Subgroups — C₅×#71440 #14313939

Order 1920.

520 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×D₂₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#5106	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#4509	C₂	1	4	yes	maximal, complemented
5	C₅×#2140	—	3	1	no	maximal, Hall
6	C₂²×C₅×D₁₂	—	2	1	no	—
7	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₅×#926	—	2	1	no	—
9	C₂×C₅×D₂₄	—	2	1	no	—
10	C₂×C₅×D₂₄	—	2	2	no	—
11	C₂×C₅×D₂₄	C₄	1	4	yes	complemented
12	C₂×C₅×D₂₄	C₂²	1	2	yes	—
13	C₃×C₅×M₃₂	—	2	2	no	—
14	C₂×C₃×C₅×C₁₆	C₂²	1	2	yes	complemented
15	C₅×#818	C₂²	1	2	yes	complemented
16	C₅×#817	—	2	1	no	—
17	#71440	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₂²×C₅×D₈	—	3	1	no	—
19	C₂×C₅×M₃₂	D₃	1	1	yes	characteristic, complemented
20	C₅×#427	—	3	1	no	—
21	C₅×#357	—	3	4	no	—
22	C₂³×C₅×D₃	—	4	1	no	—
23	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
24	C₂×C₅×D₁₂	—	2	4	no	—
25	C₂×C₅×D₁₂	—	2	2	no	—
26	C₂×C₅×D₁₂	—	4	1	no	—
27	C₂×C₅×D₁₂	—	4	2	no	—
28	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
29	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
30	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
31	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
32	C₅×#217	—	4	1	no	—
33	C₅×C₄⋊Dic₃	—	2	2	no	—
34	C₄×C₅×Dic₃	—	4	1	no	—
35	C₅×D₂₄	—	2	4	no	—
36	C₅×D₂₄	—	2	2	no	—
37	C₅×D₂₄	C₂×C₄	1	4	yes	—
38	C₃×C₅×C₁₆	—	2	2	no	—
39	C₂²×D₂₄	C₂×C₅	1	1	yes	characteristic
40	C₂×C₃×M₃₂	C₂×C₅	1	1	yes	characteristic, complemented
41	#5106	C₂×C₅	1	1	yes	characteristic, complemented
42	#4509	C₂×C₅	1	4	yes	complemented
43	C₂²×C₅×D₄	—	6	1	no	—
44	C₂×C₅×D₈	—	3	4	no	—
45	C₂×C₅×D₈	—	3	2	no	—
46	C₂×C₅×D₈	—	6	1	no	—
47	C₂×C₅×D₈	—	6	2	no	—
48	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
49	C₅×#117	—	6	1	no	—
50	C₅×M₃₂	—	2	2	no	—
51	C₂×C₅×C₁₆	C₂×D₃	1	2	yes	complemented
52	C₅×#107	—	3	2	no	—
53	C₅×#106	—	6	1	no	—
54	#2140	—	3	1	no	Sylow
55	C₂³×C₃×C₅	D₈	1	1	yes	characteristic
56	C₂²×C₅×D₃	—	4	4	no	—
57	C₂²×C₅×D₃	—	4	2	no	—
58	C₂²×C₅×D₃	—	8	4	no	—
59	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
60	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
61	C₂×C₃×C₄×C₅	SD₁₆	1	2	yes	—
62	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
63	C₂×C₅×Dic₃	—	4	2	no	—
64	C₅×D₁₂	—	2	4	no	—
65	C₅×D₁₂	—	4	4	no	—
