Subgroups — C₅×#68269 #14289760

Order 1920.

456 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×Q₈×D₁₂	C₂	1	1	yes	maximal, characteristic
2	C₅×#5021	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#4842	C₂	1	1	yes	maximal, characteristic
4	C₅×#4681	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₅×Q₈⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4486	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#4459	C₂	1	1	yes	maximal, characteristic
8	C₅×#1637	—	3	1	no	maximal, Hall
9	C₂×C₅×D₃×Q₈	—	2	1	no	—
10	C₃×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
11	C₅×#897	—	2	1	no	—
12	C₅×#897	—	2	2	no	—
13	C₄×C₅×D₁₂	—	2	1	no	—
14	C₄×C₅×D₁₂	C₂²	1	1	yes	characteristic
15	C₅×#869	—	2	1	no	—
16	C₅×#869	C₂²	1	1	yes	characteristic
17	C₅×#869	C₂²	1	1	yes	characteristic
18	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
19	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
20	C₅×#835	—	2	1	no	—
21	C₅×#821	—	2	1	no	—
22	C₅×#817	—	2	1	no	—
23	C₅×#816	—	2	2	no	—
24	C₅×#807	—	2	1	no	—
25	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
26	#68269	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
27	C₅×D₄×Q₈	—	3	1	no	—
28	C₅×#487	—	3	1	no	—
29	C₅×#477	—	3	1	no	—
30	C₅×#476	—	3	1	no	—
31	C₅×#332	—	3	1	no	—
32	C₅×Q₈⋊C₈	D₃	1	1	yes	characteristic, complemented
33	C₅×D₄⋊C₈	—	3	1	no	—
34	C₂×C₃×C₅×Q₈	D₄	1	1	yes	characteristic
35	C₅×D₃×Q₈	—	4	2	no	—
36	C₅×D₃×Q₈	—	4	2	no	—
37	C₂×C₅×D₁₂	D₄	1	1	yes	characteristic
38	C₂×C₄×C₅×D₃	—	2	2	no	—
39	C₂×C₄×C₅×D₃	—	2	1	no	—
40	C₂×C₅×Dic₆	—	2	2	no	—
41	C₂×C₅×Dic₆	—	2	1	no	—
42	C₂×C₅×Dic₆	—	2	1	no	—
43	C₂×C₃×C₅×C₈	—	2	1	no	—
44	C₂×C₃×C₅×C₈	—	2	1	no	—
45	C₃×C₅×C₄⋊C₄	—	2	1	no	—
46	C₃×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
47	C₃×C₄²×C₅	—	2	1	no	—
48	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
49	C₅×(C₂×D₃)⋊C₄	—	2	2	no	—
50	C₅×(C₂×D₃)⋊C₄	—	2	1	no	—
51	C₅×C₄⋊Dic₃	—	2	2	no	—
52	C₅×C₄⋊Dic₃	—	2	1	no	—
53	C₅×C₄⋊Dic₃	—	2	1	no	—
54	C₅×C₄⋊Dic₃	—	2	1	no	—
55	C₅×C₄⋊Dic₃	D₄	1	1	yes	characteristic
56	C₅×C₄⋊Dic₃	D₄	1	2	yes	—
57	C₂×C₅×C₃⋊C₈	—	2	1	no	—
58	Q₈×D₁₂	C₂×C₅	1	1	yes	characteristic
59	#5021	C₂×C₅	1	1	yes	characteristic, complemented
60	#4842	C₂×C₅	1	1	yes	characteristic
61	#4681	C₂×C₅	1	1	yes	characteristic, complemented
62	C₃×Q₈⋊C₈	C₂×C₅	1	1	yes	characteristic, complemented
