Subgroups — C₅×#35322 #11958042

Order 1280.

2074 conjugacy classes of subgroups.

The subgroup lattice has 300 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	2	yes	maximal
2	C₅×#2868	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#2862	C₂	1	2	yes	maximal, complemented
4	C₅×#2862	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#2621	C₂	1	4	yes	maximal
6	C₅×#2565	C₂	1	8	yes	maximal, complemented
7	C₅×#2499	C₂	1	8	yes	maximal
8	C₅×#2327	C₂	1	4	yes	maximal, complemented
9	C₄×C₅×#117	C₂	1	1	yes	maximal, characteristic
10	C₅×#533	—	2	16	no	—
11	C₂×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
12	C₂×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
13	C₂×C₅×C₄⋊Q₈	C₂²	1	4	yes	—
14	C₂×C₅×#125	C₂²	1	4	yes	—
15	C₂×C₅×C₂²⋊Q₈	C₂²	1	4	yes	—
16	C₂×C₅×C₂²⋊Q₈	C₂²	1	4	yes	—
17	C₂×C₄×C₅×Q₈	C₂²	1	4	yes	—
18	C₂×C₄×C₅×Q₈	C₂²	1	4	yes	—
19	C₂×C₅×#117	C₂²	1	4	yes	—
20	C₂×C₅×#117	C₂²	1	2	yes	—
21	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
22	C₅×#398	C₂²	1	16	yes	—
23	C₅×#397	C₂²	1	16	yes	—
24	C₅×#391	C₂²	1	8	yes	—
25	C₅×#391	C₂²	1	8	yes	—
26	C₅×#391	C₂²	1	8	yes	—
27	C₅×#387	C₂²	1	32	yes	—
28	C₅×#384	C₂²	1	8	yes	—
29	C₅×#382	C₂²	1	8	yes	—
30	C₅×#382	C₂²	1	4	yes	—
31	C₄×C₅×C₄⋊C₄	C₂²	1	4	yes	—
32	C₄×C₅×C₂²⋊C₄	C₂²	1	4	yes	—
33	C₅×#376	C₂²	1	2	yes	—
34	C₄³×C₅	C₂²	1	1	yes	characteristic
35	C₄³×C₅	C₂²	1	1	yes	characteristic
36	#35322	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
37	C₂²×C₅×Q₈	C₂³	1	4	yes	—
38	C₂²×C₅×Q₈	C₂³	1	4	yes	—
39	C₂³×C₄×C₅	C₂³	1	2	yes	—
40	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
41	C₅×C₄⋊Q₈	—	2	8	no	—
42	C₅×C₄⋊Q₈	—	2	8	no	—
43	C₅×C₄⋊Q₈	—	2	16	no	—
44	C₅×#125	—	2	16	no	—
45	C₅×C₂²⋊Q₈	—	2	16	no	—
46	C₅×C₂²⋊Q₈	—	2	16	no	—
47	C₄×C₅×Q₈	—	2	16	no	—
48	C₄×C₅×Q₈	—	2	16	no	—
49	C₅×#117	—	2	16	no	—
50	C₅×#117	—	2	8	no	—
51	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
52	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
53	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
54	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
55	C₂×C₅×C₄⋊C₄	C₂³	1	16	yes	—
56	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
57	C₂×C₅×C₄⋊C₄	C₂³	1	8	yes	—
58	C₂×C₅×C₄⋊C₄	C₂³	1	8	yes	—
59	C₂×C₅×C₂²⋊C₄	C₂³	1	4	yes	—
60	C₂×C₅×C₂²⋊C₄	C₂³	1	8	yes	—
61	C₂×C₄²×C₅	D₄	1	8	yes	—
62	C₂×C₄²×C₅	C₂³	1	2	yes	—
63	C₂×C₄²×C₅	C₂³	1	4	yes	—
64	C₂×C₄²×C₅	C₂³	1	4	yes	—
65	C₂×C₄²×C₅	C₂³	1	8	yes	—
66	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
67	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
