Subgroups — C₂²×C₅×#486 #11983322

Order 1280.

3086 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₅×#127	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#486	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₅×#102	C₂	1	4	yes	maximal, complemented
5	C₂²×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×C₅×Q₈	—	2	1	no	—
7	C₂³×C₅×D₄	—	2	1	no	—
8	C₂³×C₅×D₄	—	2	1	no	—
9	C₂³×C₅×D₄	C₂²	1	2	yes	—
10	C₂³×C₅×C₈	C₂²	1	2	yes	—
11	C₂×C₅×C₄⋊Q₈	—	2	6	no	—
12	C₂×C₅×C₄⋊Q₈	C₂²	1	12	yes	complemented
13	C₂×C₅×#127	—	2	6	no	—
14	C₂×C₅×#127	C₂²	1	12	yes	—
15	C₂²×C₅×C₄⋊C₄	—	2	1	no	—
16	C₂²×C₅×C₄⋊C₄	C₂²	1	2	yes	—
17	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
18	C₅×#486	C₂²	1	64	yes	direct product factor
19	C₂×C₅×#102	C₂²	1	48	yes	complemented
20	C₂×C₄×C₅×C₈	C₂²	1	12	yes	complemented
21	C₂²×#486	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	C₂⁵×C₅	—	2	2	no	—
23	C₂²×C₅×Q₈	—	2	12	no	—
24	C₂²×C₅×Q₈	—	4	8	no	—
25	C₂²×C₅×D₄	—	2	6	no	—
26	C₂²×C₅×D₄	—	2	12	no	—
27	C₂²×C₅×D₄	—	2	12	no	—
28	C₂²×C₅×D₄	—	2	16	no	—
29	C₂²×C₅×D₄	—	4	8	no	—
30	C₂²×C₅×D₄	—	4	8	no	—
31	C₂²×C₅×D₄	C₂³	1	12	yes	—
32	C₂³×C₄×C₅	—	2	2	no	—
33	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
34	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
35	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
36	C₂²×C₅×C₈	—	2	8	no	—
37	C₂²×C₅×C₈	C₂³	1	12	yes	—
38	C₅×C₄⋊Q₈	—	2	24	no	—
39	C₅×C₄⋊Q₈	C₂³	1	16	yes	complemented
40	C₅×#127	—	2	24	no	—
41	C₅×#127	C₂³	1	16	yes	—
42	C₂×C₅×C₄⋊C₄	—	2	12	no	—
43	C₂×C₅×C₄⋊C₄	—	2	6	no	—
44	C₂×C₅×C₄⋊C₄	C₂³	1	12	yes	—
45	C₂×C₄²×C₅	D₄	1	6	yes	—
46	C₂×C₄²×C₅	C₂³	1	6	yes	—
47	C₅×#102	C₂³	1	64	yes	complemented
48	C₄×C₅×C₈	C₂³	1	16	yes	complemented
49	C₂²×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic, complemented
50	C₂²×#127	C₂×C₅	1	1	yes	characteristic
51	C₂×#486	C₂×C₅	1	24	yes	direct product factor
52	C₂²×#102	C₂×C₅	1	4	yes	complemented
53	C₂²×C₄×C₈	C₂×C₅	1	1	yes	characteristic, complemented
54	C₂⁴×C₅	—	2	12	no	—
55	C₂⁴×C₅	—	4	8	no	—
56	C₂⁴×C₅	—	4	8	no	—
57	C₂⁴×C₅	—	4	8	no	—
58	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
59	C₂×C₅×Q₈	—	2	16	no	—
60	C₂×C₅×Q₈	—	4	48	no	—
