Group #16789008 — subgroups

Subgroups — #16789008

Order 3328.

226 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 62 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	#13130486	C₂	1	2	yes	maximal, complemented
2	#13129861	C₂	1	1	yes	maximal, characteristic, complemented
3	#13129794	C₂	1	2	yes	maximal, complemented
4	#13129753	C₂	1	1	yes	maximal, characteristic, complemented
5	#13116712	C₂	1	1	yes	maximal, characteristic
6	#11719145	—	2	2	no	—
7	C₂×C₁₃×D₁₆	C₄	1	1	yes	characteristic
8	D₈×Dic₁₃	C₂²	1	2	yes	—
9	D₄⋊#1057	—	2	2	no	—
10	#11718258	C₂²	1	1	yes	characteristic
11	#11718253	C₂²	1	2	yes	—
12	#11718251	C₂²	1	1	yes	characteristic
13	#11718154	C₄	1	2	yes	—
14	#11718097	C₄	1	1	yes	characteristic
15	#11718094	C₂²	1	1	yes	characteristic
16	C₂×#6122	—	4	2	no	—
17	C₂×C₁₃×D₈	C₂×C₄	1	2	yes	—
18	D₄×Dic₁₃	—	2	2	no	—
19	C₂²⋊#1057	—	4	2	no	—
20	C₄⋊#1057	—	2	2	no	—
21	C₄×#1057	D₄	1	2	yes	—
22	C₂×#6092	—	2	1	no	—
23	C₁₃×D₁₆	—	2	2	no	—
24	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
25	#88072	—	2	2	no	—
26	#88058	C₂×C₄	1	2	yes	—
27	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
28	C₂×#6090	C₂×C₄	1	1	yes	characteristic
29	#12089	—	13	1	no	maximal, Sylow
30	C₂×D₄×C₁₃	—	2	2	no	—
31	C₂²×Dic₁₃	—	4	2	no	—
32	#6122	—	8	4	no	—
33	C₂×#1057	—	2	2	no	—
34	C₂×#1057	—	4	2	no	—
35	C₁₃×D₈	—	2	2	no	—
36	C₁₃×D₈	—	2	2	no	—
37	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
38	C₂²⋊Dic₁₃	—	4	2	no	—
39	C₄⋊Dic₁₃	—	2	2	no	—
40	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
41	C₂×#1055	#41	1	1	yes	characteristic
42	#6092	—	4	1	no	—
43	C₁₃×C₁₆	M₁₆	1	2	yes	complemented
44	#6090	—	2	1	no	—
45	#2177	—	13	1	no	—
46	#1594	—	13	2	no	—
47	#1368	—	13	1	no	—
48	#1333	—	13	2	no	—
49	#1313	—	13	1	no	—
50	C₂³×C₁₃	—	4	2	no	—
51	D₄×C₁₃	—	4	2	no	—
52	D₄×C₁₃	—	4	2	no	—
53	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
54	C₂×Dic₁₃	—	4	2	no	—
55	C₂×Dic₁₃	—	8	2	no	—
56	C₂×Dic₁₃	—	8	2	no	—
57	C₂×Dic₁₃	AGL(1,ℤ/8)	1	2	yes	—
58	#1057	—	4	2	no	—
59	#1057	—	8	2	no	—
