Group #15777694 — subgroups

Subgroups — #15777694

Order 3328.

318 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 87 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	#13117091	C₂	1	2	yes	maximal, complemented
2	#13117059	C₂	1	2	yes	maximal, complemented
3	#13114433	C₂	1	1	yes	maximal, characteristic, complemented
4	#13112754	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#1712	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#493	—	2	1	no	—
7	C₁₃×#493	—	2	1	no	—
8	C₁₃×#493	C₂²	1	2	yes	—
9	C₂×#88067	—	2	4	no	—
10	#11718360	C₂²	1	1	yes	characteristic
11	#11718343	C₂²	1	1	yes	characteristic
12	#11718342	—	2	1	no	—
13	#11718341	—	2	1	no	—
14	C₈²×C₁₃	C₂²	1	1	yes	characteristic
15	#11718154	—	2	4	no	—
16	C₄×#6090	C₂²	1	2	yes	—
17	C₂×C₁₃×D₈	—	2	2	no	—
18	C₂×C₁₃×D₈	—	2	2	no	—
19	C₂×C₁₃×D₈	—	4	2	no	—
20	C₂×C₁₃×D₈	—	4	2	no	—
21	C₁₃×#127	—	2	2	no	—
22	C₂×Dic₅₂	—	2	2	no	—
23	#88123	—	2	1	no	—
24	#88123	—	2	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	#88067	—	4	8	no	—
29	#88058	—	2	2	no	—
30	#88057	—	4	2	no	—
31	#88056	—	4	2	no	—
32	C₂×#6090	D₄	1	4	yes	—
33	#12529	—	13	1	no	maximal, Sylow
34	C₂×D₄×C₁₃	—	4	2	no	—
35	C₂×D₄×C₁₃	—	4	2	no	—
36	C₂×D₄×C₁₃	—	4	2	no	—
37	C₂×Dic₂₆	—	4	1	no	—
38	C₂×Dic₂₆	—	4	1	no	—
39	C₁₃×D₈	—	4	4	no	—
40	C₁₃×D₈	—	4	4	no	—
41	C₁₃×D₈	—	8	4	no	—
42	C₁₃×D₈	—	8	4	no	—
43	C₂×C₈×C₁₃	D₈	1	2	yes	—
44	C₂×C₈×C₁₃	D₈	1	2	yes	—
45	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
46	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
47	C₄⋊Dic₁₃	—	4	1	no	—
48	C₄⋊Dic₁₃	—	4	1	no	—
49	C₄⋊Dic₁₃	—	4	1	no	—
50	C₄⋊Dic₁₃	—	4	1	no	—
51	Dic₅₂	—	4	4	no	—
52	#6090	—	2	4	no	—
53	#2248	—	13	2	no	—
54	#2240	—	13	2	no	—
55	#1712	D₁₃	1	1	yes	characteristic
56	#1707	—	13	1	no	—
57	#1312	—	13	1	no	—
58	C₂³×C₁₃	—	8	2	no	—
59	D₄×C₁₃	—	8	2	no	—
60	D₄×C₁₃	—	8	2	no	—
61	D₄×C₁₃	—	8	4	no	—
62	D₄×C₁₃	—	8	4	no	—
63	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
64	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
65	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
66	C₂×Dic₁₃	—	8	1	no	—
67	C₂×Dic₁₃	—	8	1	no	—
68	Dic₂₆	—	8	2	no	—
69	Dic₂₆	—	8	2	no	—
70	C₈×C₁₃	—	2	2	no	—
71	C₈×C₁₃	SD₃₂	1	4	yes	complemented
72	C₈×C₁₃	AGL(1,ℤ/8)	1	4	yes	—
73	C₂×SD₃₂	—	26	4	no	—
74	#500	—	13	1	no	—
75	#498	—	26	1	no	—
76	#494	—	13	1	no	—
77	#493	—	2	1	no	—
78	#493	—	2	1	no	—
79	#493	C₂×D₁₃	1	2	yes	—
80	#487	—	26	1	no	—
81	#357	—	26	4	no	—
82	C₄×C₁₆	—	13	2	no	—
83	C₈²	C₂×D₁₃	1	1	yes	characteristic
84	C₂²×C₁₃	—	16	2	no	—
85	C₂²×C₁₃	—	16	2	no	—
86	C₂²×C₁₃	—	16	2	no	—
87	C₂²×C₁₃	#493	1	1	yes	characteristic
88	C₄×C₁₃	#496	1	2	yes	—
89	C₄×C₁₃	C₂×SD₃₂	1	2	yes	—
90	C₄×C₁₃	#509	1	2	yes	—
91	Dic₁₃	—	16	1	no	—
92	Dic₁₃	—	16	1	no	—
93	C₂×Q₁₆	—	26	2	no	—
94	C₂×D₈	—	2	2	no	—
95	C₂×D₈	—	2	2	no	—
96	C₂×D₈	—	4	2	no	—
97	C₂×D₈	—	4	2	no	—
98	C₄⋊Q₈	—	26	1	no	—
99	C₄⋊Q₈	—	26	1	no	—
100	#127	—	2	2	no	—
101	SD₃₂	—	52	8	no	—
102	C₂×C₁₆	—	13	4	no	—
103	#107	—	26	2	no	—
104	#106	—	52	2	no	—
105	#103	—	52	2	no	—
106	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
107	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
108	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
109	C₂×C₁₃	—	32	2	no	—
110	C₂×C₁₃	#1721	1	1	yes	characteristic
111	C₂×C₁₃	#2248	1	1	yes	characteristic
112	C₂×C₁₃	#2250	1	1	yes	characteristic
113	C₂×Q₈	—	52	1	no	—
114	C₂×Q₈	—	52	1	no	—
115	C₂×D₄	—	4	2	no	—
116	C₂×D₄	—	4	2	no	—
117	C₂×D₄	—	4	2	no	—
118	Q₁₆	—	52	4	no	—
119	D₈	—	4	4	no	—
120	D₈	—	4	4	no	—
121	D₈	—	8	4	no	—
122	D₈	—	8	4	no	—
123	C₂×C₈	#6104	1	2	yes	—
124	C₂×C₈	#6104	1	2	yes	—
125	C₂×C₈	D₄×D₁₃	1	2	yes	—
126	C₄⋊C₄	—	52	1	no	—
127	C₄⋊C₄	—	52	1	no	—
128	C₄⋊C₄	—	52	1	no	—
129	C₄⋊C₄	—	52	1	no	—
130	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
131	C₁₆	—	26	4	no	—
132	C₁₃	#12529	1	1	yes	characteristic, complemented, Sylow
133	C₂³	—	8	2	no	—
134	Q₈	—	104	2	no	—
135	Q₈	—	104	2	no	—
136	D₄	—	8	4	no	—
137	D₄	—	8	4	no	—
138	D₄	—	8	2	no	—
139	D₄	—	8	2	no	—
140	C₂×C₄	—	104	1	no	—
141	C₂×C₄	—	104	1	no	—
142	C₂×C₄	C₂×#6104	1	1	yes	characteristic
143	C₂×C₄	C₂×#6104	1	1	yes	characteristic
144	C₂×C₄	#88194	1	1	yes	characteristic
145	C₈	—	2	2	no	—
146	C₈	#88067	1	4	yes	complemented
147	C₈	#88165	1	4	yes	—
148	C₂²	—	16	2	no	—
149	C₂²	—	16	2	no	—
150	C₂²	—	16	2	no	—
151	C₂²	#11718734	1	1	yes	characteristic
152	C₄	—	208	1	no	—
153	C₄	—	208	1	no	—
154	C₄	#11718808	1	2	yes	—
155	C₄	C₂×#88067	1	2	yes	—
156	C₄	#11718815	1	2	yes	—
157	C₂	—	32	2	no	—
158	C₂	#13114503	1	1	yes	characteristic
159	C₂	#13117091	1	1	yes	characteristic
160	C₂	#13117100	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13117091

