Subgroups — C₂×C₅×#2220 #11956323

Order 1280.

734 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×SD₃₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#467	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅×#466	C₂	1	1	yes	maximal, characteristic
5	C₅×#2220	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₅×#366	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₅×#358	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅×#357	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₅×Q₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₅×D₈	C₂²	1	1	yes	characteristic
11	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₂×C₅×C₂²⋊Q₈	—	2	1	no	—
13	C₂×C₅×#121	—	2	1	no	—
14	C₂×C₅×SD₃₂	—	2	4	no	—
15	C₂×C₅×SD₃₂	C₂²	1	4	yes	complemented
16	C₂×C₅×M₃₂	—	2	4	no	—
17	C₂×C₅×M₃₂	C₂²	1	4	yes	complemented
18	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
19	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
20	C₅×#467	—	2	2	no	—
21	C₅×#467	C₂²	1	4	yes	complemented
22	C₅×#466	—	2	2	no	—
23	C₅×#466	C₂²	1	4	yes	—
24	C₂×C₅×#107	C₂²	1	1	yes	characteristic
25	C₂×C₅×#107	C₂²	1	1	yes	characteristic
26	C₂×C₅×#103	—	2	1	no	—
27	C₂×C₅×#102	—	2	1	no	—
28	C₅×#366	C₂²	1	4	yes	complemented
29	C₅×#358	C₂²	1	4	yes	complemented
30	C₅×#357	C₂²	1	4	yes	complemented
31	C₂×#2220	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₂²×C₅×Q₈	—	2	1	no	—
33	C₂²×C₅×D₄	—	2	1	no	—
34	C₂²×C₅×D₄	—	4	1	no	—
35	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
36	C₂×C₅×Q₁₆	—	2	2	no	—
37	C₂×C₅×Q₁₆	—	2	1	no	—
38	C₂×C₅×Q₁₆	—	2	2	no	—
39	C₂×C₅×Q₁₆	C₂³	1	2	yes	—
40	C₂×C₅×D₈	—	2	2	no	—
41	C₂×C₅×D₈	—	2	1	no	—
42	C₂×C₅×D₈	—	2	2	no	—
43	C₂×C₅×D₈	C₂³	1	2	yes	—
44	C₂²×C₅×C₈	—	2	2	no	—
45	C₂²×C₅×C₈	—	2	2	no	—
46	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
47	C₂²×C₅×C₈	D₄	1	2	yes	—
48	C₂²×C₅×C₈	C₂³	1	2	yes	—
49	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
50	C₅×C₂²⋊Q₈	—	2	4	no	—
51	C₅×C₂²⋊Q₈	—	4	2	no	—
52	C₅×#121	—	2	4	no	—
53	C₅×#121	—	4	2	no	—
54	C₂×C₅×C₄⋊C₄	—	2	1	no	—
55	C₂×C₅×C₄⋊C₄	—	2	1	no	—
56	C₂×C₅×C₄⋊C₄	—	4	1	no	—
57	C₂×C₅×C₂²⋊C₄	—	4	1	no	—
58	C₂×C₅×C₂²⋊C₄	—	4	1	no	—
59	C₅×SD₃₂	—	2	8	no	—
60	C₅×M₃₂	—	2	8	no	—
61	C₂×C₅×C₁₆	—	2	2	no	—
62	C₂×C₅×C₁₆	D₄	1	4	yes	complemented
63	C₂×C₅×C₁₆	C₂³	1	2	yes	—
64	C₂×C₅×C₁₆	C₂³	1	2	yes	—
65	C₅×#107	—	2	1	no	—
66	C₅×#107	—	2	1	no	—
67	C₅×#107	C₂³	1	2	yes	—
68	C₅×#107	C₂³	1	2	yes	—
