Subgroups — C₅×#82028 #14478613

Order 1920.

700 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#5766	C₂	1	2	yes	maximal, complemented
2	C₂×C₃×C₅×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#5179	C₂	1	2	yes	maximal, complemented
4	C₅×#4919	C₂	1	2	yes	maximal, complemented
5	C₅×#4918	C₂	1	4	yes	maximal, complemented
6	C₅×D₃×Q₃₂	C₂	1	2	yes	maximal, complemented
7	C₅×#4904	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#4901	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#3417	—	3	1	no	maximal, Hall
10	C₅×#1007	—	2	2	no	—
11	C₂×C₃×C₅×Q₁₆	C₂²	1	2	yes	—
12	C₅×#942	—	2	2	no	—
13	C₅×#919	C₂²	1	4	yes	—
14	C₅×#918	—	2	4	no	—
15	C₅×D₃×Q₁₆	C₂²	1	4	yes	—
16	C₅×#904	C₂²	1	2	yes	—
17	C₅×#901	C₂²	1	1	yes	characteristic
18	C₃×C₅×Q₃₂	C₂²	1	2	yes	—
19	C₃×C₅×Q₃₂	C₂²	1	2	yes	—
20	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic
21	C₅×#829	C₂²	1	4	yes	—
22	C₅×#828	C₂²	1	4	yes	—
23	C₅×#812	C₂²	1	1	yes	characteristic
24	C₅×Dic₂₄	C₂²	1	1	yes	characteristic
25	C₅×#800	C₂²	1	2	yes	—
26	C₅×D₄₈	C₂²	1	1	yes	characteristic
27	C₅×#798	C₂²	1	2	yes	—
28	C₅×D₃×C₁₆	C₂²	1	2	yes	—
29	#82028	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
30	C₅×#578	—	3	2	no	—
31	C₅×#510	—	3	4	no	—
32	C₅×#510	—	3	2	no	—
33	C₅×#508	—	3	2	no	—
34	C₅×#508	—	3	1	no	—
35	C₂×C₅×Q₃₂	—	3	2	no	—
36	C₂×C₅×Q₃₂	D₃	1	1	yes	characteristic, complemented
37	C₅×#504	—	3	1	no	—
38	C₂×C₃×C₅×Q₈	—	2	2	no	—
39	C₅×#239	—	2	4	no	—
40	C₅×#239	—	4	2	no	—
41	C₅×D₃×Q₈	—	2	4	no	—
42	C₅×D₃×Q₈	—	4	2	no	—
43	C₅×#235	—	2	2	no	—
44	C₅×#235	—	4	4	no	—
45	C₅×#235	D₄	1	1	yes	characteristic
46	C₃×C₅×Q₁₆	—	2	2	no	—
47	C₃×C₅×Q₁₆	C₂³	1	4	yes	—
48	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
49	C₅×#216	—	2	4	no	—
50	C₅×#215	—	2	4	no	—
51	C₅×#208	D₄	1	1	yes	characteristic
52	C₅×Dic₁₂	C₂³	1	2	yes	—
53	C₅×D₂₄	C₂³	1	2	yes	—
54	C₅×#204	—	2	2	no	—
55	C₅×ΓL(1,𝔽₂₅)	D₄	1	2	yes	—
56	C₅×D₃×C₈	C₂³	1	2	yes	—
57	C₃×C₅×C₁₆	C₂³	1	1	yes	characteristic
58	C₃×C₅×C₁₆	C₂³	1	1	yes	characteristic
59	C₅×C₃⋊C₁₆	C₂³	1	2	yes	—
60	#5766	C₂×C₅	1	2	yes	complemented
61	C₂×C₃×Q₃₂	C₂×C₅	1	1	yes	characteristic, complemented
62	#5179	C₂×C₅	1	2	yes	complemented
63	#4919	C₂×C₅	1	2	yes	complemented
64	#4918	C₂×C₅	1	4	yes	complemented
65	D₃×Q₃₂	C₂×C₅	1	2	yes	complemented
66	#4904	C₂×C₅	1	1	yes	characteristic, complemented
67	#4901	C₂×C₅	1	1	yes	characteristic, complemented
68	C₅×2₋¹⁺⁴	—	6	2	no	—
69	C₅×#137	—	6	4	no	—
70	C₅×#137	—	6	2	no	—
