Subgroups — C₅×#83339 #14439834

Order 1920.

712 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#533	C₂	1	1	yes	maximal, characteristic
2	C₅×#5470	C₂	1	1	yes	maximal, characteristic
3	C₅×#5080	C₂	1	2	yes	maximal, complemented
4	C₅×#5078	C₂	1	2	yes	maximal, complemented
5	C₅×#5065	C₂	1	4	yes	maximal, complemented
6	C₅×#4998	C₂	1	4	yes	maximal
7	C₂×C₄×C₅×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
8	C₅×#3160	—	3	1	no	maximal, Hall
9	C₃×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₃×C₅×#125	C₂²	1	2	yes	—
11	C₃×C₅×C₂²⋊Q₈	—	2	2	no	—
12	C₃×C₄×C₅×Q₈	—	2	2	no	—
13	C₃×C₅×#117	C₂²	1	2	yes	—
14	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
15	C₅×#926	C₂²	1	2	yes	—
16	C₅×#924	—	2	2	no	—
17	C₂²×C₅×C₃⋊C₈	C₂²	1	2	yes	—
18	C₅×#870	C₂²	1	2	yes	—
19	C₅×#869	C₂²	1	2	yes	—
20	C₄×C₅×Dic₆	—	2	2	no	—
21	C₅×#808	C₂²	1	8	yes	—
22	C₅×#807	C₂²	1	8	yes	—
23	C₄×C₅×C₃⋊C₈	C₂²	1	2	yes	—
24	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
25	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
26	#83339	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
27	C₅×#533	—	3	1	no	—
28	C₅×#533	D₃	1	1	yes	characteristic
29	C₅×#500	—	3	2	no	—
30	C₅×#499	—	3	4	no	—
31	C₅×#498	—	3	2	no	—
32	C₅×#427	—	3	4	no	—
33	C₂×C₄×C₅×C₈	—	3	1	no	—
34	C₂×C₃×C₅×Q₈	—	2	2	no	—
35	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
36	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
37	C₂×C₅×Dic₆	—	2	2	no	—
38	C₃×C₅×C₄⋊C₄	—	2	4	no	—
39	C₃×C₅×C₄⋊C₄	—	2	2	no	—
40	C₃×C₅×C₄⋊C₄	C₂³	1	4	yes	—
41	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
42	C₃×C₄²×C₅	—	2	2	no	—
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×#217	—	2	2	no	—
47	C₅×C₄⋊Dic₃	—	2	2	no	—
48	C₅×C₄⋊Dic₃	C₂³	1	4	yes	—
49	C₅×Dic₃⋊C₄	—	2	4	no	—
50	C₄×C₅×Dic₃	—	2	2	no	—
51	C₂×C₅×C₃⋊C₈	Q₈	1	8	yes	complemented
52	C₂×C₅×C₃⋊C₈	C₂³	1	2	yes	—
53	C₂×C₅×C₃⋊C₈	C₂³	1	2	yes	—
54	C₃×#533	C₂×C₅	1	1	yes	characteristic
55	#5470	C₂×C₅	1	1	yes	characteristic
56	#5080	C₂×C₅	1	2	yes	complemented
57	#5078	C₂×C₅	1	2	yes	complemented
58	#5065	C₂×C₅	1	4	yes	complemented
59	#4998	C₂×C₅	1	4	yes	—
60	C₂×C₄×C₃⋊C₈	C₂×C₅	1	1	yes	characteristic
61	C₂²×C₅×C₈	—	3	2	no	—
62	C₅×C₄⋊Q₈	—	3	2	no	—
63	C₅×C₄⋊Q₈	C₂×D₃	1	2	yes	—
64	C₅×#125	—	3	2	no	—
65	C₅×#125	C₂×D₃	1	2	yes	—
66	C₅×C₂²⋊Q₈	—	2	2	no	—
67	C₅×C₂²⋊Q₈	—	6	2	no	—
68	C₄×C₅×Q₈	—	2	2	no	—
69	C₄×C₅×Q₈	—	6	2	no	—
70	C₅×#117	—	3	2	no	—
71	C₅×#117	C₂×D₃	1	2	yes	—
72	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
73	C₅×#107	—	3	8	no	—
