Subgroups — C₅×#33782 #11956502

Order 1280.

588 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3170	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#3147	C₂	1	1	yes	maximal, characteristic
3	C₅×#2251	C₂	1	2	yes	maximal, complemented
4	C₅×#2243	C₂	1	2	yes	maximal, complemented
5	C₅×#2209	C₂	1	2	yes	maximal, complemented
6	C₅×#2207	C₂	1	2	yes	maximal, complemented
7	C₅×#2189	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×#533	—	2	1	no	—
12	C₅×#532	—	2	1	no	—
13	C₂×C₅×Q₃₂	C₂²	1	4	yes	—
14	C₂×C₅×SD₃₂	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×#494	C₂²	1	1	yes	characteristic
20	C₅×#493	C₂²	1	1	yes	characteristic
21	C₅×#492	—	2	1	no	—
22	C₅×#487	—	2	1	no	—
23	C₅×#467	—	2	1	no	—
24	C₅×#467	C₂²	1	2	yes	—
25	C₄×C₅×Q₁₆	—	2	1	no	—
26	C₄×C₅×Q₁₆	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	#33782	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₂×C₅×#41	—	2	2	no	—
35	C₅×#135	—	4	4	no	—
36	C₅×#135	—	4	4	no	—
37	C₂×C₅×Q₁₆	—	2	1	no	—
38	C₂×C₅×Q₁₆	—	2	1	no	—
39	C₂×C₅×Q₁₆	C₂³	1	2	yes	—
40	C₂×C₅×Q₁₆	C₂³	1	2	yes	—
41	C₂×C₅×SD₁₆	—	2	2	no	—
42	C₂×C₅×SD₁₆	—	2	2	no	—
43	C₂×C₅×D₈	—	2	1	no	—
44	C₂×C₅×D₈	C₂³	1	2	yes	—
45	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
46	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
47	C₅×C₄⋊Q₈	—	2	1	no	—
48	C₅×C₄⋊Q₈	—	2	1	no	—
49	C₅×C₄⋊Q₈	—	2	1	no	—
50	C₅×#127	—	2	1	no	—
51	C₅×#125	—	4	1	no	—
52	C₅×#124	—	2	2	no	—
53	C₅×C₂²⋊Q₈	—	2	2	no	—
54	C₅×C₂²⋊Q₈	—	4	1	no	—
55	C₅×#121	—	4	1	no	—
56	C₅×#121	—	4	1	no	—
57	C₄×C₅×Q₈	—	2	2	no	—
58	C₄×C₅×Q₈	—	4	1	no	—
59	C₄×C₅×D₄	—	4	1	no	—
60	C₄×C₅×D₄	—	4	1	no	—
61	C₅×#117	—	4	1	no	—
62	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
63	C₅×Q₃₂	—	4	4	no	—
64	C₅×SD₃₂	—	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×#103	—	2	2	no	—
69	C₅×#103	—	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	#3170	C₂×C₅	1	1	yes	characteristic, complemented
75	#3147	C₂×C₅	1	1	yes	characteristic
76	#2251	C₂×C₅	1	2	yes	complemented
77	#2243	C₂×C₅	1	2	yes	complemented
78	#2209	C₂×C₅	1	2	yes	complemented
79	#2207	C₂×C₅	1	2	yes	complemented
80	#2189	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	C₂×C₅×Q₈	—	2	2	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	—	4	2	no	—
89	C₂×C₅×D₄	—	4	1	no	—
90	C₂×C₅×D₄	—	4	1	no	—
91	C₂²×C₄×C₅	—	4	2	no	—
92	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
93	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
94	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
95	C₅×Q₁₆	—	2	4	no	—
96	C₅×Q₁₆	—	2	2	no	—
97	C₅×Q₁₆	—	2	2	no	—
98	C₅×Q₁₆	—	4	2	no	—
99	C₅×Q₁₆	—	4	2	no	—
100	C₅×SD₁₆	—	4	4	no	—
101	C₅×SD₁₆	—	4	4	no	—
102	C₅×D₈	—	2	4	no	—
103	C₅×D₈	—	4	2	no	—
104	C₂×C₅×C₈	—	2	2	no	—
105	C₂×C₅×C₈	—	2	2	no	—
106	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
107	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
108	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
109	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
110	C₅×C₄⋊C₄	—	2	2	no	—
111	C₅×C₄⋊C₄	—	4	1	no	—
112	C₅×C₄⋊C₄	—	4	1	no	—
113	C₅×C₄⋊C₄	—	4	1	no	—
114	C₅×C₄⋊C₄	—	4	1	no	—
115	C₅×C₄⋊C₄	—	4	2	no	—
116	C₅×C₄⋊C₄	—	4	2	no	—
117	C₅×C₄⋊C₄	—	4	1	no	—
118	C₅×C₂²⋊C₄	—	4	2	no	—
119	C₅×C₂²⋊C₄	—	4	2	no	—
120	C₅×C₂²⋊C₄	—	4	2	no	—
121	C₄²×C₅	—	4	2	no	—
122	C₄²×C₅	D₈	1	1	yes	characteristic
123	C₄²×C₅	D₈	1	1	yes	characteristic
124	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
