Subgroups — C₅×#8692 #11931412

Order 1280.

358 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	C₅×#2966	C₂	1	1	yes	maximal, characteristic
2	C₄×C₅×#108	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#1568	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1529	C₂	1	2	yes	maximal
5	C₅×#1291	C₂	1	2	yes	maximal, complemented
6	C₅×#529	C₂²	1	1	yes	characteristic
7	C₅×#446	C₄	1	1	yes	characteristic
8	C₅×C₈×Q₈	C₄	1	1	yes	characteristic
9	C₅×#436	C₄	1	1	yes	characteristic
10	C₅×#435	—	2	1	no	—
11	C₅×C₈×D₄	—	2	1	no	—
12	C₅×C₈×D₄	C₄	1	1	yes	characteristic
13	C₅×#433	—	2	1	no	—
14	C₅×#432	C₂²	1	2	yes	—
15	C₅×#432	C₂²	1	1	yes	characteristic
16	C₂×C₅×#108	—	2	1	no	—
17	C₅×#405	—	2	1	no	—
18	C₄×C₅×M₁₆	C₂²	1	2	yes	—
19	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
20	C₅×#341	—	2	1	no	—
21	C₅×#335	—	2	1	no	—
22	C₅×#334	—	2	1	no	—
23	C₅×#330	—	2	2	no	—
24	C₅×#329	—	2	2	no	—
25	#8692	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₂×C₅×M₁₆	—	2	1	no	—
27	C₂×C₅×M₁₆	—	2	2	no	—
28	C₂²×C₅×C₈	—	2	1	no	—
29	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
30	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
31	C₅×#126	—	2	1	no	—
32	C₅×#125	D₄	1	1	yes	characteristic
33	C₅×#124	D₄	1	1	yes	characteristic
34	C₅×#123	—	2	1	no	—
35	C₅×C₂²⋊Q₈	Q₈	1	1	yes	characteristic
36	C₅×#121	Q₈	1	1	yes	characteristic
37	C₄×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
38	C₄×C₅×D₄	—	2	1	no	—
39	C₄×C₅×D₄	C₂×C₄	1	1	yes	characteristic
40	C₅×#117	—	2	1	no	—
41	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
42	C₅×#108	—	2	4	no	—
43	C₅×C₄⋊C₈	—	2	1	no	—
44	C₅×C₄⋊C₈	—	2	1	no	—
45	C₅×C₄⋊C₈	—	2	2	no	—
46	C₅×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
47	C₅×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
48	C₅×C₂²⋊C₈	—	2	1	no	—
49	C₅×C₂²⋊C₈	—	2	1	no	—
50	C₅×C₂²⋊C₈	—	2	1	no	—
51	C₅×C₂²⋊C₈	—	2	2	no	—
52	C₅×#97	—	2	1	no	—
53	C₅×#97	—	2	1	no	—
54	C₅×#97	—	2	2	no	—
55	C₅×#97	—	2	2	no	—
56	C₄×C₅×C₈	—	2	1	no	—
57	C₄×C₅×C₈	—	2	2	no	—
58	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
59	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
60	C₄×C₅×C₈	D₄	1	1	yes	characteristic
61	C₄×C₅×C₈	D₄	1	1	yes	characteristic
62	#2966	C₂×C₅	1	1	yes	characteristic
63	C₄×#108	C₂×C₅	1	1	yes	characteristic, complemented
64	#1568	C₂×C₅	1	1	yes	characteristic, complemented
65	#1529	C₂×C₅	1	2	yes	—
66	#1291	C₂×C₅	1	2	yes	complemented
67	C₂×C₅×Q₈	C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₅×D₄	—	2	1	no	—
69	C₂×C₅×D₄	C₄⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅	—	2	1	no	—
71	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
73	C₂²×C₄×C₅	#41	1	1	yes	characteristic
74	C₅×M₁₆	—	2	4	no	—
75	C₅×M₁₆	—	4	2	no	—
76	C₅×M₁₆	—	4	2	no	—
77	C₂×C₅×C₈	—	2	2	no	—
78	C₂×C₅×C₈	—	2	1	no	—
79	C₂×C₅×C₈	—	2	1	no	—
80	C₂×C₅×C₈	—	4	1	no	—
81	C₂×C₅×C₈	—	4	2	no	—
82	C₂×C₅×C₈	—	4	2	no	—
83	C₂×C₅×C₈	—	4	2	no	—
84	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
88	C₅×C₄⋊C₄	—	2	1	no	—
89	C₅×C₄⋊C₄	—	2	1	no	—
90	C₅×C₄⋊C₄	—	2	1	no	—
91	C₅×C₄⋊C₄	—	2	1	no	—
92	C₅×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
93	C₅×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
94	C₅×C₂²⋊C₄	—	2	1	no	—
95	C₅×C₂²⋊C₄	—	2	1	no	—
