Subgroups — C₅×#33463 #11956183

Order 1280.

652 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3414	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#443	C₂	1	2	yes	maximal, complemented
3	C₅×#2179	C₂	1	4	yes	maximal, complemented
4	C₅×#2178	C₂	1	4	yes	maximal, complemented
5	C₅×#2161	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#2143	C₂	1	2	yes	maximal, complemented
7	C₄×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅×#135	C₂²	1	2	yes	—
9	C₂×C₅×#131	—	2	2	no	—
10	C₅×#510	C₄	1	4	yes	complemented
11	C₅×#509	C₄	1	4	yes	complemented
12	C₅×#508	C₂²	1	4	yes	—
13	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
14	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
15	C₅×#443	—	2	2	no	—
16	C₅×#443	—	2	2	no	—
17	C₅×#443	C₂²	1	4	yes	complemented
18	C₅×#443	C₂²	1	4	yes	complemented
19	C₂×C₅×#108	C₂²	1	2	yes	—
20	C₂×C₅×#104	—	2	2	no	—
21	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
22	C₅×#368	C₂²	1	4	yes	complemented
23	C₅×#359	C₂²	1	8	yes	complemented
24	C₅×#346	C₂²	1	2	yes	complemented
25	C₄×C₅×C₁₆	C₂²	1	2	yes	complemented
26	#33463	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
27	C₂×C₅×#41	—	2	2	no	—
28	C₅×#135	—	2	2	no	—
29	C₅×#135	—	2	2	no	—
30	C₅×#135	C₂×C₄	1	4	yes	—
31	C₅×#135	C₂³	1	4	yes	—
32	C₂×C₅×Q₁₆	C₂×C₄	1	2	yes	—
33	C₂×C₅×SD₁₆	—	2	2	no	—
34	C₂×C₅×D₈	C₂×C₄	1	2	yes	—
35	C₅×#131	—	2	4	no	—
36	C₅×#131	—	2	4	no	—
37	C₅×#131	—	4	2	no	—
38	C₅×#131	—	4	2	no	—
39	C₂×C₅×M₁₆	—	2	2	no	—
40	C₂×C₅×M₁₆	—	4	2	no	—
41	C₂²×C₅×C₈	—	4	2	no	—
42	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
43	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
45	C₅×Q₃₂	C₂×C₄	1	4	yes	complemented
46	C₅×SD₃₂	C₂×C₄	1	8	yes	complemented
47	C₅×D₁₆	C₂×C₄	1	4	yes	complemented
48	C₅×M₃₂	—	2	2	no	—
49	C₅×M₃₂	C₂×C₄	1	4	yes	—
50	C₂×C₅×C₁₆	C₂³	1	2	yes	—
51	C₂×C₅×C₁₆	C₂³	1	2	yes	—
52	C₅×#108	—	2	2	no	—
53	C₅×#108	C₂³	1	4	yes	—
54	C₅×#104	—	2	4	no	—
55	C₅×#104	—	2	4	no	—
56	C₄×C₅×C₈	D₄	1	1	yes	characteristic
57	C₄×C₅×C₈	D₄	1	1	yes	characteristic
58	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
59	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
60	#3414	C₂×C₅	1	1	yes	characteristic
61	C₂×#443	C₂×C₅	1	2	yes	complemented
62	#2179	C₂×C₅	1	4	yes	complemented
63	#2178	C₂×C₅	1	4	yes	complemented
64	#2161	C₂×C₅	1	1	yes	characteristic, complemented
65	#2143	C₂×C₅	1	2	yes	complemented
66	C₄×M₃₂	C₂×C₅	1	1	yes	characteristic, complemented
67	C₅×#41	—	2	4	no	—
68	C₅×#41	—	2	4	no	—
69	C₅×#41	—	4	2	no	—
