Subgroups — C₅×#70852 #14291694

Order 1920.

344 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×C₈×D₄	C₂	1	1	yes	maximal, characteristic
2	C₅×#5019	C₂	1	2	yes	maximal
3	C₅×#4682	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#4486	C₂	1	2	yes	maximal, complemented
5	C₅×#4458	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1673	—	3	1	no	maximal, Hall
7	C₂²×C₃×C₅×C₈	—	2	1	no	—
8	C₃×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
9	C₂×C₅×#214	—	2	2	no	—
10	C₅×#869	C₂²	1	2	yes	—
11	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
12	C₃×C₅×C₂²⋊C₈	—	2	1	no	—
13	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
14	C₅×#818	—	2	1	no	—
15	C₅×#818	—	2	1	no	—
16	C₅×#810	—	2	2	no	—
17	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
18	#70852	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₅×#500	—	3	1	no	—
20	C₅×#461	—	3	2	no	—
21	C₅×C₈×D₄	D₃	1	1	yes	characteristic
22	C₅×#335	—	3	1	no	—
23	C₅×#332	—	3	2	no	—
24	C₂×C₃×C₅×D₄	D₄	1	1	yes	characteristic
25	C₂²×C₃×C₄×C₅	—	2	1	no	—
26	C₂×C₅×Dic₆	—	2	2	no	—
27	C₂×C₃×C₅×C₈	—	2	1	no	—
28	C₂×C₃×C₅×C₈	—	4	2	no	—
29	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
30	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
31	C₃×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
32	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
33	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
34	C₅×#214	—	4	4	no	—
35	C₅×C₄⋊Dic₃	—	2	2	no	—
36	C₅×C₄⋊Dic₃	—	2	2	no	—
37	C₂×C₅×C₃⋊C₈	—	2	2	no	—
38	C₃×C₈×D₄	C₂×C₅	1	1	yes	characteristic
39	#5019	C₂×C₅	1	2	yes	—
40	#4682	C₂×C₅	1	1	yes	characteristic, complemented
41	#4486	C₂×C₅	1	2	yes	complemented
42	#4458	C₂×C₅	1	1	yes	characteristic, complemented
43	C₂×C₅×SD₁₆	—	6	2	no	—
44	C₂²×C₅×C₈	—	2	1	no	—
45	C₅×C₄⋊Q₈	—	3	2	no	—
46	C₄×C₅×D₄	C₂×D₃	1	1	yes	characteristic
47	C₅×#107	—	6	1	no	—
48	C₅×#107	—	6	1	no	—
49	C₅×C₄⋊C₈	—	3	2	no	—
50	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
51	C₅×#103	—	6	2	no	—
52	C₅×C₂²⋊C₈	—	2	1	no	—
53	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
54	#1673	—	3	1	no	Sylow
55	C₂³×C₃×C₅	—	2	1	no	—
56	C₃×C₅×D₄	—	2	1	no	—
57	C₃×C₅×D₄	D₈	1	2	yes	—
58	C₂×C₃×C₄×C₅	—	2	1	no	—
59	C₂×C₃×C₄×C₅	—	4	1	no	—
60	C₂×C₃×C₄×C₅	—	4	1	no	—
61	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
62	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
63	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
64	C₂×C₅×Dic₃	—	4	2	no	—
65	C₅×Dic₆	—	4	2	no	—
66	C₅×Dic₆	—	4	2	no	—
67	C₃×C₅×C₈	—	2	1	no	—
68	C₃×C₅×C₈	—	4	1	no	—
69	C₃×C₅×C₈	SD₁₆	1	2	yes	complemented
70	C₅×C₃⋊C₈	—	4	2	no	—
71	C₂²×C₃×C₈	—	2	1	no	—
72	C₃×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
73	C₂×#214	—	2	2	no	—
74	#869	C₂²×C₅	1	2	yes	—
75	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
76	C₃×C₂²⋊C₈	—	2	1	no	—
77	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
78	#818	—	2	1	no	—
79	#818	—	2	1	no	—
80	#810	—	2	2	no	—
81	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	2	yes	—
82	C₂×C₅×Q₈	—	6	2	no	—
83	C₂×C₅×D₄	D₁₂	1	1	yes	characteristic
84	C₂²×C₄×C₅	—	2	1	no	—
85	C₅×SD₁₆	—	12	4	no	—
