Group #34053558 — subgroups

Subgroups — #34053558

Order 19072.

456 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 181 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	C₂²×Dic₅₉₆	C₂	1	1	yes	maximal, characteristic
2	#23750102	C₂	1	2	yes	maximal
3	#23749731	C₂	1	4	yes	maximal, complemented
4	#23749719	C₂	1	4	yes	maximal, complemented
5	#23749633	C₂	1	2	yes	maximal
6	#23749580	C₂	1	1	yes	maximal, characteristic
7	#427×C₁₄₉	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₂₉₈	C₂²	1	2	yes	—
9	C₂×Dic₅₉₆	C₄	1	8	yes	complemented
10	C₂×Dic₅₉₆	C₂²	1	2	yes	—
11	C₂×Dic₅₉₆	C₂²	1	2	yes	—
12	C₂×#14558720	C₂²	1	1	yes	characteristic
13	C₄×Dic₂₉₈	—	2	4	no	—
14	#18011876	C₂²	1	8	yes	—
15	#18011870	C₂²	1	1	yes	characteristic
16	#18011867	—	2	4	no	—
17	#18011834	C₂²	1	8	yes	—
18	#18011821	C₂²	1	2	yes	—
19	C₈×Dic₁₄₉	C₂²	1	2	yes	—
20	C₂²×C₈×C₁₄₉	C₂²	1	1	yes	characteristic
21	#117×C₁₄₉	C₂²	1	2	yes	—
22	#107×C₁₄₉	C₂²	1	2	yes	—
23	#106×C₁₄₉	C₂²	1	2	yes	—
24	C₂×Dic₂₉₈	—	2	2	no	—
25	C₂×Dic₂₉₈	—	2	4	no	—
26	C₂×Dic₂₉₈	C₂×C₄	1	8	yes	—
27	C₂×Dic₂₉₈	C₂³	1	4	yes	—
28	C₂²×Dic₁₄₉	—	2	2	no	—
29	Dic₅₉₆	C₂×C₄	1	8	yes	—
30	Dic₅₉₆	C₂×C₄	1	8	yes	—
31	#14558720	—	2	2	no	—
32	C₂×#11924746	C₂³	1	2	yes	—
33	C₄⋊Dic₁₄₉	D₄	1	2	yes	—
34	#14558715	—	2	8	no	—
35	C₂²⋊Dic₁₄₉	D₄	1	2	yes	—
36	C₄×Dic₁₄₉	—	2	8	no	—
37	C₄×Dic₁₄₉	C₂³	1	2	yes	—
38	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
39	C₂×C₈×C₁₄₉	C₂×C₄	1	4	yes	—
40	C₂×C₈×C₁₄₉	C₂³	1	1	yes	characteristic
41	C₂×C₈×C₁₄₉	C₂³	1	1	yes	characteristic
42	C₄⋊C₄×C₁₄₉	C₂³	1	4	yes	—
43	C₂²⋊C₄×C₁₄₉	—	2	2	no	—
44	C₄²×C₁₄₉	—	2	2	no	—
45	Dic₂₉₈	—	2	8	no	—
46	Dic₂₉₈	—	2	4	no	—
47	Dic₂₉₈	C₂²×C₄	1	8	yes	—
48	C₂×Dic₁₄₉	—	2	8	no	—
49	C₂×Dic₁₄₉	—	2	4	no	—
50	C₂×Dic₁₄₉	C₂×D₄	1	4	yes	—
51	#11924746	—	2	2	no	—
52	C₂³×C₁₄₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₁₄₉	—	2	4	no	—
54	C₂×C₄×C₁₄₉	C₂²×C₄	1	2	yes	—
55	C₂×C₄×C₁₄₉	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₁₄₉	#41	1	2	yes	—
57	C₂×C₄×C₁₄₉	C₂⁴	1	1	yes	characteristic
58	C₈×C₁₄₉	C₂²×C₄	1	2	yes	—
59	C₈×C₁₄₉	C₂²×C₄	1	2	yes	—
60	C₂²×C₁₄₉	—	2	2	no	—
61	C₂²×C₁₄₉	C₄×D₄	1	2	yes	—
62	C₂²×C₁₄₉	C₂²×D₄	1	1	yes	characteristic
63	C₄×C₁₄₉	—	4	4	no	—
64	C₄×C₁₄₉	C₄×D₄	1	2	yes	—
