Group #16788699 — subgroups

Subgroups — #16788699

Order 3328.

220 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 96 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	D₁₃⋊#364	C₂	1	2	yes	maximal
2	#13129872	C₂	1	2	yes	maximal, complemented
3	#13129871	C₂	1	2	yes	maximal, complemented
4	#13116429	C₂	1	1	yes	maximal, characteristic
5	D₁₃⋊M₃₂	—	2	2	no	—
6	C₂×#88038	C₄	1	1	yes	characteristic
7	C₂×#88038	C₄	1	1	yes	characteristic
8	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
9	#11718261	C₂²	1	4	yes	—
10	#11718233	—	2	2	no	—
11	C₁₃×#346	C₄	1	1	yes	characteristic
12	C₁₆×Dic₁₃	C₂²	1	1	yes	characteristic
13	C₁₆×Dic₁₃	C₂²	1	1	yes	characteristic
14	#11718053	C₄	1	1	yes	characteristic
15	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
16	C₂×C₈×D₁₃	D₄	1	1	yes	characteristic
17	C₄²×D₁₃	Q₈	1	1	yes	characteristic
18	#88106	—	2	4	no	—
19	#88098	—	4	2	no	—
20	D₁₃⋊C₁₆	—	4	2	no	—
21	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
22	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
23	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₈×Dic₁₃	Q₈	1	1	yes	characteristic
25	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
26	C₂×#6090	C₂×C₄	1	1	yes	characteristic
27	C₂×#6090	C₂×C₄	1	1	yes	characteristic
28	C₄×#1055	D₄	1	1	yes	characteristic
29	#88038	—	2	2	no	—
30	#88038	C₈	1	4	yes	—
31	#12011	—	13	1	no	maximal, Sylow
32	C₂×C₄×D₁₃	—	2	1	no	—
33	C₂×C₄×D₁₃	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
36	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
37	C₄×Dic₁₃	—	2	1	no	—
38	C₄×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
39	C₂×#1055	C₄⋊C₄	1	1	yes	characteristic
40	C₂×#1055	C₂×D₄	1	1	yes	characteristic
41	C₈×D₁₃	C₂×C₈	1	2	yes	—
42	C₈×D₁₃	M₁₆	1	2	yes	—
43	C₈×D₁₃	D₈	1	2	yes	—
44	C₈×D₁₃	Q₁₆	1	2	yes	—
45	#6092	—	4	4	no	—
46	C₁₃×C₁₆	—	2	1	no	—
47	C₁₃×C₁₆	C₂×C₈	1	2	yes	—
48	#6090	—	2	1	no	—
49	#6090	C₂×C₈	1	2	yes	—
50	C₂×#364	—	13	2	no	—
51	C₄×M₃₂	—	13	1	no	—
52	#1373	—	13	2	no	—
53	#1373	—	13	2	no	—
54	C₂²×D₁₃	#108	1	1	yes	characteristic
55	C₂×C₄×C₁₃	—	2	1	no	—
56	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂×Dic₁₃	—	2	1	no	—
58	C₂×Dic₁₃	#108	1	1	yes	characteristic
59	C₄×D₁₃	—	4	2	no	—
60	C₄×D₁₃	C₄⋊C₈	1	2	yes	—
61	C₄×D₁₃	#107	1	2	yes	—
62	C₈×C₁₃	#107	1	2	yes	—
63	C₈×C₁₃	C₂²×C₈	1	1	yes	characteristic
