Subgroups — C₅×#68090 #14293657

Order 1920.

396 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#4694	C₂	1	1	yes	maximal, characteristic
2	C₅×#4693	C₂	1	1	yes	maximal, characteristic
3	C₅×#4693	C₂	1	1	yes	maximal, characteristic
4	C₅×#4682	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#4681	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4680	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×C₅×#322	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1714	—	3	1	no	maximal, Hall
9	C₂×C₅×#204	—	2	2	no	—
10	C₂×C₅×#204	—	2	2	no	—
11	C₅×#874	C₂²	1	1	yes	characteristic
12	C₅×#869	C₂²	1	1	yes	characteristic
13	C₅×#869	C₂²	1	1	yes	characteristic
14	C₅×#869	C₂²	1	1	yes	characteristic
15	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
16	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
17	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
18	C₅×#821	—	2	2	no	—
19	C₅×#818	—	2	2	no	—
20	C₅×#817	—	2	2	no	—
21	C₅×#816	—	2	2	no	—
22	#68090	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
23	C₅×#500	—	3	1	no	—
24	C₅×#498	—	3	1	no	—
25	C₅×#492	—	3	1	no	—
26	C₅×#492	—	3	1	no	—
27	C₅×#487	—	3	1	no	—
28	C₅×#486	—	3	1	no	—
29	C₅×#322	D₃	1	1	yes	characteristic, complemented
30	C₂×C₅×D₁₂	—	2	1	no	—
31	C₂×C₅×D₁₂	—	2	1	no	—
32	C₂×C₅×D₁₂	—	2	1	no	—
33	C₂×C₅×Dic₆	—	2	1	no	—
34	C₂×C₅×Dic₆	—	2	1	no	—
35	C₂×C₅×Dic₆	—	2	1	no	—
36	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
37	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
38	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
39	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
40	C₅×C₄⋊Dic₃	—	2	1	no	—
41	C₅×C₄⋊Dic₃	—	2	1	no	—
42	C₅×C₄⋊Dic₃	—	2	1	no	—
43	C₅×C₄⋊Dic₃	—	2	1	no	—
44	C₅×C₄⋊Dic₃	—	2	1	no	—
45	C₅×C₄⋊Dic₃	—	2	1	no	—
46	C₅×#204	—	4	4	no	—
47	C₅×#204	—	4	4	no	—
48	#4694	C₂×C₅	1	1	yes	characteristic
49	#4693	C₂×C₅	1	1	yes	characteristic
50	#4693	C₂×C₅	1	1	yes	characteristic
51	#4682	C₂×C₅	1	1	yes	characteristic, complemented
52	#4681	C₂×C₅	1	1	yes	characteristic, complemented
53	#4680	C₂×C₅	1	1	yes	characteristic, complemented
54	C₃×#322	C₂×C₅	1	1	yes	characteristic, complemented
55	C₂×C₅×SD₁₆	—	6	2	no	—
56	C₂×C₅×SD₁₆	—	6	2	no	—
57	C₅×C₄⋊Q₈	—	3	1	no	—
58	C₅×C₄⋊Q₈	—	3	1	no	—
59	C₅×C₄⋊Q₈	—	3	1	no	—
60	C₅×#127	—	3	1	no	—
61	C₅×#107	—	6	2	no	—
62	C₅×#106	—	6	2	no	—
63	C₅×#103	—	6	2	no	—
64	C₅×#102	—	6	2	no	—
65	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
66	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
