Subgroups — C₅×#78467 #14438895

Order 1920.

1144 conjugacy classes of subgroups.

The subgroup lattice has 292 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×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#4699	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#4696	C₂	1	4	yes	maximal, complemented
6	C₅×#4695	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#4693	C₂	1	2	yes	maximal, complemented
8	C₅×#3147	—	3	1	no	maximal, Hall
9	C₂×C₅×#235	C₂²	1	4	yes	—
10	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
11	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
12	C₅×#930	—	2	4	no	—
13	C₂×C₅×Dic₁₂	C₂²	1	4	yes	—
14	C₅×#904	—	2	16	no	—
15	C₂×C₅×D₂₄	C₂²	1	4	yes	—
16	C₂×C₅×#204	C₂²	1	8	yes	—
17	C₅×#875	C₂²	1	4	yes	—
18	C₅×#874	C₂²	1	2	yes	—
19	C₄×C₅×D₁₂	—	2	4	no	—
20	C₅×#869	C₂²	1	2	yes	—
21	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
22	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
23	C₃×C₄×C₅×C₈	C₂²	1	2	yes	—
24	#78467	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
25	C₂×C₅×#135	—	3	4	no	—
26	C₅×#532	—	3	2	no	—
27	C₅×#495	—	3	4	no	—
28	C₅×#494	—	3	1	no	—
29	C₅×#493	—	3	1	no	—
30	C₅×#492	—	3	2	no	—
31	C₂×C₄×C₅×C₈	D₃	1	1	yes	characteristic, complemented
32	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
33	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
34	C₂×C₅×#67	—	2	4	no	—
35	C₅×#235	—	2	16	no	—
36	C₂×C₅×D₁₂	—	2	4	no	—
37	C₂×C₅×D₁₂	C₂³	1	4	yes	—
38	C₂×C₄×C₅×D₃	—	2	4	no	—
39	C₂×C₅×Dic₆	C₂³	1	4	yes	—
40	C₂×C₃×C₅×C₈	D₄	1	8	yes	complemented
41	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
42	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
43	C₃×C₄²×C₅	D₄	1	2	yes	—
44	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
45	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
46	C₅×(C₂×D₃)⋊C₄	—	2	8	no	—
47	C₅×C₄⋊Dic₃	—	2	4	no	—
48	C₅×Dic₁₂	—	2	8	no	—
49	C₅×D₂₄	—	2	8	no	—
50	C₅×#204	—	2	16	no	—
51	C₂×#904	C₂×C₅	1	4	yes	complemented
52	#5477	C₂×C₅	1	2	yes	complemented
53	C₂×C₃×C₄×C₈	C₂×C₅	1	1	yes	characteristic, complemented
54	#4699	C₂×C₅	1	1	yes	characteristic, complemented
55	#4696	C₂×C₅	1	4	yes	complemented
56	#4695	C₂×C₅	1	1	yes	characteristic, complemented
57	#4693	C₂×C₅	1	2	yes	complemented
58	C₂×C₅×#41	—	3	4	no	—
59	C₅×#135	—	6	16	no	—
60	C₂×C₅×Q₁₆	—	3	4	no	—
61	C₂×C₅×SD₁₆	—	3	8	no	—
62	C₂×C₅×D₈	—	3	4	no	—
63	C₂²×C₅×C₈	C₂×D₃	1	2	yes	—
64	C₅×C₄⋊Q₈	—	3	2	no	—
65	C₅×#127	—	3	2	no	—
66	C₅×#124	—	3	4	no	—
67	C₅×#121	—	6	4	no	—
68	C₄×C₅×D₄	—	6	4	no	—
69	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
70	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
71	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
