Subgroups — C₅×#10613505 #72223046

Order 3200.

1062 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#66096	C₂	1	4	yes	maximal, complemented
2	C₅²×#492	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#65466	C₂	1	4	yes	maximal, complemented
4	C₅×#65384	C₂	1	2	yes	maximal, complemented
5	C₅×#65366	C₂	1	2	yes	maximal, complemented
6	C₅×#64989	C₂	1	2	yes	maximal, complemented
7	C₅×#4147	C₂²	1	2	yes	—
8	C₅×#4144	C₂²	1	2	yes	complemented
9	C₂×C₅²×SD₁₆	C₂²	1	4	yes	complemented
10	C₅²×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
11	C₅²×#127	C₂²	1	1	yes	characteristic
12	C₅×#4088	C₂²	1	4	yes	complemented
13	C₅×#4078	C₂²	1	4	yes	—
14	C₅×#4062	—	2	4	no	—
15	C₅×#4061	—	2	4	no	—
16	C₅×#4060	—	2	4	no	—
17	C₅×D₅×SD₁₆	—	2	4	no	—
18	C₅×#4050	C₂²	1	2	yes	—
19	C₅×#4047	C₂²	1	4	yes	complemented
20	C₄×C₅²×C₈	C₂²	1	1	yes	characteristic, complemented
21	C₅×#3968	C₂²	1	2	yes	complemented
22	C₅×#3965	C₂²	1	2	yes	complemented
23	C₅×#3951	C₂²	1	2	yes	complemented
24	C₅×#3937	C₂²	1	4	yes	complemented
25	C₅×#3291	—	5	1	no	maximal
26	#10613505	C₅	1	1	yes	maximal, characteristic, direct product factor
27	C₂×C₅²×Q₈	C₂³	1	2	yes	—
28	C₂×C₅²×D₄	—	4	1	no	—
29	C₂×C₅²×D₄	C₂³	1	2	yes	—
30	C₂×C₅×C₂²⋊D₅	—	4	2	no	—
31	C₅×#713	—	2	4	no	—
32	C₅×Q₈×D₅	—	2	4	no	—
33	C₅×#711	—	2	4	no	—
34	C₅×D₄×D₅	—	2	4	no	—
35	C₅×#709	—	4	2	no	—
36	C₅×#709	C₂³	1	4	yes	—
37	C₅²×SD₁₆	—	2	4	no	—
38	C₅²×SD₁₆	—	2	4	no	—
39	C₂×C₅²×C₈	C₂³	1	2	yes	—
40	C₅²×C₄⋊C₄	—	4	1	no	—
41	C₄²×C₅²	C₂³	1	1	yes	characteristic
42	C₅×#689	—	2	4	no	—
43	C₅×#688	—	2	4	no	—
44	C₅×#687	—	2	4	no	—
45	C₅×#686	—	2	4	no	—
46	C₅×#681	C₂³	1	4	yes	—
47	C₅×Dic₂₀	D₄	1	2	yes	complemented
48	C₅×D₄₀	D₄	1	2	yes	complemented
49	C₅×#677	D₄	1	4	yes	complemented
50	C₅×#676	—	2	4	no	—
51	C₅×C₈×D₅	—	2	4	no	—
52	C₅×#577	—	5	4	no	—
53	#66096	C₂×C₅	1	4	yes	complemented
54	C₅×#492	—	2	2	no	—
55	C₅×#492	D₅	1	1	yes	characteristic, complemented
56	C₅×#492	C₂×C₅	1	1	yes	characteristic, complemented
57	C₅×#471	—	5	4	no	—
58	C₅×#456	—	5	2	no	—
