Subgroups — C₅×#69087 #14312746

Order 1920.

320 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×D₃×M₃₂	C₂	1	2	yes	maximal
2	C₅×#4708	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#347	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#4512	C₂	1	2	yes	maximal, complemented
5	C₅×#4466	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#2110	—	3	1	no	maximal, Hall
7	C₂×C₅×ΓL(1,𝔽₂₅)	C₄	1	1	yes	characteristic
8	C₂×C₅×D₃×C₈	C₂²	1	1	yes	characteristic
9	C₅×#872	C₄	1	1	yes	characteristic
10	C₃×C₅×M₃₂	C₂²	1	2	yes	—
11	C₃×C₅×#97	C₂²	1	1	yes	characteristic
12	C₅×#815	C₂²	1	1	yes	characteristic
13	C₅×C₈×Dic₃	C₄	1	1	yes	characteristic
14	C₅×#812	C₂²	1	2	yes	—
15	C₄×C₅×C₃⋊C₈	C₄	1	1	yes	characteristic
16	C₅×#798	C₄	1	4	yes	complemented
17	C₅×D₃×C₁₆	C₄	1	4	yes	complemented
18	#69087	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₂×C₅×M₃₂	—	3	2	no	—
20	C₅×#405	—	3	1	no	—
21	C₅×#347	—	3	2	no	—
22	C₅×#347	—	3	1	no	—
23	C₅×#347	D₃	1	1	yes	characteristic, complemented
24	C₂×C₄×C₅×D₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
27	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
28	C₅×(C₂×D₃)⋊C₄	—	2	1	no	—
29	C₅×Dic₃⋊C₄	—	2	1	no	—
30	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
32	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
33	C₅×ΓL(1,𝔽₂₅)	—	2	2	no	—
34	C₅×D₃×C₈	C₂×C₄	1	2	yes	—
35	C₅×D₃×C₈	C₂×C₄	1	2	yes	—
36	C₃×C₅×C₁₆	C₂×C₄	1	4	yes	—
37	C₅×C₃⋊C₁₆	C₂×C₄	1	4	yes	—
38	D₃×M₃₂	C₂×C₅	1	2	yes	—
39	#4708	C₂×C₅	1	1	yes	characteristic
40	C₃×#347	C₂×C₅	1	1	yes	characteristic, complemented
41	#4512	C₂×C₅	1	2	yes	complemented
42	#4466	C₂×C₅	1	1	yes	characteristic, complemented
43	C₂×C₅×M₁₆	—	3	1	no	—
44	C₂²×C₅×C₈	—	3	1	no	—
45	C₅×#117	—	3	1	no	—
46	C₅×M₃₂	—	3	2	no	—
47	C₅×M₃₂	—	3	4	no	—
48	C₅×M₃₂	C₂×D₃	1	2	yes	—
49	C₂×C₅×C₁₆	—	3	4	no	—
50	C₅×#97	—	3	1	no	—
51	C₅×#97	C₂×D₃	1	1	yes	characteristic
52	C₄×C₅×C₈	—	3	1	no	—
53	C₄×C₅×C₈	—	3	1	no	—
54	#2110	—	3	1	no	Sylow
55	C₂²×C₅×D₃	M₁₆	1	1	yes	characteristic
56	C₂×C₃×C₄×C₅	—	2	1	no	—
57	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₅×Dic₃	—	2	1	no	—
59	C₂×C₅×Dic₃	M₁₆	1	1	yes	characteristic
60	C₄×C₅×D₃	C₄²	1	2	yes	—
61	C₄×C₅×D₃	M₁₆	1	2	yes	—
62	C₃×C₅×C₈	—	2	1	no	—
63	C₃×C₅×C₈	C₂²×C₄	1	2	yes	—
64	C₅×C₃⋊C₈	—	2	1	no	—
65	C₅×C₃⋊C₈	C₄²	1	1	yes	characteristic
66	C₅×C₃⋊C₈	C₄²	1	1	yes	characteristic
67	C₂×ΓL(1,𝔽₂₅)	C₄×C₅	1	1	yes	characteristic
68	C₂×D₃×C₈	C₂²×C₅	1	1	yes	characteristic
69	#872	C₄×C₅	1	1	yes	characteristic
70	C₃×M₃₂	C₂²×C₅	1	2	yes	—
71	C₃×#97	C₂²×C₅	1	1	yes	characteristic
72	#815	C₂²×C₅	1	1	yes	characteristic
73	C₈×Dic₃	C₄×C₅	1	1	yes	characteristic
74	#812	C₂²×C₅	1	2	yes	—
75	C₄×C₃⋊C₈	C₄×C₅	1	1	yes	characteristic
76	#798	C₄×C₅	1	4	yes	complemented
