Subgroups — C₅×#68945 #14299665

Order 1920.

504 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#5745	C₂	1	1	yes	maximal, characteristic
2	C₅×#5132	C₂	1	1	yes	maximal, characteristic
3	C₅×#5125	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#5124	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#5110	C₂	1	1	yes	maximal, characteristic
6	C₃×C₅×#341	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#4552	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1858	—	3	1	no	maximal, Hall
9	C₂×C₅×#235	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
12	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₂×C₅×#208	C₂²	1	1	yes	characteristic
14	C₂×C₅×#208	C₂²	1	1	yes	characteristic
15	C₂×C₅×#208	C₂²	1	1	yes	characteristic
16	C₅×#907	—	2	2	no	—
17	C₅×D₃×M₁₆	—	2	2	no	—
18	C₅×#901	—	2	2	no	—
19	C₅×#824	C₄	1	2	yes	—
20	C₅×#823	C₄	1	2	yes	—
21	C₅×#822	—	2	2	no	—
22	C₅×(C₂×D₃)⋊C₈	—	2	2	no	—
23	C₅×#819	—	2	2	no	—
24	#68945	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
25	C₅×#568	—	3	1	no	—
26	C₅×#430	—	3	1	no	—
27	C₅×#430	—	3	1	no	—
28	C₅×#413	—	3	1	no	—
29	C₅×#408	—	3	1	no	—
30	C₅×#341	—	3	1	no	—
31	C₅×#341	D₃	1	1	yes	characteristic, complemented
32	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
33	C₂×C₅×#67	Q₈	1	2	yes	—
34	C₅×#235	—	2	1	no	—
35	C₅×#235	—	2	1	no	—
36	C₅×#235	—	2	1	no	—
37	C₅×#235	—	2	1	no	—
38	C₂×C₅×D₁₂	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×C₅×D₃	D₄	1	2	yes	—
40	C₂×C₅×Dic₆	C₂×C₄	1	1	yes	characteristic
41	C₃×C₅×M₁₆	—	2	1	no	—
42	C₃×C₅×M₁₆	—	2	1	no	—
43	C₃×C₅×M₁₆	C₂×C₄	1	2	yes	—
44	C₃×C₅×M₁₆	D₄	1	2	yes	—
45	C₂×C₃×C₅×C₈	—	2	1	no	—
46	C₂×C₃×C₅×C₈	—	2	1	no	—
47	C₂×C₃×C₅×C₈	—	2	1	no	—
48	C₂×C₃×C₅×C₈	—	2	1	no	—
49	C₂×C₃×C₅×C₈	—	2	1	no	—
50	C₅×#208	—	2	1	no	—
51	C₅×#208	—	2	1	no	—
52	C₅×#208	—	2	1	no	—
53	C₅×#208	—	2	1	no	—
54	C₅×#208	C₂×C₄	1	2	yes	—
55	C₅×#208	D₄	1	2	yes	—
56	C₂×C₅×C₃⋊C₈	—	2	1	no	—
57	C₂×C₅×C₃⋊C₈	—	2	1	no	—
58	C₂×C₅×C₃⋊C₈	—	2	1	no	—
59	C₅×ΓL(1,𝔽₂₅)	—	2	4	no	—
60	C₅×D₃×C₈	—	2	4	no	—
61	#5745	C₂×C₅	1	1	yes	characteristic
62	#5132	C₂×C₅	1	1	yes	characteristic
63	#5125	C₂×C₅	1	1	yes	characteristic, complemented
64	#5124	C₂×C₅	1	1	yes	characteristic, complemented
65	#5110	C₂×C₅	1	1	yes	characteristic
66	C₃×#341	C₂×C₅	1	1	yes	characteristic, complemented
67	#4552	C₂×C₅	1	1	yes	characteristic, complemented
68	C₂×C₅×#41	—	3	1	no	—
69	C₅×#131	—	6	2	no	—
70	C₅×#131	—	6	2	no	—
71	C₂×C₅×M₁₆	—	3	1	no	—
