Subgroups — C₅×#69072 #14312637

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	C₂×C₅×#798	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×D₃×C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₅×#4689	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#346	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#4502	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×Dic₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₅×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#2108	—	3	1	no	maximal, Hall
9	C₂×C₅×ΓL(1,𝔽₂₅)	C₄	1	1	yes	characteristic
10	C₂×C₅×D₃×C₈	C₂²	1	1	yes	characteristic
11	C₅×#872	C₄	1	1	yes	characteristic
12	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic
14	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₅×#815	C₄	1	1	yes	characteristic
16	C₅×C₈×Dic₃	C₂²	1	1	yes	characteristic
17	C₂×C₅×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
18	C₂×C₅×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
19	C₅×#803	C₄	1	1	yes	characteristic
20	C₅×#798	C₄	1	4	yes	complemented
21	C₅×D₃×C₁₆	C₄	1	4	yes	complemented
22	#69072	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
23	C₂×C₅×M₃₂	—	3	1	no	—
24	C₂²×C₅×C₁₆	—	3	1	no	—
25	C₅×#405	—	3	1	no	—
26	C₅×#346	—	3	1	no	—
27	C₅×#346	D₃	1	1	yes	characteristic, complemented
28	C₄×C₅×C₁₆	—	3	1	no	—
29	C₄×C₅×C₁₆	—	3	1	no	—
30	C₂×C₄×C₅×D₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
33	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
34	C₅×(C₂×D₃)⋊C₄	C₈	1	2	yes	—
35	C₅×Dic₃⋊C₄	C₈	1	2	yes	—
36	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
38	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
39	C₅×ΓL(1,𝔽₂₅)	C₈	1	4	yes	—
40	C₅×D₃×C₈	C₂×C₄	1	2	yes	—
41	C₅×D₃×C₈	C₂×C₄	1	2	yes	—
42	C₃×C₅×C₁₆	C₂×C₄	1	2	yes	—
43	C₃×C₅×C₁₆	C₂×C₄	1	2	yes	—
44	C₅×C₃⋊C₁₆	C₂×C₄	1	2	yes	—
45	C₅×C₃⋊C₁₆	C₂×C₄	1	2	yes	—
46	C₂×#798	C₂×C₅	1	1	yes	characteristic
47	C₂×D₃×C₁₆	C₂×C₅	1	1	yes	characteristic
48	#4689	C₂×C₅	1	1	yes	characteristic
49	C₃×#346	C₂×C₅	1	1	yes	characteristic, complemented
50	#4502	C₂×C₅	1	1	yes	characteristic, complemented
51	Dic₃×C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
52	C₄×C₃⋊C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
53	C₂×C₅×M₁₆	—	3	1	no	—
54	C₂²×C₅×C₈	—	3	1	no	—
55	C₅×#117	—	3	1	no	—
56	C₅×M₃₂	—	3	4	no	—
57	C₂×C₅×C₁₆	—	3	1	no	—
58	C₂×C₅×C₁₆	—	3	1	no	—
59	C₂×C₅×C₁₆	—	3	4	no	—
60	C₂×C₅×C₁₆	C₂×D₃	1	1	yes	characteristic
61	C₂×C₅×C₁₆	C₂×D₃	1	1	yes	characteristic
62	C₅×#97	—	3	1	no	—
63	C₅×#97	—	3	1	no	—
64	C₄×C₅×C₈	—	3	1	no	—
65	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
66	#2108	—	3	1	no	Sylow
67	C₂²×C₅×D₃	C₂×C₈	1	1	yes	characteristic
68	C₂×C₃×C₄×C₅	C₂×C₈	1	2	yes	—
69	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₅×Dic₃	C₂×C₈	1	2	yes	—
71	C₂×C₅×Dic₃	C₂×C₈	1	1	yes	characteristic
