Group #15779316 — subgroups

Subgroups — #15779316

Order 3328.

234 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 111 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	#13114960	C₂	1	2	yes	maximal
2	#13112940	C₂	1	1	yes	maximal, characteristic
3	#13112938	C₂	1	1	yes	maximal, characteristic
4	#13112932	C₂	1	2	yes	maximal
5	C₁₃×#2108	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
7	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
8	C₁₃×#405	C₂²	1	1	yes	characteristic
9	C₂×#88059	C₂²	1	2	yes	—
10	#11718767	C₂²	1	2	yes	—
11	C₁₃×#346	C₄	1	2	yes	complemented
12	C₄×C₁₃×C₁₆	C₄	1	2	yes	complemented
13	#11718148	—	2	2	no	—
14	#11718123	—	2	2	no	—
15	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
16	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
17	C₁₃×#117	D₄	1	1	yes	characteristic
18	#88180	—	2	2	no	—
19	C₂×#6099	—	2	2	no	—
20	C₁₃×M₃₂	—	2	2	no	—
21	C₂×C₁₃×C₁₆	—	2	2	no	—
22	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
23	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
24	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
25	C₁₃×#97	Q₈	1	2	yes	—
26	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
27	#88059	—	2	2	no	—
28	#88059	—	2	2	no	—
29	#88058	—	2	2	no	—
30	#88057	—	2	2	no	—
31	#12893	—	13	1	no	maximal, Sylow
32	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
33	C₁₃×M₁₆	—	2	2	no	—
34	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
35	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
37	C₂×C₈×C₁₃	#41	1	2	yes	—
38	C₂×C₈×C₁₃	#41	1	2	yes	—
39	C₁₃×C₄⋊C₄	—	2	1	no	—
40	C₁₃×C₂²⋊C₄	—	2	1	no	—
41	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
42	C₂²⋊Dic₁₃	—	4	2	no	—
43	C₄⋊Dic₁₃	—	4	2	no	—
44	C₄×Dic₁₃	—	4	2	no	—
45	#6099	—	4	2	no	—
46	#6099	—	4	2	no	—
47	C₂×#1055	—	4	2	no	—
48	C₁₃×C₁₆	—	2	1	no	—
49	C₁₃×C₁₆	—	2	1	no	—
50	C₁₃×C₁₆	—	2	2	no	—
51	#2108	D₁₃	1	1	yes	characteristic
52	#1833	—	13	2	no	—
53	#1385	—	13	1	no	—
54	#1384	—	13	1	no	—
55	#1382	—	13	2	no	—
56	C₂³×C₁₃	#135	1	1	yes	characteristic
57	C₂×C₄×C₁₃	—	2	2	no	—
58	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₁₃	#124	1	2	yes	—
60	C₂×C₄×C₁₃	#135	1	2	yes	—
61	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
62	C₂×Dic₁₃	—	8	2	no	—
63	C₈×C₁₃	—	2	2	no	—
64	C₈×C₁₃	#117	1	1	yes	characteristic
65	C₈×C₁₃	#117	1	1	yes	characteristic
66	C₈×C₁₃	#125	1	2	yes	—
67	#1055	—	8	2	no	—
68	C₂×M₃₂	C₂×D₁₃	1	1	yes	characteristic
69	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
70	C₂×#108	—	13	2	no	—
71	#427	—	13	2	no	—
72	#405	C₂×D₁₃	1	1	yes	characteristic
73	#346	Dic₁₃	1	2	yes	complemented
74	C₄×C₁₆	Dic₁₃	1	2	yes	complemented
75	#344	—	26	2	no	—
76	H(3,ℤ/4)	—	26	2	no	—
77	C₂²×C₁₃	—	4	1	no	—
78	C₂²×C₁₃	#427	1	1	yes	characteristic
79	C₂²×C₁₃	#488	1	2	yes	—
80	C₄×C₁₃	—	4	2	no	—
81	C₄×C₁₃	#382	1	1	yes	characteristic
82	C₄×C₁₃	#427	1	1	yes	characteristic
83	C₄×C₁₃	#488	1	2	yes	—
84	Dic₁₃	—	16	2	no	—
85	C₂×M₁₆	—	26	2	no	—
86	C₂×M₁₆	D₅₂	1	1	yes	characteristic
87	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
88	#117	—	26	2	no	—
89	#117	D₅₂	1	1	yes	characteristic
90	M₃₂	—	2	2	no	—
91	C₂×C₁₆	—	2	2	no	—
92	C₂×C₁₆	C₂×Dic₁₃	1	1	yes	characteristic
93	C₂×C₁₆	C₂×Dic₁₃	1	2	yes	—
94	C₂×C₁₆	C₂×Dic₁₃	1	1	yes	characteristic
95	#108	—	26	2	no	—
96	#108	—	26	2	no	—
97	#107	—	26	2	no	—
98	#106	—	26	2	no	—
99	#97	Dic₂₆	1	2	yes	—
100	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
101	C₂×C₁₃	—	2	2	no	—
102	C₂×C₁₃	—	2	1	no	—
103	C₂×C₁₃	#1835	1	1	yes	characteristic
104	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
105	M₁₆	—	2	2	no	—
106	M₁₆	—	52	2	no	—
107	M₁₆	—	52	2	no	—
108	C₂×C₈	—	52	2	no	—
109	C₂×C₈	C₄⋊Dic₁₃	1	2	yes	—
110	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
111	C₂×C₈	#6127	1	2	yes	—
112	C₂×C₈	#6127	1	2	yes	—
113	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
114	C₄⋊C₄	—	2	1	no	—
115	C₄⋊C₄	—	52	2	no	—
116	C₂²⋊C₄	—	2	1	no	—
117	C₂²⋊C₄	—	52	2	no	—
118	C₄²	—	52	2	no	—
119	C₄²	C₄⋊Dic₁₃	1	2	yes	—
120	C₁₆	—	2	1	no	—
121	C₁₆	—	2	1	no	—
122	C₁₆	—	2	2	no	—
123	C₁₃	#12893	1	1	yes	characteristic, complemented, Sylow
124	C₂³	#88158	1	1	yes	characteristic
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	104	2	no	—
127	C₂×C₄	#88129	1	2	yes	—
128	C₂×C₄	#88158	1	2	yes	—
129	C₂×C₄	#88158	1	1	yes	characteristic
130	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
131	C₈	—	2	2	no	—
132	C₈	—	104	2	no	—
133	C₈	#88124	1	2	yes	—
134	C₈	#88180	1	1	yes	characteristic
135	C₈	#88180	1	1	yes	characteristic
136	C₂²	—	4	1	no	—
137	C₂²	#11718358	1	2	yes	—
138	C₂²	#11718767	1	1	yes	characteristic
139	C₄	—	4	2	no	—
140	C₄	—	208	2	no	—
141	C₄	#11718358	1	2	yes	—
142	C₄	#11718597	1	1	yes	characteristic
143	C₄	#11718767	1	1	yes	characteristic
144	C₂	—	2	1	no	—
145	C₂	—	2	2	no	—
146	C₂	#13114967	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13114960

