Subgroups — C₅×#68395 #14291409

Order 1920.

408 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#5073	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#5072	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#4843	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#4834	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₅×#331	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4485	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#4479	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1667	—	3	1	no	maximal, Hall
9	C₃×C₅×#127	C₂²	1	1	yes	characteristic
10	C₂×C₅×#213	—	2	1	no	—
11	C₂×C₅×#213	—	2	1	no	—
12	C₂×C₅×#213	—	2	1	no	—
13	C₂×C₅×D₂₄	—	2	1	no	—
14	C₂×C₅×D₃×C₈	—	2	1	no	—
15	C₅×#874	C₂²	1	1	yes	characteristic
16	C₄×C₅×D₁₂	C₂²	1	1	yes	characteristic
17	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
19	C₅×#832	—	2	1	no	—
20	C₅×#821	—	2	1	no	—
21	C₅×(C₂×D₃)⋊C₈	—	2	1	no	—
22	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
23	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
24	#68395	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
25	C₅×#493	—	3	1	no	—
26	C₅×#459	—	3	1	no	—
27	C₅×#459	—	3	1	no	—
28	C₅×C₈×D₄	—	3	1	no	—
29	C₅×#335	—	3	1	no	—
30	C₅×#331	—	3	1	no	—
31	C₅×#331	D₃	1	1	yes	characteristic, complemented
32	C₂×C₃×C₅×D₄	—	2	1	no	—
33	C₂×C₃×C₅×D₄	—	2	1	no	—
34	C₂×C₃×C₅×D₄	—	2	1	no	—
35	C₂×C₅×D₁₂	—	2	1	no	—
36	C₂×C₅×D₁₂	—	2	1	no	—
37	C₂×C₅×D₁₂	—	2	1	no	—
38	C₂×C₅×D₁₂	D₄	1	1	yes	characteristic
39	C₂×C₄×C₅×D₃	—	2	1	no	—
40	C₂×C₃×C₅×C₈	—	2	1	no	—
41	C₂×C₃×C₅×C₈	—	2	1	no	—
42	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
43	C₅×#213	—	4	2	no	—
44	C₅×#213	—	4	2	no	—
45	C₅×#213	—	4	2	no	—
46	C₅×(C₂×D₃)⋊C₄	—	2	1	no	—
47	C₅×C₄⋊Dic₃	D₄	1	1	yes	characteristic
48	C₂×C₅×C₃⋊C₈	—	2	1	no	—
49	C₂×C₅×C₃⋊C₈	D₄	1	1	yes	characteristic
50	C₂×C₅×C₃⋊C₈	D₄	1	1	yes	characteristic
51	C₅×D₂₄	—	4	2	no	—
52	C₅×D₃×C₈	—	4	2	no	—
53	#5073	C₂×C₅	1	1	yes	characteristic, complemented
54	#5072	C₂×C₅	1	1	yes	characteristic, complemented
55	#4843	C₂×C₅	1	1	yes	characteristic, complemented
56	#4834	C₂×C₅	1	1	yes	characteristic, complemented
57	C₃×#331	C₂×C₅	1	1	yes	characteristic, complemented
58	#4485	C₂×C₅	1	1	yes	characteristic, complemented
59	#4479	C₂×C₅	1	1	yes	characteristic, complemented
60	C₂×C₅×D₈	—	6	1	no	—
61	C₂×C₅×D₈	—	6	1	no	—
62	C₂×C₅×D₈	—	6	1	no	—
63	C₂×C₅×D₈	—	6	1	no	—
64	C₂²×C₅×C₈	—	6	1	no	—
65	C₅×#127	—	3	1	no	—
66	C₅×#127	C₂×D₃	1	1	yes	characteristic
67	C₄×C₅×D₄	—	3	1	no	—
68	C₅×C₄⋊C₈	—	3	1	no	—
69	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
70	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
71	C₅×#102	—	6	1	no	—
72	C₅×#102	—	6	1	no	—
73	C₅×C₂²⋊C₈	—	6	1	no	—
74	C₄×C₅×C₈	—	3	1	no	—
75	#1667	—	3	1	no	Sylow
76	C₂³×C₃×C₅	—	4	1	no	—
