Subgroups — C₅×#32829 #11955549

Order 1280.

612 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×#2647	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#2640	C₂	1	1	yes	maximal, characteristic
3	C₅×#2095	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#2093	C₂	1	1	yes	maximal, characteristic
5	C₅×#2074	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#2073	C₂	1	1	yes	maximal, characteristic
7	C₅×#2068	C₂	1	2	yes	maximal, complemented
8	C₅×#2067	C₂	1	2	yes	maximal
9	C₅×#2057	C₂	1	2	yes	maximal, complemented
10	C₅×#2042	C₂	1	2	yes	maximal
11	C₅×#1832	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₅×#124	C₂²	1	1	yes	characteristic
14	C₂×C₅×#107	C₂²	1	2	yes	—
15	C₂×C₅×#107	C₂²	1	2	yes	—
16	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
17	C₂×C₅×#103	C₂²	1	2	yes	—
18	C₂×C₅×#103	C₂²	1	2	yes	—
19	C₂×C₅×#102	C₂²	1	2	yes	—
20	C₂×C₅×#102	C₂²	1	2	yes	—
21	C₅×#400	—	2	2	no	—
22	C₅×#398	—	2	2	no	—
23	C₅×#397	—	2	2	no	—
24	C₅×#393	—	2	2	no	—
25	C₅×#391	C₂²	1	1	yes	characteristic
26	C₅×#391	C₂²	1	1	yes	characteristic
27	C₅×#390	C₂²	1	1	yes	characteristic
28	C₅×#390	C₂²	1	1	yes	characteristic
29	C₅×#389	C₂²	1	2	yes	—
30	C₅×#389	C₂²	1	2	yes	—
31	C₅×#384	C₂²	1	1	yes	characteristic
32	C₅×#382	C₂²	1	1	yes	characteristic
33	C₅×#376	C₂²	1	1	yes	characteristic
34	C₅×#340	C₂²	1	2	yes	—
35	C₅×#340	C₂²	1	2	yes	—
36	C₅×#336	C₂²	1	2	yes	—
37	C₅×#336	C₂²	1	1	yes	characteristic
38	C₅×#336	C₂²	1	1	yes	characteristic
39	#32829	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
40	C₂²×C₅×Q₈	—	2	1	no	—
41	C₂²×C₅×Q₈	C₂³	1	1	yes	characteristic
42	C₂²×C₅×D₄	C₂³	1	1	yes	characteristic
43	C₂²×C₅×C₈	C₂³	1	2	yes	—
44	C₂²×C₅×C₈	C₂³	1	2	yes	—
45	C₅×C₄⋊Q₈	—	4	2	no	—
46	C₅×#124	—	4	2	no	—
47	C₂×C₅×C₄⋊C₄	—	2	1	no	—
48	C₂×C₅×C₄⋊C₄	—	2	2	no	—
49	C₂×C₅×C₄⋊C₄	—	2	2	no	—
50	C₂×C₅×C₄⋊C₄	—	2	2	no	—
51	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
52	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
53	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
54	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
55	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
56	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
57	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
58	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
59	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
61	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
62	C₅×#107	—	4	2	no	—
63	C₅×#107	—	4	2	no	—
64	C₅×C₄⋊C₈	—	4	2	no	—
65	C₅×#103	—	4	2	no	—
66	C₅×#103	—	4	2	no	—
67	C₅×#102	—	4	2	no	—
68	C₅×#102	—	4	2	no	—
69	C₅×#95	—	2	2	no	—
70	C₅×#95	—	2	2	no	—
71	C₅×#95	—	2	2	no	—
72	C₅×#95	—	2	2	no	—
73	C₅×#95	—	2	2	no	—
74	C₅×#95	D₄	1	2	yes	—
75	C₅×#95	D₄	1	2	yes	—
76	#2647	C₂×C₅	1	1	yes	characteristic, complemented
77	#2640	C₂×C₅	1	1	yes	characteristic
78	#2095	C₂×C₅	1	1	yes	characteristic, complemented