59	C₅×#453	—	5	2	no	—
60	C₅×#443	—	5	2	no	—
61	#65466	C₂×C₅	1	4	yes	complemented
62	#65384	C₂×C₅	1	2	yes	complemented
63	#65366	C₂×C₅	1	2	yes	complemented
64	#64989	C₂×C₅	1	2	yes	complemented
65	C₂³×C₅²	—	2	2	no	—
66	C₂²×C₅×D₅	—	4	2	no	—
67	C₅²×Q₈	—	2	4	no	—
68	C₅²×D₄	—	2	4	no	—
69	C₅²×D₄	—	8	2	no	—
70	C₂×C₄×C₅²	—	2	1	no	—
71	C₂×C₄×C₅²	—	2	2	no	—
72	C₂×C₄×C₅²	C₂⁴	1	1	yes	characteristic
73	C₅×C₂²⋊D₅	—	2	4	no	—
74	C₅×C₂²⋊D₅	—	4	4	no	—
75	C₂×C₅×Dic₅	—	4	2	no	—
76	C₅×D₂₀	—	4	2	no	—
77	C₅×D₂₀	C₂×D₄	1	4	yes	—
78	C₄×C₅×D₅	—	2	4	no	—
79	C₄×C₅×D₅	—	4	4	no	—
80	C₅×Dic₁₀	—	4	2	no	—
81	C₅×Dic₁₀	C₂×D₄	1	4	yes	—
82	C₅²×C₈	C₂×D₄	1	2	yes	—
83	C₅²×C₈	C₂×D₄	1	2	yes	—
84	C₅×#168	—	2	4	no	—
85	C₅×2₋¹⁺⁴	—	5	2	no	—
86	C₅×H(5,𝔽₂)	—	5	2	no	—
87	#4147	C₂²×C₅	1	2	yes	—
88	#4144	C₂²×C₅	1	2	yes	complemented
89	C₅×#137	—	5	4	no	—
90	C₅×#137	—	10	4	no	—
91	C₅×AGL(1,ℤ/8)	—	5	4	no	—
92	C₅×AGL(1,ℤ/8)	—	10	4	no	—
93	C₅×#135	—	5	2	no	—
94	C₅×#135	—	10	4	no	—
95	C₂×C₅×SD₁₆	—	2	8	no	—
96	C₂×C₅×SD₁₆	—	10	4	no	—
97	C₂×C₅×SD₁₆	C₂×D₅	1	4	yes	complemented
98	C₂×C₅×SD₁₆	C₂²×C₅	1	4	yes	complemented
99	C₅×#131	—	5	4	no	—
100	C₅×C₄⋊Q₈	—	2	2	no	—
101	C₅×C₄⋊Q₈	C₂×D₅	1	1	yes	characteristic, complemented
102	C₅×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic, complemented
103	C₅×#127	—	2	2	no	—
104	C₅×#127	C₂×D₅	1	1	yes	characteristic
105	C₅×#127	C₂²×C₅	1	1	yes	characteristic
106	#4088	C₂²×C₅	1	4	yes	complemented
107	#4078	C₂²×C₅	1	4	yes	—
108	#4062	—	2	4	no	—
109	#4061	—	2	4	no	—
110	#4060	—	2	4	no	—
111	D₅×SD₁₆	—	2	4	no	—
112	#4050	C₂²×C₅	1	2	yes	—
113	#4047	C₂²×C₅	1	4	yes	complemented
114	C₅×#108	—	5	2	no	—
115	C₅×#104	—	5	4	no	—
116	C₅×#101	—	5	2	no	—
117	C₅×#100	—	5	2	no	—
118	C₄×C₅×C₈	—	2	2	no	—
119	C₄×C₅×C₈	C₂×D₅	1	1	yes	characteristic, complemented
120	C₄×C₅×C₈	C₂²×C₅	1	1	yes	characteristic, complemented
121	#3968	C₂²×C₅	1	2	yes	complemented
122	#3965	C₂²×C₅	1	2	yes	complemented
123	#3951	C₂²×C₅	1	2	yes	complemented
124	#3937	C₂²×C₅	1	4	yes	complemented
125	#3291	—	5	1	no	Sylow
126	C₂²×C₅²	—	4	2	no	—