69	C₅×#103	—	2	4	no	—
70	C₅×#102	—	2	4	no	—
71	C₂²×SD₃₂	C₂×C₅	1	1	yes	characteristic, complemented
72	C₂²×M₃₂	C₂×C₅	1	1	yes	characteristic, complemented
73	C₂×#467	C₂×C₅	1	1	yes	characteristic, complemented
74	C₂×#466	C₂×C₅	1	1	yes	characteristic
75	#2220	C₂×C₅	1	8	yes	direct product factor
76	C₂×#366	C₂×C₅	1	1	yes	characteristic, complemented
77	C₂×#358	C₂×C₅	1	1	yes	characteristic, complemented
78	C₂×#357	C₂×C₅	1	1	yes	characteristic, complemented
79	C₂⁴×C₅	—	4	1	no	—
80	C₂⁴×C₅	D₈	1	1	yes	characteristic
81	C₂×C₅×Q₈	—	2	2	no	—
82	C₂×C₅×Q₈	—	4	1	no	—
83	C₂×C₅×Q₈	—	4	2	no	—
84	C₂×C₅×Q₈	—	4	2	no	—
85	C₂×C₅×D₄	—	2	2	no	—
86	C₂×C₅×D₄	—	4	4	no	—
87	C₂×C₅×D₄	—	4	2	no	—
88	C₂×C₅×D₄	—	4	1	no	—
89	C₂×C₅×D₄	—	4	2	no	—
90	C₂×C₅×D₄	—	8	4	no	—
91	C₂²×C₄×C₅	—	2	2	no	—
92	C₂²×C₄×C₅	—	2	2	no	—
93	C₂²×C₄×C₅	—	4	1	no	—
94	C₂²×C₄×C₅	—	4	1	no	—
95	C₂²×C₄×C₅	—	4	1	no	—
96	C₂²×C₄×C₅	D₈	1	2	yes	—
97	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
98	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
99	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
100	C₅×Q₁₆	—	2	2	no	—
101	C₅×Q₁₆	—	2	2	no	—
102	C₅×Q₁₆	—	4	2	no	—
103	C₅×D₈	—	2	2	no	—
104	C₅×D₈	—	2	2	no	—
105	C₅×D₈	—	4	2	no	—
106	C₂×C₅×C₈	—	2	2	no	—
107	C₂×C₅×C₈	—	2	4	no	—
108	C₂×C₅×C₈	—	2	2	no	—
109	C₂×C₅×C₈	—	2	2	no	—
110	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
111	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
112	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
113	C₂×C₅×C₈	#41	1	2	yes	—
114	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
115	C₅×C₄⋊C₄	—	2	2	no	—
116	C₅×C₄⋊C₄	—	2	2	no	—
117	C₅×C₄⋊C₄	—	4	1	no	—
118	C₅×C₄⋊C₄	—	4	1	no	—
119	C₅×C₄⋊C₄	—	4	4	no	—
120	C₅×C₂²⋊C₄	—	4	4	no	—
121	C₅×C₂²⋊C₄	—	4	4	no	—
122	C₅×C₁₆	—	2	2	no	—
123	C₅×C₁₆	C₂×D₄	1	4	yes	complemented
124	C₂²×Q₁₆	C₂²×C₅	1	1	yes	characteristic
125	C₂²×D₈	C₂²×C₅	1	1	yes	characteristic
126	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
127	C₂×C₂²⋊Q₈	—	2	1	no	—
128	C₂×#121	—	2	1	no	—
129	C₂×SD₃₂	—	2	4	no	—
130	C₂×SD₃₂	C₂²×C₅	1	4	yes	complemented
131	C₂×M₃₂	—	2	4	no	—
132	C₂×M₃₂	C₂²×C₅	1	4	yes	complemented
133	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
134	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
135	#467	—	2	2	no	—
136	#467	C₂²×C₅	1	4	yes	complemented
137	#466	—	2	2	no	—
138	#466	C₂²×C₅	1	4	yes	—
139	C₂×#107	C₂²×C₅	1	1	yes	characteristic
