Subgroups — C₅×#68434 #14289631

Order 1920.

468 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#5726	C₂	1	1	yes	maximal, characteristic
2	C₅×#5093	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#4842	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#4842	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₅×#332	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4483	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#4483	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1635	—	3	1	no	maximal, Hall
9	C₂×C₅×#239	—	2	1	no	—
10	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₅×#946	—	2	1	no	—
12	C₅×#895	—	2	1	no	—
13	C₅×#894	—	2	2	no	—
14	C₅×D₃×C₄⋊C₄	—	2	1	no	—
15	C₄×C₅×D₁₂	C₂²	1	1	yes	characteristic
16	C₄×C₅×D₁₂	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×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₅×#821	—	2	1	no	—
21	C₅×#821	—	2	1	no	—
22	C₅×#817	—	2	1	no	—
23	C₅×#817	—	2	1	no	—
24	C₅×#810	—	2	2	no	—
25	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
26	#68434	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
27	C₅×#547	—	3	1	no	—
28	C₅×#487	—	3	1	no	—
29	C₅×#476	—	3	1	no	—
30	C₅×#476	—	3	1	no	—
31	C₅×#332	D₃	1	1	yes	characteristic, complemented
32	C₅×D₄⋊C₈	—	3	1	no	—
33	C₅×D₄⋊C₈	—	3	1	no	—
34	C₂×C₃×C₅×Q₈	—	2	1	no	—
35	C₅×#239	—	4	2	no	—
36	C₅×#239	—	4	2	no	—
37	C₂×C₅×D₁₂	—	2	2	no	—
38	C₂×C₅×D₁₂	D₄	1	1	yes	characteristic
39	C₂×C₅×D₁₂	D₄	1	1	yes	characteristic
40	C₂×C₄×C₅×D₃	—	2	2	no	—
41	C₂×C₄×C₅×D₃	—	2	1	no	—
42	C₂×C₄×C₅×D₃	—	2	1	no	—
43	C₂×C₅×Dic₆	—	2	1	no	—
44	C₂×C₃×C₅×C₈	—	2	1	no	—
45	C₂×C₃×C₅×C₈	—	2	1	no	—
46	C₃×C₅×C₄⋊C₄	—	2	1	no	—
47	C₃×C₅×C₄⋊C₄	D₄	1	2	yes	—
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×(C₂×D₃)⋊C₄	—	2	1	no	—
52	C₅×C₄⋊Dic₃	—	2	1	no	—
53	C₅×C₄⋊Dic₃	—	2	1	no	—
54	C₅×C₄⋊Dic₃	D₄	1	1	yes	characteristic
55	C₅×C₄⋊Dic₃	D₄	1	1	yes	characteristic
56	C₅×Dic₃⋊C₄	—	2	2	no	—
57	C₂×C₅×C₃⋊C₈	—	2	1	no	—
58	#5726	C₂×C₅	1	1	yes	characteristic
59	#5093	C₂×C₅	1	1	yes	characteristic, complemented
60	#4842	C₂×C₅	1	1	yes	characteristic, complemented
61	#4842	C₂×C₅	1	1	yes	characteristic, complemented
62	C₃×#332	C₂×C₅	1	1	yes	characteristic, complemented
63	#4483	C₂×C₅	1	1	yes	characteristic, complemented
64	#4483	C₂×C₅	1	1	yes	characteristic, complemented
65	C₂×C₅×#41	—	6	1	no	—
66	C₅×C₄⋊Q₈	—	3	1	no	—
67	C₅×C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
68	C₅×#123	—	6	2	no	—
