Subgroups — #72274956

Order 4608.

At least 6955 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

The subgroup lattice has 134 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72274952	C₂	1	1	yes	maximal, characteristic, complemented
2	#72274951	C₂	1	1	yes	maximal, characteristic, complemented
3	#72274949	C₂	1	1	yes	maximal, characteristic, complemented
4	#72274947	C₂	1	1	yes	maximal, characteristic, complemented
5	#72274943	C₂	1	1	yes	maximal, characteristic, complemented
6	#72274940	C₂	1	1	yes	maximal, characteristic, complemented
7	#72274937	C₂	1	1	yes	maximal, characteristic, complemented
8	#72274920	C₂	1	1	yes	maximal, characteristic, complemented
9	#72274911	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#11920740	C₂	1	1	yes	maximal, characteristic, complemented
11	#72233225	C₂	1	1	yes	maximal, characteristic, complemented
12	#72231605	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃×#11714170	C₂	1	1	yes	maximal, characteristic, complemented
14	D₃×#73026	C₂	1	1	yes	maximal, characteristic, complemented
15	#72215563	C₂	1	1	yes	maximal, characteristic, complemented
16	#11923832	C₂²	1	1	yes	characteristic, complemented
17	#11923780	C₂²	1	1	yes	characteristic, complemented
18	C₂×C₃⋊#5935	C₂²	1	1	yes	characteristic, complemented
19	C₃×#87516	C₂²	1	1	yes	characteristic, complemented
20	C₂×C₃×#5934	C₂²	1	1	yes	characteristic, complemented
21	D₃×#5935	C₂²	1	1	yes	characteristic, complemented
22	D₃×#5934	C₂²	1	1	yes	characteristic, complemented
23	#11923652	C₂²	1	1	yes	characteristic, complemented
24	#11923649	C₂²	1	1	yes	characteristic, complemented
25	#11923646	C₂²	1	1	yes	characteristic, complemented
26	#11923645	C₂²	1	1	yes	characteristic, complemented
27	#11923644	C₂²	1	1	yes	characteristic, complemented
28	#11923641	C₂²	1	1	yes	characteristic, complemented
29	C₂×C₃⋊(B(4,𝔽₂)⋊C₃)	C₂²	1	1	yes	characteristic, complemented
30	#80311⋊C₃	C₂²	1	1	yes	characteristic, complemented
31	#80308⋊C₃	C₂²	1	1	yes	characteristic, complemented
32	D₃×B(4,𝔽₂)⋊C₃	C₂²	1	1	yes	characteristic, complemented
33	#80305⋊C₃	C₂²	1	1	yes	characteristic, complemented
34	#11920988	C₂²	1	1	yes	characteristic, complemented
35	#11920987	C₂²	1	1	yes	characteristic, complemented
36	#11920984	C₂²	1	1	yes	characteristic, complemented
37	#11920982	C₂²	1	1	yes	characteristic, complemented
38	C₃×#73032	C₂²	1	1	yes	characteristic, complemented
39	C₃×#73030	C₂²	1	1	yes	characteristic, complemented
