Subgroups — C₅×#68052 #14287261

Order 1920.

360 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×D₂₄	C₂	1	1	yes	maximal, characteristic
2	C₅×#4688	C₂	1	2	yes	maximal, complemented
3	C₃×C₅×C₈²	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#4459	C₂	1	2	yes	maximal, complemented
5	C₅×#4458	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1589	—	3	1	no	maximal, Hall
7	C₂×C₅×D₂₄	C₄	1	1	yes	characteristic
8	C₂×C₅×ΓL(1,𝔽₂₅)	—	2	2	no	—
9	C₄×C₅×D₁₂	C₂²	1	2	yes	—
10	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₅×#821	C₄	1	2	yes	—
14	C₅×(C₂×D₃)⋊C₈	—	2	2	no	—
15	C₅×#818	C₄	1	1	yes	characteristic
16	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
17	#68052	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₄×C₅×D₈	—	3	1	no	—
19	C₅×#436	—	3	2	no	—
20	C₅×#335	—	3	1	no	—
21	C₅×D₄⋊C₈	—	3	2	no	—
22	C₅×C₈²	D₃	1	1	yes	characteristic, complemented
23	C₂×C₅×D₁₂	C₂×C₄	1	2	yes	—
24	C₂×C₄×C₅×D₃	—	2	2	no	—
25	C₂×C₃×C₅×C₈	—	2	1	no	—
26	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
29	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
30	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
31	C₅×(C₂×D₃)⋊C₄	—	2	2	no	—
32	C₅×C₄⋊Dic₃	C₂×C₄	1	2	yes	—
33	C₂×C₅×C₃⋊C₈	—	2	2	no	—
34	C₅×D₂₄	—	2	2	no	—
35	C₅×ΓL(1,𝔽₂₅)	—	4	4	no	—
36	C₄×D₂₄	C₂×C₅	1	1	yes	characteristic
37	#4688	C₂×C₅	1	2	yes	complemented
38	C₃×C₈²	C₂×C₅	1	1	yes	characteristic, complemented
39	#4459	C₂×C₅	1	2	yes	complemented
40	#4458	C₂×C₅	1	1	yes	characteristic, complemented
41	C₂×C₅×D₈	—	3	1	no	—
42	C₂×C₅×M₁₆	—	6	2	no	—
43	C₄×C₅×D₄	—	3	2	no	—
44	C₅×#107	—	3	1	no	—
45	C₅×C₄⋊C₈	—	3	2	no	—
46	C₅×#102	—	3	2	no	—
47	C₅×C₂²⋊C₈	—	6	2	no	—
48	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
49	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
50	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
51	#1589	—	3	1	no	Sylow
52	C₂²×C₅×D₃	—	2	2	no	—
53	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
55	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
56	C₂×C₅×Dic₃	—	2	2	no	—
57	C₅×D₁₂	—	2	2	no	—
58	C₅×D₁₂	—	2	2	no	—
59	C₄×C₅×D₃	—	4	2	no	—
60	C₄×C₅×D₃	—	4	2	no	—
61	C₃×C₅×C₈	—	2	2	no	—
62	C₃×C₅×C₈	—	2	1	no	—
63	C₃×C₅×C₈	—	2	1	no	—
64	C₃×C₅×C₈	M₁₆	1	2	yes	complemented
65	C₃×C₅×C₈	D₈	1	2	yes	complemented
66	C₅×C₃⋊C₈	—	4	2	no	—
67	C₂×D₂₄	C₄×C₅	1	1	yes	characteristic
68	C₂×ΓL(1,𝔽₂₅)	—	2	2	no	—
69	C₄×D₁₂	C₂²×C₅	1	2	yes	—
70	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
71	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
72	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
73	#821	C₄×C₅	1	2	yes	—
74	(C₂×D₃)⋊C₈	—	2	2	no	—
75	#818	C₄×C₅	1	1	yes	characteristic
76	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	2	yes	—
77	C₂×C₅×D₄	—	3	2	no	—
78	C₂²×C₄×C₅	—	6	2	no	—
79	C₅×D₈	—	6	2	no	—
80	C₅×M₁₆	—	12	4	no	—
81	C₂×C₅×C₈	—	2	1	no	—
82	C₂×C₅×C₈	—	6	2	no	—
83	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
84	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
85	C₂×C₅×C₈	D₁₂	1	1	yes	characteristic
86	C₂×C₅×C₈	D₁₂	1	1	yes	characteristic
87	C₅×C₄⋊C₄	—	3	2	no	—
88	C₅×C₂²⋊C₄	—	6	2	no	—
89	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
90	C₄×D₈	—	3	1	no	—
91	#436	—	3	2	no	—
92	#335	—	3	1	no	—
93	D₄⋊C₈	—	3	2	no	—
94	C₈²	C₅×D₃	1	1	yes	characteristic, complemented
95	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
96	C₂×C₅×D₃	—	4	2	no	—
97	C₂×C₅×D₃	—	4	2	no	—
98	C₂×C₅×D₃	—	4	2	no	—
99	C₃×C₄×C₅	—	2	1	no	—
100	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
101	C₃×C₄×C₅	#131	1	1	yes	characteristic
102	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
103	C₃×C₄×C₅	#135	1	1	yes	characteristic
104	C₅×Dic₃	—	4	2	no	—
105	C₂×D₁₂	C₂×C₄×C₅	1	2	yes	—
106	C₂×C₄×D₃	—	2	2	no	—
107	C₂×C₃×C₈	—	2	1	no	—
108	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
109	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
110	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
111	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
112	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
113	(C₂×D₃)⋊C₄	—	2	2	no	—
114	C₄⋊Dic₃	C₂×C₄×C₅	1	2	yes	—
115	C₂×C₃⋊C₈	—	2	2	no	—
116	D₂₄	—	2	2	no	—
117	ΓL(1,𝔽₂₅)	—	4	4	no	—
118	C₂³×C₅	—	6	2	no	—
