Group #34045626 — subgroups

Subgroups — #34045626

Order 19072.

292 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 129 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×C₄×Dic₂₉₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#11924746	C₂	1	1	yes	maximal, characteristic
3	#23749379	C₂	1	4	yes	maximal
4	C₂×C₄⋊C₈×C₁₄₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₉₈	C₄	1	1	yes	characteristic
6	C₂²×#11924746	C₂²	1	2	yes	—
7	C₄×Dic₂₉₈	—	2	2	no	—
8	C₄×Dic₂₉₈	—	2	2	no	—
9	C₂×C₄⋊Dic₁₄₉	C₄	1	1	yes	characteristic
10	C₂×#14558715	C₄	1	2	yes	—
11	C₂×C₄×Dic₁₄₉	C₂²	1	2	yes	—
12	C₄×#11924746	—	2	2	no	—
13	C₂²×C₈×C₁₄₉	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₄₉	C₂²	1	1	yes	characteristic
15	C₄⋊C₈×C₁₄₉	—	2	2	no	—
16	C₂×Dic₂₉₈	—	2	2	no	—
17	C₂×Dic₂₉₈	—	2	2	no	—
18	C₂×Dic₂₉₈	C₈	1	4	yes	complemented
19	C₂²×Dic₁₄₉	C₂×C₄	1	2	yes	—
20	C₂²×Dic₁₄₉	C₂×C₄	1	2	yes	—
21	C₂×#11924746	—	2	2	no	—
22	C₂×#11924746	—	2	2	no	—
23	C₂×#11924746	Q₈	1	4	yes	complemented
24	C₄⋊Dic₁₄₉	—	2	2	no	—
25	#14558715	—	2	4	no	—
26	C₄×Dic₁₄₉	—	2	2	no	—
27	C₄×Dic₁₄₉	—	2	2	no	—
28	C₂²×C₄×C₁₄₉	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₁₄₉	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
31	C₂×C₈×C₁₄₉	—	2	2	no	—
32	C₂×C₈×C₁₄₉	—	2	4	no	—
33	C₄²×C₁₄₉	D₄	1	2	yes	—
34	C₄²×C₁₄₉	D₄	1	2	yes	—
35	Dic₂₉₈	—	2	8	no	—
36	C₂×Dic₁₄₉	—	2	2	no	—
37	C₂×Dic₁₄₉	—	2	2	no	—
38	C₂×Dic₁₄₉	—	2	2	no	—
39	C₂×Dic₁₄₉	—	2	4	no	—
40	C₂×Dic₁₄₉	C₂×C₈	1	4	yes	—
41	#11924746	—	2	4	no	—
42	C₂³×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₁₄₉	—	2	2	no	—
44	C₂×C₄×C₁₄₉	C₂²⋊C₄	1	2	yes	—
45	C₂×C₄×C₁₄₉	C₂²⋊C₄	1	2	yes	—
46	C₂×C₄×C₁₄₉	C₂×C₈	1	2	yes	—
47	C₂×C₄×C₁₄₉	M₁₆	1	2	yes	—
48	C₂×C₄×C₁₄₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₄₉	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₁₄₉	C₂×Q₈	1	2	yes	—
51	C₂×C₄×C₁₄₉	#41	1	2	yes	—
52	C₈×C₁₄₉	—	4	2	no	—
53	C₂²×C₁₄₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₁₄₉	C₄×Q₈	1	2	yes	—
55	C₂²×C₁₄₉	C₂²×C₈	1	1	yes	characteristic
56	C₂²×C₁₄₉	C₂×M₁₆	1	1	yes	characteristic
57	C₂²×C₁₄₉	#131	1	2	yes	—
58	C₄×C₁₄₉	—	2	2	no	—
59	C₄×C₁₄₉	C₂²⋊C₈	1	4	yes	—
60	C₄×C₁₄₉	C₂²⋊Q₈	1	2	yes	—
61	C₄×C₁₄₉	#124	1	1	yes	characteristic
62	C₄×C₁₄₉	C₄⋊Q₈	1	1	yes	characteristic
63	Dic₁₄₉	—	2	4	no	—
64	Dic₁₄₉	—	4	2	no	—
