Subgroups — C₅×#12642 #11935362

Order 1280.

430 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#443	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#359	C₂	1	1	yes	maximal, characteristic
3	C₅×#2138	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅×#346	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1772	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1382	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1381	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅×#135	C₂²	1	1	yes	characteristic
9	C₂×C₅×#131	—	2	1	no	—
10	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
12	C₅×#443	—	2	2	no	—
13	C₅×#443	C₄	1	4	yes	complemented
14	C₂×C₅×#108	C₂²	1	1	yes	characteristic
15	C₂×C₅×#108	C₂²	1	1	yes	characteristic
16	C₅×#427	C₂²	1	1	yes	characteristic
17	C₂×C₅×#104	—	2	1	no	—
18	C₅×#405	—	2	1	no	—
19	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
20	C₅×#359	C₄	1	4	yes	complemented
21	C₅×#358	C₄	1	2	yes	complemented
22	C₅×#357	C₄	1	2	yes	complemented
23	C₅×#346	—	2	2	no	—
24	C₅×#339	—	2	1	no	—
25	#12642	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₂×C₅×#41	—	2	1	no	—
27	C₅×#135	—	2	1	no	—
28	C₅×#135	—	2	1	no	—
29	C₅×#135	C₂×C₄	1	2	yes	—
30	C₅×#135	C₂×C₄	1	2	yes	—
31	C₂×C₅×Q₁₆	C₂×C₄	1	1	yes	characteristic
32	C₂×C₅×SD₁₆	—	2	1	no	—
33	C₂×C₅×D₈	C₂×C₄	1	1	yes	characteristic
34	C₅×#131	—	2	4	no	—
35	C₅×#131	—	4	2	no	—
36	C₂×C₅×M₁₆	—	2	1	no	—
37	C₂×C₅×M₁₆	—	2	1	no	—
38	C₂×C₅×M₁₆	—	4	1	no	—
39	C₂²×C₅×C₈	—	4	1	no	—
40	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
41	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
42	C₅×#117	—	2	1	no	—
43	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
44	C₂×C₅×C₁₆	—	2	1	no	—
45	C₂×C₅×C₁₆	—	2	1	no	—
46	C₂×C₅×C₁₆	—	2	1	no	—
47	C₂×C₅×C₁₆	—	2	1	no	—
48	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
49	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
50	C₅×#108	—	2	1	no	—
51	C₅×#108	—	2	1	no	—
52	C₅×#108	C₂×C₄	1	2	yes	—
53	C₅×#108	C₂×C₄	1	2	yes	—
54	C₅×#107	C₂×C₄	1	2	yes	—
55	C₅×#106	—	2	1	no	—
56	C₅×#104	—	2	4	no	—
57	C₅×#97	—	4	1	no	—
58	C₄×C₅×C₈	—	4	1	no	—
59	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
60	C₄×C₅×C₈	D₄	1	2	yes	—
61	C₂×#443	C₂×C₅	1	1	yes	characteristic
62	C₂×#359	C₂×C₅	1	1	yes	characteristic
63	#2138	C₂×C₅	1	1	yes	characteristic
64	C₂×#346	C₂×C₅	1	1	yes	characteristic, complemented
65	#1772	C₂×C₅	1	1	yes	characteristic, complemented
66	#1382	C₂×C₅	1	1	yes	characteristic, complemented
67	#1381	C₂×C₅	1	1	yes	characteristic, complemented
68	C₅×#41	—	2	2	no	—
69	C₅×#41	—	2	2	no	—
