Subgroups — C₅×#8589 #11931309

Order 1280.

496 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₅×#1842	C₂	1	1	yes	maximal, characteristic
3	C₅×#1831	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅×#332	C₂	1	2	yes	maximal
5	C₅×#1296	C₂	1	2	yes	maximal
6	C₂×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₅×Q₈	—	2	1	no	—
8	C₂×C₅×C₄⋊C₈	—	2	1	no	—
9	C₂×C₅×C₄⋊C₈	—	2	1	no	—
10	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
11	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
12	C₅×#391	—	2	1	no	—
13	C₅×#389	—	2	1	no	—
14	C₅×#389	C₄	1	2	yes	—
15	C₅×#384	C₂²	1	1	yes	characteristic
16	C₄×C₅×C₄⋊C₄	—	2	1	no	—
17	C₄×C₅×C₄⋊C₄	C₄	1	2	yes	—
18	C₄³×C₅	C₂²	1	1	yes	characteristic
19	C₅×#336	—	2	1	no	—
20	C₅×#336	—	2	1	no	—
21	C₅×#332	—	2	4	no	—
22	#8589	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
23	C₂²×C₅×Q₈	—	2	1	no	—
24	C₂²×C₅×C₈	—	2	2	no	—
25	C₂²×C₅×C₈	—	2	2	no	—
26	C₅×C₄⋊Q₈	—	2	1	no	—
27	C₅×C₄⋊Q₈	—	2	1	no	—
28	C₅×C₄⋊Q₈	D₄	1	2	yes	—
29	C₅×C₄⋊Q₈	Q₈	1	2	yes	—
30	C₄×C₅×Q₈	—	4	2	no	—
31	C₄×C₅×Q₈	—	4	2	no	—
32	C₂×C₅×C₄⋊C₄	—	2	1	no	—
33	C₂×C₅×C₄⋊C₄	—	2	2	no	—
34	C₂×C₅×C₄⋊C₄	—	2	1	no	—
35	C₂×C₅×C₄⋊C₄	—	2	1	no	—
36	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
37	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₅	—	2	2	no	—
39	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
46	C₅×C₄⋊C₈	—	2	2	no	—
47	C₅×C₄⋊C₈	—	2	2	no	—
48	C₅×C₄⋊C₈	—	2	2	no	—
49	C₅×C₄⋊C₈	—	2	2	no	—
50	C₅×C₄⋊C₈	—	4	2	no	—
51	C₅×C₄⋊C₈	—	4	2	no	—
52	C₅×#95	—	2	2	no	—
53	C₅×#95	—	2	2	no	—
54	C₄×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic
55	#1842	C₂×C₅	1	1	yes	characteristic
56	#1831	C₂×C₅	1	1	yes	characteristic
57	C₂×#332	C₂×C₅	1	2	yes	—
58	#1296	C₂×C₅	1	2	yes	—
59	C₂×C₅×Q₈	—	2	2	no	—
60	C₂×C₅×Q₈	—	2	2	no	—
61	C₂×C₅×Q₈	—	4	1	no	—
62	C₂×C₅×Q₈	—	4	1	no	—
63	C₂×C₅×Q₈	—	4	1	no	—
64	C₂×C₅×Q₈	—	4	1	no	—
65	C₂²×C₄×C₅	—	2	2	no	—
66	C₂²×C₄×C₅	—	2	2	no	—
67	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
74	C₂×C₅×C₈	—	4	2	no	—
75	C₂×C₅×C₈	—	4	2	no	—
76	C₂×C₅×C₈	—	4	2	no	—
77	C₂×C₅×C₈	—	4	2	no	—
78	C₂×C₅×C₈	—	4	2	no	—
79	C₂×C₅×C₈	—	4	2	no	—
80	C₅×C₄⋊C₄	—	2	2	no	—
81	C₅×C₄⋊C₄	—	2	2	no	—
82	C₅×C₄⋊C₄	—	2	2	no	—
83	C₅×C₄⋊C₄	—	4	2	no	—
84	C₅×C₄⋊C₄	—	4	4	no	—
85	C₅×C₄⋊C₄	—	4	1	no	—
86	C₅×C₄⋊C₄	—	4	1	no	—
87	C₅×C₄⋊C₄	—	4	2	no	—
88	C₅×C₄⋊C₄	C₄⋊C₄	1	4	yes	—
89	C₄²×C₅	—	2	2	no	—
90	C₄²×C₅	—	2	2	no	—
91	C₄²×C₅	—	2	1	no	—
92	C₄²×C₅	—	2	1	no	—
93	C₄²×C₅	—	2	2	no	—
94	C₄²×C₅	—	4	1	no	—
95	C₄²×C₅	—	4	4	no	—
96	C₄²×C₅	D₈	1	2	yes	—
97	C₄²×C₅	SD₁₆	1	2	yes	—
98	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
100	C₄²×C₅	#41	1	1	yes	characteristic
101	C₄²×C₅	#41	1	1	yes	characteristic
102	C₂×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
103	C₂×C₄×Q₈	—	2	1	no	—
104	C₂×C₄⋊C₈	—	2	1	no	—
105	C₂×C₄⋊C₈	—	2	1	no	—
106	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
107	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
108	#391	—	2	1	no	—
109	#389	—	2	1	no	—
110	#389	C₄×C₅	1	2	yes	—
111	#384	C₂²×C₅	1	1	yes	characteristic
112	C₄×C₄⋊C₄	—	2	1	no	—
113	C₄×C₄⋊C₄	C₄×C₅	1	2	yes	—
114	C₄³	C₂²×C₅	1	1	yes	characteristic
115	#336	—	2	1	no	—
116	#336	—	2	1	no	—
117	#332	—	2	4	no	—
118	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₅×Q₈	—	4	2	no	—
120	C₅×Q₈	—	4	2	no	—
