Group #15750133 — subgroups

Subgroups — #15750133

Order 3328.

298 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 75 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	C₄×#88076	C₂	1	2	yes	maximal
2	C₁₃×#1985	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
4	C₂×C₁₃×C₄⋊C₈	C₄	1	1	yes	characteristic
5	C₂×C₁₃×#97	C₄	1	1	yes	characteristic
6	C₂×#88076	—	2	2	no	—
7	C₄×#6099	—	2	2	no	—
8	C₁₃×#336	—	2	2	no	—
9	#11718122	C₄	1	4	yes	—
10	C₂²×Q₈×C₁₃	D₄	1	1	yes	characteristic
11	C₂²×C₈×C₁₃	—	2	1	no	—
12	C₂²×C₈×C₁₃	—	2	1	no	—
13	C₄×Q₈×C₁₃	—	4	1	no	—
14	C₄×Q₈×C₁₃	—	4	1	no	—
15	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
16	C₂×C₁₃×C₄⋊C₄	Q₈	1	1	yes	characteristic
17	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄²×C₁₃	D₄	1	2	yes	—
19	C₂×#6099	—	2	4	no	—
20	C₁₃×C₄⋊C₈	—	4	1	no	—
21	C₁₃×#97	—	4	1	no	—
22	#88076	—	2	8	no	—
23	#88043	—	4	2	no	—
24	C₄×#1055	—	4	2	no	—
25	#7284	—	13	1	no	maximal, Sylow
26	C₂×Q₈×C₁₃	—	2	4	no	—
27	C₂×Q₈×C₁₃	—	4	1	no	—
28	C₂²×C₄×C₁₃	C₄²	1	1	yes	characteristic
29	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
30	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
32	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₈×C₁₃	—	2	2	no	—
34	C₂×C₈×C₁₃	—	4	1	no	—
35	C₂×C₈×C₁₃	—	4	1	no	—
36	C₂×C₈×C₁₃	—	4	2	no	—
37	C₂×C₈×C₁₃	—	4	1	no	—
38	C₁₃×C₄⋊C₄	—	4	2	no	—
39	C₁₃×C₄⋊C₄	—	4	1	no	—
40	C₄²×C₁₃	—	2	1	no	—
41	C₄²×C₁₃	—	2	1	no	—
42	C₄²×C₁₃	—	2	2	no	—
43	C₄²×C₁₃	—	2	2	no	—
44	#6099	—	2	8	no	—
45	#6099	—	4	4	no	—
46	C₂×#1055	—	4	4	no	—
47	#1985	D₁₃	1	1	yes	characteristic
48	C₄×#101	—	13	2	no	—
49	C₂³×C₁₃	#95	1	1	yes	characteristic
50	Q₈×C₁₃	—	4	4	no	—
51	C₂×C₄×C₁₃	—	2	2	no	—
52	C₂×C₄×C₁₃	—	2	2	no	—
53	C₂×C₄×C₁₃	—	2	1	no	—
54	C₂×C₄×C₁₃	—	2	1	no	—
55	C₂×C₄×C₁₃	—	2	2	no	—
56	C₂×C₄×C₁₃	—	2	1	no	—
57	C₂×C₄×C₁₃	—	2	2	no	—
58	C₂×C₄×C₁₃	—	2	1	no	—
59	C₂×C₄×C₁₃	—	4	1	no	—
60	C₂×C₄×C₁₃	—	4	2	no	—
61	C₂×C₄×C₁₃	#99	1	1	yes	characteristic
62	C₂×C₄×C₁₃	#99	1	1	yes	characteristic
63	C₂×C₄×C₁₃	#104	1	4	yes	—
64	C₈×C₁₃	—	4	2	no	—
65	C₈×C₁₃	—	8	1	no	—
66	#1055	—	4	8	no	—
67	C₂×C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
68	C₂×C₄⋊C₈	Dic₁₃	1	1	yes	characteristic
69	C₂×#101	—	26	2	no	—
70	C₄×M₁₆	—	26	2	no	—
71	C₂×#97	Dic₁₃	1	1	yes	characteristic
72	#343	—	13	4	no	—