C₂

yes

maximal, complemented

#13117059

C₂

yes

maximal, complemented

#13114433

C₂

yes

maximal, characteristic, complemented

#13112754

C₂

yes

maximal, characteristic, complemented

C₁₃×#1712

C₂

yes

maximal, characteristic

C₁₃×#493

—

C₁₃×#493

—

C₁₃×#493

C₂²

yes

—

C₂×#88067

—

#11718360

C₂²

yes

characteristic

#11718343

C₂²

yes

characteristic

#11718342

—

#11718341

—

C₈²×C₁₃

C₂²

yes

characteristic

#11718154

—

C₄×#6090

C₂²

yes

—

C₂×C₁₃×D₈

—

C₂×C₁₃×D₈

—

C₂×C₁₃×D₈

—

C₂×C₁₃×D₈

—

C₁₃×#127

—

C₂×Dic₅₂

—

#88123

—

#88123

—

C₄×C₈×C₁₃

D₄

yes

characteristic

C₄×C₈×C₁₃

D₄

yes

characteristic

C₄×C₈×C₁₃

C₂³

yes

—

—

—

—

D₄

yes

—

#12529

—

maximal, Sylow

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₁₃×D₈

—

C₁₃×D₈

—

C₁₃×D₈

—

C₁₃×D₈

—

C₂×C₈×C₁₃

D₈

yes

—

C₂×C₈×C₁₃

D₈

yes

—

C₂×C₈×C₁₃

C₂×D₄

yes

—

C₄²×C₁₃

C₂×D₄

yes

characteristic

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

Dic₅₂

—

—

—

—

D₁₃

yes

characteristic

#1707

—

#1312

—

C₂³×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

C₂×C₄×C₁₃

#127

yes

characteristic

C₂×C₄×C₁₃

C₂×D₈

yes

characteristic

C₂×C₄×C₁₃

C₂×D₈

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

—

Dic₂₆

—

Dic₂₆

—

C₈×C₁₃

—

C₈×C₁₃

SD₃₂

yes

complemented

C₈×C₁₃

AGL(1,ℤ/8)

