Group #34046015 — subgroups

Subgroups — #34046015

Order 19072.

288 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 85 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×Dic₅₉₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#18011840	C₂	1	1	yes	maximal, characteristic
3	C₂×#18011834	C₂	1	2	yes	maximal
4	#23749530	C₂	1	2	yes	maximal
5	#340×C₁₄₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₉₈	C₂²	1	2	yes	—
7	C₂×Dic₅₉₆	—	2	2	no	—
8	C₂×Dic₅₉₆	—	2	4	no	—
9	C₂²×#11924746	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₁₄₉	C₂²	1	1	yes	characteristic
11	#18011840	—	2	2	no	—
12	#18011834	—	2	4	no	—
13	#18011818	—	2	2	no	—
14	#18011818	—	2	2	no	—
15	C₂²×C₈×C₁₄₉	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₁₄₉	C₂²	1	2	yes	—
17	C₂×Dic₂₉₈	—	2	2	no	—
18	C₂×Dic₂₉₈	—	2	2	no	—
19	C₂×Dic₂₉₈	—	2	2	no	—
20	C₂×Dic₂₉₈	—	2	2	no	—
21	C₂×Dic₂₉₈	—	2	2	no	—
22	C₂×Dic₂₉₈	D₄	1	4	yes	—
23	C₂²×Dic₁₄₉	—	2	2	no	—
24	C₂²×Dic₁₄₉	D₄	1	1	yes	characteristic
25	C₂²×Dic₁₄₉	D₄	1	1	yes	characteristic
26	Dic₅₉₆	—	2	8	no	—
27	C₂×#11924746	—	2	2	no	—
28	C₂×#11924746	—	2	1	no	—
29	C₄×Dic₁₄₉	—	2	1	no	—
30	C₄×Dic₁₄₉	—	2	1	no	—
31	C₂²×C₄×C₁₄₉	—	2	2	no	—
32	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
33	C₂×C₈×C₁₄₉	—	2	2	no	—
34	C₂×C₈×C₁₄₉	D₄	1	2	yes	—
35	C₄⋊C₄×C₁₄₉	—	2	2	no	—
36	C₄⋊C₄×C₁₄₉	—	2	2	no	—
37	Dic₂₉₈	—	2	4	no	—
38	Dic₂₉₈	—	2	4	no	—
39	Dic₂₉₈	—	4	2	no	—
40	Dic₂₉₈	—	4	2	no	—
41	C₂×Dic₁₄₉	—	2	1	no	—
42	C₂×Dic₁₄₉	—	2	1	no	—
43	C₂×Dic₁₄₉	—	2	4	no	—
44	C₂×Dic₁₄₉	—	2	1	no	—
45	C₂×Dic₁₄₉	—	2	2	no	—
46	C₂×Dic₁₄₉	—	4	2	no	—
47	C₂×Dic₁₄₉	—	4	2	no	—
48	C₂×Dic₁₄₉	Q₁₆	1	2	yes	complemented
49	#11924746	—	4	1	no	—
50	C₂³×C₁₄₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₁₄₉	—	4	2	no	—
52	C₂×C₄×C₁₄₉	—	4	2	no	—
53	C₂×C₄×C₁₄₉	—	4	2	no	—
54	C₂×C₄×C₁₄₉	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₁₄₉	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₁₄₉	#41	1	2	yes	—
57	C₂×C₄×C₁₄₉	#41	1	1	yes	characteristic
58	C₈×C₁₄₉	—	2	2	no	—
59	C₂²×C₁₄₉	#121	1	2	yes	—
60	C₂²×C₁₄₉	#121	1	1	yes	characteristic
61	C₂²×C₁₄₉	C₂×Q₁₆	1	1	yes	characteristic
62	C₂²×C₁₄₉	#135	1	1	yes	characteristic
63	C₂²×C₁₄₉	#137	1	2	yes	—
64	C₄×C₁₄₉	—	8	2	no	—
65	C₄×C₁₄₉	#120	1	1	yes	characteristic
66	C₄×C₁₄₉	#123	1	2	yes	—
67	C₄×C₁₄₉	#124	1	1	yes	characteristic
68	Dic₁₄₉	—	2	2	no	—
69	Dic₁₄₉	—	4	4	no	—
70	Dic₁₄₉	—	4	1	no	—
