Subgroups — C₅×#69974 #14298397

Order 1920.

544 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#926	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#407	C₂	1	1	yes	maximal, characteristic
3	C₅×#5207	C₂	1	1	yes	maximal, characteristic
4	C₅×#4554	C₂	1	4	yes	maximal, complemented
5	C₅×#1822	—	3	1	no	maximal, Hall
6	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
7	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
8	C₂×C₅×#217	C₄	1	2	yes	—
9	C₅×#926	—	2	2	no	—
10	C₅×#926	—	2	2	no	—
11	C₂×C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
12	C₂×C₄×C₅×Dic₃	C₄	1	2	yes	—
13	C₂×C₅×#208	C₂²	1	2	yes	—
14	C₃×C₅×C₂²⋊C₈	—	2	2	no	—
15	C₅×#830	—	2	2	no	—
16	#69974	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₂×C₅×#117	—	3	1	no	—
18	C₅×#407	—	3	1	no	—
19	C₅×#407	D₃	1	1	yes	characteristic
20	C₅×#343	—	3	4	no	—
21	C₂⁴×C₃×C₅	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₃×C₄×C₅	—	2	4	no	—
23	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
25	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
26	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
27	C₂²×C₅×Dic₃	C₂×C₄	1	4	yes	—
28	C₃×C₅×M₁₆	—	2	4	no	—
29	C₂×C₃×C₅×C₈	—	2	2	no	—
30	C₅×#217	—	2	4	no	—
31	C₅×C₄⋊Dic₃	—	2	2	no	—
32	C₅×C₄⋊Dic₃	—	2	2	no	—
33	C₄×C₅×Dic₃	—	2	4	no	—
34	C₅×#208	—	2	4	no	—
35	C₂×C₅×C₃⋊C₈	—	2	2	no	—
36	C₂×#926	C₂×C₅	1	1	yes	characteristic
37	C₃×#407	C₂×C₅	1	1	yes	characteristic
38	#5207	C₂×C₅	1	1	yes	characteristic
39	#4554	C₂×C₅	1	4	yes	complemented
40	C₂³×C₄×C₅	C₂×D₃	1	1	yes	characteristic
41	C₂×C₅×M₁₆	—	3	2	no	—
42	C₂×C₅×M₁₆	C₂×D₃	1	2	yes	—
43	C₅×#117	—	6	2	no	—
44	C₅×#117	—	6	2	no	—
45	C₂×C₅×C₄⋊C₄	—	3	2	no	—
46	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
47	C₂×C₄²×C₅	—	3	2	no	—
48	C₅×C₂²⋊C₈	—	2	2	no	—
49	C₅×C₂²⋊C₈	—	6	2	no	—
50	#1822	—	3	1	no	Sylow
51	C₂³×C₃×C₅	—	2	2	no	—
52	C₂³×C₃×C₅	—	4	2	no	—
53	C₂³×C₃×C₅	C₂²⋊C₄	1	2	yes	—
54	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₃×C₄×C₅	—	2	2	no	—
56	C₂×C₃×C₄×C₅	—	2	4	no	—
57	C₂×C₃×C₄×C₅	—	2	4	no	—
58	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
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	#41	1	4	yes	—
62	C₂×C₅×Dic₃	—	2	4	no	—
63	C₂×C₅×Dic₃	—	4	4	no	—
64	C₃×C₅×C₈	—	4	2	no	—
65	C₅×C₃⋊C₈	—	4	2	no	—
66	C₂³×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
67	C₂×C₃×M₁₆	C₂²×C₅	1	2	yes	—
68	C₂×#217	C₄×C₅	1	2	yes	—
69	#926	—	2	2	no	—
70	#926	—	2	2	no	—
71	C₂×C₄⋊Dic₃	C₂²×C₅	1	2	yes	—
72	C₂×C₄×Dic₃	C₄×C₅	1	2	yes	—
73	C₂×#208	C₂²×C₅	1	2	yes	—
74	C₃×C₂²⋊C₈	—	2	2	no	—
75	#830	—	2	2	no	—
76	C₂⁴×C₅	C₄×D₃	1	1	yes	characteristic
77	C₂²×C₄×C₅	—	2	4	no	—
78	C₂²×C₄×C₅	—	3	4	no	—
79	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
80	C₂²×C₄×C₅	D₁₂	1	2	yes	—
81	C₂²×C₄×C₅	#67	1	2	yes	—
