Subgroups — C₅×#33584 #11956304

Order 1280.

722 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#510	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×#509	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#495	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₅×#135	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#2211	C₂	1	4	yes	maximal, complemented
6	C₅×#2210	C₂	1	4	yes	maximal, complemented
7	C₅×#2150	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#2141	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#2140	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₅×#135	C₂²	1	1	yes	characteristic
11	C₂²×C₅×Q₁₆	C₂²	1	1	yes	characteristic
12	C₂²×C₅×SD₁₆	—	2	1	no	—
13	C₂²×C₅×D₈	C₂²	1	1	yes	characteristic
14	C₂×C₅×#124	—	2	1	no	—
15	C₄×C₅×#41	—	2	1	no	—
16	C₅×#510	—	2	4	no	—
17	C₅×#509	—	2	4	no	—
18	C₂×C₅×Q₃₂	C₂²	1	2	yes	complemented
19	C₂×C₅×SD₃₂	C₂²	1	2	yes	—
20	C₂×C₅×SD₃₂	C₂²	1	2	yes	complemented
21	C₂×C₅×D₁₆	C₂²	1	2	yes	—
22	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
23	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
24	C₅×#495	—	2	2	no	—
25	C₅×#495	C₂²	1	4	yes	—
26	C₄×C₅×Q₁₆	C₂²	1	2	yes	complemented
27	C₄×C₅×SD₁₆	—	2	2	no	—
28	C₄×C₅×D₈	C₂²	1	2	yes	complemented
29	C₅×#427	C₂²	1	1	yes	characteristic
30	C₅×#416	—	2	1	no	—
31	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
32	C₅×C₄⋊C₁₆	C₂²	1	4	yes	complemented
33	C₅×#358	C₂²	1	4	yes	complemented
34	C₅×#357	C₂²	1	4	yes	—
35	#33584	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×#135	—	2	2	no	—
40	C₅×#135	D₄	1	4	yes	—
41	C₂×C₅×Q₁₆	—	2	2	no	—
42	C₂×C₅×Q₁₆	—	2	1	no	—
43	C₂×C₅×Q₁₆	—	2	2	no	—
44	C₂×C₅×Q₁₆	C₂³	1	2	yes	—
45	C₂×C₅×Q₁₆	C₂³	1	1	yes	characteristic
46	C₂×C₅×SD₁₆	—	2	1	no	—
47	C₂×C₅×SD₁₆	—	2	4	no	—
48	C₂×C₅×SD₁₆	—	4	4	no	—
49	C₂×C₅×D₈	—	2	2	no	—
50	C₂×C₅×D₈	—	2	1	no	—
51	C₂×C₅×D₈	—	2	2	no	—
52	C₂×C₅×D₈	C₂³	1	2	yes	—
53	C₂×C₅×D₈	C₂³	1	1	yes	characteristic
54	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
55	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
56	C₅×#124	—	2	4	no	—
57	C₅×#124	—	4	1	no	—
58	C₅×#124	—	4	1	no	—
59	C₄×C₅×Q₈	—	2	2	no	—
60	C₄×C₅×D₄	—	2	2	no	—
61	C₄×C₅×D₄	—	4	2	no	—
62	C₅×#117	—	2	1	no	—
63	C₅×#117	—	4	1	no	—
64	C₂×C₅×C₂²⋊C₄	—	4	2	no	—
65	C₂×C₄²×C₅	—	4	1	no	—
66	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
67	C₅×Q₃₂	—	2	4	no	—
68	C₅×SD₃₂	—	2	4	no	—
69	C₅×SD₃₂	—	2	4	no	—
70	C₅×D₁₆	—	2	4	no	—
71	C₅×M₃₂	—	2	2	no	—
