Subgroups — C₅×#68497 #14291769

Order 1920.

320 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#5084	C₂	1	1	yes	maximal, characteristic
2	C₅×#5078	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#4842	C₂	1	1	yes	maximal, characteristic
4	C₅×#4834	C₂	1	1	yes	maximal, characteristic
5	C₃×C₅×#333	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4481	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#4479	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1675	—	3	1	no	maximal, Hall
9	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₅×D₃×C₈	—	2	1	no	—
11	C₄×C₅×D₁₂	C₂²	1	1	yes	characteristic
12	C₅×#869	C₂²	1	1	yes	characteristic
13	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₅×#821	—	2	1	no	—
16	C₅×(C₂×D₃)⋊C₈	—	2	1	no	—
17	C₅×#817	—	2	1	no	—
18	C₅×#809	—	2	1	no	—
19	C₅×#808	—	2	1	no	—
20	C₅×#807	—	2	1	no	—
21	C₅×#807	—	2	1	no	—
22	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
23	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
24	#68497	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
25	C₅×#500	—	3	1	no	—
26	C₅×#476	—	3	1	no	—
27	C₅×#476	—	3	1	no	—
28	C₅×C₈×D₄	—	3	1	no	—
29	C₅×#335	—	3	1	no	—
30	C₅×#333	—	3	1	no	—
31	C₅×#333	D₃	1	1	yes	characteristic, complemented
32	C₂×C₃×C₅×Q₈	—	2	1	no	—
33	C₂×C₅×D₁₂	D₄	1	1	yes	characteristic
34	C₂×C₄×C₅×D₃	—	2	1	no	—
35	C₂×C₅×Dic₆	—	2	1	no	—
36	C₂×C₃×C₅×C₈	—	2	1	no	—
37	C₂×C₃×C₅×C₈	—	2	1	no	—
38	C₃×C₅×C₄⋊C₄	—	2	1	no	—
39	C₃×C₅×C₄⋊C₄	—	2	1	no	—
40	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
41	C₅×(C₂×D₃)⋊C₄	—	2	1	no	—
42	C₅×C₄⋊Dic₃	—	2	1	no	—
43	C₅×C₄⋊Dic₃	—	2	1	no	—
44	C₅×C₄⋊Dic₃	D₄	1	1	yes	characteristic
45	C₂×C₅×C₃⋊C₈	—	2	1	no	—
46	C₂×C₅×C₃⋊C₈	D₄	1	1	yes	characteristic
47	C₂×C₅×C₃⋊C₈	D₄	1	1	yes	characteristic
48	C₅×D₃×C₈	—	4	2	no	—
49	#5084	C₂×C₅	1	1	yes	characteristic
50	#5078	C₂×C₅	1	1	yes	characteristic, complemented
51	#4842	C₂×C₅	1	1	yes	characteristic
52	#4834	C₂×C₅	1	1	yes	characteristic
53	C₃×#333	C₂×C₅	1	1	yes	characteristic, complemented
54	#4481	C₂×C₅	1	1	yes	characteristic, complemented
55	#4479	C₂×C₅	1	1	yes	characteristic, complemented
56	C₂²×C₅×C₈	—	6	1	no	—
57	C₅×C₄⋊Q₈	—	3	1	no	—
58	C₅×C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
59	C₄×C₅×D₄	—	3	1	no	—
60	C₅×#107	—	6	1	no	—
61	C₅×#107	—	6	1	no	—
62	C₅×#106	—	6	1	no	—
63	C₅×#106	—	6	1	no	—
64	C₅×C₄⋊C₈	—	3	1	no	—
65	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
66	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
67	C₅×#102	—	6	1	no	—
68	C₅×#102	—	6	1	no	—
69	C₅×C₂²⋊C₈	—	6	1	no	—
70	C₄×C₅×C₈	—	3	1	no	—
71	#1675	—	3	1	no	Sylow
