Group #34046100 — subgroups

Subgroups — #34046100

Order 19072.

238 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 87 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×#18011834	C₂	1	2	yes	maximal
2	#23749531	C₂	1	1	yes	maximal, characteristic
3	#23749387	C₂	1	2	yes	maximal
4	#23749380	C₂	1	1	yes	maximal, characteristic
5	#384×C₁₄₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₉₈	C₂²	1	1	yes	characteristic
7	C₂²×#11924746	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₁₄₉	C₂²	1	1	yes	characteristic
9	#18011834	—	2	4	no	—
10	#18011818	—	2	1	no	—
11	#18011818	—	2	1	no	—
12	C₂×C₄⋊C₄×C₁₄₉	—	2	1	no	—
13	C₂×C₄⋊C₄×C₁₄₉	—	2	1	no	—
14	C₂×C₄⋊C₄×C₁₄₉	C₂²	1	2	yes	—
15	C₂×C₄²×C₁₄₉	C₂²	1	1	yes	characteristic
16	C₂×Dic₂₉₈	—	2	1	no	—
17	C₂×Dic₂₉₈	—	2	2	no	—
18	C₂×Dic₂₉₈	—	2	2	no	—
19	C₂×Dic₂₉₈	D₄	1	2	yes	—
20	C₂²×Dic₁₄₉	—	2	1	no	—
21	C₂²×Dic₁₄₉	—	2	1	no	—
22	C₂×#11924746	—	2	2	no	—
23	C₂×#11924746	—	2	2	no	—
24	C₂×#11924746	—	2	2	no	—
25	C₄⋊Dic₁₄₉	—	2	2	no	—
26	C₂²×C₄×C₁₄₉	—	2	2	no	—
27	C₂²×C₄×C₁₄₉	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₄₉	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
30	C₄⋊C₄×C₁₄₉	—	2	2	no	—
31	C₄⋊C₄×C₁₄₉	—	2	2	no	—
32	C₄⋊C₄×C₁₄₉	—	4	2	no	—
33	C₄⋊C₄×C₁₄₉	—	4	2	no	—
34	C₄²×C₁₄₉	—	2	2	no	—
35	Dic₂₉₈	—	2	4	no	—
36	Dic₂₉₈	—	4	2	no	—
37	C₂×Dic₁₄₉	—	2	2	no	—
38	C₂×Dic₁₄₉	—	4	1	no	—
39	C₂×Dic₁₄₉	—	4	1	no	—
40	C₂×Dic₁₄₉	—	4	1	no	—
41	C₂×Dic₁₄₉	—	4	1	no	—
42	C₂×Dic₁₄₉	—	4	1	no	—
43	#11924746	—	4	2	no	—
44	C₂³×C₁₄₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₄₉	—	2	1	no	—
46	C₂×C₄×C₁₄₉	—	2	1	no	—
47	C₂×C₄×C₁₄₉	—	2	1	no	—
48	C₂×C₄×C₁₄₉	—	2	1	no	—
49	C₂×C₄×C₁₄₉	—	4	2	no	—
50	C₂×C₄×C₁₄₉	—	4	2	no	—
51	C₂×C₄×C₁₄₉	—	4	2	no	—
52	C₂×C₄×C₁₄₉	SD₁₆	1	2	yes	—
53	C₂×C₄×C₁₄₉	Q₁₆	1	2	yes	—
54	C₂×C₄×C₁₄₉	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₁₄₉	#41	1	1	yes	characteristic
56	C₂×C₄×C₁₄₉	#41	1	2	yes	—
57	C₂×C₄×C₁₄₉	#41	1	2	yes	—
58	C₂²×C₁₄₉	#121	1	1	yes	characteristic
59	C₂²×C₁₄₉	#123	1	1	yes	characteristic
60	C₂²×C₁₄₉	#124	1	1	yes	characteristic
61	C₂²×C₁₄₉	C₂×SD₁₆	1	1	yes	characteristic
62	C₂²×C₁₄₉	C₂×Q₁₆	1	1	yes	characteristic
63	C₂²×C₁₄₉	AGL(1,ℤ/8)	1	1	yes	characteristic
64	C₂²×C₁₄₉	#137	1	1	yes	characteristic
65	C₄×C₁₄₉	—	2	2	no	—
66	C₄×C₁₄₉	—	2	2	no	—
67	C₄×C₁₄₉	—	8	2	no	—
68	C₄×C₁₄₉	#123	1	2	yes	—
69	C₄×C₁₄₉	#126	1	2	yes	—
70	Dic₁₄₉	—	4	2	no	—
71	Dic₁₄₉	—	8	1	no	—
72	C₂×C₁₄₉	#397	1	1	yes	characteristic
73	C₂×C₁₄₉	#460	1	1	yes	characteristic