74	C₅×#106	—	3	8	no	—
75	C₄×C₅×C₈	—	3	1	no	—
76	C₄×C₅×C₈	—	3	2	no	—
77	C₄×C₅×C₈	—	3	1	no	—
78	#3160	—	3	1	no	Sylow
79	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
80	C₃×C₅×Q₈	—	4	4	no	—
81	C₂×C₃×C₄×C₅	—	2	4	no	—
82	C₂×C₃×C₄×C₅	—	2	4	no	—
83	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
84	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
85	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
86	C₂×C₃×C₄×C₅	C₂×Q₈	1	4	yes	—
87	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
88	C₂×C₅×Dic₃	—	2	4	no	—
89	C₅×Dic₆	—	4	4	no	—
90	C₅×C₃⋊C₈	C₂×Q₈	1	4	yes	—
91	C₅×C₃⋊C₈	C₂×Q₈	1	4	yes	—
92	C₃×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
93	C₃×#125	C₂²×C₅	1	2	yes	—
94	C₃×C₂²⋊Q₈	—	2	2	no	—
95	C₃×C₄×Q₈	—	2	2	no	—
96	C₃×#117	C₂²×C₅	1	2	yes	—
97	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
98	#926	C₂²×C₅	1	2	yes	—
99	#924	—	2	2	no	—
100	C₂²×C₃⋊C₈	C₂²×C₅	1	2	yes	—
101	#870	C₂²×C₅	1	2	yes	—
102	#869	C₂²×C₅	1	2	yes	—
103	C₄×Dic₆	—	2	2	no	—
104	#808	C₂²×C₅	1	8	yes	—
105	#807	C₂²×C₅	1	8	yes	—
106	C₄×C₃⋊C₈	C₂²×C₅	1	2	yes	—
107	C₄×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
108	C₄×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
109	C₂×C₅×Q₈	—	2	2	no	—
110	C₂×C₅×Q₈	—	6	2	no	—
111	C₂²×C₄×C₅	#67	1	2	yes	—
112	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
113	C₂×C₅×C₈	—	3	8	no	—
114	C₂×C₅×C₈	—	3	2	no	—
115	C₂×C₅×C₈	—	3	2	no	—
116	C₅×C₄⋊C₄	—	2	4	no	—
117	C₅×C₄⋊C₄	—	2	2	no	—
118	C₅×C₄⋊C₄	—	3	4	no	—
119	C₅×C₄⋊C₄	—	6	4	no	—
120	C₅×C₄⋊C₄	—	6	2	no	—
121	C₅×C₄⋊C₄	C₂²×D₃	1	4	yes	—
122	C₅×C₂²⋊C₄	—	2	2	no	—
123	C₅×C₂²⋊C₄	—	6	2	no	—
124	C₄²×C₅	—	2	2	no	—
125	C₄²×C₅	—	6	2	no	—
126	C₄²×C₅	#67	1	2	yes	—
127	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
128	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
129	#533	—	3	1	no	—
130	#533	C₅×D₃	1	1	yes	characteristic
131	#500	—	3	2	no	—
132	#499	—	3	4	no	—
133	#498	—	3	2	no	—
134	#427	—	3	4	no	—
135	C₂×C₄×C₈	—	3	1	no	—
136	C₂²×C₃×C₅	—	2	2	no	—
137	C₂²×C₃×C₅	C₄⋊Q₈	1	2	yes	—
138	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
139	C₃×C₄×C₅	—	2	2	no	—
140	C₃×C₄×C₅	—	4	4	no	—
141	C₃×C₄×C₅	C₄⋊Q₈	1	2	yes	—
142	C₃×C₄×C₅	#135	1	4	yes	—
143	C₃×C₄×C₅	C₂²×Q₈	1	2	yes	—
144	C₅×Dic₃	—	4	4	no	—
145	C₂×C₃×Q₈	—	2	2	no	—
146	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
147	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
148	C₂×Dic₆	—	2	2	no	—
149	C₃×C₄⋊C₄	—	2	4	no	—
150	C₃×C₄⋊C₄	—	2	2	no	—
151	C₃×C₄⋊C₄	C₂³×C₅	1	4	yes	—
