Subgroups — C₅×#65828 #13106648

Order 1600.

632 conjugacy classes of subgroups.

The subgroup lattice has 333 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅²×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#4099	C₂	1	2	yes	maximal, complemented
3	C₅×#4076	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₅×C₂²⋊D₅	C₂	1	2	yes	maximal, complemented
5	C₅×#4070	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4031	C₂	1	2	yes	maximal, complemented
7	C₅×#4030	C₂	1	2	yes	maximal, complemented
8	C₅×#4029	C₂	1	2	yes	maximal, complemented
9	C₅×#4025	C₂	1	2	yes	maximal, complemented
10	C₂⁴×C₅²	C₂²	1	1	yes	characteristic
11	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
12	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
13	C₂×C₅×C₂²⋊D₅	C₂²	1	2	yes	—
14	C₂×C₅×C₂²⋊D₅	C₂²	1	2	yes	—
15	C₂×C₅×C₂²⋊D₅	C₂²	1	2	yes	—
16	C₂×C₅×D₂₀	C₂²	1	1	yes	characteristic, complemented
17	C₂×C₄×C₅×D₅	C₂²	1	2	yes	complemented
18	C₂×C₅×Dic₁₀	C₂²	1	1	yes	characteristic, complemented
19	C₅²×C₂²⋊C₄	C₂²	1	2	yes	—
20	C₅²×C₂²⋊C₄	C₂²	1	2	yes	—
21	C₅×C₂²⋊Dic₅	C₂²	1	2	yes	—
22	C₅×C₂²⋊Dic₅	C₂²	1	2	yes	—
23	C₅×C₂²⋊Dic₅	C₂²	1	2	yes	—
24	C₅×#685	C₂²	1	2	yes	complemented
25	C₅×#685	C₂²	1	2	yes	complemented
26	C₅×C₄⋊Dic₅	C₂²	1	1	yes	characteristic, complemented
27	C₅×C₄⋊Dic₅	C₂²	1	1	yes	characteristic, complemented
28	C₅×#683	C₂²	1	2	yes	complemented
29	C₅×#683	C₂²	1	2	yes	complemented
30	C₄×C₅×Dic₅	C₂²	1	2	yes	complemented
31	C₅×#538	—	5	1	no	maximal
32	#65828	C₅	1	1	yes	maximal, characteristic, direct product factor
33	C₂³×C₅²	—	2	2	no	—
34	C₂³×C₅²	—	4	1	no	—
35	C₂³×C₅²	—	4	1	no	—
36	C₂³×C₅²	C₂³	1	2	yes	—
37	C₂³×C₅²	C₂³	1	1	yes	characteristic
38	C₂²×C₅×D₅	C₂³	1	2	yes	—
39	C₂×C₄×C₅²	—	4	1	no	—
40	C₂×C₄×C₅²	—	4	1	no	—
41	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
42	C₂×C₄×C₅²	C₂³	1	2	yes	—
43	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
44	C₅×C₂²⋊D₅	—	2	4	no	—
45	C₅×C₂²⋊D₅	—	4	2	no	—
46	C₅×C₂²⋊D₅	—	4	2	no	—
47	C₂×C₅×Dic₅	C₂³	1	2	yes	—
48	C₂×C₅×Dic₅	C₂³	1	2	yes	—
49	C₂×C₅×Dic₅	C₂³	1	2	yes	—
50	C₅×D₂₀	—	4	1	no	—
51	C₄×C₅×D₅	—	4	2	no	—
52	C₅×Dic₁₀	—	4	1	no	—
53	C₅×#126	—	5	2	no	—
54	C₅×#124	—	5	2	no	—
55	C₅×#123	—	5	2	no	—
56	C₅×C₂²⋊Q₈	—	5	1	no	—
57	C₅×#121	—	5	1	no	—
58	C₅×#121	—	5	2	no	—
59	C₅×#120	—	5	2	no	—
60	C₄×C₅×D₄	—	5	2	no	—
61	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
62	C₂×C₅×C₂²⋊C₄	D₅	1	1	yes	characteristic, complemented
