Group #34046120 — subgroups

Subgroups — #34046120

Order 19072.

256 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 81 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×#18011823	C₂	1	1	yes	maximal, characteristic
2	#23749482	C₂	1	2	yes	maximal
3	#23749388	C₂	1	1	yes	maximal, characteristic, complemented
4	#23749387	C₂	1	2	yes	maximal, complemented
5	C₂×#107×C₁₄₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×D₁₄₉	C₂²	1	1	yes	characteristic
7	C₂²×#11924746	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₁₄₉	C₂²	1	2	yes	—
9	C₂×#14558713	—	2	2	no	—
10	#18011823	—	2	2	no	—
11	#18011818	—	2	2	no	—
12	C₂²×C₈×C₁₄₉	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₁₄₉	C₂²	1	2	yes	—
14	#107×C₁₄₉	—	2	2	no	—
15	C₂³×D₁₄₉	D₄	1	1	yes	characteristic
16	C₂²×Dic₁₄₉	—	2	2	no	—
17	C₂²×Dic₁₄₉	D₄	1	1	yes	characteristic
18	C₂×C₄×D₁₄₉	—	2	1	no	—
19	C₂×C₄×D₁₄₉	—	2	1	no	—
20	C₂×C₄×D₁₄₉	—	2	2	no	—
21	C₂×C₄×D₁₄₉	—	2	2	no	—
22	C₂×#11924746	—	2	1	no	—
23	C₂×#11924746	—	2	2	no	—
24	C₄⋊Dic₁₄₉	—	2	2	no	—
25	C₄⋊Dic₁₄₉	—	2	2	no	—
26	#14558713	—	4	4	no	—
27	C₂²×C₄×C₁₄₉	—	2	2	no	—
28	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
29	C₂×C₈×C₁₄₉	—	2	2	no	—
30	C₂×C₈×C₁₄₉	D₄	1	2	yes	—
31	C₄⋊C₄×C₁₄₉	—	2	2	no	—
32	C₄⋊C₄×C₁₄₉	—	2	2	no	—
33	C₂²×D₁₄₉	—	2	1	no	—
34	C₂²×D₁₄₉	—	2	2	no	—
35	C₂²×D₁₄₉	—	2	1	no	—
36	C₂²×D₁₄₉	—	2	2	no	—
37	C₂²×D₁₄₉	Q₁₆	1	2	yes	—
38	C₂×Dic₁₄₉	—	2	1	no	—
39	C₂×Dic₁₄₉	—	2	2	no	—
40	C₂×Dic₁₄₉	—	4	2	no	—
41	C₂×Dic₁₄₉	—	4	2	no	—
42	C₂×Dic₁₄₉	—	4	2	no	—
43	C₄×D₁₄₉	—	4	1	no	—
44	C₄×D₁₄₉	—	4	1	no	—
45	C₄×D₁₄₉	—	4	2	no	—
46	#11924746	—	4	1	no	—
47	C₂³×C₁₄₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₁₄₉	—	4	2	no	—
49	C₂×C₄×C₁₄₉	—	4	2	no	—
50	C₂×C₄×C₁₄₉	—	4	2	no	—
51	C₂×C₄×C₁₄₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₁₄₉	#41	1	1	yes	characteristic
53	C₂×C₄×C₁₄₉	#41	1	2	yes	—
54	C₂×C₄×C₁₄₉	#41	1	2	yes	—
55	C₈×C₁₄₉	—	2	2	no	—
56	C₂²×C₁₄₉	#121	1	1	yes	characteristic
57	C₂²×C₁₄₉	#123	1	2	yes	—
58	C₂²×C₁₄₉	C₂×Q₁₆	1	1	yes	characteristic
59	C₂²×C₁₄₉	#135	1	1	yes	characteristic
60	C₂²×C₁₄₉	AGL(1,ℤ/8)	1	2	yes	—
61	C₂×D₁₄₉	—	2	2	no	—
62	C₂×D₁₄₉	—	2	4	no	—
63	C₂×D₁₄₉	—	4	1	no	—
64	C₂×D₁₄₉	—	4	1	no	—
65	C₂×D₁₄₉	—	4	2	no	—
66	C₄×C₁₄₉	—	8	2	no	—
67	C₄×C₁₄₉	#123	1	1	yes	characteristic
68	C₄×C₁₄₉	#124	1	1	yes	characteristic
69	C₄×C₁₄₉	#126	1	2	yes	—
70	Dic₁₄₉	—	4	1	no	—
71	Dic₁₄₉	—	8	2	no	—
72	C₂×C₁₄₉	#397	1	1	yes	characteristic
