Group #34046111 — subgroups

Subgroups — #34046111

Order 19072.

222 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 79 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×#18011836	C₂	1	2	yes	maximal
2	#23749531	C₂	1	1	yes	maximal, characteristic
3	#23749491	C₂	1	1	yes	maximal, characteristic
4	#23749388	C₂	1	2	yes	maximal
5	#336×C₁₄₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₉₈	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₁₄₉	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₁₄₉	C₂²	1	2	yes	—
9	#18011836	—	2	4	no	—
10	#18011818	—	2	2	no	—
11	#18011818	—	2	2	no	—
12	C₂²×C₈×C₁₄₉	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₄₉	C₂²	1	1	yes	characteristic
14	C₂×Dic₂₉₈	—	2	1	no	—
15	C₂×Dic₂₉₈	—	2	2	no	—
16	C₂×Dic₂₉₈	—	2	2	no	—
17	C₂×Dic₂₉₈	D₄	1	2	yes	—
18	C₂²×Dic₁₄₉	—	2	1	no	—
19	C₂²×Dic₁₄₉	—	2	2	no	—
20	C₂²×Dic₁₄₉	—	2	1	no	—
21	C₄⋊Dic₁₄₉	—	2	2	no	—
22	C₄⋊Dic₁₄₉	—	2	2	no	—
23	C₄⋊Dic₁₄₉	—	2	2	no	—
24	C₂²×C₄×C₁₄₉	D₄	1	2	yes	—
25	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
26	C₂×C₈×C₁₄₉	—	2	2	no	—
27	C₂×C₈×C₁₄₉	—	2	2	no	—
28	C₂×C₈×C₁₄₉	—	2	2	no	—
29	C₄²×C₁₄₉	—	2	2	no	—
30	Dic₂₉₈	—	2	4	no	—
31	Dic₂₉₈	—	4	2	no	—
32	C₂×Dic₁₄₉	—	2	2	no	—
33	C₂×Dic₁₄₉	—	4	1	no	—
34	C₂×Dic₁₄₉	—	4	1	no	—
35	C₂×Dic₁₄₉	—	4	2	no	—
36	C₂×Dic₁₄₉	—	4	1	no	—
37	C₂×Dic₁₄₉	—	4	2	no	—
38	C₂×Dic₁₄₉	—	4	2	no	—
39	C₂×Dic₁₄₉	—	4	2	no	—
40	C₂³×C₁₄₉	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₄₉	—	2	2	no	—
42	C₂×C₄×C₁₄₉	—	2	2	no	—
43	C₂×C₄×C₁₄₉	—	2	2	no	—
44	C₂×C₄×C₁₄₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₄₉	#41	1	2	yes	—
46	C₂×C₄×C₁₄₉	#41	1	2	yes	—
47	C₂×C₄×C₁₄₉	#41	1	1	yes	characteristic
48	C₈×C₁₄₉	—	4	2	no	—
49	C₂²×C₁₄₉	#121	1	1	yes	characteristic
50	C₂²×C₁₄₉	#123	1	1	yes	characteristic
51	C₂²×C₁₄₉	#124	1	1	yes	characteristic
52	C₂²×C₁₄₉	#135	1	2	yes	—
53	C₂²×C₁₄₉	#137	1	2	yes	—
54	C₄×C₁₄₉	—	4	2	no	—
55	C₄×C₁₄₉	#123	1	2	yes	—
56	C₄×C₁₄₉	#126	1	2	yes	—
57	Dic₁₄₉	—	4	2	no	—
58	Dic₁₄₉	—	8	2	no	—
