Subgroups — C₂×C₃×(C₁₃⋊C₃)⋊(C₇⋊D₇) #37525489

Order 22932.

600 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₇²×C₁₃⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×(C₁₃⋊C₃)⋊(C₇⋊D₇)	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₃×C₇⋊D₉₁	C₃	1	1	yes	maximal, characteristic, complemented
4	C₂×(C₁₃⋊C₃)⋊(C₇⋊D₇)	C₃	1	3	yes	maximal, direct product factor
5	C₃×C₇²×C₁₃⋊C₃	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₃×C₇⋊D₉₁	C₂×C₃	1	2	yes	complemented
7	C₂×C₇²×C₁₃⋊C₃	C₂×C₃	1	3	yes	complemented
8	(C₁₃⋊C₃)⋊(C₇⋊D₇)	C₂×C₃	1	6	yes	direct product factor
9	C₂×C₃×C₇²×C₁₃	C₂×C₃	1	1	yes	characteristic, complemented
10	C₂×C₃×(C₁₃⋊C₃)⋊D₇	—	7	8	no	maximal
11	C₂×C₇⋊D₉₁	C₃²	1	1	yes	characteristic, complemented, Hall
12	C₃×C₇²×C₁₃	C₂²×C₃	1	1	yes	characteristic, complemented
13	C₇²×C₁₃⋊C₃	C₂²×C₃	1	3	yes	complemented
14	C₂×C₃²×C₇⋊D₇	—	13	1	no	maximal, Hall
15	C₂×C₃×C₇×C₁₃⋊C₃	D₇	1	8	yes	complemented
16	C₃×(C₁₃⋊C₃)⋊D₇	—	7	16	no	—
17	C₂×C₇²×C₁₃	C₂×C₃²	1	1	yes	characteristic, complemented
18	C₇⋊D₉₁	C₂×C₃²	1	2	yes	complemented
19	C₂×C₃×D₉₁	—	7	8	no	—
20	C₂×(C₁₃⋊C₃)⋊D₇	—	7	24	no	—
21	C₂×C₃²×C₇²	—	13	1	no	—
22	C₃²×C₇⋊D₇	—	13	2	no	—
23	C₃×C₇×C₁₃⋊C₃	C₂×D₇	1	8	yes	complemented
24	C₇²×C₁₃	C₂²×C₃²	1	1	yes	characteristic, complemented, Hall
25	C₂×C₃×C₇⋊D₇	—	13	3	no	—
26	C₂×C₃×C₇⋊D₇	—	13	1	no	—
27	C₂×C₃×C₇×C₁₃	C₃×D₇	1	8	yes	complemented
28	C₃×D₉₁	—	7	16	no	—
29	C₂×C₇×C₁₃⋊C₃	C₃×D₇	1	24	yes	complemented
30	(C₁₃⋊C₃)⋊D₇	—	7	48	no	—
31	C₂×C₃×D₁₃⋊C₃	—	49	1	no	Hall
32	C₃²×C₇²	—	13	1	no	Hall
33	C₂×D₉₁	—	7	8	no	—
34	C₂×C₃×C₇²	—	13	3	no	—
35	C₂×C₃×C₇²	D₁₃⋊C₃	1	1	yes	characteristic, complemented
36	C₃×C₇⋊D₇	—	13	6	no	—
37	C₃×C₇⋊D₇	—	13	2	no	—
38	C₃×C₇×C₁₃	C₂×C₃×D₇	1	8	yes	complemented
39	C₇×C₁₃⋊C₃	C₂×C₃×D₇	1	24	yes	complemented
40	C₂×C₃²×D₇	—	91	8	no	—
41	C₂×C₃×C₁₃⋊C₃	C₇⋊D₇	1	1	yes	characteristic, complemented
42	C₃×D₁₃⋊C₃	—	49	2	no	—
43	C₂×C₇⋊D₇	—	13	1	no	Hall
44	C₂×C₇×C₁₃	C₃²×D₇	1	8	yes	complemented
