Group #16788842 — subgroups

Subgroups — #16788842

Order 3328.

206 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 80 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	#13131252	C₂	1	2	yes	maximal
2	#13129865	C₂	1	1	yes	maximal, characteristic
3	#13129803	C₂	1	2	yes	maximal
4	C₈×#6092	C₂	1	1	yes	maximal, characteristic
5	#13113642	C₂	1	1	yes	maximal, characteristic
6	#11718491	C₂²	1	2	yes	—
7	C₄×Dic₅₂	C₄	1	1	yes	characteristic
8	#11718260	—	2	2	no	—
9	#11718258	—	2	2	no	—
10	C₄⋊#6092	C₂²	1	2	yes	—
11	C₄×#6092	C₂²	1	1	yes	characteristic
12	C₄×#6092	C₂²	1	1	yes	characteristic
13	C₁₃×#335	C₄	1	1	yes	characteristic
14	#11718076	C₄	1	2	yes	—
15	C₈×#1055	C₂²	1	1	yes	characteristic
16	C₂×Dic₅₂	C₈	1	1	yes	characteristic
17	C₄×Dic₂₆	C₂×C₄	1	2	yes	—
18	C₂×#6092	—	2	1	no	—
19	C₂×#6092	—	2	2	no	—
20	C₂×#6092	D₄	1	1	yes	characteristic
21	C₂×#6092	D₄	1	1	yes	characteristic
22	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
23	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
24	#88058	C₈	1	1	yes	characteristic
25	#88056	C₈	1	2	yes	—
26	#88054	—	2	2	no	—
27	C₈×Dic₁₃	—	2	2	no	—
28	C₄×#1055	C₂×C₄	1	1	yes	characteristic
29	C₄×#1055	C₂³	1	1	yes	characteristic
30	#12054	—	13	1	no	maximal, Sylow
31	C₂×Dic₂₆	C₂×C₈	1	2	yes	—
32	C₂×C₈×C₁₃	—	2	2	no	—
33	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
34	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
35	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
36	C₄⋊Dic₁₃	C₂×C₈	1	2	yes	—
37	#6101	—	2	2	no	—
38	C₄×Dic₁₃	—	2	2	no	—
39	C₂×#1055	—	2	1	no	—
40	C₂×#1055	C₂×D₄	1	1	yes	characteristic
41	C₂×#1055	#41	1	1	yes	characteristic
42	Dic₅₂	—	2	2	no	—
43	#6092	—	2	2	no	—
44	#6092	—	2	1	no	—
45	#6092	—	4	2	no	—
46	#6092	Q₁₆	1	2	yes	complemented
47	#2183	—	13	2	no	—
48	C₈×Q₁₆	—	13	1	no	—
49	#1370	—	13	1	no	—
50	Q₈⋊C₁₆	—	13	2	no	—
51	C₈×C₁₆	—	13	1	no	—
52	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
53	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
54	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
55	C₂×Dic₁₃	—	2	2	no	—
56	C₂×Dic₁₃	—	2	2	no	—
57	Dic₂₆	—	2	2	no	—
58	Dic₂₆	—	2	2	no	—
59	C₈×C₁₃	—	2	1	no	—
60	C₈×C₁₃	—	4	2	no	—
61	C₈×C₁₃	M₃₂	1	2	yes	—
62	#1055	—	2	2	no	—
63	#1055	—	2	1	no	—
