Subgroups — C₅×#9295 #11932015

Order 1280.

594 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3208	C₂	1	1	yes	maximal, characteristic
2	C₅×#2328	C₂	1	1	yes	maximal, characteristic
3	C₅×#1995	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1545	C₂	1	1	yes	maximal, characteristic
5	C₅×#1518	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1286	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1274	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#533	C₂²	1	1	yes	characteristic
9	C₂×C₅×#127	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₅×D₄	—	2	1	no	—
11	C₅×#492	C₄	1	2	yes	complemented
12	C₅×#486	C₄	1	2	yes	complemented
13	C₅×#460	—	2	2	no	—
14	C₅×#450	—	2	2	no	—
15	C₅×#431	C₂²	1	1	yes	characteristic
16	C₅×#431	C₂²	1	1	yes	characteristic
17	C₅×#423	C₂²	1	1	yes	characteristic
18	C₅×#390	—	2	1	no	—
19	C₅×#386	—	2	2	no	—
20	C₅×#386	—	2	2	no	—
21	C₅×#382	C₂²	1	1	yes	characteristic
22	C₅×#376	C₂²	1	1	yes	characteristic
23	C₅×#339	—	2	1	no	—
24	C₅×#339	—	2	1	no	—
25	C₅×#331	—	2	2	no	—
26	C₅×#328	—	2	2	no	—
27	#9295	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
28	C₂²×C₅×D₄	—	2	1	no	—
29	C₂²×C₅×D₄	—	2	1	no	—
30	C₂²×C₅×D₄	D₄	1	2	yes	—
31	C₂³×C₄×C₅	—	2	2	no	—
32	C₂×C₅×SD₁₆	—	2	4	no	—
33	C₂×C₅×M₁₆	—	2	1	no	—
34	C₅×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic, complemented
35	C₅×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic, complemented
36	C₅×#127	—	2	1	no	—
37	C₅×#127	—	2	1	no	—
38	C₅×#127	C₂×C₄	1	2	yes	—
39	C₅×#127	D₄	1	2	yes	—
40	C₅×#125	—	2	1	no	—
41	C₅×C₂²⋊Q₈	—	2	1	no	—
42	C₅×C₂²⋊Q₈	—	2	1	no	—
43	C₄×C₅×Q₈	—	2	1	no	—
44	C₄×C₅×Q₈	—	2	1	no	—
45	C₄×C₅×D₄	—	4	2	no	—
46	C₄×C₅×D₄	—	4	2	no	—
47	C₅×#117	—	2	1	no	—
48	C₂×C₅×C₄⋊C₄	—	2	1	no	—
49	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
50	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
51	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
52	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
53	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
54	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
55	C₅×C₄⋊C₈	—	2	1	no	—
56	C₅×C₄⋊C₈	—	2	1	no	—
57	C₅×C₄⋊C₈	—	2	1	no	—
58	C₅×C₄⋊C₈	—	2	1	no	—
59	C₅×#102	—	2	4	no	—
60	C₅×C₂²⋊C₈	—	2	1	no	—
61	C₅×C₂²⋊C₈	—	2	1	no	—
62	C₄×C₅×C₈	—	2	1	no	—
63	C₄×C₅×C₈	C₂×C₄	1	2	yes	complemented
64	C₅×#95	—	2	1	no	—
65	C₅×#95	—	2	1	no	—
66	C₅×#95	—	2	1	no	—
67	C₅×#95	D₄	1	2	yes	—
68	#3208	C₂×C₅	1	1	yes	characteristic
