Subgroups — C₅×#35285 #11958005

Order 1280.

3206 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₅×#127	C₂	1	1	yes	maximal, characteristic
3	C₅×#2858	C₂	1	4	yes	maximal, complemented
4	C₅×#2641	C₂	1	4	yes	maximal, complemented
5	C₅×#2548	C₂	1	8	yes	maximal, complemented
6	C₅×#2435	C₂	1	8	yes	maximal, complemented
7	C₅×#2309	C₂	1	4	yes	maximal, complemented
8	C₄×C₅×#117	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×C₅×D₄	C₂²	1	2	yes	—
10	C₂³×C₅×D₄	C₂²	1	4	yes	—
11	C₅×#533	—	2	8	no	—
12	C₂×C₅×C₄⋊Q₈	C₂²	1	2	yes	complemented
13	C₂×C₅×#127	—	2	8	no	—
14	C₂×C₅×#127	C₂²	1	4	yes	—
15	C₂×C₅×#127	C₂²	1	2	yes	—
16	C₂×C₅×#127	C₂²	1	2	yes	—
17	C₂×C₅×#125	C₂²	1	2	yes	complemented
18	C₂×C₅×#124	C₂²	1	16	yes	complemented
19	C₂×C₅×C₂²⋊Q₈	C₂²	1	4	yes	—
20	C₂×C₄×C₅×Q₈	C₂²	1	4	yes	complemented
21	C₂×C₅×#117	C₂²	1	2	yes	—
22	C₂×C₅×#117	C₂²	1	2	yes	—
23	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
24	C₅×#394	C₂²	1	32	yes	complemented
25	C₅×#390	C₂²	1	16	yes	complemented
26	C₅×#390	C₂²	1	16	yes	complemented
27	C₅×#390	C₂²	1	16	yes	complemented
28	C₅×#379	C₂²	1	16	yes	—
29	C₄×C₅×C₄⋊C₄	C₂²	1	4	yes	complemented
30	C₄×C₅×C₂²⋊C₄	C₂²	1	4	yes	—
31	C₅×#376	C₂²	1	2	yes	complemented
32	C₄³×C₅	C₂²	1	2	yes	complemented
33	#35285	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₂⁵×C₅	C₂³	1	4	yes	—
35	C₂²×C₅×Q₈	C₂³	1	4	yes	—
36	C₂²×C₅×D₄	—	2	16	no	—
37	C₂²×C₅×D₄	—	2	8	no	—
38	C₂²×C₅×D₄	—	2	16	no	—
39	C₂²×C₅×D₄	—	4	16	no	—
40	C₂²×C₅×D₄	C₂³	1	4	yes	—
41	C₂²×C₅×D₄	C₂³	1	4	yes	—
42	C₂²×C₅×D₄	C₂³	1	8	yes	—
43	C₂²×C₅×D₄	C₂³	1	8	yes	—
44	C₂³×C₄×C₅	C₂³	1	2	yes	—
45	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
46	C₅×C₄⋊Q₈	—	2	8	no	—
47	C₅×#127	—	2	16	no	—
48	C₅×#127	—	2	8	no	—
49	C₅×#127	—	2	8	no	—
50	C₅×#125	—	2	8	no	—
51	C₅×#124	—	4	32	no	—
52	C₅×C₂²⋊Q₈	—	2	16	no	—
53	C₄×C₅×Q₈	—	2	16	no	—
54	C₅×#117	—	2	8	no	—
55	C₅×#117	—	2	8	no	—
56	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
57	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
58	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
59	C₂×C₅×C₄⋊C₄	C₂³	1	8	yes	—
60	C₂×C₅×C₂²⋊C₄	C₂³	1	4	yes	—
61	C₂×C₅×C₂²⋊C₄	C₂³	1	4	yes	—
62	C₂×C₅×C₂²⋊C₄	C₂³	1	64	yes	—
63	C₂×C₄²×C₅	D₄	1	8	yes	—
64	C₂×C₄²×C₅	C₂³	1	4	yes	—
