Subgroups — C₅×#78445 #14418475

Order 1920.

728 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#5470	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
3	C₅×#5102	C₂	1	2	yes	maximal
4	C₅×#5101	C₂	1	2	yes	maximal
5	C₅×#4924	C₂	1	1	yes	maximal, characteristic
6	C₅×#4679	C₂	1	4	yes	maximal, complemented
7	C₅×C₈×Dic₆	C₂	1	4	yes	maximal, complemented
8	C₅×#2960	—	3	1	no	maximal, Hall
9	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
10	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
12	C₅×#926	C₂²	1	2	yes	—
13	C₅×#924	C₄	1	4	yes	—
14	C₂×C₅×#208	C₂²	1	2	yes	—
15	C₅×#870	C₄	1	2	yes	—
16	C₅×#869	C₄	1	2	yes	—
17	C₄×C₅×Dic₆	C₂²	1	4	yes	—
18	C₃×C₄×C₅×C₈	C₂²	1	2	yes	—
19	C₃×C₄×C₅×C₈	C₂²	1	2	yes	—
20	C₅×#815	C₂²	1	4	yes	—
21	C₅×Dic₃⋊C₈	C₂²	1	8	yes	—
22	C₅×C₈×Dic₃	C₂²	1	4	yes	—
23	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	4	yes	—
24	#78445	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
25	C₅×#533	—	3	1	no	—
26	C₅×#446	—	3	4	no	—
27	C₅×C₈×Q₈	—	3	4	no	—
28	C₅×#424	—	3	2	no	—
29	C₅×#423	—	3	1	no	—
30	C₅×#405	—	3	2	no	—
31	C₂×C₄×C₅×C₈	D₃	1	1	yes	characteristic
32	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
33	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
34	C₂×C₅×Dic₆	C₂×C₄	1	4	yes	—
35	C₂×C₃×C₅×C₈	—	2	2	no	—
36	C₂×C₃×C₅×C₈	Q₈	1	4	yes	complemented
37	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
38	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
39	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
40	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
41	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
42	C₃×C₄²×C₅	C₂³	1	2	yes	—
43	C₅×#217	C₂×C₄	1	4	yes	—
44	C₅×C₄⋊Dic₃	C₂×C₄	1	4	yes	—
45	C₅×C₄⋊Dic₃	C₂×C₄	1	4	yes	—
46	C₅×Dic₃⋊C₄	C₂×C₄	1	8	yes	—
47	C₄×C₅×Dic₃	C₂³	1	4	yes	—
48	C₅×#208	—	2	4	no	—
49	C₂×C₅×C₃⋊C₈	C₂³	1	4	yes	—
50	#5470	C₂×C₅	1	1	yes	characteristic
51	C₂×C₃×C₄×C₈	C₂×C₅	1	1	yes	characteristic
52	#5102	C₂×C₅	1	2	yes	—
53	#5101	C₂×C₅	1	2	yes	—
54	#4924	C₂×C₅	1	1	yes	characteristic
55	#4679	C₂×C₅	1	4	yes	complemented
56	C₈×Dic₆	C₂×C₅	1	4	yes	complemented
57	C₂×C₅×M₁₆	—	3	2	no	—
58	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
59	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
60	C₅×C₄⋊Q₈	—	3	2	no	—
61	C₅×#125	—	3	2	no	—
62	C₅×C₂²⋊Q₈	—	3	4	no	—
63	C₄×C₅×Q₈	—	3	4	no	—
64	C₅×#117	—	3	2	no	—
65	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
66	C₅×C₄⋊C₈	—	3	8	no	—
67	C₅×C₄⋊C₈	—	3	4	no	—
68	C₅×#97	—	3	4	no	—
69	C₄×C₅×C₈	—	3	4	no	—
70	C₄×C₅×C₈	C₂×D₃	1	2	yes	—
71	C₄×C₅×C₈	C₂×D₃	1	2	yes	—
72	#2960	—	3	1	no	Sylow
73	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₃×C₄×C₅	—	2	4	no	—
75	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
76	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
78	C₂×C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
79	C₂×C₃×C₄×C₅	#41	1	2	yes	—
80	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
81	C₂×C₅×Dic₃	C₂²×C₄	1	8	yes	—
82	C₅×Dic₆	—	2	8	no	—
83	C₃×C₅×C₈	—	2	2	no	—
84	C₃×C₅×C₈	C₂×Q₈	1	2	yes	—
85	C₃×C₅×C₈	C₂×Q₈	1	2	yes	—
86	C₅×C₃⋊C₈	—	2	4	no	—
87	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
