Subgroups — C₅×#33793 #11956513

Order 1280.

538 conjugacy classes of subgroups.

The subgroup lattice has 294 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
2	C₂×C₅×#500	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#2254	C₂	1	2	yes	maximal, complemented
4	C₅×#2240	C₂	1	2	yes	maximal, complemented
5	C₅×#2233	C₂	1	4	yes	maximal, complemented
6	C₅×#2226	C₂	1	4	yes	maximal, complemented
7	C₅×#2162	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#532	—	2	1	no	—
9	C₂×C₅×C₄⋊Q₈	—	2	1	no	—
10	C₅×#500	—	2	1	no	—
11	C₅×#500	—	2	1	no	—
12	C₅×#500	C₂²	1	2	yes	—
13	C₅×#500	C₂²	1	2	yes	—
14	C₅×#500	C₂²	1	1	yes	characteristic
15	C₅×#493	C₂²	1	1	yes	characteristic
16	C₅×#486	—	2	1	no	—
17	C₅×#466	—	2	1	no	—
18	C₅×#466	C₂²	1	2	yes	—
19	C₄×C₅×D₈	—	2	1	no	—
20	C₄×C₅×D₈	C₂²	1	2	yes	—
21	C₂×C₅×#107	—	2	1	no	—
22	C₂×C₅×#107	C₂²	1	2	yes	—
23	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
24	C₅×#367	C₂²	1	8	yes	—
25	C₅×C₄⋊C₁₆	C₂²	1	2	yes	—
26	C₅×#357	C₂²	1	8	yes	—
27	C₅×C₂²⋊C₁₆	C₂²	1	2	yes	—
28	C₄×C₅×C₁₆	C₂²	1	2	yes	—
29	#33793	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂×C₅×#41	—	4	1	no	—
31	C₂²×C₅×Q₈	—	4	1	no	—
32	C₂×C₅×D₈	—	2	1	no	—
33	C₂×C₅×D₈	C₂³	1	2	yes	—
34	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
35	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
36	C₅×C₄⋊Q₈	—	2	1	no	—
37	C₅×C₄⋊Q₈	—	2	2	no	—
38	C₅×C₄⋊Q₈	—	2	2	no	—
39	C₅×C₄⋊Q₈	—	4	1	no	—
40	C₅×C₄⋊Q₈	—	4	1	no	—
41	C₅×#127	—	2	1	no	—
42	C₅×#124	—	4	1	no	—
43	C₅×#121	—	2	2	no	—
44	C₅×#121	—	4	1	no	—
45	C₄×C₅×D₄	—	2	2	no	—
46	C₄×C₅×D₄	—	4	1	no	—
47	C₂×C₅×C₄⋊C₄	—	2	2	no	—
48	C₂×C₅×C₄⋊C₄	—	4	1	no	—
49	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
50	C₂×C₅×C₁₆	C₂³	1	4	yes	—
51	C₅×#107	—	2	2	no	—
52	C₅×#107	—	2	2	no	—
53	C₅×#107	—	2	2	no	—
54	C₅×#107	—	2	1	no	—
55	C₅×#107	C₂³	1	4	yes	—
56	C₅×#107	C₂³	1	2	yes	—
57	C₅×#102	—	2	2	no	—
58	C₅×#102	—	2	2	no	—
59	C₄×C₅×C₈	D₄	1	1	yes	characteristic
60	C₄×C₅×C₈	D₄	1	1	yes	characteristic
61	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
62	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
63	#3169	C₂×C₅	1	1	yes	characteristic
64	C₂×#500	C₂×C₅	1	1	yes	characteristic, complemented
65	#2254	C₂×C₅	1	2	yes	complemented
66	#2240	C₂×C₅	1	2	yes	complemented
67	#2233	C₂×C₅	1	4	yes	complemented
68	#2226	C₂×C₅	1	4	yes	complemented
69	#2162	C₂×C₅	1	1	yes	characteristic, complemented
70	C₅×#41	—	8	4	no	—
71	C₂×C₅×Q₈	—	4	1	no	—
72	C₂×C₅×Q₈	—	4	2	no	—
73	C₂×C₅×Q₈	—	4	2	no	—
74	C₂×C₅×Q₈	—	8	4	no	—
75	C₂×C₅×D₄	—	2	2	no	—
76	C₂×C₅×D₄	—	4	2	no	—
77	C₂×C₅×D₄	—	4	1	no	—
78	C₂×C₅×D₄	—	4	1	no	—
79	C₂²×C₄×C₅	—	4	2	no	—
80	C₂²×C₄×C₅	—	4	2	no	—
81	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
82	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
83	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
84	C₅×D₈	—	2	4	no	—
85	C₅×D₈	—	4	2	no	—
86	C₂×C₅×C₈	—	2	2	no	—
87	C₂×C₅×C₈	—	2	2	no	—
88	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
89	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
90	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
91	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
92	C₅×C₄⋊C₄	—	2	4	no	—
93	C₅×C₄⋊C₄	—	2	2	no	—
94	C₅×C₄⋊C₄	—	4	2	no	—
