Group #15673142 — subgroups

Subgroups — #15673142

Order 2944.

296 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 173 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	C₂×C₄×C₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₄²×Dic₂₃	C₂	1	1	yes	maximal, characteristic
3	#13092171	C₂	1	4	yes	maximal
4	C₂×C₄×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₈×C₂₃	C₂²	1	2	yes	—
6	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
7	C₂×C₄×Dic₂₃	C₄	1	2	yes	—
8	C₂×C₄×Dic₂₃	C₄	1	2	yes	—
9	C₂×C₄×Dic₂₃	C₂²	1	2	yes	—
10	C₂²×C₂₃⋊C₈	C₂²	1	2	yes	—
11	C₄×C₈×C₂₃	C₄	1	4	yes	complemented
12	C₄×C₂₃⋊C₈	C₄	1	4	yes	complemented
13	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
16	C₂²×Dic₂₃	C₂×C₄	1	4	yes	—
17	C₂×C₈×C₂₃	—	2	4	no	—
18	C₂×C₈×C₂₃	C₂×C₄	1	4	yes	—
19	C₄²×C₂₃	C₂×C₄	1	2	yes	—
20	C₄²×C₂₃	C₂×C₄	1	2	yes	—
21	C₄×Dic₂₃	—	2	4	no	—
22	C₄×Dic₂₃	—	2	4	no	—
23	C₄×Dic₂₃	C₈	1	8	yes	complemented
24	C₂×C₂₃⋊C₈	—	2	4	no	—
25	C₂×C₂₃⋊C₈	C₂×C₄	1	4	yes	—
26	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
27	C₂×C₄×C₂₃	C₄²	1	2	yes	—
28	C₂×C₄×C₂₃	C₄²	1	2	yes	—
29	C₂×C₄×C₂₃	C₂×C₈	1	4	yes	—
30	C₂×C₄×C₂₃	M₁₆	1	4	yes	—
31	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₄×C₂₃	#41	1	4	yes	—
34	C₂×Dic₂₃	—	2	4	no	—
35	C₂×Dic₂₃	—	2	4	no	—
36	C₂×Dic₂₃	C₂×C₈	1	8	yes	—
37	C₈×C₂₃	—	2	4	no	—
38	C₂₃⋊C₈	—	2	4	no	—
39	#1744	—	23	1	no	maximal, Sylow
40	C₂²×C₂₃	C₂×C₄²	1	1	yes	characteristic
41	C₂²×C₂₃	#117	1	2	yes	—
42	C₂²×C₂₃	C₂²×C₈	1	2	yes	—
43	C₂²×C₂₃	C₂×M₁₆	1	2	yes	—
44	C₄×C₂₃	C₄×C₈	1	4	yes	—
45	C₄×C₂₃	#97	1	4	yes	—
46	C₄×C₂₃	#117	1	2	yes	—
47	C₄×C₂₃	#117	1	2	yes	—
48	Dic₂₃	—	2	8	no	—
49	C₂×C₄×C₈	—	23	1	no	—
50	C₂×C₄×C₈	D₂₃	1	1	yes	characteristic