66	C₅×D₁₂	—	4	2	no	—
67	C₃×C₅×C₈	C₂²⋊C₄	1	2	yes	—
68	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
69	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
70	C₂²×D₁₂	—	2	1	no	—
71	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
72	#926	—	2	1	no	—
73	C₂×D₂₄	—	2	1	no	—
74	C₂×D₂₄	—	2	2	no	—
75	C₂×D₂₄	C₄×C₅	1	4	yes	complemented
76	C₂×D₂₄	C₂²×C₅	1	2	yes	—
77	C₃×M₃₂	—	2	2	no	—
78	C₂×C₃×C₁₆	C₂²×C₅	1	2	yes	complemented
79	#818	C₂²×C₅	1	2	yes	complemented
80	#817	—	2	1	no	—
81	C₂⁴×C₅	—	12	1	no	—
82	C₂×C₅×D₄	—	6	4	no	—
83	C₂×C₅×D₄	—	6	2	no	—
84	C₂×C₅×D₄	—	12	2	no	—
85	C₂×C₅×D₄	—	12	1	no	—
86	C₂²×C₄×C₅	D₁₂	1	1	yes	characteristic
87	C₅×D₈	—	3	4	no	—
88	C₅×D₈	—	6	4	no	—
89	C₅×D₈	—	6	2	no	—
90	C₂×C₅×C₈	C₄×D₃	1	2	yes	—
91	C₂×C₅×C₈	D₁₂	1	1	yes	characteristic
92	C₂×C₅×C₈	#67	1	2	yes	—
93	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
94	C₅×C₄⋊C₄	—	6	2	no	—
95	C₅×C₂²⋊C₄	—	12	1	no	—
96	C₄²×C₅	—	12	1	no	—
97	C₅×C₁₆	—	2	2	no	—
98	C₂²×D₈	—	3	1	no	—
99	C₂×M₃₂	C₅×D₃	1	1	yes	characteristic, complemented
100	#427	—	3	1	no	—
101	#357	—	3	4	no	—
102	C₂²×C₃×C₅	—	2	2	no	—
103	C₂²×C₃×C₅	#102	1	2	yes	—
104	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
105	C₂×C₅×D₃	—	4	4	no	—
106	C₂×C₅×D₃	—	8	8	no	—
107	C₂×C₅×D₃	—	8	4	no	—
108	C₃×C₄×C₅	#102	1	2	yes	—
109	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
111	C₅×Dic₃	—	8	2	no	—
112	C₂³×D₃	—	4	1	no	—
113	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
114	C₂×D₁₂	—	2	4	no	—
115	C₂×D₁₂	—	2	2	no	—
116	C₂×D₁₂	—	4	1	no	—
117	C₂×D₁₂	—	4	2	no	—
118	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
119	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
120	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
121	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
122	#217	—	4	1	no	—
123	C₄⋊Dic₃	—	2	2	no	—
124	C₄×Dic₃	—	4	1	no	—
125	D₂₄	—	2	4	no	—
126	D₂₄	—	2	2	no	—
127	D₂₄	C₂×C₄×C₅	1	4	yes	—
128	C₃×C₁₆	—	2	2	no	—
129	C₂³×C₅	—	12	4	no	—
130	C₂³×C₅	—	12	2	no	—
131	C₂³×C₅	—	24	4	no	—
132	C₂³×C₅	D₂₄	1	1	yes	characteristic
133	C₅×D₄	—	6	4	no	—
134	C₅×D₄	—	12	4	no	—
135	C₅×D₄	—	12	2	no	—
136	C₂×C₄×C₅	—	12	2	no	—
137	C₂×C₄×C₅	#204	1	2	yes	—
138	C₂×C₄×C₅	D₂₄	1	1	yes	characteristic
139	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	2	yes	—
