Subgroups — C₂×C₅×#5085 #14438733

Order 1920.

1312 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₅×#215	C₂	1	4	yes	maximal, complemented
3	C₂×C₅×#874	C₂	1	1	yes	maximal, characteristic
4	C₅×#5085	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₄×C₅×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅×#492	—	3	1	no	maximal, Hall
7	C₂²×C₃×C₅×Q₈	C₂²	1	2	yes	—
8	C₂²×C₅×D₁₂	—	2	2	no	—
9	C₂²×C₅×D₁₂	C₂²	1	2	yes	—
10	C₃×C₅×C₄⋊Q₈	—	2	2	no	—
11	C₃×C₅×C₄⋊Q₈	C₂²	1	4	yes	complemented
12	C₂×C₃×C₅×C₄⋊C₄	—	2	2	no	—
13	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
14	C₂×C₅×#215	—	2	16	no	—
15	C₂×C₅×#215	C₂²	1	16	yes	complemented
16	C₂²×C₅×C₃⋊C₈	C₂²	1	2	yes	—
17	C₅×#874	—	2	2	no	—
18	C₅×#874	C₂²	1	4	yes	—
19	C₄×C₅×C₃⋊C₈	C₂²	1	4	yes	complemented
20	C₂×#5085	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
21	C₂²×C₅×SD₁₆	—	3	4	no	—
22	C₂×C₅×C₄⋊Q₈	D₃	1	1	yes	characteristic, complemented
23	C₂×C₅×#127	—	3	1	no	—
24	C₅×#492	—	3	8	no	—
25	C₂×C₄×C₅×C₈	—	3	1	no	—
26	C₂³×C₅×D₃	—	2	2	no	—
27	C₂×C₃×C₅×Q₈	—	2	2	no	—
28	C₂×C₃×C₅×Q₈	—	2	8	no	—
29	C₂×C₃×C₅×Q₈	C₂³	1	4	yes	—
30	C₂²×C₃×C₄×C₅	—	2	2	no	—
31	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
32	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
33	C₂×C₅×D₁₂	—	2	8	no	—
34	C₂×C₅×D₁₂	—	2	2	no	—
35	C₂×C₅×D₁₂	—	2	8	no	—
36	C₂×C₅×D₁₂	—	4	8	no	—
37	C₂×C₅×D₁₂	C₂³	1	4	yes	—
38	C₃×C₅×C₄⋊C₄	—	2	8	no	—
39	C₃×C₄²×C₅	D₄	1	2	yes	—
40	C₃×C₄²×C₅	C₂³	1	2	yes	—
41	C₅×#215	—	2	32	no	—
42	C₂×C₅×C₃⋊C₈	D₄	1	8	yes	complemented
43	C₂×C₅×C₃⋊C₈	C₂³	1	4	yes	—
44	C₂×C₃×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic, complemented
45	C₂²×#215	C₂×C₅	1	4	yes	complemented
46	C₂×#874	C₂×C₅	1	1	yes	characteristic
47	#5085	C₂×C₅	1	8	yes	direct product factor
48	C₂×C₄×C₃⋊C₈	C₂×C₅	1	1	yes	characteristic, complemented
49	C₂²×C₅×Q₈	C₂×D₃	1	2	yes	—
50	C₂²×C₅×D₄	—	3	2	no	—
51	C₂²×C₅×D₄	—	6	2	no	—
52	C₂×C₅×SD₁₆	—	3	16	no	—
53	C₂×C₅×SD₁₆	—	6	16	no	—
54	C₂²×C₅×C₈	—	3	2	no	—
55	C₅×C₄⋊Q₈	—	2	2	no	—
56	C₅×C₄⋊Q₈	C₂×D₃	1	4	yes	complemented
57	C₅×#127	—	3	4	no	—
58	C₅×#127	—	6	2	no	—
