Group #34046104 — subgroups

Subgroups — #34046104

Order 19072.

258 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 81 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×#18011832	C₂	1	2	yes	maximal
2	#23749491	C₂	1	1	yes	maximal, characteristic, complemented
3	#23749387	C₂	1	2	yes	maximal, 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	C₂²	1	2	yes	—
8	#18011832	—	2	4	no	—
9	#18011818	—	2	2	no	—
10	C₂²×D₄×C₁₄₉	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊C₄×C₁₄₉	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊C₄×C₁₄₉	—	2	2	no	—
13	C₂×C₄²×C₁₄₉	C₂²	1	1	yes	characteristic
14	C₂²×Dic₁₄₉	—	2	2	no	—
15	C₂×#11924746	—	2	2	no	—
16	C₂×#11924746	—	2	2	no	—
17	C₂×#11924746	—	2	2	no	—
18	C₄⋊Dic₁₄₉	—	2	2	no	—
19	C₄⋊Dic₁₄₉	—	2	2	no	—
20	C₂⁴×C₁₄₉	—	2	1	no	—
21	C₂×D₄×C₁₄₉	—	2	2	no	—
22	C₂×D₄×C₁₄₉	—	2	2	no	—
23	C₂×D₄×C₁₄₉	—	2	1	no	—
24	C₂×D₄×C₁₄₉	D₄	1	2	yes	—
25	C₂²×C₄×C₁₄₉	—	2	1	no	—
26	C₂²×C₄×C₁₄₉	D₄	1	2	yes	—
27	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
28	C₄⋊C₄×C₁₄₉	—	2	2	no	—
29	C₂²⋊C₄×C₁₄₉	—	4	4	no	—
30	C₄²×C₁₄₉	—	2	2	no	—
31	C₂×Dic₁₄₉	—	4	2	no	—
32	C₂×Dic₁₄₉	—	4	2	no	—
33	C₂×Dic₁₄₉	—	4	2	no	—
34	#11924746	—	4	2	no	—
35	C₂³×C₁₄₉	—	2	2	no	—
36	C₂³×C₁₄₉	—	4	1	no	—
37	C₂³×C₁₄₉	—	4	1	no	—
38	C₂³×C₁₄₉	—	4	2	no	—
39	C₂³×C₁₄₉	—	4	2	no	—
40	C₂³×C₁₄₉	C₂×D₄	1	1	yes	characteristic
41	D₄×C₁₄₉	—	2	4	no	—
42	D₄×C₁₄₉	—	4	2	no	—
43	C₂×C₄×C₁₄₉	—	2	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	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₄₉	#41	1	2	yes	—
50	C₂×C₄×C₁₄₉	#41	1	2	yes	—
51	C₂×C₄×C₁₄₉	#41	1	1	yes	characteristic
52	C₂²×C₁₄₉	—	4	2	no	—
53	C₂²×C₁₄₉	—	4	4	no	—
54	C₂²×C₁₄₉	—	8	2	no	—
55	C₂²×C₁₄₉	—	8	1	no	—
56	C₂²×C₁₄₉	—	8	1	no	—
57	C₂²×C₁₄₉	#121	1	1	yes	characteristic
58	C₂²×C₁₄₉	#123	1	1	yes	characteristic
59	C₂²×C₁₄₉	#124	1	1	yes	characteristic
60	C₂²×C₁₄₉	#135	1	2	yes	—
61	C₂²×C₁₄₉	AGL(1,ℤ/8)	1	2	yes	—
62	C₄×C₁₄₉	—	4	2	no	—
63	C₄×C₁₄₉	—	8	1	no	—
64	C₄×C₁₄₉	#123	1	2	yes	—
65	C₄×C₁₄₉	#126	1	2	yes	—
66	Dic₁₄₉	—	8	2	no	—
67	C₂×C₁₄₉	—	8	2	no	—
68	C₂×C₁₄₉	#397	1	1	yes	characteristic
69	C₂×C₁₄₉	#463	1	2	yes	—