65	C₂×C₄²×C₅	C₂³	1	8	yes	—
66	C₂×C₄²×C₅	C₂³	1	4	yes	—
67	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
68	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
69	C₂×C₄²×C₅	C₂³	1	2	yes	—
70	C₅×#95	C₂³	1	8	yes	—
71	C₂×#533	C₂×C₅	1	1	yes	characteristic, complemented
72	C₂²×#127	C₂×C₅	1	1	yes	characteristic
73	#2858	C₂×C₅	1	4	yes	complemented
74	#2641	C₂×C₅	1	4	yes	complemented
75	#2548	C₂×C₅	1	8	yes	complemented
76	#2435	C₂×C₅	1	8	yes	complemented
77	#2309	C₂×C₅	1	4	yes	complemented
78	C₄×#117	C₂×C₅	1	1	yes	characteristic, complemented
79	C₂⁴×C₅	—	2	8	no	—
80	C₂⁴×C₅	—	2	16	no	—
81	C₂⁴×C₅	—	4	16	no	—
82	C₂⁴×C₅	C₂⁴	1	8	yes	—
83	C₂⁴×C₅	C₂⁴	1	1	yes	characteristic
84	C₂×C₅×Q₈	—	2	8	no	—
85	C₂×C₅×Q₈	—	2	16	no	—
86	C₂×C₅×D₄	—	2	16	no	—
87	C₂×C₅×D₄	—	2	8	no	—
88	C₂×C₅×D₄	—	2	8	no	—
89	C₂×C₅×D₄	—	2	16	no	—
90	C₂×C₅×D₄	—	2	16	no	—
91	C₂×C₅×D₄	—	2	16	no	—
92	C₂×C₅×D₄	—	2	32	no	—
93	C₂×C₅×D₄	—	4	16	no	—
94	C₂×C₅×D₄	—	4	64	no	—
95	C₂×C₅×D₄	—	4	32	no	—
96	C₂²×C₄×C₅	—	2	8	no	—
97	C₂²×C₄×C₅	C₂×D₄	1	8	yes	—
98	C₂²×C₄×C₅	C₂×D₄	1	4	yes	—
99	C₂²×C₄×C₅	#41	1	8	yes	—
100	C₂²×C₄×C₅	C₂⁴	1	8	yes	—
101	C₂²×C₄×C₅	C₂⁴	1	8	yes	—
102	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
103	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
104	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
105	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
106	C₅×C₄⋊C₄	—	2	8	no	—
107	C₅×C₄⋊C₄	—	2	16	no	—
108	C₅×C₄⋊C₄	—	2	8	no	—
109	C₅×C₄⋊C₄	—	2	8	no	—
110	C₅×C₂²⋊C₄	—	2	8	no	—
111	C₅×C₂²⋊C₄	—	2	8	no	—
112	C₅×C₂²⋊C₄	—	4	64	no	—
113	C₄²×C₅	—	2	8	no	—
114	C₄²×C₅	—	2	8	no	—
115	C₄²×C₅	—	2	16	no	—
116	C₄²×C₅	C₂×D₄	1	8	yes	—
117	C₄²×C₅	C₂×D₄	1	4	yes	—
118	C₄²×C₅	C₂×D₄	1	4	yes	—
119	C₂³×D₄	C₂²×C₅	1	2	yes	—
120	C₂³×D₄	C₂²×C₅	1	4	yes	—
121	#533	—	2	8	no	—
122	C₂×C₄⋊Q₈	C₂²×C₅	1	2	yes	complemented
123	C₂×#127	—	2	8	no	—
124	C₂×#127	C₂²×C₅	1	2	yes	—
125	C₂×#127	C₂²×C₅	1	2	yes	—
126	C₂×#127	C₂²×C₅	1	4	yes	—
127	C₂×#125	C₂²×C₅	1	2	yes	complemented
128	C₂×#124	C₂²×C₅	1	16	yes	complemented
129	C₂×C₂²⋊Q₈	C₂²×C₅	1	4	yes	—
130	C₂×C₄×Q₈	C₂²×C₅	1	4	yes	complemented
131	C₂×#117	C₂²×C₅	1	2	yes	—
132	C₂×#117	C₂²×C₅	1	2	yes	—
133	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
134	#394	C₂²×C₅	1	32	yes	complemented