77	C₂²×C₅×D₃	—	2	1	no	—
78	C₂²×C₅×D₃	—	4	1	no	—
79	C₃×C₅×D₄	—	4	1	no	—
80	C₃×C₅×D₄	—	4	1	no	—
81	C₃×C₅×D₄	—	4	1	no	—
82	C₃×C₅×D₄	—	4	1	no	—
83	C₃×C₅×D₄	—	8	1	no	—
84	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
85	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
86	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
87	C₂×C₅×Dic₃	—	2	1	no	—
88	C₅×D₁₂	—	2	1	no	—
89	C₅×D₁₂	—	4	1	no	—
90	C₅×D₁₂	—	4	1	no	—
91	C₅×D₁₂	—	4	1	no	—
92	C₅×D₁₂	—	4	1	no	—
93	C₅×D₁₂	—	8	1	no	—
94	C₅×D₁₂	D₈	1	2	yes	complemented
95	C₄×C₅×D₃	—	4	1	no	—
96	C₄×C₅×D₃	—	4	1	no	—
97	C₃×C₅×C₈	—	4	1	no	—
98	C₃×C₅×C₈	—	4	1	no	—
99	C₅×C₃⋊C₈	—	2	1	no	—
100	C₅×C₃⋊C₈	—	4	1	no	—
101	C₅×C₃⋊C₈	D₈	1	2	yes	complemented
102	C₃×#127	C₂²×C₅	1	1	yes	characteristic
103	C₂×#213	—	2	1	no	—
104	C₂×#213	—	2	1	no	—
105	C₂×#213	—	2	1	no	—
106	C₂×D₂₄	—	2	1	no	—
107	C₂×D₃×C₈	—	2	1	no	—
108	#874	C₂²×C₅	1	1	yes	characteristic
109	C₄×D₁₂	C₂²×C₅	1	1	yes	characteristic
110	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
111	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
112	#832	—	2	1	no	—
113	#821	—	2	1	no	—
114	(C₂×D₃)⋊C₈	—	2	1	no	—
115	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	1	yes	characteristic
116	C₄×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
117	C₂×C₅×D₄	—	2	1	no	—
118	C₂×C₅×D₄	—	2	1	no	—
119	C₂×C₅×D₄	—	2	1	no	—
120	C₂×C₅×D₄	—	3	1	no	—
121	C₂×C₅×D₄	—	6	1	no	—
122	C₂×C₅×D₄	—	6	1	no	—
123	C₂×C₅×D₄	—	6	1	no	—
124	C₂²×C₄×C₅	—	6	1	no	—
125	C₅×D₈	—	12	2	no	—
126	C₅×D₈	—	12	2	no	—
127	C₅×D₈	—	12	2	no	—
128	C₅×D₈	—	12	2	no	—
129	C₂×C₅×C₈	—	2	1	no	—
130	C₂×C₅×C₈	—	2	1	no	—
131	C₂×C₅×C₈	—	3	1	no	—
132	C₂×C₅×C₈	—	3	1	no	—
133	C₂×C₅×C₈	—	6	1	no	—
134	C₂×C₅×C₈	—	12	2	no	—
135	C₅×C₄⋊C₄	—	3	1	no	—
136	C₅×C₂²⋊C₄	—	6	1	no	—
137	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
138	#493	—	3	1	no	—
139	#459	—	3	1	no	—
140	#459	—	3	1	no	—
141	C₈×D₄	—	3	1	no	—
142	#335	—	3	1	no	—
143	#331	—	3	1	no	—
144	#331	C₅×D₃	1	1	yes	characteristic, complemented
145	C₂²×C₃×C₅	—	8	1	no	—
146	C₂²×C₃×C₅	—	8	1	no	—
147	C₂²×C₃×C₅	—	8	1	no	—
148	C₂²×C₃×C₅	#121	1	1	yes	characteristic
149	C₂×C₅×D₃	—	2	2	no	—
150	C₂×C₅×D₃	—	4	1	no	—
151	C₂×C₅×D₃	—	4	1	no	—
152	C₂×C₅×D₃	—	8	1	no	—
153	C₂×C₅×D₃	—	8	1	no	—
154	C₂×C₅×D₃	—	8	1	no	—
155	C₃×C₄×C₅	—	2	1	no	—
156	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
157	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
158	C₃×C₄×C₅	#135	1	1	yes	characteristic
159	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
160	C₅×Dic₃	—	4	1	no	—
161	C₂×C₃×D₄	—	2	1	no	—
162	C₂×C₃×D₄	—	2	1	no	—
163	C₂×C₃×D₄	—	2	1	no	—
164	C₂×D₁₂	—	2	1	no	—
165	C₂×D₁₂	—	2	1	no	—
166	C₂×D₁₂	—	2	1	no	—
167	C₂×D₁₂	C₅×D₄	1	1	yes	characteristic
168	C₂×C₄×D₃	—	2	1	no	—
169	C₂×C₃×C₈	—	2	1	no	—
170	C₂×C₃×C₈	—	2	1	no	—
