Subgroups — C₅×#19338 #11942058

Order 1280.

636 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3254	C₂	1	2	yes	maximal
2	C₅×#3170	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#2931	C₂	1	1	yes	maximal, characteristic
4	C₅×#1693	C₂	1	2	yes	maximal, complemented
5	C₅×#1689	C₂	1	2	yes	maximal, complemented
6	C₅×#1679	C₂	1	4	yes	maximal, complemented
7	C₅×#1566	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1561	C₂	1	2	yes	maximal, complemented
9	C₂×C₅×#131	—	2	1	no	—
10	C₅×#533	C₂²	1	2	yes	—
11	C₅×#532	C₂²	1	1	yes	characteristic
12	C₅×#500	C₂²	1	1	yes	characteristic
13	C₅×#494	C₂²	1	1	yes	characteristic
14	C₅×#487	C₂²	1	2	yes	—
15	C₅×#477	—	2	2	no	—
16	C₅×#474	C₂²	1	4	yes	—
17	C₅×#468	—	2	4	no	—
18	C₅×#467	—	2	1	no	—
19	C₅×#467	—	2	1	no	—
20	C₅×#461	C₂²	1	4	yes	—
21	C₅×#460	—	2	2	no	—
22	C₄×C₅×Q₁₆	—	2	1	no	—
23	C₄×C₅×Q₁₆	—	2	1	no	—
24	C₅×#436	—	2	1	no	—
25	C₅×#436	C₂²	1	2	yes	—
26	C₅×C₈×D₄	—	2	1	no	—
27	C₅×C₈×D₄	C₂²	1	2	yes	—
28	C₅×#423	C₂²	1	2	yes	—
29	C₅×#423	C₂²	1	1	yes	characteristic
30	C₅×#408	—	2	1	no	—
31	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
32	C₅×#334	C₂²	1	2	yes	—
33	C₅×#334	C₂²	1	2	yes	—
34	C₅×#332	C₂²	1	4	yes	—
35	C₅×#332	C₂²	1	4	yes	—
36	#19338	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
37	C₂×C₅×#41	—	2	1	no	—
38	C₂×C₅×Q₁₆	—	2	1	no	—
39	C₂×C₅×Q₁₆	—	2	1	no	—
40	C₂×C₅×SD₁₆	—	2	4	no	—
41	C₅×#131	—	4	4	no	—
42	C₂×C₅×M₁₆	—	2	1	no	—
43	C₂×C₅×M₁₆	—	2	2	no	—
44	C₂×C₅×M₁₆	—	2	2	no	—
45	C₂²×C₅×C₈	—	2	2	no	—
46	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
47	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
48	C₅×C₄⋊Q₈	D₄	1	1	yes	characteristic
49	C₅×C₄⋊Q₈	C₂³	1	2	yes	—
50	C₅×C₄⋊Q₈	C₂³	1	2	yes	—
51	C₅×#127	D₄	1	1	yes	characteristic
52	C₅×#125	—	2	2	no	—
53	C₅×#124	—	2	1	no	—
54	C₅×C₂²⋊Q₈	—	2	2	no	—
55	C₅×C₂²⋊Q₈	—	2	2	no	—
56	C₅×#121	—	2	1	no	—
57	C₅×#121	D₄	1	2	yes	—
58	C₄×C₅×Q₈	—	2	2	no	—
59	C₄×C₅×Q₈	—	2	2	no	—
60	C₄×C₅×D₄	—	2	1	no	—
61	C₄×C₅×D₄	C₂³	1	2	yes	—
62	C₅×#117	—	2	2	no	—
63	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
64	C₅×#107	—	2	1	no	—
65	C₅×#107	—	2	1	no	—
66	C₅×#107	—	2	4	no	—
67	C₅×C₄⋊C₈	—	2	1	no	—
68	C₅×C₄⋊C₈	—	2	2	no	—
69	C₅×C₄⋊C₈	C₂³	1	2	yes	—
70	C₅×C₄⋊C₈	C₂³	1	4	yes	—
71	C₅×#103	—	2	2	no	—
72	C₅×#103	—	2	2	no	—
73	C₅×#103	—	2	4	no	—
74	C₅×#102	—	2	4	no	—
75	C₅×C₂²⋊C₈	—	2	2	no	—
76	C₅×C₂²⋊C₈	—	2	2	no	—
77	C₄×C₅×C₈	D₄	1	1	yes	characteristic
78	C₄×C₅×C₈	D₄	1	1	yes	characteristic
79	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
80	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
81	#3254	C₂×C₅	1	2	yes	—
82	#3170	C₂×C₅	1	1	yes	characteristic, complemented
83	#2931	C₂×C₅	1	1	yes	characteristic
84	#1693	C₂×C₅	1	2	yes	complemented
85	#1689	C₂×C₅	1	2	yes	complemented
