Group #34057584 — subgroups

Subgroups — #34057584

Order 19072.

324 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 125 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	#23750350	C₂	1	2	yes	maximal
2	#23750026	C₂	1	2	yes	maximal
3	#23749931	C₂	1	4	yes	maximal
4	#23749907	C₂	1	2	yes	maximal
5	C₈×Dic₂₉₈	C₂	1	1	yes	maximal, characteristic
6	C₄×Dic₅₉₆	C₂	1	1	yes	maximal, characteristic
7	#23749719	C₂	1	2	yes	maximal
8	#500×C₁₄₉	C₂	1	1	yes	maximal, characteristic
9	C₂×Dic₅₉₆	C₂²	1	1	yes	characteristic
10	#18011909	—	2	2	no	—
11	#18011902	C₂²	1	4	yes	—
12	#18011901	—	2	4	no	—
13	Q₈×Dic₁₄₉	—	2	2	no	—
14	C₄×Dic₂₉₈	C₂²	1	1	yes	characteristic
15	C₄×Dic₂₉₈	C₂²	1	2	yes	—
16	#18011876	C₂²	1	4	yes	—
17	#18011845	C₂²	1	1	yes	characteristic
18	#18011842	C₂²	1	4	yes	—
19	C₄⋊#11924746	C₂²	1	1	yes	characteristic
20	#18011840	C₂²	1	2	yes	—
21	#18011836	C₂²	1	2	yes	—
22	#18011834	C₂²	1	4	yes	—
23	C₈×Dic₁₄₉	C₂²	1	2	yes	—
24	C₄⋊Q₈×C₁₄₉	C₂²	1	2	yes	—
25	#107×C₁₄₉	C₂²	1	2	yes	—
26	#107×C₁₄₉	C₂²	1	2	yes	—
27	C₄×C₈×C₁₄₉	C₂²	1	1	yes	characteristic
28	C₂×Dic₂₉₈	D₄	1	1	yes	characteristic
29	C₂×Dic₂₉₈	C₂³	1	2	yes	—
30	Dic₅₉₆	Q₈	1	4	yes	complemented
31	C₂×#11924746	C₂³	1	2	yes	—
32	C₄⋊Dic₁₄₉	—	2	4	no	—
33	C₄⋊Dic₁₄₉	D₄	1	1	yes	characteristic
34	C₄⋊Dic₁₄₉	C₂³	1	2	yes	—
35	#14558715	—	2	4	no	—
36	#14558715	—	2	4	no	—
37	#14558715	—	2	2	no	—
38	#14558715	D₄	1	2	yes	—
39	C₄×Dic₁₄₉	—	2	4	no	—
40	C₄×Dic₁₄₉	—	2	2	no	—
41	C₄×Dic₁₄₉	C₂³	1	2	yes	—
42	C₂×Q₈×C₁₄₉	—	2	2	no	—
43	C₂×C₈×C₁₄₉	C₂³	1	1	yes	characteristic
44	C₂×C₈×C₁₄₉	C₂³	1	1	yes	characteristic
45	C₄⋊C₄×C₁₄₉	—	2	2	no	—
46	C₄⋊C₄×C₁₄₉	C₂³	1	4	yes	—
47	C₄²×C₁₄₉	C₂³	1	1	yes	characteristic
48	Dic₂₉₈	—	2	2	no	—
49	Dic₂₉₈	—	2	2	no	—
50	Dic₂₉₈	C₂×Q₈	1	4	yes	—
51	C₂×Dic₁₄₉	—	2	2	no	—
52	C₂×Dic₁₄₉	—	2	2	no	—
53	C₂×Dic₁₄₉	C₂×D₄	1	2	yes	—
54	C₂×Dic₁₄₉	C₂×D₄	1	2	yes	—
55	#11924746	—	2	2	no	—
56	Q₈×C₁₄₉	—	4	4	no	—
57	C₂×C₄×C₁₄₉	—	2	4	no	—
58	C₂×C₄×C₁₄₉	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₁₄₉	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₁₄₉	C₂⁴	1	1	yes	characteristic
61	C₈×C₁₄₉	—	2	1	no	—
62	C₈×C₁₄₉	C₂×Q₈	1	2	yes	—
63	C₂²×C₁₄₉	C₂²×D₄	1	1	yes	characteristic
64	C₄×C₁₄₉	—	2	1	no	—
65	C₄×C₁₄₉	—	4	4	no	—
66	C₄×C₁₄₉	#135	1	2	yes	—
67	C₄×C₁₄₉	C₂²×Q₈	1	1	yes	characteristic
68	C₄×C₁₄₉	2₋¹⁺⁴	1	1	yes	characteristic
