Group #30641648 — subgroups

Subgroups — #30641648

Order 16256.

552 conjugacy classes of subgroups; only the 275 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 275 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	C₂×Q₈×Dic₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×Dic₁₂₇	C₂	1	1	yes	maximal, characteristic
3	#22549774	C₂	1	3	yes	maximal
4	#22549772	C₂	1	3	yes	maximal
5	#22549741	C₂	1	3	yes	maximal
6	C₄×Dic₁₂₇⋊C₄	C₂	1	3	yes	maximal
7	C₂×C₄×Q₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
8	Q₈×Dic₁₂₇	—	2	4	no	—
9	Q₈×Dic₁₂₇	C₄	1	8	yes	complemented
10	C₂×C₄⋊Dic₁₂₇	C₂²	1	3	yes	—
11	C₂×C₄⋊Dic₁₂₇	C₂²	1	3	yes	—
12	C₂×Dic₁₂₇⋊C₄	C₂²	1	3	yes	—
13	C₂×Dic₁₂₇⋊C₄	C₂²	1	3	yes	—
14	C₂×C₄×Dic₁₂₇	C₂²	1	3	yes	—
15	C₂×C₄×Dic₁₂₇	C₂²	1	3	yes	—
16	C₁₂₇⋊#95	C₂²	1	6	yes	—
17	C₁₂₇⋊#95	C₂²	1	1	yes	characteristic
18	C₁₂₇⋊#95	C₂²	1	3	yes	—
19	C₂²×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
20	C₄×Q₈×C₁₂₇	—	2	4	no	—
21	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	3	yes	—
22	C₂×C₄²×C₁₂₇	C₂²	1	3	yes	—
23	C₂²×Dic₁₂₇	C₂³	1	3	yes	—
24	C₂²×Dic₁₂₇	C₂³	1	1	yes	characteristic
25	C₂²×Dic₁₂₇	C₂³	1	3	yes	—
26	C₂²×Dic₁₂₇	C₂³	1	1	yes	characteristic
27	C₄⋊Dic₁₂₇	—	2	6	no	—
28	C₄⋊Dic₁₂₇	C₂×C₄	1	12	yes	—
29	Dic₁₂₇⋊C₄	—	2	6	no	—
30	Dic₁₂₇⋊C₄	—	2	6	no	—
31	C₄×Dic₁₂₇	—	2	6	no	—
32	C₄×Dic₁₂₇	C₂×C₄	1	12	yes	—
33	C₂×Q₈×C₁₂₇	C₂×C₄	1	4	yes	—
34	C₂×Q₈×C₁₂₇	D₄	1	4	yes	—
35	C₂×Q₈×C₁₂₇	Q₈	1	4	yes	—
36	C₂²×C₄×C₁₂₇	C₂³	1	3	yes	—
37	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₁₂₇	C₂³	1	3	yes	—
39	C₄⋊C₄×C₁₂₇	—	2	6	no	—
40	C₄²×C₁₂₇	—	2	6	no	—
41	C₂×Dic₁₂₇	—	2	1	no	—
42	C₂×Dic₁₂₇	—	2	3	no	—
43	C₂×Dic₁₂₇	—	2	3	no	—
44	C₂×Dic₁₂₇	—	2	1	no	—
45	C₂×Dic₁₂₇	—	2	1	no	—
46	C₂×Dic₁₂₇	—	2	1	no	—
47	C₂×Dic₁₂₇	—	2	3	no	—
48	C₂×Dic₁₂₇	—	2	3	no	—
49	C₂×Dic₁₂₇	—	2	3	no	—
50	C₂×Dic₁₂₇	C₂²×C₄	1	2	yes	—
51	C₂×Dic₁₂₇	C₂²×C₄	1	6	yes	—
