Subgroups — C₅×#33784 #11956504

Order 1280.

584 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3168	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#494	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#2251	C₂	1	2	yes	maximal, complemented
4	C₅×#2245	C₂	1	2	yes	maximal, complemented
5	C₅×#2209	C₂	1	2	yes	maximal, complemented
6	C₅×#2208	C₂	1	2	yes	maximal, complemented
7	C₅×#2188	C₂	1	2	yes	maximal, complemented
8	C₅×#2185	C₂	1	2	yes	maximal
9	C₅×#2163	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₅×Q₁₆	—	2	1	no	—
11	C₂²×C₅×Q₁₆	C₂²	1	2	yes	—
12	C₂×C₅×C₄⋊Q₈	—	2	1	no	—
13	C₅×#529	—	2	1	no	—
14	C₂×C₅×Q₃₂	C₂²	1	4	yes	—
15	C₂×C₅×SD₃₂	C₂²	1	4	yes	—
16	C₅×#499	C₂²	1	1	yes	characteristic
17	C₅×#495	C₂²	1	1	yes	characteristic
18	C₅×#494	—	2	1	no	—
19	C₅×#494	—	2	1	no	—
20	C₅×#494	C₂²	1	2	yes	—
21	C₅×#494	C₂²	1	2	yes	—
22	C₅×#488	—	2	1	no	—
23	C₅×#467	C₂²	1	1	yes	characteristic
24	C₅×#466	C₂²	1	1	yes	characteristic
25	C₅×#465	—	2	1	no	—
26	C₄×C₅×Q₁₆	C₂²	1	1	yes	characteristic
27	C₄×C₅×SD₁₆	—	2	1	no	—
28	C₄×C₅×D₈	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
30	C₅×C₄⋊C₁₆	C₂²	1	2	yes	—
31	C₅×#358	C₂²	1	4	yes	—
32	C₅×#358	C₂²	1	4	yes	—
33	C₅×C₂²⋊C₁₆	C₂²	1	2	yes	—
34	C₅×#346	C₂²	1	2	yes	—
35	#33784	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
36	C₂²×C₅×Q₈	—	2	2	no	—
37	C₂×C₅×Q₁₆	—	2	2	no	—
38	C₂×C₅×Q₁₆	—	2	2	no	—
39	C₂×C₅×Q₁₆	—	2	2	no	—
40	C₂×C₅×Q₁₆	—	4	4	no	—
41	C₂×C₅×Q₁₆	—	4	4	no	—
42	C₂×C₅×Q₁₆	C₂³	1	4	yes	—
43	C₂×C₅×Q₁₆	C₂³	1	1	yes	characteristic
44	C₂×C₅×SD₁₆	—	2	1	no	—
45	C₂×C₅×D₈	C₂³	1	1	yes	characteristic
46	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
47	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
48	C₅×C₄⋊Q₈	—	2	2	no	—
49	C₅×C₄⋊Q₈	—	2	2	no	—
50	C₅×C₄⋊Q₈	—	4	1	no	—
51	C₅×C₄⋊Q₈	—	4	1	no	—
52	C₅×#126	—	4	1	no	—
53	C₅×#125	—	2	1	no	—
54	C₅×#124	—	2	1	no	—
55	C₅×#123	—	4	1	no	—
56	C₅×C₂²⋊Q₈	—	2	1	no	—
57	C₅×#121	—	2	1	no	—
58	C₄×C₅×Q₈	—	2	1	no	—
59	C₄×C₅×D₄	—	2	1	no	—
60	C₄×C₅×D₄	—	4	1	no	—
61	C₅×#117	—	4	1	no	—
62	C₂×C₅×C₄⋊C₄	—	4	1	no	—
63	C₂×C₅×C₄⋊C₄	—	4	1	no	—
64	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
65	C₅×Q₃₂	—	4	4	no	—
66	C₅×SD₃₂	—	4	4	no	—
67	C₂×C₅×C₁₆	C₂³	1	4	yes	—
68	C₅×#107	C₂³	1	1	yes	characteristic
69	C₅×#107	C₂³	1	1	yes	characteristic
70	C₅×#106	—	2	1	no	—
71	C₅×#103	—	2	1	no	—
72	C₅×#103	—	2	1	no	—
73	C₅×#102	—	2	1	no	—
74	C₅×#102	—	2	1	no	—
75	C₄×C₅×C₈	D₄	1	1	yes	characteristic
76	C₄×C₅×C₈	D₄	1	1	yes	characteristic
77	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
78	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
79	#3168	C₂×C₅	1	1	yes	characteristic
80	C₂×#494	C₂×C₅	1	1	yes	characteristic, complemented
81	#2251	C₂×C₅	1	2	yes	complemented
82	#2245	C₂×C₅	1	2	yes	complemented
83	#2209	C₂×C₅	1	2	yes	complemented
84	#2208	C₂×C₅	1	2	yes	complemented
85	#2188	C₂×C₅	1	2	yes	complemented
86	#2185	C₂×C₅	1	2	yes	—
