Group #15672538 — subgroups

Subgroups — #15672538

Order 2944.

206 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 57 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	#13093147	C₂	1	1	yes	maximal, characteristic
2	#13092796	C₂	1	1	yes	maximal, characteristic
3	#13092713	C₂	1	2	yes	maximal
4	#13092147	C₂	1	2	yes	maximal
5	C₂₃×#333	C₂	1	1	yes	maximal, characteristic
6	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
7	#10625115	—	2	2	no	—
8	C₂×#67275	—	2	2	no	—
9	#10625059	—	2	2	no	—
10	#10625058	—	2	2	no	—
11	#10625040	C₂²	1	1	yes	characteristic
12	C₄×Dic₄₆	C₂²	1	2	yes	—
13	C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
14	#10625000	—	2	2	no	—
15	#10624998	—	2	2	no	—
16	C₄×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₂₃	D₄	1	2	yes	—
18	C₂×Dic₄₆	—	2	1	no	—
19	C₂×Dic₄₆	—	2	2	no	—
20	C₂×Dic₄₆	D₄	1	2	yes	—
21	C₂×C₈×C₂₃	—	2	2	no	—
22	C₄⋊C₄×C₂₃	—	2	2	no	—
23	C₄²×C₂₃	C₂³	1	1	yes	characteristic
24	#67275	—	4	4	no	—
25	C₄⋊Dic₂₃	—	2	2	no	—
26	C₄⋊Dic₂₃	D₄	1	2	yes	—
27	Dic₂₃⋊C₄	—	2	4	no	—
28	Dic₂₃⋊C₄	—	2	2	no	—
29	C₄×Dic₂₃	—	2	2	no	—
30	C₄×Dic₂₃	—	2	2	no	—
31	C₂×C₂₃⋊C₈	—	2	1	no	—
32	Q₈×C₂₃	—	4	2	no	—
33	C₂×C₄×C₂₃	—	2	2	no	—
34	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
36	C₂×Dic₂₃	—	2	2	no	—
37	C₂×Dic₂₃	—	2	2	no	—
38	C₂×Dic₂₃	—	4	1	no	—
39	Dic₄₆	—	2	2	no	—
40	Dic₄₆	—	4	2	no	—
41	Dic₄₆	—	8	2	no	—
42	Dic₄₆	Q₁₆	1	4	yes	complemented
43	C₈×C₂₃	—	4	2	no	—
44	C₂₃⋊C₈	—	2	2	no	—
45	#1633	—	23	1	no	maximal, Sylow
46	C₂²×C₂₃	#120	1	1	yes	characteristic
47	C₄×C₂₃	—	2	1	no	—
48	C₄×C₂₃	—	4	2	no	—
49	C₄×C₂₃	C₂×Q₁₆	1	2	yes	—
50	C₄×C₂₃	#137	1	2	yes	—
51	Dic₂₃	—	2	4	no	—
52	Dic₂₃	—	4	2	no	—
53	Dic₂₃	—	8	1	no	—
54	Q₈²	—	23	1	no	—
55	#494	—	23	1	no	—
56	#477	—	23	2	no	—
57	#333	D₂₃	1	1	yes	characteristic
58	Q₈⋊C₈	—	23	2	no	—
59	C₂×C₂₃	#451	1	2	yes	—
60	C₂×C₂₃	#453	1	1	yes	characteristic
61	C₂×Q₁₆	—	46	2	no	—
62	C₄⋊Q₈	—	23	1	no	—
63	C₄⋊Q₈	—	46	2	no	—
64	C₄⋊Q₈	—	46	2	no	—
65	C₄⋊Q₈	C₂×D₂₃	1	1	yes	characteristic
66	C₄×Q₈	—	23	2	no	—
67	C₄×Q₈	—	46	2	no	—
68	#107	—	46	2	no	—
69	C₄⋊C₈	C₂×D₂₃	1	2	yes	—
70	#103	—	46	2	no	—
71	C₄×C₈	—	23	1	no	—
72	C₂₃	#1633	1	1	yes	characteristic, complemented, Sylow
73	C₂×Q₈	—	23	2	no	—
74	C₂×Q₈	—	46	2	no	—
75	C₂×Q₈	—	46	1	no	—
76	C₂×Q₈	D₉₂	1	2	yes	—
77	Q₁₆	—	92	4	no	—
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	—	46	1	no	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	—	23	2	no	—
82	C₄⋊C₄	—	46	4	no	—
83	C₄⋊C₄	—	46	2	no	—
84	C₄⋊C₄	—	46	2	no	—
85	C₄²	—	46	2	no	—
86	C₄²	—	46	2	no	—
87	C₄²	C₂²×D₂₃	1	1	yes	characteristic
88	Q₈	—	4	2	no	—
89	Q₈	—	23	4	no	—
90	Q₈	—	46	2	no	—
91	Q₈	—	92	2	no	—
92	Q₈	—	184	2	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	46	2	no	—
95	C₂×C₄	—	46	2	no	—
96	C₂×C₄	—	92	1	no	—
97	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
98	C₂×C₄	D₄×D₂₃	1	2	yes	—
99	C₈	—	4	2	no	—
100	C₈	—	46	2	no	—
101	C₂²	#10625053	1	1	yes	characteristic
102	C₄	—	2	1	no	—
103	C₄	—	4	2	no	—
104	C₄	—	46	4	no	—
105	C₄	—	92	2	no	—
106	C₄	—	184	1	no	—
107	C₄	#10625086	1	2	yes	—
108	C₄	Q₁₆×D₂₃	1	2	yes	—
109	C₂	#13092574	1	2	yes	—
110	C₂	#13092587	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13093147

