Group #16018421 — subgroups

Subgroups — #16018421

Order 3328.

482 conjugacy classes of subgroups; only the 219 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 219 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	#13125350	C₂	1	2	yes	maximal
2	#13117122	C₂	1	8	yes	maximal, complemented
3	#13116620	C₂	1	4	yes	maximal
4	C₂×C₁₃×#346	C₂	1	1	yes	maximal, characteristic
5	#11719175	—	2	2	no	—
6	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
7	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	#11718767	—	2	2	no	—
9	#11718767	C₂²	1	4	yes	—
10	#11718344	—	2	2	no	—
11	#11718344	C₂²	1	4	yes	—
12	#11718343	C₂²	1	4	yes	—
13	#11718341	—	2	2	no	—
14	C₁₃×#346	C₂²	1	4	yes	—
15	#11718097	C₂²	1	8	yes	—
16	#11718096	C₂²	1	8	yes	—
17	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
19	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
20	#88180	—	2	4	no	—
21	#88178	—	4	4	no	—
22	#88124	—	2	4	no	—
23	#88123	—	2	4	no	—
24	C₄×Dic₂₆	—	4	4	no	—
25	C₂×C₁₃×C₁₆	Q₈	1	8	yes	complemented
26	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
27	C₄×C₈×C₁₃	D₄	1	2	yes	—
28	C₄×C₈×C₁₃	C₂³	1	2	yes	—
29	#88058	—	2	4	no	—
30	#88058	C₂³	1	8	yes	—
31	#88057	—	2	4	no	—
32	#88057	—	2	4	no	—
33	#33728	—	13	1	no	maximal, Sylow
34	C₂²×C₄×C₁₃	D₈	1	2	yes	—
35	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
36	C₂×Dic₂₆	—	4	4	no	—
37	C₂×C₈×C₁₃	Q₁₆	1	4	yes	—
38	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
40	C₂×C₈×C₁₃	C₂×Q₈	1	4	yes	—
41	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
42	C₄²×C₁₃	D₈	1	2	yes	—
43	C₄²×C₁₃	C₂×D₄	1	2	yes	—
44	C₂²⋊Dic₁₃	—	4	4	no	—
45	C₄⋊Dic₁₃	—	2	8	no	—
46	C₄⋊Dic₁₃	—	4	4	no	—
47	#6101	—	4	8	no	—
48	C₄×Dic₁₃	—	4	4	no	—
49	C₁₃×C₁₆	C₂×Q₈	1	8	yes	—
50	#3160	—	13	2	no	—
51	#2255	—	13	8	no	—
52	#2158	—	13	4	no	—
53	C₂×#346	D₁₃	1	1	yes	characteristic
54	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
55	C₂×C₄×C₁₃	—	4	2	no	—
56	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
57	C₂×C₄×C₁₃	C₂×D₈	1	4	yes	—
58	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
59	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
60	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
61	C₂×Dic₁₃	—	4	8	no	—
62	Dic₂₆	—	8	8	no	—
63	C₈×C₁₃	C₄⋊Q₈	1	2	yes	—
64	C₈×C₁₃	C₂×Q₁₆	1	4	yes	—
65	C₈×C₁₃	C₂²×Q₈	1	2	yes	—
66	#533	—	26	2	no	—
67	C₂²×C₁₆	C₂×D₁₃	1	2	yes	—
68	#500	—	13	4	no	—
