Group #34045540 — subgroups

Subgroups — #34045540

Order 19072.

312 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 93 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×#18011899	C₂	1	1	yes	maximal, characteristic
2	#23749649	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#18011829	C₂	1	2	yes	maximal, complemented
4	C₂×#18011827	C₂	1	2	yes	maximal
5	C₄×M₁₆×C₁₄₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×Dic₂₉₈	C₂²	1	1	yes	characteristic
7	C₂²×D₅₉₆	C₂²	1	1	yes	characteristic
8	C₂×#14558720	C₂²	1	2	yes	—
9	#18011899	—	2	1	no	—
10	#18011899	—	2	1	no	—
11	#18011899	—	2	2	no	—
12	C₂×#14558713	C₄	1	4	yes	—
13	C₄⋊#11924746	—	2	2	no	—
14	#18011829	—	2	4	no	—
15	#18011827	—	2	4	no	—
16	C₂×M₁₆×C₁₄₉	C₂²	1	2	yes	—
17	C₂×C₄²×C₁₄₉	C₂²	1	1	yes	characteristic
18	#97×C₁₄₉	—	2	1	no	—
19	C₄×C₈×C₁₄₉	—	2	1	no	—
20	C₂³×D₁₄₉	C₂×C₄	1	2	yes	—
21	C₂×Dic₂₉₈	—	2	2	no	—
22	C₂×Dic₂₉₈	—	2	1	no	—
23	C₂×Dic₂₉₈	—	2	1	no	—
24	C₂×Dic₂₉₈	—	2	2	no	—
25	C₂×D₅₉₆	—	2	2	no	—
26	C₂×D₅₉₆	—	2	1	no	—
27	C₂×D₅₉₆	—	2	1	no	—
28	C₂×D₅₉₆	—	2	2	no	—
29	C₂²×Dic₁₄₉	C₂×C₄	1	2	yes	—
30	#14558720	—	2	2	no	—
31	#14558720	—	2	2	no	—
32	C₂×#11924746	—	2	2	no	—
33	#14558713	—	2	4	no	—
34	#14558713	—	2	4	no	—
35	C₂²×C₄×C₁₄₉	C₂×C₄	1	2	yes	—
36	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
37	M₁₆×C₁₄₉	—	2	2	no	—
38	M₁₆×C₁₄₉	D₄	1	4	yes	—
39	C₂×C₈×C₁₄₉	—	2	2	no	—
40	C₄²×C₁₄₉	D₄	1	1	yes	characteristic
41	C₄²×C₁₄₉	D₄	1	1	yes	characteristic
42	C₄²×C₁₄₉	D₄	1	2	yes	—
43	C₂²×D₁₄₉	—	2	2	no	—
44	C₂²×D₁₄₉	—	2	4	no	—
45	C₂²×D₁₄₉	—	4	4	no	—
46	Dic₂₉₈	—	4	2	no	—
47	Dic₂₉₈	—	4	1	no	—
48	Dic₂₉₈	—	4	1	no	—
49	D₅₉₆	—	4	2	no	—
50	D₅₉₆	—	4	1	no	—
51	D₅₉₆	—	4	1	no	—
52	C₂×Dic₁₄₉	—	2	2	no	—
53	C₂×Dic₁₄₉	—	2	4	no	—
54	#11924746	—	4	2	no	—
55	C₂³×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₁₄₉	—	4	2	no	—
57	C₂×C₄×C₁₄₉	C₂²⋊C₄	1	2	yes	—
58	C₂×C₄×C₁₄₉	C₂²⋊C₄	1	2	yes	—
59	C₂×C₄×C₁₄₉	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₁₄₉	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₁₄₉	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₁₄₉	#41	1	2	yes	—
63	C₈×C₁₄₉	—	2	4	no	—
64	C₂²×C₁₄₉	—	2	2	no	—
65	C₂²×C₁₄₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₁₄₉	C₄×D₄	1	2	yes	—
67	C₂×D₁₄₉	—	4	2	no	—
68	C₂×D₁₄₉	—	4	4	no	—
69	C₂×D₁₄₉	—	8	4	no	—
70	C₄×C₁₄₉	—	2	2	no	—
71	C₄×C₁₄₉	—	2	2	no	—
72	C₄×C₁₄₉	#121	1	2	yes	—
73	C₄×C₁₄₉	#124	1	1	yes	characteristic