127	C₂²×C₅²	—	8	2	no	—
128	C₂²×C₅²	H(5,𝔽₂)	1	1	yes	characteristic
129	C₂×C₅×D₅	—	2	4	no	—
130	C₂×C₅×D₅	—	16	2	no	—
131	C₄×C₅²	—	2	2	no	—
132	C₄×C₅²	—	4	2	no	—
133	C₄×C₅²	C₂²×D₄	1	2	yes	—
134	C₅×Dic₅	—	2	4	no	—
135	C₅×#41	—	5	4	no	—
136	C₅×#41	—	10	4	no	—
137	C₅×#41	—	10	4	no	—
138	C₅×#41	—	20	2	no	—
139	C₂×C₅×Q₈	—	2	4	no	—
140	C₂×C₅×Q₈	—	10	4	no	—
141	C₂×C₅×Q₈	C₂²×D₅	1	2	yes	—
142	C₂×C₅×Q₈	C₂³×C₅	1	2	yes	—
143	C₂×C₅×D₄	—	2	4	no	—
144	C₂×C₅×D₄	—	4	1	no	—
145	C₂×C₅×D₄	—	4	1	no	—
146	C₂×C₅×D₄	—	4	4	no	—
147	C₂×C₅×D₄	—	10	4	no	—
148	C₂×C₅×D₄	—	20	2	no	—
149	C₂×C₅×D₄	C₂²×D₅	1	2	yes	—
150	C₂×C₅×D₄	C₂³×C₅	1	2	yes	—
151	C₂×C₂²⋊D₅	—	4	2	no	—
152	#713	—	2	4	no	—
153	Q₈×D₅	—	2	4	no	—
154	#711	—	2	4	no	—
155	D₄×D₅	—	2	4	no	—
156	#709	—	4	2	no	—
157	#709	C₂³×C₅	1	4	yes	—
158	C₅×Q₁₆	—	5	2	no	—
159	C₅×Q₁₆	—	10	4	no	—
160	C₅×SD₁₆	—	2	4	no	—
161	C₅×SD₁₆	—	2	4	no	—
162	C₅×SD₁₆	—	2	4	no	—
163	C₅×SD₁₆	—	2	4	no	—
164	C₅×SD₁₆	—	4	8	no	—
165	C₅×SD₁₆	—	4	8	no	—
166	C₅×SD₁₆	—	5	4	no	—
167	C₅×SD₁₆	—	10	4	no	—
168	C₅×SD₁₆	—	10	4	no	—
169	C₅×D₈	—	5	2	no	—
170	C₅×D₈	—	10	4	no	—
171	C₅×M₁₆	—	5	4	no	—
172	C₅×M₁₆	—	10	4	no	—
173	C₂×C₅×C₈	—	2	4	no	—
174	C₂×C₅×C₈	—	10	4	no	—
175	C₂×C₅×C₈	C₂²×D₅	1	2	yes	—
176	C₂×C₅×C₈	C₂³×C₅	1	2	yes	—
177	C₅×C₄⋊C₄	—	4	1	no	—
178	C₅×C₄⋊C₄	—	4	1	no	—
179	C₅×C₄⋊C₄	—	4	4	no	—
180	C₄²×C₅	—	2	2	no	—
181	C₄²×C₅	C₂²×D₅	1	1	yes	characteristic
182	C₄²×C₅	C₂³×C₅	1	1	yes	characteristic
183	#689	—	2	4	no	—
184	#688	—	2	4	no	—
185	#687	—	2	4	no	—
186	#686	—	2	4	no	—
187	#681	C₂³×C₅	1	4	yes	—
188	Dic₂₀	C₅×D₄	1	2	yes	complemented
189	D₄₀	C₅×D₄	1	2	yes	complemented
190	#677	C₅×D₄	1	4	yes	complemented
191	#676	—	2	4	no	—
192	C₈×D₅	—	2	4	no	—
193	#577	—	5	4	no	—
194	#492	C₅×D₅	1	1	yes	characteristic, complemented
195	#471	—	5	4	no	—
196	#456	—	5	2	no	—
197	#453	—	5	2	no	—
198	#443	—	5	2	no	—
199	C₂×C₅²	—	2	1	no	—
200	C₂×C₅²	—	8	2	no	—
201	C₂×C₅²	D₄²	1	1	yes	characteristic
202	C₅×D₅	—	4	4	no	—
203	C₂³×C₅	—	2	2	no	—
204	C₂³×C₅	—	2	2	no	—
205	C₂³×C₅	—	4	4	no	—
