Subgroups — C₂²×C₅×#488 #11983324

Order 1280.

2310 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₅×#124	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#488	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₅×#103	C₂	1	2	yes	maximal, complemented
5	C₂²×C₅×#102	C₂	1	2	yes	maximal
6	C₂²×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×C₅×Q₈	C₂²	1	1	yes	characteristic
8	C₂³×C₅×D₄	C₂²	1	1	yes	characteristic
9	C₂³×C₅×C₈	C₂²	1	2	yes	—
10	C₂×C₅×#125	—	2	6	no	—
11	C₂×C₅×#125	C₂²	1	12	yes	complemented
12	C₂×C₅×#124	—	2	6	no	—
13	C₂×C₅×#124	C₂²	1	12	yes	—
14	C₂²×C₅×C₄⋊C₄	—	2	2	no	—
15	C₂²×C₅×C₄⋊C₄	C₂²	1	2	yes	—
16	C₂²×C₅×C₂²⋊C₄	—	2	2	no	—
17	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
18	C₅×#488	C₂²	1	64	yes	direct product factor
19	C₂×C₅×#103	C₂²	1	24	yes	complemented
20	C₂×C₅×#102	C₂²	1	24	yes	—
21	C₂×C₄×C₅×C₈	C₂²	1	12	yes	complemented
22	C₂²×#488	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
23	C₂⁵×C₅	—	2	1	no	—
24	C₂²×C₅×Q₈	—	2	3	no	—
25	C₂²×C₅×Q₈	—	2	8	no	—
26	C₂²×C₅×Q₈	C₂³	1	6	yes	—
27	C₂²×C₅×D₄	—	2	3	no	—
28	C₂²×C₅×D₄	—	2	8	no	—
29	C₂²×C₅×D₄	C₂³	1	6	yes	—
30	C₂³×C₄×C₅	—	2	1	no	—
31	C₂³×C₄×C₅	—	2	2	no	—
32	C₂³×C₄×C₅	D₄	1	2	yes	—
33	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
34	C₂²×C₅×C₈	—	2	8	no	—
35	C₂²×C₅×C₈	C₂³	1	12	yes	—
36	C₅×#125	—	2	24	no	—
37	C₅×#125	C₂³	1	16	yes	complemented
38	C₅×#124	—	2	24	no	—
39	C₅×#124	C₂³	1	16	yes	—
40	C₂×C₅×C₄⋊C₄	—	2	24	no	—
41	C₂×C₅×C₄⋊C₄	—	2	6	no	—
42	C₂×C₅×C₄⋊C₄	C₂³	1	12	yes	—
43	C₂×C₅×C₂²⋊C₄	—	2	24	no	—
44	C₂×C₄²×C₅	D₄	1	6	yes	—
45	C₂×C₄²×C₅	C₂³	1	6	yes	—
46	C₅×#103	C₂³	1	32	yes	complemented
47	C₅×#102	C₂³	1	32	yes	—
48	C₄×C₅×C₈	C₂³	1	16	yes	complemented
49	C₂²×#125	C₂×C₅	1	1	yes	characteristic, complemented
50	C₂²×#124	C₂×C₅	1	1	yes	characteristic
51	C₂×#488	C₂×C₅	1	24	yes	direct product factor
52	C₂²×#103	C₂×C₅	1	2	yes	complemented
53	C₂²×#102	C₂×C₅	1	2	yes	—
54	C₂²×C₄×C₈	C₂×C₅	1	1	yes	characteristic, complemented
55	C₂⁴×C₅	—	2	6	no	—
56	C₂⁴×C₅	—	4	4	no	—
57	C₂⁴×C₅	—	4	8	no	—
