Group #34046224 — subgroups

Subgroups — #34046224

Order 19072.

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

normal subgroups only

This group has 290 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×#18011842	C₂	1	2	yes	maximal, complemented
2	C₂×#18011832	C₂	1	2	yes	maximal
3	#23749495	C₂	1	1	yes	maximal, characteristic, complemented
4	#23749380	C₂	1	1	yes	maximal, characteristic, complemented
5	#390×C₁₄₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11924746	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₁₄₉	—	2	2	no	—
8	C₂×C₄⋊Dic₁₄₉	C₂²	1	2	yes	—
9	#18011842	—	2	4	no	—
10	#18011832	—	2	4	no	—
11	C₂²×D₄×C₁₄₉	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₄×C₁₄₉	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊C₄×C₁₄₉	—	2	2	no	—
14	C₂×C₄²×C₁₄₉	C₂²	1	1	yes	characteristic
15	C₂²×Dic₁₄₉	—	2	2	no	—
16	C₂×#11924746	—	2	2	no	—
17	C₂×#11924746	—	2	2	no	—
18	C₂×#11924746	D₄	1	4	yes	—
19	C₄⋊Dic₁₄₉	—	2	2	no	—
20	C₄⋊Dic₁₄₉	—	2	2	no	—
21	C₄⋊Dic₁₄₉	—	4	4	no	—
22	C₂⁴×C₁₄₉	—	2	1	no	—
23	C₂×D₄×C₁₄₉	—	2	2	no	—
24	C₂×D₄×C₁₄₉	—	2	2	no	—
25	C₂×D₄×C₁₄₉	—	2	1	no	—
26	C₂×D₄×C₁₄₉	Q₈	1	2	yes	—
27	C₂²×C₄×C₁₄₉	—	2	1	no	—
28	C₂²×C₄×C₁₄₉	D₄	1	2	yes	—
29	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
30	C₄⋊C₄×C₁₄₉	—	2	2	no	—
31	C₂²⋊C₄×C₁₄₉	—	4	4	no	—
32	C₄²×C₁₄₉	—	2	2	no	—
33	C₂×Dic₁₄₉	—	4	2	no	—
34	C₂×Dic₁₄₉	—	4	4	no	—
35	#11924746	—	2	4	no	—
36	C₂³×C₁₄₉	—	2	2	no	—
37	C₂³×C₁₄₉	—	4	1	no	—
38	C₂³×C₁₄₉	—	4	1	no	—
39	C₂³×C₁₄₉	—	4	2	no	—
40	C₂³×C₁₄₉	—	4	2	no	—
41	C₂³×C₁₄₉	C₂×D₄	1	1	yes	characteristic
42	D₄×C₁₄₉	—	2	4	no	—
43	D₄×C₁₄₉	—	4	2	no	—
44	C₂×C₄×C₁₄₉	—	2	2	no	—
45	C₂×C₄×C₁₄₉	—	2	2	no	—
46	C₂×C₄×C₁₄₉	—	4	1	no	—
47	C₂×C₄×C₁₄₉	—	4	2	no	—
48	C₂×C₄×C₁₄₉	SD₁₆	1	4	yes	—
49	C₂×C₄×C₁₄₉	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₁₄₉	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄×C₁₄₉	#41	1	2	yes	—
52	C₂×C₄×C₁₄₉	#41	1	1	yes	characteristic
53	C₂²×C₁₄₉	—	4	2	no	—
54	C₂²×C₁₄₉	—	4	4	no	—
55	C₂²×C₁₄₉	—	8	2	no	—
56	C₂²×C₁₄₉	—	8	1	no	—
57	C₂²×C₁₄₉	—	8	1	no	—
58	C₂²×C₁₄₉	C₂²⋊Q₈	1	1	yes	characteristic
59	C₂²×C₁₄₉	#123	1	1	yes	characteristic
60	C₂²×C₁₄₉	#127	1	1	yes	characteristic
61	C₂²×C₁₄₉	C₂×SD₁₆	1	2	yes	—
62	C₂²×C₁₄₉	AGL(1,ℤ/8)	1	2	yes	—
63	C₄×C₁₄₉	—	2	4	no	—
64	C₄×C₁₄₉	—	8	1	no	—
65	C₄×C₁₄₉	C₂²⋊Q₈	1	2	yes	—
66	C₄×C₁₄₉	#123	1	2	yes	—
67	Dic₁₄₉	—	8	2	no	—
68	C₂×C₁₄₉	—	8	2	no	—
69	C₂×C₁₄₉	#399	1	1	yes	characteristic
70	C₂×C₁₄₉	#476	1	2	yes	—
71	C₂×C₁₄₉	#481	1	2	yes	—
72	C₂×C₁₄₉	#492	1	1	yes	characteristic
73	C₂×C₁₄₉	#496	1	1	yes	characteristic
74	C₁₄₉	#2094	1	1	yes	characteristic, complemented, Sylow
75	#2094	—	149	1	no	maximal, Sylow
76	C₂×#106	—	149	2	no	—
77	C₂×#102	—	149	2	no	—
78	#390	D₁₄₉	1	1	yes	characteristic
79	#384	—	149	1	no	—
80	#336	—	149	1	no	—
81	C₂²×D₄	C₂×D₁₄₉	1	1	yes	characteristic
82	C₂²×C₈	—	149	2	no	—
83	C₂×C₄⋊C₄	—	298	2	no	—
84	C₂×C₄⋊C₄	—	149	2	no	—
85	C₂×C₄⋊C₄	C₂×D₁₄₉	1	1	yes	characteristic
86	C₂×C₂²⋊C₄	—	2	2	no	—
87	C₂×C₄²	C₂×D₁₄₉	1	1	yes	characteristic
88	#106	—	298	4	no	—
89	#102	—	298	4	no	—
90	C₂⁴	—	2	1	no	—
91	C₂×D₄	—	2	2	no	—
92	C₂×D₄	—	2	2	no	—
93	C₂×D₄	—	2	1	no	—
94	C₂×D₄	Dic₂₉₈	1	2	yes	—
95	C₂²×C₄	—	2	1	no	—
96	C₂²×C₄	—	298	2	no	—
97	C₂²×C₄	C₂²⋊D₁₄₉	1	2	yes	—
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	2	no	—
103	C₄⋊C₄	—	298	2	no	—
104	C₄⋊C₄	—	298	2	no	—
105	C₄⋊C₄	—	596	4	no	—
106	C₂²⋊C₄	—	4	4	no	—
107	C₄²	—	2	2	no	—
108	C₂³	—	2	2	no	—
109	C₂³	—	4	1	no	—
110	C₂³	—	4	2	no	—
111	C₂³	—	4	2	no	—
112	C₂³	—	4	1	no	—
113	C₂³	C₂×C₂²⋊D₁₄₉	1	1	yes	characteristic
114	D₄	—	2	4	no	—
115	D₄	—	4	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	—	596	2	no	—
121	C₂×C₄	—	596	4	no	—
122	C₂×C₄	#14558724	1	4	yes	—
123	C₂×C₄	D₄×D₁₄₉	1	2	yes	—
124	C₂×C₄	C₂×Dic₂₉₈	1	1	yes	characteristic
125	C₂×C₄	#14558735	1	2	yes	—
126	C₂×C₄	#14558736	1	1	yes	characteristic
127	C₈	—	298	4	no	—
128	C₂²	—	4	4	no	—
129	C₂²	—	4	2	no	—
130	C₂²	—	8	2	no	—
131	C₂²	—	8	1	no	—
132	C₂²	—	8	1	no	—
133	C₂²	#18011890	1	1	yes	characteristic
134	C₂²	#18011892	1	1	yes	characteristic
135	C₂²	#18011907	1	1	yes	characteristic
136	C₂²	C₂×#14558724	1	2	yes	—
137	C₂²	#18011936	1	2	yes	—
138	C₄	—	2	4	no	—
139	C₄	—	8	1	no	—
140	C₄	—	1192	2	no	—
141	C₄	#18011888	1	2	yes	—
142	C₄	#18011893	1	2	yes	—
143	C₂	—	8	2	no	—
144	C₂	#23749561	1	1	yes	characteristic
145	C₂	#23749915	1	2	yes	—
146	C₂	#23749941	1	2	yes	—
147	C₂	#23749997	1	1	yes	characteristic
148	C₂	#23750015	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#18011842