152	C₃×C₂²⋊C₄	—	2	2	no	—
153	C₃×C₄²	—	2	2	no	—
154	C₃×C₄²	C₅×D₄	1	2	yes	—
155	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
156	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
157	#217	—	2	2	no	—
158	C₄⋊Dic₃	—	2	2	no	—
159	C₄⋊Dic₃	C₂³×C₅	1	4	yes	—
160	Dic₃⋊C₄	—	2	4	no	—
161	C₄×Dic₃	—	2	2	no	—
162	C₂×C₃⋊C₈	C₅×Q₈	1	8	yes	complemented
163	C₂×C₃⋊C₈	C₂³×C₅	1	2	yes	—
164	C₂×C₃⋊C₈	C₂³×C₅	1	2	yes	—
165	C₂³×C₅	C₂×#67	1	1	yes	characteristic
166	C₅×Q₈	—	4	4	no	—
167	C₅×Q₈	—	12	4	no	—
168	C₂×C₄×C₅	—	2	4	no	—
169	C₂×C₄×C₅	—	2	4	no	—
170	C₂×C₄×C₅	—	6	4	no	—
171	C₂×C₄×C₅	D₃×Q₈	1	4	yes	—
172	C₂×C₄×C₅	C₂×#67	1	2	yes	—
173	C₂×C₄×C₅	C₂×#67	1	2	yes	—
174	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
175	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
176	C₅×C₈	—	3	4	no	—
177	C₅×C₈	—	3	4	no	—
178	C₂²×C₈	—	3	2	no	—
179	C₄⋊Q₈	—	3	2	no	—
180	C₄⋊Q₈	C₂×C₅×D₃	1	2	yes	—
181	#125	—	3	2	no	—
182	#125	C₂×C₅×D₃	1	2	yes	—
183	C₂²⋊Q₈	—	2	2	no	—
184	C₂²⋊Q₈	—	6	2	no	—
185	C₄×Q₈	—	2	2	no	—
186	C₄×Q₈	—	6	2	no	—
187	#117	—	3	2	no	—
188	#117	C₂×C₅×D₃	1	2	yes	—
189	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
190	#107	—	3	8	no	—
191	#106	—	3	8	no	—
192	C₄×C₈	—	3	1	no	—
193	C₄×C₈	—	3	2	no	—
194	C₄×C₈	—	3	1	no	—
195	C₂×C₃×C₅	—	2	2	no	—
196	C₂×C₃×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
197	C₂×C₃×C₅	C₂×#135	1	2	yes	—
198	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
199	C₃×Q₈	—	4	4	no	—
200	C₂×C₃×C₄	—	2	4	no	—
201	C₂×C₃×C₄	—	2	4	no	—
202	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
203	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
204	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
205	C₂×C₃×C₄	C₂×C₅×Q₈	1	4	yes	—
206	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
207	C₂×Dic₃	—	2	4	no	—
208	Dic₆	—	4	4	no	—
209	C₃⋊C₈	C₂×C₅×Q₈	1	4	yes	—
210	C₃⋊C₈	C₂×C₅×Q₈	1	4	yes	—
211	C₂²×C₅	—	2	2	no	—
212	C₂²×C₅	#944	1	2	yes	—
213	C₂²×C₅	C₂²×#67	1	1	yes	characteristic
214	C₄×C₅	—	2	2	no	—
215	C₄×C₅	—	4	4	no	—
216	C₄×C₅	—	12	4	no	—
217	C₄×C₅	#944	1	2	yes	—
218	C₄×C₅	#950	1	4	yes	—
219	C₄×C₅	C₂×D₃×Q₈	1	2	yes	—
220	C₂×Q₈	—	2	2	no	—
221	C₂×Q₈	—	6	2	no	—
222	C₂²×C₄	C₅×#67	1	2	yes	—
223	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
224	C₂×C₈	—	3	8	no	—
225	C₂×C₈	—	3	2	no	—
226	C₂×C₈	—	3	2	no	—
227	C₄⋊C₄	—	2	4	no	—
228	C₄⋊C₄	—	2	2	no	—
229	C₄⋊C₄	—	3	4	no	—
230	C₄⋊C₄	—	6	4	no	—
231	C₄⋊C₄	—	6	2	no	—
232	C₄⋊C₄	C₂²×C₅×D₃	1	4	yes	—
233	C₂²⋊C₄	—	2	2	no	—
234	C₂²⋊C₄	—	6	2	no	—
235	C₄²	—	2	2	no	—
