Subgroups — #72342145

Order 3072.

At least 6453 conjugacy classes of subgroups; only the 254 normal subgroups are shown.

The subgroup lattice has 190 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72342140	C₃	1	1	yes	maximal, characteristic, complemented, Sylow
2	#72342135	C₂	1	1	yes	maximal, characteristic, complemented
3	#72342129	C₂	1	2	yes	maximal, complemented
4	#72342123	C₂	1	1	yes	maximal, characteristic, complemented
5	#72342120	C₂	1	2	yes	maximal, complemented
6	#72342111	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#21483⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#72341679	C₂	1	1	yes	maximal, characteristic, complemented
9	#72236874	C₂	1	2	yes	maximal, complemented
10	#72236864	C₂	1	1	yes	maximal, characteristic, complemented
11	#72236824	C₂	1	2	yes	maximal, complemented
12	#72227318	C₂	1	1	yes	maximal, characteristic, complemented
13	#11710937	C₂²	1	2	yes	complemented
14	#11710902	C₂²	1	1	yes	characteristic, complemented
15	#11710899	C₂²	1	2	yes	complemented
16	#11710898	C₂²	1	1	yes	characteristic, complemented
17	C₂×#85666	C₂²	1	1	yes	characteristic, complemented
18	#11710352	C₂²	1	2	yes	complemented
19	#60401⋊C₃	C₂²	1	2	yes	complemented
20	#11710347	C₂²	1	2	yes	complemented
21	#60399⋊C₃	C₂²	1	2	yes	complemented
22	#60398⋊C₃	C₂²	1	2	yes	complemented
23	C₂×#3025⋊C₃	C₂²	1	1	yes	characteristic
24	#32949⋊C₃	C₂²	1	1	yes	characteristic, complemented
25	#32948⋊C₃	C₂²	1	1	yes	characteristic, complemented
26	#32947⋊C₃	C₂²	1	2	yes	complemented
27	#32897⋊C₃	C₂²	1	2	yes	complemented
28	#32896⋊C₃	C₂²	1	2	yes	complemented
29	#32895⋊C₃	C₂²	1	2	yes	complemented
30	#32894⋊C₃	C₂²	1	2	yes	complemented
31	#21483⋊C₃	C₂²	1	1	yes	characteristic, complemented
32	#21483⋊C₃	C₂²	1	1	yes	characteristic, complemented
33	#21483⋊C₃	C₂²	1	1	yes	characteristic, complemented
34	#21483⋊C₃	C₂²	1	1	yes	characteristic, complemented
35	C₂×#1791⋊C₃	C₂²	1	1	yes	characteristic, complemented
36	#10587001	C₂×C₃	1	1	yes	characteristic, complemented
37	#10574874	C₂×C₃	1	2	yes	complemented
38	#6629044	C₂×C₃	1	1	yes	characteristic, complemented
39	#6628882	C₂×C₃	1	2	yes	complemented
40	#6573734	C₂×C₃	1	1	yes	characteristic, complemented
41	#6573733	C₂×C₃	1	2	yes	complemented
42	#6573732	C₂×C₃	1	1	yes	characteristic, complemented
43	#6573731	C₂×C₃	1	1	yes	characteristic, complemented
44	#6573730	C₂×C₃	1	2	yes	complemented
45	#6573729	C₂×C₃	1	1	yes	characteristic, complemented
46	C₂×#21483	C₂×C₃	1	1	yes	characteristic, complemented
47	#87548	C₂³	1	2	yes	—
48	#87547	C₂³	1	2	yes	—
49	C₂×#5950	C₂³	1	1	yes	characteristic
50	#85666	C₂³	1	1	yes	characteristic
51	#85666	C₂³	1	1	yes	characteristic
52	#3028⋊C₃	C₂³	1	2	yes	—
53	#3027⋊C₃	C₂³	1	2	yes	—
54	#3025⋊C₃	D₄	1	1	yes	characteristic
55	#3025⋊C₃	D₄	1	1	yes	characteristic
