Group #34045994 — subgroups

Subgroups — #34045994

Order 19072.

264 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 67 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	#23749653	C₂	1	1	yes	maximal, characteristic, complemented
2	#23749621	C₂	1	2	yes	maximal, complemented
3	#23749599	C₂	1	2	yes	maximal, complemented
4	#23749382	C₂	1	1	yes	maximal, characteristic, complemented
5	#575×C₁₄₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11924746	C₂²	1	1	yes	characteristic
7	#18011870	C₂²	1	2	yes	—
8	#18011870	C₂²	1	1	yes	characteristic
9	#18011840	—	2	2	no	—
10	#18011835	—	2	2	no	—
11	#18011832	—	2	2	no	—
12	#18011826	—	2	4	no	—
13	C₂×#41×C₁₄₉	C₂²	1	2	yes	—
14	#137×C₁₄₉	—	2	2	no	—
15	AGL(1,ℤ/8)×C₁₄₉	—	2	2	no	—
16	#135×C₁₄₉	—	2	2	no	—
17	C₂×M₁₆×C₁₄₉	C₂²	1	1	yes	characteristic
18	C₂×#11924746	—	2	2	no	—
19	C₂×#11924746	—	2	1	no	—
20	C₄⋊Dic₁₄₉	—	2	2	no	—
21	C₄⋊Dic₁₄₉	—	2	1	no	—
22	C₂²⋊Dic₁₄₉	—	2	2	no	—
23	C₂²⋊Dic₁₄₉	D₄	1	2	yes	—
24	C₄×Dic₁₄₉	—	2	1	no	—
25	C₄×Dic₁₄₉	—	2	2	no	—
26	#41×C₁₄₉	—	2	2	no	—
27	#41×C₁₄₉	—	2	2	no	—
28	#41×C₁₄₉	—	2	2	no	—
29	#41×C₁₄₉	—	2	2	no	—
30	C₂×Q₈×C₁₄₉	D₄	1	2	yes	—
31	C₂×D₄×C₁₄₉	—	2	2	no	—
32	C₂×D₄×C₁₄₉	D₄	1	2	yes	—
33	C₂²×C₄×C₁₄₉	—	2	2	no	—
34	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
35	Q₁₆×C₁₄₉	—	2	2	no	—
36	SD₁₆×C₁₄₉	—	2	4	no	—
37	D₈×C₁₄₉	—	2	2	no	—
38	M₁₆×C₁₄₉	—	2	2	no	—
39	C₂×C₈×C₁₄₉	D₄	1	2	yes	—
40	C₂×Dic₁₄₉	—	2	2	no	—
41	C₂×Dic₁₄₉	—	4	2	no	—
42	#11924746	—	4	1	no	—
43	C₂³×C₁₄₉	—	2	2	no	—
44	C₂³×C₁₄₉	C₂×D₄	1	1	yes	characteristic
45	Q₈×C₁₄₉	—	2	2	no	—
46	Q₈×C₁₄₉	—	2	2	no	—
47	D₄×C₁₄₉	—	2	2	no	—
48	D₄×C₁₄₉	—	2	2	no	—
49	D₄×C₁₄₉	—	4	2	no	—
50	D₄×C₁₄₉	—	4	2	no	—
51	C₂×C₄×C₁₄₉	—	2	2	no	—
52	C₂×C₄×C₁₄₉	—	4	2	no	—
53	C₂×C₄×C₁₄₉	—	4	2	no	—
54	C₂×C₄×C₁₄₉	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₁₄₉	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₁₄₉	#41	1	1	yes	characteristic
57	C₂×C₄×C₁₄₉	#41	1	2	yes	—
58	C₈×C₁₄₉	—	2	2	no	—
59	C₂²×C₁₄₉	—	4	1	no	—
60	C₂²×C₁₄₉	—	4	2	no	—
61	C₂²×C₁₄₉	—	8	2	no	—
62	C₂²×C₁₄₉	#121	1	1	yes	characteristic
63	C₂²×C₁₄₉	#121	1	2	yes	—
64	C₄×C₁₄₉	—	4	2	no	—
65	C₄×C₁₄₉	#120	1	1	yes	characteristic