70	C₅×#41	—	4	2	no	—
71	C₂×C₅×Q₈	—	2	2	no	—
72	C₂×C₅×D₄	—	2	2	no	—
73	C₂×C₅×D₄	—	4	2	no	—
74	C₂²×C₄×C₅	—	2	1	no	—
75	C₂²×C₄×C₅	—	4	2	no	—
76	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
77	C₅×Q₁₆	—	2	2	no	—
78	C₅×Q₁₆	C₂²×C₄	1	4	yes	—
79	C₅×SD₁₆	—	2	4	no	—
80	C₅×SD₁₆	—	2	4	no	—
81	C₅×D₈	—	2	2	no	—
82	C₅×D₈	C₂²×C₄	1	4	yes	—
83	C₅×M₁₆	—	2	4	no	—
84	C₅×M₁₆	—	4	2	no	—
85	C₅×M₁₆	—	4	4	no	—
86	C₅×M₁₆	—	4	4	no	—
87	C₂×C₅×C₈	—	4	2	no	—
88	C₂×C₅×C₈	—	4	4	no	—
89	C₂×C₅×C₈	—	4	4	no	—
90	C₂×C₅×C₈	D₈	1	2	yes	—
91	C₂×C₅×C₈	D₈	1	2	yes	—
92	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
93	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
94	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
95	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
96	C₂×C₅×C₈	#41	1	2	yes	—
97	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
98	C₄²×C₅	—	2	1	no	—
99	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
101	C₅×C₁₆	—	2	2	no	—
102	C₅×C₁₆	C₂²×C₄	1	4	yes	—
103	C₂×#135	C₂²×C₅	1	2	yes	—
104	C₂×#131	—	2	2	no	—
105	#510	C₄×C₅	1	4	yes	complemented
106	#509	C₄×C₅	1	4	yes	complemented
107	#508	C₂²×C₅	1	4	yes	—
108	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
109	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
110	#443	—	2	2	no	—
111	#443	—	2	2	no	—
112	#443	C₂²×C₅	1	4	yes	complemented
113	#443	C₂²×C₅	1	4	yes	complemented
114	C₂×#108	C₂²×C₅	1	2	yes	—
115	C₂×#104	—	2	2	no	—
116	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
117	#368	C₂²×C₅	1	4	yes	complemented
118	#359	C₂²×C₅	1	8	yes	complemented
119	#346	C₂²×C₅	1	2	yes	complemented
120	C₄×C₁₆	C₂²×C₅	1	2	yes	complemented
121	C₂³×C₅	—	4	2	no	—
122	C₂³×C₅	#135	1	1	yes	characteristic
123	C₅×Q₈	—	2	4	no	—
124	C₅×Q₈	—	4	2	no	—
125	C₅×D₄	—	2	4	no	—
126	C₅×D₄	—	4	4	no	—
127	C₅×D₄	—	4	4	no	—
128	C₅×D₄	—	4	2	no	—
129	C₂×C₄×C₅	—	2	1	no	—
130	C₂×C₄×C₅	—	2	1	no	—
131	C₂×C₄×C₅	—	4	2	no	—
132	C₂×C₄×C₅	—	4	2	no	—
133	C₂×C₄×C₅	—	4	4	no	—
134	C₂×C₄×C₅	—	4	4	no	—
135	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
136	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
137	C₂×C₄×C₅	#135	1	1	yes	characteristic
138	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
139	C₅×C₈	—	4	4	no	—
140	C₅×C₈	C₄×D₄	1	2	yes	—
141	C₅×C₈	C₂×D₈	1	2	yes	—
142	C₅×C₈	C₂×D₈	1	1	yes	characteristic
143	C₅×C₈	C₂×D₈	1	1	yes	characteristic
144	C₅×C₈	C₂³×C₄	1	1	yes	characteristic
145	C₅×C₈	C₂×#41	1	1	yes	characteristic
146	C₂×#41	—	2	2	no	—
147	#135	—	2	2	no	—
148	#135	—	2	2	no	—
149	#135	C₂×C₄×C₅	1	4	yes	—