C₂

yes

maximal, complemented

C₂×#18011832

C₂

yes

maximal

#23749495

C₂

yes

maximal, characteristic, complemented

#23749380

C₂

yes

maximal, characteristic, complemented

#390×C₁₄₉

C₂

yes

maximal, characteristic

C₂²×#11924746

C₂²

yes

—

C₂×C₄⋊Dic₁₄₉

—

C₂×C₄⋊Dic₁₄₉

C₂²

yes

—

#18011842

—

#18011832

—

C₂²×D₄×C₁₄₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₄₉

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₁₄₉

—

C₂×C₄²×C₁₄₉

C₂²

yes

characteristic

C₂²×Dic₁₄₉

—

C₂×#11924746

—

C₂×#11924746

—

C₂×#11924746

D₄

yes

—

C₄⋊Dic₁₄₉

—

C₄⋊Dic₁₄₉

—

C₄⋊Dic₁₄₉

—

C₂⁴×C₁₄₉

—

C₂×D₄×C₁₄₉

—

C₂×D₄×C₁₄₉

—

C₂×D₄×C₁₄₉

—

C₂×D₄×C₁₄₉

Q₈

yes

—

C₂²×C₄×C₁₄₉

—

C₂²×C₄×C₁₄₉

D₄

yes

—

C₂²×C₄×C₁₄₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₄₉

—

C₂²⋊C₄×C₁₄₉

—

C₄²×C₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

#11924746

—

C₂³×C₁₄₉

—

C₂³×C₁₄₉

—

C₂³×C₁₄₉

—

C₂³×C₁₄₉

—

C₂³×C₁₄₉

—

C₂³×C₁₄₉

C₂×D₄

yes

characteristic

D₄×C₁₄₉

—

D₄×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

SD₁₆

yes

—

C₂×C₄×C₁₄₉

C₂×D₄

yes

—

C₂×C₄×C₁₄₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₄₉

#41

yes

—

C₂×C₄×C₁₄₉

#41

yes

characteristic

C₂²×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

C₂²⋊Q₈

yes

C₂²×C₁₄₉

yes

C₂²×C₁₄₉

yes

C₂²×C₁₄₉

C₂×SD₁₆

yes

—

C₂²×C₁₄₉

AGL(1,ℤ/8)