82	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
83	C₅×M₁₆	—	2	4	no	—
84	C₅×M₁₆	—	6	4	no	—
85	C₂×C₅×C₈	—	2	2	no	—
86	C₂×C₅×C₈	—	6	2	no	—
87	C₅×C₄⋊C₄	—	6	2	no	—
88	C₅×C₄⋊C₄	—	6	2	no	—
89	C₅×C₂²⋊C₄	—	6	4	no	—
90	C₄²×C₅	—	6	4	no	—
91	C₂×#117	—	3	1	no	—
92	#407	—	3	1	no	—
93	#407	C₅×D₃	1	1	yes	characteristic
94	#343	—	3	4	no	—
95	C₂²×C₃×C₅	—	2	2	no	—
96	C₂²×C₃×C₅	—	2	2	no	—
97	C₂²×C₃×C₅	—	2	4	no	—
98	C₂²×C₃×C₅	—	4	4	no	—
99	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂²×C₃×C₅	#117	1	2	yes	—
101	C₃×C₄×C₅	—	2	2	no	—
102	C₃×C₄×C₅	#123	1	2	yes	—
103	C₃×C₄×C₅	#123	1	2	yes	—
104	C₅×Dic₃	—	4	4	no	—
105	C₂⁴×C₃	C₂×C₄×C₅	1	1	yes	characteristic
106	C₂²×C₃×C₄	—	2	4	no	—
107	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
108	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
109	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
110	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
111	C₂²×Dic₃	C₂×C₄×C₅	1	4	yes	—
112	C₃×M₁₆	—	2	4	no	—
113	C₂×C₃×C₈	—	2	2	no	—
114	#217	—	2	4	no	—
115	C₄⋊Dic₃	—	2	2	no	—
116	C₄⋊Dic₃	—	2	2	no	—
117	C₄×Dic₃	—	2	4	no	—
118	#208	—	2	4	no	—
119	C₂×C₃⋊C₈	—	2	2	no	—
120	C₂³×C₅	—	2	2	no	—
121	C₂³×C₅	—	4	2	no	—
122	C₂³×C₅	(C₂×D₃)⋊C₄	1	2	yes	—
123	C₂³×C₅	C₂×C₄×D₃	1	1	yes	characteristic
124	C₂×C₄×C₅	—	2	2	no	—
125	C₂×C₄×C₅	—	2	4	no	—
126	C₂×C₄×C₅	—	2	4	no	—
127	C₂×C₄×C₅	—	6	4	no	—
128	C₂×C₄×C₅	—	12	4	no	—
129	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	2	yes	—
130	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
131	C₂×C₄×C₅	#237	1	4	yes	—
132	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
133	C₅×C₈	—	4	2	no	—
134	C₅×C₈	—	12	2	no	—
135	C₂³×C₄	C₂×C₅×D₃	1	1	yes	characteristic
136	C₂×M₁₆	—	3	2	no	—
137	C₂×M₁₆	C₂×C₅×D₃	1	2	yes	—
138	#117	—	6	2	no	—
139	#117	—	6	2	no	—
140	C₂×C₄⋊C₄	—	3	2	no	—
141	C₂×C₂²⋊C₄	—	3	2	no	—
142	C₂×C₄²	—	3	2	no	—
143	C₂²⋊C₈	—	2	2	no	—
144	C₂²⋊C₈	—	6	2	no	—
145	C₂×C₃×C₅	—	2	2	no	—
146	C₂×C₃×C₅	—	4	2	no	—
147	C₂×C₃×C₅	#381	1	1	yes	characteristic
148	C₂×C₃×C₅	#413	1	2	yes	—
149	C₂³×C₃	—	2	2	no	—
150	C₂³×C₃	—	4	2	no	—
151	C₂³×C₃	C₅×C₂²⋊C₄	1	2	yes	—
152	C₂³×C₃	C₂²×C₄×C₅	1	1	yes	characteristic
153	C₂×C₃×C₄	—	2	2	no	—
154	C₂×C₃×C₄	—	2	4	no	—
155	C₂×C₃×C₄	—	2	4	no	—
156	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
157	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
158	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
159	C₂×C₃×C₄	C₅×#41	1	4	yes	—
160	C₂×Dic₃	—	2	4	no	—
161	C₂×Dic₃	—	4	4	no	—
162	C₃×C₈	—	4	2	no	—
163	C₃⋊C₈	—	4	2	no	—
164	C₂²×C₅	—	2	2	no	—
165	C₂²×C₅	—	2	4	no	—
166	C₂²×C₅	—	2	2	no	—
167	C₂²×C₅	—	4	4	no	—
168	C₂²×C₅	#877	1	2	yes	—
169	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
170	C₄×C₅	—	2	2	no	—
171	C₄×C₅	—	12	4	no	—
172	C₄×C₅	#886	1	2	yes	—
173	C₄×C₅	#935	1	2	yes	—
