Subgroups — C₅×#33583 #11956303

Order 1280.

702 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#510	C₂	1	2	yes	maximal
2	C₂×C₅×#494	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₅×#135	C₂	1	1	yes	maximal, characteristic
4	C₅×#2209	C₂	1	4	yes	maximal, complemented
5	C₅×#2208	C₂	1	4	yes	maximal, complemented
6	C₅×#2150	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#2141	C₂	1	2	yes	maximal, complemented
8	C₂×C₅×#135	C₂²	1	1	yes	characteristic
9	C₂²×C₅×Q₁₆	—	2	1	no	—
10	C₂²×C₅×Q₁₆	C₂²	1	2	yes	—
11	C₂×C₅×C₄⋊Q₈	—	2	1	no	—
12	C₄×C₅×#41	—	2	1	no	—
13	C₅×#510	—	2	8	no	—
14	C₂×C₅×Q₃₂	C₂²	1	4	yes	—
15	C₂×C₅×SD₃₂	C₂²	1	4	yes	—
16	C₂×C₅×M₃₂	C₂²	1	2	yes	—
17	C₅×#494	—	2	2	no	—
18	C₅×#494	C₂²	1	4	yes	—
19	C₄×C₅×Q₁₆	C₂²	1	2	yes	—
20	C₄×C₅×SD₁₆	—	2	2	no	—
21	C₄×C₅×D₈	C₂²	1	2	yes	—
22	C₅×#427	C₂²	1	1	yes	characteristic
23	C₅×#416	—	2	1	no	—
24	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
25	C₅×C₄⋊C₁₆	C₂²	1	4	yes	—
26	C₅×#358	C₂²	1	8	yes	—
27	#33583	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
28	C₂×C₅×#41	—	2	1	no	—
29	C₂²×C₅×Q₈	—	2	2	no	—
30	C₅×#135	—	2	2	no	—
31	C₅×#135	D₄	1	4	yes	—
32	C₂×C₅×Q₁₆	—	2	4	no	—
33	C₂×C₅×Q₁₆	—	2	2	no	—
34	C₂×C₅×Q₁₆	—	2	4	no	—
35	C₂×C₅×Q₁₆	—	2	4	no	—
36	C₂×C₅×Q₁₆	—	4	4	no	—
37	C₂×C₅×Q₁₆	C₂³	1	4	yes	—
38	C₂×C₅×Q₁₆	C₂³	1	1	yes	characteristic
39	C₂×C₅×SD₁₆	—	2	1	no	—
40	C₂×C₅×D₈	C₂³	1	1	yes	characteristic
41	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
42	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
43	C₅×C₄⋊Q₈	—	2	4	no	—
44	C₅×C₄⋊Q₈	—	4	1	no	—
45	C₅×C₄⋊Q₈	—	4	1	no	—
46	C₄×C₅×Q₈	—	2	2	no	—
47	C₄×C₅×D₄	—	2	2	no	—
48	C₄×C₅×D₄	—	4	2	no	—
49	C₅×#117	—	2	1	no	—
50	C₅×#117	—	4	1	no	—
51	C₂×C₅×C₄⋊C₄	—	4	2	no	—
52	C₂×C₄²×C₅	—	4	1	no	—
53	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
54	C₅×Q₃₂	—	2	8	no	—
55	C₅×SD₃₂	—	2	8	no	—
56	C₅×M₃₂	—	2	4	no	—
57	C₂×C₅×C₁₆	C₂³	1	4	yes	—
58	C₅×#107	C₂³	1	2	yes	—
59	C₅×#106	—	2	1	no	—
60	C₅×#103	—	2	2	no	—
61	C₅×#102	—	2	2	no	—
62	C₄×C₅×C₈	D₄	1	2	yes	—
63	C₄×C₅×C₈	C₂³	1	2	yes	—
64	C₂×#510	C₂×C₅	1	2	yes	—
65	C₂×#494	C₂×C₅	1	1	yes	characteristic, complemented
66	C₄×#135	C₂×C₅	1	1	yes	characteristic
67	#2209	C₂×C₅	1	4	yes	complemented
68	#2208	C₂×C₅	1	4	yes	complemented
69	#2150	C₂×C₅	1	1	yes	characteristic, complemented
70	#2141	C₂×C₅	1	2	yes	complemented
71	C₅×#41	—	2	4	no	—
72	C₅×#41	—	4	2	no	—
73	C₂×C₅×Q₈	—	2	1	no	—