68	C₂×C₄²×C₅	C₂³	1	2	yes	—
69	C₂×C₄²×C₅	C₂³	1	2	yes	—
70	C₅×#95	C₂³	1	8	yes	—
71	C₅×#95	C₂³	1	32	yes	—
72	C₅×#95	C₂³	1	16	yes	—
73	C₂×#533	C₂×C₅	1	2	yes	—
74	#2868	C₂×C₅	1	1	yes	characteristic, complemented
75	#2862	C₂×C₅	1	1	yes	characteristic, complemented
76	#2862	C₂×C₅	1	2	yes	complemented
77	#2621	C₂×C₅	1	4	yes	—
78	#2565	C₂×C₅	1	8	yes	complemented
79	#2499	C₂×C₅	1	8	yes	—
80	#2327	C₂×C₅	1	4	yes	complemented
81	C₄×#117	C₂×C₅	1	1	yes	characteristic
82	C₂⁴×C₅	C₂⁴	1	1	yes	characteristic
83	C₂×C₅×Q₈	—	2	8	no	—
84	C₂×C₅×Q₈	—	2	8	no	—
85	C₂×C₅×Q₈	—	2	16	no	—
86	C₂×C₅×Q₈	—	2	16	no	—
87	C₂²×C₄×C₅	—	2	4	no	—
88	C₂²×C₄×C₅	—	2	8	no	—
89	C₂²×C₄×C₅	C₂×D₄	1	8	yes	—
90	C₂²×C₄×C₅	C₂×D₄	1	4	yes	—
91	C₂²×C₄×C₅	C₂⁴	1	4	yes	—
92	C₂²×C₄×C₅	C₂⁴	1	8	yes	—
93	C₂²×C₄×C₅	C₂⁴	1	8	yes	—
94	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
95	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
96	C₂²×C₄×C₅	C₂⁴	1	4	yes	—
97	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
98	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
99	C₅×C₄⋊C₄	—	2	8	no	—
100	C₅×C₄⋊C₄	—	2	8	no	—
101	C₅×C₄⋊C₄	—	2	16	no	—
102	C₅×C₄⋊C₄	—	2	16	no	—
103	C₅×C₄⋊C₄	—	2	8	no	—
104	C₅×C₄⋊C₄	—	2	8	no	—
105	C₅×C₄⋊C₄	—	2	8	no	—
106	C₅×C₄⋊C₄	—	2	32	no	—
107	C₅×C₂²⋊C₄	—	2	16	no	—
108	C₅×C₂²⋊C₄	—	2	8	no	—
109	C₄²×C₅	—	2	16	no	—
110	C₄²×C₅	—	2	4	no	—
111	C₄²×C₅	—	2	8	no	—
112	C₄²×C₅	—	2	8	no	—
113	C₄²×C₅	C₂×D₄	1	4	yes	—
114	C₄²×C₅	C₂×D₄	1	8	yes	—
115	C₄²×C₅	C₂×D₄	1	4	yes	—
116	C₄²×C₅	#41	1	8	yes	—
117	#533	—	2	16	no	—
118	C₂×C₄⋊Q₈	C₂²×C₅	1	4	yes	—
119	C₂×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
120	C₂×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
121	C₂×#125	C₂²×C₅	1	4	yes	—
122	C₂×C₂²⋊Q₈	C₂²×C₅	1	4	yes	—
123	C₂×C₂²⋊Q₈	C₂²×C₅	1	4	yes	—
124	C₂×C₄×Q₈	C₂²×C₅	1	4	yes	—
125	C₂×C₄×Q₈	C₂²×C₅	1	4	yes	—
126	C₂×#117	C₂²×C₅	1	2	yes	—
127	C₂×#117	C₂²×C₅	1	4	yes	—
128	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
129	#398	C₂²×C₅	1	16	yes	—
130	#397	C₂²×C₅	1	16	yes	—
131	#391	C₂²×C₅	1	8	yes	—
132	#391	C₂²×C₅	1	8	yes	—
133	#391	C₂²×C₅	1	8	yes	—
134	#387	C₂²×C₅	1	32	yes	—
135	#384	C₂²×C₅	1	8	yes	—
136	#382	C₂²×C₅	1	8	yes	—
137	#382	C₂²×C₅	1	4	yes	—
138	C₄×C₄⋊C₄	C₂²×C₅	1	4	yes	—
139	C₄×C₂²⋊C₄	C₂²×C₅	1	4	yes	—
140	#376	C₂²×C₅	1	2	yes	—
141	C₄³	C₂²×C₅	1	1	yes	characteristic
142	C₄³	C₂²×C₅	1	1	yes	characteristic
143	C₂³×C₅	—	2	2	no	—
144	C₂³×C₅	—	2	4	no	—
145	C₂³×C₅	C₂²×D₄	1	2	yes	—
146	C₂³×C₅	C₂⁵	1	1	yes	characteristic
147	C₅×Q₈	—	4	16	no	—
148	C₅×Q₈	—	4	16	no	—
