Subgroups — C₅×#33774 #11956494

Order 1280.

686 conjugacy classes of subgroups.

The subgroup lattice has 320 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	C₂×C₅×#493	C₂	1	1	yes	maximal, characteristic
3	C₅×#2248	C₂	1	2	yes	maximal, complemented
4	C₅×#2240	C₂	1	2	yes	maximal, complemented
5	C₅×#2207	C₂	1	4	yes	maximal, complemented
6	C₅×#2187	C₂	1	4	yes	maximal, complemented
7	C₅×#2162	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₅×D₈	—	2	1	no	—
9	C₂²×C₅×D₈	C₂²	1	2	yes	—
10	C₅×#533	—	2	1	no	—
11	C₂×C₅×#127	—	2	1	no	—
12	C₂×C₅×SD₃₂	C₂²	1	8	yes	complemented
13	C₅×#500	C₂²	1	1	yes	characteristic, complemented
14	C₅×#494	C₂²	1	1	yes	characteristic, complemented
15	C₅×#493	—	2	1	no	—
16	C₅×#493	—	2	1	no	—
17	C₅×#493	C₂²	1	2	yes	—
18	C₅×#493	C₂²	1	2	yes	—
19	C₅×#487	—	2	1	no	—
20	C₅×#467	—	2	1	no	—
21	C₅×#467	C₂²	1	2	yes	—
22	C₄×C₅×Q₁₆	—	2	1	no	—
23	C₄×C₅×Q₁₆	C₂²	1	2	yes	complemented
24	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
25	C₅×C₄⋊C₁₆	C₂²	1	2	yes	complemented
26	C₅×#357	C₂²	1	8	yes	complemented
27	C₅×C₂²⋊C₁₆	C₂²	1	2	yes	—
28	C₄×C₅×C₁₆	C₂²	1	2	yes	complemented
29	#33774	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂²×C₅×D₄	—	2	2	no	—
31	C₂²×C₅×D₄	—	4	1	no	—
32	C₂²×C₅×D₄	—	4	1	no	—
33	C₂×C₅×Q₁₆	—	2	1	no	—
34	C₂×C₅×Q₁₆	C₂³	1	2	yes	—
35	C₂×C₅×D₈	—	2	2	no	—
36	C₂×C₅×D₈	—	2	2	no	—
37	C₂×C₅×D₈	—	2	2	no	—
38	C₂×C₅×D₈	—	4	4	no	—
39	C₂×C₅×D₈	—	4	4	no	—
40	C₂×C₅×D₈	C₂³	1	4	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×#127	—	2	2	no	—
46	C₅×#127	—	2	2	no	—
47	C₅×#127	—	4	1	no	—
48	C₅×#127	—	4	1	no	—
49	C₅×#125	—	4	1	no	—
50	C₅×C₂²⋊Q₈	—	2	2	no	—
51	C₅×C₂²⋊Q₈	—	4	1	no	—
52	C₄×C₅×Q₈	—	2	2	no	—
53	C₄×C₅×Q₈	—	4	1	no	—
54	C₅×#117	—	4	1	no	—
55	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
56	C₅×SD₃₂	—	4	8	no	—
57	C₂×C₅×C₁₆	C₂³	1	4	yes	—
58	C₅×#107	—	2	1	no	—
59	C₅×#107	C₂³	1	2	yes	—
60	C₅×#103	—	2	2	no	—
61	C₅×#103	—	2	2	no	—
62	C₄×C₅×C₈	D₄	1	1	yes	characteristic
63	C₄×C₅×C₈	D₄	1	1	yes	characteristic
64	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
65	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
66	#3170	C₂×C₅	1	1	yes	characteristic, complemented
67	C₂×#493	C₂×C₅	1	1	yes	characteristic
68	#2248	C₂×C₅	1	2	yes	complemented
69	#2240	C₂×C₅	1	2	yes	complemented
70	#2207	C₂×C₅	1	4	yes	complemented
71	#2187	C₂×C₅	1	4	yes	complemented
72	#2162	C₂×C₅	1	1	yes	characteristic, complemented
73	C₂⁴×C₅	—	4	2	no	—
74	C₂×C₅×Q₈	—	2	2	no	—
75	C₂×C₅×Q₈	—	4	1	no	—
76	C₂×C₅×D₄	—	2	4	no	—
77	C₂×C₅×D₄	—	4	4	no	—
78	C₂×C₅×D₄	—	4	4	no	—
79	C₂×C₅×D₄	—	4	2	no	—
80	C₂×C₅×D₄	—	4	2	no	—
81	C₂×C₅×D₄	—	4	2	no	—
82	C₂×C₅×D₄	—	4	2	no	—
83	C₂×C₅×D₄	—	4	2	no	—
84	C₂×C₅×D₄	—	8	4	no	—
