Subgroups — C₅×#19423 #11942143

Order 1280.

630 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3209	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#3077	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#474	C₂	1	1	yes	maximal, characteristic
4	C₅×#2975	C₂	1	1	yes	maximal, characteristic
5	C₅×#1703	C₂	1	4	yes	maximal, complemented
6	C₅×#1689	C₂	1	4	yes	maximal, complemented
7	C₅×#1570	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1549	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#1542	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#533	C₂²	1	1	yes	characteristic
11	C₂×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
13	C₅×#494	C₂²	1	2	yes	complemented
14	C₅×#487	C₂²	1	2	yes	complemented
15	C₅×#478	—	2	2	no	—
16	C₅×#476	C₂²	1	2	yes	—
17	C₅×#474	—	2	2	no	—
18	C₅×#474	C₂²	1	4	yes	—
19	C₅×#474	C₂²	1	2	yes	—
20	C₅×#462	—	2	2	no	—
21	C₅×#451	—	2	2	no	—
22	C₅×#436	—	2	2	no	—
23	C₅×#436	C₂²	1	4	yes	—
24	C₅×#431	C₂²	1	1	yes	characteristic
25	C₅×#431	C₂²	1	1	yes	characteristic
26	C₅×#428	—	2	1	no	—
27	C₂×C₅×#107	—	2	1	no	—
28	C₅×#423	C₂²	1	1	yes	characteristic
29	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
30	C₅×#418	—	2	1	no	—
31	C₂×C₅×#102	—	2	1	no	—
32	C₅×#407	—	2	2	no	—
33	C₅×#334	C₂²	1	4	yes	complemented
34	C₅×#333	C₂²	1	2	yes	complemented
35	C₅×#332	C₂²	1	2	yes	complemented
36	C₅×#332	C₂²	1	4	yes	complemented
37	#19423	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
38	C₂²×C₅×Q₈	—	2	1	no	—
39	C₂²×C₅×D₄	D₄	1	1	yes	characteristic
40	C₂³×C₄×C₅	—	2	1	no	—
41	C₂×C₅×Q₁₆	—	2	4	no	—
42	C₂×C₅×M₁₆	—	2	1	no	—
43	C₂×C₅×M₁₆	—	2	4	no	—
44	C₂²×C₅×C₈	—	2	1	no	—
45	C₅×C₄⋊Q₈	—	2	1	no	—
46	C₅×C₄⋊Q₈	—	2	1	no	—
47	C₅×C₄⋊Q₈	—	2	1	no	—
48	C₅×C₄⋊Q₈	C₂³	1	2	yes	—
49	C₅×C₄⋊Q₈	C₂³	1	1	yes	characteristic
50	C₅×C₄⋊Q₈	C₂³	1	1	yes	characteristic
51	C₅×#125	—	2	1	no	—
52	C₅×C₂²⋊Q₈	—	2	1	no	—
53	C₅×C₂²⋊Q₈	—	2	1	no	—
54	C₄×C₅×Q₈	—	2	1	no	—
55	C₄×C₅×Q₈	—	2	1	no	—
56	C₄×C₅×D₄	—	2	1	no	—
57	C₄×C₅×D₄	—	2	1	no	—
58	C₄×C₅×D₄	D₄	1	2	yes	—
59	C₄×C₅×D₄	C₂³	1	2	yes	—
60	C₅×#117	—	2	1	no	—
61	C₂×C₅×C₄⋊C₄	—	2	1	no	—
62	C₂×C₅×C₄⋊C₄	—	2	1	no	—
63	C₂×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
64	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
65	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
66	C₅×#107	—	2	4	no	—
67	C₅×#107	—	2	2	no	—
68	C₅×#106	—	2	2	no	—
69	C₅×C₄⋊C₈	—	2	1	no	—