59	C₂×C₅×C₄⋊C₄	—	2	2	no	—
60	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
61	C₄×C₅×C₈	—	3	4	no	—
62	C₂×#492	—	3	1	no	Sylow
63	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₅×D₃	—	2	4	no	—
65	C₂²×C₅×D₃	—	4	4	no	—
66	C₂²×C₅×D₃	—	4	8	no	—
67	C₃×C₅×Q₈	—	2	8	no	—
68	C₃×C₅×Q₈	—	4	4	no	—
69	C₂×C₃×C₄×C₅	—	2	4	no	—
70	C₂×C₃×C₄×C₅	—	4	4	no	—
71	C₂×C₃×C₄×C₅	SD₁₆	1	8	yes	complemented
72	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
73	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
74	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
75	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
76	C₅×D₁₂	—	2	8	no	—
77	C₅×D₁₂	—	4	4	no	—
78	C₅×D₁₂	—	4	16	no	—
79	C₅×C₃⋊C₈	C₂×D₄	1	8	yes	complemented
80	C₂²×C₃×Q₈	C₂²×C₅	1	2	yes	—
81	C₂²×D₁₂	—	2	2	no	—
82	C₂²×D₁₂	C₂²×C₅	1	2	yes	—
83	C₃×C₄⋊Q₈	—	2	2	no	—
84	C₃×C₄⋊Q₈	C₂²×C₅	1	4	yes	complemented
85	C₂×C₃×C₄⋊C₄	—	2	2	no	—
86	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
87	C₂×#215	—	2	16	no	—
88	C₂×#215	C₂²×C₅	1	16	yes	complemented
89	C₂²×C₃⋊C₈	C₂²×C₅	1	2	yes	—
90	#874	—	2	2	no	—
91	#874	C₂²×C₅	1	4	yes	—
92	C₄×C₃⋊C₈	C₂²×C₅	1	4	yes	complemented
93	C₂⁴×C₅	—	6	2	no	—
94	C₂×C₅×Q₈	—	2	2	no	—
95	C₂×C₅×Q₈	—	2	8	no	—
96	C₂×C₅×Q₈	C₂²×D₃	1	4	yes	—
97	C₂×C₅×D₄	—	3	4	no	—
98	C₂×C₅×D₄	—	6	8	no	—
99	C₂×C₅×D₄	—	6	8	no	—
100	C₂×C₅×D₄	—	6	2	no	—
101	C₂×C₅×D₄	—	12	8	no	—
102	C₂²×C₄×C₅	—	2	2	no	—
103	C₂²×C₄×C₅	#67	1	2	yes	—
104	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
105	C₅×SD₁₆	—	6	32	no	—
106	C₂×C₅×C₈	—	3	8	no	—
107	C₂×C₅×C₈	—	3	4	no	—
108	C₅×C₄⋊C₄	—	2	8	no	—
109	C₄²×C₅	#67	1	2	yes	—
110	C₄²×C₅	C₂²×D₃	1	2	yes	—
111	C₂²×SD₁₆	—	3	4	no	—
112	C₂×C₄⋊Q₈	C₅×D₃	1	1	yes	characteristic, complemented
113	C₂×#127	—	3	1	no	—
114	#492	—	3	8	no	—
115	C₂×C₄×C₈	—	3	1	no	—
116	C₂²×C₃×C₅	#127	1	2	yes	—
117	C₂²×C₃×C₅	C₂×SD₁₆	1	4	yes	—
118	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₅×D₃	—	4	4	no	—
120	C₂×C₅×D₃	—	4	8	no	—
121	C₂×C₅×D₃	—	8	8	no	—
122	C₃×C₄×C₅	—	4	4	no	—
123	C₃×C₄×C₅	#127	1	2	yes	—
124	C₃×C₄×C₅	C₂×SD₁₆	1	8	yes	complemented
125	C₃×C₄×C₅	C₂²×D₄	1	2	yes	—