206	C₂³×C₅	—	20	2	no	—
207	C₂²×D₅	—	4	2	no	—
208	C₅×Q₈	—	2	4	no	—
209	C₅×Q₈	—	2	4	no	—
210	C₅×Q₈	—	4	8	no	—
211	C₅×Q₈	—	5	4	no	—
212	C₅×Q₈	—	20	2	no	—
213	C₅×D₄	—	2	4	no	—
214	C₅×D₄	—	2	4	no	—
215	C₅×D₄	—	4	8	no	—
216	C₅×D₄	—	5	4	no	—
217	C₅×D₄	—	8	2	no	—
218	C₅×D₄	—	8	2	no	—
219	C₅×D₄	—	8	8	no	—
220	C₅×D₄	—	10	4	no	—
221	C₅×D₄	—	20	2	no	—
222	C₅×D₄	—	20	4	no	—
223	C₂×C₄×C₅	—	2	4	no	—
224	C₂×C₄×C₅	—	2	1	no	—
225	C₂×C₄×C₅	—	2	1	no	—
226	C₂×C₄×C₅	—	2	2	no	—
227	C₂×C₄×C₅	—	2	2	no	—
228	C₂×C₄×C₅	—	2	2	no	—
229	C₂×C₄×C₅	—	4	4	no	—
230	C₂×C₄×C₅	—	10	4	no	—
231	C₂×C₄×C₅	—	20	4	no	—
232	C₂×C₄×C₅	—	20	2	no	—
233	C₂×C₄×C₅	C₂³×D₅	1	1	yes	characteristic
234	C₂×C₄×C₅	C₂⁴×C₅	1	1	yes	characteristic
235	C₂²⋊D₅	—	2	4	no	—
236	C₂²⋊D₅	—	4	4	no	—
237	C₂×Dic₅	—	4	2	no	—
238	D₂₀	—	4	2	no	—
239	D₂₀	C₂×C₅×D₄	1	4	yes	—
240	C₄×D₅	—	2	4	no	—
241	C₄×D₅	—	4	4	no	—
242	Dic₁₀	—	4	2	no	—
243	Dic₁₀	C₂×C₅×D₄	1	4	yes	—
244	C₅×C₈	—	2	4	no	—
245	C₅×C₈	—	2	4	no	—
246	C₅×C₈	—	10	4	no	—
247	C₅×C₈	D₄×D₅	1	2	yes	—
248	C₅×C₈	D₄×D₅	1	2	yes	—
249	C₅×C₈	C₂×C₅×D₄	1	2	yes	—
250	C₅×C₈	C₂×C₅×D₄	1	2	yes	—
251	#168	—	2	4	no	—
252	2₋¹⁺⁴	—	5	2	no	—
253	H(5,𝔽₂)	—	5	2	no	—
254	#137	—	5	4	no	—
255	#137	—	10	4	no	—
256	AGL(1,ℤ/8)	—	5	4	no	—
257	AGL(1,ℤ/8)	—	10	4	no	—
258	#135	—	5	2	no	—
259	#135	—	10	4	no	—
260	C₂×SD₁₆	—	10	4	no	—
261	C₂×SD₁₆	C₂×C₅×D₅	1	4	yes	complemented
262	#131	—	5	4	no	—
263	C₄⋊Q₈	C₂×C₅×D₅	1	1	yes	characteristic, complemented
264	#127	C₂×C₅×D₅	1	1	yes	characteristic
265	#108	—	5	2	no	—
266	#104	—	5	4	no	—
267	#101	—	5	2	no	—
268	#100	—	5	2	no	—
269	C₄×C₈	C₂×C₅×D₅	1	1	yes	characteristic, complemented
270	C₅²	#3291	1	1	yes	characteristic, complemented, Sylow
271	C₂²×C₅	—	2	2	no	—
272	C₂²×C₅	—	4	2	no	—
273	C₂²×C₅	—	4	2	no	—
274	C₂²×C₅	—	8	4	no	—
275	C₂²×C₅	—	8	2	no	—
276	C₂²×C₅	—	8	2	no	—
277	C₂²×C₅	—	8	8	no	—
278	C₂²×C₅	—	10	4	no	—
279	C₂²×C₅	—	80	2	no	—
280	C₂²×C₅	#4144	1	1	yes	characteristic
281	C₂²×C₅	C₅×H(5,𝔽₂)	1	1	yes	characteristic
282	C₂×D₅	—	2	4	no	—
283	C₂×D₅	—	16	2	no	—
284	C₄×C₅	—	2	8	no	—
