Group #15675152 — subgroups

Subgroups — #15675152

Order 2944.

172 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 97 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×#10624994	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Dic₂₃	C₂	1	1	yes	maximal, characteristic
3	Dic₂₃⋊C₁₆	C₂	1	4	yes	maximal, complemented
4	C₂×C₂₃×M₃₂	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
6	C₂×C₄×Dic₂₃	C₄	1	1	yes	characteristic
7	C₂²×C₂₃⋊C₈	C₄	1	1	yes	characteristic
8	C₂₃×M₃₂	—	2	2	no	—
9	C₂×C₁₆×C₂₃	C₂²	1	2	yes	—
10	C₈×Dic₂₃	C₄	1	2	yes	—
11	C₈×Dic₂₃	C₂²	1	2	yes	—
12	#10624994	—	2	2	no	—
13	C₂×C₂₃⋊C₁₆	C₂²	1	2	yes	—
14	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
15	C₂²×Dic₂₃	C₈	1	2	yes	—
16	C₂×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₈×C₂₃	D₄	1	2	yes	—
18	C₂×C₈×C₂₃	Q₈	1	2	yes	—
19	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
20	C₄×Dic₂₃	C₈	1	1	yes	characteristic
21	C₄×Dic₂₃	C₈	1	1	yes	characteristic
22	C₄×Dic₂₃	C₂×C₄	1	2	yes	—
23	C₂×C₂₃⋊C₈	—	2	2	no	—
24	C₂×C₂₃⋊C₈	C₂×C₄	1	2	yes	—
25	C₁₆×C₂₃	—	2	2	no	—
26	C₂₃⋊C₁₆	—	2	2	no	—
27	C₂³×C₂₃	C₂×C₈	1	1	yes	characteristic
28	C₂×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
29	C₂×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
30	C₂×C₄×C₂₃	M₁₆	1	2	yes	—
31	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
32	C₂×Dic₂₃	—	2	4	no	—
33	C₂×Dic₂₃	C₂×C₈	1	2	yes	—
34	C₂×Dic₂₃	C₂×C₈	1	2	yes	—
35	C₈×C₂₃	C₄⋊C₄	1	2	yes	—
36	C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
37	C₈×C₂₃	C₂×Q₈	1	1	yes	characteristic
38	C₂₃⋊C₈	—	2	2	no	—
39	#2150	—	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×C₄⋊C₄	1	1	yes	characteristic
45	C₄×C₂₃	C₂×M₁₆	1	1	yes	characteristic
46	Dic₂₃	—	2	2	no	—
47	Dic₂₃	M₃₂	1	4	yes	complemented
48	C₂×M₃₂	—	23	1	no	—
49	C₂×M₃₂	D₂₃	1	1	yes	characteristic
50	C₂×C₄×C₈	—	23	1	no	—
51	C₄⋊C₁₆	—	23	4	no	—
52	C₂×C₂₃	—	2	2	no	—
53	C₂×C₂₃	C₂×C₄⋊C₈	1	1	yes	characteristic
54	C₂×C₂₃	C₂×M₃₂	1	2	yes	—
55	C₂²×C₈	—	23	1	no	—
56	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
57	C₂×C₄²	—	23	1	no	—
58	M₃₂	—	2	2	no	—
59	M₃₂	—	46	2	no	—
60	C₂×C₁₆	—	23	2	no	—
61	C₂×C₁₆	C₂×D₂₃	1	2	yes	—
62	C₄×C₈	—	23	2	no	—
63	C₄×C₈	—	23	2	no	—
64	C₂₃	#2150	1	1	yes	characteristic, complemented, Sylow
65	C₂²×C₄	—	23	2	no	—
66	C₂²×C₄	C₄×D₂₃	1	1	yes	characteristic
67	C₂×C₈	—	23	2	no	—
68	C₂×C₈	—	46	2	no	—
69	C₂×C₈	Dic₄₆	1	2	yes	—
70	C₂×C₈	C₄×D₂₃	1	1	yes	characteristic
71	C₂×C₈	C₂²⋊D₂₃	1	2	yes	—
72	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
73	C₄²	—	23	1	no	—
74	C₄²	—	23	1	no	—
75	C₄²	—	23	2	no	—
76	C₁₆	—	2	2	no	—
77	C₁₆	—	46	2	no	—
78	C₂³	C₈×D₂₃	1	1	yes	characteristic
79	C₂×C₄	—	23	2	no	—
80	C₂×C₄	—	23	2	no	—
81	C₂×C₄	—	46	4	no	—
82	C₂×C₄	C₈×D₂₃	1	1	yes	characteristic
83	C₂×C₄	#67262	1	2	yes	—
84	C₂×C₄	Dic₂₃⋊C₄	1	2	yes	—
85	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
86	C₈	—	46	2	no	—
87	C₈	Dic₂₃⋊C₄	1	2	yes	—
88	C₈	C₂×Dic₄₆	1	1	yes	characteristic
89	C₈	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
90	C₂²	—	2	2	no	—
91	C₂²	Dic₂₃⋊C₈	1	2	yes	—
92	C₂²	C₂×C₈×D₂₃	1	1	yes	characteristic
93	C₄	—	23	4	no	—
94	C₄	—	46	2	no	—
95	C₄	Dic₂₃⋊C₈	1	2	yes	—
96	C₄	C₂×#67262	1	1	yes	characteristic
97	C₄	C₂×Dic₂₃⋊C₄	1	1	yes	characteristic
98	C₂	—	2	2	no	—
99	C₂	C₂×Dic₂₃⋊C₈	1	1	yes	characteristic
100	C₂	M₃₂×D₂₃	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10624994

