Subgroups — C₂²×C₄²×#1043 #72242681

Order 6400.

32376 conjugacy classes of subgroups.

The subgroup lattice has 225 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄²×C₅⋊D₅	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₄×#1043	C₂	1	6	yes	maximal
3	C₂×C₄²×#1043	C₂	1	24	yes	maximal, direct product factor
4	C₂⁴×#1043	C₂²	1	4	yes	—
5	C₂³×C₄×C₅⋊D₅	C₂²	1	3	yes	—
6	C₂²×C₄×#1043	C₄	1	96	yes	direct product factor
7	C₂²×C₄×#1043	C₂²	1	72	yes	—
8	C₂²×C₄²×C₅²	C₄	1	1	yes	characteristic, complemented
9	C₂²×C₄×#1039	C₄	1	3	yes	complemented
10	C₂×C₄²×C₅⋊D₅	C₄	1	12	yes	complemented
11	C₂×C₄²×C₅⋊D₅	C₂²	1	12	yes	—
12	C₄²×#1043	C₂²	1	64	yes	direct product factor
13	C₂²×C₄²×AGL(1,𝔽₅)	—	5	6	no	maximal
14	C₂⁴×C₅⋊D₅	C₂³	1	1	yes	characteristic
15	C₂³×#1043	C₂×C₄	1	96	yes	—
16	C₂³×#1043	C₂³	1	24	yes	—
17	C₂³×C₄×C₅²	C₂×C₄	1	3	yes	—
18	C₂³×#1039	C₂×C₄	1	3	yes	—
19	C₂²×C₄×C₅⋊D₅	C₂×C₄	1	18	yes	—
20	C₂²×C₄×C₅⋊D₅	C₂×C₄	1	48	yes	—
21	C₂²×C₄×C₅⋊D₅	C₂³	1	18	yes	—
22	C₂×C₄×#1043	C₂×C₄	1	576	yes	direct product factor
23	C₂×C₄×#1043	C₂³	1	96	yes	—
24	C₂×C₄²×C₅²	C₂×C₄	1	12	yes	complemented
25	C₂×C₄×#1039	C₂×C₄	1	36	yes	complemented
26	C₄²×C₅⋊D₅	C₂×C₄	1	48	yes	complemented
27	C₄²×C₅⋊D₅	C₂³	1	16	yes	—
28	C₂³×C₄×AGL(1,𝔽₅)	—	5	36	no	—
29	C₂²×C₄²×D₅	—	5	6	no	—
30	C₂×C₄²×AGL(1,𝔽₅)	—	5	144	no	—
31	C₂⁴×C₅²	C₂²×C₄	1	1	yes	characteristic
32	C₂³×C₅⋊D₅	C₂²×C₄	1	3	yes	—
33	C₂³×C₅⋊D₅	C₂²×C₄	1	24	yes	—
34	C₂³×C₅⋊D₅	C₂⁴	1	3	yes	—
35	C₂²×#1043	C₄²	1	256	yes	direct product factor
36	C₂²×#1043	C₂²×C₄	1	288	yes	—
37	C₂²×#1043	C₂⁴	1	16	yes	—
38	C₂²×C₄×C₅²	C₄²	1	24	yes	complemented
39	C₂²×C₄×C₅²	C₂²×C₄	1	18	yes	—
40	C₂²×#1039	C₄²	1	24	yes	complemented
41	C₂²×#1039	C₂²×C₄	1	18	yes	—
42	C₂×C₄×C₅⋊D₅	C₄²	1	144	yes	complemented
43	C₂×C₄×C₅⋊D₅	C₂²×C₄	1	36	yes	—
44	C₂×C₄×C₅⋊D₅	C₂²×C₄	1	144	yes	—
45	C₂×C₄×C₅⋊D₅	C₂⁴	1	12	yes	—
46	C₄×#1043	C₂²×C₄	1	384	yes	direct product factor
