Group #34053796 — subgroups

Subgroups — #34053796

Order 19072.

314 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 157 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	#23750182	C₂	1	1	yes	maximal, characteristic
2	#23749754	C₂	1	4	yes	maximal, complemented
3	#23749753	C₂	1	4	yes	maximal, complemented
4	#23749683	C₂	1	4	yes	maximal
5	#23749682	C₂	1	1	yes	maximal, characteristic
6	#432×C₁₄₉	C₂	1	1	yes	maximal, characteristic
7	#18011911	C₄	1	1	yes	characteristic
8	#18011911	C₄	1	1	yes	characteristic
9	#18011903	C₄	1	2	yes	—
10	#18011890	C₄	1	2	yes	—
11	#18011890	C₄	1	2	yes	—
12	C₄×Dic₂₉₈	C₂²	1	2	yes	—
13	C₄×Dic₂₉₈	C₂²	1	2	yes	—
14	#18011870	C₂²	1	2	yes	—
15	C₄⋊#11924746	C₂²	1	2	yes	—
16	#18011840	C₂²	1	8	yes	—
17	C₂²⋊#11924746	C₂²	1	2	yes	—
18	#18011822	C₂²	1	2	yes	—
19	#18011821	C₂²	1	4	yes	—
20	C₈×Dic₁₄₉	C₂²	1	4	yes	—
21	C₂×C₄²×C₁₄₉	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₁₄₉	C₂²	1	2	yes	—
23	C₂²⋊C₈×C₁₄₉	C₂²	1	2	yes	—
24	#97×C₁₄₉	C₂²	1	2	yes	—
25	C₂×Dic₂₉₈	C₂×C₄	1	2	yes	—
26	C₂×Dic₂₉₈	C₂×C₄	1	2	yes	—
27	C₂×#11924746	C₂³	1	4	yes	—
28	C₄⋊Dic₁₄₉	C₂×C₄	1	2	yes	—
29	C₄⋊Dic₁₄₉	C₂×C₄	1	2	yes	—
30	C₄⋊Dic₁₄₉	C₂×C₄	1	2	yes	—
31	C₄⋊Dic₁₄₉	C₂×C₄	1	2	yes	—
32	#14558715	C₂×C₄	1	4	yes	—
33	#14558715	C₂×C₄	1	4	yes	—
34	C₂²⋊Dic₁₄₉	C₂×C₄	1	4	yes	—
35	C₄×Dic₁₄₉	C₂³	1	4	yes	—
36	C₂²×C₄×C₁₄₉	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₄₉	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
39	C₂×C₈×C₁₄₉	C₂³	1	4	yes	—
40	C₄²×C₁₄₉	C₂×C₄	1	1	yes	characteristic
41	C₄²×C₁₄₉	C₂×C₄	1	1	yes	characteristic
42	C₄²×C₁₄₉	C₂³	1	1	yes	characteristic
43	C₄²×C₁₄₉	C₂³	1	1	yes	characteristic
44	Dic₂₉₈	—	4	2	no	—
45	Dic₂₉₈	—	4	2	no	—
46	C₂×Dic₁₄₉	C₂²×C₄	1	4	yes	—
47	C₂×Dic₁₄₉	C₂²×C₄	1	4	yes	—
48	#11924746	—	2	4	no	—
49	C₂³×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₁₄₉	—	2	2	no	—
51	C₂×C₄×C₁₄₉	—	4	1	no	—
52	C₂×C₄×C₁₄₉	—	4	1	no	—
53	C₂×C₄×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₁₄₉	C₂⁴	1	1	yes	characteristic
59	C₈×C₁₄₉	—	2	4	no	—
60	C₂²×C₁₄₉	—	2	1	no	—
61	C₂²×C₁₄₉	—	2	1	no	—
62	C₂²×C₁₄₉	—	2	1	no	—
63	C₂²×C₁₄₉	C₂³×C₄	1	1	yes	characteristic
64	C₄×C₁₄₉	—	2	1	no	—
65	C₄×C₁₄₉	—	2	1	no	—
66	C₄×C₁₄₉	—	2	1	no	—
67	C₄×C₁₄₉	—	2	1	no	—
68	C₄×C₁₄₉	—	2	1	no	—
69	C₄×C₁₄₉	2₋¹⁺⁴	1	2	yes	—
70	Dic₁₄₉	—	2	4	no	—
71	Dic₁₄₉	—	2	4	no	—