63	C₂×C₅×C₂²⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
64	#4099	C₂×C₅	1	2	yes	complemented
65	#4076	C₂×C₅	1	1	yes	characteristic, complemented
66	C₄×C₂²⋊D₅	C₂×C₅	1	2	yes	complemented
67	#4070	C₂×C₅	1	1	yes	characteristic, complemented
68	#4031	C₂×C₅	1	2	yes	complemented
69	#4030	C₂×C₅	1	2	yes	complemented
70	#4029	C₂×C₅	1	2	yes	complemented
71	#4025	C₂×C₅	1	2	yes	complemented
72	C₂²×C₅²	—	2	1	no	—
73	C₂²×C₅²	—	2	1	no	—
74	C₂²×C₅²	—	2	2	no	—
75	C₂²×C₅²	—	2	2	no	—
76	C₂²×C₅²	—	2	2	no	—
77	C₂²×C₅²	—	4	4	no	—
78	C₂²×C₅²	#41	1	2	yes	—
79	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
80	C₂×C₅×D₅	—	2	2	no	—
81	C₂×C₅×D₅	—	2	2	no	—
82	C₂×C₅×D₅	—	2	2	no	—
83	C₄×C₅²	—	2	1	no	—
84	C₄×C₅²	—	2	2	no	—
85	C₄×C₅²	—	2	1	no	—
86	C₅×Dic₅	—	2	2	no	—
87	C₅×Dic₅	—	2	2	no	—
88	C₅×Dic₅	—	2	2	no	—
89	C₂⁴×C₅	—	2	2	no	—
90	C₂⁴×C₅	C₂×D₅	1	1	yes	characteristic
91	C₂⁴×C₅	C₂²×C₅	1	1	yes	characteristic
92	C₂×C₅×Q₈	—	5	1	no	—
93	C₂×C₅×D₄	—	5	2	no	—
94	C₂×C₅×D₄	—	5	2	no	—
95	C₂×C₅×D₄	—	5	2	no	—
96	C₂×C₅×D₄	—	5	1	no	—
97	C₂²×C₄×C₅	—	2	2	no	—
98	C₂²×C₄×C₅	—	2	2	no	—
99	C₂²×C₄×C₅	—	5	2	no	—
100	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
101	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
102	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
103	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
104	C₂×C₂²⋊D₅	C₂²×C₅	1	2	yes	—
105	C₂×C₂²⋊D₅	C₂²×C₅	1	2	yes	—
106	C₂×C₂²⋊D₅	C₂²×C₅	1	2	yes	—
107	C₂×D₂₀	C₂²×C₅	1	1	yes	characteristic, complemented
108	C₂×C₄×D₅	C₂²×C₅	1	2	yes	complemented
109	C₂×Dic₁₀	C₂²×C₅	1	1	yes	characteristic, complemented
110	C₅×C₄⋊C₄	—	5	2	no	—
111	C₅×C₄⋊C₄	—	5	2	no	—
112	C₅×C₄⋊C₄	—	5	1	no	—
113	C₅×C₄⋊C₄	—	5	1	no	—
114	C₅×C₂²⋊C₄	—	2	4	no	—
115	C₅×C₂²⋊C₄	—	2	4	no	—
116	C₅×C₂²⋊C₄	—	5	2	no	—
117	C₅×C₂²⋊C₄	—	5	2	no	—
118	C₅×C₂²⋊C₄	—	5	2	no	—
119	C₅×C₂²⋊C₄	—	5	2	no	—
120	C₅×C₂²⋊C₄	—	5	2	no	—
121	C₅×C₂²⋊C₄	C₂×D₅	1	2	yes	—
122	C₅×C₂²⋊C₄	C₂×D₅	1	2	yes	—
123	C₅×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
124	C₅×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
125	C₄²×C₅	—	5	2	no	—
126	C₂²⋊Dic₅	C₂²×C₅	1	2	yes	—
127	C₂²⋊Dic₅	C₂²×C₅	1	2	yes	—
128	C₂²⋊Dic₅	C₂²×C₅	1	2	yes	—
129	#685	C₂²×C₅	1	2	yes	complemented
130	#685	C₂²×C₅	1	2	yes	complemented
131	C₄⋊Dic₅	C₂²×C₅	1	1	yes	characteristic, complemented
132	C₄⋊Dic₅	C₂²×C₅	1	1	yes	characteristic, complemented
