Subgroups — C₅×#33585 #11956305

Order 1280.

748 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#509	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₅×D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#3155	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#2948	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#2210	C₂	1	4	yes	maximal, complemented
6	C₅×#2206	C₂	1	4	yes	maximal, complemented
7	C₅×#2151	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#2140	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#2138	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₅×AGL(1,ℤ/8)	—	2	1	no	—
11	C₂×C₅×#135	C₂²	1	1	yes	characteristic
12	C₂²×C₅×D₈	C₂²	1	1	yes	characteristic
13	C₂²×C₅×D₈	C₂²	1	1	yes	characteristic
14	C₅×#535	—	2	1	no	—
15	C₅×#518	—	2	1	no	—
16	C₅×#509	—	2	4	no	—
17	C₂×C₅×SD₃₂	C₂²	1	2	yes	complemented
18	C₂×C₅×D₁₆	—	2	4	no	—
19	C₂×C₅×D₁₆	C₂²	1	4	yes	—
20	C₂×C₅×D₁₆	C₂²	1	2	yes	complemented
21	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
22	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
23	C₅×#496	—	2	2	no	—
24	C₅×#495	C₂²	1	2	yes	complemented
25	C₅×#493	C₂²	1	2	yes	complemented
26	C₅×#442	—	2	2	no	—
27	C₄×C₅×D₈	C₂²	1	4	yes	complemented
28	C₅×#427	C₂²	1	1	yes	characteristic
29	C₅×#418	—	2	1	no	—
30	C₅×#405	C₂²	1	1	yes	characteristic
31	C₅×C₄⋊C₁₆	C₂²	1	4	yes	complemented
32	C₅×#358	C₂²	1	2	yes	complemented
33	C₅×#357	C₂²	1	2	yes	complemented
34	C₅×#357	C₂²	1	4	yes	complemented
35	#33585	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
36	C₂×C₅×#41	—	2	1	no	—
37	C₂²×C₅×D₄	—	2	1	no	—
38	C₂²×C₅×D₄	—	2	1	no	—
39	C₅×AGL(1,ℤ/8)	—	4	4	no	—
40	C₅×#135	—	2	2	no	—
41	C₅×#135	—	2	2	no	—
42	C₂×C₅×Q₁₆	C₂³	1	1	yes	characteristic
43	C₂×C₅×SD₁₆	—	2	2	no	—
44	C₂×C₅×SD₁₆	—	2	1	no	—
45	C₂×C₅×D₈	—	2	1	no	—
46	C₂×C₅×D₈	—	2	2	no	—
47	C₂×C₅×D₈	—	2	2	no	—
48	C₂×C₅×D₈	—	2	1	no	—
49	C₂×C₅×D₈	—	2	2	no	—
50	C₂×C₅×D₈	—	2	2	no	—
51	C₂×C₅×D₈	D₄	1	4	yes	complemented
52	C₂×C₅×D₈	C₂³	1	1	yes	characteristic
53	C₂×C₅×D₈	C₂³	1	2	yes	—
54	C₂×C₅×D₈	C₂³	1	2	yes	—
55	C₂×C₅×M₁₆	D₄	1	1	yes	characteristic
56	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
57	C₅×#127	—	2	2	no	—
58	C₅×#124	—	2	2	no	—
59	C₅×#121	—	4	2	no	—
60	C₅×#120	—	4	2	no	—
61	C₄×C₅×D₄	—	2	4	no	—
62	C₄×C₅×D₄	—	4	2	no	—
63	C₅×#117	—	2	1	no	—
64	C₅×#117	D₄	1	1	yes	characteristic
65	C₂×C₅×C₂²⋊C₄	—	4	2	no	—
66	C₅×SD₃₂	—	2	4	no	—
67	C₅×D₁₆	—	2	8	no	—
68	C₅×D₁₆	—	2	4	no	—
