Group #34045243 — subgroups

Subgroups — #34045243

Order 19072.

308 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 89 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	#23750196	C₂	1	1	yes	maximal, characteristic
2	#23749609	C₂	1	1	yes	maximal, characteristic
3	C₂×#18011829	C₂	1	2	yes	maximal, complemented
4	#23749397	C₂	1	2	yes	maximal, complemented
5	#568×C₁₄₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₉₈	C₂²	1	1	yes	characteristic
7	C₂×#14558720	C₂²	1	2	yes	—
8	#18011909	—	2	1	no	—
9	#18011898	—	2	1	no	—
10	#18011892	C₄	1	4	yes	—
11	#18011890	—	2	2	no	—
12	#18011870	C₂²	1	1	yes	characteristic
13	#18011830	—	2	1	no	—
14	#18011829	—	2	4	no	—
15	#18011828	—	2	1	no	—
16	C₂×#41×C₁₄₉	C₂²	1	1	yes	characteristic
17	#131×C₁₄₉	—	2	4	no	—
18	C₂×M₁₆×C₁₄₉	—	2	2	no	—
19	C₂×M₁₆×C₁₄₉	C₂²	1	2	yes	—
20	C₂×Dic₂₉₈	—	2	1	no	—
21	C₂×Dic₂₉₈	—	2	2	no	—
22	C₂×Dic₂₉₈	—	2	1	no	—
23	C₂×Dic₂₉₈	—	2	2	no	—
24	C₂²×Dic₁₄₉	C₂×C₄	1	2	yes	—
25	#14558720	—	2	2	no	—
26	#14558720	—	2	2	no	—
27	C₂×#11924746	—	2	2	no	—
28	C₄⋊Dic₁₄₉	—	2	1	no	—
29	#14558715	—	2	4	no	—
30	C₂²⋊Dic₁₄₉	—	2	4	no	—
31	C₂²⋊Dic₁₄₉	C₂×C₄	1	2	yes	—
32	C₄×Dic₁₄₉	—	2	1	no	—
33	#41×C₁₄₉	—	2	2	no	—
34	#41×C₁₄₉	—	2	2	no	—
35	C₂×Q₈×C₁₄₉	D₄	1	1	yes	characteristic
36	C₂×D₄×C₁₄₉	C₂×C₄	1	2	yes	—
37	C₂×D₄×C₁₄₉	D₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₄₉	D₄	1	2	yes	—
39	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
40	M₁₆×C₁₄₉	—	2	4	no	—
41	M₁₆×C₁₄₉	—	2	2	no	—
42	M₁₆×C₁₄₉	D₄	1	4	yes	—
43	C₂×C₈×C₁₄₉	—	2	4	no	—
44	C₂×C₈×C₁₄₉	—	2	2	no	—
45	Dic₂₉₈	—	4	1	no	—
46	Dic₂₉₈	—	4	2	no	—
47	Dic₂₉₈	—	4	1	no	—
48	C₂×Dic₁₄₉	—	2	2	no	—
49	C₂×Dic₁₄₉	—	2	2	no	—
50	C₂×Dic₁₄₉	—	2	4	no	—
51	#11924746	—	4	2	no	—
52	C₂³×C₁₄₉	C₂²⋊C₄	1	2	yes	—
53	C₂³×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
54	Q₈×C₁₄₉	—	2	2	no	—
55	D₄×C₁₄₉	—	2	4	no	—
56	D₄×C₁₄₉	—	2	2	no	—
57	C₂×C₄×C₁₄₉	—	2	2	no	—
58	C₂×C₄×C₁₄₉	—	2	2	no	—
59	C₂×C₄×C₁₄₉	C₂²⋊C₄	1	2	yes	—
60	C₂×C₄×C₁₄₉	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₁₄₉	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₁₄₉	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₁₄₉	#41	1	2	yes	—
64	C₈×C₁₄₉	—	2	4	no	—
65	C₂²×C₁₄₉	—	2	2	no	—
66	C₂²×C₁₄₉	—	4	1	no	—
67	C₂²×C₁₄₉	—	4	2	no	—
68	C₂²×C₁₄₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₁₄₉	C₄×D₄	1	2	yes	—
70	C₄×C₁₄₉	—	2	2	no	—
71	C₄×C₁₄₉	#120	1	1	yes	characteristic