63	#4486	C₂×C₅	1	1	yes	characteristic, complemented
64	#4459	C₂×C₅	1	1	yes	characteristic
65	C₂²×C₅×Q₈	—	6	1	no	—
66	C₅×C₄⋊Q₈	—	3	1	no	—
67	C₅×C₄⋊Q₈	—	3	1	no	—
68	C₅×C₄⋊Q₈	—	6	1	no	—
69	C₅×C₂²⋊Q₈	—	6	2	no	—
70	C₅×C₂²⋊Q₈	—	6	1	no	—
71	C₄×C₅×Q₈	C₂×D₃	1	1	yes	characteristic
72	C₄×C₅×D₄	—	3	1	no	—
73	C₄×C₅×D₄	—	6	1	no	—
74	C₅×#107	—	6	1	no	—
75	C₅×#106	—	6	1	no	—
76	C₅×C₄⋊C₈	—	3	1	no	—
77	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
78	C₅×#103	—	6	2	no	—
79	C₅×#103	—	6	1	no	—
80	C₅×#102	—	6	1	no	—
81	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
82	#1637	—	3	1	no	Sylow
83	C₂²×C₅×D₃	—	2	1	no	—
84	C₃×C₅×Q₈	—	2	1	no	—
85	C₃×C₅×Q₈	SD₁₆	1	2	yes	—
86	C₂×C₃×C₄×C₅	—	2	1	no	—
87	C₂×C₃×C₄×C₅	—	2	1	no	—
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	C₂×D₄	1	1	yes	characteristic
91	C₂×C₅×Dic₃	—	2	1	no	—
92	C₂×C₅×Dic₃	—	2	2	no	—
93	C₂×C₅×Dic₃	—	4	1	no	—
94	C₅×D₁₂	—	2	1	no	—
95	C₅×D₁₂	Q₁₆	1	2	yes	—
96	C₄×C₅×D₃	—	4	4	no	—
97	C₄×C₅×D₃	—	4	1	no	—
98	C₄×C₅×D₃	—	4	1	no	—
99	C₅×Dic₆	—	4	4	no	—
100	C₅×Dic₆	—	4	1	no	—
101	C₅×Dic₆	—	4	1	no	—
102	C₅×Dic₆	—	4	2	no	—
103	C₃×C₅×C₈	—	4	1	no	—
104	C₃×C₅×C₈	—	4	1	no	—
105	C₅×C₃⋊C₈	—	4	1	no	—
106	C₂×D₃×Q₈	—	2	1	no	—
107	C₃×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
108	#897	—	2	1	no	—
109	#897	—	2	2	no	—
110	C₄×D₁₂	—	2	1	no	—
111	C₄×D₁₂	C₂²×C₅	1	1	yes	characteristic
112	#869	—	2	1	no	—
113	#869	C₂²×C₅	1	1	yes	characteristic
114	#869	C₂²×C₅	1	1	yes	characteristic
115	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
116	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
117	#835	—	2	1	no	—
118	#821	—	2	1	no	—
119	#817	—	2	1	no	—
120	#816	—	2	2	no	—
121	#807	—	2	1	no	—
122	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	1	yes	characteristic
123	C₂×C₅×Q₈	—	6	2	no	—
124	C₂×C₅×Q₈	—	6	1	no	—
125	C₂×C₅×Q₈	—	6	1	no	—
126	C₂×C₅×Q₈	—	12	2	no	—
127	C₂×C₅×Q₈	—	12	2	no	—
128	C₂×C₅×Q₈	D₁₂	1	1	yes	characteristic
129	C₂×C₅×D₄	—	3	1	no	—
130	C₂²×C₄×C₅	—	6	2	no	—
131	C₂²×C₄×C₅	—	6	1	no	—
132	C₂×C₅×C₈	—	2	1	no	—
133	C₂×C₅×C₈	—	2	1	no	—
134	C₂×C₅×C₈	—	6	1	no	—
135	C₅×C₄⋊C₄	—	2	1	no	—
136	C₅×C₄⋊C₄	—	3	1	no	—
137	C₅×C₄⋊C₄	—	3	2	no	—
138	C₅×C₄⋊C₄	—	6	2	no	—
139	C₅×C₄⋊C₄	—	6	1	no	—
140	C₅×C₄⋊C₄	—	6	1	no	—
141	C₅×C₄⋊C₄	—	6	1	no	—
142	C₅×C₄⋊C₄	D₁₂	1	1	yes	characteristic
143	C₅×C₂²⋊C₄	—	6	2	no	—
144	C₅×C₂²⋊C₄	—	6	1	no	—
