Subgroups — C₅×#22530 #11945250

Order 1280.

1428 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	C₂×C₄×C₅×#41	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×C₈×Q₈	C₂	1	2	yes	maximal, complemented
3	C₅×#1954	C₂	1	6	yes	maximal, complemented
4	C₅×#1797	C₂	1	3	yes	maximal, complemented
5	C₄×C₅×C₂²⋊C₈	C₂	1	3	yes	maximal, complemented
6	C₂²×C₅×#41	C₄	1	1	yes	characteristic
7	C₄×C₅×#41	—	2	8	no	—
8	C₂×C₄×C₅×Q₈	C₄	1	1	yes	characteristic
9	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₅×D₄	C₄	1	3	yes	—
11	C₂×C₄×C₅×D₄	C₂²	1	3	yes	—
12	C₂×C₅×#117	C₄	1	3	yes	—
13	C₂²×C₄²×C₅	C₂²	1	3	yes	—
14	C₅×C₈×Q₈	—	2	8	no	—
15	C₂×C₅×C₄⋊C₈	C₂²	1	6	yes	—
16	C₂×C₅×C₂²⋊C₈	C₂²	1	6	yes	—
17	C₂×C₅×C₂²⋊C₈	C₂²	1	6	yes	—
18	C₂×C₄×C₅×C₈	C₂²	1	6	yes	—
19	C₅×#336	C₂²	1	3	yes	—
20	C₅×#336	C₂²	1	1	yes	characteristic
21	#22530	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	C₂×C₅×#41	—	2	4	no	—
23	C₂×C₅×#41	—	2	4	no	—
24	C₂×C₅×#41	C₈	1	8	yes	complemented
25	C₂²×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₅×D₄	C₂×C₄	1	3	yes	—
27	C₂³×C₄×C₅	C₂×C₄	1	3	yes	—
28	C₂³×C₄×C₅	C₂×C₄	1	3	yes	—
29	C₂³×C₄×C₅	C₂³	1	3	yes	—
30	C₂²×C₅×C₈	C₂³	1	6	yes	—
31	C₂²×C₅×C₈	C₂³	1	2	yes	—
32	C₄×C₅×Q₈	—	2	4	no	—
33	C₄×C₅×Q₈	D₄	1	8	yes	—
34	C₄×C₅×D₄	—	2	12	no	—
35	C₄×C₅×D₄	—	2	12	no	—
36	C₅×#117	—	2	12	no	—
37	C₂×C₅×C₄⋊C₄	C₂×C₄	1	3	yes	—
38	C₂×C₅×C₄⋊C₄	C₂×C₄	1	3	yes	—
39	C₂×C₅×C₂²⋊C₄	C₂×C₄	1	6	yes	—
40	C₂×C₄²×C₅	—	2	12	no	—
41	C₂×C₄²×C₅	C₂×C₄	1	3	yes	—
42	C₂×C₄²×C₅	C₂×C₄	1	3	yes	—
43	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₅	C₂³	1	3	yes	—
45	C₅×C₄⋊C₈	—	2	12	no	—
46	C₅×C₂²⋊C₈	—	2	12	no	—
47	C₅×C₂²⋊C₈	—	4	6	no	—
48	C₄×C₅×C₈	—	2	12	no	—
49	C₂×C₄×#41	C₂×C₅	1	1	yes	characteristic
50	C₂×C₈×Q₈	C₂×C₅	1	2	yes	complemented
51	#1954	C₂×C₅	1	6	yes	complemented
52	#1797	C₂×C₅	1	3	yes	complemented
53	C₄×C₂²⋊C₈	C₂×C₅	1	3	yes	complemented
54	C₂⁴×C₅	C₂²×C₄	1	3	yes	—
55	C₅×#41	—	2	32	no	—
56	C₂×C₅×Q₈	C₂²⋊C₄	1	4	yes	—
57	C₂×C₅×Q₈	C₂×C₈	1	4	yes	—
58	C₂×C₅×Q₈	M₁₆	1	4	yes	—
59	C₂×C₅×D₄	—	2	6	no	—
60	C₂×C₅×D₄	—	2	6	no	—
61	C₂×C₅×D₄	C₂×C₈	1	12	yes	—
62	C₂²×C₄×C₅	—	2	6	no	—
63	C₂²×C₄×C₅	—	2	6	no	—
64	C₂²×C₄×C₅	—	2	6	no	—
65	C₂²×C₄×C₅	—	2	6	no	—
66	C₂²×C₄×C₅	—	2	6	no	—
67	C₂²×C₄×C₅	—	2	6	no	—
68	C₂²×C₄×C₅	—	2	3	no	—
69	C₂²×C₄×C₅	—	2	3	no	—
70	C₂²×C₄×C₅	—	4	3	no	—
71	C₂²×C₄×C₅	C₂×C₈	1	12	yes	—
72	C₂²×C₄×C₅	C₂²×C₄	1	3	yes	—
73	C₂²×C₄×C₅	C₂²×C₄	1	3	yes	—
74	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₅	C₂²×C₄	1	3	yes	—
