Subgroups — C₅×#68264 #14289019

Order 1920.

456 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×Q₈×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#5027	C₂	1	1	yes	maximal, characteristic
3	C₅×#4848	C₂	1	1	yes	maximal, characteristic
4	C₅×#4693	C₂	1	1	yes	maximal, characteristic
5	C₃×C₅×Q₈⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4481	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#4456	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1623	—	3	1	no	maximal, Hall
9	C₃×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
10	C₅×Q₈×Dic₃	—	2	1	no	—
11	C₂×C₅×#215	—	2	1	no	—
12	C₂×C₅×#204	—	2	2	no	—
13	C₂×C₅×#204	—	2	1	no	—
14	C₅×#888	—	2	2	no	—
15	C₅×#888	—	2	1	no	—
16	C₅×#874	C₂²	1	1	yes	characteristic
17	C₅×#869	—	2	1	no	—
18	C₅×#869	C₂²	1	1	yes	characteristic
19	C₄×C₅×Dic₆	—	2	1	no	—
20	C₄×C₅×Dic₆	C₂²	1	1	yes	characteristic
21	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
22	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
23	C₅×#821	—	2	1	no	—
24	C₅×#809	—	2	1	no	—
25	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
26	#68264	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
27	C₅×Q₈²	—	3	1	no	—
28	C₅×#492	—	3	1	no	—
29	C₅×#460	—	3	1	no	—
30	C₅×#460	—	3	1	no	—
31	C₅×#333	—	3	1	no	—
32	C₅×Q₈⋊C₈	—	3	1	no	—
33	C₅×Q₈⋊C₈	D₃	1	1	yes	characteristic, complemented
34	C₂×C₃×C₅×Q₈	D₄	1	1	yes	characteristic
35	C₂×C₅×D₁₂	—	2	1	no	—
36	C₂×C₅×D₁₂	—	2	1	no	—
37	C₂×C₅×D₁₂	—	2	1	no	—
38	C₂×C₅×Dic₆	—	2	2	no	—
39	C₂×C₅×Dic₆	D₄	1	1	yes	characteristic
40	C₂×C₅×Dic₆	D₄	1	2	yes	—
41	C₂×C₃×C₅×C₈	—	2	1	no	—
42	C₂×C₃×C₅×C₈	—	2	1	no	—
43	C₃×C₅×C₄⋊C₄	—	2	1	no	—
44	C₃×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
45	C₃×C₄²×C₅	—	2	1	no	—
46	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
47	C₅×#215	—	4	2	no	—
48	C₅×C₄⋊Dic₃	—	2	2	no	—
49	C₅×C₄⋊Dic₃	—	2	1	no	—
50	C₅×C₄⋊Dic₃	—	2	1	no	—
51	C₅×C₄⋊Dic₃	D₄	1	1	yes	characteristic
52	C₅×Dic₃⋊C₄	—	2	2	no	—
53	C₅×Dic₃⋊C₄	—	2	1	no	—
54	C₄×C₅×Dic₃	—	2	2	no	—
55	C₄×C₅×Dic₃	—	2	1	no	—
56	C₂×C₅×C₃⋊C₈	—	2	1	no	—
57	C₅×#204	—	4	4	no	—
58	C₅×#204	—	4	2	no	—
59	Q₈×Dic₆	C₂×C₅	1	1	yes	characteristic, complemented
60	#5027	C₂×C₅	1	1	yes	characteristic
61	#4848	C₂×C₅	1	1	yes	characteristic
62	#4693	C₂×C₅	1	1	yes	characteristic
63	C₃×Q₈⋊C₈	C₂×C₅	1	1	yes	characteristic, complemented
64	#4481	C₂×C₅	1	1	yes	characteristic, complemented
65	#4456	C₂×C₅	1	1	yes	characteristic, complemented
66	C₂×C₅×SD₁₆	—	6	1	no	—
67	C₂×C₅×SD₁₆	—	6	2	no	—
68	C₂×C₅×SD₁₆	—	6	1	no	—
69	C₅×C₄⋊Q₈	—	3	1	no	—
70	C₅×C₄⋊Q₈	—	6	1	no	—
71	C₅×C₄⋊Q₈	—	6	2	no	—
72	C₅×C₄⋊Q₈	—	6	1	no	—
73	C₅×#127	—	3	1	no	—
74	C₄×C₅×Q₈	—	3	1	no	—
75	C₄×C₅×Q₈	—	6	1	no	—
76	C₄×C₅×Q₈	—	6	1	no	—
77	C₄×C₅×Q₈	C₂×D₃	1	1	yes	characteristic
78	C₅×C₄⋊C₈	—	3	1	no	—
79	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