72	C₂×C₅×M₁₆	—	3	1	no	—
73	C₂×C₅×M₁₆	—	3	1	no	—
74	C₂×C₅×M₁₆	—	6	2	no	—
75	C₂×C₅×M₁₆	C₂×D₃	1	1	yes	characteristic
76	C₂×C₅×M₁₆	C₂×D₃	1	1	yes	characteristic
77	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
78	C₅×#108	—	6	2	no	—
79	C₅×#108	—	6	2	no	—
80	C₅×#101	—	3	2	no	—
81	C₅×#100	—	3	2	no	—
82	C₅×C₂²⋊C₈	—	6	2	no	—
83	#1858	—	3	1	no	Sylow
84	C₂³×C₃×C₅	C₂×Q₈	1	1	yes	characteristic
85	C₂²×C₅×D₃	C₄⋊C₄	1	2	yes	—
86	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
88	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
89	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
90	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
91	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
92	C₅×#67	—	2	2	no	—
93	C₅×#67	—	2	2	no	—
94	C₂×C₅×Dic₃	C₄⋊C₄	1	2	yes	—
95	C₅×D₁₂	—	2	1	no	—
96	C₅×D₁₂	—	2	1	no	—
97	C₄×C₅×D₃	—	2	2	no	—
98	C₄×C₅×D₃	—	2	2	no	—
99	C₅×Dic₆	—	2	1	no	—
100	C₅×Dic₆	—	2	1	no	—
101	C₃×C₅×C₈	—	2	2	no	—
102	C₃×C₅×C₈	—	2	2	no	—
103	C₃×C₅×C₈	—	4	1	no	—
104	C₅×C₃⋊C₈	—	2	2	no	—
105	C₅×C₃⋊C₈	—	2	2	no	—
106	C₅×C₃⋊C₈	—	4	1	no	—
107	C₂×#235	C₂²×C₅	1	1	yes	characteristic
108	C₂×C₃×M₁₆	C₂²×C₅	1	1	yes	characteristic
109	C₂×C₃×M₁₆	C₂²×C₅	1	1	yes	characteristic
110	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
111	C₂×#208	C₂²×C₅	1	1	yes	characteristic
112	C₂×#208	C₂²×C₅	1	1	yes	characteristic
113	C₂×#208	C₂²×C₅	1	1	yes	characteristic
114	#907	—	2	2	no	—
115	D₃×M₁₆	—	2	2	no	—
116	#901	—	2	2	no	—
117	#824	C₄×C₅	1	2	yes	—
118	#823	C₄×C₅	1	2	yes	—
119	#822	—	2	2	no	—
120	(C₂×D₃)⋊C₈	—	2	2	no	—
121	#819	—	2	2	no	—
122	C₅×#41	—	6	1	no	—
123	C₅×#41	—	6	1	no	—
124	C₅×#41	—	6	1	no	—
125	C₅×#41	—	6	1	no	—
126	C₂×C₅×Q₈	—	3	1	no	—
127	C₂×C₅×D₄	—	3	2	no	—
128	C₂×C₅×D₄	—	3	1	no	—
129	C₂²×C₄×C₅	—	3	2	no	—
130	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
131	C₅×M₁₆	—	2	1	no	—
132	C₅×M₁₆	—	2	1	no	—
133	C₅×M₁₆	—	3	2	no	—
134	C₅×M₁₆	—	3	2	no	—
135	C₅×M₁₆	—	6	4	no	—
136	C₅×M₁₆	—	6	1	no	—
137	C₅×M₁₆	—	6	1	no	—
138	C₅×M₁₆	—	6	1	no	—
139	C₅×M₁₆	—	6	1	no	—
140	C₅×M₁₆	C₄×D₃	1	2	yes	—
141	C₅×M₁₆	D₁₂	1	2	yes	—
142	C₂×C₅×C₈	—	2	1	no	—
143	C₂×C₅×C₈	—	2	1	no	—
144	C₂×C₅×C₈	—	2	1	no	—
145	C₂×C₅×C₈	—	2	1	no	—
146	C₂×C₅×C₈	—	2	1	no	—
147	C₂×C₅×C₈	—	6	4	no	—
148	C₂×C₅×C₈	—	6	1	no	—
149	C₂×C₅×C₈	—	6	1	no	—
150	C₂×C₅×C₈	—	6	1	no	—
151	#568	—	3	1	no	—
152	#430	—	3	1	no	—
153	#430	—	3	1	no	—
154	#413	—	3	1	no	—
155	#408	—	3	1	no	—
156	#341	—	3	1	no	—