96	C₅×C₂²⋊C₄	—	2	1	no	—
97	C₅×C₂²⋊C₄	—	2	1	no	—
98	C₅×C₂²⋊C₄	—	2	1	no	—
99	C₄²×C₅	—	2	1	no	—
100	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
101	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₅	#41	1	1	yes	characteristic
104	#529	C₂²×C₅	1	1	yes	characteristic
105	#446	C₄×C₅	1	1	yes	characteristic
106	C₈×Q₈	C₄×C₅	1	1	yes	characteristic
107	#436	C₄×C₅	1	1	yes	characteristic
108	#435	—	2	1	no	—
109	C₈×D₄	—	2	1	no	—
110	C₈×D₄	C₄×C₅	1	1	yes	characteristic
111	#433	—	2	1	no	—
112	#432	C₂²×C₅	1	2	yes	—
113	#432	C₂²×C₅	1	1	yes	characteristic
114	C₂×#108	—	2	1	no	—
115	#405	—	2	1	no	—
116	C₄×M₁₆	C₂²×C₅	1	2	yes	—
117	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
118	#341	—	2	1	no	—
119	#335	—	2	1	no	—
120	#334	—	2	1	no	—
121	#330	—	2	2	no	—
122	#329	—	2	2	no	—
123	C₂³×C₅	—	2	1	no	—
124	C₂³×C₅	C₂²⋊Q₈	1	1	yes	characteristic
125	C₅×Q₈	—	4	1	no	—
126	C₅×D₄	—	4	1	no	—
127	C₅×D₄	—	4	2	no	—
128	C₂×C₄×C₅	—	2	1	no	—
129	C₂×C₄×C₅	—	2	1	no	—
130	C₂×C₄×C₅	—	2	1	no	—
131	C₂×C₄×C₅	—	2	1	no	—
132	C₂×C₄×C₅	—	2	1	no	—
133	C₂×C₄×C₅	—	4	1	no	—
134	C₂×C₄×C₅	—	4	1	no	—
135	C₂×C₄×C₅	—	4	1	no	—
136	C₂×C₄×C₅	—	4	1	no	—
137	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
138	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
139	C₂×C₄×C₅	#117	1	1	yes	characteristic
140	C₂×C₄×C₅	#121	1	1	yes	characteristic
141	C₂×C₄×C₅	#121	1	1	yes	characteristic
142	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
143	C₅×C₈	—	2	1	no	—
144	C₅×C₈	—	2	1	no	—
145	C₅×C₈	—	2	1	no	—
146	C₅×C₈	—	4	4	no	—
147	C₅×C₈	—	4	1	no	—
148	C₅×C₈	—	4	1	no	—
149	C₅×C₈	—	8	2	no	—
150	C₅×C₈	#104	1	2	yes	—
151	C₂×M₁₆	—	2	1	no	—
152	C₂×M₁₆	—	2	2	no	—
153	C₂²×C₈	—	2	1	no	—
154	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
155	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
156	#126	—	2	1	no	—
157	#125	C₅×D₄	1	1	yes	characteristic
158	#124	C₅×D₄	1	1	yes	characteristic
159	#123	—	2	1	no	—
160	C₂²⋊Q₈	C₅×Q₈	1	1	yes	characteristic
161	#121	C₅×Q₈	1	1	yes	characteristic
162	C₄×Q₈	C₂×C₄×C₅	1	1	yes	characteristic
163	C₄×D₄	—	2	1	no	—
164	C₄×D₄	C₂×C₄×C₅	1	1	yes	characteristic
165	#117	—	2	1	no	—
166	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
167	#108	—	2	4	no	—
168	C₄⋊C₈	—	2	1	no	—
169	C₄⋊C₈	—	2	1	no	—
170	C₄⋊C₈	—	2	2	no	—
171	C₄⋊C₈	C₂×C₄×C₅	1	1	yes	characteristic
172	C₄⋊C₈	C₂×C₄×C₅	1	1	yes	characteristic
173	C₂²⋊C₈	—	2	1	no	—
174	C₂²⋊C₈	—	2	1	no	—
175	C₂²⋊C₈	—	2	1	no	—
176	C₂²⋊C₈	—	2	2	no	—
177	#97	—	2	1	no	—
178	#97	—	2	1	no	—
179	#97	—	2	2	no	—
180	#97	—	2	2	no	—
181	C₄×C₈	—	2	1	no	—
182	C₄×C₈	—	2	2	no	—
183	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
184	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
185	C₄×C₈	C₅×D₄	1	1	yes	characteristic
186	C₄×C₈	C₅×D₄	1	1	yes	characteristic
187	C₂²×C₅	—	2	1	no	—
188	C₂²×C₅	—	2	1	no	—
189	C₂²×C₅	—	2	1	no	—
190	C₂²×C₅	—	4	1	no	—
191	C₂²×C₅	—	4	1	no	—
192	C₂²×C₅	—	4	1	no	—
193	C₂²×C₅	#380	1	1	yes	characteristic
194	C₄×C₅	—	2	1	no	—
195	C₄×C₅	—	2	1	no	—
196	C₄×C₅	—	2	1	no	—
197	C₄×C₅	—	2	1	no	—
198	C₄×C₅	—	2	1	no	—
199	C₄×C₅	—	4	1	no	—
200	C₄×C₅	—	4	1	no	—
201	C₄×C₅	—	4	1	no	—
202	C₄×C₅	C₂×#104	1	1	yes	characteristic
203	C₄×C₅	#421	1	1	yes	characteristic
204	C₂×Q₈	C₅×C₄⋊C₄	1	1	yes	characteristic