70	C₅×#41	—	4	1	no	—
71	C₅×#41	—	4	1	no	—
72	C₂×C₅×Q₈	—	2	1	no	—
73	C₂×C₅×D₄	—	2	1	no	—
74	C₂×C₅×D₄	—	4	1	no	—
75	C₂²×C₄×C₅	—	2	1	no	—
76	C₂²×C₄×C₅	—	4	1	no	—
77	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
78	C₅×Q₁₆	—	2	1	no	—
79	C₅×Q₁₆	C₄²	1	2	yes	—
80	C₅×SD₁₆	—	2	2	no	—
81	C₅×SD₁₆	—	2	2	no	—
82	C₅×D₈	—	2	1	no	—
83	C₅×D₈	C₄²	1	2	yes	—
84	C₅×M₁₆	—	2	2	no	—
85	C₅×M₁₆	—	2	2	no	—
86	C₅×M₁₆	—	4	1	no	—
87	C₅×M₁₆	—	4	1	no	—
88	C₅×M₁₆	—	4	4	no	—
89	C₂×C₅×C₈	—	4	1	no	—
90	C₂×C₅×C₈	—	4	1	no	—
91	C₂×C₅×C₈	—	4	4	no	—
92	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
93	C₂×C₅×C₈	D₈	1	1	yes	characteristic
94	C₂×C₅×C₈	D₈	1	1	yes	characteristic
95	C₂×C₅×C₈	SD₁₆	1	2	yes	—
96	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
100	C₂×C₅×C₈	#41	1	1	yes	characteristic
101	C₂×C₅×C₈	#41	1	1	yes	characteristic
102	C₅×C₄⋊C₄	—	2	2	no	—
103	C₅×C₂²⋊C₄	—	4	1	no	—
104	C₄²×C₅	—	2	1	no	—
105	C₄²×C₅	—	4	1	no	—
106	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
107	C₅×C₁₆	—	2	2	no	—
108	C₅×C₁₆	—	2	2	no	—
109	C₂×#135	C₂²×C₅	1	1	yes	characteristic
110	C₂×#131	—	2	1	no	—
111	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
112	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
113	#443	—	2	2	no	—
114	#443	C₄×C₅	1	4	yes	complemented
115	C₂×#108	C₂²×C₅	1	1	yes	characteristic
116	C₂×#108	C₂²×C₅	1	1	yes	characteristic
117	#427	C₂²×C₅	1	1	yes	characteristic
118	C₂×#104	—	2	1	no	—
119	#405	—	2	1	no	—
120	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
121	#359	C₄×C₅	1	4	yes	complemented
122	#358	C₄×C₅	1	2	yes	complemented
123	#357	C₄×C₅	1	2	yes	complemented
124	#346	—	2	2	no	—
125	#339	—	2	1	no	—
126	C₂³×C₅	—	4	1	no	—
127	C₂³×C₅	#135	1	1	yes	characteristic
128	C₅×Q₈	—	2	2	no	—
129	C₅×Q₈	—	4	1	no	—
130	C₅×D₄	—	2	2	no	—
131	C₅×D₄	—	4	2	no	—
132	C₅×D₄	—	4	2	no	—
133	C₅×D₄	—	4	1	no	—
134	C₂×C₄×C₅	—	2	2	no	—
135	C₂×C₄×C₅	—	4	2	no	—
136	C₂×C₄×C₅	—	4	1	no	—
137	C₂×C₄×C₅	—	4	2	no	—
138	C₂×C₄×C₅	—	4	2	no	—
139	C₂×C₄×C₅	—	4	2	no	—
140	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
141	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
142	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
143	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
144	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
145	C₂×C₄×C₅	#135	1	1	yes	characteristic
146	C₅×C₈	—	4	2	no	—
147	C₅×C₈	—	4	2	no	—
148	C₅×C₈	#102	1	2	yes	—
149	C₅×C₈	#102	1	2	yes	—
150	C₅×C₈	C₂×C₄²	1	1	yes	characteristic
151	C₅×C₈	#117	1	1	yes	characteristic
152	C₅×C₈	C₄×D₄	1	1	yes	characteristic
153	C₅×C₈	C₄×D₄	1	1	yes	characteristic
