Group #16788731 — subgroups

Subgroups — #16788731

Order 3328.

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

normal subgroups only

This group has 232 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	#13130391	C₂	1	2	yes	maximal
2	#13129861	C₂	1	4	yes	maximal, complemented
3	C₄×#88038	C₂	1	1	yes	maximal, characteristic
4	#11719052	—	2	2	no	—
5	C₂×#88038	C₄	1	2	yes	—
6	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
7	#11718253	—	2	2	no	—
8	#11718253	C₂²	1	4	yes	—
9	C₄×C₁₃×C₁₆	C₄	1	1	yes	characteristic
10	#11718094	C₂²	1	2	yes	—
11	C₄×#6090	C₄	1	1	yes	characteristic
12	C₂×#6118	—	4	2	no	—
13	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
14	C₂×C₈×D₁₃	Q₈	1	1	yes	characteristic
15	C₄²×D₁₃	D₄	1	1	yes	characteristic
16	C₄⋊#1057	—	2	4	no	—
17	C₄⋊#1057	—	4	2	no	—
18	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
19	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₈×Dic₁₃	D₄	1	1	yes	characteristic
21	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
22	C₂×#6090	C₂×C₄	1	2	yes	—
23	C₄×#1055	Q₈	1	1	yes	characteristic
24	#88038	—	2	4	no	—
25	#12020	—	13	1	no	maximal, Sylow
26	C₂×C₄×D₁₃	C₄⋊C₄	1	1	yes	characteristic
27	C₂×C₄×D₁₃	SD₁₆	1	2	yes	—
28	C₂×#1057	—	4	4	no	—
29	#6118	—	8	4	no	—
30	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
32	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
33	C₄×Dic₁₃	SD₁₆	1	2	yes	—
34	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
35	C₂×#1055	C₄⋊C₄	1	1	yes	characteristic
36	C₂×#1055	C₂×Q₈	1	1	yes	characteristic
37	C₈×D₁₃	—	2	2	no	—
38	C₈×D₁₃	—	2	2	no	—
39	C₁₃×C₁₆	M₁₆	1	4	yes	complemented
40	#6090	—	2	2	no	—
41	C₄×M₃₂	—	13	1	no	—
42	#1566	—	13	2	no	—
43	#1368	—	13	4	no	—
44	C₂²×D₁₃	#106	1	1	yes	characteristic
45	C₂×C₄×C₁₃	#106	1	2	yes	—
46	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂×Dic₁₃	#106	1	1	yes	characteristic
48	C₂×Dic₁₃	C₂×SD₁₆	1	2	yes	—
49	C₄×D₁₃	—	2	4	no	—
50	C₄×D₁₃	—	2	1	no	—
51	C₄×D₁₃	—	2	1	no	—
52	#1057	—	8	4	no	—
53	C₈×C₁₃	#108	1	2	yes	—