205	C₂×D₄	—	2	1	no	—
206	C₂×D₄	C₅×C₄⋊C₄	1	1	yes	characteristic
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
209	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
210	C₂²×C₄	C₅×#41	1	1	yes	characteristic
211	M₁₆	—	2	4	no	—
212	M₁₆	—	4	2	no	—
213	M₁₆	—	4	2	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	1	no	—
216	C₂×C₈	—	2	1	no	—
217	C₂×C₈	—	4	1	no	—
218	C₂×C₈	—	4	2	no	—
219	C₂×C₈	—	4	2	no	—
220	C₂×C₈	—	4	2	no	—
221	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
222	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
223	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
224	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
225	C₄⋊C₄	—	2	1	no	—
226	C₄⋊C₄	—	2	1	no	—
227	C₄⋊C₄	—	2	1	no	—
228	C₄⋊C₄	—	2	1	no	—
229	C₄⋊C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
230	C₄⋊C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
231	C₂²⋊C₄	—	2	1	no	—
232	C₂²⋊C₄	—	2	1	no	—
233	C₂²⋊C₄	—	2	1	no	—
234	C₂²⋊C₄	—	2	1	no	—
235	C₂²⋊C₄	—	2	1	no	—
236	C₄²	—	2	1	no	—
237	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
238	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
239	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
240	C₄²	C₅×#41	1	1	yes	characteristic
241	C₂×C₅	—	4	1	no	—
242	C₂×C₅	—	8	1	no	—
243	C₂×C₅	#1806	1	1	yes	characteristic
244	C₂×C₅	#1813	1	1	yes	characteristic
245	C₂×C₅	#1814	1	1	yes	characteristic
246	C₂³	—	2	1	no	—
247	C₂³	C₅×C₂²⋊Q₈	1	1	yes	characteristic
248	Q₈	—	4	1	no	—
249	D₄	—	4	1	no	—
250	D₄	—	4	2	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	4	1	no	—
257	C₂×C₄	—	4	1	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
261	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
262	C₂×C₄	C₅×#117	1	1	yes	characteristic
263	C₂×C₄	C₅×#121	1	1	yes	characteristic
264	C₂×C₄	C₅×#121	1	1	yes	characteristic
265	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
266	C₈	—	2	1	no	—
267	C₈	—	2	1	no	—
268	C₈	—	2	1	no	—
269	C₈	—	4	4	no	—
270	C₈	—	4	1	no	—
271	C₈	—	4	1	no	—
272	C₈	—	8	2	no	—
273	C₈	C₅×#104	1	2	yes	—
274	C₅	#8692	1	1	yes	characteristic, direct product factor, Sylow
275	C₂²	—	2	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	4	1	no	—
279	C₂²	—	4	1	no	—
280	C₂²	—	4	1	no	—
281	C₂²	C₅×#380	1	1	yes	characteristic
282	C₄	—	2	1	no	—
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	4	1	no	—
288	C₄	—	4	1	no	—
289	C₄	—	4	1	no	—
290	C₄	C₂×C₅×#104	1	1	yes	characteristic
291	C₄	C₅×#421	1	1	yes	characteristic
292	C₂	—	4	1	no	—
293	C₂	—	8	1	no	—
294	C₂	C₅×#1806	1	1	yes	characteristic
295	C₂	C₅×#1813	1	1	yes	characteristic
296	C₂	C₅×#1814	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	#380
commutator	C₄²	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₅×#121	Q₈
LCS	G ▷ C₄² ▷ 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. C₄×C₅×#108	C₂	semidirect
3. C₅×#1568	C₂	semidirect
5. C₅×#1291	C₂	semidirect
25. #8692	C₅	direct (N × H)
63. C₄×#108	C₂×C₅	semidirect
64. #1568	C₂×C₅	semidirect
66. #1291	C₂×C₅	semidirect
274. C₅	#8692	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×#2966	2
C₄×C₅×#108	2
C₅×#1568	2
C₅×#1529	2
C₅×#1291	2
#8692	5

Subgroups — C5×#8692 #11931412

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8692 #11931412