Group #15777554 — subgroups

Subgroups — #15777554

Order 3328.

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

normal subgroups only

This group has 212 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	#13116989	C₂	1	2	yes	maximal
2	#13114454	C₂	1	1	yes	maximal, characteristic
3	#13113734	C₂	1	1	yes	maximal, characteristic
4	#13112893	C₂	1	2	yes	maximal
5	C₁₃×#1367	C₂	1	1	yes	maximal, characteristic
6	#11718360	—	2	1	no	—
7	C₄×Dic₅₂	C₂²	1	1	yes	characteristic
8	#11718343	—	2	1	no	—
9	#11718343	C₂²	1	2	yes	—
10	#11718340	—	2	1	no	—
11	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	—
12	C₈²×C₁₃	C₂²	1	1	yes	characteristic
13	#11718096	—	2	2	no	—
14	#11718095	—	2	2	no	—
15	#11718050	C₂²	1	1	yes	characteristic
16	#11718048	—	2	1	no	—
17	C₂×Dic₅₂	—	4	2	no	—
18	C₂×Dic₅₂	D₄	1	1	yes	characteristic
19	#88123	—	2	2	no	—
20	C₄×Dic₂₆	—	2	1	no	—
21	C₂×C₁₃×C₁₆	—	2	2	no	—
22	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
23	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
24	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
25	#88058	—	2	2	no	—
26	#88058	—	2	2	no	—
27	#88058	—	4	1	no	—
28	#88058	—	4	1	no	—
29	#88058	D₄	1	1	yes	characteristic
30	#88056	—	2	1	no	—
31	#88055	—	4	1	no	—
32	#88054	—	4	1	no	—
33	C₄⋊#1055	—	2	1	no	—
34	#12500	—	13	1	no	maximal, Sylow
35	C₂×Dic₂₆	—	2	1	no	—
36	C₂×Dic₂₆	—	4	2	no	—
37	C₂×C₈×C₁₃	—	2	1	no	—
38	C₂×C₈×C₁₃	D₈	1	1	yes	characteristic
39	C₂×C₈×C₁₃	D₈	1	1	yes	characteristic
40	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
41	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
42	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
43	C₄⋊Dic₁₃	—	2	1	no	—
44	C₄⋊Dic₁₃	—	4	2	no	—
45	C₄⋊Dic₁₃	—	4	2	no	—
46	#6101	—	4	1	no	—
47	C₄×Dic₁₃	—	4	1	no	—
48	C₂×#1055	—	4	1	no	—
49	Dic₅₂	—	2	2	no	—
50	Dic₅₂	—	8	4	no	—
51	C₁₃×C₁₆	—	4	2	no	—
52	#2227	—	13	2	no	—
53	#1713	—	13	1	no	—
54	#1591	—	13	1	no	—
55	#1367	D₁₃	1	1	yes	characteristic
56	#1365	—	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	AGL(1,ℤ/8)	1	1	yes	characteristic
60	C₂×Dic₁₃	—	4	1	no	—
61	C₂×Dic₁₃	—	4	1	no	—
62	C₂×Dic₁₃	—	8	2	no	—
63	Dic₂₆	—	4	1	no	—
64	Dic₂₆	—	4	1	no	—
65	Dic₂₆	—	8	4	no	—
66	C₈×C₁₃	—	2	2	no	—
67	C₈×C₁₃	—	2	1	no	—
68	C₈×C₁₃	—	2	1	no	—
69	C₈×C₁₃	Q₃₂	1	2	yes	—
70	C₈×C₁₃	#137	1	2	yes	—
71	#1055	—	8	1	no	—
72	#500	—	13	2	no	—
73	#500	—	26	1	no	—
74	#494	—	26	1	no	—