171	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
172	#213	—	4	2	no	—
173	#213	—	4	2	no	—
174	#213	—	4	2	no	—
175	(C₂×D₃)⋊C₄	—	2	1	no	—
176	C₄⋊Dic₃	C₅×D₄	1	1	yes	characteristic
177	C₂×C₃⋊C₈	—	2	1	no	—
178	C₂×C₃⋊C₈	C₅×D₄	1	1	yes	characteristic
179	C₂×C₃⋊C₈	C₅×D₄	1	1	yes	characteristic
180	D₂₄	—	4	2	no	—
181	D₃×C₈	—	4	2	no	—
182	C₂³×C₅	—	4	1	no	—
183	C₂³×C₅	—	6	1	no	—
184	C₂³×C₅	—	12	1	no	—
185	C₅×D₄	—	3	2	no	—
186	C₅×D₄	—	4	1	no	—
187	C₅×D₄	—	4	1	no	—
188	C₅×D₄	—	4	1	no	—
189	C₅×D₄	—	4	1	no	—
190	C₅×D₄	—	6	1	no	—
191	C₅×D₄	—	8	1	no	—
192	C₅×D₄	—	12	1	no	—
193	C₅×D₄	—	12	1	no	—
194	C₅×D₄	—	12	1	no	—
195	C₅×D₄	—	12	1	no	—
196	C₅×D₄	—	24	1	no	—
197	C₂×C₄×C₅	—	6	1	no	—
198	C₂×C₄×C₅	—	12	1	no	—
199	C₂×C₄×C₅	—	12	1	no	—
200	C₂×C₄×C₅	#235	1	1	yes	characteristic
201	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
202	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
203	C₅×C₈	—	3	2	no	—
204	C₅×C₈	—	4	1	no	—
205	C₅×C₈	—	4	1	no	—
206	C₅×C₈	—	6	1	no	—
207	C₅×C₈	—	12	1	no	—
208	C₂×D₈	—	6	1	no	—
209	C₂×D₈	—	6	1	no	—
210	C₂×D₈	—	6	1	no	—
211	C₂×D₈	—	6	1	no	—
212	C₂²×C₈	—	6	1	no	—
213	#127	—	3	1	no	—
214	#127	C₂×C₅×D₃	1	1	yes	characteristic
215	C₄×D₄	—	3	1	no	—
216	C₄⋊C₈	—	3	1	no	—
217	C₄⋊C₈	C₂×C₅×D₃	1	1	yes	characteristic
218	C₄⋊C₈	C₂×C₅×D₃	1	1	yes	characteristic
219	#102	—	6	1	no	—
220	#102	—	6	1	no	—
221	C₂²⋊C₈	—	6	1	no	—
222	C₄×C₈	—	3	1	no	—
223	C₂×C₃×C₅	—	16	1	no	—
224	C₂×C₃×C₅	#459	1	1	yes	characteristic
225	C₂×C₃×C₅	#466	1	1	yes	characteristic
226	C₂×C₃×C₅	#472	1	1	yes	characteristic
227	C₅×D₃	—	4	2	no	—
228	C₅×D₃	—	16	1	no	—
229	C₂³×C₃	—	4	1	no	—
230	C₂²×D₃	—	2	1	no	—
231	C₂²×D₃	—	4	1	no	—
232	C₃×D₄	—	4	1	no	—
233	C₃×D₄	—	4	1	no	—
234	C₃×D₄	—	4	1	no	—
235	C₃×D₄	—	4	1	no	—
236	C₃×D₄	—	8	1	no	—
237	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
238	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
239	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
240	C₂×Dic₃	—	2	1	no	—
241	D₁₂	—	2	1	no	—
242	D₁₂	—	4	1	no	—
243	D₁₂	—	4	1	no	—
244	D₁₂	—	4	1	no	—
245	D₁₂	—	4	1	no	—
246	D₁₂	—	8	1	no	—
247	D₁₂	C₅×D₈	1	2	yes	complemented
248	C₄×D₃	—	4	1	no	—
249	C₄×D₃	—	4	1	no	—
250	C₃×C₈	—	4	1	no	—
251	C₃×C₈	—	4	1	no	—
252	C₃⋊C₈	—	2	1	no	—
253	C₃⋊C₈	—	4	1	no	—
254	C₃⋊C₈	C₅×D₈	1	2	yes	complemented
255	C₂²×C₅	—	6	2	no	—
256	C₂²×C₅	—	8	1	no	—
257	C₂²×C₅	—	8	1	no	—
258	C₂²×C₅	—	8	1	no	—
259	C₂²×C₅	—	12	1	no	—
260	C₂²×C₅	—	12	1	no	—
261	C₂²×C₅	—	24	1	no	—
262	C₂²×C₅	—	24	1	no	—
263	C₂²×C₅	—	24	1	no	—
264	C₂²×C₅	#884	1	1	yes	characteristic
265	C₄×C₅	—	2	1	no	—
266	C₄×C₅	—	12	1	no	—
267	C₄×C₅	D₃×D₈	1	1	yes	characteristic
268	C₄×C₅	D₃×D₈	1	1	yes	characteristic
269	C₄×C₅	#914	1	1	yes	characteristic
270	C₄×C₅	#916	1	1	yes	characteristic
271	C₂×D₄	—	2	1	no	—
272	C₂×D₄	—	2	1	no	—
273	C₂×D₄	—	2	1	no	—
