Subgroups — C₅×#20788 #11943508

Order 1280.

608 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2875	C₂	1	1	yes	maximal, characteristic
2	C₅×#1841	C₂	1	2	yes	maximal, complemented
3	C₅×#1833	C₂	1	2	yes	maximal, complemented
4	C₅×#1827	C₂	1	2	yes	maximal
5	C₅×#1813	C₂	1	1	yes	maximal, characteristic
6	C₅×#1812	C₂	1	1	yes	maximal, characteristic
7	C₅×#1773	C₂	1	2	yes	maximal, complemented
8	C₅×#1757	C₂	1	2	yes	maximal
9	C₅×#1747	C₂	1	2	yes	maximal, complemented
10	C₂×C₅×#131	C₂²	1	1	yes	characteristic
11	C₂×C₅×#131	C₂²	1	1	yes	characteristic
12	C₄×C₅×#41	C₂²	1	1	yes	characteristic
13	C₂×C₅×#108	C₂²	1	2	yes	—
14	C₅×#427	C₂²	1	2	yes	—
15	C₅×#413	C₂²	1	2	yes	—
16	C₅×#410	C₂²	1	2	yes	—
17	C₅×#405	C₄	1	2	yes	complemented
18	C₅×#405	C₄	1	2	yes	complemented
19	C₅×#405	C₂²	1	4	yes	—
20	C₅×#405	C₂²	1	1	yes	characteristic
21	C₅×#405	C₂²	1	1	yes	characteristic
22	C₄×C₅×M₁₆	C₄	1	2	yes	complemented
23	C₄×C₅×M₁₆	C₂²	1	1	yes	characteristic
24	C₂×C₅×#97	C₄	1	2	yes	complemented
25	C₂×C₅×#97	C₂²	1	1	yes	characteristic
26	C₅×#344	C₂²	1	4	yes	—
27	C₅×#339	C₂²	1	4	yes	—
28	C₅×#338	C₂²	1	2	yes	—
29	C₅×H(3,ℤ/4)	C₂²	1	2	yes	—
30	C₅×H(3,ℤ/4)	C₂²	1	4	yes	—
31	#20788	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₂×C₅×#41	C₂³	1	1	yes	characteristic
33	C₅×#131	—	2	4	no	—
34	C₅×#131	—	2	2	no	—
35	C₅×#131	—	2	2	no	—
36	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
37	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
38	C₂×C₅×M₁₆	C₂³	1	4	yes	—
39	C₂×C₅×M₁₆	C₂³	1	1	yes	characteristic
40	C₂×C₅×M₁₆	C₂³	1	1	yes	characteristic
41	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
42	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
43	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
44	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
45	C₄×C₅×Q₈	—	2	1	no	—
46	C₄×C₅×D₄	—	2	2	no	—
47	C₄×C₅×D₄	—	2	1	no	—
48	C₅×#117	C₂×C₄	1	2	yes	—
49	C₅×#117	C₂³	1	4	yes	—
50	C₅×#117	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
53	C₅×#108	—	2	2	no	—
54	C₅×#108	—	2	2	no	—
55	C₅×#107	—	2	2	no	—
56	C₅×#106	—	2	2	no	—
57	C₅×#101	—	2	2	no	—
58	C₅×#100	—	2	2	no	—
59	C₅×#99	—	2	4	no	—
60	C₅×#97	—	2	4	no	—
61	C₅×#97	C₂×C₄	1	1	yes	characteristic
62	C₅×#97	C₂×C₄	1	2	yes	—
63	C₅×#97	C₂×C₄	1	2	yes	—
64	C₅×#97	C₂×C₄	1	2	yes	—
65	C₅×#97	C₂×C₄	1	1	yes	characteristic
66	C₅×#97	C₂×C₄	1	1	yes	characteristic
67	C₅×#97	C₂×C₄	1	1	yes	characteristic
68	C₄×C₅×C₈	—	2	4	no	—
69	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
70	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
71	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
72	#2875	C₂×C₅	1	1	yes	characteristic
73	#1841	C₂×C₅	1	2	yes	complemented
