Subgroups — C₂×C₅×#5076 #14438727

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×#127	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₅×#214	C₂	1	4	yes	maximal, complemented
3	C₂×C₅×#869	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#5076	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×D₄	—	2	2	no	—
8	C₂²×C₃×C₅×D₄	C₂²	1	2	yes	—
9	C₂²×C₅×Dic₆	C₂²	1	2	yes	—
10	C₃×C₅×#127	—	2	2	no	—
11	C₃×C₅×#127	C₂²	1	4	yes	—
12	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
13	C₂×C₅×#214	—	2	16	no	—
14	C₂×C₅×#214	C₂²	1	16	yes	complemented
15	C₂×C₅×C₄⋊Dic₃	—	2	2	no	—
16	C₂²×C₅×C₃⋊C₈	C₂²	1	2	yes	—
17	C₅×#869	—	2	2	no	—
18	C₅×#869	C₂²	1	4	yes	complemented
19	C₄×C₅×C₃⋊C₈	C₂²	1	4	yes	complemented
20	C₂×#5076	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
21	C₂²×C₅×SD₁₆	—	3	4	no	—
22	C₂×C₅×C₄⋊Q₈	—	3	1	no	—
23	C₂×C₅×#127	D₃	1	1	yes	characteristic
24	C₅×#492	—	3	8	no	—
25	C₂×C₄×C₅×C₈	—	3	1	no	—
26	C₂⁴×C₃×C₅	—	2	2	no	—
27	C₂×C₃×C₅×D₄	—	2	8	no	—
28	C₂×C₃×C₅×D₄	—	2	8	no	—
29	C₂×C₃×C₅×D₄	—	2	2	no	—
30	C₂×C₃×C₅×D₄	—	4	8	no	—
31	C₂×C₃×C₅×D₄	C₂³	1	4	yes	—
32	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
33	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
34	C₂²×C₅×Dic₃	—	2	2	no	—
35	C₂×C₅×Dic₆	—	2	8	no	—
36	C₂×C₅×Dic₆	—	2	2	no	—
37	C₂×C₅×Dic₆	C₂³	1	4	yes	—
38	C₃×C₄²×C₅	D₄	1	2	yes	—
39	C₃×C₄²×C₅	C₂³	1	2	yes	—
40	C₅×#214	—	2	32	no	—
41	C₅×C₄⋊Dic₃	—	2	8	no	—
42	C₂×C₅×C₃⋊C₈	D₄	1	8	yes	complemented
43	C₂×C₅×C₃⋊C₈	C₂³	1	4	yes	—
44	C₂×C₃×#127	C₂×C₅	1	1	yes	characteristic
45	C₂²×#214	C₂×C₅	1	4	yes	complemented
46	C₂×#869	C₂×C₅	1	1	yes	characteristic, complemented
47	#5076	C₂×C₅	1	8	yes	direct product factor
48	C₂×C₄×C₃⋊C₈	C₂×C₅	1	1	yes	characteristic, complemented
49	C₂²×C₅×Q₈	—	3	2	no	—
50	C₂²×C₅×D₄	—	2	2	no	—
51	C₂²×C₅×D₄	C₂×D₃	1	2	yes	—
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	—	3	4	no	—
56	C₅×C₄⋊Q₈	—	6	2	no	—
57	C₅×#127	—	2	2	no	—
58	C₅×#127	C₂×D₃	1	4	yes	—
59	C₂×C₅×C₄⋊C₄	—	6	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	—	2	4	no	—
64	C₂³×C₃×C₅	—	4	4	no	—
65	C₂³×C₃×C₅	—	4	8	no	—
66	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
67	C₃×C₅×D₄	—	2	8	no	—
68	C₃×C₅×D₄	—	4	4	no	—
69	C₃×C₅×D₄	—	4	16	no	—
70	C₂×C₃×C₄×C₅	SD₁₆	1	8	yes	complemented
71	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
72	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
73	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
