Subgroups — #10611105

Order 640.

402 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₅×#117	C₂	1	1	yes	maximal, characteristic
2	C₅×#433	C₂	1	1	yes	maximal, characteristic, complemented
3	#65289	C₂	1	1	yes	maximal, characteristic, complemented
4	#65128	C₂	1	1	yes	maximal, characteristic, complemented
5	#65128	C₂	1	1	yes	maximal, characteristic, complemented
6	#65124	C₂	1	1	yes	maximal, characteristic, complemented
7	#65122	C₂	1	1	yes	maximal, characteristic, complemented
8	#65015	C₂	1	1	yes	maximal, characteristic
9	D₅×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
10	#65009	C₂	1	1	yes	maximal, characteristic, complemented
11	#65009	C₂	1	1	yes	maximal, characteristic, complemented
12	#64997	C₂	1	1	yes	maximal, characteristic, complemented
13	#64994	C₂	1	1	yes	maximal, characteristic, complemented
14	#64978	C₂	1	1	yes	maximal, characteristic, complemented
15	#64974	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×C₄×D₅	C₂²	1	1	yes	characteristic
17	C₅×#117	C₂²	1	1	yes	characteristic
18	#4072	C₂²	1	1	yes	characteristic
19	C₂×#676	C₂²	1	1	yes	characteristic
20	C₂×#676	C₂²	1	1	yes	characteristic
21	C₂×C₈×D₅	C₂²	1	1	yes	characteristic
22	C₂×C₈×D₅	C₂²	1	1	yes	characteristic
23	#4038	C₄	1	1	yes	characteristic
24	#4038	C₄	1	1	yes	characteristic
25	D₅×C₄⋊C₄	C₄	1	1	yes	characteristic
26	D₅×C₄⋊C₄	C₄	1	1	yes	characteristic
27	D₅×C₂²⋊C₄	C₄	1	1	yes	characteristic
28	D₅×C₂²⋊C₄	C₄	1	1	yes	characteristic
29	#4023	C₄	1	1	yes	characteristic
30	#4023	C₄	1	1	yes	characteristic
31	#4018	C₂²	1	1	yes	characteristic
32	#4018	C₂²	1	1	yes	characteristic
33	C₄²×D₅	C₂²	1	1	yes	characteristic
34	C₄²×D₅	C₂²	1	1	yes	characteristic
35	C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
36	C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
37	C₅×C₂²⋊C₈	C₂²	1	1	yes	characteristic
38	C₅×C₂²⋊C₈	C₂²	1	1	yes	characteristic
39	C₅×#97	C₂²	1	1	yes	characteristic, complemented
40	C₄×C₅×C₈	C₂²	1	1	yes	characteristic, complemented
41	C₂²⋊#168	C₂²	1	1	yes	characteristic
42	C₂²⋊#168	C₂²	1	1	yes	characteristic
43	#3952	C₂²	1	1	yes	characteristic
44	#3952	C₂²	1	1	yes	characteristic
45	#3952	C₂²	1	1	yes	characteristic
46	#3952	C₂²	1	1	yes	characteristic
47	#3947	C₂²	1	1	yes	characteristic, complemented
48	#3947	C₂²	1	1	yes	characteristic, complemented
49	#3946	C₂²	1	1	yes	characteristic, complemented
50	#3946	C₂²	1	1	yes	characteristic, complemented
51	#3946	C₂²	1	1	yes	characteristic, complemented
52	#3946	C₂²	1	1	yes	characteristic, complemented
53	C₈×Dic₅	C₂²	1	1	yes	characteristic, complemented