56	#3025⋊C₃	C₂³	1	1	yes	characteristic
57	#3025⋊C₃	C₂³	1	1	yes	characteristic
58	C₂×#458⋊C₃	D₄	1	1	yes	characteristic
59	C₂×B(4,𝔽₂)⋊C₃	D₄	1	1	yes	characteristic
60	#1791⋊C₃	C₂³	1	1	yes	characteristic
61	#1791⋊C₃	C₂³	1	1	yes	characteristic
62	#63103	C₂²×C₃	1	2	yes	complemented
63	#63093	C₂²×C₃	1	1	yes	characteristic, complemented
64	#63092	C₂²×C₃	1	2	yes	complemented
65	#63091	C₂²×C₃	1	1	yes	characteristic, complemented
66	C₂³×#135	A₄	1	1	yes	characteristic, complemented
67	C₂×#3573	C₂²×C₃	1	1	yes	characteristic, complemented
68	#60402	C₂²×C₃	1	2	yes	complemented
69	#60401	C₂²×C₃	1	2	yes	complemented
70	#60400	C₂²×C₃	1	2	yes	—
71	#60399	C₂²×C₃	1	2	yes	—
72	#60398	C₂²×C₃	1	2	yes	complemented
73	C₂×#3025	C₂²×C₃	1	1	yes	characteristic
74	#32949	C₂²×C₃	1	1	yes	characteristic, complemented
75	#32948	C₂²×C₃	1	1	yes	characteristic, complemented
76	#32947	C₂²×C₃	1	2	yes	complemented
77	#32897	C₂²×C₃	1	2	yes	complemented
78	#32896	C₂²×C₃	1	2	yes	complemented
79	#32895	C₂²×C₃	1	2	yes	complemented
80	#32894	C₂²×C₃	1	2	yes	complemented
81	#21483	C₂²×C₃	1	1	yes	characteristic
82	#21483	C₂²×C₃	1	1	yes	characteristic
83	#21483	C₂²×C₃	1	1	yes	characteristic, complemented
84	#21483	C₂²×C₃	1	1	yes	characteristic, complemented
85	C₂×#1791	C₂²×C₃	1	1	yes	characteristic, complemented
86	#5950	C₂×D₄	1	1	yes	characteristic
87	#5950	C₂⁴	1	1	yes	characteristic
88	C₂×#997	C₂×D₄	1	1	yes	characteristic
89	#458⋊C₃	C₂×D₄	1	1	yes	characteristic
90	#458⋊C₃	C₂×D₄	1	1	yes	characteristic
91	B(4,𝔽₂)⋊C₃	C₂×D₄	1	1	yes	characteristic
92	B(4,𝔽₂)⋊C₃	C₂×D₄	1	1	yes	characteristic
93	2₋¹⁺⁶	C₂³×C₃	1	2	yes	—
94	H(7,𝔽₂)	C₂³×C₃	1	2	yes	—
95	C₂×#585	C₂³×C₃	1	1	yes	characteristic
96	C₂³×#41	C₂×A₄	1	2	yes	complemented
97	C₂²×#135	C₂×A₄	1	1	yes	characteristic
98	C₂²×#135	C₂×A₄	1	2	yes	complemented
99	C₂²×#135	C₂×A₄	1	1	yes	characteristic, complemented
100	C₂²×#135	C₂×A₄	1	1	yes	characteristic, complemented
101	C₂²×#135	C₂×A₄	1	2	yes	complemented
102	C₂²×#135	C₂×A₄	1	1	yes	characteristic, complemented
103	C₂³×Q₁₆	C₂×A₄	1	1	yes	characteristic, complemented
104	C₂³×SD₁₆	C₂×A₄	1	2	yes	complemented
105	C₂³×D₈	C₂×A₄	1	1	yes	characteristic, complemented
106	#3573	C₂³×C₃	1	1	yes	characteristic
107	#3573	C₂³×C₃	1	1	yes	characteristic
108	C₂⁴×C₈	C₂×A₄	1	1	yes	characteristic, complemented
109	#3028	C₂³×C₃	1	2	yes	—
110	#3027	C₂³×C₃	1	2	yes	—
111	#3025	C₃×D₄	1	1	yes	characteristic
112	#3025	C₃×D₄	1	1	yes	characteristic
113	#3025	C₂³×C₃	1	1	yes	characteristic
114	#3025	C₂³×C₃	1	1	yes	characteristic
115	C₂×#458	C₃×D₄	1	1	yes	characteristic
116	C₂×B(4,𝔽₂)	C₃×D₄	1	1	yes	characteristic
117	#1791	C₂³×C₃	1	1	yes	characteristic
118	#1791	C₂³×C₃	1	1	yes	characteristic
119	#997	C₂²×D₄	1	1	yes	characteristic
120	#585	C₂×C₃×D₄	1	1	yes	characteristic
121	#585	C₂⁴×C₃	1	1	yes	characteristic
122	C₂×H(5,𝔽₂)	C₂×C₃×D₄	1	1	yes	characteristic