285	C₄×C₅	—	2	2	no	—
286	C₄×C₅	—	2	2	no	—
287	C₄×C₅	—	2	4	no	—
288	C₄×C₅	—	4	2	no	—
289	C₄×C₅	—	4	2	no	—
290	C₄×C₅	—	8	4	no	—
291	C₄×C₅	—	10	4	no	—
292	C₄×C₅	C₂×D₄×D₅	1	2	yes	—
293	C₄×C₅	C₂²×C₅×D₄	1	2	yes	—
294	Dic₅	—	2	4	no	—
295	#41	—	5	4	no	—
296	#41	—	10	4	no	—
297	#41	—	10	4	no	—
298	#41	—	20	2	no	—
299	C₂×Q₈	—	10	4	no	—
300	C₂×Q₈	C₂²×C₅×D₅	1	2	yes	—
301	C₂×D₄	—	4	1	no	—
302	C₂×D₄	—	10	4	no	—
303	C₂×D₄	—	20	2	no	—
304	C₂×D₄	C₂²×C₅×D₅	1	2	yes	—
305	Q₁₆	—	5	2	no	—
306	Q₁₆	—	10	4	no	—
307	SD₁₆	—	2	4	no	—
308	SD₁₆	—	2	4	no	—
309	SD₁₆	—	5	4	no	—
310	SD₁₆	—	10	4	no	—
311	SD₁₆	—	10	4	no	—
312	D₈	—	5	2	no	—
313	D₈	—	10	4	no	—
314	M₁₆	—	5	4	no	—
315	M₁₆	—	10	4	no	—
316	C₂×C₈	—	10	4	no	—
317	C₂×C₈	C₂²×C₅×D₅	1	2	yes	—
318	C₄⋊C₄	—	4	1	no	—
319	C₄²	C₂²×C₅×D₅	1	1	yes	characteristic
320	C₂×C₅	—	2	4	no	—
321	C₂×C₅	—	2	1	no	—
322	C₂×C₅	—	2	1	no	—
323	C₂×C₅	—	2	2	no	—
324	C₂×C₅	—	8	2	no	—
325	C₂×C₅	—	8	2	no	—
326	C₂×C₅	—	16	4	no	—
327	C₂×C₅	—	20	4	no	—
328	C₂×C₅	#66051	1	1	yes	characteristic
329	C₂×C₅	C₅×D₄²	1	1	yes	characteristic
330	D₅	—	4	4	no	—
331	C₂³	—	2	2	no	—
332	C₂³	—	20	2	no	—
333	Q₈	—	2	4	no	—
334	Q₈	—	5	4	no	—
335	Q₈	—	20	2	no	—
336	D₄	—	2	4	no	—
337	D₄	—	5	4	no	—
338	D₄	—	8	2	no	—
339	D₄	—	10	4	no	—
340	D₄	—	20	2	no	—
341	D₄	—	20	4	no	—
342	C₂×C₄	—	2	1	no	—
343	C₂×C₄	—	2	2	no	—
344	C₂×C₄	—	10	4	no	—
345	C₂×C₄	—	20	4	no	—
346	C₂×C₄	—	20	2	no	—
347	C₂×C₄	C₂³×C₅×D₅	1	1	yes	characteristic
348	C₈	—	10	4	no	—
349	C₈	C₅×D₄×D₅	1	2	yes	—
350	C₈	C₅×D₄×D₅	1	2	yes	—
351	C₅	—	2	2	no	—
352	C₅	#10613505	1	1	yes	characteristic, direct product factor
353	C₅	C₅×#3291	1	1	yes	characteristic, complemented
354	C₂²	—	4	2	no	—
355	C₂²	—	8	2	no	—
356	C₂²	—	10	4	no	—
357	C₂²	—	80	2	no	—
358	C₂²	C₅×#4144	1	1	yes	characteristic
359	C₄	—	2	2	no	—
360	C₄	—	4	2	no	—
361	C₄	—	10	4	no	—
362	C₄	C₂×C₅×D₄×D₅	1	2	yes	—
363	C₂	—	2	1	no	—
364	C₂	—	8	2	no	—
365	C₂	—	20	4	no	—
366	C₂	C₅×#66051	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₅	#66051
commutator, cosocle	C₂×C₄×C₅	C₂⁴×C₅