69	C₅×M₃₂	—	2	2	no	—
70	C₂×C₅×C₁₆	—	2	2	no	—
71	C₂×C₅×C₁₆	C₂³	1	2	yes	—
72	C₂×C₅×C₁₆	C₂³	1	2	yes	—
73	C₅×#107	C₂³	1	2	yes	—
74	C₅×#106	—	2	1	no	—
75	C₅×#102	—	2	4	no	—
76	C₅×#97	D₄	1	2	yes	—
77	C₄×C₅×C₈	C₂³	1	2	yes	—
78	C₂×#509	C₂×C₅	1	1	yes	characteristic, complemented
79	C₂²×D₁₆	C₂×C₅	1	1	yes	characteristic, complemented
80	#3155	C₂×C₅	1	1	yes	characteristic, complemented
81	#2948	C₂×C₅	1	1	yes	characteristic, complemented
82	#2210	C₂×C₅	1	4	yes	complemented
83	#2206	C₂×C₅	1	4	yes	complemented
84	#2151	C₂×C₅	1	1	yes	characteristic, complemented
85	#2140	C₂×C₅	1	1	yes	characteristic, complemented
86	#2138	C₂×C₅	1	1	yes	characteristic, complemented
87	C₂⁴×C₅	—	4	1	no	—
88	C₂⁴×C₅	—	4	1	no	—
89	C₅×#41	—	4	2	no	—
90	C₅×#41	—	4	2	no	—
91	C₂×C₅×Q₈	—	2	1	no	—
92	C₂×C₅×D₄	—	2	4	no	—
93	C₂×C₅×D₄	—	2	2	no	—
94	C₂×C₅×D₄	—	2	2	no	—
95	C₂×C₅×D₄	—	2	1	no	—
96	C₂×C₅×D₄	—	4	1	no	—
97	C₂×C₅×D₄	—	4	2	no	—
98	C₂×C₅×D₄	—	4	1	no	—
99	C₂×C₅×D₄	—	4	1	no	—
100	C₂×C₅×D₄	—	4	4	no	—
101	C₂×C₅×D₄	—	4	2	no	—
102	C₂×C₅×D₄	—	4	2	no	—
103	C₂²×C₄×C₅	—	4	1	no	—
104	C₂²×C₄×C₅	—	4	4	no	—
105	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
106	C₅×Q₁₆	—	2	1	no	—
107	C₅×Q₁₆	—	2	1	no	—
108	C₅×SD₁₆	—	4	4	no	—
109	C₅×SD₁₆	—	4	2	no	—
110	C₅×D₈	—	2	4	no	—
111	C₅×D₈	—	2	2	no	—
112	C₅×D₈	—	2	1	no	—
113	C₅×D₈	—	2	2	no	—
114	C₅×D₈	—	2	2	no	—
115	C₅×D₈	—	2	1	no	—
116	C₅×D₈	—	4	4	no	—
117	C₅×D₈	—	4	2	no	—
118	C₅×D₈	C₂×D₄	1	4	yes	—
119	C₅×M₁₆	—	2	2	no	—
120	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
121	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
122	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
123	C₂×C₅×C₈	#41	1	2	yes	—
124	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
125	C₅×C₄⋊C₄	—	2	2	no	—
126	C₅×C₄⋊C₄	D₈	1	2	yes	—
127	C₅×C₂²⋊C₄	—	4	1	no	—
128	C₅×C₂²⋊C₄	—	4	4	no	—
129	C₅×C₂²⋊C₄	—	4	4	no	—
130	C₅×C₂²⋊C₄	D₈	1	2	yes	—
131	C₄²×C₅	—	4	1	no	—
132	C₄²×C₅	C₂×D₄	1	2	yes	—
133	C₅×C₁₆	—	2	2	no	—
134	C₅×C₁₆	—	2	2	no	—
135	C₂×AGL(1,ℤ/8)	—	2	1	no	—
136	C₂×#135	C₂²×C₅	1	1	yes	characteristic
137	C₂²×D₈	C₂²×C₅	1	1	yes	characteristic
138	C₂²×D₈	C₂²×C₅	1	1	yes	characteristic
139	#535	—	2	1	no	—
140	#518	—	2	1	no	—
141	#509	—	2	4	no	—
142	C₂×SD₃₂	C₂²×C₅	1	2	yes	complemented
143	C₂×D₁₆	—	2	4	no	—
144	C₂×D₁₆	C₂²×C₅	1	4	yes	—
145	C₂×D₁₆	C₂²×C₅	1	2	yes	complemented
146	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
147	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
