Subgroups — C₅×#71425 #14312749

Order 1920.

320 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₅×#5101	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#812	C₂	1	1	yes	maximal, characteristic
4	C₅×#4512	C₂	1	4	yes	maximal, complemented
5	C₅×#2110	—	3	1	no	maximal, Hall
6	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₅×#926	C₄	1	1	yes	characteristic
8	C₂×C₅×#208	C₄	1	1	yes	characteristic
9	C₃×C₅×M₃₂	C₄	1	2	yes	complemented
10	C₃×C₅×M₃₂	C₂²	1	2	yes	—
11	C₂×C₃×C₅×C₁₆	C₄	1	2	yes	complemented
12	C₅×#815	C₂²	1	2	yes	—
13	C₅×C₈×Dic₃	C₄	1	2	yes	—
14	C₅×#812	C₄	1	2	yes	complemented
15	C₅×#812	C₂²	1	2	yes	—
16	C₂×C₅×C₃⋊C₁₆	C₄	1	2	yes	complemented
17	#71425	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₂×C₅×M₃₂	—	3	1	no	—
19	C₂×C₅×M₃₂	D₃	1	1	yes	characteristic
20	C₅×#405	—	3	1	no	—
21	C₅×#347	—	3	4	no	—
22	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
23	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
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	C₂³	1	1	yes	characteristic
29	C₅×#217	—	2	1	no	—
30	C₅×C₄⋊Dic₃	—	2	1	no	—
31	C₄×C₅×Dic₃	C₂×C₄	1	2	yes	—
32	C₅×#208	—	2	1	no	—
33	C₅×#208	—	2	1	no	—
34	C₂×C₅×C₃⋊C₈	C₂×C₄	1	2	yes	—
35	C₃×C₅×C₁₆	C₂×C₄	1	4	yes	—
36	C₅×C₃⋊C₁₆	C₂×C₄	1	4	yes	—
37	C₂×C₃×M₃₂	C₂×C₅	1	1	yes	characteristic
38	#5101	C₂×C₅	1	1	yes	characteristic
39	C₂×#812	C₂×C₅	1	1	yes	characteristic
40	#4512	C₂×C₅	1	4	yes	complemented
41	C₂×C₅×M₁₆	—	3	1	no	—
42	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
43	C₅×#117	—	3	1	no	—
44	C₅×M₃₂	—	3	2	no	—
45	C₅×M₃₂	—	3	2	no	—
46	C₅×M₃₂	Dic₃	1	2	yes	complemented
47	C₅×M₃₂	C₂×D₃	1	2	yes	—
48	C₂×C₅×C₁₆	—	3	2	no	—
49	C₂×C₅×C₁₆	Dic₃	1	2	yes	complemented
50	C₅×#97	—	3	2	no	—
51	C₄×C₅×C₈	—	3	2	no	—
52	#2110	—	3	1	no	Sylow
53	C₂³×C₃×C₅	M₁₆	1	1	yes	characteristic
54	C₂×C₃×C₄×C₅	C₄²	1	1	yes	characteristic
55	C₂×C₃×C₄×C₅	C₄²	1	1	yes	characteristic
56	C₂×C₃×C₄×C₅	M₁₆	1	1	yes	characteristic
57	C₂×C₃×C₄×C₅	M₁₆	1	1	yes	characteristic
58	C₂×C₃×C₄×C₅	M₁₆	1	1	yes	characteristic
59	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₅×Dic₃	—	2	2	no	—
61	C₃×C₅×C₈	C₄²	1	1	yes	characteristic
62	C₃×C₅×C₈	C₄²	1	1	yes	characteristic
63	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
64	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
65	C₅×C₃⋊C₈	—	2	2	no	—
66	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
67	#926	C₄×C₅	1	1	yes	characteristic
68	C₂×#208	C₄×C₅	1	1	yes	characteristic
69	C₃×M₃₂	C₄×C₅	1	2	yes	complemented
70	C₃×M₃₂	C₂²×C₅	1	2	yes	—
71	C₂×C₃×C₁₆	C₄×C₅	1	2	yes	complemented
72	#815	C₂²×C₅	1	2	yes	—
73	C₈×Dic₃	C₄×C₅	1	2	yes	—
74	#812	C₄×C₅	1	2	yes	complemented
75	#812	C₂²×C₅	1	2	yes	—
76	C₂×C₃⋊C₁₆	C₄×C₅	1	2	yes	complemented
77	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
78	C₅×M₁₆	—	6	1	no	—
79	C₅×M₁₆	—	6	1	no	—
80	C₂×C₅×C₈	—	3	2	no	—
81	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
82	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
83	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
84	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
85	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
86	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
87	C₅×C₄⋊C₄	—	6	1	no	—
88	C₅×C₂²⋊C₄	—	6	1	no	—
89	C₄²×C₅	—	3	2	no	—
90	C₅×C₁₆	—	3	4	no	—
91	C₅×C₁₆	C₂×Dic₃	1	4	yes	—
92	C₂×M₃₂	—	3	1	no	—
93	C₂×M₃₂	C₅×D₃	1	1	yes	characteristic
94	#405	—	3	1	no	—
95	#347	—	3	4	no	—
96	C₂²×C₃×C₅	—	4	1	no	—
97	C₂²×C₃×C₅	#97	1	1	yes	characteristic
98	C₂²×C₃×C₅	#97	1	1	yes	characteristic
99	C₂²×C₃×C₅	C₂×M₁₆	1	1	yes	characteristic
100	C₃×C₄×C₅	#97	1	1	yes	characteristic
101	C₃×C₄×C₅	#97	1	1	yes	characteristic
102	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
103	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
104	C₅×Dic₃	—	4	2	no	—
105	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
106	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
107	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
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	C₂×C₄×C₅	1	1	yes	characteristic
111	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
112	#217	—	2	1	no	—
113	C₄⋊Dic₃	—	2	1	no	—
114	C₄×Dic₃	C₂×C₄×C₅	1	2	yes	—
115	#208	—	2	1	no	—
116	#208	—	2	1	no	—
117	C₂×C₃⋊C₈	C₂×C₄×C₅	1	2	yes	—
118	C₃×C₁₆	C₂×C₄×C₅	1	4	yes	—
119	C₃⋊C₁₆	C₂×C₄×C₅	1	4	yes	—