73	#336	—	2	2	no	—
74	C₂²×C₁₃	—	2	1	no	—
75	C₂²×C₁₃	—	2	1	no	—
76	C₂²×C₁₃	#339	1	2	yes	—
77	C₂²×C₁₃	#342	1	1	yes	characteristic
78	C₄×C₁₃	—	2	4	no	—
79	C₄×C₁₃	—	2	1	no	—
80	C₄×C₁₃	—	4	1	no	—
81	C₄×C₁₃	—	4	1	no	—
82	C₄×C₁₃	—	4	2	no	—
83	C₄×C₁₃	#355	1	1	yes	characteristic
84	C₄×C₁₃	#356	1	1	yes	characteristic
85	C₂²×Q₈	D₅₂	1	1	yes	characteristic
86	C₂×M₁₆	—	26	4	no	—
87	C₂²×C₈	—	2	1	no	—
88	C₂²×C₈	—	2	1	no	—
89	C₄×Q₈	—	4	1	no	—
90	C₄×Q₈	—	4	1	no	—
91	C₂×C₄⋊C₄	Dic₂₆	1	1	yes	characteristic
92	C₂×C₄⋊C₄	C₄×D₁₃	1	2	yes	—
93	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
94	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
95	C₄⋊C₈	—	4	1	no	—
96	#101	—	26	8	no	—
97	#97	—	4	1	no	—
98	#97	—	52	2	no	—
99	C₄×C₈	—	52	2	no	—
100	C₂×C₁₃	—	2	2	no	—
101	C₂×C₁₃	#1297	1	1	yes	characteristic
102	C₂×C₁₃	#1392	1	1	yes	characteristic
103	C₂×C₁₃	#1394	1	1	yes	characteristic
104	C₂×Q₈	—	2	4	no	—
105	C₂×Q₈	—	4	1	no	—
106	C₂²×C₄	C₄×Dic₁₃	1	1	yes	characteristic
107	C₂²×C₄	#6101	1	2	yes	—
108	C₂²×C₄	C₄⋊Dic₁₃	1	1	yes	characteristic
109	C₂²×C₄	#6103	1	2	yes	—
110	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
111	M₁₆	—	26	8	no	—
112	M₁₆	—	52	4	no	—
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	4	1	no	—
115	C₂×C₈	—	4	1	no	—
116	C₂×C₈	—	4	2	no	—
117	C₂×C₈	—	4	1	no	—
118	C₂×C₈	—	52	4	no	—
119	C₄⋊C₄	—	4	1	no	—
120	C₄⋊C₄	—	4	2	no	—
121	C₄²	—	2	1	no	—
122	C₄²	—	2	2	no	—
123	C₄²	—	2	2	no	—
124	C₄²	—	2	1	no	—
125	C₁₃	#7284	1	1	yes	characteristic, complemented, Sylow
126	C₂³	#88071	1	1	yes	characteristic
127	Q₈	—	4	4	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	4	1	no	—
137	C₂×C₄	—	4	2	no	—
138	C₂×C₄	#88065	1	4	yes	—
139	C₂×C₄	#88074	1	1	yes	characteristic
140	C₂×C₄	#88074	1	1	yes	characteristic
141	C₈	—	4	2	no	—
142	C₈	—	8	1	no	—
143	C₈	—	52	8	no	—
144	C₂²	—	2	1	no	—
145	C₂²	—	2	1	no	—
146	C₂²	#11718118	1	1	yes	characteristic
147	C₂²	#11718147	1	2	yes	—
148	C₄	—	2	1	no	—
149	C₄	—	2	4	no	—
150	C₄	—	4	1	no	—
151	C₄	—	4	2	no	—
152	C₄	—	4	1	no	—
153	C₄	#11718134	1	1	yes	characteristic
154	C₄	#11718140	1	1	yes	characteristic
155	C₂	—	2	2	no	—
156	C₂	#13112721	1	1	yes	characteristic
157	C₂	#13112959	1	1	yes	characteristic
158	C₂	#13112963	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#88076

