Subgroups — C₅×#66113 #13106933

Order 1600.

556 conjugacy classes of subgroups.

The subgroup lattice has 377 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#4150	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#4147	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅²×#137	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#4096	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₅×#689	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4064	C₂	1	2	yes	maximal, complemented
7	C₅×D₅×Q₁₆	C₂	1	2	yes	maximal, complemented
8	C₅×#4062	C₂	1	2	yes	maximal, complemented
9	C₅×#4061	C₂	1	2	yes	maximal, complemented
10	C₅×#4055	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅×#4053	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅²×#41	C₂²	1	1	yes	characteristic
13	C₂×C₅²×Q₈	C₂²	1	1	yes	characteristic
14	C₅×#713	—	2	1	no	—
15	C₅×#713	C₂²	1	2	yes	—
16	C₅×Q₈×D₅	—	2	1	no	—
17	C₅×Q₈×D₅	C₂²	1	2	yes	—
18	C₅×Q₈×D₅	C₂²	1	2	yes	—
19	C₅×#711	—	2	2	no	—
20	C₅×#711	C₂²	1	2	yes	—
21	C₅×#709	—	2	2	no	—
22	C₅×#709	—	2	1	no	—
23	C₅×#709	C₂²	1	1	yes	characteristic
24	C₅×#709	C₂²	1	1	yes	characteristic
25	C₂×C₅×Dic₁₀	—	2	1	no	—
26	C₂×C₅×Dic₁₀	C₂²	1	1	yes	characteristic
27	C₅²×Q₁₆	C₂²	1	2	yes	—
28	C₅²×SD₁₆	C₂²	1	2	yes	—
29	C₅²×M₁₆	C₂²	1	1	yes	characteristic
30	C₅×#689	C₂²	1	2	yes	—
31	C₅×#689	C₂²	1	2	yes	—
32	C₅×#689	C₂²	1	1	yes	characteristic
33	C₅×#688	C₂²	1	1	yes	characteristic
34	C₅×#687	C₂²	1	1	yes	characteristic
35	C₅×#686	C₂²	1	1	yes	characteristic
36	C₂×C₅×#168	C₂²	1	1	yes	characteristic
37	C₅×Dic₂₀	C₂²	1	2	yes	—
38	C₅×#677	C₂²	1	2	yes	—
39	C₅×#676	C₂²	1	2	yes	—
40	C₅×C₈×D₅	C₂²	1	2	yes	—
41	C₅×#578	—	5	1	no	maximal
42	#66113	C₅	1	1	yes	maximal, characteristic, direct product factor
43	C₅²×Q₈	—	2	1	no	—
44	C₅²×Q₈	C₂³	1	2	yes	—
45	C₅²×Q₈	C₂³	1	1	yes	characteristic
46	C₅²×D₄	—	2	1	no	—
47	C₅²×D₄	C₂³	1	1	yes	characteristic
48	C₂×C₄×C₅²	—	2	1	no	—
49	C₂×C₄×C₅²	—	2	1	no	—
50	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
51	C₅×C₂²⋊D₅	—	2	1	no	—
52	C₅×C₂²⋊D₅	—	2	2	no	—
53	C₅×C₂²⋊D₅	D₄	1	2	yes	—
54	C₂×C₅×Dic₅	—	2	1	no	—
55	C₂×C₅×Dic₅	—	2	2	no	—
56	C₅×D₂₀	—	2	2	no	—
57	C₅×D₂₀	D₄	1	1	yes	characteristic
58	C₅×D₂₀	C₂³	1	1	yes	characteristic
