Subgroups — C₅×#78499 #14439081

Order 1920.

1128 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#904	C₂	1	4	yes	maximal, complemented
2	C₅×#5477	C₂	1	2	yes	maximal, complemented
3	C₂×C₃×C₅×#97	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#4715	C₂	1	4	yes	maximal, complemented
5	C₅×#4712	C₂	1	4	yes	maximal, complemented
6	C₅×#3150	—	3	1	no	maximal, Hall
7	C₂×C₅×#235	C₂²	1	4	yes	—
8	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
9	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₅×#930	—	2	4	no	—
11	C₂×C₅×Dic₁₂	C₂²	1	4	yes	—
12	C₅×#904	—	2	16	no	—
13	C₂×C₅×D₂₄	C₂²	1	4	yes	—
14	C₂×C₅×#204	C₂²	1	8	yes	—
15	C₅×#875	C₂²	1	4	yes	—
16	C₅×#874	C₂²	1	2	yes	—
17	C₄×C₅×D₁₂	—	2	4	no	—
18	C₅×#869	C₂²	1	2	yes	—
19	C₃×C₅×#97	C₂²	1	4	yes	—
20	#78499	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
21	C₂×C₅×#135	—	3	4	no	—
22	C₅×#532	—	3	2	no	—
23	C₅×#497	—	3	4	no	—
24	C₅×#496	—	3	4	no	—
25	C₂×C₅×#97	D₃	1	1	yes	characteristic, complemented
26	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
27	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
28	C₂×C₅×#67	—	2	4	no	—
29	C₅×#235	—	2	16	no	—
30	C₂×C₅×D₁₂	—	2	4	no	—
31	C₂×C₅×D₁₂	C₂³	1	4	yes	—
32	C₂×C₄×C₅×D₃	—	2	4	no	—
33	C₂×C₅×Dic₆	C₂³	1	4	yes	—
34	C₂×C₃×C₅×C₈	D₄	1	8	yes	complemented
35	C₂×C₃×C₅×C₈	C₂³	1	4	yes	—
36	C₃×C₄²×C₅	D₄	1	2	yes	—
37	C₃×C₄²×C₅	C₂³	1	2	yes	—
38	C₅×(C₂×D₃)⋊C₄	—	2	8	no	—
39	C₅×C₄⋊Dic₃	—	2	4	no	—
40	C₅×Dic₁₂	—	2	8	no	—
41	C₅×D₂₄	—	2	8	no	—
42	C₅×#204	—	2	16	no	—
43	C₂×#904	C₂×C₅	1	4	yes	complemented
44	#5477	C₂×C₅	1	2	yes	complemented
45	C₂×C₃×#97	C₂×C₅	1	1	yes	characteristic, complemented
46	#4715	C₂×C₅	1	4	yes	complemented
47	#4712	C₂×C₅	1	4	yes	complemented
48	C₂×C₅×#41	—	3	4	no	—
49	C₅×#135	—	6	16	no	—
50	C₂×C₅×Q₁₆	—	3	4	no	—
51	C₂×C₅×SD₁₆	—	3	8	no	—
52	C₂×C₅×D₈	—	3	4	no	—
53	C₂²×C₅×C₈	C₂×D₃	1	2	yes	—
54	C₅×C₄⋊Q₈	—	3	2	no	—
55	C₅×#127	—	3	2	no	—
56	C₅×#124	—	3	4	no	—
57	C₅×#121	—	6	4	no	—
58	C₄×C₅×D₄	—	6	4	no	—
59	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
60	C₅×#97	C₂×D₃	1	4	yes	—
61	#3150	—	3	1	no	Sylow
62	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₅×D₃	—	2	4	no	—
64	C₂×C₃×C₄×C₅	—	4	2	no	—
65	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