121	C₅×Q₈	—	4	2	no	—
122	C₅×Q₈	—	4	2	no	—
123	C₂×C₄×C₅	—	2	1	no	—
124	C₂×C₄×C₅	—	2	1	no	—
125	C₂×C₄×C₅	—	2	1	no	—
126	C₂×C₄×C₅	—	2	1	no	—
127	C₂×C₄×C₅	—	2	1	no	—
128	C₂×C₄×C₅	—	2	1	no	—
129	C₂×C₄×C₅	—	2	1	no	—
130	C₂×C₄×C₅	—	2	1	no	—
131	C₂×C₄×C₅	—	2	4	no	—
132	C₂×C₄×C₅	—	2	1	no	—
133	C₂×C₄×C₅	—	2	1	no	—
134	C₂×C₄×C₅	—	2	1	no	—
135	C₂×C₄×C₅	—	4	2	no	—
136	C₂×C₄×C₅	—	4	2	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	#101	1	2	yes	—
141	C₂×C₄×C₅	#102	1	2	yes	—
142	C₂×C₄×C₅	#102	1	2	yes	—
143	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
144	C₂×C₄×C₅	#117	1	1	yes	characteristic
145	C₂×C₄×C₅	#121	1	1	yes	characteristic
146	C₂×C₄×C₅	#121	1	1	yes	characteristic
147	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
148	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
149	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
150	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
151	C₂×C₄×C₅	#135	1	2	yes	—
152	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
153	C₂×C₄×C₅	#137	1	1	yes	characteristic
154	C₂×C₄×C₅	#137	1	2	yes	—
155	C₅×C₈	—	8	2	no	—
156	C₅×C₈	—	8	2	no	—
157	C₂²×Q₈	—	2	1	no	—
158	C₂²×C₈	—	2	2	no	—
159	C₂²×C₈	—	2	2	no	—
160	C₄⋊Q₈	—	2	1	no	—
161	C₄⋊Q₈	—	2	1	no	—
162	C₄⋊Q₈	C₅×D₄	1	2	yes	—
163	C₄⋊Q₈	C₅×Q₈	1	2	yes	—
164	C₄×Q₈	—	4	2	no	—
165	C₄×Q₈	—	4	2	no	—
166	C₂×C₄⋊C₄	—	2	1	no	—
167	C₂×C₄⋊C₄	—	2	2	no	—
168	C₂×C₄⋊C₄	—	2	1	no	—
169	C₂×C₄⋊C₄	—	2	1	no	—
170	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
171	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
172	C₂×C₄²	—	2	2	no	—
173	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
174	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
175	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
176	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
177	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
178	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
179	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
180	C₄⋊C₈	—	2	2	no	—
181	C₄⋊C₈	—	2	2	no	—
182	C₄⋊C₈	—	2	2	no	—
183	C₄⋊C₈	—	2	2	no	—
184	C₄⋊C₈	—	4	2	no	—
185	C₄⋊C₈	—	4	2	no	—
186	#95	—	2	2	no	—
187	#95	—	2	2	no	—
188	C₂²×C₅	#380	1	1	yes	characteristic
189	C₂²×C₅	C₂×#101	1	1	yes	characteristic
190	C₂²×C₅	#413	1	1	yes	characteristic
191	C₂²×C₅	C₂×#102	1	1	yes	characteristic
192	C₂²×C₅	#416	1	1	yes	characteristic
193	C₂²×C₅	#417	1	1	yes	characteristic
194	C₂²×C₅	#419	1	1	yes	characteristic
195	C₄×C₅	—	2	2	no	—
196	C₄×C₅	—	2	1	no	—
197	C₄×C₅	—	2	1	no	—
198	C₄×C₅	—	2	2	no	—
199	C₄×C₅	—	2	2	no	—
200	C₄×C₅	—	4	4	no	—
201	C₄×C₅	—	4	1	no	—
202	C₄×C₅	—	8	2	no	—
203	C₄×C₅	#459	1	1	yes	characteristic
204	C₄×C₅	#461	1	1	yes	characteristic
205	C₄×C₅	#464	1	2	yes	—
206	C₄×C₅	#474	1	1	yes	characteristic
207	C₄×C₅	#476	1	1	yes	characteristic
208	C₄×C₅	#479	1	2	yes	—
209	C₂×Q₈	—	2	2	no	—
210	C₂×Q₈	—	2	2	no	—
211	C₂×Q₈	—	4	1	no	—
212	C₂×Q₈	—	4	1	no	—
213	C₂×Q₈	—	4	1	no	—
214	C₂×Q₈	—	4	1	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
218	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
219	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
220	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
221	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
222	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