70	C₅×C₄⋊C₈	—	2	1	no	—
71	C₅×C₄⋊C₈	D₄	1	2	yes	—
72	C₅×C₄⋊C₈	C₂³	1	2	yes	—
73	C₅×C₄⋊C₈	C₂³	1	2	yes	—
74	C₅×C₄⋊C₈	C₂³	1	2	yes	—
75	C₅×#103	—	2	4	no	—
76	C₅×#102	—	2	4	no	—
77	C₅×#102	—	2	2	no	—
78	C₅×#102	—	2	2	no	—
79	C₅×C₂²⋊C₈	—	2	1	no	—
80	C₅×C₂²⋊C₈	—	2	4	no	—
81	C₅×C₂²⋊C₈	D₄	1	2	yes	—
82	C₄×C₅×C₈	—	2	1	no	—
83	C₄×C₅×C₈	C₂³	1	2	yes	—
84	#3209	C₂×C₅	1	1	yes	characteristic, complemented
85	#3077	C₂×C₅	1	1	yes	characteristic
86	C₂×#474	C₂×C₅	1	1	yes	characteristic
87	#2975	C₂×C₅	1	1	yes	characteristic
88	#1703	C₂×C₅	1	4	yes	complemented
89	#1689	C₂×C₅	1	4	yes	complemented
90	#1570	C₂×C₅	1	1	yes	characteristic, complemented
91	#1549	C₂×C₅	1	1	yes	characteristic, complemented
92	#1542	C₂×C₅	1	1	yes	characteristic, complemented
93	C₂⁴×C₅	—	2	1	no	—
94	C₂×C₅×Q₈	—	2	1	no	—
95	C₂×C₅×Q₈	—	2	2	no	—
96	C₂×C₅×Q₈	—	2	1	no	—
97	C₂×C₅×Q₈	—	4	1	no	—
98	C₂×C₅×Q₈	—	8	2	no	—
99	C₂×C₅×D₄	—	2	1	no	—
100	C₂×C₅×D₄	—	2	2	no	—
101	C₂×C₅×D₄	—	2	2	no	—
102	C₂×C₅×D₄	C₂×D₄	1	2	yes	—
103	C₂²×C₄×C₅	—	2	1	no	—
104	C₂²×C₄×C₅	—	2	2	no	—
105	C₂²×C₄×C₅	—	2	2	no	—
106	C₂²×C₄×C₅	—	4	1	no	—
107	C₂²×C₄×C₅	—	8	2	no	—
108	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₄×C₅	#41	1	1	yes	characteristic
111	C₅×Q₁₆	—	8	4	no	—
112	C₅×M₁₆	—	4	2	no	—
113	C₅×M₁₆	—	8	4	no	—
114	C₂×C₅×C₈	—	2	2	no	—
115	C₂×C₅×C₈	—	2	2	no	—
116	C₂×C₅×C₈	—	2	2	no	—
117	C₂×C₅×C₈	—	4	2	no	—
118	C₂×C₅×C₈	C₂×D₄	1	4	yes	—
119	C₅×C₄⋊C₄	—	2	1	no	—
120	C₅×C₄⋊C₄	—	2	2	no	—
121	C₅×C₄⋊C₄	—	2	1	no	—
122	C₅×C₄⋊C₄	—	2	1	no	—
123	C₅×C₄⋊C₄	—	2	1	no	—
124	C₅×C₄⋊C₄	—	2	2	no	—
125	C₅×C₄⋊C₄	—	2	1	no	—
126	C₅×C₄⋊C₄	—	4	1	no	—
127	C₅×C₄⋊C₄	—	4	1	no	—
128	C₅×C₄⋊C₄	—	4	1	no	—
129	C₅×C₄⋊C₄	—	4	1	no	—
130	C₅×C₄⋊C₄	C₂×D₄	1	2	yes	—
131	C₅×C₂²⋊C₄	—	2	2	no	—
132	C₅×C₂²⋊C₄	—	2	2	no	—
133	C₅×C₂²⋊C₄	—	4	1	no	—
134	C₄²×C₅	—	4	1	no	—
135	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
136	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
137	C₄²×C₅	#41	1	1	yes	characteristic
138	C₄²×C₅	C₂⁴	1	1	yes	characteristic
139	#533	C₂²×C₅	1	1	yes	characteristic
140	C₂×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
141	C₂×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
142	#494	C₂²×C₅	1	2	yes	complemented
143	#487	C₂²×C₅	1	2	yes	complemented
144	#478	—	2	2	no	—
145	#476	C₂²×C₅	1	2	yes	—
146	#474	—	2	2	no	—
147	#474	C₂²×C₅	1	2	yes	—
148	#474	C₂²×C₅	1	4	yes	—
149	#462	—	2	2	no	—
150	#451	—	2	2	no	—