69	C₅×C₂²⋊Q₈	—	6	1	no	—
70	C₅×#121	—	6	1	no	—
71	C₄×C₅×D₄	—	3	1	no	—
72	C₄×C₅×D₄	—	3	1	no	—
73	C₂×C₅×C₄⋊C₄	—	6	1	no	—
74	C₅×#106	—	6	1	no	—
75	C₅×#106	—	6	1	no	—
76	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
77	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
78	C₅×#103	—	6	2	no	—
79	C₅×#102	—	6	1	no	—
80	C₅×#102	—	6	1	no	—
81	C₄×C₅×C₈	—	3	1	no	—
82	#1635	—	3	1	no	Sylow
83	C₂²×C₅×D₃	—	2	1	no	—
84	C₂²×C₅×D₃	—	2	1	no	—
85	C₃×C₅×Q₈	—	4	1	no	—
86	C₃×C₅×Q₈	—	4	1	no	—
87	C₂×C₃×C₄×C₅	—	2	2	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	1	no	—
93	C₂×C₅×Dic₃	—	4	1	no	—
94	C₅×D₁₂	—	2	1	no	—
95	C₅×D₁₂	—	2	1	no	—
96	C₅×D₁₂	—	2	1	no	—
97	C₅×D₁₂	—	4	4	no	—
98	C₅×D₁₂	SD₁₆	1	2	yes	—
99	C₄×C₅×D₃	—	4	4	no	—
100	C₄×C₅×D₃	—	4	1	no	—
101	C₄×C₅×D₃	—	4	1	no	—
102	C₄×C₅×D₃	—	4	1	no	—
103	C₄×C₅×D₃	—	4	1	no	—
104	C₅×Dic₆	—	4	2	no	—
105	C₃×C₅×C₈	—	4	1	no	—
106	C₃×C₅×C₈	—	4	1	no	—
107	C₅×C₃⋊C₈	—	4	1	no	—
108	C₂×#239	—	2	1	no	—
109	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
110	#946	—	2	1	no	—
111	#895	—	2	1	no	—
112	#894	—	2	2	no	—
113	D₃×C₄⋊C₄	—	2	1	no	—
114	C₄×D₁₂	C₂²×C₅	1	1	yes	characteristic
115	C₄×D₁₂	C₂²×C₅	1	1	yes	characteristic
116	#869	C₂²×C₅	1	1	yes	characteristic
117	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
118	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
119	#821	—	2	1	no	—
120	#821	—	2	1	no	—
121	#817	—	2	1	no	—
122	#817	—	2	1	no	—
123	#810	—	2	2	no	—
124	C₄×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
125	C₅×#41	—	12	2	no	—
126	C₅×#41	—	12	2	no	—
127	C₂×C₅×Q₈	—	2	1	no	—
128	C₂×C₅×Q₈	—	6	1	no	—
129	C₂×C₅×D₄	—	3	1	no	—
130	C₂×C₅×D₄	—	3	1	no	—
131	C₂×C₅×D₄	—	6	2	no	—
132	C₂²×C₄×C₅	—	6	2	no	—
133	C₂²×C₄×C₅	—	6	1	no	—
134	C₂²×C₄×C₅	—	6	1	no	—
135	C₂×C₅×C₈	—	2	1	no	—
136	C₂×C₅×C₈	—	2	1	no	—
137	C₂×C₅×C₈	—	6	1	no	—
138	C₅×C₄⋊C₄	—	2	1	no	—
139	C₅×C₄⋊C₄	—	3	1	no	—
140	C₅×C₄⋊C₄	—	3	1	no	—
141	C₅×C₄⋊C₄	—	6	2	no	—
142	C₅×C₄⋊C₄	—	6	1	no	—
143	C₅×C₄⋊C₄	—	6	1	no	—
144	C₅×C₄⋊C₄	D₁₂	1	2	yes	—
145	C₅×C₂²⋊C₄	—	6	2	no	—
146	C₅×C₂²⋊C₄	—	6	1	no	—
147	C₅×C₂²⋊C₄	—	6	1	no	—
148	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
149	#547	—	3	1	no	—
150	#487	—	3	1	no	—
151	#476	—	3	1	no	—
152	#476	—	3	1	no	—
153	#332	C₅×D₃	1	1	yes	characteristic, complemented
154	D₄⋊C₈	—	3	1	no	—
155	D₄⋊C₈	—	3	1	no	—
156	C₂²×C₃×C₅	#120	1	1	yes	characteristic
157	C₂×C₅×D₃	—	2	2	no	—
158	C₂×C₅×D₃	—	4	1	no	—
159	C₂×C₅×D₃	—	4	1	no	—