65	C₂×C₁₄₉	#391	1	1	yes	characteristic
66	C₂×C₁₄₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
67	C₂×C₁₄₉	#408	1	1	yes	characteristic
68	C₂×C₁₄₉	C₈×Q₈	1	2	yes	—
69	C₂×C₁₄₉	#446	1	2	yes	—
70	C₁₄₉	#1984	1	1	yes	characteristic, complemented, Sylow
71	#1984	—	149	1	no	maximal, Sylow
72	C₂×C₄×Q₈	—	149	1	no	—
73	C₂×C₄⋊C₈	D₁₄₉	1	1	yes	characteristic
74	C₂×C₄×C₈	—	149	1	no	—
75	#336	—	149	4	no	—
76	C₂²×Q₈	—	149	1	no	—
77	C₂²×C₈	—	149	2	no	—
78	C₂²×C₈	C₂×D₁₄₉	1	2	yes	—
79	C₄×Q₈	—	298	2	no	—
80	C₄×Q₈	—	298	2	no	—
81	C₂×C₄⋊C₄	—	149	2	no	—
82	C₂×C₄⋊C₄	—	149	1	no	—
83	C₂×C₄²	—	149	2	no	—
84	C₂×C₄²	C₂×D₁₄₉	1	1	yes	characteristic
85	C₄⋊C₈	—	2	2	no	—
86	C₄×C₈	—	298	2	no	—
87	C₂×Q₈	—	298	2	no	—
88	C₂×Q₈	—	298	2	no	—
89	C₂×Q₈	—	149	4	no	—
90	C₂²×C₄	—	149	2	no	—
91	C₂²×C₄	—	149	2	no	—
92	C₂²×C₄	C₄×D₁₄₉	1	1	yes	characteristic
93	C₂²×C₄	C₄×D₁₄₉	1	1	yes	characteristic
94	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	2	4	no	—
97	C₂×C₈	—	298	2	no	—
98	C₂×C₈	—	298	2	no	—
99	C₂×C₈	—	149	4	no	—
100	C₄⋊C₄	—	298	4	no	—
101	C₄⋊C₄	—	298	2	no	—
102	C₄²	—	298	2	no	—
103	C₄²	—	298	2	no	—
104	C₄²	C₂²⋊D₁₄₉	1	2	yes	—
105	C₄²	D₅₉₆	1	2	yes	—
106	C₂³	C₂×C₄×D₁₄₉	1	1	yes	characteristic
107	Q₈	—	298	8	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	298	2	no	—
110	C₂×C₄	—	298	2	no	—
111	C₂×C₄	—	298	4	no	—
112	C₂×C₄	—	298	2	no	—
113	C₂×C₄	—	149	4	no	—
114	C₂×C₄	#14558713	1	2	yes	—
115	C₂×C₄	#14558713	1	2	yes	—
116	C₂×C₄	C₈×D₁₄₉	1	2	yes	—
117	C₂×C₄	#14558719	1	2	yes	—
118	C₂×C₄	C₂×C₂²⋊D₁₄₉	1	1	yes	characteristic
119	C₂×C₄	C₂×D₅₉₆	1	1	yes	characteristic
120	C₂×C₄	Q₈×D₁₄₉	1	2	yes	—
121	C₂×C₄	#14558735	1	2	yes	—
122	C₈	—	4	2	no	—
123	C₈	—	298	4	no	—
124	C₂²	C₂×#14558713	1	1	yes	characteristic
125	C₂²	#18011876	1	2	yes	—
126	C₂²	C₂×C₈×D₁₄₉	1	1	yes	characteristic
127	C₂²	C₂×#14558719	1	1	yes	characteristic
128	C₂²	#18011919	1	2	yes	—
129	C₄	—	2	2	no	—
130	C₄	—	298	4	no	—
131	C₄	—	596	2	no	—
132	C₄	#18011823	1	4	yes	—
133	C₄	#18011891	1	2	yes	—
134	C₄	#18011898	1	1	yes	characteristic
135	C₄	#18011909	1	1	yes	characteristic
136	C₂	#23749530	1	1	yes	characteristic
137	C₂	C₂×#18011823	1	1	yes	characteristic
138	C₂	#23749593	1	1	yes	characteristic
139	C₂	#23749749	1	2	yes	—
140	C₂	#23749752	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₂₉₈

