Group #15774786 — subgroups

Subgroups — #15774786

Order 3328.

236 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 85 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	#13116667	C₂	1	2	yes	maximal, complemented
2	C₈×D₁₀₄	C₂	1	1	yes	maximal, characteristic
3	#13112790	C₂	1	2	yes	maximal, complemented
4	#13112754	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#1367	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#88038	—	2	2	no	—
7	C₄×D₁₀₄	C₄	1	1	yes	characteristic
8	C₈×D₅₂	C₂²	1	2	yes	—
9	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	—
10	C₈²×C₁₃	C₂²	1	1	yes	characteristic
11	#11718099	—	2	2	no	—
12	C₄×#6090	C₂²	1	1	yes	characteristic
13	C₄×#6090	C₂²	1	1	yes	characteristic
14	#11718051	C₄	1	2	yes	—
15	#11718050	C₄	1	1	yes	characteristic
16	C₂×D₁₀₄	C₈	1	1	yes	characteristic
17	C₂×C₈×D₁₃	—	2	2	no	—
18	C₄×D₅₂	C₂×C₄	1	2	yes	—
19	C₂×C₁₃×C₁₆	—	2	2	no	—
20	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
23	#88061	C₈	1	2	yes	—
24	#88060	—	2	2	no	—
25	#88058	C₈	1	1	yes	characteristic
26	C₂×#6090	—	2	1	no	—
27	C₂×#6090	D₄	1	2	yes	—
28	C₄⋊#1055	C₂×C₄	1	2	yes	—
29	#88038	—	4	4	no	—
30	#12064	—	13	1	no	maximal, Sylow
31	C₂×D₅₂	C₂×C₈	1	2	yes	—
32	C₂×C₄×D₁₃	—	2	2	no	—
33	C₂×C₈×C₁₃	—	2	1	no	—
34	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
35	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
36	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
37	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
38	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
39	#6103	—	2	2	no	—
40	C₄⋊Dic₁₃	C₂×C₈	1	2	yes	—
41	C₂×#1055	—	2	2	no	—
42	D₁₀₄	—	2	2	no	—
43	C₈×D₁₃	—	4	2	no	—
44	C₈×D₁₃	—	4	2	no	—
45	C₁₃×C₁₆	—	4	2	no	—
46	#6090	—	2	2	no	—
47	#6090	—	4	1	no	—
48	#2169	—	13	2	no	—
49	C₈×D₈	—	13	1	no	—
50	#1367	D₁₃	1	1	yes	characteristic, complemented
51	D₄⋊C₁₆	—	13	2	no	—
52	#1312	—	13	1	no	—
53	C₂²×D₁₃	—	2	2	no	—
54	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
55	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
56	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
57	C₂×Dic₁₃	—	2	2	no	—
58	D₅₂	—	2	2	no	—
59	D₅₂	—	2	2	no	—
60	C₄×D₁₃	—	4	2	no	—
61	C₄×D₁₃	—	4	2	no	—
62	C₈×C₁₃	—	2	1	no	—
63	C₈×C₁₃	—	2	1	no	—
64	C₈×C₁₃	—	4	1	no	—
65	C₈×C₁₃	M₃₂	1	2	yes	complemented
66	C₈×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
67	#1055	—	4	2	no	—
68	C₂×M₃₂	—	26	2	no	—
69	C₄×D₈	—	13	1	no	—
70	C₈×D₄	—	13	2	no	—
71	C₄⋊C₁₆	C₂×D₁₃	1	2	yes	—
72	C₂²⋊C₁₆	—	26	2	no	—
73	C₄×C₁₆	—	13	1	no	—
74	C₄×C₁₆	—	13	1	no	—
75	#335	—	13	1	no	—
76	D₄⋊C₈	—	13	2	no	—
77	C₈²	C₂×D₁₃	1	1	yes	characteristic
78	C₂²×C₁₃	C₈×D₄	1	1	yes	characteristic
79	C₂×D₁₃	—	4	2	no	—
80	C₂×D₁₃	—	4	2	no	—
81	C₂×D₁₃	—	4	2	no	—
82	C₄×C₁₃	—	2	1	no	—
83	C₄×C₁₃	#442	1	1	yes	characteristic
84	C₄×C₁₃	#444	1	1	yes	characteristic
85	C₄×C₁₃	C₂×M₃₂	1	1	yes	characteristic
86	C₄×C₁₃	#504	1	1	yes	characteristic
87	Dic₁₃	—	4	2	no	—
88	C₂×D₈	—	13	1	no	—
89	C₂²×C₈	—	26	2	no	—
90	C₄×D₄	—	13	2	no	—
91	M₃₂	—	52	4	no	—
92	C₂×C₁₆	—	2	2	no	—
93	C₂×C₁₆	—	13	2	no	—
94	C₂×C₁₆	—	26	1	no	—
95	#107	—	13	1	no	—
96	C₄⋊C₈	—	13	2	no	—
97	#102	—	13	2	no	—
98	C₂²⋊C₈	—	26	2	no	—
99	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
100	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
101	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
102	C₂×C₁₃	#1580	1	1	yes	characteristic
103	C₂×C₁₃	#2169	1	1	yes	characteristic
104	C₂×C₁₃	#2171	1	1	yes	characteristic
105	D₁₃	—	8	2	no	—
106	C₂×D₄	—	13	2	no	—
107	C₂²×C₄	—	26	2	no	—
108	D₈	—	26	2	no	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	26	2	no	—
111	C₂×C₈	—	52	2	no	—
112	C₂×C₈	—	52	2	no	—
113	C₂×C₈	C₈×D₁₃	1	1	yes	characteristic
114	C₂×C₈	C₈×D₁₃	1	1	yes	characteristic
115	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
116	C₂×C₈	#6131	1	1	yes	characteristic
117	C₄⋊C₄	—	13	2	no	—
118	C₂²⋊C₄	—	26	2	no	—
119	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
120	C₁₆	—	4	2	no	—
121	C₁₆	—	26	2	no	—
122	C₁₆	—	52	1	no	—
123	C₁₃	#12064	1	1	yes	characteristic, complemented, Sylow
124	C₂³	—	26	2	no	—
125	D₄	—	26	2	no	—
126	D₄	—	26	2	no	—
127	C₂×C₄	—	26	2	no	—
128	C₂×C₄	—	52	2	no	—
129	C₂×C₄	—	52	2	no	—
130	C₂×C₄	#88147	1	1	yes	characteristic
131	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
132	C₂×C₄	#88161	1	1	yes	characteristic
133	C₈	—	2	1	no	—
134	C₈	—	2	1	no	—
135	C₈	—	4	1	no	—
136	C₈	—	52	2	no	—
137	C₈	#88038	1	2	yes	complemented
138	C₈	#88168	1	2	yes	—
139	C₂²	—	52	2	no	—
140	C₂²	—	52	2	no	—
141	C₂²	—	52	2	no	—
142	C₂²	#11718495	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	52	2	no	—
145	C₄	#11718530	1	1	yes	characteristic
146	C₄	#11718555	1	1	yes	characteristic
147	C₄	C₂×#88038	1	1	yes	characteristic
148	C₄	#11718568	1	1	yes	characteristic
149	C₂	—	104	2	no	—
150	C₂	#13113659	1	1	yes	characteristic
151	C₂	#13116667	1	1	yes	characteristic
152	C₂	#13116677	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13116667