145	C₄²×C₅	—	2	1	no	—
146	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
147	D₄×Q₈	—	3	1	no	—
148	#487	—	3	1	no	—
149	#477	—	3	1	no	—
150	#476	—	3	1	no	—
151	#332	—	3	1	no	—
152	Q₈⋊C₈	C₅×D₃	1	1	yes	characteristic, complemented
153	D₄⋊C₈	—	3	1	no	—
154	C₂²×C₃×C₅	#120	1	1	yes	characteristic
155	C₂×C₅×D₃	—	2	2	no	—
156	C₂×C₅×D₃	—	4	1	no	—
157	C₂×C₅×D₃	—	4	1	no	—
158	C₃×C₄×C₅	—	2	2	no	—
159	C₃×C₄×C₅	—	2	1	no	—
160	C₃×C₄×C₅	—	4	1	no	—
161	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
162	C₃×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
163	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
164	C₃×C₄×C₅	#137	1	1	yes	characteristic
165	C₅×Dic₃	—	4	1	no	—
166	C₅×Dic₃	—	4	2	no	—
167	C₅×Dic₃	—	8	1	no	—
168	C₂×C₃×Q₈	C₅×D₄	1	1	yes	characteristic
169	D₃×Q₈	—	4	2	no	—
170	D₃×Q₈	—	4	2	no	—
171	C₂×D₁₂	C₅×D₄	1	1	yes	characteristic
172	C₂×C₄×D₃	—	2	2	no	—
173	C₂×C₄×D₃	—	2	1	no	—
174	C₂×Dic₆	—	2	2	no	—
175	C₂×Dic₆	—	2	1	no	—
176	C₂×Dic₆	—	2	1	no	—
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	(C₂×D₃)⋊C₄	—	2	2	no	—
184	(C₂×D₃)⋊C₄	—	2	1	no	—
185	C₄⋊Dic₃	—	2	2	no	—
186	C₄⋊Dic₃	—	2	1	no	—
187	C₄⋊Dic₃	—	2	1	no	—
188	C₄⋊Dic₃	—	2	1	no	—
189	C₄⋊Dic₃	C₅×D₄	1	2	yes	—
190	C₄⋊Dic₃	C₅×D₄	1	1	yes	characteristic
191	C₂×C₃⋊C₈	—	2	1	no	—
192	C₂³×C₅	—	6	1	no	—
193	C₅×Q₈	—	2	1	no	—
194	C₅×Q₈	—	12	4	no	—
195	C₅×Q₈	—	12	2	no	—
196	C₅×Q₈	—	12	1	no	—
197	C₅×Q₈	—	12	1	no	—
198	C₅×Q₈	#204	1	2	yes	—
199	C₅×D₄	—	3	2	no	—
200	C₅×D₄	—	6	1	no	—
201	C₂×C₄×C₅	—	2	1	no	—
202	C₂×C₄×C₅	—	2	1	no	—
203	C₂×C₄×C₅	—	6	2	no	—
204	C₂×C₄×C₅	—	6	1	no	—
205	C₂×C₄×C₅	—	12	4	no	—
206	C₂×C₄×C₅	—	12	1	no	—
207	C₂×C₄×C₅	—	12	1	no	—
208	C₂×C₄×C₅	—	12	1	no	—
209	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
210	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
211	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
212	C₅×C₈	—	4	1	no	—
213	C₅×C₈	—	4	1	no	—
214	C₅×C₈	—	12	1	no	—
215	C₂²×Q₈	—	6	1	no	—
216	C₄⋊Q₈	—	3	1	no	—
217	C₄⋊Q₈	—	3	1	no	—
218	C₄⋊Q₈	—	6	1	no	—
219	C₂²⋊Q₈	—	6	2	no	—
220	C₂²⋊Q₈	—	6	1	no	—
221	C₄×Q₈	C₂×C₅×D₃	1	1	yes	characteristic
222	C₄×D₄	—	3	1	no	—
223	C₄×D₄	—	6	1	no	—
224	#107	—	6	1	no	—
225	#106	—	6	1	no	—
226	C₄⋊C₈	—	3	1	no	—
227	C₄⋊C₈	C₂×C₅×D₃	1	1	yes	characteristic
228	#103	—	6	2	no	—
229	#103	—	6	1	no	—
230	#102	—	6	1	no	—
