Group #34046162 — subgroups

Subgroups — #34046162

Order 19072.

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

normal subgroups only

This group has 230 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×#18011844	C₂	1	2	yes	maximal
2	#23749495	C₂	1	1	yes	maximal, characteristic
3	#23749387	C₂	1	2	yes	maximal
4	#23749380	C₂	1	1	yes	maximal, characteristic
5	#389×C₁₄₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11924746	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₁₄₉	—	2	1	no	—
8	C₂×C₄⋊Dic₁₄₉	—	2	1	no	—
9	C₂×C₄⋊Dic₁₄₉	C₂²	1	2	yes	—
10	#18011844	—	2	4	no	—
11	C₂×C₄⋊C₄×C₁₄₉	—	2	1	no	—
12	C₂×C₄⋊C₄×C₁₄₉	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₁₄₉	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₁₄₉	C₂²	1	1	yes	characteristic
15	#95×C₁₄₉	—	2	1	no	—
16	C₂²×Dic₁₄₉	—	2	2	no	—
17	C₂×#11924746	—	2	2	no	—
18	C₂×#11924746	—	2	2	no	—
19	C₂×#11924746	Q₈	1	4	yes	—
20	C₄⋊Dic₁₄₉	—	2	2	no	—
21	C₄⋊Dic₁₄₉	—	2	2	no	—
22	C₄⋊Dic₁₄₉	—	4	2	no	—
23	C₄⋊Dic₁₄₉	—	4	2	no	—
24	C₂²×C₄×C₁₄₉	—	2	1	no	—
25	C₂²×C₄×C₁₄₉	—	2	1	no	—
26	C₂²×C₄×C₁₄₉	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₄₉	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₁₄₉	—	2	1	no	—
30	C₄⋊C₄×C₁₄₉	—	2	2	no	—
31	C₄⋊C₄×C₁₄₉	—	4	2	no	—
32	C₄⋊C₄×C₁₄₉	D₄	1	2	yes	—
33	C₄²×C₁₄₉	—	2	2	no	—
34	C₂×Dic₁₄₉	—	4	2	no	—
35	C₂×Dic₁₄₉	—	4	2	no	—
36	C₂×Dic₁₄₉	—	4	2	no	—
37	#11924746	—	2	4	no	—
38	C₂³×C₁₄₉	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₄₉	—	2	1	no	—
40	C₂×C₄×C₁₄₉	—	2	1	no	—
41	C₂×C₄×C₁₄₉	—	2	1	no	—
42	C₂×C₄×C₁₄₉	—	2	1	no	—
43	C₂×C₄×C₁₄₉	—	2	2	no	—
44	C₂×C₄×C₁₄₉	—	4	1	no	—
45	C₂×C₄×C₁₄₉	—	4	1	no	—
46	C₂×C₄×C₁₄₉	—	4	1	no	—
47	C₂×C₄×C₁₄₉	—	4	1	no	—
48	C₂×C₄×C₁₄₉	—	4	1	no	—
49	C₂×C₄×C₁₄₉	D₈	1	2	yes	—
50	C₂×C₄×C₁₄₉	Q₁₆	1	2	yes	—
51	C₂×C₄×C₁₄₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₁₄₉	C₂×Q₈	1	2	yes	—
53	C₂×C₄×C₁₄₉	#41	1	2	yes	—
54	C₂×C₄×C₁₄₉	#41	1	1	yes	characteristic
55	C₂²×C₁₄₉	#121	1	1	yes	characteristic
56	C₂²×C₁₄₉	#123	1	1	yes	characteristic
57	C₂²×C₁₄₉	C₄⋊Q₈	1	1	yes	characteristic
58	C₂²×C₁₄₉	C₂×D₈	1	1	yes	characteristic
59	C₂²×C₁₄₉	C₂×Q₁₆	1	1	yes	characteristic
60	C₂²×C₁₄₉	AGL(1,ℤ/8)	1	1	yes	characteristic
61	C₂²×C₁₄₉	#137	1	1	yes	characteristic
62	C₄×C₁₄₉	—	2	2	no	—
63	C₄×C₁₄₉	—	2	2	no	—
64	C₄×C₁₄₉	—	4	2	no	—
65	C₄×C₁₄₉	—	8	1	no	—
66	C₄×C₁₄₉	C₂²⋊Q₈	1	2	yes	—
67	C₄×C₁₄₉	#125	1	2	yes	—
68	Dic₁₄₉	—	8	2	no	—
69	C₂×C₁₄₉	#398	1	1	yes	characteristic