126	C₂³×D₃	—	2	2	no	—
127	C₂×C₃×Q₈	—	2	2	no	—
128	C₂×C₃×Q₈	—	2	8	no	—
129	C₂×C₃×Q₈	C₂³×C₅	1	4	yes	—
130	C₂²×C₃×C₄	—	2	2	no	—
131	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
132	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
133	C₂×D₁₂	—	2	8	no	—
134	C₂×D₁₂	—	2	2	no	—
135	C₂×D₁₂	—	2	8	no	—
136	C₂×D₁₂	—	4	8	no	—
137	C₂×D₁₂	C₂³×C₅	1	4	yes	—
138	C₃×C₄⋊C₄	—	2	8	no	—
139	C₃×C₄²	C₅×D₄	1	2	yes	—
140	C₃×C₄²	C₂³×C₅	1	2	yes	—
141	#215	—	2	32	no	—
142	C₂×C₃⋊C₈	C₅×D₄	1	8	yes	complemented
143	C₂×C₃⋊C₈	C₂³×C₅	1	4	yes	—
144	C₂³×C₅	—	6	4	no	—
145	C₂³×C₅	—	12	4	no	—
146	C₂³×C₅	—	12	8	no	—
147	C₂³×C₅	C₂×#67	1	1	yes	characteristic
148	C₅×Q₈	—	2	8	no	—
149	C₅×Q₈	—	4	4	no	—
150	C₅×D₄	—	6	8	no	—
151	C₅×D₄	—	12	4	no	—
152	C₅×D₄	—	12	16	no	—
153	C₂×C₄×C₅	—	2	4	no	—
154	C₂×C₄×C₅	—	4	4	no	—
155	C₂×C₄×C₅	#215	1	8	yes	complemented
156	C₂×C₄×C₅	D₃×D₄	1	4	yes	—
157	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
158	C₂×C₄×C₅	C₂×#67	1	4	yes	—
159	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
160	C₅×C₈	—	3	8	no	—
161	C₂²×Q₈	C₂×C₅×D₃	1	2	yes	—
162	C₂²×D₄	—	3	2	no	—
163	C₂²×D₄	—	6	2	no	—
164	C₂×SD₁₆	—	3	16	no	—
165	C₂×SD₁₆	—	6	16	no	—
166	C₂²×C₈	—	3	2	no	—
167	C₄⋊Q₈	—	2	2	no	—
168	C₄⋊Q₈	C₂×C₅×D₃	1	4	yes	complemented
169	#127	—	3	4	no	—
170	#127	—	6	2	no	—
171	C₂×C₄⋊C₄	—	2	2	no	—
172	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
173	C₄×C₈	—	3	4	no	—
174	C₂×C₃×C₅	#492	1	4	yes	complemented
175	C₂×C₃×C₅	C₂×#127	1	1	yes	characteristic
176	C₂×C₃×C₅	C₂²×SD₁₆	1	2	yes	—
177	C₅×D₃	—	8	4	no	—
178	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
179	C₂²×D₃	—	2	4	no	—
180	C₂²×D₃	—	4	4	no	—
181	C₂²×D₃	—	4	8	no	—
182	C₃×Q₈	—	2	8	no	—
183	C₃×Q₈	—	4	4	no	—
184	C₂×C₃×C₄	—	2	4	no	—
185	C₂×C₃×C₄	—	4	4	no	—
186	C₂×C₃×C₄	C₅×SD₁₆	1	8	yes	complemented
187	C₂×C₃×C₄	C₂×C₅×D₄	1	4	yes	—
188	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
189	C₂×C₃×C₄	C₂×C₅×D₄	1	4	yes	—
190	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
191	D₁₂	—	2	8	no	—
192	D₁₂	—	4	4	no	—
193	D₁₂	—	4	16	no	—
194	C₃⋊C₈	C₂×C₅×D₄	1	8	yes	complemented