72	C₂²×C₅×D₃	—	2	1	no	—
73	C₃×C₅×Q₈	—	8	1	no	—
74	C₂×C₃×C₄×C₅	—	4	1	no	—
75	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
76	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
77	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
78	C₂×C₅×Dic₃	—	2	1	no	—
79	C₂×C₅×Dic₃	—	4	1	no	—
80	C₅×D₁₂	—	2	1	no	—
81	C₅×D₁₂	Q₁₆	1	2	yes	—
82	C₄×C₅×D₃	—	4	1	no	—
83	C₄×C₅×D₃	—	4	1	no	—
84	C₅×Dic₆	—	8	1	no	—
85	C₃×C₅×C₈	—	4	1	no	—
86	C₃×C₅×C₈	—	4	1	no	—
87	C₅×C₃⋊C₈	—	2	1	no	—
88	C₅×C₃⋊C₈	—	4	1	no	—
89	C₅×C₃⋊C₈	SD₁₆	1	2	yes	—
90	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
91	C₂×D₃×C₈	—	2	1	no	—
92	C₄×D₁₂	C₂²×C₅	1	1	yes	characteristic
93	#869	C₂²×C₅	1	1	yes	characteristic
94	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
95	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
96	#821	—	2	1	no	—
97	(C₂×D₃)⋊C₈	—	2	1	no	—
98	#817	—	2	1	no	—
99	#809	—	2	1	no	—
100	#808	—	2	1	no	—
101	#807	—	2	1	no	—
102	#807	—	2	1	no	—
103	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	1	yes	characteristic
104	C₄×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
105	C₂×C₅×Q₈	—	2	1	no	—
106	C₂×C₅×Q₈	—	6	1	no	—
107	C₂×C₅×D₄	—	3	1	no	—
108	C₂²×C₄×C₅	—	6	1	no	—
109	C₂×C₅×C₈	—	2	1	no	—
110	C₂×C₅×C₈	—	2	1	no	—
111	C₂×C₅×C₈	—	3	1	no	—
112	C₂×C₅×C₈	—	3	1	no	—
113	C₂×C₅×C₈	—	6	1	no	—
114	C₂×C₅×C₈	—	12	2	no	—
115	C₅×C₄⋊C₄	—	2	1	no	—
116	C₅×C₄⋊C₄	—	2	1	no	—
117	C₅×C₄⋊C₄	—	3	1	no	—
118	C₅×C₄⋊C₄	—	6	1	no	—
119	C₅×C₄⋊C₄	—	6	1	no	—
120	C₅×C₂²⋊C₄	—	6	1	no	—
121	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
122	#500	—	3	1	no	—
123	#476	—	3	1	no	—
124	#476	—	3	1	no	—
125	C₈×D₄	—	3	1	no	—
126	#335	—	3	1	no	—
127	#333	—	3	1	no	—
128	#333	C₅×D₃	1	1	yes	characteristic, complemented
129	C₂²×C₃×C₅	#121	1	1	yes	characteristic
130	C₂×C₅×D₃	—	2	2	no	—
131	C₂×C₅×D₃	—	4	1	no	—
132	C₂×C₅×D₃	—	4	1	no	—
133	C₃×C₄×C₅	—	2	1	no	—
134	C₃×C₄×C₅	—	8	1	no	—
135	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
136	C₃×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
137	C₃×C₄×C₅	#135	1	1	yes	characteristic
138	C₃×C₄×C₅	#137	1	1	yes	characteristic
139	C₅×Dic₃	—	4	1	no	—
140	C₅×Dic₃	—	8	1	no	—
141	C₂×C₃×Q₈	—	2	1	no	—
142	C₂×D₁₂	C₅×D₄	1	1	yes	characteristic
143	C₂×C₄×D₃	—	2	1	no	—
144	C₂×Dic₆	—	2	1	no	—
145	C₂×C₃×C₈	—	2	1	no	—
146	C₂×C₃×C₈	—	2	1	no	—
147	C₃×C₄⋊C₄	—	2	1	no	—
148	C₃×C₄⋊C₄	—	2	1	no	—
149	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
150	(C₂×D₃)⋊C₄	—	2	1	no	—
151	C₄⋊Dic₃	—	2	1	no	—
152	C₄⋊Dic₃	—	2	1	no	—
153	C₄⋊Dic₃	C₅×D₄	1	1	yes	characteristic
154	C₂×C₃⋊C₈	—	2	1	no	—
155	C₂×C₃⋊C₈	C₅×D₄	1	1	yes	characteristic
