Group #34044773 — subgroups

Subgroups — #34044773

Order 19072.

272 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 77 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×#18011869	C₂	1	1	yes	maximal, characteristic
2	#23749588	C₂	1	2	yes	maximal
3	#23749394	C₂	1	1	yes	maximal, characteristic, complemented
4	#23749393	C₂	1	2	yes	maximal, complemented
5	#343×C₁₄₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×D₁₄₉	C₂²	1	1	yes	characteristic
7	C₂×#14558720	C₂²	1	2	yes	—
8	#18011869	—	2	4	no	—
9	C₂×C₄⋊Dic₁₄₉	C₂²	1	1	yes	characteristic
10	C₂×#14558713	C₄	1	2	yes	—
11	C₂×C₄×Dic₁₄₉	C₄	1	2	yes	—
12	#18011823	—	2	4	no	—
13	C₂×M₁₆×C₁₄₉	C₂²	1	2	yes	—
14	C₂×C₄⋊C₄×C₁₄₉	C₂²	1	1	yes	characteristic
15	C₂³×D₁₄₉	C₂×C₄	1	1	yes	characteristic
16	C₂²×Dic₁₄₉	C₂×C₄	1	1	yes	characteristic
17	C₂²×Dic₁₄₉	C₂×C₄	1	2	yes	—
18	C₂×C₄×D₁₄₉	—	2	1	no	—
19	C₂×C₄×D₁₄₉	—	2	2	no	—
20	C₂×C₄×D₁₄₉	—	2	1	no	—
21	C₂×C₄×D₁₄₉	D₄	1	4	yes	—
22	#14558720	—	2	2	no	—
23	#14558720	—	2	2	no	—
24	C₂×#11924746	—	2	2	no	—
25	C₄⋊Dic₁₄₉	—	2	2	no	—
26	#14558713	—	2	4	no	—
27	C₄×Dic₁₄₉	—	2	4	no	—
28	C₂²×C₄×C₁₄₉	C₂×C₄	1	2	yes	—
29	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
30	M₁₆×C₁₄₉	—	2	2	no	—
31	M₁₆×C₁₄₉	—	2	2	no	—
32	C₂×C₈×C₁₄₉	—	2	2	no	—
33	C₄⋊C₄×C₁₄₉	—	2	2	no	—
34	C₂²×D₁₄₉	—	2	1	no	—
35	C₂²×D₁₄₉	—	2	1	no	—
36	C₂²×D₁₄₉	—	4	1	no	—
37	C₂²×D₁₄₉	—	4	1	no	—
38	C₂²×D₁₄₉	C₂²⋊C₄	1	2	yes	—
39	C₂×Dic₁₄₉	—	2	1	no	—
40	C₂×Dic₁₄₉	—	2	1	no	—
41	C₂×Dic₁₄₉	—	2	2	no	—
42	C₂×Dic₁₄₉	—	4	2	no	—
43	C₂×Dic₁₄₉	C₂²⋊C₄	1	2	yes	—
44	C₄×D₁₄₉	—	2	4	no	—
45	C₄×D₁₄₉	—	2	4	no	—
46	#11924746	—	4	2	no	—
47	C₂³×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₁₄₉	—	2	2	no	—
49	C₂×C₄×C₁₄₉	—	4	2	no	—
50	C₂×C₄×C₁₄₉	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₁₄₉	#41	1	1	yes	characteristic
52	C₂×C₄×C₁₄₉	#41	1	2	yes	—
53	C₂×C₄×C₁₄₉	#41	1	1	yes	characteristic
54	C₈×C₁₄₉	—	4	2	no	—
55	C₂²×C₁₄₉	—	2	1	no	—
56	C₂²×C₁₄₉	—	2	1	no	—
57	C₂²×C₁₄₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₁₄₉	#117	1	1	yes	characteristic
59	C₂²×C₁₄₉	#117	1	1	yes	characteristic
60	C₂×D₁₄₉	—	2	2	no	—
61	C₂×D₁₄₉	—	2	2	no	—
62	C₂×D₁₄₉	—	2	2	no	—
63	C₂×D₁₄₉	—	4	2	no	—
64	C₂×D₁₄₉	—	4	2	no	—
65	C₄×C₁₄₉	—	4	2	no	—
66	C₄×C₁₄₉	#123	1	2	yes	—
67	C₄×C₁₄₉	#123	1	2	yes	—
68	Dic₁₄₉	—	2	2	no	—
69	Dic₁₄₉	—	4	2	no	—