67	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
68	#1714	—	3	1	no	Sylow
69	C₂²×C₅×D₃	—	4	1	no	—
70	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
71	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
72	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
73	C₂×C₅×Dic₃	—	4	1	no	—
74	C₂×C₅×Dic₃	—	4	1	no	—
75	C₂×C₅×Dic₃	—	4	1	no	—
76	C₅×D₁₂	—	4	2	no	—
77	C₅×D₁₂	—	4	2	no	—
78	C₅×D₁₂	—	4	2	no	—
79	C₅×Dic₆	—	4	2	no	—
80	C₅×Dic₆	—	4	2	no	—
81	C₅×Dic₆	—	4	2	no	—
82	C₃×C₅×C₈	—	2	2	no	—
83	C₃×C₅×C₈	—	2	2	no	—
84	C₃×C₅×C₈	SD₁₆	1	4	yes	complemented
85	C₂×#204	—	2	2	no	—
86	C₂×#204	—	2	2	no	—
87	#874	C₂²×C₅	1	1	yes	characteristic
88	#869	C₂²×C₅	1	1	yes	characteristic
89	#869	C₂²×C₅	1	1	yes	characteristic
90	#869	C₂²×C₅	1	1	yes	characteristic
91	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
92	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
93	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
94	#821	—	2	2	no	—
95	#818	—	2	2	no	—
96	#817	—	2	2	no	—
97	#816	—	2	2	no	—
98	C₂×C₅×Q₈	—	6	1	no	—
99	C₂×C₅×Q₈	—	6	1	no	—
100	C₂×C₅×Q₈	—	6	1	no	—
101	C₂×C₅×D₄	—	6	1	no	—
102	C₂×C₅×D₄	—	6	1	no	—
103	C₂×C₅×D₄	—	6	1	no	—
104	C₅×SD₁₆	—	12	4	no	—
105	C₅×SD₁₆	—	12	4	no	—
106	C₂×C₅×C₈	D₁₂	1	2	yes	—
107	C₂×C₅×C₈	D₁₂	1	2	yes	—
108	C₂×C₅×C₈	D₁₂	1	2	yes	—
109	C₅×C₄⋊C₄	—	6	1	no	—
110	C₅×C₄⋊C₄	—	6	1	no	—
111	C₅×C₄⋊C₄	—	6	1	no	—
112	C₅×C₄⋊C₄	—	6	1	no	—
113	C₅×C₄⋊C₄	—	6	1	no	—
114	C₅×C₄⋊C₄	—	6	1	no	—
115	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
116	#500	—	3	1	no	—
117	#498	—	3	1	no	—
118	#492	—	3	1	no	—
119	#492	—	3	1	no	—
120	#487	—	3	1	no	—
121	#486	—	3	1	no	—
122	#322	C₅×D₃	1	1	yes	characteristic, complemented
123	C₂²×C₃×C₅	#127	1	1	yes	characteristic
124	C₂×C₅×D₃	—	8	1	no	—
125	C₂×C₅×D₃	—	8	1	no	—
126	C₂×C₅×D₃	—	8	1	no	—
127	C₃×C₄×C₅	C₂×SD₁₆	1	2	yes	—
128	C₃×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
129	C₃×C₄×C₅	#137	1	2	yes	—
130	C₅×Dic₃	—	8	1	no	—
131	C₅×Dic₃	—	8	1	no	—
132	C₅×Dic₃	—	8	1	no	—
133	C₂×D₁₂	—	2	1	no	—
134	C₂×D₁₂	—	2	1	no	—
135	C₂×D₁₂	—	2	1	no	—
136	C₂×Dic₆	—	2	1	no	—
137	C₂×Dic₆	—	2	1	no	—
138	C₂×Dic₆	—	2	1	no	—
139	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
140	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
141	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
142	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
143	C₄⋊Dic₃	—	2	1	no	—
144	C₄⋊Dic₃	—	2	1	no	—
145	C₄⋊Dic₃	—	2	1	no	—
146	C₄⋊Dic₃	—	2	1	no	—
147	C₄⋊Dic₃	—	2	1	no	—
148	C₄⋊Dic₃	—	2	1	no	—
