Subgroups — #10611215

Order 640.

492 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#4053	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#4047	C₂	1	1	yes	maximal, characteristic, complemented
3	#65882	C₂	1	1	yes	maximal, characteristic
4	C₅×#435	C₂	1	1	yes	maximal, characteristic, complemented
5	#65417	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×C₂²⋊D₅	C₂	1	1	yes	maximal, characteristic, complemented
7	#65300	C₂	1	1	yes	maximal, characteristic, complemented
8	#65125	C₂	1	1	yes	maximal, characteristic, complemented
9	D₅×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
10	#65018	C₂	1	1	yes	maximal, characteristic, complemented
11	#65016	C₂	1	1	yes	maximal, characteristic, complemented
12	#65016	C₂	1	1	yes	maximal, characteristic, complemented
13	#65014	C₂	1	1	yes	maximal, characteristic, complemented
14	#64996	C₂	1	1	yes	maximal, characteristic, complemented
15	#64995	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#709	C₂²	1	1	yes	characteristic
17	C₂×#709	C₂²	1	1	yes	characteristic
18	C₂×C₅×M₁₆	C₂²	1	1	yes	characteristic
19	C₂²×C₅×C₈	C₂²	1	1	yes	characteristic
20	C₄×C₅×D₄	C₂²	1	1	yes	characteristic
21	#4086	C₄	1	1	yes	characteristic
22	#4084	C₄	1	1	yes	characteristic
23	C₄×C₂²⋊D₅	C₂²	1	1	yes	characteristic
24	C₄×C₂²⋊D₅	C₂²	1	1	yes	characteristic
25	C₂×#681	C₂²	1	1	yes	characteristic
26	C₂²×#168	C₂²	1	1	yes	characteristic
27	#4053	—	2	4	no	—
28	#4047	—	2	4	no	—
29	C₂×#676	C₂²	1	1	yes	characteristic
30	C₂×#676	C₂²	1	1	yes	characteristic
31	C₂×#676	C₂²	1	1	yes	characteristic
32	C₂×#676	C₂²	1	1	yes	characteristic
33	C₂×C₈×D₅	C₂²	1	1	yes	characteristic
34	C₂×C₈×D₅	C₂²	1	1	yes	characteristic
35	C₂×C₈×D₅	C₂²	1	1	yes	characteristic
36	C₂×C₈×D₅	C₂²	1	1	yes	characteristic
37	#4041	C₄	1	1	yes	characteristic
38	#4034	C₄	1	1	yes	characteristic
39	#4031	C₄	1	1	yes	characteristic
40	#4031	C₄	1	1	yes	characteristic
41	#4030	C₄	1	1	yes	characteristic
42	#4030	C₄	1	1	yes	characteristic
43	C₄×D₂₀	C₂²	1	1	yes	characteristic
44	C₄²×D₅	C₂²	1	1	yes	characteristic
45	C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
46	C₅×C₂²⋊C₈	C₂²	1	1	yes	characteristic
47	C₅×C₂²⋊C₈	C₂²	1	1	yes	characteristic
48	C₅×#97	C₂²	1	1	yes	characteristic
49	C₂²⋊#168	C₂²	1	1	yes	characteristic
50	C₂²⋊#168	C₂²	1	1	yes	characteristic
51	#3952	C₂²	1	1	yes	characteristic
52	#3952	C₂²	1	1	yes	characteristic
53	#3952	C₂²	1	1	yes	characteristic
54	#3952	C₂²	1	1	yes	characteristic
55	#3946	C₂²	1	1	yes	characteristic
56	#3946	C₂²	1	1	yes	characteristic
57	C₈×Dic₅	C₂²	1	1	yes	characteristic
58	C₈×Dic₅	C₂²	1	1	yes	characteristic
59	C₄⋊#168	C₂²	1	1	yes	characteristic
60	#3935	C₂²	1	1	yes	characteristic