54	C₈×Dic₅	C₂²	1	1	yes	characteristic, complemented
55	C₄⋊#168	C₂²	1	1	yes	characteristic, complemented
56	C₄⋊#168	C₂²	1	1	yes	characteristic, complemented
57	#3935	C₂²	1	1	yes	characteristic, complemented
58	C₄×#168	C₂²	1	1	yes	characteristic, complemented
59	#2924	—	5	1	no	maximal, Sylow
60	C₂³×D₅	C₂×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
62	C₂²×Dic₅	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄×D₅	—	2	2	no	—
64	C₂×C₄×D₅	—	2	2	no	—
65	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
70	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
71	C₂×C₄×D₅	C₂³	1	1	yes	characteristic
72	C₂×C₄×D₅	C₂³	1	1	yes	characteristic
73	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
74	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
75	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
76	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
77	C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
78	C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
79	C₅×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
80	C₅×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
81	C₄²×C₅	C₂³	1	1	yes	characteristic
82	C₄²×C₅	C₂³	1	1	yes	characteristic
83	C₂²⋊Dic₅	C₂×C₄	1	1	yes	characteristic
84	C₂²⋊Dic₅	C₂×C₄	1	1	yes	characteristic
85	#685	C₂×C₄	1	1	yes	characteristic
86	#685	C₂×C₄	1	1	yes	characteristic
87	#685	C₂×C₄	1	1	yes	characteristic
88	#685	C₂×C₄	1	1	yes	characteristic
89	C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
90	C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
91	#683	C₂×C₄	1	1	yes	characteristic
92	#683	C₂×C₄	1	1	yes	characteristic
93	#683	C₂×C₄	1	1	yes	characteristic
94	#683	C₂×C₄	1	1	yes	characteristic
95	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
96	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
97	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
98	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
99	C₄×Dic₅	C₂³	1	1	yes	characteristic
100	C₄×Dic₅	C₂³	1	1	yes	characteristic
101	C₂×#168	C₂³	1	1	yes	characteristic
102	C₂×#168	C₂³	1	1	yes	characteristic
103	C₂×#168	C₂³	1	1	yes	characteristic
104	C₂×#168	C₂³	1	1	yes	characteristic
105	#676	—	2	2	no	—
106	#676	—	2	2	no	—
107	C₈×D₅	—	2	2	no	—
108	C₈×D₅	—	2	2	no	—
109	C₂×#117	—	5	1	no	—
110	#433	—	5	1	no	—
111	#433	—	5	1	no	—
112	#433	—	5	1	no	—
113	#433	D₅	1	1	yes	characteristic, complemented
114	#432	—	5	1	no	—
115	#432	—	5	1	no	—
116	#431	—	5	1	no	—
117	#431	—	5	1	no	—
118	#424	—	5	1	no	—
119	#424	—	5	1	no	—
120	#407	—	5	1	no	—
121	C₂×C₂²⋊C₈	—	5	1	no	—
