Subgroups — C₅×#5779 #11756164

Order 960.

496 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#1010	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#1007	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×#137	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#950	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₅×#216	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#918	C₂	1	2	yes	maximal, complemented
7	C₅×D₃×Q₁₆	C₂	1	2	yes	maximal, complemented
8	C₅×#916	C₂	1	2	yes	maximal, complemented
9	C₅×#915	C₂	1	2	yes	maximal, complemented
10	C₅×#909	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅×#907	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#578	—	3	1	no	maximal, Hall
13	C₃×C₅×#41	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
15	C₅×#239	—	2	1	no	—
16	C₅×#239	C₂²	1	2	yes	—
17	C₅×D₃×Q₈	—	2	1	no	—
18	C₅×D₃×Q₈	C₂²	1	2	yes	—
19	C₅×D₃×Q₈	C₂²	1	2	yes	—
20	C₅×#237	—	2	2	no	—
21	C₅×#237	C₂²	1	2	yes	—
22	C₅×#235	—	2	2	no	—
23	C₅×#235	—	2	1	no	—
24	C₅×#235	C₂²	1	1	yes	characteristic
25	C₅×#235	C₂²	1	1	yes	characteristic
26	C₂×C₅×Dic₆	—	2	1	no	—
27	C₂×C₅×Dic₆	C₂²	1	1	yes	characteristic
28	C₃×C₅×Q₁₆	C₂²	1	2	yes	—
29	C₃×C₅×SD₁₆	C₂²	1	2	yes	—
30	C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
31	C₅×#216	C₂²	1	2	yes	—
32	C₅×#216	C₂²	1	2	yes	—
33	C₅×#216	C₂²	1	1	yes	characteristic
34	C₅×#215	C₂²	1	1	yes	characteristic
35	C₅×#214	C₂²	1	1	yes	characteristic
36	C₅×#213	C₂²	1	1	yes	characteristic
37	C₂×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
38	C₅×Dic₁₂	C₂²	1	2	yes	—
39	C₅×#204	C₂²	1	2	yes	—
40	C₅×ΓL(1,𝔽₂₅)	C₂²	1	2	yes	—
41	C₅×D₃×C₈	C₂²	1	2	yes	—
42	#5779	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
43	C₅×2₋¹⁺⁴	—	3	1	no	—
44	C₅×2₋¹⁺⁴	—	3	1	no	—
45	C₅×#137	—	3	2	no	—
46	C₅×#137	—	3	2	no	—
47	C₅×#137	—	3	1	no	—
48	C₅×#137	D₃	1	1	yes	characteristic, complemented
49	C₅×#135	—	3	1	no	—
50	C₅×#135	—	3	2	no	—
51	C₂×C₅×Q₁₆	—	3	1	no	—
52	C₂×C₅×Q₁₆	—	3	2	no	—
53	C₅×#131	—	3	1	no	—
54	C₃×C₅×Q₈	—	2	1	no	—
55	C₃×C₅×Q₈	C₂³	1	1	yes	characteristic
56	C₃×C₅×Q₈	C₂³	1	2	yes	—
57	C₃×C₅×D₄	—	2	1	no	—
58	C₃×C₅×D₄	C₂³	1	1	yes	characteristic
59	C₂×C₃×C₄×C₅	—	2	1	no	—
60	C₂×C₃×C₄×C₅	—	2	1	no	—
61	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
62	C₅×#67	—	2	1	no	—
63	C₅×#67	—	2	2	no	—
64	C₅×#67	D₄	1	2	yes	—
