Subgroups — C₂³×C₄×#10587739 #31797759

Order 17536.

3000 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×#10587739	C₂	1	2	yes	maximal
2	C₂²×C₄×#10587739	C₂	1	28	yes	maximal, direct product factor
3	C₂³×C₄×D₁₃₇	C₂	1	1	yes	maximal, characteristic
4	C₂³×#10587739	C₄	1	32	yes	direct product factor
5	C₂³×#10587739	C₂²	1	28	yes	—
6	C₂×C₄×#10587739	C₂²	1	112	yes	direct product factor
7	C₂⁴×D₁₃₇	C₂²	1	1	yes	characteristic
8	C₂³×Dic₁₃₇	C₄	1	1	yes	characteristic, complemented
9	C₂²×C₄×D₁₃₇	C₄	1	14	yes	complemented
10	C₂²×C₄×D₁₃₇	C₂²	1	14	yes	—
11	C₂³×C₄×C₁₃₇	C₄	1	1	yes	characteristic, complemented
12	C₂²×#10587739	C₂×C₄	1	224	yes	direct product factor
13	C₂²×#10587739	C₂³	1	56	yes	—
14	C₄×#10587739	C₂³	1	64	yes	direct product factor
15	C₂³×D₁₃₇	C₂×C₄	1	16	yes	—
16	C₂³×D₁₃₇	C₂×C₄	1	7	yes	—
17	C₂³×D₁₃₇	C₂³	1	7	yes	—
18	C₂²×Dic₁₃₇	C₂×C₄	1	14	yes	complemented
19	C₂×C₄×D₁₃₇	C₂×C₄	1	84	yes	complemented
20	C₂×C₄×D₁₃₇	C₂³	1	28	yes	—
21	C₂⁴×C₁₃₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₁₃₇	C₂×C₄	1	14	yes	complemented
23	C₂×#10587739	C₂²×C₄	1	224	yes	direct product factor
24	C₂×#10587739	C₂⁴	1	16	yes	—
25	C₂²×D₁₃₇	C₄²	1	56	yes	complemented
26	C₂²×D₁₃₇	C₂²×C₄	1	21	yes	—
27	C₂²×D₁₃₇	C₂²×C₄	1	56	yes	—
28	C₂²×D₁₃₇	C₂⁴	1	7	yes	—
29	C₂×Dic₁₃₇	C₂²×C₄	1	28	yes	complemented
30	C₄×D₁₃₇	C₂²×C₄	1	56	yes	complemented
31	C₄×D₁₃₇	C₂⁴	1	8	yes	—
32	C₂³×C₁₃₇	C₄²	1	8	yes	complemented
33	C₂³×C₁₃₇	C₂²×C₄	1	7	yes	—
34	C₂×C₄×C₁₃₇	C₂²×C₄	1	28	yes	complemented
35	C₂²×C₁₃₇	C₂×C₄²	1	28	yes	complemented
36	C₂²×C₁₃₇	C₂³×C₄	1	7	yes	—
37	C₂×D₁₃₇	C₂×C₄²	1	84	yes	complemented
38	C₂×D₁₃₇	C₂³×C₄	1	28	yes	—
39	C₂×D₁₃₇	C₂³×C₄	1	7	yes	—
40	C₂×D₁₃₇	C₂⁵	1	1	yes	characteristic
41	#10587739	C₂³×C₄	1	32	yes	direct product factor
42	C₄×C₁₃₇	C₂³×C₄	1	8	yes	complemented
43	Dic₁₃₇	C₂³×C₄	1	8	yes	complemented
44	C₂×C₁₃₇	C₂²×C₄²	1	14	yes	complemented
45	C₂×C₁₃₇	C₂⁴×C₄	1	1	yes	characteristic
46	D₁₃₇	C₂²×C₄²	1	14	yes	complemented
47	D₁₃₇	C₂⁴×C₄	1	2	yes	—
48	C₁₃₇	C₂³×C₄²	1	1	yes	characteristic, complemented, Sylow
49	C₂³×C₄²	—	137	1	no	maximal, Sylow
50	C₂⁴×C₄	—	137	2	no	—
51	C₂⁴×C₄	—	137	1	no	—
52	C₂²×C₄²	—	137	28	no	—
53	C₂⁵	—	137	1	no	—
54	C₂³×C₄	—	137	32	no	—
55	C₂³×C₄	—	137	28	no	—
56	C₂³×C₄	—	137	1	no	—
57	C₂³×C₄	—	137	14	no	—
58	C₂³×C₄	—	137	14	no	—
59	C₂³×C₄	#10587739	1	1	yes	characteristic, direct product factor
60	C₂×C₄²	—	137	112	no	—
61	C₂⁴	—	137	16	no	—
62	C₂⁴	—	137	7	no	—
63	C₂⁴	—	137	7	no	—
64	C₂⁴	C₂×#10587739	1	1	yes	characteristic