51	C₄³	—	23	1	no	—
52	#336	—	23	4	no	—
53	C₂×C₂₃	#376	1	1	yes	characteristic
54	C₂×C₂₃	C₂×C₄×C₈	1	1	yes	characteristic
55	C₂×C₂₃	C₂×#97	1	1	yes	characteristic
56	C₂×C₂₃	#431	1	4	yes	—
57	C₂²×C₈	—	23	2	no	—
58	C₂²×C₈	C₂×D₂₃	1	2	yes	—
59	C₂×C₄²	—	23	2	no	—
60	C₂×C₄²	—	23	2	no	—
61	C₂×C₄²	—	23	2	no	—
62	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
63	C₄×C₈	—	23	4	no	—
64	C₄×C₈	Dic₂₃	1	4	yes	complemented
65	C₂₃	#1744	1	1	yes	characteristic, complemented, Sylow
66	C₂²×C₄	—	23	4	no	—
67	C₂²×C₄	C₄×D₂₃	1	1	yes	characteristic
68	C₂²×C₄	C₄×D₂₃	1	1	yes	characteristic
69	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
70	C₂×C₈	—	2	4	no	—
71	C₂×C₈	—	23	4	no	—
72	C₂×C₈	—	46	4	no	—
73	C₂×C₈	C₂×Dic₂₃	1	4	yes	—
74	C₄²	—	23	8	no	—
75	C₄²	—	46	4	no	—
76	C₄²	—	46	4	no	—
77	C₄²	C₄×D₂₃	1	2	yes	—
78	C₄²	C₂×Dic₂₃	1	2	yes	—
79	C₂³	C₂×C₄×D₂₃	1	1	yes	characteristic
80	C₂×C₄	—	23	8	no	—
81	C₂×C₄	—	46	4	no	—
82	C₂×C₄	—	46	4	no	—
83	C₂×C₄	C₈×D₂₃	1	4	yes	—
84	C₂×C₄	#67262	1	4	yes	—
85	C₂×C₄	C₄×Dic₂₃	1	2	yes	—
86	C₂×C₄	C₄×Dic₂₃	1	2	yes	—
87	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
88	C₂×C₄	#67288	1	4	yes	—
89	C₂×C₄	C₂²×Dic₂₃	1	1	yes	characteristic
90	C₈	—	2	4	no	—
91	C₈	—	46	4	no	—
92	C₂²	#10625043	1	2	yes	—
93	C₂²	C₂×C₈×D₂₃	1	2	yes	—
94	C₂²	C₂×#67262	1	2	yes	—
95	C₂²	C₂×C₄×Dic₂₃	1	1	yes	characteristic
96	C₄	—	46	8	no	—
97	C₄	C₈×Dic₂₃	1	4	yes	—
98	C₄	#10624997	1	4	yes	—
99	C₄	#10625043	1	2	yes	—
100	C₄	#10625097	1	2	yes	—
101	C₂	#13092263	1	1	yes	characteristic
102	C₂	C₂×C₈×Dic₂₃	1	1	yes	characteristic
103	C₂	C₂×#10624997	1	1	yes	characteristic
104	C₂	#13092468	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×C₂₃⋊C₈

