Subgroups — C₅×#33586 #11956306

Order 1280.

716 conjugacy classes of subgroups.

The subgroup lattice has 354 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
2	C₂²×C₅×SD₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#3156	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×#2208	C₂	1	4	yes	maximal, complemented
7	C₅×#2151	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#2141	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#2138	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₅×#137	—	2	1	no	—
11	C₂×C₅×#135	C₂²	1	1	yes	characteristic
12	C₂²×C₅×Q₁₆	C₂²	1	1	yes	characteristic
13	C₂²×C₅×D₈	C₂²	1	1	yes	characteristic
14	C₅×#536	—	2	1	no	—
15	C₅×#518	—	2	1	no	—
16	C₅×#509	—	2	4	no	—
17	C₂×C₅×SD₃₂	—	2	4	no	—
18	C₂×C₅×SD₃₂	C₂²	1	4	yes	—
19	C₂×C₅×SD₃₂	C₂²	1	2	yes	—
20	C₂×C₅×D₁₆	C₂²	1	2	yes	—
21	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
22	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
23	C₅×#497	—	2	2	no	—
24	C₅×#495	C₂²	1	2	yes	complemented
25	C₅×#494	C₂²	1	2	yes	complemented
26	C₅×#442	—	2	2	no	—
27	C₄×C₅×D₈	C₂²	1	4	yes	—
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×#358	C₂²	1	4	yes	complemented
34	C₅×#357	C₂²	1	2	yes	complemented
35	#33586	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
36	C₂×C₅×#41	—	2	1	no	—
37	C₂²×C₅×Q₈	—	2	1	no	—
38	C₂²×C₅×D₄	—	2	1	no	—
39	C₅×#137	—	4	4	no	—
40	C₅×#135	—	2	2	no	—
41	C₅×#135	—	2	2	no	—
42	C₂×C₅×Q₁₆	—	2	2	no	—
43	C₂×C₅×Q₁₆	—	2	2	no	—
44	C₂×C₅×Q₁₆	—	2	1	no	—
45	C₂×C₅×Q₁₆	—	2	2	no	—
46	C₂×C₅×Q₁₆	C₂³	1	1	yes	characteristic
47	C₂×C₅×Q₁₆	C₂³	1	2	yes	—
48	C₂×C₅×SD₁₆	—	2	2	no	—
49	C₂×C₅×SD₁₆	—	2	1	no	—
50	C₂×C₅×D₈	—	2	1	no	—
51	C₂×C₅×D₈	—	2	2	no	—
52	C₂×C₅×D₈	D₄	1	4	yes	—
53	C₂×C₅×D₈	C₂³	1	1	yes	characteristic
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×C₄⋊Q₈	—	2	2	no	—
58	C₅×#124	—	2	2	no	—
59	C₅×#123	—	4	2	no	—
60	C₅×C₂²⋊Q₈	—	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	8	no	—
67	C₅×SD₃₂	—	2	4	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
79	C₂²×SD₃₂	C₂×C₅	1	1	yes	characteristic, complemented
80	#3156	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	#2208	C₂×C₅	1	4	yes	complemented
84	#2151	C₂×C₅	1	1	yes	characteristic, complemented
85	#2141	C₂×C₅	1	1	yes	characteristic, complemented
86	#2138	C₂×C₅	1	1	yes	characteristic, complemented
87	C₂⁴×C₅	—	4	1	no	—
88	C₅×#41	—	4	2	no	—
89	C₅×#41	—	4	2	no	—
90	C₂×C₅×Q₈	—	2	2	no	—
91	C₂×C₅×Q₈	—	2	1	no	—
92	C₂×C₅×Q₈	—	4	1	no	—
93	C₂×C₅×Q₈	—	4	2	no	—
94	C₂×C₅×Q₈	—	4	2	no	—
95	C₂×C₅×D₄	—	2	4	no	—
96	C₂×C₅×D₄	—	2	2	no	—
97	C₂×C₅×D₄	—	2	1	no	—
98	C₂×C₅×D₄	—	4	1	no	—
99	C₂×C₅×D₄	—	4	1	no	—
