Group #15777676 — subgroups

Subgroups — #15777676

Order 3328.

296 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 91 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	#13117092	C₂	1	2	yes	maximal
2	#13117064	C₂	1	2	yes	maximal, complemented
3	#13114454	C₂	1	1	yes	maximal, characteristic, complemented
4	#13114428	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1312	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#88040	—	2	4	no	—
7	#11718360	C₂²	1	2	yes	—
8	#11718356	C₂²	1	1	yes	characteristic
9	#11718355	—	2	1	no	—
10	#11718354	—	2	1	no	—
11	#11718343	—	2	1	no	—
12	#11718343	—	2	1	no	—
13	#11718343	C₂²	1	1	yes	characteristic
14	C₄×C₁₃×C₁₆	C₂²	1	2	yes	—
15	C₈²×C₁₃	C₂²	1	1	yes	characteristic
16	#11718095	—	2	4	no	—
17	C₂×Dic₅₂	—	2	2	no	—
18	C₂×Dic₅₂	—	2	2	no	—
19	C₂×D₁₀₄	—	2	2	no	—
20	C₂×#6095	—	4	2	no	—
21	#88128	—	2	1	no	—
22	#88123	—	2	2	no	—
23	#88123	—	2	1	no	—
24	C₂×C₁₃×C₁₆	D₄	1	4	yes	—
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	#88061	—	4	2	no	—
29	#88058	—	2	2	no	—
30	#88058	—	4	2	no	—
31	#88058	—	4	2	no	—
32	#88040	—	4	8	no	—
33	#12526	—	13	1	no	maximal, Sylow
34	C₂×D₅₂	—	4	1	no	—
35	C₂×D₅₂	—	4	1	no	—
36	C₂×D₅₂	—	4	1	no	—
37	C₂×Dic₂₆	—	4	2	no	—
38	C₂×Dic₂₆	—	4	1	no	—
39	C₂×C₈×C₁₃	D₈	1	2	yes	—
40	C₂×C₈×C₁₃	D₈	1	2	yes	—
41	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
42	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
43	C₄⋊Dic₁₃	—	4	1	no	—
44	C₄⋊Dic₁₃	—	4	2	no	—
45	C₄⋊Dic₁₃	—	4	2	no	—
46	C₄⋊Dic₁₃	—	4	1	no	—
47	Dic₅₂	—	4	4	no	—
48	Dic₅₂	—	4	4	no	—
49	D₁₀₄	—	4	4	no	—
50	#6095	—	8	4	no	—
51	C₁₃×C₁₆	—	2	4	no	—
52	#2248	—	13	2	no	—
53	#2241	—	13	2	no	—
54	#1713	—	13	1	no	—
55	#1706	—	13	1	no	—
56	#1312	D₁₃	1	1	yes	characteristic, complemented
57	C₂²×D₁₃	—	8	1	no	—
58	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
59	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
60	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
61	C₂×Dic₁₃	—	8	2	no	—
62	C₂×Dic₁₃	—	8	1	no	—
63	D₅₂	—	8	2	no	—
64	D₅₂	—	8	2	no	—
65	D₅₂	—	8	2	no	—
66	Dic₂₆	—	8	2	no	—
67	Dic₂₆	—	8	2	no	—
68	Dic₂₆	—	8	2	no	—
69	C₈×C₁₃	—	2	2	no	—
70	C₈×C₁₃	SD₃₂	1	4	yes	complemented
71	C₈×C₁₃	AGL(1,ℤ/8)	1	4	yes	—
72	C₂×SD₃₂	—	26	4	no	—
73	#500	—	13	1	no	—
74	#500	—	26	1	no	—
75	#500	—	26	1	no	—
76	#494	—	13	2	no	—
77	#493	—	13	1	no	—
78	#492	—	26	1	no	—
79	#486	—	26	1	no	—
80	#358	—	26	4	no	—
81	C₄×C₁₆	C₂×D₁₃	1	2	yes	—
82	C₈²	C₂×D₁₃	1	1	yes	characteristic
83	C₂²×C₁₃	#493	1	1	yes	characteristic
84	C₂×D₁₃	—	16	1	no	—
85	C₂×D₁₃	—	16	1	no	—
86	C₂×D₁₃	—	16	1	no	—
87	C₄×C₁₃	#496	1	2	yes	—
88	C₄×C₁₃	C₂×SD₃₂	1	2	yes	—
89	C₄×C₁₃	#510	1	2	yes	—
90	Dic₁₃	—	16	2	no	—
91	Dic₁₃	—	16	1	no	—
92	C₂×Q₁₆	—	26	2	no	—
93	C₂×Q₁₆	—	26	2	no	—
94	C₂×SD₁₆	—	52	2	no	—
95	C₂×D₈	—	26	2	no	—
96	C₄⋊Q₈	—	26	1	no	—
97	C₄⋊Q₈	—	26	2	no	—
98	#127	—	26	1	no	—
99	SD₃₂	—	52	8	no	—
100	C₂×C₁₆	D₅₂	1	4	yes	—
101	#107	—	26	2	no	—
102	#107	—	52	2	no	—
103	#107	—	52	2	no	—
104	#102	—	52	2	no	—
105	C₄×C₈	D₅₂	1	1	yes	characteristic
106	C₄×C₈	D₅₂	1	1	yes	characteristic
107	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
108	C₂×C₁₃	#1721	1	1	yes	characteristic
109	C₂×C₁₃	#2248	1	1	yes	characteristic
110	C₂×C₁₃	#2251	1	1	yes	characteristic
111	D₁₃	—	32	1	no	—
112	C₂×Q₈	—	52	2	no	—
113	C₂×Q₈	—	52	1	no	—
114	C₂×D₄	—	52	1	no	—
115	C₂×D₄	—	52	1	no	—
116	C₂×D₄	—	52	1	no	—
117	Q₁₆	—	52	4	no	—
118	Q₁₆	—	52	4	no	—
119	SD₁₆	—	104	4	no	—
120	D₈	—	52	4	no	—
121	C₂×C₈	D₁₀₄	1	2	yes	—
122	C₂×C₈	D₁₀₄	1	2	yes	—
123	C₂×C₈	C₂×D₅₂	1	2	yes	—
124	C₄⋊C₄	—	52	1	no	—
125	C₄⋊C₄	—	52	1	no	—
126	C₄⋊C₄	—	52	2	no	—
127	C₄⋊C₄	—	52	2	no	—
128	C₄²	C₂×D₅₂	1	1	yes	characteristic
129	C₁₆	—	2	4	no	—
130	C₁₃	#12526	1	1	yes	characteristic, complemented, Sylow
131	C₂³	—	104	1	no	—
132	Q₈	—	104	2	no	—
133	Q₈	—	104	2	no	—
134	Q₈	—	104	2	no	—
135	D₄	—	104	2	no	—
136	D₄	—	104	2	no	—
137	D₄	—	104	2	no	—
138	C₂×C₄	—	104	2	no	—
139	C₂×C₄	—	104	1	no	—
140	C₂×C₄	#88128	1	1	yes	characteristic
141	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
142	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
143	C₈	—	2	2	no	—
144	C₈	#88040	1	4	yes	complemented
145	C₈	#88162	1	4	yes	—
146	C₂²	—	208	1	no	—
147	C₂²	—	208	1	no	—
148	C₂²	—	208	1	no	—
149	C₂²	#11718356	1	1	yes	characteristic
150	C₄	—	208	2	no	—
151	C₄	—	208	1	no	—
152	C₄	#11718373	1	2	yes	—
153	C₄	C₂×#88040	1	2	yes	—
154	C₄	#11718570	1	2	yes	—
155	C₂	—	416	1	no	—
156	C₂	#13114502	1	1	yes	characteristic
157	C₂	#13117092	1	1	yes	characteristic
158	C₂	#13117107	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13117092