59	C₄×C₅×D₅	—	2	2	no	—
60	C₄×C₅×D₅	—	2	2	no	—
61	C₄×C₅×D₅	—	2	1	no	—
62	C₄×C₅×D₅	—	2	2	no	—
63	C₄×C₅×D₅	C₂³	1	2	yes	—
64	C₅×Dic₁₀	—	2	1	no	—
65	C₅×Dic₁₀	—	2	2	no	—
66	C₅×Dic₁₀	—	2	2	no	—
67	C₅×Dic₁₀	D₄	1	1	yes	characteristic
68	C₅×Dic₁₀	C₂³	1	2	yes	—
69	C₅×Dic₁₀	C₂³	1	1	yes	characteristic
70	C₅²×C₈	C₂³	1	2	yes	—
71	C₅×#168	C₂³	1	1	yes	characteristic
72	C₅×#168	C₂³	1	1	yes	characteristic
73	C₅×2₋¹⁺⁴	—	5	1	no	—
74	C₅×2₋¹⁺⁴	—	5	1	no	—
75	#4150	C₂×C₅	1	1	yes	characteristic, complemented
76	#4147	C₂×C₅	1	1	yes	characteristic, complemented
77	C₅×#137	—	2	2	no	—
78	C₅×#137	—	5	2	no	—
79	C₅×#137	—	5	2	no	—
80	C₅×#137	—	5	1	no	—
81	C₅×#137	D₅	1	1	yes	characteristic, complemented
82	C₅×#137	C₂×C₅	1	1	yes	characteristic, complemented
83	C₅×#135	—	5	1	no	—
84	C₅×#135	—	5	2	no	—
85	C₂×C₅×Q₁₆	—	5	1	no	—
86	C₂×C₅×Q₁₆	—	5	2	no	—
87	C₅×#131	—	5	1	no	—
88	#4096	C₂×C₅	1	1	yes	characteristic, complemented
89	C₂×#689	C₂×C₅	1	1	yes	characteristic, complemented
90	#4064	C₂×C₅	1	2	yes	complemented
91	D₅×Q₁₆	C₂×C₅	1	2	yes	complemented
92	#4062	C₂×C₅	1	2	yes	complemented
93	#4061	C₂×C₅	1	2	yes	complemented
94	#4055	C₂×C₅	1	1	yes	characteristic, complemented
95	#4053	C₂×C₅	1	1	yes	characteristic, complemented
96	C₂²×C₅²	—	2	1	no	—
97	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
98	C₂×C₅×D₅	—	2	1	no	—
99	C₂×C₅×D₅	C₂×D₄	1	2	yes	—
100	C₄×C₅²	—	2	1	no	—
101	C₄×C₅²	—	2	2	no	—
102	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
103	C₄×C₅²	C₂⁴	1	1	yes	characteristic
104	C₅×Dic₅	—	2	1	no	—
105	C₅×Dic₅	—	2	2	no	—
106	C₅×Dic₅	C₂×D₄	1	2	yes	—
107	C₅×#41	—	2	2	no	—
108	C₅×#41	—	5	1	no	—
109	C₅×#41	—	5	2	no	—
110	C₅×#41	—	5	2	no	—
111	C₅×#41	—	5	1	no	—
112	C₅×#41	—	10	2	no	—
113	C₅×#41	—	10	1	no	—
114	C₅×#41	—	10	2	no	—
115	C₅×#41	—	10	1	no	—
116	C₅×#41	C₂×D₅	1	1	yes	characteristic
117	C₅×#41	C₂²×C₅	1	1	yes	characteristic
118	C₂×C₅×Q₈	—	2	2	no	—
119	C₂×C₅×Q₈	—	5	2	no	—
120	C₂×C₅×Q₈	—	5	2	no	—
121	C₂×C₅×Q₈	—	5	1	no	—
122	C₂×C₅×Q₈	—	10	1	no	—
123	C₂×C₅×Q₈	—	10	1	no	—
124	C₂×C₅×Q₈	C₂×D₅	1	1	yes	characteristic
125	C₂×C₅×Q₈	C₂²×C₅	1	1	yes	characteristic
126	#713	—	2	1	no	—
127	#713	C₂²×C₅	1	2	yes	—
128	Q₈×D₅	—	2	1	no	—
129	Q₈×D₅	C₂²×C₅	1	2	yes	—