60	C₈×C₁₃	C₂×M₁₆	1	1	yes	characteristic
61	C₈×C₁₃	#131	1	1	yes	characteristic
62	#1055	—	2	1	no	—
63	C₂×D₁₆	#261	1	1	yes	characteristic
64	C₄×D₈	—	13	2	no	—
65	#436	—	26	2	no	—
66	#367	—	13	1	no	—
67	#357	—	13	2	no	—
68	#346	—	13	1	no	—
69	#346	—	13	1	no	—
70	#334	—	13	2	no	—
71	D₄⋊C₈	—	26	2	no	—
72	#322	—	13	1	no	—
73	C₂²×C₁₃	—	8	2	no	—
74	C₂²×C₁₃	—	8	2	no	—
75	C₂²×C₁₃	—	8	2	no	—
76	C₂²×C₁₃	#442	1	1	yes	characteristic
77	C₄×C₁₃	#436	1	1	yes	characteristic
78	C₄×C₁₃	#444	1	1	yes	characteristic
79	Dic₁₃	—	2	2	no	—
80	Dic₁₃	—	8	2	no	—
81	C₂×D₈	C₂×#261	1	2	yes	—
82	C₂×M₁₆	—	52	2	no	—
83	C₄×D₄	—	26	2	no	—
84	D₁₆	—	2	2	no	—
85	C₂×C₁₆	—	13	1	no	—
86	C₂×C₁₆	—	26	1	no	—
87	C₂×C₁₆	C₂×#261	1	1	yes	characteristic
88	#107	—	13	2	no	—
89	C₄⋊C₈	—	26	2	no	—
90	#102	—	26	2	no	—
91	C₂²⋊C₈	—	52	2	no	—
92	C₄×C₈	—	13	1	no	—
93	C₄×C₈	—	13	2	no	—
94	C₂×C₁₃	—	16	2	no	—
95	C₂×C₁₃	#1594	1	1	yes	characteristic
96	C₂×C₁₃	AGL(1,ℤ/16)	1	2	yes	—
97	C₂×D₄	—	2	2	no	—
98	C₂²×C₄	—	52	2	no	—
99	D₈	—	2	2	no	—
100	D₈	—	2	2	no	—
101	M₁₆	—	104	4	no	—
102	C₂×C₈	—	13	1	no	—
103	C₂×C₈	—	26	2	no	—
104	C₂×C₈	—	52	2	no	—
105	C₂×C₈	C₂²×#261	1	1	yes	characteristic
106	C₄⋊C₄	—	26	2	no	—
107	C₂²⋊C₄	—	52	2	no	—
108	C₄²	—	13	1	no	—
109	C₁₆	—	26	1	no	—
110	C₁₆	—	52	1	no	—
111	C₁₆	#6122	1	2	yes	complemented
112	C₁₃	#12089	1	1	yes	characteristic, complemented, Sylow
113	C₂³	—	4	2	no	—
114	D₄	—	4	2	no	—
115	D₄	—	4	2	no	—
116	C₂×C₄	—	13	2	no	—
117	C₂×C₄	—	52	2	no	—
118	C₂×C₄	—	104	2	no	—
119	C₂×C₄	—	104	2	no	—
120	C₂×C₄	D₄×#261	1	1	yes	characteristic
121	C₈	—	26	1	no	—
122	C₈	—	52	2	no	—
123	C₈	—	104	2	no	—
124	C₈	#88238	1	1	yes	characteristic
125	C₈	C₂×#6122	1	1	yes	characteristic
126	C₂²	—	8	2	no	—
127	C₂²	—	8	2	no	—
128	C₂²	—	8	2	no	—
129	C₂²	#11719102	1	1	yes	characteristic
130	C₄	—	26	2	no	—
131	C₄	—	104	2	no	—
132	C₄	#11719104	1	1	yes	characteristic
133	C₄	#11719145	1	1	yes	characteristic
134	C₂	—	16	2	no	—
135	C₂	#13130486	1	1	yes	characteristic
136	C₂	#13131246	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13130486