47	C₄²×C₅²	C₂²×C₄	1	16	yes	complemented
48	C₄×#1039	C₂²×C₄	1	48	yes	complemented
49	C₂⁴×AGL(1,𝔽₅)	—	5	24	no	—
50	C₂³×C₄×D₅	—	5	18	no	—
51	C₂²×C₄×AGL(1,𝔽₅)	—	5	576	no	—
52	C₂²×C₄×AGL(1,𝔽₅)	—	5	432	no	—
53	C₂²×C₄²×C₅	AGL(1,𝔽₅)	1	6	yes	complemented
54	C₂²×C₄×Dic₅	—	5	18	no	—
55	C₂×C₄²×D₅	—	5	72	no	—
56	C₂×C₄²×D₅	—	5	72	no	—
57	C₄²×AGL(1,𝔽₅)	—	5	384	no	—
58	C₂²×C₄³	—	25	1	no	Sylow
59	C₂³×C₅²	C₂×C₄²	1	12	yes	—
60	C₂³×C₅²	C₂³×C₄	1	3	yes	—
61	C₂²×C₅⋊D₅	C₂×C₄²	1	36	yes	—
62	C₂²×C₅⋊D₅	C₂×C₄²	1	64	yes	—
63	C₂²×C₅⋊D₅	C₂³×C₄	1	36	yes	—
64	C₂²×C₅⋊D₅	C₂³×C₄	1	3	yes	—
65	C₂²×C₅⋊D₅	C₂⁵	1	1	yes	characteristic
66	C₂×#1043	C₂×C₄²	1	384	yes	direct product factor
67	C₂×#1043	C₂³×C₄	1	96	yes	—
68	C₂×C₄×C₅²	C₂×C₄²	1	72	yes	complemented
69	C₂×C₄×C₅²	C₂³×C₄	1	12	yes	—
70	C₂×#1039	C₂×C₄²	1	72	yes	complemented
71	C₂×#1039	C₂³×C₄	1	12	yes	—
72	C₄×C₅⋊D₅	C₂×C₄²	1	144	yes	complemented
73	C₄×C₅⋊D₅	C₂³×C₄	1	48	yes	—
74	C₂⁴×D₅	—	5	6	no	—
75	C₂³×AGL(1,𝔽₅)	—	5	576	no	—
76	C₂³×AGL(1,𝔽₅)	—	5	144	no	—
77	C₂³×C₄×C₅	C₂×AGL(1,𝔽₅)	1	18	yes	—
78	C₂³×Dic₅	—	5	18	no	—
79	C₂²×C₄×D₅	—	5	288	no	—
80	C₂²×C₄×D₅	—	5	108	no	—
81	C₂²×C₄×D₅	—	5	108	no	—
82	C₂×C₄×AGL(1,𝔽₅)	—	5	3456	no	—
83	C₂×C₄×AGL(1,𝔽₅)	—	5	576	no	—
84	C₂×C₄²×C₅	C₂×AGL(1,𝔽₅)	1	72	yes	complemented
85	C₂×C₄×Dic₅	—	5	216	no	—
86	C₄²×D₅	—	5	288	no	—
87	C₄²×D₅	—	5	96	no	—
88	C₂³×C₄²	—	25	6	no	—
89	C₂³×C₄²	—	25	1	no	—
90	C₂×C₄³	—	25	24	no	—
91	C₂²×C₅²	C₄³	1	16	yes	complemented
92	C₂²×C₅²	C₂²×C₄²	1	18	yes	—
93	C₂²×C₅²	C₂⁴×C₄	1	1	yes	characteristic
94	C₂×C₅⋊D₅	C₄³	1	48	yes	complemented
95	C₂×C₅⋊D₅	C₂²×C₄²	1	48	yes	—
96	C₂×C₅⋊D₅	C₂²×C₄²	1	18	yes	—
97	C₂×C₅⋊D₅	C₂⁴×C₄	1	6	yes	—
98	#1043	C₂²×C₄²	1	64	yes	direct product factor
99	C₄×C₅²	C₂²×C₄²	1	24	yes	complemented
100	#1039	C₂²×C₄²	1	24	yes	complemented
101	C₂⁴×C₅	C₂²×AGL(1,𝔽₅)	1	6	yes	—
102	C₂³×D₅	—	5	144	no	—
103	C₂³×D₅	—	5	18	no	—