66	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
67	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
68	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
69	C₅×#67	—	4	8	no	—
70	C₂×C₅×Dic₃	—	2	4	no	—
71	C₅×D₁₂	—	2	8	no	—
72	C₅×D₁₂	—	4	8	no	—
73	C₄×C₅×D₃	—	4	8	no	—
74	C₅×Dic₆	—	2	8	no	—
75	C₃×C₅×C₈	C₂×D₄	1	8	yes	—
76	C₂×#235	C₂²×C₅	1	4	yes	—
77	C₂²×C₃×C₈	C₂²×C₅	1	2	yes	—
78	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
79	#930	—	2	4	no	—
80	C₂×Dic₁₂	C₂²×C₅	1	4	yes	—
81	#904	—	2	16	no	—
82	C₂×D₂₄	C₂²×C₅	1	4	yes	—
83	C₂×#204	C₂²×C₅	1	8	yes	—
84	#875	C₂²×C₅	1	4	yes	—
85	#874	C₂²×C₅	1	2	yes	—
86	C₄×D₁₂	—	2	4	no	—
87	#869	C₂²×C₅	1	2	yes	—
88	C₃×#97	C₂²×C₅	1	4	yes	—
89	C₅×#41	—	6	16	no	—
90	C₂×C₅×Q₈	—	3	4	no	—
91	C₂×C₅×D₄	—	3	4	no	—
92	C₂×C₅×D₄	—	6	4	no	—
93	C₂×C₅×D₄	—	6	4	no	—
94	C₂²×C₄×C₅	—	6	4	no	—
95	C₂²×C₄×C₅	D₁₂	1	2	yes	—
96	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
97	C₅×Q₁₆	—	6	8	no	—
98	C₅×SD₁₆	—	6	16	no	—
99	C₅×D₈	—	6	8	no	—
100	C₂×C₅×C₈	D₁₂	1	8	yes	complemented
101	C₂×C₅×C₈	C₂²×D₃	1	4	yes	—
102	C₅×C₄⋊C₄	—	6	4	no	—
103	C₅×C₂²⋊C₄	—	6	8	no	—
104	C₄²×C₅	D₁₂	1	2	yes	—
105	C₄²×C₅	C₂²×D₃	1	2	yes	—
106	C₂×#135	—	3	4	no	—
107	#532	—	3	2	no	—
108	#497	—	3	4	no	—
109	#496	—	3	4	no	—
110	C₂×#97	C₅×D₃	1	1	yes	characteristic, complemented
111	C₂²×C₃×C₅	—	2	2	no	—
112	C₂²×C₃×C₅	#127	1	2	yes	—
113	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₅×D₃	—	4	4	no	—
115	C₂×C₅×D₃	—	4	8	no	—
116	C₃×C₄×C₅	—	2	4	no	—
117	C₃×C₄×C₅	#127	1	2	yes	—
118	C₃×C₄×C₅	C₂²×D₄	1	2	yes	—
119	C₅×Dic₃	—	4	4	no	—
120	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
121	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
122	C₂×#67	—	2	4	no	—
123	#235	—	2	16	no	—
124	C₂×D₁₂	—	2	4	no	—
125	C₂×D₁₂	C₂³×C₅	1	4	yes	—
126	C₂×C₄×D₃	—	2	4	no	—
127	C₂×Dic₆	C₂³×C₅	1	4	yes	—
128	C₂×C₃×C₈	C₅×D₄	1	8	yes	complemented
129	C₂×C₃×C₈	C₂³×C₅	1	4	yes	—
130	C₃×C₄²	C₅×D₄	1	2	yes	—
131	C₃×C₄²	C₂³×C₅	1	2	yes	—
132	(C₂×D₃)⋊C₄	—	2	8	no	—
133	C₄⋊Dic₃	—	2	4	no	—
134	Dic₁₂	—	2	8	no	—
135	D₂₄	—	2	8	no	—
136	#204	—	2	16	no	—
137	C₂³×C₅	—	6	4	no	—
138	C₂³×C₅	C₂×D₁₂	1	1	yes	characteristic
139	C₅×Q₈	—	6	8	no	—
140	C₅×D₄	—	6	8	no	—
141	C₅×D₄	—	12	8	no	—