71	C₂×C₁₄₉	#394	1	1	yes	characteristic
72	C₂×C₁₄₉	#462	1	2	yes	—
73	C₂×C₁₄₉	#464	1	2	yes	—
74	C₂×C₁₄₉	#467	1	1	yes	characteristic
75	C₂×C₁₄₉	#470	1	1	yes	characteristic
76	C₁₄₉	#2056	1	1	yes	characteristic, complemented, Sylow
77	#2056	—	149	1	no	maximal, Sylow
78	C₂²×Q₁₆	—	149	1	no	—
79	C₂×C₄⋊C₈	—	149	1	no	—
80	C₂×#103	—	149	2	no	—
81	#391	—	149	2	no	—
82	#340	D₁₄₉	1	1	yes	characteristic
83	C₂²×Q₈	—	149	2	no	—
84	C₂×Q₁₆	—	298	4	no	—
85	C₂×Q₁₆	—	298	2	no	—
86	C₂²×C₈	—	149	1	no	—
87	C₂²×C₈	C₂×D₁₄₉	1	1	yes	characteristic
88	C₂×C₄⋊C₄	C₂×D₁₄₉	1	2	yes	—
89	C₂×C₄²	—	149	1	no	—
90	C₄⋊C₈	—	298	2	no	—
91	#103	—	298	4	no	—
92	#95	—	298	2	no	—
93	#95	—	298	2	no	—
94	C₂×Q₈	—	298	2	no	—
95	C₂×Q₈	—	298	2	no	—
96	C₂×Q₈	—	298	2	no	—
97	C₂×Q₈	—	298	2	no	—
98	C₂×Q₈	—	298	2	no	—
99	C₂×Q₈	—	149	4	no	—
100	C₂²×C₄	—	2	2	no	—
101	C₂²×C₄	—	298	2	no	—
102	C₂²×C₄	—	149	1	no	—
103	C₂²×C₄	—	149	1	no	—
104	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
105	Q₁₆	—	298	8	no	—
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	298	2	no	—
108	C₂×C₈	—	298	1	no	—
109	C₂×C₈	D₅₉₆	1	2	yes	—
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	—	2	2	no	—
112	C₄²	—	298	1	no	—
113	C₄²	—	298	1	no	—
114	C₂³	D₄×D₁₄₉	1	1	yes	characteristic
115	Q₈	—	298	4	no	—
116	Q₈	—	298	4	no	—
117	Q₈	—	596	2	no	—
118	Q₈	—	596	2	no	—
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	—	4	2	no	—
121	C₂×C₄	—	4	2	no	—
122	C₂×C₄	—	298	1	no	—
123	C₂×C₄	—	298	1	no	—
124	C₂×C₄	—	298	2	no	—
125	C₂×C₄	—	298	4	no	—
126	C₂×C₄	—	298	1	no	—
127	C₂×C₄	—	596	2	no	—
128	C₂×C₄	—	596	2	no	—
129	C₂×C₄	—	149	2	no	—
130	C₂×C₄	D₄×D₁₄₉	1	2	yes	—
131	C₂×C₄	C₂×D₅₉₆	1	1	yes	characteristic
132	C₂×C₄	#14558736	1	2	yes	—
133	C₂×C₄	#14558737	1	1	yes	characteristic
134	C₈	—	2	2	no	—
135	C₈	—	596	1	no	—
136	C₂²	#18011884	1	2	yes	—
137	C₂²	#18011886	1	1	yes	characteristic
138	C₂²	Q₁₆×D₁₄₉	1	1	yes	characteristic
139	C₂²	#18011930	1	1	yes	characteristic
140	C₂²	#18011940	1	2	yes	—
141	C₄	—	8	2	no	—
142	C₄	—	298	2	no	—
143	C₄	—	596	4	no	—
144	C₄	—	596	1	no	—
145	C₄	#18011881	1	1	yes	characteristic
146	C₄	#18011896	1	2	yes	—
147	C₄	#18011899	1	1	yes	characteristic
148	C₂	#23749543	1	1	yes	characteristic
149	C₂	#23749834	1	2	yes	—
150	C₂	#23749846	1	2	yes	—
151	C₂	#23749867	1	1	yes	characteristic
152	C₂	#23749884	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₅₉₆