70	C₂×C₁₄₉	#482	1	2	yes	—
71	C₂×C₁₄₉	#488	1	1	yes	characteristic
72	C₂×C₁₄₉	#490	1	1	yes	characteristic
73	C₁₄₉	#2073	1	1	yes	characteristic, complemented, Sylow
74	#2073	—	149	1	no	maximal, Sylow
75	C₂×#102	—	149	2	no	—
76	#390	D₁₄₉	1	1	yes	characteristic
77	#382	—	149	1	no	—
78	#340	—	149	2	no	—
79	#336	—	149	1	no	—
80	C₂²×D₄	C₂×D₁₄₉	1	1	yes	characteristic
81	C₂²×C₈	—	149	2	no	—
82	C₂×C₄⋊C₄	—	149	2	no	—
83	C₂×C₄⋊C₄	C₂×D₁₄₉	1	1	yes	characteristic
84	C₂×C₂²⋊C₄	—	2	2	no	—
85	C₂×C₄²	C₂×D₁₄₉	1	1	yes	characteristic
86	#102	—	298	4	no	—
87	#95	—	298	2	no	—
88	C₂⁴	—	2	1	no	—
89	C₂×D₄	—	2	2	no	—
90	C₂×D₄	—	2	2	no	—
91	C₂×D₄	—	2	1	no	—
92	C₂×D₄	D₅₉₆	1	2	yes	—
93	C₂²×C₄	—	2	1	no	—
94	C₂²×C₄	—	298	2	no	—
95	C₂²×C₄	C₂²⋊D₁₄₉	1	2	yes	—
96	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
97	C₂×C₈	—	298	2	no	—
98	C₂×C₈	—	298	2	no	—
99	C₂×C₈	—	298	2	no	—
100	C₄⋊C₄	—	2	2	no	—
101	C₄⋊C₄	—	298	2	no	—
102	C₄⋊C₄	—	298	2	no	—
103	C₂²⋊C₄	—	4	4	no	—
104	C₄²	—	2	2	no	—
105	C₂³	—	2	2	no	—
106	C₂³	—	4	2	no	—
107	C₂³	—	4	2	no	—
108	C₂³	—	4	1	no	—
109	C₂³	—	4	1	no	—
110	C₂³	C₂×C₂²⋊D₁₄₉	1	1	yes	characteristic
111	D₄	—	2	4	no	—
112	D₄	—	4	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	4	2	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	—	596	2	no	—
119	C₂×C₄	—	596	2	no	—
120	C₂×C₄	—	596	2	no	—
121	C₂×C₄	C₂×D₅₉₆	1	1	yes	characteristic
122	C₂×C₄	#14558735	1	2	yes	—
123	C₂×C₄	#14558735	1	2	yes	—
124	C₂×C₄	#14558736	1	1	yes	characteristic
125	C₈	—	596	2	no	—
126	C₂²	—	4	2	no	—
127	C₂²	—	4	4	no	—
128	C₂²	—	8	2	no	—
129	C₂²	—	8	1	no	—
130	C₂²	—	8	1	no	—
131	C₂²	#18011885	1	1	yes	characteristic
132	C₂²	#18011892	1	1	yes	characteristic
133	C₂²	#18011898	1	1	yes	characteristic
134	C₂²	#18011931	1	2	yes	—
135	C₂²	#18011937	1	2	yes	—
136	C₄	—	4	2	no	—
137	C₄	—	8	1	no	—
138	C₄	—	1192	2	no	—
139	C₄	#18011895	1	2	yes	—
140	C₄	#18011904	1	2	yes	—
141	C₂	—	8	2	no	—
142	C₂	#23749550	1	1	yes	characteristic
143	C₂	#23749840	1	2	yes	—
144	C₂	#23749946	1	2	yes	—
145	C₂	#23749979	1	1	yes	characteristic
146	C₂	#23749989	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#18011832