151	#436	—	2	2	no	—
152	#436	C₂²×C₅	1	4	yes	—
153	#431	C₂²×C₅	1	1	yes	characteristic
154	#431	C₂²×C₅	1	1	yes	characteristic
155	#428	—	2	1	no	—
156	C₂×#107	—	2	1	no	—
157	#423	C₂²×C₅	1	1	yes	characteristic
158	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
159	#418	—	2	1	no	—
160	C₂×#102	—	2	1	no	—
161	#407	—	2	2	no	—
162	#334	C₂²×C₅	1	4	yes	complemented
163	#333	C₂²×C₅	1	2	yes	complemented
164	#332	C₂²×C₅	1	2	yes	complemented
165	#332	C₂²×C₅	1	4	yes	complemented
166	C₂³×C₅	—	2	2	no	—
167	C₂³×C₅	—	4	2	no	—
168	C₂³×C₅	—	8	1	no	—
169	C₂³×C₅	—	8	1	no	—
170	C₂³×C₅	#121	1	1	yes	characteristic
171	C₅×Q₈	—	4	2	no	—
172	C₅×Q₈	—	4	4	no	—
173	C₅×Q₈	—	8	1	no	—
174	C₅×Q₈	—	8	2	no	—
175	C₅×D₄	—	2	2	no	—
176	C₅×D₄	—	2	2	no	—
177	C₅×D₄	—	4	2	no	—
178	C₂×C₄×C₅	—	2	2	no	—
179	C₂×C₄×C₅	—	2	2	no	—
180	C₂×C₄×C₅	—	4	1	no	—
181	C₂×C₄×C₅	—	4	4	no	—
182	C₂×C₄×C₅	—	4	1	no	—
183	C₂×C₄×C₅	—	4	1	no	—
184	C₂×C₄×C₅	—	4	2	no	—
185	C₂×C₄×C₅	—	8	2	no	—
186	C₂×C₄×C₅	—	8	1	no	—
187	C₂×C₄×C₅	—	8	2	no	—
188	C₂×C₄×C₅	#121	1	1	yes	characteristic
189	C₂×C₄×C₅	#121	1	1	yes	characteristic
190	C₂×C₄×C₅	#121	1	1	yes	characteristic
191	C₂×C₄×C₅	#137	1	2	yes	—
192	C₂×C₄×C₅	#137	1	2	yes	—
193	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
194	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
195	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
196	C₅×C₈	—	2	4	no	—
197	C₅×C₈	—	4	2	no	—
198	C₅×C₈	—	4	2	no	—
199	C₅×C₈	—	4	2	no	—
200	C₂²×Q₈	—	2	1	no	—
201	C₂²×D₄	C₅×D₄	1	1	yes	characteristic
202	C₂³×C₄	—	2	1	no	—
203	C₂×Q₁₆	—	2	4	no	—
204	C₂×M₁₆	—	2	1	no	—
205	C₂×M₁₆	—	2	4	no	—
206	C₂²×C₈	—	2	1	no	—
207	C₄⋊Q₈	—	2	1	no	—
208	C₄⋊Q₈	—	2	1	no	—
209	C₄⋊Q₈	—	2	1	no	—
210	C₄⋊Q₈	C₂³×C₅	1	2	yes	—
211	C₄⋊Q₈	C₂³×C₅	1	1	yes	characteristic
212	C₄⋊Q₈	C₂³×C₅	1	1	yes	characteristic
213	#125	—	2	1	no	—
214	C₂²⋊Q₈	—	2	1	no	—
215	C₂²⋊Q₈	—	2	1	no	—
216	C₄×Q₈	—	2	1	no	—
217	C₄×Q₈	—	2	1	no	—
218	C₄×D₄	—	2	1	no	—
219	C₄×D₄	—	2	1	no	—
220	C₄×D₄	C₅×D₄	1	2	yes	—
221	C₄×D₄	C₂³×C₅	1	2	yes	—
222	#117	—	2	1	no	—
223	C₂×C₄⋊C₄	—	2	1	no	—
224	C₂×C₄⋊C₄	—	2	1	no	—
225	C₂×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
226	C₂×C₂²⋊C₄	—	2	1	no	—
227	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
228	#107	—	2	4	no	—
229	#107	—	2	2	no	—
230	#106	—	2	2	no	—
231	C₄⋊C₈	—	2	1	no	—
232	C₄⋊C₈	—	2	1	no	—
233	C₄⋊C₈	C₅×D₄	1	2	yes	—
234	C₄⋊C₈	C₂³×C₅	1	2	yes	—
235	C₄⋊C₈	C₂³×C₅	1	2	yes	—
