Group #15672767 — subgroups

Subgroups — #15672767

Order 2944.

156 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 61 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	#13092797	C₂	1	1	yes	maximal, characteristic
2	#13092702	C₂	1	2	yes	maximal
3	#13092170	C₂	1	1	yes	maximal, characteristic
4	#13092164	C₂	1	2	yes	maximal
5	C₈×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic
6	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
7	C₂×Dic₉₂	—	2	1	no	—
8	C₂×Dic₉₂	—	2	1	no	—
9	#10625040	C₂²	1	2	yes	—
10	C₂₃×C₄⋊C₈	—	2	1	no	—
11	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
12	C₄×C₈×C₂₃	—	2	1	no	—
13	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
14	#10625017	—	2	2	no	—
15	#10624990	—	2	2	no	—
16	C₄⋊(C₂₃⋊C₈)	C₂²	1	2	yes	—
17	C₂×Q₈×C₂₃	D₄	1	1	yes	characteristic
18	C₂×Dic₄₆	—	2	2	no	—
19	C₂×C₈×C₂₃	—	2	1	no	—
20	C₂×C₈×C₂₃	D₄	1	1	yes	characteristic
21	C₂×C₈×C₂₃	D₄	1	1	yes	characteristic
22	C₄⋊C₄×C₂₃	—	2	1	no	—
23	C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
24	C₄²×C₂₃	—	2	1	no	—
25	C₄²×C₂₃	C₂³	1	1	yes	characteristic
26	C₄⋊Dic₂₃	—	2	2	no	—
27	C₄⋊Dic₂₃	—	2	2	no	—
28	C₂×C₂₃⋊C₈	—	2	2	no	—
29	Dic₉₂	—	4	2	no	—
30	Dic₉₂	—	4	2	no	—
31	Q₈×C₂₃	—	2	1	no	—
32	Q₈×C₂₃	Q₁₆	1	2	yes	complemented
33	C₂×C₄×C₂₃	—	2	1	no	—
34	C₂×C₄×C₂₃	—	2	1	no	—
35	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
38	C₂×Dic₂₃	—	4	2	no	—
39	Dic₄₆	—	4	2	no	—
40	Dic₄₆	—	4	2	no	—
41	C₈×C₂₃	—	2	1	no	—
42	C₈×C₂₃	—	4	1	no	—
43	C₈×C₂₃	SD₁₆	1	2	yes	—
44	C₂₃⋊C₈	—	4	2	no	—
45	#1670	—	23	1	no	maximal, Sylow
46	C₂²×C₂₃	#121	1	1	yes	characteristic
47	C₄×C₂₃	—	2	2	no	—
48	C₄×C₂₃	—	2	1	no	—
49	C₄×C₂₃	—	4	1	no	—
50	C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
51	C₄×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
52	C₄×C₂₃	#135	1	1	yes	characteristic
53	C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
54	Dic₂₃	—	8	2	no	—
55	#494	—	23	1	no	—
56	#475	—	23	2	no	—
57	C₈×Q₈	D₂₃	1	1	yes	characteristic
58	#335	—	23	1	no	—
59	#332	—	23	2	no	—
60	C₂×C₂₃	#460	1	1	yes	characteristic
61	C₂×C₂₃	#467	1	1	yes	characteristic
62	C₂×C₂₃	#473	1	1	yes	characteristic
63	C₂×Q₁₆	—	46	1	no	—
64	C₂×Q₁₆	—	46	1	no	—
65	C₄⋊Q₈	—	23	2	no	—
66	C₄×Q₈	C₂×D₂₃	1	1	yes	characteristic
67	#106	—	46	2	no	—
68	C₄⋊C₈	—	2	1	no	—
69	C₄⋊C₈	—	23	2	no	—
70	C₄⋊C₈	C₂×D₂₃	1	1	yes	characteristic
71	#103	—	46	2	no	—
72	C₄×C₈	—	2	1	no	—
73	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic
74	C₂₃	#1670	1	1	yes	characteristic, complemented, Sylow
75	C₂×Q₈	—	46	2	no	—
76	C₂×Q₈	D₉₂	1	1	yes	characteristic
77	Q₁₆	—	92	2	no	—
78	Q₁₆	—	92	2	no	—
79	C₂×C₈	—	2	1	no	—
80	C₂×C₈	—	46	2	no	—
81	C₂×C₈	C₂²⋊D₂₃	1	1	yes	characteristic
82	C₂×C₈	C₂²⋊D₂₃	1	1	yes	characteristic
83	C₄⋊C₄	—	2	1	no	—
84	C₄⋊C₄	—	46	2	no	—
85	C₄⋊C₄	—	46	2	no	—
86	C₄⋊C₄	D₉₂	1	1	yes	characteristic
87	C₄²	—	2	1	no	—
88	C₄²	C₂²×D₂₃	1	1	yes	characteristic
89	Q₈	—	2	1	no	—
90	Q₈	—	92	2	no	—
91	Q₈	—	92	2	no	—
92	Q₈	Dic₉₂	1	2	yes	complemented
93	C₂×C₄	—	2	1	no	—
94	C₂×C₄	—	2	1	no	—
95	C₂×C₄	—	92	2	no	—
96	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
97	C₂×C₄	#67288	1	1	yes	characteristic
98	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
99	C₈	—	2	1	no	—
100	C₈	—	4	1	no	—
101	C₈	—	92	2	no	—
102	C₈	#67274	1	2	yes	—
103	C₂²	#10625101	1	1	yes	characteristic
104	C₄	—	2	2	no	—
105	C₄	—	2	1	no	—
106	C₄	—	4	1	no	—
107	C₄	—	184	2	no	—
108	C₄	#10625075	1	1	yes	characteristic
109	C₄	C₂×Dic₉₂	1	1	yes	characteristic
110	C₄	C₂×#67274	1	1	yes	characteristic
111	C₄	#10625120	1	1	yes	characteristic
112	C₂	#13092617	1	1	yes	characteristic
113	C₂	#13092658	1	1	yes	characteristic
114	C₂	#13092692	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13092797

