Group #19100307 — subgroups

Subgroups — C₇×3₋¹⁺²² #19100307

Order 5103.

836 conjugacy classes of subgroups; only the 518 normal subgroups are shown.

normal subgroups only

This group has 836 conjugacy classes of subgroups — too many to draw. Only its 518 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	C₃²×C₇×3₋¹⁺²	C₃	1	2	yes	maximal
2	C₇×#6900	C₃	1	9	yes	maximal, complemented
3	C₇×#6896	C₃	1	9	yes	maximal, complemented
4	C₇×#6894	C₃	1	12	yes	maximal, complemented
5	C₇×#6891	C₃	1	2	yes	maximal
6	C₇×C₉×3₋¹⁺²	C₃	1	6	yes	maximal, complemented
7	3₋¹⁺²²	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
8	C₃⁴×C₇	C₃²	1	1	yes	characteristic
9	C₃×C₇×3₋¹⁺²	—	3	18	no	—
10	C₃×C₇×3₋¹⁺²	C₃²	1	4	yes	—
11	C₃×C₇×3₋¹⁺²	C₃²	1	6	yes	—
12	C₃×C₇×3₋¹⁺²	C₃²	1	6	yes	—
13	C₃×C₇×3₋¹⁺²	C₃²	1	6	yes	—
14	C₃×C₇×3₋¹⁺²	C₃²	1	2	yes	—
15	C₃²×C₇×C₉	C₃²	1	6	yes	—
16	C₇×C₉⋊C₉	C₃²	1	36	yes	complemented
17	C₇×C₉⋊C₉	C₃²	1	36	yes	complemented
18	C₇×C₃²⋊C₉	C₃²	1	12	yes	—
19	C₇×C₃²⋊C₉	C₃²	1	6	yes	—
20	C₇×C₉²	C₃²	1	9	yes	complemented
21	C₃²×3₋¹⁺²	C₃×C₇	1	2	yes	—
22	#6900	C₃×C₇	1	9	yes	complemented
23	#6896	C₃×C₇	1	9	yes	complemented
24	#6894	C₃×C₇	1	12	yes	complemented
25	#6891	C₃×C₇	1	2	yes	—
26	C₉×3₋¹⁺²	C₃×C₇	1	6	yes	complemented
27	C₃³×C₇	—	3	6	no	—
28	C₃³×C₇	—	9	2	no	—
29	C₃³×C₇	C₃³	1	1	yes	characteristic
30	C₃³×C₇	C₃³	1	1	yes	characteristic
31	C₃³×C₇	C₃³	1	2	yes	—
32	C₇×3₋¹⁺²	—	3	12	no	—
33	C₇×3₋¹⁺²	—	3	18	no	—
34	C₇×3₋¹⁺²	—	3	18	no	—
35	C₇×3₋¹⁺²	—	9	6	no	—
36	C₇×3₋¹⁺²	3₋¹⁺²	1	18	yes	direct product factor
37	C₃×C₇×C₉	—	3	18	no	—
38	C₃×C₇×C₉	C₃³	1	18	yes	—
39	C₃×C₇×C₉	C₃³	1	12	yes	—
40	C₃×C₇×C₉	C₃³	1	6	yes	—
41	C₃⁴	C₃²×C₇	1	1	yes	characteristic
42	C₃×3₋¹⁺²	—	3	18	no	—
43	C₃×3₋¹⁺²	C₃²×C₇	1	4	yes	—
44	C₃×3₋¹⁺²	C₃²×C₇	1	6	yes	—
45	C₃×3₋¹⁺²	C₃²×C₇	1	6	yes	—
46	C₃×3₋¹⁺²	C₃²×C₇	1	6	yes	—
47	C₃×3₋¹⁺²	C₃²×C₇	1	2	yes	—
48	C₃²×C₉	C₃²×C₇	1	6	yes	—
49	C₉⋊C₉	C₃²×C₇	1	36	yes	complemented
50	C₉⋊C₉	C₃²×C₇	1	36	yes	complemented
51	C₃²⋊C₉	C₃²×C₇	1	12	yes	—
52	C₃²⋊C₉	C₃²×C₇	1	6	yes	—
53	C₉²	C₃²×C₇	1	9	yes	complemented
54	C₃²×C₇	—	3	2	no	—
55	C₃²×C₇	—	3	4	no	—
56	C₃²×C₇	—	3	2	no	—
57	C₃²×C₇	—	3	2	no	—
58	C₃²×C₇	—	3	2	no	—
59	C₃²×C₇	—	3	2	no	—
60	C₃²×C₇	—	9	9	no	—
61	C₃²×C₇	C₃×3₋¹⁺²	1	6	yes	—
62	C₃²×C₇	C₃⁴	1	1	yes	characteristic
63	C₇×C₉	—	3	12	no	—
64	C₇×C₉	—	3	18	no	—
65	C₇×C₉	C₃×3₋¹⁺²	1	18	yes	complemented
66	C₃³	—	3	6	no	—
67	C₃³	—	9	2	no	—
68	C₃³	C₃³×C₇	1	1	yes	characteristic
69	C₃³	C₃³×C₇	1	1	yes	characteristic
70	C₃³	C₃³×C₇	1	2	yes	—
71	3₋¹⁺²	—	3	12	no	—
72	3₋¹⁺²	—	3	18	no	—
73	3₋¹⁺²	—	3	18	no	—
74	3₋¹⁺²	—	9	6	no	—
75	3₋¹⁺²	C₇×3₋¹⁺²	1	18	yes	direct product factor
76	C₃×C₉	—	3	18	no	—
77	C₃×C₉	C₃³×C₇	1	18	yes	—
78	C₃×C₉	C₃³×C₇	1	12	yes	—
79	C₃×C₉	C₃³×C₇	1	6	yes	—
80	C₃×C₇	—	3	6	no	—
81	C₃×C₇	—	9	1	no	—
82	C₃×C₇	—	9	1	no	—
83	C₃×C₇	C₃²×3₋¹⁺²	1	2	yes	—
84	C₃×C₇	3₋¹⁺⁴	1	2	yes	—
85	C₃²	—	3	2	no	—
86	C₃²	—	3	4	no	—
87	C₃²	—	3	2	no	—
88	C₃²	—	3	2	no	—
89	C₃²	—	3	2	no	—
90	C₃²	—	3	2	no	—
91	C₃²	—	9	9	no	—
92	C₃²	C₃×C₇×3₋¹⁺²	1	6	yes	—
93	C₃²	C₃⁴×C₇	1	1	yes	characteristic
94	C₉	—	3	12	no	—
95	C₉	—	3	18	no	—
96	C₉	C₃×C₇×3₋¹⁺²	1	18	yes	complemented
97	C₇	3₋¹⁺²²	1	1	yes	characteristic, direct product factor, Sylow
98	C₃	—	3	6	no	—
99	C₃	—	9	1	no	—
100	C₃	—	9	1	no	—
101	C₃	C₃²×C₇×3₋¹⁺²	1	2	yes	—
102	C₃	C₇×3₋¹⁺⁴	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×C₇×3₋¹⁺²