122	#405	—	5	1	no	—
123	#405	—	5	1	no	—
124	C₂³×C₅	C₂²×C₄	1	1	yes	characteristic
125	C₂²×D₅	—	2	2	no	—
126	C₂²×D₅	—	2	2	no	—
127	C₂²×D₅	—	4	1	no	—
128	C₂²×D₅	C₂²×C₄	1	1	yes	characteristic
129	C₂²×D₅	C₂²×C₄	1	1	yes	characteristic
130	C₂×C₄×C₅	—	2	1	no	—
131	C₂×C₄×C₅	—	2	1	no	—
132	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
133	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
134	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
135	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
136	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
137	C₂×C₄×C₅	C₂⁴	1	1	yes	characteristic
138	C₂×Dic₅	—	2	2	no	—
139	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
140	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
141	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
142	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
143	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
144	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
145	C₄×D₅	—	2	2	no	—
146	C₄×D₅	—	2	2	no	—
147	C₄×D₅	—	2	2	no	—
148	C₄×D₅	—	2	1	no	—
149	C₄×D₅	—	2	1	no	—
150	C₄×D₅	—	2	2	no	—
151	C₄×D₅	—	4	1	no	—
152	C₄×D₅	—	4	1	no	—
153	C₄×D₅	#41	1	2	yes	—
154	C₄×D₅	#41	1	2	yes	—
155	C₅×C₈	—	2	1	no	—
156	C₅×C₈	—	2	1	no	—
157	C₅×C₈	—	2	1	no	—
158	C₅×C₈	—	2	1	no	—
159	#168	—	2	1	no	—
160	#168	—	2	1	no	—
161	#168	—	2	1	no	—
162	#168	—	2	1	no	—
163	C₂³×C₄	—	5	1	no	—
164	C₂×M₁₆	—	5	1	no	—
165	C₂×M₁₆	—	5	1	no	—
166	C₂²×C₈	—	5	1	no	—
167	C₂²×C₈	—	5	1	no	—
168	#117	—	5	1	no	—
169	#117	—	5	1	no	—
170	#117	—	5	1	no	—
171	#117	—	5	1	no	—
172	#117	—	5	1	no	—
173	#117	—	5	1	no	—
174	#117	—	5	1	no	—
175	#117	C₂×D₅	1	1	yes	characteristic
176	C₂×C₄⋊C₄	—	5	1	no	—
177	C₂×C₄⋊C₄	—	5	1	no	—
178	C₂×C₂²⋊C₄	—	5	1	no	—
179	C₂×C₂²⋊C₄	—	5	1	no	—
180	C₂×C₄²	—	5	1	no	—
181	C₂×C₄²	—	5	1	no	—
182	C₄⋊C₈	—	5	1	no	—
183	C₄⋊C₈	—	5	1	no	—
184	C₄⋊C₈	—	5	1	no	—
185	C₄⋊C₈	—	5	1	no	—
186	C₄⋊C₈	—	5	1	no	—
187	C₄⋊C₈	—	5	1	no	—
188	C₄⋊C₈	C₂×D₅	1	1	yes	characteristic, complemented
189	C₄⋊C₈	C₂×D₅	1	1	yes	characteristic, complemented
190	C₂²⋊C₈	—	5	1	no	—
191	C₂²⋊C₈	—	5	1	no	—
192	C₂²⋊C₈	—	5	1	no	—
193	C₂²⋊C₈	—	5	1	no	—
194	C₂²⋊C₈	—	5	1	no	—
195	C₂²⋊C₈	—	5	1	no	—
196	C₂²⋊C₈	C₂×D₅	1	1	yes	characteristic
197	C₂²⋊C₈	C₂×D₅	1	1	yes	characteristic
198	#97	—	5	1	no	—
199	#97	—	5	1	no	—
200	#97	—	5	1	no	—
201	#97	C₂×D₅	1	1	yes	characteristic, complemented
202	C₄×C₈	—	5	1	no	—
203	C₄×C₈	—	5	1	no	—
204	C₄×C₈	—	5	1	no	—
205	C₄×C₈	C₂×D₅	1	1	yes	characteristic, complemented
206	C₂²×C₅	—	2	1	no	—
207	C₂²×C₅	—	2	1	no	—
208	C₂²×C₅	—	2	1	no	—