160	C₂×C₅×D₃	—	4	1	no	—
161	C₂×C₅×D₃	—	4	1	no	—
162	C₂×C₅×D₃	—	4	1	no	—
163	C₃×C₄×C₅	—	2	1	no	—
164	C₃×C₄×C₅	—	4	2	no	—
165	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
166	C₃×C₄×C₅	#135	1	1	yes	characteristic
167	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
168	C₃×C₄×C₅	#137	1	1	yes	characteristic
169	C₅×Dic₃	—	4	1	no	—
170	C₅×Dic₃	—	4	1	no	—
171	C₅×Dic₃	—	8	1	no	—
172	C₂×C₃×Q₈	—	2	1	no	—
173	#239	—	4	2	no	—
174	#239	—	4	2	no	—
175	C₂×D₁₂	—	2	2	no	—
176	C₂×D₁₂	C₅×D₄	1	1	yes	characteristic
177	C₂×D₁₂	C₅×D₄	1	1	yes	characteristic
178	C₂×C₄×D₃	—	2	2	no	—
179	C₂×C₄×D₃	—	2	1	no	—
180	C₂×C₄×D₃	—	2	1	no	—
181	C₂×Dic₆	—	2	1	no	—
182	C₂×C₃×C₈	—	2	1	no	—
183	C₂×C₃×C₈	—	2	1	no	—
184	C₃×C₄⋊C₄	—	2	1	no	—
185	C₃×C₄⋊C₄	C₅×D₄	1	2	yes	—
186	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
187	(C₂×D₃)⋊C₄	—	2	2	no	—
188	(C₂×D₃)⋊C₄	—	2	1	no	—
189	(C₂×D₃)⋊C₄	—	2	1	no	—
190	C₄⋊Dic₃	—	2	1	no	—
191	C₄⋊Dic₃	—	2	1	no	—
192	C₄⋊Dic₃	C₅×D₄	1	1	yes	characteristic
193	C₄⋊Dic₃	C₅×D₄	1	1	yes	characteristic
194	Dic₃⋊C₄	—	2	2	no	—
195	C₂×C₃⋊C₈	—	2	1	no	—
196	C₂³×C₅	—	6	1	no	—
197	C₂³×C₅	—	6	1	no	—
198	C₅×Q₈	—	4	1	no	—
199	C₅×Q₈	—	4	1	no	—
200	C₅×Q₈	—	12	2	no	—
201	C₅×D₄	—	3	2	no	—
202	C₅×D₄	—	6	1	no	—
203	C₅×D₄	—	6	1	no	—
204	C₅×D₄	—	6	1	no	—
205	C₅×D₄	—	12	4	no	—
206	C₂×C₄×C₅	—	2	2	no	—
207	C₂×C₄×C₅	—	6	1	no	—
208	C₂×C₄×C₅	—	6	1	no	—
209	C₂×C₄×C₅	—	12	4	no	—
210	C₂×C₄×C₅	—	12	1	no	—
211	C₂×C₄×C₅	—	12	1	no	—
212	C₂×C₄×C₅	—	12	1	no	—
213	C₂×C₄×C₅	—	12	1	no	—
214	C₂×C₄×C₅	—	12	1	no	—
215	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
216	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
217	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
218	C₅×C₈	—	4	1	no	—
219	C₅×C₈	—	4	1	no	—
220	C₅×C₈	—	12	1	no	—
221	C₂×#41	—	6	1	no	—
222	C₄⋊Q₈	—	3	1	no	—
223	C₄⋊Q₈	C₂×C₅×D₃	1	1	yes	characteristic
224	#123	—	6	2	no	—
225	C₂²⋊Q₈	—	6	1	no	—
226	#121	—	6	1	no	—
227	C₄×D₄	—	3	1	no	—
228	C₄×D₄	—	3	1	no	—
229	C₂×C₄⋊C₄	—	6	1	no	—
230	#106	—	6	1	no	—
231	#106	—	6	1	no	—
232	C₄⋊C₈	C₂×C₅×D₃	1	1	yes	characteristic
233	C₄⋊C₈	C₂×C₅×D₃	1	1	yes	characteristic
234	#103	—	6	2	no	—
235	#102	—	6	1	no	—
236	#102	—	6	1	no	—
237	C₄×C₈	—	3	1	no	—
238	C₂×C₃×C₅	#450	1	1	yes	characteristic
239	C₂×C₃×C₅	#452	1	1	yes	characteristic
240	C₂×C₃×C₅	#454	1	1	yes	characteristic
241	C₅×D₃	—	4	2	no	—
242	C₅×D₃	—	8	1	no	—
243	C₂²×D₃	—	2	1	no	—
244	C₂²×D₃	—	2	1	no	—
245	C₃×Q₈	—	4	1	no	—