73	C₂×C₁₄₉	#467	1	1	yes	characteristic
74	C₂×C₁₄₉	#469	1	1	yes	characteristic
75	C₂×C₁₄₉	#482	1	2	yes	—
76	C₂×C₁₄₉	#484	1	2	yes	—
77	D₁₄₉	—	4	2	no	—
78	C₁₄₉	#2076	1	1	yes	characteristic, complemented, Sylow
79	#2076	—	149	1	no	maximal, Sylow
80	C₂×#107	D₁₄₉	1	1	yes	characteristic, complemented
81	C₂×C₂²⋊C₈	—	149	1	no	—
82	#380	—	149	2	no	—
83	#340	—	149	1	no	—
84	#340	—	149	2	no	—
85	C₂³×C₄	—	149	1	no	—
86	C₂²×C₈	—	149	1	no	—
87	C₂²×C₈	C₂×D₁₄₉	1	1	yes	characteristic
88	C₂×C₄⋊C₄	—	149	2	no	—
89	C₂×C₄⋊C₄	C₂×D₁₄₉	1	2	yes	—
90	C₂×C₂²⋊C₄	—	298	2	no	—
91	#107	—	2	2	no	—
92	C₂²⋊C₈	—	298	2	no	—
93	#95	—	298	2	no	—
94	C₂⁴	—	149	1	no	—
95	C₂²×C₄	—	2	2	no	—
96	C₂²×C₄	—	298	1	no	—
97	C₂²×C₄	—	298	1	no	—
98	C₂²×C₄	—	298	2	no	—
99	C₂²×C₄	—	298	2	no	—
100	C₂²×C₄	—	298	2	no	—
101	C₂²×C₄	—	149	1	no	—
102	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
103	C₂×C₈	—	2	2	no	—
104	C₂×C₈	—	298	1	no	—
105	C₂×C₈	—	298	2	no	—
106	C₂×C₈	C₂²⋊D₁₄₉	1	2	yes	—
107	C₄⋊C₄	—	2	2	no	—
108	C₄⋊C₄	—	2	2	no	—
109	C₄⋊C₄	—	298	2	no	—
110	C₄⋊C₄	—	298	2	no	—
111	C₂²⋊C₄	—	596	4	no	—
112	C₂³	—	298	1	no	—
113	C₂³	—	298	2	no	—
114	C₂³	—	298	1	no	—
115	C₂³	—	298	2	no	—
116	C₂³	—	149	2	no	—
117	C₂³	D₄×D₁₄₉	1	1	yes	characteristic
118	C₂×C₄	—	4	2	no	—
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	—	4	2	no	—
121	C₂×C₄	—	298	1	no	—
122	C₂×C₄	—	298	2	no	—
123	C₂×C₄	—	596	2	no	—
124	C₂×C₄	—	596	1	no	—
125	C₂×C₄	—	596	1	no	—
126	C₂×C₄	—	596	2	no	—
127	C₂×C₄	—	596	2	no	—
128	C₂×C₄	—	596	2	no	—
129	C₂×C₄	C₂×C₂²⋊D₁₄₉	1	1	yes	characteristic
130	C₂×C₄	#14558735	1	1	yes	characteristic
131	C₂×C₄	#14558736	1	2	yes	—
132	C₂×C₄	#14558737	1	2	yes	—
133	C₈	—	2	2	no	—
134	C₈	—	596	1	no	—
135	C₂²	—	298	4	no	—
136	C₂²	—	298	2	no	—
137	C₂²	—	596	2	no	—
138	C₂²	—	596	1	no	—
139	C₂²	—	596	1	no	—
140	C₂²	#18011882	1	1	yes	characteristic
141	C₂²	#18011896	1	2	yes	—
142	C₂²	Q₁₆×D₁₄₉	1	1	yes	characteristic
143	C₂²	#18011934	1	1	yes	characteristic
144	C₂²	#18011935	1	2	yes	—
145	C₄	—	8	2	no	—
146	C₄	—	596	1	no	—
147	C₄	—	1192	2	no	—
148	C₄	#18011894	1	1	yes	characteristic
149	C₄	#18011900	1	1	yes	characteristic
150	C₄	#18011906	1	2	yes	—
151	C₂	—	596	2	no	—
152	C₂	#23749551	1	1	yes	characteristic
153	C₂	#23749863	1	1	yes	characteristic
154	C₂	#23749875	1	1	yes	characteristic
155	C₂	#23749952	1	2	yes	—
156	C₂	#23749962	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#18011823