133	#683	C₂²×C₅	1	2	yes	complemented
134	#683	C₂²×C₅	1	2	yes	complemented
135	C₄×Dic₅	C₂²×C₅	1	2	yes	complemented
136	#538	—	5	1	no	Sylow
137	C₂×C₅²	—	2	2	no	—
138	C₂×C₅²	—	4	2	no	—
139	C₂×C₅²	C₂×#41	1	1	yes	characteristic
140	C₂×C₅²	H(5,𝔽₂)	1	1	yes	characteristic
141	C₂×C₅²	H(5,𝔽₂)	1	1	yes	characteristic
142	C₅×D₅	—	4	2	no	—
143	C₂³×C₅	—	2	8	no	—
144	C₂³×C₅	—	2	2	no	—
145	C₂³×C₅	—	2	2	no	—
146	C₂³×C₅	—	2	2	no	—
147	C₂³×C₅	—	2	4	no	—
148	C₂³×C₅	—	4	1	no	—
149	C₂³×C₅	—	4	1	no	—
150	C₂³×C₅	—	4	1	no	—
151	C₂³×C₅	—	4	1	no	—
152	C₂³×C₅	—	4	4	no	—
153	C₂³×C₅	—	4	4	no	—
154	C₂³×C₅	—	5	2	no	—
155	C₂³×C₅	C₂²×D₅	1	2	yes	—
156	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
157	C₂³×C₅	C₂³×C₅	1	2	yes	—
158	C₂³×C₅	C₂³×C₅	1	1	yes	characteristic
159	C₂²×D₅	C₂³×C₅	1	2	yes	—
160	C₅×Q₈	—	20	1	no	—
161	C₅×D₄	—	10	4	no	—
162	C₅×D₄	—	20	1	no	—
163	C₅×D₄	—	20	2	no	—
164	C₅×D₄	—	20	2	no	—
165	C₂×C₄×C₅	—	2	4	no	—
166	C₂×C₄×C₅	—	2	2	no	—
167	C₂×C₄×C₅	—	2	2	no	—
168	C₂×C₄×C₅	—	4	1	no	—
169	C₂×C₄×C₅	—	4	1	no	—
170	C₂×C₄×C₅	—	4	1	no	—
171	C₂×C₄×C₅	—	4	1	no	—
172	C₂×C₄×C₅	—	4	4	no	—
173	C₂×C₄×C₅	—	4	4	no	—
174	C₂×C₄×C₅	—	5	2	no	—
175	C₂×C₄×C₅	—	5	2	no	—
176	C₂×C₄×C₅	—	5	2	no	—
177	C₂×C₄×C₅	—	20	2	no	—
178	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
179	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
180	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
181	C₂×C₄×C₅	C₂³×C₅	1	2	yes	—
182	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
183	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
184	C₂²⋊D₅	—	2	4	no	—
185	C₂²⋊D₅	—	4	2	no	—
186	C₂²⋊D₅	—	4	2	no	—
187	C₂×Dic₅	C₂³×C₅	1	2	yes	—
188	C₂×Dic₅	C₂³×C₅	1	2	yes	—
189	C₂×Dic₅	C₂³×C₅	1	2	yes	—
190	D₂₀	—	4	1	no	—
191	C₄×D₅	—	4	2	no	—
192	Dic₁₀	—	4	1	no	—
193	#126	—	5	2	no	—
194	#124	—	5	2	no	—
195	#123	—	5	2	no	—
196	C₂²⋊Q₈	—	5	1	no	—
197	#121	—	5	1	no	—
198	#121	—	5	2	no	—
199	#120	—	5	2	no	—
200	C₄×D₄	—	5	2	no	—
201	C₂×C₂²⋊C₄	C₅×D₅	1	1	yes	characteristic, complemented
202	C₅²	#538	1	1	yes	characteristic, complemented, Sylow
203	C₂²×C₅	—	2	4	no	—
204	C₂²×C₅	—	2	4	no	—
205	C₂²×C₅	—	2	1	no	—
206	C₂²×C₅	—	2	1	no	—
207	C₂²×C₅	—	2	1	no	—
208	C₂²×C₅	—	2	1	no	—
209	C₂²×C₅	—	2	8	no	—
210	C₂²×C₅	—	2	2	no	—
211	C₂²×C₅	—	2	2	no	—
212	C₂²×C₅	—	2	2	no	—
213	C₂²×C₅	—	2	2	no	—
214	C₂²×C₅	—	2	2	no	—
215	C₂²×C₅	—	2	2	no	—
216	C₂²×C₅	—	2	4	no	—
217	C₂²×C₅	—	2	2	no	—
218	C₂²×C₅	—	4	4	no	—
219	C₂²×C₅	—	4	4	no	—
