Subgroups — C₅×#33465 #11956185

Order 1280.

642 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#508	C₂	1	1	yes	maximal, characteristic
2	C₅×#2957	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	C₂×C₅×#368	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#2143	C₂	1	2	yes	maximal, complemented
7	C₅×#2108	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅×#135	C₂²	1	2	yes	—
9	C₅×#568	—	2	2	no	—
10	C₅×#508	C₄	1	2	yes	complemented
11	C₅×#508	C₄	1	2	yes	complemented
12	C₅×#508	C₂²	1	2	yes	—
13	C₅×#508	C₂²	1	2	yes	—
14	C₂×C₅×Q₃₂	C₄	1	1	yes	characteristic, complemented
15	C₂×C₅×SD₃₂	C₄	1	2	yes	complemented
16	C₂×C₅×D₁₆	C₄	1	1	yes	characteristic, complemented
17	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
18	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
19	C₅×#444	—	2	4	no	—
20	C₅×#443	C₂²	1	8	yes	complemented
21	C₂×C₅×#108	C₂²	1	2	yes	—
22	C₅×#421	—	2	2	no	—
23	C₅×#405	C₂²	1	1	yes	characteristic
24	C₅×#368	C₂²	1	2	yes	complemented
25	C₅×#368	C₂²	1	2	yes	complemented
26	C₅×#359	C₂²	1	8	yes	complemented
27	C₅×#346	C₂²	1	2	yes	complemented
28	C₄×C₅×C₁₆	C₂²	1	2	yes	complemented
29	#33465	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂×C₅×#41	—	2	2	no	—
31	C₅×#135	—	2	2	no	—
32	C₅×#135	—	2	2	no	—
33	C₅×#135	C₂×C₄	1	4	yes	—
34	C₅×#135	C₂³	1	4	yes	—
35	C₂×C₅×Q₁₆	C₂×C₄	1	2	yes	—
36	C₂×C₅×SD₁₆	—	2	2	no	—
37	C₂×C₅×D₈	C₂×C₄	1	2	yes	—
38	C₅×#131	—	2	8	no	—
39	C₅×#131	—	4	4	no	—
40	C₂×C₅×M₁₆	—	2	2	no	—
41	C₂×C₅×M₁₆	—	4	4	no	—
42	C₂×C₅×M₁₆	D₄	1	1	yes	characteristic
43	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
44	C₅×#117	D₄	1	1	yes	characteristic
45	C₅×Q₃₂	C₂×C₄	1	2	yes	—
46	C₅×Q₃₂	C₂×C₄	1	2	yes	—
47	C₅×SD₃₂	C₂×C₄	1	4	yes	—
48	C₅×SD₃₂	C₂×C₄	1	4	yes	—
49	C₅×D₁₆	C₂×C₄	1	2	yes	—
50	C₅×D₁₆	C₂×C₄	1	2	yes	—
51	C₅×M₃₂	—	2	1	no	—
52	C₅×M₃₂	—	2	1	no	—
53	C₂×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
54	C₂×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
55	C₂×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
56	C₂×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
57	C₂×C₅×C₁₆	C₂³	1	2	yes	—
58	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
59	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
60	C₅×#108	—	2	2	no	—
61	C₅×#108	C₂³	1	4	yes	—
62	C₅×#104	—	2	8	no	—
63	C₅×#97	D₄	1	2	yes	—
64	C₄×C₅×C₈	C₂³	1	2	yes	—
65	C₂×#508	C₂×C₅	1	1	yes	characteristic
66	#2957	C₂×C₅	1	2	yes	complemented
67	#2179	C₂×C₅	1	4	yes	complemented
68	#2178	C₂×C₅	1	4	yes	complemented
69	C₂×#368	C₂×C₅	1	1	yes	characteristic, complemented
70	#2143	C₂×C₅	1	2	yes	complemented
71	#2108	C₂×C₅	1	1	yes	characteristic, complemented
72	C₅×#41	—	2	4	no	—
73	C₅×#41	—	2	4	no	—