72	C₅×M₃₂	—	2	2	no	—
73	C₂×C₅×C₁₆	C₂³	1	2	yes	—
74	C₂×C₅×C₁₆	C₂³	1	2	yes	—
75	C₅×#107	C₂³	1	2	yes	—
76	C₅×#106	—	2	1	no	—
77	C₅×#103	—	2	2	no	—
78	C₅×#102	—	2	2	no	—
79	C₄×C₅×C₈	D₄	1	2	yes	—
80	C₄×C₅×C₈	C₂³	1	2	yes	—
81	C₂×#510	C₂×C₅	1	1	yes	characteristic, complemented
82	C₂×#509	C₂×C₅	1	1	yes	characteristic
83	C₂×#495	C₂×C₅	1	1	yes	characteristic, complemented
84	C₄×#135	C₂×C₅	1	1	yes	characteristic, complemented
85	#2211	C₂×C₅	1	4	yes	complemented
86	#2210	C₂×C₅	1	4	yes	complemented
87	#2150	C₂×C₅	1	1	yes	characteristic, complemented
88	#2141	C₂×C₅	1	1	yes	characteristic, complemented
89	#2140	C₂×C₅	1	1	yes	characteristic, complemented
90	C₂⁴×C₅	—	4	1	no	—
91	C₅×#41	—	2	4	no	—
92	C₅×#41	—	4	2	no	—
93	C₂×C₅×Q₈	—	2	1	no	—
94	C₂×C₅×Q₈	—	2	2	no	—
95	C₂×C₅×Q₈	—	4	1	no	—
96	C₂×C₅×Q₈	—	4	2	no	—
97	C₂×C₅×Q₈	—	4	2	no	—
98	C₂×C₅×D₄	—	2	1	no	—
99	C₂×C₅×D₄	—	2	2	no	—
100	C₂×C₅×D₄	—	4	1	no	—
101	C₂×C₅×D₄	—	4	2	no	—
102	C₂×C₅×D₄	—	4	1	no	—
103	C₂×C₅×D₄	—	4	2	no	—
104	C₂²×C₄×C₅	—	4	1	no	—
105	C₂²×C₄×C₅	—	4	1	no	—
106	C₂²×C₄×C₅	—	4	2	no	—
107	C₂²×C₄×C₅	D₈	1	2	yes	—
108	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
109	C₅×Q₁₆	—	2	2	no	—
110	C₅×Q₁₆	—	2	1	no	—
111	C₅×Q₁₆	—	2	2	no	—
112	C₅×Q₁₆	—	4	2	no	—
113	C₅×Q₁₆	C₂×D₄	1	2	yes	—
114	C₅×SD₁₆	—	2	4	no	—
115	C₅×SD₁₆	—	4	8	no	—
116	C₅×D₈	—	2	2	no	—
117	C₅×D₈	—	2	1	no	—
118	C₅×D₈	—	2	2	no	—
119	C₅×D₈	—	4	2	no	—
120	C₅×D₈	C₂×D₄	1	2	yes	—
121	C₂×C₅×C₈	—	2	2	no	—
122	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
123	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
124	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
125	C₂×C₅×C₈	#41	1	2	yes	—
126	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
127	C₅×C₄⋊C₄	—	2	2	no	—
128	C₅×C₄⋊C₄	—	4	2	no	—
129	C₅×C₂²⋊C₄	—	4	1	no	—
130	C₅×C₂²⋊C₄	—	4	4	no	—
131	C₅×C₂²⋊C₄	—	4	4	no	—
132	C₅×C₂²⋊C₄	—	4	2	no	—
133	C₄²×C₅	—	4	1	no	—
134	C₄²×C₅	—	4	2	no	—
135	C₄²×C₅	D₈	1	1	yes	characteristic
136	C₄²×C₅	D₈	1	1	yes	characteristic
137	C₄²×C₅	C₂×D₄	1	2	yes	—
138	C₅×C₁₆	—	2	2	no	—
139	C₅×C₁₆	—	2	2	no	—
140	C₂×#135	C₂²×C₅	1	1	yes	characteristic
141	C₂²×Q₁₆	C₂²×C₅	1	1	yes	characteristic
142	C₂²×SD₁₆	—	2	1	no	—
143	C₂²×D₈	C₂²×C₅	1	1	yes	characteristic
144	C₂×#124	—	2	1	no	—
145	C₄×#41	—	2	1	no	—
146	#510	—	2	4	no	—
147	#509	—	2	4	no	—
148	C₂×Q₃₂	C₂²×C₅	1	2	yes	complemented
149	C₂×SD₃₂	C₂²×C₅	1	2	yes	—
150	C₂×SD₃₂	C₂²×C₅	1	2	yes	complemented
151	C₂×D₁₆	C₂²×C₅	1	2	yes	—