65	C₄×C₁₄₉	C₂³×C₄	1	1	yes	characteristic
66	C₄×C₁₄₉	C₂×#41	1	1	yes	characteristic
67	Dic₁₄₉	—	2	8	no	—
68	Dic₁₄₉	—	2	4	no	—
69	C₂×C₁₄₉	—	2	2	no	—
70	C₂×C₁₄₉	C₂×C₄×D₄	1	1	yes	characteristic
71	C₂×C₁₄₉	#578	1	2	yes	—
72	C₁₄₉	#2950	1	1	yes	characteristic, complemented, Sylow
73	#2950	—	149	1	no	maximal, Sylow
74	C₂²×Q₁₆	—	149	1	no	—
75	#519	—	149	2	no	—
76	#441	—	149	4	no	—
77	C₄×Q₁₆	—	149	4	no	—
78	#427	D₁₄₉	1	1	yes	characteristic
79	#419	—	149	2	no	—
80	#405	—	149	1	no	—
81	C₂²×Q₈	—	149	2	no	—
82	C₂×Q₁₆	—	149	8	no	—
83	C₂×Q₁₆	—	149	2	no	—
84	C₂×Q₁₆	—	149	2	no	—
85	C₂×M₁₆	—	149	1	no	—
86	C₂²×C₈	C₂×D₁₄₉	1	1	yes	characteristic
87	C₄×Q₈	—	298	4	no	—
88	C₄×Q₈	—	149	8	no	—
89	#117	—	298	4	no	—
90	#117	—	149	1	no	—
91	#117	C₂×D₁₄₉	1	2	yes	—
92	#107	C₂×D₁₄₉	1	2	yes	—
93	#106	C₂×D₁₄₉	1	2	yes	—
94	#103	—	149	8	no	—
95	#97	—	149	2	no	—
96	C₄×C₈	—	149	2	no	—
97	C₂×Q₈	—	298	2	no	—
98	C₂×Q₈	—	298	4	no	—
99	C₂×Q₈	—	149	8	no	—
100	C₂×Q₈	—	149	4	no	—
101	C₂²×C₄	—	298	2	no	—
102	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
103	Q₁₆	—	149	8	no	—
104	Q₁₆	—	149	8	no	—
105	M₁₆	—	298	2	no	—
106	C₂×C₈	—	149	2	no	—
107	C₂×C₈	C₄×D₁₄₉	1	4	yes	—
108	C₂×C₈	C₂²×D₁₄₉	1	1	yes	characteristic
109	C₂×C₈	C₂²×D₁₄₉	1	1	yes	characteristic
110	C₄⋊C₄	—	298	8	no	—
111	C₄⋊C₄	—	149	2	no	—
112	C₄⋊C₄	C₂²×D₁₄₉	1	4	yes	—
113	C₂²⋊C₄	—	2	2	no	—
114	C₂²⋊C₄	—	149	2	no	—
115	C₄²	—	2	2	no	—
116	C₄²	—	298	8	no	—
117	C₄²	—	149	2	no	—
118	C₂³	D₄×D₁₄₉	1	1	yes	characteristic
119	Q₈	—	298	4	no	—
120	Q₈	—	298	8	no	—
121	Q₈	—	149	8	no	—
122	C₂×C₄	—	2	4	no	—
123	C₂×C₄	—	298	4	no	—
124	C₂×C₄	—	298	8	no	—
125	C₂×C₄	—	149	4	no	—
126	C₂×C₄	C₂×C₄×D₁₄₉	1	2	yes	—
127	C₂×C₄	D₄×D₁₄₉	1	1	yes	characteristic
128	C₂×C₄	#14558737	1	2	yes	—
129	C₂×C₄	C₂³×D₁₄₉	1	1	yes	characteristic
130	C₈	—	298	2	no	—
131	C₈	C₂×C₄×D₁₄₉	1	2	yes	—
132	C₈	C₂×C₄×D₁₄₉	1	2	yes	—
133	C₂²	—	2	2	no	—
134	C₂²	#18011873	1	2	yes	—
135	C₂²	C₂×D₄×D₁₄₉	1	1	yes	characteristic
136	C₄	—	4	4	no	—
137	C₄	—	298	4	no	—
138	C₄	—	298	8	no	—
139	C₄	#18011873	1	2	yes	—
140	C₄	C₂²×C₄×D₁₄₉	1	1	yes	characteristic
141	C₄	C₂×#14558737	1	1	yes	characteristic
142	C₂	—	2	2	no	—
143	C₂	C₂×#18011873	1	1	yes	characteristic
144	C₂	#23750439	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₅₉₆

