Group #15777705 — subgroups

Subgroups — #15777705

Order 3328.

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

normal subgroups only

This group has 272 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	#13117111	C₂	1	2	yes	maximal, complemented
2	#13117060	C₂	1	2	yes	maximal
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	#11718360	—	2	1	no	—
7	#11718356	C₂²	1	1	yes	characteristic
8	#11718355	—	2	1	no	—
9	#11718354	—	2	1	no	—
10	#11718343	—	2	1	no	—
11	#11718343	C₂²	1	1	yes	characteristic
12	#11718343	C₂²	1	2	yes	—
13	C₄×C₁₃×C₁₆	C₂²	1	2	yes	—
14	C₈²×C₁₃	C₂²	1	1	yes	characteristic
15	#11718101	—	2	4	no	—
16	#11718096	—	2	4	no	—
17	C₂×Dic₅₂	—	4	2	no	—
18	C₂×D₁₀₄	—	2	2	no	—
19	C₂×#6095	—	4	2	no	—
20	#88128	—	2	1	no	—
21	#88123	—	2	2	no	—
22	#88123	—	2	1	no	—
23	C₂×C₁₃×C₁₆	D₄	1	4	yes	—
24	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
25	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
26	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
27	#88061	—	4	2	no	—
28	#88058	—	2	2	no	—
29	#88058	—	2	2	no	—
30	#88058	—	2	2	no	—
31	#88058	—	4	2	no	—
32	#12531	—	13	1	no	maximal, Sylow
33	C₂×D₅₂	—	4	1	no	—
34	C₂×D₅₂	—	4	1	no	—
35	C₂×D₅₂	—	4	1	no	—
36	C₂×Dic₂₆	—	4	1	no	—
37	C₂×Dic₂₆	—	4	2	no	—
38	C₂×C₈×C₁₃	D₈	1	2	yes	—
39	C₂×C₈×C₁₃	D₈	1	2	yes	—
40	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
41	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
42	C₄⋊Dic₁₃	—	4	2	no	—
43	C₄⋊Dic₁₃	—	4	1	no	—
44	C₄⋊Dic₁₃	—	4	2	no	—
45	C₄⋊Dic₁₃	—	4	1	no	—
46	Dic₅₂	—	8	4	no	—
47	D₁₀₄	—	4	4	no	—
48	#6095	—	8	4	no	—
49	C₁₃×C₁₆	—	2	4	no	—
50	#2252	—	13	2	no	—
51	#2240	—	13	2	no	—
52	#1713	—	13	1	no	—
53	#1706	—	13	1	no	—
54	#1312	D₁₃	1	1	yes	characteristic, complemented
55	C₂²×D₁₃	—	8	1	no	—
56	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
57	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
58	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
59	C₂×Dic₁₃	—	8	2	no	—
60	C₂×Dic₁₃	—	8	1	no	—
61	D₅₂	—	8	2	no	—
62	D₅₂	—	8	2	no	—
63	D₅₂	—	8	2	no	—
64	Dic₂₆	—	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	1	no	—
70	#500	—	13	2	no	—
71	#500	—	26	1	no	—
72	#494	—	26	1	no	—
73	#493	—	13	1	no	—
74	#492	—	26	1	no	—
75	#486	—	26	1	no	—
76	#366	—	26	4	no	—
77	#357	—	26	4	no	—
78	C₄×C₁₆	C₂×D₁₃	1	2	yes	—
79	C₈²	C₂×D₁₃	1	1	yes	characteristic
80	C₂²×C₁₃	#493	1	1	yes	characteristic
81	C₂×D₁₃	—	16	1	no	—
82	C₂×D₁₃	—	16	1	no	—
83	C₂×D₁₃	—	16	1	no	—
84	C₄×C₁₃	#497	1	2	yes	—
85	C₄×C₁₃	C₂×D₁₆	1	2	yes	—
86	C₄×C₁₃	#510	1	2	yes	—
87	Dic₁₃	—	16	1	no	—
88	Dic₁₃	—	16	2	no	—
89	C₂×Q₁₆	—	52	2	no	—
90	C₂×SD₁₆	—	52	2	no	—
91	C₂×D₈	—	26	2	no	—
92	C₄⋊Q₈	—	26	1	no	—
93	C₄⋊Q₈	—	26	2	no	—
94	#127	—	26	1	no	—
95	C₂×C₁₆	D₅₂	1	4	yes	—
96	#107	—	26	2	no	—
97	#107	—	26	2	no	—
98	#107	—	26	2	no	—
99	#107	—	52	2	no	—
100	#102	—	52	2	no	—
101	C₄×C₈	D₅₂	1	1	yes	characteristic
102	C₄×C₈	D₅₂	1	1	yes	characteristic
103	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
104	C₂×C₁₃	#1722	1	1	yes	characteristic
105	C₂×C₁₃	#2246	1	1	yes	characteristic
106	C₂×C₁₃	#2251	1	1	yes	characteristic
107	D₁₃	—	32	1	no	—
108	C₂×Q₈	—	52	1	no	—
109	C₂×Q₈	—	52	2	no	—
110	C₂×D₄	—	52	1	no	—
111	C₂×D₄	—	52	1	no	—
112	C₂×D₄	—	52	1	no	—
113	Q₁₆	—	104	4	no	—
114	SD₁₆	—	104	4	no	—
115	D₈	—	52	4	no	—
116	C₂×C₈	D₁₀₄	1	2	yes	—
117	C₂×C₈	D₁₀₄	1	2	yes	—
118	C₂×C₈	C₂×D₅₂	1	2	yes	—
119	C₄⋊C₄	—	52	2	no	—
120	C₄⋊C₄	—	52	1	no	—
121	C₄⋊C₄	—	52	1	no	—
122	C₄⋊C₄	—	52	2	no	—
123	C₄²	C₂×D₅₂	1	1	yes	characteristic
124	C₁₆	—	2	4	no	—
125	C₁₃	#12531	1	1	yes	characteristic, complemented, Sylow
126	C₂³	—	104	1	no	—
127	Q₈	—	104	2	no	—
128	Q₈	—	104	4	no	—
129	D₄	—	104	2	no	—
130	D₄	—	104	2	no	—
131	D₄	—	104	2	no	—
132	C₂×C₄	—	104	1	no	—
133	C₂×C₄	—	104	2	no	—
134	C₂×C₄	#88128	1	1	yes	characteristic
135	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
136	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
137	C₈	—	2	2	no	—
138	C₈	D₂₀₈	1	4	yes	—
139	C₈	#88163	1	4	yes	—
140	C₂²	—	208	1	no	—
141	C₂²	—	208	1	no	—
142	C₂²	—	208	1	no	—
143	C₂²	#11718356	1	1	yes	characteristic
144	C₄	—	208	1	no	—
145	C₄	—	208	2	no	—
146	C₄	#11718376	1	2	yes	—
147	C₄	C₂×D₂₀₈	1	2	yes	—
148	C₄	#11718570	1	2	yes	—
149	C₂	—	416	1	no	—
150	C₂	#13114507	1	1	yes	characteristic
151	C₂	#13117086	1	1	yes	characteristic
152	C₂	#13117107	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13117111