88	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
89	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
90	#926	C₂²×C₅	1	2	yes	—
91	#924	C₄×C₅	1	4	yes	—
92	C₂×#208	C₂²×C₅	1	2	yes	—
93	#870	C₄×C₅	1	2	yes	—
94	#869	C₄×C₅	1	2	yes	—
95	C₄×Dic₆	C₂²×C₅	1	4	yes	—
96	C₃×C₄×C₈	C₂²×C₅	1	2	yes	—
97	C₃×C₄×C₈	C₂²×C₅	1	2	yes	—
98	#815	C₂²×C₅	1	4	yes	—
99	Dic₃⋊C₈	C₂²×C₅	1	8	yes	—
100	C₈×Dic₃	C₂²×C₅	1	4	yes	—
101	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	4	yes	—
102	C₂×C₅×Q₈	—	3	4	no	—
103	C₂²×C₄×C₅	C₄×D₃	1	2	yes	—
104	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
105	C₅×M₁₆	—	6	4	no	—
106	C₂×C₅×C₈	—	2	2	no	—
107	C₂×C₅×C₈	—	3	4	no	—
108	C₂×C₅×C₈	Dic₆	1	4	yes	complemented
109	C₂×C₅×C₈	C₂²×D₃	1	2	yes	—
110	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
111	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
112	C₅×C₄⋊C₄	—	3	4	no	—
113	C₅×C₄⋊C₄	—	3	8	no	—
114	C₅×C₄⋊C₄	—	3	4	no	—
115	C₅×C₂²⋊C₄	—	3	4	no	—
116	C₄²×C₅	—	3	4	no	—
117	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
118	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
119	C₄²×C₅	C₂²×D₃	1	2	yes	—
120	#533	—	3	1	no	—
121	#446	—	3	4	no	—
122	C₈×Q₈	—	3	4	no	—
123	#424	—	3	2	no	—
124	#423	—	3	1	no	—
125	#405	—	3	2	no	—
126	C₂×C₄×C₈	C₅×D₃	1	1	yes	characteristic
127	C₂²×C₃×C₅	—	2	2	no	—
128	C₂²×C₃×C₅	C₄×Q₈	1	2	yes	—
129	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
130	C₃×C₄×C₅	—	2	2	no	—
131	C₃×C₄×C₅	C₄×Q₈	1	2	yes	—
132	C₃×C₄×C₅	#131	1	4	yes	—
133	C₃×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
134	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
135	C₅×Dic₃	—	2	8	no	—
136	C₂²×C₃×C₄	C₂×C₄×C₅	1	2	yes	—
137	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
138	C₂×Dic₆	C₂×C₄×C₅	1	4	yes	—
139	C₂×C₃×C₈	—	2	2	no	—
140	C₂×C₃×C₈	C₅×Q₈	1	4	yes	complemented
141	C₂×C₃×C₈	C₂³×C₅	1	2	yes	—
142	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
143	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
144	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
145	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
146	C₃×C₄²	C₂³×C₅	1	2	yes	—
147	#217	C₂×C₄×C₅	1	4	yes	—
148	C₄⋊Dic₃	C₂×C₄×C₅	1	4	yes	—
149	C₄⋊Dic₃	C₂×C₄×C₅	1	4	yes	—
150	Dic₃⋊C₄	C₂×C₄×C₅	1	8	yes	—
151	C₄×Dic₃	C₂³×C₅	1	4	yes	—
152	#208	—	2	4	no	—
153	C₂×C₃⋊C₈	C₂³×C₅	1	4	yes	—
154	C₂³×C₅	C₂×C₄×D₃	1	1	yes	characteristic
155	C₅×Q₈	—	6	8	no	—
156	C₂×C₄×C₅	—	2	4	no	—
157	C₂×C₄×C₅	—	3	8	no	—
158	C₂×C₄×C₅	C₂×Dic₆	1	2	yes	—
159	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
160	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
161	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
162	C₂×C₄×C₅	#235	1	2	yes	—
163	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
164	C₅×C₈	—	2	2	no	—
165	C₅×C₈	—	6	4	no	—
166	C₅×C₈	C₂×Dic₆	1	2	yes	—
167	C₅×C₈	C₂×Dic₆	1	2	yes	—
168	C₂×M₁₆	—	3	2	no	—
169	C₂²×C₈	C₂×C₅×D₃	1	1	yes	characteristic
170	C₂²×C₈	C₂×C₅×D₃	1	1	yes	characteristic
171	C₄⋊Q₈	—	3	2	no	—
172	#125	—	3	2	no	—
173	C₂²⋊Q₈	—	3	4	no	—
174	C₄×Q₈	—	3	4	no	—
175	#117	—	3	2	no	—
176	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