77	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
78	C₂×C₅×C₈	—	2	2	no	—
79	C₂×C₅×C₈	—	2	2	no	—
80	C₂×C₅×C₈	—	2	6	no	—
81	C₂×C₅×C₈	—	2	6	no	—
82	C₂×C₅×C₈	—	2	6	no	—
83	C₂×C₅×C₈	#41	1	4	yes	complemented
84	C₅×C₄⋊C₄	—	2	6	no	—
85	C₅×C₄⋊C₄	C₂²⋊C₄	1	12	yes	—
86	C₅×C₂²⋊C₄	—	2	12	no	—
87	C₄²×C₅	—	2	6	no	—
88	C₄²×C₅	—	2	6	no	—
89	C₄²×C₅	—	2	2	no	—
90	C₄²×C₅	C₂×D₄	1	12	yes	—
91	C₂²×#41	C₄×C₅	1	1	yes	characteristic
92	C₄×#41	—	2	8	no	—
93	C₂×C₄×Q₈	C₄×C₅	1	1	yes	characteristic
94	C₂×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
95	C₂×C₄×D₄	C₄×C₅	1	3	yes	—
96	C₂×C₄×D₄	C₂²×C₅	1	3	yes	—
97	C₂×#117	C₄×C₅	1	3	yes	—
98	C₂²×C₄²	C₂²×C₅	1	3	yes	—
99	C₈×Q₈	—	2	8	no	—
100	C₂×C₄⋊C₈	C₂²×C₅	1	6	yes	—
101	C₂×C₂²⋊C₈	C₂²×C₅	1	6	yes	—
102	C₂×C₂²⋊C₈	C₂²×C₅	1	6	yes	—
103	C₂×C₄×C₈	C₂²×C₅	1	6	yes	—
104	#336	C₂²×C₅	1	3	yes	—
105	#336	C₂²×C₅	1	1	yes	characteristic
106	C₂³×C₅	—	2	3	no	—
107	C₂³×C₅	—	2	3	no	—
108	C₂³×C₅	—	2	3	no	—
109	C₂³×C₅	—	2	3	no	—
110	C₂³×C₅	—	2	3	no	—
111	C₂³×C₅	—	2	3	no	—
112	C₂³×C₅	C₂²×C₈	1	6	yes	—
113	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
114	C₅×Q₈	C₂²⋊C₈	1	16	yes	complemented
115	C₅×D₄	—	2	24	no	—
116	C₂×C₄×C₅	—	2	1	no	—
117	C₂×C₄×C₅	—	2	1	no	—
118	C₂×C₄×C₅	—	2	1	no	—
119	C₂×C₄×C₅	—	2	3	no	—
120	C₂×C₄×C₅	—	2	1	no	—
121	C₂×C₄×C₅	—	2	3	no	—
122	C₂×C₄×C₅	—	2	24	no	—
123	C₂×C₄×C₅	—	2	3	no	—
124	C₂×C₄×C₅	—	2	3	no	—
125	C₂×C₄×C₅	—	2	3	no	—
126	C₂×C₄×C₅	—	4	12	no	—
127	C₂×C₄×C₅	—	4	6	no	—
128	C₂×C₄×C₅	—	4	6	no	—
129	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	6	yes	—
130	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	6	yes	—
131	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
132	C₂×C₄×C₅	C₂²×C₈	1	6	yes	—
133	C₂×C₄×C₅	C₂×M₁₆	1	6	yes	—
134	C₂×C₄×C₅	#131	1	2	yes	—
135	C₂×C₄×C₅	C₂²×D₄	1	2	yes	—
136	C₂×C₄×C₅	C₂×#41	1	2	yes	—
137	C₂×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
138	C₂×C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
139	C₅×C₈	—	2	4	no	—
140	C₅×C₈	—	4	6	no	—
141	C₂×#41	—	2	4	no	—
142	C₂×#41	—	2	4	no	—
143	C₂×#41	C₅×C₈	1	8	yes	complemented
144	C₂²×Q₈	C₂×C₄×C₅	1	1	yes	characteristic
145	C₂²×D₄	C₂×C₄×C₅	1	3	yes	—
146	C₂³×C₄	C₂×C₄×C₅	1	3	yes	—
147	C₂³×C₄	C₂×C₄×C₅	1	3	yes	—
148	C₂³×C₄	C₂³×C₅	1	3	yes	—
149	C₂²×C₈	C₂³×C₅	1	6	yes	—
150	C₂²×C₈	C₂³×C₅	1	2	yes	—
151	C₄×Q₈	—	2	4	no	—
152	C₄×Q₈	C₅×D₄	1	8	yes	—
153	C₄×D₄	—	2	12	no	—
154	C₄×D₄	—	2	12	no	—
155	#117	—	2	12	no	—
156	C₂×C₄⋊C₄	C₂×C₄×C₅	1	3	yes	—
157	C₂×C₄⋊C₄	C₂×C₄×C₅	1	3	yes	—
158	C₂×C₂²⋊C₄	C₂×C₄×C₅	1	6	yes	—