45	D₉₁	—	7	16	no	—
46	C₂×C₃×D₁₃	—	49	1	no	—
47	C₂×D₁₃⋊C₃	—	49	3	no	—
48	C₃×C₇²	—	13	3	no	—
49	C₃×C₇²	C₂×D₁₃⋊C₃	1	1	yes	characteristic, complemented
50	C₂×C₃²×C₇	—	13	8	no	—
51	C₃²×D₇	—	91	16	no	—
52	C₃×C₁₃⋊C₃	C₂×C₇⋊D₇	1	1	yes	characteristic, complemented, Hall
53	C₂×C₇²	C₃×D₁₃⋊C₃	1	1	yes	characteristic, complemented
54	C₇⋊D₇	—	13	2	no	—
55	C₇×C₁₃	C₂×C₃²×D₇	1	8	yes	complemented
56	C₂×C₃×D₇	—	91	24	no	—
57	C₂×C₃×D₇	—	91	8	no	—
58	C₂×C₃×C₁₃	C₃×C₇⋊D₇	1	1	yes	characteristic, complemented
59	C₃×D₁₃	—	49	2	no	—
60	C₂×C₁₃⋊C₃	C₃×C₇⋊D₇	1	3	yes	complemented
61	D₁₃⋊C₃	—	49	6	no	—
62	C₃²×C₇	—	13	8	no	—
63	C₂×D₁₃	—	49	1	no	Hall
64	C₇²	C₂×C₃×D₁₃⋊C₃	1	1	yes	characteristic, complemented, Sylow
65	C₂×C₃×C₇	—	13	24	no	—
66	C₂×C₃×C₇	(C₁₃⋊C₃)⋊D₇	1	8	yes	complemented
67	C₃×D₇	—	91	48	no	—
68	C₃×D₇	—	91	16	no	—
69	C₃×C₁₃	C₂×C₃×C₇⋊D₇	1	1	yes	characteristic, complemented
70	C₁₃⋊C₃	C₂×C₃×C₇⋊D₇	1	3	yes	complemented
71	C₂²×C₃²	—	637	1	no	Hall
72	C₂×D₇	—	91	8	no	—
73	C₂×C₁₃	C₃²×C₇⋊D₇	1	1	yes	characteristic, complemented
74	D₁₃	—	49	2	no	—
75	C₃×C₇	—	13	24	no	—
76	C₃×C₇	C₂×(C₁₃⋊C₃)⋊D₇	1	8	yes	complemented
77	C₂×C₃²	—	13	1	no	—
78	C₂×C₃²	—	637	2	no	—
79	C₂×C₇	C₃×(C₁₃⋊C₃)⋊D₇	1	8	yes	complemented
80	D₇	—	91	16	no	—
81	C₁₃	C₂×C₃²×C₇⋊D₇	1	1	yes	characteristic, complemented, Sylow
82	C₂²×C₃	—	637	3	no	—
83	C₂²×C₃	—	637	1	no	—
84	C₃²	—	13	1	no	Sylow
85	C₇	C₂×C₃×(C₁₃⋊C₃)⋊D₇	1	8	yes	complemented
86	C₂×C₃	—	13	3	no	—
87	C₂×C₃	—	637	6	no	—
88	C₂×C₃	—	637	2	no	—
89	C₂×C₃	(C₁₃⋊C₃)⋊(C₇⋊D₇)	1	1	yes	characteristic, direct product factor
90	C₂²	—	637	1	no	Sylow
91	C₃	—	13	3	no	—
92	C₃	C₂×(C₁₃⋊C₃)⋊(C₇⋊D₇)	1	1	yes	characteristic, direct product factor
93	C₂	—	637	2	no	—
94	C₂	C₃×(C₁₃⋊C₃)⋊(C₇⋊D₇)	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₃	(C₁₃⋊C₃)⋊(C₇⋊D₇)
commutator, cosocle, hypocenter	C₇²×C₁₃	C₂²×C₃²
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₇²×C₁₃	C₂×C₃
LCS	G ▷ C₇²×C₁₃
UCS	C₁ ◁ C₂×C₃

