Subgroups — C₅×#66102 #13106922

Order 1600.

586 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×D₅×#41	C₂	1	2	yes	maximal, complemented
2	C₅²×#135	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#4096	C₂	1	2	yes	maximal, complemented
4	C₅×#4064	C₂	1	2	yes	maximal, complemented
5	C₅×#4061	C₂	1	2	yes	maximal, complemented
6	C₅×#4060	C₂	1	2	yes	maximal, complemented
7	C₅×#4057	C₂	1	2	yes	maximal, complemented
8	C₅×#4050	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₅×#676	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅²×#41	C₂²	1	2	yes	complemented
11	C₅×#713	C₂²	1	2	yes	—
12	C₅×Q₈×D₅	C₂²	1	2	yes	complemented
13	C₅×#711	—	2	2	no	—
14	C₅×#711	C₂²	1	2	yes	—
15	C₅×D₄×D₅	—	2	2	no	—
16	C₅×D₄×D₅	C₂²	1	2	yes	complemented
17	C₅×#709	—	2	2	no	—
18	C₅×#709	C₂²	1	2	yes	complemented
19	C₂×C₄×C₅×D₅	—	2	2	no	—
20	C₂×C₄×C₅×D₅	C₂²	1	1	yes	characteristic
21	C₅²×Q₁₆	C₂²	1	1	yes	characteristic, complemented
22	C₅²×SD₁₆	C₂²	1	2	yes	complemented
23	C₅²×D₈	C₂²	1	1	yes	characteristic, complemented
24	C₂×C₅²×C₈	C₂²	1	1	yes	characteristic, complemented
25	C₅×#689	C₂²	1	2	yes	complemented
26	C₅×#688	C₂²	1	2	yes	complemented
27	C₅×#687	C₂²	1	2	yes	complemented
28	C₅×#686	C₂²	1	2	yes	complemented
29	C₂×C₅×#168	C₂²	1	1	yes	characteristic, complemented
30	C₅×Dic₂₀	C₂²	1	1	yes	characteristic, complemented
31	C₅×D₄₀	C₂²	1	1	yes	characteristic, complemented
32	C₅×#677	C₂²	1	2	yes	complemented
33	C₅×#676	C₂²	1	2	yes	complemented
34	C₅×#676	C₂²	1	2	yes	complemented
35	C₅×#575	—	5	1	no	maximal
36	#66102	C₅	1	1	yes	maximal, characteristic, direct product factor
37	C₂²×C₅×D₅	—	2	2	no	—
38	C₂²×C₅×D₅	D₄	1	1	yes	characteristic
39	C₅²×Q₈	C₂³	1	2	yes	—
40	C₅²×D₄	—	2	2	no	—
41	C₅²×D₄	C₂³	1	2	yes	—
42	C₂×C₄×C₅²	—	2	2	no	—
43	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
44	C₅×C₂²⋊D₅	—	2	2	no	—
45	C₅×C₂²⋊D₅	—	2	2	no	—
46	C₅×C₂²⋊D₅	—	2	2	no	—
47	C₂×C₅×Dic₅	—	2	2	no	—
48	C₂×C₅×Dic₅	D₄	1	1	yes	characteristic
49	C₅×D₂₀	—	2	2	no	—
50	C₅×D₂₀	C₂³	1	2	yes	—
51	C₄×C₅×D₅	—	2	2	no	—
52	C₄×C₅×D₅	—	2	2	no	—
53	C₄×C₅×D₅	—	2	2	no	—