61	#2935	—	5	1	no	maximal, Sylow
62	C₂×C₅×D₄	C₂×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
64	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
65	C₂×C₂²⋊D₅	C₂×C₄	1	1	yes	characteristic
66	C₂×C₂²⋊D₅	C₂×C₄	1	1	yes	characteristic
67	C₂×C₂²⋊D₅	C₂×C₄	1	1	yes	characteristic
68	C₂×C₂²⋊D₅	C₂×C₄	1	1	yes	characteristic
69	C₂×C₂²⋊D₅	C₂×C₄	1	1	yes	characteristic
70	C₂×C₂²⋊D₅	C₂×C₄	1	1	yes	characteristic
71	#709	—	2	2	no	—
72	#709	—	2	2	no	—
73	#709	—	2	2	no	—
74	#709	—	2	2	no	—
75	C₂×D₂₀	C₂×C₄	1	1	yes	characteristic
76	C₂×D₂₀	C₂×C₄	1	1	yes	characteristic
77	C₂×D₂₀	C₂×C₄	1	1	yes	characteristic
78	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
79	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
80	C₂×C₄×D₅	C₂³	1	1	yes	characteristic
81	C₂×C₄×D₅	C₂³	1	1	yes	characteristic
82	C₂×C₄×D₅	C₂³	1	1	yes	characteristic
83	C₂×Dic₁₀	C₂×C₄	1	1	yes	characteristic
84	C₂×Dic₁₀	C₂×C₄	1	1	yes	characteristic
85	C₅×M₁₆	—	2	2	no	—
86	C₂×C₅×C₈	—	2	2	no	—
87	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
88	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
89	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
90	C₂×C₅×C₈	C₂³	1	1	yes	characteristic
91	C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
92	C₅×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
93	C₅×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
94	C₄²×C₅	C₂³	1	1	yes	characteristic
95	C₂²⋊Dic₅	C₂×C₄	1	1	yes	characteristic
96	C₂²⋊Dic₅	C₂×C₄	1	1	yes	characteristic
97	#685	C₂×C₄	1	1	yes	characteristic
98	#685	C₂×C₄	1	1	yes	characteristic
99	#685	C₂×C₄	1	1	yes	characteristic
100	#685	C₂×C₄	1	1	yes	characteristic
101	C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
102	#683	C₂×C₄	1	1	yes	characteristic
103	#683	C₂×C₄	1	1	yes	characteristic
104	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
105	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
106	C₄×Dic₅	C₂³	1	1	yes	characteristic
107	#681	—	2	2	no	—
108	C₂×#168	—	2	2	no	—
109	C₂×#168	C₂³	1	1	yes	characteristic
110	C₂×#168	C₂³	1	1	yes	characteristic
111	C₂×#168	C₂³	1	1	yes	characteristic
112	C₂×#168	C₂³	1	1	yes	characteristic
113	#676	—	2	2	no	—
114	#676	—	2	2	no	—
115	#676	—	2	2	no	—
116	#676	D₄	1	4	yes	complemented
117	C₈×D₅	—	2	2	no	—
118	C₈×D₅	—	2	2	no	—
119	C₈×D₅	—	2	2	no	—
120	C₈×D₅	—	2	2	no	—
121	C₂×#131	—	5	1	no	—
122	C₂×#131	—	5	1	no	—
123	#532	—	5	1	no	—
124	#436	—	5	1	no	—
125	#436	—	5	1	no	—
126	#435	—	5	1	no	—
127	#435	—	5	1	no	—
128	#435	—	5	1	no	—
129	#435	D₅	1	1	yes	characteristic, complemented
130	C₈×D₄	—	5	1	no	—
131	C₈×D₄	—	5	1	no	—
132	C₂×C₄⋊C₈	—	5	1	no	—
133	#408	—	5	1	no	—
134	#408	—	5	1	no	—
135	C₄×M₁₆	—	5	1	no	—