74	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
75	C₂×C₅×Dic₃	—	2	4	no	—
76	C₂×C₅×Dic₃	—	4	4	no	—
77	C₅×Dic₆	—	2	8	no	—
78	C₅×Dic₆	—	4	4	no	—
79	C₅×C₃⋊C₈	C₂×D₄	1	8	yes	complemented
80	C₂²×C₃×D₄	—	2	2	no	—
81	C₂²×C₃×D₄	C₂²×C₅	1	2	yes	—
82	C₂²×Dic₆	C₂²×C₅	1	2	yes	—
83	C₃×#127	—	2	2	no	—
84	C₃×#127	C₂²×C₅	1	4	yes	—
85	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
86	C₂×#214	—	2	16	no	—
87	C₂×#214	C₂²×C₅	1	16	yes	complemented
88	C₂×C₄⋊Dic₃	—	2	2	no	—
89	C₂²×C₃⋊C₈	C₂²×C₅	1	2	yes	—
90	#869	—	2	2	no	—
91	#869	C₂²×C₅	1	4	yes	complemented
92	C₄×C₃⋊C₈	C₂²×C₅	1	4	yes	complemented
93	C₂⁴×C₅	—	2	2	no	—
94	C₂×C₅×Q₈	—	3	4	no	—
95	C₂×C₅×Q₈	—	6	2	no	—
96	C₂×C₅×Q₈	—	6	8	no	—
97	C₂×C₅×D₄	—	2	8	no	—
98	C₂×C₅×D₄	—	2	2	no	—
99	C₂×C₅×D₄	—	2	8	no	—
100	C₂×C₅×D₄	—	4	8	no	—
101	C₂×C₅×D₄	C₂²×D₃	1	4	yes	—
102	C₂²×C₄×C₅	—	6	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	—	6	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	—	3	1	no	—
113	C₂×#127	C₅×D₃	1	1	yes	characteristic
114	#492	—	3	8	no	—
115	C₂×C₄×C₈	—	3	1	no	—
116	C₂²×C₃×C₅	—	4	4	no	—
117	C₂²×C₃×C₅	—	4	8	no	—
118	C₂²×C₃×C₅	—	8	8	no	—
119	C₂²×C₃×C₅	#127	1	2	yes	—
120	C₂²×C₃×C₅	C₂×SD₁₆	1	4	yes	—
121	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
122	C₃×C₄×C₅	#127	1	2	yes	—
123	C₃×C₄×C₅	C₂×SD₁₆	1	8	yes	complemented
124	C₃×C₄×C₅	C₂²×D₄	1	2	yes	—
125	C₅×Dic₃	—	4	4	no	—
126	C₂⁴×C₃	—	2	2	no	—
127	C₂×C₃×D₄	—	2	8	no	—
128	C₂×C₃×D₄	—	2	2	no	—
129	C₂×C₃×D₄	—	2	8	no	—
130	C₂×C₃×D₄	—	4	8	no	—
131	C₂×C₃×D₄	C₂³×C₅	1	4	yes	—
132	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
133	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
134	C₂²×Dic₃	—	2	2	no	—
135	C₂×Dic₆	—	2	8	no	—
136	C₂×Dic₆	—	2	2	no	—
137	C₂×Dic₆	C₂³×C₅	1	4	yes	—
138	C₃×C₄²	C₅×D₄	1	2	yes	—
139	C₃×C₄²	C₂³×C₅	1	2	yes	—
140	#214	—	2	32	no	—
141	C₄⋊Dic₃	—	2	8	no	—
142	C₂×C₃⋊C₈	C₅×D₄	1	8	yes	complemented
143	C₂×C₃⋊C₈	C₂³×C₅	1	4	yes	—
144	C₂³×C₅	—	2	4	no	—
145	C₂³×C₅	—	4	8	no	—
146	C₂³×C₅	—	4	4	no	—
147	C₂³×C₅	C₂×#67	1	1	yes	characteristic
148	C₅×Q₈	—	6	8	no	—
149	C₅×Q₈	—	12	4	no	—
150	C₅×D₄	—	2	8	no	—
151	C₅×D₄	—	4	16	no	—
152	C₅×D₄	—	4	4	no	—
153	C₂×C₄×C₅	—	6	4	no	—
154	C₂×C₄×C₅	—	12	4	no	—
155	C₂×C₄×C₅	#214	1	8	yes	complemented
156	C₂×C₄×C₅	D₃×D₄	1	4	yes	—
157	C₂×C₄×C₅	C₂×#67	1	4	yes	—
158	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
159	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
160	C₅×C₈	—	3	8	no	—
