Group #15777215 — subgroups

Subgroups — #15777215

Order 3328.

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

normal subgroups only

This group has 218 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	#13116978	C₂	1	2	yes	maximal
2	#13114454	C₂	1	1	yes	maximal, characteristic
3	#13112906	C₂	1	1	yes	maximal, characteristic
4	#13112892	C₂	1	2	yes	maximal
5	C₈×C₁₃×Q₁₆	C₂	1	1	yes	maximal, characteristic
6	C₈×Q₈×C₁₃	—	2	1	no	—
7	C₄×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
8	#11718360	—	2	1	no	—
9	#11718343	—	2	1	no	—
10	#11718343	C₂²	1	2	yes	—
11	C₁₃×#335	C₂²	1	1	yes	characteristic
12	C₁₃×Q₈⋊C₈	—	2	1	no	—
13	C₈²×C₁₃	C₂²	1	1	yes	characteristic
14	#11718156	—	2	2	no	—
15	#11718082	—	2	2	no	—
16	C₄⋊#6090	C₂²	1	2	yes	—
17	C₂×C₁₃×Q₁₆	D₄	1	1	yes	characteristic
18	C₄×Q₈×C₁₃	—	2	1	no	—
19	C₂×Dic₅₂	—	4	1	no	—
20	C₂×Dic₅₂	—	4	1	no	—
21	#88123	—	2	2	no	—
22	C₁₃×#107	D₄	1	1	yes	characteristic
23	C₁₃×C₄⋊C₈	—	2	1	no	—
24	C₁₃×C₄⋊C₈	—	4	1	no	—
25	C₁₃×#103	—	2	1	no	—
26	C₄×C₈×C₁₃	—	4	1	no	—
27	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
28	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
29	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
30	#88058	—	2	2	no	—
31	#88058	—	2	2	no	—
32	#88058	—	4	2	no	—
33	C₂×#6090	—	2	2	no	—
34	#12446	—	13	1	no	maximal, Sylow
35	C₂×Q₈×C₁₃	—	2	1	no	—
36	C₂×Dic₂₆	—	4	2	no	—
37	C₁₃×Q₁₆	—	2	1	no	—
38	C₁₃×Q₁₆	Q₁₆	1	2	yes	complemented
39	C₂×C₈×C₁₃	—	2	1	no	—
40	C₂×C₈×C₁₃	—	4	1	no	—
41	C₂×C₈×C₁₃	D₈	1	1	yes	characteristic
42	C₂×C₈×C₁₃	D₈	1	1	yes	characteristic
43	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
45	C₁₃×C₄⋊C₄	—	2	1	no	—
46	C₁₃×C₄⋊C₄	—	4	1	no	—
47	C₄²×C₁₃	—	4	1	no	—
48	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
49	C₄⋊Dic₁₃	—	4	2	no	—
50	C₄⋊Dic₁₃	—	4	2	no	—
51	Dic₅₂	—	8	2	no	—
52	Dic₅₂	—	8	2	no	—
53	#6090	—	4	2	no	—
54	#2225	—	13	2	no	—
55	#1713	—	13	1	no	—
56	C₈×Q₁₆	D₁₃	1	1	yes	characteristic
57	#1370	—	13	1	no	—
58	#1364	—	13	2	no	—
59	Q₈×C₁₃	—	2	2	no	—
60	Q₈×C₁₃	—	4	1	no	—
61	C₂×C₄×C₁₃	—	4	1	no	—
62	C₂×C₄×C₁₃	—	4	1	no	—
63	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
64	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
65	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
66	C₂×Dic₁₃	—	8	2	no	—
67	Dic₂₆	—	8	2	no	—
68	Dic₂₆	—	8	2	no	—
69	C₈×C₁₃	—	2	2	no	—
70	C₈×C₁₃	—	2	1	no	—
71	C₈×C₁₃	—	2	1	no	—
72	C₈×C₁₃	—	8	1	no	—
73	C₈×C₁₃	SD₃₂	1	2	yes	—
74	C₈×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
75	C₈×C₁₃	#135	1	1	yes	characteristic
76	#500	—	13	2	no	—
77	#500	—	26	1	no	—
78	#494	—	26	1	no	—
79	C₈×Q₈	—	2	1	no	—