64	#1055	C₂×Q₁₆	1	1	yes	characteristic
65	#1055	#135	1	1	yes	characteristic
66	C₈×Q₈	—	13	2	no	—
67	C₄×Q₁₆	—	13	1	no	—
68	C₄⋊C₁₆	—	13	2	no	—
69	C₄⋊C₁₆	—	26	2	no	—
70	#346	—	26	2	no	—
71	C₄×C₁₆	—	13	1	no	—
72	C₄×C₁₆	—	13	1	no	—
73	#335	#261	1	1	yes	characteristic
74	Q₈⋊C₈	—	13	2	no	—
75	C₈²	—	13	1	no	—
76	C₂²×C₁₃	C₈×D₄	1	1	yes	characteristic
77	C₄×C₁₃	—	2	1	no	—
78	C₄×C₁₃	C₄×Q₁₆	1	1	yes	characteristic
79	C₄×C₁₃	#443	1	1	yes	characteristic
80	C₄×C₁₃	C₂×M₃₂	1	1	yes	characteristic
81	C₄×C₁₃	#504	1	1	yes	characteristic
82	Dic₁₃	—	4	2	no	—
83	Dic₁₃	—	4	2	no	—
84	C₂×Q₁₆	—	13	1	no	—
85	C₄×Q₈	—	13	2	no	—
86	C₂×C₁₆	—	13	1	no	—
87	C₂×C₁₆	—	13	1	no	—
88	C₂×C₁₆	—	26	1	no	—
89	C₂×C₁₆	—	26	2	no	—
90	#107	—	13	1	no	—
91	C₄⋊C₈	—	26	2	no	—
92	C₄⋊C₈	C₂×#261	1	2	yes	—
93	#103	—	13	2	no	—
94	C₄×C₈	—	13	1	no	—
95	C₄×C₈	—	13	1	no	—
96	C₄×C₈	—	26	2	no	—
97	C₄×C₈	C₂×#261	1	1	yes	characteristic
98	C₂×C₁₃	C₈×Q₁₆	1	1	yes	characteristic
99	C₂×C₁₃	#2169	1	1	yes	characteristic
100	C₂×C₁₃	#2170	1	1	yes	characteristic
101	C₂×Q₈	—	13	2	no	—
102	Q₁₆	—	26	2	no	—
103	C₂×C₈	—	2	2	no	—
104	C₂×C₈	—	13	1	no	—
105	C₂×C₈	—	13	1	no	—
106	C₂×C₈	—	26	1	no	—
107	C₂×C₈	D₁₃⋊C₈	1	1	yes	characteristic
108	C₂×C₈	D₁₃⋊C₈	1	1	yes	characteristic
109	C₄⋊C₄	—	13	2	no	—
110	C₄⋊C₄	—	26	2	no	—
111	C₄²	—	26	2	no	—
112	C₄²	C₂²×#261	1	1	yes	characteristic
113	C₁₆	—	13	2	no	—
114	C₁₆	—	26	1	no	—
115	C₁₆	—	26	2	no	—
116	C₁₆	—	52	2	no	—
117	C₁₃	#12054	1	1	yes	characteristic, complemented, Sylow
118	Q₈	—	26	2	no	—
119	Q₈	—	26	2	no	—
120	C₂×C₄	—	26	2	no	—
121	C₂×C₄	—	26	2	no	—
122	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
123	C₂×C₄	D₄×#261	1	1	yes	characteristic
124	C₂×C₄	#88240	1	1	yes	characteristic
125	C₈	—	2	1	no	—
126	C₈	—	4	2	no	—
127	C₈	—	13	1	no	—
128	C₈	—	13	1	no	—
129	C₈	—	26	2	no	—
130	C₈	—	26	1	no	—
131	C₈	#88098	1	2	yes	—
132	C₂²	#11719073	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	52	2	no	—
135	C₄	—	52	2	no	—
136	C₄	C₂×#88098	1	1	yes	characteristic
137	C₄	#11719096	1	1	yes	characteristic
138	C₄	#11719103	1	1	yes	characteristic
139	C₄	Q₁₆×#261	1	1	yes	characteristic
140	C₂	#13130424	1	1	yes	characteristic
141	C₂	#13131222	1	1	yes	characteristic
142	C₂	#13131225	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13131252

