Group #30647309 — subgroups

Subgroups — C₂×#22549913 #30647309

Order 16256.

512 conjugacy classes of subgroups; only the 229 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 229 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	C₂²×C₈×D₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#16861167	C₂	1	2	yes	maximal
3	#22549913	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16861143	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16861140	C₂	1	2	yes	maximal, complemented
6	C₂×#107×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₄×D₁₂₇	C₂²	1	1	yes	characteristic
8	C₂²×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₈×D₁₂₇	C₄	1	8	yes	complemented
10	C₂×C₈×D₁₂₇	C₂²	1	4	yes	—
11	#16861167	—	2	4	no	—
12	#16861167	C₂²	1	8	yes	—
13	C₂×C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
14	C₂×(C₂×D₁₂₇)⋊C₄	—	2	2	no	—
15	C₂×C₄×Dic₁₂₇	—	2	2	no	—
16	#16861143	C₂²	1	4	yes	complemented
17	#16861140	C₂²	1	8	yes	complemented
18	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
20	#107×C₁₂₇	C₂²	1	4	yes	complemented
21	C₂³×D₁₂₇	D₄	1	1	yes	characteristic
22	C₂²×Dic₁₂₇	—	2	2	no	—
23	C₂²×Dic₁₂₇	D₄	1	1	yes	characteristic
24	C₂×C₄×D₁₂₇	C₂×C₄	1	4	yes	—
25	C₂×C₄×D₁₂₇	D₄	1	2	yes	—
26	C₂×C₄×D₁₂₇	Q₈	1	4	yes	—
27	C₂×C₄×D₁₂₇	C₂³	1	2	yes	—
28	C₂×C₁₂₇⋊C₈	C₂×C₄	1	4	yes	—
29	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
30	C₈×D₁₂₇	C₂×C₄	1	16	yes	complemented
31	C₄⋊Dic₁₂₇	—	2	2	no	—
32	C₄⋊Dic₁₂₇	C₂³	1	4	yes	—
33	(C₂×D₁₂₇)⋊C₄	—	2	8	no	—
34	C₄×Dic₁₂₇	—	2	8	no	—
35	C₂²×C₄×C₁₂₇	—	2	2	no	—
36	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
37	C₂×C₈×C₁₂₇	C₂×C₄	1	4	yes	—
38	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
39	C₄⋊C₄×C₁₂₇	—	2	2	no	—
40	C₄⋊C₄×C₁₂₇	C₂³	1	4	yes	—
41	C₂²×D₁₂₇	—	2	2	no	—
42	C₂²×D₁₂₇	—	2	2	no	—
43	C₂²×D₁₂₇	C₄⋊C₄	1	4	yes	—
44	C₂²×D₁₂₇	C₂×D₄	1	2	yes	—
45	C₂×Dic₁₂₇	—	2	4	no	—
46	C₂×Dic₁₂₇	—	4	4	no	—
47	C₂×Dic₁₂₇	C₄⋊C₄	1	4	yes	—
48	C₂×Dic₁₂₇	C₂×D₄	1	2	yes	—
49	C₄×D₁₂₇	C₄⋊C₄	1	8	yes	—
50	C₄×D₁₂₇	C₂²×C₄	1	4	yes	—
51	C₄×D₁₂₇	C₂×Q₈	1	4	yes	—
52	C₁₂₇⋊C₈	C₂²×C₄	1	4	yes	—
53	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₁₂₇	—	2	4	no	—
55	C₂×C₄×C₁₂₇	—	4	4	no	—
56	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
57	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
59	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
60	C₈×C₁₂₇	C₂²×C₄	1	4	yes	—
61	C₂²×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
62	C₂²×C₁₂₇	#135	1	2	yes	—
63	C₂²×C₁₂₇	#135	1	2	yes	—
64	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
65	C₂×D₁₂₇	—	2	2	no	—
66	C₂×D₁₂₇	—	2	2	no	—
67	C₂×D₁₂₇	—	2	8	no	—
68	C₂×D₁₂₇	C₂×C₄⋊C₄	1	4	yes	—
69	C₄×C₁₂₇	—	4	4	no	—
70	C₄×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