54	C₈×C₁₃	C₂×M₁₆	1	2	yes	—
55	#1055	—	2	1	no	—
56	#1055	—	2	1	no	—
57	C₂×M₃₂	—	13	2	no	—
58	#423	—	26	2	no	—
59	C₂×C₄×C₈	—	13	1	no	—
60	#346	—	13	2	no	—
61	C₄×C₁₆	—	13	1	no	—
62	C₄×C₁₆	#261	1	1	yes	characteristic
63	#334	—	13	4	no	—
64	#334	—	26	2	no	—
65	C₂²×C₁₃	C₂×#106	1	1	yes	characteristic
66	C₂×D₁₃	—	2	1	no	—
67	C₂×D₁₃	—	2	2	no	—
68	C₄×C₁₃	#365	1	4	yes	—
69	C₄×C₁₃	#423	1	1	yes	characteristic
70	C₄×C₁₃	C₂×#108	1	1	yes	characteristic
71	Dic₁₃	—	2	1	no	—
72	Dic₁₃	—	2	2	no	—
73	C₂×M₁₆	—	52	2	no	—
74	C₂²×C₈	—	13	1	no	—
75	C₂²×C₈	—	13	1	no	—
76	C₂×C₄²	—	13	1	no	—
77	M₃₂	—	26	4	no	—
78	C₂×C₁₆	—	13	2	no	—
79	C₂×C₁₆	C₂×#261	1	2	yes	—
80	C₄⋊C₈	—	26	4	no	—
81	C₄⋊C₈	—	52	2	no	—
82	C₄×C₈	—	13	1	no	—
83	C₄×C₈	—	13	1	no	—
84	C₄×C₈	—	13	1	no	—
85	C₄×C₈	C₂×#261	1	1	yes	characteristic
86	C₂×C₁₃	#1566	1	1	yes	characteristic
87	C₂×C₁₃	C₂×#365	1	2	yes	—
88	D₁₃	—	4	1	no	—
89	C₂²×C₄	—	13	2	no	—
90	C₂²×C₄	—	13	1	no	—
91	M₁₆	—	104	4	no	—
92	C₂×C₈	—	13	1	no	—
93	C₂×C₈	—	13	1	no	—
94	C₂×C₈	—	26	2	no	—
95	C₂×C₈	—	26	2	no	—
96	C₂×C₈	—	52	4	no	—
97	C₂×C₈	C₄⋊#261	1	1	yes	characteristic
98	C₂×C₈	C₂²×#261	1	1	yes	characteristic
99	C₄²	—	13	2	no	—
100	C₄²	—	13	1	no	—
101	C₄²	C₄⋊#261	1	1	yes	characteristic
102	C₁₆	—	26	2	no	—
103	C₁₆	#6118	1	4	yes	complemented
104	C₁₃	#12020	1	1	yes	characteristic, complemented, Sylow
105	C₂³	—	13	1	no	—
106	C₂×C₄	—	13	2	no	—
107	C₂×C₄	—	13	1	no	—
108	C₂×C₄	—	26	4	no	—
109	C₂×C₄	—	26	1	no	—
110	C₂×C₄	—	26	1	no	—
111	C₂×C₄	#88101	1	2	yes	—
112	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
113	C₈	—	26	1	no	—
114	C₈	—	26	1	no	—
115	C₈	—	104	4	no	—
116	C₈	#88103	1	2	yes	—
117	C₈	C₂×#6118	1	2	yes	—
118	C₂²	—	26	1	no	—
119	C₂²	—	26	2	no	—
120	C₂²	C₂×#88101	1	1	yes	characteristic
121	C₄	—	26	1	no	—
122	C₄	—	26	2	no	—
123	C₄	#11718220	1	4	yes	—
124	C₄	#11719052	1	1	yes	characteristic
125	C₄	C₂×#88103	1	1	yes	characteristic
126	C₂	—	52	1	no	—
127	C₂	#13130391	1	1	yes	characteristic
128	C₂	C₂×#11718220	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13130391