86	#1679	C₂×C₅	1	4	yes	complemented
87	#1566	C₂×C₅	1	1	yes	characteristic, complemented
88	#1561	C₂×C₅	1	2	yes	complemented
89	C₅×#41	—	4	2	no	—
90	C₅×#41	—	4	2	no	—
91	C₂×C₅×Q₈	—	2	1	no	—
92	C₂×C₅×Q₈	—	2	2	no	—
93	C₂×C₅×Q₈	—	2	2	no	—
94	C₂×C₅×D₄	—	2	2	no	—
95	C₂×C₅×D₄	—	2	1	no	—
96	C₂×C₅×D₄	—	2	1	no	—
97	C₂×C₅×D₄	C₂×D₄	1	2	yes	—
98	C₂²×C₄×C₅	—	2	2	no	—
99	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₅	#41	1	1	yes	characteristic
102	C₅×Q₁₆	—	4	2	no	—
103	C₅×Q₁₆	—	4	2	no	—
104	C₅×SD₁₆	—	8	4	no	—
105	C₅×M₁₆	—	4	2	no	—
106	C₅×M₁₆	—	4	4	no	—
107	C₅×M₁₆	—	8	2	no	—
108	C₂×C₅×C₈	—	2	2	no	—
109	C₂×C₅×C₈	—	2	2	no	—
110	C₂×C₅×C₈	—	2	4	no	—
111	C₂×C₅×C₈	—	4	1	no	—
112	C₂×C₅×C₈	—	4	4	no	—
113	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
114	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
115	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
116	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
117	C₅×C₄⋊C₄	—	2	1	no	—
118	C₅×C₄⋊C₄	—	2	2	no	—
119	C₅×C₄⋊C₄	—	2	2	no	—
120	C₅×C₄⋊C₄	—	2	2	no	—
121	C₅×C₄⋊C₄	—	2	2	no	—
122	C₅×C₄⋊C₄	—	4	2	no	—
123	C₅×C₄⋊C₄	—	4	2	no	—
124	C₅×C₄⋊C₄	C₂×D₄	1	2	yes	—
125	C₅×C₂²⋊C₄	—	2	2	no	—
126	C₅×C₂²⋊C₄	—	2	2	no	—
127	C₅×C₂²⋊C₄	—	4	2	no	—
128	C₄²×C₅	—	4	2	no	—
129	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
130	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
131	C₄²×C₅	#41	1	1	yes	characteristic
132	C₄²×C₅	C₂⁴	1	1	yes	characteristic
133	C₂×#131	—	2	1	no	—
134	#533	C₂²×C₅	1	2	yes	—
135	#532	C₂²×C₅	1	1	yes	characteristic
136	#500	C₂²×C₅	1	1	yes	characteristic
137	#494	C₂²×C₅	1	1	yes	characteristic
138	#487	C₂²×C₅	1	2	yes	—
139	#477	—	2	2	no	—
140	#474	C₂²×C₅	1	4	yes	—
141	#468	—	2	4	no	—
142	#467	—	2	1	no	—
143	#467	—	2	1	no	—
144	#461	C₂²×C₅	1	4	yes	—
145	#460	—	2	2	no	—
146	C₄×Q₁₆	—	2	1	no	—
147	C₄×Q₁₆	—	2	1	no	—
148	#436	—	2	1	no	—
149	#436	C₂²×C₅	1	2	yes	—
150	C₈×D₄	—	2	1	no	—
151	C₈×D₄	C₂²×C₅	1	2	yes	—
152	#423	C₂²×C₅	1	1	yes	characteristic
153	#423	C₂²×C₅	1	2	yes	—
154	#408	—	2	1	no	—
155	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
156	#334	C₂²×C₅	1	2	yes	—
157	#334	C₂²×C₅	1	2	yes	—
158	#332	C₂²×C₅	1	4	yes	—
159	#332	C₂²×C₅	1	4	yes	—
160	C₂³×C₅	—	2	2	no	—
161	C₂³×C₅	#121	1	1	yes	characteristic
162	C₅×Q₈	—	4	2	no	—
163	C₅×Q₈	—	4	2	no	—
164	C₅×Q₈	—	4	2	no	—
165	C₅×Q₈	—	4	2	no	—
166	C₅×Q₈	—	4	2	no	—
167	C₅×D₄	—	2	4	no	—
168	C₅×D₄	—	4	2	no	—
169	C₅×D₄	—	4	4	no	—
170	C₅×D₄	—	4	2	no	—
171	C₂×C₄×C₅	—	2	1	no	—
172	C₂×C₄×C₅	—	2	1	no	—
173	C₂×C₄×C₅	—	2	2	no	—
174	C₂×C₄×C₅	—	2	2	no	—
175	C₂×C₄×C₅	—	2	2	no	—
176	C₂×C₄×C₅	—	4	2	no	—
177	C₂×C₄×C₅	—	4	2	no	—
178	C₂×C₄×C₅	—	4	2	no	—
179	C₂×C₄×C₅	—	4	2	no	—
180	C₂×C₄×C₅	#121	1	1	yes	characteristic
181	C₂×C₄×C₅	#121	1	1	yes	characteristic
182	C₂×C₄×C₅	#121	1	1	yes	characteristic