71	C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
72	C₄×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
73	Dic₁₂₇	—	4	4	no	—
74	Dic₁₂₇	C₂×C₄⋊C₄	1	4	yes	—
75	C₂×C₁₂₇	#427	1	4	yes	complemented
76	C₂×C₁₂₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₁₂₇	C₂×#135	1	1	yes	characteristic
78	C₂×C₁₂₇	C₂×#135	1	1	yes	characteristic
79	D₁₂₇	—	2	4	no	—
80	C₂×#427	—	127	1	no	maximal, Sylow
81	C₁₂₇	C₂×#427	1	1	yes	characteristic, complemented, Sylow
82	C₂³×C₈	—	127	1	no	—
83	C₂×#117	—	127	2	no	—
84	#427	—	127	8	no	—
85	C₂×#107	—	127	1	no	—
86	C₂×#107	D₁₂₇	1	1	yes	characteristic, complemented
87	C₂×#106	—	127	2	no	—
88	C₂³×C₄	—	127	1	no	—
89	C₂²×C₈	—	127	1	no	—
90	C₂²×C₈	—	127	8	no	—
91	C₂²×C₈	—	127	4	no	—
92	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
93	#117	—	127	8	no	—
94	#117	—	254	4	no	—
95	C₂×C₄⋊C₄	—	127	2	no	—
96	C₂×C₄⋊C₄	C₂×D₁₂₇	1	2	yes	—
97	C₂×C₂²⋊C₄	—	254	2	no	—
98	C₂×C₄²	—	254	2	no	—
99	#107	—	127	4	no	—
100	#107	C₂×D₁₂₇	1	4	yes	complemented
101	#106	—	127	8	no	—
102	C₂⁴	—	127	1	no	—
103	C₂²×C₄	—	2	2	no	—
104	C₂²×C₄	—	127	1	no	—
105	C₂²×C₄	—	127	4	no	—
106	C₂²×C₄	—	127	2	no	—
107	C₂²×C₄	—	127	4	no	—
108	C₂²×C₄	—	127	2	no	—
109	C₂²×C₄	—	254	2	no	—
110	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
111	C₂×C₈	—	127	4	no	—
112	C₂×C₈	—	127	2	no	—
113	C₂×C₈	—	127	16	no	—
114	C₂×C₈	C₄×D₁₂₇	1	4	yes	—
115	C₂×C₈	C₂²×D₁₂₇	1	2	yes	—
116	C₄⋊C₄	—	2	2	no	—
117	C₄⋊C₄	—	127	4	no	—
118	C₄⋊C₄	—	254	2	no	—
119	C₄⋊C₄	C₂²×D₁₂₇	1	4	yes	—
120	C₂²⋊C₄	—	254	8	no	—
121	C₄²	—	254	8	no	—
122	C₂³	—	127	2	no	—
123	C₂³	—	127	4	no	—
124	C₂³	—	254	2	no	—
125	C₂³	—	254	2	no	—
126	C₂³	D₄×D₁₂₇	1	1	yes	characteristic
127	C₂×C₄	—	2	4	no	—
128	C₂×C₄	—	4	4	no	—
129	C₂×C₄	—	127	2	no	—
130	C₂×C₄	—	127	4	no	—
131	C₂×C₄	—	127	4	no	—
132	C₂×C₄	—	127	4	no	—
133	C₂×C₄	—	127	8	no	—
134	C₂×C₄	—	508	4	no	—
135	C₂×C₄	—	254	4	no	—
136	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
137	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
138	C₂×C₄	Q₈×D₁₂₇	1	2	yes	—
139	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
140	C₈	—	127	4	no	—
141	C₈	C₂×C₄×D₁₂₇	1	4	yes	—
142	C₂²	—	127	4	no	—
143	C₂²	—	254	2	no	—
144	C₂²	—	254	2	no	—
145	C₂²	—	254	8	no	—
146	C₂²	C₄⋊C₄×D₁₂₇	1	2	yes	—
147	C₂²	#16861228	1	2	yes	—
148	C₂²	#16861232	1	2	yes	—
149	C₂²	C₂×D₄×D₁₂₇	1	1	yes	characteristic
150	C₄	—	4	4	no	—
151	C₄	—	127	4	no	—
152	C₄	—	508	4	no	—
153	C₄	C₄⋊C₄×D₁₂₇	1	2	yes	—
154	C₄	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
155	C₄	C₂×Q₈×D₁₂₇	1	1	yes	characteristic
156	C₂	—	254	4	no	—
157	C₂	#22549913	1	4	yes	direct product factor
158	C₂	C₂×C₄⋊C₄×D₁₂₇	1	1	yes	characteristic
159	C₂	C₂×#16861228	1	1	yes	characteristic
160	C₂	C₂×#16861232	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₈×D₁₂₇

