Subgroups — C₅×#9892 #11932612

Order 1280.

432 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2864	C₂	1	1	yes	maximal, characteristic
2	C₅×#1767	C₂	1	1	yes	maximal, characteristic
3	C₄×C₅×#97	C₂	1	1	yes	maximal, characteristic
4	C₅×#1295	C₂	1	2	yes	maximal
5	C₅×#1295	C₂	1	2	yes	maximal
6	C₂×C₅×C₄⋊C₈	—	2	1	no	—
7	C₂×C₅×#97	—	2	1	no	—
8	C₂×C₅×#97	C₂²	1	2	yes	—
9	C₂×C₅×#97	C₂²	1	2	yes	—
10	C₂×C₄×C₅×C₈	—	2	1	no	—
11	C₅×#389	C₄	1	1	yes	characteristic
12	C₅×#389	C₄	1	1	yes	characteristic
13	C₅×#389	C₄	1	1	yes	characteristic
14	C₅×#389	C₄	1	1	yes	characteristic
15	C₅×#387	—	2	1	no	—
16	C₅×#387	—	2	1	no	—
17	C₅×#387	—	2	1	no	—
18	C₅×#387	—	2	1	no	—
19	C₅×#382	C₂²	1	1	yes	characteristic
20	C₅×#382	C₂²	1	1	yes	characteristic
21	C₄³×C₅	C₂²	1	1	yes	characteristic
22	C₅×#336	—	2	1	no	—
23	#9892	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂²×C₅×C₈	—	2	2	no	—
25	C₂²×C₅×C₈	—	2	2	no	—
26	C₂×C₅×C₄⋊C₄	—	2	1	no	—
27	C₂×C₅×C₄⋊C₄	—	2	1	no	—
28	C₂×C₅×C₄⋊C₄	—	2	1	no	—
29	C₂×C₅×C₄⋊C₄	—	2	1	no	—
30	C₂×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
31	C₂×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
32	C₂×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
33	C₂×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
41	C₅×C₄⋊C₈	—	2	4	no	—
42	C₅×#97	—	2	4	no	—
43	C₅×#97	—	2	2	no	—
44	C₅×#97	—	2	4	no	—
45	C₅×#97	Q₈	1	4	yes	—
46	C₄×C₅×C₈	—	2	4	no	—
47	C₅×#95	—	2	1	no	—
48	C₅×#95	—	2	1	no	—
49	C₅×#95	—	2	1	no	—
50	C₅×#95	—	2	1	no	—
51	C₅×#95	—	2	1	no	—
52	C₅×#95	—	2	1	no	—
53	C₅×#95	—	2	1	no	—
54	C₅×#95	—	2	1	no	—
55	#2864	C₂×C₅	1	1	yes	characteristic
56	#1767	C₂×C₅	1	1	yes	characteristic
57	C₄×#97	C₂×C₅	1	1	yes	characteristic
58	#1295	C₂×C₅	1	2	yes	—
59	#1295	C₂×C₅	1	2	yes	—
60	C₂²×C₄×C₅	—	2	1	no	—
61	C₂²×C₄×C₅	—	2	1	no	—
62	C₂²×C₄×C₅	—	2	1	no	—
63	C₂²×C₄×C₅	—	2	1	no	—
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	1	yes	characteristic
67	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
71	C₂×C₅×C₈	—	2	4	no	—
72	C₂×C₅×C₈	—	2	4	no	—
73	C₂×C₅×C₈	—	2	4	no	—
74	C₂×C₅×C₈	—	2	4	no	—
75	C₂×C₅×C₈	—	4	2	no	—
76	C₂×C₅×C₈	—	4	2	no	—
77	C₅×C₄⋊C₄	—	4	1	no	—
78	C₅×C₄⋊C₄	—	4	1	no	—
79	C₅×C₄⋊C₄	—	4	1	no	—
80	C₅×C₄⋊C₄	—	4	1	no	—
81	C₅×C₄⋊C₄	—	4	2	no	—
82	C₅×C₄⋊C₄	—	4	2	no	—
83	C₅×C₄⋊C₄	—	4	2	no	—
84	C₅×C₄⋊C₄	—	4	2	no	—
85	C₄²×C₅	—	2	2	no	—
86	C₄²×C₅	—	2	2	no	—
87	C₄²×C₅	—	2	2	no	—
88	C₄²×C₅	—	4	1	no	—
89	C₄²×C₅	—	4	1	no	—
90	C₄²×C₅	—	4	1	no	—
91	C₄²×C₅	C₂×Q₈	1	2	yes	—
92	C₄²×C₅	#41	1	2	yes	—
93	C₂×C₄⋊C₈	—	2	1	no	—
94	C₂×#97	—	2	1	no	—
95	C₂×#97	C₂²×C₅	1	2	yes	—
96	C₂×#97	C₂²×C₅	1	2	yes	—
97	C₂×C₄×C₈	—	2	1	no	—
98	#389	C₄×C₅	1	1	yes	characteristic
99	#389	C₄×C₅	1	1	yes	characteristic
100	#389	C₄×C₅	1	1	yes	characteristic
101	#389	C₄×C₅	1	1	yes	characteristic
102	#387	—	2	1	no	—
103	#387	—	2	1	no	—
104	#387	—	2	1	no	—
105	#387	—	2	1	no	—
106	#382	C₂²×C₅	1	1	yes	characteristic