80	C₅×#102	—	6	1	no	—
81	C₅×#102	—	6	1	no	—
82	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
83	#1623	—	3	1	no	Sylow
84	C₂²×C₅×D₃	—	4	1	no	—
85	C₃×C₅×Q₈	—	2	1	no	—
86	C₃×C₅×Q₈	SD₁₆	1	2	yes	complemented
87	C₂×C₃×C₄×C₅	—	2	1	no	—
88	C₂×C₃×C₄×C₅	—	2	1	no	—
89	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
90	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
91	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
92	C₂×C₅×Dic₃	—	2	1	no	—
93	C₂×C₅×Dic₃	—	2	2	no	—
94	C₂×C₅×Dic₃	—	2	1	no	—
95	C₅×D₁₂	—	4	2	no	—
96	C₅×D₁₂	—	4	1	no	—
97	C₅×D₁₂	—	4	1	no	—
98	C₅×D₁₂	—	4	1	no	—
99	C₅×D₁₂	—	4	1	no	—
100	C₅×Dic₆	—	2	1	no	—
101	C₅×Dic₆	—	4	2	no	—
102	C₅×Dic₆	—	8	2	no	—
103	C₅×Dic₆	SD₁₆	1	2	yes	complemented
104	C₃×C₅×C₈	—	2	2	no	—
105	C₃×C₅×C₈	—	4	1	no	—
106	C₅×C₃⋊C₈	—	4	1	no	—
107	C₃×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
108	Q₈×Dic₃	—	2	1	no	—
109	C₂×#215	—	2	1	no	—
110	C₂×#204	—	2	2	no	—
111	C₂×#204	—	2	1	no	—
112	#888	—	2	2	no	—
113	#888	—	2	1	no	—
114	#874	C₂²×C₅	1	1	yes	characteristic
115	#869	—	2	1	no	—
116	#869	C₂²×C₅	1	1	yes	characteristic
117	C₄×Dic₆	—	2	1	no	—
118	C₄×Dic₆	C₂²×C₅	1	1	yes	characteristic
119	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
120	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
121	#821	—	2	1	no	—
122	#809	—	2	1	no	—
123	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	1	yes	characteristic
124	C₂×C₅×Q₈	—	3	1	no	—
125	C₂×C₅×Q₈	—	3	2	no	—
126	C₂×C₅×Q₈	—	6	2	no	—
127	C₂×C₅×Q₈	D₁₂	1	1	yes	characteristic
128	C₂×C₅×D₄	—	6	1	no	—
129	C₂×C₅×D₄	—	6	1	no	—
130	C₂×C₅×D₄	—	6	1	no	—
131	C₅×SD₁₆	—	12	4	no	—
132	C₅×SD₁₆	—	12	2	no	—
133	C₅×SD₁₆	—	12	2	no	—
134	C₂×C₅×C₈	—	2	1	no	—
135	C₂×C₅×C₈	—	2	1	no	—
136	C₂×C₅×C₈	—	6	1	no	—
137	C₅×C₄⋊C₄	—	2	1	no	—
138	C₅×C₄⋊C₄	—	3	1	no	—
139	C₅×C₄⋊C₄	—	6	2	no	—
140	C₅×C₄⋊C₄	—	6	2	no	—
141	C₅×C₄⋊C₄	—	6	1	no	—
142	C₅×C₄⋊C₄	—	6	1	no	—
143	C₅×C₄⋊C₄	—	6	1	no	—
144	C₅×C₄⋊C₄	D₁₂	1	1	yes	characteristic
145	C₄²×C₅	—	2	1	no	—
146	C₄²×C₅	—	6	2	no	—
147	C₄²×C₅	—	6	1	no	—
148	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
149	Q₈²	—	3	1	no	—
150	#492	—	3	1	no	—
151	#460	—	3	1	no	—
152	#460	—	3	1	no	—
153	#333	—	3	1	no	—
154	Q₈⋊C₈	—	3	1	no	—
155	Q₈⋊C₈	C₅×D₃	1	1	yes	characteristic, complemented
156	C₂²×C₃×C₅	#120	1	1	yes	characteristic
157	C₂×C₅×D₃	—	8	1	no	—
158	C₂×C₅×D₃	—	8	1	no	—
159	C₂×C₅×D₃	—	8	1	no	—
160	C₃×C₄×C₅	—	2	2	no	—
161	C₃×C₄×C₅	—	2	1	no	—
162	C₃×C₄×C₅	—	4	1	no	—
163	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
164	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
165	C₃×C₄×C₅	#137	1	1	yes	characteristic
166	C₃×C₄×C₅	#137	1	1	yes	characteristic
167	C₅×Dic₃	—	2	2	no	—
168	C₅×Dic₃	—	4	2	no	—
169	C₅×Dic₃	—	4	1	no	—
170	C₂×C₃×Q₈	C₅×D₄	1	1	yes	characteristic
171	C₂×D₁₂	—	2	1	no	—
172	C₂×D₁₂	—	2	1	no	—
173	C₂×D₁₂	—	2	1	no	—
