Subgroups — C₅×#19931 #11942651

Order 1280.

554 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3387	C₂	1	2	yes	maximal
2	C₅×#3363	C₂	1	2	yes	maximal
3	C₅×#1707	C₂	1	1	yes	maximal, characteristic
4	C₅×#1692	C₂	1	2	yes	maximal
5	C₅×#1684	C₂	1	2	yes	maximal
6	C₅×#1648	C₂	1	2	yes	maximal
7	C₅×#1633	C₂	1	2	yes	maximal
8	C₅×#1591	C₂	1	1	yes	maximal, characteristic
9	C₅×C₈×Q₁₆	C₂	1	1	yes	maximal, characteristic
10	C₅×Q₈²	C₂²	1	2	yes	—
11	C₅×#557	C₂²	1	2	yes	—
12	C₅×#501	—	2	2	no	—
13	C₅×#500	C₂²	1	1	yes	characteristic
14	C₅×#498	C₂²	1	2	yes	—
15	C₅×#494	C₂²	1	1	yes	characteristic
16	C₅×#487	—	2	2	no	—
17	C₅×#487	C₂²	1	2	yes	—
18	C₅×#479	—	2	2	no	—
19	C₅×#477	—	2	2	no	—
20	C₅×#477	C₂²	1	2	yes	—
21	C₅×#477	C₂²	1	2	yes	—
22	C₅×#475	—	2	2	no	—
23	C₅×#475	C₂²	1	2	yes	—
24	C₅×#462	—	2	2	no	—
25	C₅×#462	C₂²	1	2	yes	—
26	C₅×#446	C₂²	1	2	yes	—
27	C₅×C₈×Q₈	C₂²	1	2	yes	—
28	C₅×#441	—	2	2	no	—
29	C₄×C₅×Q₁₆	—	2	2	no	—
30	C₄×C₅×Q₁₆	C₂²	1	1	yes	characteristic
31	C₄×C₅×Q₁₆	C₂²	1	1	yes	characteristic
32	C₅×#335	C₂²	1	1	yes	characteristic
33	C₅×#335	C₂²	1	1	yes	characteristic
34	C₅×#333	C₂²	1	2	yes	—
35	C₅×#332	C₂²	1	2	yes	—
36	C₅×Q₈⋊C₈	C₂²	1	2	yes	—
37	C₅×Q₈⋊C₈	C₂²	1	2	yes	—
38	C₅×C₈²	C₂²	1	1	yes	characteristic
39	#19931	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
40	C₂×C₅×Q₁₆	—	2	2	no	—
41	C₂×C₅×Q₁₆	—	4	1	no	—
42	C₂×C₅×Q₁₆	D₄	1	1	yes	characteristic
43	C₂×C₅×Q₁₆	D₄	1	1	yes	characteristic
44	C₅×C₄⋊Q₈	—	2	2	no	—
45	C₅×C₄⋊Q₈	—	2	2	no	—
46	C₅×C₄⋊Q₈	—	2	2	no	—
47	C₅×C₄⋊Q₈	—	4	2	no	—
48	C₅×C₄⋊Q₈	C₂³	1	2	yes	—
49	C₅×C₄⋊Q₈	C₂³	1	2	yes	—
50	C₅×#125	—	2	2	no	—
51	C₅×#125	—	4	2	no	—
52	C₄×C₅×Q₈	—	2	2	no	—
53	C₄×C₅×Q₈	—	2	2	no	—
54	C₄×C₅×Q₈	—	2	2	no	—
55	C₄×C₅×Q₈	—	2	2	no	—
56	C₄×C₅×Q₈	C₂³	1	2	yes	—
57	C₄×C₅×Q₈	C₂³	1	2	yes	—
58	C₅×#107	—	2	2	no	—
59	C₅×#107	—	2	2	no	—
60	C₅×#107	—	4	1	no	—
61	C₅×#107	D₄	1	1	yes	characteristic
62	C₅×#107	D₄	1	1	yes	characteristic
63	C₅×#106	—	2	2	no	—
64	C₅×#106	—	2	2	no	—
65	C₅×#106	—	2	2	no	—
66	C₅×C₄⋊C₈	—	2	2	no	—
67	C₅×C₄⋊C₈	—	2	2	no	—
68	C₅×C₄⋊C₈	C₂³	1	2	yes	—
69	C₅×C₄⋊C₈	C₂³	1	2	yes	—
70	C₅×#103	—	2	2	no	—
71	C₅×#103	—	2	2	no	—
72	C₅×#103	—	2	2	no	—
73	C₅×#103	—	2	2	no	—
74	C₅×#103	—	2	2	no	—
75	C₅×#103	—	2	2	no	—
76	C₅×#103	D₄	1	2	yes	—
77	C₅×#103	D₄	1	2	yes	—
