Subgroups — C₅×#7902 #11930622

Order 1280.

686 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#432	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#431	C₂	1	1	yes	maximal, characteristic
3	C₄×C₅×#117	C₂	1	1	yes	maximal, characteristic
4	C₅×#1279	C₂	1	4	yes	maximal, complemented
5	C₂×C₅×#117	C₄	1	2	yes	—
6	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
7	C₅×#432	—	2	2	no	—
8	C₅×#432	C₄	1	4	yes	complemented
9	C₅×#431	—	2	2	no	—
10	C₅×#431	C₄	1	4	yes	complemented
11	C₂×C₅×C₄⋊C₈	—	2	1	no	—
12	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₅×C₂²⋊C₈	—	2	1	no	—
14	C₂×C₅×C₂²⋊C₈	—	2	1	no	—
15	C₂×C₅×#97	—	2	1	no	—
16	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
17	C₄×C₅×C₄⋊C₄	C₄	1	2	yes	—
18	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
19	C₄×C₅×C₂²⋊C₄	—	2	1	no	—
20	C₄×C₅×C₂²⋊C₄	—	2	1	no	—
21	C₅×#376	—	2	1	no	—
22	C₄³×C₅	—	2	1	no	—
23	#7902	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
26	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
27	C₂²×C₅×C₈	—	2	2	no	—
28	C₂²×C₅×C₈	—	2	2	no	—
29	C₅×#117	—	2	8	no	—
30	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
31	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
32	C₂×C₄²×C₅	—	2	2	no	—
33	C₂×C₄²×C₅	—	2	2	no	—
34	C₂×C₄²×C₅	—	2	2	no	—
35	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₅	D₄	1	2	yes	—
41	C₂×C₄²×C₅	Q₈	1	2	yes	—
42	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
43	C₅×C₄⋊C₈	—	2	4	no	—
44	C₅×C₄⋊C₈	—	2	2	no	—
45	C₅×C₄⋊C₈	C₂×C₄	1	4	yes	—
46	C₅×C₂²⋊C₈	—	2	4	no	—
47	C₅×C₂²⋊C₈	—	2	4	no	—
48	C₅×#97	—	2	4	no	—
49	C₄×C₅×C₈	—	2	2	no	—
50	C₄×C₅×C₈	C₂×C₄	1	4	yes	—
51	C₅×#95	—	2	2	no	—
52	C₅×#95	—	2	2	no	—
53	C₂×#432	C₂×C₅	1	1	yes	characteristic
54	C₂×#431	C₂×C₅	1	1	yes	characteristic
55	C₄×#117	C₂×C₅	1	1	yes	characteristic
56	#1279	C₂×C₅	1	4	yes	complemented
57	C₂⁴×C₅	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₅	—	2	4	no	—
59	C₂²×C₄×C₅	—	2	4	no	—
60	C₂²×C₄×C₅	—	2	4	no	—
61	C₂²×C₄×C₅	—	4	2	no	—
62	C₂²×C₄×C₅	C₄²	1	2	yes	—
63	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
67	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
68	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
69	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
70	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
71	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
74	C₂×C₅×C₈	—	2	4	no	—
75	C₂×C₅×C₈	—	2	4	no	—
76	C₂×C₅×C₈	—	4	4	no	—
77	C₂×C₅×C₈	—	4	4	no	—
78	C₅×C₄⋊C₄	—	2	4	no	—
79	C₅×C₄⋊C₄	M₁₆	1	8	yes	—
80	C₅×C₂²⋊C₄	—	2	8	no	—
81	C₄²×C₅	—	2	8	no	—
82	C₄²×C₅	—	4	4	no	—
83	C₄²×C₅	—	4	4	no	—
84	C₄²×C₅	C₄²	1	2	yes	—
85	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
86	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
87	C₄²×C₅	C₄⋊C₄	1	2	yes	—
88	C₄²×C₅	C₄⋊C₄	1	2	yes	—
89	C₄²×C₅	C₄⋊C₄	1	2	yes	—
90	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
91	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
94	C₂×#117	C₄×C₅	1	2	yes	—
95	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
96	#432	—	2	2	no	—
97	#432	C₄×C₅	1	4	yes	complemented
98	#431	—	2	2	no	—
99	#431	C₄×C₅	1	4	yes	complemented
100	C₂×C₄⋊C₈	—	2	1	no	—