65	C₂×C₅×Dic₃	—	2	1	no	—
66	C₂×C₅×Dic₃	—	2	2	no	—
67	C₅×D₁₂	—	2	2	no	—
68	C₅×D₁₂	D₄	1	1	yes	characteristic
69	C₅×D₁₂	C₂³	1	1	yes	characteristic
70	C₄×C₅×D₃	—	2	2	no	—
71	C₄×C₅×D₃	—	2	2	no	—
72	C₄×C₅×D₃	—	2	1	no	—
73	C₄×C₅×D₃	—	2	2	no	—
74	C₄×C₅×D₃	C₂³	1	2	yes	—
75	C₅×Dic₆	—	2	1	no	—
76	C₅×Dic₆	—	2	2	no	—
77	C₅×Dic₆	—	2	2	no	—
78	C₅×Dic₆	D₄	1	1	yes	characteristic
79	C₅×Dic₆	C₂³	1	2	yes	—
80	C₅×Dic₆	C₂³	1	1	yes	characteristic
81	C₃×C₅×C₈	C₂³	1	2	yes	—
82	C₅×C₃⋊C₈	C₂³	1	1	yes	characteristic
83	C₅×C₃⋊C₈	C₂³	1	1	yes	characteristic
84	#1010	C₂×C₅	1	1	yes	characteristic, complemented
85	#1007	C₂×C₅	1	1	yes	characteristic, complemented
86	C₃×#137	C₂×C₅	1	1	yes	characteristic, complemented
87	#950	C₂×C₅	1	1	yes	characteristic, complemented
88	C₂×#216	C₂×C₅	1	1	yes	characteristic, complemented
89	#918	C₂×C₅	1	2	yes	complemented
90	D₃×Q₁₆	C₂×C₅	1	2	yes	complemented
91	#916	C₂×C₅	1	2	yes	complemented
92	#915	C₂×C₅	1	2	yes	complemented
93	#909	C₂×C₅	1	1	yes	characteristic, complemented
94	#907	C₂×C₅	1	1	yes	characteristic, complemented
95	C₅×#41	—	3	1	no	—
96	C₅×#41	—	3	2	no	—
97	C₅×#41	—	3	2	no	—
98	C₅×#41	—	3	1	no	—
99	C₅×#41	—	6	2	no	—
100	C₅×#41	—	6	1	no	—
101	C₅×#41	—	6	2	no	—
102	C₅×#41	—	6	1	no	—
103	C₅×#41	C₂×D₃	1	1	yes	characteristic
104	C₂×C₅×Q₈	—	3	2	no	—
105	C₂×C₅×Q₈	—	3	2	no	—
106	C₂×C₅×Q₈	—	3	1	no	—
107	C₂×C₅×Q₈	—	6	1	no	—
108	C₂×C₅×Q₈	—	6	1	no	—
109	C₂×C₅×Q₈	C₂×D₃	1	1	yes	characteristic
110	C₅×Q₁₆	—	3	1	no	—
111	C₅×Q₁₆	—	3	2	no	—
112	C₅×Q₁₆	—	3	2	no	—
113	C₅×Q₁₆	—	3	2	no	—
114	C₅×Q₁₆	C₂×D₃	1	2	yes	—
115	C₅×SD₁₆	—	3	1	no	—
116	C₅×SD₁₆	—	3	1	no	—
117	C₅×SD₁₆	—	3	2	no	—
118	C₅×SD₁₆	C₂×D₃	1	2	yes	—
119	C₅×D₈	—	3	1	no	—
120	C₅×M₁₆	—	3	2	no	—
121	C₅×M₁₆	C₂×D₃	1	1	yes	characteristic
122	C₂×C₅×C₈	—	3	1	no	—
123	C₂×C₅×C₈	—	3	2	no	—
124	#578	—	3	1	no	Sylow
125	C₂²×C₃×C₅	—	2	1	no	—
126	C₂²×C₃×C₅	C₂×D₄	1	1	yes	characteristic
127	C₂×C₅×D₃	—	2	1	no	—
128	C₂×C₅×D₃	C₂×D₄	1	2	yes	—
129	C₃×C₄×C₅	—	2	1	no	—
130	C₃×C₄×C₅	—	2	2	no	—
131	C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
132	C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
133	C₅×Dic₃	—	2	1	no	—
134	C₅×Dic₃	—	2	2	no	—
135	C₅×Dic₃	C₂×D₄	1	2	yes	—
136	C₃×#41	C₂²×C₅	1	1	yes	characteristic
137	C₂×C₃×Q₈	C₂²×C₅	1	1	yes	characteristic
138	#239	—	2	1	no	—
139	#239	C₂²×C₅	1	2	yes	—
140	D₃×Q₈	—	2	1	no	—