87	#2163	C₂×C₅	1	1	yes	characteristic, complemented
88	C₂×C₅×Q₈	—	2	1	no	—
89	C₂×C₅×Q₈	—	2	4	no	—
90	C₂×C₅×Q₈	—	4	4	no	—
91	C₂×C₅×Q₈	—	4	4	no	—
92	C₂×C₅×Q₈	—	4	2	no	—
93	C₂×C₅×D₄	—	2	1	no	—
94	C₂×C₅×D₄	—	4	1	no	—
95	C₂²×C₄×C₅	—	4	2	no	—
96	C₂²×C₄×C₅	—	4	1	no	—
97	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
98	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
99	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
100	C₅×Q₁₆	—	2	4	no	—
101	C₅×Q₁₆	—	2	4	no	—
102	C₅×Q₁₆	—	2	2	no	—
103	C₅×Q₁₆	—	4	4	no	—
104	C₅×Q₁₆	—	4	4	no	—
105	C₅×Q₁₆	—	4	4	no	—
106	C₅×SD₁₆	—	4	2	no	—
107	C₅×D₈	—	2	2	no	—
108	C₂×C₅×C₈	—	2	2	no	—
109	C₂×C₅×C₈	—	2	2	no	—
110	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
111	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
112	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
113	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
114	C₅×C₄⋊C₄	—	2	1	no	—
115	C₅×C₄⋊C₄	—	2	1	no	—
116	C₅×C₄⋊C₄	—	4	2	no	—
117	C₅×C₄⋊C₄	—	4	2	no	—
118	C₅×C₄⋊C₄	—	4	2	no	—
119	C₅×C₄⋊C₄	—	4	2	no	—
120	C₅×C₄⋊C₄	—	4	1	no	—
121	C₅×C₄⋊C₄	—	4	1	no	—
122	C₅×C₄⋊C₄	—	4	1	no	—
123	C₅×C₄⋊C₄	—	4	1	no	—
124	C₅×C₂²⋊C₄	—	4	1	no	—
125	C₅×C₂²⋊C₄	—	4	1	no	—
126	C₅×C₂²⋊C₄	—	4	1	no	—
127	C₅×C₂²⋊C₄	—	4	1	no	—
128	C₅×C₂²⋊C₄	—	4	1	no	—
129	C₄²×C₅	—	4	1	no	—
130	C₄²×C₅	D₈	1	1	yes	characteristic
131	C₄²×C₅	D₈	1	1	yes	characteristic
132	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
133	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
134	C₅×C₁₆	—	2	4	no	—
135	C₂²×Q₁₆	—	2	1	no	—
136	C₂²×Q₁₆	C₂²×C₅	1	2	yes	—
137	C₂×C₄⋊Q₈	—	2	1	no	—
138	#529	—	2	1	no	—
139	C₂×Q₃₂	C₂²×C₅	1	4	yes	—
140	C₂×SD₃₂	C₂²×C₅	1	4	yes	—
141	#499	C₂²×C₅	1	1	yes	characteristic
142	#495	C₂²×C₅	1	1	yes	characteristic
143	#494	—	2	1	no	—
144	#494	—	2	1	no	—
145	#494	C₂²×C₅	1	2	yes	—
146	#494	C₂²×C₅	1	2	yes	—
147	#488	—	2	1	no	—
148	#467	C₂²×C₅	1	1	yes	characteristic
149	#466	C₂²×C₅	1	1	yes	characteristic
150	#465	—	2	1	no	—
151	C₄×Q₁₆	C₂²×C₅	1	1	yes	characteristic
152	C₄×SD₁₆	—	2	1	no	—
153	C₄×D₈	C₂²×C₅	1	1	yes	characteristic
154	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
155	C₄⋊C₁₆	C₂²×C₅	1	2	yes	—
156	#358	C₂²×C₅	1	4	yes	—
157	#358	C₂²×C₅	1	4	yes	—
158	C₂²⋊C₁₆	C₂²×C₅	1	2	yes	—
159	#346	C₂²×C₅	1	2	yes	—
160	C₂³×C₅	—	4	1	no	—
161	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
162	C₅×Q₈	—	4	4	no	—
163	C₅×Q₈	—	4	1	no	—
164	C₅×Q₈	—	4	1	no	—
165	C₅×Q₈	—	4	4	no	—
166	C₅×Q₈	—	8	4	no	—
167	C₅×D₄	—	4	1	no	—
168	C₅×D₄	—	4	1	no	—
169	C₅×D₄	—	8	2	no	—
170	C₂×C₄×C₅	—	2	2	no	—
171	C₂×C₄×C₅	—	2	1	no	—
172	C₂×C₄×C₅	—	2	1	no	—
173	C₂×C₄×C₅	—	2	2	no	—
174	C₂×C₄×C₅	—	4	1	no	—
175	C₂×C₄×C₅	—	4	1	no	—
176	C₂×C₄×C₅	—	4	1	no	—
177	C₂×C₄×C₅	—	4	4	no	—
178	C₂×C₄×C₅	—	8	4	no	—
179	C₂×C₄×C₅	—	8	2	no	—
180	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
181	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
182	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
