Subgroups — #72340973

Order 3072.

At least 7346 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

The subgroup lattice has 102 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72340970	C₃	1	1	yes	maximal, characteristic, complemented, Sylow
2	#72340968	C₂	1	2	yes	maximal, complemented
3	#72340966	C₂	1	2	yes	maximal, complemented
4	#72340964	C₂	1	1	yes	maximal, characteristic, complemented
5	#72247728	C₂	1	1	yes	maximal, characteristic, complemented
6	#72237717	C₂	1	1	yes	maximal, characteristic, complemented
7	#60045⋊C₃	C₂²	1	1	yes	characteristic, complemented
8	#58990⋊C₃	C₂²	1	1	yes	characteristic, complemented
9	#58990⋊C₃	C₂²	1	1	yes	characteristic, complemented
10	#58169⋊C₃	C₂²	1	2	yes	complemented
11	#53169⋊C₃	C₂²	1	2	yes	complemented
12	#10332164	C₂×C₃	1	2	yes	complemented
13	#10320507	C₂×C₃	1	2	yes	complemented
14	#9061103	C₂×C₃	1	1	yes	characteristic, complemented
15	#8934366	C₂×C₃	1	1	yes	characteristic, complemented
16	#8108057	C₂×C₃	1	1	yes	characteristic, complemented
17	#2679⋊C₃	C₂³	1	1	yes	characteristic, complemented
18	C₂²×H(3,𝔽₄)	A₄	1	1	yes	characteristic, complemented
19	C₂²×#560	A₄	1	1	yes	characteristic, complemented
20	C₂³×#120	A₄	1	1	yes	characteristic, complemented
21	#60045	C₂²×C₃	1	1	yes	characteristic, complemented
22	#58990	C₂²×C₃	1	1	yes	characteristic, complemented
23	#58990	C₂²×C₃	1	1	yes	characteristic, complemented
24	#58169	C₂²×C₃	1	2	yes	complemented
25	#53169	C₂²×C₃	1	2	yes	complemented
26	C₂×#2842	A₄	1	2	yes	complemented
27	C₂⁷	C₂×A₄	1	1	yes	characteristic, complemented
28	C₂×H(3,𝔽₄)	C₂×A₄	1	1	yes	characteristic
29	C₂×H(3,𝔽₄)	C₂×A₄	1	2	yes	complemented
30	C₂×H(3,𝔽₄)	C₂×A₄	1	1	yes	characteristic, complemented
31	C₂×#560	C₂×A₄	1	1	yes	characteristic, complemented
32	C₂×#560	C₂×A₄	1	2	yes	complemented
33	C₂×#560	C₂×A₄	1	1	yes	characteristic, complemented
34	C₂²×#127	C₂×A₄	1	1	yes	characteristic, complemented
35	C₂²×#126	C₂×A₄	1	2	yes	complemented
36	C₂²×#120	C₂×A₄	1	2	yes	—
37	C₂²×#120	C₂×A₄	1	1	yes	characteristic
38	C₂²×#120	C₂×A₄	1	2	yes	complemented
39	C₂²×#120	C₂×A₄	1	1	yes	characteristic, complemented
40	C₂²×#120	C₂×A₄	1	2	yes	complemented
41	C₂²×#120	C₂×A₄	1	1	yes	characteristic, complemented
42	#2842	C₂×A₄	1	2	yes	complemented
43	#2842	C₂×A₄	1	2	yes	complemented
44	#2842	C₂×A₄	1	2	yes	complemented
45	#2842	C₂×A₄	1	2	yes	complemented
46	#2679	C₂³×C₃	1	1	yes	characteristic, complemented
47	C₂×#399	C₂×A₄	1	2	yes	complemented
48	C₂×#396	C₂×A₄	1	2	yes	complemented
49	C₂×#396	C₂×A₄	1	2	yes	complemented
50	C₂⁶	C₂²×A₄	1	1	yes	characteristic
51	C₂⁶	C₂²×A₄	1	2	yes	—
52	C₂⁶	C₂²×A₄	1	1	yes	characteristic, complemented
53	C₂⁶	C₂²⋊A₄	1	1	yes	characteristic
54	C₂×#127	C₂²×A₄	1	1	yes	characteristic
55	C₂×#127	C₂²×A₄	1	1	yes	characteristic, complemented
56	C₂×#126	C₂²×A₄	1	2	yes	—
57	C₂×#126	C₂²×A₄	1	2	yes	complemented
58	C₂×#120	C₂²×A₄	1	1	yes	characteristic
59	C₂×#120	C₂²×A₄	1	1	yes	characteristic
60	C₂×#120	C₂²×A₄	1	1	yes	characteristic
61	C₂×#120	C₂²×A₄	1	2	yes	—
62	C₂×#120	C₂²×A₄	1	2	yes	—
63	C₂×#120	C₂²×A₄	1	1	yes	characteristic, complemented
64	C₂×#120	C₂²×A₄	1	2	yes	complemented
65	C₂²×C₄²	C₂²×A₄	1	1	yes	characteristic, complemented