101	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
102	C₂×C₂²⋊C₈	—	2	1	no	—
103	C₂×C₂²⋊C₈	—	2	1	no	—
104	C₂×#97	—	2	1	no	—
105	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
106	C₄×C₄⋊C₄	C₄×C₅	1	2	yes	—
107	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
108	C₄×C₂²⋊C₄	—	2	1	no	—
109	C₄×C₂²⋊C₄	—	2	1	no	—
110	#376	—	2	1	no	—
111	C₄³	—	2	1	no	—
112	C₂³×C₅	—	2	2	no	—
113	C₂³×C₅	—	2	2	no	—
114	C₂³×C₅	—	2	2	no	—
115	C₂³×C₅	#95	1	2	yes	—
116	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₅	—	2	2	no	—
118	C₂×C₄×C₅	—	2	2	no	—
119	C₂×C₄×C₅	—	2	2	no	—
120	C₂×C₄×C₅	—	2	2	no	—
121	C₂×C₄×C₅	—	2	4	no	—
122	C₂×C₄×C₅	—	2	4	no	—
123	C₂×C₄×C₅	—	2	4	no	—
124	C₂×C₄×C₅	—	2	4	no	—
125	C₂×C₄×C₅	—	2	8	no	—
126	C₂×C₄×C₅	—	4	2	no	—
127	C₂×C₄×C₅	—	4	2	no	—
128	C₂×C₄×C₅	—	4	4	no	—
129	C₂×C₄×C₅	—	4	4	no	—
130	C₂×C₄×C₅	#95	1	2	yes	—
131	C₂×C₄×C₅	#95	1	2	yes	—
132	C₂×C₄×C₅	#95	1	2	yes	—
133	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
134	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
136	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
138	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
139	C₂×C₄×C₅	C₂×M₁₆	1	4	yes	—
140	C₂×C₄×C₅	#135	1	4	yes	—
141	C₅×C₈	—	4	4	no	—
142	C₅×C₈	—	4	4	no	—
143	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
144	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
145	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
146	C₂²×C₈	—	2	2	no	—
147	C₂²×C₈	—	2	2	no	—
148	#117	—	2	8	no	—
149	C₂×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
150	C₂×C₂²⋊C₄	—	2	2	no	—
151	C₂×C₄²	—	2	2	no	—
152	C₂×C₄²	—	2	2	no	—
153	C₂×C₄²	—	2	2	no	—
154	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
155	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
156	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
157	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
158	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
159	C₂×C₄²	C₅×D₄	1	2	yes	—
160	C₂×C₄²	C₅×Q₈	1	2	yes	—
161	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
162	C₄⋊C₈	—	2	4	no	—
163	C₄⋊C₈	—	2	2	no	—
164	C₄⋊C₈	C₂×C₄×C₅	1	4	yes	—
165	C₂²⋊C₈	—	2	4	no	—
166	C₂²⋊C₈	—	2	4	no	—
167	#97	—	2	4	no	—
168	C₄×C₈	—	2	2	no	—
169	C₄×C₈	C₂×C₄×C₅	1	4	yes	—
170	#95	—	2	2	no	—
171	#95	—	2	2	no	—
172	C₂²×C₅	—	2	2	no	—
173	C₂²×C₅	—	2	2	no	—
174	C₂²×C₅	—	2	2	no	—
175	C₂²×C₅	—	4	4	no	—
176	C₂²×C₅	C₂×#95	1	1	yes	characteristic
177	C₂²×C₅	#407	1	2	yes	—
178	C₂²×C₅	#416	1	2	yes	—
179	C₂²×C₅	#421	1	2	yes	—
180	C₄×C₅	—	2	2	no	—
181	C₄×C₅	—	2	1	no	—
182	C₄×C₅	—	2	2	no	—
183	C₄×C₅	—	2	1	no	—
184	C₄×C₅	—	2	2	no	—
185	C₄×C₅	—	4	8	no	—
186	C₄×C₅	C₄×M₁₆	1	2	yes	—
187	C₄×C₅	#416	1	2	yes	—
188	C₄×C₅	#423	1	2	yes	—
189	C₄×C₅	#427	1	2	yes	—
190	C₂⁴	C₅×C₂²⋊C₄	1	1	yes	characteristic
191	C₂²×C₄	—	2	4	no	—
192	C₂²×C₄	—	2	4	no	—
193	C₂²×C₄	—	2	4	no	—
194	C₂²×C₄	—	4	2	no	—
195	C₂²×C₄	C₄²×C₅	1	2	yes	—
196	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
197	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
198	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
199	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
200	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