183	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
184	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
185	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
186	C₅×C₈	—	2	1	no	—
187	C₅×C₈	—	2	1	no	—
188	C₅×C₈	AGL(1,ℤ/8)	1	2	yes	—
189	C₅×C₈	H(5,𝔽₂)	1	2	yes	—
190	C₂²×Q₈	—	2	2	no	—
191	C₂×Q₁₆	—	2	2	no	—
192	C₂×Q₁₆	—	2	2	no	—
193	C₂×Q₁₆	—	2	2	no	—
194	C₂×Q₁₆	—	4	4	no	—
195	C₂×Q₁₆	—	4	4	no	—
196	C₂×Q₁₆	C₂³×C₅	1	4	yes	—
197	C₂×Q₁₆	C₂³×C₅	1	1	yes	characteristic
198	C₂×SD₁₆	—	2	1	no	—
199	C₂×D₈	C₂³×C₅	1	1	yes	characteristic
200	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
201	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
202	C₄⋊Q₈	—	2	2	no	—
203	C₄⋊Q₈	—	2	2	no	—
204	C₄⋊Q₈	—	4	1	no	—
205	C₄⋊Q₈	—	4	1	no	—
206	#126	—	4	1	no	—
207	#125	—	2	1	no	—
208	#124	—	2	1	no	—
209	#123	—	4	1	no	—
210	C₂²⋊Q₈	—	2	1	no	—
211	#121	—	2	1	no	—
212	C₄×Q₈	—	2	1	no	—
213	C₄×D₄	—	2	1	no	—
214	C₄×D₄	—	4	1	no	—
215	#117	—	4	1	no	—
216	C₂×C₄⋊C₄	—	4	1	no	—
217	C₂×C₄⋊C₄	—	4	1	no	—
218	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
219	Q₃₂	—	4	4	no	—
220	SD₃₂	—	4	4	no	—
221	C₂×C₁₆	C₂³×C₅	1	4	yes	—
222	#107	C₂³×C₅	1	1	yes	characteristic
223	#107	C₂³×C₅	1	1	yes	characteristic
224	#106	—	2	1	no	—
225	#103	—	2	1	no	—
226	#103	—	2	1	no	—
227	#102	—	2	1	no	—
228	#102	—	2	1	no	—
229	C₄×C₈	C₅×D₄	1	1	yes	characteristic
230	C₄×C₈	C₅×D₄	1	1	yes	characteristic
231	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
232	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
233	C₂²×C₅	—	2	1	no	—
234	C₂²×C₅	—	2	1	no	—
235	C₂²×C₅	—	8	1	no	—
236	C₂²×C₅	—	8	1	no	—
237	C₂²×C₅	—	8	1	no	—
238	C₂²×C₅	#510	1	2	yes	—
239	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
240	C₄×C₅	—	2	1	no	—
241	C₄×C₅	—	2	1	no	—
242	C₄×C₅	—	2	1	no	—
243	C₄×C₅	—	2	1	no	—
244	C₄×C₅	—	8	1	no	—
245	C₄×C₅	—	8	1	no	—
246	C₄×C₅	—	8	4	no	—
247	C₄×C₅	—	8	1	no	—
248	C₄×C₅	#510	1	2	yes	—
249	C₄×C₅	#535	1	1	yes	characteristic
250	C₄×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
251	C₂×Q₈	—	2	1	no	—
252	C₂×Q₈	—	2	4	no	—
253	C₂×Q₈	—	4	4	no	—
254	C₂×Q₈	—	4	4	no	—
255	C₂×Q₈	—	4	2	no	—
256	C₂×D₄	—	2	1	no	—
257	C₂×D₄	—	4	1	no	—
258	C₂²×C₄	—	4	2	no	—
259	C₂²×C₄	—	4	1	no	—
260	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
261	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
262	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
263	Q₁₆	—	2	4	no	—
264	Q₁₆	—	2	2	no	—
265	Q₁₆	—	2	4	no	—
266	Q₁₆	—	4	4	no	—
267	Q₁₆	—	4	4	no	—
268	Q₁₆	—	4	4	no	—
269	SD₁₆	—	4	2	no	—
270	D₈	—	2	2	no	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
274	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
275	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
276	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
277	C₄⋊C₄	—	2	1	no	—
278	C₄⋊C₄	—	2	1	no	—
279	C₄⋊C₄	—	4	2	no	—
280	C₄⋊C₄	—	4	2	no	—
281	C₄⋊C₄	—	4	2	no	—
282	C₄⋊C₄	—	4	2	no	—
283	C₄⋊C₄	—	4	1	no	—
284	C₄⋊C₄	—	4	1	no	—