142	C₅×D₄	—	12	8	no	—
143	C₂×C₄×C₅	—	4	2	no	—
144	C₂×C₄×C₅	—	6	4	no	—
145	C₂×C₄×C₅	—	12	8	no	—
146	C₂×C₄×C₅	C₂×D₁₂	1	4	yes	—
147	C₂×C₄×C₅	C₂×D₁₂	1	4	yes	—
148	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
149	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
150	C₅×C₈	C₂×D₁₂	1	8	yes	—
151	C₂×#41	—	3	4	no	—
152	#135	—	6	16	no	—
153	C₂×Q₁₆	—	3	4	no	—
154	C₂×SD₁₆	—	3	8	no	—
155	C₂×D₈	—	3	4	no	—
156	C₂²×C₈	C₂×C₅×D₃	1	2	yes	—
157	C₄⋊Q₈	—	3	2	no	—
158	#127	—	3	2	no	—
159	#124	—	3	4	no	—
160	#121	—	6	4	no	—
161	C₄×D₄	—	6	4	no	—
162	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
163	#97	C₂×C₅×D₃	1	4	yes	—
164	C₂×C₃×C₅	—	2	2	no	—
165	C₂×C₃×C₅	C₂×#127	1	1	yes	characteristic
166	C₂×C₃×C₅	#575	1	2	yes	—
167	C₅×D₃	—	8	4	no	—
168	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
169	C₂²×D₃	—	2	4	no	—
170	C₂×C₃×C₄	—	4	2	no	—
171	C₂×C₃×C₄	C₂×C₅×D₄	1	4	yes	—
172	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
173	C₂×C₃×C₄	C₂×C₅×D₄	1	4	yes	—
174	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
175	#67	—	4	8	no	—
176	C₂×Dic₃	—	2	4	no	—
177	D₁₂	—	2	8	no	—
178	D₁₂	—	4	8	no	—
179	C₄×D₃	—	4	8	no	—
180	Dic₆	—	2	8	no	—
181	C₃×C₈	C₂×C₅×D₄	1	8	yes	—
182	C₂²×C₅	—	2	2	no	—
183	C₂²×C₅	—	12	4	no	—
184	C₂²×C₅	—	12	8	no	—
185	C₂²×C₅	#874	1	2	yes	—
186	C₂²×C₅	C₂²×D₁₂	1	1	yes	characteristic
187	C₄×C₅	—	2	4	no	—
188	C₄×C₅	—	12	4	no	—
189	C₄×C₅	#874	1	2	yes	—
190	C₄×C₅	C₂²×D₁₂	1	2	yes	—
191	#41	—	6	16	no	—
192	C₂×Q₈	—	3	4	no	—
193	C₂×D₄	—	3	4	no	—
194	C₂×D₄	—	6	4	no	—
195	C₂×D₄	—	6	4	no	—
196	C₂²×C₄	—	6	4	no	—
197	C₂²×C₄	C₅×D₁₂	1	2	yes	—
198	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
199	Q₁₆	—	6	8	no	—
200	SD₁₆	—	6	16	no	—
201	D₈	—	6	8	no	—
202	C₂×C₈	C₅×D₁₂	1	8	yes	complemented
203	C₂×C₈	C₂²×C₅×D₃	1	4	yes	—
204	C₄⋊C₄	—	6	4	no	—
205	C₂²⋊C₄	—	6	8	no	—
206	C₄²	C₅×D₁₂	1	2	yes	—
207	C₄²	C₂²×C₅×D₃	1	2	yes	—
208	C₃×C₅	#3150	1	1	yes	characteristic, complemented, Hall
209	C₂²×C₃	—	2	2	no	—
210	C₂²×C₃	C₅×#127	1	2	yes	—
211	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
212	C₂×D₃	—	4	4	no	—
213	C₂×D₃	—	4	8	no	—
214	C₃×C₄	—	2	4	no	—
215	C₃×C₄	C₅×#127	1	2	yes	—
216	C₃×C₄	C₂²×C₅×D₄	1	2	yes	—
217	Dic₃	—	4	4	no	—
218	C₂×C₅	—	2	2	no	—
