Group #15777703 — subgroups

Subgroups — #15777703

Order 3328.

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

normal subgroups only

This group has 272 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	#13117109	C₂	1	2	yes	maximal, complemented
2	#13117059	C₂	1	2	yes	maximal
3	#13114454	C₂	1	1	yes	maximal, characteristic, complemented
4	#13112754	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#1706	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#500	C₂²	1	1	yes	characteristic
7	C₁₃×#493	C₂²	1	1	yes	characteristic
8	C₁₃×#492	—	2	1	no	—
9	C₁₃×#486	—	2	1	no	—
10	#11718360	—	2	1	no	—
11	#11718343	—	2	1	no	—
12	#11718343	C₂²	1	2	yes	—
13	C₈²×C₁₃	C₂²	1	1	yes	characteristic
14	#11718154	—	2	4	no	—
15	#11718081	—	2	4	no	—
16	C₄×#6090	C₂²	1	2	yes	—
17	C₂×C₁₃×SD₁₆	—	4	2	no	—
18	C₂×C₁₃×D₈	—	2	2	no	—
19	C₁₃×C₄⋊Q₈	—	2	1	no	—
20	C₁₃×#127	—	2	1	no	—
21	C₂×Dic₅₂	—	4	2	no	—
22	#88123	—	2	2	no	—
23	C₁₃×#107	—	2	2	no	—
24	C₁₃×#102	—	4	2	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	#88058	—	2	2	no	—
29	#88058	—	2	2	no	—
30	#88058	—	4	2	no	—
31	C₂×#6090	D₄	1	4	yes	—
32	#12531	—	13	1	no	maximal, Sylow
33	C₂×Q₈×C₁₃	—	4	1	no	—
34	C₂×D₄×C₁₃	—	4	1	no	—
35	C₂×D₄×C₁₃	—	4	1	no	—
36	C₂×D₄×C₁₃	—	4	1	no	—
37	C₂×Dic₂₆	—	4	2	no	—
38	C₁₃×SD₁₆	—	8	4	no	—
39	C₁₃×D₈	—	4	4	no	—
40	C₂×C₈×C₁₃	D₈	1	2	yes	—
41	C₂×C₈×C₁₃	D₈	1	2	yes	—
42	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
43	C₁₃×C₄⋊C₄	—	4	1	no	—
44	C₁₃×C₄⋊C₄	—	4	1	no	—
45	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
46	C₄⋊Dic₁₃	—	4	2	no	—
47	C₄⋊Dic₁₃	—	4	2	no	—
48	Dic₅₂	—	8	4	no	—
49	#6090	—	2	4	no	—
50	#2252	—	13	2	no	—
51	#2240	—	13	2	no	—
52	#1713	—	13	1	no	—
53	#1706	D₁₃	1	1	yes	characteristic
54	#1312	—	13	1	no	—
55	C₂³×C₁₃	—	8	1	no	—
56	Q₈×C₁₃	—	8	2	no	—
57	D₄×C₁₃	—	8	2	no	—
58	D₄×C₁₃	—	8	2	no	—
59	D₄×C₁₃	—	8	2	no	—
60	C₂×C₄×C₁₃	—	8	1	no	—
61	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
62	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
63	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
64	C₂×Dic₁₃	—	8	2	no	—
65	Dic₂₆	—	8	4	no	—
66	C₈×C₁₃	—	2	2	no	—
67	C₈×C₁₃	D₁₆	1	4	yes	—
68	C₈×C₁₃	#137	1	4	yes	—
69	#500	—	13	2	no	—
70	#500	—	26	1	no	—
71	#500	C₂×D₁₃	1	1	yes	characteristic