209	C₂²×C₅	C₂³×C₄	1	1	yes	characteristic
210	C₂×D₅	—	2	1	no	—
211	C₂×D₅	—	2	1	no	—
212	C₂×D₅	—	2	1	no	—
213	C₂×D₅	—	2	1	no	—
214	C₂×D₅	—	4	4	no	—
215	C₂×D₅	#117	1	2	yes	—
216	C₂×D₅	#131	1	2	yes	—
217	C₄×C₅	—	2	1	no	—
218	C₄×C₅	—	2	1	no	—
219	C₄×C₅	—	2	1	no	—
220	C₄×C₅	—	2	1	no	—
221	C₄×C₅	—	2	1	no	—
222	C₄×C₅	C₂×#41	1	1	yes	characteristic
223	C₄×C₅	C₂×#41	1	1	yes	characteristic
224	Dic₅	—	2	1	no	—
225	Dic₅	—	2	1	no	—
226	Dic₅	—	2	1	no	—
227	Dic₅	—	2	1	no	—
228	Dic₅	—	2	1	no	—
229	Dic₅	#117	1	2	yes	—
230	C₂⁴	—	5	1	no	—
231	C₂²×C₄	—	5	1	no	—
232	C₂²×C₄	—	5	1	no	—
233	C₂²×C₄	—	5	1	no	—
234	C₂²×C₄	—	5	1	no	—
235	C₂²×C₄	—	5	1	no	—
236	C₂²×C₄	—	5	1	no	—
237	C₂²×C₄	—	5	1	no	—
238	C₂²×C₄	—	5	1	no	—
239	C₂²×C₄	—	5	1	no	—
240	C₂²×C₄	—	10	2	no	—
241	C₂²×C₄	—	10	2	no	—
242	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
243	M₁₆	—	10	2	no	—
244	M₁₆	—	10	2	no	—
245	C₂×C₈	—	5	1	no	—
246	C₂×C₈	—	5	1	no	—
247	C₂×C₈	—	5	1	no	—
248	C₂×C₈	—	5	1	no	—
249	C₂×C₈	—	10	2	no	—
250	C₂×C₈	—	10	2	no	—
251	C₂×C₈	C₂²×D₅	1	1	yes	characteristic
252	C₂×C₈	C₂²×D₅	1	1	yes	characteristic
253	C₂×C₈	C₂²×D₅	1	1	yes	characteristic
254	C₂×C₈	C₂²×D₅	1	1	yes	characteristic
255	C₄⋊C₄	—	5	1	no	—
256	C₄⋊C₄	—	5	1	no	—
257	C₄⋊C₄	—	5	1	no	—
258	C₄⋊C₄	—	5	1	no	—
259	C₄⋊C₄	—	5	1	no	—
260	C₄⋊C₄	—	5	1	no	—
261	C₄⋊C₄	C₄×D₅	1	1	yes	characteristic
262	C₄⋊C₄	C₄×D₅	1	1	yes	characteristic
263	C₂²⋊C₄	—	5	1	no	—
264	C₂²⋊C₄	—	5	1	no	—
265	C₂²⋊C₄	—	5	1	no	—
266	C₂²⋊C₄	—	5	1	no	—
267	C₂²⋊C₄	—	5	1	no	—
268	C₂²⋊C₄	—	5	1	no	—
269	C₂²⋊C₄	C₄×D₅	1	1	yes	characteristic
270	C₂²⋊C₄	C₄×D₅	1	1	yes	characteristic
271	C₄²	—	5	1	no	—
272	C₄²	—	5	1	no	—
273	C₄²	—	5	1	no	—
274	C₄²	—	5	1	no	—
275	C₄²	—	5	1	no	—
276	C₄²	—	5	1	no	—
277	C₄²	C₂²×D₅	1	1	yes	characteristic
278	C₄²	C₂²×D₅	1	1	yes	characteristic
279	C₂×C₅	—	4	1	no	—
280	C₂×C₅	C₂×#117	1	1	yes	characteristic
281	C₂×C₅	C₂×#131	1	1	yes	characteristic
282	C₂×C₅	#568	1	1	yes	characteristic
283	D₅	—	2	2	no	—
284	D₅	—	2	2	no	—
285	C₂³	—	5	1	no	—
286	C₂³	—	5	1	no	—
287	C₂³	—	10	2	no	—
288	C₂³	—	10	2	no	—
289	C₂³	—	20	1	no	—
290	C₂³	C₂×C₄×D₅	1	1	yes	characteristic
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	5	1	no	—
294	C₂×C₄	—	5	2	no	—
295	C₂×C₄	—	5	2	no	—
296	C₂×C₄	—	5	1	no	—
297	C₂×C₄	—	5	1	no	—
298	C₂×C₄	—	5	1	no	—
299	C₂×C₄	—	5	1	no	—
300	C₂×C₄	—	5	1	no	—
301	C₂×C₄	—	10	2	no	—
302	C₂×C₄	—	10	2	no	—
303	C₂×C₄	—	10	2	no	—
304	C₂×C₄	—	10	2	no	—
305	C₂×C₄	—	10	2	no	—
306	C₂×C₄	—	10	1	no	—
307	C₂×C₄	—	10	1	no	—
308	C₂×C₄	—	20	1	no	—
309	C₂×C₄	—	20	1	no	—
310	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