64	C₈×C₁₃	C₂×M₁₆	1	1	yes	characteristic
65	#1055	C₄⋊C₈	1	2	yes	—
66	#1055	C₂×D₈	1	1	yes	characteristic
67	#1055	C₂×Q₁₆	1	1	yes	characteristic
68	C₂×M₃₂	—	13	1	no	—
69	C₂×M₃₂	—	13	1	no	—
70	C₂×M₃₂	—	26	2	no	—
71	C₂×C₄×C₈	—	13	1	no	—
72	#364	—	13	4	no	—
73	#364	—	26	2	no	—
74	#346	—	13	1	no	—
75	#346	#261	1	1	yes	characteristic
76	C₄×C₁₆	—	13	1	no	—
77	C₄×C₁₆	—	13	1	no	—
78	C₂²×C₁₃	C₂×#108	1	1	yes	characteristic
79	C₂×D₁₃	—	4	1	no	—
80	C₂×D₁₃	#335	1	2	yes	—
81	C₄×C₁₃	—	4	1	no	—
82	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
83	C₄×C₁₃	C₂×#107	1	1	yes	characteristic
84	Dic₁₃	—	2	2	no	—
85	Dic₁₃	#335	1	2	yes	—
86	C₂²×C₈	—	13	1	no	—
87	C₂²×C₈	—	13	1	no	—
88	C₂×C₄²	—	13	1	no	—
89	M₃₂	—	13	4	no	—
90	M₃₂	—	26	2	no	—
91	M₃₂	—	26	4	no	—
92	M₃₂	—	52	2	no	—
93	C₂×C₁₆	—	13	1	no	—
94	C₂×C₁₆	—	13	1	no	—
95	C₂×C₁₆	—	52	2	no	—
96	C₂×C₁₆	C₂×#261	1	1	yes	characteristic
97	C₂×C₁₆	C₂×#261	1	1	yes	characteristic
98	C₄×C₈	—	13	1	no	—
99	C₄×C₈	—	13	1	no	—
100	C₄×C₈	—	13	1	no	—
101	C₄×C₈	C₂×#261	1	1	yes	characteristic
102	C₂×C₁₃	—	2	1	no	—
103	C₂×C₁₃	C₂×#335	1	1	yes	characteristic
104	D₁₃	—	2	2	no	—
105	C₂²×C₄	—	13	1	no	—
106	C₂²×C₄	—	26	1	no	—
107	C₂×C₈	—	13	1	no	—
108	C₂×C₈	—	13	1	no	—
109	C₂×C₈	—	13	2	no	—
110	C₂×C₈	—	13	2	no	—
111	C₂×C₈	—	13	2	no	—
112	C₂×C₈	—	13	2	no	—
113	C₂×C₈	C₄⋊#261	1	1	yes	characteristic
114	C₂×C₈	C₂²×#261	1	1	yes	characteristic
115	C₄²	—	13	1	no	—
116	C₄²	—	26	1	no	—
117	C₄²	C₄⋊#261	1	1	yes	characteristic
118	C₁₆	—	2	1	no	—
119	C₁₆	—	13	2	no	—
120	C₁₆	—	26	1	no	—
121	C₁₆	—	52	4	no	—
122	C₁₆	D₁₃⋊C₈	1	2	yes	—
123	C₁₃	#12011	1	1	yes	characteristic, complemented, Sylow
124	C₂³	—	13	1	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	13	1	no	—
127	C₂×C₄	—	13	2	no	—
128	C₂×C₄	—	13	2	no	—
129	C₂×C₄	—	26	1	no	—
130	C₂×C₄	—	52	2	no	—
131	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
132	C₈	—	13	2	no	—
133	C₈	—	13	1	no	—
134	C₈	—	13	1	no	—
135	C₈	#88102	1	2	yes	—
136	C₈	C₂×D₁₃⋊C₈	1	1	yes	characteristic
137	C₈	D₁₃⋊M₁₆	1	1	yes	characteristic
138	C₂²	—	13	2	no	—
139	C₂²	—	52	1	no	—
140	C₂²	D₁₃⋊#108	1	1	yes	characteristic
141	C₄	—	4	1	no	—
142	C₄	—	13	2	no	—
143	C₄	—	26	2	no	—
144	C₄	D₁₃⋊(C₄⋊C₈)	1	1	yes	characteristic
145	C₄	C₂×#88102	1	1	yes	characteristic
146	C₂	—	2	1	no	—
147	C₂	—	26	2	no	—
148	C₂	D₁₃⋊#335	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₁₃⋊#364

