Group #15777181 — subgroups

Subgroups — #15777181

Order 3328.

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

normal subgroups only

This group has 216 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	#13117089	C₂	1	1	yes	maximal, characteristic
2	#13114206	C₂	1	2	yes	maximal
3	#13112908	C₂	1	1	yes	maximal, characteristic
4	#13112886	C₂	1	2	yes	maximal
5	Q₈×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₈×Q₈×C₁₃	C₂²	1	1	yes	characteristic
7	#11718681	—	2	2	no	—
8	C₂×Dic₁₀₄	—	2	1	no	—
9	C₂×Dic₁₀₄	—	2	1	no	—
10	#11718360	C₂²	1	2	yes	—
11	C₁₃×C₄⋊C₁₆	—	2	1	no	—
12	C₁₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₄×C₁₃×C₁₆	—	2	1	no	—
14	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
15	#11718074	—	2	2	no	—
16	#11718050	C₂²	1	2	yes	—
17	C₄×Q₈×C₁₃	D₄	1	1	yes	characteristic
18	C₂×Dic₅₂	—	2	2	no	—
19	C₂×Dic₅₂	—	2	2	no	—
20	#88123	—	2	2	no	—
21	C₂×C₁₃×C₁₆	—	2	1	no	—
22	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
23	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
24	C₁₃×C₄⋊C₈	—	2	1	no	—
25	C₁₃×C₄⋊C₈	D₄	1	1	yes	characteristic
26	C₄×C₈×C₁₃	—	2	1	no	—
27	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
28	#88075	—	4	2	no	—
29	#88048	—	4	2	no	—
30	C₄⋊#1055	—	2	2	no	—
31	Dic₁₀₄	—	4	2	no	—
32	Dic₁₀₄	—	4	2	no	—
33	#12440	—	13	1	no	maximal, Sylow
34	C₂×Q₈×C₁₃	D₈	1	1	yes	characteristic
35	C₂×Dic₂₆	—	4	2	no	—
36	C₂×C₈×C₁₃	—	2	1	no	—
37	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
38	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
39	C₁₃×C₄⋊C₄	—	2	1	no	—
40	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
41	C₄²×C₁₃	—	2	1	no	—
42	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
43	C₄⋊Dic₁₃	—	4	2	no	—
44	C₄⋊Dic₁₃	—	4	2	no	—
45	C₂×#1055	—	4	2	no	—
46	Dic₅₂	—	4	2	no	—
47	Dic₅₂	—	4	2	no	—
48	Dic₅₂	—	8	2	no	—
49	C₁₃×C₁₆	—	2	1	no	—
50	C₁₃×C₁₆	—	4	1	no	—
51	C₁₃×C₁₆	SD₁₆	1	2	yes	—
52	#2247	—	13	1	no	—
53	Q₈×C₁₆	D₁₃	1	1	yes	characteristic
54	#1670	—	13	2	no	—
55	#1371	—	13	1	no	—
56	#1362	—	13	2	no	—
57	Q₈×C₁₃	—	2	1	no	—
58	Q₈×C₁₃	Q₃₂	1	2	yes	complemented
59	C₂×C₄×C₁₃	—	2	1	no	—
60	C₂×C₄×C₁₃	—	2	1	no	—
61	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
62	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
63	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
64	C₂×Dic₁₃	—	8	2	no	—
65	Dic₂₆	—	8	2	no	—
66	Dic₂₆	—	8	2	no	—
67	C₈×C₁₃	—	2	1	no	—
68	C₈×C₁₃	—	4	1	no	—
69	C₈×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
70	C₈×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
71	#1055	—	8	2	no	—
72	C₂×Q₃₂	—	26	1	no	—