74	#1833	C₂×C₅	1	2	yes	complemented
75	#1827	C₂×C₅	1	2	yes	—
76	#1813	C₂×C₅	1	1	yes	characteristic
77	#1812	C₂×C₅	1	1	yes	characteristic
78	#1773	C₂×C₅	1	2	yes	complemented
79	#1757	C₂×C₅	1	2	yes	—
80	#1747	C₂×C₅	1	2	yes	complemented
81	C₅×#41	—	2	2	no	—
82	C₅×#41	—	2	2	no	—
83	C₂×C₅×Q₈	#41	1	1	yes	characteristic
84	C₂×C₅×D₄	#41	1	2	yes	—
85	C₂×C₅×D₄	#41	1	1	yes	characteristic
86	C₂²×C₄×C₅	—	2	2	no	—
87	C₂²×C₄×C₅	—	2	1	no	—
88	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
89	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
90	C₅×M₁₆	—	2	2	no	—
91	C₅×M₁₆	—	2	2	no	—
92	C₅×M₁₆	—	2	4	no	—
93	C₅×M₁₆	—	2	4	no	—
94	C₅×M₁₆	—	2	4	no	—
95	C₅×M₁₆	—	2	4	no	—
96	C₂×C₅×C₈	—	2	2	no	—
97	C₂×C₅×C₈	—	2	2	no	—
98	C₂×C₅×C₈	—	2	4	no	—
99	C₂×C₅×C₈	—	2	4	no	—
100	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
101	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
102	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₅×C₈	#41	1	4	yes	—
107	C₂×C₅×C₈	#41	1	2	yes	—
108	C₂×C₅×C₈	#41	1	2	yes	—
109	C₅×C₄⋊C₄	—	2	2	no	—
110	C₅×C₄⋊C₄	—	2	1	no	—
111	C₅×C₄⋊C₄	—	2	4	no	—
112	C₅×C₂²⋊C₄	—	2	2	no	—
113	C₅×C₂²⋊C₄	—	2	1	no	—
114	C₅×C₂²⋊C₄	—	2	4	no	—
115	C₄²×C₅	—	2	2	no	—
116	C₄²×C₅	—	2	1	no	—
117	C₄²×C₅	—	2	4	no	—
118	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
119	C₄²×C₅	C₂²×C₄	1	2	yes	—
120	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
121	C₂×#131	C₂²×C₅	1	1	yes	characteristic
122	C₂×#131	C₂²×C₅	1	1	yes	characteristic
123	C₄×#41	C₂²×C₅	1	1	yes	characteristic
124	C₂×#108	C₂²×C₅	1	2	yes	—
125	#427	C₂²×C₅	1	2	yes	—
126	#413	C₂²×C₅	1	2	yes	—
127	#410	C₂²×C₅	1	2	yes	—
128	#405	C₄×C₅	1	2	yes	complemented
129	#405	C₄×C₅	1	2	yes	complemented
130	#405	C₂²×C₅	1	4	yes	—
131	#405	C₂²×C₅	1	1	yes	characteristic
132	#405	C₂²×C₅	1	1	yes	characteristic
133	C₄×M₁₆	C₄×C₅	1	2	yes	complemented
134	C₄×M₁₆	C₂²×C₅	1	1	yes	characteristic
135	C₂×#97	C₄×C₅	1	2	yes	complemented
136	C₂×#97	C₂²×C₅	1	1	yes	characteristic
137	#344	C₂²×C₅	1	4	yes	—
138	#339	C₂²×C₅	1	4	yes	—
139	#338	C₂²×C₅	1	2	yes	—
140	H(3,ℤ/4)	C₂²×C₅	1	2	yes	—
141	H(3,ℤ/4)	C₂²×C₅	1	4	yes	—
142	C₂³×C₅	#117	1	2	yes	—
143	C₂³×C₅	C₂×#41	1	1	yes	characteristic
144	C₅×Q₈	—	2	2	no	—
145	C₅×D₄	—	2	4	no	—
146	C₅×D₄	—	2	2	no	—
147	C₂×C₄×C₅	—	2	2	no	—
148	C₂×C₄×C₅	—	2	2	no	—
149	C₂×C₄×C₅	—	2	1	no	—
150	C₂×C₄×C₅	—	2	2	no	—
151	C₂×C₄×C₅	—	2	1	no	—
152	C₂×C₄×C₅	—	4	4	no	—
153	C₂×C₄×C₅	—	8	1	no	—
154	C₂×C₄×C₅	—	8	2	no	—
155	C₂×C₄×C₅	#117	1	2	yes	—
156	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
157	C₂×C₄×C₅	C₂×#41	1	2	yes	—
158	C₂×C₄×C₅	C₂×#41	1	2	yes	—
159	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
160	C₅×C₈	—	2	2	no	—
161	C₅×C₈	—	2	2	no	—
162	C₅×C₈	—	4	4	no	—
