Group #15777182 — subgroups

Subgroups — #15777182

Order 3328.

216 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 69 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	C₁₆×Dic₂₆	C₂	1	1	yes	maximal, characteristic
3	#13114207	C₂	1	2	yes	maximal
4	#13112884	C₂	1	2	yes	maximal
5	C₁₃×#1371	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₁₀₄	—	2	1	no	—
7	C₂×Dic₁₀₄	—	2	1	no	—
8	#11718512	—	2	2	no	—
9	#11718360	C₂²	1	2	yes	—
10	C₈×Dic₂₆	C₂²	1	1	yes	characteristic
11	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
12	C₁₃×#335	C₂²	1	2	yes	—
13	#11718093	—	2	1	no	—
14	C₁₆×Dic₁₃	—	2	1	no	—
15	#11718078	—	2	2	no	—
16	C₄⋊#6090	C₂²	1	1	yes	characteristic
17	C₂×Dic₅₂	—	2	2	no	—
18	C₂×Dic₅₂	—	2	2	no	—
19	#88123	—	2	2	no	—
20	C₄×Dic₂₆	D₄	1	1	yes	characteristic
21	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
22	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
23	C₁₃×C₄⋊C₈	—	2	2	no	—
24	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
25	#88057	—	4	2	no	—
26	#88056	—	4	2	no	—
27	#88054	—	2	1	no	—
28	C₈×Dic₁₃	—	2	1	no	—
29	C₂×#6090	—	2	1	no	—
30	C₄⋊#1055	D₄	1	1	yes	characteristic
31	Dic₁₀₄	—	4	2	no	—
32	Dic₁₀₄	—	4	2	no	—
33	#12440	—	13	1	no	maximal, Sylow
34	C₂×Dic₂₆	—	4	2	no	—
35	C₂×Dic₂₆	D₈	1	1	yes	characteristic
36	C₂×C₈×C₁₃	—	4	2	no	—
37	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
38	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
39	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
40	C₄⋊Dic₁₃	—	4	2	no	—
41	C₄⋊Dic₁₃	—	4	2	no	—
42	C₄⋊Dic₁₃	D₈	1	1	yes	characteristic
43	#6101	—	2	1	no	—
44	C₄×Dic₁₃	—	2	1	no	—
45	C₂×#1055	—	2	1	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	SD₁₆	1	2	yes	—
51	#6090	—	4	1	no	—
52	#2247	—	13	1	no	—
53	Q₈×C₁₆	—	13	1	no	—
54	#1670	—	13	2	no	—
55	#1371	D₁₃	1	1	yes	characteristic
56	#1362	—	13	2	no	—
57	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
58	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
59	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
60	C₂×Dic₁₃	—	2	1	no	—
61	C₂×Dic₁₃	—	2	1	no	—
62	C₂×Dic₁₃	—	8	2	no	—
63	Dic₂₆	—	2	1	no	—
64	Dic₂₆	—	8	2	no	—
65	Dic₂₆	—	8	2	no	—
66	Dic₂₆	Q₃₂	1	2	yes	complemented
67	C₈×C₁₃	—	2	1	no	—
68	C₈×C₁₃	—	8	2	no	—
69	C₈×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
