Group #72245419 — subgroups

Subgroups — #72245419

Order 2304.

318 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 183 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	#11772661	C₂	1	1	yes	maximal, characteristic
2	#11771294	C₂	1	1	yes	maximal, characteristic
3	C₃²×#1397	C₂	1	1	yes	maximal, characteristic
4	#10682521	—	3	4	no	maximal
5	C₂×C₃²×C₃₂	C₄	1	2	yes	—
6	C₃²×C₄×C₁₆	C₂²	1	1	yes	characteristic
7	C₂×C₃⋊(C₃⋊C₃₂)	C₄	1	2	yes	—
8	C₄×C₃⋊(C₃⋊C₁₆)	C₄	1	1	yes	characteristic
9	C₈×C₃⋊(C₃⋊C₈)	C₄	1	1	yes	characteristic
10	C₃×#1397	D₃	1	4	yes	—
11	#67508	—	3	4	no	—
12	#67498	—	3	4	no	—
13	C₂×C₃²×C₁₆	C₂×C₄	1	1	yes	characteristic
14	C₂×C₃²×C₁₆	C₂×C₄	1	1	yes	characteristic
15	C₃²×C₄×C₈	C₂×C₄	1	1	yes	characteristic
16	C₂×C₃⋊(C₃⋊C₁₆)	C₈	1	2	yes	—
17	C₄×C₃⋊(C₃⋊C₈)	C₈	1	2	yes	—
18	C₃²×C₃₂	—	2	2	no	—
19	C₃⋊(C₃⋊C₃₂)	—	2	2	no	—
20	#7332	—	9	1	no	Sylow
21	C₂×C₃×C₃₂	Dic₃	1	8	yes	—
22	C₃×C₄×C₁₆	C₂×D₃	1	4	yes	—
23	C₂×C₃⋊C₃₂	—	3	8	no	—
24	C₄×C₃⋊C₁₆	—	3	4	no	—
25	C₈×C₃⋊C₈	—	3	4	no	—
26	C₂×C₃²×C₈	C₄²	1	1	yes	characteristic
27	C₂×C₃²×C₈	C₂×C₈	1	1	yes	characteristic
28	C₃²×C₄²	C₂×C₈	1	1	yes	characteristic
29	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
30	C₃²×C₁₆	M₁₆	1	2	yes	—
31	C₃²×C₁₆	M₁₆	1	2	yes	—
32	C₃⋊(C₃⋊C₁₆)	C₁₆	1	4	yes	—
33	#1397	—	9	1	no	—
34	#1397	C₃⋊D₃	1	1	yes	characteristic
35	#1311	—	9	1	no	—
36	C₂×C₃×C₁₆	C₄×D₃	1	4	yes	—
37	C₂×C₃×C₁₆	C₂×Dic₃	1	4	yes	—
38	C₃×C₄×C₈	C₄×D₃	1	4	yes	—
39	C₂×C₃⋊C₁₆	—	3	8	no	—
40	C₄×C₃⋊C₈	—	3	8	no	—
41	C₃×C₃₂	—	2	8	no	—
42	C₃⋊C₃₂	—	6	8	no	—
43	C₂×C₃²×C₄	C₄×C₈	1	1	yes	characteristic
44	C₂×C₃²×C₄	M₃₂	1	2	yes	—
45	C₃²×C₈	#97	1	1	yes	characteristic
46	C₃²×C₈	#97	1	1	yes	characteristic
47	C₃²×C₈	C₂×C₁₆	1	2	yes	—
48	C₃⋊(C₃⋊C₈)	—	4	2	no	—
49	C₂×C₃₂	—	9	2	no	—
50	C₂×C₃₂	C₃⋊Dic₃	1	2	yes	—
51	C₄×C₁₆	—	9	1	no	—
52	C₄×C₁₆	C₂×C₃⋊D₃	1	1	yes	characteristic
53	C₈²	—	9	1	no	—
54	C₂×C₃×C₈	D₃×C₈	1	4	yes	—
55	C₂×C₃×C₈	C₄×Dic₃	1	4	yes	—
56	C₃×C₄²	D₃×C₈	1	4	yes	—
57	C₂×C₃⋊C₈	—	6	8	no	—
58	C₃×C₁₆	ΓL(1,𝔽₂₅)	1	8	yes	—
59	C₃×C₁₆	#208	1	8	yes	—
60	C₃⋊C₁₆	—	3	16	no	—
61	C₂²×C₃²	#346	1	1	yes	characteristic
62	C₃²×C₄	—	2	2	no	—
63	C₃²×C₄	#322	1	1	yes	characteristic
64	C₃²×C₄	C₄×C₁₆	1	1	yes	characteristic
65	C₂×C₁₆	—	9	2	no	—
66	C₂×C₁₆	C₄×C₃⋊D₃	1	1	yes	characteristic
67	C₂×C₁₆	C₂×C₃⋊Dic₃	1	1	yes	characteristic
68	C₄×C₈	—	9	2	no	—
69	C₄×C₈	C₄×C₃⋊D₃	1	1	yes	characteristic
70	C₃₂	—	2	2	no	—
71	C₃₂	—	18	2	no	—
72	C₂×C₃×C₄	#798	1	8	yes	—
73	C₂×C₃×C₄	C₈×Dic₃	1	4	yes	—
74	C₃×C₈	D₃×C₁₆	1	8	yes	—
75	C₃×C₈	#803	1	4	yes	—
76	C₃×C₈	#815	1	4	yes	—
77	C₃⋊C₈	—	12	8	no	—
78	C₂×C₃²	#1312	1	1	yes	characteristic
79	C₂×C₃²	#1398	1	2	yes	—
80	C₂×C₈	—	18	2	no	—
81	C₂×C₈	C₈×C₃⋊D₃	1	1	yes	characteristic
82	C₂×C₈	C₄×C₃⋊Dic₃	1	1	yes	characteristic
83	C₄²	C₈×C₃⋊D₃	1	1	yes	characteristic
84	C₁₆	—	9	4	no	—
85	C₁₆	#3743	1	2	yes	—
86	C₁₆	#3748	1	2	yes	—
87	C₂²×C₃	#4502	1	4	yes	—
88	C₃×C₄	—	2	8	no	—
89	C₃×C₄	#4454	1	4	yes	—
90	C₃×C₄	Dic₃×C₁₆	1	4	yes	—
91	C₃²	#7332	1	1	yes	characteristic, complemented, Sylow
92	C₂×C₄	#63625	1	2	yes	—
93	C₂×C₄	C₈×C₃⋊Dic₃	1	1	yes	characteristic
94	C₈	—	36	2	no	—
95	C₈	C₁₆×C₃⋊D₃	1	2	yes	—
96	C₈	#63630	1	1	yes	characteristic
97	C₈	#63642	1	1	yes	characteristic
98	C₂×C₃	#67499	1	4	yes	—
99	C₂×C₃	#67600	1	8	yes	—
100	C₂²	#10589368	1	1	yes	characteristic
101	C₄	—	2	2	no	—
102	C₄	#10589320	1	1	yes	characteristic
103	C₄	C₁₆×C₃⋊Dic₃	1	1	yes	characteristic
104	C₃	#10682521	1	4	yes	complemented
105	C₂	#11771309	1	1	yes	characteristic
106	C₂	#11772672	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11772661