yes

—

C₄×C₁₄₉

—

C₄×C₁₄₉

—

C₄×C₁₄₉

C₂²⋊Q₈

yes

—

C₄×C₁₄₉

#123

yes

—

Dic₁₄₉

—

C₂×C₁₄₉

—

C₂×C₁₄₉

#399

yes

characteristic

C₂×C₁₄₉

#476

yes

—

C₂×C₁₄₉

#481

yes

—

C₂×C₁₄₉

yes

C₂×C₁₄₉

yes

C₁₄₉

yes

characteristic, complemented, Sylow

#2094

—

149

maximal, Sylow

C₂×#106

—

149

—

C₂×#102

—

149

—

#390

D₁₄₉

yes

characteristic

#384

—

149

—

#336

—

149

—

C₂²×D₄

C₂×D₁₄₉

yes

characteristic

C₂²×C₈

—

149

—

C₂×C₄⋊C₄

—

298

—

C₂×C₄⋊C₄

—

149

—

C₂×C₄⋊C₄

C₂×D₁₄₉

yes

characteristic

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₁₄₉

yes

characteristic

#106

—

298

—

#102

—

298

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

Dic₂₉₈

yes

—

C₂²×C₄

—

C₂²×C₄

—

298

—

C₂²×C₄

C₂²⋊D₁₄₉

yes

—

C₂²×C₄

C₂²×D₁₄₉

yes

characteristic

C₂×C₈

—

298

—

100

C₂×C₈

—

298

—

101

C₂×C₈

—

149

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

298

—

104

C₄⋊C₄

—

298

—

105

C₄⋊C₄

—

596

—

106

C₂²⋊C₄

—

107

C₄²

—

108

C₂³

—

109

C₂³

—

110

C₂³

—

111

C₂³

—

112

C₂³

—

113

C₂³

C₂×C₂²⋊D₁₄₉

yes

characteristic

114

D₄

—

115

D₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

596

—

121

C₂×C₄

—

596

—

122

C₂×C₄

#14558724

yes

—

123

C₂×C₄

D₄×D₁₄₉

yes

—

124

C₂×C₄

C₂×Dic₂₉₈

yes

characteristic

125

C₂×C₄

#14558735

yes

—

126

C₂×C₄

#14558736

yes

characteristic

127

C₈

—

298

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

#18011890

yes

characteristic

134

C₂²

#18011892

yes

characteristic

135

C₂²

#18011907

yes

characteristic

136

C₂²

C₂×#14558724

yes

—

137

C₂²

#18011936

yes

—

138

C₄

—

139

C₄

—

140

C₄

—

1192

—

141

C₄

#18011888

yes

—

142

C₄

#18011893

yes

—

143

C₂

—

144

C₂

#23749561

yes

characteristic

145

C₂

#23749915

yes

—

146

C₂

#23749941

yes

—

147

C₂

#23749997

yes

characteristic

148

C₂

#23750015

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₄₉
commutator, cosocle	C₂²×C₄×C₁₄₉	C₂³
Frattini	C₂²×C₄	C₂²×D₁₄₉
Fitting	#390×C₁₄₉	C₂
socle	C₂³×C₁₄₉	C₂×D₄
prime radical	C₂²×D₄×C₁₄₉	C₂²
hypercenter	#390	D₁₄₉
hypocenter	C₁₄₉	#2094
LCS	G ▷ C₂²×C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #390

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₄₉

commutator, cosocle

C₂²×C₄×C₁₄₉

C₂³

Frattini

C₂²×C₄

C₂²×D₁₄₉

Fitting

#390×C₁₄₉

C₂

socle

C₂³×C₁₄₉

C₂×D₄

prime radical

C₂²×D₄×C₁₄₉

C₂²

D₁₄₉

C₁₄₉

G ▷ C₂²×C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #390

N	H ≅ G/N	type
1. C₂×#18011842	C₂	semidirect
3. #23749495	C₂	semidirect
4. #23749380	C₂	semidirect
74. C₁₄₉	#2094	semidirect

H ≅ G/N

type

1. C₂×#18011842

C₂

semidirect

3. #23749495

C₂

semidirect

4. #23749380

C₂

semidirect

74. C₁₄₉

#2094

semidirect

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

subgroup	index
C₂×#18011842	2
C₂×#18011832	2
#23749495	2
#23749380	2
#390×C₁₄₉	2
#2094	149

subgroup

index

#390×C₁₄₉

149

Subgroups — #34046224

Conjugacy classes of subgroups

Complements

Maximal subgroups