Group #15783555 — subgroups

Subgroups — #15783555

Order 3328.

158 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 113 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	C₄⋊#88034	C₂	1	1	yes	maximal, characteristic
2	C₄⋊#88034	C₂	1	2	yes	maximal
3	#13112970	C₂	1	2	yes	maximal
4	C₄×#88034	C₂	1	1	yes	maximal, characteristic
5	Q₈×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₈×Q₈×C₁₃	C₄	1	1	yes	characteristic
7	C₁₃×C₄⋊C₁₆	C₄	1	1	yes	characteristic
8	C₁₃×C₄⋊C₁₆	C₄	1	2	yes	—
9	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
10	C₄×C₁₃×C₁₆	C₂²	1	2	yes	—
11	C₂×#88034	C₂²	1	1	yes	characteristic
12	C₂×#88034	C₂²	1	2	yes	—
13	C₂×#88034	C₂²	1	1	yes	characteristic
14	C₄×Q₈×C₁₃	C₈	1	1	yes	characteristic
15	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
16	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
17	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
18	C₁₃×C₄⋊C₈	C₈	1	2	yes	—
19	C₁₃×C₄⋊C₈	C₈	1	1	yes	characteristic
20	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
22	#88034	—	2	1	no	—
23	#88034	—	2	2	no	—
24	#88034	Q₈	1	2	yes	complemented
25	#13660	—	13	1	no	maximal, Sylow
26	C₂×Q₈×C₁₃	C₁₆	1	1	yes	characteristic
27	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
28	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
29	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
30	C₁₃×C₄⋊C₄	C₁₆	1	2	yes	—
31	C₁₃×C₄⋊C₄	C₁₆	1	1	yes	characteristic
32	C₄²×C₁₃	C₂×C₈	1	2	yes	—
33	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
34	C₁₃×C₁₆	—	2	1	no	—
35	C₁₃×C₁₆	—	2	2	no	—
36	C₁₃×C₁₆	C₂×Q₈	1	1	yes	characteristic
37	C₁₃×C₁₆	#41	1	1	yes	characteristic
38	Q₈×C₁₆	D₁₃	1	1	yes	characteristic
39	C₄⋊C₃₂	—	13	1	no	—
40	C₄⋊C₃₂	—	13	2	no	—
41	#1397	—	13	2	no	—
42	C₄×C₃₂	—	13	1	no	—
43	Q₈×C₁₃	—	2	2	no	—
44	C₂×C₄×C₁₃	C₂×C₁₆	1	2	yes	—
45	C₂×C₄×C₁₃	C₂×C₁₆	1	1	yes	characteristic
46	C₂×C₄×C₁₃	C₂×C₁₆	1	2	yes	—
47	C₂×C₄×C₁₃	C₂×C₁₆	1	1	yes	characteristic
48	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
49	C₈×C₁₃	—	2	1	no	—
50	C₈×C₁₃	—	2	2	no	—
51	C₈×C₁₃	C₄×Q₈	1	1	yes	characteristic
52	C₈×C₁₃	#131	1	1	yes	characteristic
53	C₈×Q₈	Dic₁₃	1	1	yes	characteristic
54	C₂×C₃₂	—	13	1	no	—
55	C₂×C₃₂	—	13	1	no	—
56	C₂×C₃₂	—	13	2	no	—
57	C₄⋊C₁₆	Dic₁₃	1	1	yes	characteristic
58	C₄⋊C₁₆	Dic₁₃	1	2	yes	—
59	C₄×C₁₆	C₂×D₁₃	1	1	yes	characteristic
60	C₄×C₁₆	C₂×D₁₃	1	2	yes	—
61	C₂²×C₁₃	C₂²×C₁₆	1	1	yes	characteristic
62	C₄×C₁₃	—	2	2	no	—
63	C₄×C₁₃	—	2	2	no	—
64	C₄×C₁₃	—	2	1	no	—
65	C₄×C₁₃	M₆₄	1	2	yes	—
66	C₄×C₁₃	C₈×Q₈	1	1	yes	characteristic
67	C₄×C₁₃	#504	1	1	yes	characteristic
68	C₄×Q₈	#1055	1	1	yes	characteristic
69	C₂×C₁₆	C₂×Dic₁₃	1	1	yes	characteristic
70	C₂×C₁₆	C₂×Dic₁₃	1	2	yes	—
71	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
72	C₄⋊C₈	#1055	1	2	yes	—
73	C₄⋊C₈	#1055	1	1	yes	characteristic
74	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
75	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
76	C₃₂	—	13	2	no	—
77	C₃₂	—	26	1	no	—
78	C₃₂	—	26	2	no	—
79	C₂×C₁₃	Q₈×C₁₆	1	1	yes	characteristic
80	C₂×C₁₃	C₂×M₆₄	1	1	yes	characteristic
81	C₂×C₁₃	#2258	1	1	yes	characteristic
82	C₂×Q₈	#6090	1	1	yes	characteristic
83	C₂×C₈	C₂×#1055	1	1	yes	characteristic
84	C₂×C₈	C₂×#1055	1	2	yes	—
85	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
86	C₄⋊C₄	#6090	1	2	yes	—
87	C₄⋊C₄	#6090	1	1	yes	characteristic
88	C₄²	C₂×#1055	1	2	yes	—
89	C₄²	C₂×#1055	1	1	yes	characteristic
90	C₁₆	—	2	1	no	—
91	C₁₆	—	2	2	no	—
92	C₁₆	Q₈×D₁₃	1	1	yes	characteristic
93	C₁₆	#6131	1	1	yes	characteristic
94	C₁₃	#13660	1	1	yes	characteristic, complemented, Sylow
95	Q₈	—	2	2	no	—
96	C₂×C₄	C₂×#6090	1	2	yes	—
97	C₂×C₄	C₂×#6090	1	1	yes	characteristic
98	C₂×C₄	C₂×#6090	1	2	yes	—
99	C₂×C₄	C₂×#6090	1	1	yes	characteristic
100	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
101	C₈	—	2	1	no	—
102	C₈	—	2	2	no	—
103	C₈	Q₈×Dic₁₃	1	1	yes	characteristic
104	C₈	#88202	1	1	yes	characteristic
105	C₂²	C₂²×#6090	1	1	yes	characteristic
106	C₄	—	2	2	no	—
107	C₄	—	2	2	no	—
108	C₄	—	2	1	no	—
109	C₄	#11718091	1	2	yes	—
110	C₄	Q₈×#1055	1	1	yes	characteristic
111	C₄	#11718801	1	1	yes	characteristic
112	C₂	Q₈×#6090	1	1	yes	characteristic
113	C₂	C₂×#11718091	1	1	yes	characteristic
114	C₂	#13117130	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄⋊#88034