C₂

yes

maximal, characteristic

C₂×#18011840

C₂

yes

maximal, characteristic

C₂×#18011834

C₂

yes

maximal

#23749530

C₂

yes

maximal

#340×C₁₄₉

C₂

yes

maximal, characteristic

C₂²×Dic₂₉₈

C₂²

yes

—

C₂×Dic₅₉₆

—

C₂×Dic₅₉₆

—

C₂²×#11924746

C₂²

yes

characteristic

C₂×C₄×Dic₁₄₉

C₂²

yes

—

—

—

—

C₂²×C₈×C₁₄₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₄₉

C₂²

yes

—

C₂×Dic₂₉₈

—

C₂×Dic₂₉₈

—

C₂×Dic₂₉₈

—

C₂×Dic₂₉₈

—

C₂×Dic₂₉₈

—

C₂×Dic₂₉₈

D₄

yes

—

C₂²×Dic₁₄₉

—

C₂²×Dic₁₄₉

D₄

yes

characteristic

C₂²×Dic₁₄₉

D₄

yes

characteristic

Dic₅₉₆

—

C₂×#11924746

—

C₂×#11924746

—

C₄×Dic₁₄₉

—

C₄×Dic₁₄₉

—

C₂²×C₄×C₁₄₉

—

C₂²×C₄×C₁₄₉

C₂³

yes

characteristic

C₂×C₈×C₁₄₉

—

C₂×C₈×C₁₄₉

D₄

yes

—

C₄⋊C₄×C₁₄₉

—

C₄⋊C₄×C₁₄₉

—

Dic₂₉₈

—

Dic₂₉₈

—

Dic₂₉₈

—

Dic₂₉₈

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

Q₁₆

yes

complemented

#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

C₂×D₄

yes

characteristic

C₂×C₄×C₁₄₉

#41

yes

—

C₂×C₄×C₁₄₉

#41

yes

characteristic

C₈×C₁₄₉

—

C₂²×C₁₄₉

#121

yes

—

C₂²×C₁₄₉

#121

yes

characteristic

C₂²×C₁₄₉

C₂×Q₁₆

yes

C₂²×C₁₄₉

yes

C₂²×C₁₄₉

yes

—

C₄×C₁₄₉

—

C₄×C₁₄₉

#120

yes

characteristic

C₄×C₁₄₉

#123

yes

—

C₄×C₁₄₉

#124

yes

characteristic

Dic₁₄₉

—

Dic₁₄₉

—

Dic₁₄₉

—

C₂×C₁₄₉

#394

yes

characteristic

C₂×C₁₄₉

#462

yes

—

C₂×C₁₄₉

#464

yes

—

C₂×C₁₄₉

yes

C₂×C₁₄₉

yes

C₁₄₉

yes

characteristic, complemented, Sylow

#2056

—

149

maximal, Sylow

C₂²×Q₁₆

—

149

—

C₂×C₄⋊C₈

—

149

—

C₂×#103

—

149

—

#391

—

149

—

#340

D₁₄₉

yes

characteristic

C₂²×Q₈

—

149

—

C₂×Q₁₆

—

298

—

C₂×Q₁₆

—

298

—

C₂²×C₈

—

149

—

C₂²×C₈

C₂×D₁₄₉

yes

characteristic

C₂×C₄⋊C₄

C₂×D₁₄₉

yes

—

C₂×C₄²

—

149

—

C₄⋊C₈

—

298

—

#103

—

298

—

#95

—

298

—

#95

—

298

—

C₂×Q₈

—

298

—

C₂×Q₈

—

298

—

C₂×Q₈

—

298

—

C₂×Q₈

—

298

—

C₂×Q₈

—

298

—

C₂×Q₈

—

149

—

100

C₂²×C₄

—

101

C₂²×C₄

—

298

—

102

C₂²×C₄

—

149

—

103

C₂²×C₄

—

149

—

104

C₂²×C₄

C₂²×D₁₄₉

yes

characteristic

105

Q₁₆

—

298

—

106

C₂×C₈

—

107

C₂×C₈

—

298