70	C₂×C₁₄₉	#459	1	1	yes	characteristic
71	C₂×C₁₄₉	#462	1	1	yes	characteristic
72	C₂×C₁₄₉	#480	1	1	yes	characteristic
73	C₂×C₁₄₉	#484	1	1	yes	characteristic
74	C₂×C₁₄₉	#500	1	1	yes	characteristic
75	C₂×C₁₄₉	#501	1	1	yes	characteristic
76	C₁₄₉	#2083	1	1	yes	characteristic, complemented, Sylow
77	#2083	—	149	1	no	maximal, Sylow
78	C₂×#107	—	149	2	no	—
79	#389	D₁₄₉	1	1	yes	characteristic
80	#384	—	149	1	no	—
81	#340	—	149	2	no	—
82	#336	—	149	1	no	—
83	C₂²×C₈	—	149	2	no	—
84	C₂×C₄⋊C₄	—	2	1	no	—
85	C₂×C₄⋊C₄	—	298	1	no	—
86	C₂×C₄⋊C₄	—	298	1	no	—
87	C₂×C₄⋊C₄	—	149	2	no	—
88	C₂×C₄⋊C₄	C₂×D₁₄₉	1	1	yes	characteristic
89	C₂×C₄⋊C₄	C₂×D₁₄₉	1	1	yes	characteristic
90	C₂×C₄²	C₂×D₁₄₉	1	1	yes	characteristic
91	#107	—	298	4	no	—
92	#95	—	2	1	no	—
93	C₂²×C₄	—	2	1	no	—
94	C₂²×C₄	—	2	1	no	—
95	C₂²×C₄	—	298	2	no	—
96	C₂²×C₄	C₂²⋊D₁₄₉	1	1	yes	characteristic
97	C₂²×C₄	C₂²⋊D₁₄₉	1	1	yes	characteristic
98	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
99	C₂×C₈	—	298	2	no	—
100	C₂×C₈	—	298	2	no	—
101	C₂×C₈	—	149	4	no	—
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	4	2	no	—
105	C₄⋊C₄	—	298	2	no	—
106	C₄⋊C₄	—	298	2	no	—
107	C₄⋊C₄	—	596	2	no	—
108	C₄⋊C₄	—	596	2	no	—
109	C₄⋊C₄	D₅₉₆	1	2	yes	—
110	C₄²	—	2	2	no	—
111	C₂³	C₂×C₂²⋊D₁₄₉	1	1	yes	characteristic
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	4	1	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	—	4	1	no	—
122	C₂×C₄	—	596	2	no	—
123	C₂×C₄	—	596	2	no	—
124	C₂×C₄	—	596	2	no	—
125	C₂×C₄	#14558721	1	2	yes	—
126	C₂×C₄	#14558726	1	2	yes	—
127	C₂×C₄	C₂×D₅₉₆	1	1	yes	characteristic
128	C₂×C₄	Q₈×D₁₄₉	1	2	yes	—
129	C₂×C₄	#14558735	1	2	yes	—
130	C₂×C₄	#14558736	1	1	yes	characteristic
131	C₈	—	298	4	no	—
132	C₂²	#18011885	1	1	yes	characteristic
133	C₂²	#18011892	1	1	yes	characteristic
134	C₂²	#18011909	1	1	yes	characteristic
135	C₂²	C₂×#14558721	1	1	yes	characteristic
136	C₂²	C₂×#14558726	1	1	yes	characteristic
137	C₂²	#18011936	1	1	yes	characteristic
138	C₂²	#18011941	1	1	yes	characteristic
139	C₄	—	2	2	no	—
140	C₄	—	2	2	no	—
141	C₄	—	4	2	no	—
142	C₄	—	8	1	no	—
143	C₄	—	1192	2	no	—
144	C₄	#18011891	1	2	yes	—
145	C₄	#18011901	1	2	yes	—
146	C₂	#23749555	1	1	yes	characteristic
147	C₂	#23749820	1	1	yes	characteristic
148	C₂	#23749835	1	1	yes	characteristic
149	C₂	#23749936	1	1	yes	characteristic
150	C₂	#23749958	1	1	yes	characteristic
151	C₂	#23750028	1	1	yes	characteristic
152	C₂	#23750035	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#18011844