71	C₅×#135	—	3	4	no	—
72	C₅×#135	—	3	2	no	—
73	C₂×C₅×Q₁₆	—	3	4	no	—
74	C₂×C₅×Q₁₆	C₂×D₃	1	2	yes	—
75	C₅×#131	—	3	1	no	—
76	C₅×Q₃₂	—	3	4	no	—
77	C₅×Q₃₂	—	3	1	no	—
78	C₅×Q₃₂	C₂×D₃	1	2	yes	—
79	C₅×Q₃₂	C₂×D₃	1	2	yes	—
80	C₅×SD₃₂	—	3	4	no	—
81	C₅×SD₃₂	—	3	2	no	—
82	C₅×D₁₆	—	3	1	no	—
83	C₅×M₃₂	—	3	1	no	—
84	C₅×M₃₂	—	3	2	no	—
85	C₂×C₅×C₁₆	—	3	2	no	—
86	C₂×C₅×C₁₆	C₂×D₃	1	1	yes	characteristic
87	#3417	—	3	1	no	Sylow
88	C₃×C₅×Q₈	—	2	4	no	—
89	C₃×C₅×Q₈	—	4	2	no	—
90	C₂×C₃×C₄×C₅	—	4	2	no	—
91	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
92	C₅×#67	—	4	2	no	—
93	C₅×#67	D₈	1	2	yes	—
94	C₅×D₁₂	—	2	2	no	—
95	C₅×D₁₂	—	4	4	no	—
96	C₅×D₁₂	D₈	1	1	yes	characteristic
97	C₄×C₅×D₃	—	4	4	no	—
98	C₄×C₅×D₃	—	4	2	no	—
99	C₄×C₅×D₃	—	4	4	no	—
100	C₄×C₅×D₃	C₂×D₄	1	2	yes	—
101	C₅×Dic₆	—	2	2	no	—
102	C₅×Dic₆	—	4	4	no	—
103	C₅×Dic₆	D₈	1	1	yes	characteristic
104	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
105	C₃×C₅×C₈	C₂⁴	1	1	yes	characteristic
106	C₅×C₃⋊C₈	C₂×D₄	1	2	yes	—
107	#1007	—	2	2	no	—
108	C₂×C₃×Q₁₆	C₂²×C₅	1	2	yes	—
109	#942	—	2	2	no	—
110	#919	C₂²×C₅	1	4	yes	—
111	#918	—	2	4	no	—
112	D₃×Q₁₆	C₂²×C₅	1	4	yes	—
113	#904	C₂²×C₅	1	2	yes	—
114	#901	C₂²×C₅	1	1	yes	characteristic
115	C₃×Q₃₂	C₂²×C₅	1	2	yes	—
116	C₃×Q₃₂	C₂²×C₅	1	2	yes	—
117	C₂×C₃×C₁₆	C₂²×C₅	1	1	yes	characteristic
118	#829	C₂²×C₅	1	4	yes	—
119	#828	C₂²×C₅	1	4	yes	—
120	#812	C₂²×C₅	1	1	yes	characteristic
121	Dic₂₄	C₂²×C₅	1	1	yes	characteristic
122	#800	C₂²×C₅	1	2	yes	—
123	D₄₈	C₂²×C₅	1	1	yes	characteristic
124	#798	C₂²×C₅	1	2	yes	—
125	D₃×C₁₆	C₂²×C₅	1	2	yes	—
126	C₅×#41	—	3	1	no	—
127	C₅×#41	—	6	4	no	—
128	C₅×#41	—	6	2	no	—
129	C₅×#41	—	12	4	no	—
130	C₅×#41	—	12	2	no	—
131	C₂×C₅×Q₈	—	2	2	no	—
132	C₂×C₅×Q₈	—	6	4	no	—
133	C₂×C₅×Q₈	—	12	2	no	—
134	C₅×Q₁₆	—	2	2	no	—
135	C₅×Q₁₆	—	3	2	no	—
136	C₅×Q₁₆	—	6	4	no	—
137	C₅×Q₁₆	C₂²×D₃	1	4	yes	—
138	C₅×SD₁₆	—	6	4	no	—
139	C₅×SD₁₆	—	6	2	no	—
140	C₅×D₈	—	3	2	no	—
141	C₅×M₁₆	—	3	1	no	—
142	C₅×M₁₆	—	3	2	no	—
143	C₂×C₅×C₈	—	3	2	no	—
144	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
145	C₅×C₁₆	—	3	2	no	—
146	C₅×C₁₆	C₂²×D₃	1	1	yes	characteristic
147	C₅×C₁₆	C₂²×D₃	1	1	yes	characteristic
148	#578	—	3	2	no	—
149	#510	—	3	2	no	—
150	#510	—	3	4	no	—
151	#508	—	3	2	no	—
152	#508	—	3	1	no	—
153	C₂×Q₃₂	—	3	2	no	—
154	C₂×Q₃₂	C₅×D₃	1	1	yes	characteristic, complemented
155	#504	—	3	1	no	—
156	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
157	C₂×C₅×D₃	—	4	2	no	—
