Subgroups — C₅×#70555 #14299096

Order 1920.

496 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	C₂×C₃×C₅×#106	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#817	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×C₈×Dic₃	C₂	1	1	yes	maximal, characteristic
4	C₅×#4977	C₂	1	2	yes	maximal
5	C₂×C₅×#807	C₂	1	2	yes	maximal
6	C₅×#1846	—	3	1	no	maximal, Hall
7	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
9	C₂×C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
10	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
11	C₂²×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
12	C₃×C₅×#106	—	2	2	no	—
13	C₅×C₃⋊#95	—	2	4	no	—
14	C₅×#817	—	2	2	no	—
15	C₅×C₈×Dic₃	C₄	1	4	yes	complemented
16	C₅×#807	—	2	4	no	—
17	#70555	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₂×C₅×#107	—	3	2	no	—
19	C₂×C₅×#106	—	3	1	no	—
20	C₂×C₅×#106	D₃	1	1	yes	characteristic
21	C₂×C₄×C₅×C₈	—	3	1	no	—
22	C₅×#382	—	3	2	no	—
23	C₂²×C₃×C₄×C₅	—	2	2	no	—
24	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
25	C₂²×C₅×Dic₃	—	2	2	no	—
26	C₂²×C₅×Dic₃	D₄	1	2	yes	—
27	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
28	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
29	C₂×C₃×C₅×C₈	Q₈	1	2	yes	—
30	C₃×C₅×C₄⋊C₄	—	2	2	no	—
31	C₃×C₅×C₄⋊C₄	—	2	2	no	—
32	C₅×C₄⋊Dic₃	—	2	2	no	—
33	C₅×C₄⋊Dic₃	—	2	2	no	—
34	C₄×C₅×Dic₃	C₂×C₄	1	2	yes	—
35	C₄×C₅×Dic₃	Q₈	1	2	yes	—
36	C₂×C₅×C₃⋊C₈	C₂×C₄	1	2	yes	—
37	C₂×C₅×C₃⋊C₈	D₄	1	2	yes	—
38	C₂×C₅×C₃⋊C₈	Q₈	1	2	yes	—
39	C₂×C₃×#106	C₂×C₅	1	1	yes	characteristic
40	C₂×#817	C₂×C₅	1	1	yes	characteristic
41	C₂×C₈×Dic₃	C₂×C₅	1	1	yes	characteristic
42	#4977	C₂×C₅	1	2	yes	—
43	C₂×#807	C₂×C₅	1	2	yes	—
44	C₂²×C₅×C₈	—	3	1	no	—
45	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
46	C₂×C₅×C₄⋊C₄	—	3	2	no	—
47	C₂×C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
48	C₂×C₄²×C₅	—	3	1	no	—
49	C₅×#107	—	6	4	no	—
50	C₅×#106	—	2	2	no	—
51	C₅×#106	—	6	2	no	—
52	C₄×C₅×C₈	—	3	4	no	—
53	C₅×#95	—	6	4	no	—
54	#1846	—	3	1	no	Sylow
55	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
56	C₂×C₃×C₄×C₅	—	4	2	no	—
57	C₂×C₃×C₄×C₅	—	4	4	no	—
58	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
60	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
61	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
63	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
64	C₂×C₅×Dic₃	—	2	4	no	—
65	C₂×C₅×Dic₃	—	4	2	no	—
66	C₂×C₅×Dic₃	—	4	4	no	—
67	C₂×C₅×Dic₃	C₄⋊C₄	1	4	yes	—
68	C₃×C₅×C₈	C₄⋊C₄	1	4	yes	—
69	C₅×C₃⋊C₈	C₄⋊C₄	1	4	yes	—
70	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
71	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
72	C₂×C₄⋊Dic₃	C₂²×C₅	1	2	yes	—
73	C₂×C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
74	C₂²×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
75	C₃×#106	—	2	2	no	—
76	C₃⋊#95	—	2	4	no	—
77	#817	—	2	2	no	—
78	C₈×Dic₃	C₄×C₅	1	4	yes	complemented
79	#807	—	2	4	no	—
80	C₂²×C₄×C₅	—	2	2	no	—
81	C₂²×C₄×C₅	—	3	2	no	—
82	C₂²×C₄×C₅	—	6	2	no	—
83	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
84	C₂×C₅×C₈	—	3	2	no	—
85	C₂×C₅×C₈	—	3	2	no	—
86	C₂×C₅×C₈	—	3	2	no	—
87	C₂×C₅×C₈	Dic₆	1	2	yes	—
88	C₂×C₅×C₈	C₄×D₃	1	2	yes	—
89	C₂×C₅×C₈	#67	1	2	yes	—
