Group #34045779 — subgroups

Subgroups — #34045779

Order 19072.

474 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 99 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×#14558724	C₂	1	2	yes	maximal, complemented
2	C₂×#18011832	C₂	1	2	yes	maximal, complemented
3	#23749531	C₂	1	1	yes	maximal, characteristic, complemented
4	#23749380	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#127×C₁₄₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₉₈	C₂²	1	1	yes	characteristic
7	C₂×#14558724	—	2	4	no	—
8	C₂×#14558724	—	2	4	no	—
9	C₂×#14558724	—	2	4	no	—
10	C₂²×#11924746	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₁₄₉	C₂²	1	1	yes	characteristic
12	#18011832	—	2	4	no	—
13	#18011818	—	2	2	no	—
14	C₂²×D₄×C₁₄₉	—	2	2	no	—
15	C₂²×D₄×C₁₄₉	C₂²	1	2	yes	—
16	#127×C₁₄₉	—	2	4	no	—
17	C₂×C₄²×C₁₄₉	C₂²	1	1	yes	characteristic
18	C₂×Dic₂₉₈	—	2	2	no	—
19	C₂×Dic₂₉₈	—	2	2	no	—
20	C₂×Dic₂₉₈	—	2	1	no	—
21	C₂×Dic₂₉₈	D₄	1	2	yes	complemented
22	C₂²×Dic₁₄₉	—	2	1	no	—
23	C₂²×Dic₁₄₉	—	2	1	no	—
24	#14558724	—	2	16	no	—
25	C₂×#11924746	—	2	2	no	—
26	C₂×#11924746	—	2	2	no	—
27	C₂×#11924746	D₄	1	4	yes	complemented
28	C₄⋊Dic₁₄₉	—	2	2	no	—
29	C₂⁴×C₁₄₉	—	2	2	no	—
30	C₂×D₄×C₁₄₉	—	2	8	no	—
31	C₂×D₄×C₁₄₉	—	2	2	no	—
32	C₂×D₄×C₁₄₉	—	2	2	no	—
33	C₂×D₄×C₁₄₉	—	2	2	no	—
34	C₂×D₄×C₁₄₉	—	2	2	no	—
35	C₂×D₄×C₁₄₉	—	2	2	no	—
36	C₂×D₄×C₁₄₉	—	4	4	no	—
37	C₂×D₄×C₁₄₉	—	4	4	no	—
38	C₂×D₄×C₁₄₉	D₄	1	4	yes	—
39	C₂²×C₄×C₁₄₉	D₄	1	2	yes	—
40	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
41	C₄²×C₁₄₉	—	2	2	no	—
42	Dic₂₉₈	—	2	4	no	—
43	Dic₂₉₈	—	4	2	no	—
44	C₂×Dic₁₄₉	—	2	2	no	—
45	C₂×Dic₁₄₉	—	4	1	no	—
46	C₂×Dic₁₄₉	—	4	1	no	—
47	C₂×Dic₁₄₉	—	4	1	no	—
48	C₂×Dic₁₄₉	—	4	2	no	—
49	#11924746	—	2	4	no	—
50	C₂³×C₁₄₉	—	2	4	no	—
51	C₂³×C₁₄₉	—	4	2	no	—
52	C₂³×C₁₄₉	—	4	2	no	—
53	C₂³×C₁₄₉	—	4	4	no	—
54	C₂³×C₁₄₉	—	4	4	no	—
55	C₂³×C₁₄₉	C₂×D₄	1	1	yes	characteristic
56	D₄×C₁₄₉	—	2	4	no	—
57	D₄×C₁₄₉	—	2	4	no	—
58	D₄×C₁₄₉	—	4	16	no	—
59	D₄×C₁₄₉	—	4	2	no	—
60	D₄×C₁₄₉	—	4	2	no	—
61	C₂×C₄×C₁₄₉	—	2	2	no	—
62	C₂×C₄×C₁₄₉	—	2	2	no	—
63	C₂×C₄×C₁₄₉	SD₁₆	1	4	yes	complemented
64	C₂×C₄×C₁₄₉	C₂×D₄	1	1	yes	characteristic
65	C₂×C₄×C₁₄₉	C₂×D₄	1	2	yes	—
66	C₂×C₄×C₁₄₉	C₂×D₄	1	2	yes	—
67	C₂×C₄×C₁₄₉	#41	1	1	yes	characteristic
68	C₂²×C₁₄₉	—	4	4	no	—
69	C₂²×C₁₄₉	—	4	8	no	—
70	C₂²×C₁₄₉	—	8	4	no	—
71	C₂²×C₁₄₉	—	8	2	no	—
72	C₂²×C₁₄₉	—	8	2	no	—
73	C₂²×C₁₄₉	#120	1	1	yes	characteristic
74	C₂²×C₁₄₉	#121	1	1	yes	characteristic
75	C₂²×C₁₄₉	#127	1	1	yes	characteristic
76	C₂²×C₁₄₉	C₂×SD₁₆	1	2	yes	—
77	C₂²×C₁₄₉	AGL(1,ℤ/8)	1	2	yes	—
78	C₄×C₁₄₉	—	2	4	no	—
79	C₄×C₁₄₉	#120	1	2	yes	—
80	C₄×C₁₄₉	#121	1	2	yes	—
81	Dic₁₄₉	—	4	2	no	—
