Subgroups — C₅×#33751 #11956471

Order 1280.

548 conjugacy classes of subgroups.

The subgroup lattice has 332 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×#3169	C₂	1	1	yes	maximal, characteristic
3	C₅×#2252	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#2248	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#2241	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#2240	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#2220	C₂	1	4	yes	maximal, complemented
8	C₅×#2175	C₂	1	4	yes	maximal, complemented
9	C₄×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#533	—	2	1	no	—
11	C₅×#532	—	2	1	no	—
12	C₂×C₅×SD₃₂	C₂²	1	4	yes	—
13	C₂×C₅×M₃₂	C₂²	1	2	yes	—
14	C₅×#500	C₂²	1	1	yes	characteristic
15	C₅×#500	C₂²	1	1	yes	characteristic
16	C₅×#494	C₂²	1	1	yes	characteristic
17	C₅×#493	C₂²	1	1	yes	characteristic
18	C₅×#487	—	2	1	no	—
19	C₅×#486	—	2	1	no	—
20	C₅×#467	—	2	1	no	—
21	C₅×#467	C₂²	1	2	yes	—
22	C₅×#466	—	2	1	no	—
23	C₅×#466	C₂²	1	2	yes	—
24	C₄×C₅×Q₁₆	—	2	1	no	—
25	C₄×C₅×Q₁₆	C₂²	1	2	yes	—
26	C₄×C₅×D₈	—	2	1	no	—
27	C₄×C₅×D₈	C₂²	1	2	yes	—
28	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
29	C₅×#366	C₂²	1	4	yes	—
30	C₅×#358	C₂²	1	4	yes	—
31	C₅×#357	C₂²	1	4	yes	—
32	C₅×#346	C₂²	1	2	yes	—
33	C₄×C₅×C₁₆	C₂²	1	1	yes	characteristic
34	C₄×C₅×C₁₆	C₂²	1	1	yes	characteristic
35	#33751	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
36	C₂×C₅×#41	—	4	1	no	—
37	C₂×C₅×Q₁₆	—	2	1	no	—
38	C₂×C₅×Q₁₆	C₂³	1	2	yes	—
39	C₂×C₅×D₈	—	2	1	no	—
40	C₂×C₅×D₈	C₂³	1	2	yes	—
41	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
42	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
43	C₅×C₄⋊Q₈	—	2	1	no	—
44	C₅×C₄⋊Q₈	—	2	1	no	—
45	C₅×C₄⋊Q₈	—	2	1	no	—
46	C₅×#127	—	2	1	no	—
47	C₅×#125	—	4	1	no	—
48	C₅×#124	—	4	1	no	—
49	C₅×C₂²⋊Q₈	—	2	2	no	—
50	C₅×C₂²⋊Q₈	—	4	1	no	—
51	C₅×#121	—	2	2	no	—
52	C₅×#121	—	4	1	no	—
53	C₄×C₅×Q₈	—	2	2	no	—
54	C₄×C₅×Q₈	—	4	1	no	—
55	C₄×C₅×D₄	—	2	2	no	—
56	C₄×C₅×D₄	—	4	1	no	—
57	C₅×#117	—	4	1	no	—
58	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
59	C₅×SD₃₂	—	2	8	no	—
60	C₅×M₃₂	—	2	4	no	—
61	C₂×C₅×C₁₆	C₂³	1	2	yes	—
62	C₂×C₅×C₁₆	C₂³	1	2	yes	—
63	C₅×#107	—	2	1	no	—
64	C₅×#107	—	2	1	no	—
65	C₅×#107	C₂³	1	2	yes	—
66	C₅×#107	C₂³	1	2	yes	—
67	C₅×#103	—	2	2	no	—
68	C₅×#103	—	2	2	no	—
69	C₅×#102	—	2	2	no	—
70	C₅×#102	—	2	2	no	—
71	C₄×C₅×C₈	D₄	1	1	yes	characteristic
72	C₄×C₅×C₈	D₄	1	1	yes	characteristic
73	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
74	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
75	#3170	C₂×C₅	1	1	yes	characteristic, complemented
76	#3169	C₂×C₅	1	1	yes	characteristic
77	#2252	C₂×C₅	1	1	yes	characteristic, complemented
78	#2248	C₂×C₅	1	1	yes	characteristic, complemented
