Subgroups — C₅×#70491 #14285678

Order 1920.

368 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#423	C₂	1	1	yes	maximal, characteristic
2	C₅×#4924	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₅×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₅×#4479	C₂	1	4	yes	maximal, complemented
5	C₅×#1567	—	3	1	no	maximal, Hall
6	C₂×C₃×C₅×M₁₆	—	2	1	no	—
7	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
8	C₂×C₅×#208	—	2	1	no	—
9	C₂²×C₅×C₃⋊C₈	C₄	1	2	yes	—
10	C₃×C₅×C₄⋊C₈	—	2	1	no	—
11	C₃×C₅×C₄⋊C₈	C₂²	1	2	yes	—
12	C₅×C₄⋊(C₃⋊C₈)	—	2	1	no	—
13	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
14	C₄×C₅×C₃⋊C₈	C₄	1	1	yes	characteristic
15	C₄×C₅×C₃⋊C₈	C₄	1	1	yes	characteristic
16	C₄×C₅×C₃⋊C₈	C₂²	1	2	yes	—
17	#70491	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₅×#423	—	3	1	no	—
19	C₅×#423	D₃	1	1	yes	characteristic
20	C₂×C₄×C₅×C₈	—	3	1	no	—
21	C₅×#335	—	3	4	no	—
22	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
24	C₂²×C₃×C₄×C₅	Q₈	1	1	yes	characteristic
25	C₃×C₅×M₁₆	—	4	2	no	—
26	C₂×C₃×C₅×C₈	—	2	2	no	—
27	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
28	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
29	C₃×C₄²×C₅	Q₈	1	1	yes	characteristic
30	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
31	C₅×#208	—	4	2	no	—
32	C₂×C₅×C₃⋊C₈	—	2	4	no	—
33	C₂×C₅×C₃⋊C₈	—	2	2	no	—
34	C₂×C₅×C₃⋊C₈	C₂×C₄	1	2	yes	—
35	C₂×C₅×C₃⋊C₈	C₂×C₄	1	2	yes	—
36	C₃×#423	C₂×C₅	1	1	yes	characteristic
37	#4924	C₂×C₅	1	1	yes	characteristic
38	C₂×C₄×C₃⋊C₈	C₂×C₅	1	1	yes	characteristic
39	#4479	C₂×C₅	1	4	yes	complemented
40	C₂×C₅×M₁₆	—	2	1	no	—
41	C₂×C₅×M₁₆	—	6	1	no	—
42	C₂²×C₅×C₈	—	3	2	no	—
43	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
44	C₅×C₄⋊C₈	—	2	1	no	—
45	C₅×C₄⋊C₈	—	3	2	no	—
46	C₅×C₄⋊C₈	—	6	1	no	—
47	C₅×C₄⋊C₈	C₂×D₃	1	2	yes	—
48	C₄×C₅×C₈	—	3	1	no	—
49	C₄×C₅×C₈	—	3	1	no	—
50	C₄×C₅×C₈	—	3	2	no	—
51	#1567	—	3	1	no	Sylow
52	C₂³×C₃×C₅	C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₃×C₄×C₅	—	2	2	no	—
54	C₂×C₃×C₄×C₅	—	2	2	no	—
55	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₃×C₄×C₅	D₈	1	2	yes	—
59	C₂×C₃×C₄×C₅	Q₁₆	1	2	yes	—
60	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
62	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
63	C₃×C₅×C₈	—	4	2	no	—
64	C₅×C₃⋊C₈	—	2	2	no	—
65	C₅×C₃⋊C₈	—	4	2	no	—
66	C₅×C₃⋊C₈	M₁₆	1	4	yes	complemented
67	C₂×C₃×M₁₆	—	2	1	no	—
68	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
69	C₂×#208	—	2	1	no	—
70	C₂²×C₃⋊C₈	C₄×C₅	1	2	yes	—
71	C₃×C₄⋊C₈	—	2	1	no	—
72	C₃×C₄⋊C₈	C₂²×C₅	1	2	yes	—
73	C₄⋊(C₃⋊C₈)	—	2	1	no	—
74	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	2	yes	—
75	C₄×C₃⋊C₈	C₄×C₅	1	1	yes	characteristic
76	C₄×C₃⋊C₈	C₄×C₅	1	1	yes	characteristic
77	C₄×C₃⋊C₈	C₂²×C₅	1	2	yes	—
78	C₂²×C₄×C₅	Dic₆	1	1	yes	characteristic
79	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
80	C₂²×C₄×C₅	#67	1	1	yes	characteristic
81	C₅×M₁₆	—	4	2	no	—
82	C₅×M₁₆	—	12	2	no	—
83	C₂×C₅×C₈	—	2	2	no	—
84	C₂×C₅×C₈	—	3	2	no	—
85	C₂×C₅×C₈	—	3	2	no	—
86	C₂×C₅×C₈	—	6	4	no	—
87	C₂×C₅×C₈	—	6	2	no	—
88	C₄²×C₅	Dic₆	1	1	yes	characteristic
89	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