85	C₂×C₅×D₄	—	8	4	no	—
86	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
87	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
88	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
89	C₅×Q₁₆	—	2	4	no	—
90	C₅×Q₁₆	—	4	2	no	—
91	C₅×D₈	—	2	8	no	—
92	C₅×D₈	—	4	4	no	—
93	C₅×D₈	—	4	4	no	—
94	C₅×D₈	—	4	4	no	—
95	C₂×C₅×C₈	—	2	2	no	—
96	C₂×C₅×C₈	—	2	2	no	—
97	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
98	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
99	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
100	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
101	C₅×C₄⋊C₄	—	2	2	no	—
102	C₅×C₄⋊C₄	—	4	1	no	—
103	C₅×C₄⋊C₄	—	4	1	no	—
104	C₅×C₄⋊C₄	—	4	2	no	—
105	C₅×C₄⋊C₄	—	4	2	no	—
106	C₅×C₄⋊C₄	—	4	1	no	—
107	C₅×C₂²⋊C₄	—	4	2	no	—
108	C₄²×C₅	—	4	2	no	—
109	C₄²×C₅	D₈	1	1	yes	characteristic
110	C₄²×C₅	D₈	1	1	yes	characteristic
111	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
113	C₅×C₁₆	—	2	4	no	—
114	C₂²×D₈	—	2	1	no	—
115	C₂²×D₈	C₂²×C₅	1	2	yes	—
116	#533	—	2	1	no	—
117	C₂×#127	—	2	1	no	—
118	C₂×SD₃₂	C₂²×C₅	1	8	yes	complemented
119	#500	C₂²×C₅	1	1	yes	characteristic, complemented
120	#494	C₂²×C₅	1	1	yes	characteristic, complemented
121	#493	—	2	1	no	—
122	#493	—	2	1	no	—
123	#493	C₂²×C₅	1	2	yes	—
124	#493	C₂²×C₅	1	2	yes	—
125	#487	—	2	1	no	—
126	#467	—	2	1	no	—
127	#467	C₂²×C₅	1	2	yes	—
128	C₄×Q₁₆	—	2	1	no	—
129	C₄×Q₁₆	C₂²×C₅	1	2	yes	complemented
130	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
131	C₄⋊C₁₆	C₂²×C₅	1	2	yes	complemented
132	#357	C₂²×C₅	1	8	yes	complemented
133	C₂²⋊C₁₆	C₂²×C₅	1	2	yes	—
134	C₄×C₁₆	C₂²×C₅	1	2	yes	complemented
135	C₂³×C₅	—	4	4	no	—
136	C₂³×C₅	—	8	4	no	—
137	C₂³×C₅	—	8	4	no	—
138	C₂³×C₅	—	8	4	no	—
139	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
140	C₅×Q₈	—	4	2	no	—
141	C₅×Q₈	—	4	2	no	—
142	C₅×Q₈	—	8	2	no	—
143	C₅×D₄	—	4	4	no	—
144	C₅×D₄	—	4	4	no	—
145	C₅×D₄	—	8	4	no	—
146	C₅×D₄	—	8	4	no	—
147	C₅×D₄	—	8	4	no	—
148	C₅×D₄	—	8	4	no	—
149	C₅×D₄	—	8	4	no	—
150	C₂×C₄×C₅	—	2	1	no	—
151	C₂×C₄×C₅	—	2	1	no	—
152	C₂×C₄×C₅	—	2	2	no	—
153	C₂×C₄×C₅	—	4	2	no	—
154	C₂×C₄×C₅	—	4	2	no	—
155	C₂×C₄×C₅	SD₃₂	1	4	yes	complemented
156	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
157	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
158	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
159	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
160	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
161	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
162	C₅×C₈	—	2	1	no	—
163	C₅×C₈	—	2	1	no	—
164	C₅×C₈	AGL(1,ℤ/8)	1	2	yes	—
165	C₅×C₈	H(5,𝔽₂)	1	2	yes	—
166	C₂²×D₄	—	2	2	no	—
167	C₂²×D₄	—	4	1	no	—
168	C₂²×D₄	—	4	1	no	—
169	C₂×Q₁₆	—	2	1	no	—
