Subgroups — C₅×#8258 #11930978

Order 1280.

406 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	C₅×#2960	C₂	1	1	yes	maximal, characteristic
2	C₅×C₈×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₅×#1542	C₂	1	1	yes	maximal, characteristic
4	C₅×#1503	C₂	1	1	yes	maximal, characteristic
5	C₅×#1452	C₂	1	1	yes	maximal, characteristic
6	C₅×#1303	C₂	1	1	yes	maximal, characteristic
7	C₅×#1301	C₂	1	1	yes	maximal, characteristic
8	C₅×#533	C₂²	1	1	yes	characteristic
9	C₅×#446	—	2	1	no	—
10	C₅×#446	C₄	1	2	yes	complemented
11	C₅×C₈×Q₈	—	2	1	no	—
12	C₅×C₈×Q₈	C₄	1	2	yes	complemented
13	C₅×#431	C₂²	1	1	yes	characteristic
14	C₅×#431	C₂²	1	1	yes	characteristic
15	C₅×#424	—	2	1	no	—
16	C₅×#423	C₂²	1	1	yes	characteristic
17	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₅×#405	—	2	1	no	—
19	C₂×C₄×C₅×C₈	—	2	1	no	—
20	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
21	C₄×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₅×#336	—	2	1	no	—
23	C₅×#333	C₄	1	2	yes	complemented
24	C₅×#332	C₄	1	2	yes	complemented
25	C₅×#328	C₄	1	4	yes	—
26	C₅×#322	—	2	1	no	—
27	C₅×#322	—	2	1	no	—
28	#8258	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
29	C₂×C₅×M₁₆	—	2	1	no	—
30	C₂²×C₅×C₈	—	2	1	no	—
31	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
32	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
33	C₅×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
34	C₅×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
35	C₅×#125	—	2	1	no	—
36	C₅×C₂²⋊Q₈	—	2	1	no	—
37	C₅×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
38	C₄×C₅×Q₈	—	2	1	no	—
39	C₄×C₅×Q₈	C₂×C₄	1	2	yes	—
40	C₅×#117	—	2	1	no	—
41	C₂×C₅×C₄⋊C₄	—	2	1	no	—
42	C₂×C₄²×C₅	—	2	1	no	—
43	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
44	C₅×C₄⋊C₈	—	2	2	no	—
45	C₅×C₄⋊C₈	—	2	2	no	—
46	C₅×C₄⋊C₈	—	2	1	no	—
47	C₅×C₄⋊C₈	—	2	1	no	—
48	C₅×C₄⋊C₈	—	2	1	no	—
49	C₅×C₄⋊C₈	C₂×C₄	1	2	yes	—
50	C₅×C₄⋊C₈	C₂×C₄	1	2	yes	—
51	C₅×C₄⋊C₈	C₂×C₄	1	2	yes	—
52	C₅×C₂²⋊C₈	—	2	1	no	—
53	C₅×C₂²⋊C₈	C₂×C₄	1	2	yes	—
54	C₅×#97	—	2	2	no	—
55	C₄×C₅×C₈	—	2	2	no	—
56	C₄×C₅×C₈	—	2	1	no	—
57	C₄×C₅×C₈	—	2	1	no	—
58	C₄×C₅×C₈	—	4	2	no	—
59	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
60	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
61	C₄×C₅×C₈	D₄	1	1	yes	characteristic
62	C₄×C₅×C₈	D₄	1	1	yes	characteristic
63	C₅×#95	C₂×C₄	1	2	yes	—
64	#2960	C₂×C₅	1	1	yes	characteristic
65	C₈×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic
66	#1542	C₂×C₅	1	1	yes	characteristic
67	#1503	C₂×C₅	1	1	yes	characteristic
68	#1452	C₂×C₅	1	1	yes	characteristic
69	#1303	C₂×C₅	1	1	yes	characteristic
70	#1301	C₂×C₅	1	1	yes	characteristic
71	C₂×C₅×Q₈	—	2	1	no	—
72	C₂×C₅×Q₈	C₄²	1	2	yes	—
73	C₂²×C₄×C₅	—	2	2	no	—
74	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₅	#41	1	1	yes	characteristic
77	C₅×M₁₆	—	4	1	no	—
78	C₅×M₁₆	—	4	1	no	—
79	C₂×C₅×C₈	—	2	1	no	—
80	C₂×C₅×C₈	—	2	1	no	—
81	C₂×C₅×C₈	—	2	1	no	—
82	C₂×C₅×C₈	—	2	1	no	—
83	C₂×C₅×C₈	—	2	2	no	—
84	C₂×C₅×C₈	—	2	2	no	—
85	C₂×C₅×C₈	—	2	2	no	—
86	C₂×C₅×C₈	—	2	2	no	—
87	C₂×C₅×C₈	—	4	1	no	—
88	C₂×C₅×C₈	—	4	1	no	—
89	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
93	C₅×C₄⋊C₄	—	2	4	no	—
94	C₅×C₄⋊C₄	—	2	2	no	—
95	C₅×C₄⋊C₄	—	2	1	no	—
96	C₅×C₄⋊C₄	—	2	1	no	—
97	C₅×C₄⋊C₄	—	2	1	no	—
98	C₅×C₄⋊C₄	—	2	2	no	—
99	C₅×C₄⋊C₄	C₄²	1	2	yes	—
100	C₅×C₂²⋊C₄	—	2	2	no	—
101	C₄²×C₅	—	2	2	no	—