136	C₂³×C₅	C₂²×C₄	1	1	yes	characteristic
137	C₂³×C₅	C₂²×C₄	1	1	yes	characteristic
138	C₂²×D₅	C₂²×C₄	1	1	yes	characteristic
139	C₂²×D₅	C₂²×C₄	1	1	yes	characteristic
140	C₂²×D₅	C₂²×C₄	1	1	yes	characteristic
141	C₅×D₄	—	4	1	no	—
142	C₂×C₄×C₅	—	2	1	no	—
143	C₂×C₄×C₅	—	2	1	no	—
144	C₂×C₄×C₅	—	2	1	no	—
145	C₂×C₄×C₅	—	2	1	no	—
146	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
147	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
148	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
149	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
150	C₂×C₄×C₅	C₂⁴	1	1	yes	characteristic
151	C₂²⋊D₅	—	2	2	no	—
152	C₂²⋊D₅	—	2	2	no	—
153	C₂²⋊D₅	—	2	2	no	—
154	C₂²⋊D₅	—	2	2	no	—
155	C₂²⋊D₅	—	2	2	no	—
156	C₂²⋊D₅	—	2	2	no	—
157	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
158	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
159	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
160	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
161	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
162	D₂₀	—	2	2	no	—
163	D₂₀	—	2	2	no	—
164	D₂₀	—	4	1	no	—
165	C₄×D₅	—	2	1	no	—
166	C₄×D₅	—	2	1	no	—
167	C₄×D₅	—	2	1	no	—
168	C₄×D₅	—	2	1	no	—
169	C₄×D₅	—	2	2	no	—
170	C₄×D₅	—	4	1	no	—
171	C₄×D₅	C₂×D₄	1	2	yes	—
172	C₄×D₅	#41	1	2	yes	—
173	Dic₁₀	—	2	2	no	—
174	Dic₁₀	—	2	2	no	—
175	C₅×C₈	—	2	1	no	—
176	C₅×C₈	—	2	1	no	—
177	C₅×C₈	—	2	1	no	—
178	C₅×C₈	C₂×D₄	1	2	yes	—
179	#168	—	2	1	no	—
180	#168	—	2	1	no	—
181	#168	—	2	1	no	—
182	#168	C₂×D₄	1	2	yes	—
183	C₂×#41	—	5	1	no	—
184	C₂×#41	—	5	1	no	—
185	#131	—	10	4	no	—
186	#131	—	10	4	no	—
187	C₂×M₁₆	—	5	1	no	—
188	C₂×M₁₆	—	5	1	no	—
189	C₂×M₁₆	—	5	1	no	—
190	C₂×M₁₆	—	5	1	no	—
191	C₂×M₁₆	—	5	1	no	—
192	C₂×M₁₆	C₂×D₅	1	1	yes	characteristic
193	C₂²×C₈	—	5	1	no	—
194	C₂²×C₈	—	5	1	no	—
195	C₂²×C₈	—	5	1	no	—
196	C₂²×C₈	—	5	1	no	—
197	C₂²×C₈	—	5	1	no	—
198	C₂²×C₈	C₂×D₅	1	1	yes	characteristic
199	C₄⋊Q₈	—	5	1	no	—
200	#127	—	5	1	no	—
201	#124	—	5	1	no	—
202	#124	—	5	1	no	—
203	#121	—	5	1	no	—
204	#121	—	5	1	no	—
205	#121	—	5	1	no	—
206	#121	—	5	1	no	—
207	C₄×D₄	—	5	1	no	—
208	C₄×D₄	—	5	1	no	—
209	C₄×D₄	—	5	1	no	—
210	C₄×D₄	C₂×D₅	1	1	yes	characteristic
211	C₂×C₄²	—	5	1	no	—
212	C₄⋊C₈	—	5	1	no	—
213	C₄⋊C₈	—	5	1	no	—
214	C₄⋊C₈	—	5	1	no	—
215	C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
216	C₂²⋊C₈	—	5	1	no	—
217	C₂²⋊C₈	—	5	1	no	—
218	C₂²⋊C₈	—	5	1	no	—
219	C₂²⋊C₈	—	5	1	no	—
220	C₂²⋊C₈	—	5	1	no	—
221	C₂²⋊C₈	—	5	1	no	—
222	C₂²⋊C₈	C₂×D₅	1	1	yes	characteristic
223	C₂²⋊C₈	C₂×D₅	1	1	yes	characteristic
224	#97	—	5	1	no	—
225	#97	C₂×D₅	1	1	yes	characteristic
226	C₄×C₈	—	5	1	no	—
227	C₄×C₈	—	5	1	no	—
228	C₂²×C₅	—	2	1	no	—
229	C₂²×C₅	—	2	1	no	—
230	C₂²×C₅	—	2	1	no	—
231	C₂²×C₅	—	2	1	no	—