54	C₄×C₅×D₅	D₄	1	2	yes	—
55	C₄×C₅×D₅	C₂³	1	2	yes	—
56	C₅×Dic₁₀	—	2	2	no	—
57	C₅×Dic₁₀	C₂³	1	2	yes	—
58	C₅²×C₈	C₂³	1	1	yes	characteristic
59	C₅²×C₈	C₂³	1	1	yes	characteristic
60	C₅×#168	C₂³	1	2	yes	—
61	C₂×C₅×#41	—	5	2	no	—
62	D₅×#41	C₂×C₅	1	2	yes	complemented
63	C₅×#137	—	5	2	no	—
64	C₅×#137	—	5	2	no	—
65	C₅×AGL(1,ℤ/8)	—	5	2	no	—
66	C₅×AGL(1,ℤ/8)	—	5	2	no	—
67	C₅×#135	—	2	2	no	—
68	C₅×#135	—	5	2	no	—
69	C₅×#135	—	5	1	no	—
70	C₅×#135	D₅	1	1	yes	characteristic, complemented
71	C₅×#135	C₂×C₅	1	1	yes	characteristic, complemented
72	C₂×C₅×M₁₆	—	5	1	no	—
73	#4096	C₂×C₅	1	2	yes	complemented
74	#4064	C₂×C₅	1	2	yes	complemented
75	#4061	C₂×C₅	1	2	yes	complemented
76	#4060	C₂×C₅	1	2	yes	complemented
77	#4057	C₂×C₅	1	2	yes	complemented
78	#4050	C₂×C₅	1	1	yes	characteristic, complemented
79	C₂×#676	C₂×C₅	1	1	yes	characteristic, complemented
80	C₂²×C₅²	—	2	2	no	—
81	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
82	C₂×C₅×D₅	—	2	2	no	—
83	C₂×C₅×D₅	—	4	1	no	—
84	C₂×C₅×D₅	—	4	4	no	—
85	C₂×C₅×D₅	C₂×D₄	1	2	yes	—
86	C₄×C₅²	—	2	2	no	—
87	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
88	C₄×C₅²	C₂⁴	1	1	yes	characteristic
89	C₅×Dic₅	—	2	2	no	—
90	C₅×Dic₅	C₂×D₄	1	2	yes	—
91	C₅×#41	—	2	4	no	—
92	C₅×#41	—	5	2	no	—
93	C₅×#41	—	5	2	no	—
94	C₅×#41	—	5	2	no	—
95	C₅×#41	—	10	2	no	—
96	C₅×#41	—	10	2	no	—
97	C₅×#41	C₂×D₅	1	2	yes	complemented
98	C₅×#41	C₂²×C₅	1	2	yes	complemented
99	C₂×C₅×Q₈	—	5	2	no	—
100	C₂×C₅×D₄	—	5	2	no	—
101	C₂×C₅×D₄	—	10	2	no	—
102	C₂²×C₄×C₅	—	5	1	no	—
103	C₂²×C₄×C₅	—	10	2	no	—
104	#713	C₂²×C₅	1	2	yes	—
105	Q₈×D₅	C₂²×C₅	1	2	yes	complemented
106	#711	—	2	2	no	—
107	#711	C₂²×C₅	1	2	yes	—
108	D₄×D₅	—	2	2	no	—
109	D₄×D₅	C₂²×C₅	1	2	yes	complemented
110	#709	—	2	2	no	—
111	#709	C₂²×C₅	1	2	yes	complemented
112	C₂×C₄×D₅	—	2	2	no	—
113	C₂×C₄×D₅	C₂²×C₅	1	1	yes	characteristic
114	C₅×Q₁₆	—	2	2	no	—
115	C₅×Q₁₆	—	5	2	no	—
116	C₅×Q₁₆	—	5	1	no	—
117	C₅×Q₁₆	C₂×D₅	1	1	yes	characteristic, complemented
118	C₅×Q₁₆	C₂²×C₅	1	1	yes	characteristic, complemented
119	C₅×SD₁₆	—	2	4	no	—
120	C₅×SD₁₆	—	5	2	no	—
121	C₅×SD₁₆	—	5	2	no	—
