Subgroups — C₅×#33816 #11956536

Order 1280.

502 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×#3168	C₂	1	2	yes	maximal, complemented
2	C₅×#2229	C₂	1	2	yes	maximal, complemented
3	C₅×#2228	C₂	1	2	yes	maximal, complemented
4	C₅×#2217	C₂	1	2	yes	maximal, complemented
5	C₅×#2216	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#2215	C₂	1	1	yes	maximal, characteristic
7	C₅×#2211	C₂	1	2	yes	maximal, complemented
8	C₅×#2210	C₂	1	2	yes	maximal, complemented
9	C₂×C₅×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#529	—	2	2	no	—
11	C₂×C₅×Q₃₂	C₂²	1	1	yes	characteristic
12	C₂×C₅×SD₃₂	C₂²	1	2	yes	—
13	C₂×C₅×D₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
15	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
16	C₅×#499	C₂²	1	2	yes	—
17	C₅×#495	C₂²	1	2	yes	—
18	C₅×#488	—	2	2	no	—
19	C₅×#467	C₂²	1	2	yes	—
20	C₅×#466	C₂²	1	2	yes	—
21	C₅×#465	—	2	2	no	—
22	C₄×C₅×Q₁₆	C₂²	1	2	yes	—
23	C₄×C₅×SD₁₆	—	2	2	no	—
24	C₄×C₅×D₈	C₂²	1	2	yes	—
25	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
26	C₅×#367	C₂²	1	2	yes	—
27	C₅×#366	C₂²	1	1	yes	characteristic
28	C₅×#366	C₂²	1	1	yes	characteristic
29	C₅×C₄⋊C₁₆	C₂²	1	2	yes	—
30	C₅×C₄⋊C₁₆	C₂²	1	2	yes	—
31	C₅×#358	C₂²	1	2	yes	—
32	C₅×#358	C₂²	1	2	yes	—
33	C₅×#357	C₂²	1	2	yes	—
34	C₅×#357	C₂²	1	2	yes	—
35	#33816	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
36	C₂×C₅×Q₁₆	C₂³	1	2	yes	—
37	C₂×C₅×SD₁₆	—	2	2	no	—
38	C₂×C₅×D₈	C₂³	1	2	yes	—
39	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
40	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
41	C₅×#126	—	4	2	no	—
42	C₅×#125	—	2	2	no	—
43	C₅×#124	—	2	2	no	—
44	C₅×#123	—	4	2	no	—
45	C₅×C₂²⋊Q₈	—	2	2	no	—
46	C₅×#121	—	2	2	no	—
47	C₄×C₅×Q₈	—	2	2	no	—
48	C₄×C₅×D₄	—	2	2	no	—
49	C₄×C₅×D₄	—	4	2	no	—
50	C₅×#117	—	4	2	no	—
51	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
52	C₅×Q₃₂	—	4	1	no	—
53	C₅×SD₃₂	—	4	2	no	—
54	C₅×D₁₆	—	4	1	no	—
55	C₂×C₅×C₁₆	—	4	1	no	—
56	C₂×C₅×C₁₆	—	4	1	no	—
57	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
58	C₂×C₅×C₁₆	C₂³	1	2	yes	—
59	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
60	C₅×#107	C₂³	1	2	yes	—
61	C₅×#107	C₂³	1	2	yes	—
62	C₅×#106	—	2	2	no	—
63	C₅×#103	—	2	2	no	—
64	C₅×#103	—	2	2	no	—
65	C₅×#102	—	2	2	no	—
66	C₅×#102	—	2	2	no	—
67	C₄×C₅×C₈	D₄	1	2	yes	—
68	C₄×C₅×C₈	C₂³	1	2	yes	—
69	#3168	C₂×C₅	1	2	yes	complemented
70	#2229	C₂×C₅	1	2	yes	complemented
71	#2228	C₂×C₅	1	2	yes	complemented
72	#2217	C₂×C₅	1	2	yes	complemented
73	#2216	C₂×C₅	1	1	yes	characteristic, complemented
74	#2215	C₂×C₅	1	1	yes	characteristic
75	#2211	C₂×C₅	1	2	yes	complemented
76	#2210	C₂×C₅	1	2	yes	complemented
77	C₂×C₄⋊C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
78	C₂×C₅×Q₈	—	2	2	no	—
79	C₂×C₅×D₄	—	2	2	no	—
80	C₂×C₅×D₄	—	4	2	no	—
81	C₂²×C₄×C₅	—	4	2	no	—
82	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
83	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
84	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
85	C₅×Q₁₆	—	2	2	no	—
86	C₅×Q₁₆	—	2	2	no	—
87	C₅×SD₁₆	—	4	4	no	—
88	C₅×D₈	—	2	2	no	—
89	C₅×D₈	—	2	2	no	—
90	C₂×C₅×C₈	—	2	1	no	—
91	C₂×C₅×C₈	—	2	1	no	—
92	C₂×C₅×C₈	—	4	1	no	—
93	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
94	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