90	C₅×C₄⋊C₄	—	2	2	no	—
91	C₅×C₄⋊C₄	—	2	2	no	—
92	C₅×C₄⋊C₄	—	6	2	no	—
93	C₅×C₄⋊C₄	—	6	2	no	—
94	C₄²×C₅	—	3	2	no	—
95	C₄²×C₅	—	3	2	no	—
96	C₂×#107	—	3	2	no	—
97	C₂×#106	—	3	1	no	—
98	C₂×#106	C₅×D₃	1	1	yes	characteristic
99	C₂×C₄×C₈	—	3	1	no	—
100	#382	—	3	2	no	—
101	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂²×C₃×C₅	#127	1	1	yes	characteristic
103	C₂²×C₃×C₅	C₄⋊Q₈	1	1	yes	characteristic
104	C₂²×C₃×C₅	#135	1	4	yes	—
105	C₃×C₄×C₅	—	8	2	no	—
106	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₃×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
109	C₃×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
110	C₅×Dic₃	—	2	4	no	—
111	C₅×Dic₃	—	8	2	no	—
112	C₂²×C₃×C₄	—	2	2	no	—
113	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
114	C₂²×Dic₃	—	2	2	no	—
115	C₂²×Dic₃	C₅×D₄	1	2	yes	—
116	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
117	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
118	C₂×C₃×C₈	C₅×Q₈	1	2	yes	—
119	C₃×C₄⋊C₄	—	2	2	no	—
120	C₃×C₄⋊C₄	—	2	2	no	—
121	C₄⋊Dic₃	—	2	2	no	—
122	C₄⋊Dic₃	—	2	2	no	—
123	C₄×Dic₃	C₂×C₄×C₅	1	2	yes	—
124	C₄×Dic₃	C₅×Q₈	1	2	yes	—
125	C₂×C₃⋊C₈	C₂×C₄×C₅	1	2	yes	—
126	C₂×C₃⋊C₈	C₅×D₄	1	2	yes	—
127	C₂×C₃⋊C₈	C₅×Q₈	1	2	yes	—
128	C₂³×C₅	D₃×D₄	1	1	yes	characteristic
129	C₂×C₄×C₅	—	3	4	no	—
130	C₂×C₄×C₅	—	4	4	no	—
131	C₂×C₄×C₅	—	4	2	no	—
132	C₂×C₄×C₅	—	6	4	no	—
133	C₂×C₄×C₅	—	12	4	no	—
134	C₂×C₄×C₅	—	12	2	no	—
135	C₂×C₄×C₅	C₂×Dic₆	1	1	yes	characteristic
136	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
137	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
138	C₂×C₄×C₅	D₃×Q₈	1	1	yes	characteristic
139	C₂×C₄×C₅	D₃×Q₈	1	1	yes	characteristic
140	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
141	C₅×C₈	—	3	4	no	—
142	C₅×C₈	Dic₃⋊C₄	1	4	yes	—
143	C₂²×C₈	—	3	1	no	—
144	C₂²×C₈	C₂×C₅×D₃	1	1	yes	characteristic
145	C₂×C₄⋊C₄	—	3	2	no	—
146	C₂×C₄⋊C₄	C₂×C₅×D₃	1	2	yes	—
147	C₂×C₄²	—	3	1	no	—
148	#107	—	6	4	no	—
149	#106	—	2	2	no	—
150	#106	—	6	2	no	—
151	C₄×C₈	—	3	4	no	—
152	#95	—	6	4	no	—
153	C₂×C₃×C₅	#384	1	1	yes	characteristic
154	C₂×C₃×C₅	#427	1	2	yes	—
155	C₂×C₃×C₅	#495	1	2	yes	—
156	C₂×C₃×C₅	#499	1	2	yes	—
157	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
158	C₂×C₃×C₄	—	4	2	no	—
159	C₂×C₃×C₄	—	4	4	no	—
160	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
161	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
162	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
163	C₂×C₃×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
164	C₂×C₃×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
165	C₂×C₃×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
166	C₂×Dic₃	—	2	4	no	—
167	C₂×Dic₃	—	4	2	no	—
168	C₂×Dic₃	—	4	4	no	—
169	C₂×Dic₃	C₅×C₄⋊C₄	1	4	yes	—
170	C₃×C₈	C₅×C₄⋊C₄	1	4	yes	—
171	C₃⋊C₈	C₅×C₄⋊C₄	1	4	yes	—
172	C₂²×C₅	#888	1	1	yes	characteristic
173	C₂²×C₅	D₃×C₄⋊C₄	1	1	yes	characteristic
174	C₂²×C₅	#916	1	4	yes	—
175	C₂²×C₅	#940	1	1	yes	characteristic
176	C₄×C₅	—	6	4	no	—
177	C₄×C₅	—	8	2	no	—
178	C₄×C₅	—	24	2	no	—
179	C₄×C₅	#888	1	1	yes	characteristic
180	C₄×C₅	D₃×C₄⋊C₄	1	1	yes	characteristic
181	C₄×C₅	C₂×Dic₃⋊C₄	1	1	yes	characteristic
182	C₄×C₅	#944	1	1	yes	characteristic