C₂

yes

maximal, characteristic

#23750102

C₂

yes

maximal

#23749731

C₂

yes

maximal, complemented

#23749719

C₂

yes

maximal, complemented

#23749633

C₂

yes

maximal

#23749580

C₂

yes

maximal, characteristic

#427×C₁₄₉

C₂

yes

maximal, characteristic

C₂²×Dic₂₉₈

C₂²

yes

—

C₂×Dic₅₉₆

C₄

yes

complemented

C₂×Dic₅₉₆

C₂²

yes

—

C₂×Dic₅₉₆

C₂²

yes

—

C₂×#14558720

C₂²

yes

characteristic

C₄×Dic₂₉₈

—

#18011876

C₂²

yes

—

#18011870

C₂²

yes

characteristic

#18011867

—

#18011834

C₂²

yes

—

#18011821

C₂²

yes

—

C₈×Dic₁₄₉

C₂²

yes

—

C₂²×C₈×C₁₄₉

C₂²

yes

characteristic

#117×C₁₄₉

C₂²

yes

—

#107×C₁₄₉

C₂²

yes

—

#106×C₁₄₉

C₂²

yes

—

C₂×Dic₂₉₈

—

C₂×Dic₂₉₈

—

C₂×Dic₂₉₈

C₂×C₄

yes

—

C₂×Dic₂₉₈

C₂³

yes

—

C₂²×Dic₁₄₉

—

Dic₅₉₆

C₂×C₄

yes

—

Dic₅₉₆

C₂×C₄

yes

—

#14558720

—

C₂×#11924746

C₂³

yes

—

C₄⋊Dic₁₄₉

D₄

yes

—

#14558715

—

C₂²⋊Dic₁₄₉

D₄

yes

—

C₄×Dic₁₄₉

—

C₄×Dic₁₄₉

C₂³

yes

—

C₂²×C₄×C₁₄₉

C₂³

yes

characteristic

C₂×C₈×C₁₄₉

C₂×C₄

yes

—

C₂×C₈×C₁₄₉

C₂³

yes

characteristic

C₂×C₈×C₁₄₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₄₉

C₂³

yes

—

C₂²⋊C₄×C₁₄₉

—

C₄²×C₁₄₉

—

Dic₂₉₈

—

Dic₂₉₈

—

Dic₂₉₈

C₂²×C₄

yes

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

C₂×D₄

yes

—

#11924746

—

C₂³×C₁₄₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

C₂²×C₄

yes

—

C₂×C₄×C₁₄₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₄₉

#41

yes

—

C₂×C₄×C₁₄₉

C₂⁴

yes

characteristic

C₈×C₁₄₉

C₂²×C₄

yes

—

C₈×C₁₄₉

C₂²×C₄

yes

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

C₄×D₄

yes

—

C₂²×C₁₄₉

C₂²×D₄

yes

characteristic

C₄×C₁₄₉

—

C₄×C₁₄₉

C₄×D₄

yes

—

C₄×C₁₄₉

C₂³×C₄

yes

characteristic

C₄×C₁₄₉

C₂×#41

yes

characteristic

Dic₁₄₉

—

Dic₁₄₉

—

C₂×C₁₄₉

—

C₂×C₁₄₉

C₂×C₄×D₄

yes

characteristic

C₂×C₁₄₉

#578

yes

—

C₁₄₉

#2950

yes

characteristic, complemented, Sylow

#2950

—

149

maximal, Sylow

C₂²×Q₁₆

—

149

—

#519

—

149

—

#441

—

149

—

C₄×Q₁₆

—

149

—

#427

D₁₄₉

yes

characteristic

#419

—

149

—

#405

—

149

—

C₂²×Q₈

—

149

—

C₂×Q₁₆

—

149

—

C₂×Q₁₆

—

149

—

C₂×Q₁₆

—

149

—

C₂×M₁₆

—

149

—

C₂²×C₈

C₂×D₁₄₉

yes

characteristic

C₄×Q₈

—

298

—

C₄×Q₈

—

149

—

#117

—

298

—

#117

—

149

—

#117

C₂×D₁₄₉

yes

—

#107

C₂×D₁₄₉

yes

—

#106

C₂×D₁₄₉

yes

—

#103

—

149

—

#97

—

149

—

C₄×C₈