135	#390	C₂²×C₅	1	16	yes	complemented
136	#390	C₂²×C₅	1	16	yes	complemented
137	#390	C₂²×C₅	1	16	yes	complemented
138	#379	C₂²×C₅	1	16	yes	—
139	C₄×C₄⋊C₄	C₂²×C₅	1	4	yes	complemented
140	C₄×C₂²⋊C₄	C₂²×C₅	1	4	yes	—
141	#376	C₂²×C₅	1	2	yes	complemented
142	C₄³	C₂²×C₅	1	2	yes	complemented
143	C₂³×C₅	—	2	4	no	—
144	C₂³×C₅	—	2	8	no	—
145	C₂³×C₅	—	2	16	no	—
146	C₂³×C₅	—	2	32	no	—
147	C₂³×C₅	—	4	32	no	—
148	C₂³×C₅	—	4	16	no	—
149	C₂³×C₅	—	8	32	no	—
150	C₂³×C₅	C₂²×D₄	1	2	yes	—
151	C₂³×C₅	C₂×#41	1	4	yes	—
152	C₂³×C₅	C₂⁵	1	1	yes	characteristic
153	C₅×Q₈	—	4	16	no	—
154	C₅×D₄	—	4	32	no	—
155	C₅×D₄	—	4	16	no	—
156	C₅×D₄	—	4	16	no	—
157	C₅×D₄	—	4	32	no	—
158	C₂×C₄×C₅	—	2	4	no	—
159	C₂×C₄×C₅	—	2	16	no	—
160	C₂×C₄×C₅	—	2	8	no	—
161	C₂×C₄×C₅	—	2	8	no	—
162	C₂×C₄×C₅	—	2	16	no	—
163	C₂×C₄×C₅	—	2	16	no	—
164	C₂×C₄×C₅	#124	1	16	yes	complemented
165	C₂×C₄×C₅	C₂²×D₄	1	4	yes	—
166	C₂×C₄×C₅	C₂²×D₄	1	2	yes	—
167	C₂×C₄×C₅	C₂²×D₄	1	2	yes	—
168	C₂×C₄×C₅	C₂²×D₄	1	4	yes	—
169	C₂×C₄×C₅	C₂×#41	1	4	yes	—
170	C₂×C₄×C₅	C₂×#41	1	4	yes	—
171	C₂×C₄×C₅	H(5,𝔽₂)	1	8	yes	—
172	C₂⁵	C₂³×C₅	1	4	yes	—
173	C₂²×Q₈	C₂³×C₅	1	4	yes	—
174	C₂²×D₄	—	2	16	no	—
175	C₂²×D₄	—	2	8	no	—
176	C₂²×D₄	—	2	16	no	—
177	C₂²×D₄	—	4	16	no	—
178	C₂²×D₄	C₂³×C₅	1	8	yes	—
179	C₂²×D₄	C₂³×C₅	1	4	yes	—
180	C₂²×D₄	C₂³×C₅	1	4	yes	—
181	C₂²×D₄	C₂³×C₅	1	8	yes	—
182	C₂³×C₄	C₂³×C₅	1	2	yes	—
183	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
184	C₄⋊Q₈	—	2	8	no	—
185	#127	—	2	16	no	—
186	#127	—	2	8	no	—
187	#127	—	2	8	no	—
188	#125	—	2	8	no	—
189	#124	—	4	32	no	—
190	C₂²⋊Q₈	—	2	16	no	—
191	C₄×Q₈	—	2	16	no	—
192	#117	—	2	8	no	—
193	#117	—	2	8	no	—
194	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
195	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
196	C₂×C₄⋊C₄	C₂³×C₅	1	8	yes	—
197	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
198	C₂×C₂²⋊C₄	C₂³×C₅	1	4	yes	—
199	C₂×C₂²⋊C₄	C₂³×C₅	1	4	yes	—
200	C₂×C₂²⋊C₄	C₂³×C₅	1	64	yes	—
201	C₂×C₄²	C₅×D₄	1	8	yes	—
202	C₂×C₄²	C₂³×C₅	1	4	yes	—
203	C₂×C₄²	C₂³×C₅	1	4	yes	—
204	C₂×C₄²	C₂³×C₅	1	8	yes	—
205	C₂×C₄²	C₂³×C₅	1	2	yes	—
206	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
207	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
208	#95	C₂³×C₅	1	8	yes	—
209	C₂²×C₅	—	2	4	no	—
210	C₂²×C₅	—	2	8	no	—