141	D₃×Q₈	C₂²×C₅	1	2	yes	—
142	D₃×Q₈	C₂²×C₅	1	2	yes	—
143	#237	—	2	2	no	—
144	#237	C₂²×C₅	1	2	yes	—
145	#235	—	2	2	no	—
146	#235	—	2	1	no	—
147	#235	C₂²×C₅	1	1	yes	characteristic
148	#235	C₂²×C₅	1	1	yes	characteristic
149	C₂×Dic₆	—	2	1	no	—
150	C₂×Dic₆	C₂²×C₅	1	1	yes	characteristic
151	C₃×Q₁₆	C₂²×C₅	1	2	yes	—
152	C₃×SD₁₆	C₂²×C₅	1	2	yes	—
153	C₃×M₁₆	C₂²×C₅	1	1	yes	characteristic
154	#216	C₂²×C₅	1	2	yes	—
155	#216	C₂²×C₅	1	2	yes	—
156	#216	C₂²×C₅	1	1	yes	characteristic
157	#215	C₂²×C₅	1	1	yes	characteristic
158	#214	C₂²×C₅	1	1	yes	characteristic
159	#213	C₂²×C₅	1	1	yes	characteristic
160	C₂×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
161	Dic₁₂	C₂²×C₅	1	2	yes	—
162	#204	C₂²×C₅	1	2	yes	—
163	ΓL(1,𝔽₂₅)	C₂²×C₅	1	2	yes	—
164	D₃×C₈	C₂²×C₅	1	2	yes	—
165	C₅×Q₈	—	2	1	no	—
166	C₅×Q₈	—	3	1	no	—
167	C₅×Q₈	—	3	1	no	—
168	C₅×Q₈	—	3	2	no	—
169	C₅×Q₈	—	6	2	no	—
170	C₅×Q₈	—	6	2	no	—
171	C₅×Q₈	—	6	1	no	—
172	C₅×Q₈	C₂²×D₃	1	1	yes	characteristic
173	C₅×Q₈	C₂²×D₃	1	2	yes	—
174	C₅×D₄	—	2	1	no	—
175	C₅×D₄	—	3	2	no	—
176	C₅×D₄	—	3	1	no	—
177	C₅×D₄	—	3	1	no	—
178	C₅×D₄	—	6	2	no	—
179	C₅×D₄	—	6	2	no	—
180	C₅×D₄	—	6	1	no	—
181	C₅×D₄	C₂²×D₃	1	1	yes	characteristic
182	C₂×C₄×C₅	—	2	1	no	—
183	C₂×C₄×C₅	—	2	1	no	—
184	C₂×C₄×C₅	—	3	2	no	—
185	C₂×C₄×C₅	—	6	2	no	—
186	C₂×C₄×C₅	—	6	2	no	—
187	C₂×C₄×C₅	—	6	2	no	—
188	C₂×C₄×C₅	—	6	2	no	—
189	C₂×C₄×C₅	—	6	1	no	—
190	C₂×C₄×C₅	—	6	1	no	—
191	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
192	C₅×C₈	—	3	1	no	—
193	C₅×C₈	—	3	1	no	—
194	C₅×C₈	C₂²×D₃	1	2	yes	—
195	2₋¹⁺⁴	—	3	1	no	—
196	2₋¹⁺⁴	—	3	1	no	—
197	#137	—	3	2	no	—
198	#137	—	3	2	no	—
199	#137	—	3	1	no	—
200	#137	C₅×D₃	1	1	yes	characteristic, complemented
201	#135	—	3	1	no	—
202	#135	—	3	2	no	—
203	C₂×Q₁₆	—	3	1	no	—
204	C₂×Q₁₆	—	3	2	no	—
205	#131	—	3	1	no	—
206	C₂×C₃×C₅	—	2	1	no	—
207	C₂×C₃×C₅	—	4	1	no	—
208	C₂×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
209	C₅×D₃	—	2	2	no	—
210	C₅×D₃	—	4	1	no	—
211	C₃×Q₈	—	2	1	no	—
212	C₃×Q₈	C₂³×C₅	1	1	yes	characteristic
213	C₃×Q₈	C₂³×C₅	1	2	yes	—
214	C₃×D₄	—	2	1	no	—
215	C₃×D₄	C₂³×C₅	1	1	yes	characteristic
216	C₂×C₃×C₄	—	2	1	no	—
217	C₂×C₃×C₄	—	2	1	no	—
218	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
219	#67	—	2	1	no	—
220	#67	—	2	2	no	—
221	#67	C₅×D₄	1	2	yes	—
222	C₂×Dic₃	—	2	1	no	—
223	C₂×Dic₃	—	2	2	no	—
224	D₁₂	—	2	2	no	—
