Subgroups — C₅×#19296 #11942016

Order 1280.

680 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#500	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#3039	C₂	1	2	yes	maximal
3	C₅×#2965	C₂	1	1	yes	maximal, characteristic
4	C₅×#1694	C₂	1	4	yes	maximal, complemented
5	C₅×#1693	C₂	1	4	yes	maximal, complemented
6	C₅×#1566	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1546	C₂	1	2	yes	maximal, complemented
8	C₂×C₅×#137	—	2	2	no	—
9	C₂×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₄×C₅×#41	C₂²	1	1	yes	characteristic
11	C₅×#500	—	2	1	no	—
12	C₅×#500	—	2	1	no	—
13	C₅×#500	C₂²	1	4	yes	—
14	C₅×#462	—	2	2	no	—
15	C₅×#462	C₂²	1	4	yes	—
16	C₅×#461	—	2	2	no	—
17	C₅×#461	C₂²	1	4	yes	—
18	C₅×#446	C₂²	1	2	yes	—
19	C₅×#436	C₂²	1	2	yes	—
20	C₅×#435	—	2	2	no	—
21	C₅×#431	—	2	1	no	—
22	C₅×#431	C₂²	1	1	yes	characteristic
23	C₂×C₅×#107	—	2	2	no	—
24	C₅×#423	C₂²	1	2	yes	—
25	C₅×#419	—	2	2	no	—
26	C₄×C₅×M₁₆	—	2	1	no	—
27	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
28	C₅×#334	C₂²	1	4	yes	—
29	C₅×#332	C₂²	1	8	yes	—
30	#19296	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
31	C₂×C₅×#41	D₄	1	1	yes	characteristic
32	C₂²×C₅×Q₈	—	2	2	no	—
33	C₅×#137	—	4	8	no	—
34	C₂×C₅×Q₁₆	—	2	4	no	—
35	C₂×C₅×SD₁₆	—	2	4	no	—
36	C₂×C₅×M₁₆	—	2	2	no	—
37	C₂×C₅×M₁₆	—	2	2	no	—
38	C₂²×C₅×C₈	D₄	1	2	yes	—
39	C₅×C₄⋊Q₈	—	2	2	no	—
40	C₅×C₄⋊Q₈	—	2	2	no	—
41	C₅×C₄⋊Q₈	—	2	2	no	—
42	C₅×C₄⋊Q₈	C₂³	1	4	yes	—
43	C₄×C₅×Q₈	D₄	1	1	yes	characteristic
44	C₄×C₅×Q₈	C₂³	1	1	yes	characteristic
45	C₄×C₅×D₄	—	2	1	no	—
46	C₄×C₅×D₄	—	2	1	no	—
47	C₄×C₅×D₄	D₄	1	1	yes	characteristic
48	C₄×C₅×D₄	C₂³	1	1	yes	characteristic
49	C₅×#117	—	2	1	no	—
50	C₅×#117	D₄	1	1	yes	characteristic
51	C₂×C₅×C₄⋊C₄	—	2	2	no	—
52	C₂×C₅×C₄⋊C₄	—	2	2	no	—
53	C₂×C₄²×C₅	—	2	1	no	—
54	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
55	C₅×#107	—	2	4	no	—
56	C₅×#107	—	2	4	no	—
57	C₅×C₄⋊C₈	—	2	2	no	—
58	C₅×C₄⋊C₈	—	2	2	no	—
59	C₅×C₄⋊C₈	C₂³	1	2	yes	—
60	C₅×C₄⋊C₈	C₂³	1	4	yes	—
61	C₅×#103	—	2	4	no	—
62	C₅×#103	—	2	4	no	—
63	C₅×C₂²⋊C₈	—	2	1	no	—
64	C₅×C₂²⋊C₈	—	2	2	no	—
65	C₅×#97	—	2	2	no	—
66	C₄×C₅×C₈	—	2	1	no	—
67	C₄×C₅×C₈	D₄	1	1	yes	characteristic
68	C₄×C₅×C₈	D₄	1	1	yes	characteristic
69	C₄×C₅×C₈	C₂³	1	2	yes	—
70	C₂×#500	C₂×C₅	1	1	yes	characteristic, complemented
71	#3039	C₂×C₅	1	2	yes	—
72	#2965	C₂×C₅	1	1	yes	characteristic
73	#1694	C₂×C₅	1	4	yes	complemented
74	#1693	C₂×C₅	1	4	yes	complemented
75	#1566	C₂×C₅	1	1	yes	characteristic, complemented
76	#1546	C₂×C₅	1	2	yes	complemented
77	C₅×#41	—	2	4	no	—
78	C₂×C₅×Q₈	—	2	4	no	—
