Subgroups — C₂²×C₅×#495 #11983351

Order 1280.

3518 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	C₂³×C₅×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₅×SD₁₆	C₂	1	2	yes	maximal, complemented
3	C₂³×C₅×D₈	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₅×#124	C₂	1	2	yes	maximal, complemented
5	C₂×C₅×#495	C₂	1	24	yes	maximal, direct product factor
6	C₂²×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×C₅×Q₈	C₂²	1	2	yes	—
8	C₂³×C₅×D₄	C₂²	1	2	yes	—
9	C₂²×C₅×Q₁₆	—	2	8	no	—
10	C₂²×C₅×Q₁₆	C₂²	1	12	yes	complemented
11	C₂²×C₅×SD₁₆	—	2	16	no	—
12	C₂²×C₅×SD₁₆	C₂²	1	24	yes	complemented
13	C₂²×C₅×D₈	—	2	8	no	—
14	C₂²×C₅×D₈	C₂²	1	12	yes	—
15	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
16	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
17	C₂×C₅×#124	—	2	12	no	—
18	C₂×C₅×#124	C₂²	1	24	yes	complemented
19	C₂²×C₅×C₂²⋊C₄	—	2	4	no	—
20	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
21	C₅×#495	C₂²	1	64	yes	direct product factor
22	C₂×C₄×C₅×C₈	C₂²	1	12	yes	complemented
23	C₂²×#495	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂⁵×C₅	—	2	2	no	—
25	C₂²×C₅×Q₈	—	2	6	no	—
26	C₂²×C₅×Q₈	—	2	8	no	—
27	C₂²×C₅×Q₈	—	2	8	no	—
28	C₂²×C₅×Q₈	C₂³	1	12	yes	—
29	C₂²×C₅×D₄	—	2	8	no	—
30	C₂²×C₅×D₄	—	2	8	no	—
31	C₂²×C₅×D₄	—	2	6	no	—
32	C₂²×C₅×D₄	C₂³	1	12	yes	—
33	C₂³×C₄×C₅	—	2	2	no	—
34	C₂³×C₄×C₅	D₄	1	2	yes	—
35	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
36	C₂×C₅×Q₁₆	—	2	48	no	—
37	C₂×C₅×Q₁₆	C₂³	1	16	yes	complemented
38	C₂×C₅×SD₁₆	—	2	96	no	—
39	C₂×C₅×SD₁₆	C₂³	1	32	yes	complemented
40	C₂×C₅×D₈	—	2	48	no	—
41	C₂×C₅×D₈	C₂³	1	16	yes	—
42	C₂²×C₅×C₈	D₄	1	8	yes	—
43	C₂²×C₅×C₈	D₄	1	8	yes	—
44	C₂²×C₅×C₈	C₂³	1	6	yes	—
45	C₂²×C₅×C₈	C₂³	1	6	yes	—
46	C₅×#124	—	2	48	no	—
47	C₅×#124	C₂³	1	32	yes	complemented
48	C₂×C₅×C₂²⋊C₄	—	2	48	no	—
49	C₂×C₄²×C₅	D₄	1	6	yes	—
50	C₂×C₄²×C₅	C₂³	1	6	yes	—
51	C₄×C₅×C₈	C₂³	1	16	yes	complemented
52	C₂³×Q₁₆	C₂×C₅	1	1	yes	characteristic, complemented
53	C₂³×SD₁₆	C₂×C₅	1	2	yes	complemented
54	C₂³×D₈	C₂×C₅	1	1	yes	characteristic
55	C₂²×#124	C₂×C₅	1	2	yes	complemented
56	C₂×#495	C₂×C₅	1	24	yes	direct product factor
57	C₂²×C₄×C₈	C₂×C₅	1	1	yes	characteristic, complemented
58	C₂⁴×C₅	—	2	12	no	—
59	C₂⁴×C₅	—	4	8	no	—