274	C₂×D₄	—	3	1	no	—
275	C₂×D₄	—	6	1	no	—
276	C₂×D₄	—	6	1	no	—
277	C₂×D₄	—	6	1	no	—
278	C₂²×C₄	—	6	1	no	—
279	D₈	—	12	2	no	—
280	D₈	—	12	2	no	—
281	D₈	—	12	2	no	—
282	D₈	—	12	2	no	—
283	C₂×C₈	—	2	1	no	—
284	C₂×C₈	—	2	1	no	—
285	C₂×C₈	—	3	1	no	—
286	C₂×C₈	—	3	1	no	—
287	C₂×C₈	—	6	1	no	—
288	C₂×C₈	—	12	2	no	—
289	C₄⋊C₄	—	3	1	no	—
290	C₂²⋊C₄	—	6	1	no	—
291	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
292	C₃×C₅	#1667	1	1	yes	characteristic, complemented, Hall
293	C₂²×C₃	—	8	1	no	—
294	C₂²×C₃	—	8	1	no	—
295	C₂²×C₃	—	8	1	no	—
296	C₂²×C₃	C₅×#121	1	1	yes	characteristic
297	C₂×D₃	—	2	2	no	—
298	C₂×D₃	—	4	1	no	—
299	C₂×D₃	—	4	1	no	—
300	C₂×D₃	—	8	1	no	—
301	C₂×D₃	—	8	1	no	—
302	C₂×D₃	—	8	1	no	—
303	C₃×C₄	—	2	1	no	—
304	C₃×C₄	C₂×C₅×D₈	1	1	yes	characteristic
305	C₃×C₄	C₂×C₅×D₈	1	1	yes	characteristic
306	C₃×C₄	C₅×#135	1	1	yes	characteristic
307	C₃×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
308	Dic₃	—	4	1	no	—
309	C₂×C₅	—	12	2	no	—
310	C₂×C₅	—	16	1	no	—
311	C₂×C₅	—	48	1	no	—
312	C₂×C₅	#4749	1	1	yes	characteristic
313	C₂×C₅	#4775	1	1	yes	characteristic
314	C₂×C₅	#4786	1	1	yes	characteristic
315	C₂³	—	4	1	no	—
316	C₂³	—	6	1	no	—
317	C₂³	—	12	1	no	—
318	D₄	—	3	2	no	—
319	D₄	—	4	1	no	—
320	D₄	—	4	1	no	—
321	D₄	—	4	1	no	—
322	D₄	—	4	1	no	—
323	D₄	—	6	1	no	—
324	D₄	—	8	1	no	—
325	D₄	—	12	1	no	—
326	D₄	—	12	1	no	—
327	D₄	—	12	1	no	—
328	D₄	—	12	1	no	—
329	D₄	—	24	1	no	—
330	C₂×C₄	—	6	1	no	—
331	C₂×C₄	—	12	1	no	—
332	C₂×C₄	—	12	1	no	—
333	C₂×C₄	C₅×#235	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	—	3	2	no	—
337	C₈	—	4	1	no	—
338	C₈	—	4	1	no	—
339	C₈	—	6	1	no	—
340	C₈	—	12	1	no	—
341	C₂×C₃	—	16	1	no	—
342	C₂×C₃	C₅×#459	1	1	yes	characteristic
343	C₂×C₃	C₅×#466	1	1	yes	characteristic
344	C₂×C₃	C₅×#472	1	1	yes	characteristic
345	D₃	—	4	2	no	—
346	D₃	—	16	1	no	—
347	C₅	#68395	1	1	yes	characteristic, direct product factor, Sylow
348	C₂²	—	6	2	no	—
349	C₂²	—	8	1	no	—
350	C₂²	—	8	1	no	—
351	C₂²	—	8	1	no	—
352	C₂²	—	12	1	no	—
353	C₂²	—	12	1	no	—
354	C₂²	—	24	1	no	—
355	C₂²	—	24	1	no	—
356	C₂²	—	24	1	no	—
357	C₂²	C₅×#884	1	1	yes	characteristic
358	C₄	—	2	1	no	—
359	C₄	—	12	1	no	—
360	C₄	C₅×D₃×D₈	1	1	yes	characteristic
361	C₄	C₅×D₃×D₈	1	1	yes	characteristic
362	C₄	C₅×#914	1	1	yes	characteristic
363	C₄	C₅×#916	1	1	yes	characteristic
364	C₃	C₅×#1667	1	1	yes	characteristic, complemented, Sylow
365	C₂	—	12	2	no	—
366	C₂	—	16	1	no	—
367	C₂	—	48	1	no	—
368	C₂	C₅×#4749	1	1	yes	characteristic
369	C₂	C₅×#4775	1	1	yes	characteristic
370	C₂	C₅×#4786	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	#884
commutator, cosocle	C₃×C₄²	C₂³×C₅
Frattini	C₄²	C₂²×C₅×D₃
Fitting	C₃×C₅×#331	C₂
socle	C₂²×C₃×C₅	#121
hypercenter	C₅×#331	D₃
hypocenter	C₃	C₅×#1667
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×#331