72	C₄×C₅×C₈	C₂×D₃	1	2	yes	—
73	#3147	—	3	1	no	Sylow
74	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₅×D₃	—	2	4	no	—
76	C₂×C₃×C₄×C₅	—	2	4	no	—
77	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
78	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
79	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
80	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
81	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
82	C₅×#67	—	4	8	no	—
83	C₂×C₅×Dic₃	—	2	4	no	—
84	C₅×D₁₂	—	2	8	no	—
85	C₅×D₁₂	—	4	8	no	—
86	C₄×C₅×D₃	—	4	8	no	—
87	C₅×Dic₆	—	2	8	no	—
88	C₃×C₅×C₈	C₂×D₄	1	4	yes	—
89	C₃×C₅×C₈	C₂×D₄	1	4	yes	—
90	C₂×#235	C₂²×C₅	1	4	yes	—
91	C₂²×C₃×C₈	C₂²×C₅	1	2	yes	—
92	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
93	#930	—	2	4	no	—
94	C₂×Dic₁₂	C₂²×C₅	1	4	yes	—
95	#904	—	2	16	no	—
96	C₂×D₂₄	C₂²×C₅	1	4	yes	—
97	C₂×#204	C₂²×C₅	1	8	yes	—
98	#875	C₂²×C₅	1	4	yes	—
99	#874	C₂²×C₅	1	2	yes	—
100	C₄×D₁₂	—	2	4	no	—
101	#869	C₂²×C₅	1	2	yes	—
102	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
103	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
104	C₃×C₄×C₈	C₂²×C₅	1	2	yes	—
105	C₅×#41	—	6	16	no	—
106	C₂×C₅×Q₈	—	3	4	no	—
107	C₂×C₅×D₄	—	3	4	no	—
108	C₂×C₅×D₄	—	6	4	no	—
109	C₂×C₅×D₄	—	6	4	no	—
110	C₂²×C₄×C₅	—	6	4	no	—
111	C₂²×C₄×C₅	D₁₂	1	2	yes	—
112	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
113	C₅×Q₁₆	—	6	8	no	—
114	C₅×SD₁₆	—	6	16	no	—
115	C₅×D₈	—	6	8	no	—
116	C₂×C₅×C₈	D₁₂	1	8	yes	complemented
117	C₂×C₅×C₈	C₂²×D₃	1	2	yes	—
118	C₂×C₅×C₈	C₂²×D₃	1	2	yes	—
119	C₅×C₄⋊C₄	—	6	4	no	—
120	C₅×C₂²⋊C₄	—	6	8	no	—
121	C₄²×C₅	D₁₂	1	2	yes	—
122	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
123	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
124	C₂×#135	—	3	4	no	—
125	#532	—	3	2	no	—
126	#495	—	3	4	no	—
127	#494	—	3	1	no	—
128	#493	—	3	1	no	—
129	#492	—	3	2	no	—
130	C₂×C₄×C₈	C₅×D₃	1	1	yes	characteristic, complemented
131	C₂²×C₃×C₅	—	2	2	no	—
132	C₂²×C₃×C₅	#127	1	2	yes	—
133	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₅×D₃	—	4	4	no	—
135	C₂×C₅×D₃	—	4	8	no	—
136	C₃×C₄×C₅	—	2	2	no	—
137	C₃×C₄×C₅	#127	1	2	yes	—
138	C₃×C₄×C₅	#135	1	4	yes	complemented
139	C₃×C₄×C₅	C₂²×D₄	1	2	yes	—
140	C₅×Dic₃	—	4	4	no	—
141	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
142	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
143	C₂×#67	—	2	4	no	—
144	#235	—	2	16	no	—
145	C₂×D₁₂	—	2	4	no	—
146	C₂×D₁₂	C₂³×C₅	1	4	yes	—
147	C₂×C₄×D₃	—	2	4	no	—
148	C₂×Dic₆	C₂³×C₅	1	4	yes	—
149	C₂×C₃×C₈	C₅×D₄	1	8	yes	complemented
150	C₂×C₃×C₈	C₂³×C₅	1	2	yes	—
151	C₂×C₃×C₈	C₂³×C₅	1	2	yes	—
152	C₃×C₄²	C₅×D₄	1	2	yes	—
153	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
154	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
