Group #30649262 — subgroups

Subgroups — #30649262

Order 16256.

340 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 149 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	#22550504	C₂	1	2	yes	maximal
2	#22550284	C₂	1	4	yes	maximal, complemented
3	#22550276	C₂	1	2	yes	maximal, complemented
4	#22550273	C₂	1	2	yes	maximal, complemented
5	#22549919	C₂	1	2	yes	maximal
6	C₂×#16861141	C₂	1	1	yes	maximal, characteristic
7	#22549831	C₂	1	1	yes	maximal, characteristic
8	#427×C₁₂₇	C₂	1	1	yes	maximal, characteristic
9	C₂×#14523232	C₂²	1	1	yes	characteristic
10	#16861209	—	2	2	no	—
11	#16861208	C₂²	1	4	yes	—
12	#16861201	—	2	2	no	—
13	#16861200	C₂²	1	4	yes	—
14	#16861186	—	2	4	no	—
15	#16861168	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
17	#16861142	C₂²	1	4	yes	—
18	#16861141	C₂²	1	2	yes	—
19	#16861141	C₂²	1	2	yes	—
20	#16861140	C₂²	1	4	yes	—
21	#16861119	C₂²	1	2	yes	—
22	C₈×Dic₁₂₇	C₂²	1	2	yes	—
23	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
24	#117×C₁₂₇	C₂²	1	2	yes	—
25	#107×C₁₂₇	C₂²	1	2	yes	—
26	#106×C₁₂₇	C₂²	1	1	yes	characteristic
27	#106×C₁₂₇	C₂²	1	1	yes	characteristic
28	C₂×Dic₂₅₄	—	2	4	no	—
29	C₂²×Dic₁₂₇	—	2	2	no	—
30	#14523232	Q₈	1	4	yes	complemented
31	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
32	C₄⋊Dic₁₂₇	—	2	4	no	—
33	C₄⋊Dic₁₂₇	—	2	2	no	—
34	C₄⋊Dic₁₂₇	D₄	1	2	yes	—
35	C₄⋊Dic₁₂₇	C₂³	1	4	yes	—
36	Dic₁₂₇⋊C₄	—	2	4	no	—
37	Dic₁₂₇⋊C₄	—	2	4	no	—
38	C₂²⋊Dic₁₂₇	D₄	1	2	yes	—
39	C₄×Dic₁₂₇	C₂³	1	2	yes	—
40	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
41	C₂×C₈×C₁₂₇	Q₈	1	4	yes	complemented
42	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
43	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
44	C₄⋊C₄×C₁₂₇	C₂³	1	2	yes	—
45	C₄⋊C₄×C₁₂₇	C₂³	1	2	yes	—
46	C₂²⋊C₄×C₁₂₇	—	2	2	no	—
47	C₄²×C₁₂₇	—	2	2	no	—
48	Dic₂₅₄	—	8	4	no	—
49	C₂×Dic₁₂₇	—	2	4	no	—
50	C₂×Dic₁₂₇	—	8	2	no	—
51	C₂×Dic₁₂₇	C₂×D₄	1	4	yes	—
52	C₁₂₇⋊C₈	C₂×Q₈	1	4	yes	—
53	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₁₂₇	—	2	2	no	—
55	C₂×C₄×C₁₂₇	—	2	2	no	—
56	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
58	C₂×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
59	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
60	C₈×C₁₂₇	C₂×Q₈	1	2	yes	—
61	C₈×C₁₂₇	C₂×Q₈	1	2	yes	—
62	C₂²×C₁₂₇	—	2	2	no	—
63	C₂²×C₁₂₇	C₄⋊Q₈	1	2	yes	—
64	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
65	C₄×C₁₂₇	—	4	2	no	—
66	C₄×C₁₂₇	—	4	2	no	—
67	C₄×C₁₂₇	C₄⋊Q₈	1	2	yes	—
68	C₄×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