74	C₂×C₁₄₉	#462	1	1	yes	characteristic
75	C₂×C₁₄₉	#483	1	1	yes	characteristic
76	C₂×C₁₄₉	#484	1	1	yes	characteristic
77	C₂×C₁₄₉	#487	1	1	yes	characteristic
78	C₂×C₁₄₉	#489	1	1	yes	characteristic
79	C₁₄₉	#2072	1	1	yes	characteristic, complemented, Sylow
80	#2072	—	149	1	no	maximal, Sylow
81	C₂×#103	—	149	2	no	—
82	#391	—	149	1	no	—
83	#384	D₁₄₉	1	1	yes	characteristic
84	#340	—	149	2	no	—
85	#336	—	149	1	no	—
86	C₂²×Q₈	—	149	1	no	—
87	C₂²×C₈	—	149	2	no	—
88	C₂×C₄⋊C₄	—	2	1	no	—
89	C₂×C₄⋊C₄	—	2	1	no	—
90	C₂×C₄⋊C₄	—	149	1	no	—
91	C₂×C₄⋊C₄	C₂×D₁₄₉	1	2	yes	—
92	C₂×C₄²	C₂×D₁₄₉	1	1	yes	characteristic
93	#103	—	298	4	no	—
94	#95	—	298	1	no	—
95	#95	—	298	1	no	—
96	C₂×Q₈	—	298	1	no	—
97	C₂×Q₈	—	298	2	no	—
98	C₂×Q₈	—	298	2	no	—
99	C₂×Q₈	—	149	2	no	—
100	C₂²×C₄	—	2	2	no	—
101	C₂²×C₄	—	298	1	no	—
102	C₂²×C₄	—	298	1	no	—
103	C₂²×C₄	C₂²⋊D₁₄₉	1	1	yes	characteristic
104	C₂²×C₄	C₂²⋊D₁₄₉	1	1	yes	characteristic
105	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
106	C₂×C₈	—	298	2	no	—
107	C₂×C₈	—	298	2	no	—
108	C₂×C₈	—	298	2	no	—
109	C₄⋊C₄	—	2	2	no	—
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	—	4	2	no	—
112	C₄⋊C₄	—	4	2	no	—
113	C₄⋊C₄	—	298	2	no	—
114	C₄²	—	2	2	no	—
115	C₂³	C₂×C₂²⋊D₁₄₉	1	1	yes	characteristic
116	Q₈	—	298	4	no	—
117	Q₈	—	596	2	no	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	—	298	2	no	—
126	C₂×C₄	—	596	1	no	—
127	C₂×C₄	—	596	1	no	—
128	C₂×C₄	—	596	1	no	—
129	C₂×C₄	—	596	1	no	—
130	C₂×C₄	—	596	1	no	—
131	C₂×C₄	#14558724	1	2	yes	—
132	C₂×C₄	#14558726	1	2	yes	—
133	C₂×C₄	D₄×D₁₄₉	1	1	yes	characteristic
134	C₂×C₄	#14558735	1	1	yes	characteristic
135	C₂×C₄	#14558736	1	2	yes	—
136	C₂×C₄	#14558737	1	2	yes	—
137	C₈	—	596	2	no	—
138	C₂²	#18011882	1	1	yes	characteristic
139	C₂²	#18011894	1	1	yes	characteristic
140	C₂²	#18011900	1	1	yes	characteristic
141	C₂²	C₂×#14558724	1	1	yes	characteristic
142	C₂²	C₂×#14558726	1	1	yes	characteristic
143	C₂²	#18011937	1	1	yes	characteristic
144	C₂²	#18011942	1	1	yes	characteristic
145	C₄	—	2	2	no	—
146	C₄	—	2	2	no	—
147	C₄	—	8	2	no	—
148	C₄	—	596	2	no	—
149	C₄	—	1192	1	no	—
150	C₄	#18011896	1	2	yes	—
151	C₄	#18011906	1	2	yes	—
152	C₂	#23749551	1	1	yes	characteristic
153	C₂	#23749822	1	1	yes	characteristic
154	C₂	#23749832	1	1	yes	characteristic
155	C₂	#23749955	1	1	yes	characteristic
156	C₂	#23749960	1	1	yes	characteristic
157	C₂	#23749977	1	1	yes	characteristic
158	C₂	#23749987	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#18011834