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×C₃×C₇²×C₁₃⋊C₃	C₂	semidirect
2. C₃×(C₁₃⋊C₃)⋊(C₇⋊D₇)	C₂	direct (N × H)
3. C₂×C₃×C₇⋊D₉₁	C₃	semidirect
4. C₂×(C₁₃⋊C₃)⋊(C₇⋊D₇)	C₃	direct (N × H)
5. C₃×C₇²×C₁₃⋊C₃	C₂²	semidirect
6. C₃×C₇⋊D₉₁	C₂×C₃	semidirect
7. C₂×C₇²×C₁₃⋊C₃	C₂×C₃	semidirect
8. (C₁₃⋊C₃)⋊(C₇⋊D₇)	C₂×C₃	direct (N × H)
9. C₂×C₃×C₇²×C₁₃	C₂×C₃	semidirect
11. C₂×C₇⋊D₉₁	C₃²	semidirect
12. C₃×C₇²×C₁₃	C₂²×C₃	semidirect
13. C₇²×C₁₃⋊C₃	C₂²×C₃	semidirect
15. C₂×C₃×C₇×C₁₃⋊C₃	D₇	semidirect
17. C₂×C₇²×C₁₃	C₂×C₃²	semidirect
18. C₇⋊D₉₁	C₂×C₃²	semidirect
23. C₃×C₇×C₁₃⋊C₃	C₂×D₇	semidirect
24. C₇²×C₁₃	C₂²×C₃²	semidirect
27. C₂×C₃×C₇×C₁₃	C₃×D₇	semidirect
29. C₂×C₇×C₁₃⋊C₃	C₃×D₇	semidirect
35. C₂×C₃×C₇²	D₁₃⋊C₃	semidirect
38. C₃×C₇×C₁₃	C₂×C₃×D₇	semidirect
39. C₇×C₁₃⋊C₃	C₂×C₃×D₇	semidirect
41. C₂×C₃×C₁₃⋊C₃	C₇⋊D₇	semidirect
44. C₂×C₇×C₁₃	C₃²×D₇	semidirect
49. C₃×C₇²	C₂×D₁₃⋊C₃	semidirect
52. C₃×C₁₃⋊C₃	C₂×C₇⋊D₇	semidirect
53. C₂×C₇²	C₃×D₁₃⋊C₃	semidirect
55. C₇×C₁₃	C₂×C₃²×D₇	semidirect
58. C₂×C₃×C₁₃	C₃×C₇⋊D₇	semidirect
60. C₂×C₁₃⋊C₃	C₃×C₇⋊D₇	semidirect
64. C₇²	C₂×C₃×D₁₃⋊C₃	semidirect
66. C₂×C₃×C₇	(C₁₃⋊C₃)⋊D₇	semidirect
69. C₃×C₁₃	C₂×C₃×C₇⋊D₇	semidirect
70. C₁₃⋊C₃	C₂×C₃×C₇⋊D₇	semidirect
73. C₂×C₁₃	C₃²×C₇⋊D₇	semidirect
76. C₃×C₇	C₂×(C₁₃⋊C₃)⋊D₇	semidirect
79. C₂×C₇	C₃×(C₁₃⋊C₃)⋊D₇	semidirect
81. C₁₃	C₂×C₃²×C₇⋊D₇	semidirect
85. C₇	C₂×C₃×(C₁₃⋊C₃)⋊D₇	semidirect
89. C₂×C₃	(C₁₃⋊C₃)⋊(C₇⋊D₇)	direct (N × H)
92. C₃	C₂×(C₁₃⋊C₃)⋊(C₇⋊D₇)	direct (N × H)
94. C₂	C₃×(C₁₃⋊C₃)⋊(C₇⋊D₇)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×C₇²×C₁₃⋊C₃	2
C₃×(C₁₃⋊C₃)⋊(C₇⋊D₇)	2
C₂×C₃×C₇⋊D₉₁	3
C₂×(C₁₃⋊C₃)⋊(C₇⋊D₇)	3
C₂×C₃×(C₁₃⋊C₃)⋊D₇	7
C₂×C₃²×C₇⋊D₇	13

Subgroups — C2×C3×(C13⋊C3)⋊(C7⋊D7) #37525489

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×(C₁₃⋊C₃)⋊(C₇⋊D₇) #37525489