Group #15674363 — subgroups

Subgroups — #15674363

Order 2944.

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

normal subgroups only

This group has 252 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	C₂×C₄×C₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic
2	#13092283	C₂	1	1	yes	maximal, characteristic
3	#13092179	C₂	1	4	yes	maximal
4	C₂×C₂₃×#125	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#125	—	2	2	no	—
6	C₂₃×#125	C₄	1	4	yes	complemented
7	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
8	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
9	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
11	C₂²×C₂₃⋊C₈	C₂²	1	2	yes	—
12	C₂₃⋊#95	—	2	2	no	—
13	C₄×C₂₃⋊C₈	—	2	2	no	—
14	C₂²×C₄×C₂₃	—	2	2	no	—
15	C₂²×C₄×C₂₃	D₄	1	2	yes	—
16	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
17	C₂²×Dic₂₃	—	2	2	no	—
18	C₄⋊C₄×C₂₃	—	2	8	no	—
19	C₄⋊C₄×C₂₃	—	2	2	no	—
20	C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
21	C₄²×C₂₃	C₂×C₄	1	2	yes	—
22	C₄²×C₂₃	D₄	1	2	yes	—
23	C₄⋊Dic₂₃	—	2	4	no	—
24	C₂×C₂₃⋊C₈	—	2	2	no	—
25	C₂×C₂₃⋊C₈	—	2	2	no	—
26	C₂×C₂₃⋊C₈	Q₈	1	4	yes	—
27	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
28	C₂×C₄×C₂₃	—	2	4	no	—
29	C₂×C₄×C₂₃	—	2	4	no	—
30	C₂×C₄×C₂₃	—	4	4	no	—
31	C₂×C₄×C₂₃	C₂²⋊C₄	1	4	yes	—
32	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
35	C₂×C₄×C₂₃	#41	1	2	yes	—
36	C₂×Dic₂₃	—	4	2	no	—
37	C₂×Dic₂₃	—	4	4	no	—
38	C₂₃⋊C₈	—	2	4	no	—
39	#1991	—	23	1	no	maximal, Sylow
40	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
41	C₂²×C₂₃	#124	1	1	yes	characteristic
42	C₂²×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
43	C₂²×C₂₃	#135	1	4	yes	—
44	C₄×C₂₃	—	2	4	no	—
45	C₄×C₂₃	—	4	4	no	—
46	C₄×C₂₃	C₄×Q₈	1	2	yes	—
47	C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
48	Dic₂₃	—	8	2	no	—
49	C₂×#125	D₂₃	1	1	yes	characteristic
50	C₂×C₄×C₈	—	23	1	no	—
51	#382	—	23	1	no	—
52	#340	—	23	4	no	—
53	C₂×C₂₃	#391	1	1	yes	characteristic
54	C₂×C₂₃	#416	1	2	yes	—
55	C₂×C₂₃	#488	1	2	yes	—
56	C₂×C₂₃	#499	1	2	yes	—
57	C₂²×C₈	—	23	2	no	—
58	#125	—	2	2	no	—
59	#125	Dic₂₃	1	4	yes	complemented
60	C₂×C₄⋊C₄	—	2	2	no	—
61	C₂×C₄⋊C₄	—	23	2	no	—
62	C₂×C₄⋊C₄	C₂×D₂₃	1	2	yes	—
63	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
64	C₄×C₈	—	46	2	no	—
65	#95	—	46	2	no	—
66	C₂₃	#1991	1	1	yes	characteristic, complemented, Sylow
67	C₂²×C₄	—	2	2	no	—
68	C₂²×C₄	—	46	2	no	—
69	C₂²×C₄	C₂²⋊D₂₃	1	2	yes	—
70	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
71	C₂×C₈	—	23	4	no	—
72	C₂×C₈	—	46	2	no	—
73	C₂×C₈	—	46	2	no	—
74	C₄⋊C₄	—	2	8	no	—
75	C₄⋊C₄	—	2	2	no	—
76	C₄⋊C₄	—	46	4	no	—
77	C₄⋊C₄	C₂×Dic₂₃	1	4	yes	—
78	C₄²	C₂×Dic₂₃	1	2	yes	—
79	C₄²	C₂²⋊D₂₃	1	2	yes	—
80	C₂³	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
81	C₂×C₄	—	2	4	no	—
82	C₂×C₄	—	2	4	no	—
83	C₂×C₄	—	4	4	no	—
84	C₂×C₄	—	92	2	no	—
85	C₂×C₄	—	92	4	no	—
86	C₂×C₄	C₂²⋊Dic₂₃	1	4	yes	—
87	C₂×C₄	Q₈×D₂₃	1	2	yes	—
88	C₂×C₄	#67292	1	2	yes	—
89	C₂×C₄	C₂²×Dic₂₃	1	1	yes	characteristic
90	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
91	C₈	—	46	4	no	—
92	C₂²	#10625115	1	1	yes	characteristic
93	C₂²	#10625118	1	1	yes	characteristic
94	C₂²	#10625121	1	4	yes	—
95	C₂²	C₂×C₂²⋊Dic₂₃	1	1	yes	characteristic
96	C₄	—	2	4	no	—
97	C₄	—	4	4	no	—
98	C₄	—	184	2	no	—
99	C₄	Q₈×Dic₂₃	1	2	yes	—
100	C₄	#10625117	1	2	yes	—
101	C₂	#13092324	1	1	yes	characteristic
102	C₂	#13092413	1	2	yes	—
103	C₂	#13092774	1	2	yes	—
104	C₂	#13092816	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×C₂₃⋊C₈