77	D₃×C₁₆	C₄×C₅	1	4	yes	complemented
78	C₂²×C₄×C₅	—	3	1	no	—
79	C₅×M₁₆	—	6	2	no	—
80	C₂×C₅×C₈	—	3	1	no	—
81	C₂×C₅×C₈	—	3	2	no	—
82	C₂×C₅×C₈	—	3	2	no	—
83	C₂×C₅×C₈	—	3	1	no	—
84	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
85	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
86	C₅×C₄⋊C₄	—	6	1	no	—
87	C₅×C₂²⋊C₄	—	6	1	no	—
88	C₄²×C₅	—	3	1	no	—
89	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
90	C₅×C₁₆	—	3	4	no	—
91	C₅×C₁₆	C₄×D₃	1	4	yes	—
92	C₂×M₃₂	—	3	2	no	—
93	#405	—	3	1	no	—
94	#347	—	3	2	no	—
95	#347	—	3	1	no	—
96	#347	C₅×D₃	1	1	yes	characteristic, complemented
97	C₂²×C₃×C₅	C₂×M₁₆	1	1	yes	characteristic
98	C₂×C₅×D₃	—	4	1	no	—
99	C₂×C₅×D₃	#97	1	2	yes	—
100	C₃×C₄×C₅	—	4	1	no	—
101	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
102	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
103	C₅×Dic₃	—	4	1	no	—
104	C₅×Dic₃	#97	1	2	yes	—
105	C₂×C₄×D₃	C₂×C₄×C₅	1	1	yes	characteristic
106	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
107	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
108	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
109	(C₂×D₃)⋊C₄	—	2	1	no	—
110	Dic₃⋊C₄	—	2	1	no	—
111	C₄×Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
112	C₂×C₃⋊C₈	C₂×C₄×C₅	1	1	yes	characteristic
113	C₂×C₃⋊C₈	C₂×C₄×C₅	1	1	yes	characteristic
114	ΓL(1,𝔽₂₅)	—	2	2	no	—
115	D₃×C₈	C₂×C₄×C₅	1	2	yes	—
116	D₃×C₈	C₂×C₄×C₅	1	2	yes	—
117	C₃×C₁₆	C₂×C₄×C₅	1	4	yes	—
118	C₃⋊C₁₆	C₂×C₄×C₅	1	4	yes	—
119	C₂³×C₅	—	3	1	no	—
120	C₂×C₄×C₅	—	2	1	no	—
121	C₂×C₄×C₅	—	3	1	no	—
122	C₂×C₄×C₅	—	3	2	no	—
123	C₂×C₄×C₅	—	3	2	no	—
124	C₂×C₄×C₅	—	6	1	no	—
125	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
126	C₅×C₈	—	2	1	no	—
127	C₅×C₈	—	3	1	no	—
128	C₅×C₈	—	3	1	no	—
129	C₅×C₈	—	6	1	no	—
130	C₅×C₈	C₂×C₄×D₃	1	2	yes	—
131	C₂×M₁₆	—	3	1	no	—
132	C₂²×C₈	—	3	1	no	—
133	#117	—	3	1	no	—
134	M₃₂	—	3	2	no	—
135	M₃₂	—	3	4	no	—
136	M₃₂	C₂×C₅×D₃	1	2	yes	—
137	C₂×C₁₆	—	3	4	no	—
138	#97	—	3	1	no	—
139	#97	C₂×C₅×D₃	1	1	yes	characteristic
140	C₄×C₈	—	3	1	no	—
141	C₄×C₈	—	3	1	no	—
142	C₂×C₃×C₅	—	2	1	no	—
143	C₂×C₃×C₅	C₂×#97	1	1	yes	characteristic
144	C₅×D₃	—	2	2	no	—
145	C₂²×D₃	C₅×M₁₆	1	1	yes	characteristic
146	C₂×C₃×C₄	—	2	1	no	—
147	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
148	C₂×Dic₃	—	2	1	no	—
149	C₂×Dic₃	C₅×M₁₆	1	1	yes	characteristic
150	C₄×D₃	C₄²×C₅	1	2	yes	—
151	C₄×D₃	C₅×M₁₆	1	2	yes	—
152	C₃×C₈	—	2	1	no	—
153	C₃×C₈	C₂²×C₄×C₅	1	2	yes	—
154	C₃⋊C₈	—	2	1	no	—
155	C₃⋊C₈	C₄²×C₅	1	1	yes	characteristic
156	C₃⋊C₈	C₄²×C₅	1	1	yes	characteristic
157	C₂²×C₅	—	3	2	no	—
158	C₂²×C₅	—	12	1	no	—
159	C₂²×C₅	D₃×M₁₆	1	1	yes	characteristic
160	C₄×C₅	—	3	2	no	—
161	C₄×C₅	—	4	1	no	—
162	C₄×C₅	—	12	1	no	—
163	C₄×C₅	C₄²×D₃	1	1	yes	characteristic