246	C₃×Q₈	—	4	1	no	—
247	C₂×C₃×C₄	—	2	2	no	—
248	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
249	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
250	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
251	C₂×Dic₃	—	2	1	no	—
252	C₂×Dic₃	—	2	1	no	—
253	C₂×Dic₃	—	4	1	no	—
254	D₁₂	—	2	1	no	—
255	D₁₂	—	2	1	no	—
256	D₁₂	—	2	1	no	—
257	D₁₂	—	4	4	no	—
258	D₁₂	C₅×SD₁₆	1	2	yes	—
259	C₄×D₃	—	4	4	no	—
260	C₄×D₃	—	4	1	no	—
261	C₄×D₃	—	4	1	no	—
262	C₄×D₃	—	4	1	no	—
263	C₄×D₃	—	4	1	no	—
264	Dic₆	—	4	2	no	—
265	C₃×C₈	—	4	1	no	—
266	C₃×C₈	—	4	1	no	—
267	C₃⋊C₈	—	4	1	no	—
268	C₂²×C₅	—	6	2	no	—
269	C₂²×C₅	—	12	1	no	—
270	C₂²×C₅	—	12	1	no	—
271	C₂²×C₅	—	12	1	no	—
272	C₂²×C₅	—	12	1	no	—
273	C₂²×C₅	—	12	1	no	—
274	C₂²×C₅	#882	1	1	yes	characteristic
275	C₄×C₅	—	2	1	no	—
276	C₄×C₅	—	4	2	no	—
277	C₄×C₅	—	12	1	no	—
278	C₄×C₅	—	12	1	no	—
279	C₄×C₅	—	24	1	no	—
280	C₄×C₅	#911	1	1	yes	characteristic
281	C₄×C₅	D₃×SD₁₆	1	1	yes	characteristic
282	C₄×C₅	#916	1	1	yes	characteristic
283	C₄×C₅	#918	1	1	yes	characteristic
284	#41	—	12	2	no	—
285	#41	—	12	2	no	—
286	C₂×Q₈	—	2	1	no	—
287	C₂×Q₈	—	6	1	no	—
288	C₂×D₄	—	3	1	no	—
289	C₂×D₄	—	3	1	no	—
290	C₂×D₄	—	6	2	no	—
291	C₂²×C₄	—	6	2	no	—
292	C₂²×C₄	—	6	1	no	—
293	C₂²×C₄	—	6	1	no	—
294	C₂×C₈	—	2	1	no	—
295	C₂×C₈	—	2	1	no	—
296	C₂×C₈	—	6	1	no	—
297	C₄⋊C₄	—	2	1	no	—
298	C₄⋊C₄	—	3	1	no	—
299	C₄⋊C₄	—	3	1	no	—
300	C₄⋊C₄	—	6	2	no	—
301	C₄⋊C₄	—	6	1	no	—
302	C₄⋊C₄	—	6	1	no	—
303	C₄⋊C₄	C₅×D₁₂	1	2	yes	—
304	C₂²⋊C₄	—	6	2	no	—
305	C₂²⋊C₄	—	6	1	no	—
306	C₂²⋊C₄	—	6	1	no	—
307	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
308	C₃×C₅	#1635	1	1	yes	characteristic, complemented, Hall
309	C₂²×C₃	C₅×#120	1	1	yes	characteristic
310	C₂×D₃	—	2	2	no	—
311	C₂×D₃	—	4	1	no	—
312	C₂×D₃	—	4	1	no	—
313	C₂×D₃	—	4	1	no	—
314	C₂×D₃	—	4	1	no	—
315	C₂×D₃	—	4	1	no	—
316	C₃×C₄	—	2	1	no	—
317	C₃×C₄	—	4	2	no	—
318	C₃×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
319	C₃×C₄	C₅×#135	1	1	yes	characteristic
320	C₃×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
321	C₃×C₄	C₅×#137	1	1	yes	characteristic
322	Dic₃	—	4	1	no	—
323	Dic₃	—	4	1	no	—
324	Dic₃	—	8	1	no	—
325	C₂×C₅	—	12	2	no	—
326	C₂×C₅	—	24	1	no	—
327	C₂×C₅	#4753	1	1	yes	characteristic
328	C₂×C₅	#4776	1	1	yes	characteristic
329	C₂×C₅	#4808	1	1	yes	characteristic
330	C₂³	—	6	1	no	—
331	C₂³	—	6	1	no	—
332	Q₈	—	4	1	no	—
333	Q₈	—	4	1	no	—
334	Q₈	—	12	2	no	—
335	D₄	—	3	2	no	—
