Group #34046186 — subgroups

Subgroups — #34046186

Order 19072.

224 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 89 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×#18011845	C₂	1	1	yes	maximal, characteristic
2	C₂×#18011840	C₂	1	1	yes	maximal, characteristic
3	#23749524	C₂	1	2	yes	maximal
4	#23749387	C₂	1	2	yes	maximal
5	#340×C₁₄₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11924746	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₁₄₉	—	2	2	no	—
8	C₂×C₄⋊Dic₁₄₉	C₂²	1	2	yes	—
9	C₂×C₄×Dic₁₄₉	C₂²	1	1	yes	characteristic
10	#18011845	—	2	2	no	—
11	#18011840	—	2	2	no	—
12	#18011818	—	2	2	no	—
13	C₂²×C₈×C₁₄₉	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₁₄₉	C₂²	1	2	yes	—
15	C₂²×Dic₁₄₉	—	2	2	no	—
16	C₂²×Dic₁₄₉	D₄	1	1	yes	characteristic
17	C₂²×Dic₁₄₉	D₄	1	1	yes	characteristic
18	C₂×#11924746	—	2	1	no	—
19	C₂×#11924746	—	2	2	no	—
20	C₄⋊Dic₁₄₉	—	2	2	no	—
21	C₄⋊Dic₁₄₉	—	2	2	no	—
22	C₄⋊Dic₁₄₉	—	4	4	no	—
23	C₄×Dic₁₄₉	—	2	1	no	—
24	C₄×Dic₁₄₉	—	2	1	no	—
25	C₂²×C₄×C₁₄₉	—	2	2	no	—
26	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
27	C₂×C₈×C₁₄₉	—	2	2	no	—
28	C₂×C₈×C₁₄₉	D₄	1	2	yes	—
29	C₄⋊C₄×C₁₄₉	—	2	2	no	—
30	C₄⋊C₄×C₁₄₉	Q₈	1	4	yes	—
31	C₂×Dic₁₄₉	—	2	1	no	—
32	C₂×Dic₁₄₉	—	2	1	no	—
33	C₂×Dic₁₄₉	—	2	1	no	—
34	C₂×Dic₁₄₉	—	2	2	no	—
35	C₂×Dic₁₄₉	—	4	2	no	—
36	C₂×Dic₁₄₉	—	4	2	no	—
37	C₂×Dic₁₄₉	—	4	2	no	—
38	C₂×Dic₁₄₉	Q₁₆	1	2	yes	—
39	#11924746	—	4	1	no	—
40	C₂³×C₁₄₉	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₄₉	—	2	4	no	—
42	C₂×C₄×C₁₄₉	—	4	2	no	—
43	C₂×C₄×C₁₄₉	—	4	2	no	—
44	C₂×C₄×C₁₄₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₄₉	C₂×Q₈	1	2	yes	—
46	C₂×C₄×C₁₄₉	#41	1	2	yes	—
47	C₂×C₄×C₁₄₉	#41	1	1	yes	characteristic
48	C₈×C₁₄₉	—	2	2	no	—
49	C₂²×C₁₄₉	#121	1	1	yes	characteristic
50	C₂²×C₁₄₉	C₂²⋊Q₈	1	2	yes	—
51	C₂²×C₁₄₉	C₂×Q₁₆	1	1	yes	characteristic
52	C₂²×C₁₄₉	#135	1	1	yes	characteristic
53	C₂²×C₁₄₉	AGL(1,ℤ/8)	1	2	yes	—
54	C₄×C₁₄₉	—	4	4	no	—
55	C₄×C₁₄₉	#123	1	1	yes	characteristic
56	C₄×C₁₄₉	#125	1	2	yes	—
57	C₄×C₁₄₉	C₄⋊Q₈	1	1	yes	characteristic
58	Dic₁₄₉	—	2	2	no	—
59	Dic₁₄₉	—	4	1	no	—
