Subgroups — #72341860

Order 3072.

At least 8832 conjugacy classes of subgroups; only the 264 normal subgroups are shown.

The subgroup lattice has 136 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#3025⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
2	#72341820	C₂	1	2	yes	maximal, complemented
3	#72341793	C₂	1	2	yes	maximal, complemented
4	#72341762	C₂	1	4	yes	maximal, complemented
5	#72341563	C₂	1	2	yes	maximal, complemented
6	#72341483	C₂	1	1	yes	maximal, characteristic, complemented
7	#72280972	C₃	1	1	yes	maximal, characteristic, complemented, Sylow
8	C₂×#11710937	C₂	1	1	yes	maximal, characteristic, complemented
9	#72241384	C₂	1	2	yes	maximal, complemented
10	C₂×#87548	C₂²	1	1	yes	characteristic, complemented
11	C₂×#87547	C₂²	1	2	yes	—
12	C₂×#87547	C₂²	1	1	yes	characteristic, complemented
13	C₂²×#5950	C₂²	1	1	yes	characteristic
14	C₂²×#5950	C₂²	1	2	yes	complemented
15	C₂×#3025⋊C₃	C₂²	1	2	yes	complemented
16	C₂×#3025⋊C₃	C₂²	1	2	yes	complemented
17	C₂²×#458⋊C₃	C₂²	1	1	yes	characteristic, complemented
18	C₂²×B(4,𝔽₂)⋊C₃	C₂²	1	1	yes	characteristic, complemented
19	#32901⋊C₃	C₂²	1	4	yes	complemented
20	#32900⋊C₃	C₂²	1	4	yes	complemented
21	#32899⋊C₃	C₂²	1	4	yes	complemented
22	#32898⋊C₃	C₂²	1	4	yes	complemented
23	#20610⋊C₃	C₂²	1	2	yes	complemented
24	#20609⋊C₃	C₂²	1	2	yes	complemented
25	#20608⋊C₃	C₂²	1	2	yes	complemented
26	C₂×#63103	C₂×C₃	1	1	yes	characteristic, complemented
27	C₂²×#3025	C₂×C₃	1	1	yes	characteristic, complemented
28	#6628899	C₂×C₃	1	2	yes	complemented
29	#6628898	C₂×C₃	1	2	yes	complemented
30	#6628897	C₂×C₃	1	2	yes	complemented
31	#6628896	C₂×C₃	1	2	yes	complemented
32	#6628891	C₂×C₃	1	4	yes	complemented
33	#6370734	C₂×C₃	1	1	yes	characteristic, complemented
34	C₂×#5950	D₄	1	4	yes	complemented
35	C₂×#5950	C₂³	1	2	yes	—
36	C₂×#5950	C₂³	1	1	yes	characteristic
37	C₂×#5950	C₂³	1	1	yes	characteristic
38	C₂²×#997	C₂³	1	2	yes	—
39	C₂²×#997	C₂³	1	1	yes	characteristic
40	C₂×#458⋊C₃	C₂³	1	2	yes	complemented
41	C₂×B(4,𝔽₂)⋊C₃	C₂³	1	2	yes	complemented
42	#1790⋊C₃	C₂³	1	4	yes	complemented
43	C₂×2₋¹⁺⁶	C₂²×C₃	1	1	yes	characteristic, complemented
44	C₂×H(7,𝔽₂)	C₂²×C₃	1	2	yes	—
45	C₂×H(7,𝔽₂)	C₂²×C₃	1	1	yes	characteristic, complemented
46	C₂²×#585	C₂²×C₃	1	1	yes	characteristic
47	C₂²×#585	C₂²×C₃	1	2	yes	complemented
48	C₂²×#525	A₄	1	1	yes	characteristic, complemented
49	C₂×#3025	C₂²×C₃	1	2	yes	—
50	C₂×#3025	C₂²×C₃	1	2	yes	—
51	C₂²×#458	C₂²×C₃	1	1	yes	characteristic
52	C₂²×B(4,𝔽₂)	C₂²×C₃	1	1	yes	characteristic