95	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
96	C₅×C₄⋊C₄	—	2	2	no	—
97	C₅×C₄⋊C₄	—	2	2	no	—
98	C₅×C₄⋊C₄	—	4	2	no	—
99	C₅×C₄⋊C₄	—	4	2	no	—
100	C₅×C₄⋊C₄	—	4	2	no	—
101	C₅×C₄⋊C₄	—	4	2	no	—
102	C₅×C₂²⋊C₄	—	4	2	no	—
103	C₅×C₂²⋊C₄	—	4	2	no	—
104	C₅×C₂²⋊C₄	—	4	2	no	—
105	C₅×C₂²⋊C₄	—	4	2	no	—
106	C₅×C₂²⋊C₄	—	4	2	no	—
107	C₄²×C₅	—	4	2	no	—
108	C₄²×C₅	D₈	1	2	yes	—
109	C₄²×C₅	C₂×D₄	1	2	yes	—
110	C₅×C₁₆	—	2	1	no	—
111	C₅×C₁₆	—	2	2	no	—
112	C₅×C₁₆	—	2	1	no	—
113	#529	—	2	2	no	—
114	C₂×Q₃₂	C₂²×C₅	1	1	yes	characteristic
115	C₂×SD₃₂	C₂²×C₅	1	2	yes	—
116	C₂×D₁₆	C₂²×C₅	1	1	yes	characteristic
117	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
118	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
119	#499	C₂²×C₅	1	2	yes	—
120	#495	C₂²×C₅	1	2	yes	—
121	#488	—	2	2	no	—
122	#467	C₂²×C₅	1	2	yes	—
123	#466	C₂²×C₅	1	2	yes	—
124	#465	—	2	2	no	—
125	C₄×Q₁₆	C₂²×C₅	1	2	yes	—
126	C₄×SD₁₆	—	2	2	no	—
127	C₄×D₈	C₂²×C₅	1	2	yes	—
128	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
129	#367	C₂²×C₅	1	2	yes	—
130	#366	C₂²×C₅	1	1	yes	characteristic
131	#366	C₂²×C₅	1	1	yes	characteristic
132	C₄⋊C₁₆	C₂²×C₅	1	2	yes	—
133	C₄⋊C₁₆	C₂²×C₅	1	2	yes	—
134	#358	C₂²×C₅	1	2	yes	—
135	#358	C₂²×C₅	1	2	yes	—
136	#357	C₂²×C₅	1	2	yes	—
137	#357	C₂²×C₅	1	2	yes	—
138	C₂³×C₅	—	4	2	no	—
139	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
140	C₅×Q₈	—	4	2	no	—
141	C₅×Q₈	—	4	2	no	—
142	C₅×D₄	—	4	2	no	—
143	C₅×D₄	—	4	2	no	—
144	C₅×D₄	—	8	4	no	—
145	C₂×C₄×C₅	—	2	2	no	—
146	C₂×C₄×C₅	—	2	1	no	—
147	C₂×C₄×C₅	—	2	1	no	—
148	C₂×C₄×C₅	—	4	1	no	—
149	C₂×C₄×C₅	—	4	2	no	—
150	C₂×C₄×C₅	—	4	2	no	—
151	C₂×C₄×C₅	—	4	2	no	—
152	C₂×C₄×C₅	—	8	4	no	—
153	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
154	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
155	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
156	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
157	C₅×C₈	—	2	1	no	—
158	C₅×C₈	—	2	2	no	—
159	C₅×C₈	H(5,𝔽₂)	1	1	yes	characteristic
160	C₅×C₈	2₋¹⁺⁴	1	1	yes	characteristic
161	C₂×Q₁₆	C₂³×C₅	1	2	yes	—
162	C₂×SD₁₆	—	2	2	no	—
163	C₂×D₈	C₂³×C₅	1	2	yes	—
164	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
165	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
166	#126	—	4	2	no	—
167	#125	—	2	2	no	—
168	#124	—	2	2	no	—
169	#123	—	4	2	no	—
170	C₂²⋊Q₈	—	2	2	no	—
171	#121	—	2	2	no	—
172	C₄×Q₈	—	2	2	no	—
173	C₄×D₄	—	2	2	no	—
174	C₄×D₄	—	4	2	no	—
175	#117	—	4	2	no	—
176	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
177	Q₃₂	—	4	1	no	—
178	SD₃₂	—	4	2	no	—
179	D₁₆	—	4	1	no	—
180	C₂×C₁₆	—	4	1	no	—
181	C₂×C₁₆	—	4	1	no	—
182	C₂×C₁₆	C₂³×C₅	1	1	yes	characteristic
183	C₂×C₁₆	C₂³×C₅	1	2	yes	—
184	C₂×C₁₆	C₂³×C₅	1	1	yes	characteristic
185	#107	C₂³×C₅	1	2	yes	—
186	#107	C₂³×C₅	1	2	yes	—
187	#106	—	2	2	no	—
188	#103	—	2	2	no	—
189	#103	—	2	2	no	—
190	#102	—	2	2	no	—
191	#102	—	2	2	no	—
192	C₄×C₈	C₅×D₄	1	2	yes	—
193	C₄×C₈	C₂³×C₅	1	2	yes	—
194	C₂²×C₅	—	2	1	no	—
195	C₂²×C₅	—	2	1	no	—
196	C₂²×C₅	—	8	2	no	—
197	C₂²×C₅	—	8	2	no	—
198	C₂²×C₅	—	8	2	no	—
199	C₂²×C₅	#508	1	2	yes	—
200	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
201	C₄×C₅	—	2	2	no	—
202	C₄×C₅	—	2	2	no	—
203	C₄×C₅	—	2	1	no	—
204	C₄×C₅	—	8	2	no	—
205	C₄×C₅	—	8	2	no	—
