Subgroups — C₅×#33780 #11956500

Order 1280.

632 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×#3169	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#3147	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#2250	C₂	1	2	yes	maximal, complemented
4	C₅×#2243	C₂	1	2	yes	maximal, complemented
5	C₅×#2208	C₂	1	2	yes	maximal, complemented
6	C₅×#2206	C₂	1	2	yes	maximal, complemented
7	C₅×#2186	C₂	1	4	yes	maximal, complemented
8	C₅×#2163	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₅×#135	—	2	1	no	—
10	C₂×C₅×#135	C₂²	1	2	yes	—
11	C₅×#532	—	2	1	no	—
12	C₅×#532	—	2	1	no	—
13	C₂×C₅×SD₃₂	C₂²	1	4	yes	—
14	C₂×C₅×D₁₆	C₂²	1	4	yes	—
15	C₅×#500	C₂²	1	1	yes	characteristic
16	C₅×#495	—	2	1	no	—
17	C₅×#495	C₂²	1	2	yes	—
18	C₅×#494	C₂²	1	1	yes	characteristic
19	C₅×#493	C₂²	1	1	yes	characteristic
20	C₅×#493	C₂²	1	1	yes	characteristic
21	C₅×#492	—	2	1	no	—
22	C₅×#486	—	2	1	no	—
23	C₅×#466	—	2	1	no	—
24	C₅×#466	C₂²	1	2	yes	—
25	C₄×C₅×D₈	—	2	1	no	—
26	C₄×C₅×D₈	C₂²	1	2	yes	—
27	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
28	C₅×C₄⋊C₁₆	C₂²	1	2	yes	—
29	C₅×#358	C₂²	1	4	yes	—
30	C₅×#357	C₂²	1	4	yes	—
31	C₅×C₂²⋊C₁₆	C₂²	1	2	yes	—
32	C₅×#346	C₂²	1	2	yes	—
33	#33780	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₂×C₅×#41	—	2	2	no	—
35	C₂×C₅×#41	—	4	1	no	—
36	C₅×#135	—	4	4	no	—
37	C₅×#135	—	4	4	no	—
38	C₂×C₅×Q₁₆	—	2	1	no	—
39	C₂×C₅×Q₁₆	C₂³	1	2	yes	—
40	C₂×C₅×SD₁₆	—	2	2	no	—
41	C₂×C₅×SD₁₆	—	2	2	no	—
42	C₂×C₅×D₈	—	2	1	no	—
43	C₂×C₅×D₈	—	2	1	no	—
44	C₂×C₅×D₈	C₂³	1	2	yes	—
45	C₂×C₅×D₈	C₂³	1	2	yes	—
46	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
47	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
48	C₅×C₄⋊Q₈	—	2	1	no	—
49	C₅×C₄⋊Q₈	—	2	1	no	—
50	C₅×#127	—	2	1	no	—
51	C₅×#127	—	2	1	no	—
52	C₅×#124	—	2	2	no	—
53	C₅×#124	—	4	1	no	—
54	C₅×#121	—	2	2	no	—
55	C₅×#121	—	4	1	no	—
56	C₅×#121	—	4	1	no	—
57	C₅×#121	—	4	1	no	—
58	C₄×C₅×D₄	—	2	2	no	—
59	C₄×C₅×D₄	—	4	1	no	—
60	C₄×C₅×D₄	—	4	1	no	—
61	C₄×C₅×D₄	—	4	1	no	—
62	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
63	C₅×SD₃₂	—	4	4	no	—
64	C₅×D₁₆	—	4	4	no	—
65	C₂×C₅×C₁₆	C₂³	1	4	yes	—
66	C₅×#107	—	2	1	no	—
67	C₅×#107	C₂³	1	2	yes	—
68	C₅×#102	—	2	2	no	—
69	C₅×#102	—	2	2	no	—
70	C₄×C₅×C₈	D₄	1	1	yes	characteristic
71	C₄×C₅×C₈	D₄	1	1	yes	characteristic
72	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
73	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
74	#3169	C₂×C₅	1	1	yes	characteristic, complemented
75	#3147	C₂×C₅	1	1	yes	characteristic, complemented
76	#2250	C₂×C₅	1	2	yes	complemented
77	#2243	C₂×C₅	1	2	yes	complemented
78	#2208	C₂×C₅	1	2	yes	complemented
79	#2206	C₂×C₅	1	2	yes	complemented
80	#2186	C₂×C₅	1	4	yes	complemented