C₂

yes

maximal

#13112940

C₂

yes

maximal, characteristic

#13112938

C₂

yes

maximal, characteristic

#13112932

C₂

yes

maximal

C₁₃×#2108

C₂

yes

maximal, characteristic

C₂×C₁₃×M₃₂

C₂²

yes

characteristic

C₂²×C₁₃×C₁₆

C₂²

yes

characteristic

C₁₃×#405

C₂²

yes

characteristic

C₂×#88059

C₂²

yes

—

#11718767

C₂²

yes

—

C₁₃×#346

C₄

yes

complemented

C₄×C₁₃×C₁₆

C₄

yes

complemented

#11718148

—

#11718123

—

C₂×C₁₃×M₁₆

D₄

yes

characteristic

C₂²×C₈×C₁₃

C₂³

yes

characteristic

C₁₃×#117

D₄

yes

characteristic

#88180

—

C₂×#6099

—

C₁₃×M₃₂

—

C₂×C₁₃×C₁₆

—

C₂×C₁₃×C₁₆

C₂×C₄

yes

characteristic

C₂×C₁₃×C₁₆

C₂×C₄

yes

—

C₂×C₁₃×C₁₆

C₂×C₄

yes

characteristic

C₁₃×#97

Q₈

yes

—

C₄×C₈×C₁₃

C₂×C₄

yes

—

—

—

—

—

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₁₃×M₁₆

—

C₂×C₈×C₁₃

C₄⋊C₄

yes

—

C₂×C₈×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₁₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₁₃

#41

yes

—

C₂×C₈×C₁₃

#41

yes

—

C₁₃×C₄⋊C₄

—

C₁₃×C₂²⋊C₄

—

C₄²×C₁₃

C₄⋊C₄

yes

—

C₂²⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄×Dic₁₃

—

#6099

—

#6099

—

C₂×#1055

—

C₁₃×C₁₆

—

C₁₃×C₁₆

—

C₁₃×C₁₆

—

#2108

D₁₃

yes

—

—

—

—

C₂³×C₁₃

#135

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃

#124

yes

—

C₂×C₄×C₁₃

#135

yes

—

C₂×C₄×C₁₃

#135

yes

characteristic

C₂×Dic₁₃

—

C₈×C₁₃

—

C₈×C₁₃

yes

C₈×C₁₃

yes

C₈×C₁₃

yes

—

#1055

—

C₂×M₃₂

C₂×D₁₃

yes

characteristic

C₂²×C₁₆

C₂×D₁₃

yes

—

—

C₂×D₁₃

yes

characteristic

#346

Dic₁₃

yes

complemented

C₄×C₁₆

Dic₁₃

yes

complemented

#344

—

H(3,ℤ/4)