236	C₄²	—	6	2	no	—
237	C₄²	C₅×#67	1	2	yes	—
238	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
239	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
240	C₃×C₅	#3160	1	1	yes	characteristic, complemented, Hall
241	C₂²×C₃	—	2	2	no	—
242	C₂²×C₃	C₅×C₄⋊Q₈	1	2	yes	—
243	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
244	C₃×C₄	—	2	2	no	—
245	C₃×C₄	—	4	4	no	—
246	C₃×C₄	C₅×C₄⋊Q₈	1	2	yes	—
247	C₃×C₄	C₅×#135	1	4	yes	—
248	C₃×C₄	C₂²×C₅×Q₈	1	2	yes	—
249	Dic₃	—	4	4	no	—
250	C₂×C₅	—	2	2	no	—
251	C₂×C₅	C₂×#944	1	1	yes	characteristic
252	C₂×C₅	C₂×#950	1	2	yes	—
253	C₂³	C₂×C₅×#67	1	1	yes	characteristic
254	Q₈	—	4	4	no	—
255	Q₈	—	12	4	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	6	4	no	—
259	C₂×C₄	C₅×D₃×Q₈	1	4	yes	—
260	C₂×C₄	C₂×C₅×#67	1	2	yes	—
261	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
262	C₂×C₄	C₂×C₅×#67	1	2	yes	—
263	C₂×C₄	C₂³×C₅×D₃	1	1	yes	characteristic
264	C₈	—	3	4	no	—
265	C₈	—	3	4	no	—
266	C₂×C₃	—	2	2	no	—
267	C₂×C₃	C₂×C₅×C₄⋊Q₈	1	1	yes	characteristic
268	C₂×C₃	C₂×C₅×#135	1	2	yes	—
269	C₅	#83339	1	1	yes	characteristic, direct product factor, Sylow
270	C₂²	—	2	2	no	—
271	C₂²	C₅×#944	1	2	yes	—
272	C₂²	C₂²×C₅×#67	1	1	yes	characteristic
273	C₄	—	2	2	no	—
274	C₄	—	4	4	no	—
275	C₄	—	12	4	no	—
276	C₄	C₅×#944	1	2	yes	—
277	C₄	C₅×#950	1	4	yes	—
278	C₄	C₂×C₅×D₃×Q₈	1	2	yes	—
279	C₃	C₅×#3160	1	1	yes	characteristic, complemented, Sylow
280	C₂	—	2	2	no	—
281	C₂	C₂×C₅×#944	1	1	yes	characteristic
282	C₂	C₂×C₅×#950	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅	C₂×#67
commutator, cosocle	C₂×C₃×C₄	C₂⁴×C₅
Frattini	C₂×C₄	C₂³×C₅×D₃
Fitting	C₃×C₅×#533	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
prime radical	C₂³×C₃×C₅	C₂×D₄
hypercenter	C₅×#533	D₃
hypocenter	C₃	C₅×#3160
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ C₅×#533

Complements

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

N	H ≅ G/N	type
3. C₅×#5080	C₂	semidirect
4. C₅×#5078	C₂	semidirect
5. C₅×#5065	C₂	semidirect
26. #83339	C₅	direct (N × H)
51. C₂×C₅×C₃⋊C₈	Q₈	semidirect
56. #5080	C₂×C₅	semidirect
57. #5078	C₂×C₅	semidirect
58. #5065	C₂×C₅	semidirect
162. C₂×C₃⋊C₈	C₅×Q₈	semidirect
240. C₃×C₅	#3160	semidirect
269. C₅	#83339	direct (N × H)
279. C₃	C₅×#3160	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×#533	2
C₅×#5470	2
C₅×#5080	2
C₅×#5078	2
C₅×#5065	2
C₅×#4998	2
C₂×C₄×C₅×C₃⋊C₈	2
C₅×#3160	3
#83339	5

Subgroups — C5×#83339 #14439834

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#83339 #14439834