70	C₂×C₁₄₉	—	2	1	no	—
71	C₂×C₁₄₉	—	2	1	no	—
72	C₂×C₁₄₉	#381	1	1	yes	characteristic
73	C₂×C₁₄₉	#413	1	1	yes	characteristic
74	C₂×C₁₄₉	#413	1	1	yes	characteristic
75	D₁₄₉	—	4	2	no	—
76	C₁₄₉	#1822	1	1	yes	characteristic, complemented, Sylow
77	#1822	—	149	1	no	maximal, Sylow
78	C₂×#117	—	149	1	no	—
79	#407	—	149	2	no	—
80	#343	—	149	2	no	—
81	#343	—	149	1	no	—
82	#343	D₁₄₉	1	1	yes	characteristic, complemented
83	C₂³×C₄	—	149	1	no	—
84	C₂×M₁₆	—	149	2	no	—
85	C₂×M₁₆	C₂×D₁₄₉	1	2	yes	—
86	#117	—	298	4	no	—
87	C₂×C₄⋊C₄	—	149	1	no	—
88	C₂×C₄⋊C₄	C₂×D₁₄₉	1	1	yes	characteristic
89	C₂×C₂²⋊C₄	—	149	2	no	—
90	C₂×C₄²	—	149	2	no	—
91	C₂²⋊C₈	—	298	4	no	—
92	C₂⁴	—	149	1	no	—
93	C₂²×C₄	—	298	1	no	—
94	C₂²×C₄	—	298	2	no	—
95	C₂²×C₄	—	298	1	no	—
96	C₂²×C₄	—	149	1	no	—
97	C₂²×C₄	—	149	4	no	—
98	C₂²×C₄	—	149	2	no	—
99	C₂²×C₄	C₄×D₁₄₉	1	2	yes	—
100	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
101	M₁₆	—	2	2	no	—
102	M₁₆	—	2	2	no	—
103	M₁₆	—	298	2	no	—
104	M₁₆	—	298	2	no	—
105	C₂×C₈	—	2	2	no	—
106	C₂×C₈	—	298	2	no	—
107	C₄⋊C₄	—	2	2	no	—
108	C₄⋊C₄	—	298	2	no	—
109	C₂²⋊C₄	—	298	4	no	—
110	C₄²	—	298	4	no	—
111	C₂³	—	298	1	no	—
112	C₂³	—	298	1	no	—
113	C₂³	—	596	1	no	—
114	C₂³	—	596	1	no	—
115	C₂³	—	149	2	no	—
116	C₂³	C₂×C₄×D₁₄₉	1	1	yes	characteristic
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	4	2	no	—
119	C₂×C₄	—	298	1	no	—
120	C₂×C₄	—	298	1	no	—
121	C₂×C₄	—	298	4	no	—
122	C₂×C₄	—	298	4	no	—
123	C₂×C₄	—	298	2	no	—
124	C₂×C₄	—	596	2	no	—
125	C₂×C₄	—	149	2	no	—
126	C₂×C₄	D₄×D₁₄₉	1	2	yes	—
127	C₂×C₄	#14558735	1	1	yes	characteristic
128	C₂×C₄	#14558736	1	2	yes	—
129	C₂×C₄	#14558737	1	1	yes	characteristic
130	C₈	—	4	2	no	—
131	C₈	—	596	2	no	—
132	C₂²	—	2	1	no	—
133	C₂²	—	2	1	no	—
134	C₂²	—	298	2	no	—
135	C₂²	—	298	2	no	—
136	C₂²	—	298	2	no	—
137	C₂²	—	596	2	no	—
138	C₂²	—	596	2	no	—
139	C₂²	C₂²⋊C₄×D₁₄₉	1	1	yes	characteristic
140	C₂²	#18011868	1	1	yes	characteristic
141	C₂²	#18011869	1	1	yes	characteristic
142	C₄	—	4	2	no	—
143	C₄	—	298	2	no	—
144	C₄	—	596	2	no	—
145	C₄	#18011893	1	2	yes	—
146	C₄	#18011896	1	2	yes	—
147	C₂	—	2	1	no	—
148	C₂	—	2	1	no	—
149	C₂	—	596	2	no	—
150	C₂	#23749487	1	1	yes	characteristic
151	C₂	#23749606	1	1	yes	characteristic
152	C₂	#23749608	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#18011869