152	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
153	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
154	#495	—	2	2	no	—
155	#495	C₂²×C₅	1	4	yes	—
156	C₄×Q₁₆	C₂²×C₅	1	2	yes	complemented
157	C₄×SD₁₆	—	2	2	no	—
158	C₄×D₈	C₂²×C₅	1	2	yes	complemented
159	#427	C₂²×C₅	1	1	yes	characteristic
160	#416	—	2	1	no	—
161	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
162	C₄⋊C₁₆	C₂²×C₅	1	4	yes	complemented
163	#358	C₂²×C₅	1	4	yes	complemented
164	#357	C₂²×C₅	1	4	yes	—
165	C₂³×C₅	—	4	1	no	—
166	C₂³×C₅	—	4	2	no	—
167	C₂³×C₅	—	8	4	no	—
168	C₂³×C₅	—	8	2	no	—
169	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
170	C₅×Q₈	—	2	2	no	—
171	C₅×Q₈	—	4	1	no	—
172	C₅×Q₈	—	4	2	no	—
173	C₅×Q₈	—	4	2	no	—
174	C₅×Q₈	—	8	2	no	—
175	C₅×D₄	—	2	2	no	—
176	C₅×D₄	—	4	4	no	—
177	C₅×D₄	—	4	2	no	—
178	C₅×D₄	—	4	1	no	—
179	C₅×D₄	—	4	2	no	—
180	C₅×D₄	—	8	2	no	—
181	C₂×C₄×C₅	—	4	2	no	—
182	C₂×C₄×C₅	—	4	1	no	—
183	C₂×C₄×C₅	—	4	4	no	—
184	C₂×C₄×C₅	—	4	2	no	—
185	C₂×C₄×C₅	—	4	2	no	—
186	C₂×C₄×C₅	—	8	4	no	—
187	C₂×C₄×C₅	—	8	2	no	—
188	C₂×C₄×C₅	#121	1	2	yes	—
189	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
190	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
191	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
192	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
193	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
194	C₅×C₈	—	2	2	no	—
195	C₅×C₈	#121	1	2	yes	—
196	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
197	C₅×C₈	C₂×#41	1	1	yes	characteristic
198	C₂×#41	—	2	1	no	—
199	C₂²×Q₈	—	2	1	no	—
200	C₂²×D₄	—	2	1	no	—
201	#135	—	2	2	no	—
202	#135	C₅×D₄	1	4	yes	—
203	C₂×Q₁₆	—	2	2	no	—
204	C₂×Q₁₆	—	2	1	no	—
205	C₂×Q₁₆	—	2	2	no	—
206	C₂×Q₁₆	C₂³×C₅	1	2	yes	—
207	C₂×Q₁₆	C₂³×C₅	1	1	yes	characteristic
208	C₂×SD₁₆	—	2	1	no	—
209	C₂×SD₁₆	—	2	4	no	—
210	C₂×SD₁₆	—	4	4	no	—
211	C₂×D₈	—	2	2	no	—
212	C₂×D₈	—	2	1	no	—
213	C₂×D₈	—	2	2	no	—
214	C₂×D₈	C₂³×C₅	1	2	yes	—
215	C₂×D₈	C₂³×C₅	1	1	yes	characteristic
216	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
217	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
218	#124	—	2	4	no	—
219	#124	—	4	1	no	—
220	#124	—	4	1	no	—
221	C₄×Q₈	—	2	2	no	—
222	C₄×D₄	—	2	2	no	—
223	C₄×D₄	—	4	2	no	—
224	#117	—	2	1	no	—
225	#117	—	4	1	no	—
226	C₂×C₂²⋊C₄	—	4	2	no	—
227	C₂×C₄²	—	4	1	no	—
228	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
229	Q₃₂	—	2	4	no	—
230	SD₃₂	—	2	4	no	—
231	SD₃₂	—	2	4	no	—
232	D₁₆	—	2	4	no	—