69	C₄×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
70	Dic₁₂₇	—	2	4	no	—
71	Dic₁₂₇	—	4	4	no	—
72	C₂×C₁₂₇	—	2	2	no	—
73	C₂×C₁₂₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
74	C₂×C₁₂₇	#577	1	2	yes	—
75	#3163	—	127	1	no	maximal, Sylow
76	C₁₂₇	#3163	1	1	yes	characteristic, complemented, Sylow
77	#544	—	127	2	no	—
78	#501	—	127	4	no	—
79	#499	—	127	2	no	—
80	#498	—	127	2	no	—
81	#428	—	127	2	no	—
82	#427	D₁₂₇	1	1	yes	characteristic
83	C₂×#106	—	127	1	no	—
84	#405	—	127	1	no	—
85	C₂×M₁₆	—	127	1	no	—
86	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
87	C₄⋊Q₈	—	127	4	no	—
88	C₄⋊Q₈	—	254	2	no	—
89	#125	—	127	4	no	—
90	#125	—	254	2	no	—
91	C₂²⋊Q₈	—	254	4	no	—
92	#117	—	127	1	no	—
93	#117	C₂×D₁₂₇	1	2	yes	—
94	C₂×C₄⋊C₄	—	127	2	no	—
95	#107	—	127	4	no	—
96	#107	C₂×D₁₂₇	1	2	yes	—
97	#106	—	127	2	no	—
98	#106	—	127	2	no	—
99	#106	—	127	4	no	—
100	#106	C₂×D₁₂₇	1	1	yes	characteristic
101	#106	C₂×D₁₂₇	1	1	yes	characteristic
102	#97	—	127	2	no	—
103	C₄×C₈	—	127	2	no	—
104	C₂×Q₈	—	254	4	no	—
105	C₂²×C₄	—	254	2	no	—
106	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
107	M₁₆	—	127	4	no	—
108	C₂×C₈	—	127	2	no	—
109	C₂×C₈	Dic₂₅₄	1	4	yes	complemented
110	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
111	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
112	C₄⋊C₄	—	127	2	no	—
113	C₄⋊C₄	—	127	4	no	—
114	C₄⋊C₄	—	254	4	no	—
115	C₄⋊C₄	—	254	4	no	—
116	C₄⋊C₄	—	254	4	no	—
117	C₄⋊C₄	—	254	2	no	—
118	C₄⋊C₄	C₂²×D₁₂₇	1	2	yes	—
119	C₄⋊C₄	C₂²×D₁₂₇	1	2	yes	—
120	C₂²⋊C₄	—	2	2	no	—
121	C₂²⋊C₄	—	127	2	no	—
122	C₄²	—	2	2	no	—
123	C₄²	—	127	2	no	—
124	C₂³	D₄×D₁₂₇	1	1	yes	characteristic
125	Q₈	—	1016	4	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	127	4	no	—
129	C₂×C₄	—	1016	2	no	—
130	C₂×C₄	—	254	4	no	—
131	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
132	C₂×C₄	Q₈×D₁₂₇	1	2	yes	—
133	C₂×C₄	C₂×Dic₂₅₄	1	2	yes	—
134	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
135	C₈	—	127	4	no	—
136	C₈	C₂×Dic₂₅₄	1	2	yes	—
137	C₈	C₂×Dic₂₅₄	1	2	yes	—
138	C₂²	—	2	2	no	—
139	C₂²	#16861208	1	2	yes	—
140	C₂²	C₂×D₄×D₁₂₇	1	1	yes	characteristic
141	C₄	—	4	2	no	—
142	C₄	—	4	2	no	—
143	C₄	—	508	4	no	—
144	C₄	—	254	4	no	—
145	C₄	#16861208	1	2	yes	—
146	C₄	C₂×Q₈×D₁₂₇	1	1	yes	characteristic
147	C₄	C₂²×Dic₂₅₄	1	1	yes	characteristic
148	C₂	—	2	2	no	—
149	C₂	C₂×#16861208	1	1	yes	characteristic
150	C₂	#22550683	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22550504

