Subgroups — C₅×#9870 #11932590

Order 1280.

566 conjugacy classes of subgroups.

The subgroup lattice has 302 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	C₄²×C₅×C₈	C₂	1	1	yes	maximal, characteristic
4	C₅×#1279	C₂	1	4	yes	maximal
5	C₂×C₅×C₄⋊Q₈	C₄	1	1	yes	characteristic
6	C₂×C₄×C₅×Q₈	—	2	1	no	—
7	C₂×C₅×C₄⋊C₈	—	2	1	no	—
8	C₂×C₅×C₄⋊C₈	—	2	1	no	—
9	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
10	C₂×C₄×C₅×C₈	—	2	1	no	—
11	C₂×C₄×C₅×C₈	—	2	1	no	—
12	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
13	C₅×#391	—	2	1	no	—
14	C₅×#389	—	2	1	no	—
15	C₅×#389	C₄	1	2	yes	—
16	C₅×#384	C₄	1	1	yes	characteristic
17	C₄×C₅×C₄⋊C₄	—	2	1	no	—
18	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
19	C₄³×C₅	C₂²	1	1	yes	characteristic
20	#9870	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂²×C₅×Q₈	—	2	1	no	—
22	C₂²×C₅×C₈	—	2	2	no	—
23	C₂²×C₅×C₈	—	2	2	no	—
24	C₅×C₄⋊Q₈	—	2	2	no	—
25	C₅×C₄⋊Q₈	C₈	1	4	yes	complemented
26	C₄×C₅×Q₈	—	4	2	no	—
27	C₄×C₅×Q₈	—	4	2	no	—
28	C₂×C₅×C₄⋊C₄	—	2	1	no	—
29	C₂×C₅×C₄⋊C₄	—	2	1	no	—
30	C₂×C₅×C₄⋊C₄	—	2	1	no	—
31	C₂×C₅×C₄⋊C₄	—	2	2	no	—
32	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
33	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₅	—	2	2	no	—
35	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
42	C₅×C₄⋊C₈	—	2	4	no	—
43	C₅×C₄⋊C₈	—	4	2	no	—
44	C₅×C₄⋊C₈	—	4	2	no	—
45	C₄×C₅×C₈	—	2	4	no	—
46	C₄×C₅×C₈	—	2	4	no	—
47	C₄×C₅×C₈	—	2	2	no	—
48	C₄×C₅×C₈	Q₈	1	4	yes	complemented
49	C₅×#95	—	2	2	no	—
50	C₅×#95	—	2	2	no	—
51	C₄×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic
52	#1842	C₂×C₅	1	1	yes	characteristic
53	C₄²×C₈	C₂×C₅	1	1	yes	characteristic
54	#1279	C₂×C₅	1	4	yes	—
55	C₂×C₅×Q₈	—	2	4	no	—
56	C₂×C₅×Q₈	—	4	2	no	—
57	C₂×C₅×Q₈	—	4	2	no	—
58	C₂²×C₄×C₅	—	2	2	no	—
59	C₂²×C₄×C₅	—	2	2	no	—
60	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
67	C₂×C₅×C₈	—	2	4	no	—
68	C₂×C₅×C₈	—	2	4	no	—
69	C₂×C₅×C₈	—	2	4	no	—
70	C₂×C₅×C₈	—	4	2	no	—
71	C₂×C₅×C₈	—	4	2	no	—
72	C₂×C₅×C₈	—	4	2	no	—
73	C₅×C₄⋊C₄	—	2	4	no	—
74	C₅×C₄⋊C₄	—	2	2	no	—
75	C₅×C₄⋊C₄	—	2	4	no	—
76	C₅×C₄⋊C₄	—	4	4	no	—
77	C₅×C₄⋊C₄	—	4	2	no	—
78	C₅×C₄⋊C₄	—	4	2	no	—
79	C₅×C₄⋊C₄	C₂×C₈	1	4	yes	—
80	C₄²×C₅	—	2	2	no	—
81	C₄²×C₅	—	2	2	no	—
82	C₄²×C₅	—	2	2	no	—
83	C₄²×C₅	—	4	2	no	—
84	C₄²×C₅	—	4	2	no	—
85	C₄²×C₅	C₂×C₈	1	2	yes	—
86	C₄²×C₅	M₁₆	1	2	yes	—
87	C₄²×C₅	D₈	1	2	yes	—
88	C₄²×C₅	SD₁₆	1	2	yes	—
89	C₄²×C₅	SD₁₆	1	2	yes	—
90	C₄²×C₅	Q₁₆	1	2	yes	—
91	C₄²×C₅	C₂×Q₈	1	2	yes	—
92	C₄²×C₅	#41	1	2	yes	—
93	C₂×C₄⋊Q₈	C₄×C₅	1	1	yes	characteristic
94	C₂×C₄×Q₈	—	2	1	no	—
95	C₂×C₄⋊C₈	—	2	1	no	—
96	C₂×C₄⋊C₈	—	2	1	no	—
97	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
98	C₂×C₄×C₈	—	2	1	no	—
99	C₂×C₄×C₈	—	2	1	no	—
100	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
101	#391	—	2	1	no	—
102	#389	—	2	1	no	—
103	#389	C₄×C₅	1	2	yes	—
104	#384	C₄×C₅	1	1	yes	characteristic
105	C₄×C₄⋊C₄	—	2	1	no	—
106	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
107	C₄³	C₂²×C₅	1	1	yes	characteristic
108	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₅×Q₈	—	4	8	no	—
110	C₂×C₄×C₅	—	2	1	no	—
111	C₂×C₄×C₅	—	2	1	no	—
112	C₂×C₄×C₅	—	2	1	no	—
113	C₂×C₄×C₅	—	2	1	no	—
114	C₂×C₄×C₅	—	2	1	no	—
115	C₂×C₄×C₅	—	2	1	no	—
116	C₂×C₄×C₅	—	2	4	no	—
117	C₂×C₄×C₅	—	4	2	no	—