100	C₂×C₅×D₄	—	4	2	no	—
101	C₂²×C₄×C₅	—	4	1	no	—
102	C₂²×C₄×C₅	—	4	1	no	—
103	C₂²×C₄×C₅	—	4	4	no	—
104	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
105	C₅×Q₁₆	—	2	2	no	—
106	C₅×Q₁₆	—	2	1	no	—
107	C₅×Q₁₆	—	2	2	no	—
108	C₅×Q₁₆	—	2	1	no	—
109	C₅×Q₁₆	—	4	4	no	—
110	C₅×Q₁₆	—	4	2	no	—
111	C₅×SD₁₆	—	4	4	no	—
112	C₅×SD₁₆	—	4	2	no	—
113	C₅×D₈	—	2	4	no	—
114	C₅×D₈	—	2	1	no	—
115	C₅×D₈	—	2	2	no	—
116	C₅×D₈	—	2	1	no	—
117	C₅×D₈	C₂×D₄	1	4	yes	—
118	C₅×M₁₆	—	2	2	no	—
119	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
120	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
121	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
122	C₂×C₅×C₈	#41	1	2	yes	—
123	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
124	C₅×C₄⋊C₄	—	2	2	no	—
125	C₅×C₄⋊C₄	—	4	4	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×#137	—	2	1	no	—
136	C₂×#135	C₂²×C₅	1	1	yes	characteristic
137	C₂²×Q₁₆	C₂²×C₅	1	1	yes	characteristic
138	C₂²×D₈	C₂²×C₅	1	1	yes	characteristic
139	#536	—	2	1	no	—
140	#518	—	2	1	no	—
141	#509	—	2	4	no	—
142	C₂×SD₃₂	—	2	4	no	—
143	C₂×SD₃₂	C₂²×C₅	1	4	yes	—
144	C₂×SD₃₂	C₂²×C₅	1	2	yes	—
145	C₂×D₁₆	C₂²×C₅	1	2	yes	—
146	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
147	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
148	#497	—	2	2	no	—
149	#495	C₂²×C₅	1	2	yes	complemented
150	#494	C₂²×C₅	1	2	yes	complemented
151	#442	—	2	2	no	—
152	C₄×D₈	C₂²×C₅	1	4	yes	—
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	#358	C₂²×C₅	1	4	yes	complemented
159	#357	C₂²×C₅	1	2	yes	complemented
160	C₂³×C₅	—	4	2	no	—
161	C₂³×C₅	—	4	4	no	—
162	C₂³×C₅	—	4	1	no	—
163	C₂³×C₅	—	16	2	no	—
164	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
165	C₅×Q₈	—	4	1	no	—
166	C₅×Q₈	—	4	2	no	—
167	C₅×Q₈	—	4	2	no	—
168	C₅×Q₈	—	4	1	no	—
169	C₅×Q₈	—	8	2	no	—
170	C₅×D₄	—	2	4	no	—
171	C₅×D₄	—	4	1	no	—
172	C₅×D₄	—	4	2	no	—
173	C₅×D₄	—	4	1	no	—
174	C₅×D₄	—	4	4	no	—
175	C₅×D₄	—	8	2	no	—
176	C₂×C₄×C₅	—	4	2	no	—
177	C₂×C₄×C₅	—	4	2	no	—
178	C₂×C₄×C₅	—	4	1	no	—
179	C₂×C₄×C₅	—	8	2	no	—
180	C₂×C₄×C₅	—	8	2	no	—
181	C₂×C₄×C₅	—	16	4	no	—
182	C₂×C₄×C₅	#121	1	2	yes	—
183	C₂×C₄×C₅	C₂×D₈	1	4	yes	—
184	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
185	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
186	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
187	C₅×C₈	—	2	2	no	—
188	C₅×C₈	#121	1	2	yes	—
189	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
190	C₅×C₈	C₂×#41	1	1	yes	characteristic
191	C₂×#41	—	2	1	no	—
192	C₂²×Q₈	—	2	1	no	—
193	C₂²×D₄	—	2	1	no	—
194	#137	—	4	4	no	—
195	#135	—	2	2	no	—
196	#135	—	2	2	no	—
197	C₂×Q₁₆	—	2	2	no	—
198	C₂×Q₁₆	—	2	1	no	—
199	C₂×Q₁₆	—	2	2	no	—
