Subgroups — C₅×#74319 #14322242

Order 1920.

1848 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#893	C₂	1	6	yes	maximal, complemented
2	C₂×C₅×#874	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#384	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#4988	C₂	1	6	yes	maximal, complemented
5	C₄²×C₅×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×C₅×#127	—	3	1	no	maximal, Hall
7	C₂²×C₅×D₁₂	C₂²	1	6	yes	—
8	C₂²×C₄×C₅×D₃	C₂²	1	4	yes	—
9	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	6	yes	—
10	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
11	C₂×C₅×(C₂×D₃)⋊C₄	C₂²	1	12	yes	—
12	C₂×C₄×C₅×Dic₃	C₂²	1	3	yes	—
13	C₂×C₄×C₅×Dic₃	C₂²	1	3	yes	—
14	C₅×#893	—	2	24	no	—
15	C₅×#874	C₄	1	8	yes	complemented
16	#74319	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₂×C₅×#127	—	3	1	no	—
18	C₂×C₄×C₅×D₄	—	3	6	no	—
19	C₅×#390	—	3	6	no	—
20	C₅×#384	D₃	1	1	yes	characteristic, complemented
21	C₄³×C₅	—	3	1	no	—
22	C₂³×C₅×D₃	C₂³	1	4	yes	—
23	C₂²×C₃×C₄×C₅	C₂³	1	4	yes	—
24	C₂²×C₃×C₄×C₅	C₂³	1	3	yes	—
25	C₂²×C₅×Dic₃	C₂³	1	3	yes	—
26	C₂²×C₅×Dic₃	C₂³	1	1	yes	characteristic
27	C₂×C₅×D₁₂	—	2	12	no	—
28	C₂×C₅×D₁₂	—	2	12	no	—
29	C₂×C₅×D₁₂	C₂×C₄	1	24	yes	complemented
30	C₂×C₄×C₅×D₃	—	2	24	no	—
31	C₃×C₅×C₄⋊C₄	—	2	12	no	—
32	C₃×C₄²×C₅	C₂×C₄	1	4	yes	complemented
33	C₅×(C₂×D₃)⋊C₄	—	2	24	no	—
34	C₄×C₅×Dic₃	—	2	6	no	—
35	C₄×C₅×Dic₃	D₄	1	12	yes	complemented
36	C₂×#893	C₂×C₅	1	6	yes	complemented
37	C₂×#874	C₂×C₅	1	1	yes	characteristic
38	C₃×#384	C₂×C₅	1	1	yes	characteristic, complemented
39	#4988	C₂×C₅	1	6	yes	complemented
40	C₄²×Dic₃	C₂×C₅	1	1	yes	characteristic, complemented
41	C₂²×C₅×D₄	—	3	6	no	—
42	C₂³×C₄×C₅	—	3	4	no	—
43	C₅×#127	—	3	8	no	—
44	C₄×C₅×D₄	—	6	24	no	—
45	C₂×C₅×C₄⋊C₄	C₂×D₃	1	6	yes	—
46	C₂×C₅×C₂²⋊C₄	—	3	12	no	—
47	C₂×C₄²×C₅	—	3	3	no	—
48	C₂×C₄²×C₅	—	3	3	no	—
49	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
50	C₄×#127	—	3	1	no	Sylow
51	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
52	C₂²×C₅×D₃	—	2	12	no	—
53	C₂²×C₅×D₃	—	2	12	no	—
54	C₂²×C₅×D₃	C₂²×C₄	1	8	yes	—
55	C₂×C₃×C₄×C₅	—	2	12	no	—
56	C₂×C₃×C₄×C₅	C₂²×C₄	1	6	yes	—
57	C₂×C₃×C₄×C₅	C₂×D₄	1	6	yes	—