C₂

yes

maximal, characteristic

#13092702

C₂

yes

maximal

#13092170

C₂

yes

maximal, characteristic

#13092164

C₂

yes

maximal

C₈×Q₈×C₂₃

C₂

yes

maximal, characteristic

C₄×Q₈×C₂₃

C₂²

yes

characteristic

C₂×Dic₉₂

—

C₂×Dic₉₂

—

#10625040

C₂²

yes

—

C₂₃×C₄⋊C₈

—

C₂₃×C₄⋊C₈

C₂²

yes

characteristic

C₄×C₈×C₂₃

—

C₄×C₈×C₂₃

C₂²

yes

characteristic

#10625017

—

#10624990

—

C₄⋊(C₂₃⋊C₈)

C₂²

yes

—

C₂×Q₈×C₂₃

D₄

yes

characteristic

C₂×Dic₄₆

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

D₄

yes

characteristic

C₂×C₈×C₂₃

D₄

yes

characteristic

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

D₄

yes

characteristic

C₄²×C₂₃

—

C₄²×C₂₃

C₂³

yes

characteristic

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₂×C₂₃⋊C₈

—

Dic₉₂

—

Dic₉₂

—

Q₈×C₂₃

—

Q₈×C₂₃

Q₁₆

yes

complemented

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

#41

yes

characteristic

C₂×Dic₂₃

—

Dic₄₆

—

Dic₄₆

—

C₈×C₂₃

—

C₈×C₂₃

—

C₈×C₂₃

SD₁₆

yes

—

C₂₃⋊C₈

—

—

C₂²×C₂₃

yes

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₂×SD₁₆

yes

characteristic

C₄×C₂₃

C₂×Q₁₆

yes

C₄×C₂₃

yes

C₄×C₂₃

yes

Dic₂₃

—

#494

—

#475

—

C₈×Q₈

D₂₃

yes

characteristic

#335

—

#332

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄×Q₈

C₂×D₂₃

yes

characteristic

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₂₃

yes

characteristic

#103

—

C₄×C₈

—

C₄×C₈

C₂×D₂₃

yes

characteristic

C₂₃

#1670

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

D₉₂

yes

characteristic

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₂₃

yes

characteristic

C₂×C₈

C₂²⋊D₂₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₉₂

yes

characteristic

C₄²

—

C₄²

C₂²×D₂₃

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

Q₈

Dic₉₂

yes

complemented

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×D₉₂

yes

characteristic

C₂×C₄

#67288

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

C₈

—

100

C₈

—

101

C₈

—

102

C₈

#67274

yes

—

103

C₂²

#10625101

yes

characteristic

104

C₄

—

105

C₄

—

106

C₄

—

107

C₄

—

184

—

108

C₄

#10625075

yes

characteristic

109

C₄

C₂×Dic₉₂

yes

characteristic

110

C₄

C₂×#67274

yes

characteristic

111

C₄

#10625120

yes

characteristic

112

C₂

#13092617

yes

characteristic

113

C₂

#13092658

yes

characteristic

114

C₂

#13092692

yes

characteristic

	subgroup	G/H
center	C₂²	#10625101
commutator, cosocle	C₄²×C₂₃	C₂³
Frattini	C₄²	C₂²×D₂₃
Fitting	C₈×Q₈×C₂₃	C₂
socle, prime radical	C₂²×C₂₃	#121
hypercenter	C₈×Q₈	D₂₃
hypocenter	C₂₃	#1670
LCS	G ▷ C₄²×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈

subgroup

G/H

center

C₂²

#10625101

commutator, cosocle

C₄²×C₂₃

C₂³

Frattini

C₄²

C₂²×D₂₃

Fitting

C₈×Q₈×C₂₃

C₂

socle, prime radical

C₂²×C₂₃

#121

hypercenter

C₈×Q₈

D₂₃

hypocenter

C₂₃

#1670

LCS

G ▷ C₄²×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈

N	H ≅ G/N	type
32. Q₈×C₂₃	Q₁₆	semidirect
74. C₂₃	#1670	semidirect
92. Q₈	Dic₉₂	semidirect

H ≅ G/N

type

32. Q₈×C₂₃

Q₁₆

semidirect

74. C₂₃

#1670

semidirect

92. Q₈

Dic₉₂

semidirect

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

subgroup	index
#13092797	2
#13092702	2
#13092170	2
#13092164	2
C₈×Q₈×C₂₃	2
#1670	23

subgroup

index

C₈×Q₈×C₂₃

Subgroups — #15672767

Conjugacy classes of subgroups

Complements

Maximal subgroups