69	Dic₁₄₉	—	2	4	no	—
70	Dic₁₄₉	—	2	2	no	—
71	Dic₁₄₉	—	2	2	no	—
72	Dic₁₄₉	—	4	2	no	—
73	C₂×C₁₄₉	D₄×Q₈	1	1	yes	characteristic
74	C₂×C₁₄₉	C₂×#135	1	1	yes	characteristic
75	C₂×C₁₄₉	#578	1	1	yes	characteristic
76	C₁₄₉	#3383	1	1	yes	characteristic, complemented, Sylow
77	#3383	—	149	1	no	maximal, Sylow
78	#557	—	149	2	no	—
79	#500	D₁₄₉	1	1	yes	characteristic
80	#499	—	149	2	no	—
81	#479	—	149	4	no	—
82	#475	—	149	2	no	—
83	C₈×Q₈	—	149	1	no	—
84	C₄×Q₁₆	—	149	2	no	—
85	C₄×Q₁₆	—	149	1	no	—
86	C₂×Q₁₆	—	149	1	no	—
87	C₄⋊Q₈	—	298	2	no	—
88	C₄⋊Q₈	C₂×D₁₄₉	1	2	yes	—
89	#125	—	298	4	no	—
90	#125	—	149	4	no	—
91	C₄×Q₈	—	298	2	no	—
92	C₄×Q₈	—	149	4	no	—
93	C₄×Q₈	—	149	2	no	—
94	C₄×Q₈	—	149	1	no	—
95	#107	—	149	1	no	—
96	#107	C₂×D₁₄₉	1	2	yes	—
97	#107	C₂×D₁₄₉	1	2	yes	—
98	#106	—	149	4	no	—
99	C₄⋊C₈	—	149	2	no	—
100	C₄⋊C₈	—	149	1	no	—
101	#103	—	149	2	no	—
102	#103	—	149	4	no	—
103	C₄×C₈	—	149	2	no	—
104	C₄×C₈	C₂×D₁₄₉	1	1	yes	characteristic
105	C₂×Q₈	—	2	2	no	—
106	C₂×Q₈	—	149	2	no	—
107	C₂×Q₈	—	149	1	no	—
108	Q₁₆	—	149	4	no	—
109	C₂×C₈	—	149	2	no	—
110	C₂×C₈	C₂²×D₁₄₉	1	1	yes	characteristic
111	C₂×C₈	C₂²×D₁₄₉	1	1	yes	characteristic
112	C₄⋊C₄	—	2	2	no	—
113	C₄⋊C₄	—	298	4	no	—
114	C₄⋊C₄	—	298	4	no	—
115	C₄⋊C₄	—	298	4	no	—
116	C₄⋊C₄	—	298	2	no	—
117	C₄⋊C₄	—	149	2	no	—
118	C₄⋊C₄	—	149	1	no	—
119	C₄⋊C₄	—	149	2	no	—
120	C₄⋊C₄	C₂²×D₁₄₉	1	4	yes	—
121	C₄²	—	298	4	no	—
122	C₄²	—	298	2	no	—
123	C₄²	—	149	2	no	—
124	C₄²	C₂²×D₁₄₉	1	1	yes	characteristic
125	Q₈	—	4	4	no	—
126	Q₈	—	298	2	no	—
127	Q₈	—	298	2	no	—
128	Q₈	—	149	4	no	—
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	—	298	2	no	—
131	C₂×C₄	—	298	2	no	—
132	C₂×C₄	—	149	2	no	—
133	C₂×C₄	—	149	2	no	—
134	C₂×C₄	D₄×D₁₄₉	1	1	yes	characteristic
135	C₂×C₄	D₄×D₁₄₉	1	1	yes	characteristic
136	C₂×C₄	C₂³×D₁₄₉	1	1	yes	characteristic
137	C₈	—	2	1	no	—
138	C₈	—	298	2	no	—
139	C₈	Q₈×D₁₄₉	1	2	yes	—
140	C₂²	C₂×D₄×D₁₄₉	1	1	yes	characteristic
141	C₄	—	2	1	no	—
142	C₄	—	4	4	no	—
143	C₄	—	298	2	no	—
144	C₄	—	298	2	no	—
145	C₄	—	298	4	no	—
146	C₄	—	596	2	no	—
147	C₄	#18011930	1	2	yes	—
148	C₄	C₂×Q₈×D₁₄₉	1	1	yes	characteristic
149	C₄	#18011959	1	1	yes	characteristic
150	C₂	#23750290	1	1	yes	characteristic
151	C₂	C₂×#18011930	1	1	yes	characteristic
152	C₂	#23750438	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23750350