C₂

yes

maximal, complemented

#13117060

C₂

yes

maximal

#13114454

C₂

yes

maximal, characteristic, complemented

#13114428

C₂

yes

maximal, characteristic

C₁₃×#1312

C₂

yes

maximal, characteristic, complemented

#11718360

—

#11718356

C₂²

yes

—

—

—

C₂²

yes

characteristic

#11718343

C₂²

yes

—

C₄×C₁₃×C₁₆

C₂²

yes

—

C₈²×C₁₃

C₂²

yes

characteristic

#11718101

—

#11718096

—

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

—

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

—

C₈×C₁₃

—

C₈×C₁₃

D₁₆

yes

—

C₈×C₁₃

#137

yes

—

—

—

—

—

—

—

—

—

—

C₄×C₁₆

C₂×D₁₃

yes

—

C₈²

C₂×D₁₃

yes

characteristic

C₂²×C₁₃

#493

yes

characteristic

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

#497

yes

—

C₄×C₁₃

C₂×D₁₆

yes

—

C₄×C₁₃

#510

yes

—

Dic₁₃

—

Dic₁₃

—

C₂×Q₁₆

—

C₂×SD₁₆

—

C₂×D₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#127

—

C₂×C₁₆

D₅₂

yes

—

—

—

—

—

100

#102

—

101

C₄×C₈

D₅₂

yes

characteristic

102

C₄×C₈

D₅₂

yes

characteristic

103

C₄×C₈

C₂²×D₁₃

yes

characteristic

104

C₂×C₁₃

#1722

yes

characteristic

105

C₂×C₁₃

#2246

yes

characteristic

106

C₂×C₁₃

#2251

yes

characteristic

107

D₁₃

—

108

C₂×Q₈

—

109

C₂×Q₈

—

110

C₂×D₄

—

111

C₂×D₄

—

112

C₂×D₄

—

113

Q₁₆

—

104

—

114

SD₁₆

—

104

—

115

D₈

—

116

C₂×C₈

D₁₀₄

yes

—

117

C₂×C₈

D₁₀₄

yes

—

118

C₂×C₈

C₂×D₅₂

yes

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄²

C₂×D₅₂

yes

characteristic

124

C₁₆

—

125

C₁₃

#12531

yes

characteristic, complemented, Sylow

126

C₂³

—

104

—

127

Q₈

—

104

—

128

Q₈

—

104

—

129

D₄

—

104

—

130

D₄

—

104

—

131

D₄

—

104

—

132

C₂×C₄

—

104

—

133

C₂×C₄

—

104

—

134

C₂×C₄

#88128

yes

characteristic

135

C₂×C₄

C₂×D₁₀₄

yes

characteristic

136

C₂×C₄

C₂×D₁₀₄

yes

characteristic

137

C₈

—

138

C₈

D₂₀₈

yes

—

139

C₈

#88163

yes

—

140

C₂²

—

208

—

141

C₂²

—

208

—

142

C₂²

—

208

—

143

C₂²

#11718356

yes

characteristic

144

C₄

—

208

—

145

C₄

—

208

—

146

C₄

#11718376

yes

—

147

C₄

C₂×D₂₀₈

yes

—

148

C₄

#11718570

yes

—

149

C₂

—

416

—

150

C₂

#13114507

yes

characteristic

151

C₂

#13117086

yes

characteristic

152

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	#12531
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
1. #13117111	C₂	semidirect
3. #13114454	C₂	semidirect
5. C₁₃×#1312	C₂	semidirect
54. #1312	D₁₃	semidirect
125. C₁₃	#12531	semidirect

H ≅ G/N

type

1. #13117111

C₂

semidirect

3. #13114454

C₂

semidirect

5. C₁₃×#1312

C₂

semidirect

54. #1312

D₁₃

semidirect

125. C₁₃

#12531

semidirect

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

subgroup	index
#13117111	2
#13117060	2
#13114454	2
#13114428	2
C₁₃×#1312	2
#12531	13

subgroup

index

C₁₃×#1312

Subgroups — #15777705

Conjugacy classes of subgroups

Complements

Maximal subgroups