Subgroups — C₅×#12563 #11935283

Order 1280.

570 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₅×#108	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#427	C₂	1	1	yes	maximal, characteristic
4	C₅×#1383	C₂	1	4	yes	maximal, complemented
5	C₂²×C₅×M₁₆	—	2	1	no	—
6	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₂×C₅×#117	—	2	1	no	—
8	C₂×C₅×M₃₂	—	2	2	no	—
9	C₂×C₅×M₃₂	—	2	2	no	—
10	C₂×C₅×M₃₂	C₄	1	2	yes	complemented
11	C₂×C₅×M₃₂	C₂²	1	2	yes	—
12	C₂²×C₅×C₁₆	C₄	1	2	yes	complemented
13	C₂×C₅×#108	—	2	1	no	—
14	C₂×C₅×#108	—	2	2	no	—
15	C₂×C₅×#108	C₄	1	4	yes	complemented
16	C₂×C₅×#108	C₂²	1	2	yes	—
17	C₅×#427	—	2	2	no	—
18	C₅×#427	C₄	1	4	yes	—
19	C₂×C₅×#107	—	2	1	no	—
20	C₂×C₅×#106	C₂²	1	2	yes	—
21	#12563	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
23	C₂×C₅×M₁₆	—	2	4	no	—
24	C₂×C₅×M₁₆	—	2	2	no	—
25	C₂×C₅×M₁₆	—	4	2	no	—
26	C₂×C₅×M₁₆	—	4	1	no	—
27	C₂²×C₅×C₈	—	4	1	no	—
28	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
31	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
32	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
33	C₂²×C₅×C₈	D₄	1	2	yes	—
34	C₂²×C₅×C₈	D₄	1	2	yes	—
35	C₂²×C₅×C₈	Q₈	1	1	yes	characteristic
36	C₂²×C₅×C₈	Q₈	1	1	yes	characteristic
37	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
38	C₅×#117	—	2	4	no	—
39	C₅×#117	—	4	2	no	—
40	C₂×C₅×C₄⋊C₄	—	2	2	no	—
41	C₂×C₅×C₂²⋊C₄	—	4	1	no	—
42	C₂×C₄²×C₅	—	4	1	no	—
43	C₅×M₃₂	—	2	4	no	—
44	C₅×M₃₂	—	2	2	no	—
45	C₅×M₃₂	—	2	2	no	—
46	C₂×C₅×C₁₆	—	2	4	no	—
47	C₂×C₅×C₁₆	C₂×C₄	1	4	yes	—
48	C₅×#108	—	2	4	no	—
49	C₅×#108	—	2	2	no	—
50	C₅×#108	C₂×C₄	1	4	yes	—
51	C₅×#107	—	2	4	no	—
52	C₅×#106	—	2	2	no	—
53	C₅×#106	C₂×C₄	1	4	yes	—
54	C₂²×M₃₂	C₂×C₅	1	1	yes	characteristic
55	C₂²×#108	C₂×C₅	1	1	yes	characteristic
56	C₂×#427	C₂×C₅	1	1	yes	characteristic
57	#1383	C₂×C₅	1	4	yes	complemented
58	C₂⁴×C₅	SD₁₆	1	1	yes	characteristic
59	C₂²×C₄×C₅	—	4	2	no	—
60	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
61	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅	D₈	1	2	yes	—
65	C₂²×C₄×C₅	SD₁₆	1	1	yes	characteristic
66	C₂²×C₄×C₅	SD₁₆	1	1	yes	characteristic
67	C₂²×C₄×C₅	SD₁₆	1	1	yes	characteristic
68	C₂²×C₄×C₅	Q₁₆	1	2	yes	—
69	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
70	C₅×M₁₆	—	2	4	no	—
71	C₅×M₁₆	—	4	4	no	—
72	C₅×M₁₆	—	4	2	no	—
73	C₂×C₅×C₈	—	4	4	no	—
74	C₂×C₅×C₈	—	4	2	no	—
75	C₂×C₅×C₈	C₄²	1	2	yes	—
76	C₂×C₅×C₈	C₄²	1	2	yes	—
77	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
78	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
79	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
80	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
81	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
84	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
85	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
86	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
87	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
91	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
92	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
93	C₅×C₄⋊C₄	—	2	4	no	—
94	C₅×C₄⋊C₄	—	4	2	no	—
95	C₅×C₂²⋊C₄	—	4	4	no	—
96	C₄²×C₅	—	4	4	no	—
97	C₅×C₁₆	—	2	4	no	—
98	C₂²×M₁₆	—	2	1	no	—
99	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
100	C₂×#117	—	2	1	no	—
101	C₂×M₃₂	—	2	2	no	—
102	C₂×M₃₂	—	2	2	no	—
103	C₂×M₃₂	C₄×C₅	1	2	yes	complemented
104	C₂×M₃₂	C₂²×C₅	1	2	yes	—
105	C₂²×C₁₆	C₄×C₅	1	2	yes	complemented
106	C₂×#108	—	2	1	no	—
107	C₂×#108	—	2	2	no	—
108	C₂×#108	C₄×C₅	1	4	yes	complemented
109	C₂×#108	C₂²×C₅	1	2	yes	—
110	#427	—	2	2	no	—
111	#427	C₄×C₅	1	4	yes	—