58	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
59	C₂×C₅×Q₈	—	2	6	no	—
60	C₂×C₅×Q₈	—	2	24	no	—
61	C₂×C₅×Q₈	—	4	12	no	—
62	C₂×C₅×Q₈	C₂⁴	1	4	yes	—
63	C₂×C₅×D₄	—	2	6	no	—
64	C₂×C₅×D₄	—	2	24	no	—
65	C₂×C₅×D₄	—	4	12	no	—
66	C₂×C₅×D₄	C₂⁴	1	4	yes	—
67	C₂²×C₄×C₅	—	2	8	no	—
68	C₂²×C₄×C₅	—	2	6	no	—
69	C₂²×C₄×C₅	—	2	12	no	—
70	C₂²×C₄×C₅	—	4	4	no	—
71	C₂²×C₄×C₅	—	4	8	no	—
72	C₂²×C₄×C₅	C₂×D₄	1	12	yes	—
73	C₂²×C₄×C₅	C₂×D₄	1	3	yes	—
74	C₂²×C₄×C₅	#41	1	8	yes	—
75	C₂²×C₄×C₅	C₂⁴	1	3	yes	—
76	C₂×C₅×C₈	—	2	24	no	—
77	C₂×C₅×C₈	C₂⁴	1	8	yes	—
78	C₅×C₄⋊C₄	—	2	32	no	—
79	C₅×C₄⋊C₄	—	2	12	no	—
80	C₅×C₄⋊C₄	C₂⁴	1	8	yes	—
81	C₅×C₂²⋊C₄	—	2	32	no	—
82	C₄²×C₅	C₂×D₄	1	12	yes	—
83	C₄²×C₅	C₂⁴	1	4	yes	—
84	C₂³×Q₈	C₂²×C₅	1	1	yes	characteristic
85	C₂³×D₄	C₂²×C₅	1	1	yes	characteristic
86	C₂³×C₈	C₂²×C₅	1	2	yes	—
87	C₂×#125	—	2	6	no	—
88	C₂×#125	C₂²×C₅	1	12	yes	complemented
89	C₂×#124	—	2	6	no	—
90	C₂×#124	C₂²×C₅	1	12	yes	—
91	C₂²×C₄⋊C₄	—	2	2	no	—
92	C₂²×C₄⋊C₄	C₂²×C₅	1	2	yes	—
93	C₂²×C₂²⋊C₄	—	2	2	no	—
94	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
95	#488	C₂²×C₅	1	64	yes	direct product factor
96	C₂×#103	C₂²×C₅	1	24	yes	complemented
97	C₂×#102	C₂²×C₅	1	24	yes	—
98	C₂×C₄×C₈	C₂²×C₅	1	12	yes	complemented
99	C₂³×C₅	—	2	4	no	—
100	C₂³×C₅	—	4	12	no	—
101	C₂³×C₅	—	4	24	no	—
102	C₂³×C₅	—	8	16	no	—
103	C₂³×C₅	#124	1	4	yes	—
104	C₂³×C₅	#135	1	8	yes	—
105	C₂³×C₅	C₂²×D₄	1	3	yes	—
106	C₅×Q₈	—	2	8	no	—
107	C₅×Q₈	—	4	12	no	—
108	C₅×D₄	—	2	8	no	—
109	C₅×D₄	—	4	12	no	—
110	C₂×C₄×C₅	—	2	24	no	—
111	C₂×C₄×C₅	—	2	4	no	—
112	C₂×C₄×C₅	—	2	8	no	—
113	C₂×C₄×C₅	—	4	12	no	—
114	C₂×C₄×C₅	—	4	24	no	—
115	C₂×C₄×C₅	#124	1	12	yes	—
116	C₂×C₄×C₅	C₂²×D₄	1	8	yes	—
117	C₂×C₄×C₅	C₂²×D₄	1	3	yes	—
118	C₂×C₄×C₅	C₂×#41	1	12	yes	—
119	C₂×C₄×C₅	C₂⁵	1	1	yes	characteristic
120	C₅×C₈	—	2	8	no	—
121	C₂⁵	—	2	1	no	—
122	C₂²×Q₈	—	2	3	no	—
123	C₂²×Q₈	—	2	8	no	—
124	C₂²×Q₈	C₂³×C₅	1	6	yes	—
125	C₂²×D₄	—	2	3	no	—
126	C₂²×D₄	—	2	8	no	—
127	C₂²×D₄	C₂³×C₅	1	6	yes	—
128	C₂³×C₄	—	2	1	no	—
129	C₂³×C₄	—	2	2	no	—
130	C₂³×C₄	C₅×D₄	1	2	yes	—