154	C₂×#41	—	2	1	no	—
155	#135	—	2	1	no	—
156	#135	—	2	1	no	—
157	#135	C₂×C₄×C₅	1	2	yes	—
158	#135	C₂×C₄×C₅	1	2	yes	—
159	C₂×Q₁₆	C₂×C₄×C₅	1	1	yes	characteristic
160	C₂×SD₁₆	—	2	1	no	—
161	C₂×D₈	C₂×C₄×C₅	1	1	yes	characteristic
162	#131	—	2	4	no	—
163	#131	—	4	2	no	—
164	C₂×M₁₆	—	2	1	no	—
165	C₂×M₁₆	—	2	1	no	—
166	C₂×M₁₆	—	4	1	no	—
167	C₂²×C₈	—	4	1	no	—
168	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
169	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
170	#117	—	2	1	no	—
171	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
172	C₂×C₁₆	—	2	1	no	—
173	C₂×C₁₆	—	2	1	no	—
174	C₂×C₁₆	—	2	1	no	—
175	C₂×C₁₆	—	2	1	no	—
176	C₂×C₁₆	C₂×C₄×C₅	1	2	yes	—
177	C₂×C₁₆	C₂×C₄×C₅	1	2	yes	—
178	#108	—	2	1	no	—
179	#108	—	2	1	no	—
180	#108	C₂×C₄×C₅	1	2	yes	—
181	#108	C₂×C₄×C₅	1	2	yes	—
182	#107	C₂×C₄×C₅	1	2	yes	—
183	#106	—	2	1	no	—
184	#104	—	2	4	no	—
185	#97	—	4	1	no	—
186	C₄×C₈	—	4	1	no	—
187	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
188	C₄×C₈	C₅×D₄	1	2	yes	—
189	C₂²×C₅	—	2	2	no	—
190	C₂²×C₅	—	4	2	no	—
191	C₂²×C₅	—	8	2	no	—
192	C₂²×C₅	#416	1	1	yes	characteristic
193	C₂²×C₅	C₄×D₈	1	1	yes	characteristic
194	C₂²×C₅	C₄×SD₁₆	1	1	yes	characteristic
195	C₄×C₅	—	4	2	no	—
196	C₄×C₅	—	4	2	no	—
197	C₄×C₅	—	8	2	no	—
198	C₄×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
199	C₄×C₅	C₂×#102	1	1	yes	characteristic
200	C₄×C₅	C₄×D₈	1	1	yes	characteristic
201	C₄×C₅	C₄×SD₁₆	1	1	yes	characteristic
202	#41	—	2	2	no	—
203	#41	—	2	2	no	—
204	#41	—	4	1	no	—
205	#41	—	4	1	no	—
206	C₂×Q₈	—	2	1	no	—
207	C₂×D₄	—	2	1	no	—
208	C₂×D₄	—	4	1	no	—
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	—	4	1	no	—
211	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
212	Q₁₆	—	2	1	no	—
213	Q₁₆	C₄²×C₅	1	2	yes	—
214	SD₁₆	—	2	2	no	—
215	SD₁₆	—	2	2	no	—
216	D₈	—	2	1	no	—
217	D₈	C₄²×C₅	1	2	yes	—
218	M₁₆	—	2	2	no	—
219	M₁₆	—	2	2	no	—
220	M₁₆	—	4	1	no	—
221	M₁₆	—	4	1	no	—
222	M₁₆	—	4	4	no	—
223	C₂×C₈	—	4	1	no	—
224	C₂×C₈	—	4	1	no	—
225	C₂×C₈	—	4	4	no	—
226	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
227	C₂×C₈	C₅×D₈	1	1	yes	characteristic
228	C₂×C₈	C₅×D₈	1	1	yes	characteristic
229	C₂×C₈	C₅×SD₁₆	1	2	yes	—
230	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
231	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
232	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
233	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
234	C₂×C₈	C₅×#41	1	1	yes	characteristic
235	C₂×C₈	C₅×#41	1	1	yes	characteristic
236	C₄⋊C₄	—	2	2	no	—
237	C₂²⋊C₄	—	4	1	no	—
238	C₄²	—	2	1	no	—
239	C₄²	—	4	1	no	—