112	C₂×#107	—	2	1	no	—
113	C₂×#106	C₂²×C₅	1	2	yes	—
114	C₂³×C₅	—	4	2	no	—
115	C₂³×C₅	#102	1	2	yes	—
116	C₂³×C₅	#103	1	2	yes	—
117	C₂³×C₅	#106	1	1	yes	characteristic
118	C₂³×C₅	#106	1	1	yes	characteristic
119	C₂³×C₅	C₂×SD₁₆	1	1	yes	characteristic
120	C₂×C₄×C₅	—	4	4	no	—
121	C₂×C₄×C₅	—	8	4	no	—
122	C₂×C₄×C₅	#95	1	2	yes	—
123	C₂×C₄×C₅	#95	1	2	yes	—
124	C₂×C₄×C₅	#102	1	2	yes	—
125	C₂×C₄×C₅	#102	1	2	yes	—
126	C₂×C₄×C₅	#102	1	2	yes	—
127	C₂×C₄×C₅	#103	1	2	yes	—
128	C₂×C₄×C₅	#103	1	2	yes	—
129	C₂×C₄×C₅	#103	1	2	yes	—
130	C₂×C₄×C₅	#106	1	1	yes	characteristic
131	C₂×C₄×C₅	#106	1	1	yes	characteristic
132	C₂×C₄×C₅	#107	1	2	yes	—
133	C₂×C₄×C₅	#107	1	2	yes	—
134	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
136	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
139	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
140	C₅×C₈	—	4	4	no	—
141	C₅×C₈	#95	1	2	yes	—
142	C₅×C₈	#95	1	2	yes	—
143	C₅×C₈	C₂×C₄²	1	1	yes	characteristic
144	C₅×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
145	C₅×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
146	C₅×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
147	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
148	C₂×M₁₆	—	2	4	no	—
149	C₂×M₁₆	—	2	2	no	—
150	C₂×M₁₆	—	4	2	no	—
151	C₂×M₁₆	—	4	1	no	—
152	C₂²×C₈	—	4	1	no	—
153	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
154	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
155	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
156	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
157	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
158	C₂²×C₈	C₅×D₄	1	2	yes	—
159	C₂²×C₈	C₅×D₄	1	2	yes	—
160	C₂²×C₈	C₅×Q₈	1	1	yes	characteristic
161	C₂²×C₈	C₅×Q₈	1	1	yes	characteristic
162	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
163	#117	—	2	4	no	—
164	#117	—	4	2	no	—
165	C₂×C₄⋊C₄	—	2	2	no	—
166	C₂×C₂²⋊C₄	—	4	1	no	—
167	C₂×C₄²	—	4	1	no	—
168	M₃₂	—	2	4	no	—
169	M₃₂	—	2	2	no	—
170	M₃₂	—	2	2	no	—
171	C₂×C₁₆	—	2	4	no	—
172	C₂×C₁₆	C₂×C₄×C₅	1	4	yes	—
173	#108	—	2	4	no	—
174	#108	—	2	2	no	—
175	#108	C₂×C₄×C₅	1	4	yes	—
176	#107	—	2	4	no	—
177	#106	—	2	2	no	—
178	#106	C₂×C₄×C₅	1	4	yes	—
179	C₂²×C₅	—	2	2	no	—
180	C₂²×C₅	—	2	2	no	—
181	C₂²×C₅	—	2	2	no	—
182	C₂²×C₅	—	4	2	no	—
183	C₂²×C₅	—	4	2	no	—
184	C₂²×C₅	#340	1	2	yes	—
185	C₂²×C₅	#340	1	2	yes	—
186	C₂²×C₅	C₂×#102	1	1	yes	characteristic
187	C₂²×C₅	C₂×#103	1	1	yes	characteristic
188	C₂²×C₅	C₂×#106	1	1	yes	characteristic
189	C₄×C₅	—	8	4	no	—
190	C₄×C₅	#340	1	2	yes	—
191	C₄×C₅	#340	1	2	yes	—
192	C₄×C₅	C₂×#95	1	1	yes	characteristic
193	C₄×C₅	C₂×#102	1	1	yes	characteristic
194	C₄×C₅	C₂×#103	1	1	yes	characteristic
195	C₄×C₅	C₂×#107	1	1	yes	characteristic
196	C₂⁴	C₅×SD₁₆	1	1	yes	characteristic
197	C₂²×C₄	—	4	2	no	—
198	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
199	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
200	C₂²×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
201	C₂²×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
202	C₂²×C₄	C₅×D₈	1	2	yes	—
203	C₂²×C₄	C₅×SD₁₆	1	1	yes	characteristic
204	C₂²×C₄	C₅×SD₁₆	1	1	yes	characteristic
205	C₂²×C₄	C₅×SD₁₆	1	1	yes	characteristic
206	C₂²×C₄	C₅×Q₁₆	1	2	yes	—
207	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
208	M₁₆	—	2	4	no	—
209	M₁₆	—	4	2	no	—
210	M₁₆	—	4	4	no	—
211	C₂×C₈	—	4	2	no	—
212	C₂×C₈	—	4	4	no	—
213	C₂×C₈	C₄²×C₅	1	2	yes	—
214	C₂×C₈	C₄²×C₅	1	2	yes	—
215	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
216	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
217	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