157	#341	C₅×D₃	1	1	yes	characteristic, complemented
158	C₂²×C₃×C₅	—	4	1	no	—
159	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
160	C₂²×C₃×C₅	C₂²⋊Q₈	1	1	yes	characteristic
161	C₂²×C₃×C₅	C₂²⋊Q₈	1	1	yes	characteristic
162	C₂×C₅×D₃	—	2	2	no	—
163	C₂×C₅×D₃	—	4	2	no	—
164	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
165	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
166	C₃×C₄×C₅	#120	1	1	yes	characteristic
167	C₃×C₄×C₅	#123	1	1	yes	characteristic
168	C₅×Dic₃	—	2	2	no	—
169	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
170	C₂×#67	C₅×Q₈	1	2	yes	—
171	#235	—	2	1	no	—
172	#235	—	2	1	no	—
173	#235	—	2	1	no	—
174	#235	—	2	1	no	—
175	C₂×D₁₂	C₂×C₄×C₅	1	1	yes	characteristic
176	C₂×C₄×D₃	C₅×D₄	1	2	yes	—
177	C₂×Dic₆	C₂×C₄×C₅	1	1	yes	characteristic
178	C₃×M₁₆	—	2	1	no	—
179	C₃×M₁₆	—	2	1	no	—
180	C₃×M₁₆	C₂×C₄×C₅	1	2	yes	—
181	C₃×M₁₆	C₅×D₄	1	2	yes	—
182	C₂×C₃×C₈	—	2	1	no	—
183	C₂×C₃×C₈	—	2	1	no	—
184	C₂×C₃×C₈	—	2	1	no	—
185	C₂×C₃×C₈	—	2	1	no	—
186	C₂×C₃×C₈	—	2	1	no	—
187	#208	—	2	1	no	—
188	#208	—	2	1	no	—
189	#208	—	2	1	no	—
190	#208	—	2	1	no	—
191	#208	C₂×C₄×C₅	1	2	yes	—
192	#208	C₅×D₄	1	2	yes	—
193	C₂×C₃⋊C₈	—	2	1	no	—
194	C₂×C₃⋊C₈	—	2	1	no	—
195	C₂×C₃⋊C₈	—	2	1	no	—
196	ΓL(1,𝔽₂₅)	—	2	4	no	—
197	D₃×C₈	—	2	4	no	—
198	C₂³×C₅	—	3	2	no	—
199	C₂³×C₅	D₃×Q₈	1	1	yes	characteristic
200	C₅×Q₈	—	6	1	no	—
201	C₅×Q₈	—	6	1	no	—
202	C₅×D₄	—	6	2	no	—
203	C₅×D₄	—	6	2	no	—
204	C₅×D₄	—	6	1	no	—
205	C₅×D₄	—	6	1	no	—
206	C₂×C₄×C₅	—	3	2	no	—
207	C₂×C₄×C₅	—	6	2	no	—
208	C₂×C₄×C₅	—	6	2	no	—
209	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
210	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
211	C₂×C₄×C₅	#235	1	1	yes	characteristic
212	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
213	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
214	C₂×C₄×C₅	#237	1	1	yes	characteristic
215	C₅×C₈	—	2	2	no	—
216	C₅×C₈	—	2	2	no	—
217	C₅×C₈	—	4	1	no	—
218	C₅×C₈	—	6	2	no	—
219	C₅×C₈	—	6	2	no	—
220	C₅×C₈	—	12	1	no	—
221	C₂×#41	—	3	1	no	—
222	#131	—	6	2	no	—
223	#131	—	6	2	no	—
224	C₂×M₁₆	—	3	1	no	—
225	C₂×M₁₆	—	3	1	no	—
226	C₂×M₁₆	—	3	1	no	—
227	C₂×M₁₆	—	6	2	no	—
228	C₂×M₁₆	C₂×C₅×D₃	1	1	yes	characteristic
229	C₂×M₁₆	C₂×C₅×D₃	1	1	yes	characteristic
230	C₂²×C₈	C₂×C₅×D₃	1	1	yes	characteristic
231	#108	—	6	2	no	—
232	#108	—	6	2	no	—
233	#101	—	3	2	no	—
234	#100	—	3	2	no	—
235	C₂²⋊C₈	—	6	2	no	—
236	C₂×C₃×C₅	—	2	1	no	—
237	C₂×C₃×C₅	—	2	1	no	—
238	C₂×C₃×C₅	—	2	1	no	—
