Group #15756735 — subgroups

Subgroups — #15756735

Order 3328.

306 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 131 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	#13114973	C₂	1	2	yes	maximal
2	#13113594	C₂	1	2	yes	maximal
3	#13113592	C₂	1	1	yes	maximal, characteristic
4	C₂×#11718046	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1566	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#423	C₂²	1	2	yes	—
7	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	#11718585	C₂²	1	2	yes	—
9	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
10	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
11	C₁₃×#334	—	2	2	no	—
12	#11718150	—	2	4	no	—
13	#11718071	—	2	4	no	—
14	#11718049	—	2	2	no	—
15	#11718046	C₄	1	4	yes	—
16	C₂×C₁₃×M₁₆	—	2	2	no	—
17	C₂²×C₈×C₁₃	Q₈	1	2	yes	—
18	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
19	C₂×#6099	—	2	2	no	—
20	C₂²×#1055	D₄	1	2	yes	—
21	C₂²×#1055	Q₈	1	2	yes	—
22	C₁₃×C₄⋊C₈	—	2	2	no	—
23	C₁₃×C₄⋊C₈	—	2	2	no	—
24	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₁₃	D₄	1	2	yes	—
26	C₄⋊#1055	—	2	2	no	—
27	C₄⋊#1055	—	2	2	no	—
28	C₄×#1055	C₂×C₄	1	2	yes	—
29	C₄×#1055	C₂×C₄	1	2	yes	—
30	C₄×#1055	D₄	1	2	yes	—
31	C₄×#1055	Q₈	1	2	yes	—
32	#8827	—	13	1	no	maximal, Sylow
33	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
35	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
36	C₁₃×M₁₆	—	4	4	no	—
37	C₂×C₈×C₁₃	—	4	2	no	—
38	C₂×C₈×C₁₃	—	4	2	no	—
39	C₂×C₈×C₁₃	C₄⋊C₄	1	4	yes	—
40	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
41	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
42	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
43	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
44	#6099	—	4	4	no	—
45	C₂×#1055	—	2	4	no	—
46	C₂×#1055	—	4	2	no	—
47	C₂×#1055	—	4	2	no	—
48	C₂×#1055	C₄⋊C₄	1	4	yes	—
49	C₂×#1055	C₄⋊C₄	1	4	yes	—
50	#1837	—	13	2	no	—
51	#1567	—	13	2	no	—
52	#1566	—	13	1	no	—
53	#1566	D₁₃	1	1	yes	characteristic
54	C₂×#322	—	13	1	no	—
55	C₂³×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
56	C₂×C₄×C₁₃	—	2	2	no	—
57	C₂×C₄×C₁₃	—	2	2	no	—
58	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
62	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
63	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
64	C₂×C₄×C₁₃	#135	1	4	yes	—
65	C₈×C₁₃	—	2	4	no	—
66	C₈×C₁₃	—	8	2	no	—
67	#1055	—	2	4	no	—
68	#1055	—	2	4	no	—
69	#1055	—	8	2	no	—
70	#423	—	13	2	no	—
71	#423	C₂×D₁₃	1	2	yes	—
72	C₂×C₄×C₈	—	13	1	no	—
73	C₂×C₄×C₈	—	13	1	no	—
74	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