149	C₂×C₄×C₅	—	2	2	no	—
150	C₂×C₄×C₅	—	2	8	no	—
151	C₂×C₄×C₅	—	2	4	no	—
152	C₂×C₄×C₅	—	2	8	no	—
153	C₂×C₄×C₅	—	2	16	no	—
154	C₂×C₄×C₅	—	2	8	no	—
155	C₂×C₄×C₅	—	2	8	no	—
156	C₂×C₄×C₅	—	2	4	no	—
157	C₂×C₄×C₅	—	2	16	no	—
158	C₂×C₄×C₅	—	2	8	no	—
159	C₂×C₄×C₅	—	2	16	no	—
160	C₂×C₄×C₅	C₂²×D₄	1	4	yes	—
161	C₂×C₄×C₅	C₂²×D₄	1	2	yes	—
162	C₂×C₄×C₅	C₂²×D₄	1	2	yes	—
163	C₂×C₄×C₅	C₂²×D₄	1	4	yes	—
164	C₂×C₄×C₅	C₂×#41	1	8	yes	—
165	C₂×C₄×C₅	C₂×#41	1	4	yes	—
166	C₂×C₄×C₅	2₋¹⁺⁴	1	8	yes	—
167	C₂²×Q₈	C₂³×C₅	1	4	yes	—
168	C₂²×Q₈	C₂³×C₅	1	4	yes	—
169	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
170	C₂³×C₄	C₂³×C₅	1	2	yes	—
171	C₄⋊Q₈	—	2	8	no	—
172	C₄⋊Q₈	—	2	8	no	—
173	C₄⋊Q₈	—	2	16	no	—
174	#125	—	2	16	no	—
175	C₂²⋊Q₈	—	2	16	no	—
176	C₂²⋊Q₈	—	2	16	no	—
177	C₄×Q₈	—	2	16	no	—
178	C₄×Q₈	—	2	16	no	—
179	#117	—	2	16	no	—
180	#117	—	2	8	no	—
181	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
182	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
183	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
184	C₂×C₄⋊C₄	C₂³×C₅	1	8	yes	—
185	C₂×C₄⋊C₄	C₂³×C₅	1	8	yes	—
186	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
187	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
188	C₂×C₄⋊C₄	C₂³×C₅	1	16	yes	—
189	C₂×C₂²⋊C₄	C₂³×C₅	1	8	yes	—
190	C₂×C₂²⋊C₄	C₂³×C₅	1	4	yes	—
191	C₂×C₄²	C₅×D₄	1	8	yes	—
192	C₂×C₄²	C₂³×C₅	1	8	yes	—
193	C₂×C₄²	C₂³×C₅	1	2	yes	—
194	C₂×C₄²	C₂³×C₅	1	4	yes	—
195	C₂×C₄²	C₂³×C₅	1	4	yes	—
196	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
197	C₂×C₄²	C₂³×C₅	1	2	yes	—
198	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
199	C₂×C₄²	C₂³×C₅	1	2	yes	—
200	#95	C₂³×C₅	1	8	yes	—
201	#95	C₂³×C₅	1	32	yes	—
202	#95	C₂³×C₅	1	16	yes	—
203	C₂²×C₅	—	2	2	no	—
204	C₂²×C₅	—	2	4	no	—
205	C₂²×C₅	—	4	4	no	—
206	C₂²×C₅	C₂³×D₄	1	1	yes	characteristic
207	C₂²×C₅	C₂²×#41	1	2	yes	—
208	C₂²×C₅	C₂×2₋¹⁺⁴	1	4	yes	—
209	C₄×C₅	—	2	2	no	—
210	C₄×C₅	—	2	4	no	—
211	C₄×C₅	—	2	8	no	—
212	C₄×C₅	—	4	8	no	—
213	C₄×C₅	—	4	8	no	—
214	C₄×C₅	—	4	4	no	—
215	C₄×C₅	#532	1	4	yes	—
216	C₄×C₅	#536	1	8	yes	—
217	C₂⁴	C₂⁴×C₅	1	1	yes	characteristic
218	C₂×Q₈	—	2	8	no	—
219	C₂×Q₈	—	2	8	no	—
220	C₂×Q₈	—	2	16	no	—
221	C₂×Q₈	—	2	16	no	—
222	C₂²×C₄	—	2	4	no	—
223	C₂²×C₄	—	2	8	no	—
224	C₂²×C₄	C₂×C₅×D₄	1	8	yes	—
225	C₂²×C₄	C₂×C₅×D₄	1	4	yes	—
226	C₂²×C₄	C₂⁴×C₅	1	4	yes	—
227	C₂²×C₄	C₂⁴×C₅	1	8	yes	—
228	C₂²×C₄	C₂⁴×C₅	1	8	yes	—
229	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
230	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
231	C₂²×C₄	C₂⁴×C₅	1	4	yes	—