183	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
184	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
185	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
186	C₅×C₈	—	2	1	no	—
187	C₅×C₈	—	2	1	no	—
188	C₅×C₈	—	2	1	no	—
189	C₅×C₈	—	4	2	no	—
190	C₅×C₈	—	4	4	no	—
191	C₅×C₈	AGL(1,ℤ/8)	1	2	yes	—
192	C₂×#41	—	2	1	no	—
193	C₂×Q₁₆	—	2	1	no	—
194	C₂×Q₁₆	—	2	1	no	—
195	C₂×SD₁₆	—	2	4	no	—
196	#131	—	4	4	no	—
197	C₂×M₁₆	—	2	1	no	—
198	C₂×M₁₆	—	2	2	no	—
199	C₂×M₁₆	—	2	2	no	—
200	C₂²×C₈	—	2	2	no	—
201	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
202	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
203	C₄⋊Q₈	C₅×D₄	1	1	yes	characteristic
204	C₄⋊Q₈	C₂³×C₅	1	2	yes	—
205	C₄⋊Q₈	C₂³×C₅	1	2	yes	—
206	#127	C₅×D₄	1	1	yes	characteristic
207	#125	—	2	2	no	—
208	#124	—	2	1	no	—
209	C₂²⋊Q₈	—	2	2	no	—
210	C₂²⋊Q₈	—	2	2	no	—
211	#121	—	2	1	no	—
212	#121	C₅×D₄	1	2	yes	—
213	C₄×Q₈	—	2	2	no	—
214	C₄×Q₈	—	2	2	no	—
215	C₄×D₄	—	2	1	no	—
216	C₄×D₄	C₂³×C₅	1	2	yes	—
217	#117	—	2	2	no	—
218	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
219	#107	—	2	1	no	—
220	#107	—	2	1	no	—
221	#107	—	2	4	no	—
222	C₄⋊C₈	—	2	1	no	—
223	C₄⋊C₈	—	2	2	no	—
224	C₄⋊C₈	C₂³×C₅	1	2	yes	—
225	C₄⋊C₈	C₂³×C₅	1	4	yes	—
226	#103	—	2	2	no	—
227	#103	—	2	2	no	—
228	#103	—	2	4	no	—
229	#102	—	2	4	no	—
230	C₂²⋊C₈	—	2	2	no	—
231	C₂²⋊C₈	—	2	2	no	—
232	C₄×C₈	C₅×D₄	1	1	yes	characteristic
233	C₄×C₈	C₅×D₄	1	1	yes	characteristic
234	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
235	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
236	C₂²×C₅	—	2	1	no	—
237	C₂²×C₅	—	2	1	no	—
238	C₂²×C₅	—	2	1	no	—
239	C₂²×C₅	—	4	2	no	—
240	C₂²×C₅	—	4	2	no	—
241	C₂²×C₅	—	4	2	no	—
242	C₂²×C₅	C₂×#121	1	1	yes	characteristic
243	C₄×C₅	—	2	1	no	—
244	C₄×C₅	—	2	1	no	—
245	C₄×C₅	—	2	1	no	—
246	C₄×C₅	—	4	2	no	—
247	C₄×C₅	—	8	2	no	—
248	C₄×C₅	—	8	2	no	—
249	C₄×C₅	#473	1	2	yes	—
250	C₄×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
251	C₄×C₅	#575	1	1	yes	characteristic
252	C₄×C₅	#577	1	2	yes	—
253	#41	—	4	2	no	—
254	#41	—	4	2	no	—
255	C₂×Q₈	—	2	1	no	—
256	C₂×Q₈	—	2	2	no	—
257	C₂×Q₈	—	2	2	no	—
258	C₂×D₄	—	2	2	no	—
259	C₂×D₄	—	2	1	no	—
260	C₂×D₄	—	2	1	no	—
261	C₂×D₄	C₂×C₅×D₄	1	2	yes	—
262	C₂²×C₄	—	2	2	no	—
263	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
264	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
265	C₂²×C₄	C₅×#41	1	1	yes	characteristic
266	Q₁₆	—	4	2	no	—
267	Q₁₆	—	4	2	no	—
268	SD₁₆	—	8	4	no	—
269	M₁₆	—	4	2	no	—
270	M₁₆	—	4	4	no	—
271	M₁₆	—	8	2	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	2	4	no	—
275	C₂×C₈	—	4	1	no	—
276	C₂×C₈	—	4	4	no	—
277	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
278	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
279	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