122	C₅×SD₁₆	—	5	2	no	—
123	C₅×SD₁₆	C₂×D₅	1	2	yes	complemented
124	C₅×SD₁₆	C₂²×C₅	1	2	yes	complemented
125	C₅×D₈	—	2	2	no	—
126	C₅×D₈	—	5	2	no	—
127	C₅×D₈	—	5	1	no	—
128	C₅×D₈	C₂×D₅	1	1	yes	characteristic, complemented
129	C₅×D₈	C₂²×C₅	1	1	yes	characteristic, complemented
130	C₅×M₁₆	—	5	2	no	—
131	C₅×M₁₆	—	5	2	no	—
132	C₂×C₅×C₈	—	2	2	no	—
133	C₂×C₅×C₈	—	5	1	no	—
134	C₂×C₅×C₈	C₂×D₅	1	1	yes	characteristic, complemented
135	C₂×C₅×C₈	C₂²×C₅	1	1	yes	characteristic, complemented
136	#689	C₂²×C₅	1	2	yes	complemented
137	#688	C₂²×C₅	1	2	yes	complemented
138	#687	C₂²×C₅	1	2	yes	complemented
139	#686	C₂²×C₅	1	2	yes	complemented
140	C₂×#168	C₂²×C₅	1	1	yes	characteristic, complemented
141	Dic₂₀	C₂²×C₅	1	1	yes	characteristic, complemented
142	D₄₀	C₂²×C₅	1	1	yes	characteristic, complemented
143	#677	C₂²×C₅	1	2	yes	complemented
144	#676	C₂²×C₅	1	2	yes	complemented
145	#676	C₂²×C₅	1	2	yes	complemented
146	#575	—	5	1	no	Sylow
147	C₂×C₅²	—	2	1	no	—
148	C₂×C₅²	—	4	2	no	—
149	C₂×C₅²	C₂²×D₄	1	1	yes	characteristic
150	C₅×D₅	—	2	2	no	—
151	C₅×D₅	—	4	2	no	—
152	C₂³×C₅	—	5	1	no	—
153	C₂³×C₅	—	10	2	no	—
154	C₂²×D₅	—	2	2	no	—
155	C₂²×D₅	C₅×D₄	1	1	yes	characteristic
156	C₅×Q₈	—	2	4	no	—
157	C₅×Q₈	—	5	2	no	—
158	C₅×Q₈	—	10	2	no	—
159	C₅×Q₈	C₂²×D₅	1	2	yes	—
160	C₅×Q₈	C₂³×C₅	1	2	yes	—
161	C₅×D₄	—	2	2	no	—
162	C₅×D₄	—	2	2	no	—
163	C₅×D₄	—	2	4	no	—
164	C₅×D₄	—	4	4	no	—
165	C₅×D₄	—	5	2	no	—
166	C₅×D₄	—	10	2	no	—
167	C₅×D₄	—	10	2	no	—
168	C₅×D₄	—	10	2	no	—
169	C₅×D₄	—	10	2	no	—
170	C₅×D₄	C₂²×D₅	1	2	yes	—
171	C₅×D₄	C₂³×C₅	1	2	yes	—
172	C₂×C₄×C₅	—	2	2	no	—
173	C₂×C₄×C₅	—	2	2	no	—
174	C₂×C₄×C₅	—	2	2	no	—
175	C₂×C₄×C₅	—	4	4	no	—
176	C₂×C₄×C₅	—	5	1	no	—
177	C₂×C₄×C₅	—	5	2	no	—
178	C₂×C₄×C₅	—	5	2	no	—
179	C₂×C₄×C₅	—	10	2	no	—
180	C₂×C₄×C₅	—	10	2	no	—
181	C₂×C₄×C₅	—	10	2	no	—
182	C₂×C₄×C₅	—	10	2	no	—
183	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
184	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
185	C₂²⋊D₅	—	2	2	no	—
186	C₂²⋊D₅	—	2	2	no	—
187	C₂²⋊D₅	—	2	2	no	—
188	C₂×Dic₅	—	2	2	no	—
189	C₂×Dic₅	C₅×D₄	1	1	yes	characteristic
