Group #16791474 — subgroups

Subgroups — #16791474

Order 3328.

188 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 54 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	Q₃₂×#261	C₂	1	2	yes	maximal
2	#13129998	C₂	1	1	yes	maximal, characteristic
3	#13129986	C₂	1	2	yes	maximal
4	#13129942	C₂	1	1	yes	maximal, characteristic
5	D₁₃×Q₆₄	C₂	1	1	yes	maximal, characteristic
6	Q₁₆×#261	—	2	2	no	—
7	D₁₃×Q₃₂	C₂²	1	2	yes	—
8	#11718280	—	2	2	no	—
9	#11718221	C₂²	1	2	yes	—
10	C₁₆×#261	C₂²	1	1	yes	characteristic
11	D₁₃⋊C₃₂	C₂²	1	1	yes	characteristic
12	C₁₃×Q₆₄	C₄	1	1	yes	characteristic, complemented
13	#11718114	C₄	1	2	yes	complemented
14	Dic₂₀₈	C₄	1	1	yes	characteristic, complemented
15	D₁₃×C₃₂	C₂²	1	1	yes	characteristic
16	Q₈×#261	—	4	2	no	—
17	Q₁₆×D₁₃	—	2	2	no	—
18	#88117	—	4	2	no	—
19	#88102	—	2	2	no	—
20	C₈×#261	D₄	1	1	yes	characteristic
21	D₁₃⋊C₁₆	D₄	1	1	yes	characteristic
22	C₁₃×Q₃₂	C₂×C₄	1	2	yes	—
23	#88069	—	2	2	no	—
24	Dic₁₀₄	C₂×C₄	1	2	yes	—
25	C₁₆×D₁₃	C₂³	1	1	yes	characteristic
26	#88036	—	2	1	no	—
27	C₁₃×C₃₂	C₂×C₄	1	1	yes	characteristic
28	#88034	C₂×C₄	1	1	yes	characteristic
29	#13653	—	13	1	no	maximal, Sylow
30	Q₈×D₁₃	—	4	2	no	—
31	C₄⋊#261	—	4	2	no	—
32	C₄⋊#261	—	8	2	no	—
33	C₄×#261	—	8	2	no	—
34	C₄×#261	D₈	1	1	yes	characteristic
35	D₁₃⋊C₈	D₈	1	1	yes	characteristic
36	C₁₃×Q₁₆	—	2	2	no	—
37	#6107	—	4	2	no	—
38	Dic₅₂	—	2	2	no	—
39	C₈×D₁₃	C₂×D₄	1	1	yes	characteristic
40	#6092	—	2	1	no	—
41	C₁₃×C₁₆	C₂²×C₄	1	1	yes	characteristic
42	#6090	#41	1	1	yes	characteristic
43	C₂×Q₆₄	—	13	1	no	—
44	C₄×Q₃₂	—	13	2	no	—
45	#1424	—	13	1	no	—
46	#1416	—	13	2	no	—
47	#1397	—	13	1	no	—
48	C₂×#261	—	8	2	no	—
49	C₂×#261	D₁₆	1	2	yes	—
50	Q₈×C₁₃	—	4	2	no	—
51	C₄×D₁₃	—	8	2	no	—
52	C₄×D₁₃	C₂×D₈	1	1	yes	characteristic
53	Dic₂₆	—	4	2	no	—