107	#382	C₂²×C₅	1	1	yes	characteristic
108	C₄³	C₂²×C₅	1	1	yes	characteristic
109	#336	—	2	1	no	—
110	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₅	—	2	1	no	—
112	C₂×C₄×C₅	—	2	1	no	—
113	C₂×C₄×C₅	—	2	1	no	—
114	C₂×C₄×C₅	—	2	1	no	—
115	C₂×C₄×C₅	—	2	1	no	—
116	C₂×C₄×C₅	—	2	1	no	—
117	C₂×C₄×C₅	—	2	1	no	—
118	C₂×C₄×C₅	—	2	1	no	—
119	C₂×C₄×C₅	—	2	1	no	—
120	C₂×C₄×C₅	—	2	1	no	—
121	C₂×C₄×C₅	—	2	1	no	—
122	C₂×C₄×C₅	—	2	1	no	—
123	C₂×C₄×C₅	—	2	1	no	—
124	C₂×C₄×C₅	—	2	1	no	—
125	C₂×C₄×C₅	—	2	1	no	—
126	C₂×C₄×C₅	—	4	1	no	—
127	C₂×C₄×C₅	—	4	1	no	—
128	C₂×C₄×C₅	—	4	1	no	—
129	C₂×C₄×C₅	—	4	1	no	—
130	C₂×C₄×C₅	—	4	1	no	—
131	C₂×C₄×C₅	—	4	1	no	—
132	C₂×C₄×C₅	—	4	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	—	4	1	no	—
138	C₂×C₄×C₅	#104	1	2	yes	—
139	C₂×C₄×C₅	#104	1	2	yes	—
140	C₂×C₄×C₅	#104	1	2	yes	—
141	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
142	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
143	C₂×C₄×C₅	#124	1	1	yes	characteristic
144	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
145	C₅×C₈	—	2	8	no	—
146	C₅×C₈	—	4	4	no	—
147	C₂²×C₈	—	2	2	no	—
148	C₂²×C₈	—	2	2	no	—
149	C₂×C₄⋊C₄	—	2	1	no	—
150	C₂×C₄⋊C₄	—	2	1	no	—
151	C₂×C₄⋊C₄	—	2	1	no	—
152	C₂×C₄⋊C₄	—	2	1	no	—
153	C₂×C₄⋊C₄	C₂×C₄×C₅	1	1	yes	characteristic
154	C₂×C₄⋊C₄	C₂×C₄×C₅	1	1	yes	characteristic
155	C₂×C₄⋊C₄	C₂×C₄×C₅	1	1	yes	characteristic
156	C₂×C₄⋊C₄	C₂×C₄×C₅	1	1	yes	characteristic
157	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
158	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
159	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
160	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
161	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
162	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
163	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
164	C₄⋊C₈	—	2	4	no	—
165	#97	—	2	4	no	—
166	#97	—	2	2	no	—
167	#97	—	2	4	no	—
168	#97	C₅×Q₈	1	4	yes	—
169	C₄×C₈	—	2	4	no	—
170	#95	—	2	1	no	—
171	#95	—	2	1	no	—
172	#95	—	2	1	no	—
173	#95	—	2	1	no	—
174	#95	—	2	1	no	—
175	#95	—	2	1	no	—
176	#95	—	2	1	no	—
177	#95	—	2	1	no	—
178	C₂²×C₅	#391	1	1	yes	characteristic
179	C₂²×C₅	#413	1	1	yes	characteristic
180	C₂²×C₅	#413	1	1	yes	characteristic
181	C₂²×C₅	C₂×#104	1	1	yes	characteristic
182	C₂²×C₅	C₂×#104	1	1	yes	characteristic
183	C₂²×C₅	C₂×#104	1	1	yes	characteristic
184	C₂²×C₅	#421	1	1	yes	characteristic
185	C₄×C₅	—	2	1	no	—
186	C₄×C₅	—	2	1	no	—
187	C₄×C₅	—	2	2	no	—
188	C₄×C₅	—	2	2	no	—
189	C₄×C₅	—	2	2	no	—
190	C₄×C₅	—	2	2	no	—
191	C₄×C₅	—	2	2	no	—
192	C₄×C₅	—	4	1	no	—
193	C₄×C₅	—	8	1	no	—
194	C₄×C₅	—	8	1	no	—
195	C₄×C₅	—	8	1	no	—
196	C₄×C₅	—	8	1	no	—
197	C₂²×C₄	—	2	1	no	—
198	C₂²×C₄	—	2	1	no	—
199	C₂²×C₄	—	2	1	no	—
200	C₂²×C₄	—	2	1	no	—
201	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
202	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