72	#494	—	26	1	no	—
73	#493	C₂×D₁₃	1	1	yes	characteristic
74	#492	—	2	1	no	—
75	#486	—	2	1	no	—
76	#366	—	26	4	no	—
77	#357	—	26	4	no	—
78	C₄×C₁₆	—	13	2	no	—
79	C₈²	C₂×D₁₃	1	1	yes	characteristic
80	C₂²×C₁₃	—	16	1	no	—
81	C₂²×C₁₃	—	16	1	no	—
82	C₂²×C₁₃	—	16	1	no	—
83	C₂²×C₁₃	#493	1	1	yes	characteristic
84	C₄×C₁₃	—	16	1	no	—
85	C₄×C₁₃	#497	1	2	yes	—
86	C₄×C₁₃	C₂×D₁₆	1	2	yes	—
87	C₄×C₁₃	#510	1	2	yes	—
88	Dic₁₃	—	16	2	no	—
89	C₂×Q₁₆	—	52	2	no	—
90	C₂×SD₁₆	—	4	2	no	—
91	C₂×D₈	—	2	2	no	—
92	C₄⋊Q₈	—	2	1	no	—
93	C₄⋊Q₈	—	26	2	no	—
94	#127	—	2	1	no	—
95	C₂×C₁₆	—	13	4	no	—
96	#107	—	2	2	no	—
97	#107	—	26	2	no	—
98	#107	—	26	2	no	—
99	#107	—	52	2	no	—
100	#102	—	4	2	no	—
101	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
102	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
103	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
104	C₂×C₁₃	—	32	1	no	—
105	C₂×C₁₃	#1722	1	1	yes	characteristic
106	C₂×C₁₃	#2246	1	1	yes	characteristic
107	C₂×C₁₃	#2251	1	1	yes	characteristic
108	C₂×Q₈	—	4	1	no	—
109	C₂×Q₈	—	52	2	no	—
110	C₂×D₄	—	4	1	no	—
111	C₂×D₄	—	4	1	no	—
112	C₂×D₄	—	4	1	no	—
113	Q₁₆	—	104	4	no	—
114	SD₁₆	—	8	4	no	—
115	D₈	—	4	4	no	—
116	C₂×C₈	#6104	1	2	yes	—
117	C₂×C₈	#6104	1	2	yes	—
118	C₂×C₈	D₄×D₁₃	1	2	yes	—
119	C₄⋊C₄	—	4	1	no	—
120	C₄⋊C₄	—	4	1	no	—
121	C₄⋊C₄	—	52	2	no	—
122	C₄⋊C₄	—	52	2	no	—
123	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
124	C₁₆	—	26	4	no	—
125	C₁₃	#12531	1	1	yes	characteristic, complemented, Sylow
126	C₂³	—	8	1	no	—
127	Q₈	—	8	2	no	—
128	Q₈	—	104	4	no	—
129	D₄	—	8	2	no	—
130	D₄	—	8	2	no	—
131	D₄	—	8	2	no	—
132	C₂×C₄	—	8	1	no	—
133	C₂×C₄	—	104	2	no	—
134	C₂×C₄	C₂×#6104	1	1	yes	characteristic
135	C₂×C₄	C₂×#6104	1	1	yes	characteristic
136	C₂×C₄	#88194	1	1	yes	characteristic
137	C₈	—	2	2	no	—
138	C₈	#88066	1	4	yes	—
139	C₈	#88169	1	4	yes	—
140	C₂²	—	16	1	no	—
141	C₂²	—	16	1	no	—
142	C₂²	—	16	1	no	—
143	C₂²	#11718734	1	1	yes	characteristic
144	C₄	—	16	1	no	—
145	C₄	—	208	2	no	—
146	C₄	C₂×#88066	1	2	yes	—
147	C₄	#11718828	1	2	yes	—
148	C₄	#11718856	1	2	yes	—
149	C₂	—	32	1	no	—
150	C₂	#13114505	1	1	yes	characteristic
151	C₂	#13117085	1	1	yes	characteristic
152	C₂	#13117108	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13117109