95	C₅×C₄⋊C₄	—	4	2	no	—
96	C₅×C₄⋊C₄	—	4	2	no	—
97	C₅×C₄⋊C₄	—	4	1	no	—
98	C₅×C₂²⋊C₄	—	4	2	no	—
99	C₅×C₂²⋊C₄	—	4	2	no	—
100	C₄²×C₅	D₈	1	1	yes	characteristic
101	C₄²×C₅	D₈	1	1	yes	characteristic
102	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
104	C₅×C₁₆	—	2	4	no	—
105	#532	—	2	1	no	—
106	C₂×C₄⋊Q₈	—	2	1	no	—
107	#500	—	2	1	no	—
108	#500	—	2	1	no	—
109	#500	C₂²×C₅	1	2	yes	—
110	#500	C₂²×C₅	1	2	yes	—
111	#500	C₂²×C₅	1	1	yes	characteristic
112	#493	C₂²×C₅	1	1	yes	characteristic
113	#486	—	2	1	no	—
114	#466	—	2	1	no	—
115	#466	C₂²×C₅	1	2	yes	—
116	C₄×D₈	—	2	1	no	—
117	C₄×D₈	C₂²×C₅	1	2	yes	—
118	C₂×#107	—	2	1	no	—
119	C₂×#107	C₂²×C₅	1	2	yes	—
120	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
121	#367	C₂²×C₅	1	8	yes	—
122	C₄⋊C₁₆	C₂²×C₅	1	2	yes	—
123	#357	C₂²×C₅	1	8	yes	—
124	C₂²⋊C₁₆	C₂²×C₅	1	2	yes	—
125	C₄×C₁₆	C₂²×C₅	1	2	yes	—
126	C₂³×C₅	—	4	2	no	—
127	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
128	C₅×Q₈	—	8	4	no	—
129	C₅×Q₈	—	8	2	no	—
130	C₅×Q₈	—	8	4	no	—
131	C₅×D₄	—	4	2	no	—
132	C₅×D₄	—	4	2	no	—
133	C₅×D₄	—	8	4	no	—
134	C₅×D₄	—	8	2	no	—
135	C₅×D₄	—	8	2	no	—
136	C₂×C₄×C₅	—	2	1	no	—
137	C₂×C₄×C₅	—	2	1	no	—
138	C₂×C₄×C₅	—	2	2	no	—
139	C₂×C₄×C₅	—	4	2	no	—
140	C₂×C₄×C₅	—	4	4	no	—
141	C₂×C₄×C₅	—	8	4	no	—
142	C₂×C₄×C₅	—	8	4	no	—
143	C₂×C₄×C₅	SD₃₂	1	4	yes	—
144	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
145	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
146	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
147	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
148	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
149	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
150	C₅×C₈	—	2	1	no	—
151	C₅×C₈	—	2	1	no	—
152	C₅×C₈	#137	1	2	yes	—
153	C₅×C₈	2₋¹⁺⁴	1	2	yes	—
154	C₂×#41	—	4	1	no	—
155	C₂²×Q₈	—	4	1	no	—
156	C₂×D₈	—	2	1	no	—
157	C₂×D₈	C₂³×C₅	1	2	yes	—
158	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
159	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
160	C₄⋊Q₈	—	2	2	no	—
161	C₄⋊Q₈	—	2	2	no	—
162	C₄⋊Q₈	—	2	1	no	—
163	C₄⋊Q₈	—	4	1	no	—
164	C₄⋊Q₈	—	4	1	no	—
165	#127	—	2	1	no	—
166	#124	—	4	1	no	—
167	#121	—	2	2	no	—
168	#121	—	4	1	no	—
169	C₄×D₄	—	2	2	no	—
170	C₄×D₄	—	4	1	no	—
171	C₂×C₄⋊C₄	—	2	2	no	—
172	C₂×C₄⋊C₄	—	4	1	no	—
173	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
174	C₂×C₁₆	C₂³×C₅	1	4	yes	—
175	#107	—	2	2	no	—
176	#107	—	2	2	no	—
177	#107	—	2	2	no	—
178	#107	—	2	1	no	—
179	#107	C₂³×C₅	1	4	yes	—
180	#107	C₂³×C₅	1	2	yes	—
181	#102	—	2	2	no	—
182	#102	—	2	2	no	—
183	C₄×C₈	C₅×D₄	1	1	yes	characteristic
184	C₄×C₈	C₅×D₄	1	1	yes	characteristic
185	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
186	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
187	C₂²×C₅	—	2	1	no	—
188	C₂²×C₅	—	2	1	no	—
189	C₂²×C₅	—	8	2	no	—
190	C₂²×C₅	—	8	2	no	—
191	C₂²×C₅	—	8	2	no	—
192	C₂²×C₅	C₂×SD₃₂	1	2	yes	—
193	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
194	C₄×C₅	—	2	1	no	—
195	C₄×C₅	—	2	1	no	—
196	C₄×C₅	—	8	2	no	—
197	C₄×C₅	—	8	4	no	—
198	C₄×C₅	C₂×SD₃₂	1	2	yes	—
199	C₄×C₅	C₂×SD₃₂	1	2	yes	—
200	C₄×C₅	#509	1	2	yes	—
201	C₄×C₅	#536	1	1	yes	characteristic
202	C₄×C₅	C₂×#137	1	1	yes	characteristic
203	#41	—	8	4	no	—
204	C₂×Q₈	—	4	1	no	—
205	C₂×Q₈	—	4	2	no	—
206	C₂×Q₈	—	4	2	no	—