311	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
312	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
313	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
314	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
315	C₂×C₄	C₂³×D₅	1	1	yes	characteristic
316	C₈	—	2	1	no	—
317	C₈	—	2	1	no	—
318	C₈	—	2	1	no	—
319	C₈	—	2	1	no	—
320	C₈	—	10	1	no	—
321	C₈	—	10	1	no	—
322	C₈	—	10	1	no	—
323	C₈	—	10	1	no	—
324	C₅	#2924	1	1	yes	characteristic, complemented, Sylow
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	5	2	no	—
329	C₂²	—	5	2	no	—
330	C₂²	—	10	1	no	—
331	C₂²	—	10	1	no	—
332	C₂²	—	10	1	no	—
333	C₂²	—	10	1	no	—
334	C₂²	—	20	4	no	—
335	C₂²	C₂²×C₄×D₅	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	2	1	no	—
341	C₄	—	5	2	no	—
342	C₄	—	10	1	no	—
343	C₄	—	10	1	no	—
344	C₄	—	10	1	no	—
345	C₄	—	10	1	no	—
346	C₄	—	10	1	no	—
347	C₄	D₅×#41	1	1	yes	characteristic
348	C₄	D₅×#41	1	1	yes	characteristic
349	C₂	—	4	1	no	—
350	C₂	—	10	2	no	—
351	C₂	—	10	2	no	—
352	C₂	D₅×#117	1	1	yes	characteristic
353	C₂	D₅×#131	1	1	yes	characteristic
354	C₂	#66084	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅
commutator, socle	C₂²×C₅	C₂³×C₄
Fitting	C₅×#433	C₂
prime radical	C₂³×D₅	C₂×C₄
cosocle	C₂×C₄×C₅	C₂⁴
hypercenter	#433	D₅
hypocenter	C₅	#2924
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ #433

Complements

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

N	H ≅ G/N	type
2. C₅×#433	C₂	semidirect
3. #65289	C₂	semidirect
4. #65128	C₂	semidirect
5. #65128	C₂	semidirect
6. #65124	C₂	semidirect
7. #65122	C₂	semidirect
10. #65009	C₂	semidirect
11. #65009	C₂	semidirect
12. #64997	C₂	semidirect
13. #64994	C₂	semidirect
14. #64978	C₂	semidirect
15. #64974	C₂	semidirect
35. C₅×C₄⋊C₈	C₂²	semidirect
36. C₅×C₄⋊C₈	C₂²	semidirect
39. C₅×#97	C₂²	semidirect
40. C₄×C₅×C₈	C₂²	semidirect
47. #3947	C₂²	semidirect
48. #3947	C₂²	semidirect
49. #3946	C₂²	semidirect
50. #3946	C₂²	semidirect
51. #3946	C₂²	semidirect
52. #3946	C₂²	semidirect
53. C₈×Dic₅	C₂²	semidirect
54. C₈×Dic₅	C₂²	semidirect
55. C₄⋊#168	C₂²	semidirect
56. C₄⋊#168	C₂²	semidirect
57. #3935	C₂²	semidirect
58. C₄×#168	C₂²	semidirect
113. #433	D₅	semidirect
188. C₄⋊C₈	C₂×D₅	semidirect
189. C₄⋊C₈	C₂×D₅	semidirect
201. #97	C₂×D₅	semidirect
205. C₄×C₈	C₂×D₅	semidirect
324. C₅	#2924	semidirect

Maximal subgroups

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

subgroup	index
D₅×#117	2
C₅×#433	2
#65289	2
#65128	2
#65124	2
#65122	2
#65015	2
D₅×C₂²⋊C₈	2
#65009	2
#64997	2
#64994	2
#64978	2
#64974	2
#2924	5