177	C₄⋊C₈	—	3	8	no	—
178	C₄⋊C₈	—	3	4	no	—
179	#97	—	3	4	no	—
180	C₄×C₈	—	3	4	no	—
181	C₄×C₈	C₂×C₅×D₃	1	2	yes	—
182	C₄×C₈	C₂×C₅×D₃	1	2	yes	—
183	C₂×C₃×C₅	—	2	2	no	—
184	C₂×C₃×C₅	C₂×C₄×Q₈	1	1	yes	characteristic
185	C₂×C₃×C₅	C₂×#131	1	2	yes	—
186	C₂³×C₃	C₂²×C₄×C₅	1	1	yes	characteristic
187	C₂×C₃×C₄	—	2	4	no	—
188	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
189	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
190	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
191	C₂×C₃×C₄	C₂×C₅×Q₈	1	2	yes	—
192	C₂×C₃×C₄	C₅×#41	1	2	yes	—
193	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
194	C₂×Dic₃	C₂²×C₄×C₅	1	8	yes	—
195	Dic₆	—	2	8	no	—
196	C₃×C₈	—	2	2	no	—
197	C₃×C₈	C₂×C₅×Q₈	1	2	yes	—
198	C₃×C₈	C₂×C₅×Q₈	1	2	yes	—
199	C₃⋊C₈	—	2	4	no	—
200	C₂²×C₅	—	2	2	no	—
201	C₂²×C₅	C₄×Dic₆	1	2	yes	—
202	C₂²×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
203	C₄×C₅	—	2	2	no	—
204	C₄×C₅	—	6	8	no	—
205	C₄×C₅	C₄×Dic₆	1	2	yes	—
206	C₄×C₅	#901	1	4	yes	—
207	C₄×C₅	C₂²×Dic₆	1	1	yes	characteristic
208	C₄×C₅	C₂×#235	1	1	yes	characteristic
209	C₂×Q₈	—	3	4	no	—
210	C₂²×C₄	C₄×C₅×D₃	1	2	yes	—
211	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
212	M₁₆	—	6	4	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	3	4	no	—
215	C₂×C₈	C₅×Dic₆	1	4	yes	complemented
216	C₂×C₈	C₂²×C₅×D₃	1	2	yes	—
217	C₂×C₈	C₂²×C₅×D₃	1	1	yes	characteristic
218	C₂×C₈	C₂²×C₅×D₃	1	1	yes	characteristic
219	C₄⋊C₄	—	3	4	no	—
220	C₄⋊C₄	—	3	8	no	—
221	C₄⋊C₄	—	3	4	no	—
222	C₂²⋊C₄	—	3	4	no	—
223	C₄²	—	3	4	no	—
224	C₄²	C₄×C₅×D₃	1	1	yes	characteristic
225	C₄²	C₄×C₅×D₃	1	1	yes	characteristic
226	C₄²	C₂²×C₅×D₃	1	2	yes	—
227	C₃×C₅	#2960	1	1	yes	characteristic, complemented, Hall
228	C₂²×C₃	—	2	2	no	—
229	C₂²×C₃	C₄×C₅×Q₈	1	2	yes	—
230	C₂²×C₃	C₂³×C₄×C₅	1	1	yes	characteristic
231	C₃×C₄	—	2	2	no	—
232	C₃×C₄	C₄×C₅×Q₈	1	2	yes	—
233	C₃×C₄	C₅×#131	1	4	yes	—
234	C₃×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
235	C₃×C₄	C₂×C₅×#41	1	1	yes	characteristic
236	Dic₃	—	2	8	no	—
237	C₂×C₅	—	2	2	no	—
238	C₂×C₅	C₂×C₄×Dic₆	1	1	yes	characteristic
239	C₂×C₅	C₂×#901	1	2	yes	—
240	C₂³	C₂×C₄×C₅×D₃	1	1	yes	characteristic
241	Q₈	—	6	8	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	3	8	no	—
244	C₂×C₄	C₂×C₅×Dic₆	1	2	yes	—
245	C₂×C₄	C₂×C₄×C₅×D₃	1	2	yes	—
246	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
247	C₂×C₄	C₂×C₄×C₅×D₃	1	2	yes	—
248	C₂×C₄	C₅×#235	1	2	yes	—
249	C₂×C₄	C₂³×C₅×D₃	1	1	yes	characteristic
250	C₈	—	2	2	no	—
251	C₈	—	6	4	no	—
252	C₈	C₂×C₅×Dic₆	1	2	yes	—
253	C₈	C₂×C₅×Dic₆	1	2	yes	—
254	C₂×C₃	—	2	2	no	—
255	C₂×C₃	C₂×C₄×C₅×Q₈	1	1	yes	characteristic
256	C₂×C₃	C₂×C₅×#131	1	2	yes	—
257	C₅	#78445	1	1	yes	characteristic, direct product factor, Sylow
258	C₂²	—	2	2	no	—
259	C₂²	C₄×C₅×Dic₆	1	2	yes	—
260	C₂²	C₂²×C₄×C₅×D₃	1	1	yes	characteristic
261	C₄	—	2	2	no	—
262	C₄	—	6	8	no	—
263	C₄	C₄×C₅×Dic₆	1	2	yes	—
264	C₄	C₅×#901	1	4	yes	—
265	C₄	C₂²×C₅×Dic₆	1	1	yes	characteristic
266	C₄	C₂×C₅×#235	1	1	yes	characteristic
267	C₃	C₅×#2960	1	1	yes	characteristic, complemented, Sylow
268	C₂	—	2	2	no	—
269	C₂	C₂×C₄×C₅×Dic₆	1	1	yes	characteristic
270	C₂	C₂×C₅×#901	1	2	yes	—

