Subgroups — C₅×#8624 #11931344

Order 1280.

418 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#436	C₂	1	1	yes	maximal, characteristic
2	C₅×#1568	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#1568	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1466	C₂	1	2	yes	maximal
5	C₅×#1290	C₂	1	2	yes	maximal, complemented
6	C₂²×C₅×M₁₆	C₄	1	2	yes	—
7	C₂×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
8	C₅×#436	—	2	1	no	—
9	C₅×#436	—	2	1	no	—
10	C₅×#436	—	2	1	no	—
11	C₅×#436	—	2	1	no	—
12	C₅×#432	C₂²	1	2	yes	—
13	C₅×#432	C₂²	1	2	yes	—
14	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₅×C₂²⋊C₈	C₄	1	2	yes	—
16	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
17	C₅×#335	—	2	1	no	—
18	C₅×#335	—	2	1	no	—
19	C₅×#334	—	2	1	no	—
20	C₅×#334	—	2	1	no	—
21	C₅×#323	—	2	4	no	—
22	#8624	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
23	C₂²×C₅×D₄	Q₈	1	1	yes	characteristic
24	C₂³×C₄×C₅	C₂×C₄	1	2	yes	—
25	C₂×C₅×M₁₆	—	2	2	no	—
26	C₂×C₅×M₁₆	—	2	2	no	—
27	C₂×C₅×M₁₆	—	4	4	no	—
28	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
30	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
31	C₄×C₅×D₄	—	2	1	no	—
32	C₄×C₅×D₄	—	2	1	no	—
33	C₄×C₅×D₄	—	2	1	no	—
34	C₄×C₅×D₄	—	2	1	no	—
35	C₂×C₅×C₄⋊C₄	Q₈	1	1	yes	characteristic
36	C₂×C₅×C₂²⋊C₄	D₄	1	2	yes	—
37	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
38	C₅×C₄⋊C₈	—	2	1	no	—
39	C₅×C₄⋊C₈	—	2	1	no	—
40	C₅×C₄⋊C₈	—	2	2	no	—
41	C₅×C₄⋊C₈	—	2	2	no	—
42	C₅×C₂²⋊C₈	—	2	2	no	—
43	C₅×C₂²⋊C₈	—	2	2	no	—
44	C₅×C₂²⋊C₈	—	2	2	no	—
45	C₅×C₂²⋊C₈	—	2	2	no	—
46	C₅×#97	—	2	2	no	—
47	C₅×#97	—	2	2	no	—
48	C₄×C₅×C₈	D₄	1	1	yes	characteristic
49	C₄×C₅×C₈	D₄	1	1	yes	characteristic
50	C₄×C₅×C₈	D₄	1	1	yes	characteristic
51	C₄×C₅×C₈	D₄	1	1	yes	characteristic
52	C₂×#436	C₂×C₅	1	1	yes	characteristic
53	#1568	C₂×C₅	1	1	yes	characteristic, complemented
54	#1568	C₂×C₅	1	1	yes	characteristic, complemented
55	#1466	C₂×C₅	1	2	yes	—
56	#1290	C₂×C₅	1	2	yes	complemented
57	C₂⁴×C₅	C₄⋊C₄	1	2	yes	—
58	C₂×C₅×D₄	—	2	1	no	—
59	C₂×C₅×D₄	—	2	1	no	—
60	C₂×C₅×D₄	—	4	2	no	—
61	C₂²×C₄×C₅	—	2	2	no	—
62	C₂²×C₄×C₅	—	2	2	no	—
63	C₂²×C₄×C₅	—	4	2	no	—
64	C₂²×C₄×C₅	—	4	2	no	—
65	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
66	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
69	C₅×M₁₆	—	4	4	no	—
70	C₅×M₁₆	—	4	4	no	—
71	C₂×C₅×C₈	—	2	2	no	—
72	C₂×C₅×C₈	—	2	2	no	—
73	C₂×C₅×C₈	—	4	1	no	—
74	C₂×C₅×C₈	—	4	2	no	—
75	C₂×C₅×C₈	—	4	2	no	—
76	C₂×C₅×C₈	—	4	2	no	—
77	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
81	C₅×C₄⋊C₄	—	2	1	no	—
82	C₅×C₄⋊C₄	—	2	1	no	—
83	C₅×C₂²⋊C₄	—	2	2	no	—
84	C₅×C₂²⋊C₄	—	2	2	no	—
85	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₅	#41	1	1	yes	characteristic
88	C₄²×C₅	#41	1	1	yes	characteristic
89	C₂²×M₁₆	C₄×C₅	1	2	yes	—
90	C₂×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
91	#436	—	2	1	no	—