72	C₄×C₅×D₃	C₄²	1	2	yes	—
73	C₄×C₅×D₃	C₂×C₈	1	2	yes	—
74	C₃×C₅×C₈	C₂×C₈	1	2	yes	—
75	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
76	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
77	C₅×C₃⋊C₈	C₄²	1	2	yes	—
78	C₅×C₃⋊C₈	C₂×C₈	1	2	yes	—
79	C₂×ΓL(1,𝔽₂₅)	C₄×C₅	1	1	yes	characteristic
80	C₂×D₃×C₈	C₂²×C₅	1	1	yes	characteristic
81	#872	C₄×C₅	1	1	yes	characteristic
82	C₂×C₃×C₁₆	C₂²×C₅	1	1	yes	characteristic
83	C₂×C₃×C₁₆	C₂²×C₅	1	1	yes	characteristic
84	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
85	#815	C₄×C₅	1	1	yes	characteristic
86	C₈×Dic₃	C₂²×C₅	1	1	yes	characteristic
87	C₂×C₃⋊C₁₆	C₂²×C₅	1	1	yes	characteristic
88	C₂×C₃⋊C₁₆	C₂²×C₅	1	1	yes	characteristic
89	#803	C₄×C₅	1	1	yes	characteristic
90	#798	C₄×C₅	1	4	yes	complemented
91	D₃×C₁₆	C₄×C₅	1	4	yes	complemented
92	C₂²×C₄×C₅	—	3	1	no	—
93	C₅×M₁₆	—	3	4	no	—
94	C₂×C₅×C₈	—	3	1	no	—
95	C₂×C₅×C₈	—	3	1	no	—
96	C₂×C₅×C₈	—	3	2	no	—
97	C₂×C₅×C₈	—	3	2	no	—
98	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
99	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
100	C₅×C₄⋊C₄	—	3	2	no	—
101	C₅×C₂²⋊C₄	—	3	2	no	—
102	C₄²×C₅	—	3	1	no	—
103	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
104	C₅×C₁₆	—	3	2	no	—
105	C₅×C₁₆	—	3	2	no	—
106	C₅×C₁₆	C₄×D₃	1	2	yes	—
107	C₅×C₁₆	C₄×D₃	1	2	yes	—
108	C₂×M₃₂	—	3	1	no	—
109	C₂²×C₁₆	—	3	1	no	—
110	#405	—	3	1	no	—
111	#346	—	3	1	no	—
112	#346	C₅×D₃	1	1	yes	characteristic, complemented
113	C₄×C₁₆	—	3	1	no	—
114	C₄×C₁₆	—	3	1	no	—
115	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
116	C₂×C₅×D₃	—	2	2	no	—
117	C₂×C₅×D₃	C₄×C₈	1	2	yes	—
118	C₃×C₄×C₅	—	2	2	no	—
119	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
120	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
121	C₅×Dic₃	—	2	2	no	—
122	C₅×Dic₃	C₄×C₈	1	2	yes	—
123	C₂×C₄×D₃	C₂×C₄×C₅	1	1	yes	characteristic
124	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
125	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
126	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
127	(C₂×D₃)⋊C₄	C₅×C₈	1	2	yes	—
128	Dic₃⋊C₄	C₅×C₈	1	2	yes	—
129	C₄×Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
130	C₂×C₃⋊C₈	C₂×C₄×C₅	1	1	yes	characteristic
131	C₂×C₃⋊C₈	C₂×C₄×C₅	1	1	yes	characteristic
132	ΓL(1,𝔽₂₅)	C₅×C₈	1	4	yes	—
133	D₃×C₈	C₂×C₄×C₅	1	2	yes	—
134	D₃×C₈	C₂×C₄×C₅	1	2	yes	—
135	C₃×C₁₆	C₂×C₄×C₅	1	2	yes	—
136	C₃×C₁₆	C₂×C₄×C₅	1	2	yes	—
137	C₃⋊C₁₆	C₂×C₄×C₅	1	2	yes	—
138	C₃⋊C₁₆	C₂×C₄×C₅	1	2	yes	—
139	C₂³×C₅	—	3	1	no	—
140	C₂×C₄×C₅	—	3	1	no	—
141	C₂×C₄×C₅	—	3	2	no	—
142	C₂×C₄×C₅	—	3	2	no	—
143	C₂×C₄×C₅	—	3	2	no	—
144	C₂×C₄×C₅	D₃×C₈	1	2	yes	—
145	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
146	C₅×C₈	—	3	2	no	—
147	C₅×C₈	—	3	2	no	—
148	C₅×C₈	D₃×C₈	1	2	yes	—
149	C₅×C₈	C₂×C₄×D₃	1	1	yes	characteristic
150	C₅×C₈	C₂×C₄×D₃	1	1	yes	characteristic
151	C₂×M₁₆	—	3	1	no	—
152	C₂²×C₈	—	3	1	no	—