203	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
204	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
205	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
206	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
207	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
208	C₂×C₈	—	2	4	no	—
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	—	4	2	no	—
213	C₂×C₈	—	4	2	no	—
214	C₄⋊C₄	—	4	1	no	—
215	C₄⋊C₄	—	4	1	no	—
216	C₄⋊C₄	—	4	1	no	—
217	C₄⋊C₄	—	4	1	no	—
218	C₄⋊C₄	—	4	2	no	—
219	C₄⋊C₄	—	4	2	no	—
220	C₄⋊C₄	—	4	2	no	—
221	C₄⋊C₄	—	4	2	no	—
222	C₄²	—	2	2	no	—
223	C₄²	—	2	2	no	—
224	C₄²	—	2	2	no	—
225	C₄²	—	4	1	no	—
226	C₄²	—	4	1	no	—
227	C₄²	—	4	1	no	—
228	C₄²	C₂×C₅×Q₈	1	2	yes	—
229	C₄²	C₅×#41	1	2	yes	—
230	C₂×C₅	#1524	1	1	yes	characteristic
231	C₂×C₅	#1524	1	1	yes	characteristic
232	C₂×C₅	#1529	1	1	yes	characteristic
233	C₂×C₅	#1532	1	1	yes	characteristic
234	C₂×C₅	#1986	1	1	yes	characteristic
235	C₂×C₅	#1994	1	1	yes	characteristic
236	C₂×C₅	#1998	1	1	yes	characteristic
237	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	4	1	no	—
254	C₂×C₄	—	4	1	no	—
255	C₂×C₄	—	4	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	—	4	1	no	—
261	C₂×C₄	—	4	1	no	—
262	C₂×C₄	—	4	1	no	—
263	C₂×C₄	—	4	1	no	—
264	C₂×C₄	—	4	1	no	—
265	C₂×C₄	C₅×#104	1	2	yes	—
266	C₂×C₄	C₅×#104	1	2	yes	—
267	C₂×C₄	C₅×#104	1	2	yes	—
268	C₂×C₄	C₄×C₅×Q₈	1	2	yes	—
269	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
270	C₂×C₄	C₅×#124	1	1	yes	characteristic
271	C₂×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
272	C₈	—	2	8	no	—
273	C₈	—	4	4	no	—
274	C₅	#9892	1	1	yes	characteristic, direct product factor, Sylow
275	C₂²	C₅×#391	1	1	yes	characteristic
276	C₂²	C₅×#413	1	1	yes	characteristic
277	C₂²	C₅×#413	1	1	yes	characteristic
278	C₂²	C₂×C₅×#104	1	1	yes	characteristic
279	C₂²	C₂×C₅×#104	1	1	yes	characteristic
280	C₂²	C₂×C₅×#104	1	1	yes	characteristic
281	C₂²	C₅×#421	1	1	yes	characteristic
282	C₄	—	2	2	no	—
283	C₄	—	2	2	no	—
284	C₄	—	2	2	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	2	2	no	—
288	C₄	—	2	2	no	—
289	C₄	—	4	1	no	—
290	C₄	—	8	1	no	—
291	C₄	—	8	1	no	—
292	C₄	—	8	1	no	—
293	C₄	—	8	1	no	—
294	C₂	C₅×#1524	1	1	yes	characteristic
295	C₂	C₅×#1524	1	1	yes	characteristic
296	C₂	C₅×#1529	1	1	yes	characteristic
297	C₂	C₅×#1532	1	1	yes	characteristic
298	C₂	C₅×#1986	1	1	yes	characteristic
299	C₂	C₅×#1994	1	1	yes	characteristic
300	C₂	C₅×#1998	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₅	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₄³×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
23. #9892	C₅	direct (N × H)
274. C₅	#9892	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×#2864	2
C₅×#1767	2
C₄×C₅×#97	2
C₅×#1295	2
#9892	5

Subgroups — C5×#9892 #11932612

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#9892 #11932612