72	C₂×C₁₄₉	—	4	1	no	—
73	C₂×C₁₄₉	#519	1	1	yes	characteristic
74	C₂×C₁₄₉	#568	1	1	yes	characteristic
75	C₂×C₁₄₉	#568	1	1	yes	characteristic
76	C₁₄₉	#2987	1	1	yes	characteristic, complemented, Sylow
77	#2987	—	149	1	no	maximal, Sylow
78	#533	—	149	1	no	—
79	#446	—	149	4	no	—
80	#446	—	149	4	no	—
81	#433	—	149	4	no	—
82	#432	—	149	1	no	—
83	#432	D₁₄₉	1	1	yes	characteristic
84	C₄⋊Q₈	—	149	1	no	—
85	C₄⋊Q₈	—	149	1	no	—
86	#125	—	149	2	no	—
87	C₂²⋊Q₈	—	149	2	no	—
88	C₂²⋊Q₈	—	149	2	no	—
89	C₄×Q₈	—	149	2	no	—
90	C₄×Q₈	—	149	2	no	—
91	#117	—	149	2	no	—
92	C₂×C₄²	C₂×D₁₄₉	1	1	yes	characteristic
93	C₄⋊C₈	—	149	8	no	—
94	C₄⋊C₈	—	149	2	no	—
95	C₄⋊C₈	C₂×D₁₄₉	1	2	yes	—
96	C₂²⋊C₈	—	149	2	no	—
97	C₂²⋊C₈	C₂×D₁₄₉	1	2	yes	—
98	#97	—	149	4	no	—
99	#97	—	149	2	no	—
100	#97	C₂×D₁₄₉	1	2	yes	—
101	C₄×C₈	—	149	4	no	—
102	C₂×Q₈	—	149	2	no	—
103	C₂×Q₈	—	149	2	no	—
104	C₂²×C₄	C₄×D₁₄₉	1	1	yes	characteristic
105	C₂²×C₄	C₄×D₁₄₉	1	1	yes	characteristic
106	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
107	C₂×C₈	—	149	4	no	—
108	C₂×C₈	C₂²×D₁₄₉	1	4	yes	—
109	C₄⋊C₄	—	149	2	no	—
110	C₄⋊C₄	—	149	2	no	—
111	C₄⋊C₄	—	149	4	no	—
112	C₄⋊C₄	—	149	4	no	—
113	C₄⋊C₄	—	149	2	no	—
114	C₄⋊C₄	—	149	2	no	—
115	C₂²⋊C₄	—	149	4	no	—
116	C₄²	—	149	4	no	—
117	C₄²	C₄×D₁₄₉	1	1	yes	characteristic
118	C₄²	C₄×D₁₄₉	1	1	yes	characteristic
119	C₄²	C₂²×D₁₄₉	1	1	yes	characteristic
120	C₄²	C₂²×D₁₄₉	1	1	yes	characteristic
121	C₂³	C₂×C₄×D₁₄₉	1	1	yes	characteristic
122	Q₈	—	596	2	no	—
123	Q₈	—	596	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	149	4	no	—
128	C₂×C₄	—	149	4	no	—
129	C₂×C₄	C₂×C₄×D₁₄₉	1	1	yes	characteristic
130	C₂×C₄	C₂×C₄×D₁₄₉	1	1	yes	characteristic
131	C₂×C₄	C₂×C₄×D₁₄₉	1	1	yes	characteristic
132	C₂×C₄	C₂×C₄×D₁₄₉	1	1	yes	characteristic
133	C₂×C₄	C₂×C₄×D₁₄₉	1	1	yes	characteristic
134	C₂×C₄	C₂³×D₁₄₉	1	1	yes	characteristic
135	C₈	—	2	4	no	—
136	C₈	—	298	4	no	—
137	C₂²	—	2	1	no	—
138	C₂²	—	2	1	no	—
139	C₂²	—	2	1	no	—
140	C₂²	C₂²×C₄×D₁₄₉	1	1	yes	characteristic
141	C₄	—	2	1	no	—
142	C₄	—	2	1	no	—
143	C₄	—	2	1	no	—
144	C₄	—	2	1	no	—
145	C₄	—	2	1	no	—
146	C₄	—	298	4	no	—
147	C₄	—	298	4	no	—
148	C₄	#18011958	1	2	yes	—
149	C₂	—	4	1	no	—
150	C₂	#23750102	1	1	yes	characteristic
151	C₂	#23750391	1	1	yes	characteristic
152	C₂	#23750391	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23750182