232	C₂²×C₅	—	2	1	no	—
233	C₂²×C₅	—	2	1	no	—
234	C₂²×C₅	C₂³×C₄	1	1	yes	characteristic
235	C₂×D₅	—	2	1	no	—
236	C₂×D₅	—	2	1	no	—
237	C₂×D₅	—	2	1	no	—
238	C₂×D₅	—	2	1	no	—
239	C₂×D₅	—	2	1	no	—
240	C₂×D₅	—	2	1	no	—
241	C₂×D₅	—	2	1	no	—
242	C₂×D₅	—	2	1	no	—
243	C₂×D₅	C₄×D₄	1	2	yes	—
244	C₄×C₅	—	2	1	no	—
245	C₄×C₅	—	2	1	no	—
246	C₄×C₅	—	2	1	no	—
247	C₄×C₅	—	2	1	no	—
248	C₄×C₅	C₂²×D₄	1	1	yes	characteristic
249	C₄×C₅	C₂×#41	1	1	yes	characteristic
250	Dic₅	—	2	1	no	—
251	Dic₅	—	2	1	no	—
252	Dic₅	—	2	1	no	—
253	Dic₅	C₄×D₄	1	2	yes	—
254	Dic₅	#131	1	2	yes	complemented
255	#41	—	10	2	no	—
256	#41	—	10	2	no	—
257	#41	—	10	2	no	—
258	#41	—	10	2	no	—
259	C₂×Q₈	—	5	1	no	—
260	C₂×Q₈	—	5	1	no	—
261	C₂×D₄	—	5	1	no	—
262	C₂×D₄	—	5	1	no	—
263	C₂×D₄	—	5	1	no	—
264	C₂×D₄	—	5	1	no	—
265	C₂×D₄	—	5	1	no	—
266	C₂×D₄	—	5	1	no	—
267	C₂×D₄	—	5	1	no	—
268	C₂×D₄	—	5	1	no	—
269	C₂×D₄	—	5	1	no	—
270	C₂×D₄	C₄×D₅	1	1	yes	characteristic
271	C₂²×C₄	—	5	1	no	—
272	C₂²×C₄	—	5	1	no	—
273	C₂²×C₄	—	5	1	no	—
274	C₂²×C₄	—	5	1	no	—
275	C₂²×C₄	—	5	1	no	—
276	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
277	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
278	M₁₆	—	2	2	no	—
279	M₁₆	—	5	4	no	—
280	M₁₆	—	10	2	no	—
281	M₁₆	—	10	2	no	—
282	M₁₆	—	10	2	no	—
283	M₁₆	—	10	2	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	5	1	no	—
286	C₂×C₈	—	5	1	no	—
287	C₂×C₈	—	5	1	no	—
288	C₂×C₈	—	5	1	no	—
289	C₂×C₈	—	10	2	no	—
290	C₂×C₈	—	10	2	no	—
291	C₂×C₈	—	10	2	no	—
292	C₂×C₈	—	10	2	no	—
293	C₂×C₈	—	10	2	no	—
294	C₂×C₈	C₂²×D₅	1	1	yes	characteristic
295	C₂×C₈	C₂²×D₅	1	1	yes	characteristic
296	C₂×C₈	C₂²×D₅	1	1	yes	characteristic
297	C₂×C₈	C₂²×D₅	1	1	yes	characteristic
298	C₄⋊C₄	—	5	1	no	—
299	C₄⋊C₄	—	5	1	no	—
300	C₄⋊C₄	—	5	1	no	—
301	C₄⋊C₄	C₄×D₅	1	1	yes	characteristic
302	C₂²⋊C₄	—	5	1	no	—
303	C₂²⋊C₄	—	5	1	no	—
304	C₂²⋊C₄	—	5	1	no	—
305	C₂²⋊C₄	—	5	1	no	—
306	C₂²⋊C₄	—	5	1	no	—
307	C₂²⋊C₄	—	5	1	no	—
308	C₂²⋊C₄	C₄×D₅	1	1	yes	characteristic
309	C₂²⋊C₄	C₄×D₅	1	1	yes	characteristic
310	C₄²	—	5	1	no	—
311	C₄²	—	5	1	no	—
312	C₄²	—	5	1	no	—
313	C₄²	C₂²×D₅	1	1	yes	characteristic
314	C₂×C₅	—	4	1	no	—
315	C₂×C₅	—	4	1	no	—
316	C₂×C₅	C₂×C₄×D₄	1	1	yes	characteristic
317	C₂×C₅	C₂×#131	1	1	yes	characteristic
318	C₂×C₅	#568	1	1	yes	characteristic
319	D₅	—	2	2	no	—
320	D₅	—	4	1	no	—
321	D₅	—	4	1	no	—
322	C₂³	—	5	1	no	—
323	C₂³	—	5	1	no	—
324	C₂³	—	5	1	no	—
325	C₂³	C₂×C₄×D₅	1	1	yes	characteristic
326	C₂³	C₂×C₄×D₅	1	1	yes	characteristic
327	Q₈	—	10	2	no	—
328	Q₈	—	10	2	no	—
329	D₄	—	4	1	no	—
330	D₄	—	10	2	no	—
331	D₄	—	10	2	no	—
332	D₄	—	10	2	no	—
333	D₄	—	10	2	no	—
334	D₄	—	10	2	no	—
335	D₄	—	10	2	no	—
336	D₄	—	10	2	no	—
337	D₄	—	10	2	no	—
