Group #15674996 — subgroups

Subgroups — #15674996

Order 2944.

184 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 125 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	C₂²×C₂₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	#13092372	C₂	1	1	yes	maximal, characteristic
3	#13092192	C₂	1	2	yes	maximal, complemented
4	C₁₆×Dic₂₃	C₂	1	2	yes	maximal, complemented
5	C₂×C₂₃×M₃₂	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
7	#10625097	C₄	1	1	yes	characteristic
8	C₂×#67267	C₄	1	1	yes	characteristic
9	C₂₃×M₃₂	C₄	1	4	yes	complemented
10	C₂×C₁₆×C₂₃	C₂²	1	2	yes	—
11	#10624997	C₄	1	2	yes	—
12	C₈×Dic₂₃	C₂²	1	2	yes	—
13	C₂×C₂₃⋊C₁₆	C₄	1	4	yes	complemented
14	C₂×C₂₃⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₂×C₂₃⋊C₁₆	C₂²	1	1	yes	characteristic
16	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×C₂₃	C₂×C₄	1	2	yes	—
19	C₂×C₈×C₂₃	C₂×C₄	1	2	yes	—
20	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
21	C₂²⋊Dic₂₃	C₈	1	2	yes	—
22	C₄⋊Dic₂₃	C₈	1	2	yes	—
23	C₄×Dic₂₃	C₂×C₄	1	2	yes	—
24	#67267	C₈	1	4	yes	—
25	C₂×C₂₃⋊C₈	C₂×C₄	1	2	yes	—
26	C₁₆×C₂₃	C₂×C₄	1	4	yes	—
27	C₂₃⋊C₁₆	C₂×C₄	1	2	yes	—
28	C₂₃⋊C₁₆	C₂×C₄	1	2	yes	—
29	C₂³×C₂₃	C₂×C₈	1	1	yes	characteristic
30	C₂×C₄×C₂₃	C₄²	1	2	yes	—
31	C₂×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
32	C₂×C₄×C₂₃	C₂×C₈	1	2	yes	—
33	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
34	C₂×Dic₂₃	C₂×C₈	1	4	yes	—
35	C₈×C₂₃	C₄²	1	2	yes	—
36	C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
37	C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
38	C₂₃⋊C₈	C₂×C₈	1	4	yes	—
39	#2108	—	23	1	no	maximal, Sylow
40	C₂²×C₂₃	—	2	2	no	—
41	C₂²×C₂₃	C₄×C₈	1	2	yes	—
42	C₂²×C₂₃	C₂²×C₈	1	1	yes	characteristic
43	C₄×C₂₃	C₄×C₈	1	2	yes	—
44	C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
45	C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
46	Dic₂₃	—	2	4	no	—
47	C₂×M₃₂	D₂₃	1	1	yes	characteristic
48	C₂²×C₁₆	—	23	1	no	—
49	#405	—	23	1	no	—
50	#346	—	23	2	no	—
51	C₄×C₁₆	—	23	2	no	—
52	C₂×C₂₃	—	2	2	no	—
53	C₂×C₂₃	C₂×C₄×C₈	1	1	yes	characteristic
54	C₂×C₂₃	#504	1	2	yes	—
55	C₂×M₁₆	—	23	1	no	—
56	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
57	#117	—	23	1	no	—
58	M₃₂	Dic₂₃	1	4	yes	complemented
59	C₂×C₁₆	—	23	4	no	—
60	C₂×C₁₆	—	23	1	no	—
61	C₂×C₁₆	—	23	1	no	—
62	C₂×C₁₆	C₂×D₂₃	1	2	yes	—
63	#97	—	23	2	no	—
64	C₄×C₈	—	23	2	no	—
65	C₂₃	#2108	1	1	yes	characteristic, complemented, Sylow
66	C₂²×C₄	C₄×D₂₃	1	1	yes	characteristic
67	M₁₆	—	23	4	no	—
68	C₂×C₈	—	23	2	no	—
69	C₂×C₈	C₄×D₂₃	1	1	yes	characteristic
70	C₂×C₈	C₄×D₂₃	1	2	yes	—
71	C₂×C₈	C₂×Dic₂₃	1	2	yes	—
72	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
73	C₄⋊C₄	—	23	2	no	—
74	C₂²⋊C₄	—	23	2	no	—
75	C₄²	—	23	2	no	—
76	C₁₆	—	23	2	no	—
77	C₁₆	—	23	2	no	—
78	C₁₆	C₂×Dic₂₃	1	4	yes	—
79	C₂³	C₈×D₂₃	1	1	yes	characteristic
80	C₂×C₄	—	23	4	no	—
81	C₂×C₄	C₈×D₂₃	1	2	yes	—
82	C₂×C₄	C₈×D₂₃	1	1	yes	characteristic
83	C₂×C₄	C₄×Dic₂₃	1	2	yes	—
84	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
85	C₈	—	23	4	no	—
86	C₈	C₄×Dic₂₃	1	2	yes	—
87	C₈	C₂×C₄×D₂₃	1	1	yes	characteristic
88	C₈	C₂²×Dic₂₃	1	1	yes	characteristic
89	C₂²	—	2	2	no	—
90	C₂²	C₈×Dic₂₃	1	2	yes	—
91	C₂²	C₂×C₈×D₂₃	1	1	yes	characteristic
92	C₄	—	46	4	no	—
93	C₄	C₈×Dic₂₃	1	2	yes	—
94	C₄	C₂×C₈×D₂₃	1	1	yes	characteristic
95	C₄	C₂×C₄×Dic₂₃	1	1	yes	characteristic
96	C₂	—	2	2	no	—
97	C₂	C₂×C₈×Dic₂₃	1	1	yes	characteristic
98	C₂	#13092835	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₂₃⋊C₁₆