70	C₈×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
71	#1055	—	4	1	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	—	13	1	no	—
77	C₄⋊C₁₆	—	13	1	no	—
78	C₄⋊C₁₆	—	26	1	no	—
79	C₄×C₁₆	—	26	1	no	—
80	C₄×C₁₆	C₂×D₁₃	1	1	yes	characteristic
81	#335	C₂×D₁₃	1	2	yes	—
82	#332	—	26	2	no	—
83	C₂²×C₁₃	#466	1	1	yes	characteristic
84	C₄×C₁₃	—	2	1	no	—
85	C₄×C₁₃	#460	1	1	yes	characteristic
86	C₄×C₁₃	#472	1	1	yes	characteristic
87	C₄×C₁₃	C₂×Q₃₂	1	1	yes	characteristic
88	C₄×C₁₃	#508	1	1	yes	characteristic
89	Dic₁₃	—	2	2	no	—
90	Dic₁₃	—	4	1	no	—
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	—	13	1	no	—
96	Q₃₂	—	52	2	no	—
97	Q₃₂	—	52	2	no	—
98	C₂×C₁₆	—	26	1	no	—
99	C₂×C₁₆	D₅₂	1	1	yes	characteristic
100	C₂×C₁₆	D₅₂	1	1	yes	characteristic
101	#106	—	52	2	no	—
102	C₄⋊C₈	—	2	2	no	—
103	C₄⋊C₈	—	13	1	no	—
104	C₄⋊C₈	—	26	1	no	—
105	#103	—	52	2	no	—
106	C₄×C₈	—	26	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	—	13	1	no	—
112	C₂×Q₈	—	52	2	no	—
113	Q₁₆	—	52	2	no	—
114	Q₁₆	—	52	2	no	—
115	Q₁₆	—	104	2	no	—
116	C₂×C₈	—	4	2	no	—
117	C₂×C₈	—	26	1	no	—
118	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
119	C₂×C₈	#6131	1	1	yes	characteristic
120	C₄⋊C₄	—	13	1	no	—
121	C₄⋊C₄	—	26	1	no	—
122	C₄⋊C₄	—	52	2	no	—
123	C₄⋊C₄	—	52	2	no	—
124	C₄²	—	26	1	no	—
125	C₄²	D₄×D₁₃	1	1	yes	characteristic
126	C₁₆	—	2	1	no	—
127	C₁₆	—	52	1	no	—
128	C₁₆	#6095	1	2	yes	—
129	C₁₃	#12440	1	1	yes	characteristic, complemented, Sylow
130	Q₈	—	13	2	no	—
131	Q₈	—	26	1	no	—
132	Q₈	—	104	2	no	—
133	Q₈	—	104	2	no	—
134	C₂×C₄	—	26	1	no	—
135	C₂×C₄	—	26	1	no	—
136	C₂×C₄	—	104	2	no	—
137	C₂×C₄	#88149	1	1	yes	characteristic
138	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
139	C₂×C₄	#88173	1	1	yes	characteristic
140	C₈	—	2	1	no	—
141	C₈	—	8	2	no	—
142	C₈	—	52	1	no	—
143	C₈	C₂×#6095	1	1	yes	characteristic
144	C₈	#88162	1	1	yes	characteristic
145	C₂²	#11718544	1	1	yes	characteristic
146	C₄	—	2	1	no	—
147	C₄	—	26	2	no	—
148	C₄	—	52	1	no	—
149	C₄	—	208	2	no	—
150	C₄	#11718509	1	1	yes	characteristic
151	C₄	#11718558	1	1	yes	characteristic
152	C₄	#11718573	1	1	yes	characteristic
153	C₄	D₁₃×Q₃₂	1	1	yes	characteristic
154	C₂	#13114197	1	1	yes	characteristic
155	C₂	#13116928	1	1	yes	characteristic
156	C₂	#13116962	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13117089