73	C₂×Q₃₂	—	26	1	no	—
74	#494	—	13	2	no	—
75	#475	—	26	2	no	—
76	C₈×Q₈	C₂×D₁₃	1	1	yes	characteristic
77	C₄⋊C₁₆	—	2	1	no	—
78	C₄⋊C₁₆	C₂×D₁₃	1	1	yes	characteristic
79	C₄×C₁₆	—	2	1	no	—
80	C₄×C₁₆	C₂×D₁₃	1	1	yes	characteristic
81	#335	—	13	2	no	—
82	#332	—	26	2	no	—
83	C₂²×C₁₃	#466	1	1	yes	characteristic
84	C₄×C₁₃	—	2	2	no	—
85	C₄×C₁₃	—	2	1	no	—
86	C₄×C₁₃	—	4	1	no	—
87	C₄×C₁₃	#460	1	1	yes	characteristic
88	C₄×C₁₃	#472	1	1	yes	characteristic
89	C₄×C₁₃	C₂×Q₃₂	1	1	yes	characteristic
90	C₄×C₁₃	#508	1	1	yes	characteristic
91	Dic₁₃	—	16	2	no	—
92	C₂×Q₁₆	—	26	2	no	—
93	C₂×Q₁₆	—	26	2	no	—
94	C₄⋊Q₈	—	26	2	no	—
95	C₄×Q₈	D₅₂	1	1	yes	characteristic
96	Q₃₂	—	52	2	no	—
97	Q₃₂	—	52	2	no	—
98	C₂×C₁₆	—	2	1	no	—
99	C₂×C₁₆	C₂²⋊D₁₃	1	1	yes	characteristic
100	C₂×C₁₆	C₂²⋊D₁₃	1	1	yes	characteristic
101	#106	—	52	2	no	—
102	C₄⋊C₈	—	2	1	no	—
103	C₄⋊C₈	—	26	2	no	—
104	C₄⋊C₈	D₅₂	1	1	yes	characteristic
105	#103	—	52	2	no	—
106	C₄×C₈	—	2	1	no	—
107	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
108	C₂×C₁₃	#1668	1	1	yes	characteristic
109	C₂×C₁₃	#2216	1	1	yes	characteristic
110	C₂×C₁₃	#2222	1	1	yes	characteristic
111	C₂×Q₈	—	52	2	no	—
112	C₂×Q₈	D₁₀₄	1	1	yes	characteristic
113	Q₁₆	—	52	2	no	—
114	Q₁₆	—	52	2	no	—
115	Q₁₆	—	104	2	no	—
116	C₂×C₈	—	2	1	no	—
117	C₂×C₈	—	52	2	no	—
118	C₂×C₈	#6127	1	1	yes	characteristic
119	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
120	C₄⋊C₄	—	2	1	no	—
121	C₄⋊C₄	—	52	2	no	—
122	C₄⋊C₄	—	52	2	no	—
123	C₄⋊C₄	D₁₀₄	1	1	yes	characteristic
124	C₄²	—	2	1	no	—
125	C₄²	C₂×D₅₂	1	1	yes	characteristic
126	C₁₆	—	2	1	no	—
127	C₁₆	—	4	1	no	—
128	C₁₆	#6106	1	2	yes	—
129	C₁₃	#12440	1	1	yes	characteristic, complemented, Sylow
130	Q₈	—	2	1	no	—
131	Q₈	—	104	2	no	—
132	Q₈	—	104	2	no	—
133	Q₈	Dic₁₀₄	1	2	yes	complemented
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	104	2	no	—
137	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
138	C₂×C₄	#88158	1	1	yes	characteristic
139	C₂×C₄	#88184	1	1	yes	characteristic
140	C₈	—	2	1	no	—
141	C₈	—	4	1	no	—
142	C₈	—	104	2	no	—
143	C₈	C₂×#6106	1	1	yes	characteristic
144	C₈	#88203	1	1	yes	characteristic
145	C₂²	#11718776	1	1	yes	characteristic
146	C₄	—	2	1	no	—
147	C₄	—	2	2	no	—
148	C₄	—	4	1	no	—
149	C₄	—	208	2	no	—
150	C₄	#11718565	1	1	yes	characteristic
151	C₄	C₂×Dic₁₀₄	1	1	yes	characteristic
152	C₄	#11718688	1	1	yes	characteristic
153	C₄	#11718803	1	1	yes	characteristic
154	C₂	#13114196	1	1	yes	characteristic
155	C₂	#13116927	1	1	yes	characteristic
156	C₂	#13116961	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13117089

