Subgroups — C₂×C₃²×D₆₃₇ #37524320

Order 22932.

180 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×D₆₃₇	C₂	1	2	yes	maximal, direct product factor
2	C₂×C₃²×C₁₃×C₄₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×D₆₃₇	C₃	1	4	yes	maximal, direct product factor
4	C₃²×C₁₃×C₄₉	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₃×D₆₃₇	C₂×C₃	1	8	yes	direct product factor
6	C₂×C₃×C₁₃×C₄₉	C₂×C₃	1	4	yes	complemented
7	C₂×C₃²×D₉₁	—	7	1	no	maximal
8	C₂×D₆₃₇	C₃²	1	1	yes	characteristic, direct product factor, Hall
9	C₃×C₁₃×C₄₉	C₂²×C₃	1	4	yes	complemented
10	C₂×C₃²×D₄₉	—	13	1	no	maximal, Hall
11	C₂×C₃²×C₇×C₁₃	D₇	1	1	yes	characteristic
12	C₃²×D₉₁	—	7	2	no	—
13	C₂×C₁₃×C₄₉	C₂×C₃²	1	1	yes	characteristic, complemented
14	D₆₃₇	C₂×C₃²	1	2	yes	direct product factor
15	C₂×C₃×D₉₁	—	7	4	no	—
16	C₂×C₃²×C₄₉	D₁₃	1	1	yes	characteristic, complemented
17	C₃²×D₄₉	—	13	2	no	—
18	C₃²×C₇×C₁₃	C₂×D₇	1	1	yes	characteristic
19	C₁₃×C₄₉	C₂²×C₃²	1	1	yes	characteristic, complemented, Hall
20	C₂×C₃×D₄₉	—	13	4	no	—
21	C₂×C₃×C₇×C₁₃	C₃×D₇	1	4	yes	—
22	C₃×D₉₁	—	7	8	no	—
23	C₂×C₃²×D₁₃	—	49	1	no	Hall
24	C₃²×C₄₉	C₂×D₁₃	1	1	yes	characteristic, complemented, Hall
25	C₂×D₉₁	—	7	1	no	—
26	C₂×C₃×C₄₉	C₃×D₁₃	1	4	yes	complemented
27	C₃×D₄₉	—	13	8	no	—
28	C₃×C₇×C₁₃	C₂×C₃×D₇	1	4	yes	—
29	C₂×C₃²×D₇	—	91	1	no	—
30	C₂×C₃²×C₁₃	D₄₉	1	1	yes	characteristic, complemented
31	C₃²×D₁₃	—	49	2	no	—
32	C₂×D₄₉	—	13	1	no	Hall
33	C₂×C₇×C₁₃	C₃²×D₇	1	1	yes	characteristic
34	D₉₁	—	7	2	no	—
35	C₂×C₃×D₁₃	—	49	4	no	—
36	C₃×C₄₉	C₂×C₃×D₁₃	1	4	yes	complemented
37	C₂×C₃²×C₇	D₉₁	1	1	yes	characteristic
38	C₃²×D₇	—	91	2	no	—
39	C₃²×C₁₃	C₂×D₄₉	1	1	yes	characteristic, complemented, Hall
40	C₂×C₄₉	C₃²×D₁₃	1	1	yes	characteristic, complemented
41	D₄₉	—	13	2	no	—
42	C₇×C₁₃	C₂×C₃²×D₇	1	1	yes	characteristic
43	C₂×C₃×D₇	—	91	4	no	—
44	C₂×C₃×C₁₃	C₃×D₄₉	1	4	yes	complemented
45	C₃×D₁₃	—	49	8	no	—
46	C₃²×C₇	C₂×D₉₁	1	1	yes	characteristic
47	C₂×D₁₃	—	49	1	no	Hall
48	C₄₉	C₂×C₃²×D₁₃	1	1	yes	characteristic, complemented, Sylow
49	C₂×C₃×C₇	C₃×D₉₁	1	4	yes	—
50	C₃×D₇	—	91	8	no	—
51	C₃×C₁₃	C₂×C₃×D₄₉	1	4	yes	complemented
52	C₂²×C₃²	—	637	1	no	Hall
53	C₂×D₇	—	91	1	no	—
54	C₂×C₁₃	C₃²×D₄₉	1	1	yes	characteristic, complemented
55	D₁₃	—	49	2	no	—
56	C₃×C₇	C₂×C₃×D₉₁	1	4	yes	—
57	C₂×C₃²	—	637	2	no	—
58	C₂×C₃²	D₆₃₇	1	1	yes	characteristic, direct product factor
59	C₂×C₇	C₃²×D₉₁	1	1	yes	characteristic
60	D₇	—	91	2	no	—
61	C₁₃	C₂×C₃²×D₄₉	1	1	yes	characteristic, complemented, Sylow
62	C₂²×C₃	—	637	4	no	—
63	C₃²	C₂×D₆₃₇	1	1	yes	characteristic, direct product factor, Sylow
64	C₇	C₂×C₃²×D₉₁	1	1	yes	characteristic
65	C₂×C₃	—	637	8	no	—
66	C₂×C₃	C₃×D₆₃₇	1	4	yes	direct product factor
67	C₂²	—	637	1	no	Sylow
68	C₃	C₂×C₃×D₆₃₇	1	4	yes	direct product factor
69	C₂	—	637	2	no	—
70	C₂	C₃²×D₆₃₇	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₃²	D₆₃₇
commutator, cosocle, hypocenter	C₁₃×C₄₉	C₂²×C₃²
Frattini	C₇	C₂×C₃²×D₉₁
Fitting	C₂×C₃²×C₁₃×C₄₉	C₂
socle	C₂×C₃²×C₇×C₁₃	D₇
LCS	G ▷ C₁₃×C₄₉
UCS	C₁ ◁ C₂×C₃²