74	C₄×C₁₄₉	#127	1	1	yes	characteristic
75	Dic₁₄₉	—	4	2	no	—
76	C₂×C₁₄₉	—	2	2	no	—
77	C₂×C₁₄₉	#390	1	1	yes	characteristic
78	C₂×C₁₄₉	#413	1	2	yes	—
79	D₁₄₉	—	8	2	no	—
80	C₁₄₉	#1967	1	1	yes	characteristic, complemented, Sylow
81	#1967	—	149	1	no	maximal, Sylow
82	C₂×#124	—	149	1	no	—
83	#423	—	149	1	no	—
84	C₂×#101	—	149	2	no	—
85	C₂×#100	—	149	2	no	—
86	C₄×M₁₆	D₁₄₉	1	1	yes	characteristic, complemented
87	C₂²×Q₈	—	149	1	no	—
88	C₂²×D₄	—	149	1	no	—
89	C₂×M₁₆	—	149	2	no	—
90	C₂×M₁₆	C₂×D₁₄₉	1	2	yes	—
91	#124	—	298	1	no	—
92	#124	—	298	1	no	—
93	#124	—	298	2	no	—
94	C₂×C₂²⋊C₄	—	149	4	no	—
95	C₂×C₄²	C₂×D₁₄₉	1	1	yes	characteristic
96	C₄⋊C₈	—	298	2	no	—
97	#101	—	298	4	no	—
98	#100	—	298	4	no	—
99	#97	—	2	1	no	—
100	C₄×C₈	—	2	1	no	—
101	C₂⁴	—	149	2	no	—
102	C₂×Q₈	—	298	1	no	—
103	C₂×Q₈	—	298	2	no	—
104	C₂×Q₈	—	298	1	no	—
105	C₂×Q₈	—	298	2	no	—
106	C₂×D₄	—	298	1	no	—
107	C₂×D₄	—	298	2	no	—
108	C₂×D₄	—	298	1	no	—
109	C₂×D₄	—	298	2	no	—
110	C₂²×C₄	—	149	2	no	—
111	C₂²×C₄	C₄×D₁₄₉	1	2	yes	—
112	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
113	M₁₆	—	2	2	no	—
114	M₁₆	—	298	2	no	—
115	M₁₆	—	298	2	no	—
116	M₁₆	D₅₉₆	1	4	yes	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	298	2	no	—
119	C₂²⋊C₄	—	298	4	no	—
120	C₂²⋊C₄	—	298	4	no	—
121	C₄²	C₂²⋊D₁₄₉	1	1	yes	characteristic
122	C₄²	C₂²⋊D₁₄₉	1	1	yes	characteristic
123	C₄²	D₅₉₆	1	2	yes	—
124	C₂³	—	298	2	no	—
125	C₂³	—	298	4	no	—
126	C₂³	—	596	4	no	—
127	C₂³	C₂×C₄×D₁₄₉	1	1	yes	characteristic
128	Q₈	—	596	1	no	—
129	Q₈	—	596	2	no	—
130	Q₈	—	596	1	no	—
131	D₄	—	596	1	no	—
132	D₄	—	596	2	no	—
133	D₄	—	596	1	no	—
134	C₂×C₄	—	4	2	no	—
135	C₂×C₄	—	298	2	no	—
136	C₂×C₄	—	298	4	no	—
137	C₂×C₄	#14558713	1	2	yes	—
138	C₂×C₄	#14558713	1	2	yes	—
139	C₂×C₄	C₂×C₂²⋊D₁₄₉	1	1	yes	characteristic
140	C₂×C₄	C₂×D₅₉₆	1	2	yes	—
141	C₂×C₄	C₂×D₅₉₆	1	1	yes	characteristic
142	C₂×C₄	#14558736	1	2	yes	—
143	C₈	—	2	4	no	—
144	C₈	—	596	2	no	—
145	C₂²	—	2	2	no	—
146	C₂²	—	596	2	no	—
147	C₂²	—	596	4	no	—
148	C₂²	—	1192	4	no	—
149	C₂²	C₂×#14558713	1	1	yes	characteristic
150	C₂²	C₄×D₅₉₆	1	2	yes	—
151	C₄	—	2	2	no	—
152	C₄	—	2	2	no	—
153	C₄	—	596	2	no	—
154	C₄	#18011885	1	2	yes	—
155	C₄	#18011899	1	1	yes	characteristic
156	C₄	#18011908	1	1	yes	characteristic
157	C₂	—	2	2	no	—
158	C₂	—	1192	2	no	—
159	C₂	#23749527	1	1	yes	characteristic
160	C₂	#23749607	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#18011899