150	#135	C₂³×C₅	1	4	yes	—
151	C₂×Q₁₆	C₂×C₄×C₅	1	2	yes	—
152	C₂×SD₁₆	—	2	2	no	—
153	C₂×D₈	C₂×C₄×C₅	1	2	yes	—
154	#131	—	2	4	no	—
155	#131	—	2	4	no	—
156	#131	—	4	2	no	—
157	#131	—	4	2	no	—
158	C₂×M₁₆	—	2	2	no	—
159	C₂×M₁₆	—	4	2	no	—
160	C₂²×C₈	—	4	2	no	—
161	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
162	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
163	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
164	Q₃₂	C₂×C₄×C₅	1	4	yes	complemented
165	SD₃₂	C₂×C₄×C₅	1	8	yes	complemented
166	D₁₆	C₂×C₄×C₅	1	4	yes	complemented
167	M₃₂	—	2	2	no	—
168	M₃₂	C₂×C₄×C₅	1	4	yes	—
169	C₂×C₁₆	C₂³×C₅	1	2	yes	—
170	C₂×C₁₆	C₂³×C₅	1	2	yes	—
171	#108	—	2	2	no	—
172	#108	C₂³×C₅	1	4	yes	—
173	#104	—	2	4	no	—
174	#104	—	2	4	no	—
175	C₄×C₈	C₅×D₄	1	1	yes	characteristic
176	C₄×C₈	C₅×D₄	1	1	yes	characteristic
177	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
178	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
179	C₂²×C₅	—	4	1	no	—
180	C₂²×C₅	—	4	4	no	—
181	C₂²×C₅	—	8	4	no	—
182	C₂²×C₅	C₄×D₈	1	2	yes	—
183	C₂²×C₅	C₂×#135	1	1	yes	characteristic
184	C₄×C₅	—	2	2	no	—
185	C₄×C₅	—	4	1	no	—
186	C₄×C₅	—	4	4	no	—
187	C₄×C₅	C₄×D₈	1	2	yes	—
188	C₄×C₅	C₂×C₄×D₄	1	1	yes	characteristic
189	C₄×C₅	C₂²×D₈	1	1	yes	characteristic
190	#41	—	2	4	no	—
191	#41	—	2	4	no	—
192	#41	—	4	2	no	—
193	#41	—	4	2	no	—
194	C₂×Q₈	—	2	2	no	—
195	C₂×D₄	—	2	2	no	—
196	C₂×D₄	—	4	2	no	—
197	C₂²×C₄	—	2	1	no	—
198	C₂²×C₄	—	4	2	no	—
199	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
200	Q₁₆	—	2	2	no	—
201	Q₁₆	C₂²×C₄×C₅	1	4	yes	—
202	SD₁₆	—	2	4	no	—
203	SD₁₆	—	2	4	no	—
204	D₈	—	2	2	no	—
205	D₈	C₂²×C₄×C₅	1	4	yes	—
206	M₁₆	—	2	4	no	—
207	M₁₆	—	4	2	no	—
208	M₁₆	—	4	4	no	—
209	M₁₆	—	4	4	no	—
210	C₂×C₈	—	4	2	no	—
211	C₂×C₈	—	4	4	no	—
212	C₂×C₈	—	4	4	no	—
213	C₂×C₈	C₅×D₈	1	2	yes	—
214	C₂×C₈	C₅×D₈	1	2	yes	—
215	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
216	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
217	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
218	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
219	C₂×C₈	C₅×#41	1	2	yes	—
220	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
221	C₄²	—	2	1	no	—
222	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
223	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
224	C₁₆	—	2	2	no	—
225	C₁₆	C₂²×C₄×C₅	1	4	yes	—
226	C₂×C₅	—	2	1	no	—
227	C₂×C₅	—	2	2	no	—
228	C₂×C₅	—	8	4	no	—
229	C₂×C₅	C₂×C₄×D₈	1	1	yes	characteristic
230	C₂³	—	4	2	no	—
231	C₂³	C₅×#135	1	1	yes	characteristic
232	Q₈	—	2	4	no	—