79	#2093	C₂×C₅	1	1	yes	characteristic
80	#2074	C₂×C₅	1	1	yes	characteristic, complemented
81	#2073	C₂×C₅	1	1	yes	characteristic
82	#2068	C₂×C₅	1	2	yes	complemented
83	#2067	C₂×C₅	1	2	yes	—
84	#2057	C₂×C₅	1	2	yes	complemented
85	#2042	C₂×C₅	1	2	yes	—
86	#1832	C₂×C₅	1	1	yes	characteristic, complemented
87	C₂⁴×C₅	—	2	1	no	—
88	C₂×C₅×Q₈	—	2	2	no	—
89	C₂×C₅×Q₈	—	2	1	no	—
90	C₂×C₅×Q₈	—	2	2	no	—
91	C₂×C₅×Q₈	—	2	1	no	—
92	C₂×C₅×Q₈	—	4	2	no	—
93	C₂×C₅×Q₈	—	8	2	no	—
94	C₂×C₅×D₄	—	2	2	no	—
95	C₂×C₅×D₄	—	2	1	no	—
96	C₂×C₅×D₄	—	2	2	no	—
97	C₂×C₅×D₄	—	2	1	no	—
98	C₂²×C₄×C₅	—	2	1	no	—
99	C₂²×C₄×C₅	—	2	1	no	—
100	C₂²×C₄×C₅	—	2	1	no	—
101	C₂²×C₄×C₅	—	2	2	no	—
102	C₂²×C₄×C₅	—	2	2	no	—
103	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
106	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
107	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
108	C₂×C₅×C₈	—	2	2	no	—
109	C₂×C₅×C₈	—	2	2	no	—
110	C₂×C₅×C₈	—	2	2	no	—
111	C₂×C₅×C₈	—	2	2	no	—
112	C₂×C₅×C₈	—	2	2	no	—
113	C₂×C₅×C₈	—	2	2	no	—
114	C₅×C₄⋊C₄	—	2	2	no	—
115	C₅×C₄⋊C₄	—	2	2	no	—
116	C₅×C₄⋊C₄	—	2	2	no	—
117	C₅×C₄⋊C₄	—	2	2	no	—
118	C₅×C₄⋊C₄	—	2	2	no	—
119	C₅×C₄⋊C₄	—	2	2	no	—
120	C₅×C₄⋊C₄	—	4	2	no	—
121	C₅×C₄⋊C₄	—	8	2	no	—
122	C₅×C₄⋊C₄	—	8	2	no	—
123	C₅×C₄⋊C₄	—	8	2	no	—
124	C₅×C₂²⋊C₄	—	4	2	no	—
125	C₅×C₂²⋊C₄	—	4	2	no	—
126	C₅×C₂²⋊C₄	—	8	2	no	—
127	C₅×C₂²⋊C₄	—	8	2	no	—
128	C₄²×C₅	—	2	2	no	—
129	C₄²×C₅	—	2	1	no	—
130	C₄²×C₅	—	2	2	no	—
131	C₄²×C₅	—	2	1	no	—
132	C₂×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
133	C₂×#124	C₂²×C₅	1	1	yes	characteristic
134	C₂×#107	C₂²×C₅	1	2	yes	—
135	C₂×#107	C₂²×C₅	1	2	yes	—
136	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
137	C₂×#103	C₂²×C₅	1	2	yes	—
138	C₂×#103	C₂²×C₅	1	2	yes	—
139	C₂×#102	C₂²×C₅	1	2	yes	—
140	C₂×#102	C₂²×C₅	1	2	yes	—
141	#400	—	2	2	no	—
142	#398	—	2	2	no	—
143	#397	—	2	2	no	—
144	#393	—	2	2	no	—
145	#391	C₂²×C₅	1	1	yes	characteristic
146	#391	C₂²×C₅	1	1	yes	characteristic
147	#390	C₂²×C₅	1	1	yes	characteristic
148	#390	C₂²×C₅	1	1	yes	characteristic
149	#389	C₂²×C₅	1	2	yes	—
150	#389	C₂²×C₅	1	2	yes	—
151	#384	C₂²×C₅	1	1	yes	characteristic
152	#382	C₂²×C₅	1	1	yes	characteristic
153	#376	C₂²×C₅	1	1	yes	characteristic
154	#340	C₂²×C₅	1	2	yes	—
155	#340	C₂²×C₅	1	2	yes	—
156	#336	C₂²×C₅	1	2	yes	—
157	#336	C₂²×C₅	1	1	yes	characteristic
158	#336	C₂²×C₅	1	1	yes	characteristic
159	C₂³×C₅	—	4	1	no	—
160	C₂³×C₅	—	4	1	no	—
161	C₂³×C₅	—	4	1	no	—
162	C₂³×C₅	—	4	2	no	—
163	C₂³×C₅	—	4	2	no	—
164	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
165	C₅×Q₈	—	4	2	no	—
166	C₅×Q₈	—	4	2	no	—
167	C₅×Q₈	—	8	4	no	—
168	C₅×D₄	—	4	2	no	—
169	C₅×D₄	—	4	2	no	—
170	C₂×C₄×C₅	—	2	1	no	—
171	C₂×C₄×C₅	—	2	1	no	—
172	C₂×C₄×C₅	—	2	2	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	—	2	2	no	—
177	C₂×C₄×C₅	—	2	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	1	no	—
182	C₂×C₄×C₅	—	4	1	no	—