74	C₅×#41	—	4	2	no	—
75	C₅×#41	—	4	2	no	—
76	C₂×C₅×Q₈	—	2	2	no	—
77	C₂×C₅×D₄	—	2	2	no	—
78	C₂×C₅×D₄	—	4	2	no	—
79	C₂²×C₄×C₅	—	4	2	no	—
80	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
81	C₅×Q₁₆	—	2	2	no	—
82	C₅×Q₁₆	C₂²×C₄	1	4	yes	—
83	C₅×SD₁₆	—	2	4	no	—
84	C₅×SD₁₆	—	2	4	no	—
85	C₅×D₈	—	2	2	no	—
86	C₅×D₈	C₂²×C₄	1	4	yes	—
87	C₅×M₁₆	—	2	4	no	—
88	C₅×M₁₆	—	4	2	no	—
89	C₅×M₁₆	—	4	8	no	—
90	C₅×M₁₆	D₈	1	2	yes	—
91	C₅×M₁₆	D₈	1	2	yes	—
92	C₂×C₅×C₈	—	4	2	no	—
93	C₂×C₅×C₈	—	4	8	no	—
94	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
97	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
98	C₂×C₅×C₈	#41	1	1	yes	characteristic
99	C₂×C₅×C₈	#41	1	1	yes	characteristic
100	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
101	C₅×C₄⋊C₄	—	2	1	no	—
102	C₅×C₂²⋊C₄	—	2	1	no	—
103	C₄²×C₅	C₂×D₄	1	2	yes	—
104	C₅×C₁₆	—	2	2	no	—
105	C₅×C₁₆	C₂²×C₄	1	1	yes	characteristic
106	C₅×C₁₆	C₂²×C₄	1	1	yes	characteristic
107	C₅×C₁₆	C₂²×C₄	1	1	yes	characteristic
108	C₅×C₁₆	C₂²×C₄	1	1	yes	characteristic
109	C₂×#135	C₂²×C₅	1	2	yes	—
110	#568	—	2	2	no	—
111	#508	C₄×C₅	1	2	yes	complemented
112	#508	C₄×C₅	1	2	yes	complemented
113	#508	C₂²×C₅	1	2	yes	—
114	#508	C₂²×C₅	1	2	yes	—
115	C₂×Q₃₂	C₄×C₅	1	1	yes	characteristic, complemented
116	C₂×SD₃₂	C₄×C₅	1	2	yes	complemented
117	C₂×D₁₆	C₄×C₅	1	1	yes	characteristic, complemented
118	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
119	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
120	#444	—	2	4	no	—
121	#443	C₂²×C₅	1	8	yes	complemented
122	C₂×#108	C₂²×C₅	1	2	yes	—
123	#421	—	2	2	no	—
124	#405	C₂²×C₅	1	1	yes	characteristic
125	#368	C₂²×C₅	1	2	yes	complemented
126	#368	C₂²×C₅	1	2	yes	complemented
127	#359	C₂²×C₅	1	8	yes	complemented
128	#346	C₂²×C₅	1	2	yes	complemented
129	C₄×C₁₆	C₂²×C₅	1	2	yes	complemented
130	C₂³×C₅	—	4	2	no	—
131	C₂³×C₅	#135	1	1	yes	characteristic
132	C₅×Q₈	—	2	4	no	—
133	C₅×Q₈	—	4	2	no	—
134	C₅×D₄	—	2	4	no	—
135	C₅×D₄	—	4	4	no	—
136	C₅×D₄	—	4	4	no	—
137	C₅×D₄	—	4	2	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	—	4	4	no	—
142	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
143	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
144	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
145	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
146	C₂×C₄×C₅	#135	1	1	yes	characteristic
147	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
148	C₅×C₈	—	4	4	no	—
149	C₅×C₈	C₄×D₄	1	1	yes	characteristic
150	C₅×C₈	C₄×D₄	1	1	yes	characteristic
151	C₅×C₈	C₂×D₈	1	4	yes	—
152	C₅×C₈	C₂³×C₄	1	1	yes	characteristic
153	C₅×C₈	C₂×#41	1	1	yes	characteristic
154	C₂×#41	—	2	2	no	—
155	#135	—	2	2	no	—
156	#135	—	2	2	no	—
157	#135	C₂×C₄×C₅	1	4	yes	—