yes

—

C₂×SD₃₂

—

—

—

—

—

—

C₂×D₁₃

yes

—

#487

—

#357

—

C₄×C₁₆

—

C₈²

C₂×D₁₃

yes

characteristic

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

#493

yes

characteristic

C₄×C₁₃

#496

yes

—

C₄×C₁₃

C₂×SD₃₂

yes

—

C₄×C₁₃

#509

yes

—

Dic₁₃

—

Dic₁₃

—

C₂×Q₁₆

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

100

#127

—

101

SD₃₂

—

102

C₂×C₁₆

—

103

#107

—

104

#106

—

105

#103

—

106

C₄×C₈

C₂²⋊D₁₃

yes

characteristic

107

C₄×C₈

C₂²⋊D₁₃

yes

characteristic

108

C₄×C₈

C₂²×D₁₃

yes

characteristic

109

C₂×C₁₃

—

110

C₂×C₁₃

#1721

yes

characteristic

111

C₂×C₁₃

#2248

yes

characteristic

112

C₂×C₁₃

#2250

yes

characteristic

113

C₂×Q₈

—

114

C₂×Q₈

—

115

C₂×D₄

—

116

C₂×D₄

—

117

C₂×D₄

—

118

Q₁₆

—

119

D₈

—

120

D₈

—

121

D₈

—

122

D₈

—

123

C₂×C₈

#6104

yes

—

124

C₂×C₈

#6104

yes

—

125

C₂×C₈

D₄×D₁₃

yes

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄²

C₂×C₂²⋊D₁₃

yes

characteristic

131

C₁₆

—

132

C₁₃

#12529

yes

characteristic, complemented, Sylow

133

C₂³

—

134

Q₈

—

104

—

135

Q₈

—

104

—

136

D₄

—

137

D₄

—

138

D₄

—

139

D₄

—

140

C₂×C₄

—

104

—

141

C₂×C₄

—

104

—

142

C₂×C₄

C₂×#6104

yes

characteristic

143

C₂×C₄

C₂×#6104

yes

characteristic

144

C₂×C₄

#88194

yes

characteristic

145

C₈

—

146

C₈

#88067

yes

complemented

147

C₈

#88165

yes

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

#11718734

yes

characteristic

152

C₄

—

208

—

153

C₄

—

208

—

154

C₄

#11718808

yes

—

155

C₄

C₂×#88067

yes

—

156

C₄

#11718815

yes

—

157

C₂

—

158

C₂

#13114503

yes

characteristic

159

C₂

#13117091

yes

characteristic

160

C₂

#13117100

yes

characteristic

	subgroup	G/H
center	C₂²	#11718734
commutator, cosocle	C₄×C₈×C₁₃	C₂³
Frattini	C₄×C₈	C₂²×D₁₃
Fitting, prime radical	C₁₃×#1712	C₂
socle	C₂²×C₁₃	#493
hypercenter	#1712	D₁₃
hypocenter	C₁₃	#12529
LCS	G ▷ C₄×C₈×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1712

subgroup

G/H

center

C₂²

#11718734

commutator, cosocle

C₄×C₈×C₁₃

C₂³

Frattini

C₄×C₈

C₂²×D₁₃

Fitting, prime radical

C₁₃×#1712

C₂

C₂²×C₁₃

D₁₃

C₁₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1712

N	H ≅ G/N	type
1. #13117091	C₂	semidirect
2. #13117059	C₂	semidirect
3. #13114433	C₂	semidirect
4. #13112754	C₂	semidirect
71. C₈×C₁₃	SD₃₂	semidirect
132. C₁₃	#12529	semidirect
146. C₈	#88067	semidirect

H ≅ G/N

type

1. #13117091

C₂

semidirect

2. #13117059

C₂

semidirect

3. #13114433

C₂

semidirect

4. #13112754

C₂

semidirect

71. C₈×C₁₃

SD₃₂

semidirect

132. C₁₃

#12529

semidirect

146. C₈

#88067

semidirect

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

subgroup	index
#13117091	2
#13117059	2
#13114433	2
#13112754	2
C₁₃×#1712	2
#12529	13

subgroup

index

C₁₃×#1712

Subgroups — #15777694

Conjugacy classes of subgroups

Complements

Maximal subgroups