C₂

yes

maximal, characteristic

C₄²×Dic₂₃

C₂

yes

maximal, characteristic

#13092171

C₂

yes

maximal

C₂×C₄×C₈×C₂₃

C₂

yes

maximal, characteristic

C₂²×C₈×C₂₃

C₂²

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂×C₄×Dic₂₃

C₄

yes

—

C₂×C₄×Dic₂₃

C₄

yes

—

C₂×C₄×Dic₂₃

C₂²

yes

—

C₂²×C₂₃⋊C₈

C₂²

yes

—

C₄×C₈×C₂₃

C₄

yes

complemented

C₄×C₂₃⋊C₈

C₄

yes

complemented

C₂²×C₄×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

C₂×C₄

yes

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

C₂×C₄

yes

—

C₄²×C₂₃

C₂×C₄

yes

—

C₄²×C₂₃

C₂×C₄

yes

—

C₄×Dic₂₃

—

C₄×Dic₂₃

—

C₄×Dic₂₃

C₈

yes

complemented

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

C₂×C₄

yes

—

C₂³×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₄²

yes

—

C₂×C₄×C₂₃

C₄²

yes

—

C₂×C₄×C₂₃

C₂×C₈

yes

—

C₂×C₄×C₂₃

M₁₆

yes

—

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

#41

yes

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

C₂×C₈

yes

—

C₈×C₂₃

—

C₂₃⋊C₈

—

#1744

—

maximal, Sylow

C₂²×C₂₃

C₂×C₄²

yes

characteristic

C₂²×C₂₃

#117

yes

—

C₂²×C₂₃

C₂²×C₈

yes

—

C₂²×C₂₃

C₂×M₁₆

yes

—

C₄×C₂₃

C₄×C₈

yes

—

C₄×C₂₃

#97

yes

—

C₄×C₂₃

#117

yes

—

C₄×C₂₃

#117

yes

—

Dic₂₃

—

C₂×C₄×C₈

—

C₂×C₄×C₈

D₂₃

yes

characteristic

C₄³

—

#336

—

C₂×C₂₃

#376

yes

characteristic

C₂×C₂₃

C₂×C₄×C₈

yes

characteristic

C₂×C₂₃

C₂×#97

yes

characteristic

C₂×C₂₃

#431

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×D₂₃

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

C₄×C₈

—

C₄×C₈

Dic₂₃

yes

complemented

C₂₃

#1744

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

C₄×D₂₃

yes

characteristic

C₂²×C₄

C₄×D₂₃

yes

characteristic

C₂²×C₄

C₂²×D₂₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×Dic₂₃

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₄×D₂₃

yes

—

C₄²

C₂×Dic₂₃

yes

—

C₂³

C₂×C₄×D₂₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₈×D₂₃

yes

—

C₂×C₄

#67262

yes

—

C₂×C₄

C₄×Dic₂₃

yes

—

C₂×C₄

C₄×Dic₂₃

yes

—

C₂×C₄

C₂×C₄×D₂₃

yes

characteristic

C₂×C₄

#67288

yes

—

C₂×C₄

C₂²×Dic₂₃

yes

characteristic

C₈

—

C₈

—

C₂²

#10625043

yes

—

C₂²

C₂×C₈×D₂₃

yes

—

C₂²

C₂×#67262

yes

—

C₂²

C₂×C₄×Dic₂₃

yes

characteristic

C₄

—

C₄

C₈×Dic₂₃

yes

—

C₄

#10624997

yes

—

C₄

#10625043

yes

—

100

C₄

#10625097

yes

—

101

C₂

#13092263

yes

characteristic

102

C₂

C₂×C₈×Dic₂₃

yes

characteristic

103

C₂

C₂×#10624997

yes

characteristic

104

C₂

#13092468

yes

—

	subgroup	G/H
center	C₂×C₄²	C₂×D₂₃
commutator	C₂×C₂₃	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂×C₄×C₈×C₂₃	C₂
socle, prime radical	C₂³×C₂₃	C₂²×C₄
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	C₂×C₄×C₈	D₂₃
hypocenter	C₂₃	#1744
LCS	G ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂×C₄²

C₂×D₂₃

commutator

C₂×C₂₃

C₂×C₄×C₈

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂×C₄×C₈×C₂₃

C₂

socle, prime radical

C₂³×C₂₃

C₂²×C₄

cosocle

C₂²×C₄×C₂₃

C₂³

hypercenter

C₂×C₄×C₈

D₂₃

C₂₃

G ▷ C₂×C₂₃ ▷ C₂₃

C₁ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
11. C₄×C₈×C₂₃	C₄	semidirect
12. C₄×C₂₃⋊C₈	C₄	semidirect
23. C₄×Dic₂₃	C₈	semidirect
64. C₄×C₈	Dic₂₃	semidirect
65. C₂₃	#1744	semidirect

H ≅ G/N

type

11. C₄×C₈×C₂₃

C₄

semidirect

12. C₄×C₂₃⋊C₈

C₄

semidirect

23. C₄×Dic₂₃

C₈

semidirect

64. C₄×C₈

Dic₂₃

semidirect

65. C₂₃

#1744

semidirect

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

subgroup	index
C₂×C₄×C₂₃⋊C₈	2
C₄²×Dic₂₃	2
#13092171	2
C₂×C₄×C₈×C₂₃	2
#1744	23

subgroup

index

C₂×C₄×C₂₃⋊C₈

C₄²×Dic₂₃

#13092171

C₂×C₄×C₈×C₂₃

#1744

Subgroups — #15673142

Conjugacy classes of subgroups

Complements

Maximal subgroups