82	Dic₁₄₉	—	8	1	no	—
83	C₂×C₁₄₉	—	8	4	no	—
84	C₂×C₁₄₉	#392	1	1	yes	characteristic
85	C₂×C₁₄₉	#450	1	2	yes	—
86	C₂×C₁₄₉	#460	1	2	yes	—
87	C₂×C₁₄₉	#492	1	1	yes	characteristic
88	C₂×C₁₄₉	#496	1	1	yes	characteristic
89	C₁₄₉	#2014	1	1	yes	characteristic, complemented, Sylow
90	#2014	—	149	1	no	maximal, Sylow
91	C₂²×SD₁₆	—	149	2	no	—
92	C₂×#127	D₁₄₉	1	1	yes	characteristic
93	C₂×#102	—	149	2	no	—
94	#391	—	149	1	no	—
95	#336	—	149	1	no	—
96	C₂²×Q₈	—	149	1	no	—
97	C₂²×D₄	—	2	2	no	—
98	C₂²×D₄	C₂×D₁₄₉	1	2	yes	—
99	C₂×SD₁₆	—	298	4	no	—
100	C₂×SD₁₆	—	298	4	no	—
101	C₂×SD₁₆	—	298	4	no	—
102	C₂²×C₈	—	149	2	no	—
103	#127	—	2	4	no	—
104	C₂×C₄⋊C₄	—	149	1	no	—
105	C₂×C₄²	C₂×D₁₄₉	1	1	yes	characteristic
106	#102	—	298	4	no	—
107	#95	—	298	2	no	—
108	C₂⁴	—	2	2	no	—
109	C₂×Q₈	—	298	1	no	—
110	C₂×Q₈	—	298	2	no	—
111	C₂×Q₈	—	298	2	no	—
112	C₂×Q₈	—	149	2	no	—
113	C₂×D₄	—	2	8	no	—
114	C₂×D₄	—	2	2	no	—
115	C₂×D₄	—	2	2	no	—
116	C₂×D₄	—	2	2	no	—
117	C₂×D₄	—	2	2	no	—
118	C₂×D₄	—	2	2	no	—
119	C₂×D₄	—	4	4	no	—
120	C₂×D₄	—	4	4	no	—
121	C₂×D₄	C₂²⋊D₁₄₉	1	4	yes	—
122	C₂²×C₄	—	298	1	no	—
123	C₂²×C₄	—	298	1	no	—
124	C₂²×C₄	C₂²⋊D₁₄₉	1	2	yes	—
125	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
126	SD₁₆	—	298	16	no	—
127	C₂×C₈	—	298	2	no	—
128	C₂×C₈	—	298	2	no	—
129	C₂×C₈	—	149	4	no	—
130	C₄⋊C₄	—	298	2	no	—
131	C₄²	—	2	2	no	—
132	C₂³	—	2	4	no	—
133	C₂³	—	4	2	no	—
134	C₂³	—	4	4	no	—
135	C₂³	—	4	4	no	—
136	C₂³	—	4	2	no	—
137	C₂³	C₂×C₂²⋊D₁₄₉	1	1	yes	characteristic
138	Q₈	—	298	4	no	—
139	Q₈	—	596	2	no	—
140	D₄	—	2	4	no	—
141	D₄	—	2	4	no	—
142	D₄	—	4	16	no	—
143	D₄	—	4	2	no	—
144	D₄	—	4	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	298	2	no	—
148	C₂×C₄	—	596	1	no	—
149	C₂×C₄	—	596	1	no	—
150	C₂×C₄	—	596	1	no	—
151	C₂×C₄	—	596	2	no	—
152	C₂×C₄	#14558724	1	4	yes	complemented
153	C₂×C₄	D₄×D₁₄₉	1	1	yes	characteristic
154	C₂×C₄	D₄×D₁₄₉	1	2	yes	—
155	C₂×C₄	C₂×C₂²⋊D₁₄₉	1	2	yes	—
156	C₂×C₄	#14558735	1	1	yes	characteristic
157	C₈	—	298	4	no	—
158	C₂²	—	4	8	no	—
159	C₂²	—	4	4	no	—
160	C₂²	—	8	2	no	—
161	C₂²	—	8	2	no	—
162	C₂²	—	8	4	no	—
163	C₂²	#18011880	1	1	yes	characteristic
164	C₂²	#18011882	1	1	yes	characteristic
165	C₂²	#18011907	1	1	yes	characteristic
166	C₂²	C₂×#14558724	1	2	yes	—
167	C₂²	#18011937	1	2	yes	—
168	C₄	—	2	4	no	—
169	C₄	—	596	2	no	—
170	C₄	—	1192	1	no	—
171	C₄	#18011879	1	2	yes	—
172	C₄	#18011883	1	2	yes	—
173	C₂	—	8	4	no	—
174	C₂	#23749533	1	1	yes	characteristic
175	C₂	#23749776	1	2	yes	—
176	C₂	#23749822	1	2	yes	—
177	C₂	#23749997	1	1	yes	characteristic
178	C₂	#23750012	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#14558724