75	#446	—	26	1	no	—
76	C₄×Q₁₆	—	13	1	no	—
77	#366	—	26	2	no	—
78	C₄⋊C₁₆	C₂×D₁₃	1	2	yes	—
79	#358	—	26	2	no	—
80	#335	—	13	1	no	—
81	Q₈⋊C₈	—	26	1	no	—
82	C₈²	C₂×D₁₃	1	1	yes	characteristic
83	C₂²×C₁₃	#459	1	1	yes	characteristic
84	C₄×C₁₃	—	2	1	no	—
85	C₄×C₁₃	#470	1	1	yes	characteristic
86	C₄×C₁₃	#473	1	1	yes	characteristic
87	C₄×C₁₃	C₂×Q₃₂	1	1	yes	characteristic
88	C₄×C₁₃	#510	1	1	yes	characteristic
89	Dic₁₃	—	8	1	no	—
90	Dic₁₃	—	8	1	no	—
91	Dic₁₃	—	16	2	no	—
92	C₂×Q₁₆	—	13	1	no	—
93	C₂×Q₁₆	—	52	2	no	—
94	C₄⋊Q₈	—	26	2	no	—
95	C₄×Q₈	—	26	1	no	—
96	C₂×C₁₆	—	2	2	no	—
97	#107	—	13	1	no	—
98	#107	—	26	2	no	—
99	#107	—	26	2	no	—
100	#107	—	52	1	no	—
101	#107	—	52	1	no	—
102	C₄⋊C₈	—	26	1	no	—
103	C₄⋊C₈	—	52	1	no	—
104	#103	—	26	1	no	—
105	#97	—	52	1	no	—
106	C₄×C₈	D₅₂	1	1	yes	characteristic
107	C₄×C₈	D₅₂	1	1	yes	characteristic
108	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
109	C₂×C₁₃	#1689	1	1	yes	characteristic
110	C₂×C₁₃	#2209	1	1	yes	characteristic
111	C₂×C₁₃	#2213	1	1	yes	characteristic
112	C₂×Q₈	—	26	1	no	—
113	C₂×Q₈	—	52	2	no	—
114	Q₁₆	—	26	2	no	—
115	Q₁₆	—	104	4	no	—
116	C₂×C₈	—	2	1	no	—
117	C₂×C₈	—	52	1	no	—
118	C₂×C₈	D₁₀₄	1	1	yes	characteristic
119	C₂×C₈	D₁₀₄	1	1	yes	characteristic
120	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
121	C₂×C₈	#6131	1	1	yes	characteristic
122	C₄⋊C₄	—	26	1	no	—
123	C₄⋊C₄	—	52	1	no	—
124	C₄⋊C₄	—	52	2	no	—
125	C₄⋊C₄	—	52	2	no	—
126	C₄²	—	52	1	no	—
127	C₄²	C₂×D₅₂	1	1	yes	characteristic
128	C₁₆	—	4	2	no	—
129	C₁₃	#12500	1	1	yes	characteristic, complemented, Sylow
130	Q₈	—	52	1	no	—
131	Q₈	—	52	1	no	—
132	Q₈	—	104	4	no	—
133	C₂×C₄	—	52	1	no	—
134	C₂×C₄	—	52	1	no	—
135	C₂×C₄	—	104	2	no	—
136	C₂×C₄	#88149	1	1	yes	characteristic
137	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
138	C₂×C₄	#88162	1	1	yes	characteristic
139	C₈	—	2	2	no	—
140	C₈	—	2	1	no	—
141	C₈	—	2	1	no	—
142	C₈	—	104	1	no	—
143	C₈	Dic₁₀₄	1	2	yes	—
144	C₈	#88169	1	2	yes	—
145	C₂²	#11718504	1	1	yes	characteristic
146	C₄	—	2	1	no	—
147	C₄	—	104	1	no	—
148	C₄	—	104	1	no	—
149	C₄	—	208	2	no	—
150	C₄	#11718529	1	1	yes	characteristic
151	C₄	#11718557	1	1	yes	characteristic
152	C₄	C₂×Dic₁₀₄	1	1	yes	characteristic
153	C₄	#11718570	1	1	yes	characteristic
154	C₂	#13114326	1	1	yes	characteristic
155	C₂	#13116887	1	1	yes	characteristic
156	C₂	#13116913	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13116989

