Subgroups — C₅×#69750 #14294864

Order 1920.

592 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
2	C₄²×C₅×Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₅×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₅×#4550	C₂	1	4	yes	maximal
5	C₅×#1744	—	3	1	no	maximal, Hall
6	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
7	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₅×Dic₃	C₄	1	2	yes	—
9	C₂×C₄×C₅×Dic₃	C₄	1	2	yes	—
10	C₂×C₄×C₅×Dic₃	C₂²	1	2	yes	—
11	C₂²×C₅×C₃⋊C₈	C₂²	1	2	yes	—
12	C₃×C₄×C₅×C₈	C₄	1	4	yes	complemented
13	C₄×C₅×C₃⋊C₈	C₄	1	4	yes	complemented
14	#69750	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
15	C₂×C₄×C₅×C₈	—	3	1	no	—
16	C₂×C₄×C₅×C₈	D₃	1	1	yes	characteristic
17	C₄³×C₅	—	3	1	no	—
18	C₅×#336	—	3	4	no	—
19	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
22	C₂²×C₅×Dic₃	C₂×C₄	1	4	yes	—
23	C₂×C₃×C₅×C₈	—	2	4	no	—
24	C₂×C₃×C₅×C₈	C₂×C₄	1	4	yes	—
25	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
26	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
27	C₄×C₅×Dic₃	—	2	4	no	—
28	C₄×C₅×Dic₃	—	2	4	no	—
29	C₄×C₅×Dic₃	C₈	1	8	yes	complemented
30	C₂×C₅×C₃⋊C₈	—	2	4	no	—
31	C₂×C₅×C₃⋊C₈	C₂×C₄	1	4	yes	—
32	C₂×C₃×C₄×C₈	C₂×C₅	1	1	yes	characteristic
33	C₄²×Dic₃	C₂×C₅	1	1	yes	characteristic
34	C₂×C₄×C₃⋊C₈	C₂×C₅	1	1	yes	characteristic
35	#4550	C₂×C₅	1	4	yes	—
36	C₂²×C₅×C₈	—	3	2	no	—
37	C₂²×C₅×C₈	C₂×D₃	1	2	yes	—
38	C₂×C₄²×C₅	—	3	2	no	—
39	C₂×C₄²×C₅	—	3	2	no	—
40	C₂×C₄²×C₅	—	3	2	no	—
41	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
42	C₄×C₅×C₈	—	3	4	no	—
43	C₄×C₅×C₈	Dic₃	1	4	yes	complemented
44	#1744	—	3	1	no	Sylow
45	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₃×C₄×C₅	C₄²	1	2	yes	—
47	C₂×C₃×C₄×C₅	C₄²	1	2	yes	—
48	C₂×C₃×C₄×C₅	C₂×C₈	1	4	yes	—
49	C₂×C₃×C₄×C₅	M₁₆	1	4	yes	—
50	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₃×C₄×C₅	#41	1	4	yes	—
53	C₂×C₅×Dic₃	—	2	4	no	—
54	C₂×C₅×Dic₃	—	2	4	no	—
55	C₂×C₅×Dic₃	C₂×C₈	1	8	yes	—
56	C₃×C₅×C₈	—	2	4	no	—
57	C₅×C₃⋊C₈	—	2	4	no	—
58	C₂²×C₃×C₈	C₂²×C₅	1	2	yes	—
59	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
60	C₂×C₄×Dic₃	C₄×C₅	1	2	yes	—
61	C₂×C₄×Dic₃	C₄×C₅	1	2	yes	—
62	C₂×C₄×Dic₃	C₂²×C₅	1	2	yes	—
63	C₂²×C₃⋊C₈	C₂²×C₅	1	2	yes	—
64	C₃×C₄×C₈	C₄×C₅	1	4	yes	complemented
65	C₄×C₃⋊C₈	C₄×C₅	1	4	yes	complemented
66	C₂²×C₄×C₅	—	3	4	no	—
67	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
68	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
69	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
70	C₂×C₅×C₈	—	2	4	no	—
71	C₂×C₅×C₈	—	3	4	no	—
72	C₂×C₅×C₈	—	6	4	no	—
73	C₂×C₅×C₈	C₂×Dic₃	1	4	yes	—
74	C₄²×C₅	—	3	8	no	—
75	C₄²×C₅	—	6	4	no	—
76	C₄²×C₅	—	6	4	no	—
77	C₄²×C₅	C₄×D₃	1	2	yes	—
78	C₄²×C₅	C₂×Dic₃	1	2	yes	—