C₂

yes

maximal

#13129865

C₂

yes

maximal, characteristic

#13129803

C₂

yes

maximal

C₈×#6092

C₂

yes

maximal, characteristic

#13113642

C₂

yes

maximal, characteristic

#11718491

C₂²

yes

—

C₄×Dic₅₂

C₄

yes

—

—

C₂²

yes

—

C₄×#6092

C₂²

yes

characteristic

C₄×#6092

C₂²

yes

characteristic

C₁₃×#335

C₄

yes

characteristic

#11718076

C₄

yes

—

C₈×#1055

C₂²

yes

characteristic

C₂×Dic₅₂

C₈

yes

characteristic

C₄×Dic₂₆

C₂×C₄

yes

—

C₂×#6092

—

C₂×#6092

—

C₂×#6092

D₄

yes

characteristic

C₂×#6092

D₄

yes

characteristic

C₁₃×C₄⋊C₈

C₂×C₄

yes

—

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

#88058

C₈

yes

characteristic

#88056

C₈

yes

—

#88054

—

C₈×Dic₁₃

—

C₄×#1055

C₂×C₄

yes

characteristic

C₄×#1055

C₂³

yes

characteristic

#12054

—

maximal, Sylow

C₂×Dic₂₆

C₂×C₈

yes

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₂×C₈

yes

characteristic

C₂×C₈×C₁₃

C₂×C₈

yes

characteristic

C₄²×C₁₃

C₂²×C₄

yes

characteristic

C₄⋊Dic₁₃

C₂×C₈

yes

—

#6101

—

C₄×Dic₁₃

—

C₂×#1055

—

C₂×#1055

C₂×D₄

yes

yes

Dic₅₂

—

—

—

—

Q₁₆

yes

complemented

#2183

—

C₈×Q₁₆

—

#1370

—

Q₈⋊C₁₆

—

C₈×C₁₆

—

C₂×C₄×C₁₃

C₄×D₄

yes

characteristic

C₂×C₄×C₁₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₁₃

#131

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

—

Dic₂₆

—

Dic₂₆

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

M₃₂

yes

—

#1055

—

#1055

—

#1055

C₂×Q₁₆

yes

yes

C₈×Q₈

—

C₄×Q₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

—

#335

#261

yes

characteristic

Q₈⋊C₈

—

C₈²

—

C₂²×C₁₃

C₈×D₄

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

C₄×Q₁₆

yes

characteristic

C₄×C₁₃

#443

yes

characteristic

C₄×C₁₃

C₂×M₃₂

yes

characteristic

C₄×C₁₃

#504

yes

characteristic

Dic₁₃

—

Dic₁₃

—

C₂×Q₁₆

—

C₄×Q₈

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

C₂×#261

yes

—

#103

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×#261

yes

characteristic

C₂×C₁₃

C₈×Q₁₆

yes

characteristic

C₂×C₁₃

#2169

yes

characteristic

100

C₂×C₁₃

#2170

yes

characteristic

101

C₂×Q₈

—

102

Q₁₆

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

D₁₃⋊C₈

yes

characteristic

108

C₂×C₈

D₁₃⋊C₈

yes

characteristic

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄²

—

112

C₄²

C₂²×#261

yes

characteristic

113

C₁₆

—

114

C₁₆

—

115

C₁₆

—

116

C₁₆

—

117

C₁₃

#12054

yes

characteristic, complemented, Sylow

118

Q₈

—

119

Q₈

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

C₂×D₁₃⋊C₈

yes

characteristic

123

C₂×C₄

D₄×#261

yes

characteristic

124

C₂×C₄

#88240

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

—

129

C₈

—

130

C₈

—

131

C₈

#88098

yes

—

132

C₂²

#11719073

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

C₂×#88098

yes

characteristic

137

C₄

#11719096

yes

characteristic

138

C₄

#11719103

yes

characteristic

139

C₄

Q₁₆×#261

yes

characteristic

140

C₂

#13130424

yes

characteristic

141

C₂

#13131222

yes

characteristic

142

C₂

#13131225

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×#261
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₄²	C₂²×#261
Fitting	C₁₃×#335	C₄
socle, prime radical	C₂²×C₁₃	C₈×D₄
cosocle	C₄×#1055	C₂³
hypercenter	#335	#261
hypocenter	C₁₃	#12054
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #335

subgroup

G/H

center

C₂×C₄

D₄×#261

commutator

C₂×C₄×C₁₃

C₂²×C₈

Frattini

C₄²

C₂²×#261

Fitting

C₁₃×#335

C₄

socle, prime radical

C₂²×C₁₃

C₈×D₄

C₂³

C₁₃

G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ #335

N	H ≅ G/N	type
46. #6092	Q₁₆	semidirect
117. C₁₃	#12054	semidirect

H ≅ G/N

type

46. #6092

Q₁₆

semidirect

117. C₁₃

#12054

semidirect

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

subgroup	index
#13131252	2
#13129865	2
#13129803	2
C₈×#6092	2
#13113642	2
#12054	13

subgroup

index

Subgroups — #16788842

Conjugacy classes of subgroups

Complements

Maximal subgroups