190	D₂₀	—	2	2	no	—
191	D₂₀	C₂³×C₅	1	2	yes	—
192	C₄×D₅	—	2	2	no	—
193	C₄×D₅	—	2	2	no	—
194	C₄×D₅	—	2	2	no	—
195	C₄×D₅	C₅×D₄	1	2	yes	—
196	C₄×D₅	C₂³×C₅	1	2	yes	—
197	Dic₁₀	—	2	2	no	—
198	Dic₁₀	C₂³×C₅	1	2	yes	—
199	C₅×C₈	—	2	2	no	—
200	C₅×C₈	—	2	2	no	—
201	C₅×C₈	—	5	2	no	—
202	C₅×C₈	C₂²×D₅	1	1	yes	characteristic
203	C₅×C₈	C₂²×D₅	1	1	yes	characteristic
204	C₅×C₈	C₂³×C₅	1	1	yes	characteristic
205	C₅×C₈	C₂³×C₅	1	1	yes	characteristic
206	#168	C₂³×C₅	1	2	yes	—
207	C₂×#41	—	5	2	no	—
208	#137	—	5	2	no	—
209	#137	—	5	2	no	—
210	AGL(1,ℤ/8)	—	5	2	no	—
211	AGL(1,ℤ/8)	—	5	2	no	—
212	#135	—	5	2	no	—
213	#135	—	5	1	no	—
214	#135	C₅×D₅	1	1	yes	characteristic, complemented
215	C₂×M₁₆	—	5	1	no	—
216	C₅²	#575	1	1	yes	characteristic, complemented, Sylow
217	C₂²×C₅	—	2	2	no	—
218	C₂²×C₅	—	2	2	no	—
219	C₂²×C₅	—	2	2	no	—
220	C₂²×C₅	—	4	4	no	—
221	C₂²×C₅	—	5	2	no	—
222	C₂²×C₅	—	10	2	no	—
223	C₂²×C₅	—	20	1	no	—
224	C₂²×C₅	—	20	4	no	—
225	C₂²×C₅	D₄×D₅	1	1	yes	characteristic
226	C₂²×C₅	C₂×C₅×D₄	1	1	yes	characteristic
227	C₂×D₅	—	2	2	no	—
228	C₂×D₅	—	4	1	no	—
229	C₂×D₅	—	4	4	no	—
230	C₂×D₅	C₂×C₅×D₄	1	2	yes	—
231	C₄×C₅	—	2	2	no	—
232	C₄×C₅	—	2	2	no	—
233	C₄×C₅	—	2	2	no	—
234	C₄×C₅	—	2	2	no	—
235	C₄×C₅	—	4	4	no	—
236	C₄×C₅	—	5	2	no	—
237	C₄×C₅	—	10	2	no	—
238	C₄×C₅	D₄×D₅	1	1	yes	characteristic
239	C₄×C₅	C₂×C₅×D₄	1	1	yes	characteristic
240	C₄×C₅	C₂³×D₅	1	1	yes	characteristic
241	C₄×C₅	C₂⁴×C₅	1	1	yes	characteristic
242	Dic₅	—	2	2	no	—
243	Dic₅	C₂×C₅×D₄	1	2	yes	—
244	#41	—	5	2	no	—
245	#41	—	5	2	no	—
246	#41	—	5	2	no	—
247	#41	—	10	2	no	—
248	#41	—	10	2	no	—
249	#41	C₂×C₅×D₅	1	2	yes	complemented
250	C₂×Q₈	—	5	2	no	—
251	C₂×D₄	—	5	2	no	—
252	C₂×D₄	—	10	2	no	—
253	C₂²×C₄	—	5	1	no	—
254	C₂²×C₄	—	10	2	no	—
255	Q₁₆	—	5	2	no	—
256	Q₁₆	—	5	1	no	—
257	Q₁₆	C₂×C₅×D₅	1	1	yes	characteristic, complemented
258	SD₁₆	—	5	2	no	—
259	SD₁₆	—	5	2	no	—
260	SD₁₆	—	5	2	no	—
261	SD₁₆	C₂×C₅×D₅	1	2	yes	complemented
262	D₈	—	5	2	no	—
263	D₈	—	5	1	no	—
264	D₈	C₂×C₅×D₅	1	1	yes	characteristic, complemented
265	M₁₆	—	5	2	no	—
266	M₁₆	—	5	2	no	—
267	C₂×C₈	—	5	1	no	—