52	C₂×Dic₁₂₇	#41	1	2	yes	—
53	C₂³×C₁₂₇	C₂⁴	1	1	yes	characteristic
54	Q₈×C₁₂₇	C₄⋊C₄	1	16	yes	complemented
55	C₂×C₄×C₁₂₇	—	2	1	no	—
56	C₂×C₄×C₁₂₇	—	2	1	no	—
57	C₂×C₄×C₁₂₇	—	2	3	no	—
58	C₂×C₄×C₁₂₇	—	2	3	no	—
59	C₂×C₄×C₁₂₇	—	2	3	no	—
60	C₂×C₄×C₁₂₇	C₂²×C₄	1	6	yes	—
61	C₂×C₄×C₁₂₇	C₂×D₄	1	6	yes	—
62	C₂×C₄×C₁₂₇	C₂×Q₈	1	6	yes	—
63	C₂×C₄×C₁₂₇	#41	1	2	yes	—
64	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
65	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
66	C₂²×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
67	C₂²×C₁₂₇	C₂×#41	1	1	yes	characteristic
68	C₂²×C₁₂₇	C₂×#41	1	1	yes	characteristic
69	C₂²×C₁₂₇	2₋¹⁺⁴	1	1	yes	characteristic
70	C₂²×C₁₂₇	2₋¹⁺⁴	1	1	yes	characteristic
71	C₄×C₁₂₇	—	2	2	no	—
72	C₄×C₁₂₇	—	4	3	no	—
73	C₄×C₁₂₇	C₂×C₄⋊C₄	1	12	yes	—
74	Dic₁₂₇	—	2	6	no	—
75	Dic₁₂₇	—	2	2	no	—
76	Dic₁₂₇	—	2	2	no	—
77	Dic₁₂₇	—	4	3	no	—
78	C₂×C₁₂₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₁₂₇	C₄×#41	1	1	yes	characteristic
80	C₂×C₁₂₇	#519	1	1	yes	characteristic
81	C₂×C₁₂₇	#548	1	1	yes	characteristic
82	C₂×C₁₂₇	#548	1	1	yes	characteristic
83	C₂×C₁₂₇	#557	1	1	yes	characteristic
84	C₂×C₁₂₇	#557	1	1	yes	characteristic
85	#2351	—	127	1	no	maximal, Sylow
86	C₁₂₇	#2351	1	1	yes	characteristic, complemented, Sylow
87	C₂×C₄×Q₈	—	127	1	no	—
88	C₂×C₄×Q₈	—	127	1	no	—
89	C₂×C₄×Q₈	D₁₂₇	1	1	yes	characteristic
90	#389	—	127	3	no	—
91	#389	—	127	3	no	—
92	#382	—	127	3	no	—
93	C₄×C₄⋊C₄	—	127	3	no	—
94	C₂²×Q₈	C₂×D₁₂₇	1	1	yes	characteristic
95	C₄×Q₈	—	2	4	no	—
96	C₄×Q₈	—	127	8	no	—
97	C₄×Q₈	—	254	4	no	—
98	C₂×C₄⋊C₄	—	127	3	no	—
99	C₂×C₄⋊C₄	—	127	3	no	—
100	C₂×C₄⋊C₄	—	127	3	no	—
101	C₂×C₄⋊C₄	—	127	3	no	—
102	C₂×C₄⋊C₄	C₂×D₁₂₇	1	3	yes	—
103	C₂×C₄²	—	127	3	no	—
104	C₂×C₄²	—	127	3	no	—
105	C₂×C₄²	C₂×D₁₂₇	1	3	yes	—
106	#95	—	127	1	no	—
107	#95	—	127	3	no	—
108	#95	—	127	6	no	—
109	C₂×Q₈	C₄×D₁₂₇	1	4	yes	—