174	C₂×Dic₆	—	2	2	no	—
175	C₂×Dic₆	C₅×D₄	1	1	yes	characteristic
176	C₂×Dic₆	C₅×D₄	1	2	yes	—
177	C₂×C₃×C₈	—	2	1	no	—
178	C₂×C₃×C₈	—	2	1	no	—
179	C₃×C₄⋊C₄	—	2	1	no	—
180	C₃×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
181	C₃×C₄²	—	2	1	no	—
182	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
183	#215	—	4	2	no	—
184	C₄⋊Dic₃	—	2	2	no	—
185	C₄⋊Dic₃	—	2	1	no	—
186	C₄⋊Dic₃	—	2	1	no	—
187	C₄⋊Dic₃	C₅×D₄	1	1	yes	characteristic
188	Dic₃⋊C₄	—	2	2	no	—
189	Dic₃⋊C₄	—	2	1	no	—
190	C₄×Dic₃	—	2	2	no	—
191	C₄×Dic₃	—	2	1	no	—
192	C₂×C₃⋊C₈	—	2	1	no	—
193	#204	—	4	4	no	—
194	#204	—	4	2	no	—
195	C₂³×C₅	—	12	1	no	—
196	C₅×Q₈	—	2	1	no	—
197	C₅×Q₈	—	3	2	no	—
198	C₅×Q₈	—	6	1	no	—
199	C₅×Q₈	—	12	2	no	—
200	C₅×Q₈	—	24	2	no	—
201	C₅×Q₈	#204	1	2	yes	complemented
202	C₅×D₄	—	12	2	no	—
203	C₅×D₄	—	12	1	no	—
204	C₅×D₄	—	12	1	no	—
205	C₅×D₄	—	12	1	no	—
206	C₅×D₄	—	12	1	no	—
207	C₂×C₄×C₅	—	2	1	no	—
208	C₂×C₄×C₅	—	2	1	no	—
209	C₂×C₄×C₅	—	6	1	no	—
210	C₂×C₄×C₅	—	6	1	no	—
211	C₂×C₄×C₅	—	6	2	no	—
212	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
213	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
214	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
215	C₅×C₈	—	2	2	no	—
216	C₅×C₈	—	4	1	no	—
217	C₅×C₈	—	12	1	no	—
218	C₂×SD₁₆	—	6	1	no	—
219	C₂×SD₁₆	—	6	2	no	—
220	C₂×SD₁₆	—	6	1	no	—
221	C₄⋊Q₈	—	3	1	no	—
222	C₄⋊Q₈	—	6	1	no	—
223	C₄⋊Q₈	—	6	2	no	—
224	C₄⋊Q₈	—	6	1	no	—
225	#127	—	3	1	no	—
226	C₄×Q₈	—	3	1	no	—
227	C₄×Q₈	—	6	1	no	—
228	C₄×Q₈	—	6	1	no	—
229	C₄×Q₈	C₂×C₅×D₃	1	1	yes	characteristic
230	C₄⋊C₈	—	3	1	no	—
231	C₄⋊C₈	C₂×C₅×D₃	1	1	yes	characteristic
232	#102	—	6	1	no	—
233	#102	—	6	1	no	—
234	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
235	C₂×C₃×C₅	#448	1	1	yes	characteristic
236	C₂×C₃×C₅	#448	1	1	yes	characteristic
237	C₂×C₃×C₅	#453	1	1	yes	characteristic
238	C₅×D₃	—	16	1	no	—
239	C₂²×D₃	—	4	1	no	—
240	C₃×Q₈	—	2	1	no	—
241	C₃×Q₈	C₅×SD₁₆	1	2	yes	complemented
242	C₂×C₃×C₄	—	2	1	no	—
243	C₂×C₃×C₄	—	2	1	no	—
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×Dic₃	—	2	1	no	—
248	C₂×Dic₃	—	2	2	no	—
249	C₂×Dic₃	—	2	1	no	—
250	D₁₂	—	4	2	no	—
251	D₁₂	—	4	1	no	—
252	D₁₂	—	4	1	no	—
253	D₁₂	—	4	1	no	—
254	D₁₂	—	4	1	no	—
255	Dic₆	—	2	1	no	—
256	Dic₆	—	4	2	no	—
257	Dic₆	—	8	2	no	—
258	Dic₆	C₅×SD₁₆	1	2	yes	complemented
259	C₃×C₈	—	2	2	no	—
260	C₃×C₈	—	4	1	no	—
261	C₃⋊C₈	—	4	1	no	—
262	C₂²×C₅	—	24	1	no	—
263	C₂²×C₅	—	24	1	no	—
264	C₂²×C₅	—	24	1	no	—
265	C₂²×C₅	#882	1	1	yes	characteristic
266	C₄×C₅	—	2	2	no	—
267	C₄×C₅	—	2	1	no	—
268	C₄×C₅	—	4	1	no	—
269	C₄×C₅	—	6	2	no	—
270	C₄×C₅	—	12	2	no	—
271	C₄×C₅	—	12	1	no	—
272	C₄×C₅	C₂×#204	1	1	yes	characteristic
273	C₄×C₅	#909	1	1	yes	characteristic
274	C₄×C₅	D₃×SD₁₆	1	1	yes	characteristic
275	C₄×C₅	#918	1	1	yes	characteristic
276	C₂×Q₈	—	3	1	no	—
277	C₂×Q₈	—	3	2	no	—
278	C₂×Q₈	—	6	2	no	—
