Group #15777222 — subgroups

Subgroups — #15777222

Order 3328.

218 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 71 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	#13116985	C₂	1	2	yes	maximal
2	C₈×Dic₅₂	C₂	1	1	yes	maximal, characteristic
3	#13112906	C₂	1	1	yes	maximal, characteristic
4	#13112894	C₂	1	2	yes	maximal
5	C₁₃×#1713	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#500	—	2	1	no	—
7	C₁₃×#500	C₂²	1	2	yes	—
8	C₁₃×#494	—	2	1	no	—
9	C₄×Dic₅₂	C₂²	1	1	yes	characteristic
10	C₈×Dic₂₆	—	2	1	no	—
11	C₈²×C₁₃	C₂²	1	1	yes	characteristic
12	#11718084	—	2	2	no	—
13	#11718081	—	2	2	no	—
14	C₄⋊#6090	C₂²	1	2	yes	—
15	#11718050	C₂²	1	1	yes	characteristic
16	#11718048	—	2	1	no	—
17	C₂×C₁₃×Q₁₆	—	4	2	no	—
18	C₁₃×C₄⋊Q₈	—	2	2	no	—
19	C₂×Dic₅₂	D₄	1	1	yes	characteristic
20	C₄×Dic₂₆	—	2	1	no	—
21	C₁₃×#107	—	2	2	no	—
22	C₁₃×#107	—	2	2	no	—
23	C₁₃×#107	—	4	1	no	—
24	C₁₃×#107	—	4	1	no	—
25	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
26	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
27	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
28	#88058	D₄	1	1	yes	characteristic
29	#88056	—	2	1	no	—
30	#88054	—	4	1	no	—
31	C₈×Dic₁₃	—	4	1	no	—
32	C₂×#6090	—	2	2	no	—
33	C₄⋊#1055	—	2	1	no	—
34	#12448	—	13	1	no	maximal, Sylow
35	C₂×Q₈×C₁₃	—	4	2	no	—
36	C₂×Dic₂₆	—	2	1	no	—
37	C₁₃×Q₁₆	—	8	4	no	—
38	C₂×C₈×C₁₃	—	2	1	no	—
39	C₂×C₈×C₁₃	D₈	1	1	yes	characteristic
40	C₂×C₈×C₁₃	D₈	1	1	yes	characteristic
41	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
43	C₁₃×C₄⋊C₄	—	4	2	no	—
44	C₁₃×C₄⋊C₄	—	4	2	no	—
45	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
46	C₄⋊Dic₁₃	—	2	1	no	—
47	#6101	—	4	1	no	—
48	C₄×Dic₁₃	—	4	1	no	—
49	C₂×#1055	—	4	1	no	—
50	Dic₅₂	—	2	1	no	—
51	Dic₅₂	Q₁₆	1	2	yes	complemented
52	#6090	—	4	2	no	—
53	#2227	—	13	2	no	—
54	#1713	D₁₃	1	1	yes	characteristic
55	C₈×Q₁₆	—	13	1	no	—
56	#1370	—	13	1	no	—
57	#1365	—	13	2	no	—
58	Q₈×C₁₃	—	8	4	no	—
59	C₂×C₄×C₁₃	—	8	2	no	—
60	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
61	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
62	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
63	C₂×Dic₁₃	—	4	1	no	—
64	C₂×Dic₁₃	—	4	1	no	—
65	Dic₂₆	—	2	2	no	—
66	Dic₂₆	—	4	1	no	—
67	C₈×C₁₃	—	2	2	no	—
68	C₈×C₁₃	—	2	1	no	—
69	C₈×C₁₃	—	2	1	no	—
70	C₈×C₁₃	Q₃₂	1	2	yes	—
71	C₈×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
72	C₈×C₁₃	#135	1	1	yes	characteristic
73	#1055	—	8	1	no	—
74	#500	—	2	1	no	—
75	#500	C₂×D₁₃	1	2	yes	—
76	#494	—	2	1	no	—
77	C₈×Q₈	—	26	1	no	—