110	C₂×Q₈	C₂²⋊D₁₂₇	1	4	yes	—
111	C₂×Q₈	Dic₂₅₄	1	4	yes	—
112	C₂²×C₄	—	127	1	no	—
113	C₂²×C₄	—	127	3	no	—
114	C₂²×C₄	—	127	1	no	—
115	C₂²×C₄	—	127	3	no	—
116	C₂²×C₄	C₂²×D₁₂₇	1	3	yes	—
117	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
118	C₂²×C₄	C₂²×D₁₂₇	1	3	yes	—
119	C₄⋊C₄	—	2	6	no	—
120	C₄⋊C₄	—	127	12	no	—
121	C₄⋊C₄	—	254	6	no	—
122	C₄⋊C₄	—	254	6	no	—
123	C₄⋊C₄	—	254	6	no	—
124	C₄²	—	2	6	no	—
125	C₄²	—	127	12	no	—
126	C₄²	—	254	6	no	—
127	C₂³	C₂³×D₁₂₇	1	1	yes	characteristic
128	Q₈	Dic₁₂₇⋊C₄	1	16	yes	complemented
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	2	3	no	—
132	C₂×C₄	—	2	3	no	—
133	C₂×C₄	—	2	3	no	—
134	C₂×C₄	—	127	2	no	—
135	C₂×C₄	—	127	6	no	—
136	C₂×C₄	—	127	2	no	—
137	C₂×C₄	—	254	3	no	—
138	C₂×C₄	—	254	3	no	—
139	C₂×C₄	—	254	3	no	—
140	C₂×C₄	—	254	1	no	—
141	C₂×C₄	—	254	3	no	—
142	C₂×C₄	—	254	3	no	—
143	C₂×C₄	—	254	1	no	—
144	C₂×C₄	—	254	1	no	—
145	C₂×C₄	—	254	1	no	—
146	C₂×C₄	C₂×C₄×D₁₂₇	1	6	yes	—
147	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	6	yes	—
148	C₂×C₄	C₂×Dic₂₅₄	1	6	yes	—
149	C₂×C₄	#14523249	1	2	yes	—
150	C₂²	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
151	C₂²	C₂²×C₂²⋊D₁₂₇	1	1	yes	characteristic
152	C₂²	C₂²×Dic₂₅₄	1	1	yes	characteristic
153	C₂²	#41×D₁₂₇	1	1	yes	characteristic
154	C₂²	C₂×#14523249	1	1	yes	characteristic
155	C₂²	#16861256	1	1	yes	characteristic
156	C₂²	#16861257	1	1	yes	characteristic
157	C₄	—	2	2	no	—
158	C₄	—	4	3	no	—
159	C₄	—	508	3	no	—
160	C₄	—	254	2	no	—
161	C₄	—	254	6	no	—
162	C₄	—	254	2	no	—
163	C₄	C₂×Dic₁₂₇⋊C₄	1	12	yes	—
164	C₂	C₂²×Dic₁₂₇⋊C₄	1	1	yes	characteristic
165	C₂	C₄×#14523249	1	1	yes	characteristic
166	C₂	#22550352	1	1	yes	characteristic
167	C₂	#22550533	1	1	yes	characteristic
168	C₂	#22550537	1	1	yes	characteristic
169	C₂	#22550602	1	1	yes	characteristic
170	C₂	#22550605	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×Dic₁₂₇