123	C₂²×#41	C₂²×A₄	1	2	yes	—
124	C₂²×#41	C₂²×A₄	1	2	yes	—
125	C₂²×#41	C₂²×A₄	1	2	yes	—
126	C₂²×#41	C₂²×A₄	1	2	yes	—
127	C₂³×Q₈	C₂²×A₄	1	2	yes	—
128	C₂³×D₄	C₂²×A₄	1	2	yes	—
129	C₂⁴×C₄	C₂²×A₄	1	1	yes	characteristic
130	C₂²×Q₁₆	C₂²×A₄	1	1	yes	characteristic
131	C₂²×Q₁₆	C₂²×A₄	1	1	yes	characteristic
132	C₂²×Q₁₆	C₂²×A₄	1	2	yes	—
133	C₂²×SD₁₆	C₂²×A₄	1	2	yes	—
134	C₂²×SD₁₆	C₂²×A₄	1	2	yes	—
135	C₂²×SD₁₆	C₂²×A₄	1	2	yes	—
136	C₂²×SD₁₆	C₂²×A₄	1	2	yes	—
137	C₂²×D₈	C₂²×A₄	1	1	yes	characteristic
138	C₂²×D₈	C₂²×A₄	1	1	yes	characteristic
139	C₂²×D₈	C₂²×A₄	1	2	yes	—
140	C₂³×C₈	C₂²×A₄	1	1	yes	characteristic
141	C₂³×C₈	C₂²×A₄	1	1	yes	characteristic
142	C₂³×C₈	C₂²×A₄	1	1	yes	characteristic
143	C₂³×C₈	C₂²×A₄	1	1	yes	characteristic
144	C₂³×C₈	C₂²×A₄	1	1	yes	characteristic
145	C₂³×C₈	C₂²×A₄	1	1	yes	characteristic
146	#458	C₂×C₃×D₄	1	1	yes	characteristic
147	#458	C₂×C₃×D₄	1	1	yes	characteristic
148	B(4,𝔽₂)	C₂×C₃×D₄	1	1	yes	characteristic
149	B(4,𝔽₂)	C₂×C₃×D₄	1	1	yes	characteristic
150	C₂⁵	D₄×A₄	1	1	yes	characteristic
151	H(5,𝔽₂)	C₂²×C₃×D₄	1	1	yes	characteristic
152	C₂²×Q₈	C₂³×A₄	1	2	yes	—
153	C₂²×Q₈	C₂³×A₄	1	2	yes	—
154	C₂²×D₄	C₂³×A₄	1	2	yes	—
155	C₂²×D₄	C₂³×A₄	1	2	yes	—
156	C₂³×C₄	D₄×A₄	1	1	yes	characteristic
157	C₂³×C₄	D₄×A₄	1	1	yes	characteristic
158	C₂³×C₄	D₄×A₄	1	1	yes	characteristic
159	C₂³×C₄	C₂³×A₄	1	1	yes	characteristic
160	C₂³×C₄	C₂³×A₄	1	1	yes	characteristic
161	C₂³×C₄	C₂³×A₄	1	1	yes	characteristic
162	#135	#120⋊C₃	1	1	yes	characteristic
163	C₂²×C₈	C₂³×A₄	1	1	yes	characteristic
164	C₂²×C₈	C₂³×A₄	1	1	yes	characteristic
165	C₂²×C₈	C₂³×A₄	1	1	yes	characteristic
166	C₂²×C₈	C₂³×A₄	1	1	yes	characteristic
167	C₂⁴	C₂×D₄×A₄	1	1	yes	characteristic
168	C₂⁴	C₂×D₄×A₄	1	1	yes	characteristic
169	C₂⁴	C₂×D₄×A₄	1	1	yes	characteristic
170	#41	C₂×#120⋊C₃	1	2	yes	—
171	C₂²×C₄	C₂×D₄×A₄	1	1	yes	characteristic
172	C₂²×C₄	C₂×D₄×A₄	1	1	yes	characteristic
173	C₂²×C₄	C₂×D₄×A₄	1	1	yes	characteristic
174	C₂²×C₄	C₂⁴×A₄	1	1	yes	characteristic
175	Q₁₆	C₂×#120⋊C₃	1	1	yes	characteristic
176	SD₁₆	C₂×#120⋊C₃	1	2	yes	—
177	D₈	C₂×#120⋊C₃	1	1	yes	characteristic
178	C₂×C₈	C₂×#120⋊C₃	1	1	yes	characteristic
179	C₂³	C₂²×D₄×A₄	1	1	yes	characteristic
180	Q₈	C₂²×#120⋊C₃	1	2	yes	—
181	D₄	C₂²×#120⋊C₃	1	2	yes	—
182	C₂×C₄	C₂²×#120⋊C₃	1	1	yes	characteristic
183	C₈	C₂²×#120⋊C₃	1	1	yes	characteristic
184	C₈	C₂²×#120⋊C₃	1	1	yes	characteristic
185	C₂²	D₄×#120⋊C₃	1	1	yes	characteristic
186	C₄	D₄×#120⋊C₃	1	1	yes	characteristic
187	C₄	C₂³×#120⋊C₃	1	1	yes	characteristic
188	C₂	C₂×D₄×#120⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×#120⋊C₃
commutator, cosocle	#585	C₂⁴×C₃
Frattini	C₂²×C₄	C₂⁴×A₄
Fitting	#72342140	C₃
socle	C₂	C₂×D₄×#120⋊C₃
hypercenter	#135	#120⋊C₃
hypocenter	H(5,𝔽₂)	C₂²×C₃×D₄
LCS	G ▷ #585 ▷ H(5,𝔽₂)
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #135