170	C₂×Q₁₆	C₂³×C₅	1	2	yes	—
171	C₂×D₈	—	2	2	no	—
172	C₂×D₈	—	2	2	no	—
173	C₂×D₈	—	2	2	no	—
174	C₂×D₈	—	4	4	no	—
175	C₂×D₈	—	4	4	no	—
176	C₂×D₈	C₂³×C₅	1	4	yes	—
177	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
178	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
179	C₄⋊Q₈	—	2	1	no	—
180	C₄⋊Q₈	—	2	1	no	—
181	#127	—	2	2	no	—
182	#127	—	2	2	no	—
183	#127	—	4	1	no	—
184	#127	—	4	1	no	—
185	#125	—	4	1	no	—
186	C₂²⋊Q₈	—	2	2	no	—
187	C₂²⋊Q₈	—	4	1	no	—
188	C₄×Q₈	—	2	2	no	—
189	C₄×Q₈	—	4	1	no	—
190	#117	—	4	1	no	—
191	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
192	SD₃₂	—	4	8	no	—
193	C₂×C₁₆	C₂³×C₅	1	4	yes	—
194	#107	—	2	1	no	—
195	#107	C₂³×C₅	1	2	yes	—
196	#103	—	2	2	no	—
197	#103	—	2	2	no	—
198	C₄×C₈	C₅×D₄	1	1	yes	characteristic
199	C₄×C₈	C₅×D₄	1	1	yes	characteristic
200	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
201	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
202	C₂²×C₅	—	2	1	no	—
203	C₂²×C₅	—	2	1	no	—
204	C₂²×C₅	—	8	4	no	—
205	C₂²×C₅	—	8	4	no	—
206	C₂²×C₅	—	8	4	no	—
207	C₂²×C₅	—	16	8	no	—
208	C₂²×C₅	C₂×SD₃₂	1	2	yes	—
209	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
210	C₄×C₅	—	2	1	no	—
211	C₄×C₅	—	2	1	no	—
212	C₄×C₅	—	8	2	no	—
213	C₄×C₅	—	8	2	no	—
214	C₄×C₅	C₂×SD₃₂	1	2	yes	—
215	C₄×C₅	C₂×SD₃₂	1	2	yes	—
216	C₄×C₅	#509	1	2	yes	—
217	C₄×C₅	#535	1	1	yes	characteristic
218	C₄×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
219	C₂⁴	—	4	2	no	—
220	C₂×Q₈	—	2	2	no	—
221	C₂×Q₈	—	4	1	no	—
222	C₂×D₄	—	2	4	no	—
223	C₂×D₄	—	4	4	no	—
224	C₂×D₄	—	4	4	no	—
225	C₂×D₄	—	4	2	no	—
226	C₂×D₄	—	4	2	no	—
227	C₂×D₄	—	4	2	no	—
228	C₂×D₄	—	4	2	no	—
229	C₂×D₄	—	4	2	no	—
230	C₂×D₄	—	8	4	no	—
231	C₂×D₄	—	8	4	no	—
232	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
233	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
234	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
235	Q₁₆	—	2	4	no	—
236	Q₁₆	—	4	2	no	—
237	D₈	—	2	8	no	—
238	D₈	—	4	4	no	—
239	D₈	—	4	4	no	—
240	D₈	—	4	4	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
244	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
245	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
246	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	4	1	no	—
249	C₄⋊C₄	—	4	1	no	—
250	C₄⋊C₄	—	4	2	no	—
251	C₄⋊C₄	—	4	2	no	—
252	C₄⋊C₄	—	4	1	no	—
253	C₂²⋊C₄	—	4	2	no	—
254	C₄²	—	4	2	no	—
255	C₄²	C₅×D₈	1	1	yes	characteristic
256	C₄²	C₅×D₈	1	1	yes	characteristic
257	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
258	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
259	C₁₆	—	2	4	no	—
260	C₂×C₅	—	2	2	no	—
261	C₂×C₅	—	16	4	no	—
262	C₂×C₅	#3205	1	1	yes	characteristic
263	C₂×C₅	C₂²×SD₃₂	1	1	yes	characteristic
