈×C₁₂₇⋊C₈

D₄×Q₈×C₁₂₇

127

Subgroups — #30651372

Conjugacy classes of subgroups

Complements

Maximal subgroups