336	D₄	—	6	1	no	—
337	D₄	—	6	1	no	—
338	D₄	—	6	1	no	—
339	D₄	—	12	4	no	—
340	C₂×C₄	—	2	2	no	—
341	C₂×C₄	—	6	1	no	—
342	C₂×C₄	—	6	1	no	—
343	C₂×C₄	—	12	4	no	—
344	C₂×C₄	—	12	1	no	—
345	C₂×C₄	—	12	1	no	—
346	C₂×C₄	—	12	1	no	—
347	C₂×C₄	—	12	1	no	—
348	C₂×C₄	—	12	1	no	—
349	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
350	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
351	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
352	C₈	—	4	1	no	—
353	C₈	—	4	1	no	—
354	C₈	—	12	1	no	—
355	C₂×C₃	C₅×#450	1	1	yes	characteristic
356	C₂×C₃	C₅×#452	1	1	yes	characteristic
357	C₂×C₃	C₅×#454	1	1	yes	characteristic
358	D₃	—	4	2	no	—
359	D₃	—	8	1	no	—
360	C₅	#68434	1	1	yes	characteristic, direct product factor, Sylow
361	C₂²	—	6	2	no	—
362	C₂²	—	12	1	no	—
363	C₂²	—	12	1	no	—
364	C₂²	—	12	1	no	—
365	C₂²	—	12	1	no	—
366	C₂²	—	12	1	no	—
367	C₂²	C₅×#882	1	1	yes	characteristic
368	C₄	—	2	1	no	—
369	C₄	—	4	2	no	—
370	C₄	—	12	1	no	—
371	C₄	—	12	1	no	—
372	C₄	—	24	1	no	—
373	C₄	C₅×#911	1	1	yes	characteristic
374	C₄	C₅×D₃×SD₁₆	1	1	yes	characteristic
375	C₄	C₅×#916	1	1	yes	characteristic
376	C₄	C₅×#918	1	1	yes	characteristic
377	C₃	C₅×#1635	1	1	yes	characteristic, complemented, Sylow
378	C₂	—	12	2	no	—
379	C₂	—	24	1	no	—
380	C₂	C₅×#4753	1	1	yes	characteristic
381	C₂	C₅×#4776	1	1	yes	characteristic
382	C₂	C₅×#4808	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×#332	C₂
socle	C₂²×C₃×C₅	#120
prime radical	C₅×#5726	C₂
hypercenter	C₅×#332	D₃
hypocenter	C₃	C₅×#1635
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×#332

Complements

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

N	H ≅ G/N	type
2. C₅×#5093	C₂	semidirect
3. C₅×#4842	C₂	semidirect
4. C₅×#4842	C₂	semidirect
5. C₃×C₅×#332	C₂	semidirect
6. C₅×#4483	C₂	semidirect
7. C₅×#4483	C₂	semidirect
26. #68434	C₅	direct (N × H)
31. C₅×#332	D₃	semidirect
59. #5093	C₂×C₅	semidirect
60. #4842	C₂×C₅	semidirect
61. #4842	C₂×C₅	semidirect
62. C₃×#332	C₂×C₅	semidirect
63. #4483	C₂×C₅	semidirect
64. #4483	C₂×C₅	semidirect
153. #332	C₅×D₃	semidirect
308. C₃×C₅	#1635	semidirect
360. C₅	#68434	direct (N × H)
377. C₃	C₅×#1635	semidirect

Maximal subgroups

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

subgroup	index
C₅×#5726	2
C₅×#5093	2
C₅×#4842	2
C₃×C₅×#332	2
C₅×#4483	2
C₅×#1635	3
#68434	5

Subgroups — C5×#68434 #14289631

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#68434 #14289631