69	#2328	C₂×C₅	1	1	yes	characteristic
70	#1995	C₂×C₅	1	1	yes	characteristic, complemented
71	#1545	C₂×C₅	1	1	yes	characteristic
72	#1518	C₂×C₅	1	1	yes	characteristic, complemented
73	#1286	C₂×C₅	1	1	yes	characteristic, complemented
74	#1274	C₂×C₅	1	1	yes	characteristic, complemented
75	C₂⁴×C₅	—	2	2	no	—
76	C₂×C₅×Q₈	—	2	1	no	—
77	C₂×C₅×Q₈	—	2	1	no	—
78	C₂×C₅×D₄	—	2	2	no	—
79	C₂×C₅×D₄	—	2	2	no	—
80	C₂×C₅×D₄	—	2	2	no	—
81	C₂×C₅×D₄	—	2	2	no	—
82	C₂×C₅×D₄	—	2	2	no	—
83	C₂×C₅×D₄	—	4	1	no	—
84	C₂×C₅×D₄	—	4	1	no	—
85	C₂×C₅×D₄	—	4	1	no	—
86	C₂×C₅×D₄	—	4	1	no	—
87	C₂×C₅×D₄	—	8	2	no	—
88	C₂×C₅×D₄	—	8	2	no	—
89	C₂×C₅×D₄	C₂²⋊C₄	1	4	yes	—
90	C₂²×C₄×C₅	—	2	1	no	—
91	C₂²×C₄×C₅	—	2	1	no	—
92	C₂²×C₄×C₅	—	2	2	no	—
93	C₂²×C₄×C₅	—	4	4	no	—
94	C₂²×C₄×C₅	—	8	2	no	—
95	C₂²×C₄×C₅	—	8	2	no	—
96	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
99	C₅×SD₁₆	—	8	4	no	—
100	C₅×M₁₆	—	4	1	no	—
101	C₅×M₁₆	—	4	1	no	—
102	C₂×C₅×C₈	—	2	2	no	—
103	C₂×C₅×C₈	—	4	1	no	—
104	C₂×C₅×C₈	—	4	1	no	—
105	C₅×C₄⋊C₄	—	2	1	no	—
106	C₅×C₄⋊C₄	—	2	1	no	—
107	C₅×C₄⋊C₄	—	2	1	no	—
108	C₅×C₄⋊C₄	—	2	1	no	—
109	C₅×C₄⋊C₄	—	4	1	no	—
110	C₅×C₄⋊C₄	—	4	1	no	—
111	C₅×C₄⋊C₄	—	4	1	no	—
112	C₅×C₄⋊C₄	—	4	1	no	—
113	C₅×C₂²⋊C₄	—	4	4	no	—
114	C₅×C₂²⋊C₄	—	4	1	no	—
115	C₅×C₂²⋊C₄	—	8	2	no	—
116	C₅×C₂²⋊C₄	—	8	2	no	—
117	C₄²×C₅	—	2	1	no	—
118	C₄²×C₅	—	2	1	no	—
119	C₄²×C₅	—	2	1	no	—
120	C₄²×C₅	—	4	1	no	—
121	C₄²×C₅	SD₁₆	1	2	yes	complemented
122	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
123	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
124	C₄²×C₅	#41	1	1	yes	characteristic
125	C₄²×C₅	#41	1	1	yes	characteristic
126	#533	C₂²×C₅	1	1	yes	characteristic
127	C₂×#127	C₂²×C₅	1	1	yes	characteristic
128	C₂×C₄×D₄	—	2	1	no	—
129	#492	C₄×C₅	1	2	yes	complemented
130	#486	C₄×C₅	1	2	yes	complemented
131	#460	—	2	2	no	—
132	#450	—	2	2	no	—
133	#431	C₂²×C₅	1	1	yes	characteristic
134	#431	C₂²×C₅	1	1	yes	characteristic
135	#423	C₂²×C₅	1	1	yes	characteristic
136	#390	—	2	1	no	—
137	#386	—	2	2	no	—
138	#386	—	2	2	no	—
139	#382	C₂²×C₅	1	1	yes	characteristic
140	#376	C₂²×C₅	1	1	yes	characteristic
141	#339	—	2	1	no	—
142	#339	—	2	1	no	—
143	#331	—	2	2	no	—
144	#328	—	2	2	no	—
145	C₂³×C₅	—	2	4	no	—
146	C₂³×C₅	—	4	2	no	—
147	C₂³×C₅	—	4	2	no	—
148	C₂³×C₅	—	4	2	no	—
149	C₂³×C₅	—	4	2	no	—
150	C₂³×C₅	—	4	2	no	—
151	C₂³×C₅	—	4	2	no	—
152	C₂³×C₅	#120	1	1	yes	characteristic
153	C₅×Q₈	—	4	2	no	—
154	C₅×Q₈	—	8	1	no	—
155	C₅×D₄	—	4	4	no	—
156	C₅×D₄	—	4	2	no	—
157	C₅×D₄	—	4	2	no	—