53	#32901	C₂²×C₃	1	4	yes	complemented
54	#32900	C₂²×C₃	1	4	yes	complemented
55	#32899	C₂²×C₃	1	4	yes	—
56	#32898	C₂²×C₃	1	4	yes	—
57	#20610	C₂²×C₃	1	2	yes	complemented
58	#20609	C₂²×C₃	1	2	yes	—
59	#20608	C₂²×C₃	1	2	yes	complemented
60	#5950	C₂×D₄	1	2	yes	complemented
61	#5950	C₂×D₄	1	2	yes	complemented
62	C₂×#997	C₂×D₄	1	2	yes	—
63	C₂×#997	C₂⁴	1	1	yes	characteristic
64	C₂×#585	C₃×D₄	1	4	yes	complemented
65	C₂×#585	C₂³×C₃	1	2	yes	—
66	C₂×#585	C₂³×C₃	1	1	yes	characteristic
67	C₂×#585	C₂³×C₃	1	1	yes	characteristic
68	C₂²×H(5,𝔽₂)	C₂³×C₃	1	2	yes	—
69	C₂²×H(5,𝔽₂)	C₂³×C₃	1	1	yes	characteristic
70	C₂³×#41	C₂×A₄	1	1	yes	characteristic
71	C₂⁵×C₄	C₂×A₄	1	1	yes	characteristic, complemented
72	C₂²×#123	C₂×A₄	1	2	yes	complemented
73	C₂²×C₂²⋊Q₈	C₂×A₄	1	2	yes	complemented
74	C₂²×#121	C₂×A₄	1	2	yes	complemented
75	C₂²×#120	C₂×A₄	1	2	yes	complemented
76	C₂²×C₄×D₄	C₂×A₄	1	4	yes	complemented
77	C₂²×#117	C₂×A₄	1	1	yes	characteristic, complemented
78	C₂×#458	C₂³×C₃	1	2	yes	—
79	C₂×B(4,𝔽₂)	C₂³×C₃	1	2	yes	—
80	#1790	C₂³×C₃	1	4	yes	—
81	#997	C₂²×D₄	1	1	yes	characteristic
82	C₂⁶	C₂²×A₄	1	1	yes	characteristic
83	#585	C₂×C₃×D₄	1	2	yes	—
84	#585	C₂×C₃×D₄	1	2	yes	—
85	C₂×H(5,𝔽₂)	C₂×C₃×D₄	1	2	yes	—
86	C₂×H(5,𝔽₂)	C₂⁴×C₃	1	1	yes	characteristic
87	C₂³×Q₈	C₂²×A₄	1	1	yes	characteristic
88	C₂³×D₄	C₂²×A₄	1	2	yes	—
89	C₂³×D₄	C₂²×A₄	1	1	yes	characteristic
90	C₂³×D₄	C₂²×A₄	1	4	yes	—
91	C₂⁴×C₄	C₂²×A₄	1	2	yes	—
92	C₂⁴×C₄	C₂²×A₄	1	2	yes	—
93	C₂⁴×C₄	C₂²×A₄	1	2	yes	—
94	C₂⁴×C₄	C₂²×A₄	1	1	yes	characteristic
95	C₂⁴×C₄	C₂²×A₄	1	1	yes	characteristic
96	C₂²×C₄⋊C₄	C₂²×A₄	1	2	yes	—
97	C₂²×C₄⋊C₄	C₂²×A₄	1	4	yes	—
98	C₂²×C₂²⋊C₄	C₂²×A₄	1	2	yes	—
99	C₂²×C₂²⋊C₄	C₂²×A₄	1	4	yes	—
100	C₂²×C₂²⋊C₄	C₂²×A₄	1	4	yes	—
101	C₂²×C₄²	C₂²×A₄	1	2	yes	—
102	C₂⁵	C₂³×A₄	1	2	yes	—
103	C₂⁵	C₂³×A₄	1	2	yes	—
104	C₂⁵	C₂³×A₄	1	1	yes	characteristic
105	H(5,𝔽₂)	C₂²×C₃×D₄	1	1	yes	characteristic
106	C₂×#41	#120⋊C₃	1	1	yes	characteristic
107	C₂³×C₄	D₄×A₄	1	4	yes	—
108	C₂³×C₄	C₂³×A₄	1	2	yes	—
109	C₂³×C₄	C₂³×A₄	1	2	yes	—
110	C₂³×C₄	C₂³×A₄	1	1	yes	characteristic
111	C₂³×C₄	C₂³×A₄	1	1	yes	characteristic
112	C₂³×C₄	C₂³×A₄	1	4	yes	—
113	C₂⁴	C₂×D₄×A₄	1	2	yes	—
114	C₂⁴	C₂⁴×A₄	1	1	yes	characteristic
115	C₂×Q₈	C₂×#120⋊C₃	1	1	yes	characteristic
116	C₂×D₄	C₂×#120⋊C₃	1	2	yes	—
117	C₂×D₄	C₂×#120⋊C₃	1	1	yes	characteristic
118	C₂²×C₄	C₂×#120⋊C₃	1	2	yes	—
119	C₂²×C₄	C₂×#120⋊C₃	1	1	yes	characteristic
120	C₂²×C₄	C₂×D₄×A₄	1	2	yes	—
121	C₂²×C₄	C₂×D₄×A₄	1	2	yes	—
122	C₂³	C₂²×#120⋊C₃	1	2	yes	—
