Subgroups — C₅×#70742 #14280562

Order 1920.

392 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#431	C₂	1	1	yes	maximal, characteristic
2	C₅×#4925	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₅×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₅×#4455	C₂	1	2	yes	maximal, complemented
5	C₅×#4454	C₂	1	2	yes	maximal, complemented
6	C₅×#1452	—	3	1	no	maximal, Hall
7	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
8	C₂²×C₅×C₃⋊C₈	C₄	1	2	yes	—
9	C₃×C₅×C₄⋊C₈	C₄	1	2	yes	—
10	C₃×C₅×C₂²⋊C₈	C₄	1	2	yes	—
11	C₃×C₄×C₅×C₈	C₂²	1	2	yes	—
12	C₅×#830	C₄	1	2	yes	—
13	C₅×C₄⋊(C₃⋊C₈)	C₄	1	2	yes	—
14	C₄×C₅×C₃⋊C₈	C₄	1	2	yes	—
15	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
16	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
17	C₄×C₅×C₃⋊C₈	C₂²	1	2	yes	—
18	#70742	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₅×#431	—	3	1	no	—
20	C₅×#431	D₃	1	1	yes	characteristic
21	C₂×C₄×C₅×C₈	—	3	1	no	—
22	C₅×#322	—	3	2	no	—
23	C₅×#322	—	3	2	no	—
24	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃×C₅×C₈	C₂×C₄	1	4	yes	—
28	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
29	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
30	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
31	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
32	C₂×C₅×C₃⋊C₈	—	2	4	no	—
33	C₂×C₅×C₃⋊C₈	C₂×C₄	1	2	yes	—
34	C₂×C₅×C₃⋊C₈	C₂×C₄	1	2	yes	—
35	C₂×C₅×C₃⋊C₈	C₂×C₄	1	4	yes	—
36	C₃×#431	C₂×C₅	1	1	yes	characteristic
37	#4925	C₂×C₅	1	1	yes	characteristic
38	C₂×C₄×C₃⋊C₈	C₂×C₅	1	1	yes	characteristic
39	#4455	C₂×C₅	1	2	yes	complemented
40	#4454	C₂×C₅	1	2	yes	complemented
41	C₂²×C₅×C₈	—	3	2	no	—
42	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
43	C₅×C₄⋊C₈	—	3	2	no	—
44	C₅×C₄⋊C₈	Dic₃	1	2	yes	—
45	C₅×C₂²⋊C₈	—	3	2	no	—
46	C₅×C₂²⋊C₈	Dic₃	1	2	yes	—
47	C₄×C₅×C₈	—	3	2	no	—
48	C₄×C₅×C₈	—	3	1	no	—
49	C₄×C₅×C₈	—	3	1	no	—
50	C₄×C₅×C₈	—	3	2	no	—
51	C₄×C₅×C₈	C₂×D₃	1	2	yes	—
52	#1452	—	3	1	no	Sylow
53	C₂³×C₃×C₅	C₄²	1	1	yes	characteristic
54	C₂×C₃×C₄×C₅	—	2	2	no	—
55	C₂×C₃×C₄×C₅	—	2	2	no	—
56	C₂×C₃×C₄×C₅	C₄²	1	1	yes	characteristic
57	C₂×C₃×C₄×C₅	C₄²	1	1	yes	characteristic
58	C₂×C₃×C₄×C₅	C₄²	1	1	yes	characteristic
59	C₂×C₃×C₄×C₅	M₁₆	1	4	yes	—
60	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
63	C₃×C₅×C₈	—	2	4	no	—
64	C₅×C₃⋊C₈	—	2	2	no	—
65	C₅×C₃⋊C₈	—	2	2	no	—
66	C₅×C₃⋊C₈	—	2	4	no	—
67	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
68	C₂²×C₃⋊C₈	C₄×C₅	1	2	yes	—
69	C₃×C₄⋊C₈	C₄×C₅	1	2	yes	—
70	C₃×C₂²⋊C₈	C₄×C₅	1	2	yes	—
71	C₃×C₄×C₈	C₂²×C₅	1	2	yes	—
72	#830	C₄×C₅	1	2	yes	—
73	C₄⋊(C₃⋊C₈)	C₄×C₅	1	2	yes	—
74	C₄×C₃⋊C₈	C₄×C₅	1	2	yes	—
75	C₄×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
76	C₄×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
77	C₄×C₃⋊C₈	C₂²×C₅	1	2	yes	—
78	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
79	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
80	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
81	C₂×C₅×C₈	—	3	2	no	—
82	C₂×C₅×C₈	—	3	2	no	—