90	C₄²×C₅	#67	1	1	yes	characteristic
91	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
92	#423	—	3	1	no	—
93	#423	C₅×D₃	1	1	yes	characteristic
94	C₂×C₄×C₈	—	3	1	no	—
95	#335	—	3	4	no	—
96	C₂²×C₃×C₅	—	2	1	no	—
97	C₂²×C₃×C₅	—	2	1	no	—
98	C₂²×C₃×C₅	#107	1	2	yes	—
99	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₃×C₄×C₅	—	2	1	no	—
101	C₃×C₄×C₅	—	2	1	no	—
102	C₃×C₄×C₅	#107	1	2	yes	—
103	C₃×C₄×C₅	#108	1	2	yes	—
104	C₃×C₄×C₅	C₂×M₁₆	1	2	yes	—
105	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
106	C₃×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
107	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
108	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
109	C₂²×C₃×C₄	C₅×Q₈	1	1	yes	characteristic
110	C₃×M₁₆	—	4	2	no	—
111	C₂×C₃×C₈	—	2	2	no	—
112	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
113	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
114	C₃×C₄²	C₅×Q₈	1	1	yes	characteristic
115	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
116	#208	—	4	2	no	—
117	C₂×C₃⋊C₈	—	2	4	no	—
118	C₂×C₃⋊C₈	—	2	2	no	—
119	C₂×C₃⋊C₈	C₂×C₄×C₅	1	2	yes	—
120	C₂×C₃⋊C₈	C₂×C₄×C₅	1	2	yes	—
121	C₂³×C₅	Dic₃⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₅	—	2	2	no	—
123	C₂×C₄×C₅	—	2	2	no	—
124	C₂×C₄×C₅	Dic₃⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₅	Dic₃⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₅	Dic₃⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₅	#213	1	2	yes	—
128	C₂×C₄×C₅	#216	1	2	yes	—
129	C₂×C₄×C₅	C₂×Dic₆	1	1	yes	characteristic
130	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
131	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
132	C₅×C₈	—	3	4	no	—
133	C₅×C₈	—	4	2	no	—
134	C₅×C₈	—	6	2	no	—
135	C₅×C₈	—	12	2	no	—
136	C₂×M₁₆	—	2	1	no	—
137	C₂×M₁₆	—	6	1	no	—
138	C₂²×C₈	—	3	2	no	—
139	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
140	C₄⋊C₈	—	2	1	no	—
141	C₄⋊C₈	—	3	2	no	—
142	C₄⋊C₈	—	6	1	no	—
143	C₄⋊C₈	C₂×C₅×D₃	1	2	yes	—
144	C₄×C₈	—	3	1	no	—
145	C₄×C₈	—	3	1	no	—
146	C₄×C₈	—	3	2	no	—
147	C₂×C₃×C₅	—	2	2	no	—
148	C₂×C₃×C₅	#423	1	1	yes	characteristic
149	C₂×C₃×C₅	C₂×#107	1	1	yes	characteristic
150	C₂×C₃×C₅	C₂×#108	1	1	yes	characteristic
151	C₂³×C₃	C₅×C₄⋊C₄	1	1	yes	characteristic
152	C₂×C₃×C₄	—	2	2	no	—
153	C₂×C₃×C₄	—	2	2	no	—
154	C₂×C₃×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
155	C₂×C₃×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
156	C₂×C₃×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
157	C₂×C₃×C₄	C₅×D₈	1	2	yes	—
158	C₂×C₃×C₄	C₅×Q₁₆	1	2	yes	—
159	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
160	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
161	C₂×C₃×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
162	C₃×C₈	—	4	2	no	—
163	C₃⋊C₈	—	2	2	no	—
164	C₃⋊C₈	—	4	2	no	—
165	C₃⋊C₈	C₅×M₁₆	1	4	yes	complemented
166	C₂²×C₅	—	2	1	no	—
167	C₂²×C₅	—	2	1	no	—
168	C₂²×C₅	#807	1	2	yes	—
169	C₂²×C₅	C₂×Dic₃⋊C₄	1	1	yes	characteristic
170	C₄×C₅	—	2	1	no	—
171	C₄×C₅	—	2	1	no	—
172	C₄×C₅	#807	1	2	yes	—
173	C₄×C₅	#822	1	2	yes	—
174	C₄×C₅	D₃×M₁₆	1	2	yes	—
175	C₄×C₅	C₂×#213	1	1	yes	characteristic
176	C₄×C₅	C₂×#216	1	1	yes	characteristic