264	C₂×C₅	C₂×#509	1	1	yes	characteristic
265	C₂³	—	4	4	no	—
266	C₂³	—	8	4	no	—
267	C₂³	—	8	4	no	—
268	C₂³	—	8	4	no	—
269	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
270	Q₈	—	4	2	no	—
271	Q₈	—	4	2	no	—
272	Q₈	—	8	2	no	—
273	D₄	—	4	4	no	—
274	D₄	—	4	4	no	—
275	D₄	—	8	4	no	—
276	D₄	—	8	4	no	—
277	D₄	—	8	4	no	—
278	D₄	—	8	4	no	—
279	D₄	—	8	4	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	C₅×SD₃₂	1	4	yes	complemented
286	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
287	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
288	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
289	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
290	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
291	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
292	C₈	—	2	1	no	—
293	C₈	—	2	1	no	—
294	C₈	C₅×AGL(1,ℤ/8)	1	2	yes	—
295	C₈	C₅×H(5,𝔽₂)	1	2	yes	—
296	C₅	#33774	1	1	yes	characteristic, direct product factor, Sylow
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	8	4	no	—
300	C₂²	—	8	4	no	—
301	C₂²	—	8	4	no	—
302	C₂²	—	16	8	no	—
303	C₂²	C₂×C₅×SD₃₂	1	2	yes	—
304	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	8	2	no	—
308	C₄	—	8	2	no	—
309	C₄	C₂×C₅×SD₃₂	1	2	yes	—
310	C₄	C₂×C₅×SD₃₂	1	2	yes	—
311	C₄	C₅×#509	1	2	yes	—
312	C₄	C₅×#535	1	1	yes	characteristic
313	C₄	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
314	C₂	—	2	2	no	—
315	C₂	—	16	4	no	—
316	C₂	C₅×#3205	1	1	yes	characteristic
317	C₂	C₂²×C₅×SD₃₂	1	1	yes	characteristic
318	C₂	C₂×C₅×#509	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅
prime radical	C₂×C₅×#493	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×#2248	C₂	semidirect
4. C₅×#2240	C₂	semidirect
5. C₅×#2207	C₂	semidirect
6. C₅×#2187	C₂	semidirect
7. C₅×#2162	C₂	semidirect
12. C₂×C₅×SD₃₂	C₂²	semidirect
13. C₅×#500	C₂²	semidirect
14. C₅×#494	C₂²	semidirect
23. C₄×C₅×Q₁₆	C₂²	semidirect
25. C₅×C₄⋊C₁₆	C₂²	semidirect
26. C₅×#357	C₂²	semidirect
28. C₄×C₅×C₁₆	C₂²	semidirect
29. #33774	C₅	direct (N × H)
66. #3170	C₂×C₅	semidirect
68. #2248	C₂×C₅	semidirect
69. #2240	C₂×C₅	semidirect
70. #2207	C₂×C₅	semidirect
71. #2187	C₂×C₅	semidirect
72. #2162	C₂×C₅	semidirect
118. C₂×SD₃₂	C₂²×C₅	semidirect
119. #500	C₂²×C₅	semidirect
120. #494	C₂²×C₅	semidirect
129. C₄×Q₁₆	C₂²×C₅	semidirect
131. C₄⋊C₁₆	C₂²×C₅	semidirect
132. #357	C₂²×C₅	semidirect
134. C₄×C₁₆	C₂²×C₅	semidirect
155. C₂×C₄×C₅	SD₃₂	semidirect
285. C₂×C₄	C₅×SD₃₂	semidirect
296. C₅	#33774	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
C₂×C₅×#493	2
C₅×#2248	2
C₅×#2240	2
C₅×#2207	2
C₅×#2187	2
C₅×#2162	2
#33774	5

Subgroups — C5×#33774 #11956494

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33774 #11956494