338	D₄	—	20	1	no	—
339	C₂×C₄	—	2	1	no	—
340	C₂×C₄	—	2	1	no	—
341	C₂×C₄	—	2	1	no	—
342	C₂×C₄	—	2	1	no	—
343	C₂×C₄	—	5	1	no	—
344	C₂×C₄	—	5	1	no	—
345	C₂×C₄	—	5	2	no	—
346	C₂×C₄	—	5	2	no	—
347	C₂×C₄	—	5	1	no	—
348	C₂×C₄	—	5	1	no	—
349	C₂×C₄	—	5	1	no	—
350	C₂×C₄	—	10	2	no	—
351	C₂×C₄	—	10	1	no	—
352	C₂×C₄	—	10	1	no	—
353	C₂×C₄	—	10	1	no	—
354	C₂×C₄	—	10	1	no	—
355	C₂×C₄	—	20	1	no	—
356	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
357	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
358	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
359	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
360	C₂×C₄	C₂³×D₅	1	1	yes	characteristic
361	C₈	—	2	1	no	—
362	C₈	—	2	1	no	—
363	C₈	—	2	1	no	—
364	C₈	—	5	2	no	—
365	C₈	—	10	1	no	—
366	C₈	—	10	1	no	—
367	C₈	—	10	1	no	—
368	C₈	D₄×D₅	1	2	yes	—
369	C₅	#2935	1	1	yes	characteristic, complemented, Sylow
370	C₂²	—	2	1	no	—
371	C₂²	—	2	1	no	—
372	C₂²	—	2	1	no	—
373	C₂²	—	2	1	no	—
374	C₂²	—	2	1	no	—
375	C₂²	—	2	1	no	—
376	C₂²	—	5	2	no	—
377	C₂²	—	10	1	no	—
378	C₂²	—	10	1	no	—
379	C₂²	—	10	1	no	—
380	C₂²	—	10	1	no	—
381	C₂²	—	10	1	no	—
382	C₂²	—	10	1	no	—
383	C₂²	—	10	1	no	—
384	C₂²	—	10	1	no	—
385	C₂²	C₂²×C₄×D₅	1	1	yes	characteristic
386	C₄	—	2	1	no	—
387	C₄	—	2	1	no	—
388	C₄	—	2	1	no	—
389	C₄	—	2	1	no	—
390	C₄	—	5	2	no	—
391	C₄	—	5	2	no	—
392	C₄	—	10	1	no	—
393	C₄	—	10	1	no	—
394	C₄	—	10	1	no	—
395	C₄	C₂×D₄×D₅	1	1	yes	characteristic
396	C₄	D₅×#41	1	1	yes	characteristic
397	C₂	—	4	1	no	—
398	C₂	—	4	1	no	—
399	C₂	—	10	2	no	—
400	C₂	—	20	1	no	—
401	C₂	—	20	1	no	—
402	C₂	C₄×D₄×D₅	1	1	yes	characteristic
403	C₂	#66079	1	1	yes	characteristic
404	C₂	D₅×#131	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅
commutator, socle	C₂²×C₅	C₂³×C₄
Fitting	C₅×#435	C₂
prime radical	#65882	C₂
cosocle	C₂×C₄×C₅	C₂⁴
hypercenter	#435	D₅
hypocenter	C₅	#2935
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ #435

Complements

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

N	H ≅ G/N	type
1. C₂×#4053	C₂	semidirect
2. C₂×#4047	C₂	semidirect
4. C₅×#435	C₂	semidirect
5. #65417	C₂	semidirect
6. C₈×C₂²⋊D₅	C₂	semidirect
7. #65300	C₂	semidirect
8. #65125	C₂	semidirect
9. D₅×C₄⋊C₈	C₂	semidirect
10. #65018	C₂	semidirect
11. #65016	C₂	semidirect
12. #65016	C₂	semidirect
13. #65014	C₂	semidirect
14. #64996	C₂	semidirect
15. #64995	C₂	semidirect
116. #676	D₄	semidirect
129. #435	D₅	semidirect
254. Dic₅	#131	semidirect
369. C₅	#2935	semidirect

Maximal subgroups

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

subgroup	index
C₂×#4053	2
C₂×#4047	2
#65882	2
C₅×#435	2
#65417	2
C₈×C₂²⋊D₅	2
#65300	2
#65125	2
D₅×C₄⋊C₈	2
#65018	2
#65016	2
#65014	2
#64996	2
#64995	2
#2935	5