159	C₂×C₄²	—	2	12	no	—
160	C₂×C₄²	C₂×C₄×C₅	1	3	yes	—
161	C₂×C₄²	C₂×C₄×C₅	1	3	yes	—
162	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
163	C₂×C₄²	C₂³×C₅	1	3	yes	—
164	C₄⋊C₈	—	2	12	no	—
165	C₂²⋊C₈	—	2	12	no	—
166	C₂²⋊C₈	—	4	6	no	—
167	C₄×C₈	—	2	12	no	—
168	C₂²×C₅	—	2	6	no	—
169	C₂²×C₅	—	2	6	no	—
170	C₂²×C₅	—	2	6	no	—
171	C₂²×C₅	—	4	3	no	—
172	C₂²×C₅	—	4	3	no	—
173	C₂²×C₅	—	4	3	no	—
174	C₂²×C₅	—	4	3	no	—
175	C₂²×C₅	C₂²×C₂²⋊C₄	1	1	yes	characteristic
176	C₂²×C₅	C₄×#41	1	1	yes	characteristic
177	C₂²×C₅	#520	1	1	yes	characteristic
178	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
179	C₂²×C₅	C₂²×M₁₆	1	1	yes	characteristic
180	C₂²×C₅	C₂×#131	1	1	yes	characteristic
181	C₂²×C₅	#568	1	1	yes	characteristic
182	C₄×C₅	—	2	2	no	—
183	C₄×C₅	—	2	2	no	—
184	C₄×C₅	—	2	6	no	—
185	C₄×C₅	—	4	3	no	—
186	C₄×C₅	C₂×C₂²⋊C₈	1	12	yes	—
187	C₄×C₅	#548	1	2	yes	—
188	C₄×C₅	#550	1	2	yes	—
189	C₂⁴	C₂²×C₄×C₅	1	3	yes	—
190	#41	—	2	32	no	—
191	C₂×Q₈	C₅×C₂²⋊C₄	1	4	yes	—
192	C₂×Q₈	C₂×C₅×C₈	1	4	yes	—
193	C₂×Q₈	C₅×M₁₆	1	4	yes	—
194	C₂×D₄	—	2	6	no	—
195	C₂×D₄	—	2	6	no	—
196	C₂×D₄	C₂×C₅×C₈	1	12	yes	—
197	C₂²×C₄	—	2	6	no	—
198	C₂²×C₄	—	2	6	no	—
199	C₂²×C₄	—	2	6	no	—
200	C₂²×C₄	—	2	6	no	—
201	C₂²×C₄	—	2	6	no	—
202	C₂²×C₄	—	2	6	no	—
203	C₂²×C₄	—	2	3	no	—
204	C₂²×C₄	—	2	3	no	—
205	C₂²×C₄	—	4	3	no	—
206	C₂²×C₄	C₂×C₅×C₈	1	12	yes	—
207	C₂²×C₄	C₂²×C₄×C₅	1	3	yes	—
208	C₂²×C₄	C₂²×C₄×C₅	1	3	yes	—
209	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
210	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
211	C₂²×C₄	C₂²×C₄×C₅	1	3	yes	—
212	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	6	no	—
216	C₂×C₈	—	2	6	no	—
217	C₂×C₈	—	2	6	no	—
218	C₂×C₈	C₅×#41	1	4	yes	complemented
219	C₄⋊C₄	—	2	6	no	—
220	C₄⋊C₄	C₅×C₂²⋊C₄	1	12	yes	—
221	C₂²⋊C₄	—	2	12	no	—
222	C₄²	—	2	6	no	—
223	C₄²	—	2	6	no	—
224	C₄²	—	2	2	no	—
225	C₄²	C₂×C₅×D₄	1	12	yes	—
226	C₂×C₅	—	4	6	no	—
227	C₂×C₅	#2341	1	1	yes	characteristic
228	C₂×C₅	C₂²×C₂²⋊C₈	1	1	yes	characteristic
229	C₂×C₅	#2880	1	1	yes	characteristic
230	C₂×C₅	C₈×#41	1	1	yes	characteristic
231	C₂×C₅	#2971	1	1	yes	characteristic
232	C₂×C₅	#2988	1	1	yes	characteristic
233	C₂×C₅	#2991	1	1	yes	characteristic
234	C₂³	—	2	3	no	—
235	C₂³	—	2	3	no	—
236	C₂³	—	2	3	no	—
237	C₂³	—	2	3	no	—
238	C₂³	—	2	3	no	—
239	C₂³	—	2	3	no	—
240	C₂³	C₂²×C₅×C₈	1	6	yes	—
241	C₂³	C₂³×C₄×C₅	1	1	yes	characteristic
242	Q₈	C₅×C₂²⋊C₈	1	16	yes	complemented
243	D₄	—	2	24	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	3	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	3	no	—