C₂

yes

maximal, characteristic

C₄⋊#88034

C₂

yes

maximal

#13112970

C₂

yes

maximal

C₄×#88034

C₂

yes

maximal, characteristic

Q₈×C₁₃×C₁₆

C₂

yes

maximal, characteristic

C₈×Q₈×C₁₃

C₄

yes

characteristic

C₁₃×C₄⋊C₁₆

C₄

yes

characteristic

C₁₃×C₄⋊C₁₆

C₄

yes

—

C₄×C₁₃×C₁₆

C₂²

yes

characteristic

C₄×C₁₃×C₁₆

C₂²

yes

—

C₂×#88034

C₂²

yes

characteristic

C₂×#88034

C₂²

yes

—

C₂×#88034

C₂²

yes

characteristic

C₄×Q₈×C₁₃

C₈

yes

characteristic

C₂×C₁₃×C₁₆

C₂×C₄

yes

characteristic

C₂×C₁₃×C₁₆

C₂×C₄

yes

—

C₂×C₁₃×C₁₆

C₂³

yes

characteristic

C₁₃×C₄⋊C₈

C₈

yes

—

C₁₃×C₄⋊C₈

C₈

yes

characteristic

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

C₄×C₈×C₁₃

C₂×C₄

yes

—

#88034

—

#88034

—

#88034

Q₈

yes

complemented

#13660

—

maximal, Sylow

C₂×Q₈×C₁₃

C₁₆

yes

characteristic

C₂×C₈×C₁₃

C₂×C₈

yes

characteristic

C₂×C₈×C₁₃

C₂×C₈

yes

—

C₂×C₈×C₁₃

C₂²×C₄

yes

characteristic

C₁₃×C₄⋊C₄

C₁₆

yes

—

C₁₃×C₄⋊C₄

C₁₆

yes

characteristic

C₄²×C₁₃

C₂×C₈

yes

—

C₄²×C₁₃

C₂×C₈

yes

characteristic

C₁₃×C₁₆

—

C₁₃×C₁₆

—

C₁₃×C₁₆

C₂×Q₈

yes

characteristic

C₁₃×C₁₆

#41

yes

characteristic

Q₈×C₁₆

D₁₃

yes

characteristic

C₄⋊C₃₂

—

C₄⋊C₃₂

—

#1397

—

C₄×C₃₂

—

Q₈×C₁₃

—

C₂×C₄×C₁₃

C₂×C₁₆

yes

—

C₂×C₄×C₁₃

C₂×C₁₆

yes

characteristic

C₂×C₄×C₁₃

C₂×C₁₆

yes

—

C₂×C₄×C₁₃

C₂×C₁₆

yes

characteristic

C₂×C₄×C₁₃

C₂²×C₈

yes

characteristic

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

C₄×Q₈

yes

characteristic

C₈×C₁₃

#131

yes

characteristic

C₈×Q₈

Dic₁₃

yes

characteristic

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

C₄⋊C₁₆

Dic₁₃

yes

characteristic

C₄⋊C₁₆

Dic₁₃

yes

—

C₄×C₁₆

C₂×D₁₃

yes

characteristic

C₄×C₁₆

C₂×D₁₃

yes

—

C₂²×C₁₃

C₂²×C₁₆

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

M₆₄

yes

—

C₄×C₁₃

C₈×Q₈

yes

C₄×C₁₃

yes

C₄×Q₈

yes

C₂×C₁₆

C₂×Dic₁₃

yes

characteristic

C₂×C₁₆

C₂×Dic₁₃

yes

—

C₂×C₁₆

C₂²×D₁₃

yes