74	C₂×C₅×Q₈	—	2	4	no	—
75	C₂×C₅×Q₈	—	4	2	no	—
76	C₂×C₅×Q₈	—	4	4	no	—
77	C₂×C₅×Q₈	—	4	4	no	—
78	C₂×C₅×D₄	—	2	1	no	—
79	C₂×C₅×D₄	—	4	1	no	—
80	C₂²×C₄×C₅	—	4	1	no	—
81	C₂²×C₄×C₅	—	4	2	no	—
82	C₂²×C₄×C₅	—	4	2	no	—
83	C₂²×C₄×C₅	D₈	1	2	yes	—
84	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
85	C₅×Q₁₆	—	2	4	no	—
86	C₅×Q₁₆	—	2	1	no	—
87	C₅×Q₁₆	—	2	4	no	—
88	C₅×Q₁₆	—	4	8	no	—
89	C₅×Q₁₆	—	4	4	no	—
90	C₅×Q₁₆	C₂×D₄	1	2	yes	—
91	C₅×SD₁₆	—	2	4	no	—
92	C₅×D₈	—	2	1	no	—
93	C₅×D₈	C₂×D₄	1	2	yes	—
94	C₂×C₅×C₈	—	2	2	no	—
95	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
96	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
97	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
98	C₂×C₅×C₈	#41	1	2	yes	—
99	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
100	C₅×C₄⋊C₄	—	2	2	no	—
101	C₅×C₄⋊C₄	—	4	4	no	—
102	C₅×C₄⋊C₄	—	4	4	no	—
103	C₅×C₄⋊C₄	—	4	2	no	—
104	C₅×C₂²⋊C₄	—	4	1	no	—
105	C₅×C₂²⋊C₄	—	4	2	no	—
106	C₄²×C₅	—	4	1	no	—
107	C₄²×C₅	—	4	2	no	—
108	C₄²×C₅	D₈	1	1	yes	characteristic
109	C₄²×C₅	D₈	1	1	yes	characteristic
110	C₄²×C₅	C₂×D₄	1	2	yes	—
111	C₅×C₁₆	—	2	4	no	—
112	C₂×#135	C₂²×C₅	1	1	yes	characteristic
113	C₂²×Q₁₆	—	2	1	no	—
114	C₂²×Q₁₆	C₂²×C₅	1	2	yes	—
115	C₂×C₄⋊Q₈	—	2	1	no	—
116	C₄×#41	—	2	1	no	—
117	#510	—	2	8	no	—
118	C₂×Q₃₂	C₂²×C₅	1	4	yes	—
119	C₂×SD₃₂	C₂²×C₅	1	4	yes	—
120	C₂×M₃₂	C₂²×C₅	1	2	yes	—
121	#494	—	2	2	no	—
122	#494	C₂²×C₅	1	4	yes	—
123	C₄×Q₁₆	C₂²×C₅	1	2	yes	—
124	C₄×SD₁₆	—	2	2	no	—
125	C₄×D₈	C₂²×C₅	1	2	yes	—
126	#427	C₂²×C₅	1	1	yes	characteristic
127	#416	—	2	1	no	—
128	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
129	C₄⋊C₁₆	C₂²×C₅	1	4	yes	—
130	#358	C₂²×C₅	1	8	yes	—
131	C₂³×C₅	—	4	1	no	—
132	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
133	C₅×Q₈	—	2	2	no	—
134	C₅×Q₈	—	4	1	no	—
135	C₅×Q₈	—	4	4	no	—
136	C₅×Q₈	—	4	4	no	—
137	C₅×Q₈	—	8	4	no	—
138	C₅×D₄	—	2	2	no	—
139	C₅×D₄	—	4	4	no	—
140	C₅×D₄	—	4	1	no	—
141	C₂×C₄×C₅	—	2	4	no	—
142	C₂×C₄×C₅	—	4	1	no	—
143	C₂×C₄×C₅	—	4	4	no	—
144	C₂×C₄×C₅	—	4	2	no	—
145	C₂×C₄×C₅	—	4	4	no	—
146	C₂×C₄×C₅	—	8	4	no	—
147	C₂×C₄×C₅	—	8	4	no	—
148	C₂×C₄×C₅	#121	1	2	yes	—
149	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
150	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
151	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
152	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
153	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