161	C₂²×Q₈	—	3	2	no	—
162	C₂²×D₄	—	2	2	no	—
163	C₂²×D₄	C₂×C₅×D₃	1	2	yes	—
164	C₂×SD₁₆	—	3	16	no	—
165	C₂×SD₁₆	—	6	16	no	—
166	C₂²×C₈	—	3	2	no	—
167	C₄⋊Q₈	—	3	4	no	—
168	C₄⋊Q₈	—	6	2	no	—
169	#127	—	2	2	no	—
170	#127	C₂×C₅×D₃	1	4	yes	—
171	C₂×C₄⋊C₄	—	6	2	no	—
172	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
173	C₄×C₈	—	3	4	no	—
174	C₂×C₃×C₅	—	8	4	no	—
175	C₂×C₃×C₅	#492	1	4	yes	complemented
176	C₂×C₃×C₅	C₂×#127	1	1	yes	characteristic
177	C₂×C₃×C₅	C₂²×SD₁₆	1	2	yes	—
178	C₂³×C₃	—	2	4	no	—
179	C₂³×C₃	—	4	4	no	—
180	C₂³×C₃	—	4	8	no	—
181	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
182	C₃×D₄	—	2	8	no	—
183	C₃×D₄	—	4	4	no	—
184	C₃×D₄	—	4	16	no	—
185	C₂×C₃×C₄	C₅×SD₁₆	1	8	yes	complemented
186	C₂×C₃×C₄	C₂×C₅×D₄	1	4	yes	—
187	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
188	C₂×C₃×C₄	C₂×C₅×D₄	1	4	yes	—
189	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
190	C₂×Dic₃	—	2	4	no	—
191	C₂×Dic₃	—	4	4	no	—
192	Dic₆	—	2	8	no	—
193	Dic₆	—	4	4	no	—
194	C₃⋊C₈	C₂×C₅×D₄	1	8	yes	complemented
195	C₂²×C₅	—	4	8	no	—
196	C₂²×C₅	—	4	4	no	—
197	C₂²×C₅	—	8	8	no	—
198	C₂²×C₅	C₂×#214	1	4	yes	—
199	C₂²×C₅	#940	1	2	yes	—
200	C₂²×C₅	C₂²×#67	1	1	yes	characteristic
201	C₄×C₅	—	12	4	no	—
202	C₄×C₅	C₂×#214	1	8	yes	complemented
203	C₄×C₅	#940	1	2	yes	—
204	C₄×C₅	C₂×D₃×D₄	1	2	yes	—
205	C₂⁴	—	2	2	no	—
206	C₂×Q₈	—	3	4	no	—
207	C₂×Q₈	—	6	2	no	—
208	C₂×Q₈	—	6	8	no	—
209	C₂×D₄	—	2	8	no	—
210	C₂×D₄	—	2	2	no	—
211	C₂×D₄	—	2	8	no	—
212	C₂×D₄	—	4	8	no	—
213	C₂×D₄	C₂²×C₅×D₃	1	4	yes	—
214	C₂²×C₄	—	6	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	—	6	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	—	4	4	no	—
225	C₂²×C₃	—	4	8	no	—
226	C₂²×C₃	—	8	8	no	—
227	C₂²×C₃	C₅×#127	1	2	yes	—
228	C₂²×C₃	C₂×C₅×SD₁₆	1	4	yes	—
229	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
230	C₃×C₄	C₅×#127	1	2	yes	—
231	C₃×C₄	C₂×C₅×SD₁₆	1	8	yes	complemented
232	C₃×C₄	C₂²×C₅×D₄	1	2	yes	—
233	Dic₃	—	4	4	no	—
234	C₂×C₅	—	8	4	no	—
235	C₂×C₅	#5076	1	4	yes	direct product factor
236	C₂×C₅	C₂²×#214	1	2	yes	—
237	C₂×C₅	C₂×#940	1	1	yes	characteristic
238	C₂³	—	2	4	no	—
239	C₂³	—	4	8	no	—
240	C₂³	—	4	4	no	—
241	C₂³	C₂×C₅×#67	1	1	yes	characteristic
242	Q₈	—	6	8	no	—
243	Q₈	—	12	4	no	—
244	D₄	—	2	8	no	—
245	D₄	—	4	16	no	—
246	D₄	—	4	4	no	—
247	C₂×C₄	—	6	4	no	—
248	C₂×C₄	—	12	4	no	—
249	C₂×C₄	C₅×#214	1	8	yes	complemented
250	C₂×C₄	C₅×D₃×D₄	1	4	yes	—
251	C₂×C₄	C₂×C₅×#67	1	4	yes	—