C₂

yes

maximal, characteristic

C₂×C₄×#11924746

C₂

yes

maximal, characteristic

#23749379

C₂

yes

maximal

C₂×C₄⋊C₈×C₁₄₉

C₂

yes

maximal, characteristic

C₂²×Dic₂₉₈

C₄

yes

characteristic

C₂²×#11924746

C₂²

yes

—

C₄×Dic₂₉₈

—

C₄×Dic₂₉₈

—

C₂×C₄⋊Dic₁₄₉

C₄

yes

characteristic

C₂×#14558715

C₄

yes

—

C₂×C₄×Dic₁₄₉

C₂²

yes

—

C₄×#11924746

—

C₂²×C₈×C₁₄₉

C₂²

yes

—

C₂×C₄²×C₁₄₉

C₂²

yes

characteristic

C₄⋊C₈×C₁₄₉

—

C₂×Dic₂₉₈

—

C₂×Dic₂₉₈

—

C₂×Dic₂₉₈

C₈

yes

complemented

C₂²×Dic₁₄₉

C₂×C₄

yes

—

C₂²×Dic₁₄₉

C₂×C₄

yes

—

C₂×#11924746

—

C₂×#11924746

—

C₂×#11924746

Q₈

yes

complemented

C₄⋊Dic₁₄₉

—

#14558715

—

C₄×Dic₁₄₉

—

C₄×Dic₁₄₉

—

C₂²×C₄×C₁₄₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₄₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₄₉

C₂³

yes

characteristic

C₂×C₈×C₁₄₉

—

C₂×C₈×C₁₄₉

—

C₄²×C₁₄₉

D₄

yes

—

C₄²×C₁₄₉

D₄

yes

—

Dic₂₉₈

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

C₂×C₈

yes

—

#11924746

—

C₂³×C₁₄₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₄₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₄₉

C₂×C₈

yes

—

C₂×C₄×C₁₄₉

M₁₆

yes

—

C₂×C₄×C₁₄₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₄₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₄₉