C₂

yes

maximal

C₁₃×#1985

C₂

yes

maximal, characteristic

C₂×C₄×Q₈×C₁₃

C₂²

yes

characteristic

C₂×C₁₃×C₄⋊C₈

C₄

yes

characteristic

C₂×C₁₃×#97

C₄

yes

characteristic

C₂×#88076

—

C₄×#6099

—

C₁₃×#336

—

#11718122

C₄

yes

—

C₂²×Q₈×C₁₃

D₄

yes

characteristic

C₂²×C₈×C₁₃

—

C₂²×C₈×C₁₃

—

C₄×Q₈×C₁₃

—

C₄×Q₈×C₁₃

—

C₂×C₁₃×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₁₃×C₄⋊C₄

Q₈

yes

characteristic

C₂×C₄²×C₁₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₁₃

D₄

yes

—

C₂×#6099

—

C₁₃×C₄⋊C₈

—

C₁₃×#97

—

—

—

—

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂²×C₄×C₁₃

C₄²

yes

characteristic

C₂²×C₄×C₁₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₃

C₄⋊C₄

yes

—

C₂²×C₄×C₁₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

—

—

—

—

D₁₃

yes

characteristic

C₄×#101

—

C₂³×C₁₃

#95

yes

characteristic

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

—

C₈×C₁₃

—

C₈×C₁₃

—

#1055

—

C₂×C₄×Q₈

C₂×D₁₃

yes

characteristic

C₂×C₄⋊C₈

Dic₁₃

yes

characteristic

C₂×#101

—

C₄×M₁₆

—

C₂×#97

Dic₁₃

yes

characteristic

#343

—

#336

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

#339

yes

—

C₂²×C₁₃

#342

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

yes

C₄×C₁₃

yes

C₂²×Q₈

D₅₂

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

Dic₂₆

yes

characteristic

C₂×C₄⋊C₄

C₄×D₁₃

yes

—

C₂×C₄²

C₂×Dic₁₃

yes

characteristic

C₂×C₄²

C₂²⋊D₁₃

yes

—

C₄⋊C₈

—

#101

—

#97

—

#97

—

C₄×C₈

—

100

C₂×C₁₃

—

101

C₂×C₁₃

#1297

yes

characteristic

102

C₂×C₁₃

#1392

yes

characteristic

103

C₂×C₁₃

#1394

yes

characteristic

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂²×C₄

C₄×Dic₁₃

yes

characteristic

107

C₂²×C₄

#6101

yes

—

108

C₂²×C₄

C₄⋊Dic₁₃

yes

characteristic

109

C₂²×C₄

#6103

yes

—

110

C₂²×C₄

C₂²⋊Dic₁₃

yes

characteristic

111

M₁₆

—

112

M₁₆

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄²

—

122

C₄²

—

123

C₄²

—

124

C₄²

—

125

C₁₃

#7284

yes

characteristic, complemented, Sylow

126

C₂³

#88071

yes

characteristic

127

Q₈

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

#88065

yes

—

139

C₂×C₄

#88074

yes

characteristic

140

C₂×C₄

#88074

yes

characteristic

141

C₈

—

142

C₈

—

143

C₈

—

144

C₂²

—

145

C₂²

—

146

C₂²

#11718118

yes

characteristic

147

C₂²

#11718147

yes

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

#11718134

yes

characteristic

154

C₄

#11718140

yes

characteristic

155

C₂

—

156

C₂

#13112721

yes

characteristic

157

C₂

#13112959

yes

characteristic

158

C₂

#13112963

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#88074
commutator	C₂²×C₄×C₁₃	C₄²
Frattini	C₂×C₄×Q₈	C₂×D₁₃
Fitting	C₁₃×#1985	C₂
socle	C₂²×C₁₃	#342
prime radical	C₂³×C₁₃	#95
cosocle	C₂×C₄×Q₈×C₁₃	C₂²
hypercenter	#1985	D₁₃
hypocenter	C₁₃	#7284
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂³×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈ ◁ #1985

subgroup

G/H

center

C₂×C₄

#88074

commutator

C₂²×C₄×C₁₃

C₄²

Frattini

C₂×C₄×Q₈

C₂×D₁₃

Fitting

C₁₃×#1985

C₂

C₂²×C₁₃

C₂³×C₁₃

C₂×C₄×Q₈×C₁₃

C₂²

D₁₃

C₁₃

G ▷ C₂²×C₄×C₁₃ ▷ C₂³×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈ ◁ #1985

N	H ≅ G/N	type
125. C₁₃	#7284	semidirect

H ≅ G/N

type

125. C₁₃

#7284

semidirect

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

subgroup	index
C₄×#88076	2
C₁₃×#1985	2
#7284	13

subgroup

index

C₄×#88076

C₁₃×#1985

#7284

Subgroups — #15750133

Conjugacy classes of subgroups

Complements

Maximal subgroups