206	C₄×C₅	—	8	2	no	—
207	C₄×C₅	#537	1	1	yes	characteristic
208	C₄×C₅	#575	1	1	yes	characteristic
209	C₂×Q₈	—	2	2	no	—
210	C₂×D₄	—	2	2	no	—
211	C₂×D₄	—	4	2	no	—
212	C₂²×C₄	—	4	2	no	—
213	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
214	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
215	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
216	Q₁₆	—	2	2	no	—
217	Q₁₆	—	2	2	no	—
218	SD₁₆	—	4	4	no	—
219	D₈	—	2	2	no	—
220	D₈	—	2	2	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	4	1	no	—
224	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
225	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
226	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	—	2	2	no	—
229	C₄⋊C₄	—	4	2	no	—
230	C₄⋊C₄	—	4	2	no	—
231	C₄⋊C₄	—	4	2	no	—
232	C₄⋊C₄	—	4	2	no	—
233	C₂²⋊C₄	—	4	2	no	—
234	C₂²⋊C₄	—	4	2	no	—
235	C₂²⋊C₄	—	4	2	no	—
236	C₂²⋊C₄	—	4	2	no	—
237	C₂²⋊C₄	—	4	2	no	—
238	C₄²	—	4	2	no	—
239	C₄²	C₅×D₈	1	2	yes	—
240	C₄²	C₂×C₅×D₄	1	2	yes	—
241	C₁₆	—	2	1	no	—
242	C₁₆	—	2	2	no	—
243	C₁₆	—	2	1	no	—
244	C₂×C₅	—	2	2	no	—
245	C₂×C₅	—	16	2	no	—
246	C₂×C₅	#3245	1	1	yes	characteristic
247	C₂×C₅	C₂×#508	1	1	yes	characteristic
248	C₂×C₅	#3414	1	1	yes	characteristic
249	C₂³	—	4	2	no	—
250	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
251	Q₈	—	4	2	no	—
252	Q₈	—	4	2	no	—
253	D₄	—	4	2	no	—
254	D₄	—	4	2	no	—
255	D₄	—	8	4	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	8	4	no	—
264	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
265	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
266	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
267	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
268	C₈	—	2	1	no	—
269	C₈	—	2	2	no	—
270	C₈	C₅×H(5,𝔽₂)	1	1	yes	characteristic
271	C₈	C₅×2₋¹⁺⁴	1	1	yes	characteristic
272	C₅	#33816	1	1	yes	characteristic, direct product factor, Sylow
273	C₂²	—	2	1	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	8	2	no	—
276	C₂²	—	8	2	no	—
277	C₂²	—	8	2	no	—
278	C₂²	C₅×#508	1	2	yes	—
279	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
280	C₄	—	2	1	no	—
281	C₄	—	2	2	no	—
282	C₄	—	2	2	no	—
283	C₄	—	8	2	no	—
284	C₄	—	8	2	no	—
285	C₄	—	8	2	no	—
286	C₄	C₅×#537	1	1	yes	characteristic
287	C₄	C₅×#575	1	1	yes	characteristic
288	C₂	—	2	2	no	—
289	C₂	—	16	2	no	—
290	C₂	C₅×#3245	1	1	yes	characteristic
291	C₂	C₂×C₅×#508	1	1	yes	characteristic
292	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×#2215	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×#3168	C₂	semidirect
2. C₅×#2229	C₂	semidirect
3. C₅×#2228	C₂	semidirect
4. C₅×#2217	C₂	semidirect
5. C₅×#2216	C₂	semidirect
7. C₅×#2211	C₂	semidirect
8. C₅×#2210	C₂	semidirect
9. C₂×C₅×C₄⋊C₁₆	C₂	semidirect
35. #33816	C₅	direct (N × H)
69. #3168	C₂×C₅	semidirect
70. #2229	C₂×C₅	semidirect
71. #2228	C₂×C₅	semidirect
72. #2217	C₂×C₅	semidirect
73. #2216	C₂×C₅	semidirect
75. #2211	C₂×C₅	semidirect
76. #2210	C₂×C₅	semidirect
77. C₂×C₄⋊C₁₆	C₂×C₅	semidirect
272. C₅	#33816	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×#3168	2
C₅×#2229	2
C₅×#2228	2
C₅×#2217	2
C₅×#2216	2
C₅×#2215	2
C₅×#2211	2
C₅×#2210	2
C₂×C₅×C₄⋊C₁₆	2
#33816	5

Subgroups — C5×#33816 #11956536

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33816 #11956536