60	C₂⁴×C₅	—	4	16	no	—
61	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
62	C₂×C₅×Q₈	—	2	12	no	—
63	C₂×C₅×Q₈	—	2	24	no	—
64	C₂×C₅×Q₈	—	2	24	no	—
65	C₂×C₅×Q₈	—	4	24	no	—
66	C₂×C₅×Q₈	C₂⁴	1	8	yes	—
67	C₂×C₅×D₄	—	2	24	no	—
68	C₂×C₅×D₄	—	2	24	no	—
69	C₂×C₅×D₄	—	2	12	no	—
70	C₂×C₅×D₄	—	4	24	no	—
71	C₂×C₅×D₄	C₂⁴	1	8	yes	—
72	C₂²×C₄×C₅	—	2	8	no	—
73	C₂²×C₄×C₅	—	2	12	no	—
74	C₂²×C₄×C₅	—	4	8	no	—
75	C₂²×C₄×C₅	C₂×D₄	1	12	yes	—
76	C₂²×C₄×C₅	C₂×D₄	1	3	yes	—
77	C₂²×C₄×C₅	C₂×D₄	1	4	yes	—
78	C₂²×C₄×C₅	C₂×D₄	1	4	yes	—
79	C₂²×C₄×C₅	C₂⁴	1	3	yes	—
80	C₅×Q₁₆	—	2	32	no	—
81	C₅×SD₁₆	—	2	64	no	—
82	C₅×D₈	—	2	32	no	—
83	C₂×C₅×C₈	C₂×D₄	1	24	yes	—
84	C₂×C₅×C₈	C₂×D₄	1	24	yes	—
85	C₂×C₅×C₈	C₂⁴	1	4	yes	—
86	C₂×C₅×C₈	C₂⁴	1	4	yes	—
87	C₅×C₂²⋊C₄	—	2	64	no	—
88	C₄²×C₅	C₂×D₄	1	12	yes	—
89	C₄²×C₅	C₂⁴	1	4	yes	—
90	C₂³×Q₈	C₂²×C₅	1	2	yes	—
91	C₂³×D₄	C₂²×C₅	1	2	yes	—
92	C₂²×Q₁₆	—	2	8	no	—
93	C₂²×Q₁₆	C₂²×C₅	1	12	yes	complemented
94	C₂²×SD₁₆	—	2	16	no	—
95	C₂²×SD₁₆	C₂²×C₅	1	24	yes	complemented
96	C₂²×D₈	—	2	8	no	—
97	C₂²×D₈	C₂²×C₅	1	12	yes	—
98	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
99	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
100	C₂×#124	—	2	12	no	—
101	C₂×#124	C₂²×C₅	1	24	yes	complemented
102	C₂²×C₂²⋊C₄	—	2	4	no	—
103	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
104	#495	C₂²×C₅	1	64	yes	direct product factor
105	C₂×C₄×C₈	C₂²×C₅	1	12	yes	complemented
106	C₂³×C₅	—	2	8	no	—
107	C₂³×C₅	—	4	24	no	—
108	C₂³×C₅	—	4	48	no	—
109	C₂³×C₅	—	8	32	no	—
110	C₂³×C₅	#127	1	4	yes	—
111	C₂³×C₅	#135	1	8	yes	—
112	C₂³×C₅	C₂²×D₄	1	3	yes	—
113	C₅×Q₈	—	2	8	no	—
114	C₅×Q₈	—	2	8	no	—
115	C₅×Q₈	—	4	24	no	—
116	C₅×D₄	—	2	8	no	—
117	C₅×D₄	—	2	8	no	—
118	C₅×D₄	—	4	24	no	—
119	C₂×C₄×C₅	—	2	24	no	—
120	C₂×C₄×C₅	—	2	8	no	—
121	C₂×C₄×C₅	—	4	24	no	—
122	C₂×C₄×C₅	#127	1	12	yes	—
123	C₂×C₄×C₅	C₂²×D₄	1	8	yes	—
124	C₂×C₄×C₅	C₂²×D₄	1	3	yes	—
125	C₂×C₄×C₅	C₂²×D₄	1	6	yes	—
126	C₂×C₄×C₅	C₂²×D₄	1	6	yes	—
127	C₂×C₄×C₅	C₂⁵	1	1	yes	characteristic
128	C₅×C₈	C₂²×D₄	1	8	yes	—
129	C₅×C₈	C₂²×D₄	1	8	yes	—
130	C₂⁵	—	2	2	no	—
131	C₂²×Q₈	—	2	6	no	—
132	C₂²×Q₈	—	2	8	no	—
133	C₂²×Q₈	—	2	8	no	—
134	C₂²×Q₈	C₂³×C₅	1	12	yes	—
135	C₂²×D₄	—	2	8	no	—
136	C₂²×D₄	—	2	8	no	—
137	C₂²×D₄	—	2	6	no	—
138	C₂²×D₄	C₂³×C₅	1	12	yes	—