223	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
224	C₂×C₈	—	4	2	no	—
225	C₂×C₈	—	4	2	no	—
226	C₂×C₈	—	4	2	no	—
227	C₂×C₈	—	4	2	no	—
228	C₂×C₈	—	4	2	no	—
229	C₂×C₈	—	4	2	no	—
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	—	4	2	no	—
234	C₄⋊C₄	—	4	1	no	—
235	C₄⋊C₄	—	4	1	no	—
236	C₄⋊C₄	—	4	2	no	—
237	C₄⋊C₄	—	4	4	no	—
238	C₄⋊C₄	C₅×C₄⋊C₄	1	4	yes	—
239	C₄²	—	2	2	no	—
240	C₄²	—	2	1	no	—
241	C₄²	—	2	1	no	—
242	C₄²	—	2	2	no	—
243	C₄²	—	2	2	no	—
244	C₄²	—	4	1	no	—
245	C₄²	—	4	4	no	—
246	C₄²	C₅×D₈	1	2	yes	—
247	C₄²	C₅×SD₁₆	1	2	yes	—
248	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
249	C₄²	C₂×C₅×Q₈	1	1	yes	characteristic
250	C₄²	C₅×#41	1	1	yes	characteristic
251	C₄²	C₅×#41	1	1	yes	characteristic
252	C₂×C₅	#1540	1	1	yes	characteristic
253	C₂×C₅	#1546	1	1	yes	characteristic
254	C₂×C₅	#1552	1	1	yes	characteristic
255	C₂×C₅	#1559	1	1	yes	characteristic
256	C₂×C₅	#1801	1	1	yes	characteristic
257	C₂×C₅	#1802	1	1	yes	characteristic
258	C₂×C₅	#1805	1	1	yes	characteristic
259	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
260	Q₈	—	4	2	no	—
261	Q₈	—	4	2	no	—
262	Q₈	—	4	2	no	—
263	Q₈	—	4	2	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	C₅×#101	1	2	yes	—
282	C₂×C₄	C₅×#102	1	2	yes	—
283	C₂×C₄	C₅×#102	1	2	yes	—
284	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
285	C₂×C₄	C₅×#117	1	1	yes	characteristic
286	C₂×C₄	C₅×#121	1	1	yes	characteristic
287	C₂×C₄	C₅×#121	1	1	yes	characteristic
288	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
289	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
290	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
291	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
292	C₂×C₄	C₅×#135	1	2	yes	—
293	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
294	C₂×C₄	C₅×#137	1	1	yes	characteristic
295	C₂×C₄	C₅×#137	1	2	yes	—
296	C₈	—	8	2	no	—
297	C₈	—	8	2	no	—
298	C₅	#8589	1	1	yes	characteristic, direct product factor, Sylow
299	C₂²	C₅×#380	1	1	yes	characteristic
300	C₂²	C₂×C₅×#101	1	1	yes	characteristic
301	C₂²	C₅×#413	1	1	yes	characteristic
302	C₂²	C₂×C₅×#102	1	1	yes	characteristic
303	C₂²	C₅×#416	1	1	yes	characteristic
304	C₂²	C₅×#417	1	1	yes	characteristic
305	C₂²	C₅×#419	1	1	yes	characteristic
306	C₄	—	2	2	no	—
307	C₄	—	2	1	no	—
308	C₄	—	2	1	no	—
309	C₄	—	2	2	no	—
310	C₄	—	2	2	no	—
311	C₄	—	4	4	no	—
312	C₄	—	4	1	no	—
313	C₄	—	8	2	no	—
314	C₄	C₅×#459	1	1	yes	characteristic
315	C₄	C₅×#461	1	1	yes	characteristic
316	C₄	C₅×#464	1	2	yes	—
317	C₄	C₅×#474	1	1	yes	characteristic
318	C₄	C₅×#476	1	1	yes	characteristic
319	C₄	C₅×#479	1	2	yes	—
320	C₂	C₅×#1540	1	1	yes	characteristic
321	C₂	C₅×#1546	1	1	yes	characteristic
322	C₂	C₅×#1552	1	1	yes	characteristic
323	C₂	C₅×#1559	1	1	yes	characteristic
324	C₂	C₅×#1801	1	1	yes	characteristic
325	C₂	C₅×#1802	1	1	yes	characteristic
326	C₂	C₅×#1805	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₅	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₄³×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
22. #8589	C₅	direct (N × H)
298. C₅	#8589	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₅×C₄⋊Q₈	2
C₅×#1842	2
C₅×#1831	2
C₂×C₅×#332	2
C₅×#1296	2
#8589	5

Subgroups — C5×#8589 #11931309

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8589 #11931309