211	C₂²×C₅	—	4	8	no	—
212	C₂²×C₅	—	4	16	no	—
213	C₂²×C₅	—	4	32	no	—
214	C₂²×C₅	—	8	32	no	—
215	C₂²×C₅	C₂×#124	1	4	yes	—
216	C₂²×C₅	C₂³×D₄	1	1	yes	characteristic
217	C₂²×C₅	C₂²×#41	1	2	yes	—
218	C₂²×C₅	C₂×H(5,𝔽₂)	1	4	yes	—
219	C₄×C₅	—	2	4	no	—
220	C₄×C₅	—	4	8	no	—
221	C₄×C₅	—	4	8	no	—
222	C₄×C₅	C₂×#124	1	4	yes	—
223	C₄×C₅	C₂×#124	1	4	yes	—
224	C₄×C₅	#535	1	8	yes	—
225	C₂⁴	—	2	8	no	—
226	C₂⁴	—	2	16	no	—
227	C₂⁴	—	4	16	no	—
228	C₂⁴	C₂⁴×C₅	1	8	yes	—
229	C₂⁴	C₂⁴×C₅	1	1	yes	characteristic
230	C₂×Q₈	—	2	8	no	—
231	C₂×Q₈	—	2	16	no	—
232	C₂×D₄	—	2	16	no	—
233	C₂×D₄	—	2	8	no	—
234	C₂×D₄	—	2	8	no	—
235	C₂×D₄	—	2	16	no	—
236	C₂×D₄	—	2	16	no	—
237	C₂×D₄	—	2	16	no	—
238	C₂×D₄	—	2	32	no	—
239	C₂×D₄	—	4	16	no	—
240	C₂×D₄	—	4	64	no	—
241	C₂×D₄	—	4	32	no	—
242	C₂²×C₄	—	2	8	no	—
243	C₂²×C₄	C₂×C₅×D₄	1	8	yes	—
244	C₂²×C₄	C₂×C₅×D₄	1	4	yes	—
245	C₂²×C₄	C₅×#41	1	8	yes	—
246	C₂²×C₄	C₂⁴×C₅	1	8	yes	—
247	C₂²×C₄	C₂⁴×C₅	1	8	yes	—
248	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
249	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
250	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
251	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
252	C₄⋊C₄	—	2	8	no	—
253	C₄⋊C₄	—	2	16	no	—
254	C₄⋊C₄	—	2	8	no	—
255	C₄⋊C₄	—	2	8	no	—
256	C₂²⋊C₄	—	2	8	no	—
257	C₂²⋊C₄	—	2	8	no	—
258	C₂²⋊C₄	—	4	64	no	—
259	C₄²	—	2	8	no	—
260	C₄²	—	2	8	no	—
261	C₄²	—	2	16	no	—
262	C₄²	C₂×C₅×D₄	1	8	yes	—
263	C₄²	C₂×C₅×D₄	1	4	yes	—
264	C₄²	C₂×C₅×D₄	1	4	yes	—
265	C₂×C₅	—	2	4	no	—
266	C₂×C₅	—	8	8	no	—
267	C₂×C₅	C₂²×#124	1	1	yes	characteristic
268	C₂×C₅	C₂×#535	1	2	yes	—
269	C₂×C₅	#3508	1	4	yes	—
270	C₂³	—	2	4	no	—
271	C₂³	—	2	8	no	—
272	C₂³	—	2	16	no	—
273	C₂³	—	2	32	no	—
274	C₂³	—	4	32	no	—
275	C₂³	—	4	16	no	—
276	C₂³	—	8	32	no	—
277	C₂³	C₂²×C₅×D₄	1	2	yes	—
278	C₂³	C₂×C₅×#41	1	4	yes	—
279	C₂³	C₂⁵×C₅	1	1	yes	characteristic
280	Q₈	—	4	16	no	—
281	D₄	—	4	32	no	—
282	D₄	—	4	16	no	—
283	D₄	—	4	16	no	—
284	D₄	—	4	32	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	16	no	—
287	C₂×C₄	—	2	8	no	—
288	C₂×C₄	—	2	8	no	—
289	C₂×C₄	—	2	16	no	—
290	C₂×C₄	—	2	16	no	—
291	C₂×C₄	C₅×#124	1	16	yes	complemented
292	C₂×C₄	C₂²×C₅×D₄	1	4	yes	—
293	C₂×C₄	C₂²×C₅×D₄	1	2	yes	—
294	C₂×C₄	C₂²×C₅×D₄	1	2	yes	—
295	C₂×C₄	C₂²×C₅×D₄	1	4	yes	—