C₂

yes

maximal

#23749531

C₂

yes

maximal, characteristic

#23749387

C₂

yes

maximal

#23749380

C₂

yes

maximal, characteristic

#384×C₁₄₉

C₂

yes

maximal, characteristic

C₂²×Dic₂₉₈

C₂²

yes

characteristic

C₂²×#11924746

C₂²

yes

—

C₂×C₄⋊Dic₁₄₉

C₂²

yes

—

—

—

C₂×C₄⋊C₄×C₁₄₉

—

C₂×C₄⋊C₄×C₁₄₉

—

C₂×C₄⋊C₄×C₁₄₉

C₂²

yes

—

C₂×C₄²×C₁₄₉

C₂²

yes

characteristic

C₂×Dic₂₉₈

—

C₂×Dic₂₉₈

—

C₂×Dic₂₉₈

—

C₂×Dic₂₉₈

D₄

yes

—

C₂²×Dic₁₄₉

—

C₂²×Dic₁₄₉

—

C₂×#11924746

—

C₂×#11924746

—

C₂×#11924746

—

C₄⋊Dic₁₄₉

—

C₂²×C₄×C₁₄₉

—

C₂²×C₄×C₁₄₉

D₄

yes

characteristic

C₂²×C₄×C₁₄₉

D₄

yes

characteristic

C₂²×C₄×C₁₄₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₄₉

—

C₄⋊C₄×C₁₄₉

—

C₄⋊C₄×C₁₄₉

—

C₄⋊C₄×C₁₄₉

—

C₄²×C₁₄₉

—

Dic₂₉₈

—

Dic₂₉₈

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

#11924746

—

C₂³×C₁₄₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

SD₁₆

yes

—

C₂×C₄×C₁₄₉

Q₁₆

yes

—

C₂×C₄×C₁₄₉

C₂×D₄

yes

C₂×C₄×C₁₄₉

yes

C₂×C₄×C₁₄₉

yes

—

C₂×C₄×C₁₄₉

#41

yes

—

C₂²×C₁₄₉

yes

C₂²×C₁₄₉

yes

C₂²×C₁₄₉

yes

C₂²×C₁₄₉

C₂×SD₁₆

yes

characteristic

C₂²×C₁₄₉

C₂×Q₁₆

yes

C₂²×C₁₄₉

yes

C₂²×C₁₄₉

yes

C₄×C₁₄₉

—

C₄×C₁₄₉

—

C₄×C₁₄₉

—

C₄×C₁₄₉

#123

yes

—

C₄×C₁₄₉

#126

yes

—

Dic₁₄₉

—

Dic₁₄₉

—

C₂×C₁₄₉

yes

C₂×C₁₄₉

yes

C₂×C₁₄₉

yes

C₂×C₁₄₉

yes

C₂×C₁₄₉

yes

C₂×C₁₄₉

yes

C₂×C₁₄₉

yes

C₁₄₉

yes

characteristic, complemented, Sylow

#2072

—

149

maximal, Sylow

C₂×#103

—

149

—

#391

—

149

—

#384

D₁₄₉

yes

characteristic

#340

—

149

—

#336

—

149

—

C₂²×Q₈

—

149

—

C₂²×C₈

—

149

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

149

—

C₂×C₄⋊C₄

C₂×D₁₄₉

yes

—

C₂×C₄²

C₂×D₁₄₉

yes

characteristic

#103

—

298

—

#95

—

298

—

#95

—

298

—

C₂×Q₈

—

298

—

C₂×Q₈

—

298

—

C₂×Q₈

—

298

—

C₂×Q₈

—

149

—

100

C₂²×C₄

—

101

C₂²×C₄

—

298

—

102

C₂²×C₄

—

298

—

103

C₂²×C₄

C₂²⋊D₁₄₉

yes

characteristic

104

C₂²×C₄

C₂²⋊D₁₄₉

yes

characteristic

105

C₂²×C₄

C₂²×D₁₄₉

yes

characteristic

106

C₂×C₈

—

298

—

107

C₂×C₈

—

298

—

108