54	Dic₂₆	—	8	2	no	—
55	#1057	—	2	1	no	—
56	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
57	#1055	#135	1	1	yes	characteristic
58	C₂×Q₃₂	—	13	2	no	—
59	C₄×Q₁₆	—	26	2	no	—
60	Q₆₄	—	13	2	no	—
61	Q₆₄	—	13	1	no	—
62	Q₆₄	#261	1	1	yes	characteristic, complemented
63	C₂×C₃₂	—	13	1	no	—
64	C₂×C₃₂	—	13	1	no	—
65	#366	—	13	2	no	—
66	#358	—	26	2	no	—
67	C₄×C₁₆	—	13	1	no	—
68	C₂×D₁₃	C₂×D₁₆	1	1	yes	characteristic
69	#261	—	2	2	no	—
70	#261	—	16	2	no	—
71	C₄×C₁₃	—	8	2	no	—
72	C₄×C₁₃	C₄×D₈	1	1	yes	characteristic
73	Dic₁₃	—	8	2	no	—
74	Dic₁₃	#508	1	1	yes	characteristic
75	C₂×Q₁₆	—	26	2	no	—
76	C₄×Q₈	—	52	2	no	—
77	Q₃₂	—	13	2	no	—
78	Q₃₂	—	26	2	no	—
79	Q₃₂	C₂×#261	1	2	yes	—
80	C₂×C₁₆	—	13	1	no	—
81	C₂×C₁₆	—	13	1	no	—
82	#107	—	26	2	no	—
83	#103	—	52	2	no	—
84	C₄×C₈	—	13	1	no	—
85	C₃₂	—	13	1	no	—
86	C₃₂	—	26	1	no	—
87	C₃₂	C₂×#261	1	1	yes	characteristic
88	C₂×C₁₃	C₄×D₁₆	1	1	yes	characteristic
89	D₁₃	#2264	1	2	yes	—
90	C₂×Q₈	—	52	2	no	—
91	Q₁₆	—	2	2	no	—
92	Q₁₆	—	26	2	no	—
93	Q₁₆	—	52	2	no	—
94	C₂×C₈	—	13	1	no	—
95	C₂×C₈	—	13	1	no	—
96	C₄⋊C₄	—	52	2	no	—
97	C₄⋊C₄	—	104	2	no	—
98	C₄²	—	13	1	no	—
99	C₄²	—	104	2	no	—
100	C₁₆	—	13	1	no	—
101	C₁₆	—	26	1	no	—
102	C₁₆	C₂²×#261	1	1	yes	characteristic
103	C₁₃	#13653	1	1	yes	characteristic, complemented, Sylow
104	Q₈	—	4	2	no	—
105	Q₈	—	52	2	no	—
106	Q₈	—	104	2	no	—
107	C₂×C₄	—	13	1	no	—
108	C₂×C₄	—	13	2	no	—
109	C₂×C₄	—	104	2	no	—
110	C₂×C₄	—	104	2	no	—
111	C₈	—	13	1	no	—
112	C₈	—	26	1	no	—
113	C₈	D₄×#261	1	1	yes	characteristic
114	C₂²	—	13	1	no	—
115	C₄	—	8	2	no	—
116	C₄	—	13	1	no	—
117	C₄	—	26	2	no	—
118	C₄	—	104	2	no	—
119	C₄	—	208	2	no	—
120	C₄	D₈×#261	1	1	yes	characteristic
121	C₂	—	13	2	no	—
122	C₂	D₁₆×#261	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₃₂×#261

