Group #16791491 — subgroups

Subgroups — #16791491

Order 3328.

166 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 92 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	#13129994	C₂	1	2	yes	maximal
2	C₄⋊#88036	C₂	1	1	yes	maximal, characteristic
3	#13129941	C₂	1	2	yes	maximal
4	C₄×#88036	C₂	1	1	yes	maximal, characteristic
5	#13116738	C₂	1	1	yes	maximal, characteristic
6	C₈×Dic₂₆	C₄	1	1	yes	characteristic
7	C₂×#88036	C₂²	1	2	yes	—
8	C₂×#88036	C₂²	1	1	yes	characteristic
9	C₂×#88036	C₂²	1	1	yes	characteristic
10	C₁₃×C₄⋊C₁₆	C₄	1	1	yes	characteristic
11	#11718093	C₄	1	2	yes	—
12	C₁₆×Dic₁₃	C₂²	1	2	yes	—
13	C₄×#6090	C₂²	1	1	yes	characteristic
14	C₄×Dic₂₆	C₈	1	1	yes	characteristic
15	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
16	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
17	#88054	C₈	1	2	yes	—
18	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
19	C₂×#6090	C₂×C₄	1	1	yes	characteristic
20	C₂×#6090	C₂³	1	1	yes	characteristic
21	C₄⋊#1055	C₈	1	1	yes	characteristic
22	#88036	—	2	2	no	—
23	#88036	—	2	1	no	—
24	#88036	Q₈	1	2	yes	complemented
25	Q₈×C₃₂	—	13	1	no	maximal, Sylow
26	C₂×Dic₂₆	C₁₆	1	1	yes	characteristic
27	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
28	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
29	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
30	C₄⋊Dic₁₃	C₁₆	1	1	yes	characteristic
31	#6101	C₁₆	1	2	yes	—
32	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
33	C₂×#1055	C₂×C₈	1	2	yes	—
34	C₁₃×C₁₆	—	2	2	no	—
35	#6090	—	2	1	no	—
36	#6090	C₂×Q₈	1	1	yes	characteristic
37	#6090	#41	1	1	yes	characteristic
38	Q₈×C₁₆	—	13	1	no	—
39	C₄⋊C₃₂	—	13	1	no	—
40	C₄⋊C₃₂	—	13	2	no	—
41	C₄×C₃₂	—	13	1	no	—
42	C₄×C₃₂	—	13	2	no	—
43	C₂×C₄×C₁₃	C₂×C₁₆	1	1	yes	characteristic
44	C₂×C₄×C₁₃	C₂×C₁₆	1	1	yes	characteristic
45	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
46	C₂×Dic₁₃	C₂×C₁₆	1	2	yes	—
47	C₂×Dic₁₃	C₂×C₁₆	1	2	yes	—
48	Dic₂₆	C₃₂	1	4	yes	complemented