58	C₂×C₃×C₄×C₅	#41	1	6	yes	—
59	C₂×C₅×Dic₃	—	2	6	no	—
60	C₂×C₅×Dic₃	C₂×D₄	1	6	yes	—
61	C₂×C₅×Dic₃	C₂×D₄	1	6	yes	—
62	C₅×D₁₂	—	2	48	no	—
63	C₄×C₅×D₃	—	4	16	no	—
64	C₂²×D₁₂	C₂²×C₅	1	6	yes	—
65	C₂²×C₄×D₃	C₂²×C₅	1	4	yes	—
66	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	6	yes	—
67	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
68	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	12	yes	—
69	C₂×C₄×Dic₃	C₂²×C₅	1	3	yes	—
70	C₂×C₄×Dic₃	C₂²×C₅	1	3	yes	—
71	#893	—	2	24	no	—
72	#874	C₄×C₅	1	8	yes	complemented
73	C₂⁴×C₅	—	3	4	no	—
74	C₂×C₅×D₄	—	3	24	no	—
75	C₂×C₅×D₄	—	6	12	no	—
76	C₂×C₅×D₄	—	6	12	no	—
77	C₂²×C₄×C₅	—	3	1	no	—
78	C₂²×C₄×C₅	—	3	3	no	—
79	C₂²×C₄×C₅	—	6	24	no	—
80	C₂²×C₄×C₅	C₂²×D₃	1	4	yes	—
81	C₂²×C₄×C₅	C₂²×D₃	1	3	yes	—
82	C₅×C₄⋊C₄	—	2	12	no	—
83	C₅×C₂²⋊C₄	—	6	24	no	—
84	C₄²×C₅	—	3	12	no	—
85	C₄²×C₅	—	6	6	no	—
86	C₄²×C₅	C₄×D₃	1	4	yes	complemented
87	C₂×#127	—	3	1	no	—
88	C₂×C₄×D₄	—	3	6	no	—
89	#390	—	3	6	no	—
90	#384	C₅×D₃	1	1	yes	characteristic, complemented
91	C₄³	—	3	1	no	—
92	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
93	C₂²×C₃×C₅	C₂²×D₄	1	3	yes	—
94	C₂²×C₃×C₅	C₂×#41	1	3	yes	—
95	C₂×C₅×D₃	—	2	24	no	—
96	C₂×C₅×D₃	—	4	4	no	—
97	C₂×C₅×D₃	—	4	12	no	—
98	C₃×C₄×C₅	—	4	4	no	—
99	C₃×C₄×C₅	C₄×D₄	1	12	yes	complemented
100	C₅×Dic₃	—	2	6	no	—
101	C₅×Dic₃	#127	1	4	yes	complemented
102	C₂³×D₃	C₂³×C₅	1	4	yes	—
103	C₂²×C₃×C₄	C₂³×C₅	1	4	yes	—
104	C₂²×C₃×C₄	C₂³×C₅	1	3	yes	—
105	C₂²×Dic₃	C₂³×C₅	1	3	yes	—
106	C₂²×Dic₃	C₂³×C₅	1	1	yes	characteristic
107	C₂×D₁₂	—	2	12	no	—
108	C₂×D₁₂	—	2	12	no	—
109	C₂×D₁₂	C₂×C₄×C₅	1	24	yes	complemented
110	C₂×C₄×D₃	—	2	24	no	—
111	C₃×C₄⋊C₄	—	2	12	no	—
112	C₃×C₄²	C₂×C₄×C₅	1	4	yes	complemented
113	(C₂×D₃)⋊C₄	—	2	24	no	—
114	C₄×Dic₃	—	2	6	no	—
115	C₄×Dic₃	C₅×D₄	1	12	yes	complemented
116	C₂³×C₅	—	3	8	no	—
117	C₂³×C₅	—	6	12	no	—
118	C₂³×C₅	—	6	12	no	—
119	C₂³×C₅	C₂³×D₃	1	1	yes	characteristic
120	C₅×D₄	—	6	48	no	—
121	C₂×C₄×C₅	—	2	12	no	—
122	C₂×C₄×C₅	—	3	6	no	—
123	C₂×C₄×C₅	—	3	6	no	—
124	C₂×C₄×C₅	—	6	6	no	—
125	C₂×C₄×C₅	—	12	16	no	—
126	C₂×C₄×C₅	C₂×C₄×D₃	1	6	yes	—
127	C₂×C₄×C₅	D₃×D₄	1	6	yes	—
128	C₂×C₄×C₅	#239	1	6	yes	—