218	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
219	C₂×C₈	C₅×C₄⋊C₄	1	2	yes	—
220	C₂×C₈	C₅×C₄⋊C₄	1	2	yes	—
221	C₂×C₈	C₅×C₄⋊C₄	1	2	yes	—
222	C₂×C₈	C₅×C₄⋊C₄	1	2	yes	—
223	C₂×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
224	C₂×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
225	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
226	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
227	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
228	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
229	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
230	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	—	4	2	no	—
233	C₂²⋊C₄	—	4	4	no	—
234	C₄²	—	4	4	no	—
235	C₁₆	—	2	4	no	—
236	C₂×C₅	—	2	1	no	—
237	C₂×C₅	—	2	2	no	—
238	C₂×C₅	—	2	2	no	—
239	C₂×C₅	—	2	1	no	—
240	C₂×C₅	C₂×#340	1	1	yes	characteristic
241	C₂×C₅	#2155	1	2	yes	—
242	C₂³	—	4	2	no	—
243	C₂³	C₅×#102	1	2	yes	—
244	C₂³	C₅×#103	1	2	yes	—
245	C₂³	C₅×#106	1	1	yes	characteristic
246	C₂³	C₅×#106	1	1	yes	characteristic
247	C₂³	C₂×C₅×SD₁₆	1	1	yes	characteristic
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	—	8	4	no	—
250	C₂×C₄	C₅×#95	1	2	yes	—
251	C₂×C₄	C₅×#95	1	2	yes	—
252	C₂×C₄	C₅×#102	1	2	yes	—
253	C₂×C₄	C₅×#102	1	2	yes	—
254	C₂×C₄	C₅×#102	1	2	yes	—
255	C₂×C₄	C₅×#103	1	2	yes	—
256	C₂×C₄	C₅×#103	1	2	yes	—
257	C₂×C₄	C₅×#103	1	2	yes	—
258	C₂×C₄	C₅×#106	1	1	yes	characteristic
259	C₂×C₄	C₅×#106	1	1	yes	characteristic
260	C₂×C₄	C₅×#107	1	2	yes	—
261	C₂×C₄	C₅×#107	1	2	yes	—
262	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
263	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
264	C₂×C₄	C₂×C₅×C₄⋊C₄	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	C₂×C₅×Q₁₆	1	1	yes	characteristic
268	C₈	—	4	4	no	—
269	C₈	C₅×#95	1	2	yes	—
270	C₈	C₅×#95	1	2	yes	—
271	C₈	C₂×C₄²×C₅	1	1	yes	characteristic
272	C₈	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
273	C₈	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
274	C₈	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
275	C₅	#12563	1	1	yes	characteristic, direct product factor, Sylow
276	C₂²	—	2	2	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	4	2	no	—
280	C₂²	—	4	2	no	—
281	C₂²	C₅×#340	1	2	yes	—
282	C₂²	C₅×#340	1	2	yes	—
283	C₂²	C₂×C₅×#102	1	1	yes	characteristic
284	C₂²	C₂×C₅×#103	1	1	yes	characteristic
285	C₂²	C₂×C₅×#106	1	1	yes	characteristic
286	C₄	—	8	4	no	—
287	C₄	C₅×#340	1	2	yes	—
288	C₄	C₅×#340	1	2	yes	—
289	C₄	C₂×C₅×#95	1	1	yes	characteristic
290	C₄	C₂×C₅×#102	1	1	yes	characteristic
291	C₄	C₂×C₅×#103	1	1	yes	characteristic
292	C₄	C₂×C₅×#107	1	1	yes	characteristic
293	C₂	—	2	1	no	—
294	C₂	—	2	2	no	—
295	C₂	—	2	2	no	—
296	C₂	—	2	1	no	—
297	C₂	C₂×C₅×#340	1	1	yes	characteristic
298	C₂	C₅×#2155	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅	#106
commutator	C₈	C₂×C₄²×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂²×C₅	C₂×#106
prime radical	C₂⁴×C₅	SD₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂³×C₄×C₅ ◁ C₂³×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅×#1383	C₂	semidirect
10. C₂×C₅×M₃₂	C₄	semidirect
12. C₂²×C₅×C₁₆	C₄	semidirect
15. C₂×C₅×#108	C₄	semidirect
21. #12563	C₅	direct (N × H)
57. #1383	C₂×C₅	semidirect
103. C₂×M₃₂	C₄×C₅	semidirect
105. C₂²×C₁₆	C₄×C₅	semidirect
108. C₂×#108	C₄×C₅	semidirect
275. C₅	#12563	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₅×M₃₂	2
C₂²×C₅×#108	2
C₂×C₅×#427	2
C₅×#1383	2
#12563	5

Subgroups — C5×#12563 #11935283

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#12563 #11935283