156	C₂×C₃⋊C₈	C₅×D₄	1	1	yes	characteristic
157	D₃×C₈	—	4	2	no	—
158	C₂³×C₅	—	6	1	no	—
159	C₅×Q₈	—	8	1	no	—
160	C₅×Q₈	—	24	1	no	—
161	C₅×D₄	—	3	2	no	—
162	C₅×D₄	—	6	1	no	—
163	C₂×C₄×C₅	—	4	1	no	—
164	C₂×C₄×C₅	—	6	1	no	—
165	C₂×C₄×C₅	—	12	1	no	—
166	C₂×C₄×C₅	—	12	1	no	—
167	C₂×C₄×C₅	—	12	1	no	—
168	C₂×C₄×C₅	#235	1	1	yes	characteristic
169	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
170	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
171	C₅×C₈	—	3	2	no	—
172	C₅×C₈	—	4	1	no	—
173	C₅×C₈	—	4	1	no	—
174	C₅×C₈	—	6	1	no	—
175	C₅×C₈	—	12	1	no	—
176	C₂²×C₈	—	6	1	no	—
177	C₄⋊Q₈	—	3	1	no	—
178	C₄⋊Q₈	C₂×C₅×D₃	1	1	yes	characteristic
179	C₄×D₄	—	3	1	no	—
180	#107	—	6	1	no	—
181	#107	—	6	1	no	—
182	#106	—	6	1	no	—
183	#106	—	6	1	no	—
184	C₄⋊C₈	—	3	1	no	—
185	C₄⋊C₈	C₂×C₅×D₃	1	1	yes	characteristic
186	C₄⋊C₈	C₂×C₅×D₃	1	1	yes	characteristic
187	#102	—	6	1	no	—
188	#102	—	6	1	no	—
189	C₂²⋊C₈	—	6	1	no	—
190	C₄×C₈	—	3	1	no	—
191	C₂×C₃×C₅	#461	1	1	yes	characteristic
192	C₂×C₃×C₅	#467	1	1	yes	characteristic
193	C₂×C₃×C₅	#472	1	1	yes	characteristic
194	C₅×D₃	—	4	2	no	—
195	C₂²×D₃	—	2	1	no	—
196	C₃×Q₈	—	8	1	no	—
197	C₂×C₃×C₄	—	4	1	no	—
198	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
199	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
200	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
201	C₂×Dic₃	—	2	1	no	—
202	C₂×Dic₃	—	4	1	no	—
203	D₁₂	—	2	1	no	—
204	D₁₂	C₅×Q₁₆	1	2	yes	—
205	C₄×D₃	—	4	1	no	—
206	C₄×D₃	—	4	1	no	—
207	Dic₆	—	8	1	no	—
208	C₃×C₈	—	4	1	no	—
209	C₃×C₈	—	4	1	no	—
210	C₃⋊C₈	—	2	1	no	—
211	C₃⋊C₈	—	4	1	no	—
212	C₃⋊C₈	C₅×SD₁₆	1	2	yes	—
213	C₂²×C₅	—	6	2	no	—
214	C₂²×C₅	—	12	1	no	—
215	C₂²×C₅	—	12	1	no	—
216	C₂²×C₅	#884	1	1	yes	characteristic
217	C₄×C₅	—	2	1	no	—
218	C₄×C₅	—	8	1	no	—
219	C₄×C₅	—	12	1	no	—
220	C₄×C₅	—	24	1	no	—
221	C₄×C₅	D₃×SD₁₆	1	1	yes	characteristic
222	C₄×C₅	#916	1	1	yes	characteristic
223	C₄×C₅	D₃×Q₁₆	1	1	yes	characteristic
224	C₄×C₅	#918	1	1	yes	characteristic
225	C₂×Q₈	—	2	1	no	—
226	C₂×Q₈	—	6	1	no	—
227	C₂×D₄	—	3	1	no	—
228	C₂²×C₄	—	6	1	no	—
229	C₂×C₈	—	2	1	no	—
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	3	1	no	—
232	C₂×C₈	—	3	1	no	—
233	C₂×C₈	—	6	1	no	—
234	C₂×C₈	—	12	2	no	—
235	C₄⋊C₄	—	2	1	no	—
236	C₄⋊C₄	—	2	1	no	—
237	C₄⋊C₄	—	3	1	no	—
238	C₄⋊C₄	—	6	1	no	—
239	C₄⋊C₄	—	6	1	no	—
240	C₂²⋊C₄	—	6	1	no	—
241	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
242	C₃×C₅	#1675	1	1	yes	characteristic, complemented, Hall
243	C₂²×C₃	C₅×#121	1	1	yes	characteristic
244	C₂×D₃	—	2	2	no	—
245	C₂×D₃	—	4	1	no	—
246	C₂×D₃	—	4	1	no	—
247	C₃×C₄	—	2	1	no	—