129	C₂²×D₄	—	3	6	no	—
130	C₂³×C₄	—	3	4	no	—
131	#127	—	3	8	no	—
132	C₄×D₄	—	6	24	no	—
133	C₂×C₄⋊C₄	C₂×C₅×D₃	1	6	yes	—
134	C₂×C₂²⋊C₄	—	3	12	no	—
135	C₂×C₄²	—	3	3	no	—
136	C₂×C₄²	—	3	3	no	—
137	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
138	C₂×C₃×C₅	C₂×C₄×D₄	1	3	yes	—
139	C₂×C₃×C₅	C₂×#127	1	1	yes	characteristic
140	C₂×C₃×C₅	#532	1	3	yes	—
141	C₅×D₃	—	4	8	no	—
142	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
143	C₂²×D₃	—	2	12	no	—
144	C₂²×D₃	—	2	12	no	—
145	C₂²×D₃	C₂²×C₄×C₅	1	8	yes	—
146	C₂×C₃×C₄	—	2	12	no	—
147	C₂×C₃×C₄	C₂²×C₄×C₅	1	6	yes	—
148	C₂×C₃×C₄	C₂×C₅×D₄	1	6	yes	—
149	C₂×C₃×C₄	C₅×#41	1	6	yes	—
150	C₂×Dic₃	—	2	6	no	—
151	C₂×Dic₃	C₂×C₅×D₄	1	6	yes	—
152	C₂×Dic₃	C₂×C₅×D₄	1	6	yes	—
153	D₁₂	—	2	48	no	—
154	C₄×D₃	—	4	16	no	—
155	C₂²×C₅	—	6	24	no	—
156	C₂²×C₅	—	12	4	no	—
157	C₂²×C₅	—	12	12	no	—
158	C₂²×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
159	C₂²×C₅	C₂×D₃×D₄	1	3	yes	—
160	C₂²×C₅	C₂×#239	1	3	yes	—
161	C₄×C₅	—	3	4	no	—
162	C₄×C₅	—	4	4	no	—
163	C₄×C₅	—	6	6	no	—
164	C₄×C₅	#893	1	12	yes	complemented
165	C₂⁴	—	3	4	no	—
166	C₂×D₄	—	3	24	no	—
167	C₂×D₄	—	6	12	no	—
168	C₂×D₄	—	6	12	no	—
169	C₂²×C₄	—	3	1	no	—
170	C₂²×C₄	—	3	3	no	—
171	C₂²×C₄	—	6	24	no	—
172	C₂²×C₄	C₂²×C₅×D₃	1	4	yes	—
173	C₂²×C₄	C₂²×C₅×D₃	1	3	yes	—
174	C₄⋊C₄	—	2	12	no	—
175	C₂²⋊C₄	—	6	24	no	—
176	C₄²	—	3	12	no	—
177	C₄²	—	6	6	no	—
178	C₄²	C₄×C₅×D₃	1	4	yes	complemented
179	C₃×C₅	C₄×#127	1	1	yes	characteristic, complemented, Hall
180	C₂²×C₃	C₂³×C₄×C₅	1	1	yes	characteristic
181	C₂²×C₃	C₂²×C₅×D₄	1	3	yes	—
182	C₂²×C₃	C₂×C₅×#41	1	3	yes	—
183	C₂×D₃	—	2	24	no	—
184	C₂×D₃	—	4	4	no	—
185	C₂×D₃	—	4	12	no	—
186	C₃×C₄	—	4	4	no	—
187	C₃×C₄	C₄×C₅×D₄	1	12	yes	complemented
188	Dic₃	—	2	6	no	—
189	Dic₃	C₅×#127	1	4	yes	complemented
190	C₂×C₅	—	12	8	no	—
191	C₂×C₅	C₂×#893	1	3	yes	—
192	C₂×C₅	D₃×#127	1	1	yes	characteristic
193	C₂×C₅	#5725	1	3	yes	—
194	C₂³	—	3	8	no	—
195	C₂³	—	6	12	no	—
196	C₂³	—	6	12	no	—
197	C₂³	C₂³×C₅×D₃	1	1	yes	characteristic
198	D₄	—	6	48	no	—
199	C₂×C₄	—	2	12	no	—
200	C₂×C₄	—	3	6	no	—
201	C₂×C₄	—	3	6	no	—
202	C₂×C₄	—	6	6	no	—
203	C₂×C₄	—	12	16	no	—
204	C₂×C₄	C₂×C₄×C₅×D₃	1	6	yes	—