153	#117	—	3	1	no	—
154	M₃₂	—	3	4	no	—
155	C₂×C₁₆	—	3	1	no	—
156	C₂×C₁₆	—	3	1	no	—
157	C₂×C₁₆	—	3	4	no	—
158	C₂×C₁₆	C₂×C₅×D₃	1	1	yes	characteristic
159	C₂×C₁₆	C₂×C₅×D₃	1	1	yes	characteristic
160	#97	—	3	1	no	—
161	#97	—	3	1	no	—
162	C₄×C₈	—	3	1	no	—
163	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
164	C₂×C₃×C₅	C₂×C₄×C₈	1	1	yes	characteristic
165	C₂×C₃×C₅	#504	1	2	yes	—
166	C₅×D₃	—	2	2	no	—
167	C₂²×D₃	C₂×C₅×C₈	1	1	yes	characteristic
168	C₂×C₃×C₄	C₂×C₅×C₈	1	2	yes	—
169	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
170	C₂×Dic₃	C₂×C₅×C₈	1	2	yes	—
171	C₂×Dic₃	C₂×C₅×C₈	1	1	yes	characteristic
172	C₄×D₃	C₄²×C₅	1	2	yes	—
173	C₄×D₃	C₂×C₅×C₈	1	2	yes	—
174	C₃×C₈	C₂×C₅×C₈	1	2	yes	—
175	C₃×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
176	C₃×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
177	C₃⋊C₈	C₄²×C₅	1	2	yes	—
178	C₃⋊C₈	C₂×C₅×C₈	1	2	yes	—
179	C₂²×C₅	—	3	2	no	—
180	C₂²×C₅	—	6	2	no	—
181	C₂²×C₅	C₂×D₃×C₈	1	1	yes	characteristic
182	C₄×C₅	—	2	2	no	—
183	C₄×C₅	—	3	2	no	—
184	C₄×C₅	—	6	2	no	—
185	C₄×C₅	C₄²×D₃	1	1	yes	characteristic
186	C₄×C₅	C₂×D₃×C₈	1	1	yes	characteristic
187	C₂²×C₄	—	3	1	no	—
188	M₁₆	—	3	4	no	—
189	C₂×C₈	—	3	1	no	—
190	C₂×C₈	—	3	1	no	—
191	C₂×C₈	—	3	2	no	—
192	C₂×C₈	—	3	2	no	—
193	C₂×C₈	C₄×C₅×D₃	1	1	yes	characteristic
194	C₂×C₈	C₂²×C₅×D₃	1	1	yes	characteristic
195	C₄⋊C₄	—	3	2	no	—
196	C₂²⋊C₄	—	3	2	no	—
197	C₄²	—	3	1	no	—
198	C₄²	C₄×C₅×D₃	1	1	yes	characteristic
199	C₁₆	—	3	2	no	—
200	C₁₆	—	3	2	no	—
201	C₁₆	C₄×C₅×D₃	1	2	yes	—
202	C₁₆	C₄×C₅×D₃	1	2	yes	—
203	C₃×C₅	#2108	1	1	yes	characteristic, complemented, Hall
204	C₂²×C₃	C₂²×C₅×C₈	1	1	yes	characteristic
205	C₂×D₃	—	2	2	no	—
206	C₂×D₃	C₄×C₅×C₈	1	2	yes	—
207	C₃×C₄	—	2	2	no	—
208	C₃×C₄	C₂×C₄²×C₅	1	1	yes	characteristic
209	C₃×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
210	Dic₃	—	2	2	no	—
211	Dic₃	C₄×C₅×C₈	1	2	yes	—
212	C₂×C₅	—	6	2	no	—
213	C₂×C₅	C₄×D₃×C₈	1	1	yes	characteristic
214	C₂×C₅	#4907	1	2	yes	—
215	C₂³	—	3	1	no	—
216	C₂×C₄	—	3	1	no	—
217	C₂×C₄	—	3	2	no	—
218	C₂×C₄	—	3	2	no	—
219	C₂×C₄	—	3	2	no	—
220	C₂×C₄	C₅×D₃×C₈	1	2	yes	—
221	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
222	C₈	—	3	2	no	—
223	C₈	—	3	2	no	—
224	C₈	C₅×D₃×C₈	1	2	yes	—
225	C₈	C₂×C₄×C₅×D₃	1	1	yes	characteristic
226	C₈	C₂×C₄×C₅×D₃	1	1	yes	characteristic
227	C₂×C₃	C₂×C₄×C₅×C₈	1	1	yes	characteristic
228	C₂×C₃	C₅×#504	1	2	yes	—
229	D₃	—	2	2	no	—
230	C₅	#69072	1	1	yes	characteristic, direct product factor, Sylow
231	C₂²	—	3	2	no	—
232	C₂²	—	6	2	no	—
233	C₂²	C₂×C₅×D₃×C₈	1	1	yes	characteristic
234	C₄	—	2	2	no	—
235	C₄	—	3	2	no	—
236	C₄	—	6	2	no	—
237	C₄	C₄²×C₅×D₃	1	1	yes	characteristic
238	C₄	C₂×C₅×D₃×C₈	1	1	yes	characteristic
239	C₃	C₅×#2108	1	1	yes	characteristic, complemented, Sylow
240	C₂	—	6	2	no	—
241	C₂	C₄×C₅×D₃×C₈	1	1	yes	characteristic
242	C₂	C₅×#4907	1	2	yes	—