65	C₂²×C₄	—	137	56	no	—
66	C₂²×C₄	—	137	224	no	—
67	C₂²×C₄	—	137	14	no	—
68	C₂²×C₄	—	137	84	no	—
69	C₂²×C₄	—	137	28	no	—
70	C₂²×C₄	C₂×#10587739	1	14	yes	direct product factor
71	C₄²	—	137	64	no	—
72	C₂³	—	137	56	no	—
73	C₂³	—	137	56	no	—
74	C₂³	—	137	7	no	—
75	C₂³	—	137	21	no	—
76	C₂³	C₄×#10587739	1	8	yes	direct product factor
77	C₂³	C₂²×#10587739	1	7	yes	—
78	C₂×C₄	—	137	16	no	—
79	C₂×C₄	—	137	224	no	—
80	C₂×C₄	—	137	28	no	—
81	C₂×C₄	—	137	8	no	—
82	C₂×C₄	—	137	56	no	—
83	C₂×C₄	C₂²×#10587739	1	28	yes	direct product factor
84	C₂²	—	137	28	no	—
85	C₂²	—	137	84	no	—
86	C₂²	—	137	1	no	—
87	C₂²	—	137	7	no	—
88	C₂²	C₂×C₄×#10587739	1	28	yes	direct product factor
89	C₂²	C₂³×#10587739	1	7	yes	—
90	C₄	—	137	32	no	—
91	C₄	—	137	8	no	—
92	C₄	C₂³×#10587739	1	8	yes	direct product factor
93	C₂	—	137	2	no	—
94	C₂	—	137	14	no	—
95	C₂	C₂²×C₄×#10587739	1	14	yes	direct product factor
96	C₂	C₂⁴×#10587739	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂³×C₄	#10587739
commutator, hypocenter	C₁₃₇	C₂³×C₄²
Frattini	C₂	C₂⁴×#10587739
Fitting	C₂³×C₄×C₁₃₇	C₄
socle	C₂⁴×C₁₃₇	C₂×C₄
prime radical	C₂⁴×D₁₃₇	C₂²
cosocle	C₂×D₁₃₇	C₂⁵
LCS	G ▷ C₁₃₇
UCS	C₁ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄×#10587739	C₂	direct (N × H)
4. C₂³×#10587739	C₄	direct (N × H)
6. C₂×C₄×#10587739	C₂²	direct (N × H)
8. C₂³×Dic₁₃₇	C₄	semidirect
9. C₂²×C₄×D₁₃₇	C₄	semidirect
11. C₂³×C₄×C₁₃₇	C₄	semidirect
12. C₂²×#10587739	C₂×C₄	direct (N × H)
14. C₄×#10587739	C₂³	direct (N × H)
18. C₂²×Dic₁₃₇	C₂×C₄	semidirect
19. C₂×C₄×D₁₃₇	C₂×C₄	semidirect
22. C₂²×C₄×C₁₃₇	C₂×C₄	semidirect
23. C₂×#10587739	C₂²×C₄	direct (N × H)
25. C₂²×D₁₃₇	C₄²	semidirect
29. C₂×Dic₁₃₇	C₂²×C₄	semidirect
30. C₄×D₁₃₇	C₂²×C₄	semidirect
32. C₂³×C₁₃₇	C₄²	semidirect
34. C₂×C₄×C₁₃₇	C₂²×C₄	semidirect
35. C₂²×C₁₃₇	C₂×C₄²	semidirect
37. C₂×D₁₃₇	C₂×C₄²	semidirect
41. #10587739	C₂³×C₄	direct (N × H)
42. C₄×C₁₃₇	C₂³×C₄	semidirect
43. Dic₁₃₇	C₂³×C₄	semidirect
44. C₂×C₁₃₇	C₂²×C₄²	semidirect
46. D₁₃₇	C₂²×C₄²	semidirect
48. C₁₃₇	C₂³×C₄²	semidirect
59. C₂³×C₄	#10587739	direct (N × H)
70. C₂²×C₄	C₂×#10587739	direct (N × H)
76. C₂³	C₄×#10587739	direct (N × H)
83. C₂×C₄	C₂²×#10587739	direct (N × H)
88. C₂²	C₂×C₄×#10587739	direct (N × H)
92. C₄	C₂³×#10587739	direct (N × H)
95. C₂	C₂²×C₄×#10587739	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×#10587739	2
C₂²×C₄×#10587739	2
C₂³×C₄×D₁₃₇	2
C₂³×C₄²	137

Subgroups — C23×C4×#10587739 #31797759

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₄×#10587739 #31797759