131	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
132	C₂²×C₈	—	2	8	no	—
133	C₂²×C₈	C₂³×C₅	1	12	yes	—
134	#125	—	2	24	no	—
135	#125	C₂³×C₅	1	16	yes	complemented
136	#124	—	2	24	no	—
137	#124	C₂³×C₅	1	16	yes	—
138	C₂×C₄⋊C₄	—	2	24	no	—
139	C₂×C₄⋊C₄	—	2	6	no	—
140	C₂×C₄⋊C₄	C₂³×C₅	1	12	yes	—
141	C₂×C₂²⋊C₄	—	2	24	no	—
142	C₂×C₄²	C₅×D₄	1	6	yes	—
143	C₂×C₄²	C₂³×C₅	1	6	yes	—
144	#103	C₂³×C₅	1	32	yes	complemented
145	#102	C₂³×C₅	1	32	yes	—
146	C₄×C₈	C₂³×C₅	1	16	yes	complemented
147	C₂²×C₅	—	4	4	no	—
148	C₂²×C₅	—	4	8	no	—
149	C₂²×C₅	—	8	24	no	—
150	C₂²×C₅	#488	1	16	yes	direct product factor
151	C₂²×C₅	C₂×#124	1	6	yes	—
152	C₂²×C₅	C₂×#135	1	12	yes	—
153	C₂²×C₅	C₂³×D₄	1	1	yes	characteristic
154	C₄×C₅	—	2	8	no	—
155	C₄×C₅	—	4	4	no	—
156	C₄×C₅	—	4	8	no	—
157	C₄×C₅	C₂×#124	1	6	yes	—
158	C₄×C₅	C₂²×#41	1	2	yes	—
159	C₂⁴	—	2	6	no	—
160	C₂⁴	—	4	4	no	—
161	C₂⁴	—	4	8	no	—
162	C₂⁴	C₂×C₅×D₄	1	1	yes	characteristic
163	C₂×Q₈	—	2	6	no	—
164	C₂×Q₈	—	2	24	no	—
165	C₂×Q₈	—	4	12	no	—
166	C₂×Q₈	C₂⁴×C₅	1	4	yes	—
167	C₂×D₄	—	2	6	no	—
168	C₂×D₄	—	2	24	no	—
169	C₂×D₄	—	4	12	no	—
170	C₂×D₄	C₂⁴×C₅	1	4	yes	—
171	C₂²×C₄	—	2	8	no	—
172	C₂²×C₄	—	2	6	no	—
173	C₂²×C₄	—	2	12	no	—
174	C₂²×C₄	—	4	4	no	—
175	C₂²×C₄	—	4	8	no	—
176	C₂²×C₄	C₂×C₅×D₄	1	12	yes	—
177	C₂²×C₄	C₂×C₅×D₄	1	3	yes	—
178	C₂²×C₄	C₅×#41	1	8	yes	—
179	C₂²×C₄	C₂⁴×C₅	1	3	yes	—
180	C₂×C₈	—	2	24	no	—
181	C₂×C₈	C₂⁴×C₅	1	8	yes	—
182	C₄⋊C₄	—	2	32	no	—
183	C₄⋊C₄	—	2	12	no	—
184	C₄⋊C₄	C₂⁴×C₅	1	8	yes	—
185	C₂²⋊C₄	—	2	32	no	—
186	C₄²	C₂×C₅×D₄	1	12	yes	—
187	C₄²	C₂⁴×C₅	1	4	yes	—
188	C₂×C₅	—	8	4	no	—
189	C₂×C₅	C₂×#488	1	12	yes	direct product factor
190	C₂×C₅	C₂²×#124	1	1	yes	characteristic
191	C₂×C₅	C₂²×#135	1	2	yes	—
192	C₂³	—	2	4	no	—
193	C₂³	—	4	12	no	—
194	C₂³	—	4	24	no	—
195	C₂³	—	8	16	no	—
196	C₂³	C₅×#124	1	4	yes	—
197	C₂³	C₅×#135	1	8	yes	—
198	C₂³	C₂²×C₅×D₄	1	3	yes	—
199	Q₈	—	2	8	no	—
200	Q₈	—	4	12	no	—
201	D₄	—	2	8	no	—
202	D₄	—	4	12	no	—
203	C₂×C₄	—	2	24	no	—
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	—	2	8	no	—
206	C₂×C₄	—	4	12	no	—
207	C₂×C₄	—	4	24	no	—