C₂

yes

maximal, complemented

C₂×#18011832

C₂

yes

maximal, complemented

#23749531

C₂

yes

maximal, characteristic, complemented

#23749380

C₂

yes

maximal, characteristic, complemented

C₂×#127×C₁₄₉

C₂

yes

maximal, characteristic

C₂²×Dic₂₉₈

C₂²

yes

—

—

—

C₂²

yes

—

C₂×C₄⋊Dic₁₄₉

C₂²

yes

characteristic

#18011832

—

#18011818

—

C₂²×D₄×C₁₄₉

—

C₂²×D₄×C₁₄₉

C₂²

yes

—

#127×C₁₄₉

—

C₂×C₄²×C₁₄₉

C₂²

yes

characteristic

C₂×Dic₂₉₈

—

C₂×Dic₂₉₈

—

C₂×Dic₂₉₈

—

C₂×Dic₂₉₈

D₄

yes

complemented

C₂²×Dic₁₄₉

—

C₂²×Dic₁₄₉

—

—

—

—

D₄

yes

complemented

C₄⋊Dic₁₄₉

—

C₂⁴×C₁₄₉

—

C₂×D₄×C₁₄₉

—

C₂×D₄×C₁₄₉

—

C₂×D₄×C₁₄₉

—

C₂×D₄×C₁₄₉

—

C₂×D₄×C₁₄₉

—

C₂×D₄×C₁₄₉

—

C₂×D₄×C₁₄₉

—

C₂×D₄×C₁₄₉

—

C₂×D₄×C₁₄₉

D₄

yes

—

C₂²×C₄×C₁₄₉

D₄

yes

—

C₂²×C₄×C₁₄₉

C₂³

yes

characteristic

C₄²×C₁₄₉

—

Dic₂₉₈

—

Dic₂₉₈

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

#11924746

—

C₂³×C₁₄₉

—

C₂³×C₁₄₉

—

C₂³×C₁₄₉

—

C₂³×C₁₄₉

—

C₂³×C₁₄₉

—

C₂³×C₁₄₉

C₂×D₄

yes

characteristic

D₄×C₁₄₉

—

D₄×C₁₄₉

—

D₄×C₁₄₉

—

D₄×C₁₄₉

—

D₄×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

SD₁₆

yes

complemented

C₂×C₄×C₁₄₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₄₉

C₂×D₄

yes

—

C₂×C₄×C₁₄₉

C₂×D₄

yes

—

C₂×C₄×C₁₄₉

#41

yes

characteristic

C₂²×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

yes

C₂²×C₁₄₉

yes

C₂²×C₁₄₉

yes

C₂²×C₁₄₉

C₂×SD₁₆

yes

—

C₂²×C₁₄₉

AGL(1,ℤ/8)