81	#2163	C₂×C₅	1	1	yes	characteristic, complemented
82	C₅×#41	—	4	4	no	—
83	C₅×#41	—	4	4	no	—
84	C₅×#41	—	8	4	no	—
85	C₂×C₅×Q₈	—	2	2	no	—
86	C₂×C₅×Q₈	—	4	1	no	—
87	C₂×C₅×D₄	—	2	2	no	—
88	C₂×C₅×D₄	—	2	2	no	—
89	C₂×C₅×D₄	—	4	2	no	—
90	C₂×C₅×D₄	—	4	2	no	—
91	C₂×C₅×D₄	—	4	1	no	—
92	C₂×C₅×D₄	—	4	1	no	—
93	C₂×C₅×D₄	—	4	1	no	—
94	C₂×C₅×D₄	—	4	1	no	—
95	C₂²×C₄×C₅	—	4	2	no	—
96	C₂²×C₄×C₅	—	4	2	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	—	4	2	no	—
102	C₅×SD₁₆	—	4	4	no	—
103	C₅×SD₁₆	—	4	4	no	—
104	C₅×D₈	—	2	4	no	—
105	C₅×D₈	—	2	2	no	—
106	C₅×D₈	—	2	2	no	—
107	C₅×D₈	—	4	2	no	—
108	C₅×D₈	—	4	2	no	—
109	C₂×C₅×C₈	—	2	2	no	—
110	C₂×C₅×C₈	—	2	2	no	—
111	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
112	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
113	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
114	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
115	C₅×C₄⋊C₄	—	2	2	no	—
116	C₅×C₄⋊C₄	—	4	1	no	—
117	C₅×C₄⋊C₄	—	4	1	no	—
118	C₅×C₄⋊C₄	—	4	1	no	—
119	C₅×C₂²⋊C₄	—	4	2	no	—
120	C₅×C₂²⋊C₄	—	4	2	no	—
121	C₅×C₂²⋊C₄	—	4	2	no	—
122	C₅×C₂²⋊C₄	—	4	2	no	—
123	C₄²×C₅	D₈	1	1	yes	characteristic
124	C₄²×C₅	D₈	1	1	yes	characteristic
125	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
126	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
127	C₅×C₁₆	—	2	4	no	—
128	C₂×#135	—	2	1	no	—
129	C₂×#135	C₂²×C₅	1	2	yes	—
130	#532	—	2	1	no	—
131	#532	—	2	1	no	—
132	C₂×SD₃₂	C₂²×C₅	1	4	yes	—
133	C₂×D₁₆	C₂²×C₅	1	4	yes	—
134	#500	C₂²×C₅	1	1	yes	characteristic
135	#495	—	2	1	no	—
136	#495	C₂²×C₅	1	2	yes	—
137	#494	C₂²×C₅	1	1	yes	characteristic
138	#493	C₂²×C₅	1	1	yes	characteristic
139	#493	C₂²×C₅	1	1	yes	characteristic
140	#492	—	2	1	no	—
141	#486	—	2	1	no	—
142	#466	—	2	1	no	—
143	#466	C₂²×C₅	1	2	yes	—
144	C₄×D₈	—	2	1	no	—
145	C₄×D₈	C₂²×C₅	1	2	yes	—
146	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
147	C₄⋊C₁₆	C₂²×C₅	1	2	yes	—
148	#358	C₂²×C₅	1	4	yes	—
149	#357	C₂²×C₅	1	4	yes	—
150	C₂²⋊C₁₆	C₂²×C₅	1	2	yes	—
151	#346	C₂²×C₅	1	2	yes	—
152	C₂³×C₅	—	4	2	no	—
153	C₂³×C₅	—	4	2	no	—
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	—	4	2	no	—
161	C₅×D₄	—	4	2	no	—
162	C₅×D₄	—	8	2	no	—
163	C₅×D₄	—	8	2	no	—
164	C₅×D₄	—	8	2	no	—
165	C₅×D₄	—	8	4	no	—
166	C₅×D₄	—	8	4	no	—
167	C₅×D₄	—	8	2	no	—
168	C₂×C₄×C₅	—	2	1	no	—
169	C₂×C₄×C₅	—	2	1	no	—
170	C₂×C₄×C₅	—	2	2	no	—
171	C₂×C₄×C₅	—	4	1	no	—
172	C₂×C₄×C₅	—	4	2	no	—
173	C₂×C₄×C₅	—	4	2	no	—
174	C₂×C₄×C₅	—	8	4	no	—
175	C₂×C₄×C₅	—	8	4	no	—
176	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
177	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
178	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
179	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
180	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