80	C₄×Q₁₆	C₂×D₁₃	1	1	yes	characteristic
81	#367	—	26	2	no	—
82	C₄⋊C₁₆	—	13	2	no	—
83	#358	—	26	2	no	—
84	#335	C₂×D₁₃	1	1	yes	characteristic
85	Q₈⋊C₈	—	2	1	no	—
86	C₈²	C₂×D₁₃	1	1	yes	characteristic
87	C₂²×C₁₃	#459	1	1	yes	characteristic
88	C₄×C₁₃	—	2	1	no	—
89	C₄×C₁₃	—	4	2	no	—
90	C₄×C₁₃	—	8	1	no	—
91	C₄×C₁₃	#467	1	1	yes	characteristic
92	C₄×C₁₃	#473	1	1	yes	characteristic
93	C₄×C₁₃	C₂×SD₃₂	1	1	yes	characteristic
94	C₄×C₁₃	#509	1	1	yes	characteristic
95	Dic₁₃	—	16	2	no	—
96	C₂×Q₁₆	—	52	1	no	—
97	C₂×Q₁₆	—	52	1	no	—
98	C₂×Q₁₆	D₅₂	1	1	yes	characteristic
99	C₄⋊Q₈	—	26	2	no	—
100	C₄×Q₈	—	2	1	no	—
101	C₂×C₁₆	—	26	2	no	—
102	#107	—	26	2	no	—
103	#107	—	26	2	no	—
104	#107	—	52	2	no	—
105	#107	D₅₂	1	1	yes	characteristic
106	C₄⋊C₈	—	2	1	no	—
107	C₄⋊C₈	—	4	1	no	—
108	#103	—	2	1	no	—
109	C₄×C₈	—	4	1	no	—
110	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
111	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
112	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
113	C₂×C₁₃	#1669	1	1	yes	characteristic
114	C₂×C₁₃	#2207	1	1	yes	characteristic
115	C₂×C₁₃	#2214	1	1	yes	characteristic
116	C₂×Q₈	—	2	1	no	—
117	C₂×Q₈	—	52	2	no	—
118	Q₁₆	—	2	1	no	—
119	Q₁₆	—	104	2	no	—
120	Q₁₆	—	104	2	no	—
121	Q₁₆	Dic₅₂	1	2	yes	complemented
122	C₂×C₈	—	2	1	no	—
123	C₂×C₈	—	4	1	no	—
124	C₂×C₈	#6104	1	1	yes	characteristic
125	C₂×C₈	#6104	1	1	yes	characteristic
126	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
127	C₂×C₈	#6127	1	1	yes	characteristic
128	C₄⋊C₄	—	2	1	no	—
129	C₄⋊C₄	—	4	1	no	—
130	C₄⋊C₄	—	52	2	no	—
131	C₄⋊C₄	—	52	2	no	—
132	C₄²	—	4	1	no	—
133	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
134	C₁₆	—	52	2	no	—
135	C₁₃	#12446	1	1	yes	characteristic, complemented, Sylow
136	Q₈	—	2	2	no	—
137	Q₈	—	4	1	no	—
138	Q₈	—	104	2	no	—
139	Q₈	—	104	2	no	—
140	C₂×C₄	—	4	1	no	—
141	C₂×C₄	—	4	1	no	—
142	C₂×C₄	—	104	2	no	—
143	C₂×C₄	#88184	1	1	yes	characteristic
144	C₂×C₄	C₂×#6104	1	1	yes	characteristic
145	C₂×C₄	#88203	1	1	yes	characteristic
146	C₈	—	2	2	no	—
147	C₈	—	2	1	no	—
148	C₈	—	2	1	no	—
149	C₈	—	8	1	no	—
150	C₈	#88068	1	2	yes	—
151	C₈	#88158	1	1	yes	characteristic
152	C₈	C₂×Dic₅₂	1	1	yes	characteristic
153	C₂²	#11718676	1	1	yes	characteristic
154	C₄	—	2	1	no	—
155	C₄	—	4	2	no	—
156	C₄	—	8	1	no	—
157	C₄	—	208	2	no	—
158	C₄	#11718777	1	1	yes	characteristic
159	C₄	#11718804	1	1	yes	characteristic
160	C₄	C₂×#88068	1	1	yes	characteristic
161	C₄	#11718854	1	1	yes	characteristic
162	C₂	#13114201	1	1	yes	characteristic
163	C₂	#13116876	1	1	yes	characteristic
164	C₂	#13116917	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13116978