177	C₂²×C₄	C₅×Dic₆	1	1	yes	characteristic
178	C₂²×C₄	C₄×C₅×D₃	1	1	yes	characteristic
179	C₂²×C₄	C₅×#67	1	1	yes	characteristic
180	M₁₆	—	4	2	no	—
181	M₁₆	—	12	2	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	3	2	no	—
184	C₂×C₈	—	3	2	no	—
185	C₂×C₈	—	6	4	no	—
186	C₂×C₈	—	6	2	no	—
187	C₄²	C₅×Dic₆	1	1	yes	characteristic
188	C₄²	C₄×C₅×D₃	1	1	yes	characteristic
189	C₄²	C₅×#67	1	1	yes	characteristic
190	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
191	C₃×C₅	#1567	1	1	yes	characteristic, complemented, Hall
192	C₂²×C₃	—	2	1	no	—
193	C₂²×C₃	—	2	1	no	—
194	C₂²×C₃	C₅×#107	1	2	yes	—
195	C₂²×C₃	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
196	C₃×C₄	—	2	1	no	—
197	C₃×C₄	—	2	1	no	—
198	C₃×C₄	C₅×#107	1	2	yes	—
199	C₃×C₄	C₅×#108	1	2	yes	—
200	C₃×C₄	C₂×C₅×M₁₆	1	2	yes	—
201	C₃×C₄	C₂×C₅×D₈	1	1	yes	characteristic
202	C₃×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
203	C₂×C₅	—	2	2	no	—
204	C₂×C₅	C₂×#807	1	1	yes	characteristic
205	C₂×C₅	#5118	1	1	yes	characteristic
206	C₂×C₅	C₂×#822	1	1	yes	characteristic
207	C₂³	C₅×Dic₃⋊C₄	1	1	yes	characteristic
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	C₅×Dic₃⋊C₄	1	1	yes	characteristic
211	C₂×C₄	C₅×Dic₃⋊C₄	1	1	yes	characteristic
212	C₂×C₄	C₅×Dic₃⋊C₄	1	1	yes	characteristic
213	C₂×C₄	C₅×#213	1	2	yes	—
214	C₂×C₄	C₅×#216	1	2	yes	—
215	C₂×C₄	C₂×C₅×Dic₆	1	1	yes	characteristic
216	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
217	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
218	C₈	—	3	4	no	—
219	C₈	—	4	2	no	—
220	C₈	—	6	2	no	—
221	C₈	—	12	2	no	—
222	C₂×C₃	—	2	2	no	—
223	C₂×C₃	C₅×#423	1	1	yes	characteristic
224	C₂×C₃	C₂×C₅×#107	1	1	yes	characteristic
225	C₂×C₃	C₂×C₅×#108	1	1	yes	characteristic
226	C₅	#70491	1	1	yes	characteristic, direct product factor, Sylow
227	C₂²	—	2	1	no	—
228	C₂²	—	2	1	no	—
229	C₂²	C₅×#807	1	2	yes	—
230	C₂²	C₂×C₅×Dic₃⋊C₄	1	1	yes	characteristic
231	C₄	—	2	1	no	—
232	C₄	—	2	1	no	—
233	C₄	C₅×#807	1	2	yes	—
234	C₄	C₅×#822	1	2	yes	—
235	C₄	C₅×D₃×M₁₆	1	2	yes	—
236	C₄	C₂×C₅×#213	1	1	yes	characteristic
237	C₄	C₂×C₅×#216	1	1	yes	characteristic
238	C₃	C₅×#1567	1	1	yes	characteristic, complemented, Sylow
239	C₂	—	2	2	no	—
240	C₂	C₂×C₅×#807	1	1	yes	characteristic
241	C₂	C₅×#5118	1	1	yes	characteristic
242	C₂	C₂×C₅×#822	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	C₂×#67
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₄²	C₂²×C₅×D₃
Fitting	C₃×C₅×#423	C₂
socle	C₂²×C₃×C₅	C₂×C₄⋊C₄
prime radical	C₂³×C₃×C₅	C₄⋊C₄
cosocle	C₃×C₄²	C₂³×C₅
hypercenter	C₅×#423	D₃
hypocenter	C₃	C₅×#1567
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ C₅×#423

Complements

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

N	H ≅ G/N	type
4. C₅×#4479	C₂	semidirect
17. #70491	C₅	direct (N × H)
39. #4479	C₂×C₅	semidirect
66. C₅×C₃⋊C₈	M₁₆	semidirect
165. C₃⋊C₈	C₅×M₁₆	semidirect
191. C₃×C₅	#1567	semidirect
226. C₅	#70491	direct (N × H)
238. C₃	C₅×#1567	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×#423	2
C₅×#4924	2
C₂×C₄×C₅×C₃⋊C₈	2
C₅×#4479	2
C₅×#1567	3
#70491	5

Subgroups — C5×#70491 #14285678

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#70491 #14285678