C₂

yes

maximal

#23749495

C₂

yes

maximal, characteristic

#23749387

C₂

yes

maximal

#23749380

C₂

yes

maximal, characteristic

#389×C₁₄₉

C₂

yes

maximal, characteristic

C₂²×#11924746

C₂²

yes

—

C₂×C₄⋊Dic₁₄₉

—

C₂×C₄⋊Dic₁₄₉

—

C₂×C₄⋊Dic₁₄₉

C₂²

yes

—

#18011844

—

C₂×C₄⋊C₄×C₁₄₉

—

C₂×C₄⋊C₄×C₁₄₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₄₉

C₂²

yes

characteristic

C₂×C₄²×C₁₄₉

C₂²

yes

characteristic

#95×C₁₄₉

—

C₂²×Dic₁₄₉

—

C₂×#11924746

—

C₂×#11924746

—

C₂×#11924746

Q₈

yes

—

C₄⋊Dic₁₄₉

—

C₄⋊Dic₁₄₉

—

C₄⋊Dic₁₄₉

—

C₄⋊Dic₁₄₉

—

C₂²×C₄×C₁₄₉

—

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

D₄

yes

—

C₄²×C₁₄₉

—

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

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

D₈

yes

—

C₂×C₄×C₁₄₉

Q₁₆

yes

—

C₂×C₄×C₁₄₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₄₉

C₂×Q₈

yes

—

C₂×C₄×C₁₄₉

#41

yes

—

C₂×C₄×C₁₄₉

yes

C₂²×C₁₄₉

yes

C₂²×C₁₄₉

yes

C₂²×C₁₄₉

C₄⋊Q₈

yes

characteristic

C₂²×C₁₄₉

C₂×D₈

yes

characteristic

C₂²×C₁₄₉

C₂×Q₁₆

yes

C₂²×C₁₄₉

yes

C₂²×C₁₄₉

yes

C₄×C₁₄₉

—

C₄×C₁₄₉

—

C₄×C₁₄₉

—

C₄×C₁₄₉

—

C₄×C₁₄₉

C₂²⋊Q₈

yes

—

C₄×C₁₄₉

#125

yes

—

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

#2083

—

149

maximal, Sylow

C₂×#107

—

149

—

#389

D₁₄₉

yes

characteristic

#384

—

149

—

#340

—

149

—

#336

—

149

—

C₂²×C₈

—

149

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

298

—

C₂×C₄⋊C₄

—

298

—

C₂×C₄⋊C₄

—

149

—

C₂×C₄⋊C₄

C₂×D₁₄₉

yes

characteristic

C₂×C₄⋊C₄

C₂×D₁₄₉

yes

characteristic

C₂×C₄²

C₂×D₁₄₉

yes

characteristic

#107

—

298

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

298

—

C₂²×C₄

C₂²⋊D₁₄₉

yes

characteristic

C₂²×C₄

C₂²⋊D₁₄₉

yes

characteristic

C₂²×C₄

C₂²×D₁₄₉

yes

characteristic

C₂×C₈

—

298

—

100

C₂×C₈

—

298

—

101

C₂×C₈

—

149

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

298

—

106

C₄⋊C₄

—

298

—

107

C₄⋊C₄

—

596

—

108

C₄⋊C₄

—

596

—

109

C₄⋊C₄

D₅₉₆

yes

—

110

C₄²

—

111

C₂³

C₂×C₂²⋊D₁₄₉

yes

characteristic

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

596

—

123

C₂×C₄

—

596

—

124

C₂×C₄

—

596

—

125

C₂×C₄

#14558721

yes

—

126

C₂×C₄

#14558726

yes

—

127

C₂×C₄

C₂×D₅₉₆

yes

characteristic

128

C₂×C₄

Q₈×D₁₄₉

yes

—

129

C₂×C₄

#14558735

yes

—

130

C₂×C₄

#14558736

yes

characteristic

131

C₈

—

298

—

132

C₂²

#18011885

yes

characteristic

133

C₂²

#18011892

yes

characteristic

134

C₂²

#18011909

yes

characteristic

135

C₂²

C₂×#14558721

yes

characteristic

136

C₂²

C₂×#14558726

yes

characteristic

137

C₂²

#18011936

yes

characteristic

138

C₂²

#18011941

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

1192

—

144

C₄

#18011891

yes

—

145

C₄

#18011901

yes

—

146

C₂

#23749555

yes

characteristic

147

C₂

#23749820

yes

characteristic

148

C₂

#23749835

yes

characteristic

149

C₂

#23749936

yes

characteristic

150

C₂

#23749958

yes

characteristic

151

C₂

#23750028

yes

characteristic

152

C₂

#23750035

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₄₉
commutator, cosocle	C₂²×C₄×C₁₄₉	C₂³
Frattini	C₂²×C₄	C₂²×D₁₄₉
Fitting	#389×C₁₄₉	C₂
socle, prime radical	C₂³×C₁₄₉	C₂×D₄
hypercenter	#389	D₁₄₉
hypocenter	C₁₄₉	#2083
LCS	G ▷ C₂²×C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #389

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₄₉

commutator, cosocle

C₂²×C₄×C₁₄₉

C₂³

Frattini

C₂²×C₄

C₂²×D₁₄₉

Fitting

#389×C₁₄₉

C₂

socle, prime radical

C₂³×C₁₄₉

C₂×D₄

D₁₄₉

C₁₄₉

G ▷ C₂²×C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #389

N	H ≅ G/N	type
76. C₁₄₉	#2083	semidirect

H ≅ G/N

type

76. C₁₄₉

#2083

semidirect

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

subgroup	index
C₂×#18011844	2
#23749495	2
#23749387	2
#23749380	2
#389×C₁₄₉	2
#2083	149

subgroup

index

#389×C₁₄₉

149

Subgroups — #34046162

Conjugacy classes of subgroups

Complements

Maximal subgroups