158	C₅×D₄	—	4	2	no	—
159	C₅×D₄	—	4	2	no	—
160	C₅×D₄	—	8	2	no	—
161	C₅×D₄	—	8	2	no	—
162	C₅×D₄	—	8	2	no	—
163	C₂×C₄×C₅	—	2	2	no	—
164	C₂×C₄×C₅	—	4	1	no	—
165	C₂×C₄×C₅	—	4	1	no	—
166	C₂×C₄×C₅	—	4	1	no	—
167	C₂×C₄×C₅	—	4	1	no	—
168	C₂×C₄×C₅	—	4	1	no	—
169	C₂×C₄×C₅	—	4	1	no	—
170	C₂×C₄×C₅	—	4	2	no	—
171	C₂×C₄×C₅	—	4	2	no	—
172	C₂×C₄×C₅	—	4	1	no	—
173	C₂×C₄×C₅	—	4	1	no	—
174	C₂×C₄×C₅	—	4	2	no	—
175	C₂×C₄×C₅	—	4	1	no	—
176	C₂×C₄×C₅	—	8	8	no	—
177	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
178	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
179	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
180	C₂×C₄×C₅	#121	1	1	yes	characteristic
181	C₂×C₄×C₅	#121	1	1	yes	characteristic
182	C₂×C₄×C₅	#123	1	1	yes	characteristic
183	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
184	C₂×C₄×C₅	#135	1	1	yes	characteristic
185	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
186	C₂×C₄×C₅	#137	1	1	yes	characteristic
187	C₅×C₈	—	4	2	no	—
188	C₅×C₈	—	8	1	no	—
189	C₅×C₈	—	8	1	no	—
190	C₂²×D₄	—	2	1	no	—
191	C₂²×D₄	—	2	1	no	—
192	C₂²×D₄	C₅×D₄	1	2	yes	—
193	C₂³×C₄	—	2	2	no	—
194	C₂×SD₁₆	—	2	4	no	—
195	C₂×M₁₆	—	2	1	no	—
196	C₄⋊Q₈	C₂×C₄×C₅	1	1	yes	characteristic, complemented
197	C₄⋊Q₈	C₂×C₄×C₅	1	1	yes	characteristic, complemented
198	#127	—	2	1	no	—
199	#127	—	2	1	no	—
200	#127	C₂×C₄×C₅	1	2	yes	—
201	#127	C₅×D₄	1	2	yes	—
202	#125	—	2	1	no	—
203	C₂²⋊Q₈	—	2	1	no	—
204	C₂²⋊Q₈	—	2	1	no	—
205	C₄×Q₈	—	2	1	no	—
206	C₄×Q₈	—	2	1	no	—
207	C₄×D₄	—	4	2	no	—
208	C₄×D₄	—	4	2	no	—
209	#117	—	2	1	no	—
210	C₂×C₄⋊C₄	—	2	1	no	—
211	C₂×C₂²⋊C₄	—	2	2	no	—
212	C₂×C₂²⋊C₄	—	2	2	no	—
213	C₂×C₂²⋊C₄	—	2	2	no	—
214	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
215	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
216	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
217	C₄⋊C₈	—	2	1	no	—
218	C₄⋊C₈	—	2	1	no	—
219	C₄⋊C₈	—	2	1	no	—
220	C₄⋊C₈	—	2	1	no	—
221	#102	—	2	4	no	—
222	C₂²⋊C₈	—	2	1	no	—
223	C₂²⋊C₈	—	2	1	no	—
224	C₄×C₈	—	2	1	no	—
225	C₄×C₈	C₂×C₄×C₅	1	2	yes	complemented
226	#95	—	2	1	no	—
227	#95	—	2	1	no	—
228	#95	—	2	1	no	—
229	#95	C₅×D₄	1	2	yes	—
230	C₂²×C₅	—	2	1	no	—
231	C₂²×C₅	—	2	1	no	—
232	C₂²×C₅	—	4	4	no	—
233	C₂²×C₅	—	4	4	no	—
234	C₂²×C₅	—	4	4	no	—
235	C₂²×C₅	—	8	2	no	—
236	C₂²×C₅	—	8	2	no	—
237	C₂²×C₅	—	8	2	no	—
238	C₂²×C₅	—	8	2	no	—
239	C₂²×C₅	#386	1	1	yes	characteristic
240	C₂²×C₅	#450	1	1	yes	characteristic
241	C₂²×C₅	#452	1	1	yes	characteristic
242	C₄×C₅	—	2	1	no	—