208	C₂×C₄	C₅×#124	1	12	yes	—
209	C₂×C₄	C₂²×C₅×D₄	1	8	yes	—
210	C₂×C₄	C₂²×C₅×D₄	1	3	yes	—
211	C₂×C₄	C₂×C₅×#41	1	12	yes	—
212	C₂×C₄	C₂⁵×C₅	1	1	yes	characteristic
213	C₈	—	2	8	no	—
214	C₅	C₂²×#488	1	1	yes	characteristic, direct product factor, Sylow
215	C₂²	—	4	4	no	—
216	C₂²	—	4	8	no	—
217	C₂²	—	8	24	no	—
218	C₂²	C₅×#488	1	16	yes	direct product factor
219	C₂²	C₂×C₅×#124	1	6	yes	—
220	C₂²	C₂×C₅×#135	1	12	yes	—
221	C₂²	C₂³×C₅×D₄	1	1	yes	characteristic
222	C₄	—	2	8	no	—
223	C₄	—	4	4	no	—
224	C₄	—	4	8	no	—
225	C₄	C₂×C₅×#124	1	6	yes	—
226	C₄	C₂²×C₅×#41	1	2	yes	—
227	C₂	—	8	4	no	—
228	C₂	C₂×C₅×#488	1	12	yes	direct product factor
229	C₂	C₂²×C₅×#124	1	1	yes	characteristic
230	C₂	C₂²×C₅×#135	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₅	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅
prime radical	C₂³×C₅×D₄	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅ ◁ C₂²×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₅×#125	C₂	semidirect
3. C₂×C₅×#488	C₂	direct (N × H)
4. C₂²×C₅×#103	C₂	semidirect
6. C₂²×C₄×C₅×C₈	C₂	semidirect
11. C₂×C₅×#125	C₂²	semidirect
18. C₅×#488	C₂²	direct (N × H)
19. C₂×C₅×#103	C₂²	semidirect
21. C₂×C₄×C₅×C₈	C₂²	semidirect
22. C₂²×#488	C₅	direct (N × H)
37. C₅×#125	C₂³	semidirect
46. C₅×#103	C₂³	semidirect
48. C₄×C₅×C₈	C₂³	semidirect
49. C₂²×#125	C₂×C₅	semidirect
51. C₂×#488	C₂×C₅	direct (N × H)
52. C₂²×#103	C₂×C₅	semidirect
54. C₂²×C₄×C₈	C₂×C₅	semidirect
88. C₂×#125	C₂²×C₅	semidirect
95. #488	C₂²×C₅	direct (N × H)
96. C₂×#103	C₂²×C₅	semidirect
98. C₂×C₄×C₈	C₂²×C₅	semidirect
135. #125	C₂³×C₅	semidirect
144. #103	C₂³×C₅	semidirect
146. C₄×C₈	C₂³×C₅	semidirect
150. C₂²×C₅	#488	direct (N × H)
189. C₂×C₅	C₂×#488	direct (N × H)
214. C₅	C₂²×#488	direct (N × H)
218. C₂²	C₅×#488	direct (N × H)
228. C₂	C₂×C₅×#488	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×C₅×#125	2
C₂²×C₅×#124	2
C₂×C₅×#488	2
C₂²×C₅×#103	2
C₂²×C₅×#102	2
C₂²×C₄×C₅×C₈	2
C₂²×#488	5

Subgroups — C22×C5×#488 #11983324

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅×#488 #11983324