C₂

yes

maximal, characteristic

C₂×C₈×Dic₂₃

C₂

yes

maximal, characteristic

Dic₂₃⋊C₁₆

C₂

yes

maximal, complemented

C₂×C₂₃×M₃₂

C₂

yes

maximal, characteristic

C₂²×C₈×C₂₃

C₂²

yes

characteristic

C₂×C₄×Dic₂₃

C₄

yes

characteristic

C₂²×C₂₃⋊C₈

C₄

yes

characteristic

C₂₃×M₃₂

—

C₂×C₁₆×C₂₃

C₂²

yes

—

C₈×Dic₂₃

C₄

yes

—

C₈×Dic₂₃

C₂²

yes

—

#10624994

—

C₂×C₂₃⋊C₁₆

C₂²

yes

—

C₂²×C₄×C₂₃

C₂×C₄

yes

characteristic

C₂²×Dic₂₃

C₈

yes

—

C₂×C₈×C₂₃

C₂×C₄

yes

characteristic

C₂×C₈×C₂₃

D₄

yes

—

C₂×C₈×C₂₃

Q₈

yes

—

C₂×C₈×C₂₃

C₂³

yes

characteristic

C₄×Dic₂₃

C₈

yes

characteristic

C₄×Dic₂₃

C₈

yes

characteristic

C₄×Dic₂₃

C₂×C₄

yes

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

C₂×C₄

yes

—

C₁₆×C₂₃

—

C₂₃⋊C₁₆

—

C₂³×C₂₃

C₂×C₈

yes

characteristic

C₂×C₄×C₂₃

C₄⋊C₄

yes

—

C₂×C₄×C₂₃

C₂×C₈

yes

characteristic

C₂×C₄×C₂₃

M₁₆

yes

—

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×Dic₂₃

—

C₂×Dic₂₃

C₂×C₈

yes

—

C₂×Dic₂₃

C₂×C₈

yes

—

C₈×C₂₃

C₄⋊C₄

yes

—

C₈×C₂₃

C₂×D₄

yes

characteristic

C₈×C₂₃

C₂×Q₈

yes

characteristic

C₂₃⋊C₈

—

#2150

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

C₄⋊C₈

yes

—

C₂²×C₂₃

C₂²×C₈

yes

characteristic

C₄×C₂₃

C₄⋊C₈

yes

—

C₄×C₂₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₂₃

C₂×M₁₆

yes

characteristic

Dic₂₃

—

Dic₂₃

M₃₂

yes

complemented

C₂×M₃₂

—

C₂×M₃₂

D₂₃

yes

characteristic

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₂×C₂₃

—

C₂×C₂₃

C₂×C₄⋊C₈

yes

characteristic

C₂×C₂₃

C₂×M₃₂

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×D₂₃

yes

characteristic

C₂×C₄²

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₂₃

yes

—

C₄×C₈

—

C₄×C₈

—

C₂₃

#2150

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

C₄×D₂₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₄₆

yes

—

C₂×C₈

C₄×D₂₃

yes

characteristic

C₂×C₈

C₂²⋊D₂₃

yes

—

C₂×C₈

C₂²×D₂₃

yes

characteristic

C₄²

—

C₄²

—

C₄²

—

C₁₆

—

C₁₆

—

C₂³

C₈×D₂₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₈×D₂₃

yes

characteristic

C₂×C₄

#67262

yes

—

C₂×C₄

Dic₂₃⋊C₄

yes

—

C₂×C₄

C₂×C₄×D₂₃

yes

characteristic

C₈

—

C₈

Dic₂₃⋊C₄

yes

—

C₈

C₂×Dic₄₆

yes

characteristic

C₈

C₂×C₂²⋊D₂₃

yes

characteristic

C₂²

—

C₂²

Dic₂₃⋊C₈

yes

—

C₂²

C₂×C₈×D₂₃

yes

characteristic

C₄

—

C₄

—

C₄

Dic₂₃⋊C₈

yes

—

C₄

C₂×#67262

yes

characteristic

C₄

C₂×Dic₂₃⋊C₄

yes

characteristic

C₂

—

C₂

C₂×Dic₂₃⋊C₈

yes

characteristic

100

C₂

M₃₂×D₂₃

yes

—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₂₃
commutator, socle	C₂²×C₂₃	C₂²×C₈
Fitting	C₂×C₂₃×M₃₂	C₂
prime radical	C₂³×C₂₃	C₂×C₈
cosocle	C₂×C₈×C₂₃	C₂³
hypercenter	C₂×M₃₂	D₂₃
hypocenter	C₂₃	#2150
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₂₃

commutator, socle

C₂²×C₂₃

C₂²×C₈

Fitting

C₂×C₂₃×M₃₂

C₂

prime radical

C₂³×C₂₃

C₂×C₈

cosocle

C₂×C₈×C₂₃

C₂³

hypercenter

C₂×M₃₂

D₂₃

C₂₃

G ▷ C₂²×C₂₃ ▷ C₂₃

C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
3. Dic₂₃⋊C₁₆	C₂	semidirect
47. Dic₂₃	M₃₂	semidirect
64. C₂₃	#2150	semidirect

H ≅ G/N

type

3. Dic₂₃⋊C₁₆

C₂

semidirect

47. Dic₂₃

M₃₂

semidirect

64. C₂₃

#2150

semidirect

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

subgroup	index
C₂×#10624994	2
C₂×C₈×Dic₂₃	2
Dic₂₃⋊C₁₆	2
C₂×C₂₃×M₃₂	2
#2150	23

subgroup

index

C₂×#10624994

C₂×C₈×Dic₂₃

Dic₂₃⋊C₁₆

C₂×C₂₃×M₃₂

#2150

Subgroups — #15675152

Conjugacy classes of subgroups

Complements

Maximal subgroups