C₂

yes

maximal, characteristic

#11771294

C₂

yes

maximal, characteristic

C₃²×#1397

C₂

yes

maximal, characteristic

#10682521

—

maximal

C₂×C₃²×C₃₂

C₄

yes

—

C₃²×C₄×C₁₆

C₂²

yes

characteristic

C₂×C₃⋊(C₃⋊C₃₂)

C₄

yes

—

C₄×C₃⋊(C₃⋊C₁₆)

C₄

yes

characteristic

C₈×C₃⋊(C₃⋊C₈)

C₄

yes

characteristic

C₃×#1397

D₃

yes

—

#67508

—

#67498

—

C₂×C₃²×C₁₆

C₂×C₄

yes

characteristic

C₂×C₃²×C₁₆

C₂×C₄

yes

characteristic

C₃²×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₃⋊(C₃⋊C₁₆)

C₈

yes

—

C₄×C₃⋊(C₃⋊C₈)

C₈

yes

—

C₃²×C₃₂

—

C₃⋊(C₃⋊C₃₂)

—

#7332

—

Sylow

C₂×C₃×C₃₂

Dic₃

yes

—

C₃×C₄×C₁₆

C₂×D₃

yes

—

C₂×C₃⋊C₃₂

—

C₄×C₃⋊C₁₆

—

C₈×C₃⋊C₈

—

C₂×C₃²×C₈

C₄²

yes

characteristic

C₂×C₃²×C₈

C₂×C₈

yes

characteristic

C₃²×C₄²

C₂×C₈

yes

characteristic

C₂×C₃⋊(C₃⋊C₈)

—

C₃²×C₁₆

M₁₆

yes

—

C₃²×C₁₆

M₁₆

yes

—

C₃⋊(C₃⋊C₁₆)