104	C₂³×D₅	—	5	18	no	—
105	C₂²×AGL(1,𝔽₅)	—	5	96	no	—
106	C₂²×AGL(1,𝔽₅)	—	5	1728	no	—
107	C₂²×AGL(1,𝔽₅)	—	5	1536	no	—
108	C₂²×C₄×C₅	C₄×AGL(1,𝔽₅)	1	144	yes	complemented
109	C₂²×C₄×C₅	C₂²×AGL(1,𝔽₅)	1	108	yes	—
110	C₂²×Dic₅	—	5	144	no	—
111	C₂²×Dic₅	—	5	108	no	—
112	C₂×C₄×D₅	—	5	864	no	—
113	C₂×C₄×D₅	—	5	864	no	—
114	C₂×C₄×D₅	—	5	72	no	—
115	C₂×C₄×D₅	—	5	216	no	—
116	C₄×AGL(1,𝔽₅)	—	5	2304	no	—
117	C₄²×C₅	C₂²×AGL(1,𝔽₅)	1	96	yes	complemented
118	C₄×Dic₅	—	5	288	no	—
119	C₂⁴×C₄	—	25	4	no	—
120	C₂⁴×C₄	—	25	3	no	—
121	C₂²×C₄²	—	25	96	no	—
122	C₂²×C₄²	—	25	72	no	—
123	C₂²×C₄²	—	25	3	no	—
124	C₂²×C₄²	—	25	12	no	—
125	C₂²×C₄²	—	25	12	no	—
126	C₂²×C₄²	#1043	1	1	yes	characteristic, direct product factor
127	C₄³	—	25	64	no	—
128	C₂×C₅²	C₂×C₄³	1	12	yes	complemented
129	C₂×C₅²	C₂³×C₄²	1	3	yes	—
130	C₅⋊D₅	C₂×C₄³	1	12	yes	complemented
131	C₅⋊D₅	C₂³×C₄²	1	4	yes	—
132	C₂³×C₅	C₂×C₄×AGL(1,𝔽₅)	1	72	yes	—
133	C₂³×C₅	C₂³×AGL(1,𝔽₅)	1	18	yes	—
134	C₂²×D₅	—	5	384	no	—
135	C₂²×D₅	—	5	216	no	—
136	C₂²×D₅	—	5	216	no	—
137	C₂²×D₅	—	5	18	no	—
138	C₂²×D₅	—	5	6	no	—
139	C₂×AGL(1,𝔽₅)	—	5	2304	no	—
140	C₂×AGL(1,𝔽₅)	—	5	576	no	—
141	C₂×C₄×C₅	C₂×C₄×AGL(1,𝔽₅)	1	432	yes	complemented
142	C₂×C₄×C₅	C₂³×AGL(1,𝔽₅)	1	72	yes	—
143	C₂×Dic₅	—	5	432	no	—
144	C₂×Dic₅	—	5	72	no	—
145	C₄×D₅	—	5	864	no	—
146	C₄×D₅	—	5	288	no	—
147	C₂⁵	—	25	1	no	—
148	C₂³×C₄	—	25	96	no	—
149	C₂³×C₄	—	25	24	no	—
150	C₂³×C₄	—	25	3	no	—
151	C₂³×C₄	—	25	48	no	—
152	C₂³×C₄	—	25	18	no	—
153	C₂³×C₄	—	25	18	no	—
154	C₂³×C₄	C₂×#1043	1	3	yes	—
155	C₂×C₄²	—	25	96	no	—
156	C₂×C₄²	—	25	576	no	—
157	C₂×C₄²	—	25	36	no	—
158	C₂×C₄²	—	25	16	no	—
159	C₂×C₄²	—	25	48	no	—
160	C₂×C₄²	C₂×#1043	1	12	yes	direct product factor
161	C₅²	C₂²×C₄³	1	1	yes	characteristic, complemented, Sylow
162	C₂²×C₅	C₄²×AGL(1,𝔽₅)	1	96	yes	complemented
163	C₂²×C₅	C₂²×C₄×AGL(1,𝔽₅)	1	108	yes	—
164	C₂²×C₅	C₂⁴×AGL(1,𝔽₅)	1	6	yes	—