C₂

yes

maximal, characteristic

C₁₆×Dic₂₆

C₂

yes

maximal, characteristic

#13114207

C₂

yes

maximal

#13112884

C₂

yes

maximal

C₁₃×#1371

C₂

yes

maximal, characteristic

C₂×Dic₁₀₄

—

C₂×Dic₁₀₄

—

#11718512

—

#11718360

C₂²

yes

—

C₈×Dic₂₆

C₂²

yes

characteristic

C₄×C₁₃×C₁₆

C₂²

yes

characteristic

C₁₃×#335

C₂²

yes

—

#11718093

—

C₁₆×Dic₁₃

—

#11718078

—

C₄⋊#6090

C₂²

yes

characteristic

C₂×Dic₅₂

—

C₂×Dic₅₂

—

#88123

—

C₄×Dic₂₆

D₄

yes

characteristic

C₂×C₁₃×C₁₆

D₄

yes

characteristic

C₂×C₁₃×C₁₆

D₄

yes

characteristic

C₁₃×C₄⋊C₈

—

C₄×C₈×C₁₃

C₂³

yes

—

—

—

C₈×Dic₁₃

—

C₂×#6090

—

C₄⋊#1055

D₄

yes

characteristic

Dic₁₀₄

—

Dic₁₀₄

—

#12440

—

maximal, Sylow

C₂×Dic₂₆

—

C₂×Dic₂₆

D₈

yes

characteristic

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₂×D₄

yes

characteristic

C₂×C₈×C₁₃

#41

yes

characteristic

C₄²×C₁₃

C₂×D₄

yes

characteristic

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

D₈

yes

characteristic

#6101

—

C₄×Dic₁₃

—

C₂×#1055

—

Dic₅₂

—

Dic₅₂

—

Dic₅₂

—

C₁₃×C₁₆

—

C₁₃×C₁₆

SD₁₆

yes

—

#6090

—

#2247

—

Q₈×C₁₆

—

#1670

—

#1371

D₁₃

yes

characteristic

#1362

—

C₂×C₄×C₁₃

#121

yes

characteristic

C₂×C₄×C₁₃

C₂×D₈

yes

characteristic

C₂×C₄×C₁₃

#135

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

Dic₂₆

—

Dic₂₆

—

Dic₂₆

—

Dic₂₆

Q₃₂

yes

complemented

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

C₂×SD₁₆

yes

C₈×C₁₃

yes

—

C₂×Q₃₂

—

C₂×Q₃₂

—

#494

—

#475

—

C₈×Q₈

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₁₃

yes

characteristic

#335

C₂×D₁₃

yes

—

#332

—

C₂²×C₁₃

#466

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

yes

C₄×C₁₃

yes

C₄×C₁₃

C₂×Q₃₂

yes

characteristic

C₄×C₁₃

#508

yes

characteristic

Dic₁₃

—

Dic₁₃

—

Dic₁₃

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄×Q₈

—

Q₃₂

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

D₅₂

yes

characteristic

100

C₂×C₁₆

D₅₂

yes

characteristic

101

#106

—

102

C₄⋊C₈

—

103

C₄⋊C₈

—

104

C₄⋊C₈

—

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

—

113

Q₁₆

—

114

Q₁₆

—

115

Q₁₆

—

104

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

C₂×D₅₂

yes

characteristic

119

C₂×C₈

#6131

yes

characteristic

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄²

—

125

C₄²

D₄×D₁₃

yes

characteristic

126

C₁₆

—

127

C₁₆

—

128

C₁₆

#6095

yes

—

129

C₁₃

#12440

yes

characteristic, complemented, Sylow

130

Q₈

—

131

Q₈

—

132

Q₈

—

104

—

133

Q₈

—

104

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

104

—

137

C₂×C₄

#88149

yes

characteristic

138

C₂×C₄

D₈×D₁₃

yes

characteristic

139

C₂×C₄

#88173

yes

characteristic

140

C₈

—

141

C₈

—

142

C₈

—

143

C₈

C₂×#6095

yes

characteristic

144

C₈

#88162

yes

characteristic

145

C₂²

#11718544

yes

characteristic

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

208

—

150

C₄

#11718509

yes

characteristic

151

C₄

#11718558

yes

characteristic

152

C₄

#11718573

yes

characteristic

153

C₄

D₁₃×Q₃₂

yes

characteristic

154

C₂

#13114197

yes

characteristic

155

C₂

#13116928

yes

characteristic

156

C₂

#13116962

yes

characteristic

	subgroup	G/H
center	C₂²	#11718544
commutator, cosocle	C₄×C₈×C₁₃	C₂³
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₁₃×#1371	C₂
socle, prime radical	C₂²×C₁₃	#466
hypercenter	#1371	D₁₃
hypocenter	C₁₃	#12440
LCS	G ▷ C₄×C₈×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1371

subgroup

G/H

center

C₂²

#11718544

commutator, cosocle

C₄×C₈×C₁₃

C₂³

Frattini

C₄×C₈

C₂²×D₁₃

Fitting

C₁₃×#1371

C₂

C₂²×C₁₃

D₁₃

C₁₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1371

N	H ≅ G/N	type
66. Dic₂₆	Q₃₂	semidirect
129. C₁₃	#12440	semidirect

H ≅ G/N

type

66. Dic₂₆

Q₃₂

semidirect

129. C₁₃

#12440

semidirect

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

subgroup	index
#13117089	2
C₁₆×Dic₂₆	2
#13114207	2
#13112884	2
C₁₃×#1371	2
#12440	13

subgroup

index

C₁₆×Dic₂₆

C₁₃×#1371

Subgroups — #15777182

Conjugacy classes of subgroups

Complements

Maximal subgroups