158	#135	C₂³×C₅	1	4	yes	—
159	C₂×Q₁₆	C₂×C₄×C₅	1	2	yes	—
160	C₂×SD₁₆	—	2	2	no	—
161	C₂×D₈	C₂×C₄×C₅	1	2	yes	—
162	#131	—	2	8	no	—
163	#131	—	4	4	no	—
164	C₂×M₁₆	—	2	2	no	—
165	C₂×M₁₆	—	4	4	no	—
166	C₂×M₁₆	C₅×D₄	1	1	yes	characteristic
167	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
168	#117	C₅×D₄	1	1	yes	characteristic
169	Q₃₂	C₂×C₄×C₅	1	2	yes	—
170	Q₃₂	C₂×C₄×C₅	1	2	yes	—
171	SD₃₂	C₂×C₄×C₅	1	4	yes	—
172	SD₃₂	C₂×C₄×C₅	1	4	yes	—
173	D₁₆	C₂×C₄×C₅	1	2	yes	—
174	D₁₆	C₂×C₄×C₅	1	2	yes	—
175	M₃₂	—	2	1	no	—
176	M₃₂	—	2	1	no	—
177	C₂×C₁₆	C₂×C₄×C₅	1	1	yes	characteristic
178	C₂×C₁₆	C₂×C₄×C₅	1	1	yes	characteristic
179	C₂×C₁₆	C₂×C₄×C₅	1	1	yes	characteristic
180	C₂×C₁₆	C₂×C₄×C₅	1	1	yes	characteristic
181	C₂×C₁₆	C₂³×C₅	1	2	yes	—
182	C₂×C₁₆	C₂³×C₅	1	1	yes	characteristic
183	C₂×C₁₆	C₂³×C₅	1	1	yes	characteristic
184	#108	—	2	2	no	—
185	#108	C₂³×C₅	1	4	yes	—
186	#104	—	2	8	no	—
187	#97	C₅×D₄	1	2	yes	—
188	C₄×C₈	C₂³×C₅	1	2	yes	—
189	C₂²×C₅	—	4	1	no	—
190	C₂²×C₅	—	4	4	no	—
191	C₂²×C₅	—	16	2	no	—
192	C₂²×C₅	C₄×D₈	1	1	yes	characteristic
193	C₂²×C₅	C₄×D₈	1	1	yes	characteristic
194	C₂²×C₅	C₂×#135	1	1	yes	characteristic
195	C₄×C₅	—	4	2	no	—
196	C₄×C₅	—	4	4	no	—
197	C₄×C₅	C₄×D₈	1	1	yes	characteristic
198	C₄×C₅	C₄×D₈	1	1	yes	characteristic
199	C₄×C₅	C₂×C₄×D₄	1	1	yes	characteristic
200	C₄×C₅	C₂²×D₈	1	1	yes	characteristic
201	#41	—	2	4	no	—
202	#41	—	2	4	no	—
203	#41	—	4	2	no	—
204	#41	—	4	2	no	—
205	C₂×Q₈	—	2	2	no	—
206	C₂×D₄	—	2	2	no	—
207	C₂×D₄	—	4	2	no	—
208	C₂²×C₄	—	4	2	no	—
209	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
210	Q₁₆	—	2	2	no	—
211	Q₁₆	C₂²×C₄×C₅	1	4	yes	—
212	SD₁₆	—	2	4	no	—
213	SD₁₆	—	2	4	no	—
214	D₈	—	2	2	no	—
215	D₈	C₂²×C₄×C₅	1	4	yes	—
216	M₁₆	—	2	4	no	—
217	M₁₆	—	4	2	no	—
218	M₁₆	—	4	8	no	—
219	M₁₆	C₅×D₈	1	2	yes	—
220	M₁₆	C₅×D₈	1	2	yes	—
221	C₂×C₈	—	4	2	no	—
222	C₂×C₈	—	4	8	no	—
223	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
224	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
225	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
226	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
227	C₂×C₈	C₅×#41	1	1	yes	characteristic
228	C₂×C₈	C₅×#41	1	1	yes	characteristic
229	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
230	C₄⋊C₄	—	2	1	no	—
231	C₂²⋊C₄	—	2	1	no	—
232	C₄²	C₂×C₅×D₄	1	2	yes	—
233	C₁₆	—	2	2	no	—
234	C₁₆	C₂²×C₄×C₅	1	1	yes	characteristic
235	C₁₆	C₂²×C₄×C₅	1	1	yes	characteristic
236	C₁₆	C₂²×C₄×C₅	1	1	yes	characteristic
237	C₁₆	C₂²×C₄×C₅	1	1	yes	characteristic
238	C₂×C₅	—	2	1	no	—
239	C₂×C₅	—	2	1	no	—
240	C₂×C₅	—	2	1	no	—
241	C₂×C₅	—	8	4	no	—
242	C₂×C₅	C₂×C₄×D₈	1	1	yes	characteristic
243	C₂³	—	4	2	no	—
244	C₂³	C₅×#135	1	1	yes	characteristic
245	Q₈	—	2	4	no	—