140	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
141	C₅×C₈	(C₂×D₃)⋊C₄	1	2	yes	—
142	C₅×C₈	C₂×C₄×D₃	1	1	yes	characteristic
143	C₅×C₈	C₂×#67	1	1	yes	characteristic
144	C₂²×D₄	—	6	1	no	—
145	C₂×D₈	—	3	4	no	—
146	C₂×D₈	—	3	2	no	—
147	C₂×D₈	—	6	1	no	—
148	C₂×D₈	—	6	2	no	—
149	C₂²×C₈	C₂×C₅×D₃	1	1	yes	characteristic
150	#117	—	6	1	no	—
151	M₃₂	—	2	2	no	—
152	C₂×C₁₆	C₂×C₅×D₃	1	2	yes	complemented
153	#107	—	3	2	no	—
154	#106	—	6	1	no	—
155	C₂×C₃×C₅	—	2	2	no	—
156	C₂×C₃×C₅	C₂×#102	1	1	yes	characteristic
157	C₂×C₃×C₅	#509	1	2	yes	—
158	C₅×D₃	—	8	4	no	—
159	C₂³×C₃	C₅×D₈	1	1	yes	characteristic
160	C₂²×D₃	—	4	2	no	—
161	C₂²×D₃	—	4	4	no	—
162	C₂²×D₃	—	8	4	no	—
163	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
164	C₂×C₃×C₄	C₅×D₈	1	1	yes	characteristic
165	C₂×C₃×C₄	C₅×SD₁₆	1	2	yes	—
166	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
167	C₂×Dic₃	—	4	2	no	—
168	D₁₂	—	2	4	no	—
169	D₁₂	—	4	2	no	—
170	D₁₂	—	4	4	no	—
171	C₃×C₈	C₅×C₂²⋊C₄	1	2	yes	—
172	C₃×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
173	C₃×C₈	C₂×C₅×D₄	1	1	yes	characteristic
174	C₂²×C₅	—	2	2	no	—
175	C₂²×C₅	—	12	4	no	—
176	C₂²×C₅	—	24	8	no	—
177	C₂²×C₅	—	24	4	no	—
178	C₂²×C₅	#821	1	2	yes	—
179	C₂²×C₅	C₂×D₂₄	1	1	yes	characteristic
180	C₄×C₅	—	24	2	no	—
181	C₄×C₅	#821	1	2	yes	—
182	C₄×C₅	C₂×#204	1	1	yes	characteristic
183	C₄×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
184	C₂⁴	—	12	1	no	—
185	C₂×D₄	—	6	4	no	—
186	C₂×D₄	—	6	2	no	—
187	C₂×D₄	—	12	2	no	—
188	C₂×D₄	—	12	1	no	—
189	C₂²×C₄	C₅×D₁₂	1	1	yes	characteristic
190	D₈	—	3	4	no	—
191	D₈	—	6	4	no	—
192	D₈	—	6	2	no	—
193	C₂×C₈	C₄×C₅×D₃	1	2	yes	—
194	C₂×C₈	C₅×D₁₂	1	1	yes	characteristic
195	C₂×C₈	C₅×#67	1	2	yes	—
196	C₂×C₈	C₂²×C₅×D₃	1	1	yes	characteristic
197	C₄⋊C₄	—	6	2	no	—
198	C₂²⋊C₄	—	12	1	no	—
199	C₄²	—	12	1	no	—
200	C₁₆	—	2	2	no	—
201	C₃×C₅	#2140	1	1	yes	characteristic, complemented, Hall
202	C₂²×C₃	—	2	2	no	—
203	C₂²×C₃	C₅×#102	1	2	yes	—
204	C₂²×C₃	C₂×C₅×D₈	1	1	yes	characteristic
205	C₂×D₃	—	4	4	no	—
206	C₂×D₃	—	8	8	no	—
207	C₂×D₃	—	8	4	no	—
208	C₃×C₄	C₅×#102	1	2	yes	—
209	C₃×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
210	C₃×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
211	Dic₃	—	8	2	no	—
212	C₂×C₅	—	2	2	no	—
213	C₂×C₅	—	24	4	no	—
214	C₂×C₅	#4908	1	2	yes	—