78	C₄×Q₁₆	—	13	1	no	—
79	#366	—	26	2	no	—
80	C₄⋊C₁₆	—	13	2	no	—
81	#358	—	26	2	no	—
82	#335	—	13	1	no	—
83	Q₈⋊C₈	—	26	1	no	—
84	C₈²	C₂×D₁₃	1	1	yes	characteristic
85	C₂²×C₁₃	#459	1	1	yes	characteristic
86	C₄×C₁₃	—	2	1	no	—
87	C₄×C₁₃	—	16	2	no	—
88	C₄×C₁₃	#467	1	1	yes	characteristic
89	C₄×C₁₃	#473	1	1	yes	characteristic
90	C₄×C₁₃	C₂×Q₃₂	1	1	yes	characteristic
91	C₄×C₁₃	#510	1	1	yes	characteristic
92	Dic₁₃	—	4	2	no	—
93	Dic₁₃	—	8	1	no	—
94	C₂×Q₁₆	—	4	2	no	—
95	C₂×Q₁₆	—	13	1	no	—
96	C₄⋊Q₈	—	2	2	no	—
97	C₄×Q₈	—	26	1	no	—
98	C₂×C₁₆	—	26	2	no	—
99	#107	—	2	2	no	—
100	#107	—	2	2	no	—
101	#107	—	4	1	no	—
102	#107	—	4	1	no	—
103	#107	—	13	1	no	—
104	C₄⋊C₈	—	26	1	no	—
105	C₄⋊C₈	—	52	1	no	—
106	#103	—	26	1	no	—
107	C₄×C₈	—	52	1	no	—
108	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
109	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
110	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
111	C₂×C₁₃	#1669	1	1	yes	characteristic
112	C₂×C₁₃	#2209	1	1	yes	characteristic
113	C₂×C₁₃	#2213	1	1	yes	characteristic
114	C₂×Q₈	—	4	2	no	—
115	C₂×Q₈	—	26	1	no	—
116	Q₁₆	—	8	4	no	—
117	Q₁₆	—	13	2	no	—
118	Q₁₆	—	26	1	no	—
119	C₂×C₈	—	2	1	no	—
120	C₂×C₈	—	52	1	no	—
121	C₂×C₈	#6104	1	1	yes	characteristic
122	C₂×C₈	#6104	1	1	yes	characteristic
123	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
124	C₂×C₈	#6129	1	1	yes	characteristic
125	C₄⋊C₄	—	4	2	no	—
126	C₄⋊C₄	—	4	2	no	—
127	C₄⋊C₄	—	26	1	no	—
128	C₄⋊C₄	—	52	1	no	—
129	C₄²	—	52	1	no	—
130	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
131	C₁₆	—	52	2	no	—
132	C₁₃	#12448	1	1	yes	characteristic, complemented, Sylow
133	Q₈	—	8	4	no	—
134	Q₈	—	26	2	no	—
135	Q₈	—	52	1	no	—
136	C₂×C₄	—	8	2	no	—
137	C₂×C₄	—	52	1	no	—
138	C₂×C₄	—	52	1	no	—
139	C₂×C₄	C₂×#6104	1	1	yes	characteristic
140	C₂×C₄	#88186	1	1	yes	characteristic
141	C₂×C₄	#88192	1	1	yes	characteristic
142	C₈	—	2	2	no	—
143	C₈	—	2	1	no	—
144	C₈	—	2	1	no	—
145	C₈	—	104	1	no	—
146	C₈	#88069	1	2	yes	—
147	C₈	Q₁₆×D₁₃	1	1	yes	characteristic
148	C₈	#88173	1	1	yes	characteristic
149	C₂²	#11718733	1	1	yes	characteristic
150	C₄	—	2	1	no	—
151	C₄	—	16	2	no	—
152	C₄	—	52	2	no	—
153	C₄	—	104	1	no	—
154	C₄	#11718840	1	1	yes	characteristic
155	C₄	C₂×#88069	1	1	yes	characteristic
156	C₄	#11718849	1	1	yes	characteristic
157	C₄	#11718853	1	1	yes	characteristic
158	C₂	#13114198	1	1	yes	characteristic
159	C₂	#13116883	1	1	yes	characteristic
160	C₂	#13116909	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13116985