205	C₂×C₄	C₅×D₃×D₄	1	6	yes	—
206	C₂×C₄	C₅×#239	1	6	yes	—
207	C₂×C₃	C₂×C₄×C₅×D₄	1	3	yes	—
208	C₂×C₃	C₂×C₅×#127	1	1	yes	characteristic
209	C₂×C₃	C₅×#532	1	3	yes	—
210	D₃	—	4	8	no	—
211	C₅	#74319	1	1	yes	characteristic, direct product factor, Sylow
212	C₂²	—	6	24	no	—
213	C₂²	—	12	4	no	—
214	C₂²	—	12	12	no	—
215	C₂²	C₂²×C₄×C₅×D₃	1	1	yes	characteristic
216	C₂²	C₂×C₅×D₃×D₄	1	3	yes	—
217	C₂²	C₂×C₅×#239	1	3	yes	—
218	C₄	—	3	4	no	—
219	C₄	—	4	4	no	—
220	C₄	—	6	6	no	—
221	C₄	C₅×#893	1	12	yes	complemented
222	C₃	C₄×C₅×#127	1	1	yes	characteristic, complemented, Sylow
223	C₂	—	12	8	no	—
224	C₂	C₂×C₅×#893	1	3	yes	—
225	C₂	C₅×D₃×#127	1	1	yes	characteristic
226	C₂	C₅×#5725	1	3	yes	—

	subgroup	G/H
center	C₂³×C₅	C₂³×D₃
commutator	C₂²×C₃	C₂³×C₄×C₅
Frattini	C₂³	C₂³×C₅×D₃
Fitting	C₃×C₅×#384	C₂
socle	C₂³×C₃×C₅	C₂⁴
prime radical	C₂×C₅×#874	C₂
cosocle	C₂³×C₃	C₂⁴×C₅
hypercenter	C₅×#384	D₃
hypocenter	C₃	C₄×C₅×#127
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₅ ◁ C₅×#384

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×#893	C₂	semidirect
3. C₃×C₅×#384	C₂	semidirect
4. C₅×#4988	C₂	semidirect
5. C₄²×C₅×Dic₃	C₂	semidirect
15. C₅×#874	C₄	semidirect
16. #74319	C₅	direct (N × H)
20. C₅×#384	D₃	semidirect
29. C₂×C₅×D₁₂	C₂×C₄	semidirect
32. C₃×C₄²×C₅	C₂×C₄	semidirect
35. C₄×C₅×Dic₃	D₄	semidirect
36. C₂×#893	C₂×C₅	semidirect
38. C₃×#384	C₂×C₅	semidirect
39. #4988	C₂×C₅	semidirect
40. C₄²×Dic₃	C₂×C₅	semidirect
72. #874	C₄×C₅	semidirect
86. C₄²×C₅	C₄×D₃	semidirect
90. #384	C₅×D₃	semidirect
99. C₃×C₄×C₅	C₄×D₄	semidirect
101. C₅×Dic₃	#127	semidirect
109. C₂×D₁₂	C₂×C₄×C₅	semidirect
112. C₃×C₄²	C₂×C₄×C₅	semidirect
115. C₄×Dic₃	C₅×D₄	semidirect
164. C₄×C₅	#893	semidirect
178. C₄²	C₄×C₅×D₃	semidirect
179. C₃×C₅	C₄×#127	semidirect
187. C₃×C₄	C₄×C₅×D₄	semidirect
189. Dic₃	C₅×#127	semidirect
211. C₅	#74319	direct (N × H)
221. C₄	C₅×#893	semidirect
222. C₃	C₄×C₅×#127	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#893	2
C₂×C₅×#874	2
C₃×C₅×#384	2
C₅×#4988	2
C₄²×C₅×Dic₃	2
C₄×C₅×#127	3
#74319	5

Subgroups — C5×#74319 #14322242

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#74319 #14322242