139	C₂³×C₄	—	2	2	no	—
140	C₂³×C₄	C₅×D₄	1	2	yes	—
141	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
142	C₂×Q₁₆	—	2	48	no	—
143	C₂×Q₁₆	C₂³×C₅	1	16	yes	complemented
144	C₂×SD₁₆	—	2	96	no	—
145	C₂×SD₁₆	C₂³×C₅	1	32	yes	complemented
146	C₂×D₈	—	2	48	no	—
147	C₂×D₈	C₂³×C₅	1	16	yes	—
148	C₂²×C₈	C₅×D₄	1	8	yes	—
149	C₂²×C₈	C₅×D₄	1	8	yes	—
150	C₂²×C₈	C₂³×C₅	1	6	yes	—
151	C₂²×C₈	C₂³×C₅	1	6	yes	—
152	#124	—	2	48	no	—
153	#124	C₂³×C₅	1	32	yes	complemented
154	C₂×C₂²⋊C₄	—	2	48	no	—
155	C₂×C₄²	C₅×D₄	1	6	yes	—
156	C₂×C₄²	C₂³×C₅	1	6	yes	—
157	C₄×C₈	C₂³×C₅	1	16	yes	complemented
158	C₂²×C₅	—	4	8	no	—
159	C₂²×C₅	—	4	16	no	—
160	C₂²×C₅	—	8	48	no	—
161	C₂²×C₅	#495	1	16	yes	direct product factor
162	C₂²×C₅	C₂×#127	1	6	yes	—
163	C₂²×C₅	C₂×#135	1	12	yes	—
164	C₂²×C₅	C₂³×D₄	1	1	yes	characteristic
165	C₄×C₅	—	2	8	no	—
166	C₄×C₅	—	4	8	no	—
167	C₄×C₅	C₂×#127	1	6	yes	—
168	C₄×C₅	C₂³×D₄	1	1	yes	characteristic
169	C₄×C₅	C₂³×D₄	1	1	yes	characteristic
170	C₂⁴	—	2	12	no	—
171	C₂⁴	—	4	8	no	—
172	C₂⁴	—	4	16	no	—
173	C₂⁴	C₂×C₅×D₄	1	1	yes	characteristic
174	C₂×Q₈	—	2	12	no	—
175	C₂×Q₈	—	2	24	no	—
176	C₂×Q₈	—	2	24	no	—
177	C₂×Q₈	—	4	24	no	—
178	C₂×Q₈	C₂⁴×C₅	1	8	yes	—
179	C₂×D₄	—	2	24	no	—
180	C₂×D₄	—	2	24	no	—
181	C₂×D₄	—	2	12	no	—
182	C₂×D₄	—	4	24	no	—
183	C₂×D₄	C₂⁴×C₅	1	8	yes	—
184	C₂²×C₄	—	2	8	no	—
185	C₂²×C₄	—	2	12	no	—
186	C₂²×C₄	—	4	8	no	—
187	C₂²×C₄	C₂×C₅×D₄	1	12	yes	—
188	C₂²×C₄	C₂×C₅×D₄	1	3	yes	—
189	C₂²×C₄	C₂×C₅×D₄	1	4	yes	—
190	C₂²×C₄	C₂×C₅×D₄	1	4	yes	—
191	C₂²×C₄	C₂⁴×C₅	1	3	yes	—
192	Q₁₆	—	2	32	no	—
193	SD₁₆	—	2	64	no	—
194	D₈	—	2	32	no	—
195	C₂×C₈	C₂×C₅×D₄	1	24	yes	—
196	C₂×C₈	C₂×C₅×D₄	1	24	yes	—
197	C₂×C₈	C₂⁴×C₅	1	4	yes	—
198	C₂×C₈	C₂⁴×C₅	1	4	yes	—
199	C₂²⋊C₄	—	2	64	no	—
200	C₄²	C₂×C₅×D₄	1	12	yes	—
201	C₄²	C₂⁴×C₅	1	4	yes	—
202	C₂×C₅	—	8	8	no	—
203	C₂×C₅	C₂×#495	1	12	yes	direct product factor
204	C₂×C₅	C₂²×#127	1	1	yes	characteristic
205	C₂×C₅	C₂²×#135	1	2	yes	—
206	C₂³	—	2	8	no	—
207	C₂³	—	4	24	no	—
208	C₂³	—	4	48	no	—
209	C₂³	—	8	32	no	—
210	C₂³	C₅×#127	1	4	yes	—
211	C₂³	C₅×#135	1	8	yes	—
212	C₂³	C₂²×C₅×D₄	1	3	yes	—
213	Q₈	—	2	8	no	—
214	Q₈	—	2	8	no	—
215	Q₈	—	4	24	no	—
216	D₄	—	2	8	no	—
217	D₄	—	2	8	no	—
218	D₄	—	4	24	no	—
219	C₂×C₄	—	2	24	no	—