148	#496	—	2	2	no	—
149	#495	C₂²×C₅	1	2	yes	complemented
150	#493	C₂²×C₅	1	2	yes	complemented
151	#442	—	2	2	no	—
152	C₄×D₈	C₂²×C₅	1	4	yes	complemented
153	#427	C₂²×C₅	1	1	yes	characteristic
154	#418	—	2	1	no	—
155	#405	C₂²×C₅	1	1	yes	characteristic
156	C₄⋊C₁₆	C₂²×C₅	1	4	yes	complemented
157	#358	C₂²×C₅	1	2	yes	complemented
158	#357	C₂²×C₅	1	2	yes	complemented
159	#357	C₂²×C₅	1	4	yes	complemented
160	C₂³×C₅	—	4	2	no	—
161	C₂³×C₅	—	4	4	no	—
162	C₂³×C₅	—	4	2	no	—
163	C₂³×C₅	—	4	1	no	—
164	C₂³×C₅	—	8	2	no	—
165	C₂³×C₅	—	16	2	no	—
166	C₂³×C₅	—	16	2	no	—
167	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
168	C₅×Q₈	—	4	1	no	—
169	C₅×Q₈	—	4	1	no	—
170	C₅×D₄	—	2	4	no	—
171	C₅×D₄	—	4	2	no	—
172	C₅×D₄	—	4	1	no	—
173	C₅×D₄	—	4	2	no	—
174	C₅×D₄	—	4	1	no	—
175	C₅×D₄	—	4	2	no	—
176	C₅×D₄	—	4	4	no	—
177	C₅×D₄	—	8	2	no	—
178	C₅×D₄	—	8	2	no	—
179	C₅×D₄	—	16	4	no	—
180	C₂×C₄×C₅	—	4	2	no	—
181	C₂×C₄×C₅	—	4	1	no	—
182	C₂×C₄×C₅	—	8	2	no	—
183	C₂×C₄×C₅	—	16	4	no	—
184	C₂×C₄×C₅	#121	1	2	yes	—
185	C₂×C₄×C₅	C₂×D₈	1	4	yes	—
186	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
187	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
188	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
189	C₅×C₈	—	2	2	no	—
190	C₅×C₈	#121	1	2	yes	—
191	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
192	C₅×C₈	C₂×#41	1	1	yes	characteristic
193	C₂×#41	—	2	1	no	—
194	C₂²×D₄	—	2	1	no	—
195	C₂²×D₄	—	2	1	no	—
196	AGL(1,ℤ/8)	—	4	4	no	—
197	#135	—	2	2	no	—
198	#135	—	2	2	no	—
199	C₂×Q₁₆	C₂³×C₅	1	1	yes	characteristic
200	C₂×SD₁₆	—	2	1	no	—
201	C₂×SD₁₆	—	2	2	no	—
202	C₂×D₈	—	2	1	no	—
203	C₂×D₈	—	2	2	no	—
204	C₂×D₈	—	2	1	no	—
205	C₂×D₈	—	2	2	no	—
206	C₂×D₈	—	2	2	no	—
207	C₂×D₈	—	2	2	no	—
208	C₂×D₈	C₅×D₄	1	4	yes	complemented
209	C₂×D₈	C₂³×C₅	1	2	yes	—
210	C₂×D₈	C₂³×C₅	1	1	yes	characteristic
211	C₂×D₈	C₂³×C₅	1	2	yes	—
212	C₂×M₁₆	C₅×D₄	1	1	yes	characteristic
213	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
214	#127	—	2	2	no	—
215	#124	—	2	2	no	—
216	#121	—	4	2	no	—
217	#120	—	4	2	no	—
218	C₄×D₄	—	2	4	no	—
219	C₄×D₄	—	4	2	no	—
220	#117	—	2	1	no	—
221	#117	C₅×D₄	1	1	yes	characteristic
222	C₂×C₂²⋊C₄	—	4	2	no	—
223	SD₃₂	—	2	4	no	—
224	D₁₆	—	2	8	no	—
225	D₁₆	—	2	4	no	—
226	M₃₂	—	2	2	no	—
227	C₂×C₁₆	—	2	2	no	—
228	C₂×C₁₆	C₂³×C₅	1	2	yes	—
229	C₂×C₁₆	C₂³×C₅	1	2	yes	—
230	#107	C₂³×C₅	1	2	yes	—
231	#106	—	2	1	no	—
232	#102	—	2	4	no	—
233	#97	C₅×D₄	1	2	yes	—