C₂

yes

maximal, complemented

#13117059

C₂

yes

maximal

#13114454

C₂

yes

maximal, characteristic, complemented

#13112754

C₂

yes

maximal, characteristic, complemented

C₁₃×#1706

C₂

yes

maximal, characteristic

C₁₃×#500

C₂²

yes

characteristic

C₁₃×#493

C₂²

yes

characteristic

C₁₃×#492

—

C₁₃×#486

—

#11718360

—

#11718343

—

#11718343

C₂²

yes

—

C₈²×C₁₃

C₂²

yes

—

—

C₂²

yes

—

C₂×C₁₃×SD₁₆

—

C₂×C₁₃×D₈

—

C₁₃×C₄⋊Q₈

—

C₁₃×#127

—

C₂×Dic₅₂

—

#88123

—

C₁₃×#107

—

C₁₃×#102

—

C₄×C₈×C₁₃

D₄

yes

characteristic

C₄×C₈×C₁₃

D₄

yes

characteristic

C₄×C₈×C₁₃

C₂³

yes

—

—

—

D₄

yes

—

#12531

—

maximal, Sylow

C₂×Q₈×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×Dic₂₆

—

C₁₃×SD₁₆

—

C₁₃×D₈

—

C₂×C₈×C₁₃

D₈

yes

—

C₂×C₈×C₁₃

D₈

yes

—

C₂×C₈×C₁₃

C₂×D₄

yes

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

C₂×D₄

yes

characteristic

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

Dic₅₂

—

—

—

—

—

D₁₃

yes

characteristic

#1312

—

C₂³×C₁₃

—

Q₈×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#127

yes

characteristic

C₂×C₄×C₁₃

C₂×D₈

yes

characteristic

C₂×C₄×C₁₃

C₂×D₈

yes

characteristic

C₂×Dic₁₃

—

Dic₂₆

—

C₈×C₁₃

—

C₈×C₁₃

D₁₆

yes

—

C₈×C₁₃

#137

yes

—

#500

—

#500

—

#500

C₂×D₁₃

yes

characteristic

#494

—

#493

C₂×D₁₃

yes

—

—

—

—

C₄×C₁₆

—

C₈²

C₂×D₁₃

yes

characteristic

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

#493

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

#497

yes

—

C₄×C₁₃

C₂×D₁₆

yes

—

C₄×C₁₃

#510

yes

—

Dic₁₃

—

C₂×Q₁₆

—

C₂×SD₁₆

—

C₂×D₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#127

—

C₂×C₁₆

—

—

—

—

—

100

#102

—

101

C₄×C₈

C₂²⋊D₁₃

yes

characteristic

102

C₄×C₈

C₂²⋊D₁₃

yes

characteristic

103

C₄×C₈

C₂²×D₁₃

yes

characteristic

104

C₂×C₁₃

—

105

C₂×C₁₃

#1722

yes

characteristic

106

C₂×C₁₃

#2246

yes

characteristic

107

C₂×C₁₃

#2251

yes

characteristic

108

C₂×Q₈

—

109

C₂×Q₈

—

110

C₂×D₄

—

111

C₂×D₄

—

112

C₂×D₄

—

113

Q₁₆

—

104

—

114

SD₁₆

—

115

D₈

—

116

C₂×C₈

#6104

yes

—

117

C₂×C₈

#6104

yes

—

118

C₂×C₈

D₄×D₁₃

yes

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄²

C₂×C₂²⋊D₁₃

yes

characteristic

124

C₁₆

—

125

C₁₃

#12531

yes

characteristic, complemented, Sylow

126

C₂³

—

127

Q₈

—

128

Q₈

—

104

—

129

D₄

—

130

D₄

—

131

D₄

—

132

C₂×C₄

—

133

C₂×C₄

—

104

—

134

C₂×C₄

C₂×#6104

yes

characteristic

135

C₂×C₄

C₂×#6104

yes

characteristic

136

C₂×C₄

#88194

yes

characteristic

137

C₈

—

138

C₈

#88066

yes

—

139

C₈

#88169

yes

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

#11718734

yes

characteristic

144

C₄

—

145

C₄

—

208

—

146

C₄

C₂×#88066

yes

—

147

C₄

#11718828

yes

—

148

C₄

#11718856

yes

—

149

C₂

—

150

C₂

#13114505

yes

characteristic

151

C₂

#13117085

yes

characteristic

152

C₂

#13117108

yes

characteristic

	subgroup	G/H
center	C₂²	#11718734
commutator, cosocle	C₄×C₈×C₁₃	C₂³
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₁₃×#1706	C₂
socle	C₂²×C₁₃	#493
prime radical	C₁₃×#493	C₂²
hypercenter	#1706	D₁₃
hypocenter	C₁₃	#12531
LCS	G ▷ C₄×C₈×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1706

subgroup

G/H

center

C₂²

#11718734

commutator, cosocle

C₄×C₈×C₁₃

C₂³

Frattini

C₄×C₈

C₂²×D₁₃

Fitting

C₁₃×#1706

C₂

socle

C₂²×C₁₃

#493

prime radical

C₁₃×#493

C₂²

D₁₃

C₁₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1706

N	H ≅ G/N	type
1. #13117109	C₂	semidirect
3. #13114454	C₂	semidirect
4. #13112754	C₂	semidirect
125. C₁₃	#12531	semidirect

H ≅ G/N

type

1. #13117109

C₂

semidirect

3. #13114454

C₂

semidirect

4. #13112754

C₂

semidirect

125. C₁₃

#12531

semidirect

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

subgroup	index
#13117109	2
#13117059	2
#13114454	2
#13112754	2
C₁₃×#1706	2
#12531	13

subgroup

index

C₁₃×#1706

Subgroups — #15777703

Conjugacy classes of subgroups

Complements

Maximal subgroups