220	C₂²×C₅	—	4	16	no	—
221	C₂²×C₅	—	4	4	no	—
222	C₂²×C₅	—	4	4	no	—
223	C₂²×C₅	—	10	2	no	—
224	C₂²×C₅	—	10	2	no	—
225	C₂²×C₅	—	10	2	no	—
226	C₂²×C₅	#709	1	2	yes	—
227	C₂²×C₅	C₅×#41	1	2	yes	—
228	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
229	C₂²×C₅	C₂⁴×C₅	1	1	yes	characteristic
230	C₂×D₅	—	2	2	no	—
231	C₂×D₅	—	2	2	no	—
232	C₂×D₅	—	2	2	no	—
233	C₄×C₅	—	2	2	no	—
234	C₄×C₅	—	2	1	no	—
235	C₄×C₅	—	2	1	no	—
236	C₄×C₅	—	2	1	no	—
237	C₄×C₅	—	2	2	no	—
238	C₄×C₅	—	2	1	no	—
239	C₄×C₅	—	4	4	no	—
240	C₄×C₅	—	4	2	no	—
241	C₄×C₅	—	4	2	no	—
242	C₄×C₅	—	10	2	no	—
243	C₄×C₅	—	10	2	no	—
244	C₄×C₅	—	10	2	no	—
245	Dic₅	—	2	2	no	—
246	Dic₅	—	2	2	no	—
247	Dic₅	—	2	2	no	—
248	C₂⁴	C₂×C₅×D₅	1	1	yes	characteristic
249	C₂×Q₈	—	5	1	no	—
250	C₂×D₄	—	5	2	no	—
251	C₂×D₄	—	5	2	no	—
252	C₂×D₄	—	5	2	no	—
253	C₂×D₄	—	5	1	no	—
254	C₂²×C₄	—	5	2	no	—
255	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
256	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
257	C₄⋊C₄	—	5	2	no	—
258	C₄⋊C₄	—	5	2	no	—
259	C₄⋊C₄	—	5	1	no	—
260	C₄⋊C₄	—	5	1	no	—
261	C₂²⋊C₄	—	5	2	no	—
262	C₂²⋊C₄	—	5	2	no	—
263	C₂²⋊C₄	—	5	2	no	—
264	C₂²⋊C₄	—	5	2	no	—
265	C₂²⋊C₄	—	5	2	no	—
266	C₂²⋊C₄	C₂×C₅×D₅	1	2	yes	—
267	C₂²⋊C₄	C₂×C₅×D₅	1	2	yes	—
268	C₄²	—	5	2	no	—
269	C₂×C₅	—	2	8	no	—
270	C₂×C₅	—	2	2	no	—
271	C₂×C₅	—	2	2	no	—
272	C₂×C₅	—	2	2	no	—
273	C₂×C₅	—	2	2	no	—
274	C₂×C₅	—	2	2	no	—
275	C₂×C₅	—	4	8	no	—
276	C₂×C₅	—	4	2	no	—
277	C₂×C₅	—	4	2	no	—
278	C₂×C₅	—	20	2	no	—
279	C₂×C₅	C₂×#709	1	1	yes	characteristic
280	C₂×C₅	#4144	1	1	yes	characteristic
281	C₂×C₅	#4144	1	1	yes	characteristic
282	C₂×C₅	C₂×C₅×#41	1	1	yes	characteristic
283	C₂×C₅	C₅×H(5,𝔽₂)	1	1	yes	characteristic
284	C₂×C₅	C₅×H(5,𝔽₂)	1	1	yes	characteristic
285	D₅	—	4	2	no	—
286	C₂³	—	2	2	no	—
287	C₂³	—	4	1	no	—
288	C₂³	—	4	1	no	—
289	C₂³	—	5	2	no	—
290	C₂³	C₂²×C₅×D₅	1	2	yes	—
291	C₂³	C₂²×C₅×D₅	1	1	yes	characteristic
292	Q₈	—	20	1	no	—
293	D₄	—	10	4	no	—
294	D₄	—	20	1	no	—
295	D₄	—	20	2	no	—
296	D₄	—	20	2	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	5	2	no	—
300	C₂×C₄	—	5	2	no	—
301	C₂×C₄	—	5	2	no	—
302	C₂×C₄	—	20	2	no	—
303	C₂×C₄	C₂²×C₅×D₅	1	2	yes	—
304	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
305	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
306	C₅	—	2	2	no	—
307	C₅	#65828	1	1	yes	characteristic, direct product factor
308	C₅	C₅×#538	1	1	yes	characteristic, complemented