40	C₃×#73027	C₂²	1	1	yes	characteristic, complemented
41	C₃×#73026	C₂²	1	1	yes	characteristic, complemented
42	#11920741	C₂²	1	1	yes	characteristic, complemented
43	#11920740	C₂²	1	1	yes	characteristic, complemented
44	#11920740	C₂²	1	1	yes	characteristic, complemented
45	#11920740	C₂²	1	1	yes	characteristic, complemented
46	#11920740	C₂²	1	1	yes	characteristic, complemented
47	#11920739	C₂²	1	1	yes	characteristic, complemented
48	#11920737	C₂²	1	1	yes	characteristic, complemented
49	#11920735	C₂²	1	1	yes	characteristic, complemented
50	C₃×#3026⋊C₃	C₂²	1	1	yes	characteristic, complemented
51	#11714170	D₃	1	1	yes	characteristic, complemented
52	#11661939	D₃	1	1	yes	characteristic, complemented
53	C₂×C₃×#997	C₂³	1	1	yes	characteristic, complemented
54	D₃×#997	C₂³	1	1	yes	characteristic, complemented
55	C₃⋊#5950	C₂³	1	1	yes	characteristic, complemented
56	C₃⋊#5935	C₂³	1	1	yes	characteristic
57	C₃⋊#5935	C₂³	1	1	yes	characteristic, complemented
58	C₃⋊#5934	C₂³	1	1	yes	characteristic, complemented
59	C₃⋊#5932	C₂³	1	1	yes	characteristic, complemented
60	C₃×#5935	C₂³	1	1	yes	characteristic, complemented
61	C₃×#5934	C₂³	1	1	yes	characteristic
62	C₃×#5934	C₂³	1	1	yes	characteristic, complemented
63	C₃×#5933	C₂³	1	1	yes	characteristic, complemented
64	C₃×#458⋊C₃	C₂³	1	1	yes	characteristic, complemented
65	C₃×B(4,𝔽₂)⋊C₃	C₂³	1	1	yes	characteristic, complemented
66	C₃⋊(B(4,𝔽₂)⋊C₃)	C₂³	1	1	yes	characteristic
67	C₃⋊(B(4,𝔽₂)⋊C₃)	C₂³	1	1	yes	characteristic, complemented
68	#87568	C₂×D₃	1	1	yes	characteristic, complemented
69	#87516	C₂×D₃	1	1	yes	characteristic, complemented
70	C₂×#5934	C₂×D₃	1	1	yes	characteristic, complemented
71	C₃×#3026	C₂×D₃	1	1	yes	characteristic, complemented
72	C₂×#5244	C₂×D₃	1	1	yes	characteristic, complemented
73	#80311	C₂×D₃	1	1	yes	characteristic, complemented
74	#80308	C₂×D₃	1	1	yes	characteristic, complemented
75	D₃×B(4,𝔽₂)	C₂×D₃	1	1	yes	characteristic, complemented
76	#80305	C₂×D₃	1	1	yes	characteristic, complemented
77	#3026⋊C₃	C₂×D₃	1	1	yes	characteristic, complemented
78	#73032	C₂×D₃	1	1	yes	characteristic, complemented
79	#73030	C₂×D₃	1	1	yes	characteristic, complemented
80	#73027	C₂×D₃	1	1	yes	characteristic, complemented
81	#73026	C₂×D₃	1	1	yes	characteristic, complemented
82	C₃×#997	C₂⁴	1	1	yes	characteristic
83	C₂×C₃×H(5,𝔽₂)	C₂²×D₃	1	1	yes	characteristic, complemented
84	#5966	C₂²×D₃	1	1	yes	characteristic, complemented
85	D₃×H(5,𝔽₂)	C₂²×D₃	1	1	yes	characteristic, complemented
86	C₂²×D₃×D₄	S₄	1	1	yes	characteristic, complemented