69	#499	—	13	4	no	—
70	#499	—	26	2	no	—
71	#498	—	26	2	no	—
72	#427	—	13	4	no	—
73	#427	—	26	2	no	—
74	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
75	#367	—	13	8	no	—
76	#366	—	13	8	no	—
77	#346	C₂×D₁₃	1	4	yes	—
78	C₂²×C₁₃	—	2	2	no	—
79	C₂²×C₁₃	#500	1	2	yes	—
80	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
81	C₄×C₁₃	—	2	4	no	—
82	C₄×C₁₃	#500	1	2	yes	—
83	C₄×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
84	C₄×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
85	Dic₁₃	—	8	8	no	—
86	C₂²×C₈	D₅₂	1	1	yes	characteristic
87	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
88	C₄⋊Q₈	—	26	4	no	—
89	#125	—	26	4	no	—
90	C₂²⋊Q₈	—	52	4	no	—
91	C₄×Q₈	—	52	4	no	—
92	#117	—	26	4	no	—
93	C₂×C₄²	D₅₂	1	1	yes	characteristic
94	C₂×C₁₆	Dic₂₆	1	8	yes	complemented
95	C₂×C₁₆	C₂²×D₁₃	1	4	yes	—
96	#107	—	13	8	no	—
97	#107	—	26	4	no	—
98	#106	—	26	4	no	—
99	#106	—	26	4	no	—
100	C₄×C₈	D₅₂	1	2	yes	—
101	C₄×C₈	C₂²×D₁₃	1	2	yes	—
102	C₂×C₁₃	—	2	2	no	—
103	C₂×C₁₃	C₂×#500	1	1	yes	characteristic
104	C₂×C₁₃	#3414	1	2	yes	—
105	C₂×Q₈	—	52	4	no	—
106	C₂²×C₄	D₁₀₄	1	2	yes	—
107	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
108	C₂×C₈	Dic₅₂	1	4	yes	—
109	C₂×C₈	C₂×Dic₂₆	1	4	yes	—
110	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
111	C₂×C₈	C₂×D₅₂	1	2	yes	—
112	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
113	C₄⋊C₄	—	26	8	no	—
114	C₄⋊C₄	—	52	8	no	—
115	C₄⋊C₄	—	52	4	no	—
116	C₂²⋊C₄	—	52	4	no	—
117	C₄²	—	52	4	no	—
118	C₄²	D₁₀₄	1	2	yes	—
119	C₄²	C₂×D₅₂	1	2	yes	—
120	C₁₆	C₂×Dic₂₆	1	8	yes	—
121	C₁₃	#33728	1	1	yes	characteristic, complemented, Sylow
122	C₂³	C₂×D₁₀₄	1	1	yes	characteristic
123	Q₈	—	104	8	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	—	52	8	no	—
126	C₂×C₄	#88123	1	2	yes	—
127	C₂×C₄	C₂×D₁₀₄	1	4	yes	—
128	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
129	C₂×C₄	C₂×Dic₅₂	1	2	yes	—
130	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
131	C₈	#88123	1	2	yes	—
132	C₈	C₂×Dic₅₂	1	4	yes	—
133	C₈	C₂²×Dic₂₆	1	2	yes	—
134	C₂²	—	2	2	no	—
135	C₂²	#11718343	1	2	yes	—
136	C₂²	C₂²×D₁₀₄	1	1	yes	characteristic
137	C₄	—	2	4	no	—
138	C₄	—	104	8	no	—
139	C₄	#11718343	1	2	yes	—
140	C₄	C₂×#88123	1	1	yes	characteristic
141	C₄	C₂²×Dic₅₂	1	1	yes	characteristic
142	C₂	—	2	2	no	—
143	C₂	C₂×#11718343	1	1	yes	characteristic
144	C₂	#13127972	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13125350

