Subgroups — #72275070

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	#72275067	C₂	1	1	yes	maximal, characteristic, complemented
2	#72275065	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#73026	C₂	1	1	yes	maximal, characteristic, complemented
4	#72275061	C₂	1	1	yes	maximal, characteristic, complemented
5	#72275051	C₂	1	1	yes	maximal, characteristic, complemented
6	#72275039	C₂	1	1	yes	maximal, characteristic, complemented
7	#72274967	C₂	1	1	yes	maximal, characteristic, complemented
8	#72274963	C₂	1	1	yes	maximal, characteristic, complemented
9	#72274952	C₂	1	1	yes	maximal, characteristic, complemented
10	#72274922	C₂	1	1	yes	maximal, characteristic, complemented
11	#72233229	C₂	1	1	yes	maximal, characteristic, complemented
12	#72231608	C₂	1	1	yes	maximal, characteristic, complemented
13	#72231605	C₂	1	1	yes	maximal, characteristic, complemented
14	D₃×#73026	C₂	1	1	yes	maximal, characteristic, complemented
15	#72215533	C₂	1	1	yes	maximal, characteristic, complemented
16	#11923832	C₂²	1	1	yes	characteristic, complemented
17	#11923781	C₂²	1	1	yes	characteristic, complemented
18	#11923780	C₂²	1	1	yes	characteristic, complemented
19	C₂×C₃×#5935	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	#11923650	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	#11923642	C₂²	1	1	yes	characteristic, complemented
28	#11923641	C₂²	1	1	yes	characteristic, complemented
29	#84110⋊C₃	C₂²	1	1	yes	characteristic, complemented
30	#80307⋊C₃	C₂²	1	1	yes	characteristic, complemented
31	D₃×B(4,𝔽₂)⋊C₃	C₂²	1	1	yes	characteristic, complemented
32	#80305⋊C₃	C₂²	1	1	yes	characteristic, complemented
33	#80304⋊C₃	C₂²	1	1	yes	characteristic, complemented
34	#11920984	C₂²	1	1	yes	characteristic, complemented
35	#11920983	C₂²	1	1	yes	characteristic, complemented
36	#11920982	C₂²	1	1	yes	characteristic, complemented
37	#11920981	C₂²	1	1	yes	characteristic, complemented
38	C₃×#73026	C₂²	1	1	yes	characteristic, complemented
39	C₃×#73026	C₂²	1	1	yes	characteristic, complemented
40	C₃×#73026	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	#11920737	C₂²	1	1	yes	characteristic, complemented
45	#11920736	C₂²	1	1	yes	characteristic, complemented
46	#11920733	C₂²	1	1	yes	characteristic, complemented
47	#11920732	C₂²	1	1	yes	characteristic, complemented
48	#11920729	C₂²	1	1	yes	characteristic, complemented
49	#11920728	C₂²	1	1	yes	characteristic, complemented
50	C₂×C₃×B(4,𝔽₂)⋊C₃	C₂²	1	1	yes	characteristic, complemented
51	C₂×#73026	D₃	1	1	yes	characteristic, complemented
52	#11661935	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, complemented
57	C₃⋊#5934	C₂³	1	1	yes	characteristic, complemented
58	C₃⋊#5933	C₂³	1	1	yes	characteristic, complemented
59	C₃⋊#5932	C₂³	1	1	yes	characteristic, complemented
60	C₃×#5935	C₂³	1	1	yes	characteristic
61	C₃×#5935	C₂³	1	1	yes	characteristic, complemented
62	C₃×#5934	C₂³	1	1	yes	characteristic