174	C₂⁴	C₄×C₅×D₃	1	1	yes	characteristic
175	C₂²×C₄	—	2	4	no	—
176	C₂²×C₄	—	3	4	no	—
177	C₂²×C₄	C₄×C₅×D₃	1	1	yes	characteristic
178	C₂²×C₄	C₅×D₁₂	1	2	yes	—
179	C₂²×C₄	C₅×#67	1	2	yes	—
180	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
181	M₁₆	—	2	4	no	—
182	M₁₆	—	6	4	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	6	2	no	—
185	C₄⋊C₄	—	6	2	no	—
186	C₄⋊C₄	—	6	2	no	—
187	C₂²⋊C₄	—	6	4	no	—
188	C₄²	—	6	4	no	—
189	C₃×C₅	#1822	1	1	yes	characteristic, complemented, Hall
190	C₂²×C₃	—	2	2	no	—
191	C₂²×C₃	—	2	2	no	—
192	C₂²×C₃	—	2	4	no	—
193	C₂²×C₃	—	4	4	no	—
194	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
195	C₂²×C₃	C₅×#117	1	2	yes	—
196	C₃×C₄	—	2	2	no	—
197	C₃×C₄	C₅×#123	1	2	yes	—
198	C₃×C₄	C₅×#123	1	2	yes	—
199	Dic₃	—	4	4	no	—
200	C₂×C₅	—	2	2	no	—
201	C₂×C₅	—	4	2	no	—
202	C₂×C₅	#4943	1	1	yes	characteristic
203	C₂×C₅	#5132	1	2	yes	—
204	C₂³	—	2	2	no	—
205	C₂³	—	4	2	no	—
206	C₂³	C₅×(C₂×D₃)⋊C₄	1	2	yes	—
207	C₂³	C₂×C₄×C₅×D₃	1	1	yes	characteristic
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	4	no	—
210	C₂×C₄	—	2	4	no	—
211	C₂×C₄	—	6	4	no	—
212	C₂×C₄	—	12	4	no	—
213	C₂×C₄	C₅×(C₂×D₃)⋊C₄	1	2	yes	—
214	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
215	C₂×C₄	C₅×#237	1	4	yes	—
216	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
217	C₈	—	4	2	no	—
218	C₈	—	12	2	no	—
219	C₂×C₃	—	2	2	no	—
220	C₂×C₃	—	4	2	no	—
221	C₂×C₃	C₅×#381	1	1	yes	characteristic
222	C₂×C₃	C₅×#413	1	2	yes	—
223	C₅	#69974	1	1	yes	characteristic, direct product factor, Sylow
224	C₂²	—	2	2	no	—
225	C₂²	—	2	2	no	—
226	C₂²	—	2	4	no	—
227	C₂²	—	4	4	no	—
228	C₂²	C₅×#877	1	2	yes	—
229	C₂²	C₂×C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
230	C₄	—	2	2	no	—
231	C₄	—	12	4	no	—
232	C₄	C₅×#886	1	2	yes	—
233	C₄	C₅×#935	1	2	yes	—
234	C₃	C₅×#1822	1	1	yes	characteristic, complemented, Sylow
235	C₂	—	2	2	no	—
236	C₂	—	4	2	no	—
237	C₂	C₅×#4943	1	1	yes	characteristic
238	C₂	C₅×#5132	1	2	yes	—

	subgroup	G/H
center	C₂²×C₅	C₂×(C₂×D₃)⋊C₄
commutator	C₂³×C₃	C₂²×C₄×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₃×C₅×#407	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₃×C₅	C₂×C₄
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₅×#407	D₃
hypocenter	C₃	C₅×#1822
LCS	G ▷ C₂³×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₄×C₅ ◁ C₅×#407

Complements

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

N	H ≅ G/N	type
4. C₅×#4554	C₂	semidirect
16. #69974	C₅	direct (N × H)
39. #4554	C₂×C₅	semidirect
189. C₃×C₅	#1822	semidirect
223. C₅	#69974	direct (N × H)
234. C₃	C₅×#1822	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#926	2
C₃×C₅×#407	2
C₅×#5207	2
C₅×#4554	2
C₅×#1822	3
#69974	5

Subgroups — C5×#69974 #14298397

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#69974 #14298397