220	C₂×C₄	—	2	8	no	—
221	C₂×C₄	—	4	24	no	—
222	C₂×C₄	C₅×#127	1	12	yes	—
223	C₂×C₄	C₂²×C₅×D₄	1	8	yes	—
224	C₂×C₄	C₂²×C₅×D₄	1	3	yes	—
225	C₂×C₄	C₂²×C₅×D₄	1	6	yes	—
226	C₂×C₄	C₂²×C₅×D₄	1	6	yes	—
227	C₂×C₄	C₂⁵×C₅	1	1	yes	characteristic
228	C₈	C₂²×C₅×D₄	1	8	yes	—
229	C₈	C₂²×C₅×D₄	1	8	yes	—
230	C₅	C₂²×#495	1	1	yes	characteristic, direct product factor, Sylow
231	C₂²	—	4	8	no	—
232	C₂²	—	4	16	no	—
233	C₂²	—	8	48	no	—
234	C₂²	C₅×#495	1	16	yes	direct product factor
235	C₂²	C₂×C₅×#127	1	6	yes	—
236	C₂²	C₂×C₅×#135	1	12	yes	—
237	C₂²	C₂³×C₅×D₄	1	1	yes	characteristic
238	C₄	—	2	8	no	—
239	C₄	—	4	8	no	—
240	C₄	C₂×C₅×#127	1	6	yes	—
241	C₄	C₂³×C₅×D₄	1	1	yes	characteristic
242	C₄	C₂³×C₅×D₄	1	1	yes	characteristic
243	C₂	—	8	8	no	—
244	C₂	C₂×C₅×#495	1	12	yes	direct product factor
245	C₂	C₂²×C₅×#127	1	1	yes	characteristic
246	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×D₈	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. C₂³×C₅×Q₁₆	C₂	semidirect
2. C₂³×C₅×SD₁₆	C₂	semidirect
4. C₂²×C₅×#124	C₂	semidirect
5. C₂×C₅×#495	C₂	direct (N × H)
6. C₂²×C₄×C₅×C₈	C₂	semidirect
10. C₂²×C₅×Q₁₆	C₂²	semidirect
12. C₂²×C₅×SD₁₆	C₂²	semidirect
18. C₂×C₅×#124	C₂²	semidirect
21. C₅×#495	C₂²	direct (N × H)
22. C₂×C₄×C₅×C₈	C₂²	semidirect
23. C₂²×#495	C₅	direct (N × H)
37. C₂×C₅×Q₁₆	C₂³	semidirect
39. C₂×C₅×SD₁₆	C₂³	semidirect
47. C₅×#124	C₂³	semidirect
51. C₄×C₅×C₈	C₂³	semidirect
52. C₂³×Q₁₆	C₂×C₅	semidirect
53. C₂³×SD₁₆	C₂×C₅	semidirect
55. C₂²×#124	C₂×C₅	semidirect
56. C₂×#495	C₂×C₅	direct (N × H)
57. C₂²×C₄×C₈	C₂×C₅	semidirect
93. C₂²×Q₁₆	C₂²×C₅	semidirect
95. C₂²×SD₁₆	C₂²×C₅	semidirect
101. C₂×#124	C₂²×C₅	semidirect
104. #495	C₂²×C₅	direct (N × H)
105. C₂×C₄×C₈	C₂²×C₅	semidirect
143. C₂×Q₁₆	C₂³×C₅	semidirect
145. C₂×SD₁₆	C₂³×C₅	semidirect
153. #124	C₂³×C₅	semidirect
157. C₄×C₈	C₂³×C₅	semidirect
161. C₂²×C₅	#495	direct (N × H)
203. C₂×C₅	C₂×#495	direct (N × H)
230. C₅	C₂²×#495	direct (N × H)
234. C₂²	C₅×#495	direct (N × H)
244. C₂	C₂×C₅×#495	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₅×Q₁₆	2
C₂³×C₅×SD₁₆	2
C₂³×C₅×D₈	2
C₂²×C₅×#124	2
C₂×C₅×#495	2
C₂²×C₄×C₅×C₈	2
C₂²×#495	5

Subgroups — C22×C5×#495 #11983351

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅×#495 #11983351