Subgroups — C₅×#63837 #13087184

Order 1440.

306 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#627	C₂	1	1	yes	maximal, characteristic
2	C₅×#3750	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×Dic₃⋊C₄	C₂	1	1	yes	maximal, characteristic
4	C₃×C₄×C₅×Dic₃	C₂	1	1	yes	maximal, characteristic
5	C₅×Dic₃⋊Dic₃	C₂	1	1	yes	maximal, characteristic
6	C₅×Dic₃²	C₂	1	1	yes	maximal, characteristic
7	C₅×Dic₃²	C₂	1	1	yes	maximal, characteristic
8	C₅×#887	—	3	1	no	maximal
9	C₄×C₅×Dic₆	—	3	1	no	maximal
10	C₂×C₃²×C₄×C₅	C₂²	1	1	yes	characteristic
11	C₂×C₅×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
12	C₂×C₅×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₅×Dic₃	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₅×Dic₃	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₅×Dic₃	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₅×Dic₃	C₂²	1	1	yes	characteristic
17	C₅×#627	C₄	1	4	yes	complemented
18	#63837	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₂×C₅×Dic₆	—	3	1	no	—
20	C₂×C₅×Dic₆	—	3	1	no	—
21	C₃×C₅×C₄⋊C₄	—	3	1	no	—
22	C₃×C₄²×C₅	—	3	1	no	—
23	C₅×C₄⋊Dic₃	—	3	1	no	—
24	C₅×Dic₃⋊C₄	—	3	1	no	—
25	C₅×Dic₃⋊C₄	—	3	1	no	—
26	C₅×Dic₃⋊C₄	—	3	1	no	—
27	C₅×Dic₃⋊C₄	—	6	1	no	—
28	C₅×Dic₃⋊C₄	D₃	1	1	yes	characteristic
29	C₄×C₅×Dic₃	—	3	1	no	—
30	C₄×C₅×Dic₃	—	3	1	no	—
31	C₄×C₅×Dic₃	—	3	1	no	—
32	C₄×C₅×Dic₃	—	3	1	no	—
33	C₄×C₅×Dic₃	D₃	1	1	yes	characteristic
34	C₂²×C₃²×C₅	C₂³	1	1	yes	characteristic
35	C₃²×C₄×C₅	—	2	1	no	—
36	C₅×C₃⋊Dic₃	—	2	1	no	—
37	C₅×C₃⋊Dic₃	C₂×C₄	1	2	yes	—
38	C₃×C₅×Dic₃	—	2	1	no	—
39	C₃×C₅×Dic₃	C₂×C₄	1	2	yes	—
40	C₃×C₅×Dic₃	C₂×C₄	1	2	yes	—
41	C₃×C₅×Dic₃	Q₈	1	2	yes	complemented
42	C₄×C₅×Q₈	—	9	1	no	Hall
43	C₂×#627	C₂×C₅	1	1	yes	characteristic
44	#3750	C₂×C₅	1	1	yes	characteristic
45	C₃×Dic₃⋊C₄	C₂×C₅	1	1	yes	characteristic
46	C₃×C₄×Dic₃	C₂×C₅	1	1	yes	characteristic
47	Dic₃⋊Dic₃	C₂×C₅	1	1	yes	characteristic
48	Dic₃²	C₂×C₅	1	1	yes	characteristic
49	Dic₃²	C₂×C₅	1	1	yes	characteristic
50	C₂×C₃×C₄×C₅	—	2	1	no	—
51	C₂×C₃×C₄×C₅	—	3	1	no	—
52	C₂×C₃×C₄×C₅	—	3	1	no	—
53	C₂×C₃×C₄×C₅	—	3	1	no	—
54	C₂×C₃×C₄×C₅	—	3	1	no	—
55	C₂×C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic
56	C₂×C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic
57	C₂×C₅×Dic₃	—	3	1	no	—