130	Q₈×D₅	C₂²×C₅	1	2	yes	—
131	#711	—	2	2	no	—
132	#711	C₂²×C₅	1	2	yes	—
133	#709	—	2	2	no	—
134	#709	—	2	1	no	—
135	#709	C₂²×C₅	1	1	yes	characteristic
136	#709	C₂²×C₅	1	1	yes	characteristic
137	C₂×Dic₁₀	—	2	1	no	—
138	C₂×Dic₁₀	C₂²×C₅	1	1	yes	characteristic
139	C₅×Q₁₆	—	2	4	no	—
140	C₅×Q₁₆	—	5	1	no	—
141	C₅×Q₁₆	—	5	2	no	—
142	C₅×Q₁₆	—	5	2	no	—
143	C₅×Q₁₆	—	5	2	no	—
144	C₅×Q₁₆	C₂×D₅	1	2	yes	—
145	C₅×Q₁₆	C₂²×C₅	1	2	yes	—
146	C₅×SD₁₆	—	2	4	no	—
147	C₅×SD₁₆	—	5	1	no	—
148	C₅×SD₁₆	—	5	1	no	—
149	C₅×SD₁₆	—	5	2	no	—
150	C₅×SD₁₆	C₂×D₅	1	2	yes	—
151	C₅×SD₁₆	C₂²×C₅	1	2	yes	—
152	C₅×D₈	—	5	1	no	—
153	C₅×M₁₆	—	2	2	no	—
154	C₅×M₁₆	—	5	2	no	—
155	C₅×M₁₆	C₂×D₅	1	1	yes	characteristic
156	C₅×M₁₆	C₂²×C₅	1	1	yes	characteristic
157	C₂×C₅×C₈	—	5	1	no	—
158	C₂×C₅×C₈	—	5	2	no	—
159	#689	C₂²×C₅	1	1	yes	characteristic
160	#689	C₂²×C₅	1	2	yes	—
161	#689	C₂²×C₅	1	2	yes	—
162	#688	C₂²×C₅	1	1	yes	characteristic
163	#687	C₂²×C₅	1	1	yes	characteristic
164	#686	C₂²×C₅	1	1	yes	characteristic
165	C₂×#168	C₂²×C₅	1	1	yes	characteristic
166	Dic₂₀	C₂²×C₅	1	2	yes	—
167	#677	C₂²×C₅	1	2	yes	—
168	#676	C₂²×C₅	1	2	yes	—
169	C₈×D₅	C₂²×C₅	1	2	yes	—
170	#578	—	5	1	no	Sylow
171	C₂×C₅²	—	2	1	no	—
172	C₂×C₅²	—	4	1	no	—
173	C₂×C₅²	C₂²×D₄	1	1	yes	characteristic
174	C₅×D₅	—	2	2	no	—
175	C₅×D₅	—	4	1	no	—
176	C₅×Q₈	—	2	1	no	—
177	C₅×Q₈	—	2	1	no	—
178	C₅×Q₈	—	2	4	no	—
179	C₅×Q₈	—	2	2	no	—
180	C₅×Q₈	—	4	2	no	—
181	C₅×Q₈	—	5	1	no	—
182	C₅×Q₈	—	5	2	no	—
183	C₅×Q₈	—	5	1	no	—
184	C₅×Q₈	—	10	2	no	—
185	C₅×Q₈	—	10	2	no	—
186	C₅×Q₈	—	10	1	no	—
187	C₅×Q₈	C₂²×D₅	1	1	yes	characteristic
188	C₅×Q₈	C₂²×D₅	1	2	yes	—
189	C₅×Q₈	C₂³×C₅	1	1	yes	characteristic
190	C₅×Q₈	C₂³×C₅	1	2	yes	—
191	C₅×D₄	—	2	1	no	—
192	C₅×D₄	—	2	1	no	—
193	C₅×D₄	—	2	2	no	—
194	C₅×D₄	—	4	2	no	—
195	C₅×D₄	—	5	2	no	—
196	C₅×D₄	—	5	1	no	—
197	C₅×D₄	—	5	1	no	—
198	C₅×D₄	—	10	2	no	—
199	C₅×D₄	—	10	2	no	—
200	C₅×D₄	—	10	1	no	—
201	C₅×D₄	C₂²×D₅	1	1	yes	characteristic
202	C₅×D₄	C₂³×C₅	1	1	yes	characteristic
203	C₂×C₄×C₅	—	2	1	no	—
204	C₂×C₄×C₅	—	2	1	no	—
205	C₂×C₄×C₅	—	2	1	no	—
206	C₂×C₄×C₅	—	2	1	no	—
207	C₂×C₄×C₅	—	2	2	no	—
208	C₂×C₄×C₅	—	4	2	no	—
209	C₂×C₄×C₅	—	4	2	no	—
210	C₂×C₄×C₅	—	5	2	no	—
211	C₂×C₄×C₅	—	10	2	no	—
212	C₂×C₄×C₅	—	10	2	no	—