C₂×Q₈

yes

—

C₂×C₄×C₁₄₉

#41

yes

—

C₈×C₁₄₉

—

C₂²×C₁₄₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₄₉

C₄×Q₈

yes

—

C₂²×C₁₄₉

C₂²×C₈

yes

characteristic

C₂²×C₁₄₉

C₂×M₁₆

yes

characteristic

C₂²×C₁₄₉

#131

yes

—

C₄×C₁₄₉

—

C₄×C₁₄₉

C₂²⋊C₈

yes

—

C₄×C₁₄₉

C₂²⋊Q₈

yes

—

C₄×C₁₄₉

#124

yes

characteristic

C₄×C₁₄₉

C₄⋊Q₈

yes

characteristic

Dic₁₄₉

—

Dic₁₄₉

—

C₂×C₁₄₉

#391

yes

characteristic

C₂×C₁₄₉

C₂×C₂²⋊C₈

yes

characteristic

C₂×C₁₄₉

#408

yes

characteristic

C₂×C₁₄₉

C₈×Q₈

yes

—

C₂×C₁₄₉

#446

yes

—

C₁₄₉

#1984

yes

characteristic, complemented, Sylow

#1984

—

149

maximal, Sylow

C₂×C₄×Q₈

—

149

—

C₂×C₄⋊C₈

D₁₄₉

yes

characteristic

C₂×C₄×C₈

—

149

—

#336

—

149

—

C₂²×Q₈

—

149

—

C₂²×C₈

—

149

—

C₂²×C₈

C₂×D₁₄₉

yes

—

C₄×Q₈

—

298

—

C₄×Q₈

—

298

—

C₂×C₄⋊C₄

—

149

—

C₂×C₄⋊C₄

—

149

—

C₂×C₄²

—

149

—

C₂×C₄²

C₂×D₁₄₉

yes

characteristic

C₄⋊C₈

—

C₄×C₈

—

298

—

C₂×Q₈

—

298

—

C₂×Q₈

—

298

—

C₂×Q₈

—

149

—

C₂²×C₄

—

149

—

C₂²×C₄

—

149

—

C₂²×C₄

C₄×D₁₄₉

yes

characteristic

C₂²×C₄

C₄×D₁₄₉

yes

characteristic

C₂²×C₄

C₂²×D₁₄₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

298

—

C₂×C₈

—

298

—

C₂×C₈

—

149

—

100

C₄⋊C₄

—

298

—

101

C₄⋊C₄

—

298

—

102

C₄²

—

298

—

103

C₄²

—

298

—

104

C₄²

C₂²⋊D₁₄₉

yes

—

105

C₄²

D₅₉₆

yes

—

106

C₂³

C₂×C₄×D₁₄₉

yes

characteristic

107

Q₈

—

298

—

108

C₂×C₄

—

109

C₂×C₄

—

298

—

110

C₂×C₄

—

298

—

111

C₂×C₄

—

298

—

112

C₂×C₄

—

298

—

113

C₂×C₄

—

149

—

114

C₂×C₄

#14558713

yes

—

115

C₂×C₄

#14558713

yes

—

116

C₂×C₄

C₈×D₁₄₉

yes

—

117

C₂×C₄

#14558719

yes

—

118

C₂×C₄

C₂×C₂²⋊D₁₄₉

yes

characteristic

119

C₂×C₄

C₂×D₅₉₆

yes

characteristic

120

C₂×C₄

Q₈×D₁₄₉

yes

—

121

C₂×C₄

#14558735

yes

—

122

C₈

—

123

C₈

—

298

—

124

C₂²

C₂×#14558713

yes

characteristic

125

C₂²

#18011876

yes

—

126

C₂²

C₂×C₈×D₁₄₉

yes

characteristic

127

C₂²

C₂×#14558719

yes

characteristic

128

C₂²

#18011919

yes

—

129

C₄

—

130

C₄

—

298

—

131

C₄

—

596

—

132

C₄

#18011823

yes

—

133

C₄

#18011891

yes

—

134

C₄

#18011898

yes

characteristic

135

C₄

#18011909

yes

characteristic

136

C₂

#23749530

yes

characteristic

137

C₂

C₂×#18011823

yes

characteristic

138

C₂

#23749593

yes

characteristic

139

C₂

#23749749

yes

—

140

C₂

#23749752

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₄₉
commutator	C₂²×C₁₄₉	C₂²×C₈
Fitting	C₂×C₄⋊C₈×C₁₄₉	C₂
socle, prime radical	C₂³×C₁₄₉	C₂²×C₄
cosocle	C₂²×C₄×C₁₄₉	C₂³
hypercenter	C₂×C₄⋊C₈	D₁₄₉
hypocenter	C₁₄₉	#1984
LCS	G ▷ C₂²×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₄₉

commutator

C₂²×C₁₄₉

C₂²×C₈

Fitting

C₂×C₄⋊C₈×C₁₄₉

C₂

socle, prime radical

C₂³×C₁₄₉

C₂²×C₄

cosocle

C₂²×C₄×C₁₄₉

C₂³

hypercenter

C₂×C₄⋊C₈

D₁₄₉

C₁₄₉

G ▷ C₂²×C₁₄₉ ▷ C₁₄₉

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
18. C₂×Dic₂₉₈	C₈	semidirect
23. C₂×#11924746	Q₈	semidirect
70. C₁₄₉	#1984	semidirect

H ≅ G/N

type

18. C₂×Dic₂₉₈

C₈

semidirect

23. C₂×#11924746

Q₈

semidirect

70. C₁₄₉

#1984

semidirect

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

subgroup	index
C₂×C₄×Dic₂₉₈	2
C₂×C₄×#11924746	2
#23749379	2
C₂×C₄⋊C₈×C₁₄₉	2
#1984	149

subgroup

index

C₂×C₄×Dic₂₉₈

C₂×C₄×#11924746

#23749379

C₂×C₄⋊C₈×C₁₄₉

#1984

149

Subgroups — #34045626

Conjugacy classes of subgroups

Complements

Maximal subgroups