215	C₂×C₅	C₂×#821	1	1	yes	characteristic
216	C₂³	—	12	2	no	—
217	C₂³	—	12	4	no	—
218	C₂³	—	24	4	no	—
219	C₂³	C₅×D₂₄	1	1	yes	characteristic
220	D₄	—	6	4	no	—
221	D₄	—	12	2	no	—
222	D₄	—	12	4	no	—
223	C₂×C₄	—	12	2	no	—
224	C₂×C₄	C₅×#204	1	2	yes	—
225	C₂×C₄	C₅×D₂₄	1	1	yes	characteristic
226	C₂×C₄	C₅×(C₂×D₃)⋊C₄	1	2	yes	—
227	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
228	C₈	C₅×(C₂×D₃)⋊C₄	1	2	yes	—
229	C₈	C₂×C₄×C₅×D₃	1	1	yes	characteristic
230	C₈	C₂×C₅×#67	1	1	yes	characteristic
231	C₂×C₃	—	2	2	no	—
232	C₂×C₃	C₂×C₅×#102	1	1	yes	characteristic
233	C₂×C₃	C₅×#509	1	2	yes	—
234	D₃	—	8	4	no	—
235	C₅	#71440	1	1	yes	characteristic, direct product factor, Sylow
236	C₂²	—	2	2	no	—
237	C₂²	—	12	4	no	—
238	C₂²	—	24	8	no	—
239	C₂²	—	24	4	no	—
240	C₂²	C₅×#821	1	2	yes	—
241	C₂²	C₂×C₅×D₂₄	1	1	yes	characteristic
242	C₄	—	24	2	no	—
243	C₄	C₅×#821	1	2	yes	—
244	C₄	C₂×C₅×#204	1	1	yes	characteristic
245	C₄	C₂×C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
246	C₃	C₅×#2140	1	1	yes	characteristic, complemented, Sylow
247	C₂	—	2	2	no	—
248	C₂	—	24	4	no	—
249	C₂	C₅×#4908	1	2	yes	—
250	C₂	C₂×C₅×#821	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	C₂×D₂₄
commutator	C₃×C₈	C₂²×C₄×C₅
Frattini	C₂×C₈	C₂²×C₅×D₃
Fitting	C₂×C₃×C₅×M₃₂	C₂
socle	C₂²×C₃×C₅	C₂×D₈
prime radical	C₂²×C₅×D₂₄	C₂
cosocle	C₂×C₃×C₈	C₂³×C₅
hypercenter	C₂×C₅×M₃₂	D₃
hypocenter	C₃	C₅×#2140
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₂²×C₅×C₈ ◁ C₂×C₅×M₃₂

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₅×M₃₂	C₂	semidirect
3. C₅×#5106	C₂	semidirect
4. C₅×#4509	C₂	semidirect
11. C₂×C₅×D₂₄	C₄	semidirect
14. C₂×C₃×C₅×C₁₆	C₂²	semidirect
15. C₅×#818	C₂²	semidirect
17. #71440	C₅	direct (N × H)
19. C₂×C₅×M₃₂	D₃	semidirect
40. C₂×C₃×M₃₂	C₂×C₅	semidirect
41. #5106	C₂×C₅	semidirect
42. #4509	C₂×C₅	semidirect
51. C₂×C₅×C₁₆	C₂×D₃	semidirect
75. C₂×D₂₄	C₄×C₅	semidirect
78. C₂×C₃×C₁₆	C₂²×C₅	semidirect
79. #818	C₂²×C₅	semidirect
99. C₂×M₃₂	C₅×D₃	semidirect
152. C₂×C₁₆	C₂×C₅×D₃	semidirect
201. C₃×C₅	#2140	semidirect
235. C₅	#71440	direct (N × H)
246. C₃	C₅×#2140	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₅×D₂₄	2
C₂×C₃×C₅×M₃₂	2
C₅×#5106	2
C₅×#4509	2
C₅×#2140	3
#71440	5

Subgroups — C5×#71440 #14313939

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#71440 #14313939