87	C₂×#997	C₂²×D₃	1	1	yes	characteristic, complemented
88	#5935	C₂²×D₃	1	1	yes	characteristic, complemented
89	#5934	C₂²×D₃	1	1	yes	characteristic
90	#5934	C₂²×D₃	1	1	yes	characteristic, complemented
91	#5933	C₂²×D₃	1	1	yes	characteristic, complemented
92	C₃×#458	C₂²×D₃	1	1	yes	characteristic, complemented
93	C₃×B(4,𝔽₂)	C₂²×D₃	1	1	yes	characteristic, complemented
94	#5244	C₂²×D₃	1	1	yes	characteristic
95	#5244	C₂²×D₃	1	1	yes	characteristic, complemented
96	#458⋊C₃	C₂²×D₃	1	1	yes	characteristic, complemented
97	B(4,𝔽₂)⋊C₃	C₂²×D₃	1	1	yes	characteristic, complemented
98	#3026	D₃²	1	1	yes	characteristic, complemented
99	C₂⁴×D₃	C₂×S₄	1	1	yes	characteristic, complemented
100	C₂⁴×D₃	C₂×S₄	1	1	yes	characteristic, complemented
101	C₃×H(5,𝔽₂)	C₂³×D₃	1	1	yes	characteristic
102	C₂²×C₃×D₄	C₂×S₄	1	1	yes	characteristic, complemented
103	C₂²×#67	C₂×S₄	1	1	yes	characteristic
104	C₂²×#67	C₂×S₄	1	1	yes	characteristic, complemented
105	C₂²×D₁₂	C₂×S₄	1	1	yes	characteristic, complemented
106	C₂²×C₄×D₃	C₂×S₄	1	1	yes	characteristic, complemented
107	#997	C₂³×D₃	1	1	yes	characteristic
108	C₂×H(5,𝔽₂)	C₂×D₃²	1	1	yes	characteristic, complemented
109	#458	C₂×D₃²	1	1	yes	characteristic, complemented
110	B(4,𝔽₂)	C₂×D₃²	1	1	yes	characteristic, complemented
111	C₂⁴×C₃	C₂²×S₄	1	1	yes	characteristic
112	C₂⁴×C₃	C₂²×S₄	1	1	yes	characteristic
113	C₂³×D₃	C₂²×S₄	1	1	yes	characteristic
114	C₂³×D₃	C₂²×S₄	1	1	yes	characteristic
115	C₂³×D₃	C₂²×S₄	1	1	yes	characteristic
116	C₂²×C₃×C₄	C₂²×S₄	1	1	yes	characteristic
117	C₂²×Dic₃	C₂²×S₄	1	1	yes	characteristic
118	H(5,𝔽₂)	C₂²×D₃²	1	1	yes	characteristic
119	C₂²×D₄	D₃×S₄	1	1	yes	characteristic, complemented
120	C₂³×C₃	C₂³×S₄	1	1	yes	characteristic
121	#67	#1020⋊C₂	1	1	yes	characteristic
122	C₂⁴	C₂×D₃×S₄	1	1	yes	characteristic
123	C₂⁴	C₂×D₃×S₄	1	1	yes	characteristic
124	C₂²×C₄	C₂×D₃×S₄	1	1	yes	characteristic
125	C₂²×C₃	C₂×#1020⋊C₂	1	1	yes	characteristic
126	C₂×D₃	C₂×#1020⋊C₂	1	1	yes	characteristic
127	Dic₃	C₂×#1020⋊C₂	1	1	yes	characteristic
128	C₂³	C₂²×D₃×S₄	1	1	yes	characteristic
129	C₂×C₃	C₂²×#1020⋊C₂	1	1	yes	characteristic
130	C₂²	D₃×#1020⋊C₂	1	1	yes	characteristic
131	C₃	#72217027	1	1	yes	characteristic, complemented
132	C₂	C₂×D₃×#1020⋊C₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₃×#1020⋊C₂
commutator, cosocle, hypocenter	C₃×#997	C₂⁴
Frattini	C₂³	C₂²×D₃×S₄
Fitting	C₃×#3026	C₂×D₃
socle	C₂×C₃	C₂²×#1020⋊C₂
hypercenter	C₂²	D₃×#1020⋊C₂
LCS	G ▷ C₃×#997
UCS	C₁ ◁ C₂ ◁ C₂²