C₂

yes

maximal, characteristic

#13092796

C₂

yes

maximal, characteristic

#13092713

C₂

yes

maximal

#13092147

C₂

yes

maximal

C₂₃×#333

C₂

yes

maximal, characteristic

C₂₃×C₄⋊Q₈

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₄×Dic₄₆

C₂²

yes

—

C₂₃×C₄⋊C₈

C₂²

yes

—

#10625000

—

#10624998

—

C₄×C₂₃⋊C₈

C₂²

yes

characteristic

C₂×Q₈×C₂₃

D₄

yes

—

C₂×Dic₄₆

—

C₂×Dic₄₆

—

C₂×Dic₄₆

D₄

yes

—

C₂×C₈×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄²×C₂₃

C₂³

yes

characteristic

#67275

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

D₄

yes

—

Dic₂₃⋊C₄

—

Dic₂₃⋊C₄

—

C₄×Dic₂₃

—

C₄×Dic₂₃

—

C₂×C₂₃⋊C₈

—

Q₈×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

Dic₄₆

—

Dic₄₆

—

Dic₄₆

—

Dic₄₆

Q₁₆

yes

complemented

C₈×C₂₃

—

C₂₃⋊C₈

—

—

C₂²×C₂₃

yes

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₂×Q₁₆

yes

—

C₄×C₂₃

#137

yes

—

Dic₂₃

—

Dic₂₃

—

Dic₂₃

—

Q₈²

—

#494

—

#477

—

#333

D₂₃

yes

characteristic

Q₈⋊C₈

—

C₂×C₂₃

#451

yes

—

C₂×C₂₃

#453

yes

characteristic

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₂₃

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

#107

—

C₄⋊C₈

C₂×D₂₃

yes

—

#103

—

C₄×C₈

—

C₂₃

#1633

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

D₉₂

yes

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂²×D₂₃

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

Q₈

—

Q₈

—

184

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×D₉₂

yes

characteristic

C₂×C₄

D₄×D₂₃

yes

—

C₈

—

100

C₈

—

101

C₂²

#10625053

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

—

184

—

107

C₄

#10625086

yes

—

108

C₄

Q₁₆×D₂₃

yes

—

109

C₂

#13092574

yes

—

110

C₂

#13092587

yes

characteristic

	subgroup	G/H
center	C₂²	#10625053
commutator, cosocle	C₄²×C₂₃	C₂³
Frattini	C₄²	C₂²×D₂₃
Fitting	C₂₃×#333	C₂
socle, prime radical	C₂²×C₂₃	#120
hypercenter	#333	D₂₃
hypocenter	C₂₃	#1633
LCS	G ▷ C₄²×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #333

subgroup

G/H

center

C₂²

#10625053

commutator, cosocle

C₄²×C₂₃

C₂³

Frattini

C₄²

C₂²×D₂₃

Fitting

C₂₃×#333

C₂

C₂²×C₂₃

D₂₃

C₂₃

G ▷ C₄²×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #333

N	H ≅ G/N	type
42. Dic₄₆	Q₁₆	semidirect
72. C₂₃	#1633	semidirect

H ≅ G/N

type

42. Dic₄₆

Q₁₆

semidirect

72. C₂₃

#1633

semidirect

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

subgroup	index
#13093147	2
#13092796	2
#13092713	2
#13092147	2
C₂₃×#333	2
#1633	23

subgroup

index

C₂₃×#333

Subgroups — #15672538

Conjugacy classes of subgroups

Complements

Maximal subgroups