149	#204	—	4	4	no	—
150	#204	—	4	4	no	—
151	C₂³×C₅	—	12	1	no	—
152	C₅×Q₈	—	12	2	no	—
153	C₅×Q₈	—	12	2	no	—
154	C₅×Q₈	—	12	2	no	—
155	C₅×D₄	—	12	2	no	—
156	C₅×D₄	—	12	2	no	—
157	C₅×D₄	—	12	2	no	—
158	C₂×C₄×C₅	—	12	1	no	—
159	C₂×C₄×C₅	—	12	1	no	—
160	C₂×C₄×C₅	—	12	1	no	—
161	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
162	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
163	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
164	C₅×C₈	—	2	2	no	—
165	C₅×C₈	—	2	2	no	—
166	C₅×C₈	#204	1	4	yes	complemented
167	C₂×SD₁₆	—	6	2	no	—
168	C₂×SD₁₆	—	6	2	no	—
169	C₄⋊Q₈	—	3	1	no	—
170	C₄⋊Q₈	—	3	1	no	—
171	C₄⋊Q₈	—	3	1	no	—
172	#127	—	3	1	no	—
173	#107	—	6	2	no	—
174	#106	—	6	2	no	—
175	#103	—	6	2	no	—
176	#102	—	6	2	no	—
177	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
178	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
179	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
180	C₂×C₃×C₅	#492	1	1	yes	characteristic
181	C₂×C₃×C₅	#496	1	1	yes	characteristic
182	C₂×C₃×C₅	#497	1	1	yes	characteristic
183	C₅×D₃	—	16	1	no	—
184	C₂²×D₃	—	4	1	no	—
185	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
186	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
187	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
188	C₂×Dic₃	—	4	1	no	—
189	C₂×Dic₃	—	4	1	no	—
190	C₂×Dic₃	—	4	1	no	—
191	D₁₂	—	4	2	no	—
192	D₁₂	—	4	2	no	—
193	D₁₂	—	4	2	no	—
194	Dic₆	—	4	2	no	—
195	Dic₆	—	4	2	no	—
196	Dic₆	—	4	2	no	—
197	C₃×C₈	—	2	2	no	—
198	C₃×C₈	—	2	2	no	—
199	C₃×C₈	C₅×SD₁₆	1	4	yes	complemented
200	C₂²×C₅	—	24	1	no	—
201	C₂²×C₅	—	24	1	no	—
202	C₂²×C₅	—	24	1	no	—
203	C₂²×C₅	#874	1	1	yes	characteristic
204	C₄×C₅	—	24	1	no	—
205	C₄×C₅	—	24	1	no	—
206	C₄×C₅	—	24	1	no	—
207	C₄×C₅	C₂×#204	1	2	yes	—
208	C₄×C₅	#908	1	2	yes	—
209	C₄×C₅	#909	1	2	yes	—
210	C₂×Q₈	—	6	1	no	—
211	C₂×Q₈	—	6	1	no	—
212	C₂×Q₈	—	6	1	no	—
213	C₂×D₄	—	6	1	no	—
214	C₂×D₄	—	6	1	no	—
215	C₂×D₄	—	6	1	no	—
216	SD₁₆	—	12	4	no	—
217	SD₁₆	—	12	4	no	—
218	C₂×C₈	C₅×D₁₂	1	2	yes	—
219	C₂×C₈	C₅×D₁₂	1	2	yes	—
220	C₂×C₈	C₅×D₁₂	1	2	yes	—
221	C₄⋊C₄	—	6	1	no	—
222	C₄⋊C₄	—	6	1	no	—
223	C₄⋊C₄	—	6	1	no	—
224	C₄⋊C₄	—	6	1	no	—
225	C₄⋊C₄	—	6	1	no	—
226	C₄⋊C₄	—	6	1	no	—
227	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
228	C₃×C₅	#1714	1	1	yes	characteristic, complemented, Hall
229	C₂²×C₃	C₅×#127	1	1	yes	characteristic
230	C₂×D₃	—	8	1	no	—
231	C₂×D₃	—	8	1	no	—
232	C₂×D₃	—	8	1	no	—
233	C₃×C₄	C₂×C₅×SD₁₆	1	2	yes	—
234	C₃×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