C₂

yes

maximal

#13129998

C₂

yes

maximal, characteristic

#13129986

C₂

yes

maximal

#13129942

C₂

yes

maximal, characteristic

D₁₃×Q₆₄

C₂

yes

maximal, characteristic

Q₁₆×#261

—

D₁₃×Q₃₂

C₂²

yes

—

#11718280

—

#11718221

C₂²

yes

—

C₁₆×#261

C₂²

yes

characteristic

D₁₃⋊C₃₂

C₂²

yes

characteristic

C₁₃×Q₆₄

C₄

yes

characteristic, complemented

#11718114

C₄

yes

complemented

Dic₂₀₈

C₄

yes

characteristic, complemented

D₁₃×C₃₂

C₂²

yes

characteristic

Q₈×#261

—

Q₁₆×D₁₃

—

#88117

—

#88102

—

C₈×#261

D₄

yes

characteristic

D₁₃⋊C₁₆

D₄

yes

characteristic

C₁₃×Q₃₂

C₂×C₄

yes

—

#88069

—

Dic₁₀₄

C₂×C₄

yes

—

C₁₆×D₁₃

C₂³

yes

characteristic

#88036

—

C₁₃×C₃₂

C₂×C₄

yes

characteristic

#88034

C₂×C₄

yes

characteristic

#13653

—

maximal, Sylow

Q₈×D₁₃

—

—

—

—

D₈

yes

characteristic

D₁₃⋊C₈

D₈

yes

characteristic

C₁₃×Q₁₆

—

#6107

—

Dic₅₂

—

C₈×D₁₃

C₂×D₄

yes

characteristic

#6092

—

C₁₃×C₁₆

C₂²×C₄

yes

yes

C₂×Q₆₄

—

C₄×Q₃₂

—

—

—

—

—

D₁₆

yes

—

Q₈×C₁₃

—

C₄×D₁₃

—

C₄×D₁₃

C₂×D₈

yes

characteristic

Dic₂₆

—

Dic₂₆

—

#1057

—

C₈×C₁₃

C₄×D₄

yes

yes

C₂×Q₃₂

—

C₄×Q₁₆

—

Q₆₄

—

Q₆₄

—

Q₆₄

#261

yes

characteristic, complemented

C₂×C₃₂

—

C₂×C₃₂

—

#366

—

#358

—

C₄×C₁₆

—

C₂×D₁₃

C₂×D₁₆

yes

characteristic

#261

—

#261

—

C₄×C₁₃

—

C₄×C₁₃

C₄×D₈

yes

characteristic

Dic₁₃

—

Dic₁₃

#508

yes

characteristic

C₂×Q₁₆

—

C₄×Q₈

—

Q₃₂

—

Q₃₂

—

Q₃₂

C₂×#261

yes

—

C₂×C₁₆

—

C₂×C₁₆

—

#107

—

#103

—

C₄×C₈

—

C₃₂

—

C₃₂

—

C₃₂

C₂×#261

yes

characteristic

C₂×C₁₃

C₄×D₁₆

yes

characteristic

D₁₃

#2264

yes

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

104

—

C₄²

—

C₄²

—

104

—

100

C₁₆

—

101

C₁₆

—

102

C₁₆

C₂²×#261

yes

characteristic

103

C₁₃

#13653

yes

characteristic, complemented, Sylow

104

Q₈

—

105

Q₈

—

106

Q₈

—

104

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

104

—

110

C₂×C₄

—

104

—

111

C₈

—

112

C₈

—

113

C₈

D₄×#261

yes

characteristic

114

C₂²

—

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

104

—

119

C₄

—

208

—

120

C₄

D₈×#261

yes

characteristic

121

C₂

—

122

C₂

D₁₆×#261

yes

characteristic

	subgroup	G/H
center	C₂	D₁₆×#261
commutator	C₁₃×C₁₆	C₂²×C₄
Frattini	C₁₆	C₂²×#261
Fitting	C₁₃×Q₆₄	C₄
socle	C₂×C₁₃	C₄×D₁₆
prime radical	C₂×D₁₃	C₂×D₁₆
cosocle	C₁₆×D₁₃	C₂³
hypercenter	Q₆₄	#261
hypocenter	C₁₃	#13653
LCS	G ▷ C₁₃×C₁₆ ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ Q₆₄

subgroup

G/H

center

C₂

D₁₆×#261

commutator

C₁₃×C₁₆

C₂²×C₄

Frattini

C₁₆

C₂²×#261

Fitting

C₁₃×Q₆₄

C₄

socle

C₂×C₁₃

C₄×D₁₆

prime radical

C₂×D₁₃

C₂×D₁₆

cosocle

C₁₆×D₁₃

C₂³

Q₆₄

C₁₃

G ▷ C₁₃×C₁₆ ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ Q₆₄

N	H ≅ G/N	type
12. C₁₃×Q₆₄	C₄	semidirect
13. #11718114	C₄	semidirect
14. Dic₂₀₈	C₄	semidirect
62. Q₆₄	#261	semidirect
103. C₁₃	#13653	semidirect

H ≅ G/N

type

12. C₁₃×Q₆₄

C₄

semidirect

13. #11718114

C₄

semidirect

14. Dic₂₀₈

C₄

semidirect

62. Q₆₄

#261

semidirect

103. C₁₃

#13653

semidirect

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

subgroup	index
Q₃₂×#261	2
#13129998	2
#13129986	2
#13129942	2
D₁₃×Q₆₄	2
#13653	13

subgroup

index

Q₃₂×#261

D₁₃×Q₆₄

Subgroups — #16791474

Conjugacy classes of subgroups

Complements

Maximal subgroups