233	M₃₂	—	2	2	no	—
234	M₃₂	—	2	2	no	—
235	C₂×C₁₆	C₂³×C₅	1	2	yes	—
236	C₂×C₁₆	C₂³×C₅	1	2	yes	—
237	#107	C₂³×C₅	1	2	yes	—
238	#106	—	2	1	no	—
239	#103	—	2	2	no	—
240	#102	—	2	2	no	—
241	C₄×C₈	C₅×D₄	1	2	yes	—
242	C₄×C₈	C₂³×C₅	1	2	yes	—
243	C₂²×C₅	—	2	2	no	—
244	C₂²×C₅	—	4	2	no	—
245	C₂²×C₅	—	8	2	no	—
246	C₂²×C₅	—	8	4	no	—
247	C₂²×C₅	—	8	2	no	—
248	C₂²×C₅	—	16	4	no	—
249	C₂²×C₅	#459	1	2	yes	—
250	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
251	C₄×C₅	—	2	2	no	—
252	C₄×C₅	—	2	2	no	—
253	C₄×C₅	—	4	2	no	—
254	C₄×C₅	—	8	2	no	—
255	C₄×C₅	—	8	2	no	—
256	C₄×C₅	#459	1	2	yes	—
257	C₄×C₅	C₂×#121	1	1	yes	characteristic
258	C₄×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
259	C₂⁴	—	4	1	no	—
260	#41	—	2	4	no	—
261	#41	—	4	2	no	—
262	C₂×Q₈	—	2	1	no	—
263	C₂×Q₈	—	2	2	no	—
264	C₂×Q₈	—	4	1	no	—
265	C₂×Q₈	—	4	2	no	—
266	C₂×Q₈	—	4	2	no	—
267	C₂×D₄	—	2	1	no	—
268	C₂×D₄	—	2	2	no	—
269	C₂×D₄	—	4	1	no	—
270	C₂×D₄	—	4	2	no	—
271	C₂×D₄	—	4	1	no	—
272	C₂×D₄	—	4	2	no	—
273	C₂²×C₄	—	4	1	no	—
274	C₂²×C₄	—	4	1	no	—
275	C₂²×C₄	—	4	2	no	—
276	C₂²×C₄	C₅×D₈	1	2	yes	—
277	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
278	Q₁₆	—	2	2	no	—
279	Q₁₆	—	2	1	no	—
280	Q₁₆	—	2	2	no	—
281	Q₁₆	—	4	2	no	—
282	Q₁₆	C₂×C₅×D₄	1	2	yes	—
283	SD₁₆	—	2	4	no	—
284	SD₁₆	—	4	8	no	—
285	D₈	—	2	2	no	—
286	D₈	—	2	1	no	—
287	D₈	—	2	2	no	—
288	D₈	—	4	2	no	—
289	D₈	C₂×C₅×D₄	1	2	yes	—
290	C₂×C₈	—	2	2	no	—
291	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
292	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
293	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
294	C₂×C₈	C₅×#41	1	2	yes	—
295	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
296	C₄⋊C₄	—	2	2	no	—
297	C₄⋊C₄	—	4	2	no	—
298	C₂²⋊C₄	—	4	1	no	—
299	C₂²⋊C₄	—	4	4	no	—
300	C₂²⋊C₄	—	4	4	no	—
301	C₂²⋊C₄	—	4	2	no	—
302	C₄²	—	4	1	no	—
303	C₄²	—	4	2	no	—
304	C₄²	C₅×D₈	1	1	yes	characteristic
305	C₄²	C₅×D₈	1	1	yes	characteristic
306	C₄²	C₂×C₅×D₄	1	2	yes	—
307	C₁₆	—	2	2	no	—
308	C₁₆	—	2	2	no	—
309	C₂×C₅	—	2	2	no	—
310	C₂×C₅	—	8	2	no	—
311	C₂×C₅	—	16	2	no	—
312	C₂×C₅	C₂×#459	1	1	yes	characteristic
313	C₂×C₅	#3414	1	2	yes	—
314	C₂³	—	4	1	no	—
315	C₂³	—	4	2	no	—
316	C₂³	—	8	4	no	—
317	C₂³	—	8	2	no	—
318	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
319	Q₈	—	2	2	no	—
320	Q₈	—	4	1	no	—
321	Q₈	—	4	2	no	—
322	Q₈	—	4	2	no	—
323	Q₈	—	8	2	no	—
324	D₄	—	2	2	no	—
325	D₄	—	4	4	no	—
326	D₄	—	4	2	no	—
327	D₄	—	4	1	no	—
328	D₄	—	4	2	no	—