C₃

yes

maximal

C₇×#6900

C₃

yes

maximal, complemented

C₇×#6896

C₃

yes

maximal, complemented

C₇×#6894

C₃

yes

maximal, complemented

C₇×#6891

C₃

yes

maximal

C₇×C₉×3₋¹⁺²

C₃

yes

maximal, complemented

3₋¹⁺²²

C₇

yes

maximal, characteristic, direct product factor, Sylow

C₃⁴×C₇

C₃²

yes

characteristic

C₃×C₇×3₋¹⁺²

—

C₃×C₇×3₋¹⁺²

C₃²

yes

—

C₃×C₇×3₋¹⁺²

C₃²

yes

—

C₃×C₇×3₋¹⁺²

C₃²

yes

—

C₃×C₇×3₋¹⁺²

C₃²

yes

—

C₃×C₇×3₋¹⁺²

C₃²

yes

—

C₃²×C₇×C₉

C₃²

yes

—

C₇×C₉⋊C₉

C₃²

yes

complemented

C₇×C₉⋊C₉

C₃²

yes

complemented

C₇×C₃²⋊C₉

C₃²

yes

—

C₇×C₃²⋊C₉

C₃²

yes

—

C₇×C₉²

C₃²

yes

complemented

C₃²×3₋¹⁺²

C₃×C₇

yes

—

#6900

C₃×C₇

yes

complemented

#6896

C₃×C₇

yes

complemented

#6894

C₃×C₇

yes

complemented

#6891

C₃×C₇

yes

—

C₉×3₋¹⁺²

C₃×C₇

yes

complemented

C₃³×C₇

—

C₃³×C₇

—

C₃³×C₇

C₃³

yes

characteristic

C₃³×C₇

C₃³

yes

characteristic

C₃³×C₇

C₃³

yes

—

C₇×3₋¹⁺²

—

C₇×3₋¹⁺²

—

C₇×3₋¹⁺²

—

C₇×3₋¹⁺²

—

C₇×3₋¹⁺²

3₋¹⁺²

yes

direct product factor

C₃×C₇×C₉

—

C₃×C₇×C₉

C₃³

yes

—

C₃×C₇×C₉

C₃³

yes

—

C₃×C₇×C₉

C₃³

yes

—

C₃⁴

C₃²×C₇

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃²×C₇

yes

—

C₃×3₋¹⁺²

C₃²×C₇

yes

—

C₃×3₋¹⁺²

C₃²×C₇

yes

—

C₃×3₋¹⁺²

C₃²×C₇

yes

—

C₃×3₋¹⁺²

C₃²×C₇

yes

—

C₃²×C₉

C₃²×C₇

yes

—

C₉⋊C₉

C₃²×C₇

yes

complemented

C₉⋊C₉

C₃²×C₇

yes

complemented

C₃²⋊C₉

C₃²×C₇

yes

—

C₃²⋊C₉

C₃²×C₇

yes

—

C₉²

C₃²×C₇

yes

complemented

C₃²×C₇

—

C₃²×C₇

—

C₃²×C₇

—

C₃²×C₇

—

C₃²×C₇

—

C₃²×C₇

—

C₃²×C₇

—

C₃²×C₇

C₃×3₋¹⁺²

yes

—

C₃²×C₇

C₃⁴

yes

characteristic

C₇×C₉

—

C₇×C₉

—

C₇×C₉

C₃×3₋¹⁺²

yes

complemented

C₃³

—

C₃³

—

C₃³

C₃³×C₇

yes

characteristic

C₃³

C₃³×C₇

yes

characteristic

C₃³

C₃³×C₇

yes

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

C₇×3₋¹⁺²

yes

direct product factor

C₃×C₉

—

C₃×C₉

C₃³×C₇

yes

—

C₃×C₉

C₃³×C₇

yes

—

C₃×C₉

C₃³×C₇

yes

—

C₃×C₇

—

C₃×C₇

—

C₃×C₇

—

C₃×C₇

C₃²×3₋¹⁺²

yes

—