49	C₈×C₁₃	—	2	1	no	—
50	C₈×C₁₃	C₄×Q₈	1	1	yes	characteristic
51	C₈×C₁₃	#131	1	1	yes	characteristic
52	#1055	—	2	2	no	—
53	C₈×Q₈	—	13	1	no	—
54	C₂×C₃₂	—	13	1	no	—
55	C₂×C₃₂	—	13	2	no	—
56	C₂×C₃₂	—	13	1	no	—
57	C₄⋊C₁₆	—	13	2	no	—
58	C₄⋊C₁₆	#261	1	1	yes	characteristic
59	C₄×C₁₆	—	13	2	no	—
60	C₄×C₁₆	—	13	1	no	—
61	C₂²×C₁₃	C₂²×C₁₆	1	1	yes	characteristic
62	C₄×C₁₃	—	2	1	no	—
63	C₄×C₁₃	C₂×C₃₂	1	2	yes	—
64	C₄×C₁₃	C₈×Q₈	1	1	yes	characteristic
65	C₄×C₁₃	#504	1	1	yes	characteristic
66	Dic₁₃	—	2	2	no	—
67	Dic₁₃	C₂×C₃₂	1	4	yes	—
68	C₄×Q₈	—	13	1	no	—
69	C₂×C₁₆	—	13	1	no	—
70	C₂×C₁₆	—	13	1	no	—
71	C₂×C₁₆	C₂×#261	1	2	yes	—
72	C₄⋊C₈	—	13	2	no	—
73	C₄⋊C₈	—	13	1	no	—
74	C₄×C₈	—	13	2	no	—
75	C₄×C₈	C₂×#261	1	1	yes	characteristic
76	C₃₂	—	13	2	no	—
77	C₃₂	—	26	2	no	—
78	C₃₂	—	26	1	no	—
79	C₂×C₁₃	Q₈×C₁₆	1	1	yes	characteristic
80	C₂×C₁₃	C₂²×C₃₂	1	1	yes	characteristic
81	C₂×C₁₃	#2258	1	1	yes	characteristic
82	C₂×Q₈	—	13	1	no	—
83	C₂×C₈	—	13	2	no	—
84	C₂×C₈	D₁₃⋊C₈	1	1	yes	characteristic
85	C₂×C₈	C₂²×#261	1	1	yes	characteristic
86	C₄⋊C₄	—	13	2	no	—
87	C₄⋊C₄	—	13	1	no	—
88	C₄²	—	13	2	no	—
89	C₄²	D₁₃⋊C₈	1	1	yes	characteristic
90	C₁₆	—	2	2	no	—
91	C₁₆	—	13	1	no	—
92	C₁₆	—	13	1	no	—
93	C₁₆	—	26	1	no	—
94	C₁₃	Q₈×C₃₂	1	1	yes	characteristic, complemented, Sylow
95	Q₈	—	13	4	no	—
96	C₂×C₄	—	13	2	no	—
97	C₂×C₄	—	13	2	no	—
98	C₂×C₄	D₁₃⋊C₁₆	1	1	yes	characteristic
99	C₂×C₄	D₁₃⋊C₁₆	1	1	yes	characteristic
100	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
101	C₈	—	2	1	no	—
102	C₈	—	26	2	no	—
103	C₈	#88238	1	1	yes	characteristic
104	C₈	Q₈×#261	1	1	yes	characteristic
105	C₂²	C₂×D₁₃⋊C₁₆	1	1	yes	characteristic
106	C₄	—	2	1	no	—
107	C₄	—	13	4	no	—
108	C₄	—	26	2	no	—
109	C₄	D₁₃⋊C₃₂	1	2	yes	—
110	C₄	#11719074	1	1	yes	characteristic
111	C₄	#11719096	1	1	yes	characteristic
112	C₂	#13131250	1	1	yes	characteristic
113	C₂	C₂×D₁₃⋊C₃₂	1	1	yes	characteristic
114	C₂	#13131260	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13129994