79	#2241	C₂×C₅	1	1	yes	characteristic, complemented
80	#2240	C₂×C₅	1	1	yes	characteristic, complemented
81	#2220	C₂×C₅	1	4	yes	complemented
82	#2175	C₂×C₅	1	4	yes	complemented
83	C₄×M₃₂	C₂×C₅	1	1	yes	characteristic, complemented
84	C₅×#41	—	8	4	no	—
85	C₂×C₅×Q₈	—	2	2	no	—
86	C₂×C₅×Q₈	—	4	1	no	—
87	C₂×C₅×Q₈	—	4	1	no	—
88	C₂×C₅×D₄	—	2	2	no	—
89	C₂×C₅×D₄	—	4	2	no	—
90	C₂×C₅×D₄	—	4	1	no	—
91	C₂×C₅×D₄	—	4	1	no	—
92	C₂²×C₄×C₅	—	4	2	no	—
93	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
94	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
95	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
96	C₅×Q₁₆	—	2	4	no	—
97	C₅×Q₁₆	—	4	2	no	—
98	C₅×D₈	—	2	4	no	—
99	C₅×D₈	—	4	2	no	—
100	C₂×C₅×C₈	—	2	2	no	—
101	C₂×C₅×C₈	—	2	2	no	—
102	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
103	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
104	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
105	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
106	C₅×C₄⋊C₄	—	2	2	no	—
107	C₅×C₄⋊C₄	—	2	2	no	—
108	C₅×C₄⋊C₄	—	4	1	no	—
109	C₅×C₄⋊C₄	—	4	1	no	—
110	C₅×C₄⋊C₄	—	4	1	no	—
111	C₅×C₄⋊C₄	—	4	2	no	—
112	C₅×C₄⋊C₄	—	4	2	no	—
113	C₅×C₄⋊C₄	—	4	1	no	—
114	C₅×C₂²⋊C₄	—	4	2	no	—
115	C₅×C₂²⋊C₄	—	4	2	no	—
116	C₅×C₂²⋊C₄	—	4	2	no	—
117	C₄²×C₅	—	4	2	no	—
118	C₄²×C₅	D₈	1	1	yes	characteristic
119	C₄²×C₅	D₈	1	1	yes	characteristic
120	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
121	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
122	C₅×C₁₆	—	2	2	no	—
123	C₅×C₁₆	#41	1	4	yes	complemented
124	#533	—	2	1	no	—
125	#532	—	2	1	no	—
126	C₂×SD₃₂	C₂²×C₅	1	4	yes	—
127	C₂×M₃₂	C₂²×C₅	1	2	yes	—
128	#500	C₂²×C₅	1	1	yes	characteristic
129	#500	C₂²×C₅	1	1	yes	characteristic
130	#494	C₂²×C₅	1	1	yes	characteristic
131	#493	C₂²×C₅	1	1	yes	characteristic
132	#487	—	2	1	no	—
133	#486	—	2	1	no	—
134	#467	—	2	1	no	—
135	#467	C₂²×C₅	1	2	yes	—
136	#466	—	2	1	no	—
137	#466	C₂²×C₅	1	2	yes	—
138	C₄×Q₁₆	—	2	1	no	—
139	C₄×Q₁₆	C₂²×C₅	1	2	yes	—
140	C₄×D₈	—	2	1	no	—
141	C₄×D₈	C₂²×C₅	1	2	yes	—
142	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
143	#366	C₂²×C₅	1	4	yes	—
144	#358	C₂²×C₅	1	4	yes	—
145	#357	C₂²×C₅	1	4	yes	—
146	#346	C₂²×C₅	1	2	yes	—
147	C₄×C₁₆	C₂²×C₅	1	1	yes	characteristic
148	C₄×C₁₆	C₂²×C₅	1	1	yes	characteristic
149	C₂³×C₅	—	4	2	no	—
150	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
151	C₅×Q₈	—	4	2	no	—
152	C₅×Q₈	—	4	2	no	—
153	C₅×Q₈	—	8	2	no	—
154	C₅×Q₈	—	8	2	no	—
155	C₅×D₄	—	4	2	no	—
156	C₅×D₄	—	4	2	no	—
157	C₅×D₄	—	8	4	no	—
158	C₅×D₄	—	8	2	no	—
159	C₅×D₄	—	8	2	no	—
160	C₂×C₄×C₅	—	2	1	no	—
161	C₂×C₄×C₅	—	2	1	no	—
162	C₂×C₄×C₅	—	2	2	no	—
163	C₂×C₄×C₅	—	2	2	no	—
164	C₂×C₄×C₅	—	4	2	no	—
165	C₂×C₄×C₅	—	4	2	no	—
166	C₂×C₄×C₅	—	4	2	no	—
167	C₂×C₄×C₅	—	8	4	no	—