C₂

yes

maximal, complemented

C₈×D₁₀₄

C₂

yes

maximal, characteristic

#13112790

C₂

yes

maximal, complemented

#13112754

C₂

yes

maximal, characteristic, complemented

C₁₃×#1367

C₂

yes

maximal, characteristic, complemented

C₂×#88038

—

C₄×D₁₀₄

C₄

yes

characteristic

C₈×D₅₂

C₂²

yes

—

C₁₃×C₄⋊C₁₆

C₂²

yes

—

C₈²×C₁₃

C₂²

yes

characteristic

#11718099

—

C₄×#6090

C₂²

yes

characteristic

C₄×#6090

C₂²

yes

characteristic

#11718051

C₄

yes

—

#11718050

C₄

yes

characteristic

C₂×D₁₀₄

C₈

yes

characteristic

C₂×C₈×D₁₃

—

C₄×D₅₂

C₂×C₄

yes

—

C₂×C₁₃×C₁₆

—

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

C₄×C₈×C₁₃

C₂³

yes

characteristic

#88061

C₈

yes

—

#88060

—

#88058

C₈

yes

characteristic

C₂×#6090

—

C₂×#6090

D₄

yes

—

C₄⋊#1055

C₂×C₄

yes

—

#88038

—

#12064

—

maximal, Sylow

C₂×D₅₂

C₂×C₈

yes

—

C₂×C₄×D₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₂×C₈

yes

characteristic

C₂×C₈×C₁₃

C₂×C₈

yes

characteristic

C₂×C₈×C₁₃

C₂×D₄

yes

characteristic

C₂×C₈×C₁₃

#41

yes

characteristic

C₄²×C₁₃

C₂²×C₄

yes

characteristic

#6103

—

C₄⋊Dic₁₃

C₂×C₈

yes

—

C₂×#1055

—

D₁₀₄

—

C₈×D₁₃

—

C₈×D₁₃

—

C₁₃×C₁₆

—

#6090

—

#6090

—

#2169

—

C₈×D₈

—

#1367

D₁₃

yes

characteristic, complemented

D₄⋊C₁₆

—

#1312

—

C₂²×D₁₃

—

C₂×C₄×C₁₃

C₄×D₄

yes

characteristic

C₂×C₄×C₁₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₁₃

#131

yes

characteristic

C₂×Dic₁₃

—

D₅₂

—

D₅₂

—

C₄×D₁₃

—

C₄×D₁₃

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

M₃₂

yes

complemented

C₈×C₁₃

AGL(1,ℤ/8)