154	C₅×C₈	—	2	2	no	—
155	C₅×C₈	#121	1	2	yes	—
156	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
157	C₅×C₈	C₂×#41	1	1	yes	characteristic
158	C₂×#41	—	2	1	no	—
159	C₂²×Q₈	—	2	2	no	—
160	#135	—	2	2	no	—
161	#135	C₅×D₄	1	4	yes	—
162	C₂×Q₁₆	—	2	4	no	—
163	C₂×Q₁₆	—	2	4	no	—
164	C₂×Q₁₆	—	2	2	no	—
165	C₂×Q₁₆	—	2	4	no	—
166	C₂×Q₁₆	—	4	4	no	—
167	C₂×Q₁₆	C₂³×C₅	1	4	yes	—
168	C₂×Q₁₆	C₂³×C₅	1	1	yes	characteristic
169	C₂×SD₁₆	—	2	1	no	—
170	C₂×D₈	C₂³×C₅	1	1	yes	characteristic
171	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
172	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
173	C₄⋊Q₈	—	2	4	no	—
174	C₄⋊Q₈	—	4	1	no	—
175	C₄⋊Q₈	—	4	1	no	—
176	C₄×Q₈	—	2	2	no	—
177	C₄×D₄	—	2	2	no	—
178	C₄×D₄	—	4	2	no	—
179	#117	—	2	1	no	—
180	#117	—	4	1	no	—
181	C₂×C₄⋊C₄	—	4	2	no	—
182	C₂×C₄²	—	4	1	no	—
183	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
184	Q₃₂	—	2	8	no	—
185	SD₃₂	—	2	8	no	—
186	M₃₂	—	2	4	no	—
187	C₂×C₁₆	C₂³×C₅	1	4	yes	—
188	#107	C₂³×C₅	1	2	yes	—
189	#106	—	2	1	no	—
190	#103	—	2	2	no	—
191	#102	—	2	2	no	—
192	C₄×C₈	C₅×D₄	1	2	yes	—
193	C₄×C₈	C₂³×C₅	1	2	yes	—
194	C₂²×C₅	—	2	2	no	—
195	C₂²×C₅	—	4	2	no	—
196	C₂²×C₅	—	8	2	no	—
197	C₂²×C₅	#459	1	2	yes	—
198	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
199	C₄×C₅	—	2	2	no	—
200	C₄×C₅	—	4	2	no	—
201	C₄×C₅	—	8	2	no	—
202	C₄×C₅	—	8	4	no	—
203	C₄×C₅	#459	1	2	yes	—
204	C₄×C₅	#510	1	4	yes	complemented
205	C₄×C₅	C₂×#121	1	1	yes	characteristic
206	C₄×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
207	#41	—	2	4	no	—
208	#41	—	4	2	no	—
209	C₂×Q₈	—	2	1	no	—
210	C₂×Q₈	—	2	4	no	—
211	C₂×Q₈	—	4	2	no	—
212	C₂×Q₈	—	4	4	no	—
213	C₂×Q₈	—	4	4	no	—
214	C₂×D₄	—	2	1	no	—
215	C₂×D₄	—	4	1	no	—
216	C₂²×C₄	—	4	1	no	—
217	C₂²×C₄	—	4	2	no	—
218	C₂²×C₄	—	4	2	no	—
219	C₂²×C₄	C₅×D₈	1	2	yes	—
220	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
221	Q₁₆	—	2	4	no	—
222	Q₁₆	—	2	1	no	—
223	Q₁₆	—	2	4	no	—
224	Q₁₆	—	4	8	no	—
225	Q₁₆	—	4	4	no	—
226	Q₁₆	C₂×C₅×D₄	1	2	yes	—
227	SD₁₆	—	2	4	no	—
228	D₈	—	2	1	no	—
229	D₈	C₂×C₅×D₄	1	2	yes	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
232	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
233	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
234	C₂×C₈	C₅×#41	1	2	yes	—
235	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	4	4	no	—
238	C₄⋊C₄	—	4	4	no	—
239	C₄⋊C₄	—	4	2	no	—
240	C₂²⋊C₄	—	4	1	no	—
241	C₂²⋊C₄	—	4	2	no	—