279	C₂×Q₈	C₅×D₁₂	1	1	yes	characteristic
280	C₂×D₄	—	6	1	no	—
281	C₂×D₄	—	6	1	no	—
282	C₂×D₄	—	6	1	no	—
283	SD₁₆	—	12	4	no	—
284	SD₁₆	—	12	2	no	—
285	SD₁₆	—	12	2	no	—
286	C₂×C₈	—	2	1	no	—
287	C₂×C₈	—	2	1	no	—
288	C₂×C₈	—	6	1	no	—
289	C₄⋊C₄	—	2	1	no	—
290	C₄⋊C₄	—	3	1	no	—
291	C₄⋊C₄	—	6	2	no	—
292	C₄⋊C₄	—	6	2	no	—
293	C₄⋊C₄	—	6	1	no	—
294	C₄⋊C₄	—	6	1	no	—
295	C₄⋊C₄	—	6	1	no	—
296	C₄⋊C₄	C₅×D₁₂	1	1	yes	characteristic
297	C₄²	—	2	1	no	—
298	C₄²	—	6	2	no	—
299	C₄²	—	6	1	no	—
300	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
301	C₃×C₅	#1623	1	1	yes	characteristic, complemented, Hall
302	C₂²×C₃	C₅×#120	1	1	yes	characteristic
303	C₂×D₃	—	8	1	no	—
304	C₂×D₃	—	8	1	no	—
305	C₂×D₃	—	8	1	no	—
306	C₃×C₄	—	2	2	no	—
307	C₃×C₄	—	2	1	no	—
308	C₃×C₄	—	4	1	no	—
309	C₃×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
310	C₃×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
311	C₃×C₄	C₅×#137	1	1	yes	characteristic
312	C₃×C₄	C₅×#137	1	1	yes	characteristic
313	Dic₃	—	2	2	no	—
314	Dic₃	—	4	2	no	—
315	Dic₃	—	4	1	no	—
316	C₂×C₅	—	48	1	no	—
317	C₂×C₅	#4738	1	1	yes	characteristic
318	C₂×C₅	#4806	1	1	yes	characteristic
319	C₂×C₅	#4822	1	1	yes	characteristic
320	C₂³	—	12	1	no	—
321	Q₈	—	2	1	no	—
322	Q₈	—	3	2	no	—
323	Q₈	—	6	1	no	—
324	Q₈	—	12	2	no	—
325	Q₈	—	24	2	no	—
326	Q₈	C₅×#204	1	2	yes	complemented
327	D₄	—	12	2	no	—
328	D₄	—	12	1	no	—
329	D₄	—	12	1	no	—
330	D₄	—	12	1	no	—
331	D₄	—	12	1	no	—
332	C₂×C₄	—	2	1	no	—
333	C₂×C₄	—	2	1	no	—
334	C₂×C₄	—	6	1	no	—
335	C₂×C₄	—	6	1	no	—
336	C₂×C₄	—	6	2	no	—
337	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
338	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
339	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
340	C₈	—	2	2	no	—
341	C₈	—	4	1	no	—
342	C₈	—	12	1	no	—
343	C₂×C₃	C₅×#448	1	1	yes	characteristic
344	C₂×C₃	C₅×#448	1	1	yes	characteristic
345	C₂×C₃	C₅×#453	1	1	yes	characteristic
346	D₃	—	16	1	no	—
347	C₅	#68264	1	1	yes	characteristic, direct product factor, Sylow
348	C₂²	—	24	1	no	—
349	C₂²	—	24	1	no	—
350	C₂²	—	24	1	no	—
351	C₂²	C₅×#882	1	1	yes	characteristic
352	C₄	—	2	2	no	—
353	C₄	—	2	1	no	—
354	C₄	—	4	1	no	—
355	C₄	—	6	2	no	—
356	C₄	—	12	2	no	—
357	C₄	—	12	1	no	—
358	C₄	C₂×C₅×#204	1	1	yes	characteristic
359	C₄	C₅×#909	1	1	yes	characteristic
360	C₄	C₅×D₃×SD₁₆	1	1	yes	characteristic
361	C₄	C₅×#918	1	1	yes	characteristic
362	C₃	C₅×#1623	1	1	yes	characteristic, complemented, Sylow
363	C₂	—	48	1	no	—
364	C₂	C₅×#4738	1	1	yes	characteristic
365	C₂	C₅×#4806	1	1	yes	characteristic
366	C₂	C₅×#4822	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	#882
commutator, cosocle	C₃×C₄²	C₂³×C₅
Frattini	C₄²	C₂²×C₅×D₃
Fitting	C₃×C₅×Q₈⋊C₈	C₂
socle	C₂²×C₃×C₅	#120
prime radical	C₅×#874	C₂²
hypercenter	C₅×Q₈⋊C₈	D₃
hypocenter	C₃	C₅×#1623
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×Q₈⋊C₈