231	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
232	C₂×C₃×C₅	#450	1	1	yes	characteristic
233	C₂×C₃×C₅	#451	1	1	yes	characteristic
234	C₂×C₃×C₅	#454	1	1	yes	characteristic
235	C₅×D₃	—	4	2	no	—
236	C₂²×D₃	—	2	1	no	—
237	C₃×Q₈	—	2	1	no	—
238	C₃×Q₈	C₅×SD₁₆	1	2	yes	—
239	C₂×C₃×C₄	—	2	1	no	—
240	C₂×C₃×C₄	—	2	1	no	—
241	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
242	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
243	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
244	C₂×Dic₃	—	2	1	no	—
245	C₂×Dic₃	—	2	2	no	—
246	C₂×Dic₃	—	4	1	no	—
247	D₁₂	—	2	1	no	—
248	D₁₂	C₅×Q₁₆	1	2	yes	—
249	C₄×D₃	—	4	4	no	—
250	C₄×D₃	—	4	1	no	—
251	C₄×D₃	—	4	1	no	—
252	Dic₆	—	4	4	no	—
253	Dic₆	—	4	1	no	—
254	Dic₆	—	4	1	no	—
255	Dic₆	—	4	2	no	—
256	C₃×C₈	—	4	1	no	—
257	C₃×C₈	—	4	1	no	—
258	C₃⋊C₈	—	4	1	no	—
259	C₂²×C₅	—	6	2	no	—
260	C₂²×C₅	—	12	1	no	—
261	C₂²×C₅	—	12	1	no	—
262	C₂²×C₅	#882	1	1	yes	characteristic
263	C₄×C₅	—	2	2	no	—
264	C₄×C₅	—	2	1	no	—
265	C₄×C₅	—	4	1	no	—
266	C₄×C₅	—	12	2	no	—
267	C₄×C₅	—	12	1	no	—
268	C₄×C₅	—	24	1	no	—
269	C₄×C₅	C₂×#204	1	1	yes	characteristic
270	C₄×C₅	#908	1	1	yes	characteristic
271	C₄×C₅	#915	1	1	yes	characteristic
272	C₄×C₅	D₃×Q₁₆	1	1	yes	characteristic
273	C₂×Q₈	—	6	2	no	—
274	C₂×Q₈	—	6	1	no	—
275	C₂×Q₈	—	6	1	no	—
276	C₂×Q₈	—	12	2	no	—
277	C₂×Q₈	—	12	2	no	—
278	C₂×Q₈	C₅×D₁₂	1	1	yes	characteristic
279	C₂×D₄	—	3	1	no	—
280	C₂²×C₄	—	6	2	no	—
281	C₂²×C₄	—	6	1	no	—
282	C₂×C₈	—	2	1	no	—
283	C₂×C₈	—	2	1	no	—
284	C₂×C₈	—	6	1	no	—
285	C₄⋊C₄	—	2	1	no	—
286	C₄⋊C₄	—	3	2	no	—
287	C₄⋊C₄	—	3	1	no	—
288	C₄⋊C₄	—	6	2	no	—
289	C₄⋊C₄	—	6	1	no	—
290	C₄⋊C₄	—	6	1	no	—
291	C₄⋊C₄	—	6	1	no	—
292	C₄⋊C₄	C₅×D₁₂	1	1	yes	characteristic
293	C₂²⋊C₄	—	6	2	no	—
294	C₂²⋊C₄	—	6	1	no	—
295	C₄²	—	2	1	no	—
296	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
297	C₃×C₅	#1637	1	1	yes	characteristic, complemented, Hall
298	C₂²×C₃	C₅×#120	1	1	yes	characteristic
299	C₂×D₃	—	2	2	no	—
300	C₂×D₃	—	4	1	no	—
301	C₂×D₃	—	4	1	no	—
302	C₃×C₄	—	2	2	no	—
303	C₃×C₄	—	2	1	no	—
304	C₃×C₄	—	4	1	no	—
305	C₃×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
306	C₃×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
307	C₃×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
308	C₃×C₄	C₅×#137	1	1	yes	characteristic
309	Dic₃	—	4	1	no	—
310	Dic₃	—	4	2	no	—
311	Dic₃	—	8	1	no	—
312	C₂×C₅	—	12	2	no	—