72	C₄×C₁₄₉	#121	1	2	yes	—
73	C₄×C₁₄₉	#123	1	1	yes	characteristic
74	Dic₁₄₉	—	4	2	no	—
75	Dic₁₄₉	—	4	2	no	—
76	C₂×C₁₄₉	—	2	1	no	—
77	C₂×C₁₄₉	—	2	2	no	—
78	C₂×C₁₄₉	—	4	2	no	—
79	C₂×C₁₄₉	#386	1	1	yes	characteristic
80	C₁₄₉	#1915	1	1	yes	characteristic, complemented, Sylow
81	#1915	—	149	1	no	maximal, Sylow
82	#568	D₁₄₉	1	1	yes	characteristic
83	#536	—	149	1	no	—
84	#413	—	149	1	no	—
85	C₂×#101	—	149	2	no	—
86	#344	—	149	2	no	—
87	C₂×#41	C₂×D₁₄₉	1	1	yes	characteristic
88	C₂²×Q₈	—	149	1	no	—
89	#131	—	2	4	no	—
90	C₂×M₁₆	—	2	2	no	—
91	C₂×M₁₆	—	149	2	no	—
92	C₂×M₁₆	C₂×D₁₄₉	1	2	yes	—
93	C₄⋊Q₈	—	298	1	no	—
94	#124	—	298	1	no	—
95	#123	—	149	4	no	—
96	C₂²⋊Q₈	—	298	2	no	—
97	#117	—	149	1	no	—
98	#101	—	298	1	no	—
99	#101	—	298	4	no	—
100	#100	—	298	1	no	—
101	#41	—	2	2	no	—
102	#41	—	2	2	no	—
103	C₂×Q₈	—	298	2	no	—
104	C₂×Q₈	—	298	1	no	—
105	C₂×Q₈	—	298	1	no	—
106	C₂×Q₈	—	298	2	no	—
107	C₂×Q₈	D₅₉₆	1	1	yes	characteristic
108	C₂×D₄	C₄×D₁₄₉	1	2	yes	—
109	C₂×D₄	D₅₉₆	1	1	yes	characteristic
110	C₂²×C₄	—	149	2	no	—
111	C₂²×C₄	C₂²⋊D₁₄₉	1	2	yes	—
112	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
113	M₁₆	—	2	4	no	—
114	M₁₆	—	2	2	no	—
115	M₁₆	—	298	2	no	—
116	M₁₆	—	298	2	no	—
117	M₁₆	C₂²⋊D₁₄₉	1	4	yes	—
118	C₂×C₈	—	2	4	no	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	298	2	no	—
121	C₄⋊C₄	—	298	1	no	—
122	C₄⋊C₄	—	298	4	no	—
123	C₂²⋊C₄	—	298	4	no	—
124	C₂²⋊C₄	—	149	2	no	—
125	C₄²	—	298	1	no	—
126	C₂³	#14558713	1	2	yes	—
127	C₂³	C₂×C₄×D₁₄₉	1	1	yes	characteristic
128	Q₈	—	2	2	no	—
129	Q₈	—	596	1	no	—
130	Q₈	—	596	2	no	—
131	Q₈	—	596	1	no	—
132	D₄	—	2	4	no	—
133	D₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	298	2	no	—
137	C₂×C₄	—	298	2	no	—
138	C₂×C₄	—	298	4	no	—
139	C₂×C₄	#14558713	1	2	yes	—
140	C₂×C₄	C₂×C₂²⋊D₁₄₉	1	2	yes	—
141	C₂×C₄	C₂×C₂²⋊D₁₄₉	1	1	yes	characteristic
142	C₂×C₄	C₂×D₅₉₆	1	1	yes	characteristic
143	C₂×C₄	#14558736	1	2	yes	—
144	C₈	—	2	4	no	—
145	C₈	—	596	2	no	—
146	C₂²	—	2	2	no	—
147	C₂²	—	4	1	no	—
148	C₂²	—	4	2	no	—
149	C₂²	C₂×#14558713	1	1	yes	characteristic
150	C₂²	C₄×C₂²⋊D₁₄₉	1	2	yes	—
151	C₄	—	2	2	no	—
152	C₄	—	596	2	no	—
153	C₄	—	596	2	no	—
154	C₄	#18011880	1	1	yes	characteristic
155	C₄	#18011885	1	2	yes	—
156	C₄	#18011894	1	1	yes	characteristic
157	C₂	—	2	1	no	—
158	C₂	—	2	2	no	—
159	C₂	—	4	2	no	—
160	C₂	#23749504	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23750196