125	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
126	C₅×C₁₆	—	2	4	no	—
127	C₂×#135	—	2	1	no	—
128	C₂×#135	C₂²×C₅	1	2	yes	—
129	#533	—	2	1	no	—
130	#532	—	2	1	no	—
131	C₂×Q₃₂	C₂²×C₅	1	4	yes	—
132	C₂×SD₃₂	C₂²×C₅	1	4	yes	—
133	#500	C₂²×C₅	1	1	yes	characteristic
134	#495	—	2	1	no	—
135	#495	C₂²×C₅	1	2	yes	—
136	#494	C₂²×C₅	1	1	yes	characteristic
137	#494	C₂²×C₅	1	1	yes	characteristic
138	#493	C₂²×C₅	1	1	yes	characteristic
139	#492	—	2	1	no	—
140	#487	—	2	1	no	—
141	#467	—	2	1	no	—
142	#467	C₂²×C₅	1	2	yes	—
143	C₄×Q₁₆	—	2	1	no	—
144	C₄×Q₁₆	C₂²×C₅	1	2	yes	—
145	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
146	C₄⋊C₁₆	C₂²×C₅	1	2	yes	—
147	#358	C₂²×C₅	1	4	yes	—
148	#357	C₂²×C₅	1	4	yes	—
149	C₂²⋊C₁₆	C₂²×C₅	1	2	yes	—
150	#346	C₂²×C₅	1	2	yes	—
151	C₂³×C₅	—	4	2	no	—
152	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
153	C₅×Q₈	—	4	2	no	—
154	C₅×Q₈	—	4	2	no	—
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	—	8	2	no	—
161	C₅×D₄	—	8	4	no	—
162	C₅×D₄	—	8	2	no	—
163	C₂×C₄×C₅	—	2	1	no	—
164	C₂×C₄×C₅	—	2	1	no	—
165	C₂×C₄×C₅	—	2	2	no	—
166	C₂×C₄×C₅	—	4	1	no	—
167	C₂×C₄×C₅	—	4	2	no	—
168	C₂×C₄×C₅	—	4	2	no	—
169	C₂×C₄×C₅	—	4	2	no	—
170	C₂×C₄×C₅	—	8	4	no	—
171	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
172	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
173	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
174	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
175	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
176	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
177	C₅×C₈	—	2	1	no	—
178	C₅×C₈	—	2	1	no	—
179	C₅×C₈	AGL(1,ℤ/8)	1	2	yes	—
180	C₅×C₈	H(5,𝔽₂)	1	2	yes	—
181	C₂×#41	—	2	2	no	—
182	#135	—	4	4	no	—
183	#135	—	4	4	no	—
184	C₂×Q₁₆	—	2	1	no	—
185	C₂×Q₁₆	—	2	1	no	—
186	C₂×Q₁₆	C₂³×C₅	1	2	yes	—
187	C₂×Q₁₆	C₂³×C₅	1	2	yes	—
188	C₂×SD₁₆	—	2	2	no	—
189	C₂×SD₁₆	—	2	2	no	—
190	C₂×D₈	—	2	1	no	—
191	C₂×D₈	C₂³×C₅	1	2	yes	—
192	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
193	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
194	C₄⋊Q₈	—	2	1	no	—
195	C₄⋊Q₈	—	2	1	no	—
196	C₄⋊Q₈	—	2	1	no	—
197	#127	—	2	1	no	—
198	#125	—	4	1	no	—
199	#124	—	2	2	no	—
200	C₂²⋊Q₈	—	2	2	no	—
201	C₂²⋊Q₈	—	4	1	no	—
202	#121	—	4	1	no	—
203	#121	—	4	1	no	—
204	C₄×Q₈	—	2	2	no	—
205	C₄×Q₈	—	4	1	no	—
206	C₄×D₄	—	4	1	no	—
207	C₄×D₄	—	4	1	no	—
208	#117	—	4	1	no	—
209	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
210	Q₃₂	—	4	4	no	—
211	SD₃₂	—	4	4	no	—
212	C₂×C₁₆	C₂³×C₅	1	4	yes	—
213	#107	—	2	1	no	—
214	#107	C₂³×C₅	1	2	yes	—
215	#103	—	2	2	no	—
216	#103	—	2	2	no	—
217	C₄×C₈	C₅×D₄	1	1	yes	characteristic
218	C₄×C₈	C₅×D₄	1	1	yes	characteristic
219	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
220	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
221	C₂²×C₅	—	2	1	no	—
222	C₂²×C₅	—	2	1	no	—
223	C₂²×C₅	—	2	1	no	—
224	C₂²×C₅	—	8	2	no	—
225	C₂²×C₅	—	8	2	no	—
226	C₂²×C₅	—	8	2	no	—
227	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
228	C₄×C₅	—	2	1	no	—
229	C₄×C₅	—	2	1	no	—
230	C₄×C₅	—	2	1	no	—
231	C₄×C₅	—	2	1	no	—
232	C₄×C₅	—	8	2	no	—
233	C₄×C₅	—	8	2	no	—
234	C₄×C₅	—	8	2	no	—
235	C₄×C₅	#510	1	2	yes	—
236	C₄×C₅	#535	1	1	yes	characteristic
237	C₄×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
238	#41	—	4	4	no	—
239	#41	—	4	4	no	—
240	C₂×Q₈	—	2	2	no	—
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	—	4	1	no	—
243	C₂×D₄	—	2	2	no	—