92	#436	—	2	1	no	—
93	#436	—	2	1	no	—
94	#436	—	2	1	no	—
95	#432	C₂²×C₅	1	2	yes	—
96	#432	C₂²×C₅	1	2	yes	—
97	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
98	C₂×C₂²⋊C₈	C₄×C₅	1	2	yes	—
99	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
100	#335	—	2	1	no	—
101	#335	—	2	1	no	—
102	#334	—	2	1	no	—
103	#334	—	2	1	no	—
104	#323	—	2	4	no	—
105	C₂³×C₅	—	2	2	no	—
106	C₂³×C₅	—	2	4	no	—
107	C₂³×C₅	—	4	2	no	—
108	C₂³×C₅	—	4	2	no	—
109	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₅×D₄	—	4	2	no	—
111	C₅×D₄	—	4	2	no	—
112	C₂×C₄×C₅	—	2	2	no	—
113	C₂×C₄×C₅	—	2	1	no	—
114	C₂×C₄×C₅	—	2	1	no	—
115	C₂×C₄×C₅	—	2	2	no	—
116	C₂×C₄×C₅	—	4	1	no	—
117	C₂×C₄×C₅	—	4	4	no	—
118	C₂×C₄×C₅	—	4	2	no	—
119	C₂×C₄×C₅	—	4	2	no	—
120	C₂×C₄×C₅	—	4	2	no	—
121	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₅	#117	1	1	yes	characteristic
123	C₂×C₄×C₅	#121	1	1	yes	characteristic
124	C₂×C₄×C₅	#121	1	1	yes	characteristic
125	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
126	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
127	C₅×C₈	—	2	1	no	—
128	C₅×C₈	—	2	1	no	—
129	C₅×C₈	—	2	1	no	—
130	C₅×C₈	—	4	2	no	—
131	C₅×C₈	—	8	2	no	—
132	C₅×C₈	—	8	2	no	—
133	C₅×C₈	#100	1	2	yes	—
134	C₂²×D₄	C₅×Q₈	1	1	yes	characteristic
135	C₂³×C₄	C₂×C₄×C₅	1	2	yes	—
136	C₂×M₁₆	—	2	2	no	—
137	C₂×M₁₆	—	2	2	no	—
138	C₂×M₁₆	—	4	4	no	—
139	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
140	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
141	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
142	C₄×D₄	—	2	1	no	—
143	C₄×D₄	—	2	1	no	—
144	C₄×D₄	—	2	1	no	—
145	C₄×D₄	—	2	1	no	—
146	C₂×C₄⋊C₄	C₅×Q₈	1	1	yes	characteristic
147	C₂×C₂²⋊C₄	C₅×D₄	1	2	yes	—
148	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
149	C₄⋊C₈	—	2	1	no	—
150	C₄⋊C₈	—	2	1	no	—
151	C₄⋊C₈	—	2	2	no	—
152	C₄⋊C₈	—	2	2	no	—
153	C₂²⋊C₈	—	2	2	no	—
154	C₂²⋊C₈	—	2	2	no	—
155	C₂²⋊C₈	—	2	2	no	—
156	C₂²⋊C₈	—	2	2	no	—
157	#97	—	2	2	no	—
158	#97	—	2	2	no	—
159	C₄×C₈	C₅×D₄	1	1	yes	characteristic
160	C₄×C₈	C₅×D₄	1	1	yes	characteristic
161	C₄×C₈	C₅×D₄	1	1	yes	characteristic
162	C₄×C₈	C₅×D₄	1	1	yes	characteristic
163	C₂²×C₅	—	2	1	no	—
164	C₂²×C₅	—	2	1	no	—
165	C₂²×C₅	—	4	2	no	—
166	C₂²×C₅	—	4	2	no	—
167	C₂²×C₅	—	4	2	no	—
168	C₂²×C₅	—	8	4	no	—
169	C₂²×C₅	#380	1	1	yes	characteristic
170	C₂²×C₅	#430	1	2	yes	—
171	C₄×C₅	—	2	1	no	—
172	C₄×C₅	—	2	1	no	—
173	C₄×C₅	—	2	1	no	—
174	C₄×C₅	—	2	1	no	—
175	C₄×C₅	—	2	1	no	—
176	C₄×C₅	—	4	2	no	—
177	C₄×C₅	C₂×#100	1	1	yes	characteristic
178	C₄×C₅	#413	1	1	yes	characteristic
179	C₂⁴	C₅×C₄⋊C₄	1	2	yes	—
180	C₂×D₄	—	2	1	no	—
181	C₂×D₄	—	2	1	no	—
182	C₂×D₄	—	4	2	no	—
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	—	2	2	no	—
185	C₂²×C₄	—	4	2	no	—
186	C₂²×C₄	—	4	2	no	—
187	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
188	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