58	C₂×C₅×Dic₃	—	3	1	no	—
59	C₂×C₅×Dic₃	—	3	1	no	—
60	C₂×C₅×Dic₃	—	3	1	no	—
61	C₂×C₅×Dic₃	—	6	1	no	—
62	C₂×C₅×Dic₃	—	6	1	no	—
63	C₂×C₅×Dic₃	C₂×D₃	1	1	yes	characteristic
64	C₂×C₅×Dic₃	C₂×D₃	1	1	yes	characteristic
65	C₂×C₅×Dic₃	C₂×D₃	1	1	yes	characteristic
66	C₂×C₅×Dic₃	C₂×D₃	1	1	yes	characteristic
67	C₅×Dic₆	—	3	4	no	—
68	C₅×Dic₆	—	3	4	no	—
69	#887	—	3	1	no	—
70	C₄×Dic₆	—	3	1	no	—
71	C₂×C₃²×C₅	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₃²×C₅	C₂×Q₈	1	1	yes	characteristic
73	C₂×C₃²×C₅	#41	1	1	yes	characteristic
74	C₂×C₅×Q₈	—	9	1	no	—
75	C₅×C₄⋊C₄	—	3	1	no	—
76	C₅×C₄⋊C₄	—	9	1	no	—
77	C₅×C₄⋊C₄	—	9	1	no	—
78	C₄²×C₅	—	3	1	no	—
79	C₄²×C₅	—	9	1	no	—
80	C₄²×C₅	—	9	1	no	—
81	C₂×C₃²×C₄	C₂²×C₅	1	1	yes	characteristic
82	C₂×C₃⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
83	C₂×C₃⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
84	C₂×C₃×Dic₃	C₂²×C₅	1	1	yes	characteristic
85	C₂×C₃×Dic₃	C₂²×C₅	1	1	yes	characteristic
86	C₂×C₃×Dic₃	C₂²×C₅	1	1	yes	characteristic
87	C₂×C₃×Dic₃	C₂²×C₅	1	1	yes	characteristic
88	#627	C₄×C₅	1	4	yes	complemented
89	C₂²×C₃×C₅	—	2	1	no	—
90	C₂²×C₃×C₅	C₂²×D₃	1	1	yes	characteristic
91	C₂²×C₃×C₅	C₂²×D₃	1	1	yes	characteristic
92	C₃×C₄×C₅	—	2	1	no	—
93	C₃×C₄×C₅	—	2	1	no	—
94	C₃×C₄×C₅	—	3	2	no	—
95	C₃×C₄×C₅	—	3	2	no	—
96	C₃×C₄×C₅	—	3	2	no	—
97	C₃×C₄×C₅	—	4	1	no	—
98	C₃×C₄×C₅	—	6	1	no	—
99	C₅×Dic₃	—	2	1	no	—
100	C₅×Dic₃	—	3	2	no	—
101	C₅×Dic₃	—	3	2	no	—
102	C₅×Dic₃	—	6	2	no	—
103	C₅×Dic₃	—	6	1	no	—
104	C₅×Dic₃	—	6	1	no	—
105	C₅×Dic₃	—	12	1	no	—
106	C₅×Dic₃	Dic₆	1	2	yes	complemented
107	C₅×Dic₃	C₄×D₃	1	2	yes	—
108	C₅×Dic₃	C₄×D₃	1	2	yes	—
109	C₂×Dic₆	—	3	1	no	—
110	C₂×Dic₆	—	3	1	no	—
111	C₃×C₄⋊C₄	—	3	1	no	—
112	C₃×C₄²	—	3	1	no	—
113	C₄⋊Dic₃	—	3	1	no	—
114	Dic₃⋊C₄	—	3	1	no	—
115	Dic₃⋊C₄	—	3	1	no	—
116	Dic₃⋊C₄	—	3	1	no	—
117	Dic₃⋊C₄	—	6	1	no	—
118	Dic₃⋊C₄	C₅×D₃	1	1	yes	characteristic
119	C₄×Dic₃	—	3	1	no	—
120	C₄×Dic₃	—	3	1	no	—
121	C₄×Dic₃	—	3	1	no	—
122	C₄×Dic₃	—	3	1	no	—
123	C₄×Dic₃	C₅×D₃	1	1	yes	characteristic
124	C₃²×C₅	C₄×Q₈	1	1	yes	characteristic, complemented, Hall
125	C₅×Q₈	—	9	4	no	—
126	C₂×C₄×C₅	—	3	1	no	—
127	C₂×C₄×C₅	—	3	1	no	—
128	C₂×C₄×C₅	—	3	1	no	—
129	C₂×C₄×C₅	—	3	1	no	—
130	C₂×C₄×C₅	—	9	1	no	—
131	C₂×C₄×C₅	—	9	1	no	—
132	C₂×C₄×C₅	D₃²	1	1	yes	characteristic