Frattini	C₂×C₄	C₂³×C₅×D₅
Fitting	C₅²×#492	C₂
socle	C₂×C₅²	D₄²
hypercenter	C₅×#492	D₅
hypocenter	C₅	C₅×#3291
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₅×#492

Complements

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

N	H ≅ G/N	type
1. C₅×#66096	C₂	semidirect
2. C₅²×#492	C₂	semidirect
3. C₅×#65466	C₂	semidirect
4. C₅×#65384	C₂	semidirect
5. C₅×#65366	C₂	semidirect
6. C₅×#64989	C₂	semidirect
8. C₅×#4144	C₂²	semidirect
9. C₂×C₅²×SD₁₆	C₂²	semidirect
10. C₅²×C₄⋊Q₈	C₂²	semidirect
12. C₅×#4088	C₂²	semidirect
19. C₅×#4047	C₂²	semidirect
20. C₄×C₅²×C₈	C₂²	semidirect
21. C₅×#3968	C₂²	semidirect
22. C₅×#3965	C₂²	semidirect
23. C₅×#3951	C₂²	semidirect
24. C₅×#3937	C₂²	semidirect
26. #10613505	C₅	direct (N × H)
47. C₅×Dic₂₀	D₄	semidirect
48. C₅×D₄₀	D₄	semidirect
49. C₅×#677	D₄	semidirect
53. #66096	C₂×C₅	semidirect
55. C₅×#492	D₅	semidirect
56. C₅×#492	C₂×C₅	semidirect
61. #65466	C₂×C₅	semidirect
62. #65384	C₂×C₅	semidirect
63. #65366	C₂×C₅	semidirect
64. #64989	C₂×C₅	semidirect
88. #4144	C₂²×C₅	semidirect
97. C₂×C₅×SD₁₆	C₂×D₅	semidirect
98. C₂×C₅×SD₁₆	C₂²×C₅	semidirect
101. C₅×C₄⋊Q₈	C₂×D₅	semidirect
102. C₅×C₄⋊Q₈	C₂²×C₅	semidirect
106. #4088	C₂²×C₅	semidirect
113. #4047	C₂²×C₅	semidirect
119. C₄×C₅×C₈	C₂×D₅	semidirect
120. C₄×C₅×C₈	C₂²×C₅	semidirect
121. #3968	C₂²×C₅	semidirect
122. #3965	C₂²×C₅	semidirect
123. #3951	C₂²×C₅	semidirect
124. #3937	C₂²×C₅	semidirect
188. Dic₂₀	C₅×D₄	semidirect
189. D₄₀	C₅×D₄	semidirect
190. #677	C₅×D₄	semidirect
194. #492	C₅×D₅	semidirect
261. C₂×SD₁₆	C₂×C₅×D₅	semidirect
263. C₄⋊Q₈	C₂×C₅×D₅	semidirect
269. C₄×C₈	C₂×C₅×D₅	semidirect
270. C₅²	#3291	semidirect
352. C₅	#10613505	direct (N × H)
353. C₅	C₅×#3291	semidirect

Maximal subgroups

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

subgroup	index
C₅×#66096	2
C₅²×#492	2
C₅×#65466	2
C₅×#65384	2
C₅×#65366	2
C₅×#64989	2
C₅×#3291	5
#10613505	5

Subgroups — C5×#10613505 #72223046

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#10613505 #72223046