239	C₂×C₃×C₅	#385	1	1	yes	characteristic
240	C₅×D₃	—	4	2	no	—
241	C₂³×C₃	C₂×C₅×Q₈	1	1	yes	characteristic
242	C₂²×D₃	C₅×C₄⋊C₄	1	2	yes	—
243	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
244	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
245	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
246	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
247	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
248	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
249	#67	—	2	2	no	—
250	#67	—	2	2	no	—
251	C₂×Dic₃	C₅×C₄⋊C₄	1	2	yes	—
252	D₁₂	—	2	1	no	—
253	D₁₂	—	2	1	no	—
254	C₄×D₃	—	2	2	no	—
255	C₄×D₃	—	2	2	no	—
256	Dic₆	—	2	1	no	—
257	Dic₆	—	2	1	no	—
258	C₃×C₈	—	2	2	no	—
259	C₃×C₈	—	2	2	no	—
260	C₃×C₈	—	4	1	no	—
261	C₃⋊C₈	—	2	2	no	—
262	C₃⋊C₈	—	2	2	no	—
263	C₃⋊C₈	—	4	1	no	—
264	C₂²×C₅	—	4	1	no	—
265	C₂²×C₅	—	6	2	no	—
266	C₂²×C₅	—	12	2	no	—
267	C₂²×C₅	D₃×C₄⋊C₄	1	1	yes	characteristic
268	C₂²×C₅	#896	1	1	yes	characteristic
269	C₂²×C₅	#897	1	1	yes	characteristic
270	C₄×C₅	—	6	2	no	—
271	C₄×C₅	C₄×D₁₂	1	1	yes	characteristic
272	C₄×C₅	#67⋊C₄	1	1	yes	characteristic
273	C₄×C₅	#882	1	1	yes	characteristic
274	C₄×C₅	#883	1	1	yes	characteristic
275	#41	—	6	1	no	—
276	#41	—	6	1	no	—
277	#41	—	6	1	no	—
278	#41	—	6	1	no	—
279	C₂×Q₈	—	3	1	no	—
280	C₂×D₄	—	3	2	no	—
281	C₂×D₄	—	3	1	no	—
282	C₂²×C₄	—	3	2	no	—
283	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
284	M₁₆	—	2	1	no	—
285	M₁₆	—	2	1	no	—
286	M₁₆	—	3	2	no	—
287	M₁₆	—	3	2	no	—
288	M₁₆	—	6	4	no	—
289	M₁₆	—	6	1	no	—
290	M₁₆	—	6	1	no	—
291	M₁₆	—	6	1	no	—
292	M₁₆	—	6	1	no	—
293	M₁₆	C₄×C₅×D₃	1	2	yes	—
294	M₁₆	C₅×D₁₂	1	2	yes	—
295	C₂×C₈	—	2	1	no	—
296	C₂×C₈	—	2	1	no	—
297	C₂×C₈	—	2	1	no	—
298	C₂×C₈	—	2	1	no	—
299	C₂×C₈	—	2	1	no	—
300	C₂×C₈	—	6	4	no	—
301	C₂×C₈	—	6	1	no	—
302	C₂×C₈	—	6	1	no	—
303	C₂×C₈	—	6	1	no	—
304	C₃×C₅	#1858	1	1	yes	characteristic, complemented, Hall
305	C₂²×C₃	—	4	1	no	—
306	C₂²×C₃	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
307	C₂²×C₃	C₅×C₂²⋊Q₈	1	1	yes	characteristic
308	C₂²×C₃	C₅×C₂²⋊Q₈	1	1	yes	characteristic
309	C₂×D₃	—	2	2	no	—
310	C₂×D₃	—	4	2	no	—
311	C₃×C₄	C₄×C₅×D₄	1	1	yes	characteristic
312	C₃×C₄	C₄×C₅×D₄	1	1	yes	characteristic
313	C₃×C₄	C₅×#120	1	1	yes	characteristic
314	C₃×C₄	C₅×#123	1	1	yes	characteristic
315	Dic₃	—	2	2	no	—
316	C₂×C₅	—	2	1	no	—
317	C₂×C₅	—	2	1	no	—
318	C₂×C₅	—	2	1	no	—
319	C₂×C₅	—	12	2	no	—
320	C₂×C₅	#4668	1	1	yes	characteristic
321	C₂³	—	3	2	no	—