181	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
182	C₅×C₈	—	2	1	no	—
183	C₅×C₈	—	2	1	no	—
184	C₅×C₈	AGL(1,ℤ/8)	1	2	yes	—
185	C₅×C₈	H(5,𝔽₂)	1	2	yes	—
186	C₂×#41	—	2	2	no	—
187	C₂×#41	—	4	1	no	—
188	#135	—	4	4	no	—
189	#135	—	4	4	no	—
190	C₂×Q₁₆	—	2	1	no	—
191	C₂×Q₁₆	C₂³×C₅	1	2	yes	—
192	C₂×SD₁₆	—	2	2	no	—
193	C₂×SD₁₆	—	2	2	no	—
194	C₂×D₈	—	2	1	no	—
195	C₂×D₈	—	2	1	no	—
196	C₂×D₈	C₂³×C₅	1	2	yes	—
197	C₂×D₈	C₂³×C₅	1	2	yes	—
198	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
199	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
200	C₄⋊Q₈	—	2	1	no	—
201	C₄⋊Q₈	—	2	1	no	—
202	#127	—	2	1	no	—
203	#127	—	2	1	no	—
204	#124	—	2	2	no	—
205	#124	—	4	1	no	—
206	#121	—	2	2	no	—
207	#121	—	4	1	no	—
208	#121	—	4	1	no	—
209	#121	—	4	1	no	—
210	C₄×D₄	—	2	2	no	—
211	C₄×D₄	—	4	1	no	—
212	C₄×D₄	—	4	1	no	—
213	C₄×D₄	—	4	1	no	—
214	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
215	SD₃₂	—	4	4	no	—
216	D₁₆	—	4	4	no	—
217	C₂×C₁₆	C₂³×C₅	1	4	yes	—
218	#107	—	2	1	no	—
219	#107	C₂³×C₅	1	2	yes	—
220	#102	—	2	2	no	—
221	#102	—	2	2	no	—
222	C₄×C₈	C₅×D₄	1	1	yes	characteristic
223	C₄×C₈	C₅×D₄	1	1	yes	characteristic
224	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
225	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
226	C₂²×C₅	—	2	1	no	—
227	C₂²×C₅	—	2	1	no	—
228	C₂²×C₅	—	2	1	no	—
229	C₂²×C₅	—	8	2	no	—
230	C₂²×C₅	—	8	2	no	—
231	C₂²×C₅	—	8	2	no	—
232	C₂²×C₅	—	8	2	no	—
233	C₂²×C₅	—	8	2	no	—
234	C₂²×C₅	—	8	2	no	—
235	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
236	C₄×C₅	—	2	1	no	—
237	C₄×C₅	—	2	1	no	—
238	C₄×C₅	—	2	1	no	—
239	C₄×C₅	—	2	1	no	—
240	C₄×C₅	—	8	2	no	—
241	C₄×C₅	—	8	2	no	—
242	C₄×C₅	#509	1	2	yes	—
243	C₄×C₅	#535	1	1	yes	characteristic
244	C₄×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
245	#41	—	4	4	no	—
246	#41	—	4	4	no	—
247	#41	—	8	4	no	—
248	C₂×Q₈	—	2	2	no	—
249	C₂×Q₈	—	4	1	no	—
250	C₂×D₄	—	2	2	no	—
251	C₂×D₄	—	2	2	no	—
252	C₂×D₄	—	4	2	no	—
253	C₂×D₄	—	4	2	no	—
254	C₂×D₄	—	4	1	no	—
255	C₂×D₄	—	4	1	no	—
256	C₂×D₄	—	4	1	no	—
257	C₂×D₄	—	4	1	no	—
258	C₂²×C₄	—	4	2	no	—
259	C₂²×C₄	—	4	2	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	—	4	2	no	—
265	SD₁₆	—	4	4	no	—
266	SD₁₆	—	4	4	no	—
267	D₈	—	2	2	no	—
268	D₈	—	2	4	no	—
269	D₈	—	2	2	no	—
270	D₈	—	4	2	no	—
271	D₈	—	4	2	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
275	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
276	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
277	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	—	4	1	no	—
280	C₄⋊C₄	—	4	1	no	—
281	C₄⋊C₄	—	4	1	no	—
282	C₂²⋊C₄	—	4	2	no	—
283	C₂²⋊C₄	—	4	2	no	—
284	C₂²⋊C₄	—	4	2	no	—
285	C₂²⋊C₄	—	4	2	no	—
286	C₄²	C₅×D₈	1	1	yes	characteristic