C₂

yes

maximal

C₈×Dic₅₂

C₂

yes

maximal, characteristic

#13112906

C₂

yes

maximal, characteristic

#13112894

C₂

yes

maximal

C₁₃×#1713

C₂

yes

maximal, characteristic

C₁₃×#500

—

C₁₃×#500

C₂²

yes

—

C₁₃×#494

—

C₄×Dic₅₂

C₂²

yes

characteristic

C₈×Dic₂₆

—

C₈²×C₁₃

C₂²

yes

—

—

C₂²

yes

—

#11718050

C₂²

yes

characteristic

#11718048

—

C₂×C₁₃×Q₁₆

—

C₁₃×C₄⋊Q₈

—

C₂×Dic₅₂

D₄

yes

characteristic

C₄×Dic₂₆

—

C₁₃×#107

—

C₁₃×#107

—

C₁₃×#107

—

C₁₃×#107

—

C₄×C₈×C₁₃

D₄

yes

characteristic

C₄×C₈×C₁₃

D₄

yes

characteristic

C₄×C₈×C₁₃

C₂³

yes

characteristic

#88058

D₄

yes

characteristic

#88056

—

#88054

—

C₈×Dic₁₃

—

—

—

—

C₂×Q₈×C₁₃

—

C₂×Dic₂₆

—

C₁₃×Q₁₆

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

D₈

yes

characteristic

C₂×C₈×C₁₃

D₈

yes

characteristic

C₂×C₈×C₁₃

C₂×D₄

yes

characteristic

C₂×C₈×C₁₃

#41

yes

characteristic

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

C₂×D₄

yes

characteristic

C₄⋊Dic₁₃

—

#6101

—

C₄×Dic₁₃

—

C₂×#1055

—

Dic₅₂

—

Dic₅₂

Q₁₆

yes

—

—

D₁₃

yes

characteristic

C₈×Q₁₆

—

#1370

—

#1365

—

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#121

yes

characteristic

C₂×C₄×C₁₃

C₂×D₈

yes

characteristic

C₂×C₄×C₁₃

AGL(1,ℤ/8)

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

—

Dic₂₆

—

Dic₂₆

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

Q₃₂

yes

—

C₈×C₁₃

C₂×Q₁₆

yes

C₈×C₁₃

yes

—

—

C₂×D₁₃

yes

—

#494

—

C₈×Q₈

—

C₄×Q₁₆

—

#366

—

C₄⋊C₁₆

—

#358

—

#335

—

Q₈⋊C₈

—

C₈²

C₂×D₁₃

yes

characteristic

C₂²×C₁₃

#459

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

yes

C₄×C₁₃

yes

C₄×C₁₃

C₂×Q₃₂

yes

characteristic

C₄×C₁₃

#510

yes

characteristic

Dic₁₃

—

Dic₁₃

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄×Q₈

—

C₂×C₁₆

—

#107

—

100

#107

—

101

#107

—

102

#107

—

103

#107

—

104

C₄⋊C₈

—

105

C₄⋊C₈

—

106

#103

—

107

C₄×C₈

—

108

C₄×C₈

C₂²⋊D₁₃

yes

characteristic

109

C₄×C₈

C₂²⋊D₁₃

yes

characteristic

110

C₄×C₈

C₂²×D₁₃

yes

characteristic

111

C₂×C₁₃

#1669

yes

characteristic

112

C₂×C₁₃

#2209

yes

characteristic

113

C₂×C₁₃

#2213

yes

characteristic

114

C₂×Q₈

—

115

C₂×Q₈

—

116

Q₁₆

—

117

Q₁₆

—

118

Q₁₆

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

#6104

yes

characteristic

122

C₂×C₈

#6104

yes

characteristic

123

C₂×C₈

D₄×D₁₃

yes

characteristic

124

C₂×C₈

#6129

yes

characteristic

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄²

—

130

C₄²

C₂×C₂²⋊D₁₃

yes

characteristic

131

C₁₆

—

132

C₁₃

#12448

yes

characteristic, complemented, Sylow

133

Q₈

—

134

Q₈

—

135

Q₈

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

C₂×#6104

yes

characteristic

140

C₂×C₄

#88186

yes

characteristic

141

C₂×C₄

#88192

yes

characteristic

142

C₈

—

143

C₈

—

144

C₈

—

145

C₈

—

104

—

146

C₈

#88069

yes

—

147

C₈

Q₁₆×D₁₃

yes

characteristic

148

C₈

#88173

yes

characteristic

149

C₂²

#11718733

yes

characteristic

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

104

—

154

C₄

#11718840

yes

characteristic

155

C₄

C₂×#88069

yes

characteristic

156

C₄

#11718849

yes

characteristic

157

C₄

#11718853

yes

characteristic

158

C₂

#13114198

yes

characteristic

159

C₂

#13116883

yes

characteristic

160

C₂

#13116909

yes

characteristic

	subgroup	G/H
center	C₂²	#11718733
commutator, cosocle	C₄×C₈×C₁₃	C₂³
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₁₃×#1713	C₂
socle, prime radical	C₂²×C₁₃	#459
hypercenter	#1713	D₁₃
hypocenter	C₁₃	#12448
LCS	G ▷ C₄×C₈×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1713

subgroup

G/H

center

C₂²

#11718733

commutator, cosocle

C₄×C₈×C₁₃

C₂³

Frattini

C₄×C₈

C₂²×D₁₃

Fitting

C₁₃×#1713

C₂

C₂²×C₁₃

D₁₃

C₁₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1713

N	H ≅ G/N	type
51. Dic₅₂	Q₁₆	semidirect
132. C₁₃	#12448	semidirect

H ≅ G/N

type

51. Dic₅₂

Q₁₆

semidirect

132. C₁₃

#12448

semidirect

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

subgroup	index
#13116985	2
C₈×Dic₅₂	2
#13112906	2
#13112894	2
C₁₃×#1713	2
#12448	13

subgroup

index

C₈×Dic₅₂

C₁₃×#1713

Subgroups — #15777222

Conjugacy classes of subgroups

Complements

Maximal subgroups