123	C₂³	C₂²×#120⋊C₃	1	1	yes	characteristic
124	C₂³	C₂²×D₄×A₄	1	1	yes	characteristic
125	C₂×C₄	AΣL(1,𝔽₆₄)	1	4	yes	—
126	C₂×C₄	C₂²×#120⋊C₃	1	2	yes	—
127	C₂×C₄	C₂²×#120⋊C₃	1	1	yes	characteristic
128	C₂×C₄	C₂²×#120⋊C₃	1	1	yes	characteristic
129	C₂²	C₂×AΣL(1,𝔽₆₄)	1	2	yes	—
130	C₂²	C₂³×#120⋊C₃	1	1	yes	characteristic
131	C₄	C₂×AΣL(1,𝔽₆₄)	1	2	yes	—
132	C₄	C₂×AΣL(1,𝔽₆₄)	1	2	yes	—
133	C₂	#72236824	1	2	yes	—
134	C₂	C₂²×AΣL(1,𝔽₆₄)	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×#120⋊C₃
commutator, cosocle	C₂×H(5,𝔽₂)	C₂⁴×C₃
Frattini	C₂⁴	C₂⁴×A₄
Fitting	#72280972	C₃
socle	C₂²	C₂³×#120⋊C₃
hypercenter	C₂×#41	#120⋊C₃
hypocenter	H(5,𝔽₂)	C₂²×C₃×D₄
LCS	G ▷ C₂×H(5,𝔽₂) ▷ H(5,𝔽₂)
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
1. C₂²×#3025⋊C₃	C₂	semidirect
2. #72341820	C₂	semidirect
3. #72341793	C₂	semidirect
4. #72341762	C₂	semidirect
5. #72341563	C₂	semidirect
6. #72341483	C₂	semidirect
7. #72280972	C₃	semidirect
8. C₂×#11710937	C₂	semidirect
9. #72241384	C₂	semidirect
10. C₂×#87548	C₂²	semidirect
12. C₂×#87547	C₂²	semidirect
14. C₂²×#5950	C₂²	semidirect
15. C₂×#3025⋊C₃	C₂²	semidirect
16. C₂×#3025⋊C₃	C₂²	semidirect
17. C₂²×#458⋊C₃	C₂²	semidirect
18. C₂²×B(4,𝔽₂)⋊C₃	C₂²	semidirect
19. #32901⋊C₃	C₂²	semidirect
20. #32900⋊C₃	C₂²	semidirect
21. #32899⋊C₃	C₂²	semidirect
22. #32898⋊C₃	C₂²	semidirect
23. #20610⋊C₃	C₂²	semidirect
24. #20609⋊C₃	C₂²	semidirect
25. #20608⋊C₃	C₂²	semidirect
26. C₂×#63103	C₂×C₃	semidirect
27. C₂²×#3025	C₂×C₃	semidirect
28. #6628899	C₂×C₃	semidirect
29. #6628898	C₂×C₃	semidirect
30. #6628897	C₂×C₃	semidirect
31. #6628896	C₂×C₃	semidirect
32. #6628891	C₂×C₃	semidirect
33. #6370734	C₂×C₃	semidirect
34. C₂×#5950	D₄	semidirect
40. C₂×#458⋊C₃	C₂³	semidirect
41. C₂×B(4,𝔽₂)⋊C₃	C₂³	semidirect
42. #1790⋊C₃	C₂³	semidirect
43. C₂×2₋¹⁺⁶	C₂²×C₃	semidirect
45. C₂×H(7,𝔽₂)	C₂²×C₃	semidirect
47. C₂²×#585	C₂²×C₃	semidirect
48. C₂²×#525	A₄	semidirect
53. #32901	C₂²×C₃	semidirect
54. #32900	C₂²×C₃	semidirect
57. #20610	C₂²×C₃	semidirect
59. #20608	C₂²×C₃	semidirect
60. #5950	C₂×D₄	semidirect
61. #5950	C₂×D₄	semidirect
64. C₂×#585	C₃×D₄	semidirect
71. C₂⁵×C₄	C₂×A₄	semidirect
72. C₂²×#123	C₂×A₄	semidirect
73. C₂²×C₂²⋊Q₈	C₂×A₄	semidirect
74. C₂²×#121	C₂×A₄	semidirect
75. C₂²×#120	C₂×A₄	semidirect
76. C₂²×C₄×D₄	C₂×A₄	semidirect
77. C₂²×#117	C₂×A₄	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#3025⋊C₃	2
#72341820	2
#72341793	2
#72341762	2
#72341563	2
#72341483	2
#72280972	3
C₂×#11710937	2
#72241384	2
A₄×#525	4