158	C₂×C₅×D₃	C₂×D₈	1	2	yes	—
159	C₃×C₄×C₅	—	4	4	no	—
160	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
161	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
162	C₅×Dic₃	—	4	2	no	—
163	C₅×Dic₃	C₂×D₈	1	2	yes	—
164	C₂×C₃×Q₈	—	2	2	no	—
165	#239	—	2	4	no	—
166	#239	—	4	2	no	—
167	D₃×Q₈	—	2	4	no	—
168	D₃×Q₈	—	4	2	no	—
169	#235	—	2	2	no	—
170	#235	—	4	4	no	—
171	#235	C₅×D₄	1	1	yes	characteristic
172	C₃×Q₁₆	—	2	2	no	—
173	C₃×Q₁₆	C₂³×C₅	1	4	yes	—
174	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
175	#216	—	2	4	no	—
176	#215	—	2	4	no	—
177	#208	C₅×D₄	1	1	yes	characteristic
178	Dic₁₂	C₂³×C₅	1	2	yes	—
179	D₂₄	C₂³×C₅	1	2	yes	—
180	#204	—	2	2	no	—
181	ΓL(1,𝔽₂₅)	C₅×D₄	1	2	yes	—
182	D₃×C₈	C₂³×C₅	1	2	yes	—
183	C₃×C₁₆	C₂³×C₅	1	1	yes	characteristic
184	C₃×C₁₆	C₂³×C₅	1	1	yes	characteristic
185	C₃⋊C₁₆	C₂³×C₅	1	2	yes	—
186	C₅×Q₈	—	2	4	no	—
187	C₅×Q₈	—	3	1	no	—
188	C₅×Q₈	—	4	2	no	—
189	C₅×Q₈	—	6	2	no	—
190	C₅×Q₈	—	12	4	no	—
191	C₅×D₄	—	3	2	no	—
192	C₅×D₄	—	3	1	no	—
193	C₅×D₄	—	6	2	no	—
194	C₅×D₄	—	12	4	no	—
195	C₅×D₄	—	12	2	no	—
196	C₂×C₄×C₅	—	3	2	no	—
197	C₂×C₄×C₅	—	4	2	no	—
198	C₂×C₄×C₅	—	12	4	no	—
199	C₂×C₄×C₅	—	12	4	no	—
200	C₂×C₄×C₅	—	12	2	no	—
201	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
202	C₅×C₈	—	3	2	no	—
203	C₅×C₈	D₃×D₄	1	1	yes	characteristic
204	C₅×C₈	C₂³×D₃	1	1	yes	characteristic
205	2₋¹⁺⁴	—	6	2	no	—
206	#137	—	6	4	no	—
207	#137	—	6	2	no	—
208	#135	—	3	4	no	—
209	#135	—	3	2	no	—
210	C₂×Q₁₆	—	3	4	no	—
211	C₂×Q₁₆	C₂×C₅×D₃	1	2	yes	—
212	#131	—	3	1	no	—
213	Q₃₂	—	3	4	no	—
214	Q₃₂	—	3	1	no	—
215	Q₃₂	C₂×C₅×D₃	1	2	yes	—
216	Q₃₂	C₂×C₅×D₃	1	2	yes	—
217	SD₃₂	—	3	4	no	—
218	SD₃₂	—	3	2	no	—
219	D₁₆	—	3	1	no	—
220	M₃₂	—	3	1	no	—
221	M₃₂	—	3	2	no	—
222	C₂×C₁₆	—	3	2	no	—
223	C₂×C₁₆	C₂×C₅×D₃	1	1	yes	characteristic
224	C₂×C₃×C₅	—	2	1	no	—
225	C₂×C₃×C₅	C₂²×D₈	1	1	yes	characteristic
226	C₅×D₃	—	2	2	no	—
227	C₅×D₃	—	8	2	no	—
228	C₃×Q₈	—	2	4	no	—
229	C₃×Q₈	—	4	2	no	—
230	C₂×C₃×C₄	—	4	2	no	—
231	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
232	#67	—	4	2	no	—
233	#67	C₅×D₈	1	2	yes	—
234	D₁₂	—	2	2	no	—
235	D₁₂	—	4	4	no	—
236	D₁₂	C₅×D₈	1	1	yes	characteristic
237	C₄×D₃	—	4	4	no	—
238	C₄×D₃	—	4	2	no	—
239	C₄×D₃	—	4	4	no	—
240	C₄×D₃	C₂×C₅×D₄	1	2	yes	—
241	Dic₆	—	2	2	no	—
242	Dic₆	—	4	4	no	—
243	Dic₆	C₅×D₈	1	1	yes	characteristic
244	C₃×C₈	C₂×C₅×D₄	1	1	yes	characteristic
245	C₃×C₈	C₂⁴×C₅	1	1	yes	characteristic
246	C₃⋊C₈	C₂×C₅×D₄	1	2	yes	—
247	C₂²×C₅	—	3	2	no	—