Complements

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

N	H ≅ G/N	type
1. C₅×#5073	C₂	semidirect
2. C₅×#5072	C₂	semidirect
3. C₅×#4843	C₂	semidirect
4. C₅×#4834	C₂	semidirect
5. C₃×C₅×#331	C₂	semidirect
6. C₅×#4485	C₂	semidirect
7. C₅×#4479	C₂	semidirect
24. #68395	C₅	direct (N × H)
31. C₅×#331	D₃	semidirect
53. #5073	C₂×C₅	semidirect
54. #5072	C₂×C₅	semidirect
55. #4843	C₂×C₅	semidirect
56. #4834	C₂×C₅	semidirect
57. C₃×#331	C₂×C₅	semidirect
58. #4485	C₂×C₅	semidirect
59. #4479	C₂×C₅	semidirect
94. C₅×D₁₂	D₈	semidirect
101. C₅×C₃⋊C₈	D₈	semidirect
144. #331	C₅×D₃	semidirect
247. D₁₂	C₅×D₈	semidirect
254. C₃⋊C₈	C₅×D₈	semidirect
292. C₃×C₅	#1667	semidirect
347. C₅	#68395	direct (N × H)
364. C₃	C₅×#1667	semidirect

Maximal subgroups

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

subgroup	index
C₅×#5073	2
C₅×#5072	2
C₅×#4843	2
C₅×#4834	2
C₃×C₅×#331	2
C₅×#4485	2
C₅×#4479	2
C₅×#1667	3
#68395	5

Subgroups — C5×#68395 #14291409

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#68395 #14291409