Subgroups — C₅×#81891 #14439831

Order 1920.

712 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×#5728	C₂	1	2	yes	maximal
2	C₃×C₅×#498	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#5078	C₂	1	2	yes	maximal, complemented
4	C₅×#4860	C₂	1	4	yes	maximal
5	C₅×#4858	C₂	1	4	yes	maximal, complemented
6	C₄×C₅×D₃×C₈	C₂	1	1	yes	maximal, characteristic
7	C₅×#4680	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#3160	—	3	1	no	maximal, Hall
9	C₃×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₅×#946	—	2	2	no	—
11	C₅×Q₈×Dic₃	—	2	2	no	—
12	C₂×C₅×D₃×C₈	C₂²	1	2	yes	—
13	C₅×#897	—	2	2	no	—
14	C₅×#892	C₂²	1	4	yes	—
15	C₅×#890	C₂²	1	4	yes	—
16	C₅×#887	—	2	2	no	—
17	C₄²×C₅×D₃	C₂²	1	1	yes	characteristic
18	C₅×#869	C₂²	1	2	yes	—
19	C₃×C₅×#106	C₂²	1	4	yes	—
20	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
21	C₅×#817	C₂²	1	4	yes	—
22	C₅×C₈×Dic₃	C₂²	1	2	yes	—
23	C₅×#807	C₂²	1	8	yes	—
24	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
25	#81891	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
26	C₅×#533	—	3	2	no	—
27	C₅×#500	—	3	2	no	—
28	C₅×#499	—	3	4	no	—
29	C₅×#498	—	3	1	no	—
30	C₅×#498	D₃	1	1	yes	characteristic, complemented
31	C₅×#427	—	3	4	no	—
32	C₂×C₄×C₅×C₈	—	3	1	no	—
33	C₂×C₃×C₅×Q₈	—	2	2	no	—
34	C₂×C₄×C₅×D₃	D₄	1	2	yes	—
35	C₂×C₄×C₅×D₃	C₂³	1	1	yes	characteristic
36	C₂×C₅×Dic₆	—	2	2	no	—
37	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
38	C₃×C₅×C₄⋊C₄	—	2	2	no	—
39	C₃×C₅×C₄⋊C₄	C₂³	1	4	yes	—
40	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
41	C₅×(C₂×D₃)⋊C₄	—	2	4	no	—
42	C₅×C₄⋊Dic₃	—	2	2	no	—
43	C₅×C₄⋊Dic₃	—	2	4	no	—
44	C₅×C₄⋊Dic₃	C₂³	1	4	yes	—
45	C₅×Dic₃⋊C₄	—	2	4	no	—
46	C₄×C₅×Dic₃	—	2	4	no	—
47	C₄×C₅×Dic₃	D₄	1	2	yes	—
48	C₄×C₅×Dic₃	C₂³	1	1	yes	characteristic
49	C₂×C₅×C₃⋊C₈	C₂³	1	2	yes	—
50	C₅×D₃×C₈	Q₈	1	8	yes	complemented
51	#5728	C₂×C₅	1	2	yes	—
52	C₃×#498	C₂×C₅	1	1	yes	characteristic, complemented
53	#5078	C₂×C₅	1	2	yes	complemented
54	#4860	C₂×C₅	1	4	yes	—
55	#4858	C₂×C₅	1	4	yes	complemented
56	C₄×D₃×C₈	C₂×C₅	1	1	yes	characteristic
57	#4680	C₂×C₅	1	1	yes	characteristic, complemented
58	C₂²×C₅×C₈	—	3	2	no	—
59	C₅×C₄⋊Q₈	—	3	2	no	—
60	C₅×C₄⋊Q₈	C₂×D₃	1	2	yes	—
61	C₅×#125	—	3	4	no	—
62	C₅×C₂²⋊Q₈	—	6	2	no	—
63	C₅×C₂²⋊Q₈	—	6	2	no	—
64	C₄×C₅×Q₈	—	6	2	no	—
65	C₄×C₅×Q₈	—	6	2	no	—
66	C₅×#117	—	3	4	no	—