60	Dic₁₄₉	—	8	2	no	—
61	C₂×C₁₄₉	#398	1	1	yes	characteristic
62	C₂×C₁₄₉	#467	1	1	yes	characteristic
63	C₂×C₁₄₉	#469	1	1	yes	characteristic
64	C₂×C₁₄₉	#477	1	2	yes	—
65	C₂×C₁₄₉	#478	1	2	yes	—
66	C₁₄₉	#2087	1	1	yes	characteristic, complemented, Sylow
67	#2087	—	149	1	no	maximal, Sylow
68	C₂×#107	—	149	1	no	—
69	C₂×C₄⋊C₈	—	149	1	no	—
70	#389	—	149	2	no	—
71	#340	—	149	2	no	—
72	#340	D₁₄₉	1	1	yes	characteristic
73	C₂²×C₈	—	149	1	no	—
74	C₂²×C₈	C₂×D₁₄₉	1	1	yes	characteristic
75	C₂×C₄⋊C₄	—	298	2	no	—
76	C₂×C₄⋊C₄	—	149	2	no	—
77	C₂×C₄⋊C₄	C₂×D₁₄₉	1	2	yes	—
78	C₂×C₄²	—	149	1	no	—
79	#107	—	298	2	no	—
80	C₄⋊C₈	—	298	2	no	—
81	#95	—	298	2	no	—
82	C₂²×C₄	—	2	2	no	—
83	C₂²×C₄	—	298	2	no	—
84	C₂²×C₄	—	149	1	no	—
85	C₂²×C₄	—	149	1	no	—
86	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	298	1	no	—
89	C₂×C₈	—	298	2	no	—
90	C₂×C₈	D₅₉₆	1	2	yes	—
91	C₄⋊C₄	—	2	2	no	—
92	C₄⋊C₄	—	298	2	no	—
93	C₄⋊C₄	—	298	2	no	—
94	C₄⋊C₄	—	596	4	no	—
95	C₄⋊C₄	Dic₂₉₈	1	4	yes	—
96	C₄²	—	298	1	no	—
97	C₄²	—	298	1	no	—
98	C₂³	D₄×D₁₄₉	1	1	yes	characteristic
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	298	1	no	—
103	C₂×C₄	—	298	1	no	—
104	C₂×C₄	—	298	2	no	—
105	C₂×C₄	—	298	1	no	—
106	C₂×C₄	—	596	2	no	—
107	C₂×C₄	—	596	2	no	—
108	C₂×C₄	—	596	2	no	—
109	C₂×C₄	—	149	2	no	—
110	C₂×C₄	C₂×D₅₉₆	1	1	yes	characteristic
111	C₂×C₄	C₂×Dic₂₉₈	1	2	yes	—
112	C₂×C₄	#14558735	1	2	yes	—
113	C₂×C₄	#14558737	1	1	yes	characteristic
114	C₈	—	2	2	no	—
115	C₈	—	596	1	no	—
116	C₂²	#18011886	1	1	yes	characteristic
117	C₂²	#18011888	1	2	yes	—
118	C₂²	Q₁₆×D₁₄₉	1	1	yes	characteristic
119	C₂²	#18011930	1	1	yes	characteristic
120	C₂²	#18011938	1	2	yes	—
121	C₄	—	4	4	no	—
122	C₄	—	298	2	no	—
123	C₄	—	596	1	no	—
124	C₄	—	1192	2	no	—
125	C₄	#18011895	1	1	yes	characteristic
126	C₄	#18011902	1	2	yes	—
127	C₄	#18011911	1	1	yes	characteristic
128	C₂	#23749557	1	1	yes	characteristic
129	C₂	#23749867	1	1	yes	characteristic
130	C₂	#23749879	1	1	yes	characteristic
131	C₂	#23749918	1	2	yes	—
132	C₂	#23749923	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#18011845