236	C₄⋊C₈	C₂³×C₅	1	2	yes	—
237	#103	—	2	4	no	—
238	#102	—	2	4	no	—
239	#102	—	2	2	no	—
240	#102	—	2	2	no	—
241	C₂²⋊C₈	—	2	1	no	—
242	C₂²⋊C₈	—	2	4	no	—
243	C₂²⋊C₈	C₅×D₄	1	2	yes	—
244	C₄×C₈	—	2	1	no	—
245	C₄×C₈	C₂³×C₅	1	2	yes	—
246	C₂²×C₅	—	2	1	no	—
247	C₂²×C₅	—	2	1	no	—
248	C₂²×C₅	—	4	2	no	—
249	C₂²×C₅	—	4	2	no	—
250	C₂²×C₅	—	4	2	no	—
251	C₂²×C₅	—	8	4	no	—
252	C₂²×C₅	#470	1	2	yes	—
253	C₂²×C₅	C₂×#121	1	1	yes	characteristic
254	C₄×C₅	—	2	1	no	—
255	C₄×C₅	—	2	1	no	—
256	C₄×C₅	—	2	1	no	—
257	C₄×C₅	—	4	2	no	—
258	C₄×C₅	—	8	2	no	—
259	C₄×C₅	—	8	1	no	—
260	C₄×C₅	—	8	1	no	—
261	C₄×C₅	#470	1	2	yes	—
262	C₄×C₅	C₂×#137	1	1	yes	characteristic
263	C₄×C₅	C₂×#137	1	1	yes	characteristic
264	C₄×C₅	#576	1	2	yes	—
265	C₂⁴	—	2	1	no	—
266	C₂×Q₈	—	2	2	no	—
267	C₂×Q₈	—	2	1	no	—
268	C₂×Q₈	—	2	1	no	—
269	C₂×Q₈	—	4	1	no	—
270	C₂×Q₈	—	8	2	no	—
271	C₂×D₄	—	2	1	no	—
272	C₂×D₄	—	2	2	no	—
273	C₂×D₄	—	2	2	no	—
274	C₂×D₄	C₂×C₅×D₄	1	2	yes	—
275	C₂²×C₄	—	2	1	no	—
276	C₂²×C₄	—	2	2	no	—
277	C₂²×C₄	—	2	2	no	—
278	C₂²×C₄	—	4	1	no	—
279	C₂²×C₄	—	8	2	no	—
280	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
281	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
282	C₂²×C₄	C₅×#41	1	1	yes	characteristic
283	Q₁₆	—	8	4	no	—
284	M₁₆	—	4	2	no	—
285	M₁₆	—	8	4	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	—	2	2	no	—
288	C₂×C₈	—	2	2	no	—
289	C₂×C₈	—	4	2	no	—
290	C₂×C₈	C₂×C₅×D₄	1	4	yes	—
291	C₄⋊C₄	—	2	1	no	—
292	C₄⋊C₄	—	2	1	no	—
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	—	2	1	no	—
295	C₄⋊C₄	—	2	1	no	—
296	C₄⋊C₄	—	2	2	no	—
297	C₄⋊C₄	—	2	1	no	—
298	C₄⋊C₄	—	4	1	no	—
299	C₄⋊C₄	—	4	1	no	—
300	C₄⋊C₄	—	4	1	no	—
301	C₄⋊C₄	—	4	1	no	—
302	C₄⋊C₄	C₂×C₅×D₄	1	2	yes	—
303	C₂²⋊C₄	—	2	2	no	—
304	C₂²⋊C₄	—	2	2	no	—
305	C₂²⋊C₄	—	4	1	no	—
306	C₄²	—	4	1	no	—
307	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
308	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
309	C₄²	C₅×#41	1	1	yes	characteristic
310	C₄²	C₂⁴×C₅	1	1	yes	characteristic
311	C₂×C₅	—	2	2	no	—
312	C₂×C₅	—	8	2	no	—
313	C₂×C₅	#3041	1	1	yes	characteristic
314	C₂×C₅	C₂×#470	1	1	yes	characteristic
315	C₂×C₅	#3069	1	1	yes	characteristic
316	C₂³	—	2	2	no	—
317	C₂³	—	4	2	no	—
318	C₂³	—	8	1	no	—
319	C₂³	—	8	1	no	—
320	C₂³	C₅×#121	1	1	yes	characteristic
321	Q₈	—	4	2	no	—
322	Q₈	—	4	4	no	—
323	Q₈	—	8	1	no	—
324	Q₈	—	8	2	no	—
325	D₄	—	2	2	no	—
326	D₄	—	2	2	no	—