C₂

yes

maximal

#13129861

C₂

yes

maximal, complemented

C₄×#88038

C₂

yes

maximal, characteristic

#11719052

—

C₂×#88038

C₄

yes

—

C₄×C₈×D₁₃

C₂²

yes

characteristic

#11718253

—

#11718253

C₂²

yes

—

C₄×C₁₃×C₁₆

C₄

yes

characteristic

#11718094

C₂²

yes

—

C₄×#6090

C₄

yes

characteristic

C₂×#6118

—

C₂×C₈×D₁₃

C₂×C₄

yes

characteristic

C₂×C₈×D₁₃

Q₈

yes

characteristic

C₄²×D₁₃

D₄

yes

characteristic

C₄⋊#1057

—

C₄⋊#1057

—

C₂×C₁₃×C₁₆

C₂×C₄

yes

—

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

C₈×Dic₁₃

D₄

yes

characteristic

C₈×Dic₁₃

C₂³

yes

characteristic

C₂×#6090

C₂×C₄

yes

—

C₄×#1055

Q₈

yes

—

—

C₂×C₄×D₁₃

C₄⋊C₄

yes

characteristic

C₂×C₄×D₁₃

SD₁₆

yes

—

C₂×#1057

—

#6118

—

C₂×C₈×C₁₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₃

C₂²×C₄

yes

characteristic

C₄²×C₁₃

C₄⋊C₄

yes

characteristic

C₄×Dic₁₃

SD₁₆

yes

—

C₄×Dic₁₃

C₂×D₄

yes

characteristic

C₂×#1055

C₄⋊C₄

yes

characteristic

C₂×#1055

C₂×Q₈

yes

characteristic

C₈×D₁₃

—

C₈×D₁₃

—

C₁₃×C₁₆

M₁₆

yes

complemented

#6090

—

C₄×M₃₂

—

#1566

—

#1368

—

C₂²×D₁₃

#106

yes

characteristic

C₂×C₄×C₁₃

#106

yes

—

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂×Dic₁₃

#106

yes

characteristic

C₂×Dic₁₃

C₂×SD₁₆

yes

—

C₄×D₁₃

—

C₄×D₁₃

—

C₄×D₁₃

—

#1057

—

C₈×C₁₃

#108

yes

—

C₈×C₁₃

C₂×M₁₆

yes

—

#1055

—

#1055

—

C₂×M₃₂

—

#423

—

C₂×C₄×C₈

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

yes

—

—

C₂²×C₁₃

C₂×#106

yes

characteristic

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

#365

yes

—

C₄×C₁₃

yes

C₄×C₁₃

yes

Dic₁₃

—

Dic₁₃

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₂×#261

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

—

D₁₃

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

104

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊#261

yes

characteristic

C₂×C₈

C₂²×#261

yes

characteristic

C₄²

—

100

C₄²

—

101

C₄²

C₄⋊#261

yes

characteristic

102

C₁₆

—

103

C₁₆

#6118

yes

complemented

104

C₁₃

#12020

yes

characteristic, complemented, Sylow

105

C₂³

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

#88101

yes

—

112

C₂×C₄

C₂×C₄⋊#261

yes

characteristic

113

C₈

—

114

C₈

—

115

C₈

—

104

—

116

C₈

#88103

yes

—

117

C₈

C₂×#6118

yes

—

118

C₂²

—

119

C₂²

—

120

C₂²

C₂×#88101

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

#11718220

yes

—

124

C₄

#11719052

yes

characteristic

125

C₄

C₂×#88103

yes

characteristic

126

C₂

—

127

C₂

#13130391

yes

characteristic

128

C₂

C₂×#11718220

yes

—

	subgroup	G/H
center	C₂²	C₂×#88101
commutator	C₂×C₈×C₁₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×#261
Fitting	C₄×C₁₃×C₁₆	C₄
socle	C₂²×C₁₃	C₂×#106
prime radical	C₂²×D₁₃	#106
cosocle	C₈×Dic₁₃	C₂³
hypercenter	C₄×C₁₆	#261
hypocenter	C₁₃	#12020
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

subgroup

G/H

center

C₂²

C₂×#88101

commutator

C₂×C₈×C₁₃

C₂²×C₄

Frattini

C₂×C₈

C₂²×#261

Fitting

C₄×C₁₃×C₁₆

C₄

C₂²×C₁₃

C₂²×D₁₃

C₈×Dic₁₃

C₂³

C₄×C₁₆

C₁₃

G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

N	H ≅ G/N	type
2. #13129861	C₂	semidirect
39. C₁₃×C₁₆	M₁₆	semidirect
103. C₁₆	#6118	semidirect
104. C₁₃	#12020	semidirect

H ≅ G/N

type

2. #13129861

C₂

semidirect

39. C₁₃×C₁₆

M₁₆

semidirect

103. C₁₆

#6118

semidirect

104. C₁₃

#12020

semidirect

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

subgroup	index
#13130391	2
#13129861	2
C₄×#88038	2
#12020	13

subgroup

index

Subgroups — #16788731

Conjugacy classes of subgroups

Complements

Maximal subgroups