219	C₂×C₅	—	24	4	no	—
220	C₂×C₅	C₂×#874	1	1	yes	characteristic
221	C₂×C₅	#5748	1	2	yes	—
222	C₂³	—	6	4	no	—
223	C₂³	C₂×C₅×D₁₂	1	1	yes	characteristic
224	Q₈	—	6	8	no	—
225	D₄	—	6	8	no	—
226	D₄	—	12	8	no	—
227	D₄	—	12	8	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	6	4	no	—
230	C₂×C₄	—	12	8	no	—
231	C₂×C₄	C₂×C₅×D₁₂	1	4	yes	—
232	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
233	C₂×C₄	C₂×C₅×D₁₂	1	4	yes	—
234	C₂×C₄	C₂³×C₅×D₃	1	1	yes	characteristic
235	C₈	C₂×C₅×D₁₂	1	8	yes	—
236	C₂×C₃	—	2	2	no	—
237	C₂×C₃	C₂×C₅×#127	1	1	yes	characteristic
238	C₂×C₃	C₅×#575	1	2	yes	—
239	D₃	—	8	4	no	—
240	C₅	#78499	1	1	yes	characteristic, direct product factor, Sylow
241	C₂²	—	2	2	no	—
242	C₂²	—	12	4	no	—
243	C₂²	—	12	8	no	—
244	C₂²	C₅×#874	1	2	yes	—
245	C₂²	C₂²×C₅×D₁₂	1	1	yes	characteristic
246	C₄	—	2	4	no	—
247	C₄	—	12	4	no	—
248	C₄	C₅×#874	1	2	yes	—
249	C₄	C₂²×C₅×D₁₂	1	2	yes	—
250	C₃	C₅×#3150	1	1	yes	characteristic, complemented, Sylow
251	C₂	—	2	2	no	—
252	C₂	—	24	4	no	—
253	C₂	C₂×C₅×#874	1	1	yes	characteristic
254	C₂	C₅×#5748	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅	C₂×D₁₂
commutator, cosocle	C₂×C₃×C₄	C₂⁴×C₅
Frattini	C₂×C₄	C₂³×C₅×D₃
Fitting	C₂×C₃×C₅×#97	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
hypercenter	C₂×C₅×#97	D₃
hypocenter	C₃	C₅×#3150
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₅×#97

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×#904	C₂	semidirect
2. C₅×#5477	C₂	semidirect
3. C₂×C₃×C₅×#97	C₂	semidirect
4. C₅×#4715	C₂	semidirect
5. C₅×#4712	C₂	semidirect
20. #78499	C₅	direct (N × H)
25. C₂×C₅×#97	D₃	semidirect
34. C₂×C₃×C₅×C₈	D₄	semidirect
43. C₂×#904	C₂×C₅	semidirect
44. #5477	C₂×C₅	semidirect
45. C₂×C₃×#97	C₂×C₅	semidirect
46. #4715	C₂×C₅	semidirect
47. #4712	C₂×C₅	semidirect
100. C₂×C₅×C₈	D₁₂	semidirect
110. C₂×#97	C₅×D₃	semidirect
128. C₂×C₃×C₈	C₅×D₄	semidirect
202. C₂×C₈	C₅×D₁₂	semidirect
208. C₃×C₅	#3150	semidirect
240. C₅	#78499	direct (N × H)
250. C₃	C₅×#3150	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#904	2
C₅×#5477	2
C₂×C₃×C₅×#97	2
C₅×#4715	2
C₅×#4712	2
C₅×#3150	3
#78499	5

Subgroups — C5×#78499 #14439081

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#78499 #14439081