163	C₅×C₈	#117	1	1	yes	characteristic
164	C₅×C₈	#117	1	1	yes	characteristic
165	C₅×C₈	#117	1	2	yes	—
166	C₅×C₈	#117	1	2	yes	—
167	C₅×C₈	#117	1	2	yes	—
168	C₂×#41	C₂³×C₅	1	1	yes	characteristic
169	#131	—	2	4	no	—
170	#131	—	2	2	no	—
171	#131	—	2	2	no	—
172	C₂×M₁₆	C₂×C₄×C₅	1	2	yes	—
173	C₂×M₁₆	C₂×C₄×C₅	1	2	yes	—
174	C₂×M₁₆	C₂³×C₅	1	4	yes	—
175	C₂×M₁₆	C₂³×C₅	1	1	yes	characteristic
176	C₂×M₁₆	C₂³×C₅	1	1	yes	characteristic
177	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
178	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
179	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
180	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
181	C₄×Q₈	—	2	1	no	—
182	C₄×D₄	—	2	2	no	—
183	C₄×D₄	—	2	1	no	—
184	#117	C₂×C₄×C₅	1	2	yes	—
185	#117	C₂³×C₅	1	4	yes	—
186	#117	C₂³×C₅	1	1	yes	characteristic
187	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
188	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
189	#108	—	2	2	no	—
190	#108	—	2	2	no	—
191	#107	—	2	2	no	—
192	#106	—	2	2	no	—
193	#101	—	2	2	no	—
194	#100	—	2	2	no	—
195	#99	—	2	4	no	—
196	#97	—	2	4	no	—
197	#97	C₂×C₄×C₅	1	1	yes	characteristic
198	#97	C₂×C₄×C₅	1	2	yes	—
199	#97	C₂×C₄×C₅	1	2	yes	—
200	#97	C₂×C₄×C₅	1	2	yes	—
201	#97	C₂×C₄×C₅	1	1	yes	characteristic
202	#97	C₂×C₄×C₅	1	1	yes	characteristic
203	#97	C₂×C₄×C₅	1	1	yes	characteristic
204	C₄×C₈	—	2	4	no	—
205	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
206	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
207	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
208	C₂²×C₅	—	2	2	no	—
209	C₂²×C₅	—	4	1	no	—
210	C₂²×C₅	—	4	2	no	—
211	C₂²×C₅	C₂×#117	1	1	yes	characteristic
212	C₂²×C₅	#529	1	2	yes	—
213	C₄×C₅	—	2	2	no	—
214	C₄×C₅	—	4	2	no	—
215	C₄×C₅	—	4	1	no	—
216	C₄×C₅	—	4	1	no	—
217	C₄×C₅	—	8	4	no	—
218	C₄×C₅	C₂×#117	1	1	yes	characteristic
219	C₄×C₅	C₂×#117	1	1	yes	characteristic
220	C₄×C₅	#529	1	2	yes	—
221	#41	—	2	2	no	—
222	#41	—	2	2	no	—
223	C₂×Q₈	C₅×#41	1	1	yes	characteristic
224	C₂×D₄	C₅×#41	1	2	yes	—
225	C₂×D₄	C₅×#41	1	1	yes	characteristic
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	1	no	—
228	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
229	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
230	M₁₆	—	2	2	no	—
231	M₁₆	—	2	2	no	—
232	M₁₆	—	2	4	no	—
233	M₁₆	—	2	4	no	—
234	M₁₆	—	2	4	no	—
235	M₁₆	—	2	4	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	—	2	4	no	—
240	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
241	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
242	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
243	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
244	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