165	C₂×D₅	—	5	36	no	—
166	C₂×D₅	—	5	108	no	—
167	C₂×D₅	—	5	288	no	—
168	C₂×D₅	—	5	288	no	—
169	AGL(1,𝔽₅)	—	5	384	no	—
170	C₄×C₅	C₂²×C₄×AGL(1,𝔽₅)	1	144	yes	complemented
171	Dic₅	—	5	144	no	—
172	C₂⁴	—	25	24	no	—
173	C₂⁴	—	25	3	no	—
174	C₂⁴	—	25	3	no	—
175	C₂⁴	C₂²×#1043	1	1	yes	characteristic
176	C₂²×C₄	—	25	16	no	—
177	C₂²×C₄	—	25	288	no	—
178	C₂²×C₄	—	25	256	no	—
179	C₂²×C₄	—	25	24	no	—
180	C₂²×C₄	—	25	18	no	—
181	C₂²×C₄	—	25	144	no	—
182	C₂²×C₄	—	25	144	no	—
183	C₂²×C₄	—	25	12	no	—
184	C₂²×C₄	—	25	36	no	—
185	C₂²×C₄	C₄×#1043	1	24	yes	direct product factor
186	C₂²×C₄	C₂²×#1043	1	18	yes	—
187	C₄²	—	25	384	no	—
188	C₄²	—	25	48	no	—
189	C₄²	C₂²×#1043	1	16	yes	direct product factor
190	C₂×C₅	C₂×C₄²×AGL(1,𝔽₅)	1	72	yes	complemented
191	C₂×C₅	C₂³×C₄×AGL(1,𝔽₅)	1	18	yes	—
192	D₅	—	5	24	no	—
193	D₅	—	5	72	no	—
194	C₂³	—	25	36	no	—
195	C₂³	—	25	36	no	—
196	C₂³	—	25	64	no	—
197	C₂³	—	25	1	no	—
198	C₂³	—	25	3	no	—
199	C₂³	C₂×C₄×#1043	1	12	yes	—
200	C₂³	C₂³×#1043	1	3	yes	—
201	C₂×C₄	—	25	96	no	—
202	C₂×C₄	—	25	384	no	—
203	C₂×C₄	—	25	72	no	—
204	C₂×C₄	—	25	12	no	—
205	C₂×C₄	—	25	48	no	—
206	C₂×C₄	—	25	144	no	—
207	C₂×C₄	C₂×C₄×#1043	1	72	yes	direct product factor
208	C₂×C₄	C₂³×#1043	1	12	yes	—
209	C₅	C₂²×C₄²×AGL(1,𝔽₅)	1	6	yes	complemented
210	C₂²	—	25	6	no	—
211	C₂²	—	25	18	no	—
212	C₂²	—	25	48	no	—
213	C₂²	—	25	48	no	—
214	C₂²	C₄²×#1043	1	16	yes	direct product factor
215	C₂²	C₂²×C₄×#1043	1	18	yes	—
216	C₂²	C₂⁴×#1043	1	1	yes	characteristic
217	C₄	—	25	64	no	—
218	C₄	—	25	24	no	—
219	C₄	C₂²×C₄×#1043	1	24	yes	direct product factor
220	C₂	—	25	4	no	—
221	C₂	—	25	12	no	—
222	C₂	C₂×C₄²×#1043	1	12	yes	direct product factor
223	C₂	C₂³×C₄×#1043	1	3	yes	—

	subgroup	G/H
center, hypercenter	C₂²×C₄²	#1043
commutator, hypocenter	C₅²	C₂²×C₄³
Frattini	C₂²	C₂⁴×#1043
Fitting	C₂²×C₄²×C₅²	C₄
socle	C₂⁴×C₅²	C₂²×C₄
prime radical	C₂⁴×C₅⋊D₅	C₂³