195	C₂²×C₅	—	12	4	no	—
196	C₂²×C₅	—	12	8	no	—
197	C₂²×C₅	—	24	8	no	—
198	C₂²×C₅	#940	1	2	yes	—
199	C₂²×C₅	C₂×#215	1	4	yes	—
200	C₂²×C₅	C₂²×#67	1	1	yes	characteristic
201	C₄×C₅	—	4	4	no	—
202	C₄×C₅	#940	1	2	yes	—
203	C₄×C₅	C₂×#215	1	8	yes	complemented
204	C₄×C₅	C₂×D₃×D₄	1	2	yes	—
205	C₂⁴	—	6	2	no	—
206	C₂×Q₈	—	2	2	no	—
207	C₂×Q₈	—	2	8	no	—
208	C₂×Q₈	C₂²×C₅×D₃	1	4	yes	—
209	C₂×D₄	—	3	4	no	—
210	C₂×D₄	—	6	8	no	—
211	C₂×D₄	—	6	8	no	—
212	C₂×D₄	—	6	2	no	—
213	C₂×D₄	—	12	8	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	C₅×#67	1	2	yes	—
216	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
217	SD₁₆	—	6	32	no	—
218	C₂×C₈	—	3	8	no	—
219	C₂×C₈	—	3	4	no	—
220	C₄⋊C₄	—	2	8	no	—
221	C₄²	C₅×#67	1	2	yes	—
222	C₄²	C₂²×C₅×D₃	1	2	yes	—
223	C₃×C₅	C₂×#492	1	1	yes	characteristic, complemented, Hall
224	C₂²×C₃	C₅×#127	1	2	yes	—
225	C₂²×C₃	C₂×C₅×SD₁₆	1	4	yes	—
226	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
227	C₂×D₃	—	4	4	no	—
228	C₂×D₃	—	4	8	no	—
229	C₂×D₃	—	8	8	no	—
230	C₃×C₄	—	4	4	no	—
231	C₃×C₄	C₅×#127	1	2	yes	—
232	C₃×C₄	C₂×C₅×SD₁₆	1	8	yes	complemented
233	C₃×C₄	C₂²×C₅×D₄	1	2	yes	—
234	C₂×C₅	—	24	4	no	—
235	C₂×C₅	#5085	1	4	yes	direct product factor
236	C₂×C₅	C₂×#940	1	1	yes	characteristic
237	C₂×C₅	C₂²×#215	1	2	yes	—
238	C₂³	—	6	4	no	—
239	C₂³	—	12	4	no	—
240	C₂³	—	12	8	no	—
241	C₂³	C₂×C₅×#67	1	1	yes	characteristic
242	Q₈	—	2	8	no	—
243	Q₈	—	4	4	no	—
244	D₄	—	6	8	no	—
245	D₄	—	12	4	no	—
246	D₄	—	12	16	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	C₅×#215	1	8	yes	complemented
250	C₂×C₄	C₅×D₃×D₄	1	4	yes	—
251	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
252	C₂×C₄	C₂×C₅×#67	1	4	yes	—
253	C₂×C₄	C₂³×C₅×D₃	1	1	yes	characteristic
254	C₈	—	3	8	no	—
255	C₂×C₃	C₅×#492	1	4	yes	complemented
256	C₂×C₃	C₂×C₅×#127	1	1	yes	characteristic
257	C₂×C₃	C₂²×C₅×SD₁₆	1	2	yes	—
258	D₃	—	8	4	no	—
259	C₅	C₂×#5085	1	1	yes	characteristic, direct product factor, Sylow
260	C₂²	—	12	4	no	—
261	C₂²	—	12	8	no	—
262	C₂²	—	24	8	no	—
263	C₂²	C₅×#940	1	2	yes	—
264	C₂²	C₂×C₅×#215	1	4	yes	—
265	C₂²	C₂²×C₅×#67	1	1	yes	characteristic