234	C₄×C₈	C₂³×C₅	1	2	yes	—
235	C₂²×C₅	—	2	2	no	—
236	C₂²×C₅	—	4	4	no	—
237	C₂²×C₅	—	8	4	no	—
238	C₂²×C₅	—	8	2	no	—
239	C₂²×C₅	—	8	4	no	—
240	C₂²×C₅	—	8	1	no	—
241	C₂²×C₅	—	8	2	no	—
242	C₂²×C₅	—	16	4	no	—
243	C₂²×C₅	—	16	4	no	—
244	C₂²×C₅	#459	1	2	yes	—
245	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
246	C₄×C₅	—	2	4	no	—
247	C₄×C₅	—	8	2	no	—
248	C₄×C₅	—	8	1	no	—
249	C₄×C₅	#459	1	2	yes	—
250	C₄×C₅	C₂×#121	1	1	yes	characteristic
251	C₄×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
252	C₂⁴	—	4	1	no	—
253	C₂⁴	—	4	1	no	—
254	#41	—	4	2	no	—
255	#41	—	4	2	no	—
256	C₂×Q₈	—	2	1	no	—
257	C₂×D₄	—	2	4	no	—
258	C₂×D₄	—	2	2	no	—
259	C₂×D₄	—	2	2	no	—
260	C₂×D₄	—	2	1	no	—
261	C₂×D₄	—	4	1	no	—
262	C₂×D₄	—	4	2	no	—
263	C₂×D₄	—	4	1	no	—
264	C₂×D₄	—	4	1	no	—
265	C₂×D₄	—	4	4	no	—
266	C₂×D₄	—	4	2	no	—
267	C₂×D₄	—	4	2	no	—
268	C₂²×C₄	—	4	1	no	—
269	C₂²×C₄	—	4	4	no	—
270	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
271	Q₁₆	—	2	1	no	—
272	Q₁₆	—	2	1	no	—
273	SD₁₆	—	4	4	no	—
274	SD₁₆	—	4	2	no	—
275	D₈	—	2	4	no	—
276	D₈	—	2	2	no	—
277	D₈	—	2	1	no	—
278	D₈	—	2	2	no	—
279	D₈	—	2	2	no	—
280	D₈	—	2	1	no	—
281	D₈	—	4	4	no	—
282	D₈	—	4	2	no	—
283	D₈	C₂×C₅×D₄	1	4	yes	—
284	M₁₆	—	2	2	no	—
285	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
286	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
287	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
288	C₂×C₈	C₅×#41	1	2	yes	—
289	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	C₅×D₈	1	2	yes	—
292	C₂²⋊C₄	—	4	1	no	—
293	C₂²⋊C₄	—	4	4	no	—
294	C₂²⋊C₄	—	4	4	no	—
295	C₂²⋊C₄	C₅×D₈	1	2	yes	—
296	C₄²	—	4	1	no	—
297	C₄²	C₂×C₅×D₄	1	2	yes	—
298	C₁₆	—	2	2	no	—
299	C₁₆	—	2	2	no	—
300	C₂×C₅	—	2	2	no	—
301	C₂×C₅	—	8	4	no	—
302	C₂×C₅	—	16	1	no	—
303	C₂×C₅	—	16	2	no	—
304	C₂×C₅	C₂×#459	1	1	yes	characteristic
305	C₂×C₅	#3415	1	2	yes	—
306	C₂³	—	4	2	no	—
307	C₂³	—	4	4	no	—
308	C₂³	—	4	2	no	—
309	C₂³	—	4	1	no	—
310	C₂³	—	8	2	no	—
311	C₂³	—	16	2	no	—
312	C₂³	—	16	2	no	—
313	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
314	Q₈	—	4	1	no	—
315	Q₈	—	4	1	no	—
316	D₄	—	2	4	no	—
317	D₄	—	4	4	no	—
318	D₄	—	4	2	no	—
319	D₄	—	4	1	no	—
320	D₄	—	4	2	no	—
321	D₄	—	4	1	no	—
322	D₄	—	4	2	no	—
323	D₄	—	8	2	no	—
324	D₄	—	8	2	no	—
325	D₄	—	16	4	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	4	1	no	—
328	C₂×C₄	—	8	2	no	—
329	C₂×C₄	—	16	4	no	—
330	C₂×C₄	C₅×#121	1	2	yes	—
331	C₂×C₄	C₂×C₅×D₈	1	4	yes	—