102	C₄²×C₅	—	4	1	no	—
103	C₄²×C₅	—	4	1	no	—
104	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
105	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
106	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₅	#41	1	1	yes	characteristic
108	#533	C₂²×C₅	1	1	yes	characteristic
109	#446	—	2	1	no	—
110	#446	C₄×C₅	1	2	yes	complemented
111	C₈×Q₈	—	2	1	no	—
112	C₈×Q₈	C₄×C₅	1	2	yes	complemented
113	#431	C₂²×C₅	1	1	yes	characteristic
114	#431	C₂²×C₅	1	1	yes	characteristic
115	#424	—	2	1	no	—
116	#423	C₂²×C₅	1	1	yes	characteristic
117	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
118	#405	—	2	1	no	—
119	C₂×C₄×C₈	—	2	1	no	—
120	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
121	C₄×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
122	#336	—	2	1	no	—
123	#333	C₄×C₅	1	2	yes	complemented
124	#332	C₄×C₅	1	2	yes	complemented
125	#328	C₄×C₅	1	4	yes	—
126	#322	—	2	1	no	—
127	#322	—	2	1	no	—
128	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
129	C₅×Q₈	—	4	2	no	—
130	C₅×Q₈	—	4	2	no	—
131	C₂×C₄×C₅	—	2	2	no	—
132	C₂×C₄×C₅	—	2	4	no	—
133	C₂×C₄×C₅	—	2	2	no	—
134	C₂×C₄×C₅	—	2	2	no	—
135	C₂×C₄×C₅	—	4	1	no	—
136	C₂×C₄×C₅	—	4	2	no	—
137	C₂×C₄×C₅	—	4	2	no	—
138	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
139	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
140	C₂×C₄×C₅	#117	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	C₄×D₄	1	1	yes	characteristic
144	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
145	C₂×C₄×C₅	#137	1	1	yes	characteristic
146	C₂×C₄×C₅	#137	1	1	yes	characteristic
147	C₂×C₄×C₅	#137	1	1	yes	characteristic
148	C₅×C₈	—	2	1	no	—
149	C₅×C₈	—	2	1	no	—
150	C₅×C₈	—	2	1	no	—
151	C₅×C₈	—	2	1	no	—
152	C₅×C₈	—	4	2	no	—
153	C₅×C₈	—	4	2	no	—
154	C₅×C₈	—	4	2	no	—
155	C₅×C₈	—	4	2	no	—
156	C₂×M₁₆	—	2	1	no	—
157	C₂²×C₈	—	2	1	no	—
158	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
159	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
160	C₄⋊Q₈	C₂×C₄×C₅	1	1	yes	characteristic
161	C₄⋊Q₈	C₂×C₄×C₅	1	1	yes	characteristic
162	#125	—	2	1	no	—
163	C₂²⋊Q₈	—	2	1	no	—
164	C₂²⋊Q₈	C₂×C₄×C₅	1	2	yes	—
165	C₄×Q₈	—	2	1	no	—
166	C₄×Q₈	C₂×C₄×C₅	1	2	yes	—
167	#117	—	2	1	no	—
168	C₂×C₄⋊C₄	—	2	1	no	—
169	C₂×C₄²	—	2	1	no	—
170	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
171	C₄⋊C₈	—	2	2	no	—
172	C₄⋊C₈	—	2	2	no	—
173	C₄⋊C₈	—	2	1	no	—
174	C₄⋊C₈	—	2	1	no	—
175	C₄⋊C₈	—	2	1	no	—
176	C₄⋊C₈	C₂×C₄×C₅	1	2	yes	—
177	C₄⋊C₈	C₂×C₄×C₅	1	2	yes	—
178	C₄⋊C₈	C₂×C₄×C₅	1	2	yes	—
179	C₂²⋊C₈	—	2	1	no	—
180	C₂²⋊C₈	C₂×C₄×C₅	1	2	yes	—
181	#97	—	2	2	no	—
182	C₄×C₈	—	2	2	no	—
183	C₄×C₈	—	2	1	no	—
184	C₄×C₈	—	2	1	no	—
185	C₄×C₈	—	4	2	no	—
186	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
187	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
188	C₄×C₈	C₅×D₄	1	1	yes	characteristic
189	C₄×C₈	C₅×D₄	1	1	yes	characteristic
190	#95	C₂×C₄×C₅	1	2	yes	—
191	C₂²×C₅	—	2	1	no	—
192	C₂²×C₅	—	2	1	no	—
193	C₂²×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
194	C₂²×C₅	#440	1	1	yes	characteristic
195	C₂²×C₅	#441	1	1	yes	characteristic
196	C₄×C₅	—	2	1	no	—
197	C₄×C₅	—	2	1	no	—
198	C₄×C₅	—	2	1	no	—
199	C₄×C₅	—	4	4	no	—
200	C₄×C₅	—	4	2	no	—
201	C₄×C₅	—	4	2	no	—
202	C₄×C₅	#417	1	1	yes	characteristic
203	C₄×C₅	#419	1	1	yes	characteristic
204	C₄×C₅	#440	1	1	yes	characteristic
205	C₄×C₅	#441	1	1	yes	characteristic
206	C₄×C₅	#443	1	2	yes	—