67	C₂×C₄²×C₅	—	3	1	no	—
68	C₅×#107	—	3	8	no	—
69	C₅×#106	—	3	4	no	—
70	C₅×#106	C₂×D₃	1	4	yes	—
71	C₄×C₅×C₈	—	3	2	no	—
72	C₄×C₅×C₈	—	3	1	no	—
73	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
74	#3160	—	3	1	no	Sylow
75	C₂²×C₅×D₃	C₂×D₄	1	1	yes	characteristic
76	C₃×C₅×Q₈	—	4	4	no	—
77	C₂×C₃×C₄×C₅	—	2	4	no	—
78	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
79	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
80	C₂×C₅×Dic₃	—	2	4	no	—
81	C₂×C₅×Dic₃	C₂×D₄	1	2	yes	—
82	C₂×C₅×Dic₃	C₂×D₄	1	1	yes	characteristic
83	C₄×C₅×D₃	—	2	4	no	—
84	C₄×C₅×D₃	C₂×Q₈	1	4	yes	—
85	C₅×Dic₆	—	4	4	no	—
86	C₃×C₅×C₈	C₂×Q₈	1	4	yes	—
87	C₅×C₃⋊C₈	C₂×Q₈	1	4	yes	—
88	C₃×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
89	#946	—	2	2	no	—
90	Q₈×Dic₃	—	2	2	no	—
91	C₂×D₃×C₈	C₂²×C₅	1	2	yes	—
92	#897	—	2	2	no	—
93	#892	C₂²×C₅	1	4	yes	—
94	#890	C₂²×C₅	1	4	yes	—
95	#887	—	2	2	no	—
96	C₄²×D₃	C₂²×C₅	1	1	yes	characteristic
97	#869	C₂²×C₅	1	2	yes	—
98	C₃×#106	C₂²×C₅	1	4	yes	—
99	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
100	#817	C₂²×C₅	1	4	yes	—
101	C₈×Dic₃	C₂²×C₅	1	2	yes	—
102	#807	C₂²×C₅	1	8	yes	—
103	C₄×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
104	C₂×C₅×Q₈	—	2	2	no	—
105	C₂×C₅×Q₈	—	6	2	no	—
106	C₂²×C₄×C₅	—	3	1	no	—
107	C₂²×C₄×C₅	—	3	2	no	—
108	C₂×C₅×C₈	—	3	2	no	—
109	C₂×C₅×C₈	—	3	8	no	—
110	C₂×C₅×C₈	C₂²×D₃	1	2	yes	—
111	C₅×C₄⋊C₄	—	2	2	no	—
112	C₅×C₄⋊C₄	—	3	4	no	—
113	C₅×C₄⋊C₄	—	6	4	no	—
114	C₅×C₄⋊C₄	—	6	4	no	—
115	C₅×C₄⋊C₄	—	6	2	no	—
116	C₅×C₄⋊C₄	C₂²×D₃	1	4	yes	—
117	C₅×C₂²⋊C₄	—	6	4	no	—
118	C₄²×C₅	—	3	2	no	—
119	C₄²×C₅	—	3	1	no	—
120	C₄²×C₅	—	6	4	no	—
121	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
122	#533	—	3	2	no	—
123	#500	—	3	2	no	—
124	#499	—	3	4	no	—
125	#498	—	3	1	no	—
126	#498	C₅×D₃	1	1	yes	characteristic, complemented
127	#427	—	3	4	no	—
128	C₂×C₄×C₈	—	3	1	no	—
129	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
130	C₂×C₅×D₃	—	2	2	no	—
131	C₂×C₅×D₃	C₄⋊Q₈	1	2	yes	—
132	C₃×C₄×C₅	—	4	4	no	—
133	C₃×C₄×C₅	#135	1	4	yes	—
134	C₃×C₄×C₅	C₂²×Q₈	1	2	yes	—
135	C₅×Dic₃	—	2	2	no	—
136	C₅×Dic₃	—	4	4	no	—
137	C₅×Dic₃	C₄⋊Q₈	1	2	yes	—
138	C₂×C₃×Q₈	—	2	2	no	—
139	C₂×C₄×D₃	C₅×D₄	1	2	yes	—
140	C₂×C₄×D₃	C₂³×C₅	1	1	yes	characteristic
141	C₂×Dic₆	—	2	2	no	—
142	C₂×C₃×C₈	C₂³×C₅	1	2	yes	—
143	C₃×C₄⋊C₄	—	2	2	no	—
144	C₃×C₄⋊C₄	C₂³×C₅	1	4	yes	—
