Subgroups — C₅×#82556 #14412012

Order 1920.

1432 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×#1007	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#519	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#5812	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#5006	C₂	1	12	yes	maximal, complemented
5	C₅×#2884	—	3	1	no	maximal, Hall
6	C₂²×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
7	C₅×#1007	—	2	4	no	—
8	C₅×#1007	—	2	4	no	—
9	C₂×C₅×#239	C₄	1	4	yes	complemented
10	C₂×C₅×D₃×Q₈	C₄	1	4	yes	complemented
11	C₂×C₅×#235	C₂²	1	6	yes	—
12	C₃×C₄×C₅×Q₈	—	2	4	no	—
13	C₃×C₅×#117	C₂²	1	6	yes	—
14	C₅×Q₈×Dic₃	—	2	4	no	—
15	C₅×#926	C₂²	1	6	yes	—
16	C₅×#806	C₂²	1	16	yes	—
17	#82556	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₂×C₅×2₋¹⁺⁴	—	3	1	no	—
19	C₅×#519	—	3	1	no	—
20	C₅×#519	D₃	1	1	yes	characteristic, complemented
21	C₅×#410	—	3	12	no	—
22	C₂×C₃×C₅×Q₈	C₂×C₄	1	4	yes	—
23	C₂×C₃×C₅×Q₈	D₄	1	4	yes	—
24	C₂×C₃×C₅×Q₈	D₄	1	4	yes	—
25	C₂²×C₃×C₄×C₅	C₂³	1	3	yes	—
26	C₂×C₅×#67	C₂³	1	4	yes	—
27	C₅×#239	—	2	16	no	—
28	C₅×D₃×Q₈	—	2	16	no	—
29	C₅×#235	—	2	12	no	—
30	C₅×#235	—	2	12	no	—
31	C₂×C₅×D₁₂	C₂×C₄	1	6	yes	—
32	C₂×C₄×C₅×D₃	C₂×C₄	1	12	yes	—
33	C₂×C₅×Dic₆	C₂×C₄	1	6	yes	—
34	C₃×C₅×C₄⋊C₄	—	2	6	no	—
35	C₃×C₅×C₂²⋊C₄	C₂³	1	4	yes	—
36	C₃×C₄²×C₅	—	2	6	no	—
37	C₅×#217	C₂³	1	4	yes	—
38	C₅×C₄⋊Dic₃	—	2	6	no	—
39	C₄×C₅×Dic₃	—	2	6	no	—
40	C₂×#1007	C₂×C₅	1	1	yes	characteristic
41	C₃×#519	C₂×C₅	1	1	yes	characteristic, complemented
42	#5812	C₂×C₅	1	1	yes	characteristic, complemented
43	#5006	C₂×C₅	1	12	yes	complemented
44	C₅×2₋¹⁺⁴	—	6	4	no	—
45	C₅×2₋¹⁺⁴	—	6	4	no	—
46	C₂×C₅×#41	—	3	4	no	—
47	C₂×C₅×#41	—	3	6	no	—
48	C₂²×C₅×Q₈	—	3	4	no	—
49	C₂²×C₅×Q₈	C₂×D₃	1	1	yes	characteristic
50	C₄×C₅×Q₈	—	2	4	no	—
51	C₄×C₅×Q₈	—	6	4	no	—
52	C₅×#117	—	3	6	no	—
53	C₅×#117	C₂×D₃	1	6	yes	—
54	C₅×#99	—	3	16	no	—
55	#2884	—	3	1	no	Sylow
56	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
57	C₂²×C₅×D₃	C₂²×C₄	1	4	yes	—
58	C₃×C₅×Q₈	C₂²⋊C₄	1	16	yes	—
59	C₂×C₃×C₄×C₅	—	2	4	no	—
60	C₂×C₃×C₄×C₅	C₂²×C₄	1	6	yes	—
61	C₂×C₃×C₄×C₅	C₂×D₄	1	6	yes	—
62	C₂×C₃×C₄×C₅	C₂×D₄	1	6	yes	—
63	C₅×#67	—	2	4	no	—