140	C₂×#107	C₂²×C₅	1	1	yes	characteristic
141	C₂×#103	—	2	1	no	—
142	C₂×#102	—	2	1	no	—
143	#366	C₂²×C₅	1	4	yes	complemented
144	#358	C₂²×C₅	1	4	yes	complemented
145	#357	C₂²×C₅	1	4	yes	complemented
146	C₂³×C₅	—	2	2	no	—
147	C₂³×C₅	—	2	2	no	—
148	C₂³×C₅	—	2	2	no	—
149	C₂³×C₅	—	4	2	no	—
150	C₂³×C₅	—	8	2	no	—
151	C₂³×C₅	—	8	2	no	—
152	C₂³×C₅	—	8	2	no	—
153	C₂³×C₅	C₂×D₈	1	2	yes	—
154	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
155	C₅×Q₈	—	4	2	no	—
156	C₅×Q₈	—	4	2	no	—
157	C₅×Q₈	—	8	2	no	—
158	C₅×D₄	—	4	2	no	—
159	C₅×D₄	—	4	2	no	—
160	C₅×D₄	—	8	8	no	—
161	C₅×D₄	—	8	2	no	—
162	C₂×C₄×C₅	—	2	2	no	—
163	C₂×C₄×C₅	—	2	2	no	—
164	C₂×C₄×C₅	—	2	2	no	—
165	C₂×C₄×C₅	—	2	4	no	—
166	C₂×C₄×C₅	—	4	2	no	—
167	C₂×C₄×C₅	—	4	2	no	—
168	C₂×C₄×C₅	—	4	2	no	—
169	C₂×C₄×C₅	—	8	2	no	—
170	C₂×C₄×C₅	—	8	2	no	—
171	C₂×C₄×C₅	—	8	2	no	—
172	C₂×C₄×C₅	#121	1	2	yes	—
173	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
174	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
175	C₂×C₄×C₅	#135	1	2	yes	—
176	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
177	C₅×C₈	—	2	2	no	—
178	C₅×C₈	#121	1	2	yes	—
179	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
180	C₅×C₈	C₂×#41	1	1	yes	characteristic
181	C₂²×Q₈	—	2	1	no	—
182	C₂²×D₄	—	2	1	no	—
183	C₂²×D₄	—	4	1	no	—
184	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
185	C₂×Q₁₆	—	2	2	no	—
186	C₂×Q₁₆	—	2	1	no	—
187	C₂×Q₁₆	—	2	2	no	—
188	C₂×Q₁₆	C₂³×C₅	1	2	yes	—
189	C₂×D₈	—	2	2	no	—
190	C₂×D₈	—	2	1	no	—
191	C₂×D₈	—	2	2	no	—
192	C₂×D₈	C₂³×C₅	1	2	yes	—
193	C₂²×C₈	—	2	2	no	—
194	C₂²×C₈	—	2	2	no	—
195	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
196	C₂²×C₈	C₅×D₄	1	2	yes	—
197	C₂²×C₈	C₂³×C₅	1	2	yes	—
198	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
199	C₂²⋊Q₈	—	2	4	no	—
200	C₂²⋊Q₈	—	4	2	no	—
201	#121	—	2	4	no	—
202	#121	—	4	2	no	—
203	C₂×C₄⋊C₄	—	2	1	no	—
204	C₂×C₄⋊C₄	—	2	1	no	—
205	C₂×C₄⋊C₄	—	4	1	no	—
206	C₂×C₂²⋊C₄	—	4	1	no	—
207	C₂×C₂²⋊C₄	—	4	1	no	—
208	SD₃₂	—	2	8	no	—
209	M₃₂	—	2	8	no	—
210	C₂×C₁₆	—	2	2	no	—
211	C₂×C₁₆	C₅×D₄	1	4	yes	complemented
212	C₂×C₁₆	C₂³×C₅	1	2	yes	—
213	C₂×C₁₆	C₂³×C₅	1	2	yes	—
214	#107	—	2	1	no	—
215	#107	—	2	1	no	—
216	#107	C₂³×C₅	1	2	yes	—
217	#107	C₂³×C₅	1	2	yes	—
218	#103	—	2	4	no	—
219	#102	—	2	4	no	—
220	C₂²×C₅	—	2	2	no	—
221	C₂²×C₅	—	2	2	no	—
222	C₂²×C₅	—	2	2	no	—
223	C₂²×C₅	—	4	4	no	—
224	C₂²×C₅	—	8	2	no	—
225	C₂²×C₅	—	8	2	no	—
226	C₂²×C₅	—	8	2	no	—
227	C₂²×C₅	—	16	4	no	—