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	1	no	—
248	C₄×C₅	—	8	2	no	—
249	C₄×C₅	—	8	1	no	—
250	C₄×C₅	—	8	1	no	—
251	C₄×C₅	C₄×SD₁₆	1	1	yes	characteristic
252	C₄×C₅	#440	1	1	yes	characteristic
253	C₄×C₅	#461	1	1	yes	characteristic
254	C₄×C₅	#463	1	1	yes	characteristic
255	C₂⁴	—	2	2	no	—
256	C₂×Q₈	—	2	1	no	—
257	C₂×Q₈	—	2	1	no	—
258	C₂×D₄	—	2	2	no	—
259	C₂×D₄	—	2	2	no	—
260	C₂×D₄	—	2	2	no	—
261	C₂×D₄	—	2	2	no	—
262	C₂×D₄	—	2	2	no	—
263	C₂×D₄	—	4	1	no	—
264	C₂×D₄	—	4	1	no	—
265	C₂×D₄	—	4	1	no	—
266	C₂×D₄	—	4	1	no	—
267	C₂×D₄	—	8	2	no	—
268	C₂×D₄	—	8	2	no	—
269	C₂×D₄	C₅×C₂²⋊C₄	1	4	yes	—
270	C₂²×C₄	—	2	1	no	—
271	C₂²×C₄	—	2	1	no	—
272	C₂²×C₄	—	2	2	no	—
273	C₂²×C₄	—	4	4	no	—
274	C₂²×C₄	—	8	2	no	—
275	C₂²×C₄	—	8	2	no	—
276	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
277	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
278	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
279	SD₁₆	—	8	4	no	—
280	M₁₆	—	4	1	no	—
281	M₁₆	—	4	1	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	4	1	no	—
284	C₂×C₈	—	4	1	no	—
285	C₄⋊C₄	—	2	1	no	—
286	C₄⋊C₄	—	2	1	no	—
287	C₄⋊C₄	—	2	1	no	—
288	C₄⋊C₄	—	2	1	no	—
289	C₄⋊C₄	—	4	1	no	—
290	C₄⋊C₄	—	4	1	no	—
291	C₄⋊C₄	—	4	1	no	—
292	C₄⋊C₄	—	4	1	no	—
293	C₂²⋊C₄	—	4	4	no	—
294	C₂²⋊C₄	—	4	1	no	—
295	C₂²⋊C₄	—	8	2	no	—
296	C₂²⋊C₄	—	8	2	no	—
297	C₄²	—	2	1	no	—
298	C₄²	—	2	1	no	—
299	C₄²	—	2	1	no	—
300	C₄²	—	4	1	no	—
301	C₄²	C₅×SD₁₆	1	2	yes	complemented
302	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
303	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
304	C₄²	C₅×#41	1	1	yes	characteristic
305	C₄²	C₅×#41	1	1	yes	characteristic
306	C₂×C₅	—	2	1	no	—
307	C₂×C₅	—	2	1	no	—
308	C₂×C₅	—	8	4	no	—
309	C₂×C₅	#1877	1	1	yes	characteristic
310	C₂×C₅	#1904	1	1	yes	characteristic
311	C₂×C₅	#1911	1	1	yes	characteristic
312	C₂³	—	2	4	no	—
313	C₂³	—	4	2	no	—
314	C₂³	—	4	2	no	—
315	C₂³	—	4	2	no	—
316	C₂³	—	4	2	no	—
317	C₂³	—	4	2	no	—
318	C₂³	—	4	2	no	—
319	C₂³	C₅×#120	1	1	yes	characteristic
320	Q₈	—	4	2	no	—
321	Q₈	—	8	1	no	—
322	D₄	—	4	4	no	—
323	D₄	—	4	2	no	—
324	D₄	—	4	2	no	—
325	D₄	—	4	2	no	—
326	D₄	—	4	2	no	—
327	D₄	—	8	2	no	—
328	D₄	—	8	2	no	—
329	D₄	—	8	2	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	4	1	no	—
332	C₂×C₄	—	4	1	no	—
333	C₂×C₄	—	4	1	no	—
334	C₂×C₄	—	4	1	no	—
335	C₂×C₄	—	4	1	no	—
336	C₂×C₄	—	4	1	no	—
337	C₂×C₄	—	4	2	no	—
338	C₂×C₄	—	4	2	no	—
339	C₂×C₄	—	4	1	no	—
340	C₂×C₄	—	4	1	no	—
341	C₂×C₄	—	4	2	no	—
342	C₂×C₄	—	4	1	no	—