C₂×C₈

—

298

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

298

—

114

C₄²

—

115

C₂³

C₂×C₂²⋊D₁₄₉

yes

characteristic

116

Q₈

—

298

—

117

Q₈

—

596

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

298

—

126

C₂×C₄

—

596

—

127

C₂×C₄

—

596

—

128

C₂×C₄

—

596

—

129

C₂×C₄

—

596

—

130

C₂×C₄

—

596

—

131

C₂×C₄

#14558724

yes

—

132

C₂×C₄

#14558726

yes

—

133

C₂×C₄

D₄×D₁₄₉

yes

characteristic

134

C₂×C₄

#14558735

yes

characteristic

135

C₂×C₄

#14558736

yes

—

136

C₂×C₄

#14558737

yes

—

137

C₈

—

596

—

138

C₂²

#18011882

yes

characteristic

139

C₂²

#18011894

yes

characteristic

140

C₂²

#18011900

yes

characteristic

141

C₂²

C₂×#14558724

yes

characteristic

142

C₂²

C₂×#14558726

yes

characteristic

143

C₂²

#18011937

yes

characteristic

144

C₂²

#18011942

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

596

—

149

C₄

—

1192

—

150

C₄

#18011896

yes

—

151

C₄

#18011906

yes

—

152

C₂

#23749551

yes

characteristic

153

C₂

#23749822

yes

characteristic

154

C₂

#23749832

yes

characteristic

155

C₂

#23749955

yes

characteristic

156

C₂

#23749960

yes

characteristic

157

C₂

#23749977

yes

characteristic

158

C₂

#23749987

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₄₉
commutator, cosocle	C₂²×C₄×C₁₄₉	C₂³
Frattini	C₂²×C₄	C₂²×D₁₄₉
Fitting	#384×C₁₄₉	C₂
socle, prime radical	C₂³×C₁₄₉	C₂×D₄
hypercenter	#384	D₁₄₉
hypocenter	C₁₄₉	#2072
LCS	G ▷ C₂²×C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₄₉

commutator, cosocle

C₂²×C₄×C₁₄₉

C₂³

Frattini

C₂²×C₄

C₂²×D₁₄₉

Fitting

#384×C₁₄₉

C₂

socle, prime radical

C₂³×C₁₄₉

C₂×D₄

D₁₄₉

C₁₄₉

G ▷ C₂²×C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
79. C₁₄₉	#2072	semidirect

H ≅ G/N

type

79. C₁₄₉

#2072

semidirect

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

subgroup	index
C₂×#18011834	2
#23749531	2
#23749387	2
#23749380	2
#384×C₁₄₉	2
#2072	149

subgroup

index

#384×C₁₄₉

149

Subgroups — #34046100

Conjugacy classes of subgroups

Complements

Maximal subgroups