yes

—

#1055

—

C₂×M₃₂

—

C₄×D₈

—

C₈×D₄

—

C₄⋊C₁₆

C₂×D₁₃

yes

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

#335

—

D₄⋊C₈

—

C₈²

C₂×D₁₃

yes

characteristic

C₂²×C₁₃

C₈×D₄

yes

characteristic

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

—

C₄×C₁₃

yes

C₄×C₁₃

yes

C₄×C₁₃

C₂×M₃₂

yes

characteristic

C₄×C₁₃

#504

yes

characteristic

Dic₁₃

—

C₂×D₈

—

C₂²×C₈

—

C₄×D₄

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#107

—

C₄⋊C₈

—

#102

—

C₂²⋊C₈

—

C₄×C₈

C₄×D₁₃

yes

characteristic

100

C₄×C₈

C₄×D₁₃

yes

characteristic

101

C₄×C₈

C₂²×D₁₃

yes

characteristic

102

C₂×C₁₃

#1580

yes

characteristic

103

C₂×C₁₃

#2169

yes

characteristic

104

C₂×C₁₃

#2171

yes

characteristic

105

D₁₃

—

106

C₂×D₄

—

107

C₂²×C₄

—

108

D₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

C₈×D₁₃

yes

characteristic

114

C₂×C₈

C₈×D₁₃

yes

characteristic

115

C₂×C₈

D₄×D₁₃

yes

characteristic

116

C₂×C₈

#6131

yes

characteristic

117

C₄⋊C₄

—

118

C₂²⋊C₄

—

119

C₄²

C₂×C₄×D₁₃

yes

characteristic

120

C₁₆

—

121

C₁₆

—

122

C₁₆

—

123

C₁₃

#12064

yes

characteristic, complemented, Sylow

124

C₂³

—

125

D₄

—

126

D₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

#88147

yes

characteristic

131

C₂×C₄

C₂×C₈×D₁₃

yes

characteristic

132

C₂×C₄

#88161

yes

characteristic

133

C₈

—

134

C₈

—

135

C₈

—

136

C₈

—

137

C₈

#88038

yes

complemented

138

C₈

#88168

yes

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

#11718495

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

#11718530

yes

characteristic

146

C₄

#11718555

yes

characteristic

147

C₄

C₂×#88038

yes

characteristic

148

C₄

#11718568

yes

characteristic

149

C₂

—

104

—

150

C₂

#13113659

yes

characteristic

151

C₂

#13116667

yes

characteristic

152

C₂

#13116677

yes

characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₁₃×#1367	C₂
socle	C₂²×C₁₃	C₈×D₄
prime radical	C₂×D₁₀₄	C₈
cosocle	C₄×C₈×C₁₃	C₂³
hypercenter	#1367	D₁₃
hypocenter	C₁₃	#12064
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1367

subgroup

G/H

center

C₂×C₈

D₄×D₁₃

commutator

C₂×C₄×C₁₃

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₁₃

Fitting

C₁₃×#1367

C₂

socle

C₂²×C₁₃

C₈×D₄

prime radical

C₂×D₁₀₄

C₈

cosocle

C₄×C₈×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1367

N	H ≅ G/N	type
1. #13116667	C₂	semidirect
3. #13112790	C₂	semidirect
4. #13112754	C₂	semidirect
5. C₁₃×#1367	C₂	semidirect
50. #1367	D₁₃	semidirect
65. C₈×C₁₃	M₃₂	semidirect
123. C₁₃	#12064	semidirect
137. C₈	#88038	semidirect

H ≅ G/N

type

1. #13116667

C₂

semidirect

3. #13112790

C₂

semidirect

4. #13112754

C₂

semidirect

5. C₁₃×#1367

C₂

semidirect

50. #1367

D₁₃

semidirect

65. C₈×C₁₃

M₃₂

semidirect

123. C₁₃

#12064

semidirect

137. C₈

#88038

semidirect

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

subgroup	index
#13116667	2
C₈×D₁₀₄	2
#13112790	2
#13112754	2
C₁₃×#1367	2
#12064	13

subgroup

index

C₈×D₁₀₄

C₁₃×#1367

Subgroups — #15774786

Conjugacy classes of subgroups

Complements

Maximal subgroups