235	C₃×C₄	C₅×#137	1	2	yes	—
236	Dic₃	—	8	1	no	—
237	Dic₃	—	8	1	no	—
238	Dic₃	—	8	1	no	—
239	C₂×C₅	—	48	1	no	—
240	C₂×C₅	#4693	1	1	yes	characteristic
241	C₂×C₅	#4712	1	1	yes	characteristic
242	C₂×C₅	#4715	1	1	yes	characteristic
243	C₂³	—	12	1	no	—
244	Q₈	—	12	2	no	—
245	Q₈	—	12	2	no	—
246	Q₈	—	12	2	no	—
247	D₄	—	12	2	no	—
248	D₄	—	12	2	no	—
249	D₄	—	12	2	no	—
250	C₂×C₄	—	12	1	no	—
251	C₂×C₄	—	12	1	no	—
252	C₂×C₄	—	12	1	no	—
253	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
254	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
255	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
256	C₈	—	2	2	no	—
257	C₈	—	2	2	no	—
258	C₈	C₅×#204	1	4	yes	complemented
259	C₂×C₃	C₅×#492	1	1	yes	characteristic
260	C₂×C₃	C₅×#496	1	1	yes	characteristic
261	C₂×C₃	C₅×#497	1	1	yes	characteristic
262	D₃	—	16	1	no	—
263	C₅	#68090	1	1	yes	characteristic, direct product factor, Sylow
264	C₂²	—	24	1	no	—
265	C₂²	—	24	1	no	—
266	C₂²	—	24	1	no	—
267	C₂²	C₅×#874	1	1	yes	characteristic
268	C₄	—	24	1	no	—
269	C₄	—	24	1	no	—
270	C₄	—	24	1	no	—
271	C₄	C₂×C₅×#204	1	2	yes	—
272	C₄	C₅×#908	1	2	yes	—
273	C₄	C₅×#909	1	2	yes	—
274	C₃	C₅×#1714	1	1	yes	characteristic, complemented, Sylow
275	C₂	—	48	1	no	—
276	C₂	C₅×#4693	1	1	yes	characteristic
277	C₂	C₅×#4712	1	1	yes	characteristic
278	C₂	C₅×#4715	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	#874
commutator, cosocle	C₃×C₄²	C₂³×C₅
Frattini	C₄²	C₂²×C₅×D₃
Fitting	C₃×C₅×#322	C₂
socle	C₂²×C₃×C₅	#127
prime radical	C₅×#874	C₂²
hypercenter	C₅×#322	D₃
hypocenter	C₃	C₅×#1714
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×#322

Complements

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

N	H ≅ G/N	type
4. C₅×#4682	C₂	semidirect
5. C₅×#4681	C₂	semidirect
6. C₅×#4680	C₂	semidirect
7. C₃×C₅×#322	C₂	semidirect
22. #68090	C₅	direct (N × H)
29. C₅×#322	D₃	semidirect
51. #4682	C₂×C₅	semidirect
52. #4681	C₂×C₅	semidirect
53. #4680	C₂×C₅	semidirect
54. C₃×#322	C₂×C₅	semidirect
84. C₃×C₅×C₈	SD₁₆	semidirect
122. #322	C₅×D₃	semidirect
166. C₅×C₈	#204	semidirect
199. C₃×C₈	C₅×SD₁₆	semidirect
228. C₃×C₅	#1714	semidirect
258. C₈	C₅×#204	semidirect
263. C₅	#68090	direct (N × H)
274. C₃	C₅×#1714	semidirect

Maximal subgroups

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

subgroup	index
C₅×#4694	2
C₅×#4693	2
C₅×#4682	2
C₅×#4681	2
C₅×#4680	2
C₃×C₅×#322	2
C₅×#1714	3
#68090	5

Subgroups — C5×#68090 #14293657

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#68090 #14293657