242	C₄²	—	4	1	no	—
243	C₄²	—	4	2	no	—
244	C₄²	C₅×D₈	1	1	yes	characteristic
245	C₄²	C₅×D₈	1	1	yes	characteristic
246	C₄²	C₂×C₅×D₄	1	2	yes	—
247	C₁₆	—	2	4	no	—
248	C₂×C₅	—	2	2	no	—
249	C₂×C₅	—	8	2	no	—
250	C₂×C₅	C₂×#459	1	1	yes	characteristic
251	C₂×C₅	C₂×#510	1	2	yes	—
252	C₂³	—	4	1	no	—
253	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
254	Q₈	—	2	2	no	—
255	Q₈	—	4	1	no	—
256	Q₈	—	4	4	no	—
257	Q₈	—	4	4	no	—
258	Q₈	—	8	4	no	—
259	D₄	—	2	2	no	—
260	D₄	—	4	4	no	—
261	D₄	—	4	1	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	4	1	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	4	4	no	—
267	C₂×C₄	—	8	4	no	—
268	C₂×C₄	—	8	4	no	—
269	C₂×C₄	C₅×#121	1	2	yes	—
270	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
271	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
272	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
273	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
274	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
275	C₈	—	2	2	no	—
276	C₈	C₅×#121	1	2	yes	—
277	C₈	C₂²×C₅×D₄	1	1	yes	characteristic
278	C₈	C₂×C₅×#41	1	1	yes	characteristic
279	C₅	#33583	1	1	yes	characteristic, direct product factor, Sylow
280	C₂²	—	2	2	no	—
281	C₂²	—	4	2	no	—
282	C₂²	—	8	2	no	—
283	C₂²	C₅×#459	1	2	yes	—
284	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
285	C₄	—	2	2	no	—
286	C₄	—	4	2	no	—
287	C₄	—	8	4	no	—
288	C₄	—	8	2	no	—
289	C₄	C₅×#459	1	2	yes	—
290	C₄	C₅×#510	1	4	yes	complemented
291	C₄	C₂×C₅×#121	1	1	yes	characteristic
292	C₄	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
293	C₂	—	2	2	no	—
294	C₂	—	8	2	no	—
295	C₂	C₂×C₅×#459	1	1	yes	characteristic
296	C₂	C₂×C₅×#510	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅
prime radical	C₂×C₅×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#494	C₂	semidirect
4. C₅×#2209	C₂	semidirect
5. C₅×#2208	C₂	semidirect
6. C₅×#2150	C₂	semidirect
7. C₅×#2141	C₂	semidirect
27. #33583	C₅	direct (N × H)
65. C₂×#494	C₂×C₅	semidirect
67. #2209	C₂×C₅	semidirect
68. #2208	C₂×C₅	semidirect
69. #2150	C₂×C₅	semidirect
70. #2141	C₂×C₅	semidirect
204. C₄×C₅	#510	semidirect
279. C₅	#33583	direct (N × H)
290. C₄	C₅×#510	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#510	2
C₂×C₅×#494	2
C₄×C₅×#135	2
C₅×#2209	2
C₅×#2208	2
C₅×#2150	2
C₅×#2141	2
#33583	5

Subgroups — C5×#33583 #11956303

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33583 #11956303