75	#341	—	26	4	no	—
76	#335	—	26	4	no	—
77	#334	—	2	2	no	—
78	#334	—	26	2	no	—
79	#322	—	13	4	no	—
80	C₂²×C₁₃	—	2	1	no	—
81	C₂²×C₁₃	—	2	1	no	—
82	C₂²×C₁₃	#384	1	1	yes	characteristic
83	C₂²×C₁₃	#499	1	2	yes	—
84	C₄×C₁₃	—	2	1	no	—
85	C₄×C₁₃	—	2	1	no	—
86	C₄×C₁₃	#427	1	2	yes	—
87	C₄×C₁₃	#430	1	2	yes	—
88	C₄×C₁₃	#430	1	2	yes	—
89	C₄×C₁₃	#495	1	2	yes	—
90	C₂×M₁₆	—	2	2	no	—
91	C₂×M₁₆	—	26	2	no	—
92	C₂²×C₈	—	13	2	no	—
93	C₂²×C₈	—	13	2	no	—
94	C₂²×C₈	Dic₂₆	1	2	yes	—
95	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
96	C₄⋊C₈	—	2	2	no	—
97	C₄⋊C₈	—	2	2	no	—
98	C₄⋊C₈	—	26	2	no	—
99	C₄⋊C₈	—	26	2	no	—
100	C₄×C₈	—	13	2	no	—
101	C₄×C₈	—	13	2	no	—
102	C₄×C₈	—	13	2	no	—
103	C₄×C₈	—	13	2	no	—
104	C₄×C₈	C₄×D₁₃	1	2	yes	—
105	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
106	C₂×C₁₃	—	2	2	no	—
107	C₂×C₁₃	#1846	1	1	yes	characteristic
108	C₂×C₁₃	#1849	1	1	yes	characteristic
109	C₂×C₁₃	#1849	1	1	yes	characteristic
110	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
111	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
112	C₂²×C₄	Q₈×D₁₃	1	1	yes	characteristic
113	M₁₆	—	4	4	no	—
114	M₁₆	—	52	4	no	—
115	C₂×C₈	—	4	2	no	—
116	C₂×C₈	—	4	2	no	—
117	C₂×C₈	—	13	4	no	—
118	C₂×C₈	—	13	4	no	—
119	C₂×C₈	—	26	4	no	—
120	C₂×C₈	—	52	2	no	—
121	C₂×C₈	—	52	2	no	—
122	C₂×C₈	#6101	1	4	yes	—
123	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
124	C₄²	D₄×D₁₃	1	1	yes	characteristic
125	C₄²	Q₈×D₁₃	1	1	yes	characteristic
126	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
127	C₁₃	#8827	1	1	yes	characteristic, complemented, Sylow
128	C₂³	#88142	1	1	yes	characteristic
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	#88142	1	1	yes	characteristic
132	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
133	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
134	C₂×C₄	#88170	1	4	yes	—
135	C₂×C₄	C₂×#6101	1	1	yes	characteristic
136	C₂×C₄	#88194	1	1	yes	characteristic
137	C₂×C₄	#88198	1	1	yes	characteristic
138	C₈	—	2	4	no	—
139	C₈	—	8	2	no	—
140	C₈	—	26	4	no	—
141	C₈	—	26	4	no	—
142	C₈	—	104	2	no	—
143	C₂²	—	2	1	no	—
144	C₂²	—	2	1	no	—
145	C₂²	#11718519	1	2	yes	—
146	C₂²	#11718637	1	1	yes	characteristic
147	C₄	—	2	1	no	—
148	C₄	—	2	1	no	—
149	C₄	#11718521	1	2	yes	—
150	C₄	#11718554	1	2	yes	—
151	C₄	#11718785	1	2	yes	—
152	C₄	#11718829	1	2	yes	—
153	C₂	—	2	2	no	—
154	C₂	#13114997	1	1	yes	characteristic
155	C₂	#13115009	1	1	yes	characteristic
156	C₂	#13115010	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13114973