C₂

yes

maximal

#23750026

C₂

yes

maximal

#23749931

C₂

yes

maximal

#23749907

C₂

yes

maximal

C₈×Dic₂₉₈

C₂

yes

maximal, characteristic

C₄×Dic₅₉₆

C₂

yes

maximal, characteristic

#23749719

C₂

yes

maximal

#500×C₁₄₉

C₂

yes

maximal, characteristic

C₂×Dic₅₉₆

C₂²

yes

characteristic

#18011909

—

#18011902

C₂²

yes

—

#18011901

—

Q₈×Dic₁₄₉

—

C₄×Dic₂₉₈

C₂²

yes

characteristic

C₄×Dic₂₉₈

C₂²

yes

—

#18011876

C₂²

yes

—

#18011845

C₂²

yes

characteristic

#18011842

C₂²

yes

—

C₄⋊#11924746

C₂²

yes

characteristic

#18011840

C₂²

yes

—

#18011836

C₂²

yes

—

#18011834

C₂²

yes

—

C₈×Dic₁₄₉

C₂²

yes

—

C₄⋊Q₈×C₁₄₉

C₂²

yes

—

#107×C₁₄₉

C₂²

yes

—

#107×C₁₄₉

C₂²

yes

—

C₄×C₈×C₁₄₉

C₂²

yes

characteristic

C₂×Dic₂₉₈

D₄

yes

characteristic

C₂×Dic₂₉₈

C₂³

yes

—

Dic₅₉₆

Q₈

yes

complemented

C₂×#11924746

C₂³

yes

—

C₄⋊Dic₁₄₉

—

C₄⋊Dic₁₄₉

D₄

yes

characteristic

C₄⋊Dic₁₄₉

C₂³

yes

—

—

—

—

D₄

yes

—

C₄×Dic₁₄₉

—

C₄×Dic₁₄₉

—

C₄×Dic₁₄₉

C₂³

yes

—

C₂×Q₈×C₁₄₉

—

C₂×C₈×C₁₄₉

C₂³

yes

characteristic

C₂×C₈×C₁₄₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₄₉

—

C₄⋊C₄×C₁₄₉

C₂³

yes

—

C₄²×C₁₄₉

C₂³

yes

characteristic

Dic₂₉₈

—

Dic₂₉₈

—

Dic₂₉₈

C₂×Q₈

yes

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

C₂×D₄

yes

—

C₂×Dic₁₄₉

C₂×D₄

yes

—

#11924746

—

Q₈×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₄₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₄₉