248	C₂²×C₅	—	12	2	no	—
249	C₂²×C₅	D₃×D₈	1	1	yes	characteristic
250	C₄×C₅	—	3	2	no	—
251	C₄×C₅	—	4	4	no	—
252	C₄×C₅	—	12	2	no	—
253	C₄×C₅	D₃×D₈	1	1	yes	characteristic
254	C₄×C₅	C₂×D₃×D₄	1	1	yes	characteristic
255	#41	—	3	1	no	—
256	#41	—	6	4	no	—
257	#41	—	6	2	no	—
258	#41	—	12	4	no	—
259	#41	—	12	2	no	—
260	C₂×Q₈	—	2	2	no	—
261	C₂×Q₈	—	6	4	no	—
262	C₂×Q₈	—	12	2	no	—
263	Q₁₆	—	2	2	no	—
264	Q₁₆	—	3	2	no	—
265	Q₁₆	—	6	4	no	—
266	Q₁₆	C₂²×C₅×D₃	1	4	yes	—
267	SD₁₆	—	6	4	no	—
268	SD₁₆	—	6	2	no	—
269	D₈	—	3	2	no	—
270	M₁₆	—	3	1	no	—
271	M₁₆	—	3	2	no	—
272	C₂×C₈	—	3	2	no	—
273	C₂×C₈	C₂²×C₅×D₃	1	1	yes	characteristic
274	C₁₆	—	3	2	no	—
275	C₁₆	C₂²×C₅×D₃	1	1	yes	characteristic
276	C₁₆	C₂²×C₅×D₃	1	1	yes	characteristic
277	C₃×C₅	#3417	1	1	yes	characteristic, complemented, Hall
278	C₂²×C₃	C₂×C₅×D₈	1	1	yes	characteristic
279	C₂×D₃	—	4	2	no	—
280	C₂×D₃	C₂×C₅×D₈	1	2	yes	—
281	C₃×C₄	—	4	4	no	—
282	C₃×C₄	C₂×C₅×D₈	1	1	yes	characteristic
283	C₃×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
284	Dic₃	—	4	2	no	—
285	Dic₃	C₂×C₅×D₈	1	2	yes	—
286	C₂×C₅	—	2	1	no	—
287	C₂×C₅	—	6	2	no	—
288	C₂×C₅	—	24	2	no	—
289	C₂×C₅	C₂×D₃×D₈	1	1	yes	characteristic
290	Q₈	—	2	4	no	—
291	Q₈	—	3	1	no	—
292	Q₈	—	4	2	no	—
293	Q₈	—	6	2	no	—
294	Q₈	—	12	4	no	—
295	D₄	—	3	2	no	—
296	D₄	—	3	1	no	—
297	D₄	—	6	2	no	—
298	D₄	—	12	4	no	—
299	D₄	—	12	2	no	—
300	C₂×C₄	—	3	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	12	4	no	—
303	C₂×C₄	—	12	4	no	—
304	C₂×C₄	—	12	2	no	—
305	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
306	C₈	—	3	2	no	—
307	C₈	C₅×D₃×D₄	1	1	yes	characteristic
308	C₈	C₂³×C₅×D₃	1	1	yes	characteristic
309	C₂×C₃	—	2	1	no	—
310	C₂×C₃	C₂²×C₅×D₈	1	1	yes	characteristic
311	D₃	—	2	2	no	—
312	D₃	—	8	2	no	—
313	C₅	#82028	1	1	yes	characteristic, direct product factor, Sylow
314	C₂²	—	3	2	no	—
315	C₂²	—	12	2	no	—
316	C₂²	C₅×D₃×D₈	1	1	yes	characteristic
317	C₄	—	3	2	no	—
318	C₄	—	4	4	no	—
319	C₄	—	12	2	no	—
320	C₄	C₅×D₃×D₈	1	1	yes	characteristic
321	C₄	C₂×C₅×D₃×D₄	1	1	yes	characteristic
322	C₃	C₅×#3417	1	1	yes	characteristic, complemented, Sylow
323	C₂	—	2	1	no	—
324	C₂	—	6	2	no	—
325	C₂	—	24	2	no	—
326	C₂	C₂×C₅×D₃×D₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₅	C₂×D₃×D₈
commutator, cosocle	C₃×C₈	C₂⁴×C₅
Frattini	C₈	C₂³×C₅×D₃
Fitting	C₂×C₃×C₅×Q₃₂	C₂
socle	C₂×C₃×C₅	C₂²×D₈
hypercenter	C₂×C₅×Q₃₂	D₃
hypocenter	C₃	C₅×#3417
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×C₈ ◁ C₂×C₅×Q₃₂