C₂

yes

maximal

#13113594

C₂

yes

maximal

#13113592

C₂

yes

maximal, characteristic

C₂×#11718046

C₂

yes

maximal, characteristic

C₁₃×#1566

C₂

yes

maximal, characteristic

C₁₃×#423

C₂²

yes

—

C₂×C₄×C₈×C₁₃

C₂²

yes

characteristic

#11718585

C₂²

yes

—

C₂×C₄×#1055

C₂²

yes

characteristic

C₂×C₄×#1055

C₂²

yes

characteristic

C₁₃×#334

—

—

—

—

C₄

yes

—

C₂×C₁₃×M₁₆

—

C₂²×C₈×C₁₃

Q₈

yes

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂×#6099

—

C₂²×#1055

D₄

yes

—

C₂²×#1055

Q₈

yes

—

C₁₃×C₄⋊C₈

—

C₁₃×C₄⋊C₈

—

C₄×C₈×C₁₃

C₂×C₄

yes

—

C₄×C₈×C₁₃

D₄

yes

—

C₄⋊#1055

—

C₄⋊#1055

—

C₄×#1055

C₂×C₄

yes

—

C₄×#1055

C₂×C₄

yes

—

C₄×#1055

D₄

yes

—

C₄×#1055

Q₈

yes

—

#8827

—

maximal, Sylow

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₃

C₂×Q₈

yes

characteristic

C₂²×C₄×C₁₃

C₂×Q₈

yes

characteristic

C₁₃×M₁₆

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₄⋊C₄

yes

—

C₄²×C₁₃

C₂²×C₄

yes

characteristic

C₄²×C₁₃

C₂×D₄

yes

characteristic

C₄²×C₁₃

C₂×D₄

yes

characteristic

C₄²×C₁₃

C₂×Q₈

yes

—

—

—

—

C₄⋊C₄

yes

—

C₂×#1055

C₄⋊C₄

yes

—

—

—

—

D₁₃

yes

characteristic

C₂×#322

—

C₂³×C₁₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃

#127

yes

characteristic

C₂×C₄×C₁₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₁₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₁₃

#135

yes

—

C₈×C₁₃

—

C₈×C₁₃

—

—

—

—

—

C₂×D₁₃

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₁₃

yes

—

—

—

—

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

#384

yes

characteristic

C₂²×C₁₃

#499

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#427

yes

—

C₄×C₁₃

#430

yes

—

C₄×C₁₃

#430

yes

—

C₄×C₁₃

#495

yes

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

Dic₂₆

yes

—

C₂×C₄²

C₂²×D₁₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

100

C₄×C₈

—

101

C₄×C₈

—

102

C₄×C₈

—

103

C₄×C₈

—

104

C₄×C₈

C₄×D₁₃

yes

—

105

C₄×C₈

C₂²⋊D₁₃

yes

—

106

C₂×C₁₃

—

107

C₂×C₁₃

#1846

yes

characteristic

108

C₂×C₁₃

#1849

yes

characteristic

109

C₂×C₁₃

#1849

yes

characteristic

110

C₂²×C₄

C₂×Dic₂₆

yes

characteristic

111

C₂²×C₄

D₄×D₁₃

yes

characteristic

112

C₂²×C₄

Q₈×D₁₃

yes

characteristic

113

M₁₆

—

114

M₁₆

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

#6101

yes

—

123

C₄²

C₂×C₄×D₁₃

yes

characteristic

124

C₄²

D₄×D₁₃

yes

characteristic

125

C₄²

Q₈×D₁₃

yes

characteristic

126

C₄²

C₂×C₂²⋊D₁₃

yes

characteristic

127

C₁₃

#8827

yes

characteristic, complemented, Sylow

128

C₂³

#88142

yes

characteristic

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

#88142

yes

characteristic

132

C₂×C₄

C₄⋊C₄×D₁₃

yes

characteristic

133

C₂×C₄

C₄⋊C₄×D₁₃

yes

characteristic

134

C₂×C₄

#88170

yes

—

135

C₂×C₄

C₂×#6101

yes

characteristic

136

C₂×C₄

#88194

yes

characteristic

137

C₂×C₄

#88198

yes

characteristic

138

C₈

—

139

C₈

—

140

C₈

—

141

C₈

—

142

C₈

—

104

—

143

C₂²

—

144

C₂²

—

145

C₂²

#11718519

yes

—

146

C₂²

#11718637

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

#11718521

yes

—

150

C₄

#11718554

yes

—

151

C₄

#11718785

yes

—

152

C₄

#11718829

yes

—

153

C₂

—

154

C₂

#13114997

yes

characteristic

155

C₂

#13115009

yes

characteristic

156

C₂

#13115010

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#88194
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1566	C₂
socle	C₂²×C₁₃	#384
prime radical	C₂³×C₁₃	C₄⋊Q₈
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1566	D₁₃
hypocenter	C₁₃	#8827
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

subgroup

G/H

center

C₂×C₄

#88194

commutator

C₄²×C₁₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₁₃×#1566

C₂

socle

C₂²×C₁₃

#384

prime radical

C₂³×C₁₃

C₄⋊Q₈

cosocle

C₂×C₄²×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

N	H ≅ G/N	type
127. C₁₃	#8827	semidirect

H ≅ G/N

type

127. C₁₃

#8827

semidirect

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

subgroup	index
#13114973	2
#13113594	2
#13113592	2
C₂×#11718046	2
C₁₃×#1566	2
#8827	13

subgroup

index

C₁₃×#1566

Subgroups — #15756735

Conjugacy classes of subgroups

Complements

Maximal subgroups