201	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
202	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
203	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
204	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
205	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
206	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
207	C₂×C₈	—	2	4	no	—
208	C₂×C₈	—	2	4	no	—
209	C₂×C₈	—	4	4	no	—
210	C₂×C₈	—	4	4	no	—
211	C₄⋊C₄	—	2	4	no	—
212	C₄⋊C₄	C₅×M₁₆	1	8	yes	—
213	C₂²⋊C₄	—	2	8	no	—
214	C₄²	—	2	8	no	—
215	C₄²	—	4	4	no	—
216	C₄²	—	4	4	no	—
217	C₄²	C₄²×C₅	1	2	yes	—
218	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
219	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
220	C₄²	C₅×C₄⋊C₄	1	2	yes	—
221	C₄²	C₅×C₄⋊C₄	1	2	yes	—
222	C₄²	C₅×C₄⋊C₄	1	2	yes	—
223	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
224	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
225	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
226	C₄²	C₂×C₅×Q₈	1	1	yes	characteristic
227	C₂×C₅	—	4	2	no	—
228	C₂×C₅	#1488	1	4	yes	—
229	C₂×C₅	#1728	1	1	yes	characteristic
230	C₂×C₅	#1733	1	1	yes	characteristic
231	C₂×C₅	#1735	1	1	yes	characteristic
232	C₂³	—	2	2	no	—
233	C₂³	—	2	2	no	—
234	C₂³	—	2	2	no	—
235	C₂³	C₅×#95	1	2	yes	—
236	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	2	8	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	C₅×#95	1	2	yes	—
251	C₂×C₄	C₅×#95	1	2	yes	—
252	C₂×C₄	C₅×#95	1	2	yes	—
253	C₂×C₄	C₂×C₄²×C₅	1	1	yes	characteristic
254	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
255	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
256	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
257	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
258	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
259	C₂×C₄	C₂×C₅×M₁₆	1	4	yes	—
260	C₂×C₄	C₅×#135	1	4	yes	—
261	C₈	—	4	4	no	—
262	C₈	—	4	4	no	—
263	C₅	#7902	1	1	yes	characteristic, direct product factor, Sylow
264	C₂²	—	2	2	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	4	4	no	—
268	C₂²	C₂×C₅×#95	1	1	yes	characteristic
269	C₂²	C₅×#407	1	2	yes	—
270	C₂²	C₅×#416	1	2	yes	—
271	C₂²	C₅×#421	1	2	yes	—
272	C₄	—	2	2	no	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	2	no	—
276	C₄	—	2	2	no	—
277	C₄	—	4	8	no	—
278	C₄	C₄×C₅×M₁₆	1	2	yes	—
279	C₄	C₅×#416	1	2	yes	—
280	C₄	C₅×#423	1	2	yes	—
281	C₄	C₅×#427	1	2	yes	—
282	C₂	—	4	2	no	—
283	C₂	C₅×#1488	1	4	yes	—
284	C₂	C₅×#1728	1	1	yes	characteristic
285	C₂	C₅×#1733	1	1	yes	characteristic
286	C₂	C₅×#1735	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
socle	C₂³×C₅	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₅	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ C₂²×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅×#1279	C₂	semidirect
8. C₅×#432	C₄	semidirect
10. C₅×#431	C₄	semidirect
23. #7902	C₅	direct (N × H)
56. #1279	C₂×C₅	semidirect
97. #432	C₄×C₅	semidirect
99. #431	C₄×C₅	semidirect
263. C₅	#7902	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×#432	2
C₂×C₅×#431	2
C₄×C₅×#117	2
C₅×#1279	2
#7902	5

Subgroups — C5×#7902 #11930622

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#7902 #11930622