63	C₃×#5934	C₂³	1	1	yes	characteristic, complemented
64	C₃×B(4,𝔽₂)⋊C₃	C₂³	1	1	yes	characteristic
65	C₃×B(4,𝔽₂)⋊C₃	C₂³	1	1	yes	characteristic, complemented
66	C₃⋊(B(4,𝔽₂)⋊C₃)	C₂³	1	1	yes	characteristic, complemented
67	C₃⋊(#458⋊C₃)	C₂³	1	1	yes	characteristic, complemented
68	#87568	C₂×D₃	1	1	yes	characteristic, complemented
69	C₂×#5935	C₂×D₃	1	1	yes	characteristic, complemented
70	C₂×#5934	C₂×D₃	1	1	yes	characteristic, complemented
71	C₂×C₃×B(4,𝔽₂)	C₂×D₃	1	1	yes	characteristic, complemented
72	#84110	C₂×D₃	1	1	yes	characteristic, complemented
73	#80307	C₂×D₃	1	1	yes	characteristic, complemented
74	D₃×B(4,𝔽₂)	C₂×D₃	1	1	yes	characteristic, complemented
75	#80305	C₂×D₃	1	1	yes	characteristic, complemented
76	#80304	C₂×D₃	1	1	yes	characteristic, complemented
77	C₂×B(4,𝔽₂)⋊C₃	C₂×D₃	1	1	yes	characteristic, complemented
78	#73026	C₂×D₃	1	1	yes	characteristic, complemented
79	#73026	C₂×D₃	1	1	yes	characteristic, complemented
80	#73026	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	C₂³×#67	S₄	1	1	yes	characteristic, complemented
85	#5966	C₂²×D₃	1	1	yes	characteristic, complemented
86	D₃×H(5,𝔽₂)	C₂²×D₃	1	1	yes	characteristic, complemented
87	C₂×#997	C₂²×D₃	1	1	yes	characteristic, complemented
88	#5935	C₂²×D₃	1	1	yes	characteristic
89	#5935	C₂²×D₃	1	1	yes	characteristic, complemented
90	#5934	C₂²×D₃	1	1	yes	characteristic
91	#5934	C₂²×D₃	1	1	yes	characteristic, complemented
92	C₃×B(4,𝔽₂)	C₂²×D₃	1	1	yes	characteristic
93	C₃×B(4,𝔽₂)	C₂²×D₃	1	1	yes	characteristic, complemented
94	#5244	C₂²×D₃	1	1	yes	characteristic, complemented
95	#5243	C₂²×D₃	1	1	yes	characteristic, complemented
96	B(4,𝔽₂)⋊C₃	C₂²×D₃	1	1	yes	characteristic
97	B(4,𝔽₂)⋊C₃	C₂²×D₃	1	1	yes	characteristic, complemented
98	C₂×B(4,𝔽₂)	D₃²	1	1	yes	characteristic, complemented
99	C₂⁵×C₃	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×#67	C₂×S₄	1	1	yes	characteristic
103	C₂²×#67	C₂×S₄	1	1	yes	characteristic, complemented
104	C₂²×#67	C₂×S₄	1	1	yes	characteristic, complemented
105	C₂²×#67	C₂×S₄	1	1	yes	characteristic, complemented
106	C₂³×Dic₃	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	B(4,𝔽₂)	C₂×D₃²	1	1	yes	characteristic
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	C₂⁴×C₃	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×Dic₃	C₂²×S₄	1	1	yes	characteristic
117	C₂²×Dic₃	C₂²×S₄	1	1	yes	characteristic
118	C₂⁵	D₃×S₄	1	1	yes	characteristic, complemented
119	H(5,𝔽₂)	C₂²×D₃²	1	1	yes	characteristic
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	C₂⁴	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	C₂×C₃×B(4,𝔽₂)	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. #72275067	C₂	semidirect
2. #72275065	C₂	semidirect
3. C₂×C₃×#73026	C₂	semidirect
4. #72275061	C₂	semidirect
5. #72275051	C₂	semidirect
6. #72275039	C₂	semidirect
7. #72274967	C₂	semidirect
8. #72274963	C₂	semidirect
9. #72274952	C₂	semidirect
10. #72274922	C₂	semidirect