155	(C₂×D₃)⋊C₄	—	2	8	no	—
156	C₄⋊Dic₃	—	2	4	no	—
157	Dic₁₂	—	2	8	no	—
158	D₂₄	—	2	8	no	—
159	#204	—	2	16	no	—
160	C₂³×C₅	—	6	4	no	—
161	C₂³×C₅	C₂×D₁₂	1	1	yes	characteristic
162	C₅×Q₈	—	6	8	no	—
163	C₅×D₄	—	6	8	no	—
164	C₅×D₄	—	12	8	no	—
165	C₅×D₄	—	12	8	no	—
166	C₂×C₄×C₅	—	2	4	no	—
167	C₂×C₄×C₅	—	6	4	no	—
168	C₂×C₄×C₅	—	12	8	no	—
169	C₂×C₄×C₅	C₂×D₁₂	1	2	yes	—
170	C₂×C₄×C₅	C₂×D₁₂	1	4	yes	—
171	C₂×C₄×C₅	C₂×D₁₂	1	2	yes	—
172	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
173	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
174	C₅×C₈	C₂×D₁₂	1	4	yes	—
175	C₅×C₈	C₂×D₁₂	1	4	yes	—
176	C₂×#41	—	3	4	no	—
177	#135	—	6	16	no	—
178	C₂×Q₁₆	—	3	4	no	—
179	C₂×SD₁₆	—	3	8	no	—
180	C₂×D₈	—	3	4	no	—
181	C₂²×C₈	C₂×C₅×D₃	1	2	yes	—
182	C₄⋊Q₈	—	3	2	no	—
183	#127	—	3	2	no	—
184	#124	—	3	4	no	—
185	#121	—	6	4	no	—
186	C₄×D₄	—	6	4	no	—
187	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
188	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
189	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
190	C₄×C₈	C₂×C₅×D₃	1	2	yes	—
191	C₂×C₃×C₅	—	2	2	no	—
192	C₂×C₃×C₅	C₂×#127	1	1	yes	characteristic
193	C₂×C₃×C₅	C₂×#135	1	2	yes	—
194	C₅×D₃	—	8	4	no	—
195	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
196	C₂²×D₃	—	2	4	no	—
197	C₂×C₃×C₄	—	2	4	no	—
198	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
199	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
200	C₂×C₃×C₄	C₂×C₅×D₄	1	4	yes	—
201	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
202	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
203	#67	—	4	8	no	—
204	C₂×Dic₃	—	2	4	no	—
205	D₁₂	—	2	8	no	—
206	D₁₂	—	4	8	no	—
207	C₄×D₃	—	4	8	no	—
208	Dic₆	—	2	8	no	—
209	C₃×C₈	C₂×C₅×D₄	1	4	yes	—
210	C₃×C₈	C₂×C₅×D₄	1	4	yes	—
211	C₂²×C₅	—	2	2	no	—
212	C₂²×C₅	—	12	4	no	—
213	C₂²×C₅	—	12	8	no	—
214	C₂²×C₅	#874	1	2	yes	—
215	C₂²×C₅	C₂²×D₁₂	1	1	yes	characteristic
216	C₄×C₅	—	2	2	no	—
217	C₄×C₅	—	12	4	no	—
218	C₄×C₅	#874	1	2	yes	—
219	C₄×C₅	#904	1	4	yes	complemented
220	C₄×C₅	C₂²×D₁₂	1	2	yes	—
221	#41	—	6	16	no	—
222	C₂×Q₈	—	3	4	no	—
223	C₂×D₄	—	3	4	no	—
224	C₂×D₄	—	6	4	no	—
225	C₂×D₄	—	6	4	no	—
226	C₂²×C₄	—	6	4	no	—
227	C₂²×C₄	C₅×D₁₂	1	2	yes	—
228	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
229	Q₁₆	—	6	8	no	—
230	SD₁₆	—	6	16	no	—
231	D₈	—	6	8	no	—
232	C₂×C₈	C₅×D₁₂	1	8	yes	complemented
233	C₂×C₈	C₂²×C₅×D₃	1	2	yes	—
234	C₂×C₈	C₂²×C₅×D₃	1	2	yes	—
235	C₄⋊C₄	—	6	4	no	—
236	C₂²⋊C₄	—	6	8	no	—
237	C₄²	C₅×D₁₂	1	2	yes	—
238	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
239	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
240	C₃×C₅	#3147	1	1	yes	characteristic, complemented, Hall
241	C₂²×C₃	—	2	2	no	—
242	C₂²×C₃	C₅×#127	1	2	yes	—