200	C₂×Q₁₆	—	2	2	no	—
201	C₂×Q₁₆	C₂³×C₅	1	2	yes	—
202	C₂×Q₁₆	C₂³×C₅	1	1	yes	characteristic
203	C₂×SD₁₆	—	2	1	no	—
204	C₂×SD₁₆	—	2	2	no	—
205	C₂×D₈	—	2	1	no	—
206	C₂×D₈	—	2	2	no	—
207	C₂×D₈	C₅×D₄	1	4	yes	—
208	C₂×D₈	C₂³×C₅	1	1	yes	characteristic
209	C₂×D₈	C₂³×C₅	1	2	yes	—
210	C₂×M₁₆	C₅×D₄	1	1	yes	characteristic
211	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
212	C₄⋊Q₈	—	2	2	no	—
213	#124	—	2	2	no	—
214	#123	—	4	2	no	—
215	C₂²⋊Q₈	—	4	2	no	—
216	C₄×D₄	—	2	4	no	—
217	C₄×D₄	—	4	2	no	—
218	#117	—	2	1	no	—
219	#117	C₅×D₄	1	1	yes	characteristic
220	C₂×C₂²⋊C₄	—	4	2	no	—
221	SD₃₂	—	2	8	no	—
222	SD₃₂	—	2	4	no	—
223	D₁₆	—	2	4	no	—
224	M₃₂	—	2	2	no	—
225	C₂×C₁₆	—	2	2	no	—
226	C₂×C₁₆	C₂³×C₅	1	2	yes	—
227	C₂×C₁₆	C₂³×C₅	1	2	yes	—
228	#107	C₂³×C₅	1	2	yes	—
229	#106	—	2	1	no	—
230	#102	—	2	4	no	—
231	#97	C₅×D₄	1	2	yes	—
232	C₄×C₈	C₂³×C₅	1	2	yes	—
233	C₂²×C₅	—	2	2	no	—
234	C₂²×C₅	—	4	4	no	—
235	C₂²×C₅	—	8	4	no	—
236	C₂²×C₅	—	8	2	no	—
237	C₂²×C₅	—	8	1	no	—
238	C₂²×C₅	—	16	4	no	—
239	C₂²×C₅	#459	1	2	yes	—
240	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
241	C₄×C₅	—	2	4	no	—
242	C₄×C₅	—	8	2	no	—
243	C₄×C₅	—	8	1	no	—
244	C₄×C₅	—	8	2	no	—
245	C₄×C₅	#459	1	2	yes	—
246	C₄×C₅	C₂×#121	1	1	yes	characteristic
247	C₄×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
248	C₂⁴	—	4	1	no	—
249	#41	—	4	2	no	—
250	#41	—	4	2	no	—
251	C₂×Q₈	—	2	2	no	—
252	C₂×Q₈	—	2	1	no	—
253	C₂×Q₈	—	4	1	no	—
254	C₂×Q₈	—	4	2	no	—
255	C₂×Q₈	—	4	2	no	—
256	C₂×D₄	—	2	4	no	—
257	C₂×D₄	—	2	2	no	—
258	C₂×D₄	—	2	1	no	—
259	C₂×D₄	—	4	1	no	—
260	C₂×D₄	—	4	1	no	—
261	C₂×D₄	—	4	2	no	—
262	C₂²×C₄	—	4	1	no	—
263	C₂²×C₄	—	4	1	no	—
264	C₂²×C₄	—	4	4	no	—
265	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
266	Q₁₆	—	2	2	no	—
267	Q₁₆	—	2	1	no	—
268	Q₁₆	—	2	2	no	—
269	Q₁₆	—	2	1	no	—
270	Q₁₆	—	4	4	no	—
271	Q₁₆	—	4	2	no	—
272	SD₁₆	—	4	4	no	—
273	SD₁₆	—	4	2	no	—
274	D₈	—	2	4	no	—
275	D₈	—	2	1	no	—
276	D₈	—	2	2	no	—
277	D₈	—	2	1	no	—
278	D₈	C₂×C₅×D₄	1	4	yes	—
279	M₁₆	—	2	2	no	—
280	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
281	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
282	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
283	C₂×C₈	C₅×#41	1	2	yes	—
284	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
285	C₄⋊C₄	—	2	2	no	—
286	C₄⋊C₄	—	4	4	no	—
287	C₄⋊C₄	C₅×D₈	1	2	yes	—
288	C₂²⋊C₄	—	4	1	no	—
289	C₂²⋊C₄	—	4	4	no	—
290	C₂²⋊C₄	—	4	4	no	—
291	C₂²⋊C₄	C₅×D₈	1	2	yes	—
292	C₄²	—	4	1	no	—
293	C₄²	C₂×C₅×D₄	1	2	yes	—
294	C₁₆	—	2	2	no	—
295	C₁₆	—	2	2	no	—
296	C₂×C₅	—	2	2	no	—
297	C₂×C₅	—	8	4	no	—
298	C₂×C₅	—	16	1	no	—
299	C₂×C₅	C₂×#459	1	1	yes	characteristic
300	C₂×C₅	#3416	1	2	yes	—