329	D₄	—	8	2	no	—
330	C₂×C₄	—	4	2	no	—
331	C₂×C₄	—	4	1	no	—
332	C₂×C₄	—	4	4	no	—
333	C₂×C₄	—	4	2	no	—
334	C₂×C₄	—	4	2	no	—
335	C₂×C₄	—	8	4	no	—
336	C₂×C₄	—	8	2	no	—
337	C₂×C₄	C₅×#121	1	2	yes	—
338	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
339	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
340	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
341	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
342	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
343	C₈	—	2	2	no	—
344	C₈	C₅×#121	1	2	yes	—
345	C₈	C₂²×C₅×D₄	1	1	yes	characteristic
346	C₈	C₂×C₅×#41	1	1	yes	characteristic
347	C₅	#33584	1	1	yes	characteristic, direct product factor, Sylow
348	C₂²	—	2	2	no	—
349	C₂²	—	4	2	no	—
350	C₂²	—	8	2	no	—
351	C₂²	—	8	2	no	—
352	C₂²	—	8	4	no	—
353	C₂²	—	16	4	no	—
354	C₂²	C₅×#459	1	2	yes	—
355	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
356	C₄	—	2	2	no	—
357	C₄	—	2	2	no	—
358	C₄	—	4	2	no	—
359	C₄	—	8	2	no	—
360	C₄	—	8	2	no	—
361	C₄	C₅×#459	1	2	yes	—
362	C₄	C₂×C₅×#121	1	1	yes	characteristic
363	C₄	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
364	C₂	—	2	2	no	—
365	C₂	—	8	2	no	—
366	C₂	—	16	2	no	—
367	C₂	C₂×C₅×#459	1	1	yes	characteristic
368	C₂	C₅×#3414	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 ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×#510	C₂	semidirect
3. C₂×C₅×#495	C₂	semidirect
4. C₄×C₅×#135	C₂	semidirect
5. C₅×#2211	C₂	semidirect
6. C₅×#2210	C₂	semidirect
7. C₅×#2150	C₂	semidirect
8. C₅×#2141	C₂	semidirect
9. C₅×#2140	C₂	semidirect
18. C₂×C₅×Q₃₂	C₂²	semidirect
20. C₂×C₅×SD₃₂	C₂²	semidirect
26. C₄×C₅×Q₁₆	C₂²	semidirect
28. C₄×C₅×D₈	C₂²	semidirect
32. C₅×C₄⋊C₁₆	C₂²	semidirect
33. C₅×#358	C₂²	semidirect
35. #33584	C₅	direct (N × H)
81. C₂×#510	C₂×C₅	semidirect
83. C₂×#495	C₂×C₅	semidirect
84. C₄×#135	C₂×C₅	semidirect
85. #2211	C₂×C₅	semidirect
86. #2210	C₂×C₅	semidirect
87. #2150	C₂×C₅	semidirect
88. #2141	C₂×C₅	semidirect
89. #2140	C₂×C₅	semidirect
148. C₂×Q₃₂	C₂²×C₅	semidirect
150. C₂×SD₃₂	C₂²×C₅	semidirect
156. C₄×Q₁₆	C₂²×C₅	semidirect
158. C₄×D₈	C₂²×C₅	semidirect
162. C₄⋊C₁₆	C₂²×C₅	semidirect
163. #358	C₂²×C₅	semidirect
347. C₅	#33584	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×#510	2
C₂×C₅×#509	2
C₂×C₅×#495	2
C₄×C₅×#135	2
C₅×#2211	2
C₅×#2210	2
C₅×#2150	2
C₅×#2141	2
C₅×#2140	2
#33584	5

Subgroups — C5×#33584 #11956304

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33584 #11956304