322	C₂³	C₅×D₃×Q₈	1	1	yes	characteristic
323	Q₈	—	6	1	no	—
324	Q₈	—	6	1	no	—
325	D₄	—	6	2	no	—
326	D₄	—	6	2	no	—
327	D₄	—	6	1	no	—
328	D₄	—	6	1	no	—
329	C₂×C₄	—	3	2	no	—
330	C₂×C₄	—	6	2	no	—
331	C₂×C₄	—	6	2	no	—
332	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
333	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
334	C₂×C₄	C₅×#235	1	1	yes	characteristic
335	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
336	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
337	C₂×C₄	C₅×#237	1	1	yes	characteristic
338	C₈	—	2	2	no	—
339	C₈	—	2	2	no	—
340	C₈	—	4	1	no	—
341	C₈	—	6	2	no	—
342	C₈	—	6	2	no	—
343	C₈	—	12	1	no	—
344	C₂×C₃	—	2	1	no	—
345	C₂×C₃	—	2	1	no	—
346	C₂×C₃	—	2	1	no	—
347	C₂×C₃	C₅×#385	1	1	yes	characteristic
348	D₃	—	4	2	no	—
349	C₅	#68945	1	1	yes	characteristic, direct product factor, Sylow
350	C₂²	—	4	1	no	—
351	C₂²	—	6	2	no	—
352	C₂²	—	12	2	no	—
353	C₂²	C₅×D₃×C₄⋊C₄	1	1	yes	characteristic
354	C₂²	C₅×#896	1	1	yes	characteristic
355	C₂²	C₅×#897	1	1	yes	characteristic
356	C₄	—	6	2	no	—
357	C₄	C₄×C₅×D₁₂	1	1	yes	characteristic
358	C₄	C₅×#67⋊C₄	1	1	yes	characteristic
359	C₄	C₅×#882	1	1	yes	characteristic
360	C₄	C₅×#883	1	1	yes	characteristic
361	C₃	C₅×#1858	1	1	yes	characteristic, complemented, Sylow
362	C₂	—	2	1	no	—
363	C₂	—	2	1	no	—
364	C₂	—	2	1	no	—
365	C₂	—	12	2	no	—
366	C₂	C₅×#4668	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅	#882
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₃×C₅×#341	C₂
socle	C₂×C₃×C₅	#385
prime radical	C₂×C₅×#235	C₂²
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₅×#341	D₃
hypocenter	C₃	C₅×#1858
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#341

Complements

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

N	H ≅ G/N	type
3. C₅×#5125	C₂	semidirect
4. C₅×#5124	C₂	semidirect
6. C₃×C₅×#341	C₂	semidirect
7. C₅×#4552	C₂	semidirect
24. #68945	C₅	direct (N × H)
31. C₅×#341	D₃	semidirect
63. #5125	C₂×C₅	semidirect
64. #5124	C₂×C₅	semidirect
66. C₃×#341	C₂×C₅	semidirect
67. #4552	C₂×C₅	semidirect
157. #341	C₅×D₃	semidirect
304. C₃×C₅	#1858	semidirect
349. C₅	#68945	direct (N × H)
361. C₃	C₅×#1858	semidirect

Maximal subgroups

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

subgroup	index
C₅×#5745	2
C₅×#5132	2
C₅×#5125	2
C₅×#5124	2
C₅×#5110	2
C₃×C₅×#341	2
C₅×#4552	2
C₅×#1858	3
#68945	5

Subgroups — C5×#68945 #14299665

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#68945 #14299665