	subgroup	G/H
center	C₂×C₅×C₈	C₂²×D₃
commutator	C₂²×C₃	C₂³×C₄×C₅
Frattini	C₂×C₄	C₂³×C₅×D₃
Fitting	C₂×C₃×C₄×C₅×C₈	C₂
socle	C₂²×C₃×C₅	C₂³×C₄
prime radical	C₂³×C₃×C₅	C₂²×C₄
cosocle	C₂×C₃×C₄	C₂⁴×C₅
hypercenter	C₂×C₄×C₅×C₈	D₃
hypocenter	C₃	C₅×#2960
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅×C₈ ◁ C₂×C₄×C₅×C₈

Complements

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

N	H ≅ G/N	type
6. C₅×#4679	C₂	semidirect
7. C₅×C₈×Dic₆	C₂	semidirect
24. #78445	C₅	direct (N × H)
36. C₂×C₃×C₅×C₈	Q₈	semidirect
55. #4679	C₂×C₅	semidirect
56. C₈×Dic₆	C₂×C₅	semidirect
108. C₂×C₅×C₈	Dic₆	semidirect
140. C₂×C₃×C₈	C₅×Q₈	semidirect
215. C₂×C₈	C₅×Dic₆	semidirect
227. C₃×C₅	#2960	semidirect
257. C₅	#78445	direct (N × H)
267. C₃	C₅×#2960	semidirect

Maximal subgroups

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

subgroup	index
C₅×#5470	2
C₂×C₃×C₄×C₅×C₈	2
C₅×#5102	2
C₅×#5101	2
C₅×#4924	2
C₅×#4679	2
C₅×C₈×Dic₆	2
C₅×#2960	3
#78445	5

Subgroups — C5×#78445 #14418475

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#78445 #14418475