301	C₂³	—	4	2	no	—
302	C₂³	—	4	4	no	—
303	C₂³	—	4	1	no	—
304	C₂³	—	16	2	no	—
305	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
306	Q₈	—	4	1	no	—
307	Q₈	—	4	2	no	—
308	Q₈	—	4	2	no	—
309	Q₈	—	4	1	no	—
310	Q₈	—	8	2	no	—
311	D₄	—	2	4	no	—
312	D₄	—	4	4	no	—
313	D₄	—	4	1	no	—
314	D₄	—	4	2	no	—
315	D₄	—	4	1	no	—
316	D₄	—	8	2	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	4	1	no	—
320	C₂×C₄	—	8	2	no	—
321	C₂×C₄	—	8	2	no	—
322	C₂×C₄	—	16	4	no	—
323	C₂×C₄	C₅×#121	1	2	yes	—
324	C₂×C₄	C₂×C₅×D₈	1	4	yes	—
325	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
326	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
327	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
328	C₈	—	2	2	no	—
329	C₈	C₅×#121	1	2	yes	—
330	C₈	C₂²×C₅×D₄	1	1	yes	characteristic
331	C₈	C₂×C₅×#41	1	1	yes	characteristic
332	C₅	#33586	1	1	yes	characteristic, direct product factor, Sylow
333	C₂²	—	2	2	no	—
334	C₂²	—	4	4	no	—
335	C₂²	—	8	4	no	—
336	C₂²	—	8	1	no	—
337	C₂²	—	8	2	no	—
338	C₂²	—	16	4	no	—
339	C₂²	C₅×#459	1	2	yes	—
340	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
341	C₄	—	2	4	no	—
342	C₄	—	8	1	no	—
343	C₄	—	8	2	no	—
344	C₄	—	8	2	no	—
345	C₄	C₅×#459	1	2	yes	—
346	C₄	C₂×C₅×#121	1	1	yes	characteristic
347	C₄	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
348	C₂	—	2	2	no	—
349	C₂	—	8	4	no	—
350	C₂	—	16	1	no	—
351	C₂	C₂×C₅×#459	1	1	yes	characteristic
352	C₂	C₅×#3416	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅
prime radical	C₂×C₅×#509	C₂
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
2. C₂²×C₅×SD₃₂	C₂	semidirect
3. C₅×#3156	C₂	semidirect
4. C₅×#2948	C₂	semidirect
5. C₅×#2210	C₂	semidirect
6. C₅×#2208	C₂	semidirect
7. C₅×#2151	C₂	semidirect
8. C₅×#2141	C₂	semidirect
9. C₅×#2138	C₂	semidirect
24. C₅×#495	C₂²	semidirect
25. C₅×#494	C₂²	semidirect
31. C₅×C₄⋊C₁₆	C₂²	semidirect
32. C₅×#358	C₂²	semidirect
33. C₅×#358	C₂²	semidirect
34. C₅×#357	C₂²	semidirect
35. #33586	C₅	direct (N × H)
79. C₂²×SD₃₂	C₂×C₅	semidirect
80. #3156	C₂×C₅	semidirect
81. #2948	C₂×C₅	semidirect
82. #2210	C₂×C₅	semidirect
83. #2208	C₂×C₅	semidirect
84. #2151	C₂×C₅	semidirect
85. #2141	C₂×C₅	semidirect
86. #2138	C₂×C₅	semidirect
149. #495	C₂²×C₅	semidirect
150. #494	C₂²×C₅	semidirect
156. C₄⋊C₁₆	C₂²×C₅	semidirect
157. #358	C₂²×C₅	semidirect
158. #358	C₂²×C₅	semidirect
159. #357	C₂²×C₅	semidirect
332. C₅	#33586	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×SD₃₂	2
C₅×#3156	2
C₅×#2948	2
C₅×#2210	2
C₅×#2208	2
C₅×#2151	2
C₅×#2141	2
C₅×#2138	2
#33586	5

Subgroups — C5×#33586 #11956306

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33586 #11956306