characteristic

C₄⋊C₈

#1055

yes

—

C₄⋊C₈

#1055

yes

characteristic

C₄×C₈

C₂×Dic₁₃

yes

characteristic

C₄×C₈

C₂×Dic₁₃

yes

—

C₃₂

—

C₃₂

—

C₃₂

—

C₂×C₁₃

Q₈×C₁₆

yes

characteristic

C₂×C₁₃

C₂×M₆₄

yes

C₂×C₁₃

yes

C₂×Q₈

yes

C₂×C₈

yes

C₂×C₈

yes

—

C₂×C₈

C₂²×Dic₁₃

yes

characteristic

C₄⋊C₄

#6090

yes

—

C₄⋊C₄

#6090

yes

characteristic

C₄²

C₂×#1055

yes

—

C₄²

C₂×#1055

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

Q₈×D₁₃

yes

C₁₆

yes

C₁₃

yes

characteristic, complemented, Sylow

Q₈

—

C₂×C₄

C₂×#6090

yes

—

C₂×C₄

C₂×#6090

yes

characteristic

C₂×C₄

C₂×#6090

yes

—

C₂×C₄

C₂×#6090

yes

characteristic

100

C₂×C₄

C₂²×#1055

yes

characteristic

101

C₈

—

102

C₈

—

103

C₈

Q₈×Dic₁₃

yes

characteristic

104

C₈

#88202

yes

characteristic

105

C₂²

C₂²×#6090

yes

characteristic

106

C₄

—

107

C₄

—

108

C₄

—

109

C₄

#11718091

yes

—

110

C₄

Q₈×#1055

yes

characteristic

111

C₄

#11718801

yes

characteristic

112

C₂

Q₈×#6090

yes

characteristic

113

C₂

C₂×#11718091

yes

characteristic

114

C₂

#13117130

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₁₃
commutator, socle, prime radical	C₂²×C₁₃	C₂²×C₁₆
Fitting	Q₈×C₁₃×C₁₆	C₂
cosocle	C₂×C₁₃×C₁₆	C₂³
hypercenter	Q₈×C₁₆	D₁₃
hypocenter	C₁₃	#13660
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₁₆ ◁ Q₈×C₁₆

subgroup

G/H

center, Frattini

C₂×C₁₆

C₂²×D₁₃

commutator, socle, prime radical

C₂²×C₁₃

C₂²×C₁₆

Fitting

Q₈×C₁₃×C₁₆

C₂

cosocle

C₂×C₁₃×C₁₆

C₂³

hypercenter

Q₈×C₁₆

D₁₃

C₁₃

G ▷ C₂²×C₁₃ ▷ C₁₃

C₁ ◁ C₂×C₁₆ ◁ Q₈×C₁₆

N	H ≅ G/N	type
24. #88034	Q₈	semidirect
94. C₁₃	#13660	semidirect

H ≅ G/N

type

24. #88034

Q₈

semidirect

94. C₁₃

#13660

semidirect

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

subgroup	index
C₄⋊#88034	2
#13112970	2
C₄×#88034	2
Q₈×C₁₃×C₁₆	2
#13660	13

subgroup

index

Q₈×C₁₃×C₁₆

Subgroups — #15783555

Conjugacy classes of subgroups

Complements

Maximal subgroups