yes

—

C₄×C₁₄₉

—

C₄×C₁₄₉

#120

yes

—

C₄×C₁₄₉

#121

yes

—

Dic₁₄₉

—

Dic₁₄₉

—

C₂×C₁₄₉

—

C₂×C₁₄₉

#392

yes

characteristic

C₂×C₁₄₉

#450

yes

—

C₂×C₁₄₉

#460

yes

—

C₂×C₁₄₉

yes

C₂×C₁₄₉

yes

C₁₄₉

yes

characteristic, complemented, Sylow

#2014

—

149

maximal, Sylow

C₂²×SD₁₆

—

149

—

C₂×#127

D₁₄₉

yes

characteristic

C₂×#102

—

149

—

#391

—

149

—

#336

—

149

—

C₂²×Q₈

—

149

—

C₂²×D₄

—

C₂²×D₄

C₂×D₁₄₉

yes

—

C₂×SD₁₆

—

298

—

100

C₂×SD₁₆

—

298

—

101

C₂×SD₁₆

—

298

—

102

C₂²×C₈

—

149

—

103

#127

—

104

C₂×C₄⋊C₄

—

149

—

105

C₂×C₄²

C₂×D₁₄₉

yes

characteristic

106

#102

—

298

—

107

#95

—

298

—

108

C₂⁴

—

109

C₂×Q₈

—

298

—

110

C₂×Q₈

—

298

—

111

C₂×Q₈

—

298

—

112

C₂×Q₈

—

149

—

113

C₂×D₄

—

114

C₂×D₄

—

115

C₂×D₄

—

116

C₂×D₄

—

117

C₂×D₄

—

118

C₂×D₄

—

119

C₂×D₄

—

120

C₂×D₄

—

121

C₂×D₄

C₂²⋊D₁₄₉

yes

—

122

C₂²×C₄

—

298

—

123

C₂²×C₄

—

298

—

124

C₂²×C₄

C₂²⋊D₁₄₉

yes

—

125

C₂²×C₄

C₂²×D₁₄₉

yes

characteristic

126

SD₁₆

—

298

—

127

C₂×C₈

—

298

—

128

C₂×C₈

—

298

—

129

C₂×C₈

—

149

—

130

C₄⋊C₄

—

298

—

131

C₄²

—

132

C₂³

—

133

C₂³

—

134

C₂³

—

135

C₂³

—

136

C₂³

—

137

C₂³

C₂×C₂²⋊D₁₄₉

yes

characteristic

138

Q₈

—

298

—

139

Q₈

—

596

—

140

D₄

—

141

D₄

—

142

D₄

—

143

D₄

—

144

D₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

298

—

148

C₂×C₄

—

596

—

149

C₂×C₄

—

596

—

150

C₂×C₄

—

596

—

151

C₂×C₄

—

596

—

152

C₂×C₄

#14558724

yes

complemented

153

C₂×C₄

D₄×D₁₄₉

yes

characteristic

154

C₂×C₄

D₄×D₁₄₉

yes

—

155

C₂×C₄

C₂×C₂²⋊D₁₄₉

yes

—

156

C₂×C₄

#14558735

yes

characteristic

157

C₈

—

298

—

158

C₂²

—

159

C₂²

—

160

C₂²

—

161

C₂²

—

162

C₂²

—

163

C₂²

#18011880

yes

characteristic

164

C₂²

#18011882

yes

characteristic

165

C₂²

#18011907

yes

characteristic

166

C₂²

C₂×#14558724

yes

—

167

C₂²

#18011937

yes

—

168

C₄

—

169

C₄

—

596

—

170

C₄

—

1192

—

171

C₄

#18011879

yes

—

172

C₄

#18011883

yes

—

173

C₂

—

174

C₂

#23749533

yes

characteristic

175

C₂

#23749776

yes

—

176

C₂

#23749822

yes

—

177

C₂

#23749997

yes

characteristic

178

C₂

#23750012

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₄₉
commutator, cosocle	C₂²×C₄×C₁₄₉	C₂³
Frattini	C₂²×C₄	C₂²×D₁₄₉
Fitting, prime radical	C₂×#127×C₁₄₉	C₂
socle	C₂³×C₁₄₉	C₂×D₄
hypercenter	C₂×#127	D₁₄₉
hypocenter	C₁₄₉	#2014
LCS	G ▷ C₂²×C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#127

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₄₉

commutator, cosocle

C₂²×C₄×C₁₄₉

C₂³

Frattini

C₂²×C₄

C₂²×D₁₄₉

Fitting, prime radical

C₂×#127×C₁₄₉

C₂

socle

C₂³×C₁₄₉

C₂×D₄

D₁₄₉

C₁₄₉

G ▷ C₂²×C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#127

N	H ≅ G/N	type
1. C₂²×#14558724	C₂	semidirect
2. C₂×#18011832	C₂	semidirect
3. #23749531	C₂	semidirect
4. #23749380	C₂	semidirect
21. C₂×Dic₂₉₈	D₄	semidirect
27. C₂×#11924746	D₄	semidirect
63. C₂×C₄×C₁₄₉	SD₁₆	semidirect
89. C₁₄₉	#2014	semidirect
152. C₂×C₄	#14558724	semidirect

H ≅ G/N

type

1. C₂²×#14558724

C₂

semidirect

2. C₂×#18011832

C₂

semidirect

3. #23749531

C₂

semidirect

4. #23749380

C₂

semidirect

21. C₂×Dic₂₉₈

D₄

semidirect

27. C₂×#11924746

D₄

semidirect

63. C₂×C₄×C₁₄₉

SD₁₆

semidirect

89. C₁₄₉

#2014

semidirect

152. C₂×C₄

#14558724

semidirect

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

subgroup	index
C₂²×#14558724	2
C₂×#18011832	2
#23749531	2
#23749380	2
C₂×#127×C₁₄₉	2
#2014	149

subgroup

index

C₂×#127×C₁₄₉

149

Subgroups — #34045779

Conjugacy classes of subgroups

Complements

Maximal subgroups