64	C₅×#67	—	2	4	no	—
65	C₂×C₅×Dic₃	—	2	4	no	—
66	C₂×C₅×Dic₃	C₂²×C₄	1	4	yes	—
67	C₅×D₁₂	—	2	12	no	—
68	C₄×C₅×D₃	—	2	24	no	—
69	C₅×Dic₆	—	2	12	no	—
70	C₂²×C₃×Q₈	C₂²×C₅	1	1	yes	characteristic
71	#1007	—	2	4	no	—
72	#1007	—	2	4	no	—
73	C₂×#239	C₄×C₅	1	4	yes	complemented
74	C₂×D₃×Q₈	C₄×C₅	1	4	yes	complemented
75	C₂×#235	C₂²×C₅	1	6	yes	—
76	C₃×C₄×Q₈	—	2	4	no	—
77	C₃×#117	C₂²×C₅	1	6	yes	—
78	Q₈×Dic₃	—	2	4	no	—
79	#926	C₂²×C₅	1	6	yes	—
80	#806	C₂²×C₅	1	16	yes	—
81	C₅×#41	—	6	16	no	—
82	C₅×#41	—	6	12	no	—
83	C₅×#41	—	6	12	no	—
84	C₂×C₅×Q₈	—	3	6	no	—
85	C₂×C₅×Q₈	—	6	16	no	—
86	C₂×C₅×Q₈	C₄×D₃	1	4	yes	—
87	C₂×C₅×Q₈	D₁₂	1	4	yes	—
88	C₂×C₅×Q₈	#67	1	4	yes	—
89	C₂×C₅×D₄	—	3	4	no	—
90	C₂×C₅×D₄	—	3	6	no	—
91	C₂²×C₄×C₅	—	3	12	no	—
92	C₂²×C₄×C₅	C₂²×D₃	1	3	yes	—
93	C₅×C₄⋊C₄	—	2	6	no	—
94	C₅×C₄⋊C₄	—	6	6	no	—
95	C₅×C₂²⋊C₄	—	3	4	no	—
96	C₅×C₂²⋊C₄	C₂²×D₃	1	4	yes	—
97	C₄²×C₅	—	2	6	no	—
98	C₄²×C₅	—	6	6	no	—
99	C₂×2₋¹⁺⁴	—	3	1	no	—
100	#519	—	3	1	no	—
101	#519	C₅×D₃	1	1	yes	characteristic, complemented
102	#410	—	3	12	no	—
103	C₂²×C₃×C₅	—	4	1	no	—
104	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
105	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
106	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
107	C₂×C₅×D₃	—	2	4	no	—
108	C₂×C₅×D₃	—	4	4	no	—
109	C₃×C₄×C₅	—	4	4	no	—
110	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	12	yes	—
111	C₅×Dic₃	—	2	4	no	—
112	C₅×Dic₃	—	4	4	no	—
113	C₂×C₃×Q₈	C₂×C₄×C₅	1	4	yes	—
114	C₂×C₃×Q₈	C₅×D₄	1	4	yes	—
115	C₂×C₃×Q₈	C₅×D₄	1	4	yes	—
116	C₂²×C₃×C₄	C₂³×C₅	1	3	yes	—
117	C₂×#67	C₂³×C₅	1	4	yes	—
118	#239	—	2	16	no	—
119	D₃×Q₈	—	2	16	no	—
120	#235	—	2	12	no	—
121	#235	—	2	12	no	—
122	C₂×D₁₂	C₂×C₄×C₅	1	6	yes	—
123	C₂×C₄×D₃	C₂×C₄×C₅	1	12	yes	—
124	C₂×Dic₆	C₂×C₄×C₅	1	6	yes	—
125	C₃×C₄⋊C₄	—	2	6	no	—
126	C₃×C₂²⋊C₄	C₂³×C₅	1	4	yes	—
127	C₃×C₄²	—	2	6	no	—
128	#217	C₂³×C₅	1	4	yes	—
129	C₄⋊Dic₃	—	2	6	no	—
130	C₄×Dic₃	—	2	6	no	—
131	C₂³×C₅	—	3	4	no	—
132	C₂³×C₅	C₂³×D₃	1	1	yes	characteristic
133	C₅×Q₈	—	6	12	no	—
134	C₅×Q₈	(C₂×D₃)⋊C₄	1	16	yes	—
135	C₅×D₄	—	6	4	no	—
136	C₅×D₄	—	6	4	no	—
137	C₅×D₄	—	6	12	no	—
138	C₂×C₄×C₅	—	2	4	no	—
139	C₂×C₄×C₅	—	3	4	no	—