C₂

yes

maximal

#22550284

C₂

yes

maximal, complemented

#22550276

C₂

yes

maximal, complemented

#22550273

C₂

yes

maximal, complemented

#22549919

C₂

yes

maximal

C₂×#16861141

C₂

yes

maximal, characteristic

#22549831

C₂

yes

maximal, characteristic

#427×C₁₂₇

C₂

yes

maximal, characteristic

C₂×#14523232

C₂²

yes

characteristic

#16861209

—

#16861208

C₂²

yes

—

#16861201

—

#16861200

C₂²

yes

—

#16861186

—

#16861168

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₂₇

C₂²

yes

—

#16861142

C₂²

yes

—

#16861141

C₂²

yes

—

#16861141

C₂²

yes

—

#16861140

C₂²

yes

—

#16861119

C₂²

yes

—

C₈×Dic₁₂₇

C₂²

yes

—

C₂²×C₈×C₁₂₇

C₂²

yes

characteristic

#117×C₁₂₇

C₂²

yes

—

#107×C₁₂₇

C₂²

yes

—

#106×C₁₂₇

C₂²

yes

characteristic

#106×C₁₂₇

C₂²

yes

characteristic

C₂×Dic₂₅₄

—

C₂²×Dic₁₂₇

—

#14523232

Q₈

yes

complemented

C₂×C₁₂₇⋊C₈

C₂³

yes

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

D₄

yes

—

C₄⋊Dic₁₂₇

C₂³

yes

—

Dic₁₂₇⋊C₄

—

Dic₁₂₇⋊C₄

—

C₂²⋊Dic₁₂₇

D₄

yes

—

C₄×Dic₁₂₇

C₂³

yes

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

C₂×C₈×C₁₂₇

Q₈

yes

complemented

C₂×C₈×C₁₂₇

C₂³

yes

characteristic

C₂×C₈×C₁₂₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₂₇

C₂³

yes

—

C₄⋊C₄×C₁₂₇

C₂³

yes

—

C₂²⋊C₄×C₁₂₇

—

C₄²×C₁₂₇

—

Dic₂₅₄

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

C₂×D₄

yes

—

C₁₂₇⋊C₈

C₂×Q₈

yes

—

C₂³×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×Q₈

yes

—

C₂×C₄×C₁₂₇

C₂×Q₈

yes

—

C₂×C₄×C₁₂₇

C₂⁴

yes

characteristic

C₈×C₁₂₇

C₂×Q₈

yes

—

C₈×C₁₂₇

C₂×Q₈

yes

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₄⋊Q₈

yes

—

C₂²×C₁₂₇

C₂²×D₄

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₄⋊Q₈

yes

—

C₄×C₁₂₇

C₂²×Q₈

yes

characteristic

C₄×C₁₂₇

C₂²×Q₈

yes

characteristic

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₁₂₇

#577

yes

—

#3163

—

127

maximal, Sylow

C₁₂₇

#3163

yes

characteristic, complemented, Sylow

#544

—

127

—

#501

—

127

—

#499

—

127

—

#498

—

127

—

#428

—

127

—

#427

D₁₂₇

yes

characteristic

C₂×#106

—

127

—

#405

—

127

—

C₂×M₁₆

—

127

—

C₂²×C₈

C₂×D₁₂₇

yes

characteristic

C₄⋊Q₈

—

127

—

C₄⋊Q₈

—

254

—

#125

—

127

—

#125

—

254

—

C₂²⋊Q₈

—

254

—

#117

—

127

—

#117

C₂×D₁₂₇

yes

—

C₂×C₄⋊C₄

—

127

—

#107

—

127

—

#107

C₂×D₁₂₇

yes

—

#106

—

127

—

#106

—

127

—

#106

—

127

—

100

#106

C₂×D₁₂₇

yes

characteristic

101

#106

C₂×D₁₂₇

yes

characteristic

102

#97

—

127

—

103

C₄×C₈

—