119	C₅×D₄	—	6	2	no	—
120	C₅×D₄	—	6	2	no	—
121	C₂×C₄×C₅	—	6	2	no	—
122	C₂×C₄×C₅	—	12	2	no	—
123	C₂×C₄×C₅	—	12	2	no	—
124	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
125	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
126	C₂×C₄×C₅	#235	1	1	yes	characteristic
127	C₅×C₈	—	2	2	no	—
128	C₅×C₈	—	2	1	no	—
129	C₅×C₈	—	2	1	no	—
130	C₅×C₈	—	12	2	no	—
131	C₅×C₈	ΓL(1,𝔽₂₅)	1	2	yes	complemented
132	C₅×C₈	D₂₄	1	2	yes	complemented
133	C₂×D₈	—	3	1	no	—
134	C₂×M₁₆	—	6	2	no	—
135	C₄×D₄	—	3	2	no	—
136	#107	—	3	1	no	—
137	C₄⋊C₈	—	3	2	no	—
138	#102	—	3	2	no	—
139	C₂²⋊C₈	—	6	2	no	—
140	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
141	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
142	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
143	C₂×C₃×C₅	#436	1	1	yes	characteristic
144	C₂×C₃×C₅	C₄×D₈	1	1	yes	characteristic
145	C₂×C₃×C₅	#443	1	1	yes	characteristic
146	C₅×D₃	—	8	2	no	—
147	C₂²×D₃	—	2	2	no	—
148	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
149	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
150	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
151	C₂×Dic₃	—	2	2	no	—
152	D₁₂	—	2	2	no	—
153	D₁₂	—	2	2	no	—
154	C₄×D₃	—	4	2	no	—
155	C₄×D₃	—	4	2	no	—
156	C₃×C₈	—	2	2	no	—
157	C₃×C₈	—	2	1	no	—
158	C₃×C₈	—	2	1	no	—
159	C₃×C₈	C₅×M₁₆	1	2	yes	complemented
160	C₃×C₈	C₅×D₈	1	2	yes	complemented
161	C₃⋊C₈	—	4	2	no	—
162	C₂²×C₅	—	12	2	no	—
163	C₂²×C₅	—	12	2	no	—
164	C₂²×C₅	—	12	2	no	—
165	C₂²×C₅	C₄×D₁₂	1	1	yes	characteristic
166	C₄×C₅	—	2	1	no	—
167	C₄×C₅	—	12	2	no	—
168	C₄×C₅	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
169	C₄×C₅	#901	1	1	yes	characteristic
170	C₄×C₅	C₂×D₂₄	1	1	yes	characteristic
171	C₄×C₅	#904	1	1	yes	characteristic
172	C₂×D₄	—	3	2	no	—
173	C₂²×C₄	—	6	2	no	—
174	D₈	—	6	2	no	—
175	M₁₆	—	12	4	no	—
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	6	2	no	—
178	C₂×C₈	C₄×C₅×D₃	1	1	yes	characteristic
179	C₂×C₈	C₄×C₅×D₃	1	1	yes	characteristic
180	C₂×C₈	C₅×D₁₂	1	1	yes	characteristic
181	C₂×C₈	C₅×D₁₂	1	1	yes	characteristic
182	C₄⋊C₄	—	3	2	no	—
183	C₂²⋊C₄	—	6	2	no	—
184	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
185	C₃×C₅	#1589	1	1	yes	characteristic, complemented, Hall
186	C₂²×C₃	C₄×C₅×D₄	1	1	yes	characteristic
187	C₂×D₃	—	4	2	no	—
188	C₂×D₃	—	4	2	no	—
189	C₂×D₃	—	4	2	no	—
190	C₃×C₄	—	2	1	no	—
191	C₃×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
192	C₃×C₄	C₅×#131	1	1	yes	characteristic
193	C₃×C₄	C₂×C₅×D₈	1	1	yes	characteristic
194	C₃×C₄	C₅×#135	1	1	yes	characteristic
195	Dic₃	—	4	2	no	—
196	C₂×C₅	—	24	2	no	—
197	C₂×C₅	#4688	1	1	yes	characteristic
198	C₂×C₅	C₄×D₂₄	1	1	yes	characteristic
199	C₂×C₅	#4700	1	1	yes	characteristic
200	C₂³	—	6	2	no	—
201	D₄	—	6	2	no	—
202	D₄	—	6	2	no	—
203	C₂×C₄	—	6	2	no	—
204	C₂×C₄	—	12	2	no	—
205	C₂×C₄	—	12	2	no	—
206	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
207	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
208	C₂×C₄	C₅×#235	1	1	yes	characteristic
209	C₈	—	2	2	no	—
210	C₈	—	2	1	no	—
211	C₈	—	2	1	no	—
212	C₈	—	12	2	no	—
213	C₈	C₅×ΓL(1,𝔽₂₅)	1	2	yes	complemented
214	C₈	C₅×D₂₄	1	2	yes	complemented
215	C₂×C₃	C₅×#436	1	1	yes	characteristic
216	C₂×C₃	C₄×C₅×D₈	1	1	yes	characteristic
217	C₂×C₃	C₅×#443	1	1	yes	characteristic
218	D₃	—	8	2	no	—
219	C₅	#68052	1	1	yes	characteristic, direct product factor, Sylow
220	C₂²	—	12	2	no	—
221	C₂²	—	12	2	no	—
222	C₂²	—	12	2	no	—
223	C₂²	C₄×C₅×D₁₂	1	1	yes	characteristic
224	C₄	—	2	1	no	—
225	C₄	—	12	2	no	—
226	C₄	C₂×C₅×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
227	C₄	C₅×#901	1	1	yes	characteristic
228	C₄	C₂×C₅×D₂₄	1	1	yes	characteristic
229	C₄	C₅×#904	1	1	yes	characteristic
230	C₃	C₅×#1589	1	1	yes	characteristic, complemented, Sylow
231	C₂	—	24	2	no	—
232	C₂	C₅×#4688	1	1	yes	characteristic
233	C₂	C₄×C₅×D₂₄	1	1	yes	characteristic
234	C₂	C₅×#4700	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	C₂×D₁₂
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₄²	C₂²×C₅×D₃
Fitting	C₃×C₅×C₈²	C₂
socle	C₂²×C₃×C₅	C₄×D₄
prime radical	C₂×C₅×D₂₄	C₄
cosocle	C₃×C₄²	C₂³×C₅
hypercenter	C₅×C₈²	D₃
hypocenter	C₃	C₅×#1589
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₄×C₅×C₈ ◁ C₅×C₈²

