Group #41006090 — subgroups

Subgroups — #41006090

Order 26973.

236 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 49 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	#22582553	C₃	1	2	yes	maximal
2	#22582552	C₃	1	2	yes	maximal
3	C₃₇⋊#6862	C₃	1	4	yes	maximal, complemented
4	C₃₇⋊#6862	C₃	1	4	yes	maximal, complemented
5	C₃₇×#6862	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃³×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
7	C₃×#11758605	—	3	2	no	—
8	C₃×#11758606	—	3	2	no	—
9	C₃²⋊#66524	C₃²	1	4	yes	—
10	C₃²⋊#66524	C₃²	1	4	yes	—
11	#15688502	—	3	2	no	—
12	#15688502	—	3	2	no	—
13	#15688502	—	3	2	no	—
14	#15688502	—	3	2	no	—
15	C₃₇×C₃²⋊C₉	C₃²	1	2	yes	—
16	C₃₇×C₃²⋊C₉	C₃²	1	2	yes	—
17	C₃³×C₃₇	C₃³	1	1	yes	characteristic
18	C₃²×C₃₇⋊C₃	—	9	2	no	—
19	C₃²×C₃₇⋊C₃	—	9	2	no	—
20	C₃²×C₃₇⋊C₃	H(3,𝔽₃)	1	1	yes	characteristic
21	C₃²×C₃₇⋊C₃	H(3,𝔽₃)	1	1	yes	characteristic
22	C₃²×C₃₇⋊C₃	H(3,𝔽₃)	1	1	yes	characteristic
23	C₃×C₉×C₃₇	—	3	2	no	—
24	C₃×C₉×C₃₇	—	3	2	no	—
25	C₃×#66524	—	3	4	no	—
26	C₃×#66524	—	3	4	no	—
27	#11758606	—	9	6	no	—
28	#11758605	—	9	6	no	—
29	#10624670	—	37	1	no	maximal, Sylow
30	C₃²×C₃₇	—	3	2	no	—
31	C₃²×C₃₇	—	3	2	no	—
32	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
33	C₃×C₃₇⋊C₃	—	3	2	no	—
34	C₃×C₃₇⋊C₃	—	3	2	no	—
35	C₃×C₃₇⋊C₃	—	3	2	no	—
36	C₃×C₃₇⋊C₃	—	3	2	no	—
37	C₃×C₃₇⋊C₃	—	3	2	no	—
38	C₃×C₃₇⋊C₃	—	3	2	no	—
39	C₃×C₃₇⋊C₃	—	9	9	no	—
40	C₉×C₃₇	—	9	2	no	—
41	C₉×C₃₇	—	9	2	no	—
42	#66524	—	9	4	no	—
43	#66524	—	9	4	no	—
44	#6891	—	37	2	no	—
45	#6891	—	37	2	no	—
46	#6862	—	37	4	no	—
47	#6862	—	37	4	no	—
48	#6862	C₃₇⋊C₃	1	1	yes	characteristic, complemented
49	C₃×C₃₇	—	9	1	no	—
50	C₃×C₃₇	#6910	1	2	yes	—
51	C₃×C₃₇	#6910	1	2	yes	—
52	C₃₇⋊C₃	—	9	1	no	—
53	C₃₇⋊C₃	—	9	1	no	—
54	C₃₇⋊C₃	—	9	1	no	—
55	C₃⁴	—	37	1	no	—
56	C₃×3₋¹⁺²	—	111	2	no	—
57	C₃×3₋¹⁺²	—	111	2	no	—
58	C₃²⋊C₉	—	37	4	no	—
59	C₃²⋊C₉	—	37	4	no	—
60	C₃²⋊C₉	—	111	2	no	—
61	C₃²⋊C₉	—	111	2	no	—
62	C₃²⋊C₉	—	111	2	no	—
63	C₃²⋊C₉	—	111	2	no	—
64	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	2	yes	—
65	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	2	yes	—
66	C₃₇	#10624670	1	1	yes	characteristic, complemented, Sylow
67	C₃³	—	37	1	no	—
68	C₃³	—	37	1	no	—
69	C₃³	—	37	1	no	—
70	C₃³	—	333	2	no	—
71	C₃³	—	333	2	no	—
72	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
73	3₋¹⁺²	—	333	6	no	—
74	3₋¹⁺²	—	333	6	no	—
75	C₃×C₉	—	3	2	no	—
76	C₃×C₉	—	3	2	no	—
77	C₃×C₉	—	111	4	no	—
78	C₃×C₉	—	111	4	no	—
79	C₃²	—	3	2	no	—
80	C₃²	—	3	2	no	—
81	C₃²	—	333	9	no	—
82	C₃²	—	111	2	no	—
83	C₃²	—	111	2	no	—
84	C₃²	—	111	2	no	—
85	C₃²	—	111	2	no	—
86	C₃²	—	111	2	no	—
87	C₃²	—	111	2	no	—
88	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic
89	C₉	—	9	2	no	—
90	C₉	—	9	2	no	—
91	C₉	—	333	4	no	—
92	C₉	—	333	4	no	—
93	C₃	—	9	1	no	—
94	C₃	—	333	1	no	—
95	C₃	—	333	1	no	—
96	C₃	—	333	1	no	—
97	C₃	#22582640	1	2	yes	—
98	C₃	#22582641	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22582553