133	C₂²×C₃²	C₂³×C₅	1	1	yes	characteristic
134	C₃²×C₄	—	2	1	no	—
135	C₃⋊Dic₃	—	2	1	no	—
136	C₃⋊Dic₃	C₂×C₄×C₅	1	2	yes	—
137	C₃×Dic₃	—	2	1	no	—
138	C₃×Dic₃	C₂×C₄×C₅	1	2	yes	—
139	C₃×Dic₃	C₂×C₄×C₅	1	2	yes	—
140	C₃×Dic₃	C₅×Q₈	1	2	yes	complemented
141	C₄×Q₈	—	9	1	no	Sylow
142	C₂×C₃×C₅	—	2	1	no	—
143	C₂×C₃×C₅	—	2	1	no	—
144	C₂×C₃×C₅	—	2	1	no	—
145	C₂×C₃×C₅	C₂×Dic₆	1	1	yes	characteristic
146	C₂×C₃×C₅	C₂×C₄×D₃	1	1	yes	characteristic
147	C₂×C₃×C₅	C₂×C₄×D₃	1	1	yes	characteristic
148	C₂×C₃×C₅	#235	1	1	yes	characteristic
149	C₂×C₃×C₅	#237	1	1	yes	characteristic
150	C₂×C₃×C₅	D₃×Q₈	1	1	yes	characteristic
151	C₂×C₃×C₄	—	2	1	no	—
152	C₂×C₃×C₄	—	3	1	no	—
153	C₂×C₃×C₄	—	3	1	no	—
154	C₂×C₃×C₄	—	3	1	no	—
155	C₂×C₃×C₄	—	3	1	no	—
156	C₂×C₃×C₄	C₂×C₅×D₃	1	1	yes	characteristic
157	C₂×C₃×C₄	C₂×C₅×D₃	1	1	yes	characteristic
158	C₂×Dic₃	—	3	1	no	—
159	C₂×Dic₃	—	3	1	no	—
160	C₂×Dic₃	—	3	1	no	—
161	C₂×Dic₃	—	3	1	no	—
162	C₂×Dic₃	—	6	1	no	—
163	C₂×Dic₃	—	6	1	no	—
164	C₂×Dic₃	C₂×C₅×D₃	1	1	yes	characteristic
165	C₂×Dic₃	C₂×C₅×D₃	1	1	yes	characteristic
166	C₂×Dic₃	C₂×C₅×D₃	1	1	yes	characteristic
167	C₂×Dic₃	C₂×C₅×D₃	1	1	yes	characteristic
168	Dic₆	—	3	4	no	—
169	Dic₆	—	3	4	no	—
170	C₂²×C₅	C₂×D₃²	1	1	yes	characteristic
171	C₄×C₅	—	2	1	no	—
172	C₄×C₅	—	3	2	no	—
173	C₄×C₅	—	3	2	no	—
174	C₄×C₅	—	3	2	no	—
175	C₄×C₅	—	6	1	no	—
176	C₄×C₅	—	9	2	no	—
177	C₄×C₅	—	18	1	no	—
178	C₂×C₃²	C₂²×C₄×C₅	1	1	yes	characteristic
179	C₂×C₃²	C₂×C₅×Q₈	1	1	yes	characteristic
180	C₂×C₃²	C₅×#41	1	1	yes	characteristic
181	C₂×Q₈	—	9	1	no	—
182	C₄⋊C₄	—	3	1	no	—
183	C₄⋊C₄	—	9	1	no	—
184	C₄⋊C₄	—	9	1	no	—
185	C₄²	—	3	1	no	—
186	C₄²	—	9	1	no	—
187	C₄²	—	9	1	no	—
188	C₃×C₅	—	2	1	no	—
189	C₃×C₅	C₄×Dic₆	1	1	yes	characteristic, complemented
190	C₃×C₅	#887	1	1	yes	characteristic, complemented
191	C₂²×C₃	—	2	1	no	—
192	C₂²×C₃	C₂²×C₅×D₃	1	1	yes	characteristic
193	C₂²×C₃	C₂²×C₅×D₃	1	1	yes	characteristic
194	C₃×C₄	—	2	1	no	—
195	C₃×C₄	—	2	1	no	—
196	C₃×C₄	—	3	2	no	—
197	C₃×C₄	—	3	2	no	—
198	C₃×C₄	—	3	2	no	—
199	C₃×C₄	—	4	1	no	—
200	C₃×C₄	—	6	1	no	—
201	Dic₃	—	2	1	no	—
202	Dic₃	—	3	2	no	—
203	Dic₃	—	3	2	no	—
204	Dic₃	—	6	2	no	—
205	Dic₃	—	6	1	no	—
206	Dic₃	—	6	1	no	—
207	Dic₃	—	12	1	no	—
208	Dic₃	C₅×Dic₆	1	2	yes	complemented
209	Dic₃	C₄×C₅×D₃	1	2	yes	—