Complements

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

N	H ≅ G/N	type
1. C₅×Q₈×Dic₆	C₂	semidirect
5. C₃×C₅×Q₈⋊C₈	C₂	semidirect
6. C₅×#4481	C₂	semidirect
7. C₅×#4456	C₂	semidirect
26. #68264	C₅	direct (N × H)
33. C₅×Q₈⋊C₈	D₃	semidirect
59. Q₈×Dic₆	C₂×C₅	semidirect
63. C₃×Q₈⋊C₈	C₂×C₅	semidirect
64. #4481	C₂×C₅	semidirect
65. #4456	C₂×C₅	semidirect
86. C₃×C₅×Q₈	SD₁₆	semidirect
103. C₅×Dic₆	SD₁₆	semidirect
155. Q₈⋊C₈	C₅×D₃	semidirect
201. C₅×Q₈	#204	semidirect
241. C₃×Q₈	C₅×SD₁₆	semidirect
258. Dic₆	C₅×SD₁₆	semidirect
301. C₃×C₅	#1623	semidirect
326. Q₈	C₅×#204	semidirect
347. C₅	#68264	direct (N × H)
362. C₃	C₅×#1623	semidirect

Maximal subgroups

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

subgroup	index
C₅×Q₈×Dic₆	2
C₅×#5027	2
C₅×#4848	2
C₅×#4693	2
C₃×C₅×Q₈⋊C₈	2
C₅×#4481	2
C₅×#4456	2
C₅×#1623	3
#68264	5

Subgroups — C5×#68264 #14289019

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#68264 #14289019