79	C₂×C₅×Q₈	—	4	2	no	—
80	C₂×C₅×Q₈	—	4	4	no	—
81	C₂×C₅×Q₈	—	4	4	no	—
82	C₂×C₅×Q₈	C₂×D₄	1	1	yes	characteristic
83	C₂×C₅×D₄	—	2	1	no	—
84	C₂×C₅×D₄	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₅	—	2	1	no	—
86	C₂²×C₄×C₅	—	2	1	no	—
87	C₂²×C₄×C₅	—	2	1	no	—
88	C₂²×C₄×C₅	—	4	2	no	—
89	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₅	#41	1	1	yes	characteristic
92	C₅×Q₁₆	—	4	8	no	—
93	C₅×SD₁₆	—	4	8	no	—
94	C₅×M₁₆	—	4	4	no	—
95	C₅×M₁₆	—	4	4	no	—
96	C₂×C₅×C₈	—	2	4	no	—
97	C₂×C₅×C₈	—	2	4	no	—
98	C₂×C₅×C₈	—	2	2	no	—
99	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
100	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
101	C₅×C₄⋊C₄	—	2	1	no	—
102	C₅×C₄⋊C₄	—	2	1	no	—
103	C₅×C₄⋊C₄	—	2	4	no	—
104	C₅×C₄⋊C₄	—	2	4	no	—
105	C₅×C₄⋊C₄	—	4	2	no	—
106	C₅×C₄⋊C₄	—	4	2	no	—
107	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
108	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
109	C₅×C₂²⋊C₄	—	2	1	no	—
110	C₅×C₂²⋊C₄	—	2	1	no	—
111	C₅×C₂²⋊C₄	—	2	1	no	—
112	C₄²×C₅	—	2	1	no	—
113	C₄²×C₅	—	2	1	no	—
114	C₄²×C₅	—	2	1	no	—
115	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₅	#41	1	1	yes	characteristic
118	C₄²×C₅	C₂⁴	1	1	yes	characteristic
119	C₂×#137	—	2	2	no	—
120	C₂×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
121	C₄×#41	C₂²×C₅	1	1	yes	characteristic
122	#500	—	2	1	no	—
123	#500	—	2	1	no	—
124	#500	C₂²×C₅	1	4	yes	—
125	#462	—	2	2	no	—
126	#462	C₂²×C₅	1	4	yes	—
127	#461	—	2	2	no	—
128	#461	C₂²×C₅	1	4	yes	—
129	#446	C₂²×C₅	1	2	yes	—
130	#436	C₂²×C₅	1	2	yes	—
131	#435	—	2	2	no	—
132	#431	—	2	1	no	—
133	#431	C₂²×C₅	1	1	yes	characteristic
134	C₂×#107	—	2	2	no	—
135	#423	C₂²×C₅	1	2	yes	—
136	#419	—	2	2	no	—
137	C₄×M₁₆	—	2	1	no	—
138	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
139	#334	C₂²×C₅	1	4	yes	—
140	#332	C₂²×C₅	1	8	yes	—
141	C₂³×C₅	—	2	1	no	—
142	C₂³×C₅	#121	1	1	yes	characteristic
143	C₅×Q₈	—	2	2	no	—
144	C₅×Q₈	—	4	4	no	—
145	C₅×Q₈	—	4	4	no	—
146	C₅×Q₈	—	8	4	no	—
147	C₅×D₄	—	2	2	no	—
148	C₅×D₄	—	4	2	no	—
149	C₂×C₄×C₅	—	2	2	no	—
150	C₂×C₄×C₅	—	2	2	no	—
151	C₂×C₄×C₅	—	2	1	no	—
152	C₂×C₄×C₅	—	2	1	no	—
153	C₂×C₄×C₅	—	2	1	no	—
154	C₂×C₄×C₅	—	4	2	no	—
155	C₂×C₄×C₅	—	4	2	no	—
156	C₂×C₄×C₅	—	4	2	no	—
157	C₂×C₄×C₅	—	4	4	no	—
158	C₂×C₄×C₅	—	8	4	no	—
159	C₂×C₄×C₅	#121	1	1	yes	characteristic
160	C₂×C₄×C₅	#121	1	1	yes	characteristic
161	C₂×C₄×C₅	#121	1	1	yes	characteristic
162	C₂×C₄×C₅	AGL(1,ℤ/8)	1	4	yes	—
163	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
164	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