59	Dic₁₄₉	—	8	1	no	—
60	C₂×C₁₄₉	#397	1	1	yes	characteristic
61	C₂×C₁₄₉	#464	1	2	yes	—
62	C₂×C₁₄₉	#485	1	2	yes	—
63	C₂×C₁₄₉	#488	1	1	yes	characteristic
64	C₂×C₁₄₉	#491	1	1	yes	characteristic
65	C₁₄₉	#2074	1	1	yes	characteristic, complemented, Sylow
66	#2074	—	149	1	no	maximal, Sylow
67	C₂×#103	—	149	2	no	—
68	#391	—	149	1	no	—
69	#382	—	149	1	no	—
70	#340	—	149	2	no	—
71	#336	D₁₄₉	1	1	yes	characteristic
72	C₂²×Q₈	—	149	1	no	—
73	C₂²×C₈	C₂×D₁₄₉	1	2	yes	—
74	C₂×C₄⋊C₄	—	149	1	no	—
75	C₂×C₄⋊C₄	—	149	2	no	—
76	C₂×C₄²	C₂×D₁₄₉	1	1	yes	characteristic
77	#103	—	298	4	no	—
78	#95	—	298	2	no	—
79	#95	—	298	2	no	—
80	C₂×Q₈	—	298	1	no	—
81	C₂×Q₈	—	298	2	no	—
82	C₂×Q₈	—	298	2	no	—
83	C₂×Q₈	—	149	2	no	—
84	C₂²×C₄	—	298	1	no	—
85	C₂²×C₄	—	298	1	no	—
86	C₂²×C₄	—	298	2	no	—
87	C₂²×C₄	D₅₉₆	1	2	yes	—
88	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	2	2	no	—
92	C₄⋊C₄	—	298	2	no	—
93	C₄⋊C₄	—	298	2	no	—
94	C₄⋊C₄	—	298	2	no	—
95	C₄²	—	2	2	no	—
96	C₂³	C₂×D₅₉₆	1	1	yes	characteristic
97	Q₈	—	298	4	no	—
98	Q₈	—	596	2	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	2	2	no	—
102	C₂×C₄	—	298	2	no	—
103	C₂×C₄	—	596	1	no	—
104	C₂×C₄	—	596	1	no	—
105	C₂×C₄	—	596	2	no	—
106	C₂×C₄	—	596	2	no	—
107	C₂×C₄	—	596	2	no	—
108	C₂×C₄	—	596	2	no	—
109	C₂×C₄	—	596	1	no	—
110	C₂×C₄	D₄×D₁₄₉	1	1	yes	characteristic
111	C₂×C₄	#14558735	1	2	yes	—
112	C₂×C₄	#14558736	1	2	yes	—
113	C₂×C₄	#14558737	1	1	yes	characteristic
114	C₈	—	4	2	no	—
115	C₂²	#18011886	1	1	yes	characteristic
116	C₂²	#18011895	1	1	yes	characteristic
117	C₂²	#18011899	1	1	yes	characteristic
118	C₂²	#18011933	1	2	yes	—
119	C₂²	#18011944	1	2	yes	—
120	C₄	—	4	2	no	—
121	C₄	—	596	2	no	—
122	C₄	—	1192	1	no	—
123	C₄	—	1192	2	no	—
124	C₄	#18011893	1	2	yes	—
125	C₄	#18011906	1	2	yes	—
126	C₂	#23749552	1	1	yes	characteristic
127	C₂	#23749848	1	2	yes	—
128	C₂	#23749967	1	2	yes	—
129	C₂	#23749980	1	1	yes	characteristic
130	C₂	#23749994	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#18011836