C₂

yes

maximal

C₄⋊#88036

C₂

yes

maximal, characteristic

#13129941

C₂

yes

maximal

C₄×#88036

C₂

yes

maximal, characteristic

#13116738

C₂

yes

maximal, characteristic

C₈×Dic₂₆

C₄

yes

characteristic

C₂×#88036

C₂²

yes

—

C₂×#88036

C₂²

yes

characteristic

C₂×#88036

C₂²

yes

characteristic

C₁₃×C₄⋊C₁₆

C₄

yes

characteristic

#11718093

C₄

yes

—

C₁₆×Dic₁₃

C₂²

yes

—

C₄×#6090

C₂²

yes

characteristic

C₄×Dic₂₆

C₈

yes

characteristic

C₂×C₁₃×C₁₆

C₂×C₄

yes

—

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

#88054

C₈

yes

—

C₈×Dic₁₃

C₂×C₄

yes

—

C₂×#6090

C₂×C₄

yes

characteristic

C₂×#6090

C₂³

yes

characteristic

C₄⋊#1055

C₈

yes

—

—

Q₈

yes

complemented

Q₈×C₃₂

—

maximal, Sylow

C₂×Dic₂₆

C₁₆

yes

characteristic

C₂×C₈×C₁₃

C₂×C₈

yes

characteristic

C₂×C₈×C₁₃

C₂²×C₄

yes

characteristic

C₄²×C₁₃

C₂×C₈

yes

characteristic

C₄⋊Dic₁₃

C₁₆

yes

characteristic

#6101

C₁₆

yes

—

C₄×Dic₁₃

C₂×C₈

yes

—

C₂×#1055

C₂×C₈

yes

—

C₁₃×C₁₆

—

#6090

—

#6090

C₂×Q₈

yes

yes

Q₈×C₁₆

—

C₄⋊C₃₂

—

C₄⋊C₃₂

—

C₄×C₃₂

—

C₄×C₃₂

—

C₂×C₄×C₁₃

C₂×C₁₆

yes

characteristic

C₂×C₄×C₁₃

C₂×C₁₆

yes

characteristic

C₂×C₄×C₁₃

C₂²×C₈

yes

characteristic

C₂×Dic₁₃

C₂×C₁₆

yes

—

C₂×Dic₁₃

C₂×C₁₆

yes

—

Dic₂₆

C₃₂

yes

complemented

C₈×C₁₃

—

C₈×C₁₃

C₄×Q₈

yes

C₈×C₁₃

yes

—

C₈×Q₈

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

C₄⋊C₁₆

—

C₄⋊C₁₆

#261

yes

characteristic

C₄×C₁₆

—

C₄×C₁₆

—

C₂²×C₁₃

C₂²×C₁₆

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

C₂×C₃₂

yes

—

C₄×C₁₃

C₈×Q₈

yes

characteristic

C₄×C₁₃

#504

yes

characteristic

Dic₁₃

—

Dic₁₃

C₂×C₃₂

yes

—

C₄×Q₈

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×#261

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

C₂×#261

yes

characteristic

C₃₂

—

C₃₂

—

C₃₂

—

C₂×C₁₃

Q₈×C₁₆

yes

characteristic

C₂×C₁₃

C₂²×C₃₂

yes

characteristic

C₂×C₁₃

#2258

yes

characteristic

C₂×Q₈

—

C₂×C₈

—

C₂×C₈

D₁₃⋊C₈

yes

characteristic

C₂×C₈

C₂²×#261

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

D₁₃⋊C₈

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

—

C₁₃

Q₈×C₃₂

yes

characteristic, complemented, Sylow

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

D₁₃⋊C₁₆

yes

characteristic

C₂×C₄

D₁₃⋊C₁₆

yes

characteristic

100

C₂×C₄

C₂×D₁₃⋊C₈

yes

characteristic

101

C₈

—

102

C₈

—

103

C₈

#88238

yes

characteristic

104

C₈

Q₈×#261

yes

characteristic

105

C₂²

C₂×D₁₃⋊C₁₆

yes

characteristic

106

C₄

—

107

C₄

—

108

C₄

—

109

C₄

D₁₃⋊C₃₂

yes

—

110

C₄

#11719074

yes

characteristic

111

C₄

#11719096

yes

characteristic

112

C₂

#13131250

yes

characteristic

113

C₂

C₂×D₁₃⋊C₃₂

yes

characteristic

114

C₂

#13131260

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×#261
commutator	C₂×C₁₃	C₂²×C₃₂
Fitting	C₁₃×C₄⋊C₁₆	C₄
socle, prime radical	C₂²×C₁₃	C₂²×C₁₆
cosocle	C₂×#6090	C₂³
hypercenter	C₄⋊C₁₆	#261
hypocenter	C₁₃	Q₈×C₃₂
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄⋊C₁₆

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×#261

commutator

C₂×C₁₃

C₂²×C₃₂

Fitting

C₁₃×C₄⋊C₁₆

C₄

socle, prime radical

C₂²×C₁₃

C₂²×C₁₆

C₂³

C₄⋊C₁₆

C₁₃

Q₈×C₃₂

LCS

G ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂×C₈ ◁ C₄⋊C₁₆

N	H ≅ G/N	type
24. #88036	Q₈	semidirect
48. Dic₂₆	C₃₂	semidirect
94. C₁₃	Q₈×C₃₂	semidirect

H ≅ G/N

type

24. #88036

Q₈

semidirect

48. Dic₂₆

C₃₂

semidirect

94. C₁₃

Q₈×C₃₂

semidirect

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

subgroup	index
#13129994	2
C₄⋊#88036	2
#13129941	2
C₄×#88036	2
#13116738	2
Q₈×C₃₂	13

subgroup

index

Q₈×C₃₂

Subgroups — #16791491

Conjugacy classes of subgroups

Complements

Maximal subgroups