127

—

104

C₂×Q₈

—

254

—

105

C₂²×C₄

—

254

—

106

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

107

M₁₆

—

127

—

108

C₂×C₈

—

127

—

109

C₂×C₈

Dic₂₅₄

yes

complemented

110

C₂×C₈

C₂²×D₁₂₇

yes

characteristic

111

C₂×C₈

C₂²×D₁₂₇

yes

characteristic

112

C₄⋊C₄

—

127

—

113

C₄⋊C₄

—

127

—

114

C₄⋊C₄

—

254

—

115

C₄⋊C₄

—

254

—

116

C₄⋊C₄

—

254

—

117

C₄⋊C₄

—

254

—

118

C₄⋊C₄

C₂²×D₁₂₇

yes

—

119

C₄⋊C₄

C₂²×D₁₂₇

yes

—

120

C₂²⋊C₄

—

121

C₂²⋊C₄

—

127

—

122

C₄²

—

123

C₄²

—

127

—

124

C₂³

D₄×D₁₂₇

yes

characteristic

125

Q₈

—

1016

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

127

—

129

C₂×C₄

—

1016

—

130

C₂×C₄

—

254

—

131

C₂×C₄

D₄×D₁₂₇

yes

characteristic

132

C₂×C₄

Q₈×D₁₂₇

yes

—

133

C₂×C₄

C₂×Dic₂₅₄

yes

—

134

C₂×C₄

C₂³×D₁₂₇

yes

characteristic

135

C₈

—

127

—

136

C₈

C₂×Dic₂₅₄

yes

—

137

C₈

C₂×Dic₂₅₄

yes

—

138

C₂²

—

139

C₂²

#16861208

yes

—

140

C₂²

C₂×D₄×D₁₂₇

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

508

—

144

C₄

—

254

—

145

C₄

#16861208

yes

—

146

C₄

C₂×Q₈×D₁₂₇

yes

characteristic

147

C₄

C₂²×Dic₂₅₄

yes

characteristic

148

C₂

—

149

C₂

C₂×#16861208

yes

characteristic

150

C₂

#22550683

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₂₇
commutator, cosocle	C₂×C₄×C₁₂₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	#427×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂²×D₄
prime radical	C₂³×C₁₂₇	C₂×D₄
hypercenter	#427	D₁₂₇
hypocenter	C₁₂₇	#3163
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂²

C₂×D₄×D₁₂₇

commutator, cosocle

C₂×C₄×C₁₂₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₂₇

Fitting

#427×C₁₂₇

C₂

socle

C₂²×C₁₂₇

C₂²×D₄

prime radical

C₂³×C₁₂₇

C₂×D₄

D₁₂₇

C₁₂₇

G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
2. #22550284	C₂	semidirect
3. #22550276	C₂	semidirect
4. #22550273	C₂	semidirect
30. #14523232	Q₈	semidirect
41. C₂×C₈×C₁₂₇	Q₈	semidirect
76. C₁₂₇	#3163	semidirect
109. C₂×C₈	Dic₂₅₄	semidirect

H ≅ G/N

type

2. #22550284

C₂

semidirect

3. #22550276

C₂

semidirect

4. #22550273

C₂

semidirect

30. #14523232

Q₈

semidirect

41. C₂×C₈×C₁₂₇

Q₈

semidirect

76. C₁₂₇

#3163

semidirect

109. C₂×C₈

Dic₂₅₄

semidirect

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

subgroup	index
#22550504	2
#22550284	2
#22550276	2
#22550273	2
#22549919	2
C₂×#16861141	2
#22549831	2
#427×C₁₂₇	2
#3163	127

subgroup

index

#427×C₁₂₇

127

Subgroups — #30649262

Conjugacy classes of subgroups

Complements

Maximal subgroups