cosocle	C₂²×C₅⋊D₅	C₂⁵
LCS	G ▷ C₅²
UCS	C₁ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
3. C₂×C₄²×#1043	C₂	direct (N × H)
6. C₂²×C₄×#1043	C₄	direct (N × H)
8. C₂²×C₄²×C₅²	C₄	semidirect
9. C₂²×C₄×#1039	C₄	semidirect
10. C₂×C₄²×C₅⋊D₅	C₄	semidirect
12. C₄²×#1043	C₂²	direct (N × H)
22. C₂×C₄×#1043	C₂×C₄	direct (N × H)
24. C₂×C₄²×C₅²	C₂×C₄	semidirect
25. C₂×C₄×#1039	C₂×C₄	semidirect
26. C₄²×C₅⋊D₅	C₂×C₄	semidirect
35. C₂²×#1043	C₄²	direct (N × H)
38. C₂²×C₄×C₅²	C₄²	semidirect
40. C₂²×#1039	C₄²	semidirect
42. C₂×C₄×C₅⋊D₅	C₄²	semidirect
46. C₄×#1043	C₂²×C₄	direct (N × H)
47. C₄²×C₅²	C₂²×C₄	semidirect
48. C₄×#1039	C₂²×C₄	semidirect
53. C₂²×C₄²×C₅	AGL(1,𝔽₅)	semidirect
66. C₂×#1043	C₂×C₄²	direct (N × H)
68. C₂×C₄×C₅²	C₂×C₄²	semidirect
70. C₂×#1039	C₂×C₄²	semidirect
72. C₄×C₅⋊D₅	C₂×C₄²	semidirect
84. C₂×C₄²×C₅	C₂×AGL(1,𝔽₅)	semidirect
91. C₂²×C₅²	C₄³	semidirect
94. C₂×C₅⋊D₅	C₄³	semidirect
98. #1043	C₂²×C₄²	direct (N × H)
99. C₄×C₅²	C₂²×C₄²	semidirect
100. #1039	C₂²×C₄²	semidirect
108. C₂²×C₄×C₅	C₄×AGL(1,𝔽₅)	semidirect
117. C₄²×C₅	C₂²×AGL(1,𝔽₅)	semidirect
126. C₂²×C₄²	#1043	direct (N × H)
128. C₂×C₅²	C₂×C₄³	semidirect
130. C₅⋊D₅	C₂×C₄³	semidirect
141. C₂×C₄×C₅	C₂×C₄×AGL(1,𝔽₅)	semidirect
160. C₂×C₄²	C₂×#1043	direct (N × H)
161. C₅²	C₂²×C₄³	semidirect
162. C₂²×C₅	C₄²×AGL(1,𝔽₅)	semidirect
170. C₄×C₅	C₂²×C₄×AGL(1,𝔽₅)	semidirect
185. C₂²×C₄	C₄×#1043	direct (N × H)
189. C₄²	C₂²×#1043	direct (N × H)
190. C₂×C₅	C₂×C₄²×AGL(1,𝔽₅)	semidirect
207. C₂×C₄	C₂×C₄×#1043	direct (N × H)
209. C₅	C₂²×C₄²×AGL(1,𝔽₅)	semidirect
214. C₂²	C₄²×#1043	direct (N × H)
219. C₄	C₂²×C₄×#1043	direct (N × H)
222. C₂	C₂×C₄²×#1043	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄²×C₅⋊D₅	2
C₂³×C₄×#1043	2
C₂×C₄²×#1043	2
C₂²×C₄²×AGL(1,𝔽₅)	5

Subgroups — C22×C42×#1043 #72242681

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄²×#1043 #72242681