120	C₂³×C₅	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
121	C₂×C₄×C₅	—	6	2	no	—
122	C₂×C₄×C₅	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
123	C₂×C₄×C₅	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
124	C₂×C₄×C₅	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
125	C₂×C₄×C₅	C₄×Dic₃	1	1	yes	characteristic
126	C₂×C₄×C₅	C₄×Dic₃	1	1	yes	characteristic
127	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
128	C₅×C₈	—	6	2	no	—
129	C₅×C₈	C₄×Dic₃	1	1	yes	characteristic
130	C₅×C₈	C₄×Dic₃	1	1	yes	characteristic
131	C₅×C₈	C₂×C₄×D₃	1	1	yes	characteristic
132	C₅×C₈	C₂²×Dic₃	1	1	yes	characteristic
133	C₂×M₁₆	—	3	1	no	—
134	C₂²×C₈	C₂×C₅×D₃	1	1	yes	characteristic
135	#117	—	3	1	no	—
136	M₃₂	—	3	2	no	—
137	M₃₂	—	3	2	no	—
138	M₃₂	C₅×Dic₃	1	2	yes	complemented
139	M₃₂	C₂×C₅×D₃	1	2	yes	—
140	C₂×C₁₆	—	3	2	no	—
141	C₂×C₁₆	C₅×Dic₃	1	2	yes	complemented
142	#97	—	3	2	no	—
143	C₄×C₈	—	3	2	no	—
144	C₂×C₃×C₅	—	2	1	no	—
145	C₂×C₃×C₅	—	2	1	no	—
146	C₂×C₃×C₅	—	2	1	no	—
147	C₂×C₃×C₅	C₂×#97	1	1	yes	characteristic
148	C₂³×C₃	C₅×M₁₆	1	1	yes	characteristic
149	C₂×C₃×C₄	C₄²×C₅	1	1	yes	characteristic
150	C₂×C₃×C₄	C₄²×C₅	1	1	yes	characteristic
151	C₂×C₃×C₄	C₅×M₁₆	1	1	yes	characteristic
152	C₂×C₃×C₄	C₅×M₁₆	1	1	yes	characteristic
153	C₂×C₃×C₄	C₅×M₁₆	1	1	yes	characteristic
154	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
155	C₂×Dic₃	—	2	2	no	—
156	C₃×C₈	C₄²×C₅	1	1	yes	characteristic
157	C₃×C₈	C₄²×C₅	1	1	yes	characteristic
158	C₃×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
159	C₃×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
160	C₃⋊C₈	—	2	2	no	—
161	C₂²×C₅	—	4	1	no	—
162	C₂²×C₅	#815	1	1	yes	characteristic
163	C₂²×C₅	#815	1	1	yes	characteristic
164	C₂²×C₅	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
165	C₄×C₅	—	12	2	no	—
166	C₄×C₅	#815	1	1	yes	characteristic
167	C₄×C₅	#815	1	1	yes	characteristic
168	C₄×C₅	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
169	C₄×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
170	C₂²×C₄	C₄×C₅×D₃	1	1	yes	characteristic
171	M₁₆	—	6	1	no	—
172	M₁₆	—	6	1	no	—
173	C₂×C₈	—	3	2	no	—
174	C₂×C₈	C₄×C₅×D₃	1	1	yes	characteristic
175	C₂×C₈	C₄×C₅×D₃	1	1	yes	characteristic
176	C₂×C₈	C₄×C₅×D₃	1	1	yes	characteristic
177	C₂×C₈	C₂×C₅×Dic₃	1	1	yes	characteristic
178	C₂×C₈	C₂×C₅×Dic₃	1	1	yes	characteristic
179	C₂×C₈	C₂²×C₅×D₃	1	1	yes	characteristic
180	C₄⋊C₄	—	6	1	no	—
181	C₂²⋊C₄	—	6	1	no	—
182	C₄²	—	3	2	no	—
183	C₁₆	—	3	4	no	—
184	C₁₆	C₂×C₅×Dic₃	1	4	yes	—
185	C₃×C₅	#2110	1	1	yes	characteristic, complemented, Hall
186	C₂²×C₃	—	4	1	no	—
187	C₂²×C₃	C₅×#97	1	1	yes	characteristic
188	C₂²×C₃	C₅×#97	1	1	yes	characteristic
189	C₂²×C₃	C₂×C₅×M₁₆	1	1	yes	characteristic
190	C₃×C₄	C₅×#97	1	1	yes	characteristic
191	C₃×C₄	C₅×#97	1	1	yes	characteristic
192	C₃×C₄	C₂×C₄²×C₅	1	1	yes	characteristic
193	C₃×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
194	Dic₃	—	4	2	no	—
195	C₂×C₅	—	2	1	no	—
196	C₂×C₅	—	2	1	no	—
197	C₂×C₅	—	2	1	no	—
198	C₂×C₅	C₂×#815	1	1	yes	characteristic
199	C₂³	C₅×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
200	C₂×C₄	—	6	2	no	—
201	C₂×C₄	C₅×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
202	C₂×C₄	C₅×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
203	C₂×C₄	C₅×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
204	C₂×C₄	C₄×C₅×Dic₃	1	1	yes	characteristic
205	C₂×C₄	C₄×C₅×Dic₃	1	1	yes	characteristic
206	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
207	C₈	—	6	2	no	—
208	C₈	C₄×C₅×Dic₃	1	1	yes	characteristic
209	C₈	C₄×C₅×Dic₃	1	1	yes	characteristic
210	C₈	C₂×C₄×C₅×D₃	1	1	yes	characteristic
211	C₈	C₂²×C₅×Dic₃	1	1	yes	characteristic
212	C₂×C₃	—	2	1	no	—
213	C₂×C₃	—	2	1	no	—
214	C₂×C₃	—	2	1	no	—
215	C₂×C₃	C₂×C₅×#97	1	1	yes	characteristic
216	C₅	#71425	1	1	yes	characteristic, direct product factor, Sylow
217	C₂²	—	4	1	no	—
218	C₂²	C₅×#815	1	1	yes	characteristic
219	C₂²	C₅×#815	1	1	yes	characteristic
220	C₂²	C₂×C₅×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
221	C₄	—	12	2	no	—
222	C₄	C₅×#815	1	1	yes	characteristic
223	C₄	C₅×#815	1	1	yes	characteristic
224	C₄	C₂×C₅×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
225	C₄	C₂×C₄×C₅×Dic₃	1	1	yes	characteristic
226	C₃	C₅×#2110	1	1	yes	characteristic, complemented, Sylow
227	C₂	—	2	1	no	—
228	C₂	—	2	1	no	—
229	C₂	—	2	1	no	—
230	C₂	C₂×C₅×#815	1	1	yes	characteristic