C₂

yes

maximal

#13117122

C₂

yes

maximal, complemented

#13116620

C₂

yes

maximal

C₂×C₁₃×#346

C₂

yes

maximal, characteristic

#11719175

—

C₂²×C₁₃×C₁₆

C₂²

yes

—

C₂×C₄×C₈×C₁₃

C₂²

yes

characteristic

#11718767

—

#11718767

C₂²

yes

—

#11718344

—

#11718344

C₂²

yes

—

#11718343

C₂²

yes

—

#11718341

—

C₁₃×#346

C₂²

yes

—

#11718097

C₂²

yes

—

#11718096

C₂²

yes

—

C₂²×C₈×C₁₃

D₄

yes

characteristic

C₂²×C₈×C₁₃

C₂³

yes

characteristic

C₂×C₄²×C₁₃

D₄

yes

—

—

—

—

C₄×Dic₂₆

—

C₂×C₁₃×C₁₆

Q₈

yes

complemented

C₂×C₁₃×C₁₆

C₂³

yes

—

C₄×C₈×C₁₃

D₄

yes

—

C₄×C₈×C₁₃

C₂³

yes

—

#88058

—

#88058

C₂³

yes

—

—

—

—

C₂²×C₄×C₁₃

D₈

yes

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂×Dic₂₆

—

C₂×C₈×C₁₃

Q₁₆

yes

—

C₂×C₈×C₁₃

C₂×D₄

yes

characteristic

C₂×C₈×C₁₃

C₂×D₄

yes

—

C₂×C₈×C₁₃

C₂×Q₈

yes

—

C₂×C₈×C₁₃

C₂⁴

yes

characteristic

C₄²×C₁₃

D₈

yes

—

C₄²×C₁₃

C₂×D₄

yes

—

C₂²⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

#6101

—

C₄×Dic₁₃

—

C₁₃×C₁₆

C₂×Q₈

yes

—

—

—

—

D₁₃

yes

characteristic

C₂³×C₁₃

C₂×D₈

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₄⋊Q₈

yes

—

C₂×C₄×C₁₃

C₂×D₈

yes

—

C₂×C₄×C₁₃

C₂×D₈

yes

characteristic

C₂×C₄×C₁₃

C₂×Q₁₆

yes

—

C₂×C₄×C₁₃

C₂²×D₄

yes

characteristic

C₂×Dic₁₃

—

Dic₂₆

—

C₈×C₁₃

C₄⋊Q₈

yes

—

C₈×C₁₃

C₂×Q₁₆

yes

—

C₈×C₁₃

C₂²×Q₈

yes

—

#533

—

C₂²×C₁₆

C₂×D₁₃

yes

—

—

—

—

—

—

—

C₂×C₄×C₈

C₂×D₁₃

yes

—

—

C₂×D₁₃

yes

—

C₂²×C₁₃

—

C₂²×C₁₃

#500

yes

—

C₂²×C₁₃

C₂²×D₈

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

#500

yes

—

C₄×C₁₃

C₂×C₄⋊Q₈

yes

characteristic

C₄×C₁₃

C₂²×Q₁₆

yes

characteristic

Dic₁₃

—

C₂²×C₈

D₅₂

yes

characteristic

C₂²×C₈

C₂²×D₁₃

yes

characteristic

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

D₅₂

yes

characteristic

C₂×C₁₆

Dic₂₆

yes

complemented

C₂×C₁₆

C₂²×D₁₃

yes

—

—

—

—

—

100

C₄×C₈

D₅₂

yes

—

101

C₄×C₈

C₂²×D₁₃

yes

—

102

C₂×C₁₃

—

103

C₂×C₁₃

C₂×#500

yes

characteristic

104

C₂×C₁₃

#3414

yes

—

105

C₂×Q₈

—

106

C₂²×C₄

D₁₀₄

yes

—

107

C₂²×C₄

C₂×D₅₂

yes

characteristic

108

C₂×C₈

Dic₅₂

yes

—

109

C₂×C₈

C₂×Dic₂₆

yes

—

110

C₂×C₈

C₂×D₅₂

yes

characteristic

111

C₂×C₈

C₂×D₅₂

yes

—

112

C₂×C₈

C₂³×D₁₃

yes

characteristic

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₂²⋊C₄

—

117

C₄²

—

118

C₄²

D₁₀₄

yes

—

119

C₄²

C₂×D₅₂

yes

—

120

C₁₆

C₂×Dic₂₆

yes

—

121

C₁₃

#33728

yes

characteristic, complemented, Sylow

122

C₂³

C₂×D₁₀₄

yes

characteristic

123

Q₈

—

104

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

#88123

yes

—

127

C₂×C₄

C₂×D₁₀₄

yes

—

128

C₂×C₄

C₂×D₁₀₄

yes

characteristic

129

C₂×C₄

C₂×Dic₅₂

yes

—

130

C₂×C₄

C₂²×D₅₂

yes

characteristic

131

C₈

#88123

yes

—

132

C₈

C₂×Dic₅₂

yes

—

133

C₈

C₂²×Dic₂₆

yes

—

134

C₂²

—

135

C₂²

#11718343

yes

—

136

C₂²

C₂²×D₁₀₄

yes

characteristic

137

C₄

—

138

C₄

—

104

—

139

C₄

#11718343

yes

—

140

C₄

C₂×#88123

yes

characteristic

141

C₄

C₂²×Dic₅₂

yes

characteristic

142

C₂

—

143

C₂

C₂×#11718343

yes

characteristic

144

C₂

#13127972

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₁₀₄
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂×C₁₃×#346	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂³×C₁₃	C₂×D₈
hypercenter	C₂×#346	D₁₃
hypocenter	C₁₃	#33728
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

subgroup

G/H

center

C₂×C₄

C₂×D₁₀₄

commutator, cosocle

C₂×C₈×C₁₃

C₂⁴

Frattini

C₂×C₈

C₂³×D₁₃

Fitting

C₂×C₁₃×#346

C₂

socle

C₂²×C₁₃

C₂²×D₈

prime radical

C₂³×C₁₃

C₂×D₈

D₁₃

C₁₃

G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

N	H ≅ G/N	type
2. #13117122	C₂	semidirect
25. C₂×C₁₃×C₁₆	Q₈	semidirect
94. C₂×C₁₆	Dic₂₆	semidirect
121. C₁₃	#33728	semidirect

H ≅ G/N

type

2. #13117122

C₂

semidirect

25. C₂×C₁₃×C₁₆

Q₈

semidirect

94. C₂×C₁₆

Dic₂₆

semidirect

121. C₁₃

#33728

semidirect

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

subgroup	index
#13125350	2
#13117122	2
#13116620	2
C₂×C₁₃×#346	2
#33728	13

subgroup

index

C₂×C₁₃×#346

Subgroups — #16018421

Conjugacy classes of subgroups

Complements

Maximal subgroups