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	—	3	2	no	—
185	C₂²×C₄	—	6	2	no	—
186	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
187	C₂×C₈	—	3	2	no	—
188	C₂×C₈	—	3	2	no	—
189	C₂×C₈	—	3	2	no	—
190	C₂×C₈	C₅×Dic₆	1	2	yes	—
191	C₂×C₈	C₄×C₅×D₃	1	2	yes	—
192	C₂×C₈	C₅×#67	1	2	yes	—
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	—	6	2	no	—
196	C₄⋊C₄	—	6	2	no	—
197	C₄²	—	3	2	no	—
198	C₄²	—	3	2	no	—
199	C₃×C₅	#1846	1	1	yes	characteristic, complemented, Hall
200	C₂²×C₃	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
201	C₂²×C₃	C₅×#127	1	1	yes	characteristic
202	C₂²×C₃	C₅×C₄⋊Q₈	1	1	yes	characteristic
203	C₂²×C₃	C₅×#135	1	4	yes	—
204	C₃×C₄	—	8	2	no	—
205	C₃×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
206	C₃×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
207	C₃×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
208	C₃×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
209	Dic₃	—	2	4	no	—
210	Dic₃	—	8	2	no	—
211	C₂×C₅	#4858	1	2	yes	—
212	C₂×C₅	#4860	1	2	yes	—
213	C₂×C₅	#4975	1	1	yes	characteristic
214	C₂×C₅	#5168	1	2	yes	—
215	C₂³	C₅×D₃×D₄	1	1	yes	characteristic
216	C₂×C₄	—	3	4	no	—
217	C₂×C₄	—	4	4	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	—	6	4	no	—
220	C₂×C₄	—	12	4	no	—
221	C₂×C₄	—	12	2	no	—
222	C₂×C₄	C₂×C₅×Dic₆	1	1	yes	characteristic
223	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
224	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
225	C₂×C₄	C₅×D₃×Q₈	1	1	yes	characteristic
226	C₂×C₄	C₅×D₃×Q₈	1	1	yes	characteristic
227	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
228	C₈	—	3	4	no	—
229	C₈	C₅×Dic₃⋊C₄	1	4	yes	—
230	C₂×C₃	C₅×#384	1	1	yes	characteristic
231	C₂×C₃	C₅×#427	1	2	yes	—
232	C₂×C₃	C₅×#495	1	2	yes	—
233	C₂×C₃	C₅×#499	1	2	yes	—
234	C₅	#70555	1	1	yes	characteristic, direct product factor, Sylow
235	C₂²	C₅×#888	1	1	yes	characteristic
236	C₂²	C₅×D₃×C₄⋊C₄	1	1	yes	characteristic
237	C₂²	C₅×#916	1	4	yes	—
238	C₂²	C₅×#940	1	1	yes	characteristic
239	C₄	—	6	4	no	—
240	C₄	—	8	2	no	—
241	C₄	—	24	2	no	—
242	C₄	C₅×#888	1	1	yes	characteristic
243	C₄	C₅×D₃×C₄⋊C₄	1	1	yes	characteristic
244	C₄	C₂×C₅×Dic₃⋊C₄	1	1	yes	characteristic
245	C₄	C₅×#944	1	1	yes	characteristic
246	C₃	C₅×#1846	1	1	yes	characteristic, complemented, Sylow
247	C₂	C₅×#4858	1	2	yes	—
248	C₂	C₅×#4860	1	2	yes	—
249	C₂	C₅×#4975	1	1	yes	characteristic
250	C₂	C₅×#5168	1	2	yes	—

	subgroup	G/H
center	C₂³×C₅	D₃×D₄
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₂×C₃×C₅×#106	C₂
socle, prime radical	C₂³×C₃×C₅	C₂×D₄
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₂×C₅×#106	D₃
hypocenter	C₃	C₅×#1846
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×#106

Complements

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

N	H ≅ G/N	type
15. C₅×C₈×Dic₃	C₄	semidirect
17. #70555	C₅	direct (N × H)
78. C₈×Dic₃	C₄×C₅	semidirect
199. C₃×C₅	#1846	semidirect
234. C₅	#70555	direct (N × H)
246. C₃	C₅×#1846	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₅×#106	2
C₂×C₅×#817	2
C₂×C₅×C₈×Dic₃	2
C₅×#4977	2
C₂×C₅×#807	2
C₅×#1846	3
#70555	5

Subgroups — C5×#70555 #14299096

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#70555 #14299096