Complements

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

N	H ≅ G/N	type
1. #72274952	C₂	semidirect
2. #72274951	C₂	semidirect
3. #72274949	C₂	semidirect
4. #72274947	C₂	semidirect
5. #72274943	C₂	semidirect
6. #72274940	C₂	semidirect
7. #72274937	C₂	semidirect
8. #72274920	C₂	semidirect
9. #72274911	C₂	semidirect
10. C₂×#11920740	C₂	semidirect
11. #72233225	C₂	semidirect
12. #72231605	C₂	semidirect
13. C₃×#11714170	C₂	semidirect
14. D₃×#73026	C₂	semidirect
15. #72215563	C₂	semidirect
16. #11923832	C₂²	semidirect
17. #11923780	C₂²	semidirect
18. C₂×C₃⋊#5935	C₂²	semidirect
19. C₃×#87516	C₂²	semidirect
20. C₂×C₃×#5934	C₂²	semidirect
21. D₃×#5935	C₂²	semidirect
22. D₃×#5934	C₂²	semidirect
23. #11923652	C₂²	semidirect
24. #11923649	C₂²	semidirect
25. #11923646	C₂²	semidirect
26. #11923645	C₂²	semidirect
27. #11923644	C₂²	semidirect
28. #11923641	C₂²	semidirect
29. C₂×C₃⋊(B(4,𝔽₂)⋊C₃)	C₂²	semidirect
30. #80311⋊C₃	C₂²	semidirect
31. #80308⋊C₃	C₂²	semidirect
32. D₃×B(4,𝔽₂)⋊C₃	C₂²	semidirect
33. #80305⋊C₃	C₂²	semidirect
34. #11920988	C₂²	semidirect
35. #11920987	C₂²	semidirect
36. #11920984	C₂²	semidirect
37. #11920982	C₂²	semidirect
38. C₃×#73032	C₂²	semidirect
39. C₃×#73030	C₂²	semidirect
40. C₃×#73027	C₂²	semidirect
41. C₃×#73026	C₂²	semidirect
42. #11920741	C₂²	semidirect
43. #11920740	C₂²	semidirect
44. #11920740	C₂²	semidirect
45. #11920740	C₂²	semidirect
46. #11920740	C₂²	semidirect
47. #11920739	C₂²	semidirect
48. #11920737	C₂²	semidirect
49. #11920735	C₂²	semidirect
50. C₃×#3026⋊C₃	C₂²	semidirect
51. #11714170	D₃	semidirect
52. #11661939	D₃	semidirect
53. C₂×C₃×#997	C₂³	semidirect
54. D₃×#997	C₂³	semidirect
55. C₃⋊#5950	C₂³	semidirect
57. C₃⋊#5935	C₂³	semidirect
58. C₃⋊#5934	C₂³	semidirect
59. C₃⋊#5932	C₂³	semidirect
60. C₃×#5935	C₂³	semidirect
62. C₃×#5934	C₂³	semidirect
63. C₃×#5933	C₂³	semidirect
64. C₃×#458⋊C₃	C₂³	semidirect
65. C₃×B(4,𝔽₂)⋊C₃	C₂³	semidirect
67. C₃⋊(B(4,𝔽₂)⋊C₃)	C₂³	semidirect
68. #87568	C₂×D₃	semidirect
69. #87516	C₂×D₃	semidirect
70. C₂×#5934	C₂×D₃	semidirect
71. C₃×#3026	C₂×D₃	semidirect
72. C₂×#5244	C₂×D₃	semidirect
73. #80311	C₂×D₃	semidirect
74. #80308	C₂×D₃	semidirect
75. D₃×B(4,𝔽₂)	C₂×D₃	semidirect
76. #80305	C₂×D₃	semidirect
77. #3026⋊C₃	C₂×D₃	semidirect
78. #73032	C₂×D₃	semidirect
79. #73030	C₂×D₃	semidirect
80. #73027	C₂×D₃	semidirect
81. #73026	C₂×D₃	semidirect
83. C₂×C₃×H(5,𝔽₂)	C₂²×D₃	semidirect
84. #5966	C₂²×D₃	semidirect
85. D₃×H(5,𝔽₂)	C₂²×D₃	semidirect
86. C₂²×D₃×D₄	S₄	semidirect
87. C₂×#997	C₂²×D₃	semidirect
88. #5935	C₂²×D₃	semidirect
90. #5934	C₂²×D₃	semidirect
91. #5933	C₂²×D₃	semidirect
92. C₃×#458	C₂²×D₃	semidirect
93. C₃×B(4,𝔽₂)	C₂²×D₃	semidirect
95. #5244	C₂²×D₃	semidirect
96. #458⋊C₃	C₂²×D₃	semidirect
97. B(4,𝔽₂)⋊C₃	C₂²×D₃	semidirect
98. #3026	D₃²	semidirect
99. C₂⁴×D₃	C₂×S₄	semidirect
100. C₂⁴×D₃	C₂×S₄	semidirect
102. C₂²×C₃×D₄	C₂×S₄	semidirect
104. C₂²×#67	C₂×S₄	semidirect
105. C₂²×D₁₂	C₂×S₄	semidirect
106. C₂²×C₄×D₃	C₂×S₄	semidirect
108. C₂×H(5,𝔽₂)	C₂×D₃²	semidirect
109. #458	C₂×D₃²	semidirect
110. B(4,𝔽₂)	C₂×D₃²	semidirect
119. C₂²×D₄	D₃×S₄	semidirect
131. C₃	#72217027	semidirect

Maximal subgroups

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

subgroup	index
#72274952	2
#72274951	2
#72274949	2
#72274947	2
#72274943	2
#72274940	2
#72274937	2
#72274920	2
#72274911	2
C₂×#11920740	2
#72255218	3
#72233225	2
#72231605	2
C₃×#11714170	2
D₃×#73026	2
#72217027	3
#72215563	2
D₃×D₄×S₄	4