268	C₂×C₈	C₂×C₅×D₅	1	1	yes	characteristic, complemented
269	C₂×C₅	—	2	1	no	—
270	C₂×C₅	—	2	1	no	—
271	C₂×C₅	—	2	2	no	—
272	C₂×C₅	—	4	2	no	—
273	C₂×C₅	—	4	2	no	—
274	C₂×C₅	—	4	2	no	—
275	C₂×C₅	—	8	4	no	—
276	C₂×C₅	—	10	2	no	—
277	C₂×C₅	—	20	2	no	—
278	C₂×C₅	C₂×D₄×D₅	1	1	yes	characteristic
279	C₂×C₅	C₂²×C₅×D₄	1	1	yes	characteristic
280	D₅	—	2	2	no	—
281	D₅	—	4	2	no	—
282	C₂³	—	5	1	no	—
283	C₂³	—	10	2	no	—
284	Q₈	—	5	2	no	—
285	Q₈	—	10	2	no	—
286	Q₈	C₂²×C₅×D₅	1	2	yes	—
287	D₄	—	2	2	no	—
288	D₄	—	5	2	no	—
289	D₄	—	10	2	no	—
290	D₄	—	10	2	no	—
291	D₄	—	10	2	no	—
292	D₄	—	10	2	no	—
293	D₄	C₂²×C₅×D₅	1	2	yes	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	5	1	no	—
296	C₂×C₄	—	5	2	no	—
297	C₂×C₄	—	5	2	no	—
298	C₂×C₄	—	10	2	no	—
299	C₂×C₄	—	10	2	no	—
300	C₂×C₄	—	10	2	no	—
301	C₂×C₄	—	10	2	no	—
302	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
303	C₈	—	5	2	no	—
304	C₈	C₂²×C₅×D₅	1	1	yes	characteristic
305	C₈	C₂²×C₅×D₅	1	1	yes	characteristic
306	C₅	—	2	2	no	—
307	C₅	#66102	1	1	yes	characteristic, direct product factor
308	C₅	C₅×#575	1	1	yes	characteristic, complemented
309	C₂²	—	2	2	no	—
310	C₂²	—	5	2	no	—
311	C₂²	—	10	2	no	—
312	C₂²	—	20	1	no	—
313	C₂²	—	20	4	no	—
314	C₂²	C₅×D₄×D₅	1	1	yes	characteristic
315	C₄	—	2	2	no	—
316	C₄	—	5	2	no	—
317	C₄	—	10	2	no	—
318	C₄	C₅×D₄×D₅	1	1	yes	characteristic
319	C₄	C₂³×C₅×D₅	1	1	yes	characteristic
320	C₂	—	2	1	no	—
321	C₂	—	4	2	no	—
322	C₂	—	10	2	no	—
323	C₂	—	20	2	no	—
324	C₂	C₂×C₅×D₄×D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅	D₄×D₅
commutator, cosocle	C₄×C₅	C₂⁴×C₅
Frattini	C₄	C₂³×C₅×D₅
Fitting	C₅²×#135	C₂
socle	C₂×C₅²	C₂²×D₄
hypercenter	C₅×#135	D₅
hypocenter	C₅	C₅×#575
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄×C₅ ◁ C₂×C₄×C₅ ◁ C₅×#135

Complements

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

N	H ≅ G/N	type
1. C₅×D₅×#41	C₂	semidirect
2. C₅²×#135	C₂	semidirect
3. C₅×#4096	C₂	semidirect
4. C₅×#4064	C₂	semidirect
5. C₅×#4061	C₂	semidirect
6. C₅×#4060	C₂	semidirect
7. C₅×#4057	C₂	semidirect
8. C₅×#4050	C₂	semidirect
9. C₂×C₅×#676	C₂	semidirect