C₂

yes

maximal, complemented

#13129861

C₂

yes

maximal, characteristic, complemented

#13129794

C₂

yes

maximal, complemented

#13129753

C₂

yes

maximal, characteristic, complemented

#13116712

C₂

yes

maximal, characteristic

#11719145

—

C₂×C₁₃×D₁₆

C₄

yes

characteristic

D₈×Dic₁₃

C₂²

yes

—

D₄⋊#1057

—

#11718258

C₂²

yes

characteristic

#11718253

C₂²

yes

—

#11718251

C₂²

yes

characteristic

#11718154

C₄

yes

—

#11718097

C₄

yes

characteristic

#11718094

C₂²

yes

characteristic

C₂×#6122

—

C₂×C₁₃×D₈

C₂×C₄

yes

—

D₄×Dic₁₃

—

C₂²⋊#1057

—

C₄⋊#1057

—

C₄×#1057

D₄

yes

—

C₂×#6092

—

C₁₃×D₁₆

—

C₂×C₁₃×C₁₆

C₂×C₄

yes

characteristic

#88072

—

#88058

C₂×C₄

yes

—

C₈×Dic₁₃

C₂³

yes

characteristic

C₂×#6090

C₂×C₄

yes

characteristic

#12089

—

maximal, Sylow

C₂×D₄×C₁₃

—

C₂²×Dic₁₃

—

#6122

—

C₂×#1057

—

C₂×#1057

—

C₁₃×D₈

—

C₁₃×D₈

—

C₂×C₈×C₁₃

C₂²×C₄

yes

characteristic

C₂²⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄×Dic₁₃

C₂×D₄

yes

yes

—

C₁₃×C₁₆

M₁₆

yes

—

—

—

—

—

—

C₂³×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

C₂×C₄×C₁₃

C₄×D₄

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

AGL(1,ℤ/8)

yes

—

#1057

—

#1057

—

C₈×C₁₃

C₂×M₁₆

yes

C₈×C₁₃

yes

—

C₂×D₁₆

#261

yes

characteristic

C₄×D₈

—

—

—

—

—

—

—

D₄⋊C₈

—

#322

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

yes

C₄×C₁₃

yes

C₄×C₁₃

yes

Dic₁₃

—

Dic₁₃

—

C₂×D₈

C₂×#261

yes

—

C₂×M₁₆

—

C₄×D₄

—

D₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×#261

yes

characteristic

#107

—

C₄⋊C₈

—

#102

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₂×C₁₃

—

C₂×C₁₃

#1594

yes

characteristic

C₂×C₁₃

AGL(1,ℤ/16)

yes

—

C₂×D₄

—

C₂²×C₄

—

D₈

—

100

D₈

—

101

M₁₆

—

104

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

C₂²×#261

yes

characteristic

106

C₄⋊C₄

—

107

C₂²⋊C₄

—

108

C₄²

—

109

C₁₆

—

110

C₁₆

—

111

C₁₆

#6122

yes

complemented

112

C₁₃

#12089

yes

characteristic, complemented, Sylow

113

C₂³

—

114

D₄

—

115

D₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

104

—

119

C₂×C₄

—

104

—

120

C₂×C₄

D₄×#261

yes

characteristic

121

C₈

—

122

C₈

—

123

C₈

—

104

—

124

C₈

#88238

yes

characteristic

125

C₈

C₂×#6122

yes

characteristic

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

#11719102

yes

characteristic

130

C₄

—

131

C₄

—

104

—

132

C₄

#11719104

yes

characteristic

133

C₄

#11719145

yes

characteristic

134

C₂

—

135

C₂

#13130486

yes

characteristic

136

C₂

#13131246

yes

—

	subgroup	G/H
center	C₂²	#11719102
commutator	C₂×C₈×C₁₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×#261
Fitting, prime radical	C₂×C₁₃×D₁₆	C₄
socle	C₂²×C₁₃	#442
cosocle	C₈×Dic₁₃	C₂³
hypercenter	C₂×D₁₆	#261
hypocenter	C₁₃	#12089
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×D₁₆

subgroup

G/H

center

C₂²

#11719102

commutator

C₂×C₈×C₁₃

C₂²×C₄

Frattini

C₂×C₈

C₂²×#261

Fitting, prime radical

C₂×C₁₃×D₁₆

C₄

socle

C₂²×C₁₃

#442

cosocle

C₈×Dic₁₃

C₂³

C₂×D₁₆

C₁₃

G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×D₁₆

N	H ≅ G/N	type
1. #13130486	C₂	semidirect
2. #13129861	C₂	semidirect
3. #13129794	C₂	semidirect
4. #13129753	C₂	semidirect
43. C₁₃×C₁₆	M₁₆	semidirect
111. C₁₆	#6122	semidirect
112. C₁₃	#12089	semidirect

H ≅ G/N

type

1. #13130486

C₂

semidirect

2. #13129861

C₂

semidirect

3. #13129794

C₂

semidirect

4. #13129753

C₂

semidirect

43. C₁₃×C₁₆

M₁₆

semidirect

111. C₁₆

#6122

semidirect

112. C₁₃

#12089

semidirect

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

subgroup	index
#13130486	2
#13129861	2
#13129794	2
#13129753	2
#13116712	2
#12089	13

subgroup

index

Subgroups — #16789008

Conjugacy classes of subgroups

Complements

Maximal subgroups