207	C₂×Q₈	—	8	4	no	—
208	C₂×D₄	—	2	2	no	—
209	C₂×D₄	—	4	2	no	—
210	C₂×D₄	—	4	1	no	—
211	C₂×D₄	—	4	1	no	—
212	C₂²×C₄	—	4	2	no	—
213	C₂²×C₄	—	4	2	no	—
214	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
215	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
217	D₈	—	2	4	no	—
218	D₈	—	4	2	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
222	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
223	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
224	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
225	C₄⋊C₄	—	2	2	no	—
226	C₄⋊C₄	—	2	4	no	—
227	C₄⋊C₄	—	4	2	no	—
228	C₄⋊C₄	—	4	2	no	—
229	C₄⋊C₄	—	4	2	no	—
230	C₄⋊C₄	—	4	1	no	—
231	C₂²⋊C₄	—	4	2	no	—
232	C₂²⋊C₄	—	4	2	no	—
233	C₄²	C₅×D₈	1	1	yes	characteristic
234	C₄²	C₅×D₈	1	1	yes	characteristic
235	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
236	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
237	C₁₆	—	2	4	no	—
238	C₂×C₅	—	2	2	no	—
239	C₂×C₅	—	16	2	no	—
240	C₂×C₅	#3226	1	1	yes	characteristic
241	C₂×C₅	C₂²×SD₃₂	1	1	yes	characteristic
242	C₂×C₅	C₂×#509	1	1	yes	characteristic
243	C₂³	—	4	2	no	—
244	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
245	Q₈	—	8	4	no	—
246	Q₈	—	8	2	no	—
247	Q₈	—	8	4	no	—
248	D₄	—	4	2	no	—
249	D₄	—	4	2	no	—
250	D₄	—	8	4	no	—
251	D₄	—	8	2	no	—
252	D₄	—	8	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	4	no	—
258	C₂×C₄	—	8	4	no	—
259	C₂×C₄	—	8	4	no	—
260	C₂×C₄	C₅×SD₃₂	1	4	yes	—
261	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
262	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
263	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
264	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
265	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
266	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
267	C₈	—	2	1	no	—
268	C₈	—	2	1	no	—
269	C₈	C₅×#137	1	2	yes	—
270	C₈	C₅×2₋¹⁺⁴	1	2	yes	—
271	C₅	#33793	1	1	yes	characteristic, direct product factor, Sylow
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	8	2	no	—
275	C₂²	—	8	2	no	—
276	C₂²	—	8	2	no	—
277	C₂²	C₂×C₅×SD₃₂	1	2	yes	—
278	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
279	C₄	—	2	1	no	—
280	C₄	—	2	1	no	—
281	C₄	—	8	2	no	—
282	C₄	—	8	4	no	—
283	C₄	C₂×C₅×SD₃₂	1	2	yes	—
284	C₄	C₂×C₅×SD₃₂	1	2	yes	—
285	C₄	C₅×#509	1	2	yes	—
286	C₄	C₅×#536	1	1	yes	characteristic
287	C₄	C₂×C₅×#137	1	1	yes	characteristic
288	C₂	—	2	2	no	—
289	C₂	—	16	2	no	—
290	C₂	C₅×#3226	1	1	yes	characteristic
291	C₂	C₂²×C₅×SD₃₂	1	1	yes	characteristic
292	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	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
2. C₂×C₅×#500	C₂	semidirect
3. C₅×#2254	C₂	semidirect
4. C₅×#2240	C₂	semidirect
5. C₅×#2233	C₂	semidirect
6. C₅×#2226	C₂	semidirect
7. C₅×#2162	C₂	semidirect
29. #33793	C₅	direct (N × H)
64. C₂×#500	C₂×C₅	semidirect
65. #2254	C₂×C₅	semidirect
66. #2240	C₂×C₅	semidirect
67. #2233	C₂×C₅	semidirect
68. #2226	C₂×C₅	semidirect
69. #2162	C₂×C₅	semidirect
271. C₅	#33793	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
C₂×C₅×#500	2
C₅×#2254	2
C₅×#2240	2
C₅×#2233	2
C₅×#2226	2
C₅×#2162	2
#33793	5

Subgroups — C5×#33793 #11956513

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33793 #11956513