232	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
233	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
234	C₄⋊C₄	—	2	8	no	—
235	C₄⋊C₄	—	2	8	no	—
236	C₄⋊C₄	—	2	16	no	—
237	C₄⋊C₄	—	2	16	no	—
238	C₄⋊C₄	—	2	8	no	—
239	C₄⋊C₄	—	2	8	no	—
240	C₄⋊C₄	—	2	8	no	—
241	C₄⋊C₄	—	2	32	no	—
242	C₂²⋊C₄	—	2	16	no	—
243	C₂²⋊C₄	—	2	8	no	—
244	C₄²	—	2	16	no	—
245	C₄²	—	2	4	no	—
246	C₄²	—	2	8	no	—
247	C₄²	—	2	8	no	—
248	C₄²	C₂×C₅×D₄	1	4	yes	—
249	C₄²	C₂×C₅×D₄	1	8	yes	—
250	C₄²	C₂×C₅×D₄	1	4	yes	—
251	C₄²	C₅×#41	1	8	yes	—
252	C₂×C₅	—	4	2	no	—
253	C₂×C₅	C₂×#532	1	1	yes	characteristic
254	C₂×C₅	C₂×#536	1	2	yes	—
255	C₂×C₅	#3509	1	4	yes	—
256	C₂³	—	2	2	no	—
257	C₂³	—	2	4	no	—
258	C₂³	C₂²×C₅×D₄	1	2	yes	—
259	C₂³	C₂⁵×C₅	1	1	yes	characteristic
260	Q₈	—	4	16	no	—
261	Q₈	—	4	16	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	8	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	8	no	—
266	C₂×C₄	—	2	16	no	—
267	C₂×C₄	—	2	8	no	—
268	C₂×C₄	—	2	8	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	16	no	—
271	C₂×C₄	—	2	8	no	—
272	C₂×C₄	—	2	16	no	—
273	C₂×C₄	C₂²×C₅×D₄	1	4	yes	—
274	C₂×C₄	C₂²×C₅×D₄	1	2	yes	—
275	C₂×C₄	C₂²×C₅×D₄	1	2	yes	—
276	C₂×C₄	C₂²×C₅×D₄	1	4	yes	—
277	C₂×C₄	C₂×C₅×#41	1	8	yes	—
278	C₂×C₄	C₂×C₅×#41	1	4	yes	—
279	C₂×C₄	C₅×2₋¹⁺⁴	1	8	yes	—
280	C₅	#35322	1	1	yes	characteristic, direct product factor, Sylow
281	C₂²	—	2	2	no	—
282	C₂²	—	2	4	no	—
283	C₂²	—	4	4	no	—
284	C₂²	C₂³×C₅×D₄	1	1	yes	characteristic
285	C₂²	C₂²×C₅×#41	1	2	yes	—
286	C₂²	C₂×C₅×2₋¹⁺⁴	1	4	yes	—
287	C₄	—	2	2	no	—
288	C₄	—	2	4	no	—
289	C₄	—	2	8	no	—
290	C₄	—	4	8	no	—
291	C₄	—	4	8	no	—
292	C₄	—	4	4	no	—
293	C₄	C₅×#532	1	4	yes	—
294	C₄	C₅×#536	1	8	yes	—
295	C₂	—	4	2	no	—
296	C₂	C₂×C₅×#532	1	1	yes	characteristic
297	C₂	C₂×C₅×#536	1	2	yes	—
298	C₂	C₅×#3509	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₅	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₅
prime radical	C₂⁴×C₅	C₂⁴
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
2. C₅×#2868	C₂	semidirect
3. C₅×#2862	C₂	semidirect
4. C₅×#2862	C₂	semidirect
6. C₅×#2565	C₂	semidirect
8. C₅×#2327	C₂	semidirect
36. #35322	C₅	direct (N × H)
74. #2868	C₂×C₅	semidirect
75. #2862	C₂×C₅	semidirect
76. #2862	C₂×C₅	semidirect
78. #2565	C₂×C₅	semidirect
80. #2327	C₂×C₅	semidirect
280. C₅	#35322	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
C₅×#2868	2
C₅×#2862	2
C₅×#2621	2
C₅×#2565	2
C₅×#2499	2
C₅×#2327	2
C₄×C₅×#117	2
#35322	5

Subgroups — C5×#35322 #11958042

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#35322 #11958042