79	C₂×C₄×C₈	—	3	1	no	—
80	C₂×C₄×C₈	C₅×D₃	1	1	yes	characteristic
81	C₄³	—	3	1	no	—
82	#336	—	3	4	no	—
83	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
84	C₂²×C₃×C₅	#117	1	2	yes	—
85	C₂²×C₃×C₅	C₂²×C₈	1	2	yes	—
86	C₂²×C₃×C₅	C₂×M₁₆	1	2	yes	—
87	C₃×C₄×C₅	C₄×C₈	1	4	yes	—
88	C₃×C₄×C₅	#97	1	4	yes	—
89	C₃×C₄×C₅	#117	1	2	yes	—
90	C₃×C₄×C₅	#117	1	2	yes	—
91	C₅×Dic₃	—	2	8	no	—
92	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
93	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
94	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
95	C₂²×Dic₃	C₂×C₄×C₅	1	4	yes	—
96	C₂×C₃×C₈	—	2	4	no	—
97	C₂×C₃×C₈	C₂×C₄×C₅	1	4	yes	—
98	C₃×C₄²	C₂×C₄×C₅	1	2	yes	—
99	C₃×C₄²	C₂×C₄×C₅	1	2	yes	—
100	C₄×Dic₃	—	2	4	no	—
101	C₄×Dic₃	—	2	4	no	—
102	C₄×Dic₃	C₅×C₈	1	8	yes	complemented
103	C₂×C₃⋊C₈	—	2	4	no	—
104	C₂×C₃⋊C₈	C₂×C₄×C₅	1	4	yes	—
105	C₂³×C₅	C₂×C₄×D₃	1	1	yes	characteristic
106	C₂×C₄×C₅	—	3	8	no	—
107	C₂×C₄×C₅	—	6	4	no	—
108	C₂×C₄×C₅	—	6	4	no	—
109	C₂×C₄×C₅	D₃×C₈	1	4	yes	—
110	C₂×C₄×C₅	ΓL(1,𝔽₂₅)	1	4	yes	—
111	C₂×C₄×C₅	C₄×Dic₃	1	2	yes	—
112	C₂×C₄×C₅	C₄×Dic₃	1	2	yes	—
113	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
114	C₂×C₄×C₅	#235	1	4	yes	—
115	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
116	C₅×C₈	—	2	4	no	—
117	C₅×C₈	—	6	4	no	—
118	C₂²×C₈	—	3	2	no	—
119	C₂²×C₈	C₂×C₅×D₃	1	2	yes	—
120	C₂×C₄²	—	3	2	no	—
121	C₂×C₄²	—	3	2	no	—
122	C₂×C₄²	—	3	2	no	—
123	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
124	C₄×C₈	—	3	4	no	—
125	C₄×C₈	C₅×Dic₃	1	4	yes	complemented
126	C₂×C₃×C₅	#376	1	1	yes	characteristic
127	C₂×C₃×C₅	C₂×C₄×C₈	1	1	yes	characteristic
128	C₂×C₃×C₅	C₂×#97	1	1	yes	characteristic
129	C₂×C₃×C₅	#431	1	4	yes	—
130	C₂³×C₃	C₂²×C₄×C₅	1	1	yes	characteristic
131	C₂×C₃×C₄	C₄²×C₅	1	2	yes	—
132	C₂×C₃×C₄	C₄²×C₅	1	2	yes	—
133	C₂×C₃×C₄	C₂×C₅×C₈	1	4	yes	—
134	C₂×C₃×C₄	C₅×M₁₆	1	4	yes	—
135	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
136	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
137	C₂×C₃×C₄	C₅×#41	1	4	yes	—
138	C₂×Dic₃	—	2	4	no	—
139	C₂×Dic₃	—	2	4	no	—
140	C₂×Dic₃	C₂×C₅×C₈	1	8	yes	—
141	C₃×C₈	—	2	4	no	—
142	C₃⋊C₈	—	2	4	no	—
143	C₂²×C₅	#872	1	2	yes	—
144	C₂²×C₅	C₂×D₃×C₈	1	2	yes	—
145	C₂²×C₅	C₂×ΓL(1,𝔽₂₅)	1	2	yes	—
146	C₂²×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
147	C₄×C₅	—	6	8	no	—
148	C₄×C₅	C₈×Dic₃	1	4	yes	—
149	C₄×C₅	#815	1	4	yes	—
150	C₄×C₅	#872	1	2	yes	—
151	C₄×C₅	#926	1	2	yes	—
152	C₂²×C₄	—	3	4	no	—
153	C₂²×C₄	C₄×C₅×D₃	1	1	yes	characteristic
154	C₂²×C₄	C₄×C₅×D₃	1	1	yes	characteristic
155	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
156	C₂×C₈	—	2	4	no	—
157	C₂×C₈	—	3	4	no	—
158	C₂×C₈	—	6	4	no	—
159	C₂×C₈	C₂×C₅×Dic₃	1	4	yes	—
160	C₄²	—	3	8	no	—