C₂

yes

maximal, characteristic

#23749588

C₂

yes

maximal

#23749394

C₂

yes

maximal, characteristic, complemented

#23749393

C₂

yes

maximal, complemented

#343×C₁₄₉

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₁₄₉

C₂²

yes

characteristic

C₂×#14558720

C₂²

yes

—

#18011869

—

C₂×C₄⋊Dic₁₄₉

C₂²

yes

characteristic

C₂×#14558713

C₄

yes

—

C₂×C₄×Dic₁₄₉

C₄

yes

—

#18011823

—

C₂×M₁₆×C₁₄₉

C₂²

yes

—

C₂×C₄⋊C₄×C₁₄₉

C₂²

yes

characteristic

C₂³×D₁₄₉

C₂×C₄

yes

characteristic

C₂²×Dic₁₄₉

C₂×C₄

yes

characteristic

C₂²×Dic₁₄₉

C₂×C₄

yes

—

C₂×C₄×D₁₄₉

—

C₂×C₄×D₁₄₉

—

C₂×C₄×D₁₄₉

—

C₂×C₄×D₁₄₉

D₄

yes

—

#14558720

—

#14558720

—

C₂×#11924746

—

C₄⋊Dic₁₄₉

—

#14558713

—

C₄×Dic₁₄₉

—

C₂²×C₄×C₁₄₉

C₂×C₄

yes

—

C₂²×C₄×C₁₄₉

C₂³

yes

characteristic

M₁₆×C₁₄₉

—

M₁₆×C₁₄₉

—

C₂×C₈×C₁₄₉

—

C₄⋊C₄×C₁₄₉

—

C₂²×D₁₄₉

—

C₂²×D₁₄₉

—

C₂²×D₁₄₉

—

C₂²×D₁₄₉

—

C₂²×D₁₄₉

C₂²⋊C₄

yes

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

C₂²⋊C₄

yes

—

C₄×D₁₄₉

—

C₄×D₁₄₉

—

#11924746

—

C₂³×C₁₄₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

C₂×D₄

yes

—

C₂×C₄×C₁₄₉

#41

yes

characteristic

C₂×C₄×C₁₄₉

#41

yes

—

C₂×C₄×C₁₄₉

#41

yes

characteristic

C₈×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

C₂×C₂²⋊C₄

yes

C₂²×C₁₄₉

yes

C₂²×C₁₄₉

yes

C₂×D₁₄₉

—

C₂×D₁₄₉

—

C₂×D₁₄₉

—

C₂×D₁₄₉

—

C₂×D₁₄₉

—

C₄×C₁₄₉

—

C₄×C₁₄₉

#123

yes

—

C₄×C₁₄₉

#123

yes

—

Dic₁₄₉

—

Dic₁₄₉

—

C₂×C₁₄₉

—

C₂×C₁₄₉

—

C₂×C₁₄₉

yes

C₂×C₁₄₉

yes

C₂×C₁₄₉

yes

D₁₄₉

—

C₁₄₉

#1822

yes

characteristic, complemented, Sylow

#1822

—

149

maximal, Sylow

C₂×#117

—

149

—

#407

—

149

—

#343

—

149

—

#343

—

149

—

#343

D₁₄₉

yes

characteristic, complemented

C₂³×C₄

—

149

—

C₂×M₁₆

—

149

—

C₂×M₁₆

C₂×D₁₄₉

yes

—

#117

—

298

—

C₂×C₄⋊C₄

—

149

—

C₂×C₄⋊C₄

C₂×D₁₄₉

yes

characteristic

C₂×C₂²⋊C₄

—

149

—

C₂×C₄²

—

149

—

C₂²⋊C₈

—

298

—

C₂⁴

—

149

—

C₂²×C₄

—

298

—

C₂²×C₄

—

298

—

C₂²×C₄

—

298

—

C₂²×C₄

—

149

—

C₂²×C₄

—

149

—

C₂²×C₄

—

149

—

C₂²×C₄

C₄×D₁₄₉

yes

—

100

C₂²×C₄

C₂²×D₁₄₉

yes

characteristic

101

M₁₆

—

102

M₁₆

—

103

M₁₆

—

298

—

104

M₁₆

—

298

—

105

C₂×C₈

—

106

C₂×C₈

—

298

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

298

—

109

C₂²⋊C₄

—