C₂

yes

maximal, characteristic

#23749609

C₂

yes

maximal, characteristic

C₂×#18011829

C₂

yes

maximal, complemented

#23749397

C₂

yes

maximal, complemented

#568×C₁₄₉

C₂

yes

maximal, characteristic

C₂²×Dic₂₉₈

C₂²

yes

characteristic

C₂×#14558720

C₂²

yes

—

#18011909

—

#18011898

—

#18011892

C₄

yes

—

#18011890

—

#18011870

C₂²

yes

—

—

—

C₂×#41×C₁₄₉

C₂²

yes

characteristic

#131×C₁₄₉

—

C₂×M₁₆×C₁₄₉

—

C₂×M₁₆×C₁₄₉

C₂²

yes

—

C₂×Dic₂₉₈

—

C₂×Dic₂₉₈

—

C₂×Dic₂₉₈

—

C₂×Dic₂₉₈

—

C₂²×Dic₁₄₉

C₂×C₄

yes

—

#14558720

—

#14558720

—

C₂×#11924746

—

C₄⋊Dic₁₄₉

—

#14558715

—

C₂²⋊Dic₁₄₉

—

C₂²⋊Dic₁₄₉

C₂×C₄

yes

—

C₄×Dic₁₄₉

—

#41×C₁₄₉

—

#41×C₁₄₉

—

C₂×Q₈×C₁₄₉

D₄

yes

characteristic

C₂×D₄×C₁₄₉

C₂×C₄

yes

—

C₂×D₄×C₁₄₉

D₄

yes

characteristic

C₂²×C₄×C₁₄₉

D₄

yes

—

C₂²×C₄×C₁₄₉

C₂³

yes

characteristic

M₁₆×C₁₄₉

—

M₁₆×C₁₄₉

—

M₁₆×C₁₄₉

D₄

yes

—

C₂×C₈×C₁₄₉

—

C₂×C₈×C₁₄₉

—

Dic₂₉₈

—

Dic₂₉₈

—

Dic₂₉₈

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

#11924746

—

C₂³×C₁₄₉

C₂²⋊C₄

yes

—

C₂³×C₁₄₉

C₂²×C₄

yes

characteristic

Q₈×C₁₄₉

—

D₄×C₁₄₉

—

D₄×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₄₉

C₂×D₄

yes

—

C₂×C₄×C₁₄₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₄₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₄₉