183	C₂×C₄×C₅	—	4	2	no	—
184	C₂×C₄×C₅	—	4	1	no	—
185	C₂×C₄×C₅	—	4	2	no	—
186	C₂×C₄×C₅	—	4	1	no	—
187	C₂×C₄×C₅	—	4	2	no	—
188	C₂×C₄×C₅	—	4	2	no	—
189	C₂×C₄×C₅	—	4	2	no	—
190	C₂×C₄×C₅	—	4	2	no	—
191	C₂×C₄×C₅	—	4	2	no	—
192	C₂×C₄×C₅	—	4	2	no	—
193	C₂×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
194	C₂×C₄×C₅	H(5,𝔽₂)	1	2	yes	—
195	C₂×C₄×C₅	2₋¹⁺⁴	1	2	yes	—
196	C₂×C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
197	C₅×C₈	—	4	2	no	—
198	C₅×C₈	—	4	2	no	—
199	C₂²×Q₈	—	2	1	no	—
200	C₂²×Q₈	C₂³×C₅	1	1	yes	characteristic
201	C₂²×D₄	C₂³×C₅	1	1	yes	characteristic
202	C₂²×C₈	C₂³×C₅	1	2	yes	—
203	C₂²×C₈	C₂³×C₅	1	2	yes	—
204	C₄⋊Q₈	—	4	2	no	—
205	#124	—	4	2	no	—
206	C₂×C₄⋊C₄	—	2	1	no	—
207	C₂×C₄⋊C₄	—	2	2	no	—
208	C₂×C₄⋊C₄	—	2	2	no	—
209	C₂×C₄⋊C₄	—	2	2	no	—
210	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
211	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
212	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
213	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
214	C₂×C₂²⋊C₄	—	2	1	no	—
215	C₂×C₂²⋊C₄	—	2	1	no	—
216	C₂×C₂²⋊C₄	—	2	2	no	—
217	C₂×C₂²⋊C₄	—	2	2	no	—
218	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
219	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
220	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
221	#107	—	4	2	no	—
222	#107	—	4	2	no	—
223	C₄⋊C₈	—	4	2	no	—
224	#103	—	4	2	no	—
225	#103	—	4	2	no	—
226	#102	—	4	2	no	—
227	#102	—	4	2	no	—
228	#95	—	2	2	no	—
229	#95	—	2	2	no	—
230	#95	—	2	2	no	—
231	#95	—	2	2	no	—
232	#95	—	2	2	no	—
233	#95	C₅×D₄	1	2	yes	—
234	#95	C₅×D₄	1	2	yes	—
235	C₂²×C₅	—	8	1	no	—
236	C₂²×C₅	—	8	1	no	—
237	C₂²×C₅	—	8	1	no	—
238	C₂²×C₅	—	8	2	no	—
239	C₂²×C₅	—	8	2	no	—
240	C₂²×C₅	#535	1	1	yes	characteristic
241	C₂²×C₅	#536	1	1	yes	characteristic
242	C₂²×C₅	#537	1	1	yes	characteristic
243	C₂²×C₅	#576	1	2	yes	—
244	C₂²×C₅	#578	1	2	yes	—
245	C₄×C₅	—	4	1	no	—
246	C₄×C₅	—	4	1	no	—
247	C₄×C₅	—	4	2	no	—
248	C₄×C₅	—	4	2	no	—
249	C₄×C₅	—	8	1	no	—
250	C₄×C₅	—	8	2	no	—
251	C₄×C₅	—	8	1	no	—
252	C₄×C₅	—	8	1	no	—
253	C₄×C₅	—	8	2	no	—
254	C₄×C₅	#562	1	2	yes	—
255	C₄×C₅	#563	1	2	yes	—
256	C₂⁴	—	2	1	no	—
257	C₂×Q₈	—	2	1	no	—
258	C₂×Q₈	—	2	2	no	—
259	C₂×Q₈	—	2	2	no	—
260	C₂×Q₈	—	2	1	no	—
261	C₂×Q₈	—	4	2	no	—
262	C₂×Q₈	—	8	2	no	—
263	C₂×D₄	—	2	1	no	—
264	C₂×D₄	—	2	2	no	—
265	C₂×D₄	—	2	2	no	—
266	C₂×D₄	—	2	1	no	—
267	C₂²×C₄	—	2	1	no	—
268	C₂²×C₄	—	2	2	no	—
269	C₂²×C₄	—	2	1	no	—
270	C₂²×C₄	—	2	1	no	—
271	C₂²×C₄	—	2	2	no	—
272	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
273	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
274	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
275	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
276	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	—	2	2	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	2	2	no	—
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	—	2	2	no	—