140	C₂×C₄×C₅	—	6	24	no	—
141	C₂×C₄×C₅	—	6	4	no	—
142	C₂×C₄×C₅	C₂×C₄×D₃	1	6	yes	—
143	C₂×C₄×C₅	C₂×D₁₂	1	6	yes	—
144	C₂×C₄×C₅	C₂×#67	1	6	yes	—
145	2₋¹⁺⁴	—	6	4	no	—
146	2₋¹⁺⁴	—	6	4	no	—
147	C₂×#41	—	3	4	no	—
148	C₂×#41	—	3	6	no	—
149	C₂²×Q₈	—	3	4	no	—
150	C₂²×Q₈	C₂×C₅×D₃	1	1	yes	characteristic
151	C₄×Q₈	—	2	4	no	—
152	C₄×Q₈	—	6	4	no	—
153	#117	—	3	6	no	—
154	#117	C₂×C₅×D₃	1	6	yes	—
155	#99	—	3	16	no	—
156	C₂×C₃×C₅	—	2	1	no	—
157	C₂×C₃×C₅	—	2	1	no	—
158	C₂×C₃×C₅	—	2	1	no	—
159	C₂×C₃×C₅	C₂²×C₂²⋊C₄	1	1	yes	characteristic
160	C₅×D₃	—	4	4	no	—
161	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
162	C₂²×D₃	C₂²×C₄×C₅	1	4	yes	—
163	C₃×Q₈	C₅×C₂²⋊C₄	1	16	yes	—
164	C₂×C₃×C₄	—	2	4	no	—
165	C₂×C₃×C₄	C₂²×C₄×C₅	1	6	yes	—
166	C₂×C₃×C₄	C₂×C₅×D₄	1	6	yes	—
167	C₂×C₃×C₄	C₂×C₅×D₄	1	6	yes	—
168	#67	—	2	4	no	—
169	#67	—	2	4	no	—
170	C₂×Dic₃	—	2	4	no	—
171	C₂×Dic₃	C₂²×C₄×C₅	1	4	yes	—
172	D₁₂	—	2	12	no	—
173	C₄×D₃	—	2	24	no	—
174	Dic₆	—	2	12	no	—
175	C₂²×C₅	—	4	1	no	—
176	C₂²×C₅	—	6	4	no	—
177	C₂²×C₅	—	12	4	no	—
178	C₂²×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
179	C₂²×C₅	C₂²×D₁₂	1	1	yes	characteristic
180	C₂²×C₅	C₂²×#67	1	1	yes	characteristic
181	C₄×C₅	—	4	4	no	—
182	C₄×C₅	—	6	4	no	—
183	C₄×C₅	—	12	4	no	—
184	C₄×C₅	C₂×(C₂×D₃)⋊C₄	1	12	yes	—
185	#41	—	6	16	no	—
186	#41	—	6	12	no	—
187	#41	—	6	12	no	—
188	C₂×Q₈	—	3	6	no	—
189	C₂×Q₈	—	6	16	no	—
190	C₂×Q₈	C₄×C₅×D₃	1	4	yes	—
191	C₂×Q₈	C₅×D₁₂	1	4	yes	—
192	C₂×Q₈	C₅×#67	1	4	yes	—
193	C₂×D₄	—	3	4	no	—
194	C₂×D₄	—	3	6	no	—
195	C₂²×C₄	—	3	12	no	—
196	C₂²×C₄	C₂²×C₅×D₃	1	3	yes	—
197	C₄⋊C₄	—	2	6	no	—
198	C₄⋊C₄	—	6	6	no	—
199	C₂²⋊C₄	—	3	4	no	—
200	C₂²⋊C₄	C₂²×C₅×D₃	1	4	yes	—
201	C₄²	—	2	6	no	—
202	C₄²	—	6	6	no	—
203	C₃×C₅	#2884	1	1	yes	characteristic, complemented, Hall
204	C₂²×C₃	—	4	1	no	—
205	C₂²×C₃	C₂³×C₄×C₅	1	1	yes	characteristic
206	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
207	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
208	C₂×D₃	—	2	4	no	—
209	C₂×D₃	—	4	4	no	—
210	C₃×C₄	—	4	4	no	—
211	C₃×C₄	C₂×C₅×C₂²⋊C₄	1	12	yes	—
212	Dic₃	—	2	4	no	—
213	Dic₃	—	4	4	no	—
214	C₂×C₅	—	2	1	no	—
215	C₂×C₅	—	2	1	no	—
216	C₂×C₅	—	2	1	no	—