C₂

yes

maximal, characteristic

#23749649

C₂

yes

maximal, characteristic, complemented

C₂×#18011829

C₂

yes

maximal, complemented

C₂×#18011827

C₂

yes

maximal

C₄×M₁₆×C₁₄₉

C₂

yes

maximal, characteristic, complemented

C₂²×Dic₂₉₈

C₂²

yes

characteristic

C₂²×D₅₉₆

C₂²

yes

characteristic

C₂×#14558720

C₂²

yes

—

—

—

—

C₄

yes

—

C₄⋊#11924746

—

#18011829

—

#18011827

—

C₂×M₁₆×C₁₄₉

C₂²

yes

—

C₂×C₄²×C₁₄₉

C₂²

yes

characteristic

#97×C₁₄₉

—

C₄×C₈×C₁₄₉

—

C₂³×D₁₄₉

C₂×C₄

yes

—

C₂×Dic₂₉₈

—

C₂×Dic₂₉₈

—

C₂×Dic₂₉₈

—

C₂×Dic₂₉₈

—

C₂×D₅₉₆

—

C₂×D₅₉₆

—

C₂×D₅₉₆

—

C₂×D₅₉₆

—

C₂²×Dic₁₄₉

C₂×C₄

yes

—

—

—

—

—

—

C₂²×C₄×C₁₄₉

C₂×C₄

yes

—

C₂²×C₄×C₁₄₉

C₂³

yes

characteristic

M₁₆×C₁₄₉

—

M₁₆×C₁₄₉

D₄

yes

—

C₂×C₈×C₁₄₉

—

C₄²×C₁₄₉

D₄

yes

characteristic

C₄²×C₁₄₉

D₄

yes

characteristic

C₄²×C₁₄₉

D₄

yes

—

C₂²×D₁₄₉

—

C₂²×D₁₄₉

—

C₂²×D₁₄₉

—

Dic₂₉₈

—

Dic₂₉₈

—

Dic₂₉₈

—

D₅₉₆

—

D₅₉₆

—

D₅₉₆

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

#11924746

—

C₂³×C₁₄₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₄₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₄₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₄₉