—

C₂²×C₁₃

—

C₂²×C₁₃

#427

yes

characteristic

C₂²×C₁₃

#488

yes

—

C₄×C₁₃

—

C₄×C₁₃

yes

C₄×C₁₃

yes

C₄×C₁₃

yes

—

Dic₁₃

—

C₂×M₁₆

—

C₂×M₁₆

D₅₂

yes

characteristic

C₂²×C₈

C₂²×D₁₃

yes

characteristic

#117

—

#117

D₅₂

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₂×Dic₁₃

yes

characteristic

C₂×C₁₆

C₂×Dic₁₃

yes

—

C₂×C₁₆

C₂×Dic₁₃

yes

—

—

—

—

Dic₂₆

yes

—

100

C₄×C₈

C₂×Dic₁₃

yes

—

101

C₂×C₁₃

—

102

C₂×C₁₃

—

103

C₂×C₁₃

#1835

yes

characteristic

104

C₂²×C₄

C₂×D₅₂

yes

characteristic

105

M₁₆

—

106

M₁₆

—

107

M₁₆

—

108

C₂×C₈

—

109

C₂×C₈

C₄⋊Dic₁₃

yes

—

110

C₂×C₈

C₂×Dic₂₆

yes

characteristic

111

C₂×C₈

#6127

yes

—

112

C₂×C₈

#6127

yes

—

113

C₂×C₈

C₂²×Dic₁₃

yes

characteristic

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₂²⋊C₄

—

117

C₂²⋊C₄

—

118

C₄²

—

119

C₄²

C₄⋊Dic₁₃

yes

—

120

C₁₆

—

121

C₁₆

—

122

C₁₆

—

123

C₁₃

#12893

yes

characteristic, complemented, Sylow

124

C₂³

#88158

yes

characteristic

125

C₂×C₄

—

126

C₂×C₄

—

104

—

127

C₂×C₄

#88129

yes

—

128

C₂×C₄

#88158

yes

—

129

C₂×C₄

#88158

yes

characteristic

130

C₂×C₄

C₂×C₄⋊Dic₁₃

yes

characteristic

131

C₈

—

132

C₈

—

104

—

133

C₈

#88124

yes

—

134

C₈

#88180

yes

characteristic

135

C₈

#88180

yes

characteristic

136

C₂²

—

137

C₂²

#11718358

yes

—

138

C₂²

#11718767

yes

characteristic

139

C₄

—

140

C₄

—

208

—

141

C₄

#11718358

yes

—

142

C₄

#11718597

yes

characteristic

143

C₄

#11718767

yes

characteristic

144

C₂

—

145

C₂

—

146

C₂

#13114967

yes

characteristic

	subgroup	G/H
center	C₄	#11718767
commutator	C₂×C₈×C₁₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₁₃×#2108	C₂
socle	C₂×C₁₃	#1835
prime radical	C₂³×C₁₃	#135
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	#2108	D₁₃
hypocenter	C₁₃	#12893
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ #117 ◁ #405 ◁ #2108

subgroup

G/H

center

C₄

#11718767

commutator

C₂×C₈×C₁₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₁₃

Fitting

C₁₃×#2108

C₂

C₂×C₁₃

C₂³×C₁₃

C₂²×C₈×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₄ ◁ #117 ◁ #405 ◁ #2108

N	H ≅ G/N	type
11. C₁₃×#346	C₄	semidirect
12. C₄×C₁₃×C₁₆	C₄	semidirect
73. #346	Dic₁₃	semidirect
74. C₄×C₁₆	Dic₁₃	semidirect
123. C₁₃	#12893	semidirect

H ≅ G/N

type

11. C₁₃×#346

C₄

semidirect

12. C₄×C₁₃×C₁₆

C₄

semidirect

73. #346

Dic₁₃

semidirect

74. C₄×C₁₆

Dic₁₃

semidirect

123. C₁₃

#12893

semidirect

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

subgroup	index
#13114960	2
#13112940	2
#13112938	2
#13112932	2
C₁₃×#2108	2
#12893	13

subgroup

index

C₁₃×#2108

Subgroups — #15779316

Conjugacy classes of subgroups

Complements

Maximal subgroups