C₂

yes

maximal, characteristic

#13092283

C₂

yes

maximal, characteristic

#13092179

C₂

yes

maximal

C₂×C₂₃×#125

C₂

yes

maximal, characteristic

C₂₃×#125

—

C₂₃×#125

C₄

yes

complemented

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄⋊C₄×C₂₃

C₂²

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₃

C₂²

yes

—

C₂²×C₂₃⋊C₈

C₂²

yes

—

C₂₃⋊#95

—

C₄×C₂₃⋊C₈

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

C₂×C₄

yes

—

C₄²×C₂₃

C₂×C₄

yes

—

C₄²×C₂₃

D₄

yes

—

C₄⋊Dic₂₃

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

Q₈

yes

—

C₂³×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×Q₈

yes

—

C₂×C₄×C₂₃

#41

yes

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂₃⋊C₈

—

#1991

—

maximal, Sylow

C₂²×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₃

#124

yes

characteristic

C₂²×C₂₃

C₄⋊Q₈

yes

characteristic

C₂²×C₂₃

#135

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₄×Q₈

yes

—

C₄×C₂₃

C₂²⋊Q₈

yes

—

Dic₂₃

—

C₂×#125

D₂₃

yes

characteristic

C₂×C₄×C₈

—

#382

—

#340

—

C₂×C₂₃

#391

yes

characteristic

C₂×C₂₃

#416

yes

—

C₂×C₂₃

#488

yes

—

C₂×C₂₃

#499

yes

—

C₂²×C₈

—

#125

—

#125

Dic₂₃

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₂₃

yes

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

C₄×C₈

—

#95

—

C₂₃

#1991

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₂₃

yes

—

C₂²×C₄

C₂²×D₂₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×Dic₂₃

yes

—

C₄²

C₂×Dic₂₃

yes

—

C₄²

C₂²⋊D₂₃

yes

—

C₂³

C₂×C₂²⋊D₂₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊Dic₂₃

yes

—

C₂×C₄

Q₈×D₂₃

yes

—

C₂×C₄

#67292

yes

—

C₂×C₄

C₂²×Dic₂₃

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

C₈

—

C₂²

yes

C₂²

yes

C₂²

yes

—

C₂²

C₂×C₂²⋊Dic₂₃

yes

characteristic

C₄

—

C₄

—

C₄

—

184

—

C₄

Q₈×Dic₂₃

yes

—

100

C₄

#10625117

yes

—

101

C₂

#13092324

yes

characteristic

102

C₂

#13092413

yes

—

103

C₂

#13092774

yes

—

104

C₂

#13092816

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₂₃
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂×C₂₃×#125	C₂
socle, prime radical	C₂³×C₂₃	C₂×D₄
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	C₂×#125	D₂₃
hypocenter	C₂₃	#1991
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₂₃

commutator

C₂×C₄×C₂₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂×C₂₃×#125

C₂

socle, prime radical

C₂³×C₂₃

C₂×D₄

cosocle

C₂²×C₄×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
6. C₂₃×#125	C₄	semidirect
59. #125	Dic₂₃	semidirect
66. C₂₃	#1991	semidirect

H ≅ G/N

type

6. C₂₃×#125

C₄

semidirect

59. #125

Dic₂₃

semidirect

66. C₂₃

#1991

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
#13092283	2
#13092179	2
C₂×C₂₃×#125	2
#1991	23

subgroup

index

C₂×C₄×C₂₃⋊C₈

#13092283

#13092179

C₂×C₂₃×#125

#1991

Subgroups — #15674363

Conjugacy classes of subgroups

Complements

Maximal subgroups