C₂

yes

maximal, characteristic

C₂×#18011840

C₂

yes

maximal, characteristic

#23749524

C₂

yes

maximal

#23749387

C₂

yes

maximal

#340×C₁₄₉

C₂

yes

maximal, characteristic

C₂²×#11924746

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₄₉

—

C₂×C₄⋊Dic₁₄₉

C₂²

yes

—

C₂×C₄×Dic₁₄₉

C₂²

yes

—

—

—

C₂²×C₈×C₁₄₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₄₉

C₂²

yes

—

C₂²×Dic₁₄₉

—

C₂²×Dic₁₄₉

D₄

yes

characteristic

C₂²×Dic₁₄₉

D₄

yes

characteristic

C₂×#11924746

—

C₂×#11924746

—

C₄⋊Dic₁₄₉

—

C₄⋊Dic₁₄₉

—

C₄⋊Dic₁₄₉

—

C₄×Dic₁₄₉

—

C₄×Dic₁₄₉

—

C₂²×C₄×C₁₄₉

—

C₂²×C₄×C₁₄₉

C₂³

yes

characteristic

C₂×C₈×C₁₄₉

—

C₂×C₈×C₁₄₉

D₄

yes

—

C₄⋊C₄×C₁₄₉

—

C₄⋊C₄×C₁₄₉

Q₈

yes

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

Q₁₆

yes

—

#11924746

—

C₂³×C₁₄₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₄₉

C₂×Q₈

yes

—

C₂×C₄×C₁₄₉

#41

yes

—

C₂×C₄×C₁₄₉

#41

yes

characteristic

C₈×C₁₄₉

—

C₂²×C₁₄₉

#121

yes

characteristic

C₂²×C₁₄₉

C₂²⋊Q₈

yes

—

C₂²×C₁₄₉

C₂×Q₁₆

yes

C₂²×C₁₄₉

yes

C₂²×C₁₄₉

yes

—

C₄×C₁₄₉

—

C₄×C₁₄₉

#123

yes

characteristic

C₄×C₁₄₉

#125

yes

—

C₄×C₁₄₉

C₄⋊Q₈

yes

characteristic

Dic₁₄₉

—

Dic₁₄₉

—

Dic₁₄₉

—

C₂×C₁₄₉

yes

C₂×C₁₄₉

yes

C₂×C₁₄₉

yes

C₂×C₁₄₉

yes

—

C₂×C₁₄₉

#478

yes

—

C₁₄₉

#2087

yes

characteristic, complemented, Sylow

#2087

—

149

maximal, Sylow

C₂×#107

—

149

—

C₂×C₄⋊C₈

—

149

—

#389

—

149

—

#340

—

149

—

#340

D₁₄₉

yes

characteristic

C₂²×C₈

—

149

—

C₂²×C₈

C₂×D₁₄₉

yes

characteristic

C₂×C₄⋊C₄

—

298

—

C₂×C₄⋊C₄

—

149

—

C₂×C₄⋊C₄

C₂×D₁₄₉

yes

—

C₂×C₄²

—

149

—

#107

—

298

—

C₄⋊C₈

—

298

—

#95

—

298

—

C₂²×C₄

—

C₂²×C₄

—

298

—

C₂²×C₄

—

149

—

C₂²×C₄

—

149

—

C₂²×C₄

C₂²×D₁₄₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

298

—

C₂×C₈

—

298

—

C₂×C₈

D₅₉₆

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

298

—

C₄⋊C₄

—

298

—

C₄⋊C₄

—

596

—

C₄⋊C₄

Dic₂₉₈

yes

—

C₄²

—

298

—

C₄²

—

298

—

C₂³

D₄×D₁₄₉

yes

characteristic

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

298

—

103

C₂×C₄

—

298

—

104

C₂×C₄

—

298

—

105

C₂×C₄

—

298

—

106

C₂×C₄

—

596

—

107

C₂×C₄

—

596

—

108

C₂×C₄

—

596

—

109

C₂×C₄

—

149

—

110

C₂×C₄

C₂×D₅₉₆

yes

characteristic

111

C₂×C₄

C₂×Dic₂₉₈

yes

—

112

C₂×C₄

#14558735

yes

—

113

C₂×C₄

#14558737

yes

characteristic

114

C₈

—

115

C₈

—

596

—

116

C₂²

#18011886

yes

characteristic

117

C₂²

#18011888

yes

—

118

C₂²

Q₁₆×D₁₄₉

yes

characteristic

119

C₂²

#18011930

yes

characteristic

120

C₂²

#18011938

yes

—

121

C₄

—

122

C₄

—

298

—

123

C₄

—

596

—

124

C₄

—

1192

—

125

C₄

#18011895

yes

characteristic

126

C₄

#18011902

yes

—

127

C₄

#18011911

yes

characteristic

128

C₂

#23749557

yes

characteristic

129

C₂

#23749867

yes

characteristic

130

C₂

#23749879

yes

characteristic

131

C₂

#23749918

yes

—

132

C₂

#23749923

yes

—

	subgroup	G/H
center	C₂³	D₄×D₁₄₉
commutator, cosocle	C₂²×C₄×C₁₄₉	C₂³
Frattini	C₂²×C₄	C₂²×D₁₄₉
Fitting	#340×C₁₄₉	C₂
socle, prime radical	C₂³×C₁₄₉	C₂×D₄
hypercenter	#340	D₁₄₉
hypocenter	C₁₄₉	#2087
LCS	G ▷ C₂²×C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

subgroup

G/H

center

C₂³

D₄×D₁₄₉

commutator, cosocle

C₂²×C₄×C₁₄₉

C₂³

Frattini

C₂²×C₄

C₂²×D₁₄₉

Fitting

#340×C₁₄₉

C₂

socle, prime radical

C₂³×C₁₄₉

C₂×D₄

D₁₄₉

C₁₄₉

G ▷ C₂²×C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

N	H ≅ G/N	type
66. C₁₄₉	#2087	semidirect

H ≅ G/N

type

66. C₁₄₉

#2087

semidirect

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

subgroup	index
C₂×#18011845	2
C₂×#18011840	2
#23749524	2
#23749387	2
#340×C₁₄₉	2
#2087	149

subgroup

index

#340×C₁₄₉

149

Subgroups — #34046186

Conjugacy classes of subgroups

Complements

Maximal subgroups