—

108

C₂×C₈

—

298

—

109

C₂×C₈

D₅₉₆

yes

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄²

—

298

—

113

C₄²

—

298

—

114

C₂³

D₄×D₁₄₉

yes

characteristic

115

Q₈

—

298

—

116

Q₈

—

298

—

117

Q₈

—

596

—

118

Q₈

—

596

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

298

—

123

C₂×C₄

—

298

—

124

C₂×C₄

—

298

—

125

C₂×C₄

—

298

—

126

C₂×C₄

—

298

—

127

C₂×C₄

—

596

—

128

C₂×C₄

—

596

—

129

C₂×C₄

—

149

—

130

C₂×C₄

D₄×D₁₄₉

yes

—

131

C₂×C₄

C₂×D₅₉₆

yes

characteristic

132

C₂×C₄

#14558736

yes

—

133

C₂×C₄

#14558737

yes

characteristic

134

C₈

—

135

C₈

—

596

—

136

C₂²

#18011884

yes

—

137

C₂²

#18011886

yes

characteristic

138

C₂²

Q₁₆×D₁₄₉

yes

characteristic

139

C₂²

#18011930

yes

characteristic

140

C₂²

#18011940

yes

—

141

C₄

—

142

C₄

—

298

—

143

C₄

—

596

—

144

C₄

—

596

—

145

C₄

#18011881

yes

characteristic

146

C₄

#18011896

yes

—

147

C₄

#18011899

yes

characteristic

148

C₂

#23749543

yes

characteristic

149

C₂

#23749834

yes

—

150

C₂

#23749846

yes

—

151

C₂

#23749867

yes

characteristic

152

C₂

#23749884

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₁₄₉
commutator, cosocle	C₂²×C₄×C₁₄₉	C₂³
Frattini	C₂²×C₄	C₂²×D₁₄₉
Fitting	#340×C₁₄₉	C₂
socle, prime radical	C₂³×C₁₄₉	C₂×D₄
hypercenter	#340	D₁₄₉
hypocenter	C₁₄₉	#2056
LCS	G ▷ C₂²×C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

subgroup

G/H

center

C₂³

D₄×D₁₄₉

commutator, cosocle

C₂²×C₄×C₁₄₉

C₂³

Frattini

C₂²×C₄

C₂²×D₁₄₉

Fitting

#340×C₁₄₉

C₂

socle, prime radical

C₂³×C₁₄₉

C₂×D₄

D₁₄₉

C₁₄₉

G ▷ C₂²×C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

N	H ≅ G/N	type
48. C₂×Dic₁₄₉	Q₁₆	semidirect
76. C₁₄₉	#2056	semidirect

H ≅ G/N

type

48. C₂×Dic₁₄₉

Q₁₆

semidirect

76. C₁₄₉

#2056

semidirect

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

subgroup	index
C₂²×Dic₅₉₆	2
C₂×#18011840	2
C₂×#18011834	2
#23749530	2
#340×C₁₄₉	2
#2056	149

subgroup

index

C₂²×Dic₅₉₆

#340×C₁₄₉

149

Subgroups — #34046015

Conjugacy classes of subgroups

Complements

Maximal subgroups