C₃

yes

maximal

#22582552

C₃

yes

maximal

C₃₇⋊#6862

C₃

yes

maximal, complemented

C₃₇⋊#6862

C₃

yes

maximal, complemented

C₃₇×#6862

C₃

yes

maximal, characteristic, complemented

C₃³×C₃₇⋊C₃

C₃²

yes

—

—

C₃²

yes

—

C₃²⋊#66524

C₃²

yes

—

—

—

—

—

C₃₇×C₃²⋊C₉

C₃²

yes

—

C₃₇×C₃²⋊C₉

C₃²

yes

—

C₃³×C₃₇

C₃³

yes

characteristic

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

—

C₃²×C₃₇⋊C₃

yes

C₃²×C₃₇⋊C₃

yes

C₃²×C₃₇⋊C₃

yes

C₃×C₉×C₃₇

—

C₃×C₉×C₃₇

—

—

—

—

—

—

C₃²×C₃₇

—

C₃²×C₃₇

—

C₃²×C₃₇

C₃×H(3,𝔽₃)

yes

characteristic

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₃×C₃₇⋊C₃

—

C₉×C₃₇

—

C₉×C₃₇

—

—

—

—

—

—

—

C₃₇⋊C₃

yes

characteristic, complemented

C₃×C₃₇

—

C₃×C₃₇

#6910

yes

—

C₃×C₃₇

#6910

yes

—

C₃₇⋊C₃

—

C₃₇⋊C₃

—

C₃₇⋊C₃

—

C₃⁴

—

C₃×3₋¹⁺²

—

111

—

C₃×3₋¹⁺²

—

111

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

111

—

C₃²⋊C₉

—

111

—

C₃²⋊C₉

—

111

—

C₃²⋊C₉

—

111

—

C₃²⋊C₉

C₃×C₃₇⋊C₃

yes

—

C₃²⋊C₉

C₃×C₃₇⋊C₃

yes

—

C₃₇

#10624670

yes

characteristic, complemented, Sylow

C₃³

—

C₃³

—

C₃³

—

C₃³

—

333

—

C₃³

—

333

—

C₃³

C₃²×C₃₇⋊C₃

yes

characteristic

3₋¹⁺²

—

333

—

3₋¹⁺²

—

333

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

111

—

C₃×C₉

—

111

—

C₃²

—

C₃²

—

C₃²

—

333

—

C₃²

—

111

—

C₃²

—

111

—

C₃²

—

111

—

C₃²

—

111

—

C₃²

—

111

—

C₃²

—

111

—

C₃²

H(3,𝔽₃)×C₃₇⋊C₃

yes

characteristic

C₉

—

C₉

—

C₉

—

333

—

C₉

—

333

—

C₃

—

C₃

—

333

—

C₃

—

333

—

C₃

—

333

—

C₃

#22582640

yes

—

C₃

#22582641

yes

—

	subgroup	G/H
center	C₃²	H(3,𝔽₃)×C₃₇⋊C₃
commutator, cosocle	C₃³×C₃₇	C₃³
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting	C₃₇×#6862	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
prime radical	C₃³×C₃₇⋊C₃	C₃²
hypercenter	#6862	C₃₇⋊C₃
hypocenter	C₃₇	#10624670
LCS	G ▷ C₃³×C₃₇ ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ #6862

subgroup

G/H

center

C₃²

H(3,𝔽₃)×C₃₇⋊C₃

commutator, cosocle

C₃³×C₃₇

C₃³

Frattini

C₃³

C₃²×C₃₇⋊C₃

Fitting

C₃₇×#6862

C₃

socle

C₃²×C₃₇

C₃×H(3,𝔽₃)

prime radical

C₃³×C₃₇⋊C₃

C₃²

C₃₇⋊C₃

C₃₇

G ▷ C₃³×C₃₇ ▷ C₃²×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₃² ◁ C₃³ ◁ #6862

N	H ≅ G/N	type
3. C₃₇⋊#6862	C₃	semidirect
4. C₃₇⋊#6862	C₃	semidirect
5. C₃₇×#6862	C₃	semidirect
48. #6862	C₃₇⋊C₃	semidirect
66. C₃₇	#10624670	semidirect

H ≅ G/N

type

3. C₃₇⋊#6862

C₃

semidirect

4. C₃₇⋊#6862

C₃

semidirect

5. C₃₇×#6862

C₃

semidirect

48. #6862

C₃₇⋊C₃

semidirect

66. C₃₇

#10624670

semidirect

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

subgroup	index
#22582553	3
#22582552	3
C₃₇⋊#6862	3
C₃₇×#6862	3
#10624670	37

subgroup

index

#22582553

#22582552

C₃₇⋊#6862

C₃₇×#6862

#10624670

Subgroups — #41006090

Conjugacy classes of subgroups

Complements

Maximal subgroups