66	C₄×C₁₄₉	#123	1	2	yes	—
67	C₄×C₁₄₉	#124	1	1	yes	characteristic
68	Dic₁₄₉	—	4	2	no	—
69	Dic₁₄₉	—	8	2	no	—
70	C₂×C₁₄₉	—	2	1	no	—
71	C₂×C₁₄₉	—	2	2	no	—
72	C₂×C₁₄₉	—	8	2	no	—
73	C₂×C₁₄₉	#394	1	1	yes	characteristic
74	C₁₄₉	#2053	1	1	yes	characteristic, complemented, Sylow
75	#2053	—	149	1	no	maximal, Sylow
76	#575	D₁₄₉	1	1	yes	characteristic
77	#424	—	149	1	no	—
78	#416	—	149	2	no	—
79	#410	—	149	2	no	—
80	H(3,ℤ/4)	—	149	1	no	—
81	C₂×#41	C₂×D₁₄₉	1	2	yes	—
82	#137	—	2	2	no	—
83	AGL(1,ℤ/8)	—	2	2	no	—
84	#135	—	2	2	no	—
85	C₂×M₁₆	C₂×D₁₄₉	1	1	yes	characteristic
86	C₂²×C₈	—	149	1	no	—
87	#117	—	149	2	no	—
88	#117	—	149	1	no	—
89	C₄⋊C₈	—	298	2	no	—
90	#103	—	298	2	no	—
91	#102	—	298	2	no	—
92	#99	—	298	4	no	—
93	#41	—	2	2	no	—
94	#41	—	2	2	no	—
95	#41	—	2	2	no	—
96	#41	—	2	2	no	—
97	C₂×Q₈	C₂²⋊D₁₄₉	1	2	yes	—
98	C₂×D₄	—	2	2	no	—
99	C₂×D₄	C₂²⋊D₁₄₉	1	2	yes	—
100	C₂²×C₄	—	2	2	no	—
101	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
102	Q₁₆	—	2	2	no	—
103	SD₁₆	—	2	4	no	—
104	D₈	—	2	2	no	—
105	M₁₆	—	2	2	no	—
106	C₂×C₈	—	298	2	no	—
107	C₂×C₈	—	298	1	no	—
108	C₂×C₈	C₂²⋊D₁₄₉	1	2	yes	—
109	C₄⋊C₄	—	298	1	no	—
110	C₄⋊C₄	—	298	2	no	—
111	C₂²⋊C₄	—	298	2	no	—
112	C₂²⋊C₄	—	149	2	no	—
113	C₄²	—	298	1	no	—
114	C₄²	—	298	2	no	—
115	C₂³	—	2	2	no	—
116	C₂³	D₄×D₁₄₉	1	1	yes	characteristic
117	Q₈	—	2	2	no	—
118	Q₈	—	2	2	no	—
119	D₄	—	2	2	no	—
120	D₄	—	2	2	no	—
121	D₄	—	4	2	no	—
122	D₄	—	4	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	—	4	2	no	—
126	C₂×C₄	—	298	2	no	—
127	C₂×C₄	—	596	2	no	—
128	C₂×C₄	C₂×C₂²⋊D₁₄₉	1	2	yes	—
129	C₂×C₄	C₂×C₂²⋊D₁₄₉	1	1	yes	characteristic
130	C₂×C₄	#14558735	1	1	yes	characteristic
131	C₂×C₄	#14558735	1	2	yes	—
132	C₈	—	2	2	no	—
133	C₈	—	596	1	no	—
134	C₂²	—	4	1	no	—
135	C₂²	—	4	2	no	—
136	C₂²	—	8	2	no	—
137	C₂²	#18011882	1	1	yes	characteristic
138	C₂²	#18011883	1	2	yes	—
139	C₄	—	4	2	no	—
140	C₄	—	596	2	no	—
141	C₄	—	1192	2	no	—
142	C₄	#18011880	1	1	yes	characteristic
143	C₄	#18011892	1	2	yes	—
144	C₄	#18011898	1	1	yes	characteristic
145	C₂	—	2	1	no	—
146	C₂	—	2	2	no	—
147	C₂	—	8	2	no	—
148	C₂	#23749539	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23749653