	subgroup	G/H
center	C₅×C₈	C₄×Dic₃
commutator	C₃×C₄	C₂×C₄²×C₅
Frattini	C₂×C₈	C₂²×C₅×D₃
Fitting	C₂×C₃×C₅×M₃₂	C₂
socle	C₂×C₃×C₅	C₂×#97
prime radical	C₂³×C₃×C₅	M₁₆
cosocle	C₂×C₃×C₈	C₂³×C₅
hypercenter	C₂×C₅×M₃₂	D₃
hypocenter	C₃	C₅×#2110
LCS	G ▷ C₃×C₄ ▷ C₃
UCS	C₁ ◁ C₅×C₈ ◁ C₂×C₅×M₃₂

Complements

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

N	H ≅ G/N	type
4. C₅×#4512	C₂	semidirect
9. C₃×C₅×M₃₂	C₄	semidirect
11. C₂×C₃×C₅×C₁₆	C₄	semidirect
14. C₅×#812	C₄	semidirect
16. C₂×C₅×C₃⋊C₁₆	C₄	semidirect
17. #71425	C₅	direct (N × H)
40. #4512	C₂×C₅	semidirect
46. C₅×M₃₂	Dic₃	semidirect
49. C₂×C₅×C₁₆	Dic₃	semidirect
69. C₃×M₃₂	C₄×C₅	semidirect
71. C₂×C₃×C₁₆	C₄×C₅	semidirect
74. #812	C₄×C₅	semidirect
76. C₂×C₃⋊C₁₆	C₄×C₅	semidirect
138. M₃₂	C₅×Dic₃	semidirect
141. C₂×C₁₆	C₅×Dic₃	semidirect
185. C₃×C₅	#2110	semidirect
216. C₅	#71425	direct (N × H)
226. 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
C₂×C₃×C₅×M₃₂	2
C₅×#5101	2
C₂×C₅×#812	2
C₅×#4512	2
C₅×#2110	3
#71425	5

Subgroups — C5×#71425 #14312749

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#71425 #14312749