78	C₅×#97	—	2	2	no	—
79	C₄×C₅×C₈	—	2	2	no	—
80	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
81	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
82	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
83	#3387	C₂×C₅	1	2	yes	—
84	#3363	C₂×C₅	1	2	yes	—
85	#1707	C₂×C₅	1	1	yes	characteristic
86	#1692	C₂×C₅	1	2	yes	—
87	#1684	C₂×C₅	1	2	yes	—
88	#1648	C₂×C₅	1	2	yes	—
89	#1633	C₂×C₅	1	2	yes	—
90	#1591	C₂×C₅	1	1	yes	characteristic
91	C₈×Q₁₆	C₂×C₅	1	1	yes	characteristic
92	C₂×C₅×Q₈	—	2	2	no	—
93	C₂×C₅×Q₈	—	2	2	no	—
94	C₂×C₅×Q₈	—	4	2	no	—
95	C₂×C₅×Q₈	C₂×D₄	1	2	yes	—
96	C₂×C₅×Q₈	C₂×D₄	1	2	yes	—
97	C₅×Q₁₆	—	4	1	no	—
98	C₅×Q₁₆	—	4	4	no	—
99	C₅×Q₁₆	—	8	2	no	—
100	C₅×Q₁₆	Q₁₆	1	4	yes	complemented
101	C₂×C₅×C₈	—	2	1	no	—
102	C₂×C₅×C₈	—	2	2	no	—
103	C₂×C₅×C₈	—	2	2	no	—
104	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
105	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
106	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
107	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
108	C₅×C₄⋊C₄	—	2	2	no	—
109	C₅×C₄⋊C₄	—	2	2	no	—
110	C₅×C₄⋊C₄	—	2	2	no	—
111	C₅×C₄⋊C₄	—	2	2	no	—
112	C₅×C₄⋊C₄	—	2	2	no	—
113	C₅×C₄⋊C₄	—	2	2	no	—
114	C₅×C₄⋊C₄	—	4	2	no	—
115	C₅×C₄⋊C₄	—	4	2	no	—
116	C₅×C₄⋊C₄	—	4	2	no	—
117	C₅×C₄⋊C₄	—	4	2	no	—
118	C₅×C₄⋊C₄	—	4	2	no	—
119	C₅×C₄⋊C₄	C₂×D₄	1	2	yes	—
120	C₅×C₄⋊C₄	C₂×D₄	1	2	yes	—
121	C₄²×C₅	—	2	2	no	—
122	C₄²×C₅	—	2	2	no	—
123	C₄²×C₅	—	4	2	no	—
124	C₄²×C₅	—	4	2	no	—
125	C₄²×C₅	C₂⁴	1	1	yes	characteristic
126	Q₈²	C₂²×C₅	1	2	yes	—
127	#557	C₂²×C₅	1	2	yes	—
128	#501	—	2	2	no	—
129	#500	C₂²×C₅	1	1	yes	characteristic
130	#498	C₂²×C₅	1	2	yes	—
131	#494	C₂²×C₅	1	1	yes	characteristic
132	#487	—	2	2	no	—
133	#487	C₂²×C₅	1	2	yes	—
134	#479	—	2	2	no	—
135	#477	—	2	2	no	—
136	#477	C₂²×C₅	1	2	yes	—
137	#477	C₂²×C₅	1	2	yes	—
138	#475	—	2	2	no	—
139	#475	C₂²×C₅	1	2	yes	—
140	#462	—	2	2	no	—
141	#462	C₂²×C₅	1	2	yes	—
142	#446	C₂²×C₅	1	2	yes	—
143	C₈×Q₈	C₂²×C₅	1	2	yes	—
144	#441	—	2	2	no	—
145	C₄×Q₁₆	—	2	2	no	—
146	C₄×Q₁₆	C₂²×C₅	1	1	yes	characteristic
147	C₄×Q₁₆	C₂²×C₅	1	1	yes	characteristic
148	#335	C₂²×C₅	1	1	yes	characteristic
149	#335	C₂²×C₅	1	1	yes	characteristic
150	#333	C₂²×C₅	1	2	yes	—
151	#332	C₂²×C₅	1	2	yes	—
152	Q₈⋊C₈	C₂²×C₅	1	2	yes	—