165	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
166	C₅×C₈	—	2	2	no	—
167	C₅×C₈	—	4	4	no	—
168	C₅×C₈	—	4	2	no	—
169	C₅×C₈	#137	1	4	yes	complemented
170	C₂×#41	C₅×D₄	1	1	yes	characteristic
171	C₂²×Q₈	—	2	2	no	—
172	#137	—	4	8	no	—
173	C₂×Q₁₆	—	2	4	no	—
174	C₂×SD₁₆	—	2	4	no	—
175	C₂×M₁₆	—	2	2	no	—
176	C₂×M₁₆	—	2	2	no	—
177	C₂²×C₈	C₅×D₄	1	2	yes	—
178	C₄⋊Q₈	—	2	2	no	—
179	C₄⋊Q₈	—	2	2	no	—
180	C₄⋊Q₈	—	2	2	no	—
181	C₄⋊Q₈	C₂³×C₅	1	4	yes	—
182	C₄×Q₈	C₅×D₄	1	1	yes	characteristic
183	C₄×Q₈	C₂³×C₅	1	1	yes	characteristic
184	C₄×D₄	—	2	1	no	—
185	C₄×D₄	—	2	1	no	—
186	C₄×D₄	C₅×D₄	1	1	yes	characteristic
187	C₄×D₄	C₂³×C₅	1	1	yes	characteristic
188	#117	—	2	1	no	—
189	#117	C₅×D₄	1	1	yes	characteristic
190	C₂×C₄⋊C₄	—	2	2	no	—
191	C₂×C₄⋊C₄	—	2	2	no	—
192	C₂×C₄²	—	2	1	no	—
193	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
194	#107	—	2	4	no	—
195	#107	—	2	4	no	—
196	C₄⋊C₈	—	2	2	no	—
197	C₄⋊C₈	—	2	2	no	—
198	C₄⋊C₈	C₂³×C₅	1	2	yes	—
199	C₄⋊C₈	C₂³×C₅	1	4	yes	—
200	#103	—	2	4	no	—
201	#103	—	2	4	no	—
202	C₂²⋊C₈	—	2	1	no	—
203	C₂²⋊C₈	—	2	2	no	—
204	#97	—	2	2	no	—
205	C₄×C₈	—	2	1	no	—
206	C₄×C₈	C₅×D₄	1	1	yes	characteristic
207	C₄×C₈	C₅×D₄	1	1	yes	characteristic
208	C₄×C₈	C₂³×C₅	1	2	yes	—
209	C₂²×C₅	—	2	1	no	—
210	C₂²×C₅	—	2	1	no	—
211	C₂²×C₅	—	4	1	no	—
212	C₂²×C₅	—	4	1	no	—
213	C₂²×C₅	—	4	1	no	—
214	C₂²×C₅	#469	1	2	yes	—
215	C₂²×C₅	C₂×#121	1	1	yes	characteristic
216	C₄×C₅	—	2	1	no	—
217	C₄×C₅	—	2	1	no	—
218	C₄×C₅	—	4	1	no	—
219	C₄×C₅	—	4	1	no	—
220	C₄×C₅	—	4	1	no	—
221	C₄×C₅	—	8	4	no	—
222	C₄×C₅	#469	1	2	yes	—
223	C₄×C₅	#473	1	2	yes	—
224	C₄×C₅	C₂×AGL(1,ℤ/8)	1	2	yes	—
225	C₄×C₅	C₂×#137	1	2	yes	—
226	#41	—	2	4	no	—
227	C₂×Q₈	—	2	4	no	—
228	C₂×Q₈	—	4	2	no	—
229	C₂×Q₈	—	4	4	no	—
230	C₂×Q₈	—	4	4	no	—
231	C₂×Q₈	C₂×C₅×D₄	1	1	yes	characteristic
232	C₂×D₄	—	2	1	no	—
233	C₂×D₄	C₂×C₅×D₄	1	1	yes	characteristic
234	C₂²×C₄	—	2	1	no	—
235	C₂²×C₄	—	2	1	no	—
236	C₂²×C₄	—	2	1	no	—
237	C₂²×C₄	—	4	2	no	—
238	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
239	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
240	C₂²×C₄	C₅×#41	1	1	yes	characteristic
241	Q₁₆	—	4	8	no	—
242	SD₁₆	—	4	8	no	—
243	M₁₆	—	4	4	no	—
244	M₁₆	—	4	4	no	—
245	C₂×C₈	—	2	4	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	4	no	—
248	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
249	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
250	C₄⋊C₄	—	2	1	no	—
251	C₄⋊C₄	—	2	1	no	—
252	C₄⋊C₄	—	2	4	no	—
253	C₄⋊C₄	—	2	4	no	—
254	C₄⋊C₄	—	4	2	no	—
255	C₄⋊C₄	—	4	2	no	—
256	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