217	C₂×C₅	—	12	4	no	—
218	C₂×C₅	C₂²×(C₂×D₃)⋊C₄	1	1	yes	characteristic
219	C₂³	—	3	4	no	—
220	C₂³	C₂³×C₅×D₃	1	1	yes	characteristic
221	Q₈	—	6	12	no	—
222	Q₈	C₅×(C₂×D₃)⋊C₄	1	16	yes	—
223	D₄	—	6	4	no	—
224	D₄	—	6	4	no	—
225	D₄	—	6	12	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	3	4	no	—
228	C₂×C₄	—	6	24	no	—
229	C₂×C₄	—	6	4	no	—
230	C₂×C₄	C₂×C₄×C₅×D₃	1	6	yes	—
231	C₂×C₄	C₂×C₅×D₁₂	1	6	yes	—
232	C₂×C₄	C₂×C₅×#67	1	6	yes	—
233	C₂×C₃	—	2	1	no	—
234	C₂×C₃	—	2	1	no	—
235	C₂×C₃	—	2	1	no	—
236	C₂×C₃	C₂²×C₅×C₂²⋊C₄	1	1	yes	characteristic
237	D₃	—	4	4	no	—
238	C₅	#82556	1	1	yes	characteristic, direct product factor, Sylow
239	C₂²	—	4	1	no	—
240	C₂²	—	6	4	no	—
241	C₂²	—	12	4	no	—
242	C₂²	C₂²×C₄×C₅×D₃	1	1	yes	characteristic
243	C₂²	C₂²×C₅×D₁₂	1	1	yes	characteristic
244	C₂²	C₂²×C₅×#67	1	1	yes	characteristic
245	C₄	—	4	4	no	—
246	C₄	—	6	4	no	—
247	C₄	—	12	4	no	—
248	C₄	C₂×C₅×(C₂×D₃)⋊C₄	1	12	yes	—
249	C₃	C₅×#2884	1	1	yes	characteristic, complemented, Sylow
250	C₂	—	2	1	no	—
251	C₂	—	2	1	no	—
252	C₂	—	2	1	no	—
253	C₂	—	12	4	no	—
254	C₂	C₂²×C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₅	C₂²×(C₂×D₃)⋊C₄
commutator	C₂²×C₃	C₂³×C₄×C₅
Frattini	C₂³	C₂³×C₅×D₃
Fitting	C₃×C₅×#519	C₂
socle	C₂×C₃×C₅	C₂²×C₂²⋊C₄
prime radical	C₂×C₅×#1007	C₂
cosocle	C₂³×C₃	C₂⁴×C₅
hypercenter	C₅×#519	D₃
hypocenter	C₃	C₅×#2884
LCS	G ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₂²×C₅×Q₈ ◁ C₅×#519

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#519	C₂	semidirect
3. C₅×#5812	C₂	semidirect
4. C₅×#5006	C₂	semidirect
9. C₂×C₅×#239	C₄	semidirect
10. C₂×C₅×D₃×Q₈	C₄	semidirect
17. #82556	C₅	direct (N × H)
20. C₅×#519	D₃	semidirect
41. C₃×#519	C₂×C₅	semidirect
42. #5812	C₂×C₅	semidirect
43. #5006	C₂×C₅	semidirect
73. C₂×#239	C₄×C₅	semidirect
74. C₂×D₃×Q₈	C₄×C₅	semidirect
101. #519	C₅×D₃	semidirect
203. C₃×C₅	#2884	semidirect
238. C₅	#82556	direct (N × H)
249. C₃	C₅×#2884	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#1007	2
C₃×C₅×#519	2
C₅×#5812	2
C₅×#5006	2
C₅×#2884	3
#82556	5

Subgroups — C5×#82556 #14412012

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#82556 #14412012