Complements

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

N	H ≅ G/N	type
1. C₃²×D₆₃₇	C₂	direct (N × H)
2. C₂×C₃²×C₁₃×C₄₉	C₂	semidirect
3. C₂×C₃×D₆₃₇	C₃	direct (N × H)
4. C₃²×C₁₃×C₄₉	C₂²	semidirect
5. C₃×D₆₃₇	C₂×C₃	direct (N × H)
6. C₂×C₃×C₁₃×C₄₉	C₂×C₃	semidirect
8. C₂×D₆₃₇	C₃²	direct (N × H)
9. C₃×C₁₃×C₄₉	C₂²×C₃	semidirect
13. C₂×C₁₃×C₄₉	C₂×C₃²	semidirect
14. D₆₃₇	C₂×C₃²	direct (N × H)
16. C₂×C₃²×C₄₉	D₁₃	semidirect
19. C₁₃×C₄₉	C₂²×C₃²	semidirect
24. C₃²×C₄₉	C₂×D₁₃	semidirect
26. C₂×C₃×C₄₉	C₃×D₁₃	semidirect
30. C₂×C₃²×C₁₃	D₄₉	semidirect
36. C₃×C₄₉	C₂×C₃×D₁₃	semidirect
39. C₃²×C₁₃	C₂×D₄₉	semidirect
40. C₂×C₄₉	C₃²×D₁₃	semidirect
44. C₂×C₃×C₁₃	C₃×D₄₉	semidirect
48. C₄₉	C₂×C₃²×D₁₃	semidirect
51. C₃×C₁₃	C₂×C₃×D₄₉	semidirect
54. C₂×C₁₃	C₃²×D₄₉	semidirect
58. C₂×C₃²	D₆₃₇	direct (N × H)
61. C₁₃	C₂×C₃²×D₄₉	semidirect
63. C₃²	C₂×D₆₃₇	direct (N × H)
66. C₂×C₃	C₃×D₆₃₇	direct (N × H)
68. C₃	C₂×C₃×D₆₃₇	direct (N × H)
70. C₂	C₃²×D₆₃₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃²×D₆₃₇	2
C₂×C₃²×C₁₃×C₄₉	2
C₂×C₃×D₆₃₇	3
C₂×C₃²×D₉₁	7
C₂×C₃²×D₄₉	13

Subgroups — C2×C32×D637 #37524320

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃²×D₆₃₇ #37524320