285	C₄⋊C₄	—	4	1	no	—
286	C₄⋊C₄	—	4	1	no	—
287	C₂²⋊C₄	—	4	1	no	—
288	C₂²⋊C₄	—	4	1	no	—
289	C₂²⋊C₄	—	4	1	no	—
290	C₂²⋊C₄	—	4	1	no	—
291	C₂²⋊C₄	—	4	1	no	—
292	C₄²	—	4	1	no	—
293	C₄²	C₅×D₈	1	1	yes	characteristic
294	C₄²	C₅×D₈	1	1	yes	characteristic
295	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
296	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
297	C₁₆	—	2	4	no	—
298	C₂×C₅	—	2	2	no	—
299	C₂×C₅	—	16	1	no	—
300	C₂×C₅	#3205	1	1	yes	characteristic
301	C₂×C₅	C₂×#510	1	1	yes	characteristic
302	C₂×C₅	C₂×#510	1	1	yes	characteristic
303	C₂³	—	4	1	no	—
304	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
305	Q₈	—	4	4	no	—
306	Q₈	—	4	1	no	—
307	Q₈	—	4	1	no	—
308	Q₈	—	4	4	no	—
309	Q₈	—	8	4	no	—
310	D₄	—	4	1	no	—
311	D₄	—	4	1	no	—
312	D₄	—	8	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	4	1	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	4	1	no	—
320	C₂×C₄	—	4	4	no	—
321	C₂×C₄	—	8	4	no	—
322	C₂×C₄	—	8	2	no	—
323	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
324	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
325	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
326	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
327	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
328	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
329	C₈	—	2	1	no	—
330	C₈	—	2	1	no	—
331	C₈	C₅×AGL(1,ℤ/8)	1	2	yes	—
332	C₈	C₅×H(5,𝔽₂)	1	2	yes	—
333	C₅	#33784	1	1	yes	characteristic, direct product factor, Sylow
334	C₂²	—	2	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	8	1	no	—
337	C₂²	—	8	1	no	—
338	C₂²	—	8	1	no	—
339	C₂²	C₅×#510	1	2	yes	—
340	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
341	C₄	—	2	1	no	—
342	C₄	—	2	1	no	—
343	C₄	—	2	1	no	—
344	C₄	—	2	1	no	—
345	C₄	—	8	1	no	—
346	C₄	—	8	1	no	—
347	C₄	—	8	4	no	—
348	C₄	—	8	1	no	—
349	C₄	C₅×#510	1	2	yes	—
350	C₄	C₅×#535	1	1	yes	characteristic
351	C₄	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
352	C₂	—	2	2	no	—
353	C₂	—	16	1	no	—
354	C₂	C₅×#3205	1	1	yes	characteristic
355	C₂	C₂×C₅×#510	1	1	yes	characteristic
356	C₂	C₂×C₅×#510	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅
prime radical	C₅×#466	C₂²
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
2. C₂×C₅×#494	C₂	semidirect
3. C₅×#2251	C₂	semidirect
4. C₅×#2245	C₂	semidirect
5. C₅×#2209	C₂	semidirect
6. C₅×#2208	C₂	semidirect
7. C₅×#2188	C₂	semidirect
9. C₅×#2163	C₂	semidirect
35. #33784	C₅	direct (N × H)
80. C₂×#494	C₂×C₅	semidirect
81. #2251	C₂×C₅	semidirect
82. #2245	C₂×C₅	semidirect
83. #2209	C₂×C₅	semidirect
84. #2208	C₂×C₅	semidirect
85. #2188	C₂×C₅	semidirect
87. #2163	C₂×C₅	semidirect
333. C₅	#33784	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×#3168	2
C₂×C₅×#494	2
C₅×#2251	2
C₅×#2245	2
C₅×#2209	2
C₅×#2208	2
C₅×#2188	2
C₅×#2185	2
C₅×#2163	2
#33784	5

Subgroups — C5×#33784 #11956504

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33784 #11956504