164	C₄×C₅	D₃×M₁₆	1	1	yes	characteristic
165	C₂²×C₄	—	3	1	no	—
166	M₁₆	—	6	2	no	—
167	C₂×C₈	—	3	1	no	—
168	C₂×C₈	—	3	2	no	—
169	C₂×C₈	—	3	2	no	—
170	C₂×C₈	—	3	1	no	—
171	C₂×C₈	C₄×C₅×D₃	1	1	yes	characteristic
172	C₂×C₈	C₂²×C₅×D₃	1	1	yes	characteristic
173	C₄⋊C₄	—	6	1	no	—
174	C₂²⋊C₄	—	6	1	no	—
175	C₄²	—	3	1	no	—
176	C₄²	C₄×C₅×D₃	1	1	yes	characteristic
177	C₁₆	—	3	4	no	—
178	C₁₆	C₄×C₅×D₃	1	4	yes	—
179	C₃×C₅	#2110	1	1	yes	characteristic, complemented, Hall
180	C₂²×C₃	C₂×C₅×M₁₆	1	1	yes	characteristic
181	C₂×D₃	—	4	1	no	—
182	C₂×D₃	C₅×#97	1	2	yes	—
183	C₃×C₄	—	4	1	no	—
184	C₃×C₄	C₂×C₄²×C₅	1	1	yes	characteristic
185	C₃×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
186	Dic₃	—	4	1	no	—
187	Dic₃	C₅×#97	1	2	yes	—
188	C₂×C₅	—	2	1	no	—
189	C₂×C₅	—	6	2	no	—
190	C₂×C₅	D₃×#97	1	1	yes	characteristic
191	C₂³	—	3	1	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	3	1	no	—
194	C₂×C₄	—	3	2	no	—
195	C₂×C₄	—	3	2	no	—
196	C₂×C₄	—	6	1	no	—
197	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
198	C₈	—	2	1	no	—
199	C₈	—	3	1	no	—
200	C₈	—	3	1	no	—
201	C₈	—	6	1	no	—
202	C₈	C₂×C₄×C₅×D₃	1	2	yes	—
203	C₂×C₃	—	2	1	no	—
204	C₂×C₃	C₂×C₅×#97	1	1	yes	characteristic
205	D₃	—	2	2	no	—
206	C₅	#69087	1	1	yes	characteristic, direct product factor, Sylow
207	C₂²	—	3	2	no	—
208	C₂²	—	12	1	no	—
209	C₂²	C₅×D₃×M₁₆	1	1	yes	characteristic
210	C₄	—	3	2	no	—
211	C₄	—	4	1	no	—
212	C₄	—	12	1	no	—
213	C₄	C₄²×C₅×D₃	1	1	yes	characteristic
214	C₄	C₅×D₃×M₁₆	1	1	yes	characteristic
215	C₃	C₅×#2110	1	1	yes	characteristic, complemented, Sylow
216	C₂	—	2	1	no	—
217	C₂	—	6	2	no	—
218	C₂	C₅×D₃×#97	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅	C₄²×D₃
commutator	C₃×C₄	C₂×C₄²×C₅
Frattini	C₂×C₈	C₂²×C₅×D₃
Fitting	C₃×C₅×#347	C₂
socle	C₂×C₃×C₅	C₂×#97
prime radical	C₂²×C₅×D₃	M₁₆
cosocle	C₂×C₃×C₈	C₂³×C₅
hypercenter	C₅×#347	D₃
hypocenter	C₃	C₅×#2110
LCS	G ▷ C₃×C₄ ▷ C₃
UCS	C₁ ◁ C₄×C₅ ◁ C₅×#347

Complements

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

N	H ≅ G/N	type
3. C₃×C₅×#347	C₂	semidirect
4. C₅×#4512	C₂	semidirect
5. C₅×#4466	C₂	semidirect
16. C₅×#798	C₄	semidirect
17. C₅×D₃×C₁₆	C₄	semidirect
18. #69087	C₅	direct (N × H)
23. C₅×#347	D₃	semidirect
40. C₃×#347	C₂×C₅	semidirect
41. #4512	C₂×C₅	semidirect
42. #4466	C₂×C₅	semidirect
76. #798	C₄×C₅	semidirect
77. D₃×C₁₆	C₄×C₅	semidirect
96. #347	C₅×D₃	semidirect
179. C₃×C₅	#2110	semidirect
206. C₅	#69087	direct (N × H)
215. C₃	C₅×#2110	semidirect

Maximal subgroups

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

subgroup	index
C₅×D₃×M₃₂	2
C₅×#4708	2
C₃×C₅×#347	2
C₅×#4512	2
C₅×#4466	2
C₅×#2110	3
#69087	5

Subgroups — C5×#69087 #14312746

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#69087 #14312746