61	C₂×C₅×D₄	—	2	12	no	—
62	C₂×C₅×D₄	—	2	16	no	—
63	C₂×C₅×D₄	—	2	16	no	—
64	C₂×C₅×D₄	—	2	48	no	—
65	C₂×C₅×D₄	—	4	24	no	—
66	C₂×C₅×D₄	—	4	48	no	—
67	C₂×C₅×D₄	—	4	48	no	—
68	C₂×C₅×D₄	C₂⁴	1	8	yes	—
69	C₂²×C₄×C₅	—	2	12	no	—
70	C₂²×C₄×C₅	—	4	8	no	—
71	C₂²×C₄×C₅	D₈	1	8	yes	—
72	C₂²×C₄×C₅	SD₁₆	1	8	yes	—
73	C₂²×C₄×C₅	C₂×D₄	1	3	yes	—
74	C₂²×C₄×C₅	C₂×D₄	1	6	yes	—
75	C₂²×C₄×C₅	C₂×D₄	1	6	yes	—
76	C₂²×C₄×C₅	#41	1	8	yes	—
77	C₂²×C₄×C₅	C₂⁴	1	3	yes	—
78	C₂×C₅×C₈	—	2	24	no	—
79	C₂×C₅×C₈	C₂⁴	1	8	yes	—
80	C₅×C₄⋊C₄	—	2	16	no	—
81	C₅×C₄⋊C₄	—	2	12	no	—
82	C₅×C₄⋊C₄	C₂⁴	1	8	yes	—
83	C₄²×C₅	C₂×D₄	1	12	yes	—
84	C₄²×C₅	C₂⁴	1	4	yes	—
85	C₂³×Q₈	—	2	1	no	—
86	C₂³×D₄	—	2	1	no	—
87	C₂³×D₄	—	2	1	no	—
88	C₂³×D₄	C₂²×C₅	1	2	yes	—
89	C₂³×C₈	C₂²×C₅	1	2	yes	—
90	C₂×C₄⋊Q₈	—	2	6	no	—
91	C₂×C₄⋊Q₈	C₂²×C₅	1	12	yes	complemented
92	C₂×#127	—	2	6	no	—
93	C₂×#127	C₂²×C₅	1	12	yes	—
94	C₂²×C₄⋊C₄	—	2	1	no	—
95	C₂²×C₄⋊C₄	C₂²×C₅	1	2	yes	—
96	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
97	#486	C₂²×C₅	1	64	yes	direct product factor
98	C₂×#102	C₂²×C₅	1	48	yes	complemented
99	C₂×C₄×C₈	C₂²×C₅	1	12	yes	complemented
100	C₂³×C₅	—	2	8	no	—
101	C₂³×C₅	—	4	24	no	—
102	C₂³×C₅	—	4	24	no	—
103	C₂³×C₅	—	4	24	no	—
104	C₂³×C₅	—	8	32	no	—
105	C₂³×C₅	#124	1	4	yes	—
106	C₂³×C₅	C₂×D₈	1	4	yes	—
107	C₂³×C₅	C₂×SD₁₆	1	4	yes	—
108	C₂³×C₅	C₂²×D₄	1	3	yes	—
109	C₅×Q₈	—	4	32	no	—
110	C₅×D₄	—	2	16	no	—
111	C₅×D₄	—	4	24	no	—
112	C₅×D₄	—	4	32	no	—
113	C₅×D₄	—	4	32	no	—
114	C₂×C₄×C₅	—	2	8	no	—
115	C₂×C₄×C₅	—	4	24	no	—
116	C₂×C₄×C₅	#124	1	12	yes	—
117	C₂×C₄×C₅	C₂×D₈	1	24	yes	—
118	C₂×C₄×C₅	C₂×SD₁₆	1	24	yes	—
119	C₂×C₄×C₅	C₂²×D₄	1	3	yes	—
120	C₂×C₄×C₅	C₂²×D₄	1	4	yes	—
121	C₂×C₄×C₅	C₂²×D₄	1	4	yes	—
122	C₂×C₄×C₅	C₂×#41	1	12	yes	—
123	C₂×C₄×C₅	C₂⁵	1	1	yes	characteristic
124	C₅×C₈	—	2	8	no	—
125	C₂⁵	—	2	2	no	—
126	C₂²×Q₈	—	2	12	no	—
127	C₂²×Q₈	—	4	8	no	—
128	C₂²×D₄	—	2	6	no	—
129	C₂²×D₄	—	2	12	no	—
130	C₂²×D₄	—	2	12	no	—
131	C₂²×D₄	—	2	16	no	—
132	C₂²×D₄	—	4	8	no	—
133	C₂²×D₄	—	4	8	no	—
134	C₂²×D₄	C₂³×C₅	1	12	yes	—
135	C₂³×C₄	—	2	2	no	—
136	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
137	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