118	C₂×C₄×C₅	—	4	2	no	—
119	C₂×C₄×C₅	—	4	2	no	—
120	C₂×C₄×C₅	—	4	2	no	—
121	C₂×C₄×C₅	—	4	2	no	—
122	C₂×C₄×C₅	C₂²⋊C₈	1	2	yes	—
123	C₂×C₄×C₅	C₂²⋊C₈	1	2	yes	—
124	C₂×C₄×C₅	#102	1	2	yes	—
125	C₂×C₄×C₅	#102	1	2	yes	—
126	C₂×C₄×C₅	#103	1	2	yes	—
127	C₂×C₄×C₅	#103	1	2	yes	—
128	C₂×C₄×C₅	#104	1	2	yes	—
129	C₂×C₄×C₅	#104	1	2	yes	—
130	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
131	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
132	C₂×C₄×C₅	#124	1	1	yes	characteristic
133	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
134	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
135	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
136	C₂×C₄×C₅	#131	1	2	yes	—
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×SD₁₆	1	1	yes	characteristic
140	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
141	C₅×C₈	—	2	8	no	—
142	C₅×C₈	—	8	2	no	—
143	C₂²×Q₈	—	2	1	no	—
144	C₂²×C₈	—	2	2	no	—
145	C₂²×C₈	—	2	2	no	—
146	C₄⋊Q₈	—	2	2	no	—
147	C₄⋊Q₈	C₅×C₈	1	4	yes	complemented
148	C₄×Q₈	—	4	2	no	—
149	C₄×Q₈	—	4	2	no	—
150	C₂×C₄⋊C₄	—	2	1	no	—
151	C₂×C₄⋊C₄	—	2	1	no	—
152	C₂×C₄⋊C₄	—	2	2	no	—
153	C₂×C₄⋊C₄	—	2	1	no	—
154	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
155	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
156	C₂×C₄²	—	2	2	no	—
157	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
158	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
159	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
160	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
161	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
162	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
163	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
164	C₄⋊C₈	—	2	4	no	—
165	C₄⋊C₈	—	4	2	no	—
166	C₄⋊C₈	—	4	2	no	—
167	C₄×C₈	—	2	4	no	—
168	C₄×C₈	—	2	4	no	—
169	C₄×C₈	—	2	2	no	—
170	C₄×C₈	C₅×Q₈	1	4	yes	complemented
171	#95	—	2	2	no	—
172	#95	—	2	2	no	—
173	C₂²×C₅	#391	1	1	yes	characteristic
174	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
175	C₂²×C₅	#408	1	1	yes	characteristic
176	C₂²×C₅	C₂×#102	1	1	yes	characteristic
177	C₂²×C₅	C₂×#103	1	1	yes	characteristic
178	C₂²×C₅	C₂×#104	1	1	yes	characteristic
179	C₂²×C₅	C₂×#104	1	1	yes	characteristic
180	C₄×C₅	—	2	2	no	—
181	C₄×C₅	—	2	2	no	—
182	C₄×C₅	—	2	2	no	—
183	C₄×C₅	—	4	4	no	—
184	C₄×C₅	—	8	2	no	—
185	C₄×C₅	D₄⋊C₈	1	4	yes	—
186	C₄×C₅	Q₈⋊C₈	1	4	yes	—
187	C₄×C₅	C₈×Q₈	1	2	yes	—
188	C₄×C₅	#446	1	2	yes	—
189	C₄×C₅	#486	1	1	yes	characteristic
190	C₄×C₅	#487	1	1	yes	characteristic
191	C₄×C₅	#498	1	1	yes	characteristic
192	C₄×C₅	#500	1	1	yes	characteristic
193	C₂×Q₈	—	2	4	no	—
194	C₂×Q₈	—	4	2	no	—
195	C₂×Q₈	—	4	2	no	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
199	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
200	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
201	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
202	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
203	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
204	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	—	2	4	no	—
207	C₂×C₈	—	2	4	no	—
208	C₂×C₈	—	4	2	no	—
209	C₂×C₈	—	4	2	no	—
210	C₂×C₈	—	4	2	no	—
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	2	4	no	—
213	C₄⋊C₄	—	2	4	no	—