C₂

yes

maximal, characteristic

#23749482

C₂

yes

maximal

#23749388

C₂

yes

maximal, characteristic, complemented

#23749387

C₂

yes

maximal, complemented

C₂×#107×C₁₄₉

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₁₄₉

C₂²

yes

characteristic

C₂²×#11924746

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₄₉

C₂²

yes

—

C₂×#14558713

—

#18011823

—

#18011818

—

C₂²×C₈×C₁₄₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₄₉

C₂²

yes

—

#107×C₁₄₉

—

C₂³×D₁₄₉

D₄

yes

characteristic

C₂²×Dic₁₄₉

—

C₂²×Dic₁₄₉

D₄

yes

characteristic

C₂×C₄×D₁₄₉

—

C₂×C₄×D₁₄₉

—

C₂×C₄×D₁₄₉

—

C₂×C₄×D₁₄₉

—

C₂×#11924746

—

C₂×#11924746

—

C₄⋊Dic₁₄₉

—

C₄⋊Dic₁₄₉

—

#14558713

—

C₂²×C₄×C₁₄₉

—

C₂²×C₄×C₁₄₉

C₂³

yes

characteristic

C₂×C₈×C₁₄₉

—

C₂×C₈×C₁₄₉

D₄

yes

—

C₄⋊C₄×C₁₄₉

—

C₄⋊C₄×C₁₄₉

—

C₂²×D₁₄₉

—

C₂²×D₁₄₉

—

C₂²×D₁₄₉

—

C₂²×D₁₄₉

—

C₂²×D₁₄₉

Q₁₆

yes

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₄×D₁₄₉

—

C₄×D₁₄₉

—

C₄×D₁₄₉

—

#11924746

—

C₂³×C₁₄₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

C₂×D₄

yes

C₂×C₄×C₁₄₉

yes

C₂×C₄×C₁₄₉

yes

—

C₂×C₄×C₁₄₉

#41

yes

—

C₈×C₁₄₉

—

C₂²×C₁₄₉

#121

yes

characteristic

C₂²×C₁₄₉

#123

yes

—

C₂²×C₁₄₉

C₂×Q₁₆

yes

C₂²×C₁₄₉

yes

C₂²×C₁₄₉

yes

—

C₂×D₁₄₉

—

C₂×D₁₄₉

—

C₂×D₁₄₉

—

C₂×D₁₄₉

—

C₂×D₁₄₉

—

C₄×C₁₄₉

—

C₄×C₁₄₉

yes

C₄×C₁₄₉

yes

C₄×C₁₄₉

yes

—

Dic₁₄₉

—

Dic₁₄₉

—

C₂×C₁₄₉

yes

C₂×C₁₄₉

yes

C₂×C₁₄₉

yes

C₂×C₁₄₉

yes

—

C₂×C₁₄₉

#484

yes

—

D₁₄₉

—

C₁₄₉

#2076

yes

characteristic, complemented, Sylow

#2076

—

149

maximal, Sylow

C₂×#107

D₁₄₉

yes

characteristic, complemented

C₂×C₂²⋊C₈

—

149

—

#380

—

149

—

#340

—

149

—

#340

—

149

—

C₂³×C₄

—

149

—

C₂²×C₈

—

149

—

C₂²×C₈

C₂×D₁₄₉

yes

characteristic

C₂×C₄⋊C₄

—

149

—

C₂×C₄⋊C₄

C₂×D₁₄₉

yes

—

C₂×C₂²⋊C₄

—

298

—

#107

—

C₂²⋊C₈

—

298

—

#95

—

298

—

C₂⁴

—

149

—

C₂²×C₄

—

C₂²×C₄

—

298

—

C₂²×C₄

—

298

—

C₂²×C₄

—

298

—

C₂²×C₄

—

298

—

100

C₂²×C₄

—

298

—

101

C₂²×C₄

—

149

—

102

C₂²×C₄

C₂²×D₁₄₉

yes

characteristic

103

C₂×C₈

—

104

C₂×C₈

—

298

—

105

C₂×C₈

—

298

—

106

C₂×C₈

C₂²⋊D₁₄₉

yes

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

298

—

110

C₄⋊C₄

—

298

—

111

C₂²⋊C₄

—

596

—

112

C₂³

—

298