	subgroup	G/H
center	C₂×C₅×C₈	C₂²×D₃
commutator	C₂×C₃	C₂×C₄×C₅×C₈
Frattini	C₂×C₈	C₂²×C₅×D₃
Fitting	C₃×C₅×#346	C₂
socle	C₂²×C₃×C₅	C₂²×C₈
prime radical	C₂²×C₅×D₃	C₂×C₈
cosocle	C₂×C₃×C₈	C₂³×C₅
hypercenter	C₅×#346	D₃
hypocenter	C₃	C₅×#2108
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅×C₈ ◁ C₅×#346

Complements

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

N	H ≅ G/N	type
4. C₃×C₅×#346	C₂	semidirect
5. C₅×#4502	C₂	semidirect
6. C₅×Dic₃×C₁₆	C₂	semidirect
7. C₄×C₅×C₃⋊C₁₆	C₂	semidirect
20. C₅×#798	C₄	semidirect
21. C₅×D₃×C₁₆	C₄	semidirect
22. #69072	C₅	direct (N × H)
27. C₅×#346	D₃	semidirect
49. C₃×#346	C₂×C₅	semidirect
50. #4502	C₂×C₅	semidirect
51. Dic₃×C₁₆	C₂×C₅	semidirect
52. C₄×C₃⋊C₁₆	C₂×C₅	semidirect
90. #798	C₄×C₅	semidirect
91. D₃×C₁₆	C₄×C₅	semidirect
112. #346	C₅×D₃	semidirect
203. C₃×C₅	#2108	semidirect
230. C₅	#69072	direct (N × H)
239. C₃	C₅×#2108	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#798	2
C₂×C₅×D₃×C₁₆	2
C₅×#4689	2
C₃×C₅×#346	2
C₅×#4502	2
C₅×Dic₃×C₁₆	2
C₄×C₅×C₃⋊C₁₆	2
C₅×#2108	3
#69072	5

Subgroups — C5×#69072 #14312637

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#69072 #14312637