161	C₄²	—	6	4	no	—
162	C₄²	—	6	4	no	—
163	C₄²	C₄×C₅×D₃	1	2	yes	—
164	C₄²	C₂×C₅×Dic₃	1	2	yes	—
165	C₃×C₅	#1744	1	1	yes	characteristic, complemented, Hall
166	C₂²×C₃	C₂×C₄²×C₅	1	1	yes	characteristic
167	C₂²×C₃	C₅×#117	1	2	yes	—
168	C₂²×C₃	C₂²×C₅×C₈	1	2	yes	—
169	C₂²×C₃	C₂×C₅×M₁₆	1	2	yes	—
170	C₃×C₄	C₄×C₅×C₈	1	4	yes	—
171	C₃×C₄	C₅×#97	1	4	yes	—
172	C₃×C₄	C₅×#117	1	2	yes	—
173	C₃×C₄	C₅×#117	1	2	yes	—
174	Dic₃	—	2	8	no	—
175	C₂×C₅	#4685	1	4	yes	—
176	C₂×C₅	#4932	1	1	yes	characteristic
177	C₂×C₅	C₂×C₈×Dic₃	1	1	yes	characteristic
178	C₂×C₅	C₂×#815	1	1	yes	characteristic
179	C₂³	C₂×C₄×C₅×D₃	1	1	yes	characteristic
180	C₂×C₄	—	3	8	no	—
181	C₂×C₄	—	6	4	no	—
182	C₂×C₄	—	6	4	no	—
183	C₂×C₄	C₅×D₃×C₈	1	4	yes	—
184	C₂×C₄	C₅×ΓL(1,𝔽₂₅)	1	4	yes	—
185	C₂×C₄	C₄×C₅×Dic₃	1	2	yes	—
186	C₂×C₄	C₄×C₅×Dic₃	1	2	yes	—
187	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
188	C₂×C₄	C₅×#235	1	4	yes	—
189	C₂×C₄	C₂²×C₅×Dic₃	1	1	yes	characteristic
190	C₈	—	2	4	no	—
191	C₈	—	6	4	no	—
192	C₂×C₃	C₅×#376	1	1	yes	characteristic
193	C₂×C₃	C₂×C₄×C₅×C₈	1	1	yes	characteristic
194	C₂×C₃	C₂×C₅×#97	1	1	yes	characteristic
195	C₂×C₃	C₅×#431	1	4	yes	—
196	C₅	#69750	1	1	yes	characteristic, direct product factor, Sylow
197	C₂²	C₅×#872	1	2	yes	—
198	C₂²	C₂×C₅×D₃×C₈	1	2	yes	—
199	C₂²	C₂×C₅×ΓL(1,𝔽₂₅)	1	2	yes	—
200	C₂²	C₂×C₄×C₅×Dic₃	1	1	yes	characteristic
201	C₄	—	6	8	no	—
202	C₄	C₅×C₈×Dic₃	1	4	yes	—
203	C₄	C₅×#815	1	4	yes	—
204	C₄	C₅×#872	1	2	yes	—
205	C₄	C₅×#926	1	2	yes	—
206	C₃	C₅×#1744	1	1	yes	characteristic, complemented, Sylow
207	C₂	C₅×#4685	1	4	yes	—
208	C₂	C₅×#4932	1	1	yes	characteristic
209	C₂	C₂×C₅×C₈×Dic₃	1	1	yes	characteristic
210	C₂	C₂×C₅×#815	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²×C₅	C₂×D₃
commutator	C₂×C₃	C₂×C₄×C₅×C₈
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₂×C₃×C₄×C₅×C₈	C₂
socle, prime radical	C₂³×C₃×C₅	C₂²×C₄
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₂×C₄×C₅×C₈	D₃
hypocenter	C₃	C₅×#1744
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×C₈

Complements

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

N	H ≅ G/N	type
12. C₃×C₄×C₅×C₈	C₄	semidirect
13. C₄×C₅×C₃⋊C₈	C₄	semidirect
14. #69750	C₅	direct (N × H)
29. C₄×C₅×Dic₃	C₈	semidirect
43. C₄×C₅×C₈	Dic₃	semidirect
64. C₃×C₄×C₈	C₄×C₅	semidirect
65. C₄×C₃⋊C₈	C₄×C₅	semidirect
102. C₄×Dic₃	C₅×C₈	semidirect
125. C₄×C₈	C₅×Dic₃	semidirect
165. C₃×C₅	#1744	semidirect
196. C₅	#69750	direct (N × H)
206. C₃	C₅×#1744	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄×C₅×C₈	2
C₄²×C₅×Dic₃	2
C₂×C₄×C₅×C₃⋊C₈	2
C₅×#4550	2
C₅×#1744	3
#69750	5

Subgroups — C5×#69750 #14294864

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#69750 #14294864