280	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
281	C₄⋊C₄	—	2	1	no	—
282	C₄⋊C₄	—	2	2	no	—
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	—	2	2	no	—
285	C₄⋊C₄	—	2	2	no	—
286	C₄⋊C₄	—	4	2	no	—
287	C₄⋊C₄	—	4	2	no	—
288	C₄⋊C₄	C₂×C₅×D₄	1	2	yes	—
289	C₂²⋊C₄	—	2	2	no	—
290	C₂²⋊C₄	—	2	2	no	—
291	C₂²⋊C₄	—	4	2	no	—
292	C₄²	—	4	2	no	—
293	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
294	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
295	C₄²	C₅×#41	1	1	yes	characteristic
296	C₄²	C₂⁴×C₅	1	1	yes	characteristic
297	C₂×C₅	—	4	1	no	—
298	C₂×C₅	—	8	2	no	—
299	C₂×C₅	#3036	1	1	yes	characteristic
300	C₂×C₅	#3060	1	1	yes	characteristic
301	C₂×C₅	C₂×#473	1	1	yes	characteristic
302	C₂³	—	2	2	no	—
303	C₂³	C₅×#121	1	1	yes	characteristic
304	Q₈	—	4	2	no	—
305	Q₈	—	4	2	no	—
306	Q₈	—	4	2	no	—
307	Q₈	—	4	2	no	—
308	Q₈	—	4	2	no	—
309	D₄	—	2	4	no	—
310	D₄	—	4	2	no	—
311	D₄	—	4	4	no	—
312	D₄	—	4	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	C₅×#121	1	1	yes	characteristic
323	C₂×C₄	C₅×#121	1	1	yes	characteristic
324	C₂×C₄	C₅×#121	1	1	yes	characteristic
325	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
326	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
327	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
328	C₈	—	2	1	no	—
329	C₈	—	2	1	no	—
330	C₈	—	2	1	no	—
331	C₈	—	4	2	no	—
332	C₈	—	4	4	no	—
333	C₈	C₅×AGL(1,ℤ/8)	1	2	yes	—
334	C₅	#19338	1	1	yes	characteristic, direct product factor, Sylow
335	C₂²	—	2	1	no	—
336	C₂²	—	2	1	no	—
337	C₂²	—	2	1	no	—
338	C₂²	—	4	2	no	—
339	C₂²	—	4	2	no	—
340	C₂²	—	4	2	no	—
341	C₂²	C₂×C₅×#121	1	1	yes	characteristic
342	C₄	—	2	1	no	—
343	C₄	—	2	1	no	—
344	C₄	—	2	1	no	—
345	C₄	—	4	2	no	—
346	C₄	—	8	2	no	—
347	C₄	—	8	2	no	—
348	C₄	C₅×#473	1	2	yes	—
349	C₄	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
350	C₄	C₅×#575	1	1	yes	characteristic
351	C₄	C₅×#577	1	2	yes	—
352	C₂	—	4	1	no	—
353	C₂	—	8	2	no	—
354	C₂	C₅×#3036	1	1	yes	characteristic
355	C₂	C₅×#3060	1	1	yes	characteristic
356	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	C₅×#532	C₂²
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
2. C₅×#3170	C₂	semidirect
4. C₅×#1693	C₂	semidirect
5. C₅×#1689	C₂	semidirect
6. C₅×#1679	C₂	semidirect
7. C₅×#1566	C₂	semidirect
8. C₅×#1561	C₂	semidirect
36. #19338	C₅	direct (N × H)
82. #3170	C₂×C₅	semidirect
84. #1693	C₂×C₅	semidirect
85. #1689	C₂×C₅	semidirect
86. #1679	C₂×C₅	semidirect
87. #1566	C₂×C₅	semidirect
88. #1561	C₂×C₅	semidirect
334. C₅	#19338	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×#3254	2
C₅×#3170	2
C₅×#2931	2
C₅×#1693	2
C₅×#1689	2
C₅×#1679	2
C₅×#1566	2
C₅×#1561	2
#19338	5

Subgroups — C5×#19338 #11942058

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#19338 #11942058