Complements

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

N	H ≅ G/N	type
2. C₅×#4688	C₂	semidirect
3. C₃×C₅×C₈²	C₂	semidirect
4. C₅×#4459	C₂	semidirect
5. C₅×#4458	C₂	semidirect
17. #68052	C₅	direct (N × H)
22. C₅×C₈²	D₃	semidirect
37. #4688	C₂×C₅	semidirect
38. C₃×C₈²	C₂×C₅	semidirect
39. #4459	C₂×C₅	semidirect
40. #4458	C₂×C₅	semidirect
64. C₃×C₅×C₈	M₁₆	semidirect
65. C₃×C₅×C₈	D₈	semidirect
94. C₈²	C₅×D₃	semidirect
131. C₅×C₈	ΓL(1,𝔽₂₅)	semidirect
132. C₅×C₈	D₂₄	semidirect
159. C₃×C₈	C₅×M₁₆	semidirect
160. C₃×C₈	C₅×D₈	semidirect
185. C₃×C₅	#1589	semidirect
213. C₈	C₅×ΓL(1,𝔽₂₅)	semidirect
214. C₈	C₅×D₂₄	semidirect
219. C₅	#68052	direct (N × H)
230. C₃	C₅×#1589	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₅×D₂₄	2
C₅×#4688	2
C₃×C₅×C₈²	2
C₅×#4459	2
C₅×#4458	2
C₅×#1589	3
#68052	5

Subgroups — C5×#68052 #14287261

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#68052 #14287261