287	C₄²	C₅×D₈	1	1	yes	characteristic
288	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
289	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
290	C₁₆	—	2	4	no	—
291	C₂×C₅	—	4	1	no	—
292	C₂×C₅	—	16	2	no	—
293	C₂×C₅	—	16	2	no	—
294	C₂×C₅	#3205	1	1	yes	characteristic
295	C₂×C₅	C₂×#509	1	1	yes	characteristic
296	C₂×C₅	#3414	1	1	yes	characteristic
297	C₂³	—	4	2	no	—
298	C₂³	—	4	2	no	—
299	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
300	Q₈	—	4	2	no	—
301	Q₈	—	4	2	no	—
302	Q₈	—	8	2	no	—
303	D₄	—	4	2	no	—
304	D₄	—	4	2	no	—
305	D₄	—	4	2	no	—
306	D₄	—	4	2	no	—
307	D₄	—	8	2	no	—
308	D₄	—	8	2	no	—
309	D₄	—	8	2	no	—
310	D₄	—	8	4	no	—
311	D₄	—	8	4	no	—
312	D₄	—	8	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	4	1	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	8	4	no	—
320	C₂×C₄	—	8	4	no	—
321	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
322	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
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	C₂²×C₅×D₄	1	1	yes	characteristic
327	C₈	—	2	1	no	—
328	C₈	—	2	1	no	—
329	C₈	C₅×AGL(1,ℤ/8)	1	2	yes	—
330	C₈	C₅×H(5,𝔽₂)	1	2	yes	—
331	C₅	#33780	1	1	yes	characteristic, direct product factor, Sylow
332	C₂²	—	2	1	no	—
333	C₂²	—	2	1	no	—
334	C₂²	—	2	1	no	—
335	C₂²	—	8	2	no	—
336	C₂²	—	8	2	no	—
337	C₂²	—	8	2	no	—
338	C₂²	—	8	2	no	—
339	C₂²	—	8	2	no	—
340	C₂²	—	8	2	no	—
341	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
342	C₄	—	2	1	no	—
343	C₄	—	2	1	no	—
344	C₄	—	2	1	no	—
345	C₄	—	2	1	no	—
346	C₄	—	8	2	no	—
347	C₄	—	8	2	no	—
348	C₄	C₅×#509	1	2	yes	—
349	C₄	C₅×#535	1	1	yes	characteristic
350	C₄	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
351	C₂	—	4	1	no	—
352	C₂	—	16	2	no	—
353	C₂	—	16	2	no	—
354	C₂	C₅×#3205	1	1	yes	characteristic
355	C₂	C₂×C₅×#509	1	1	yes	characteristic
356	C₂	C₅×#3414	1	1	yes	characteristic

	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 ◁ 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. C₅×#3169	C₂	semidirect
2. C₅×#3147	C₂	semidirect
3. C₅×#2250	C₂	semidirect
4. C₅×#2243	C₂	semidirect
5. C₅×#2208	C₂	semidirect
6. C₅×#2206	C₂	semidirect
7. C₅×#2186	C₂	semidirect
8. C₅×#2163	C₂	semidirect
33. #33780	C₅	direct (N × H)
74. #3169	C₂×C₅	semidirect
75. #3147	C₂×C₅	semidirect
76. #2250	C₂×C₅	semidirect
77. #2243	C₂×C₅	semidirect
78. #2208	C₂×C₅	semidirect
79. #2206	C₂×C₅	semidirect
80. #2186	C₂×C₅	semidirect
81. #2163	C₂×C₅	semidirect
331. C₅	#33780	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×#3169	2
C₅×#3147	2
C₅×#2250	2
C₅×#2243	2
C₅×#2208	2
C₅×#2206	2
C₅×#2186	2
C₅×#2163	2
#33780	5

Subgroups — C5×#33780 #11956500

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33780 #11956500