228	C₂²×C₅	#466	1	2	yes	—
229	C₂²×C₅	#509	1	2	yes	—
230	C₂²×C₅	#510	1	2	yes	—
231	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
232	C₄×C₅	—	2	2	no	—
233	C₄×C₅	—	8	2	no	—
234	C₄×C₅	—	8	2	no	—
235	C₄×C₅	—	8	2	no	—
236	C₄×C₅	#466	1	2	yes	—
237	C₄×C₅	C₂×#121	1	1	yes	characteristic
238	C₄×C₅	C₂×#135	1	1	yes	characteristic
239	C₂⁴	—	4	1	no	—
240	C₂⁴	C₅×D₈	1	1	yes	characteristic
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	—	4	1	no	—
243	C₂×Q₈	—	4	2	no	—
244	C₂×Q₈	—	4	2	no	—
245	C₂×D₄	—	2	2	no	—
246	C₂×D₄	—	4	4	no	—
247	C₂×D₄	—	4	2	no	—
248	C₂×D₄	—	4	1	no	—
249	C₂×D₄	—	4	2	no	—
250	C₂×D₄	—	8	4	no	—
251	C₂²×C₄	—	2	2	no	—
252	C₂²×C₄	—	2	2	no	—
253	C₂²×C₄	—	4	1	no	—
254	C₂²×C₄	—	4	1	no	—
255	C₂²×C₄	—	4	1	no	—
256	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
257	C₂²×C₄	C₅×D₈	1	2	yes	—
258	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
259	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
260	Q₁₆	—	2	2	no	—
261	Q₁₆	—	2	2	no	—
262	Q₁₆	—	4	2	no	—
263	D₈	—	2	2	no	—
264	D₈	—	2	2	no	—
265	D₈	—	4	2	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	2	4	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
271	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
272	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
273	C₂×C₈	C₅×#41	1	2	yes	—
274	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	—	4	1	no	—
278	C₄⋊C₄	—	4	1	no	—
279	C₄⋊C₄	—	4	4	no	—
280	C₂²⋊C₄	—	4	4	no	—
281	C₂²⋊C₄	—	4	4	no	—
282	C₁₆	—	2	2	no	—
283	C₁₆	C₂×C₅×D₄	1	4	yes	complemented
284	C₂×C₅	—	4	2	no	—
285	C₂×C₅	—	16	2	no	—
286	C₂×C₅	#2220	1	4	yes	direct product factor
287	C₂×C₅	C₂×#466	1	1	yes	characteristic
288	C₂×C₅	C₂×#509	1	1	yes	characteristic
289	C₂×C₅	C₂×#510	1	1	yes	characteristic
290	C₂³	—	2	2	no	—
291	C₂³	—	2	2	no	—
292	C₂³	—	2	2	no	—
293	C₂³	—	4	2	no	—
294	C₂³	—	8	2	no	—
295	C₂³	—	8	2	no	—
296	C₂³	—	8	2	no	—
297	C₂³	C₂×C₅×D₈	1	2	yes	—
298	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
299	Q₈	—	4	2	no	—
300	Q₈	—	4	2	no	—
301	Q₈	—	8	2	no	—
302	D₄	—	4	2	no	—
303	D₄	—	4	2	no	—
304	D₄	—	8	8	no	—
305	D₄	—	8	2	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	4	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	8	2	no	—
314	C₂×C₄	—	8	2	no	—
315	C₂×C₄	—	8	2	no	—
316	C₂×C₄	C₅×#121	1	2	yes	—
317	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
318	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