—

113

C₂³

—

298

—

114

C₂³

—

298

—

115

C₂³

—

298

—

116

C₂³

—

149

—

117

C₂³

D₄×D₁₄₉

yes

characteristic

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

298

—

122

C₂×C₄

—

298

—

123

C₂×C₄

—

596

—

124

C₂×C₄

—

596

—

125

C₂×C₄

—

596

—

126

C₂×C₄

—

596

—

127

C₂×C₄

—

596

—

128

C₂×C₄

—

596

—

129

C₂×C₄

C₂×C₂²⋊D₁₄₉

yes

characteristic

130

C₂×C₄

#14558735

yes

characteristic

131

C₂×C₄

#14558736

yes

—

132

C₂×C₄

#14558737

yes

—

133

C₈

—

134

C₈

—

596

—

135

C₂²

—

298

—

136

C₂²

—

298

—

137

C₂²

—

596

—

138

C₂²

—

596

—

139

C₂²

—

596

—

140

C₂²

#18011882

yes

characteristic

141

C₂²

#18011896

yes

—

142

C₂²

Q₁₆×D₁₄₉

yes

characteristic

143

C₂²

#18011934

yes

characteristic

144

C₂²

#18011935

yes

—

145

C₄

—

146

C₄

—

596

—

147

C₄

—

1192

—

148

C₄

#18011894

yes

characteristic

149

C₄

#18011900

yes

characteristic

150

C₄

#18011906

yes

—

151

C₂

—

596

—

152

C₂

#23749551

yes

characteristic

153

C₂

#23749863

yes

characteristic

154

C₂

#23749875

yes

characteristic

155

C₂

#23749952

yes

—

156

C₂

#23749962

yes

—

	subgroup	G/H
center	C₂³	D₄×D₁₄₉
commutator, cosocle	C₂²×C₄×C₁₄₉	C₂³
Frattini	C₂²×C₄	C₂²×D₁₄₉
Fitting	C₂×#107×C₁₄₉	C₂
socle	C₂³×C₁₄₉	C₂×D₄
prime radical	C₂³×D₁₄₉	D₄
hypercenter	C₂×#107	D₁₄₉
hypocenter	C₁₄₉	#2076
LCS	G ▷ C₂²×C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

subgroup

G/H

center

C₂³

D₄×D₁₄₉

commutator, cosocle

C₂²×C₄×C₁₄₉

C₂³

Frattini

C₂²×C₄

C₂²×D₁₄₉

Fitting

C₂×#107×C₁₄₉

C₂

socle

C₂³×C₁₄₉

C₂×D₄

prime radical

C₂³×D₁₄₉

D₄

D₁₄₉

C₁₄₉

G ▷ C₂²×C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

N	H ≅ G/N	type
3. #23749388	C₂	semidirect
4. #23749387	C₂	semidirect
5. C₂×#107×C₁₄₉	C₂	semidirect
78. C₁₄₉	#2076	semidirect
80. C₂×#107	D₁₄₉	semidirect

H ≅ G/N

type

3. #23749388

C₂

semidirect

4. #23749387

C₂

semidirect

5. C₂×#107×C₁₄₉

C₂

semidirect

78. C₁₄₉

#2076

semidirect

80. C₂×#107

D₁₄₉

semidirect

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

subgroup	index
C₂×#18011823	2
#23749482	2
#23749388	2
#23749387	2
C₂×#107×C₁₄₉	2
#2076	149

subgroup

index

C₂×#107×C₁₄₉

149

Subgroups — #34046120

Conjugacy classes of subgroups

Complements

Maximal subgroups