83	C₂×C₅×C₈	—	3	4	no	—
84	C₂×C₅×C₈	—	6	4	no	—
85	C₂×C₅×C₈	C₂×Dic₃	1	4	yes	—
86	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
87	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
88	C₄²×C₅	C₂×Dic₃	1	1	yes	characteristic
89	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
90	#431	—	3	1	no	—
91	#431	C₅×D₃	1	1	yes	characteristic
92	C₂×C₄×C₈	—	3	1	no	—
93	#322	—	3	2	no	—
94	#322	—	3	2	no	—
95	C₂²×C₃×C₅	—	2	1	no	—
96	C₂²×C₃×C₅	—	2	1	no	—
97	C₂²×C₃×C₅	#97	1	2	yes	—
98	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
99	C₃×C₄×C₅	—	2	1	no	—
100	C₃×C₄×C₅	—	2	1	no	—
101	C₃×C₄×C₅	#97	1	2	yes	—
102	C₃×C₄×C₅	C₂×M₁₆	1	2	yes	—
103	C₃×C₄×C₅	#131	1	2	yes	—
104	C₃×C₄×C₅	#131	1	2	yes	—
105	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
106	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
107	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
108	C₂×C₃×C₈	C₂×C₄×C₅	1	4	yes	—
109	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
110	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
111	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
112	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
113	C₂×C₃⋊C₈	—	2	4	no	—
114	C₂×C₃⋊C₈	C₂×C₄×C₅	1	2	yes	—
115	C₂×C₃⋊C₈	C₂×C₄×C₅	1	2	yes	—
116	C₂×C₃⋊C₈	C₂×C₄×C₅	1	4	yes	—
117	C₂³×C₅	C₄×Dic₃	1	1	yes	characteristic
118	C₂×C₄×C₅	—	2	2	no	—
119	C₂×C₄×C₅	—	2	2	no	—
120	C₂×C₄×C₅	ΓL(1,𝔽₂₅)	1	4	yes	—
121	C₂×C₄×C₅	C₄×Dic₃	1	1	yes	characteristic
122	C₂×C₄×C₅	C₄×Dic₃	1	1	yes	characteristic
123	C₂×C₄×C₅	C₄×Dic₃	1	1	yes	characteristic
124	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
125	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
126	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
127	C₅×C₈	—	2	4	no	—
128	C₅×C₈	—	6	2	no	—
129	C₅×C₈	—	6	2	no	—
130	C₅×C₈	—	6	4	no	—
131	C₂²×C₈	—	3	2	no	—
132	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
133	C₄⋊C₈	—	3	2	no	—
134	C₄⋊C₈	C₅×Dic₃	1	2	yes	—
135	C₂²⋊C₈	—	3	2	no	—
136	C₂²⋊C₈	C₅×Dic₃	1	2	yes	—
137	C₄×C₈	—	3	2	no	—
138	C₄×C₈	—	3	1	no	—
139	C₄×C₈	—	3	1	no	—
140	C₄×C₈	—	3	2	no	—
141	C₄×C₈	C₂×C₅×D₃	1	2	yes	—
142	C₂×C₃×C₅	—	2	2	no	—
143	C₂×C₃×C₅	C₂×#97	1	1	yes	characteristic
144	C₂×C₃×C₅	#405	1	1	yes	characteristic
145	C₂×C₃×C₅	#405	1	1	yes	characteristic
146	C₂³×C₃	C₄²×C₅	1	1	yes	characteristic
147	C₂×C₃×C₄	—	2	2	no	—
148	C₂×C₃×C₄	—	2	2	no	—
149	C₂×C₃×C₄	C₄²×C₅	1	1	yes	characteristic
150	C₂×C₃×C₄	C₄²×C₅	1	1	yes	characteristic
151	C₂×C₃×C₄	C₄²×C₅	1	1	yes	characteristic
152	C₂×C₃×C₄	C₅×M₁₆	1	4	yes	—
153	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
154	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
155	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
156	C₃×C₈	—	2	4	no	—
157	C₃⋊C₈	—	2	2	no	—
158	C₃⋊C₈	—	2	2	no	—
159	C₃⋊C₈	—	2	4	no	—
160	C₂²×C₅	—	2	1	no	—
161	C₂²×C₅	—	2	1	no	—
162	C₂²×C₅	#815	1	2	yes	—
163	C₂²×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
164	C₄×C₅	—	2	1	no	—
165	C₄×C₅	—	2	1	no	—
166	C₄×C₅	#815	1	2	yes	—
167	C₄×C₅	C₂×ΓL(1,𝔽₂₅)	1	2	yes	—