252	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
253	C₂×C₄	C₂³×C₅×D₃	1	1	yes	characteristic
254	C₈	—	3	8	no	—
255	C₂×C₃	—	8	4	no	—
256	C₂×C₃	C₅×#492	1	4	yes	complemented
257	C₂×C₃	C₂×C₅×#127	1	1	yes	characteristic
258	C₂×C₃	C₂²×C₅×SD₁₆	1	2	yes	—
259	C₅	C₂×#5076	1	1	yes	characteristic, direct product factor, Sylow
260	C₂²	—	4	8	no	—
261	C₂²	—	4	4	no	—
262	C₂²	—	8	8	no	—
263	C₂²	C₂×C₅×#214	1	4	yes	—
264	C₂²	C₅×#940	1	2	yes	—
265	C₂²	C₂²×C₅×#67	1	1	yes	characteristic
266	C₄	—	12	4	no	—
267	C₄	C₂×C₅×#214	1	8	yes	complemented
268	C₄	C₅×#940	1	2	yes	—
269	C₄	C₂×C₅×D₃×D₄	1	2	yes	—
270	C₃	C₂×C₅×#492	1	1	yes	characteristic, complemented, Sylow
271	C₂	—	8	4	no	—
272	C₂	C₅×#5076	1	4	yes	direct product factor
273	C₂	C₂²×C₅×#214	1	2	yes	—
274	C₂	C₂×C₅×#940	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₅	C₂×#67
commutator, cosocle	C₂×C₃×C₄	C₂⁴×C₅
Frattini	C₂×C₄	C₂³×C₅×D₃
Fitting, prime radical	C₂×C₃×C₅×#127	C₂
socle	C₂³×C₃×C₅	C₂×D₄
hypercenter	C₂×C₅×#127	D₃
hypocenter	C₃	C₂×C₅×#492
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₅×#127

Complements

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

N	H ≅ G/N	type
2. C₂²×C₅×#214	C₂	semidirect
3. C₂×C₅×#869	C₂	semidirect
4. C₅×#5076	C₂	direct (N × H)
5. C₂×C₄×C₅×C₃⋊C₈	C₂	semidirect
14. C₂×C₅×#214	C₂²	semidirect
18. C₅×#869	C₂²	semidirect
19. C₄×C₅×C₃⋊C₈	C₂²	semidirect
20. C₂×#5076	C₅	direct (N × H)
42. C₂×C₅×C₃⋊C₈	D₄	semidirect
45. C₂²×#214	C₂×C₅	semidirect
46. C₂×#869	C₂×C₅	semidirect
47. #5076	C₂×C₅	direct (N × H)
48. C₂×C₄×C₃⋊C₈	C₂×C₅	semidirect
70. C₂×C₃×C₄×C₅	SD₁₆	semidirect
79. C₅×C₃⋊C₈	C₂×D₄	semidirect
87. C₂×#214	C₂²×C₅	semidirect
91. #869	C₂²×C₅	semidirect
92. C₄×C₃⋊C₈	C₂²×C₅	semidirect
123. C₃×C₄×C₅	C₂×SD₁₆	semidirect
142. C₂×C₃⋊C₈	C₅×D₄	semidirect
155. C₂×C₄×C₅	#214	semidirect
175. C₂×C₃×C₅	#492	semidirect
185. C₂×C₃×C₄	C₅×SD₁₆	semidirect
194. C₃⋊C₈	C₂×C₅×D₄	semidirect
202. C₄×C₅	C₂×#214	semidirect
223. C₃×C₅	C₂×#492	semidirect
231. C₃×C₄	C₂×C₅×SD₁₆	semidirect
235. C₂×C₅	#5076	direct (N × H)
249. C₂×C₄	C₅×#214	semidirect
256. C₂×C₃	C₅×#492	semidirect
259. C₅	C₂×#5076	direct (N × H)
267. C₄	C₂×C₅×#214	semidirect
270. C₃	C₂×C₅×#492	semidirect
272. C₂	C₅×#5076	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×#127	2
C₂²×C₅×#214	2
C₂×C₅×#869	2
C₅×#5076	2
C₂×C₄×C₅×C₃⋊C₈	2
C₂×C₅×#492	3
C₂×#5076	5

Subgroups — C2×C5×#5076 #14438727

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#5076 #14438727