Complements

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

N	H ≅ G/N	type
1. C₅×#5766	C₂	semidirect
2. C₂×C₃×C₅×Q₃₂	C₂	semidirect
3. C₅×#5179	C₂	semidirect
4. C₅×#4919	C₂	semidirect
5. C₅×#4918	C₂	semidirect
6. C₅×D₃×Q₃₂	C₂	semidirect
7. C₅×#4904	C₂	semidirect
8. C₅×#4901	C₂	semidirect
29. #82028	C₅	direct (N × H)
36. C₂×C₅×Q₃₂	D₃	semidirect
60. #5766	C₂×C₅	semidirect
61. C₂×C₃×Q₃₂	C₂×C₅	semidirect
62. #5179	C₂×C₅	semidirect
63. #4919	C₂×C₅	semidirect
64. #4918	C₂×C₅	semidirect
65. D₃×Q₃₂	C₂×C₅	semidirect
66. #4904	C₂×C₅	semidirect
67. #4901	C₂×C₅	semidirect
154. C₂×Q₃₂	C₅×D₃	semidirect
277. C₃×C₅	#3417	semidirect
313. C₅	#82028	direct (N × H)
322. C₃	C₅×#3417	semidirect

Maximal subgroups

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

subgroup	index
C₅×#5766	2
C₂×C₃×C₅×Q₃₂	2
C₅×#5179	2
C₅×#4919	2
C₅×#4918	2
C₅×D₃×Q₃₂	2
C₅×#4904	2
C₅×#4901	2
C₅×#3417	3
#82028	5

Subgroups — C5×#82028 #14478613

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#82028 #14478613