138	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
139	C₂²×C₈	—	2	8	no	—
140	C₂²×C₈	C₂³×C₅	1	12	yes	—
141	C₄⋊Q₈	—	2	24	no	—
142	C₄⋊Q₈	C₂³×C₅	1	16	yes	complemented
143	#127	—	2	24	no	—
144	#127	C₂³×C₅	1	16	yes	—
145	C₂×C₄⋊C₄	—	2	12	no	—
146	C₂×C₄⋊C₄	—	2	6	no	—
147	C₂×C₄⋊C₄	C₂³×C₅	1	12	yes	—
148	C₂×C₄²	C₅×D₄	1	6	yes	—
149	C₂×C₄²	C₂³×C₅	1	6	yes	—
150	#102	C₂³×C₅	1	64	yes	complemented
151	C₄×C₈	C₂³×C₅	1	16	yes	complemented
152	C₂²×C₅	—	4	8	no	—
153	C₂²×C₅	—	4	8	no	—
154	C₂²×C₅	—	4	8	no	—
155	C₂²×C₅	—	8	48	no	—
156	C₂²×C₅	#486	1	16	yes	direct product factor
157	C₂²×C₅	C₂×#124	1	6	yes	—
158	C₂²×C₅	C₂²×D₈	1	6	yes	—
159	C₂²×C₅	C₂²×SD₁₆	1	6	yes	—
160	C₂²×C₅	C₂³×D₄	1	1	yes	characteristic
161	C₄×C₅	—	4	8	no	—
162	C₄×C₅	C₂×#124	1	6	yes	—
163	C₄×C₅	C₂²×D₈	1	8	yes	—
164	C₄×C₅	C₂²×SD₁₆	1	8	yes	—
165	C₄×C₅	C₂²×#41	1	2	yes	—
166	C₂⁴	—	2	12	no	—
167	C₂⁴	—	4	8	no	—
168	C₂⁴	—	4	8	no	—
169	C₂⁴	—	4	8	no	—
170	C₂⁴	C₂×C₅×D₄	1	1	yes	characteristic
171	C₂×Q₈	—	2	16	no	—
172	C₂×Q₈	—	4	48	no	—
173	C₂×D₄	—	2	12	no	—
174	C₂×D₄	—	2	16	no	—
175	C₂×D₄	—	2	16	no	—
176	C₂×D₄	—	2	48	no	—
177	C₂×D₄	—	4	24	no	—
178	C₂×D₄	—	4	48	no	—
179	C₂×D₄	—	4	48	no	—
180	C₂×D₄	C₂⁴×C₅	1	8	yes	—
181	C₂²×C₄	—	2	12	no	—
182	C₂²×C₄	—	4	8	no	—
183	C₂²×C₄	C₅×D₈	1	8	yes	—
184	C₂²×C₄	C₅×SD₁₆	1	8	yes	—
185	C₂²×C₄	C₂×C₅×D₄	1	3	yes	—
186	C₂²×C₄	C₂×C₅×D₄	1	6	yes	—
187	C₂²×C₄	C₂×C₅×D₄	1	6	yes	—
188	C₂²×C₄	C₅×#41	1	8	yes	—
189	C₂²×C₄	C₂⁴×C₅	1	3	yes	—
190	C₂×C₈	—	2	24	no	—
191	C₂×C₈	C₂⁴×C₅	1	8	yes	—
192	C₄⋊C₄	—	2	16	no	—
193	C₄⋊C₄	—	2	12	no	—
194	C₄⋊C₄	C₂⁴×C₅	1	8	yes	—
195	C₄²	C₂×C₅×D₄	1	12	yes	—
196	C₄²	C₂⁴×C₅	1	4	yes	—
197	C₂×C₅	—	8	8	no	—
198	C₂×C₅	C₂×#486	1	12	yes	direct product factor
199	C₂×C₅	C₂²×#124	1	1	yes	characteristic
200	C₂×C₅	C₂³×D₈	1	1	yes	characteristic
201	C₂×C₅	C₂³×SD₁₆	1	1	yes	characteristic
202	C₂³	—	2	8	no	—
203	C₂³	—	4	24	no	—
204	C₂³	—	4	24	no	—
205	C₂³	—	4	24	no	—
206	C₂³	—	8	32	no	—
207	C₂³	C₅×#124	1	4	yes	—
208	C₂³	C₂×C₅×D₈	1	4	yes	—
209	C₂³	C₂×C₅×SD₁₆	1	4	yes	—
210	C₂³	C₂²×C₅×D₄	1	3	yes	—
211	Q₈	—	4	32	no	—
212	D₄	—	2	16	no	—
213	D₄	—	4	24	no	—
214	D₄	—	4	32	no	—
215	D₄	—	4	32	no	—
216	C₂×C₄	—	2	8	no	—
217	C₂×C₄	—	4	24	no	—
218	C₂×C₄	C₅×#124	1	12	yes	—