C₂

yes

maximal

#13117064

C₂

yes

maximal, complemented

#13114454

C₂

yes

maximal, characteristic, complemented

#13114428

C₂

yes

maximal, characteristic

C₁₃×#1312

C₂

yes

maximal, characteristic, complemented

C₂×#88040

—

#11718360

C₂²

yes

—

#11718356

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₄×C₁₃×C₁₆

C₂²

yes

—

C₈²×C₁₃

C₂²

yes

characteristic

#11718095

—

C₂×Dic₅₂

—

C₂×Dic₅₂

—

C₂×D₁₀₄

—

—

—

—

—

C₂×C₁₃×C₁₆

D₄

yes

—

C₄×C₈×C₁₃

D₄

yes

characteristic

C₄×C₈×C₁₃

D₄

yes

characteristic

C₄×C₈×C₁₃

C₂³

yes

—

—

—

—

—

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

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

—

Dic₅₂

—

D₁₀₄

—

#6095

—

C₁₃×C₁₆

—

—

—

—

—

D₁₃

yes

characteristic, complemented

C₂²×D₁₃

—

C₂×C₄×C₁₃

#127

yes

characteristic

C₂×C₄×C₁₃

C₂×D₈

yes

characteristic

C₂×C₄×C₁₃

C₂×D₈

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

—

D₅₂

—

D₅₂

—

D₅₂

—

Dic₂₆

—

Dic₂₆

—

Dic₂₆

—

C₈×C₁₃

—

C₈×C₁₃

SD₃₂

yes

complemented

C₈×C₁₃

AGL(1,ℤ/8)