145	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
146	(C₂×D₃)⋊C₄	—	2	4	no	—
147	C₄⋊Dic₃	—	2	2	no	—
148	C₄⋊Dic₃	—	2	4	no	—
149	C₄⋊Dic₃	C₂³×C₅	1	4	yes	—
150	Dic₃⋊C₄	—	2	4	no	—
151	C₄×Dic₃	—	2	4	no	—
152	C₄×Dic₃	C₅×D₄	1	2	yes	—
153	C₄×Dic₃	C₂³×C₅	1	1	yes	characteristic
154	C₂×C₃⋊C₈	C₂³×C₅	1	2	yes	—
155	D₃×C₈	C₅×Q₈	1	8	yes	complemented
156	C₂³×C₅	—	3	1	no	—
157	C₅×Q₈	—	4	4	no	—
158	C₅×Q₈	—	12	4	no	—
159	C₂×C₄×C₅	—	2	4	no	—
160	C₂×C₄×C₅	—	3	1	no	—
161	C₂×C₄×C₅	—	3	4	no	—
162	C₂×C₄×C₅	—	3	2	no	—
163	C₂×C₄×C₅	—	6	4	no	—
164	C₂×C₄×C₅	—	6	4	no	—
165	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
166	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
167	C₅×C₈	—	3	4	no	—
168	C₅×C₈	D₃×Q₈	1	4	yes	—
169	C₂²×C₈	—	3	2	no	—
170	C₄⋊Q₈	—	3	2	no	—
171	C₄⋊Q₈	C₂×C₅×D₃	1	2	yes	—
172	#125	—	3	4	no	—
173	C₂²⋊Q₈	—	6	2	no	—
174	C₂²⋊Q₈	—	6	2	no	—
175	C₄×Q₈	—	6	2	no	—
176	C₄×Q₈	—	6	2	no	—
177	#117	—	3	4	no	—
178	C₂×C₄²	—	3	1	no	—
179	#107	—	3	8	no	—
180	#106	—	3	4	no	—
181	#106	C₂×C₅×D₃	1	4	yes	—
182	C₄×C₈	—	3	2	no	—
183	C₄×C₈	—	3	1	no	—
184	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
185	C₂×C₃×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
186	C₂×C₃×C₅	C₂×#135	1	2	yes	—
187	C₅×D₃	—	2	2	no	—
188	C₂²×D₃	C₂×C₅×D₄	1	1	yes	characteristic
189	C₃×Q₈	—	4	4	no	—
190	C₂×C₃×C₄	—	2	4	no	—
191	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
192	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
193	C₂×Dic₃	—	2	4	no	—
194	C₂×Dic₃	C₂×C₅×D₄	1	2	yes	—
195	C₂×Dic₃	C₂×C₅×D₄	1	1	yes	characteristic
196	C₄×D₃	—	2	4	no	—
197	C₄×D₃	C₂×C₅×Q₈	1	4	yes	—
198	Dic₆	—	4	4	no	—
199	C₃×C₈	C₂×C₅×Q₈	1	4	yes	—
200	C₃⋊C₈	C₂×C₅×Q₈	1	4	yes	—
201	C₂²×C₅	—	3	2	no	—
202	C₂²×C₅	—	6	2	no	—
203	C₂²×C₅	C₂×D₃×D₄	1	1	yes	characteristic
204	C₄×C₅	—	3	2	no	—
205	C₄×C₅	—	4	4	no	—
206	C₄×C₅	—	6	2	no	—
207	C₄×C₅	—	12	4	no	—
208	C₄×C₅	#916	1	4	yes	—
209	C₄×C₅	C₂×D₃×Q₈	1	2	yes	—
210	C₂×Q₈	—	2	2	no	—
211	C₂×Q₈	—	6	2	no	—
212	C₂²×C₄	—	3	1	no	—
213	C₂²×C₄	—	3	2	no	—
214	C₂×C₈	—	3	2	no	—
215	C₂×C₈	—	3	8	no	—
216	C₂×C₈	C₂²×C₅×D₃	1	2	yes	—
217	C₄⋊C₄	—	2	2	no	—
218	C₄⋊C₄	—	3	4	no	—
219	C₄⋊C₄	—	6	4	no	—
220	C₄⋊C₄	—	6	4	no	—
221	C₄⋊C₄	—	6	2	no	—
222	C₄⋊C₄	C₂²×C₅×D₃	1	4	yes	—
223	C₂²⋊C₄	—	6	4	no	—
224	C₄²	—	3	2	no	—
225	C₄²	—	3	1	no	—
226	C₄²	—	6	4	no	—
227	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