248	C₃×C₄	—	8	1	no	—
249	C₃×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
250	C₃×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
251	C₃×C₄	C₅×#135	1	1	yes	characteristic
252	C₃×C₄	C₅×#137	1	1	yes	characteristic
253	Dic₃	—	4	1	no	—
254	Dic₃	—	8	1	no	—
255	C₂×C₅	—	12	2	no	—
256	C₂×C₅	#4749	1	1	yes	characteristic
257	C₂×C₅	#4813	1	1	yes	characteristic
258	C₂×C₅	#4818	1	1	yes	characteristic
259	C₂³	—	6	1	no	—
260	Q₈	—	8	1	no	—
261	Q₈	—	24	1	no	—
262	D₄	—	3	2	no	—
263	D₄	—	6	1	no	—
264	C₂×C₄	—	4	1	no	—
265	C₂×C₄	—	6	1	no	—
266	C₂×C₄	—	12	1	no	—
267	C₂×C₄	—	12	1	no	—
268	C₂×C₄	—	12	1	no	—
269	C₂×C₄	C₅×#235	1	1	yes	characteristic
270	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
271	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
272	C₈	—	3	2	no	—
273	C₈	—	4	1	no	—
274	C₈	—	4	1	no	—
275	C₈	—	6	1	no	—
276	C₈	—	12	1	no	—
277	C₂×C₃	C₅×#461	1	1	yes	characteristic
278	C₂×C₃	C₅×#467	1	1	yes	characteristic
279	C₂×C₃	C₅×#472	1	1	yes	characteristic
280	D₃	—	4	2	no	—
281	C₅	#68497	1	1	yes	characteristic, direct product factor, Sylow
282	C₂²	—	6	2	no	—
283	C₂²	—	12	1	no	—
284	C₂²	—	12	1	no	—
285	C₂²	C₅×#884	1	1	yes	characteristic
286	C₄	—	2	1	no	—
287	C₄	—	8	1	no	—
288	C₄	—	12	1	no	—
289	C₄	—	24	1	no	—
290	C₄	C₅×D₃×SD₁₆	1	1	yes	characteristic
291	C₄	C₅×#916	1	1	yes	characteristic
292	C₄	C₅×D₃×Q₁₆	1	1	yes	characteristic
293	C₄	C₅×#918	1	1	yes	characteristic
294	C₃	C₅×#1675	1	1	yes	characteristic, complemented, Sylow
295	C₂	—	12	2	no	—
296	C₂	C₅×#4749	1	1	yes	characteristic
297	C₂	C₅×#4813	1	1	yes	characteristic
298	C₂	C₅×#4818	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×#333	C₂
socle	C₂²×C₃×C₅	#121
prime radical	C₂×C₅×D₁₂	D₄
hypercenter	C₅×#333	D₃
hypocenter	C₃	C₅×#1675
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×#333

Complements

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

N	H ≅ G/N	type
2. C₅×#5078	C₂	semidirect
5. C₃×C₅×#333	C₂	semidirect
6. C₅×#4481	C₂	semidirect
7. C₅×#4479	C₂	semidirect
24. #68497	C₅	direct (N × H)
31. C₅×#333	D₃	semidirect
50. #5078	C₂×C₅	semidirect
53. C₃×#333	C₂×C₅	semidirect
54. #4481	C₂×C₅	semidirect
55. #4479	C₂×C₅	semidirect
128. #333	C₅×D₃	semidirect
242. C₃×C₅	#1675	semidirect
281. C₅	#68497	direct (N × H)
294. C₃	C₅×#1675	semidirect

Maximal subgroups

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

subgroup	index
C₅×#5084	2
C₅×#5078	2
C₅×#4842	2
C₅×#4834	2
C₃×C₅×#333	2
C₅×#4481	2
C₅×#4479	2
C₅×#1675	3
#68497	5

Subgroups — C5×#68497 #14291769

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#68497 #14291769