C₂

yes

maximal, characteristic

#23749754

C₂

yes

maximal, complemented

#23749753

C₂

yes

maximal, complemented

#23749683

C₂

yes

maximal

#23749682

C₂

yes

maximal, characteristic

#432×C₁₄₉

C₂

yes

maximal, characteristic

#18011911

C₄

yes

characteristic

#18011911

C₄

yes

characteristic

#18011903

C₄

yes

—

#18011890

C₄

yes

—

#18011890

C₄

yes

—

C₄×Dic₂₉₈

C₂²

yes

—

C₄×Dic₂₉₈

C₂²

yes

—

#18011870

C₂²

yes

—

C₄⋊#11924746

C₂²

yes

—

#18011840

C₂²

yes

—

C₂²⋊#11924746

C₂²

yes

—

#18011822

C₂²

yes

—

#18011821

C₂²

yes

—

C₈×Dic₁₄₉

C₂²

yes

—

C₂×C₄²×C₁₄₉

C₂²

yes

characteristic

C₄⋊C₈×C₁₄₉

C₂²

yes

—

C₂²⋊C₈×C₁₄₉

C₂²

yes

—

#97×C₁₄₉

C₂²

yes

—

C₂×Dic₂₉₈

C₂×C₄

yes

—

C₂×Dic₂₉₈

C₂×C₄

yes

—

C₂×#11924746

C₂³

yes

—

C₄⋊Dic₁₄₉

C₂×C₄

yes

—

C₄⋊Dic₁₄₉

C₂×C₄

yes

—

C₄⋊Dic₁₄₉

C₂×C₄

yes

—

C₄⋊Dic₁₄₉

C₂×C₄

yes

—

#14558715

C₂×C₄

yes

—

#14558715

C₂×C₄

yes

—

C₂²⋊Dic₁₄₉

C₂×C₄

yes

—

C₄×Dic₁₄₉

C₂³

yes

—

C₂²×C₄×C₁₄₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₄₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₄₉

C₂³

yes

characteristic

C₂×C₈×C₁₄₉

C₂³

yes

—

C₄²×C₁₄₉

C₂×C₄

yes

characteristic

C₄²×C₁₄₉

C₂×C₄

yes

characteristic

C₄²×C₁₄₉

C₂³

yes

characteristic

C₄²×C₁₄₉

C₂³

yes

characteristic

Dic₂₉₈

—

Dic₂₉₈

—

C₂×Dic₁₄₉

C₂²×C₄

yes

—

C₂×Dic₁₄₉

C₂²×C₄

yes

—

#11924746

—

C₂³×C₁₄₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₄₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₄₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₄₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₄₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₄₉