C₃×C₇

3₋¹⁺⁴

yes

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

C₃×C₇×3₋¹⁺²

yes

—

C₃²

C₃⁴×C₇

yes

characteristic

C₉

—

C₉

—

C₉

C₃×C₇×3₋¹⁺²

yes

complemented

C₇

3₋¹⁺²²

yes

characteristic, direct product factor, Sylow

C₃

—

C₃

—

100

C₃

—

101

C₃

C₃²×C₇×3₋¹⁺²

yes

—

102

C₃

C₇×3₋¹⁺⁴

yes

—

	subgroup	G/H
center, socle	C₃²×C₇	C₃⁴
commutator, Frattini	C₃²	C₃⁴×C₇
LCS	G ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃²×C₇ ◁ G

subgroup

G/H

center, socle

C₃²×C₇

C₃⁴

commutator, Frattini

C₃²

C₃⁴×C₇

LCS

G ▷ C₃² ▷ C₁

UCS

C₁ ◁ C₃²×C₇ ◁ G

N	H ≅ G/N	type
2. C₇×#6900	C₃	semidirect
3. C₇×#6896	C₃	semidirect
4. C₇×#6894	C₃	semidirect
6. C₇×C₉×3₋¹⁺²	C₃	semidirect
7. 3₋¹⁺²²	C₇	direct (N × H)
16. C₇×C₉⋊C₉	C₃²	semidirect
17. C₇×C₉⋊C₉	C₃²	semidirect
20. C₇×C₉²	C₃²	semidirect
22. #6900	C₃×C₇	semidirect
23. #6896	C₃×C₇	semidirect
24. #6894	C₃×C₇	semidirect
26. C₉×3₋¹⁺²	C₃×C₇	semidirect
36. C₇×3₋¹⁺²	3₋¹⁺²	direct (N × H)
49. C₉⋊C₉	C₃²×C₇	semidirect
50. C₉⋊C₉	C₃²×C₇	semidirect
53. C₉²	C₃²×C₇	semidirect
65. C₇×C₉	C₃×3₋¹⁺²	semidirect
75. 3₋¹⁺²	C₇×3₋¹⁺²	direct (N × H)
96. C₉	C₃×C₇×3₋¹⁺²	semidirect
97. C₇	3₋¹⁺²²	direct (N × H)

H ≅ G/N

type

2. C₇×#6900

C₃

semidirect

3. C₇×#6896

C₃

semidirect

4. C₇×#6894

C₃

semidirect

6. C₇×C₉×3₋¹⁺²

C₃

semidirect

7. 3₋¹⁺²²

C₇

direct (N × H)

16. C₇×C₉⋊C₉

C₃²

semidirect

17. C₇×C₉⋊C₉

C₃²

semidirect

20. C₇×C₉²

C₃²

semidirect

22. #6900

C₃×C₇

semidirect

23. #6896

C₃×C₇

semidirect

24. #6894

C₃×C₇

semidirect

26. C₉×3₋¹⁺²

C₃×C₇

semidirect

36. C₇×3₋¹⁺²

3₋¹⁺²

direct (N × H)

49. C₉⋊C₉

C₃²×C₇

semidirect

50. C₉⋊C₉

C₃²×C₇

semidirect

53. C₉²

C₃²×C₇

semidirect

65. C₇×C₉

C₃×3₋¹⁺²

semidirect

75. 3₋¹⁺²

C₇×3₋¹⁺²

direct (N × H)

96. C₉

C₃×C₇×3₋¹⁺²

semidirect

97. C₇

3₋¹⁺²²

direct (N × H)

Subgroups — C₇×3₋¹⁺²² #19100307

Conjugacy classes of subgroups

Complements

Subgroups — C7×3−1+22 #19100307

Conjugacy classes of subgroups

Complements

Subgroups — C₇×3₋¹⁺²² #19100307