228	C₃×C₅	#3160	1	1	yes	characteristic, complemented, Hall
229	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
230	C₂×D₃	—	2	2	no	—
231	C₂×D₃	C₅×C₄⋊Q₈	1	2	yes	—
232	C₃×C₄	—	4	4	no	—
233	C₃×C₄	C₅×#135	1	4	yes	—
234	C₃×C₄	C₂²×C₅×Q₈	1	2	yes	—
235	Dic₃	—	2	2	no	—
236	Dic₃	—	4	4	no	—
237	Dic₃	C₅×C₄⋊Q₈	1	2	yes	—
238	C₂×C₅	—	6	2	no	—
239	C₂×C₅	D₃×C₄⋊Q₈	1	1	yes	characteristic
240	C₂×C₅	C₂×#916	1	2	yes	—
241	C₂³	—	3	1	no	—
242	Q₈	—	4	4	no	—
243	Q₈	—	12	4	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	3	1	no	—
246	C₂×C₄	—	3	4	no	—
247	C₂×C₄	—	3	2	no	—
248	C₂×C₄	—	6	4	no	—
249	C₂×C₄	—	6	4	no	—
250	C₂×C₄	C₅×D₃×D₄	1	2	yes	—
251	C₂×C₄	C₂³×C₅×D₃	1	1	yes	characteristic
252	C₈	—	3	4	no	—
253	C₈	C₅×D₃×Q₈	1	4	yes	—
254	C₂×C₃	C₂×C₅×C₄⋊Q₈	1	1	yes	characteristic
255	C₂×C₃	C₂×C₅×#135	1	2	yes	—
256	D₃	—	2	2	no	—
257	C₅	#81891	1	1	yes	characteristic, direct product factor, Sylow
258	C₂²	—	3	2	no	—
259	C₂²	—	6	2	no	—
260	C₂²	C₂×C₅×D₃×D₄	1	1	yes	characteristic
261	C₄	—	3	2	no	—
262	C₄	—	4	4	no	—
263	C₄	—	6	2	no	—
264	C₄	—	12	4	no	—
265	C₄	C₅×#916	1	4	yes	—
266	C₄	C₂×C₅×D₃×Q₈	1	2	yes	—
267	C₃	C₅×#3160	1	1	yes	characteristic, complemented, Sylow
268	C₂	—	6	2	no	—
269	C₂	C₅×D₃×C₄⋊Q₈	1	1	yes	characteristic
270	C₂	C₂×C₅×#916	1	2	yes	—

	subgroup	G/H
center	C₂²×C₅	C₂×D₃×D₄
commutator, cosocle	C₂×C₃×C₄	C₂⁴×C₅
Frattini	C₂×C₄	C₂³×C₅×D₃
Fitting	C₃×C₅×#498	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
prime radical	C₂²×C₅×D₃	C₂×D₄
hypercenter	C₅×#498	D₃
hypocenter	C₃	C₅×#3160
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×#498

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#498	C₂	semidirect
3. C₅×#5078	C₂	semidirect
5. C₅×#4858	C₂	semidirect
7. C₅×#4680	C₂	semidirect
25. #81891	C₅	direct (N × H)
30. C₅×#498	D₃	semidirect
50. C₅×D₃×C₈	Q₈	semidirect
52. C₃×#498	C₂×C₅	semidirect
53. #5078	C₂×C₅	semidirect
55. #4858	C₂×C₅	semidirect
57. #4680	C₂×C₅	semidirect
126. #498	C₅×D₃	semidirect
155. D₃×C₈	C₅×Q₈	semidirect
228. C₃×C₅	#3160	semidirect
257. C₅	#81891	direct (N × H)
267. 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
C₅×#5728	2
C₃×C₅×#498	2
C₅×#5078	2
C₅×#4860	2
C₅×#4858	2
C₄×C₅×D₃×C₈	2
C₅×#4680	2
C₅×#3160	3
#81891	5

Subgroups — C5×#81891 #14439831

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#81891 #14439831