C₂⁴

yes

characteristic

C₈×C₁₄₉

—

C₈×C₁₄₉

C₂×Q₈

yes

—

C₂²×C₁₄₉

C₂²×D₄

yes

characteristic

C₄×C₁₄₉

—

C₄×C₁₄₉

—

C₄×C₁₄₉

#135

yes

—

C₄×C₁₄₉

C₂²×Q₈

yes

characteristic

C₄×C₁₄₉

2₋¹⁺⁴

yes

characteristic

Dic₁₄₉

—

Dic₁₄₉

—

Dic₁₄₉

—

Dic₁₄₉

—

C₂×C₁₄₉

D₄×Q₈

yes

C₂×C₁₄₉

yes

C₂×C₁₄₉

yes

C₁₄₉

yes

characteristic, complemented, Sylow

#3383

—

149

maximal, Sylow

#557

—

149

—

#500

D₁₄₉

yes

characteristic

#499

—

149

—

#479

—

149

—

#475

—

149

—

C₈×Q₈

—

149

—

C₄×Q₁₆

—

149

—

C₄×Q₁₆

—

149

—

C₂×Q₁₆

—

149

—

C₄⋊Q₈

—

298

—

C₄⋊Q₈

C₂×D₁₄₉

yes

—

#125

—

298

—

#125

—

149

—

C₄×Q₈

—

298

—

C₄×Q₈

—

149

—

C₄×Q₈

—

149

—

C₄×Q₈

—

149

—

#107

—

149

—

#107

C₂×D₁₄₉

yes

—

#107

C₂×D₁₄₉

yes

—

#106

—

149

—

C₄⋊C₈

—

149

—

100

C₄⋊C₈

—

149

—

101

#103

—

149

—

102

#103

—

149

—

103

C₄×C₈

—

149

—

104

C₄×C₈

C₂×D₁₄₉

yes

characteristic

105

C₂×Q₈

—

106

C₂×Q₈

—

149

—

107

C₂×Q₈

—

149

—

108

Q₁₆

—

149

—

109

C₂×C₈

—

149

—

110

C₂×C₈

C₂²×D₁₄₉

yes

characteristic

111

C₂×C₈

C₂²×D₁₄₉

yes

characteristic

112

C₄⋊C₄

—

113

C₄⋊C₄

—

298

—

114

C₄⋊C₄

—

298

—

115

C₄⋊C₄

—

298

—

116

C₄⋊C₄

—

298

—

117

C₄⋊C₄

—

149

—

118

C₄⋊C₄

—

149

—

119

C₄⋊C₄

—

149

—

120

C₄⋊C₄

C₂²×D₁₄₉

yes

—

121

C₄²

—

298

—

122

C₄²

—

298

—

123

C₄²

—

149

—

124

C₄²

C₂²×D₁₄₉

yes

characteristic

125

Q₈

—

126

Q₈

—

298

—

127

Q₈

—

298

—

128

Q₈

—

149

—

129

C₂×C₄

—

130

C₂×C₄

—

298

—

131

C₂×C₄

—

298

—

132

C₂×C₄

—

149

—

133

C₂×C₄

—

149

—

134

C₂×C₄

D₄×D₁₄₉

yes

characteristic

135

C₂×C₄

D₄×D₁₄₉

yes

characteristic

136

C₂×C₄

C₂³×D₁₄₉

yes

characteristic

137

C₈

—

138

C₈

—

298

—

139

C₈

Q₈×D₁₄₉

yes

—

140

C₂²

C₂×D₄×D₁₄₉

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

298

—

144

C₄

—

298

—

145

C₄

—

298

—

146

C₄

—

596

—

147

C₄

#18011930

yes

—

148

C₄

C₂×Q₈×D₁₄₉

yes

characteristic

149

C₄

#18011959

yes

characteristic

150

C₂

#23750290

yes

characteristic

151

C₂

C₂×#18011930

yes

characteristic

152

C₂

#23750438

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₄₉
commutator, cosocle	C₂×C₄×C₁₄₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₄₉
Fitting	#500×C₁₄₉	C₂
socle, prime radical	C₂²×C₁₄₉	C₂²×D₄
hypercenter	#500	D₁₄₉
hypocenter	C₁₄₉	#3383
LCS	G ▷ C₂×C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

subgroup

G/H

center

C₂²

C₂×D₄×D₁₄₉

commutator, cosocle

C₂×C₄×C₁₄₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₄₉

Fitting

#500×C₁₄₉

C₂

socle, prime radical

C₂²×C₁₄₉

C₂²×D₄

D₁₄₉

C₁₄₉

G ▷ C₂×C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #500

N	H ≅ G/N	type
30. Dic₅₉₆	Q₈	semidirect
76. C₁₄₉	#3383	semidirect

H ≅ G/N

type

30. Dic₅₉₆

Q₈

semidirect

76. C₁₄₉

#3383

semidirect

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

subgroup	index
#23750350	2
#23750026	2
#23749931	2
#23749907	2
C₈×Dic₂₉₈	2
C₄×Dic₅₉₆	2
#23749719	2
#500×C₁₄₉	2
#3383	149

subgroup

index

C₈×Dic₂₉₈

C₄×Dic₅₉₆

#23749719

#500×C₁₄₉

#3383

149

Subgroups — #34057584

Conjugacy classes of subgroups

Complements

Maximal subgroups