168	C₄×C₅	#907	1	2	yes	—
169	C₄×C₅	#948	1	2	yes	—
170	C₂²×C₄	C₄×C₅×D₃	1	1	yes	characteristic
171	C₂²×C₄	C₄×C₅×D₃	1	1	yes	characteristic
172	C₂²×C₄	C₂×C₅×Dic₃	1	1	yes	characteristic
173	C₂×C₈	—	3	2	no	—
174	C₂×C₈	—	3	2	no	—
175	C₂×C₈	—	3	4	no	—
176	C₂×C₈	—	6	4	no	—
177	C₂×C₈	C₂×C₅×Dic₃	1	4	yes	—
178	C₄²	C₄×C₅×D₃	1	1	yes	characteristic
179	C₄²	C₄×C₅×D₃	1	1	yes	characteristic
180	C₄²	C₂×C₅×Dic₃	1	1	yes	characteristic
181	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
182	C₃×C₅	#1452	1	1	yes	characteristic, complemented, Hall
183	C₂²×C₃	—	2	1	no	—
184	C₂²×C₃	—	2	1	no	—
185	C₂²×C₃	C₅×#97	1	2	yes	—
186	C₂²×C₃	C₂×C₄²×C₅	1	1	yes	characteristic
187	C₃×C₄	—	2	1	no	—
188	C₃×C₄	—	2	1	no	—
189	C₃×C₄	C₅×#97	1	2	yes	—
190	C₃×C₄	C₂×C₅×M₁₆	1	2	yes	—
191	C₃×C₄	C₅×#131	1	2	yes	—
192	C₃×C₄	C₅×#131	1	2	yes	—
193	C₂×C₅	—	2	2	no	—
194	C₂×C₅	#4997	1	1	yes	characteristic
195	C₂×C₅	C₂×#815	1	1	yes	characteristic
196	C₂×C₅	#5122	1	1	yes	characteristic
197	C₂³	C₄×C₅×Dic₃	1	1	yes	characteristic
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	C₅×ΓL(1,𝔽₂₅)	1	4	yes	—
201	C₂×C₄	C₄×C₅×Dic₃	1	1	yes	characteristic
202	C₂×C₄	C₄×C₅×Dic₃	1	1	yes	characteristic
203	C₂×C₄	C₄×C₅×Dic₃	1	1	yes	characteristic
204	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
205	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
206	C₂×C₄	C₂²×C₅×Dic₃	1	1	yes	characteristic
207	C₈	—	2	4	no	—
208	C₈	—	6	2	no	—
209	C₈	—	6	2	no	—
210	C₈	—	6	4	no	—
211	C₂×C₃	—	2	2	no	—
212	C₂×C₃	C₂×C₅×#97	1	1	yes	characteristic
213	C₂×C₃	C₅×#405	1	1	yes	characteristic
214	C₂×C₃	C₅×#405	1	1	yes	characteristic
215	C₅	#70742	1	1	yes	characteristic, direct product factor, Sylow
216	C₂²	—	2	1	no	—
217	C₂²	—	2	1	no	—
218	C₂²	C₅×#815	1	2	yes	—
219	C₂²	C₂×C₄×C₅×Dic₃	1	1	yes	characteristic
220	C₄	—	2	1	no	—
221	C₄	—	2	1	no	—
222	C₄	C₅×#815	1	2	yes	—
223	C₄	C₂×C₅×ΓL(1,𝔽₂₅)	1	2	yes	—
224	C₄	C₅×#907	1	2	yes	—
225	C₄	C₅×#948	1	2	yes	—
226	C₃	C₅×#1452	1	1	yes	characteristic, complemented, Sylow
227	C₂	—	2	2	no	—
228	C₂	C₅×#4997	1	1	yes	characteristic
229	C₂	C₂×C₅×#815	1	1	yes	characteristic
230	C₂	C₅×#5122	1	1	yes	characteristic

	subgroup	G/H
center	C₄²×C₅	C₂²×D₃
commutator	C₂²×C₃	C₂×C₄²×C₅
Frattini	C₄²	C₂²×C₅×D₃
Fitting	C₃×C₅×#431	C₂
socle	C₂²×C₃×C₅	C₂×C₄²
prime radical	C₂³×C₃×C₅	C₄²
cosocle	C₃×C₄²	C₂³×C₅
hypercenter	C₅×#431	D₃
hypocenter	C₃	C₅×#1452
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₄²×C₅ ◁ C₅×#431

Complements

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

N	H ≅ G/N	type
4. C₅×#4455	C₂	semidirect
5. C₅×#4454	C₂	semidirect
18. #70742	C₅	direct (N × H)
39. #4455	C₂×C₅	semidirect
40. #4454	C₂×C₅	semidirect
182. C₃×C₅	#1452	semidirect
215. C₅	#70742	direct (N × H)
226. C₃	C₅×#1452	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×#431	2
C₅×#4925	2
C₂×C₄×C₅×C₃⋊C₈	2
C₅×#4455	2
C₅×#4454	2
C₅×#1452	3
#70742	5

Subgroups — C5×#70742 #14280562

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#70742 #14280562