296	C₂×C₄	C₂×C₅×#41	1	4	yes	—
297	C₂×C₄	C₂×C₅×#41	1	4	yes	—
298	C₂×C₄	C₅×H(5,𝔽₂)	1	8	yes	—
299	C₅	#35285	1	1	yes	characteristic, direct product factor, Sylow
300	C₂²	—	2	4	no	—
301	C₂²	—	2	8	no	—
302	C₂²	—	4	8	no	—
303	C₂²	—	4	16	no	—
304	C₂²	—	4	32	no	—
305	C₂²	—	8	32	no	—
306	C₂²	C₂×C₅×#124	1	4	yes	—
307	C₂²	C₂³×C₅×D₄	1	1	yes	characteristic
308	C₂²	C₂²×C₅×#41	1	2	yes	—
309	C₂²	C₂×C₅×H(5,𝔽₂)	1	4	yes	—
310	C₄	—	2	4	no	—
311	C₄	—	4	8	no	—
312	C₄	—	4	8	no	—
313	C₄	C₂×C₅×#124	1	4	yes	—
314	C₄	C₂×C₅×#124	1	4	yes	—
315	C₄	C₅×#535	1	8	yes	—
316	C₂	—	2	4	no	—
317	C₂	—	8	8	no	—
318	C₂	C₂²×C₅×#124	1	1	yes	characteristic
319	C₂	C₂×C₅×#535	1	2	yes	—
320	C₂	C₅×#3508	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₅	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₅
prime radical	C₂²×C₅×#127	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×#533	C₂	semidirect
3. C₅×#2858	C₂	semidirect
4. C₅×#2641	C₂	semidirect
5. C₅×#2548	C₂	semidirect
6. C₅×#2435	C₂	semidirect
7. C₅×#2309	C₂	semidirect
8. C₄×C₅×#117	C₂	semidirect
12. C₂×C₅×C₄⋊Q₈	C₂²	semidirect
17. C₂×C₅×#125	C₂²	semidirect
18. C₂×C₅×#124	C₂²	semidirect
20. C₂×C₄×C₅×Q₈	C₂²	semidirect
24. C₅×#394	C₂²	semidirect
25. C₅×#390	C₂²	semidirect
26. C₅×#390	C₂²	semidirect
27. C₅×#390	C₂²	semidirect
29. C₄×C₅×C₄⋊C₄	C₂²	semidirect
31. C₅×#376	C₂²	semidirect
32. C₄³×C₅	C₂²	semidirect
33. #35285	C₅	direct (N × H)
71. C₂×#533	C₂×C₅	semidirect
73. #2858	C₂×C₅	semidirect
74. #2641	C₂×C₅	semidirect
75. #2548	C₂×C₅	semidirect
76. #2435	C₂×C₅	semidirect
77. #2309	C₂×C₅	semidirect
78. C₄×#117	C₂×C₅	semidirect
122. C₂×C₄⋊Q₈	C₂²×C₅	semidirect
127. C₂×#125	C₂²×C₅	semidirect
128. C₂×#124	C₂²×C₅	semidirect
130. C₂×C₄×Q₈	C₂²×C₅	semidirect
134. #394	C₂²×C₅	semidirect
135. #390	C₂²×C₅	semidirect
136. #390	C₂²×C₅	semidirect
137. #390	C₂²×C₅	semidirect
139. C₄×C₄⋊C₄	C₂²×C₅	semidirect
141. #376	C₂²×C₅	semidirect
142. C₄³	C₂²×C₅	semidirect
164. C₂×C₄×C₅	#124	semidirect
291. C₂×C₄	C₅×#124	semidirect
299. C₅	#35285	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×#533	2
C₂²×C₅×#127	2
C₅×#2858	2
C₅×#2641	2
C₅×#2548	2
C₅×#2435	2
C₅×#2309	2
C₄×C₅×#117	2
#35285	5

Subgroups — C5×#35285 #11958005

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#35285 #11958005