243	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
244	C₂×D₃	—	4	4	no	—
245	C₂×D₃	—	4	8	no	—
246	C₃×C₄	—	2	2	no	—
247	C₃×C₄	C₅×#127	1	2	yes	—
248	C₃×C₄	C₅×#135	1	4	yes	complemented
249	C₃×C₄	C₂²×C₅×D₄	1	2	yes	—
250	Dic₃	—	4	4	no	—
251	C₂×C₅	—	2	2	no	—
252	C₂×C₅	—	24	4	no	—
253	C₂×C₅	C₂×#874	1	1	yes	characteristic
254	C₂×C₅	C₂×#904	1	2	yes	—
255	C₂³	—	6	4	no	—
256	C₂³	C₂×C₅×D₁₂	1	1	yes	characteristic
257	Q₈	—	6	8	no	—
258	D₄	—	6	8	no	—
259	D₄	—	12	8	no	—
260	D₄	—	12	8	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	6	4	no	—
263	C₂×C₄	—	12	8	no	—
264	C₂×C₄	C₂×C₅×D₁₂	1	2	yes	—
265	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
266	C₂×C₄	C₂×C₅×D₁₂	1	4	yes	—
267	C₂×C₄	C₂×C₅×D₁₂	1	2	yes	—
268	C₂×C₄	C₂³×C₅×D₃	1	1	yes	characteristic
269	C₈	C₂×C₅×D₁₂	1	4	yes	—
270	C₈	C₂×C₅×D₁₂	1	4	yes	—
271	C₂×C₃	—	2	2	no	—
272	C₂×C₃	C₂×C₅×#127	1	1	yes	characteristic
273	C₂×C₃	C₂×C₅×#135	1	2	yes	—
274	D₃	—	8	4	no	—
275	C₅	#78467	1	1	yes	characteristic, direct product factor, Sylow
276	C₂²	—	2	2	no	—
277	C₂²	—	12	4	no	—
278	C₂²	—	12	8	no	—
279	C₂²	C₅×#874	1	2	yes	—
280	C₂²	C₂²×C₅×D₁₂	1	1	yes	characteristic
281	C₄	—	2	2	no	—
282	C₄	—	12	4	no	—
283	C₄	C₅×#874	1	2	yes	—
284	C₄	C₅×#904	1	4	yes	complemented
285	C₄	C₂²×C₅×D₁₂	1	2	yes	—
286	C₃	C₅×#3147	1	1	yes	characteristic, complemented, Sylow
287	C₂	—	2	2	no	—
288	C₂	—	24	4	no	—
289	C₂	C₂×C₅×#874	1	1	yes	characteristic
290	C₂	C₂×C₅×#904	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×C₄×C₅×C₈	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
hypercenter	C₂×C₄×C₅×C₈	D₃
hypocenter	C₃	C₅×#3147
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×C₈

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×C₄×C₅×C₈	C₂	semidirect
4. C₅×#4699	C₂	semidirect
5. C₅×#4696	C₂	semidirect
6. C₅×#4695	C₂	semidirect
7. C₅×#4693	C₂	semidirect
24. #78467	C₅	direct (N × H)
31. C₂×C₄×C₅×C₈	D₃	semidirect
40. C₂×C₃×C₅×C₈	D₄	semidirect
51. C₂×#904	C₂×C₅	semidirect
52. #5477	C₂×C₅	semidirect
53. C₂×C₃×C₄×C₈	C₂×C₅	semidirect
54. #4699	C₂×C₅	semidirect
55. #4696	C₂×C₅	semidirect
56. #4695	C₂×C₅	semidirect
57. #4693	C₂×C₅	semidirect
116. C₂×C₅×C₈	D₁₂	semidirect
130. C₂×C₄×C₈	C₅×D₃	semidirect
138. C₃×C₄×C₅	#135	semidirect
149. C₂×C₃×C₈	C₅×D₄	semidirect
219. C₄×C₅	#904	semidirect
232. C₂×C₈	C₅×D₁₂	semidirect
240. C₃×C₅	#3147	semidirect
248. C₃×C₄	C₅×#135	semidirect
275. C₅	#78467	direct (N × H)
284. C₄	C₅×#904	semidirect
286. C₃	C₅×#3147	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×C₄×C₅×C₈	2
C₅×#4699	2
C₅×#4696	2
C₅×#4695	2
C₅×#4693	2
C₅×#3147	3
#78467	5

Subgroups — C5×#78467 #14438895

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#78467 #14438895