309	C₂²	—	2	1	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	2	no	—
312	C₂²	—	2	2	no	—
313	C₂²	—	2	2	no	—
314	C₂²	—	4	4	no	—
315	C₂²	—	10	2	no	—
316	C₂²	—	10	2	no	—
317	C₂²	—	10	2	no	—
318	C₂²	C₅×#709	1	2	yes	—
319	C₂²	C₂³×C₅×D₅	1	1	yes	characteristic
320	C₄	—	2	2	no	—
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	10	2	no	—
324	C₄	—	10	2	no	—
325	C₄	—	10	2	no	—
326	C₂	—	2	2	no	—
327	C₂	—	4	2	no	—
328	C₂	—	20	2	no	—
329	C₂	C₂×C₅×#709	1	1	yes	characteristic
330	C₂	C₅×#4144	1	1	yes	characteristic
331	C₂	C₅×#4144	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	C₂³×D₅
commutator, cosocle	C₂²×C₅	C₂⁴×C₅
Frattini	C₂²	C₂³×C₅×D₅
Fitting	C₂×C₅²×C₂²⋊C₄	C₂
socle	C₂²×C₅²	C₂⁴
hypercenter	C₂×C₅×C₂²⋊C₄	D₅
hypocenter	C₅	C₅×#538
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₅×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂×C₅²×C₂²⋊C₄	C₂	semidirect
2. C₅×#4099	C₂	semidirect
3. C₅×#4076	C₂	semidirect
4. C₄×C₅×C₂²⋊D₅	C₂	semidirect
5. C₅×#4070	C₂	semidirect
6. C₅×#4031	C₂	semidirect
7. C₅×#4030	C₂	semidirect
8. C₅×#4029	C₂	semidirect
9. C₅×#4025	C₂	semidirect
16. C₂×C₅×D₂₀	C₂²	semidirect
17. C₂×C₄×C₅×D₅	C₂²	semidirect
18. C₂×C₅×Dic₁₀	C₂²	semidirect
24. C₅×#685	C₂²	semidirect
25. C₅×#685	C₂²	semidirect
26. C₅×C₄⋊Dic₅	C₂²	semidirect
27. C₅×C₄⋊Dic₅	C₂²	semidirect
28. C₅×#683	C₂²	semidirect
29. C₅×#683	C₂²	semidirect
30. C₄×C₅×Dic₅	C₂²	semidirect
32. #65828	C₅	direct (N × H)
62. C₂×C₅×C₂²⋊C₄	D₅	semidirect
63. C₂×C₅×C₂²⋊C₄	C₂×C₅	semidirect
64. #4099	C₂×C₅	semidirect
65. #4076	C₂×C₅	semidirect
66. C₄×C₂²⋊D₅	C₂×C₅	semidirect
67. #4070	C₂×C₅	semidirect
68. #4031	C₂×C₅	semidirect
69. #4030	C₂×C₅	semidirect
70. #4029	C₂×C₅	semidirect
71. #4025	C₂×C₅	semidirect
107. C₂×D₂₀	C₂²×C₅	semidirect
108. C₂×C₄×D₅	C₂²×C₅	semidirect
109. C₂×Dic₁₀	C₂²×C₅	semidirect
129. #685	C₂²×C₅	semidirect
130. #685	C₂²×C₅	semidirect
131. C₄⋊Dic₅	C₂²×C₅	semidirect
132. C₄⋊Dic₅	C₂²×C₅	semidirect
133. #683	C₂²×C₅	semidirect
134. #683	C₂²×C₅	semidirect
135. C₄×Dic₅	C₂²×C₅	semidirect
201. C₂×C₂²⋊C₄	C₅×D₅	semidirect
202. C₅²	#538	semidirect
307. C₅	#65828	direct (N × H)
308. C₅	C₅×#538	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅²×C₂²⋊C₄	2
C₅×#4099	2
C₅×#4076	2
C₄×C₅×C₂²⋊D₅	2
C₅×#4070	2
C₅×#4031	2
C₅×#4030	2
C₅×#4029	2
C₅×#4025	2
C₅×#538	5
#65828	5

Subgroups — C5×#65828 #13106648

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#65828 #13106648