245	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
246	C₂×C₈	C₅×#41	1	4	yes	—
247	C₂×C₈	C₅×#41	1	2	yes	—
248	C₂×C₈	C₅×#41	1	2	yes	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	2	1	no	—
251	C₄⋊C₄	—	2	4	no	—
252	C₂²⋊C₄	—	2	2	no	—
253	C₂²⋊C₄	—	2	1	no	—
254	C₂²⋊C₄	—	2	4	no	—
255	C₄²	—	2	2	no	—
256	C₄²	—	2	1	no	—
257	C₄²	—	2	4	no	—
258	C₄²	C₂²×C₄×C₅	1	2	yes	—
259	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
260	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
261	C₂×C₅	—	2	1	no	—
262	C₂×C₅	—	2	2	no	—
263	C₂×C₅	—	4	2	no	—
264	C₂×C₅	#2283	1	1	yes	characteristic
265	C₂³	C₅×#117	1	2	yes	—
266	C₂³	C₂×C₅×#41	1	1	yes	characteristic
267	Q₈	—	2	2	no	—
268	D₄	—	2	4	no	—
269	D₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	8	1	no	—
277	C₂×C₄	—	8	2	no	—
278	C₂×C₄	C₅×#117	1	2	yes	—
279	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
280	C₂×C₄	C₂×C₅×#41	1	2	yes	—
281	C₂×C₄	C₂×C₅×#41	1	2	yes	—
282	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
283	C₈	—	2	2	no	—
284	C₈	—	2	2	no	—
285	C₈	—	4	4	no	—
286	C₈	C₅×#117	1	1	yes	characteristic
287	C₈	C₅×#117	1	1	yes	characteristic
288	C₈	C₅×#117	1	2	yes	—
289	C₈	C₅×#117	1	2	yes	—
290	C₈	C₅×#117	1	2	yes	—
291	C₅	#20788	1	1	yes	characteristic, direct product factor, Sylow
292	C₂²	—	2	2	no	—
293	C₂²	—	4	1	no	—
294	C₂²	—	4	2	no	—
295	C₂²	C₂×C₅×#117	1	1	yes	characteristic
296	C₂²	C₅×#529	1	2	yes	—
297	C₄	—	2	2	no	—
298	C₄	—	4	1	no	—
299	C₄	—	4	2	no	—
300	C₄	—	4	1	no	—
301	C₄	—	8	4	no	—
302	C₄	C₂×C₅×#117	1	1	yes	characteristic
303	C₄	C₂×C₅×#117	1	1	yes	characteristic
304	C₄	C₅×#529	1	2	yes	—
305	C₂	—	2	1	no	—
306	C₂	—	2	2	no	—
307	C₂	—	4	2	no	—
308	C₂	C₅×#2283	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅	C₂×#117
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle	C₂×C₅	#2283
prime radical	C₂×C₅×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₅×#405 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#1841	C₂	semidirect
3. C₅×#1833	C₂	semidirect
7. C₅×#1773	C₂	semidirect
9. C₅×#1747	C₂	semidirect
17. C₅×#405	C₄	semidirect
18. C₅×#405	C₄	semidirect
22. C₄×C₅×M₁₆	C₄	semidirect
24. C₂×C₅×#97	C₄	semidirect
31. #20788	C₅	direct (N × H)
73. #1841	C₂×C₅	semidirect
74. #1833	C₂×C₅	semidirect
78. #1773	C₂×C₅	semidirect
80. #1747	C₂×C₅	semidirect
128. #405	C₄×C₅	semidirect
129. #405	C₄×C₅	semidirect
133. C₄×M₁₆	C₄×C₅	semidirect
135. C₂×#97	C₄×C₅	semidirect
291. C₅	#20788	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×#2875	2
C₅×#1841	2
C₅×#1833	2
C₅×#1827	2
C₅×#1813	2
C₅×#1812	2
C₅×#1773	2
C₅×#1757	2
C₅×#1747	2
#20788	5

Subgroups — C5×#20788 #11943508

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#20788 #11943508