343	C₂×C₄	—	8	8	no	—
344	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
345	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
346	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
347	C₂×C₄	C₅×#121	1	1	yes	characteristic
348	C₂×C₄	C₅×#121	1	1	yes	characteristic
349	C₂×C₄	C₅×#123	1	1	yes	characteristic
350	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
351	C₂×C₄	C₅×#135	1	1	yes	characteristic
352	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
353	C₂×C₄	C₅×#137	1	1	yes	characteristic
354	C₈	—	4	2	no	—
355	C₈	—	8	1	no	—
356	C₈	—	8	1	no	—
357	C₅	#9295	1	1	yes	characteristic, direct product factor, Sylow
358	C₂²	—	2	1	no	—
359	C₂²	—	2	1	no	—
360	C₂²	—	4	4	no	—
361	C₂²	—	4	4	no	—
362	C₂²	—	4	4	no	—
363	C₂²	—	8	2	no	—
364	C₂²	—	8	2	no	—
365	C₂²	—	8	2	no	—
366	C₂²	—	8	2	no	—
367	C₂²	C₅×#386	1	1	yes	characteristic
368	C₂²	C₅×#450	1	1	yes	characteristic
369	C₂²	C₅×#452	1	1	yes	characteristic
370	C₄	—	2	1	no	—
371	C₄	—	2	1	no	—
372	C₄	—	2	1	no	—
373	C₄	—	2	1	no	—
374	C₄	—	4	2	no	—
375	C₄	—	8	2	no	—
376	C₄	—	8	1	no	—
377	C₄	—	8	1	no	—
378	C₄	—	8	1	no	—
379	C₄	C₄×C₅×SD₁₆	1	1	yes	characteristic
380	C₄	C₅×#440	1	1	yes	characteristic
381	C₄	C₅×#461	1	1	yes	characteristic
382	C₄	C₅×#463	1	1	yes	characteristic
383	C₂	—	2	1	no	—
384	C₂	—	2	1	no	—
385	C₂	—	8	4	no	—
386	C₂	C₅×#1877	1	1	yes	characteristic
387	C₂	C₅×#1904	1	1	yes	characteristic
388	C₂	C₅×#1911	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	#386
commutator	C₄²	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₂×C₅×#127	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
3. C₅×#1995	C₂	semidirect
5. C₅×#1518	C₂	semidirect
6. C₅×#1286	C₂	semidirect
7. C₅×#1274	C₂	semidirect
11. C₅×#492	C₄	semidirect
12. C₅×#486	C₄	semidirect
27. #9295	C₅	direct (N × H)
34. C₅×C₄⋊Q₈	C₂×C₄	semidirect
35. C₅×C₄⋊Q₈	C₂×C₄	semidirect
63. C₄×C₅×C₈	C₂×C₄	semidirect
70. #1995	C₂×C₅	semidirect
72. #1518	C₂×C₅	semidirect
73. #1286	C₂×C₅	semidirect
74. #1274	C₂×C₅	semidirect
121. C₄²×C₅	SD₁₆	semidirect
129. #492	C₄×C₅	semidirect
130. #486	C₄×C₅	semidirect
196. C₄⋊Q₈	C₂×C₄×C₅	semidirect
197. C₄⋊Q₈	C₂×C₄×C₅	semidirect
225. C₄×C₈	C₂×C₄×C₅	semidirect
301. C₄²	C₅×SD₁₆	semidirect
357. C₅	#9295	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×#3208	2
C₅×#2328	2
C₅×#1995	2
C₅×#1545	2
C₅×#1518	2
C₅×#1286	2
C₅×#1274	2
#9295	5

Subgroups — C5×#9295 #11932015

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#9295 #11932015