213	C₂×C₄×C₅	—	10	2	no	—
214	C₂×C₄×C₅	—	10	2	no	—
215	C₂×C₄×C₅	—	10	1	no	—
216	C₂×C₄×C₅	—	10	1	no	—
217	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
218	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
219	C₂²⋊D₅	—	2	1	no	—
220	C₂²⋊D₅	—	2	2	no	—
221	C₂²⋊D₅	C₅×D₄	1	2	yes	—
222	C₂×Dic₅	—	2	1	no	—
223	C₂×Dic₅	—	2	2	no	—
224	D₂₀	—	2	2	no	—
225	D₂₀	C₅×D₄	1	1	yes	characteristic
226	D₂₀	C₂³×C₅	1	1	yes	characteristic
227	C₄×D₅	—	2	2	no	—
228	C₄×D₅	—	2	2	no	—
229	C₄×D₅	—	2	1	no	—
230	C₄×D₅	—	2	2	no	—
231	C₄×D₅	C₂³×C₅	1	2	yes	—
232	Dic₁₀	—	2	1	no	—
233	Dic₁₀	—	2	2	no	—
234	Dic₁₀	—	2	2	no	—
235	Dic₁₀	C₅×D₄	1	1	yes	characteristic
236	Dic₁₀	C₂³×C₅	1	2	yes	—
237	Dic₁₀	C₂³×C₅	1	1	yes	characteristic
238	C₅×C₈	—	2	4	no	—
239	C₅×C₈	—	5	1	no	—
240	C₅×C₈	—	5	1	no	—
241	C₅×C₈	C₂²×D₅	1	2	yes	—
242	C₅×C₈	C₂³×C₅	1	2	yes	—
243	#168	C₂³×C₅	1	1	yes	characteristic
244	#168	C₂³×C₅	1	1	yes	characteristic
245	2₋¹⁺⁴	—	5	1	no	—
246	2₋¹⁺⁴	—	5	1	no	—
247	#137	—	5	2	no	—
248	#137	—	5	2	no	—
249	#137	—	5	1	no	—
250	#137	C₅×D₅	1	1	yes	characteristic, complemented
251	#135	—	5	1	no	—
252	#135	—	5	2	no	—
253	C₂×Q₁₆	—	5	1	no	—
254	C₂×Q₁₆	—	5	2	no	—
255	#131	—	5	1	no	—
256	C₅²	#578	1	1	yes	characteristic, complemented, Sylow
257	C₂²×C₅	—	2	1	no	—
258	C₂²×C₅	—	2	1	no	—
259	C₂²×C₅	—	2	2	no	—
260	C₂²×C₅	—	4	2	no	—
261	C₂²×C₅	—	5	2	no	—
262	C₂²×C₅	—	10	1	no	—
263	C₂²×C₅	D₄×D₅	1	1	yes	characteristic
264	C₂²×C₅	C₂×C₅×D₄	1	1	yes	characteristic
265	C₂×D₅	—	2	1	no	—
266	C₂×D₅	C₂×C₅×D₄	1	2	yes	—
267	C₄×C₅	—	2	1	no	—
268	C₄×C₅	—	2	2	no	—
269	C₄×C₅	—	2	1	no	—
270	C₄×C₅	—	2	2	no	—
271	C₄×C₅	—	2	2	no	—
272	C₄×C₅	—	2	2	no	—
273	C₄×C₅	—	4	2	no	—
274	C₄×C₅	—	4	4	no	—
275	C₄×C₅	—	5	2	no	—
276	C₄×C₅	—	10	1	no	—
277	C₄×C₅	—	10	2	no	—
278	C₄×C₅	D₄×D₅	1	1	yes	characteristic
279	C₄×C₅	C₂×C₅×D₄	1	1	yes	characteristic
280	C₄×C₅	C₂³×D₅	1	1	yes	characteristic
281	C₄×C₅	C₂⁴×C₅	1	1	yes	characteristic
282	Dic₅	—	2	1	no	—
283	Dic₅	—	2	2	no	—
284	Dic₅	C₂×C₅×D₄	1	2	yes	—
285	#41	—	5	1	no	—
286	#41	—	5	2	no	—
287	#41	—	5	2	no	—
288	#41	—	5	1	no	—
289	#41	—	10	2	no	—
290	#41	—	10	1	no	—
291	#41	—	10	2	no	—
292	#41	—	10	1	no	—
293	#41	C₂×C₅×D₅	1	1	yes	characteristic
294	C₂×Q₈	—	5	2	no	—
295	C₂×Q₈	—	5	2	no	—