285	C₄⋊C₄	—	2	2	no	—
286	C₄⋊C₄	—	2	2	no	—
287	C₄⋊C₄	—	2	2	no	—
288	C₄⋊C₄	—	2	2	no	—
289	C₄⋊C₄	—	4	2	no	—
290	C₄⋊C₄	—	8	2	no	—
291	C₄⋊C₄	—	8	2	no	—
292	C₄⋊C₄	—	8	2	no	—
293	C₂²⋊C₄	—	4	2	no	—
294	C₂²⋊C₄	—	4	2	no	—
295	C₂²⋊C₄	—	8	2	no	—
296	C₂²⋊C₄	—	8	2	no	—
297	C₄²	—	2	1	no	—
298	C₄²	—	2	2	no	—
299	C₄²	—	2	2	no	—
300	C₄²	—	2	1	no	—
301	C₂×C₅	—	16	1	no	—
302	C₂×C₅	#2820	1	1	yes	characteristic
303	C₂×C₅	#3220	1	1	yes	characteristic
304	C₂×C₅	#3223	1	1	yes	characteristic
305	C₂×C₅	#3243	1	2	yes	—
306	C₂×C₅	#3262	1	2	yes	—
307	C₂³	—	4	1	no	—
308	C₂³	—	4	1	no	—
309	C₂³	—	4	1	no	—
310	C₂³	—	4	2	no	—
311	C₂³	—	4	2	no	—
312	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
313	Q₈	—	4	2	no	—
314	Q₈	—	4	2	no	—
315	Q₈	—	8	4	no	—
316	D₄	—	4	2	no	—
317	D₄	—	4	2	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	4	1	no	—
329	C₂×C₄	—	4	1	no	—
330	C₂×C₄	—	4	1	no	—
331	C₂×C₄	—	4	2	no	—
332	C₂×C₄	—	4	1	no	—
333	C₂×C₄	—	4	2	no	—
334	C₂×C₄	—	4	1	no	—
335	C₂×C₄	—	4	2	no	—
336	C₂×C₄	—	4	2	no	—
337	C₂×C₄	—	4	2	no	—
338	C₂×C₄	—	4	2	no	—
339	C₂×C₄	—	4	2	no	—
340	C₂×C₄	—	4	2	no	—
341	C₂×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
342	C₂×C₄	C₅×H(5,𝔽₂)	1	2	yes	—
343	C₂×C₄	C₅×2₋¹⁺⁴	1	2	yes	—
344	C₂×C₄	C₅×2₋¹⁺⁴	1	1	yes	characteristic
345	C₈	—	4	2	no	—
346	C₈	—	4	2	no	—
347	C₅	#32829	1	1	yes	characteristic, direct product factor, Sylow
348	C₂²	—	8	2	no	—
349	C₂²	—	8	2	no	—
350	C₂²	—	8	1	no	—
351	C₂²	—	8	1	no	—
352	C₂²	—	8	1	no	—
353	C₂²	C₅×#535	1	1	yes	characteristic
354	C₂²	C₅×#536	1	1	yes	characteristic
355	C₂²	C₅×#537	1	1	yes	characteristic
356	C₂²	C₅×#576	1	2	yes	—
357	C₂²	C₅×#578	1	2	yes	—
358	C₄	—	4	2	no	—
359	C₄	—	4	2	no	—
360	C₄	—	4	1	no	—
361	C₄	—	4	1	no	—
362	C₄	—	8	2	no	—
363	C₄	—	8	1	no	—
364	C₄	—	8	1	no	—
365	C₄	—	8	1	no	—
366	C₄	—	8	2	no	—
367	C₄	C₅×#562	1	2	yes	—
368	C₄	C₅×#563	1	2	yes	—
369	C₂	—	16	1	no	—
370	C₂	C₅×#2820	1	1	yes	characteristic
371	C₂	C₅×#3220	1	1	yes	characteristic
372	C₂	C₅×#3223	1	1	yes	characteristic
373	C₂	C₅×#3243	1	2	yes	—
374	C₂	C₅×#3262	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
prime radical	C₂²×C₅×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₅×#376 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅×#2647	C₂	semidirect
3. C₅×#2095	C₂	semidirect
5. C₅×#2074	C₂	semidirect
7. C₅×#2068	C₂	semidirect
9. C₅×#2057	C₂	semidirect
11. C₅×#1832	C₂	semidirect
39. #32829	C₅	direct (N × H)
76. #2647	C₂×C₅	semidirect
78. #2095	C₂×C₅	semidirect
80. #2074	C₂×C₅	semidirect
82. #2068	C₂×C₅	semidirect
84. #2057	C₂×C₅	semidirect
86. #1832	C₂×C₅	semidirect
347. C₅	#32829	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×#2647	2
C₅×#2640	2
C₅×#2095	2
C₅×#2093	2
C₅×#2074	2
C₅×#2073	2
C₅×#2068	2
C₅×#2067	2
C₅×#2057	2
C₅×#2042	2
C₅×#1832	2
#32829	5

Subgroups — C5×#32829 #11955549

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#32829 #11955549