313	C₂×C₅	#4731	1	1	yes	characteristic
314	C₂×C₅	#4809	1	1	yes	characteristic
315	C₂×C₅	#4827	1	1	yes	characteristic
316	C₂³	—	6	1	no	—
317	Q₈	—	2	1	no	—
318	Q₈	—	12	4	no	—
319	Q₈	—	12	2	no	—
320	Q₈	—	12	1	no	—
321	Q₈	—	12	1	no	—
322	Q₈	C₅×#204	1	2	yes	—
323	D₄	—	3	2	no	—
324	D₄	—	6	1	no	—
325	C₂×C₄	—	2	1	no	—
326	C₂×C₄	—	2	1	no	—
327	C₂×C₄	—	6	2	no	—
328	C₂×C₄	—	6	1	no	—
329	C₂×C₄	—	12	4	no	—
330	C₂×C₄	—	12	1	no	—
331	C₂×C₄	—	12	1	no	—
332	C₂×C₄	—	12	1	no	—
333	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
334	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
335	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
336	C₈	—	4	1	no	—
337	C₈	—	4	1	no	—
338	C₈	—	12	1	no	—
339	C₂×C₃	C₅×#450	1	1	yes	characteristic
340	C₂×C₃	C₅×#451	1	1	yes	characteristic
341	C₂×C₃	C₅×#454	1	1	yes	characteristic
342	D₃	—	4	2	no	—
343	C₅	#68269	1	1	yes	characteristic, direct product factor, Sylow
344	C₂²	—	6	2	no	—
345	C₂²	—	12	1	no	—
346	C₂²	—	12	1	no	—
347	C₂²	C₅×#882	1	1	yes	characteristic
348	C₄	—	2	2	no	—
349	C₄	—	2	1	no	—
350	C₄	—	4	1	no	—
351	C₄	—	12	2	no	—
352	C₄	—	12	1	no	—
353	C₄	—	24	1	no	—
354	C₄	C₂×C₅×#204	1	1	yes	characteristic
355	C₄	C₅×#908	1	1	yes	characteristic
356	C₄	C₅×#915	1	1	yes	characteristic
357	C₄	C₅×D₃×Q₁₆	1	1	yes	characteristic
358	C₃	C₅×#1637	1	1	yes	characteristic, complemented, Sylow
359	C₂	—	12	2	no	—
360	C₂	C₅×#4731	1	1	yes	characteristic
361	C₂	C₅×#4809	1	1	yes	characteristic
362	C₂	C₅×#4827	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	C₂×C₅×D₁₂	D₄
hypercenter	C₅×Q₈⋊C₈	D₃
hypocenter	C₃	C₅×#1637
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
2. C₅×#5021	C₂	semidirect
4. C₅×#4681	C₂	semidirect
5. C₃×C₅×Q₈⋊C₈	C₂	semidirect
6. C₅×#4486	C₂	semidirect
26. #68269	C₅	direct (N × H)
32. C₅×Q₈⋊C₈	D₃	semidirect
59. #5021	C₂×C₅	semidirect
61. #4681	C₂×C₅	semidirect
62. C₃×Q₈⋊C₈	C₂×C₅	semidirect
63. #4486	C₂×C₅	semidirect
152. Q₈⋊C₈	C₅×D₃	semidirect
297. C₃×C₅	#1637	semidirect
343. C₅	#68269	direct (N × H)
358. C₃	C₅×#1637	semidirect

Maximal subgroups

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

subgroup	index
C₅×Q₈×D₁₂	2
C₅×#5021	2
C₅×#4842	2
C₅×#4681	2
C₃×C₅×Q₈⋊C₈	2
C₅×#4486	2
C₅×#4459	2
C₅×#1637	3
#68269	5

Subgroups — C5×#68269 #14289760

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#68269 #14289760