C₂

yes

maximal, characteristic

#13092372

C₂

yes

maximal, characteristic

#13092192

C₂

yes

maximal, complemented

C₁₆×Dic₂₃

C₂

yes

maximal, complemented

C₂×C₂₃×M₃₂

C₂

yes

maximal, characteristic

C₂²×C₈×C₂₃

C₂²

yes

characteristic

#10625097

C₄

yes

characteristic

C₂×#67267

C₄

yes

characteristic

C₂₃×M₃₂

C₄

yes

complemented

C₂×C₁₆×C₂₃

C₂²

yes

—

#10624997

C₄

yes

—

C₈×Dic₂₃

C₂²

yes

—

C₂×C₂₃⋊C₁₆

C₄

yes

complemented

C₂×C₂₃⋊C₁₆

C₂²

yes

characteristic

C₂×C₂₃⋊C₁₆

C₂²

yes

characteristic

C₂²×C₄×C₂₃

C₂×C₄

yes

characteristic

C₂×C₈×C₂₃

C₂×C₄

yes

characteristic

C₂×C₈×C₂₃

C₂×C₄

yes

—

C₂×C₈×C₂₃

C₂×C₄

yes

—

C₂×C₈×C₂₃

C₂³

yes

characteristic

C₂²⋊Dic₂₃

C₈

yes

—

C₄⋊Dic₂₃

C₈

yes

—

C₄×Dic₂₃

C₂×C₄

yes

—

#67267

C₈

yes

—

C₂×C₂₃⋊C₈

C₂×C₄

yes

—

C₁₆×C₂₃

C₂×C₄

yes

—

C₂₃⋊C₁₆

C₂×C₄

yes

—

C₂₃⋊C₁₆

C₂×C₄

yes

—

C₂³×C₂₃

C₂×C₈

yes

characteristic

C₂×C₄×C₂₃

C₄²

yes

—

C₂×C₄×C₂₃

C₂×C₈

yes

characteristic

C₂×C₄×C₂₃

C₂×C₈

yes

—

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×Dic₂₃

C₂×C₈

yes

—

C₈×C₂₃

C₄²

yes

—

C₈×C₂₃

C₂²×C₄

yes

characteristic

C₈×C₂₃

C₂²×C₄

yes

characteristic

C₂₃⋊C₈

C₂×C₈

yes

—

#2108

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

C₄×C₈

yes

—

C₂²×C₂₃

C₂²×C₈

yes

characteristic

C₄×C₂₃

C₄×C₈

yes

—

C₄×C₂₃

C₂×C₄²

yes

characteristic

C₄×C₂₃

C₂²×C₈

yes

characteristic

Dic₂₃

—

C₂×M₃₂

D₂₃

yes

characteristic

C₂²×C₁₆

—

#405

—

#346

—

C₄×C₁₆

—

C₂×C₂₃

—

C₂×C₂₃

C₂×C₄×C₈

yes

characteristic

C₂×C₂₃

#504

yes

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₂₃

yes

characteristic

#117

—

M₃₂

Dic₂₃

yes

complemented

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₂₃

yes

—

#97

—

C₄×C₈

—

C₂₃

#2108

yes

characteristic, complemented, Sylow

C₂²×C₄

C₄×D₂₃

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