C₂

yes

maximal

#13129872

C₂

yes

maximal, complemented

#13129871

C₂

yes

maximal, complemented

#13116429

C₂

yes

maximal, characteristic

D₁₃⋊M₃₂

—

C₂×#88038

C₄

yes

characteristic

C₂×#88038

C₄

yes

characteristic

C₄×C₈×D₁₃

C₂²

yes

characteristic

#11718261

C₂²

yes

—

#11718233

—

C₁₃×#346

C₄

yes

characteristic

C₁₆×Dic₁₃

C₂²

yes

characteristic

C₁₆×Dic₁₃

C₂²

yes

characteristic

#11718053

C₄

yes

characteristic

C₂×C₈×D₁₃

C₂×C₄

yes

characteristic

C₂×C₈×D₁₃

D₄

yes

characteristic

C₄²×D₁₃

Q₈

yes

characteristic

#88106

—

#88098

—

D₁₃⋊C₁₆

—

C₂×C₁₃×C₁₆

C₂×C₄

yes

characteristic

C₂×C₁₃×C₁₆

C₂×C₄

yes

characteristic

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

C₈×Dic₁₃

Q₈

yes

characteristic

C₈×Dic₁₃

C₂³

yes

characteristic

C₂×#6090

C₂×C₄

yes

characteristic

C₂×#6090

C₂×C₄

yes

characteristic

C₄×#1055

D₄

yes

characteristic

#88038

—

#88038

C₈

yes

—

#12011

—

maximal, Sylow

C₂×C₄×D₁₃

—

C₂×C₄×D₁₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₃

C₂²×C₄

yes

characteristic

C₄²×C₁₃

C₄⋊C₄

yes

characteristic

C₄×Dic₁₃

—

C₄×Dic₁₃

C₂×Q₈

yes

characteristic

C₂×#1055

C₄⋊C₄

yes

characteristic

C₂×#1055

C₂×D₄

yes

characteristic

C₈×D₁₃

C₂×C₈

yes

—

C₈×D₁₃

M₁₆

yes

—

C₈×D₁₃

D₈

yes

—

C₈×D₁₃

Q₁₆

yes

—

#6092

—

C₁₃×C₁₆

—

C₁₃×C₁₆

C₂×C₈

yes

—

#6090

—

#6090

C₂×C₈

yes

—

C₂×#364

—

C₄×M₃₂

—

#1373

—

#1373

—

C₂²×D₁₃

#108

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

#108

yes

characteristic

C₄×D₁₃

—

C₄×D₁₃

C₄⋊C₈

yes

—

C₄×D₁₃

#107

yes

—

C₈×C₁₃

#107

yes

—

C₈×C₁₃

C₂²×C₈

yes

characteristic

C₈×C₁₃

C₂×M₁₆

yes

characteristic

#1055

C₄⋊C₈

yes

—

#1055

C₂×D₈

yes

characteristic

#1055

C₂×Q₁₆

yes

characteristic

C₂×M₃₂

—

C₂×M₃₂

—

C₂×M₃₂

—

C₂×C₄×C₈

—

—

—

—

yes

C₄×C₁₆

—

C₄×C₁₆

—

C₂²×C₁₃

C₂×#108

yes

characteristic

C₂×D₁₃

—

C₂×D₁₃

#335

yes

—

C₄×C₁₃

—

C₄×C₁₃

C₂×C₄⋊C₈

yes

characteristic

C₄×C₁₃

C₂×#107

yes

characteristic

Dic₁₃

—

Dic₁₃

#335

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

yes

C₂×C₁₆

yes

C₄×C₈

—

C₄×C₈

—

100

C₄×C₈

—

101

C₄×C₈

C₂×#261

yes

characteristic

102

C₂×C₁₃

—

103

C₂×C₁₃

C₂×#335

yes

characteristic

104

D₁₃

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

C₄⋊#261

yes

characteristic

114

C₂×C₈

C₂²×#261

yes

characteristic

115

C₄²

—

116

C₄²

—

117

C₄²

C₄⋊#261

yes

characteristic

118

C₁₆

—

119

C₁₆

—

120

C₁₆

—

121

C₁₆

—

122

C₁₆

D₁₃⋊C₈

yes

—

123

C₁₃

#12011

yes

characteristic, complemented, Sylow

124

C₂³

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

C₂×C₄⋊#261

yes

characteristic

132

C₈

—

133

C₈

—

134

C₈

—

135

C₈

#88102

yes

—

136

C₈

C₂×D₁₃⋊C₈

yes

characteristic

137

C₈

D₁₃⋊M₁₆

yes

characteristic

138

C₂²

—

139

C₂²

—

140

C₂²

D₁₃⋊#108

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

D₁₃⋊(C₄⋊C₈)

yes

characteristic

145

C₄

C₂×#88102

yes

characteristic

146

C₂

—

147

C₂

—

148

C₂

D₁₃⋊#335

yes

characteristic

	subgroup	G/H
center	C₄	C₂×#88102
commutator	C₈×C₁₃	C₂²×C₈
Frattini	C₂×C₈	C₂²×#261
Fitting	C₁₃×#346	C₄
socle	C₂×C₁₃	C₂×#335
prime radical	C₂²×D₁₃	#108
cosocle	C₈×Dic₁₃	C₂³
hypercenter	#346	#261
hypocenter	C₁₃	#12011
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #346

subgroup

G/H

center

C₄

C₂×#88102

commutator

C₈×C₁₃

C₂²×C₈

Frattini

C₂×C₈

C₂²×#261

Fitting

C₁₃×#346

C₄

C₂×C₁₃

C₂²×D₁₃

C₈×Dic₁₃

C₂³

C₁₃

G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #346

N	H ≅ G/N	type
2. #13129872	C₂	semidirect
3. #13129871	C₂	semidirect
123. C₁₃	#12011	semidirect

H ≅ G/N

type

2. #13129872

C₂

semidirect

3. #13129871

C₂

semidirect

123. C₁₃

#12011

semidirect

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

subgroup	index
D₁₃⋊#364	2
#13129872	2
#13129871	2
#13116429	2
#12011	13

subgroup

index

D₁₃⋊#364

Subgroups — #16788699

Conjugacy classes of subgroups

Complements

Maximal subgroups