Complements

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

N	H ≅ G/N	type
1. #72342140	C₃	semidirect
2. #72342135	C₂	semidirect
3. #72342129	C₂	semidirect
4. #72342123	C₂	semidirect
5. #72342120	C₂	semidirect
6. #72342111	C₂	semidirect
7. C₂×#21483⋊C₃	C₂	semidirect
8. #72341679	C₂	semidirect
9. #72236874	C₂	semidirect
10. #72236864	C₂	semidirect
11. #72236824	C₂	semidirect
12. #72227318	C₂	semidirect
13. #11710937	C₂²	semidirect
14. #11710902	C₂²	semidirect
15. #11710899	C₂²	semidirect
16. #11710898	C₂²	semidirect
17. C₂×#85666	C₂²	semidirect
18. #11710352	C₂²	semidirect
19. #60401⋊C₃	C₂²	semidirect
20. #11710347	C₂²	semidirect
21. #60399⋊C₃	C₂²	semidirect
22. #60398⋊C₃	C₂²	semidirect
24. #32949⋊C₃	C₂²	semidirect
25. #32948⋊C₃	C₂²	semidirect
26. #32947⋊C₃	C₂²	semidirect
27. #32897⋊C₃	C₂²	semidirect
28. #32896⋊C₃	C₂²	semidirect
29. #32895⋊C₃	C₂²	semidirect
30. #32894⋊C₃	C₂²	semidirect
31. #21483⋊C₃	C₂²	semidirect
32. #21483⋊C₃	C₂²	semidirect
33. #21483⋊C₃	C₂²	semidirect
34. #21483⋊C₃	C₂²	semidirect
35. C₂×#1791⋊C₃	C₂²	semidirect
36. #10587001	C₂×C₃	semidirect
37. #10574874	C₂×C₃	semidirect
38. #6629044	C₂×C₃	semidirect
39. #6628882	C₂×C₃	semidirect
40. #6573734	C₂×C₃	semidirect
41. #6573733	C₂×C₃	semidirect
42. #6573732	C₂×C₃	semidirect
43. #6573731	C₂×C₃	semidirect
44. #6573730	C₂×C₃	semidirect
45. #6573729	C₂×C₃	semidirect
46. C₂×#21483	C₂×C₃	semidirect
62. #63103	C₂²×C₃	semidirect
63. #63093	C₂²×C₃	semidirect
64. #63092	C₂²×C₃	semidirect
65. #63091	C₂²×C₃	semidirect
66. C₂³×#135	A₄	semidirect
67. C₂×#3573	C₂²×C₃	semidirect
68. #60402	C₂²×C₃	semidirect
69. #60401	C₂²×C₃	semidirect
72. #60398	C₂²×C₃	semidirect
74. #32949	C₂²×C₃	semidirect
75. #32948	C₂²×C₃	semidirect
76. #32947	C₂²×C₃	semidirect
77. #32897	C₂²×C₃	semidirect
78. #32896	C₂²×C₃	semidirect
79. #32895	C₂²×C₃	semidirect
80. #32894	C₂²×C₃	semidirect
83. #21483	C₂²×C₃	semidirect
84. #21483	C₂²×C₃	semidirect
85. C₂×#1791	C₂²×C₃	semidirect
96. C₂³×#41	C₂×A₄	semidirect
98. C₂²×#135	C₂×A₄	semidirect
99. C₂²×#135	C₂×A₄	semidirect
100. C₂²×#135	C₂×A₄	semidirect
101. C₂²×#135	C₂×A₄	semidirect
102. C₂²×#135	C₂×A₄	semidirect
103. C₂³×Q₁₆	C₂×A₄	semidirect
104. C₂³×SD₁₆	C₂×A₄	semidirect
105. C₂³×D₈	C₂×A₄	semidirect
108. C₂⁴×C₈	C₂×A₄	semidirect

Maximal subgroups

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

subgroup	index
#72342140	3
#72342135	2
#72342129	2
#72342123	2
#72342120	2
#72342111	2
C₂×#21483⋊C₃	2
#72341679	2
#72236874	2
#72236864	2
#72236824	2
#72227318	2
C₂×A₄×#135	4