257	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
258	C₂²⋊C₄	—	2	1	no	—
259	C₂²⋊C₄	—	2	1	no	—
260	C₂²⋊C₄	—	2	1	no	—
261	C₄²	—	2	1	no	—
262	C₄²	—	2	1	no	—
263	C₄²	—	2	1	no	—
264	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
265	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
266	C₄²	C₅×#41	1	1	yes	characteristic
267	C₄²	C₂⁴×C₅	1	1	yes	characteristic
268	C₂×C₅	—	2	2	no	—
269	C₂×C₅	—	8	1	no	—
270	C₂×C₅	#3039	1	1	yes	characteristic
271	C₂×C₅	C₂×#469	1	1	yes	characteristic
272	C₂×C₅	C₂×#473	1	1	yes	characteristic
273	C₂³	—	2	1	no	—
274	C₂³	C₅×#121	1	1	yes	characteristic
275	Q₈	—	2	2	no	—
276	Q₈	—	4	4	no	—
277	Q₈	—	4	4	no	—
278	Q₈	—	8	4	no	—
279	D₄	—	2	2	no	—
280	D₄	—	4	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	4	no	—
290	C₂×C₄	—	8	4	no	—
291	C₂×C₄	C₅×#121	1	1	yes	characteristic
292	C₂×C₄	C₅×#121	1	1	yes	characteristic
293	C₂×C₄	C₅×#121	1	1	yes	characteristic
294	C₂×C₄	C₅×AGL(1,ℤ/8)	1	4	yes	—
295	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
296	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
297	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
298	C₈	—	2	2	no	—
299	C₈	—	4	4	no	—
300	C₈	—	4	2	no	—
301	C₈	C₅×#137	1	4	yes	complemented
302	C₅	#19296	1	1	yes	characteristic, direct product factor, Sylow
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	4	1	no	—
306	C₂²	—	4	1	no	—
307	C₂²	—	4	1	no	—
308	C₂²	C₅×#469	1	2	yes	—
309	C₂²	C₂×C₅×#121	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	4	1	no	—
313	C₄	—	4	1	no	—
314	C₄	—	4	1	no	—
315	C₄	—	8	4	no	—
316	C₄	C₅×#469	1	2	yes	—
317	C₄	C₅×#473	1	2	yes	—
318	C₄	C₂×C₅×AGL(1,ℤ/8)	1	2	yes	—
319	C₄	C₂×C₅×#137	1	2	yes	—
320	C₂	—	2	2	no	—
321	C₂	—	8	1	no	—
322	C₂	C₅×#3039	1	1	yes	characteristic
323	C₂	C₂×C₅×#469	1	1	yes	characteristic
324	C₂	C₂×C₅×#473	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅
prime radical	C₂×C₅×#41	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×#500	C₂	semidirect
4. C₅×#1694	C₂	semidirect
5. C₅×#1693	C₂	semidirect
6. C₅×#1566	C₂	semidirect
7. C₅×#1546	C₂	semidirect
30. #19296	C₅	direct (N × H)
70. C₂×#500	C₂×C₅	semidirect
73. #1694	C₂×C₅	semidirect
74. #1693	C₂×C₅	semidirect
75. #1566	C₂×C₅	semidirect
76. #1546	C₂×C₅	semidirect
169. C₅×C₈	#137	semidirect
301. C₈	C₅×#137	semidirect
302. C₅	#19296	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×C₅×#500	2
C₅×#3039	2
C₅×#2965	2
C₅×#1694	2
C₅×#1693	2
C₅×#1566	2
C₅×#1546	2
#19296	5

Subgroups — C5×#19296 #11942016

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#19296 #11942016