319	C₂×C₄	C₅×#135	1	2	yes	—
320	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
321	C₈	—	2	2	no	—
322	C₈	C₅×#121	1	2	yes	—
323	C₈	C₂²×C₅×D₄	1	1	yes	characteristic
324	C₈	C₂×C₅×#41	1	1	yes	characteristic
325	C₅	C₂×#2220	1	1	yes	characteristic, direct product factor, Sylow
326	C₂²	—	2	2	no	—
327	C₂²	—	2	2	no	—
328	C₂²	—	2	2	no	—
329	C₂²	—	4	4	no	—
330	C₂²	—	8	2	no	—
331	C₂²	—	8	2	no	—
332	C₂²	—	8	2	no	—
333	C₂²	—	16	4	no	—
334	C₂²	C₅×#466	1	2	yes	—
335	C₂²	C₅×#509	1	2	yes	—
336	C₂²	C₅×#510	1	2	yes	—
337	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
338	C₄	—	2	2	no	—
339	C₄	—	8	2	no	—
340	C₄	—	8	2	no	—
341	C₄	—	8	2	no	—
342	C₄	C₅×#466	1	2	yes	—
343	C₄	C₂×C₅×#121	1	1	yes	characteristic
344	C₄	C₂×C₅×#135	1	1	yes	characteristic
345	C₂	—	4	2	no	—
346	C₂	—	16	2	no	—
347	C₂	C₅×#2220	1	4	yes	direct product factor
348	C₂	C₂×C₅×#466	1	1	yes	characteristic
349	C₂	C₂×C₅×#509	1	1	yes	characteristic
350	C₂	C₂×C₅×#510	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅
prime radical	C₂×C₅×#466	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂³×C₄×C₅ ◁ C₂³×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₅×SD₃₂	C₂	semidirect
2. C₂²×C₅×M₃₂	C₂	semidirect
3. C₂×C₅×#467	C₂	semidirect
5. C₅×#2220	C₂	direct (N × H)
6. C₂×C₅×#366	C₂	semidirect
7. C₂×C₅×#358	C₂	semidirect
8. C₂×C₅×#357	C₂	semidirect
15. C₂×C₅×SD₃₂	C₂²	semidirect
17. C₂×C₅×M₃₂	C₂²	semidirect
21. C₅×#467	C₂²	semidirect
28. C₅×#366	C₂²	semidirect
29. C₅×#358	C₂²	semidirect
30. C₅×#357	C₂²	semidirect
31. C₂×#2220	C₅	direct (N × H)
62. C₂×C₅×C₁₆	D₄	semidirect
71. C₂²×SD₃₂	C₂×C₅	semidirect
72. C₂²×M₃₂	C₂×C₅	semidirect
73. C₂×#467	C₂×C₅	semidirect
75. #2220	C₂×C₅	direct (N × H)
76. C₂×#366	C₂×C₅	semidirect
77. C₂×#358	C₂×C₅	semidirect
78. C₂×#357	C₂×C₅	semidirect
123. C₅×C₁₆	C₂×D₄	semidirect
130. C₂×SD₃₂	C₂²×C₅	semidirect
132. C₂×M₃₂	C₂²×C₅	semidirect
136. #467	C₂²×C₅	semidirect
143. #366	C₂²×C₅	semidirect
144. #358	C₂²×C₅	semidirect
145. #357	C₂²×C₅	semidirect
211. C₂×C₁₆	C₅×D₄	semidirect
283. C₁₆	C₂×C₅×D₄	semidirect
286. C₂×C₅	#2220	direct (N × H)
325. C₅	C₂×#2220	direct (N × H)
347. C₂	C₅×#2220	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₅×SD₃₂	2
C₂²×C₅×M₃₂	2
C₂×C₅×#467	2
C₂×C₅×#466	2
C₅×#2220	2
C₂×C₅×#366	2
C₂×C₅×#358	2
C₂×C₅×#357	2
C₂×#2220	5

Subgroups — C2×C5×#2220 #11956323

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#2220 #11956323