C₂

yes

maximal, characteristic

C₂×Q₈×Dic₁₂₇

C₂

yes

maximal, characteristic

#22549774

C₂

yes

maximal

#22549772

C₂

yes

maximal

#22549741

C₂

yes

maximal

C₄×Dic₁₂₇⋊C₄

C₂

yes

maximal

C₂×C₄×Q₈×C₁₂₇

C₂

yes

maximal, characteristic

Q₈×Dic₁₂₇

—

Q₈×Dic₁₂₇

C₄

yes

complemented

C₂×C₄⋊Dic₁₂₇

C₂²

yes

—

C₂×C₄⋊Dic₁₂₇

C₂²

yes

—

C₂×Dic₁₂₇⋊C₄

C₂²

yes

—

C₂×Dic₁₂₇⋊C₄

C₂²

yes

—

C₂×C₄×Dic₁₂₇

C₂²

yes

—

C₂×C₄×Dic₁₂₇

C₂²

yes

—

C₁₂₇⋊#95

C₂²

yes

—

C₁₂₇⋊#95

C₂²

yes

characteristic

C₁₂₇⋊#95

C₂²

yes

—

C₂²×Q₈×C₁₂₇

C₂²

yes

characteristic

C₄×Q₈×C₁₂₇

—

C₂×C₄⋊C₄×C₁₂₇

C₂²

yes

—

C₂×C₄²×C₁₂₇

C₂²

yes

—

C₂²×Dic₁₂₇

C₂³

yes

—

C₂²×Dic₁₂₇

C₂³

yes

characteristic

C₂²×Dic₁₂₇

C₂³

yes

—

C₂²×Dic₁₂₇

C₂³

yes

characteristic

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

C₂×C₄

yes

—

Dic₁₂₇⋊C₄

—

Dic₁₂₇⋊C₄

—

C₄×Dic₁₂₇

—

C₄×Dic₁₂₇

C₂×C₄

yes

—

C₂×Q₈×C₁₂₇

C₂×C₄

yes

—

C₂×Q₈×C₁₂₇

D₄

yes

—

C₂×Q₈×C₁₂₇

Q₈

yes

—

C₂²×C₄×C₁₂₇

C₂³

yes

—

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

C₂²×C₄×C₁₂₇

C₂³

yes

—

C₄⋊C₄×C₁₂₇

—

C₄²×C₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

C₂²×C₄

yes

—

C₂×Dic₁₂₇

C₂²×C₄

yes

—

C₂×Dic₁₂₇

#41

yes

—

C₂³×C₁₂₇

C₂⁴

yes

characteristic

Q₈×C₁₂₇

C₄⋊C₄

yes

complemented

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

—

C₂×C₄×C₁₂₇

C₂×D₄

yes

—

C₂×C₄×C₁₂₇

C₂×Q₈

yes

—

C₂×C₄×C₁₂₇

#41

yes

—

C₂²×C₁₂₇

C₂³×C₄

yes

characteristic

C₂²×C₁₂₇

C₂²×D₄

yes

characteristic

C₂²×C₁₂₇

C₂²×Q₈

yes

characteristic

C₂²×C₁₂₇

C₂×#41

yes

characteristic

C₂²×C₁₂₇

C₂×#41

yes

characteristic

C₂²×C₁₂₇

2₋¹⁺⁴

yes

characteristic

C₂²×C₁₂₇

2₋¹⁺⁴

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂×C₄⋊C₄

yes

—

Dic₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

C₂²×C₄⋊C₄

yes

characteristic

C₂×C₁₂₇

C₄×#41

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

—

127

maximal, Sylow

C₁₂₇

#2351

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

—

127

—

C₂×C₄×Q₈

—

127

—

C₂×C₄×Q₈

D₁₂₇

yes

characteristic

#389

—

127

—

#389

—

127

—

#382

—

127

—

C₄×C₄⋊C₄

—

127

—

C₂²×Q₈

C₂×D₁₂₇

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

127

—

C₄×Q₈

—

254

—

C₂×C₄⋊C₄

—

127

—

C₂×C₄⋊C₄

—

127

—

100

C₂×C₄⋊C₄

—

127

—

101

C₂×C₄⋊C₄

—

127

—

102

C₂×C₄⋊C₄

C₂×D₁₂₇

yes

—

103

C₂×C₄²

—

127

—

104

C₂×C₄²

—

127

—

105

C₂×C₄²

C₂×D₁₂₇

yes

—

106

#95

—

127

—

107

#95

—

127

—

108

#95

—

127

—

109

C₂×Q₈

C₄×D₁₂₇

yes

—

110

C₂×Q₈

C₂²⋊D₁₂₇

yes

—

111

C₂×Q₈

Dic₂₅₄

yes

—

112

C₂²×C₄

—

127

—

113

C₂²×C₄

—

127

—