C₂

yes

maximal, characteristic

C₂×#16861167

C₂

yes

maximal

#22549913

C₂

yes

maximal, direct product factor

C₂×#16861143

C₂

yes

maximal, characteristic, complemented

C₂×#16861140

C₂

yes

maximal, complemented

C₂×#107×C₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₁₂₇

C₂²

yes

characteristic

C₂²×C₁₂₇⋊C₈

C₂²

yes

characteristic

C₂×C₈×D₁₂₇

C₄

yes

complemented

C₂×C₈×D₁₂₇

C₂²

yes

—

#16861167

—

#16861167

C₂²

yes

—

C₂×C₄⋊Dic₁₂₇

C₂²

yes

—

C₂×(C₂×D₁₂₇)⋊C₄

—

C₂×C₄×Dic₁₂₇

—

#16861143

C₂²

yes

complemented

#16861140

C₂²

yes

complemented

C₂²×C₈×C₁₂₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₂₇

C₂²

yes

—

#107×C₁₂₇

C₂²

yes

complemented

C₂³×D₁₂₇

D₄

yes

characteristic

C₂²×Dic₁₂₇

—

C₂²×Dic₁₂₇

D₄

yes

characteristic

C₂×C₄×D₁₂₇

C₂×C₄

yes

—

C₂×C₄×D₁₂₇

D₄

yes

—

C₂×C₄×D₁₂₇

Q₈

yes

—

C₂×C₄×D₁₂₇

C₂³

yes

—

C₂×C₁₂₇⋊C₈

C₂×C₄

yes

—

C₂×C₁₂₇⋊C₈

C₂³

yes

—

C₈×D₁₂₇

C₂×C₄

yes

complemented

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

C₂³

yes

—

(C₂×D₁₂₇)⋊C₄

—

C₄×Dic₁₂₇

—

C₂²×C₄×C₁₂₇

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

C₂×C₈×C₁₂₇

C₂×C₄

yes

—

C₂×C₈×C₁₂₇

C₂³

yes

—

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

C₂³

yes

—

C₂²×D₁₂₇

—

C₂²×D₁₂₇

—

C₂²×D₁₂₇

C₄⋊C₄

yes

—

C₂²×D₁₂₇

C₂×D₄

yes

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

C₄⋊C₄

yes

—

C₂×Dic₁₂₇

C₂×D₄

yes

—

C₄×D₁₂₇

C₄⋊C₄

yes

—

C₄×D₁₂₇

C₂²×C₄

yes

—

C₄×D₁₂₇

C₂×Q₈

yes

—

C₁₂₇⋊C₈

C₂²×C₄

yes

—

C₂³×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×Q₈

yes

—

C₂×C₄×C₁₂₇

C₂⁴

yes

characteristic

C₈×C₁₂₇

C₂²×C₄

yes

—

C₂²×C₁₂₇

C₂×C₄⋊C₄

yes

—

C₂²×C₁₂₇

#135

yes

—

C₂²×C₁₂₇

#135

yes

—

C₂²×C₁₂₇

C₂²×D₄

yes

characteristic

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

C₂×C₄⋊C₄

yes

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂×C₄⋊C₄

yes

—

C₄×C₁₂₇

C₂³×C₄

yes

characteristic

C₄×C₁₂₇

C₂²×Q₈

yes

characteristic

Dic₁₂₇

—

Dic₁₂₇

C₂×C₄⋊C₄

yes

—

C₂×C₁₂₇

#427

yes

complemented

C₂×C₁₂₇

C₂²×C₄⋊C₄

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

D₁₂₇

—

C₂×#427

—

127

maximal, Sylow

C₁₂₇

C₂×#427

yes

characteristic, complemented, Sylow

C₂³×C₈

—

127

—

C₂×#117

—

127

—

#427

—

127

—

C₂×#107

—

127

—

C₂×#107

D₁₂₇

yes

characteristic, complemented

C₂×#106

—

127

—

C₂³×C₄

—

127

—

C₂²×C₈

—

127

—

C₂²×C₈

—

127

—

C₂²×C₈

—

127

—

C₂²×C₈

C₂×D₁₂₇

yes

characteristic

#117

—

127

—

#117

—

254

—

C₂×C₄⋊C₄

—

127

—

C₂×C₄⋊C₄

C₂×D₁₂₇

yes

—

C₂×C₂²⋊C₄

—

254

—

C₂×C₄²

—

254

—

#107

—

127

—

100

#107

C₂×D₁₂₇

yes

complemented

101

#106

—

127

—

102

C₂⁴

—

127

—

103

C₂²×C₄

—

104

C₂²×C₄

—

127

—

105

C₂²×C₄

—

127

—

106

C₂²×C₄

—

127

—

107

C₂²×C₄

—

127

—

108

C₂²×C₄

—

127

—

109

C₂²×C₄

—

254

—

110

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

111

C₂×C₈

—

127

—

112

C₂×C₈

—

127

—

113

C₂×C₈

—

127

—

114

C₂×C₈

C₄×D₁₂₇

yes

—

115

C₂×C₈

C₂²×D₁₂₇

yes

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