86	C₂×C₅×C₈	—	2	1	no	—
87	C₂×C₅×C₈	—	4	2	no	—
88	C₂×C₅×C₈	—	6	2	no	—
89	C₂×C₅×C₈	#67	1	1	yes	characteristic
90	C₂×C₅×C₈	#67	1	1	yes	characteristic
91	C₅×C₄⋊C₄	—	6	2	no	—
92	C₅×C₄⋊C₄	—	6	2	no	—
93	C₅×C₄⋊C₄	D₁₂	1	1	yes	characteristic
94	C₅×C₂²⋊C₄	—	2	1	no	—
95	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
96	#500	—	3	1	no	—
97	#461	—	3	2	no	—
98	C₈×D₄	C₅×D₃	1	1	yes	characteristic
99	#335	—	3	1	no	—
100	#332	—	3	2	no	—
101	C₂²×C₃×C₅	—	2	2	no	—
102	C₂²×C₃×C₅	—	4	1	no	—
103	C₂²×C₃×C₅	—	4	1	no	—
104	C₂²×C₃×C₅	#121	1	1	yes	characteristic
105	C₃×C₄×C₅	—	2	1	no	—
106	C₃×C₄×C₅	—	4	1	no	—
107	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
108	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
109	C₃×C₄×C₅	#135	1	1	yes	characteristic
110	C₃×C₄×C₅	#137	1	1	yes	characteristic
111	C₅×Dic₃	—	8	2	no	—
112	C₂×C₃×D₄	C₅×D₄	1	1	yes	characteristic
113	C₂²×C₃×C₄	—	2	1	no	—
114	C₂×Dic₆	—	2	2	no	—
115	C₂×C₃×C₈	—	2	1	no	—
116	C₂×C₃×C₈	—	4	2	no	—
117	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
118	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
119	C₃×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
120	C₃×C₂²⋊C₄	—	2	1	no	—
121	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
122	#214	—	4	4	no	—
123	C₄⋊Dic₃	—	2	2	no	—
124	C₄⋊Dic₃	—	2	2	no	—
125	C₂×C₃⋊C₈	—	2	2	no	—
126	C₂³×C₅	—	2	1	no	—
127	C₅×Q₈	—	12	2	no	—
128	C₅×Q₈	—	12	2	no	—
129	C₅×D₄	—	2	1	no	—
130	C₅×D₄	D₂₄	1	2	yes	—
131	C₂×C₄×C₅	—	2	1	no	—
132	C₂×C₄×C₅	—	4	1	no	—
133	C₂×C₄×C₅	—	4	1	no	—
134	C₂×C₄×C₅	—	12	2	no	—
135	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
136	C₂×C₄×C₅	#235	1	1	yes	characteristic
137	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
138	C₅×C₈	—	2	1	no	—
139	C₅×C₈	—	4	1	no	—
140	C₅×C₈	—	12	2	no	—
141	C₅×C₈	#214	1	2	yes	complemented
142	C₂×SD₁₆	—	6	2	no	—
143	C₂²×C₈	—	2	1	no	—
144	C₄⋊Q₈	—	3	2	no	—
145	C₄×D₄	C₂×C₅×D₃	1	1	yes	characteristic
146	#107	—	6	1	no	—
147	#107	—	6	1	no	—
148	C₄⋊C₈	—	3	2	no	—
149	C₄⋊C₈	C₂×C₅×D₃	1	1	yes	characteristic
150	#103	—	6	2	no	—
151	C₂²⋊C₈	—	2	1	no	—
152	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
153	C₂×C₃×C₅	—	4	2	no	—
154	C₂×C₃×C₅	#461	1	1	yes	characteristic
155	C₂×C₃×C₅	#466	1	1	yes	characteristic
156	C₂×C₃×C₅	#473	1	1	yes	characteristic
157	C₂³×C₃	—	2	1	no	—
158	C₃×D₄	—	2	1	no	—
159	C₃×D₄	C₅×D₈	1	2	yes	—
160	C₂×C₃×C₄	—	2	1	no	—
161	C₂×C₃×C₄	—	4	1	no	—
162	C₂×C₃×C₄	—	4	1	no	—
163	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
164	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
165	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
166	C₂×Dic₃	—	4	2	no	—
167	Dic₆	—	4	2	no	—
168	Dic₆	—	4	2	no	—
169	C₃×C₈	—	2	1	no	—
170	C₃×C₈	—	4	1	no	—
171	C₃×C₈	C₅×SD₁₆	1	2	yes	complemented
172	C₃⋊C₈	—	4	2	no	—
173	C₂²×C₅	—	2	2	no	—
174	C₂²×C₅	—	4	1	no	—
175	C₂²×C₅	—	4	1	no	—
176	C₂²×C₅	#930	1	1	yes	characteristic
177	C₄×C₅	—	2	1	no	—
178	C₄×C₅	—	4	1	no	—
179	C₄×C₅	—	24	2	no	—
180	C₄×C₅	C₂×D₂₄	1	1	yes	characteristic
181	C₄×C₅	#904	1	1	yes	characteristic
182	C₄×C₅	C₂×#214	1	1	yes	characteristic
183	C₄×C₅	#951	1	1	yes	characteristic
184	C₂×Q₈	—	6	2	no	—