C₂

yes

maximal

#23749491

C₂

yes

maximal, characteristic, complemented

#23749387

C₂

yes

maximal, complemented

#23749380

C₂

yes

maximal, characteristic, complemented

#390×C₁₄₉

C₂

yes

maximal, characteristic

C₂²×#11924746

C₂²

yes

—

C₂×C₄⋊Dic₁₄₉

C₂²

yes

—

#18011832

—

#18011818

—

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

—

C₄⋊Dic₁₄₉

—

C₄⋊Dic₁₄₉

—

C₂⁴×C₁₄₉

—

C₂×D₄×C₁₄₉

—

C₂×D₄×C₁₄₉

—

C₂×D₄×C₁₄₉

—

C₂×D₄×C₁₄₉

D₄

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

—

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

—

C₂×C₄×C₁₄₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₄₉

#41

yes

—

C₂×C₄×C₁₄₉

#41

yes

—

C₂×C₄×C₁₄₉

#41

yes

characteristic

C₂²×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

yes

C₂²×C₁₄₉

yes

C₂²×C₁₄₉

yes

C₂²×C₁₄₉

yes

—

C₂²×C₁₄₉

AGL(1,ℤ/8)

yes

—

C₄×C₁₄₉

—

C₄×C₁₄₉

—

C₄×C₁₄₉

#123

yes

—

C₄×C₁₄₉

#126

yes

—

Dic₁₄₉

—

C₂×C₁₄₉

—

C₂×C₁₄₉

#397

yes

characteristic

C₂×C₁₄₉

#463

yes

—

C₂×C₁₄₉

#482

yes

—

C₂×C₁₄₉

yes

C₂×C₁₄₉

yes

C₁₄₉

yes

characteristic, complemented, Sylow

#2073

—

149

maximal, Sylow

C₂×#102

—

149

—

#390

D₁₄₉

yes

characteristic

#382

—

149

—

#340

—

149

—

#336

—

149

—

C₂²×D₄

C₂×D₁₄₉

yes

characteristic

C₂²×C₈

—

149

—

C₂×C₄⋊C₄

—

149

—

C₂×C₄⋊C₄

C₂×D₁₄₉

yes

characteristic

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₁₄₉

yes

characteristic

#102

—

298

—

#95

—

298

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

D₅₉₆

yes

—

C₂²×C₄

—

C₂²×C₄

—

298

—

C₂²×C₄

C₂²⋊D₁₄₉

yes

—

C₂²×C₄

C₂²×D₁₄₉

yes

characteristic

C₂×C₈

—

298

—

C₂×C₈

—

298

—

C₂×C₈

—

298

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

298

—

102

C₄⋊C₄

—

298

—

103

C₂²⋊C₄

—

104

C₄²

—

105

C₂³

—

106

C₂³

—

107

C₂³

—

108

C₂³

—

109

C₂³

—

110

C₂³

C₂×C₂²⋊D₁₄₉

yes

characteristic

111

D₄

—

112

D₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

596

—

119

C₂×C₄

—

596

—

120

C₂×C₄

—

596

—

121

C₂×C₄

C₂×D₅₉₆

yes

characteristic

122

C₂×C₄

#14558735

yes

—

123

C₂×C₄

#14558735

yes

—

124

C₂×C₄

#14558736

yes

characteristic

125

C₈

—

596

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

#18011885

yes

characteristic

132

C₂²

#18011892

yes

characteristic

133

C₂²

#18011898

yes

characteristic

134

C₂²

#18011931

yes

—

135

C₂²

#18011937

yes

—

136

C₄

—

137

C₄

—

138

C₄

—

1192

—

139

C₄

#18011895

yes

—

140

C₄

#18011904

yes

—

141

C₂

—

142

C₂

#23749550

yes

characteristic

143

C₂

#23749840

yes

—

144

C₂

#23749946

yes

—

145

C₂

#23749979

yes

characteristic

146

C₂

#23749989

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	#2073
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
2. #23749491	C₂	semidirect
3. #23749387	C₂	semidirect
4. #23749380	C₂	semidirect
73. C₁₄₉	#2073	semidirect

H ≅ G/N

type

2. #23749491

C₂

semidirect

3. #23749387

C₂

semidirect

4. #23749380

C₂

semidirect

73. C₁₄₉

#2073

semidirect

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

subgroup	index
C₂×#18011832	2
#23749491	2
#23749387	2
#23749380	2
#390×C₁₄₉	2
#2073	149

subgroup

index

#390×C₁₄₉

149

Subgroups — #34046104

Conjugacy classes of subgroups

Complements

Maximal subgroups