11. #72233229	C₂	semidirect
12. #72231608	C₂	semidirect
13. #72231605	C₂	semidirect
14. D₃×#73026	C₂	semidirect
15. #72215533	C₂	semidirect
16. #11923832	C₂²	semidirect
17. #11923781	C₂²	semidirect
18. #11923780	C₂²	semidirect
19. C₂×C₃×#5935	C₂²	semidirect
20. C₂×C₃×#5934	C₂²	semidirect
21. D₃×#5935	C₂²	semidirect
22. D₃×#5934	C₂²	semidirect
23. #11923650	C₂²	semidirect
24. #11923649	C₂²	semidirect
25. #11923646	C₂²	semidirect
26. #11923645	C₂²	semidirect
27. #11923642	C₂²	semidirect
28. #11923641	C₂²	semidirect
29. #84110⋊C₃	C₂²	semidirect
30. #80307⋊C₃	C₂²	semidirect
31. D₃×B(4,𝔽₂)⋊C₃	C₂²	semidirect
32. #80305⋊C₃	C₂²	semidirect
33. #80304⋊C₃	C₂²	semidirect
34. #11920984	C₂²	semidirect
35. #11920983	C₂²	semidirect
36. #11920982	C₂²	semidirect
37. #11920981	C₂²	semidirect
38. C₃×#73026	C₂²	semidirect
39. C₃×#73026	C₂²	semidirect
40. C₃×#73026	C₂²	semidirect
41. C₃×#73026	C₂²	semidirect
42. #11920741	C₂²	semidirect
43. #11920740	C₂²	semidirect
44. #11920737	C₂²	semidirect
45. #11920736	C₂²	semidirect
46. #11920733	C₂²	semidirect
47. #11920732	C₂²	semidirect
48. #11920729	C₂²	semidirect
49. #11920728	C₂²	semidirect
50. C₂×C₃×B(4,𝔽₂)⋊C₃	C₂²	semidirect
51. C₂×#73026	D₃	semidirect
52. #11661935	D₃	semidirect
53. C₂×C₃×#997	C₂³	semidirect
54. D₃×#997	C₂³	semidirect
55. C₃⋊#5950	C₂³	semidirect
56. C₃⋊#5935	C₂³	semidirect
57. C₃⋊#5934	C₂³	semidirect
58. C₃⋊#5933	C₂³	semidirect
59. C₃⋊#5932	C₂³	semidirect
61. C₃×#5935	C₂³	semidirect
63. C₃×#5934	C₂³	semidirect
65. C₃×B(4,𝔽₂)⋊C₃	C₂³	semidirect
66. C₃⋊(B(4,𝔽₂)⋊C₃)	C₂³	semidirect
67. C₃⋊(#458⋊C₃)	C₂³	semidirect
68. #87568	C₂×D₃	semidirect
69. C₂×#5935	C₂×D₃	semidirect
70. C₂×#5934	C₂×D₃	semidirect
71. C₂×C₃×B(4,𝔽₂)	C₂×D₃	semidirect
72. #84110	C₂×D₃	semidirect
73. #80307	C₂×D₃	semidirect
74. D₃×B(4,𝔽₂)	C₂×D₃	semidirect
75. #80305	C₂×D₃	semidirect
76. #80304	C₂×D₃	semidirect
77. C₂×B(4,𝔽₂)⋊C₃	C₂×D₃	semidirect
78. #73026	C₂×D₃	semidirect
79. #73026	C₂×D₃	semidirect
80. #73026	C₂×D₃	semidirect
81. #73026	C₂×D₃	semidirect
83. C₂×C₃×H(5,𝔽₂)	C₂²×D₃	semidirect
84. C₂³×#67	S₄	semidirect
85. #5966	C₂²×D₃	semidirect
86. D₃×H(5,𝔽₂)	C₂²×D₃	semidirect
87. C₂×#997	C₂²×D₃	semidirect
89. #5935	C₂²×D₃	semidirect
91. #5934	C₂²×D₃	semidirect
93. C₃×B(4,𝔽₂)	C₂²×D₃	semidirect
94. #5244	C₂²×D₃	semidirect
95. #5243	C₂²×D₃	semidirect
97. B(4,𝔽₂)⋊C₃	C₂²×D₃	semidirect
98. C₂×B(4,𝔽₂)	D₃²	semidirect
99. C₂⁵×C₃	C₂×S₄	semidirect
100. C₂⁴×D₃	C₂×S₄	semidirect
103. C₂²×#67	C₂×S₄	semidirect
104. C₂²×#67	C₂×S₄	semidirect
105. C₂²×#67	C₂×S₄	semidirect
106. C₂³×Dic₃	C₂×S₄	semidirect
108. C₂×H(5,𝔽₂)	C₂×D₃²	semidirect
110. B(4,𝔽₂)	C₂×D₃²	semidirect
118. C₂⁵	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
#72275067	2
#72275065	2
C₂×C₃×#73026	2
#72275061	2
#72275051	2
#72275039	2
#72274967	2
#72274963	2
#72274952	2
#72274922	2
#72255219	3
#72233229	2
#72231608	2
#72231605	2
D₃×#73026	2
#72217027	3
#72215533	2
C₂×#67×S₄	4