225	D₁₂	C₅×D₄	1	1	yes	characteristic
226	D₁₂	C₂³×C₅	1	1	yes	characteristic
227	C₄×D₃	—	2	2	no	—
228	C₄×D₃	—	2	2	no	—
229	C₄×D₃	—	2	1	no	—
230	C₄×D₃	—	2	2	no	—
231	C₄×D₃	C₂³×C₅	1	2	yes	—
232	Dic₆	—	2	1	no	—
233	Dic₆	—	2	2	no	—
234	Dic₆	—	2	2	no	—
235	Dic₆	C₅×D₄	1	1	yes	characteristic
236	Dic₆	C₂³×C₅	1	2	yes	—
237	Dic₆	C₂³×C₅	1	1	yes	characteristic
238	C₃×C₈	C₂³×C₅	1	2	yes	—
239	C₃⋊C₈	C₂³×C₅	1	1	yes	characteristic
240	C₃⋊C₈	C₂³×C₅	1	1	yes	characteristic
241	C₂²×C₅	—	2	1	no	—
242	C₂²×C₅	—	3	2	no	—
243	C₂²×C₅	—	6	1	no	—
244	C₂²×C₅	D₃×D₄	1	1	yes	characteristic
245	C₄×C₅	—	2	1	no	—
246	C₄×C₅	—	2	2	no	—
247	C₄×C₅	—	3	2	no	—
248	C₄×C₅	—	6	1	no	—
249	C₄×C₅	—	6	2	no	—
250	C₄×C₅	D₃×D₄	1	1	yes	characteristic
251	C₄×C₅	C₂³×D₃	1	1	yes	characteristic
252	#41	—	3	1	no	—
253	#41	—	3	2	no	—
254	#41	—	3	2	no	—
255	#41	—	3	1	no	—
256	#41	—	6	2	no	—
257	#41	—	6	1	no	—
258	#41	—	6	2	no	—
259	#41	—	6	1	no	—
260	#41	C₂×C₅×D₃	1	1	yes	characteristic
261	C₂×Q₈	—	3	2	no	—
262	C₂×Q₈	—	3	2	no	—
263	C₂×Q₈	—	3	1	no	—
264	C₂×Q₈	—	6	1	no	—
265	C₂×Q₈	—	6	1	no	—
266	C₂×Q₈	C₂×C₅×D₃	1	1	yes	characteristic
267	Q₁₆	—	3	1	no	—
268	Q₁₆	—	3	2	no	—
269	Q₁₆	—	3	2	no	—
270	Q₁₆	—	3	2	no	—
271	Q₁₆	C₂×C₅×D₃	1	2	yes	—
272	SD₁₆	—	3	1	no	—
273	SD₁₆	—	3	1	no	—
274	SD₁₆	—	3	2	no	—
275	SD₁₆	C₂×C₅×D₃	1	2	yes	—
276	D₈	—	3	1	no	—
277	M₁₆	—	3	2	no	—
278	M₁₆	C₂×C₅×D₃	1	1	yes	characteristic
279	C₂×C₈	—	3	1	no	—
280	C₂×C₈	—	3	2	no	—
281	C₃×C₅	#578	1	1	yes	characteristic, complemented, Hall
282	C₂²×C₃	—	2	1	no	—
283	C₂²×C₃	C₂×C₅×D₄	1	1	yes	characteristic
284	C₂×D₃	—	2	1	no	—
285	C₂×D₃	C₂×C₅×D₄	1	2	yes	—
286	C₃×C₄	—	2	1	no	—
287	C₃×C₄	—	2	2	no	—
288	C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
289	C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
290	Dic₃	—	2	1	no	—
291	Dic₃	—	2	2	no	—
292	Dic₃	C₂×C₅×D₄	1	2	yes	—
293	C₂×C₅	—	2	1	no	—
294	C₂×C₅	—	4	1	no	—
295	C₂×C₅	—	6	2	no	—
296	C₂×C₅	—	12	1	no	—
297	C₂×C₅	C₂×D₃×D₄	1	1	yes	characteristic
298	Q₈	—	2	1	no	—
299	Q₈	—	3	1	no	—
300	Q₈	—	3	1	no	—
301	Q₈	—	3	2	no	—
302	Q₈	—	6	2	no	—
303	Q₈	—	6	2	no	—
304	Q₈	—	6	1	no	—
305	Q₈	C₂²×C₅×D₃	1	1	yes	characteristic
306	Q₈	C₂²×C₅×D₃	1	2	yes	—
307	D₄	—	2	1	no	—
308	D₄	—	3	2	no	—
309	D₄	—	3	1	no	—
310	D₄	—	3	1	no	—
311	D₄	—	6	2	no	—
312	D₄	—	6	2	no	—
313	D₄	—	6	1	no	—
314	D₄	C₂²×C₅×D₃	1	1	yes	characteristic