332	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
333	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
334	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
335	C₈	—	2	2	no	—
336	C₈	C₅×#121	1	2	yes	—
337	C₈	C₂²×C₅×D₄	1	1	yes	characteristic
338	C₈	C₂×C₅×#41	1	1	yes	characteristic
339	C₅	#33585	1	1	yes	characteristic, direct product factor, Sylow
340	C₂²	—	2	2	no	—
341	C₂²	—	4	4	no	—
342	C₂²	—	8	4	no	—
343	C₂²	—	8	1	no	—
344	C₂²	—	8	2	no	—
345	C₂²	—	8	2	no	—
346	C₂²	—	8	4	no	—
347	C₂²	—	16	4	no	—
348	C₂²	—	16	4	no	—
349	C₂²	C₅×#459	1	2	yes	—
350	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
351	C₄	—	2	4	no	—
352	C₄	—	8	1	no	—
353	C₄	—	8	2	no	—
354	C₄	C₅×#459	1	2	yes	—
355	C₄	C₂×C₅×#121	1	1	yes	characteristic
356	C₄	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
357	C₂	—	2	2	no	—
358	C₂	—	8	4	no	—
359	C₂	—	16	2	no	—
360	C₂	—	16	1	no	—
361	C₂	C₂×C₅×#459	1	1	yes	characteristic
362	C₂	C₅×#3415	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₅×#117 ◁ C₅×#405 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×#509	C₂	semidirect
2. C₂²×C₅×D₁₆	C₂	semidirect
3. C₅×#3155	C₂	semidirect
4. C₅×#2948	C₂	semidirect
5. C₅×#2210	C₂	semidirect
6. C₅×#2206	C₂	semidirect
7. C₅×#2151	C₂	semidirect
8. C₅×#2140	C₂	semidirect
9. C₅×#2138	C₂	semidirect
17. C₂×C₅×SD₃₂	C₂²	semidirect
20. C₂×C₅×D₁₆	C₂²	semidirect
24. C₅×#495	C₂²	semidirect
25. C₅×#493	C₂²	semidirect
27. C₄×C₅×D₈	C₂²	semidirect
31. C₅×C₄⋊C₁₆	C₂²	semidirect
32. C₅×#358	C₂²	semidirect
33. C₅×#357	C₂²	semidirect
34. C₅×#357	C₂²	semidirect
35. #33585	C₅	direct (N × H)
51. C₂×C₅×D₈	D₄	semidirect
78. C₂×#509	C₂×C₅	semidirect
79. C₂²×D₁₆	C₂×C₅	semidirect
80. #3155	C₂×C₅	semidirect
81. #2948	C₂×C₅	semidirect
82. #2210	C₂×C₅	semidirect
83. #2206	C₂×C₅	semidirect
84. #2151	C₂×C₅	semidirect
85. #2140	C₂×C₅	semidirect
86. #2138	C₂×C₅	semidirect
142. C₂×SD₃₂	C₂²×C₅	semidirect
145. C₂×D₁₆	C₂²×C₅	semidirect
149. #495	C₂²×C₅	semidirect
150. #493	C₂²×C₅	semidirect
152. C₄×D₈	C₂²×C₅	semidirect
156. C₄⋊C₁₆	C₂²×C₅	semidirect
157. #358	C₂²×C₅	semidirect
158. #357	C₂²×C₅	semidirect
159. #357	C₂²×C₅	semidirect
208. C₂×D₈	C₅×D₄	semidirect
339. C₅	#33585	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅×#509	2
C₂²×C₅×D₁₆	2
C₅×#3155	2
C₅×#2948	2
C₅×#2210	2
C₅×#2206	2
C₅×#2151	2
C₅×#2140	2
C₅×#2138	2
#33585	5

Subgroups — C5×#33585 #11956305

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33585 #11956305