114

C₂²×C₄

—

127

—

115

C₂²×C₄

—

127

—

116

C₂²×C₄

C₂²×D₁₂₇

yes

—

117

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

118

C₂²×C₄

C₂²×D₁₂₇

yes

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

127

—

121

C₄⋊C₄

—

254

—

122

C₄⋊C₄

—

254

—

123

C₄⋊C₄

—

254

—

124

C₄²

—

125

C₄²

—

127

—

126

C₄²

—

254

—

127

C₂³

C₂³×D₁₂₇

yes

characteristic

128

Q₈

Dic₁₂₇⋊C₄

yes

complemented

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

127

—

135

C₂×C₄

—

127

—

136

C₂×C₄

—

127

—

137

C₂×C₄

—

254

—

138

C₂×C₄

—

254

—

139

C₂×C₄

—

254

—

140

C₂×C₄

—

254

—

141

C₂×C₄

—

254

—

142

C₂×C₄

—

254

—

143

C₂×C₄

—

254

—

144

C₂×C₄

—

254

—

145

C₂×C₄

—

254

—

146

C₂×C₄

C₂×C₄×D₁₂₇

yes

—

147

C₂×C₄

C₂×C₂²⋊D₁₂₇

yes

—

148

C₂×C₄

C₂×Dic₂₅₄

yes

—

149

C₂×C₄

#14523249

yes

—

150

C₂²

C₂²×C₄×D₁₂₇

yes

characteristic

151

C₂²

C₂²×C₂²⋊D₁₂₇

yes

characteristic

152

C₂²

C₂²×Dic₂₅₄

yes

characteristic

153

C₂²

#41×D₁₂₇

yes

characteristic

154

C₂²

C₂×#14523249

yes

characteristic

155

C₂²

#16861256

yes

characteristic

156

C₂²

#16861257

yes

characteristic

157

C₄

—

158

C₄

—

159

C₄

—

508

—

160

C₄

—

254

—

161

C₄

—

254

—

162

C₄

—

254

—

163

C₄

C₂×Dic₁₂₇⋊C₄

yes

—

164

C₂

C₂²×Dic₁₂₇⋊C₄

yes

characteristic

165

C₂

C₄×#14523249

yes

characteristic

166

C₂

#22550352

yes

characteristic

167

C₂

#22550533

yes

characteristic

168

C₂

#22550537

yes

characteristic

169

C₂

#22550602

yes

characteristic

170

C₂

#22550605

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₂₇
commutator	C₂²×C₁₂₇	C₂³×C₄
Fitting	C₂×C₄×Q₈×C₁₂₇	C₂
socle, prime radical, cosocle	C₂³×C₁₂₇	C₂⁴
hypercenter	C₂×C₄×Q₈	D₁₂₇
hypocenter	C₁₂₇	#2351
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₂₇

commutator

C₂²×C₁₂₇

C₂³×C₄

Fitting

C₂×C₄×Q₈×C₁₂₇

C₂

socle, prime radical, cosocle

C₂³×C₁₂₇

C₂⁴

hypercenter

C₂×C₄×Q₈

D₁₂₇

C₁₂₇

G ▷ C₂²×C₁₂₇ ▷ C₁₂₇

C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
9. Q₈×Dic₁₂₇	C₄	semidirect
54. Q₈×C₁₂₇	C₄⋊C₄	semidirect
86. C₁₂₇	#2351	semidirect
128. Q₈	Dic₁₂₇⋊C₄	semidirect

H ≅ G/N

type

9. Q₈×Dic₁₂₇

C₄

semidirect

54. Q₈×C₁₂₇

C₄⋊C₄

semidirect

86. C₁₂₇

#2351

semidirect

128. Q₈

Dic₁₂₇⋊C₄

semidirect

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

subgroup	index
C₂×Q₈×Dic₁₂₇	2
#22549774	2
#22549772	2
#22549741	2
C₄×Dic₁₂₇⋊C₄	2
C₂×C₄×Q₈×C₁₂₇	2
#2351	127

subgroup

index

C₂×Q₈×Dic₁₂₇

#22549774

#22549772

#22549741

C₄×Dic₁₂₇⋊C₄

C₂×C₄×Q₈×C₁₂₇

#2351

127

Subgroups — #30641648

Conjugacy classes of subgroups

Complements

Maximal subgroups