C₂

yes

maximal

#13114454

C₂

yes

maximal, characteristic

#13112906

C₂

yes

maximal, characteristic

#13112892

C₂

yes

maximal

C₈×C₁₃×Q₁₆

C₂

yes

maximal, characteristic

C₈×Q₈×C₁₃

—

C₄×C₁₃×Q₁₆

C₂²

yes

—

—

C₂²

yes

—

C₁₃×#335

C₂²

yes

characteristic

C₁₃×Q₈⋊C₈

—

C₈²×C₁₃

C₂²

yes

—

—

C₂²

yes

—

C₂×C₁₃×Q₁₆

D₄

yes

characteristic

C₄×Q₈×C₁₃

—

C₂×Dic₅₂

—

C₂×Dic₅₂

—

#88123

—

C₁₃×#107

D₄

yes

characteristic

C₁₃×C₄⋊C₈

—

C₁₃×C₄⋊C₈

—

C₁₃×#103

—

C₄×C₈×C₁₃

—

C₄×C₈×C₁₃

D₄

yes

characteristic

C₄×C₈×C₁₃

D₄

yes

characteristic

C₄×C₈×C₁₃

C₂³

yes

—

—

—

—

—

C₂×Q₈×C₁₃

—

C₂×Dic₂₆

—

C₁₃×Q₁₆

—

C₁₃×Q₁₆

Q₁₆

yes

complemented

C₂×C₈×C₁₃

—

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

—

C₄²×C₁₃

C₂×D₄

yes

characteristic

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

Dic₅₂

—

Dic₅₂

—

#6090

—

#2225

—

#1713

—

C₈×Q₁₆

D₁₃

yes

characteristic

#1370

—

#1364

—

Q₈×C₁₃

—

Q₈×C₁₃

—

C₂×C₄×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

—

Dic₂₆

—

Dic₂₆

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

SD₃₂

yes

—

C₈×C₁₃

C₂×Q₁₆

yes

C₈×C₁₃

yes

—

—

—

C₈×Q₈

—

C₄×Q₁₆

C₂×D₁₃

yes

characteristic

#367

—

C₄⋊C₁₆

—

#358

—

#335

C₂×D₁₃

yes

characteristic

Q₈⋊C₈

—

C₈²

C₂×D₁₃

yes

characteristic

C₂²×C₁₃

#459

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

yes

C₄×C₁₃

yes

C₄×C₁₃

C₂×SD₃₂

yes

characteristic

C₄×C₁₃

#509

yes

characteristic

Dic₁₃

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

D₅₂

yes

characteristic

C₄⋊Q₈

—

100

C₄×Q₈

—

101

C₂×C₁₆

—

102

#107

—

103

#107

—

104

#107

—

105

#107

D₅₂

yes

characteristic

106

C₄⋊C₈

—

107

C₄⋊C₈

—

108

#103

—

109

C₄×C₈

—

110

C₄×C₈

C₂²⋊D₁₃

yes

characteristic

111

C₄×C₈

C₂²⋊D₁₃

yes

characteristic

112

C₄×C₈

C₂²×D₁₃

yes

characteristic

113

C₂×C₁₃

#1669

yes

characteristic

114

C₂×C₁₃

#2207

yes

characteristic

115

C₂×C₁₃

#2214

yes

characteristic

116

C₂×Q₈

—

117

C₂×Q₈

—

118

Q₁₆

—

119

Q₁₆

—

104

—

120

Q₁₆

—

104

—

121

Q₁₆

Dic₅₂

yes

complemented

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

#6104

yes

characteristic

125

C₂×C₈

#6104

yes

characteristic

126

C₂×C₈

C₂×D₅₂

yes

characteristic

127

C₂×C₈

#6127

yes

characteristic

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄²

—

133

C₄²

C₂×C₂²⋊D₁₃

yes

characteristic

134

C₁₆

—

135

C₁₃

#12446

yes

characteristic, complemented, Sylow

136

Q₈

—

137

Q₈

—

138

Q₈

—

104

—

139

Q₈

—

104

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

104

—

143

C₂×C₄

#88184

yes

characteristic

144

C₂×C₄

C₂×#6104

yes

characteristic

145

C₂×C₄

#88203

yes

characteristic

146

C₈

—

147

C₈

—

148

C₈

—

149

C₈

—

150

C₈

#88068

yes

—

151

C₈

#88158

yes

characteristic

152

C₈

C₂×Dic₅₂

yes

characteristic

153

C₂²

#11718676

yes

characteristic

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

208

—

158

C₄

#11718777

yes

characteristic

159

C₄

#11718804

yes

characteristic

160

C₄

C₂×#88068

yes

characteristic

161

C₄

#11718854

yes

characteristic

162

C₂

#13114201

yes

characteristic

163

C₂

#13116876

yes

characteristic

164

C₂

#13116917

yes

characteristic

	subgroup	G/H
center	C₂²	#11718676
commutator, cosocle	C₄×C₈×C₁₃	C₂³
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₈×C₁₃×Q₁₆	C₂
socle, prime radical	C₂²×C₁₃	#459
hypercenter	C₈×Q₁₆	D₁₃
hypocenter	C₁₃	#12446
LCS	G ▷ C₄×C₈×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×Q₁₆

subgroup

G/H

center

C₂²

#11718676

commutator, cosocle

C₄×C₈×C₁₃

C₂³

Frattini

C₄×C₈

C₂²×D₁₃

Fitting

C₈×C₁₃×Q₁₆

C₂

socle, prime radical

C₂²×C₁₃

#459

hypercenter

C₈×Q₁₆

D₁₃

hypocenter

C₁₃

#12446

LCS

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×Q₁₆

N	H ≅ G/N	type
38. C₁₃×Q₁₆	Q₁₆	semidirect
121. Q₁₆	Dic₅₂	semidirect
135. C₁₃	#12446	semidirect

H ≅ G/N

type

38. C₁₃×Q₁₆

Q₁₆

semidirect

121. Q₁₆

Dic₅₂

semidirect

135. C₁₃

#12446

semidirect

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

subgroup	index
#13116978	2
#13114454	2
#13112906	2
#13112892	2
C₈×C₁₃×Q₁₆	2
#12446	13

subgroup

index

C₈×C₁₃×Q₁₆

Subgroups — #15777215

Conjugacy classes of subgroups

Complements

Maximal subgroups