240	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
241	C₁₆	—	2	2	no	—
242	C₁₆	—	2	2	no	—
243	C₂×C₅	—	2	1	no	—
244	C₂×C₅	—	2	1	no	—
245	C₂×C₅	—	8	2	no	—
246	C₂×C₅	C₄×#102	1	1	yes	characteristic
247	C₂×C₅	#2179	1	2	yes	—
248	C₂³	—	4	1	no	—
249	C₂³	C₅×#135	1	1	yes	characteristic
250	Q₈	—	2	2	no	—
251	Q₈	—	4	1	no	—
252	D₄	—	2	2	no	—
253	D₄	—	4	2	no	—
254	D₄	—	4	2	no	—
255	D₄	—	4	1	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
263	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
264	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
265	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
266	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
267	C₂×C₄	C₅×#135	1	1	yes	characteristic
268	C₈	—	4	2	no	—
269	C₈	—	4	2	no	—
270	C₈	C₅×#102	1	2	yes	—
271	C₈	C₅×#102	1	2	yes	—
272	C₈	C₂×C₄²×C₅	1	1	yes	characteristic
273	C₈	C₅×#117	1	1	yes	characteristic
274	C₈	C₄×C₅×D₄	1	1	yes	characteristic
275	C₈	C₄×C₅×D₄	1	1	yes	characteristic
276	C₅	#12642	1	1	yes	characteristic, direct product factor, Sylow
277	C₂²	—	2	2	no	—
278	C₂²	—	4	2	no	—
279	C₂²	—	8	2	no	—
280	C₂²	C₅×#416	1	1	yes	characteristic
281	C₂²	C₄×C₅×D₈	1	1	yes	characteristic
282	C₂²	C₄×C₅×SD₁₆	1	1	yes	characteristic
283	C₄	—	4	2	no	—
284	C₄	—	4	2	no	—
285	C₄	—	8	2	no	—
286	C₄	C₄×C₅×C₂²⋊C₄	1	1	yes	characteristic
287	C₄	C₂×C₅×#102	1	1	yes	characteristic
288	C₄	C₄×C₅×D₈	1	1	yes	characteristic
289	C₄	C₄×C₅×SD₁₆	1	1	yes	characteristic
290	C₂	—	2	1	no	—
291	C₂	—	2	1	no	—
292	C₂	—	8	2	no	—
293	C₂	C₄×C₅×#102	1	1	yes	characteristic
294	C₂	C₅×#2179	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅	C₂×D₈
commutator	C₈	C₂×C₄²×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂²×C₅	C₄×D₈
prime radical	C₂×C₅×#135	C₂²
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂²×C₅×C₈ ◁ C₂×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂×C₅×#346	C₂	semidirect
5. C₅×#1772	C₂	semidirect
6. C₅×#1382	C₂	semidirect
7. C₅×#1381	C₂	semidirect
13. C₅×#443	C₄	semidirect
20. C₅×#359	C₄	semidirect
21. C₅×#358	C₄	semidirect
22. C₅×#357	C₄	semidirect
25. #12642	C₅	direct (N × H)
64. C₂×#346	C₂×C₅	semidirect
65. #1772	C₂×C₅	semidirect
66. #1382	C₂×C₅	semidirect
67. #1381	C₂×C₅	semidirect
114. #443	C₄×C₅	semidirect
121. #359	C₄×C₅	semidirect
122. #358	C₄×C₅	semidirect
123. #357	C₄×C₅	semidirect
276. C₅	#12642	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×#443	2
C₂×C₅×#359	2
C₅×#2138	2
C₂×C₅×#346	2
C₅×#1772	2
C₅×#1382	2
C₅×#1381	2
#12642	5

Subgroups — C5×#12642 #11935362

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#12642 #11935362