C₂

yes

maximal

#23749531

C₂

yes

maximal, characteristic

#23749491

C₂

yes

maximal, characteristic

#23749388

C₂

yes

maximal

#336×C₁₄₉

C₂

yes

maximal, characteristic

C₂²×Dic₂₉₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₄₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₄₉

C₂²

yes

—

#18011836

—

#18011818

—

#18011818

—

C₂²×C₈×C₁₄₉

C₂²

yes

—

C₂×C₄²×C₁₄₉

C₂²

yes

characteristic

C₂×Dic₂₉₈

—

C₂×Dic₂₉₈

—

C₂×Dic₂₉₈

—

C₂×Dic₂₉₈

D₄

yes

—

C₂²×Dic₁₄₉

—

C₂²×Dic₁₄₉

—

C₂²×Dic₁₄₉

—

C₄⋊Dic₁₄₉

—

C₄⋊Dic₁₄₉

—

C₄⋊Dic₁₄₉

—

C₂²×C₄×C₁₄₉

D₄

yes

—

C₂²×C₄×C₁₄₉

C₂³

yes

characteristic

C₂×C₈×C₁₄₉

—

C₂×C₈×C₁₄₉

—

C₂×C₈×C₁₄₉

—

C₄²×C₁₄₉

—

Dic₂₉₈

—

Dic₂₉₈

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

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

#41

yes

—

C₂×C₄×C₁₄₉

#41

yes

—

C₂×C₄×C₁₄₉

#41

yes

characteristic

C₈×C₁₄₉

—

C₂²×C₁₄₉

yes

C₂²×C₁₄₉

yes

C₂²×C₁₄₉

yes

C₂²×C₁₄₉

yes

—

C₂²×C₁₄₉

#137

yes

—

C₄×C₁₄₉

—

C₄×C₁₄₉

#123

yes

—

C₄×C₁₄₉

#126

yes

—

Dic₁₄₉

—

Dic₁₄₉

—

Dic₁₄₉

—

C₂×C₁₄₉

#397

yes

characteristic

C₂×C₁₄₉

#464

yes

—

C₂×C₁₄₉

#485

yes

—

C₂×C₁₄₉

yes

C₂×C₁₄₉

yes

C₁₄₉

yes

characteristic, complemented, Sylow

#2074

—

149

maximal, Sylow

C₂×#103

—

149

—

#391

—

149

—

#382

—

149

—

#340

—

149

—

#336

D₁₄₉

yes

characteristic

C₂²×Q₈

—

149

—

C₂²×C₈

C₂×D₁₄₉

yes

—

C₂×C₄⋊C₄

—

149

—

C₂×C₄⋊C₄

—

149

—

C₂×C₄²

C₂×D₁₄₉

yes

characteristic

#103

—

298

—

#95

—

298

—

#95

—

298

—

C₂×Q₈

—

298

—

C₂×Q₈

—

298

—

C₂×Q₈

—

298

—

C₂×Q₈

—

149

—

C₂²×C₄

—

298

—

C₂²×C₄

—

298

—

C₂²×C₄

—

298

—

C₂²×C₄

D₅₉₆

yes

—

C₂²×C₄

C₂²×D₁₄₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

298

—

C₄⋊C₄

—

298

—

C₄⋊C₄

—

298

—

C₄²

—

C₂³

C₂×D₅₉₆

yes

characteristic

Q₈

—

298

—

Q₈

—

596

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

298

—

103

C₂×C₄

—

596

—

104

C₂×C₄

—

596

—

105

C₂×C₄

—

596

—

106

C₂×C₄

—

596

—

107

C₂×C₄

—

596

—

108

C₂×C₄

—

596

—

109

C₂×C₄

—

596

—

110

C₂×C₄

D₄×D₁₄₉

yes

characteristic

111

C₂×C₄

#14558735

yes

—

112

C₂×C₄

#14558736

yes

—

113

C₂×C₄

#14558737

yes

characteristic

114

C₈

—

115

C₂²

#18011886

yes

characteristic

116

C₂²

#18011895

yes

characteristic

117

C₂²

#18011899

yes

characteristic

118

C₂²

#18011933

yes

—

119

C₂²

#18011944

yes

—

120

C₄

—

121

C₄

—

596

—

122

C₄

—

1192

—

123

C₄

—

1192

—

124

C₄

#18011893

yes

—

125

C₄

#18011906

yes

—

126

C₂

#23749552

yes

characteristic

127

C₂

#23749848

yes

—

128

C₂

#23749967

yes

—

129

C₂

#23749980

yes

characteristic

130

C₂

#23749994

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₅₉₆
commutator, cosocle	C₂²×C₄×C₁₄₉	C₂³
Frattini	C₂²×C₄	C₂²×D₁₄₉
Fitting	#336×C₁₄₉	C₂
socle, prime radical	C₂³×C₁₄₉	C₂×D₄
hypercenter	#336	D₁₄₉
hypocenter	C₁₄₉	#2074
LCS	G ▷ C₂²×C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×D₅₉₆

commutator, cosocle

C₂²×C₄×C₁₄₉

C₂³

Frattini

C₂²×C₄

C₂²×D₁₄₉

Fitting

#336×C₁₄₉

C₂

socle, prime radical

C₂³×C₁₄₉

C₂×D₄

D₁₄₉

C₁₄₉

G ▷ C₂²×C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
65. C₁₄₉	#2074	semidirect

H ≅ G/N

type

65. C₁₄₉

#2074

semidirect

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

subgroup	index
C₂×#18011836	2
#23749531	2
#23749491	2
#23749388	2
#336×C₁₄₉	2
#2074	149

subgroup

index

#336×C₁₄₉

149

Subgroups — #34046111

Conjugacy classes of subgroups

Complements

Maximal subgroups