214	C₄⋊C₄	—	4	4	no	—
215	C₄⋊C₄	—	4	2	no	—
216	C₄⋊C₄	—	4	2	no	—
217	C₄⋊C₄	C₂×C₅×C₈	1	4	yes	—
218	C₄²	—	2	2	no	—
219	C₄²	—	2	2	no	—
220	C₄²	—	2	2	no	—
221	C₄²	—	4	2	no	—
222	C₄²	—	4	2	no	—
223	C₄²	C₂×C₅×C₈	1	2	yes	—
224	C₄²	C₅×M₁₆	1	2	yes	—
225	C₄²	C₅×D₈	1	2	yes	—
226	C₄²	C₅×SD₁₆	1	2	yes	—
227	C₄²	C₅×SD₁₆	1	2	yes	—
228	C₄²	C₅×Q₁₆	1	2	yes	—
229	C₄²	C₂×C₅×Q₈	1	2	yes	—
230	C₄²	C₅×#41	1	2	yes	—
231	C₂×C₅	C₂×D₄⋊C₈	1	1	yes	characteristic
232	C₂×C₅	C₂×Q₈⋊C₈	1	1	yes	characteristic
233	C₂×C₅	#1492	1	1	yes	characteristic
234	C₂×C₅	#1493	1	1	yes	characteristic
235	C₂×C₅	#1984	1	1	yes	characteristic
236	C₂×C₅	#1990	1	1	yes	characteristic
237	C₂×C₅	#1994	1	1	yes	characteristic
238	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
239	Q₈	—	4	8	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	C₅×C₂²⋊C₈	1	2	yes	—
253	C₂×C₄	C₅×C₂²⋊C₈	1	2	yes	—
254	C₂×C₄	C₅×#102	1	2	yes	—
255	C₂×C₄	C₅×#102	1	2	yes	—
256	C₂×C₄	C₅×#103	1	2	yes	—
257	C₂×C₄	C₅×#103	1	2	yes	—
258	C₂×C₄	C₅×#104	1	2	yes	—
259	C₂×C₄	C₅×#104	1	2	yes	—
260	C₂×C₄	C₄×C₅×Q₈	1	2	yes	—
261	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
262	C₂×C₄	C₅×#124	1	1	yes	characteristic
263	C₂×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
264	C₂×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
265	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
266	C₂×C₄	C₅×#131	1	2	yes	—
267	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
268	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
269	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
270	C₂×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
271	C₈	—	2	8	no	—
272	C₈	—	8	2	no	—
273	C₅	#9870	1	1	yes	characteristic, direct product factor, Sylow
274	C₂²	C₅×#391	1	1	yes	characteristic
275	C₂²	C₂×C₅×C₂²⋊C₈	1	1	yes	characteristic
276	C₂²	C₅×#408	1	1	yes	characteristic
277	C₂²	C₂×C₅×#102	1	1	yes	characteristic
278	C₂²	C₂×C₅×#103	1	1	yes	characteristic
279	C₂²	C₂×C₅×#104	1	1	yes	characteristic
280	C₂²	C₂×C₅×#104	1	1	yes	characteristic
281	C₄	—	2	2	no	—
282	C₄	—	2	2	no	—
283	C₄	—	2	2	no	—
284	C₄	—	4	4	no	—
285	C₄	—	8	2	no	—
286	C₄	C₅×D₄⋊C₈	1	4	yes	—
287	C₄	C₅×Q₈⋊C₈	1	4	yes	—
288	C₄	C₅×C₈×Q₈	1	2	yes	—
289	C₄	C₅×#446	1	2	yes	—
290	C₄	C₅×#486	1	1	yes	characteristic
291	C₄	C₅×#487	1	1	yes	characteristic
292	C₄	C₅×#498	1	1	yes	characteristic
293	C₄	C₅×#500	1	1	yes	characteristic
294	C₂	C₂×C₅×D₄⋊C₈	1	1	yes	characteristic
295	C₂	C₂×C₅×Q₈⋊C₈	1	1	yes	characteristic
296	C₂	C₅×#1492	1	1	yes	characteristic
297	C₂	C₅×#1493	1	1	yes	characteristic
298	C₂	C₅×#1984	1	1	yes	characteristic
299	C₂	C₅×#1990	1	1	yes	characteristic
300	C₂	C₅×#1994	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅	C₂×D₄
commutator	C₂×C₄	C₂²×C₅×C₈
Frattini, cosocle	C₂×C₄²	C₂³×C₅
socle, prime radical	C₂³×C₅	C₂×C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ C₄³×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
20. #9870	C₅	direct (N × H)
25. C₅×C₄⋊Q₈	C₈	semidirect
48. C₄×C₅×C₈	Q₈	semidirect
147. C₄⋊Q₈	C₅×C₈	semidirect
170. C₄×C₈	C₅×Q₈	semidirect
273. C₅	#9870	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
C₄²×C₅×C₈	2
C₅×#1279	2
#9870	5

Subgroups — C5×#9870 #11932590

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#9870 #11932590