C₁₆

yes

—

#1397

—

#1397

C₃⋊D₃

yes

characteristic

#1311

—

C₂×C₃×C₁₆

C₄×D₃

yes

—

C₂×C₃×C₁₆

C₂×Dic₃

yes

—

C₃×C₄×C₈

C₄×D₃

yes

—

C₂×C₃⋊C₁₆

—

C₄×C₃⋊C₈

—

C₃×C₃₂

—

C₃⋊C₃₂

—

C₂×C₃²×C₄

C₄×C₈

yes

characteristic

C₂×C₃²×C₄

M₃₂

yes

—

C₃²×C₈

yes

C₃²×C₈

yes

C₃²×C₈

C₂×C₁₆

yes

—

C₃⋊(C₃⋊C₈)

—

C₂×C₃₂

—

C₂×C₃₂

C₃⋊Dic₃

yes

—

C₄×C₁₆

—

C₄×C₁₆

C₂×C₃⋊D₃

yes

characteristic

C₈²

—

C₂×C₃×C₈

D₃×C₈

yes

—

C₂×C₃×C₈

C₄×Dic₃

yes

—

C₃×C₄²

D₃×C₈

yes

—

C₂×C₃⋊C₈

—

C₃×C₁₆

ΓL(1,𝔽₂₅)

yes

—

C₃×C₁₆

#208

yes

—

C₃⋊C₁₆

—

C₂²×C₃²

#346

yes

characteristic

C₃²×C₄

—

C₃²×C₄

#322

yes

characteristic

C₃²×C₄

C₄×C₁₆

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

C₄×C₃⋊D₃

yes

characteristic

C₂×C₁₆

C₂×C₃⋊Dic₃

yes

characteristic

C₄×C₈

—

C₄×C₈

C₄×C₃⋊D₃

yes

characteristic

C₃₂

—

C₃₂

—

C₂×C₃×C₄

#798

yes

—

C₂×C₃×C₄

C₈×Dic₃

yes

—

C₃×C₈

D₃×C₁₆

yes

—

C₃×C₈

#803

yes

—

C₃×C₈

#815

yes

—

C₃⋊C₈

—

C₂×C₃²

#1312

yes

characteristic

C₂×C₃²

#1398

yes

—

C₂×C₈

—

C₂×C₈

C₈×C₃⋊D₃

yes

characteristic

C₂×C₈

C₄×C₃⋊Dic₃

yes

characteristic

C₄²

C₈×C₃⋊D₃

yes

characteristic

C₁₆

—

C₁₆

#3743

yes

—

C₁₆

#3748

yes

—

C₂²×C₃

#4502

yes

—

C₃×C₄

—

C₃×C₄

#4454

yes

—

C₃×C₄

Dic₃×C₁₆

yes

—

C₃²

#7332

yes

characteristic, complemented, Sylow

C₂×C₄

#63625

yes

—

C₂×C₄

C₈×C₃⋊Dic₃

yes

characteristic

C₈

—

C₈

C₁₆×C₃⋊D₃

yes

—

C₈

yes

C₈

yes

C₂×C₃

yes

—

C₂×C₃

#67600

yes

—

100

C₂²

#10589368

yes

characteristic

101

C₄

—

102

C₄

#10589320

yes

characteristic

103

C₄

C₁₆×C₃⋊Dic₃

yes

characteristic

104

C₃

#10682521

yes

complemented

105

C₂

#11771309

yes

characteristic

106

C₂

#11772672

yes

—

	subgroup	G/H
center	C₂×C₈	C₄×C₃⋊Dic₃
commutator	C₃²×C₄	C₄×C₁₆
Frattini	C₄×C₁₆	C₂×C₃⋊D₃
Fitting	C₃²×#1397	C₂
socle, prime radical	C₂²×C₃²	#346
cosocle	C₃²×C₄×C₁₆	C₂²
hypercenter	#1397	C₃⋊D₃
hypocenter	C₃²	#7332
LCS	G ▷ C₃²×C₄ ▷ C₃²
UCS	C₁ ◁ C₂×C₈ ◁ #1397

subgroup

G/H

center

C₂×C₈

C₄×C₃⋊Dic₃

commutator

C₃²×C₄

C₄×C₁₆

Frattini

C₄×C₁₆

C₂×C₃⋊D₃

Fitting

C₃²×#1397

C₂

socle, prime radical

C₂²×C₃²

#346

cosocle

C₃²×C₄×C₁₆

C₂²

C₃⋊D₃

C₃²

G ▷ C₃²×C₄ ▷ C₃²

C₁ ◁ C₂×C₈ ◁ #1397

N	H ≅ G/N	type
91. C₃²	#7332	semidirect
104. C₃	#10682521	semidirect

H ≅ G/N

type

91. C₃²

#7332

semidirect

104. C₃

#10682521

semidirect

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

subgroup	index
#11772661	2
#11771294	2
C₃²×#1397	2
#10682521	3

subgroup

index

#11772661

#11771294

C₃²×#1397

#10682521

Subgroups — #72245419

Conjugacy classes of subgroups

Complements

Maximal subgroups