168	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
169	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
170	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
171	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
172	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
173	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
174	C₅×C₈	—	2	1	no	—
175	C₅×C₈	—	2	1	no	—
176	C₅×C₈	#135	1	2	yes	—
177	C₅×C₈	C₂×#41	1	2	yes	—
178	C₂×#41	—	4	1	no	—
179	C₂×Q₁₆	—	2	1	no	—
180	C₂×Q₁₆	C₂³×C₅	1	2	yes	—
181	C₂×D₈	—	2	1	no	—
182	C₂×D₈	C₂³×C₅	1	2	yes	—
183	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
184	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
185	C₄⋊Q₈	—	2	1	no	—
186	C₄⋊Q₈	—	2	1	no	—
187	C₄⋊Q₈	—	2	1	no	—
188	#127	—	2	1	no	—
189	#125	—	4	1	no	—
190	#124	—	4	1	no	—
191	C₂²⋊Q₈	—	2	2	no	—
192	C₂²⋊Q₈	—	4	1	no	—
193	#121	—	2	2	no	—
194	#121	—	4	1	no	—
195	C₄×Q₈	—	2	2	no	—
196	C₄×Q₈	—	4	1	no	—
197	C₄×D₄	—	2	2	no	—
198	C₄×D₄	—	4	1	no	—
199	#117	—	4	1	no	—
200	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
201	SD₃₂	—	2	8	no	—
202	M₃₂	—	2	4	no	—
203	C₂×C₁₆	C₂³×C₅	1	2	yes	—
204	C₂×C₁₆	C₂³×C₅	1	2	yes	—
205	#107	—	2	1	no	—
206	#107	—	2	1	no	—
207	#107	C₂³×C₅	1	2	yes	—
208	#107	C₂³×C₅	1	2	yes	—
209	#103	—	2	2	no	—
210	#103	—	2	2	no	—
211	#102	—	2	2	no	—
212	#102	—	2	2	no	—
213	C₄×C₈	C₅×D₄	1	1	yes	characteristic
214	C₄×C₈	C₅×D₄	1	1	yes	characteristic
215	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
216	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
217	C₂²×C₅	—	2	1	no	—
218	C₂²×C₅	—	2	1	no	—
219	C₂²×C₅	—	2	1	no	—
220	C₂²×C₅	—	8	2	no	—
221	C₂²×C₅	—	8	2	no	—
222	C₂²×C₅	—	8	2	no	—
223	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
224	C₄×C₅	—	2	1	no	—
225	C₄×C₅	—	2	1	no	—
226	C₄×C₅	—	2	1	no	—
227	C₄×C₅	—	8	2	no	—
228	C₄×C₅	—	8	2	no	—
229	C₄×C₅	—	8	2	no	—
230	C₄×C₅	#509	1	2	yes	—
231	C₄×C₅	#510	1	2	yes	—
232	C₄×C₅	#532	1	1	yes	characteristic
233	C₄×C₅	C₂×#135	1	1	yes	characteristic
234	#41	—	8	4	no	—
235	C₂×Q₈	—	2	2	no	—
236	C₂×Q₈	—	4	1	no	—
237	C₂×Q₈	—	4	1	no	—
238	C₂×D₄	—	2	2	no	—
239	C₂×D₄	—	4	2	no	—
240	C₂×D₄	—	4	1	no	—
241	C₂×D₄	—	4	1	no	—
242	C₂²×C₄	—	4	2	no	—
243	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
244	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
245	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
246	Q₁₆	—	2	4	no	—
247	Q₁₆	—	4	2	no	—
248	D₈	—	2	4	no	—
249	D₈	—	4	2	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
253	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
254	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
255	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	4	1	no	—
259	C₄⋊C₄	—	4	1	no	—
260	C₄⋊C₄	—	4	1	no	—
261	C₄⋊C₄	—	4	2	no	—