266	C₄	—	4	4	no	—
267	C₄	C₅×#940	1	2	yes	—
268	C₄	C₂×C₅×#215	1	8	yes	complemented
269	C₄	C₂×C₅×D₃×D₄	1	2	yes	—
270	C₃	C₂×C₅×#492	1	1	yes	characteristic, complemented, Sylow
271	C₂	—	24	4	no	—
272	C₂	C₅×#5085	1	4	yes	direct product factor
273	C₂	C₂×C₅×#940	1	1	yes	characteristic
274	C₂	C₂²×C₅×#215	1	2	yes	—

	subgroup	G/H
center	C₂³×C₅	C₂×#67
commutator, cosocle	C₂×C₃×C₄	C₂⁴×C₅
Frattini	C₂×C₄	C₂³×C₅×D₃
Fitting	C₂×C₃×C₅×C₄⋊Q₈	C₂
socle	C₂³×C₃×C₅	C₂×D₄
prime radical	C₂×C₅×#874	C₂
hypercenter	C₂×C₅×C₄⋊Q₈	D₃
hypocenter	C₃	C₂×C₅×#492
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₅×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅×C₄⋊Q₈	C₂	semidirect
2. C₂²×C₅×#215	C₂	semidirect
4. C₅×#5085	C₂	direct (N × H)
5. C₂×C₄×C₅×C₃⋊C₈	C₂	semidirect
11. C₃×C₅×C₄⋊Q₈	C₂²	semidirect
15. C₂×C₅×#215	C₂²	semidirect
19. C₄×C₅×C₃⋊C₈	C₂²	semidirect
20. C₂×#5085	C₅	direct (N × H)
22. C₂×C₅×C₄⋊Q₈	D₃	semidirect
42. C₂×C₅×C₃⋊C₈	D₄	semidirect
44. C₂×C₃×C₄⋊Q₈	C₂×C₅	semidirect
45. C₂²×#215	C₂×C₅	semidirect
47. #5085	C₂×C₅	direct (N × H)
48. C₂×C₄×C₃⋊C₈	C₂×C₅	semidirect
56. C₅×C₄⋊Q₈	C₂×D₃	semidirect
71. C₂×C₃×C₄×C₅	SD₁₆	semidirect
79. C₅×C₃⋊C₈	C₂×D₄	semidirect
84. C₃×C₄⋊Q₈	C₂²×C₅	semidirect
88. C₂×#215	C₂²×C₅	semidirect
92. C₄×C₃⋊C₈	C₂²×C₅	semidirect
112. C₂×C₄⋊Q₈	C₅×D₃	semidirect
124. C₃×C₄×C₅	C₂×SD₁₆	semidirect
142. C₂×C₃⋊C₈	C₅×D₄	semidirect
155. C₂×C₄×C₅	#215	semidirect
168. C₄⋊Q₈	C₂×C₅×D₃	semidirect
174. C₂×C₃×C₅	#492	semidirect
186. C₂×C₃×C₄	C₅×SD₁₆	semidirect
194. C₃⋊C₈	C₂×C₅×D₄	semidirect
203. C₄×C₅	C₂×#215	semidirect
223. C₃×C₅	C₂×#492	semidirect
232. C₃×C₄	C₂×C₅×SD₁₆	semidirect
235. C₂×C₅	#5085	direct (N × H)
249. C₂×C₄	C₅×#215	semidirect
255. C₂×C₃	C₅×#492	semidirect
259. C₅	C₂×#5085	direct (N × H)
268. C₄	C₂×C₅×#215	semidirect
270. C₃	C₂×C₅×#492	semidirect
272. C₂	C₅×#5085	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×C₅×C₄⋊Q₈	2
C₂²×C₅×#215	2
C₂×C₅×#874	2
C₅×#5085	2
C₂×C₄×C₅×C₃⋊C₈	2
C₂×C₅×#492	3
C₂×#5085	5

Subgroups — C2×C5×#5085 #14438733

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#5085 #14438733