66	#399	C₂²×A₄	1	2	yes	—
67	#399	C₂²×A₄	1	2	yes	complemented
68	#396	C₂²×A₄	1	2	yes	—
69	#396	C₂²×A₄	1	2	yes	—
70	#396	C₂²×A₄	1	2	yes	complemented
71	#396	C₂²×A₄	1	2	yes	complemented
72	C₂×#95	C₂²×A₄	1	2	yes	complemented
73	C₂⁵	#997	1	8	yes	—
74	C₂⁵	C₂³×A₄	1	1	yes	characteristic
75	C₂⁵	C₂³×A₄	1	1	yes	characteristic
76	C₂⁵	C₂×C₂²⋊A₄	1	1	yes	characteristic
77	C₂⁵	C₂×C₂²⋊A₄	1	2	yes	—
78	C₂⁵	C₂×C₂²⋊A₄	1	2	yes	—
79	C₂⁵	C₂×C₂²⋊A₄	1	1	yes	characteristic
80	C₂⁵	C₂×C₂²⋊A₄	1	1	yes	characteristic
81	C₂×C₄²	C₂³×A₄	1	1	yes	characteristic
82	#95	C₂³×A₄	1	2	yes	—
83	C₂⁴	C₂×#997	1	4	yes	—
84	C₂⁴	C₂×#997	1	8	yes	—
85	C₂⁴	C₂×#997	1	8	yes	—
86	C₂⁴	C₂×#997	1	4	yes	—
87	C₂⁴	C₂×#997	1	4	yes	—
88	C₂⁴	C₂²×C₂²⋊A₄	1	1	yes	characteristic
89	C₂⁴	C₂²×C₂²⋊A₄	1	2	yes	—
90	C₂⁴	C₂²×C₂²⋊A₄	1	1	yes	characteristic
91	C₂⁴	C₂²×C₂²⋊A₄	1	1	yes	characteristic
92	C₂⁴	C₂²×C₂²⋊A₄	1	2	yes	—
93	C₂³	C₂²×#997	1	2	yes	—
94	C₂³	C₂²×#997	1	4	yes	—
95	C₂³	C₂²×#997	1	4	yes	—
96	C₂³	C₂²×#997	1	2	yes	—
97	C₂³	C₂²×#997	1	2	yes	—
98	C₂³	C₂³×C₂²⋊A₄	1	1	yes	characteristic
99	C₂²	C₂³×#997	1	1	yes	characteristic
100	C₂	#72262063	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂	#72262063
commutator, cosocle, hypocenter	#2679	C₂³×C₃
Frattini, socle	C₂³	C₂³×C₂²⋊A₄
Fitting	#72340970	C₃
LCS	G ▷ #2679
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
1. #72340970	C₃	semidirect
2. #72340968	C₂	semidirect
3. #72340966	C₂	semidirect
4. #72340964	C₂	semidirect
5. #72247728	C₂	semidirect
6. #72237717	C₂	semidirect
7. #60045⋊C₃	C₂²	semidirect
8. #58990⋊C₃	C₂²	semidirect
9. #58990⋊C₃	C₂²	semidirect
10. #58169⋊C₃	C₂²	semidirect
11. #53169⋊C₃	C₂²	semidirect
12. #10332164	C₂×C₃	semidirect
13. #10320507	C₂×C₃	semidirect
14. #9061103	C₂×C₃	semidirect
15. #8934366	C₂×C₃	semidirect
16. #8108057	C₂×C₃	semidirect
17. #2679⋊C₃	C₂³	semidirect
18. C₂²×H(3,𝔽₄)	A₄	semidirect
19. C₂²×#560	A₄	semidirect
20. C₂³×#120	A₄	semidirect
21. #60045	C₂²×C₃	semidirect
22. #58990	C₂²×C₃	semidirect
23. #58990	C₂²×C₃	semidirect
24. #58169	C₂²×C₃	semidirect
25. #53169	C₂²×C₃	semidirect
26. C₂×#2842	A₄	semidirect
27. C₂⁷	C₂×A₄	semidirect
29. C₂×H(3,𝔽₄)	C₂×A₄	semidirect
30. C₂×H(3,𝔽₄)	C₂×A₄	semidirect
31. C₂×#560	C₂×A₄	semidirect
32. C₂×#560	C₂×A₄	semidirect
33. C₂×#560	C₂×A₄	semidirect
34. C₂²×#127	C₂×A₄	semidirect
35. C₂²×#126	C₂×A₄	semidirect
38. C₂²×#120	C₂×A₄	semidirect
39. C₂²×#120	C₂×A₄	semidirect
40. C₂²×#120	C₂×A₄	semidirect
41. C₂²×#120	C₂×A₄	semidirect
42. #2842	C₂×A₄	semidirect
43. #2842	C₂×A₄	semidirect
44. #2842	C₂×A₄	semidirect
45. #2842	C₂×A₄	semidirect
46. #2679	C₂³×C₃	semidirect
47. C₂×#399	C₂×A₄	semidirect
48. C₂×#396	C₂×A₄	semidirect
49. C₂×#396	C₂×A₄	semidirect
52. C₂⁶	C₂²×A₄	semidirect
55. C₂×#127	C₂²×A₄	semidirect
57. C₂×#126	C₂²×A₄	semidirect
63. C₂×#120	C₂²×A₄	semidirect
64. C₂×#120	C₂²×A₄	semidirect
65. C₂²×C₄²	C₂²×A₄	semidirect
67. #399	C₂²×A₄	semidirect
70. #396	C₂²×A₄	semidirect
71. #396	C₂²×A₄	semidirect
72. C₂×#95	C₂²×A₄	semidirect

Maximal subgroups

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

subgroup	index
#72340970	3
#72340968	2
#72340966	2
#72340964	2
#72247728	2
#72237717	2
C₂²×#5450	4
C₂²×#560⋊C₃	4
C₂³×#120⋊C₃	4
C₂×#2842⋊C₃	4