244	C₂×D₄	—	4	2	no	—
245	C₂×D₄	—	4	1	no	—
246	C₂×D₄	—	4	1	no	—
247	C₂²×C₄	—	4	2	no	—
248	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
249	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
250	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
251	Q₁₆	—	2	2	no	—
252	Q₁₆	—	2	2	no	—
253	Q₁₆	—	2	4	no	—
254	Q₁₆	—	4	2	no	—
255	Q₁₆	—	4	2	no	—
256	SD₁₆	—	4	4	no	—
257	SD₁₆	—	4	4	no	—
258	D₈	—	2	4	no	—
259	D₈	—	4	2	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
263	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
264	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
265	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	4	1	no	—
268	C₄⋊C₄	—	4	1	no	—
269	C₄⋊C₄	—	4	1	no	—
270	C₄⋊C₄	—	4	1	no	—
271	C₄⋊C₄	—	4	2	no	—
272	C₄⋊C₄	—	4	2	no	—
273	C₄⋊C₄	—	4	1	no	—
274	C₂²⋊C₄	—	4	2	no	—
275	C₂²⋊C₄	—	4	2	no	—
276	C₂²⋊C₄	—	4	2	no	—
277	C₄²	—	4	2	no	—
278	C₄²	C₅×D₈	1	1	yes	characteristic
279	C₄²	C₅×D₈	1	1	yes	characteristic
280	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
281	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
282	C₁₆	—	2	4	no	—
283	C₂×C₅	—	4	1	no	—
284	C₂×C₅	—	16	2	no	—
285	C₂×C₅	#3205	1	1	yes	characteristic
286	C₂×C₅	C₂×#510	1	1	yes	characteristic
287	C₂×C₅	#3414	1	1	yes	characteristic
288	C₂³	—	4	2	no	—
289	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
290	Q₈	—	4	2	no	—
291	Q₈	—	4	2	no	—
292	Q₈	—	4	2	no	—
293	Q₈	—	4	2	no	—
294	Q₈	—	8	2	no	—
295	D₄	—	4	2	no	—
296	D₄	—	4	2	no	—
297	D₄	—	8	2	no	—
298	D₄	—	8	4	no	—
299	D₄	—	8	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	8	4	no	—
308	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
309	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
310	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
311	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
312	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
313	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
314	C₈	—	2	1	no	—
315	C₈	—	2	1	no	—
316	C₈	C₅×AGL(1,ℤ/8)	1	2	yes	—
317	C₈	C₅×H(5,𝔽₂)	1	2	yes	—
318	C₅	#33782	1	1	yes	characteristic, direct product factor, Sylow
319	C₂²	—	2	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	8	2	no	—
323	C₂²	—	8	2	no	—
324	C₂²	—	8	2	no	—
325	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	8	2	no	—
331	C₄	—	8	2	no	—
332	C₄	—	8	2	no	—
333	C₄	C₅×#510	1	2	yes	—
334	C₄	C₅×#535	1	1	yes	characteristic
335	C₄	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
336	C₂	—	4	1	no	—
337	C₂	—	16	2	no	—
338	C₂	C₅×#3205	1	1	yes	characteristic
339	C₂	C₂×C₅×#510	1	1	yes	characteristic
340	C₂	C₅×#3414	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅
prime radical	C₅×#3147	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. C₅×#3170	C₂	semidirect
3. C₅×#2251	C₂	semidirect
4. C₅×#2243	C₂	semidirect
5. C₅×#2209	C₂	semidirect
6. C₅×#2207	C₂	semidirect
7. C₅×#2189	C₂	semidirect
8. C₅×#2163	C₂	semidirect
33. #33782	C₅	direct (N × H)
74. #3170	C₂×C₅	semidirect
76. #2251	C₂×C₅	semidirect
77. #2243	C₂×C₅	semidirect
78. #2209	C₂×C₅	semidirect
79. #2207	C₂×C₅	semidirect
80. #2189	C₂×C₅	semidirect
81. #2163	C₂×C₅	semidirect
318. C₅	#33782	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×#3170	2
C₅×#3147	2
C₅×#2251	2
C₅×#2243	2
C₅×#2209	2
C₅×#2207	2
C₅×#2189	2
C₅×#2163	2
#33782	5

Subgroups — C5×#33782 #11956502

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33782 #11956502