327	D₄	—	4	2	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	4	1	no	—
331	C₂×C₄	—	4	4	no	—
332	C₂×C₄	—	4	1	no	—
333	C₂×C₄	—	4	1	no	—
334	C₂×C₄	—	4	2	no	—
335	C₂×C₄	—	8	2	no	—
336	C₂×C₄	—	8	1	no	—
337	C₂×C₄	—	8	2	no	—
338	C₂×C₄	C₅×#121	1	1	yes	characteristic
339	C₂×C₄	C₅×#121	1	1	yes	characteristic
340	C₂×C₄	C₅×#121	1	1	yes	characteristic
341	C₂×C₄	C₅×#137	1	2	yes	—
342	C₂×C₄	C₅×#137	1	2	yes	—
343	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
344	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
345	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
346	C₈	—	2	4	no	—
347	C₈	—	4	2	no	—
348	C₈	—	4	2	no	—
349	C₈	—	4	2	no	—
350	C₅	#19423	1	1	yes	characteristic, direct product factor, Sylow
351	C₂²	—	2	1	no	—
352	C₂²	—	2	1	no	—
353	C₂²	—	4	2	no	—
354	C₂²	—	4	2	no	—
355	C₂²	—	4	2	no	—
356	C₂²	—	8	4	no	—
357	C₂²	C₅×#470	1	2	yes	—
358	C₂²	C₂×C₅×#121	1	1	yes	characteristic
359	C₄	—	2	1	no	—
360	C₄	—	2	1	no	—
361	C₄	—	2	1	no	—
362	C₄	—	4	2	no	—
363	C₄	—	8	2	no	—
364	C₄	—	8	1	no	—
365	C₄	—	8	1	no	—
366	C₄	C₅×#470	1	2	yes	—
367	C₄	C₂×C₅×#137	1	1	yes	characteristic
368	C₄	C₂×C₅×#137	1	1	yes	characteristic
369	C₄	C₅×#576	1	2	yes	—
370	C₂	—	2	2	no	—
371	C₂	—	8	2	no	—
372	C₂	C₅×#3041	1	1	yes	characteristic
373	C₂	C₂×C₅×#470	1	1	yes	characteristic
374	C₂	C₅×#3069	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅
prime radical	C₂²×C₅×D₄	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. C₅×#3209	C₂	semidirect
5. C₅×#1703	C₂	semidirect
6. C₅×#1689	C₂	semidirect
7. C₅×#1570	C₂	semidirect
8. C₅×#1549	C₂	semidirect
9. C₅×#1542	C₂	semidirect
13. C₅×#494	C₂²	semidirect
14. C₅×#487	C₂²	semidirect
33. C₅×#334	C₂²	semidirect
34. C₅×#333	C₂²	semidirect
35. C₅×#332	C₂²	semidirect
36. C₅×#332	C₂²	semidirect
37. #19423	C₅	direct (N × H)
84. #3209	C₂×C₅	semidirect
88. #1703	C₂×C₅	semidirect
89. #1689	C₂×C₅	semidirect
90. #1570	C₂×C₅	semidirect
91. #1549	C₂×C₅	semidirect
92. #1542	C₂×C₅	semidirect
142. #494	C₂²×C₅	semidirect
143. #487	C₂²×C₅	semidirect
162. #334	C₂²×C₅	semidirect
163. #333	C₂²×C₅	semidirect
164. #332	C₂²×C₅	semidirect
165. #332	C₂²×C₅	semidirect
350. C₅	#19423	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×#3209	2
C₅×#3077	2
C₂×C₅×#474	2
C₅×#2975	2
C₅×#1703	2
C₅×#1689	2
C₅×#1570	2
C₅×#1549	2
C₅×#1542	2
#19423	5

Subgroups — C5×#19423 #11942143

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#19423 #11942143