262	C₄⋊C₄	—	4	2	no	—
263	C₄⋊C₄	—	4	1	no	—
264	C₂²⋊C₄	—	4	2	no	—
265	C₂²⋊C₄	—	4	2	no	—
266	C₂²⋊C₄	—	4	2	no	—
267	C₄²	—	4	2	no	—
268	C₄²	C₅×D₈	1	1	yes	characteristic
269	C₄²	C₅×D₈	1	1	yes	characteristic
270	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
271	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
272	C₁₆	—	2	2	no	—
273	C₁₆	C₅×#41	1	4	yes	complemented
274	C₂×C₅	—	4	1	no	—
275	C₂×C₅	—	16	2	no	—
276	C₂×C₅	#3169	1	1	yes	characteristic
277	C₂×C₅	C₂×#509	1	1	yes	characteristic
278	C₂×C₅	C₂×#510	1	1	yes	characteristic
279	C₂³	—	4	2	no	—
280	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
281	Q₈	—	4	2	no	—
282	Q₈	—	4	2	no	—
283	Q₈	—	8	2	no	—
284	Q₈	—	8	2	no	—
285	D₄	—	4	2	no	—
286	D₄	—	4	2	no	—
287	D₄	—	8	4	no	—
288	D₄	—	8	2	no	—
289	D₄	—	8	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	8	4	no	—
298	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
299	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
300	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
301	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
302	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
303	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
304	C₈	—	2	1	no	—
305	C₈	—	2	1	no	—
306	C₈	C₅×#135	1	2	yes	—
307	C₈	C₂×C₅×#41	1	2	yes	—
308	C₅	#33751	1	1	yes	characteristic, direct product factor, Sylow
309	C₂²	—	2	1	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	8	2	no	—
313	C₂²	—	8	2	no	—
314	C₂²	—	8	2	no	—
315	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	2	1	no	—
319	C₄	—	8	2	no	—
320	C₄	—	8	2	no	—
321	C₄	—	8	2	no	—
322	C₄	C₅×#509	1	2	yes	—
323	C₄	C₅×#510	1	2	yes	—
324	C₄	C₅×#532	1	1	yes	characteristic
325	C₄	C₂×C₅×#135	1	1	yes	characteristic
326	C₂	—	4	1	no	—
327	C₂	—	16	2	no	—
328	C₂	C₅×#3169	1	1	yes	characteristic
329	C₂	C₂×C₅×#509	1	1	yes	characteristic
330	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×#3169	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×#2252	C₂	semidirect
4. C₅×#2248	C₂	semidirect
5. C₅×#2241	C₂	semidirect
6. C₅×#2240	C₂	semidirect
7. C₅×#2220	C₂	semidirect
8. C₅×#2175	C₂	semidirect
9. C₄×C₅×M₃₂	C₂	semidirect
35. #33751	C₅	direct (N × H)
75. #3170	C₂×C₅	semidirect
77. #2252	C₂×C₅	semidirect
78. #2248	C₂×C₅	semidirect
79. #2241	C₂×C₅	semidirect
80. #2240	C₂×C₅	semidirect
81. #2220	C₂×C₅	semidirect
82. #2175	C₂×C₅	semidirect
83. C₄×M₃₂	C₂×C₅	semidirect
123. C₅×C₁₆	#41	semidirect
273. C₁₆	C₅×#41	semidirect
308. C₅	#33751	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×#3169	2
C₅×#2252	2
C₅×#2248	2
C₅×#2241	2
C₅×#2240	2
C₅×#2220	2
C₅×#2175	2
C₄×C₅×M₃₂	2
#33751	5

Subgroups — C5×#33751 #11956471

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33751 #11956471