C₂⁴

yes

characteristic

C₈×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

C₂³×C₄

yes

characteristic

C₄×C₁₄₉

—

C₄×C₁₄₉

—

C₄×C₁₄₉

—

C₄×C₁₄₉

—

C₄×C₁₄₉

—

C₄×C₁₄₉

2₋¹⁺⁴

yes

—

Dic₁₄₉

—

Dic₁₄₉

—

C₂×C₁₄₉

—

C₂×C₁₄₉

yes

C₂×C₁₄₉

yes

C₂×C₁₄₉

yes

C₁₄₉

yes

characteristic, complemented, Sylow

#2987

—

149

maximal, Sylow

#533

—

149

—

#446

—

149

—

#446

—

149

—

#433

—

149

—

#432

—

149

—

#432

D₁₄₉

yes

characteristic

C₄⋊Q₈

—

149

—

C₄⋊Q₈

—

149

—

#125

—

149

—

C₂²⋊Q₈

—

149

—

C₂²⋊Q₈

—

149

—

C₄×Q₈

—

149

—

C₄×Q₈

—

149

—

#117

—

149

—

C₂×C₄²

C₂×D₁₄₉

yes

characteristic

C₄⋊C₈

—

149

—

C₄⋊C₈

—

149

—

C₄⋊C₈

C₂×D₁₄₉

yes

—

C₂²⋊C₈

—

149

—

C₂²⋊C₈

C₂×D₁₄₉

yes

—

#97

—

149

—

#97

—

149

—

100

#97

C₂×D₁₄₉

yes

—

101

C₄×C₈

—

149

—

102

C₂×Q₈

—

149

—

103

C₂×Q₈

—

149

—

104

C₂²×C₄

C₄×D₁₄₉

yes

characteristic

105

C₂²×C₄

C₄×D₁₄₉

yes

characteristic

106

C₂²×C₄

C₂²×D₁₄₉

yes

characteristic

107

C₂×C₈

—

149

—

108

C₂×C₈

C₂²×D₁₄₉

yes

—

109

C₄⋊C₄

—

149

—

110

C₄⋊C₄

—

149

—

111

C₄⋊C₄

—

149

—

112

C₄⋊C₄

—

149

—

113

C₄⋊C₄

—

149

—

114

C₄⋊C₄

—

149

—

115

C₂²⋊C₄

—

149

—

116

C₄²

—

149

—

117

C₄²

C₄×D₁₄₉

yes

characteristic

118

C₄²

C₄×D₁₄₉

yes

characteristic

119

C₄²

C₂²×D₁₄₉

yes

characteristic

120

C₄²

C₂²×D₁₄₉

yes

characteristic

121

C₂³

C₂×C₄×D₁₄₉

yes

characteristic

122

Q₈

—

596

—

123

Q₈

—

596

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

149

—

128

C₂×C₄

—

149

—

129

C₂×C₄

C₂×C₄×D₁₄₉

yes

characteristic

130

C₂×C₄

C₂×C₄×D₁₄₉

yes

characteristic

131

C₂×C₄

C₂×C₄×D₁₄₉

yes

characteristic

132

C₂×C₄

C₂×C₄×D₁₄₉

yes

characteristic

133

C₂×C₄

C₂×C₄×D₁₄₉

yes

characteristic

134

C₂×C₄

C₂³×D₁₄₉

yes

characteristic

135

C₈

—

136

C₈

—

298

—

137

C₂²

—

138

C₂²

—

139

C₂²

—

140

C₂²

C₂²×C₄×D₁₄₉

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

298

—

147

C₄

—

298

—

148

C₄

#18011958

yes

—

149

C₂

—

150

C₂

#23750102

yes

characteristic

151

C₂

#23750391

yes

characteristic

152

C₂

#23750391

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₄₉
commutator, socle	C₂²×C₁₄₉	C₂³×C₄
Fitting	#432×C₁₄₉	C₂
prime radical	C₂³×C₁₄₉	C₂²×C₄
cosocle	C₂×C₄×C₁₄₉	C₂⁴
hypercenter	#432	D₁₄₉
hypocenter	C₁₄₉	#2987
LCS	G ▷ C₂²×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂×C₄ ◁ #432

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₁₄₉

commutator, socle

C₂²×C₁₄₉

C₂³×C₄

Fitting

#432×C₁₄₉

C₂

prime radical

C₂³×C₁₄₉

C₂²×C₄

cosocle

C₂×C₄×C₁₄₉

C₂⁴

D₁₄₉

C₁₄₉

G ▷ C₂²×C₁₄₉ ▷ C₁₄₉

C₁ ◁ C₂×C₄ ◁ #432

N	H ≅ G/N	type
2. #23749754	C₂	semidirect
3. #23749753	C₂	semidirect
76. C₁₄₉	#2987	semidirect

H ≅ G/N

type

2. #23749754

C₂

semidirect

3. #23749753

C₂

semidirect

76. C₁₄₉

#2987

semidirect

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

subgroup	index
#23750182	2
#23749754	2
#23749753	2
#23749683	2
#23749682	2
#432×C₁₄₉	2
#2987	149

subgroup

index

#432×C₁₄₉

149

Subgroups — #34053796

Conjugacy classes of subgroups

Complements

Maximal subgroups