C₄×D₂₃

yes

characteristic

C₂×C₈

C₄×D₂₃

yes

—

C₂×C₈

C₂×Dic₂₃

yes

—

C₂×C₈

C₂²×D₂₃

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₁₆

—

C₁₆

—

C₁₆

C₂×Dic₂₃

yes

—

C₂³

C₈×D₂₃

yes

characteristic

C₂×C₄

—

C₂×C₄

C₈×D₂₃

yes

—

C₂×C₄

C₈×D₂₃

yes

characteristic

C₂×C₄

C₄×Dic₂₃

yes

—

C₂×C₄

C₂×C₄×D₂₃

yes

characteristic

C₈

—

C₈

C₄×Dic₂₃

yes

—

C₈

C₂×C₄×D₂₃

yes

characteristic

C₈

C₂²×Dic₂₃

yes

characteristic

C₂²

—

C₂²

C₈×Dic₂₃

yes

—

C₂²

C₂×C₈×D₂₃

yes

characteristic

C₄

—

C₄

C₈×Dic₂₃

yes

—

C₄

C₂×C₈×D₂₃

yes

characteristic

C₄

C₂×C₄×Dic₂₃

yes

characteristic

C₂

—

C₂

C₂×C₈×Dic₂₃

yes

characteristic

C₂

#13092835

yes

—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₂₃
commutator	C₂×C₂₃	C₂×C₄×C₈
Fitting	C₂×C₂₃×M₃₂	C₂
socle	C₂²×C₂₃	C₂²×C₈
prime radical	C₂³×C₂₃	C₂×C₈
cosocle	C₂×C₈×C₂₃	C₂³
hypercenter	C₂×M₃₂	D₂₃
hypocenter	C₂₃	#2108
LCS	G ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₂₃

commutator

C₂×C₂₃

C₂×C₄×C₈

Fitting

C₂×C₂₃×M₃₂

C₂

socle

C₂²×C₂₃

C₂²×C₈

prime radical

C₂³×C₂₃

C₂×C₈

cosocle

C₂×C₈×C₂₃

C₂³

hypercenter

C₂×M₃₂

D₂₃

C₂₃

G ▷ C₂×C₂₃ ▷ C₂₃

C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
3. #13092192	C₂	semidirect
4. C₁₆×Dic₂₃	C₂	semidirect
9. C₂₃×M₃₂	C₄	semidirect
13. C₂×C₂₃⋊C₁₆	C₄	semidirect
58. M₃₂	Dic₂₃	semidirect
65. C₂₃	#2108	semidirect

H ≅ G/N

type

3. #13092192

C₂

semidirect

4. C₁₆×Dic₂₃

C₂

semidirect

9. C₂₃×M₃₂

C₄

semidirect

13. C₂×C₂₃⋊C₁₆

C₄

semidirect

58. M₃₂

Dic₂₃

semidirect

65. C₂₃

#2108

semidirect

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

subgroup	index
C₂²×C₂₃⋊C₁₆	2
#13092372	2
#13092192	2
C₁₆×Dic₂₃	2
C₂×C₂₃×M₃₂	2
#2108	23

subgroup

index

C₂²×C₂₃⋊C₁₆

#13092372

#13092192

C₁₆×Dic₂₃

C₂×C₂₃×M₃₂

#2108

Subgroups — #15674996

Conjugacy classes of subgroups

Complements

Maximal subgroups