185	C₂×D₄	C₅×D₁₂	1	1	yes	characteristic
186	C₂²×C₄	—	2	1	no	—
187	SD₁₆	—	12	4	no	—
188	C₂×C₈	—	2	1	no	—
189	C₂×C₈	—	4	2	no	—
190	C₂×C₈	—	6	2	no	—
191	C₂×C₈	C₅×#67	1	1	yes	characteristic
192	C₂×C₈	C₅×#67	1	1	yes	characteristic
193	C₄⋊C₄	—	6	2	no	—
194	C₄⋊C₄	—	6	2	no	—
195	C₄⋊C₄	C₅×D₁₂	1	1	yes	characteristic
196	C₂²⋊C₄	—	2	1	no	—
197	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
198	C₃×C₅	#1673	1	1	yes	characteristic, complemented, Hall
199	C₂²×C₃	—	2	2	no	—
200	C₂²×C₃	—	4	1	no	—
201	C₂²×C₃	—	4	1	no	—
202	C₂²×C₃	C₅×#121	1	1	yes	characteristic
203	C₃×C₄	—	2	1	no	—
204	C₃×C₄	—	4	1	no	—
205	C₃×C₄	C₂×C₅×D₈	1	1	yes	characteristic
206	C₃×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
207	C₃×C₄	C₅×#135	1	1	yes	characteristic
208	C₃×C₄	C₅×#137	1	1	yes	characteristic
209	Dic₃	—	8	2	no	—
210	C₂×C₅	—	4	2	no	—
211	C₂×C₅	#5019	1	1	yes	characteristic
212	C₂×C₅	#5115	1	1	yes	characteristic
213	C₂×C₅	#5143	1	1	yes	characteristic
214	C₂³	—	2	1	no	—
215	Q₈	—	12	2	no	—
216	Q₈	—	12	2	no	—
217	D₄	—	2	1	no	—
218	D₄	C₅×D₂₄	1	2	yes	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	4	1	no	—
221	C₂×C₄	—	4	1	no	—
222	C₂×C₄	—	12	2	no	—
223	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
224	C₂×C₄	C₅×#235	1	1	yes	characteristic
225	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
226	C₈	—	2	1	no	—
227	C₈	—	4	1	no	—
228	C₈	—	12	2	no	—
229	C₈	C₅×#214	1	2	yes	complemented
230	C₂×C₃	—	4	2	no	—
231	C₂×C₃	C₅×#461	1	1	yes	characteristic
232	C₂×C₃	C₅×#466	1	1	yes	characteristic
233	C₂×C₃	C₅×#473	1	1	yes	characteristic
234	C₅	#70852	1	1	yes	characteristic, direct product factor, Sylow
235	C₂²	—	2	2	no	—
236	C₂²	—	4	1	no	—
237	C₂²	—	4	1	no	—
238	C₂²	C₅×#930	1	1	yes	characteristic
239	C₄	—	2	1	no	—
240	C₄	—	4	1	no	—
241	C₄	—	24	2	no	—
242	C₄	C₂×C₅×D₂₄	1	1	yes	characteristic
243	C₄	C₅×#904	1	1	yes	characteristic
244	C₄	C₂×C₅×#214	1	1	yes	characteristic
245	C₄	C₅×#951	1	1	yes	characteristic
246	C₃	C₅×#1673	1	1	yes	characteristic, complemented, Sylow
247	C₂	—	4	2	no	—
248	C₂	C₅×#5019	1	1	yes	characteristic
249	C₂	C₅×#5115	1	1	yes	characteristic
250	C₂	C₅×#5143	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	#930
commutator, cosocle	C₃×C₄²	C₂³×C₅
Frattini	C₄²	C₂²×C₅×D₃
Fitting	C₃×C₅×C₈×D₄	C₂
socle	C₂²×C₃×C₅	#121
prime radical	C₂×C₃×C₅×D₄	D₄
hypercenter	C₅×C₈×D₄	D₃
hypocenter	C₃	C₅×#1673
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×C₈×D₄

Complements

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

N	H ≅ G/N	type
3. C₅×#4682	C₂	semidirect
4. C₅×#4486	C₂	semidirect
5. C₅×#4458	C₂	semidirect
18. #70852	C₅	direct (N × H)
40. #4682	C₂×C₅	semidirect
41. #4486	C₂×C₅	semidirect
42. #4458	C₂×C₅	semidirect
69. C₃×C₅×C₈	SD₁₆	semidirect
141. C₅×C₈	#214	semidirect
171. C₃×C₈	C₅×SD₁₆	semidirect
198. C₃×C₅	#1673	semidirect
229. C₈	C₅×#214	semidirect
234. C₅	#70852	direct (N × H)
246. C₃	C₅×#1673	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×C₈×D₄	2
C₅×#5019	2
C₅×#4682	2
C₅×#4486	2
C₅×#4458	2
C₅×#1673	3
#70852	5

Subgroups — C5×#70852 #14291694

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#70852 #14291694