#41

yes

—

C₈×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₄₉

C₄×D₄

yes

—

C₄×C₁₄₉

—

C₄×C₁₄₉

#120

yes

characteristic

C₄×C₁₄₉

#121

yes

—

C₄×C₁₄₉

#123

yes

characteristic

Dic₁₄₉

—

Dic₁₄₉

—

C₂×C₁₄₉

—

C₂×C₁₄₉

—

C₂×C₁₄₉

—

C₂×C₁₄₉

#386

yes

characteristic

C₁₄₉

#1915

yes

characteristic, complemented, Sylow

#1915

—

149

maximal, Sylow

#568

D₁₄₉

yes

characteristic

#536

—

149

—

#413

—

149

—

C₂×#101

—

149

—

#344

—

149

—

C₂×#41

C₂×D₁₄₉

yes

characteristic

C₂²×Q₈

—

149

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

149

—

C₂×M₁₆

C₂×D₁₄₉

yes

—

C₄⋊Q₈

—

298

—

#124

—

298

—

#123

—

149

—

C₂²⋊Q₈

—

298

—

#117

—

149

—

#101

—

298

—

#101

—

298

—

100

#100

—

298

—

101

#41

—

102

#41

—

103

C₂×Q₈

—

298

—

104

C₂×Q₈

—

298

—

105

C₂×Q₈

—

298

—

106

C₂×Q₈

—

298

—

107

C₂×Q₈

D₅₉₆

yes

characteristic

108

C₂×D₄

C₄×D₁₄₉

yes

—

109

C₂×D₄

D₅₉₆

yes

characteristic

110

C₂²×C₄

—

149

—

111

C₂²×C₄

C₂²⋊D₁₄₉

yes

—

112

C₂²×C₄

C₂²×D₁₄₉

yes

characteristic

113

M₁₆

—

114

M₁₆

—

115

M₁₆

—

298

—

116

M₁₆

—

298

—

117

M₁₆

C₂²⋊D₁₄₉

yes

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

298

—

121

C₄⋊C₄

—

298

—

122

C₄⋊C₄

—

298

—

123

C₂²⋊C₄

—

298

—

124

C₂²⋊C₄

—

149

—

125

C₄²

—

298

—

126

C₂³

#14558713

yes

—

127

C₂³

C₂×C₄×D₁₄₉

yes

characteristic

128

Q₈

—

129

Q₈

—

596

—

130

Q₈

—

596

—

131

Q₈

—

596

—

132

D₄

—

133

D₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

298

—

137

C₂×C₄

—

298

—

138

C₂×C₄

—

298

—

139

C₂×C₄

#14558713

yes

—

140

C₂×C₄

C₂×C₂²⋊D₁₄₉

yes

—

141

C₂×C₄

C₂×C₂²⋊D₁₄₉

yes

characteristic

142

C₂×C₄

C₂×D₅₉₆

yes

characteristic

143

C₂×C₄

#14558736

yes

—

144

C₈

—

145

C₈

—

596

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

C₂×#14558713

yes

characteristic

150

C₂²

C₄×C₂²⋊D₁₄₉

yes

—

151

C₄

—

152

C₄

—

596

—

153

C₄

—

596

—

154

C₄

#18011880

yes

characteristic

155

C₄

#18011885

yes

—

156

C₄

#18011894

yes

characteristic

157

C₂

—

158

C₂

—

159

C₂

—

160

C₂

#23749504

yes

characteristic

	subgroup	G/H
center	C₂	#23749504
commutator	C₂³×C₁₄₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₄₉
Fitting	#568×C₁₄₉	C₂
socle	C₂×C₁₄₉	#386
prime radical	C₂×#41×C₁₄₉	C₂²
cosocle	C₂²×C₄×C₁₄₉	C₂³
hypercenter	#568	D₁₄₉
hypocenter	C₁₄₉	#1915
LCS	G ▷ C₂³×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #568

subgroup

G/H

center

C₂

#23749504

commutator

C₂³×C₁₄₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₄₉

Fitting

#568×C₁₄₉

C₂

socle

C₂×C₁₄₉

#386

prime radical

C₂×#41×C₁₄₉

C₂²

cosocle

C₂²×C₄×C₁₄₉

C₂³

D₁₄₉

C₁₄₉

G ▷ C₂³×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #568

N	H ≅ G/N	type
3. C₂×#18011829	C₂	semidirect
4. #23749397	C₂	semidirect
80. C₁₄₉	#1915	semidirect

H ≅ G/N

type

3. C₂×#18011829

C₂

semidirect

4. #23749397

C₂

semidirect

80. C₁₄₉

#1915

semidirect

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

subgroup	index
#23750196	2
#23749609	2
C₂×#18011829	2
#23749397	2
#568×C₁₄₉	2
#1915	149

subgroup

index

#568×C₁₄₉

149

Subgroups — #34045243

Conjugacy classes of subgroups

Complements

Maximal subgroups