250	C₂×C₄	—	2	24	no	—
251	C₂×C₄	—	2	3	no	—
252	C₂×C₄	—	2	3	no	—
253	C₂×C₄	—	2	3	no	—
254	C₂×C₄	—	4	12	no	—
255	C₂×C₄	—	4	6	no	—
256	C₂×C₄	—	4	6	no	—
257	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	6	yes	—
258	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	6	yes	—
259	C₂×C₄	C₂²×C₅×C₈	1	2	yes	—
260	C₂×C₄	C₂²×C₅×C₈	1	6	yes	—
261	C₂×C₄	C₂×C₅×M₁₆	1	6	yes	—
262	C₂×C₄	C₅×#131	1	2	yes	—
263	C₂×C₄	C₂²×C₅×D₄	1	2	yes	—
264	C₂×C₄	C₂×C₅×#41	1	2	yes	—
265	C₂×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
266	C₂×C₄	C₅×2₋¹⁺⁴	1	1	yes	characteristic
267	C₈	—	2	4	no	—
268	C₈	—	4	6	no	—
269	C₅	#22530	1	1	yes	characteristic, direct product factor, Sylow
270	C₂²	—	2	6	no	—
271	C₂²	—	2	6	no	—
272	C₂²	—	2	6	no	—
273	C₂²	—	4	3	no	—
274	C₂²	—	4	3	no	—
275	C₂²	—	4	3	no	—
276	C₂²	—	4	3	no	—
277	C₂²	C₂²×C₅×C₂²⋊C₄	1	1	yes	characteristic
278	C₂²	C₄×C₅×#41	1	1	yes	characteristic
279	C₂²	C₅×#520	1	1	yes	characteristic
280	C₂²	C₂³×C₅×C₈	1	1	yes	characteristic
281	C₂²	C₂²×C₅×M₁₆	1	1	yes	characteristic
282	C₂²	C₂×C₅×#131	1	1	yes	characteristic
283	C₂²	C₅×#568	1	1	yes	characteristic
284	C₄	—	2	2	no	—
285	C₄	—	2	2	no	—
286	C₄	—	2	6	no	—
287	C₄	—	4	3	no	—
288	C₄	C₂×C₅×C₂²⋊C₈	1	12	yes	—
289	C₄	C₅×#548	1	2	yes	—
290	C₄	C₅×#550	1	2	yes	—
291	C₂	—	4	6	no	—
292	C₂	C₅×#2341	1	1	yes	characteristic
293	C₂	C₂²×C₅×C₂²⋊C₈	1	1	yes	characteristic
294	C₂	C₅×#2880	1	1	yes	characteristic
295	C₂	C₅×C₈×#41	1	1	yes	characteristic
296	C₂	C₅×#2971	1	1	yes	characteristic
297	C₂	C₅×#2988	1	1	yes	characteristic
298	C₂	C₅×#2991	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅	C₂⁴
commutator	C₂²	C₂³×C₅×C₈
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle	C₂³×C₅	C₂³×C₄
prime radical	C₂²×C₅×#41	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×C₈×Q₈	C₂	semidirect
3. C₅×#1954	C₂	semidirect
4. C₅×#1797	C₂	semidirect
5. C₄×C₅×C₂²⋊C₈	C₂	semidirect
21. #22530	C₅	direct (N × H)
24. C₂×C₅×#41	C₈	semidirect
50. C₂×C₈×Q₈	C₂×C₅	semidirect
51. #1954	C₂×C₅	semidirect
52. #1797	C₂×C₅	semidirect
53. C₄×C₂²⋊C₈	C₂×C₅	semidirect
83. C₂×C₅×C₈	#41	semidirect
114. C₅×Q₈	C₂²⋊C₈	semidirect
143. C₂×#41	C₅×C₈	semidirect
218. C₂×C₈	C₅×#41	semidirect
242. Q₈	C₅×C₂²⋊C₈	semidirect
269. C₅	#22530	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×C₄×C₅×#41	2
C₂×C₅×C₈×Q₈	2
C₅×#1954	2
C₅×#1797	2
C₄×C₅×C₂²⋊C₈	2
#22530	5

Subgroups — C5×#22530 #11945250

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#22530 #11945250