C₂

yes

maximal, characteristic, complemented

#23749621

C₂

yes

maximal, complemented

#23749599

C₂

yes

maximal, complemented

#23749382

C₂

yes

maximal, characteristic, complemented

#575×C₁₄₉

C₂

yes

maximal, characteristic

C₂²×#11924746

C₂²

yes

characteristic

#18011870

C₂²

yes

—

#18011870

C₂²

yes

—

—

—

—

C₂×#41×C₁₄₉

C₂²

yes

—

#137×C₁₄₉

—

AGL(1,ℤ/8)×C₁₄₉

—

#135×C₁₄₉

—

C₂×M₁₆×C₁₄₉

C₂²

yes

characteristic

C₂×#11924746

—

C₂×#11924746

—

C₄⋊Dic₁₄₉

—

C₄⋊Dic₁₄₉

—

C₂²⋊Dic₁₄₉

—

C₂²⋊Dic₁₄₉

D₄

yes

—

C₄×Dic₁₄₉

—

C₄×Dic₁₄₉

—

#41×C₁₄₉

—

#41×C₁₄₉

—

#41×C₁₄₉

—

#41×C₁₄₉

—

C₂×Q₈×C₁₄₉

D₄

yes

—

C₂×D₄×C₁₄₉

—

C₂×D₄×C₁₄₉

D₄

yes

—

C₂²×C₄×C₁₄₉

—

C₂²×C₄×C₁₄₉

C₂³

yes

characteristic

Q₁₆×C₁₄₉

—

SD₁₆×C₁₄₉

—

D₈×C₁₄₉

—

M₁₆×C₁₄₉

—

C₂×C₈×C₁₄₉

D₄

yes

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

#11924746

—

C₂³×C₁₄₉

—

C₂³×C₁₄₉

C₂×D₄

yes

characteristic

Q₈×C₁₄₉

—

Q₈×C₁₄₉

—

D₄×C₁₄₉

—

D₄×C₁₄₉

—

D₄×C₁₄₉

—

D₄×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

C₂×D₄

yes

—

C₂×C₄×C₁₄₉

C₂×D₄

yes

C₂×C₄×C₁₄₉

yes

C₂×C₄×C₁₄₉

yes

—

C₈×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

#121

yes

characteristic

C₂²×C₁₄₉

#121

yes

—

C₄×C₁₄₉

—

C₄×C₁₄₉

#120

yes

characteristic

C₄×C₁₄₉

#123

yes

—

C₄×C₁₄₉

#124

yes

characteristic

Dic₁₄₉

—

Dic₁₄₉

—

C₂×C₁₄₉

—

C₂×C₁₄₉

—

C₂×C₁₄₉

—

C₂×C₁₄₉

#394

yes

characteristic

C₁₄₉

#2053

yes

characteristic, complemented, Sylow

#2053

—

149

maximal, Sylow

#575

D₁₄₉

yes

characteristic

#424

—

149

—

#416

—

149

—

#410

—

149

—

H(3,ℤ/4)

—

149

—

C₂×#41

C₂×D₁₄₉

yes

—

#137

—

AGL(1,ℤ/8)

—

#135

—

C₂×M₁₆

C₂×D₁₄₉

yes

characteristic

C₂²×C₈

—

149

—

#117

—

149

—

#117

—

149

—

C₄⋊C₈

—

298

—

#103

—

298

—

#102

—

298

—

#99

—

298

—

—

—

—

—

C₂×Q₈

C₂²⋊D₁₄₉

yes

—

C₂×D₄

—

C₂×D₄

C₂²⋊D₁₄₉

yes

—

100

C₂²×C₄

—

101

C₂²×C₄

C₂²×D₁₄₉

yes

characteristic

102

Q₁₆

—

103

SD₁₆

—

104

D₈

—

105

M₁₆

—

106

C₂×C₈

—

298

—

107

C₂×C₈

—

298

—

108

C₂×C₈

C₂²⋊D₁₄₉

yes

—

109

C₄⋊C₄

—

298

—

110

C₄⋊C₄

—

298

—

111

C₂²⋊C₄

—

298

—

112

C₂²⋊C₄

—

149

—

113

C₄²

—

298

—

114

C₄²

—

298

—

115

C₂³

—

116

C₂³

D₄×D₁₄₉

yes

characteristic

117

Q₈

—

118

Q₈

—

119

D₄

—

120

D₄

—

121

D₄

—

122

D₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

298

—

127

C₂×C₄

—

596

—

128

C₂×C₄

C₂×C₂²⋊D₁₄₉

yes

—

129

C₂×C₄

C₂×C₂²⋊D₁₄₉

yes

characteristic

130

C₂×C₄

#14558735

yes

characteristic

131

C₂×C₄

#14558735

yes

—

132

C₈

—

133

C₈

—

596

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

#18011882

yes

characteristic

138

C₂²

#18011883

yes

—

139

C₄

—

140

C₄

—

596

—

141

C₄

—

1192

—

142

C₄

#18011880

yes

characteristic

143

C₄

#18011892

yes

—

144

C₄

#18011898

yes

characteristic

145

C₂

—

146

C₂

—

147

C₂

—

148

C₂

#23749539

yes

characteristic

	subgroup	G/H
center	C₄	#18011898
commutator, cosocle	C₂²×C₄×C₁₄₉	C₂³
Frattini	C₂²×C₄	C₂²×D₁₄₉
Fitting, prime radical	#575×C₁₄₉	C₂
socle	C₂×C₁₄₉	#394
hypercenter	#575	D₁₄₉
hypocenter	C₁₄₉	#2053
LCS	G ▷ C₂²×C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #575

subgroup

G/H

center

C₄

#18011898

commutator, cosocle

C₂²×C₄×C₁₄₉

C₂³

Frattini

C₂²×C₄

C₂²×D₁₄₉

Fitting, prime radical

#575×C₁₄₉

C₂

C₂×C₁₄₉

D₁₄₉

C₁₄₉

G ▷ C₂²×C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #575

N	H ≅ G/N	type
1. #23749653	C₂	semidirect
2. #23749621	C₂	semidirect
3. #23749599	C₂	semidirect
4. #23749382	C₂	semidirect
74. C₁₄₉	#2053	semidirect

H ≅ G/N

type

1. #23749653

C₂

semidirect

2. #23749621

C₂

semidirect

3. #23749599

C₂

semidirect

4. #23749382

C₂

semidirect

74. C₁₄₉

#2053

semidirect

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

subgroup	index
#23749653	2
#23749621	2
#23749599	2
#23749382	2
#575×C₁₄₉	2
#2053	149

subgroup

index

#575×C₁₄₉

149

Subgroups — #34045994

Conjugacy classes of subgroups

Complements

Maximal subgroups