296	C₂×Q₈	—	5	1	no	—
297	C₂×Q₈	—	10	1	no	—
298	C₂×Q₈	—	10	1	no	—
299	C₂×Q₈	C₂×C₅×D₅	1	1	yes	characteristic
300	Q₁₆	—	5	1	no	—
301	Q₁₆	—	5	2	no	—
302	Q₁₆	—	5	2	no	—
303	Q₁₆	—	5	2	no	—
304	Q₁₆	C₂×C₅×D₅	1	2	yes	—
305	SD₁₆	—	5	1	no	—
306	SD₁₆	—	5	1	no	—
307	SD₁₆	—	5	2	no	—
308	SD₁₆	C₂×C₅×D₅	1	2	yes	—
309	D₈	—	5	1	no	—
310	M₁₆	—	5	2	no	—
311	M₁₆	C₂×C₅×D₅	1	1	yes	characteristic
312	C₂×C₈	—	5	1	no	—
313	C₂×C₈	—	5	2	no	—
314	C₂×C₅	—	2	1	no	—
315	C₂×C₅	—	2	1	no	—
316	C₂×C₅	—	2	2	no	—
317	C₂×C₅	—	4	2	no	—
318	C₂×C₅	—	4	1	no	—
319	C₂×C₅	—	4	1	no	—
320	C₂×C₅	—	8	2	no	—
321	C₂×C₅	—	10	2	no	—
322	C₂×C₅	—	20	1	no	—
323	C₂×C₅	C₂×D₄×D₅	1	1	yes	characteristic
324	C₂×C₅	C₂²×C₅×D₄	1	1	yes	characteristic
325	D₅	—	2	2	no	—
326	D₅	—	4	1	no	—
327	Q₈	—	2	1	no	—
328	Q₈	—	5	1	no	—
329	Q₈	—	5	1	no	—
330	Q₈	—	5	2	no	—
331	Q₈	—	10	2	no	—
332	Q₈	—	10	2	no	—
333	Q₈	—	10	1	no	—
334	Q₈	C₂²×C₅×D₅	1	1	yes	characteristic
335	Q₈	C₂²×C₅×D₅	1	2	yes	—
336	D₄	—	2	1	no	—
337	D₄	—	5	2	no	—
338	D₄	—	5	1	no	—
339	D₄	—	5	1	no	—
340	D₄	—	10	2	no	—
341	D₄	—	10	2	no	—
342	D₄	—	10	1	no	—
343	D₄	C₂²×C₅×D₅	1	1	yes	characteristic
344	C₂×C₄	—	2	1	no	—
345	C₂×C₄	—	2	1	no	—
346	C₂×C₄	—	5	2	no	—
347	C₂×C₄	—	10	2	no	—
348	C₂×C₄	—	10	2	no	—
349	C₂×C₄	—	10	2	no	—
350	C₂×C₄	—	10	2	no	—
351	C₂×C₄	—	10	1	no	—
352	C₂×C₄	—	10	1	no	—
353	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
354	C₈	—	5	1	no	—
355	C₈	—	5	1	no	—
356	C₈	C₂²×C₅×D₅	1	2	yes	—
357	C₅	—	2	2	no	—
358	C₅	#66113	1	1	yes	characteristic, direct product factor
359	C₅	C₅×#578	1	1	yes	characteristic, complemented
360	C₂²	—	2	1	no	—
361	C₂²	—	5	2	no	—
362	C₂²	—	10	1	no	—
363	C₂²	C₅×D₄×D₅	1	1	yes	characteristic
364	C₄	—	2	1	no	—
365	C₄	—	2	2	no	—
366	C₄	—	5	2	no	—
367	C₄	—	10	1	no	—
368	C₄	—	10	2	no	—
369	C₄	C₅×D₄×D₅	1	1	yes	characteristic
370	C₄	C₂³×C₅×D₅	1	1	yes	characteristic
371	C₂	—	2	1	no	—
372	C₂	—	4	1	no	—
373	C₂	—	10	2	no	—
374	C₂	—	20	1	no	—
375	C₂	C₂×C₅×D₄×D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₅	C₂×D₄×D₅
commutator, cosocle	C₄×C₅	C₂⁴×C₅
Frattini	C₄	C₂³×C₅×D₅
Fitting	C₅²×#137	C₂
socle	C₂×C₅²	C₂²×D₄
hypercenter	C₅×#137	D₅
hypocenter	C₅	C₅×#578
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₅×#137