189	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
190	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
191	M₁₆	—	4	4	no	—
192	M₁₆	—	4	4	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	4	1	no	—
196	C₂×C₈	—	4	2	no	—
197	C₂×C₈	—	4	2	no	—
198	C₂×C₈	—	4	2	no	—
199	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
200	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
201	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
202	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
203	C₄⋊C₄	—	2	1	no	—
204	C₄⋊C₄	—	2	1	no	—
205	C₂²⋊C₄	—	2	2	no	—
206	C₂²⋊C₄	—	2	2	no	—
207	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
208	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
209	C₄²	C₅×#41	1	1	yes	characteristic
210	C₄²	C₅×#41	1	1	yes	characteristic
211	C₂×C₅	—	2	2	no	—
212	C₂×C₅	—	8	2	no	—
213	C₂×C₅	#1800	1	1	yes	characteristic
214	C₂×C₅	#1811	1	1	yes	characteristic
215	C₂×C₅	#1812	1	1	yes	characteristic
216	C₂³	—	2	2	no	—
217	C₂³	—	2	4	no	—
218	C₂³	—	4	2	no	—
219	C₂³	—	4	2	no	—
220	C₂³	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
221	D₄	—	4	2	no	—
222	D₄	—	4	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	4	1	no	—
228	C₂×C₄	—	4	4	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
233	C₂×C₄	C₅×#117	1	1	yes	characteristic
234	C₂×C₄	C₅×#121	1	1	yes	characteristic
235	C₂×C₄	C₅×#121	1	1	yes	characteristic
236	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
237	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
238	C₈	—	2	1	no	—
239	C₈	—	2	1	no	—
240	C₈	—	2	1	no	—
241	C₈	—	4	2	no	—
242	C₈	—	8	2	no	—
243	C₈	—	8	2	no	—
244	C₈	C₅×#100	1	2	yes	—
245	C₅	#8624	1	1	yes	characteristic, direct product factor, Sylow
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	—	4	2	no	—
249	C₂²	—	4	2	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	8	4	no	—
252	C₂²	C₅×#380	1	1	yes	characteristic
253	C₂²	C₅×#430	1	2	yes	—
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	2	1	no	—
258	C₄	—	2	1	no	—
259	C₄	—	4	2	no	—
260	C₄	C₂×C₅×#100	1	1	yes	characteristic
261	C₄	C₅×#413	1	1	yes	characteristic
262	C₂	—	2	2	no	—
263	C₂	—	8	2	no	—
264	C₂	C₅×#1800	1	1	yes	characteristic
265	C₂	C₅×#1811	1	1	yes	characteristic
266	C₂	C₅×#1812	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	#380
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₂²×C₅×D₄	Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#1568	C₂	semidirect
3. C₅×#1568	C₂	semidirect
5. C₅×#1290	C₂	semidirect
22. #8624	C₅	direct (N × H)
53. #1568	C₂×C₅	semidirect
54. #1568	C₂×C₅	semidirect
56. #1290	C₂×C₅	semidirect
245. C₅	#8624	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅×#436	2
C₅×#1568	2
C₅×#1466	2
C₅×#1290	2
#8624	5

Subgroups — C5×#8624 #11931344

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8624 #11931344