C₂×D₄

yes

—

C₂×C₄×C₁₄₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₄₉

#41

yes

—

C₈×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₄₉

C₄×D₄

yes

—

C₂×D₁₄₉

—

C₂×D₁₄₉

—

C₂×D₁₄₉

—

C₄×C₁₄₉

—

C₄×C₁₄₉

—

C₄×C₁₄₉

#121

yes

—

C₄×C₁₄₉

yes

C₄×C₁₄₉

yes

Dic₁₄₉

—

C₂×C₁₄₉

—

C₂×C₁₄₉

#390

yes

characteristic

C₂×C₁₄₉

#413

yes

—

D₁₄₉

—

C₁₄₉

#1967

yes

characteristic, complemented, Sylow

#1967

—

149

maximal, Sylow

C₂×#124

—

149

—

#423

—

149

—

C₂×#101

—

149

—

C₂×#100

—

149

—

C₄×M₁₆

D₁₄₉

yes

characteristic, complemented

C₂²×Q₈

—

149

—

C₂²×D₄

—

149

—

C₂×M₁₆

—

149

—

C₂×M₁₆

C₂×D₁₄₉

yes

—

#124

—

298

—

#124

—

298

—

#124

—

298

—

C₂×C₂²⋊C₄

—

149

—

C₂×C₄²

C₂×D₁₄₉

yes

characteristic

C₄⋊C₈

—

298

—

#101

—

298

—

#100

—

298

—

#97

—

100

C₄×C₈

—

101

C₂⁴

—

149

—

102

C₂×Q₈

—

298

—

103

C₂×Q₈

—

298

—

104

C₂×Q₈

—

298

—

105

C₂×Q₈

—

298

—

106

C₂×D₄

—

298

—

107

C₂×D₄

—

298

—

108

C₂×D₄

—

298

—

109

C₂×D₄

—

298

—

110

C₂²×C₄

—

149

—

111

C₂²×C₄

C₄×D₁₄₉

yes

—

112

C₂²×C₄

C₂²×D₁₄₉

yes

characteristic

113

M₁₆

—

114

M₁₆

—

298

—

115

M₁₆

—

298

—

116

M₁₆

D₅₉₆

yes

—

117

C₂×C₈

—

118

C₂×C₈

—

298

—

119

C₂²⋊C₄

—

298

—

120

C₂²⋊C₄

—

298

—

121

C₄²

C₂²⋊D₁₄₉

yes

characteristic

122

C₄²

C₂²⋊D₁₄₉

yes

characteristic

123

C₄²

D₅₉₆

yes

—

124

C₂³

—

298

—

125

C₂³

—

298

—

126

C₂³

—

596

—

127

C₂³

C₂×C₄×D₁₄₉

yes

characteristic

128

Q₈

—

596

—

129

Q₈

—

596

—

130

Q₈

—

596

—

131

D₄

—

596

—

132

D₄

—

596

—

133

D₄

—

596

—

134

C₂×C₄

—

135

C₂×C₄

—

298

—

136

C₂×C₄

—

298

—

137

C₂×C₄

#14558713

yes

—

138

C₂×C₄

#14558713

yes

—

139

C₂×C₄

C₂×C₂²⋊D₁₄₉

yes

characteristic

140

C₂×C₄

C₂×D₅₉₆

yes

—

141

C₂×C₄

C₂×D₅₉₆

yes

characteristic

142

C₂×C₄

#14558736

yes

—

143

C₈

—

144

C₈

—

596

—

145

C₂²

—

146

C₂²

—

596

—

147

C₂²

—

596

—

148

C₂²

—

1192

—

149

C₂²

C₂×#14558713

yes

characteristic

150

C₂²

C₄×D₅₉₆

yes

—

151

C₄

—

152

C₄

—

153

C₄

—

596

—

154

C₄

#18011885

yes

—

155

C₄

#18011899

yes

characteristic

156

C₄

#18011908

yes

characteristic

157

C₂

—

158

C₂

—

1192

—

159

C₂

#23749527

yes

characteristic

160

C₂

#23749607

yes

—

	subgroup	G/H
center	C₂²	C₂×#14558713
commutator	C₂³×C₁₄₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₄₉
Fitting	C₄×M₁₆×C₁₄₉	C₂
socle	C₂²×C₁₄₉	C₂×C₂²⋊C₄
prime radical	C₂²×D₅₉₆	C₂²
cosocle	C₂²×C₄×C₁₄₉	C₂³
hypercenter	C₄×M₁₆	D₁₄₉
hypocenter	C₁₄₉	#1967
LCS	G ▷ C₂³×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₂²

C₂×#14558713

commutator

C₂³×C₁₄₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₄₉

Fitting

C₄×M₁₆×C₁₄₉

C₂

socle

C₂²×C₁₄₉

C₂×C₂²⋊C₄

prime radical

C₂²×D₅₉₆

C₂²

cosocle

C₂²×C₄×C₁₄₉

C₂³

hypercenter

C₄×M₁₆

D₁₄₉

hypocenter

C₁₄₉

#1967

LCS

G ▷ C₂³×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
2. #23749649	C₂	semidirect
3. C₂×#18011829	C₂	semidirect
5. C₄×M₁₆×C₁₄₉	C₂	semidirect
80. C₁₄₉	#1967	semidirect
86. C₄×M₁₆	D₁₄₉	semidirect

H ≅ G/N

type

2. #23749649

C₂

semidirect

3. C₂×#18011829

C₂

semidirect

5. C₄×M₁₆×C₁₄₉

C₂

semidirect

80. C₁₄₉

#1967

semidirect

86. C₄×M₁₆

D₁₄₉

semidirect

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

subgroup	index
C₂×#18011899	2
#23749649	2
C₂×#18011829	2
C₂×#18011827	2
C₄×M₁₆×C₁₄₉	2
#1967	149

subgroup

index

C₄×M₁₆×C₁₄₉

149

Subgroups — #34045540

Conjugacy classes of subgroups

Complements

Maximal subgroups