Complements

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

N	H ≅ G/N	type
1. C₅×#4150	C₂	semidirect
2. C₅×#4147	C₂	semidirect
3. C₅²×#137	C₂	semidirect
4. C₅×#4096	C₂	semidirect
5. C₂×C₅×#689	C₂	semidirect
6. C₅×#4064	C₂	semidirect
7. C₅×D₅×Q₁₆	C₂	semidirect
8. C₅×#4062	C₂	semidirect
9. C₅×#4061	C₂	semidirect
10. C₅×#4055	C₂	semidirect
11. C₅×#4053	C₂	semidirect
42. #66113	C₅	direct (N × H)
75. #4150	C₂×C₅	semidirect
76. #4147	C₂×C₅	semidirect
81. C₅×#137	D₅	semidirect
82. C₅×#137	C₂×C₅	semidirect
88. #4096	C₂×C₅	semidirect
89. C₂×#689	C₂×C₅	semidirect
90. #4064	C₂×C₅	semidirect
91. D₅×Q₁₆	C₂×C₅	semidirect
92. #4062	C₂×C₅	semidirect
93. #4061	C₂×C₅	semidirect
94. #4055	C₂×C₅	semidirect
95. #4053	C₂×C₅	semidirect
250. #137	C₅×D₅	semidirect
256. C₅²	#578	semidirect
358. C₅	#66113	direct (N × H)
359. C₅	C₅×#578	semidirect

Maximal subgroups

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

subgroup	index
C₅×#4150	2
C₅×#4147	2
C₅²×#137	2
C₅×#4096	2
C₂×C₅×#689	2
C₅×#4064	2
C₅×D₅×Q₁₆	2
C₅×#4062	2
C₅×#4061	2
C₅×#4055	2
C₅×#4053	2
C₅×#578	5
#66113	5

Subgroups — C5×#66113 #13106933

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#66113 #13106933