—

149

—

C₂×Q₈

—

298

—

C₂×Q₈

—

298

—

C₂×Q₈

—

149

—

100

C₂×Q₈

—

149

—

101

C₂²×C₄

—

298

—

102

C₂²×C₄

C₂²×D₁₄₉

yes

characteristic

103

Q₁₆

—

149

—

104

Q₁₆

—

149

—

105

M₁₆

—

298

—

106

C₂×C₈

—

149

—

107

C₂×C₈

C₄×D₁₄₉

yes

—

108

C₂×C₈

C₂²×D₁₄₉

yes

characteristic

109

C₂×C₈

C₂²×D₁₄₉

yes

characteristic

110

C₄⋊C₄

—

298

—

111

C₄⋊C₄

—

149

—

112

C₄⋊C₄

C₂²×D₁₄₉

yes

—

113

C₂²⋊C₄

—

114

C₂²⋊C₄

—

149

—

115

C₄²

—

116

C₄²

—

298

—

117

C₄²

—

149

—

118

C₂³

D₄×D₁₄₉

yes

characteristic

119

Q₈

—

298

—

120

Q₈

—

298

—

121

Q₈

—

149

—

122

C₂×C₄

—

123

C₂×C₄

—

298

—

124

C₂×C₄

—

298

—

125

C₂×C₄

—

149

—

126

C₂×C₄

C₂×C₄×D₁₄₉

yes

—

127

C₂×C₄

D₄×D₁₄₉

yes

characteristic

128

C₂×C₄

#14558737

yes

—

129

C₂×C₄

C₂³×D₁₄₉

yes

characteristic

130

C₈

—

298

—

131

C₈

C₂×C₄×D₁₄₉

yes

—

132

C₈

C₂×C₄×D₁₄₉

yes

—

133

C₂²

—

134

C₂²

#18011873

yes

—

135

C₂²

C₂×D₄×D₁₄₉

yes

characteristic

136

C₄

—

137

C₄

—

298

—

138

C₄

—

298

—

139

C₄

#18011873

yes

—

140

C₄

C₂²×C₄×D₁₄₉

yes

characteristic

141

C₄

C₂×#14558737

yes

characteristic

142

C₂

—

143

C₂

C₂×#18011873

yes

characteristic

144

C₂

#23750439

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₄₉
commutator	C₄×C₁₄₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₄₉
Fitting	#427×C₁₄₉	C₂
socle	C₂²×C₁₄₉	C₂²×D₄
prime radical	C₂³×C₁₄₉	C₂×D₄
cosocle	C₂×C₄×C₁₄₉	C₂⁴
hypercenter	#427	D₁₄₉
hypocenter	C₁₄₉	#2950
LCS	G ▷ C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂²

C₂×D₄×D₁₄₉

commutator

C₄×C₁₄₉

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₁₄₉

Fitting

#427×C₁₄₉

C₂

socle

C₂²×C₁₄₉

C₂²×D₄

prime radical

C₂³×C₁₄₉

C₂×D₄

cosocle

C₂×C₄×C₁₄₉

C₂⁴

D₁₄₉

C₁₄₉

G ▷ C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
3. #23749731	C₂	semidirect
4. #23749719	C₂	semidirect
9. C₂×Dic₅₉₆	C₄	semidirect
72. C₁₄₉	#2950	semidirect

H ≅ G/N

type

3. #23749731

C₂

semidirect

4. #23749719

C₂

semidirect

9. C₂×Dic₅₉₆

C₄

semidirect

72. C₁₄₉

#2950

semidirect

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

subgroup	index
C₂²×Dic₅₉₆	2
#23750102	2
#23749731	2
#23749719	2
#23749633	2
#23749580	2
#427×C₁₄₉	2
#2950	149

subgroup

index

C₂²×Dic₅₉₆

#427×C₁₄₉

149

Subgroups — #34053558

Conjugacy classes of subgroups

Complements

Maximal subgroups