207	C₂×Q₈	—	2	1	no	—
208	C₂×Q₈	C₄²×C₅	1	2	yes	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
211	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
212	C₂²×C₄	C₅×#41	1	1	yes	characteristic
213	M₁₆	—	4	1	no	—
214	M₁₆	—	4	1	no	—
215	C₂×C₈	—	2	1	no	—
216	C₂×C₈	—	2	1	no	—
217	C₂×C₈	—	2	1	no	—
218	C₂×C₈	—	2	1	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	4	1	no	—
224	C₂×C₈	—	4	1	no	—
225	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
226	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
227	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
228	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
229	C₄⋊C₄	—	2	4	no	—
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	—	2	1	no	—
232	C₄⋊C₄	—	2	1	no	—
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	C₄²×C₅	1	2	yes	—
236	C₂²⋊C₄	—	2	2	no	—
237	C₄²	—	2	2	no	—
238	C₄²	—	4	1	no	—
239	C₄²	—	4	1	no	—
240	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
241	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
242	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
243	C₄²	C₅×#41	1	1	yes	characteristic
244	C₂×C₅	—	2	1	no	—
245	C₂×C₅	—	2	1	no	—
246	C₂×C₅	#1757	1	1	yes	characteristic
247	C₂×C₅	#1763	1	1	yes	characteristic
248	C₂×C₅	#1764	1	1	yes	characteristic
249	C₂³	C₄×C₅×D₄	1	1	yes	characteristic
250	Q₈	—	4	2	no	—
251	Q₈	—	4	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	4	1	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	C₂×C₄²×C₅	1	1	yes	characteristic
260	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
261	C₂×C₄	C₅×#117	1	1	yes	characteristic
262	C₂×C₄	C₄×C₅×D₄	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	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
266	C₂×C₄	C₅×#137	1	1	yes	characteristic
267	C₂×C₄	C₅×#137	1	1	yes	characteristic
268	C₂×C₄	C₅×#137	1	1	yes	characteristic
269	C₈	—	2	1	no	—
270	C₈	—	2	1	no	—
271	C₈	—	2	1	no	—
272	C₈	—	2	1	no	—
273	C₈	—	4	2	no	—
274	C₈	—	4	2	no	—
275	C₈	—	4	2	no	—
276	C₈	—	4	2	no	—
277	C₅	#8258	1	1	yes	characteristic, direct product factor, Sylow
278	C₂²	—	2	1	no	—
279	C₂²	—	2	1	no	—
280	C₂²	C₄×C₅×C₂²⋊C₄	1	1	yes	characteristic
281	C₂²	C₅×#440	1	1	yes	characteristic
282	C₂²	C₅×#441	1	1	yes	characteristic
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	4	4	no	—
287	C₄	—	4	2	no	—
288	C₄	—	4	2	no	—
289	C₄	C₅×#417	1	1	yes	characteristic
290	C₄	C₅×#419	1	1	yes	characteristic
291	C₄	C₅×#440	1	1	yes	characteristic
292	C₄	C₅×#441	1	1	yes	characteristic
293	C₄	C₅×#443	1	2	yes	—
294	C₂	—	2	1	no	—
295	C₂	—	2	1	no	—
296	C₂	C₅×#1757	1	1	yes	characteristic
297	C₂	C₅×#1763	1	1	yes	characteristic
298	C₂	C₅×#1764	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄²×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
socle	C₂²×C₅	C₄×C₂²⋊C₄
prime radical	C₂³×C₅	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
10. C₅×#446	C₄	semidirect
12. C₅×C₈×Q₈	C₄	semidirect
23. C₅×#333	C₄	semidirect
24. C₅×#332	C₄	semidirect
28. #8258	C₅	direct (N × H)
110. #446	C₄×C₅	semidirect
112. C₈×Q₈	C₄×C₅	semidirect
123. #333	C₄×C₅	semidirect
124. #332	C₄×C₅	semidirect
277. C₅	#8258	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×#2960	2
C₅×C₈×C₄⋊C₄	2
C₅×#1542	2
C₅×#1503	2
C₅×#1452	2
C₅×#1303	2
C₅×#1301	2
#8258	5

Subgroups — C5×#8258 #11930978

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8258 #11930978