10. C₅²×#41	C₂²	semidirect
12. C₅×Q₈×D₅	C₂²	semidirect
16. C₅×D₄×D₅	C₂²	semidirect
18. C₅×#709	C₂²	semidirect
21. C₅²×Q₁₆	C₂²	semidirect
22. C₅²×SD₁₆	C₂²	semidirect
23. C₅²×D₈	C₂²	semidirect
24. C₂×C₅²×C₈	C₂²	semidirect
25. C₅×#689	C₂²	semidirect
26. C₅×#688	C₂²	semidirect
27. C₅×#687	C₂²	semidirect
28. C₅×#686	C₂²	semidirect
29. C₂×C₅×#168	C₂²	semidirect
30. C₅×Dic₂₀	C₂²	semidirect
31. C₅×D₄₀	C₂²	semidirect
32. C₅×#677	C₂²	semidirect
33. C₅×#676	C₂²	semidirect
34. C₅×#676	C₂²	semidirect
36. #66102	C₅	direct (N × H)
62. D₅×#41	C₂×C₅	semidirect
70. C₅×#135	D₅	semidirect
71. C₅×#135	C₂×C₅	semidirect
73. #4096	C₂×C₅	semidirect
74. #4064	C₂×C₅	semidirect
75. #4061	C₂×C₅	semidirect
76. #4060	C₂×C₅	semidirect
77. #4057	C₂×C₅	semidirect
78. #4050	C₂×C₅	semidirect
79. C₂×#676	C₂×C₅	semidirect
97. C₅×#41	C₂×D₅	semidirect
98. C₅×#41	C₂²×C₅	semidirect
105. Q₈×D₅	C₂²×C₅	semidirect
109. D₄×D₅	C₂²×C₅	semidirect
111. #709	C₂²×C₅	semidirect
117. C₅×Q₁₆	C₂×D₅	semidirect
118. C₅×Q₁₆	C₂²×C₅	semidirect
123. C₅×SD₁₆	C₂×D₅	semidirect
124. C₅×SD₁₆	C₂²×C₅	semidirect
128. C₅×D₈	C₂×D₅	semidirect
129. C₅×D₈	C₂²×C₅	semidirect
134. C₂×C₅×C₈	C₂×D₅	semidirect
135. C₂×C₅×C₈	C₂²×C₅	semidirect
136. #689	C₂²×C₅	semidirect
137. #688	C₂²×C₅	semidirect
138. #687	C₂²×C₅	semidirect
139. #686	C₂²×C₅	semidirect
140. C₂×#168	C₂²×C₅	semidirect
141. Dic₂₀	C₂²×C₅	semidirect
142. D₄₀	C₂²×C₅	semidirect
143. #677	C₂²×C₅	semidirect
144. #676	C₂²×C₅	semidirect
145. #676	C₂²×C₅	semidirect
214. #135	C₅×D₅	semidirect
216. C₅²	#575	semidirect
249. #41	C₂×C₅×D₅	semidirect
257. Q₁₆	C₂×C₅×D₅	semidirect
261. SD₁₆	C₂×C₅×D₅	semidirect
264. D₈	C₂×C₅×D₅	semidirect
268. C₂×C₈	C₂×C₅×D₅	semidirect
307. C₅	#66102	direct (N × H)
308. C₅	C₅×#575	semidirect

Maximal subgroups

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

subgroup	index
C₅×D₅×#41	2
C₅²×#135	2
C₅×#4096	2
C₅×#4064	2
C₅×#4061	2
C₅×#4060	2
C₅×#4057	2
C₅×#4050	2
C₂×C₅×#676	2
C₅×#575	5
#66102	5

Subgroups — C5×#66102 #13106922

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#66102 #13106922