C₂

yes

maximal, characteristic

#13114206

C₂

yes

maximal

#13112908

C₂

yes

maximal, characteristic

#13112886

C₂

yes

maximal

Q₈×C₁₃×C₁₆

C₂

yes

maximal, characteristic

C₈×Q₈×C₁₃

C₂²

yes

characteristic

#11718681

—

C₂×Dic₁₀₄

—

C₂×Dic₁₀₄

—

#11718360

C₂²

yes

—

C₁₃×C₄⋊C₁₆

—

C₁₃×C₄⋊C₁₆

C₂²

yes

characteristic

C₄×C₁₃×C₁₆

—

C₄×C₁₃×C₁₆

C₂²

yes

characteristic

#11718074

—

#11718050

C₂²

yes

—

C₄×Q₈×C₁₃

D₄

yes

characteristic

C₂×Dic₅₂

—

C₂×Dic₅₂

—

#88123

—

C₂×C₁₃×C₁₆

—

C₂×C₁₃×C₁₆

D₄

yes

characteristic

C₂×C₁₃×C₁₆

D₄

yes

characteristic

C₁₃×C₄⋊C₈

—

C₁₃×C₄⋊C₈

D₄

yes

characteristic

C₄×C₈×C₁₃

—

C₄×C₈×C₁₃

C₂³

yes

—

—

—

Dic₁₀₄

—

Dic₁₀₄

—

#12440

—

maximal, Sylow

C₂×Q₈×C₁₃

D₈

yes

characteristic

C₂×Dic₂₆

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₂×D₄

yes

characteristic

C₂×C₈×C₁₃

#41

yes

characteristic

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

D₈

yes

characteristic

C₄²×C₁₃

—

C₄²×C₁₃

C₂×D₄

yes

characteristic

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₂×#1055

—

Dic₅₂

—

Dic₅₂

—

Dic₅₂

—

C₁₃×C₁₆

—

C₁₃×C₁₆

—

C₁₃×C₁₆

SD₁₆

yes

—

#2247

—

Q₈×C₁₆

D₁₃

yes

—

—

—

Q₈×C₁₃

—

Q₈×C₁₃

Q₃₂

yes

complemented

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#121

yes

characteristic

C₂×C₄×C₁₃

C₂×D₈

yes

characteristic

C₂×C₄×C₁₃

#135

yes

characteristic

C₂×Dic₁₃

—

Dic₂₆

—

Dic₂₆

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

C₂×SD₁₆

yes

C₈×C₁₃

yes

—

C₂×Q₃₂

—

C₂×Q₃₂

—

#494

—

#475

—

C₈×Q₈

C₂×D₁₃

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

C₂×D₁₃

yes

characteristic

C₄×C₁₆

—

C₄×C₁₆

C₂×D₁₃

yes

characteristic

#335

—

#332

—

C₂²×C₁₃

#466

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

yes

C₄×C₁₃

yes

C₄×C₁₃

C₂×Q₃₂

yes

characteristic

C₄×C₁₃

#508

yes

characteristic

Dic₁₃

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄×Q₈

D₅₂

yes

characteristic

Q₃₂

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₁₃

yes

characteristic

100

C₂×C₁₆

C₂²⋊D₁₃

yes

characteristic

101

#106

—

102

C₄⋊C₈

—

103

C₄⋊C₈

—

104

C₄⋊C₈

D₅₂

yes

characteristic

105

#103

—

106

C₄×C₈

—

107

C₄×C₈

C₂²×D₁₃

yes

characteristic

108

C₂×C₁₃

#1668

yes

characteristic

109

C₂×C₁₃

#2216

yes

characteristic

110

C₂×C₁₃

#2222

yes

characteristic

111

C₂×Q₈

—

112

C₂×Q₈

D₁₀₄

yes

characteristic

113

Q₁₆

—

114

Q₁₆

—

115

Q₁₆

—

104

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

#6127

yes

characteristic

119

C₂×C₈

C₂×C₂²⋊D₁₃

yes

characteristic

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

D₁₀₄

yes

characteristic

124

C₄²

—

125

C₄²

C₂×D₅₂

yes

characteristic

126

C₁₆

—

127

C₁₆

—

128

C₁₆

#6106

yes

—

129

C₁₃

#12440

yes

characteristic, complemented, Sylow

130

Q₈

—

131

Q₈

—

104

—

132

Q₈

—

104

—

133

Q₈

Dic₁₀₄

yes

complemented

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

104

—

137

C₂×C₄

C₂×D₁₀₄

yes

characteristic

138

C₂×C₄

#88158

yes

characteristic

139

C₂×C₄

#88184

yes

characteristic

140

C₈

—

141

C₈

—

142

C₈

—

104

—

143

C₈

C₂×#6106

yes

characteristic

144

C₈

#88203

yes

characteristic

145

C₂²

#11718776

yes

characteristic

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

208

—

150

C₄

#11718565

yes

characteristic

151

C₄

C₂×Dic₁₀₄

yes

characteristic

152

C₄

#11718688

yes

characteristic

153

C₄

#11718803

yes

characteristic

154

C₂

#13114196

yes

characteristic

155

C₂

#13116927

yes

characteristic

156

C₂

#13116961

yes

characteristic

	subgroup	G/H
center	C₂²	#11718776
commutator, cosocle	C₄×C₈×C₁₃	C₂³
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	Q₈×C₁₃×C₁₆	C₂
socle, prime radical	C₂²×C₁₃	#466
hypercenter	Q₈×C₁₆	D₁₃
hypocenter	C₁₃	#12440
LCS	G ▷ C₄×C₈×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ Q₈×C₁₆

subgroup

G/H

center

C₂²

#11718776

commutator, cosocle

C₄×C₈×C₁₃

C₂³

Frattini

C₄×C₈

C₂²×D₁₃

Fitting

Q₈×C₁₃×C₁₆

C₂

socle, prime radical

C₂²×C₁₃

#466

hypercenter

Q₈×C₁₆

D₁₃

hypocenter

C₁₃

#12440

LCS

G ▷ C₄×C₈×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ Q₈×C₁₆

N	H ≅ G/N	type
58. Q₈×C₁₃	Q₃₂	semidirect
129. C₁₃	#12440	semidirect
133. Q₈	Dic₁₀₄	semidirect

H ≅ G/N

type

58. Q₈×C₁₃

Q₃₂

semidirect

129. C₁₃

#12440

semidirect

133. Q₈

Dic₁₀₄

semidirect

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

subgroup	index
#13117089	2
#13114206	2
#13112908	2
#13112886	2
Q₈×C₁₃×C₁₆	2
#12440	13

subgroup

index

Q₈×C₁₃×C₁₆

Subgroups — #15777181

Conjugacy classes of subgroups

Complements

Maximal subgroups