233	Q₈	—	4	2	no	—
234	D₄	—	2	4	no	—
235	D₄	—	4	4	no	—
236	D₄	—	4	4	no	—
237	D₄	—	4	2	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
245	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
246	C₂×C₄	C₅×#135	1	1	yes	characteristic
247	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
248	C₈	—	4	4	no	—
249	C₈	C₄×C₅×D₄	1	2	yes	—
250	C₈	C₂×C₅×D₈	1	1	yes	characteristic
251	C₈	C₂×C₅×D₈	1	2	yes	—
252	C₈	C₂×C₅×D₈	1	1	yes	characteristic
253	C₈	C₂³×C₄×C₅	1	1	yes	characteristic
254	C₈	C₂×C₅×#41	1	1	yes	characteristic
255	C₅	#33463	1	1	yes	characteristic, direct product factor, Sylow
256	C₂²	—	4	1	no	—
257	C₂²	—	4	4	no	—
258	C₂²	—	8	4	no	—
259	C₂²	C₄×C₅×D₈	1	2	yes	—
260	C₂²	C₂×C₅×#135	1	1	yes	characteristic
261	C₄	—	2	2	no	—
262	C₄	—	4	1	no	—
263	C₄	—	4	4	no	—
264	C₄	C₄×C₅×D₈	1	2	yes	—
265	C₄	C₂×C₄×C₅×D₄	1	1	yes	characteristic
266	C₄	C₂²×C₅×D₈	1	1	yes	characteristic
267	C₂	—	2	1	no	—
268	C₂	—	2	2	no	—
269	C₂	—	8	4	no	—
270	C₂	C₂×C₄×C₅×D₈	1	1	yes	characteristic

	subgroup	G/H
center	C₅×C₈	C₂×D₈
commutator	C₈	C₂³×C₄×C₅
Frattini, cosocle	C₂×C₈	C₂⁴×C₅
socle	C₂×C₅	C₂×C₄×D₈
prime radical	C₅×#3414	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₅×C₈ ◁ C₂²×C₅×C₈ ◁ C₂×C₄×C₅×C₈ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₂×C₅×#443	C₂	semidirect
3. C₅×#2179	C₂	semidirect
4. C₅×#2178	C₂	semidirect
5. C₅×#2161	C₂	semidirect
6. C₅×#2143	C₂	semidirect
7. C₄×C₅×M₃₂	C₂	semidirect
10. C₅×#510	C₄	semidirect
11. C₅×#509	C₄	semidirect
17. C₅×#443	C₂²	semidirect
18. C₅×#443	C₂²	semidirect
22. C₅×#368	C₂²	semidirect
23. C₅×#359	C₂²	semidirect
24. C₅×#346	C₂²	semidirect
25. C₄×C₅×C₁₆	C₂²	semidirect
26. #33463	C₅	direct (N × H)
45. C₅×Q₃₂	C₂×C₄	semidirect
46. C₅×SD₃₂	C₂×C₄	semidirect
47. C₅×D₁₆	C₂×C₄	semidirect
61. C₂×#443	C₂×C₅	semidirect
62. #2179	C₂×C₅	semidirect
63. #2178	C₂×C₅	semidirect
64. #2161	C₂×C₅	semidirect
65. #2143	C₂×C₅	semidirect
66. C₄×M₃₂	C₂×C₅	semidirect
105. #510	C₄×C₅	semidirect
106. #509	C₄×C₅	semidirect
112. #443	C₂²×C₅	semidirect
113. #443	C₂²×C₅	semidirect
117. #368	C₂²×C₅	semidirect
118. #359	C₂²×C₅	semidirect
119. #346	C₂²×C₅	semidirect
120. C₄×C₁₆	C₂²×C₅	semidirect
164. Q₃₂	C₂×C₄×C₅	semidirect
165. SD₃₂	C₂×C₄×C₅	semidirect
166. D₁₆	C₂×C₄×C₅	semidirect
255. C₅	#33463	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×#3414	2
C₂×C₅×#443	2
C₅×#2179	2
C₅×#2178	2
C₅×#2161	2
C₅×#2143	2
C₄×C₅×M₃₂	2
#33463	5

Subgroups — C5×#33463 #11956183

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33463 #11956183