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	3	2	no	—
318	C₂×C₄	—	6	2	no	—
319	C₂×C₄	—	6	2	no	—
320	C₂×C₄	—	6	2	no	—
321	C₂×C₄	—	6	2	no	—
322	C₂×C₄	—	6	1	no	—
323	C₂×C₄	—	6	1	no	—
324	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
325	C₈	—	3	1	no	—
326	C₈	—	3	1	no	—
327	C₈	C₂²×C₅×D₃	1	2	yes	—
328	C₂×C₃	—	2	1	no	—
329	C₂×C₃	—	4	1	no	—
330	C₂×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
331	D₃	—	2	2	no	—
332	D₃	—	4	1	no	—
333	C₅	#5779	1	1	yes	characteristic, direct product factor, Sylow
334	C₂²	—	2	1	no	—
335	C₂²	—	3	2	no	—
336	C₂²	—	6	1	no	—
337	C₂²	C₅×D₃×D₄	1	1	yes	characteristic
338	C₄	—	2	1	no	—
339	C₄	—	2	2	no	—
340	C₄	—	3	2	no	—
341	C₄	—	6	1	no	—
342	C₄	—	6	2	no	—
343	C₄	C₅×D₃×D₄	1	1	yes	characteristic
344	C₄	C₂³×C₅×D₃	1	1	yes	characteristic
345	C₃	C₅×#578	1	1	yes	characteristic, complemented, Sylow
346	C₂	—	2	1	no	—
347	C₂	—	4	1	no	—
348	C₂	—	6	2	no	—
349	C₂	—	12	1	no	—
350	C₂	C₂×C₅×D₃×D₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₅	C₂×D₃×D₄
commutator, cosocle	C₃×C₄	C₂⁴×C₅
Frattini	C₄	C₂³×C₅×D₃
Fitting	C₃×C₅×#137	C₂
socle	C₂×C₃×C₅	C₂²×D₄
hypercenter	C₅×#137	D₃
hypocenter	C₃	C₅×#578
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₅×#137

Complements

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

N	H ≅ G/N	type
1. C₅×#1010	C₂	semidirect
2. C₅×#1007	C₂	semidirect
3. C₃×C₅×#137	C₂	semidirect
4. C₅×#950	C₂	semidirect
5. C₂×C₅×#216	C₂	semidirect
6. C₅×#918	C₂	semidirect
7. C₅×D₃×Q₁₆	C₂	semidirect
8. C₅×#916	C₂	semidirect
9. C₅×#915	C₂	semidirect
10. C₅×#909	C₂	semidirect
11. C₅×#907	C₂	semidirect
42. #5779	C₅	direct (N × H)
48. C₅×#137	D₃	semidirect
84. #1010	C₂×C₅	semidirect
85. #1007	C₂×C₅	semidirect
86. C₃×#137	C₂×C₅	semidirect
87. #950	C₂×C₅	semidirect
88. C₂×#216	C₂×C₅	semidirect
89. #918	C₂×C₅	semidirect
90. D₃×Q₁₆	C₂×C₅	semidirect
91. #916	C₂×C₅	semidirect
92. #915	C₂×C₅	semidirect
93. #909	C₂×C₅	semidirect
94. #907	C₂×C₅	semidirect
200. #137	C₅×D₃	semidirect
281. C₃×C₅	#578	semidirect
333. C₅	#5779	direct (N × H)
345. C₃	C₅×#578	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₅×#1010	2
C₅×#1007	2
C₃×C₅×#137	2
C₅×#950	2
C₂×C₅×#216	2
C₅×#918	2
C₅×D₃×Q₁₆	2
C₅×#916	2
C₅×#915	2
C₅×#909	2
C₅×#907	2
C₅×#578	3
#5779	5

Subgroups — C5×#5779 #11756164

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#5779 #11756164