298

—

110

C₄²

—

298

—

111

C₂³

—

298

—

112

C₂³

—

298

—

113

C₂³

—

596

—

114

C₂³

—

596

—

115

C₂³

—

149

—

116

C₂³

C₂×C₄×D₁₄₉

yes

characteristic

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

298

—

120

C₂×C₄

—

298

—

121

C₂×C₄

—

298

—

122

C₂×C₄

—

298

—

123

C₂×C₄

—

298

—

124

C₂×C₄

—

596

—

125

C₂×C₄

—

149

—

126

C₂×C₄

D₄×D₁₄₉

yes

—

127

C₂×C₄

#14558735

yes

characteristic

128

C₂×C₄

#14558736

yes

—

129

C₂×C₄

#14558737

yes

characteristic

130

C₈

—

131

C₈

—

596

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

298

—

135

C₂²

—

298

—

136

C₂²

—

298

—

137

C₂²

—

596

—

138

C₂²

—

596

—

139

C₂²

C₂²⋊C₄×D₁₄₉

yes

characteristic

140

C₂²

#18011868

yes

characteristic

141

C₂²

#18011869

yes

characteristic

142

C₄

—

143

C₄

—

298

—

144

C₄

—

596

—

145

C₄

#18011893

yes

—

146

C₄

#18011896

yes

—

147

C₂

—

148

C₂

—

149

C₂

—

596

—

150

C₂

#23749487

yes

characteristic

151

C₂

#23749606

yes

characteristic

152

C₂

#23749608

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₁₄₉
commutator	C₂³×C₁₄₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₄₉
Fitting	#343×C₁₄₉	C₂
socle	C₂²×C₁₄₉	C₂×C₂²⋊C₄
prime radical	C₂³×D₁₄₉	C₂×C₄
cosocle	C₂²×C₄×C₁₄₉	C₂³
hypercenter	#343	D₁₄₉
hypocenter	C₁₄₉	#1822
LCS	G ▷ C₂³×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #343

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₁₄₉

commutator

C₂³×C₁₄₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₄₉

Fitting

#343×C₁₄₉

C₂

socle

C₂²×C₁₄₉

C₂×C₂²⋊C₄

prime radical

C₂³×D₁₄₉

C₂×C₄

cosocle

C₂²×C₄×C₁₄₉

C₂³

D₁₄₉

C₁₄₉

G ▷ C₂³×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #343

N	H ≅ G/N	type
3. #23749394	C₂	semidirect
4. #23749393	C₂	semidirect
5. #343×C₁₄₉	C₂	semidirect
76. C₁₄₉	#1822	semidirect
82. #343	D₁₄₉	semidirect

H ≅ G/N

type

3. #23749394

C₂

semidirect

4. #23749393

C₂

semidirect

5. #343×C₁₄₉

C₂

semidirect

76. C₁₄₉

#1822

semidirect

82. #343

D₁₄₉

semidirect

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

subgroup	index
C₂×#18011869	2
#23749588	2
#23749394	2
#23749393	2
#343×C₁₄₉	2
#1822	149

subgroup

index

#343×C₁₄₉

149

Subgroups — #34044773

Conjugacy classes of subgroups

Complements

Maximal subgroups