219	C₂×C₄	C₂×C₅×D₈	1	24	yes	—
220	C₂×C₄	C₂×C₅×SD₁₆	1	24	yes	—
221	C₂×C₄	C₂²×C₅×D₄	1	3	yes	—
222	C₂×C₄	C₂²×C₅×D₄	1	4	yes	—
223	C₂×C₄	C₂²×C₅×D₄	1	4	yes	—
224	C₂×C₄	C₂×C₅×#41	1	12	yes	—
225	C₂×C₄	C₂⁵×C₅	1	1	yes	characteristic
226	C₈	—	2	8	no	—
227	C₅	C₂²×#486	1	1	yes	characteristic, direct product factor, Sylow
228	C₂²	—	4	8	no	—
229	C₂²	—	4	8	no	—
230	C₂²	—	4	8	no	—
231	C₂²	—	8	48	no	—
232	C₂²	C₅×#486	1	16	yes	direct product factor
233	C₂²	C₂×C₅×#124	1	6	yes	—
234	C₂²	C₂²×C₅×D₈	1	6	yes	—
235	C₂²	C₂²×C₅×SD₁₆	1	6	yes	—
236	C₂²	C₂³×C₅×D₄	1	1	yes	characteristic
237	C₄	—	4	8	no	—
238	C₄	C₂×C₅×#124	1	6	yes	—
239	C₄	C₂²×C₅×D₈	1	8	yes	—
240	C₄	C₂²×C₅×SD₁₆	1	8	yes	—
241	C₄	C₂²×C₅×#41	1	2	yes	—
242	C₂	—	8	8	no	—
243	C₂	C₂×C₅×#486	1	12	yes	direct product factor
244	C₂	C₂²×C₅×#124	1	1	yes	characteristic
245	C₂	C₂³×C₅×D₈	1	1	yes	characteristic
246	C₂	C₂³×C₅×SD₁₆	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₅	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅
prime radical	C₂²×C₅×#127	C₂
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×C₄⋊Q₈	C₂	semidirect
3. C₂×C₅×#486	C₂	direct (N × H)
4. C₂²×C₅×#102	C₂	semidirect
5. C₂²×C₄×C₅×C₈	C₂	semidirect
12. C₂×C₅×C₄⋊Q₈	C₂²	semidirect
18. C₅×#486	C₂²	direct (N × H)
19. C₂×C₅×#102	C₂²	semidirect
20. C₂×C₄×C₅×C₈	C₂²	semidirect
21. C₂²×#486	C₅	direct (N × H)
39. C₅×C₄⋊Q₈	C₂³	semidirect
47. C₅×#102	C₂³	semidirect
48. C₄×C₅×C₈	C₂³	semidirect
49. C₂²×C₄⋊Q₈	C₂×C₅	semidirect
51. C₂×#486	C₂×C₅	direct (N × H)
52. C₂²×#102	C₂×C₅	semidirect
53. C₂²×C₄×C₈	C₂×C₅	semidirect
91. C₂×C₄⋊Q₈	C₂²×C₅	semidirect
97. #486	C₂²×C₅	direct (N × H)
98. C₂×#102	C₂²×C₅	semidirect
99. C₂×C₄×C₈	C₂²×C₅	semidirect
142. C₄⋊Q₈	C₂³×C₅	semidirect
150. #102	C₂³×C₅	semidirect
151. C₄×C₈	C₂³×C₅	semidirect
156. C₂²×C₅	#486	direct (N × H)
198. C₂×C₅	C₂×#486	direct (N × H)
227. C₅	C₂²×#486	direct (N × H)
232. C₂²	C₅×#486	direct (N × H)
243. C₂	C₂×C₅×#486	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×C₄⋊Q₈	2
C₂²×C₅×#127	2
C₂×C₅×#486	2
C₂²×C₅×#102	2
C₂²×C₄×C₅×C₈	2
C₂²×#486	5

Subgroups — C22×C5×#486 #11983322

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅×#486 #11983322