yes

—

C₂×SD₃₂

—

—

—

—

—

—

—

—

—

C₄×C₁₆

C₂×D₁₃

yes

—

C₈²

C₂×D₁₃

yes

characteristic

C₂²×C₁₃

#493

yes

characteristic

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

#496

yes

—

C₄×C₁₃

C₂×SD₃₂

yes

—

C₄×C₁₃

#510

yes

—

Dic₁₃

—

Dic₁₃

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×SD₁₆

—

C₂×D₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#127

—

SD₃₂

—

100

C₂×C₁₆

D₅₂

yes

—

101

#107

—

102

#107

—

103

#107

—

104

#102

—

105

C₄×C₈

D₅₂

yes

characteristic

106

C₄×C₈

D₅₂

yes

characteristic

107

C₄×C₈

C₂²×D₁₃

yes

characteristic

108

C₂×C₁₃

#1721

yes

characteristic

109

C₂×C₁₃

#2248

yes

characteristic

110

C₂×C₁₃

#2251

yes

characteristic

111

D₁₃

—

112

C₂×Q₈

—

113

C₂×Q₈

—

114

C₂×D₄

—

115

C₂×D₄

—

116

C₂×D₄

—

117

Q₁₆

—

118

Q₁₆

—

119

SD₁₆

—

104

—

120

D₈

—

121

C₂×C₈

D₁₀₄

yes

—

122

C₂×C₈

D₁₀₄

yes

—

123

C₂×C₈

C₂×D₅₂

yes

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄²

C₂×D₅₂

yes

characteristic

129

C₁₆

—

130

C₁₃

#12526

yes

characteristic, complemented, Sylow

131

C₂³

—

104

—

132

Q₈

—

104

—

133

Q₈

—

104

—

134

Q₈

—

104

—

135

D₄

—

104

—

136

D₄

—

104

—

137

D₄

—

104

—

138

C₂×C₄

—

104

—

139

C₂×C₄

—

104

—

140

C₂×C₄

#88128

yes

characteristic

141

C₂×C₄

C₂×D₁₀₄

yes

characteristic

142

C₂×C₄

C₂×D₁₀₄

yes

characteristic

143

C₈

—

144

C₈

#88040

yes

complemented

145

C₈

#88162

yes

—

146

C₂²

—

208

—

147

C₂²

—

208

—

148

C₂²

—

208

—

149

C₂²

#11718356

yes

characteristic

150

C₄

—

208

—

151

C₄

—

208

—

152

C₄

#11718373

yes

—

153

C₄

C₂×#88040

yes

—

154

C₄

#11718570

yes

—

155

C₂

—

416

—

156

C₂

#13114502

yes

characteristic

157

C₂

#13117092

yes

characteristic

158

C₂

#13117107

yes

characteristic

	subgroup	G/H
center	C₂²	#11718356
commutator, cosocle	C₄×C₈×C₁₃	C₂³
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₁₃×#1312	C₂
socle	C₂²×C₁₃	#493
prime radical	#11718356	C₂²
hypercenter	#1312	D₁₃
hypocenter	C₁₃	#12526
LCS	G ▷ C₄×C₈×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1312

subgroup

G/H

center

C₂²

#11718356

commutator, cosocle

C₄×C₈×C₁₃

C₂³

Frattini

C₄×C₈

C₂²×D₁₃

Fitting

C₁₃×#1312

C₂

C₂²×C₁₃

C₂²

D₁₃

C₁₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1312

N	H ≅ G/N	type
2. #13117064	C₂	semidirect
3. #13114454	C₂	semidirect
5. C₁₃×#1312	C₂	semidirect
56. #1312	D₁₃	semidirect
70. C₈×C₁₃	SD₃₂	semidirect
130. C₁₃	#12526	semidirect
144. C₈	#88040	semidirect

H ≅ G/N

type

2. #13117064

C₂

semidirect

3. #13114454

C₂

semidirect

5. C₁₃×#1312

C₂

semidirect

56. #1312

D₁₃

semidirect

70. C₈×C₁₃

SD₃₂

semidirect

130. C₁₃

#12526

semidirect

144. C₈

#88040

semidirect

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

subgroup	index
#13117092	2
#13117064	2
#13114454	2
#13114428	2
C₁₃×#1312	2
#12526	13

subgroup

index

C₁₃×#1312

Subgroups — #15777676

Conjugacy classes of subgroups

Complements

Maximal subgroups