C₂

yes

maximal

#13114454

C₂

yes

maximal, characteristic

#13113734

C₂

yes

maximal, characteristic

#13112893

C₂

yes

maximal

C₁₃×#1367

C₂

yes

maximal, characteristic

#11718360

—

C₄×Dic₅₂

C₂²

yes

characteristic

#11718343

—

#11718343

C₂²

yes

—

#11718340

—

C₁₃×C₄⋊C₁₆

C₂²

yes

—

C₈²×C₁₃

C₂²

yes

—

—

C₂²

yes

characteristic

#11718048

—

C₂×Dic₅₂

—

C₂×Dic₅₂

D₄

yes

characteristic

#88123

—

C₄×Dic₂₆

—

C₂×C₁₃×C₁₆

—

C₄×C₈×C₁₃

D₄

yes

characteristic

C₄×C₈×C₁₃

D₄

yes

characteristic

C₄×C₈×C₁₃

C₂³

yes

—

—

—

—

D₄

yes

—

—

—

—

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

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

C₄²×C₁₃

C₂×D₄

yes

characteristic

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

#6101

—

C₄×Dic₁₃

—

C₂×#1055

—

Dic₅₂

—

Dic₅₂

—

C₁₃×C₁₆

—

—

—

—

D₁₃

yes

characteristic

#1365

—

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

—

C₂×Dic₁₃

—

Dic₂₆

—

Dic₂₆

—

Dic₂₆

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

Q₃₂

yes

—

C₈×C₁₃

#137

yes

—

—

—

—

—

—

C₄×Q₁₆

—

#366

—

C₄⋊C₁₆

C₂×D₁₃

yes

—

#358

—

#335

—

Q₈⋊C₈

—

C₈²

C₂×D₁₃

yes

characteristic

C₂²×C₁₃

#459

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

yes

C₄×C₁₃

yes

C₄×C₁₃

C₂×Q₃₂

yes

characteristic

C₄×C₁₃

#510

yes

characteristic

Dic₁₃

—

Dic₁₃

—

Dic₁₃

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄×Q₈

—

C₂×C₁₆

—

#107

—

#107

—

#107

—

100

#107

—

101

#107

—

102

C₄⋊C₈

—

103

C₄⋊C₈

—

104

#103

—

105

#97

—

106

C₄×C₈

D₅₂

yes

characteristic

107

C₄×C₈

D₅₂

yes

characteristic

108

C₄×C₈

C₂²×D₁₃

yes

characteristic

109

C₂×C₁₃

#1689

yes

characteristic

110

C₂×C₁₃

#2209

yes

characteristic

111

C₂×C₁₃

#2213

yes

characteristic

112

C₂×Q₈

—

113

C₂×Q₈

—

114

Q₁₆

—

115

Q₁₆

—

104

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

D₁₀₄

yes

characteristic

119

C₂×C₈

D₁₀₄

yes

characteristic

120

C₂×C₈

D₄×D₁₃

yes

characteristic

121

C₂×C₈

#6131

yes

characteristic

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄²

—

127

C₄²

C₂×D₅₂

yes

characteristic

128

C₁₆

—

129

C₁₃

#12500

yes

characteristic, complemented, Sylow

130

Q₈

—

131

Q₈

—

132

Q₈

—

104

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

104

—

136

C₂×C₄

#88149

yes

characteristic

137

C₂×C₄

C₂×D₁₀₄

yes

characteristic

138

C₂×C₄

#88162

yes

characteristic

139

C₈

—

140

C₈

—

141

C₈

—

142

C₈

—

104

—

143

C₈

Dic₁₀₄

yes

—

144

C₈

#88169

yes

—

145

C₂²

#11718504

yes

characteristic

146

C₄

—

147

C₄

—

104

—

148

C₄

—

104

—

149

C₄

—

208

—

150

C₄

#11718529

yes

characteristic

151

C₄

#11718557

yes

characteristic

152

C₄

C₂×Dic₁₀₄

yes

characteristic

153

C₄

#11718570

yes

characteristic

154

C₂

#13114326

yes

characteristic

155

C₂

#13116887

yes

characteristic

156

C₂

#13116913

yes

characteristic

	subgroup	G/H
center	C₂²	#11718504
commutator, cosocle	C₄×C₈×C₁₃	C₂³
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₁₃×#1367	C₂
socle, prime radical	C₂²×C₁₃	#459
hypercenter	#1367	D₁₃
hypocenter	C₁₃	#12500
LCS	G ▷ C₄×C₈×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1367

subgroup

G/H

center

C₂²

#11718504

commutator, cosocle

C₄×C₈×C₁₃

C₂³

Frattini

C₄×C₈

C₂²×D₁₃

Fitting

C₁₃×#1367

C₂

C₂²×C₁₃

D₁₃

C₁₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1367

N	H ≅ G/N	type
129. C₁₃	#12500	semidirect

H ≅ G/N

type

129. C₁₃

#12500

semidirect

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

subgroup	index
#13116989	2
#13114454	2
#13113734	2
#13112893	2
C₁₃×#1367	2
#12500	13

subgroup

index

C₁₃×#1367

Subgroups — #15777554

Conjugacy classes of subgroups

Complements

Maximal subgroups