210	Dic₃	C₄×C₅×D₃	1	2	yes	—
211	C₂×C₅	D₃×Dic₆	1	1	yes	characteristic
212	C₂×C₅	C₄×D₃²	1	1	yes	characteristic
213	C₂×C₅	#3804	1	1	yes	characteristic
214	C₃²	C₄×C₅×Q₈	1	1	yes	characteristic, complemented, Sylow
215	Q₈	—	9	4	no	—
216	C₂×C₄	—	3	1	no	—
217	C₂×C₄	—	3	1	no	—
218	C₂×C₄	—	3	1	no	—
219	C₂×C₄	—	3	1	no	—
220	C₂×C₄	—	9	1	no	—
221	C₂×C₄	—	9	1	no	—
222	C₂×C₄	C₅×D₃²	1	1	yes	characteristic
223	C₂×C₃	—	2	1	no	—
224	C₂×C₃	—	2	1	no	—
225	C₂×C₃	—	2	1	no	—
226	C₂×C₃	C₂×C₅×Dic₆	1	1	yes	characteristic
227	C₂×C₃	C₂×C₄×C₅×D₃	1	1	yes	characteristic
228	C₂×C₃	C₂×C₄×C₅×D₃	1	1	yes	characteristic
229	C₂×C₃	C₅×#235	1	1	yes	characteristic
230	C₂×C₃	C₅×#237	1	1	yes	characteristic
231	C₂×C₃	C₅×D₃×Q₈	1	1	yes	characteristic
232	C₅	#63837	1	1	yes	characteristic, direct product factor, Sylow
233	C₂²	C₂×C₅×D₃²	1	1	yes	characteristic
234	C₄	—	2	1	no	—
235	C₄	—	3	2	no	—
236	C₄	—	3	2	no	—
237	C₄	—	3	2	no	—
238	C₄	—	6	1	no	—
239	C₄	—	9	2	no	—
240	C₄	—	18	1	no	—
241	C₃	—	2	1	no	—
242	C₃	C₄×C₅×Dic₆	1	1	yes	characteristic, complemented
243	C₃	C₅×#887	1	1	yes	characteristic, complemented
244	C₂	C₅×D₃×Dic₆	1	1	yes	characteristic
245	C₂	C₄×C₅×D₃²	1	1	yes	characteristic
246	C₂	C₅×#3804	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	C₂×D₃²
commutator	C₂×C₃²	C₂²×C₄×C₅
Frattini	C₂²	C₂×C₅×D₃²
Fitting	C₂×C₃²×C₄×C₅	C₂²
socle, prime radical	C₂²×C₃²×C₅	C₂³
cosocle	C₂²×C₃²	C₂³×C₅
hypercenter	C₂×C₄×C₅	D₃²
hypocenter	C₃²	C₄×C₅×Q₈
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄×C₅

Complements

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

N	H ≅ G/N	type
17. C₅×#627	C₄	semidirect
18. #63837	C₅	direct (N × H)
41. C₃×C₅×Dic₃	Q₈	semidirect
88. #627	C₄×C₅	semidirect
106. C₅×Dic₃	Dic₆	semidirect
124. C₃²×C₅	C₄×Q₈	semidirect
140. C₃×Dic₃	C₅×Q₈	semidirect
189. C₃×C₅	C₄×Dic₆	semidirect
190. C₃×C₅	#887	semidirect
208. Dic₃	C₅×Dic₆	semidirect
214. C₃²	C₄×C₅×Q₈	semidirect
232. C₅	#63837	direct (N × H)
242. C₃	C₄×C₅×Dic₆	semidirect
243. C₃	C₅×#887	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#627	2
C₅×#3750	2
C₃×C₅×Dic₃⋊C₄	2
C₃×C₄×C₅×Dic₃	2
C₅×Dic₃⋊Dic₃	2
C₅×Dic₃²	2
C₅×#887	3
C₄×C₅×Dic₆	3
#63837	5

Subgroups — C5×#63837 #13087184

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#63837 #13087184