246	Q₈	—	4	2	no	—
247	D₄	—	2	4	no	—
248	D₄	—	4	4	no	—
249	D₄	—	4	4	no	—
250	D₄	—	4	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
256	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
257	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
258	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
259	C₂×C₄	C₅×#135	1	1	yes	characteristic
260	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
261	C₈	—	4	4	no	—
262	C₈	C₄×C₅×D₄	1	1	yes	characteristic
263	C₈	C₄×C₅×D₄	1	1	yes	characteristic
264	C₈	C₂×C₅×D₈	1	4	yes	—
265	C₈	C₂³×C₄×C₅	1	1	yes	characteristic
266	C₈	C₂×C₅×#41	1	1	yes	characteristic
267	C₅	#33465	1	1	yes	characteristic, direct product factor, Sylow
268	C₂²	—	4	1	no	—
269	C₂²	—	4	4	no	—
270	C₂²	—	16	2	no	—
271	C₂²	C₄×C₅×D₈	1	1	yes	characteristic
272	C₂²	C₄×C₅×D₈	1	1	yes	characteristic
273	C₂²	C₂×C₅×#135	1	1	yes	characteristic
274	C₄	—	4	2	no	—
275	C₄	—	4	4	no	—
276	C₄	C₄×C₅×D₈	1	1	yes	characteristic
277	C₄	C₄×C₅×D₈	1	1	yes	characteristic
278	C₄	C₂×C₄×C₅×D₄	1	1	yes	characteristic
279	C₄	C₂²×C₅×D₈	1	1	yes	characteristic
280	C₂	—	2	1	no	—
281	C₂	—	2	1	no	—
282	C₂	—	2	1	no	—
283	C₂	—	8	4	no	—
284	C₂	C₂×C₄×C₅×D₈	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅	C₂²×D₈
commutator	C₈	C₂³×C₄×C₅
Frattini, cosocle	C₂×C₈	C₂⁴×C₅
socle	C₂×C₅	C₂×C₄×D₈
prime radical	C₂×C₅×#508	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂×C₅×M₁₆ ◁ C₅×#405 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#2957	C₂	semidirect
3. C₅×#2179	C₂	semidirect
4. C₅×#2178	C₂	semidirect
5. C₂×C₅×#368	C₂	semidirect
6. C₅×#2143	C₂	semidirect
7. C₅×#2108	C₂	semidirect
10. C₅×#508	C₄	semidirect
11. C₅×#508	C₄	semidirect
14. C₂×C₅×Q₃₂	C₄	semidirect
15. C₂×C₅×SD₃₂	C₄	semidirect
16. C₂×C₅×D₁₆	C₄	semidirect
20. C₅×#443	C₂²	semidirect
24. C₅×#368	C₂²	semidirect
25. C₅×#368	C₂²	semidirect
26. C₅×#359	C₂²	semidirect
27. C₅×#346	C₂²	semidirect
28. C₄×C₅×C₁₆	C₂²	semidirect
29. #33465	C₅	direct (N × H)
66. #2957	C₂×C₅	semidirect
67. #2179	C₂×C₅	semidirect
68. #2178	C₂×C₅	semidirect
69. C₂×#368	C₂×C₅	semidirect
70. #2143	C₂×C₅	semidirect
71. #2108	C₂×C₅	semidirect
111. #508	C₄×C₅	semidirect
112. #508	C₄×C₅	semidirect
115. C₂×Q₃₂	C₄×C₅	semidirect
116. C₂×SD₃₂	C₄×C₅	semidirect
117. C₂×D₁₆	C₄×C₅	semidirect
121. #443	C₂²×C₅	semidirect
125. #368	C₂²×C₅	semidirect
126. #368	C₂²×C₅	semidirect
127. #359	C₂²×C₅	semidirect
128. #346	C₂²×C₅	semidirect
129. C₄×C₁₆	C₂²×C₅	semidirect
267. C₅	#33465	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₅×#508	2
C₅×#2957	2
C₅×#2179	2
C₅×#2178	2
C₂×C₅×#368	2
C₅×#2143	2
C₅×#2108	2
#33465	5

Subgroups — C5×#33465 #11956185

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33465 #11956185