127

—

118

C₄⋊C₄

—

254

—

119

C₄⋊C₄

C₂²×D₁₂₇

yes

—

120

C₂²⋊C₄

—

254

—

121

C₄²

—

254

—

122

C₂³

—

127

—

123

C₂³

—

127

—

124

C₂³

—

254

—

125

C₂³

—

254

—

126

C₂³

D₄×D₁₂₇

yes

characteristic

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

127

—

130

C₂×C₄

—

127

—

131

C₂×C₄

—

127

—

132

C₂×C₄

—

127

—

133

C₂×C₄

—

127

—

134

C₂×C₄

—

508

—

135

C₂×C₄

—

254

—

136

C₂×C₄

C₂×C₄×D₁₂₇

yes

—

137

C₂×C₄

D₄×D₁₂₇

yes

characteristic

138

C₂×C₄

Q₈×D₁₂₇

yes

—

139

C₂×C₄

C₂³×D₁₂₇

yes

characteristic

140

C₈

—

127

—

141

C₈

C₂×C₄×D₁₂₇

yes

—

142

C₂²

—

127

—

143

C₂²

—

254

—

144

C₂²

—

254

—

145

C₂²

—

254

—

146

C₂²

C₄⋊C₄×D₁₂₇

yes

—

147

C₂²

#16861228

yes

—

148

C₂²

#16861232

yes

—

149

C₂²

C₂×D₄×D₁₂₇

yes

characteristic

150

C₄

—

151

C₄

—

127

—

152

C₄

—

508

—

153

C₄

C₄⋊C₄×D₁₂₇

yes

—

154

C₄

C₂²×C₄×D₁₂₇

yes

characteristic

155

C₄

C₂×Q₈×D₁₂₇

yes

characteristic

156

C₂

—

254

—

157

C₂

#22549913

yes

direct product factor

158

C₂

C₂×C₄⋊C₄×D₁₂₇

yes

characteristic

159

C₂

C₂×#16861228

yes

characteristic

160

C₂

C₂×#16861232

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₁₂₇
commutator	C₄×C₁₂₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	C₂×#107×C₁₂₇	C₂
socle	C₂³×C₁₂₇	C₂×D₄
prime radical	C₂³×D₁₂₇	D₄
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	C₂×#107	D₁₂₇
hypocenter	C₁₂₇	C₂×#427
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

subgroup

G/H

center

C₂³

D₄×D₁₂₇

commutator

C₄×C₁₂₇

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₁₂₇

Fitting

C₂×#107×C₁₂₇

C₂

socle

C₂³×C₁₂₇

C₂×D₄

prime radical

C₂³×D₁₂₇

D₄

cosocle

C₂×C₄×C₁₂₇

C₂⁴

D₁₂₇

C₁₂₇

G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

N	H ≅ G/N	type
3. #22549913	C₂	direct (N × H)
4. C₂×#16861143	C₂	semidirect
5. C₂×#16861140	C₂	semidirect
6. C₂×#107×C₁₂₇	C₂	semidirect
9. C₂×C₈×D₁₂₇	C₄	semidirect
16. #16861143	C₂²	semidirect
17. #16861140	C₂²	semidirect
20. #107×C₁₂₇	C₂²	semidirect
30. C₈×D₁₂₇	C₂×C₄	semidirect
75. C₂×C₁₂₇	#427	semidirect
81. C₁₂₇	C₂×#427	semidirect
86. C₂×#107	D₁₂₇	semidirect
100. #107	C₂×D₁₂₇	semidirect
157. C₂	#22549913	direct (N × H)

H ≅ G/N

type

3. #22549913

C₂

direct (N × H)

4. C₂×#16861143

C₂

semidirect

5. C₂×#16861140

C₂

semidirect

6. C₂×#107×C₁₂₇

C₂

semidirect

9. C₂×C₈×D₁₂₇

C₄

semidirect

16. #16861143

C₂²

semidirect

17. #16861140

C₂²

semidirect

20. #107×C₁₂₇

C₂²

semidirect

30. C₈×D₁₂₇

C₂×C₄

semidirect

75. C₂×C₁₂₇

#427

semidirect

81. C₁₂₇

C₂×#427

semidirect

86. C₂×#107

D₁₂₇

semidirect

100. #107

C₂×D₁₂₇

semidirect

157. C₂

#22549913

direct (N × H)

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

subgroup	index
C₂²×C₈×D₁₂₇	2
C₂×#16861167	2
#22549913	2
C₂×#16861143	2
C₂×#16861140	2
C₂×#107×C₁₂₇	2
C₂×#427	127

subgroup

index

C₂²×C₈×D₁₂₇

C₂×#107×C₁₂₇

127

Subgroups — C₂×#22549913 #30647309

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#22549913 #30647309

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22549913 #30647309