153	Q₈⋊C₈	C₂²×C₅	1	2	yes	—
154	C₈²	C₂²×C₅	1	1	yes	characteristic
155	C₅×Q₈	—	2	2	no	—
156	C₅×Q₈	—	2	2	no	—
157	C₅×Q₈	—	2	2	no	—
158	C₅×Q₈	—	4	2	no	—
159	C₅×Q₈	—	4	2	no	—
160	C₅×Q₈	—	8	2	no	—
161	C₅×Q₈	—	16	2	no	—
162	C₅×Q₈	C₂×Q₁₆	1	4	yes	—
163	C₂×C₄×C₅	—	2	2	no	—
164	C₂×C₄×C₅	—	2	2	no	—
165	C₂×C₄×C₅	—	2	2	no	—
166	C₂×C₄×C₅	—	2	2	no	—
167	C₂×C₄×C₅	—	4	2	no	—
168	C₂×C₄×C₅	—	4	2	no	—
169	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
170	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
171	C₂×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
172	C₅×C₈	—	2	2	no	—
173	C₅×C₈	—	2	1	no	—
174	C₅×C₈	—	2	1	no	—
175	C₅×C₈	—	2	1	no	—
176	C₅×C₈	—	4	2	no	—
177	C₅×C₈	—	4	2	no	—
178	C₅×C₈	C₂×Q₁₆	1	2	yes	—
179	C₂×Q₁₆	—	2	2	no	—
180	C₂×Q₁₆	—	4	1	no	—
181	C₂×Q₁₆	C₅×D₄	1	1	yes	characteristic
182	C₂×Q₁₆	C₅×D₄	1	1	yes	characteristic
183	C₄⋊Q₈	—	2	2	no	—
184	C₄⋊Q₈	—	2	2	no	—
185	C₄⋊Q₈	—	2	2	no	—
186	C₄⋊Q₈	—	4	2	no	—
187	C₄⋊Q₈	C₂³×C₅	1	2	yes	—
188	C₄⋊Q₈	C₂³×C₅	1	2	yes	—
189	#125	—	2	2	no	—
190	#125	—	4	2	no	—
191	C₄×Q₈	—	2	2	no	—
192	C₄×Q₈	—	2	2	no	—
193	C₄×Q₈	—	2	2	no	—
194	C₄×Q₈	—	2	2	no	—
195	C₄×Q₈	C₂³×C₅	1	2	yes	—
196	C₄×Q₈	C₂³×C₅	1	2	yes	—
197	#107	—	2	2	no	—
198	#107	—	2	2	no	—
199	#107	—	4	1	no	—
200	#107	C₅×D₄	1	1	yes	characteristic
201	#107	C₅×D₄	1	1	yes	characteristic
202	#106	—	2	2	no	—
203	#106	—	2	2	no	—
204	#106	—	2	2	no	—
205	C₄⋊C₈	—	2	2	no	—
206	C₄⋊C₈	—	2	2	no	—
207	C₄⋊C₈	C₂³×C₅	1	2	yes	—
208	C₄⋊C₈	C₂³×C₅	1	2	yes	—
209	#103	—	2	2	no	—
210	#103	—	2	2	no	—
211	#103	—	2	2	no	—
212	#103	—	2	2	no	—
213	#103	—	2	2	no	—
214	#103	—	2	2	no	—
215	#103	C₅×D₄	1	2	yes	—
216	#103	C₅×D₄	1	2	yes	—
217	#97	—	2	2	no	—
218	C₄×C₈	—	2	2	no	—
219	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
220	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
221	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
222	C₂²×C₅	D₄²	1	1	yes	characteristic
223	C₄×C₅	—	2	4	no	—
224	C₄×C₅	—	2	1	no	—
225	C₄×C₅	—	4	2	no	—
226	C₄×C₅	—	4	2	no	—
227	C₄×C₅	—	4	2	no	—
228	C₄×C₅	—	8	2	no	—
229	C₄×C₅	—	8	2	no	—
230	C₄×C₅	C₂²×Q₁₆	1	1	yes	characteristic
231	C₄×C₅	#575	1	1	yes	characteristic
232	C₄×C₅	#578	1	1	yes	characteristic
233	C₄×C₅	#578	1	1	yes	characteristic
234	C₂×Q₈	—	2	2	no	—
235	C₂×Q₈	—	2	2	no	—
236	C₂×Q₈	—	4	2	no	—
237	C₂×Q₈	C₂×C₅×D₄	1	2	yes	—
238	C₂×Q₈	C₂×C₅×D₄	1	2	yes	—
239	Q₁₆	—	4	1	no	—
240	Q₁₆	—	4	4	no	—
241	Q₁₆	—	8	2	no	—
242	Q₁₆	C₅×Q₁₆	1	4	yes	complemented
243	C₂×C₈	—	2	1	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
247	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
248	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
249	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	4	2	no	—
257	C₄⋊C₄	—	4	2	no	—
258	C₄⋊C₄	—	4	2	no	—
259	C₄⋊C₄	—	4	2	no	—
260	C₄⋊C₄	—	4	2	no	—
261	C₄⋊C₄	C₂×C₅×D₄	1	2	yes	—
262	C₄⋊C₄	C₂×C₅×D₄	1	2	yes	—
263	C₄²	—	2	2	no	—
264	C₄²	—	2	2	no	—
265	C₄²	—	4	2	no	—
266	C₄²	—	4	2	no	—
267	C₄²	C₂⁴×C₅	1	1	yes	characteristic
268	C₂×C₅	#3277	1	1	yes	characteristic
269	C₂×C₅	D₄×Q₁₆	1	1	yes	characteristic
270	C₂×C₅	#3291	1	1	yes	characteristic
271	Q₈	—	2	2	no	—
272	Q₈	—	2	2	no	—
273	Q₈	—	2	2	no	—
274	Q₈	—	4	2	no	—
275	Q₈	—	4	2	no	—
276	Q₈	—	8	2	no	—
277	Q₈	—	16	2	no	—
278	Q₈	C₂×C₅×Q₁₆	1	4	yes	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
286	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
287	C₂×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
288	C₈	—	2	2	no	—
289	C₈	—	2	1	no	—
290	C₈	—	2	1	no	—
291	C₈	—	2	1	no	—
292	C₈	—	4	2	no	—
293	C₈	—	4	2	no	—
294	C₈	C₂×C₅×Q₁₆	1	2	yes	—
295	C₅	#19931	1	1	yes	characteristic, direct product factor, Sylow
296	C₂²	C₅×D₄²	1	1	yes	characteristic
297	C₄	—	2	4	no	—
298	C₄	—	2	1	no	—
299	C₄	—	4	2	no	—
300	C₄	—	4	2	no	—
301	C₄	—	4	2	no	—
302	C₄	—	8	2	no	—
303	C₄	—	8	2	no	—
304	C₄	C₂²×C₅×Q₁₆	1	1	yes	characteristic
305	C₄	C₅×#575	1	1	yes	characteristic
306	C₄	C₅×#578	1	1	yes	characteristic
307	C₄	C₅×#578	1	1	yes	characteristic
308	C₂	C₅×#3277	1	1	yes	characteristic
309	C₂	C₅×D₄×Q₁₆	1	1	yes	characteristic
310	C₂	C₅×#3291	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₅	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
39. #19931	C₅	direct (N × H)
100. C₅×Q₁₆	Q₁₆	semidirect
242. Q₁₆	C₅×Q₁₆	semidirect
295. C₅	#19931	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×#3387	2
C₅×#3363	2
C₅×#1707	2
C₅×#1692	2
C₅×#1684	2
C₅×#1648	2
C₅×#1633	2
C₅×#1591	2
C₅×C₈×Q₁₆	2
#19931	5

Subgroups — C5×#19931 #11942651

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#19931 #11942651