Subgroups — C₃×#65885 #11755562

Order 960.

584 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#709	C₂	1	2	yes	maximal, complemented
2	C₃×C₄×C₅×D₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#4084	C₂	1	2	yes	maximal, complemented
4	C₃×#4070	C₂	1	2	yes	maximal, complemented
5	C₃×D₅×C₄⋊C₄	C₂	1	2	yes	maximal, complemented
6	C₃×#4029	C₂	1	4	yes	maximal, complemented
7	C₃×C₄×D₂₀	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#4015	C₂	1	1	yes	maximal, characteristic, complemented
9	#65885	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₄×C₅	C₂²	1	2	yes	—
12	C₂×C₃×C₂²⋊D₅	C₂²	1	4	yes	—
13	C₃×#709	—	2	8	no	—
14	C₂×C₃×D₂₀	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₄×D₅	C₂²	1	2	yes	—
16	C₂×C₃×C₄×D₅	C₂²	1	4	yes	—
17	C₂×C₃×Dic₁₀	C₂²	1	2	yes	—
18	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
20	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
21	C₃×C₂²⋊Dic₅	C₂²	1	4	yes	—
22	C₃×#685	C₂²	1	2	yes	—
23	C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
24	C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
25	C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
26	C₃×#683	C₂²	1	4	yes	—
27	C₃×#547	—	5	1	no	maximal, Hall
28	C₂×#709	C₂×C₃	1	2	yes	complemented
29	C₄×C₅×D₄	C₂×C₃	1	1	yes	characteristic, complemented
30	#4084	C₂×C₃	1	2	yes	complemented
31	#4070	C₂×C₃	1	2	yes	complemented
32	D₅×C₄⋊C₄	C₂×C₃	1	2	yes	complemented
33	#4029	C₂×C₃	1	4	yes	complemented
34	C₄×D₂₀	C₂×C₃	1	1	yes	characteristic, complemented
35	#4015	C₂×C₃	1	1	yes	characteristic, complemented
36	C₂³×C₃×C₅	C₂³	1	2	yes	—
37	C₂²×C₃×D₅	C₂³	1	2	yes	—
38	C₃×C₅×D₄	—	2	2	no	—
39	C₂×C₃×C₄×C₅	—	2	4	no	—
40	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
41	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
42	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
43	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
44	C₃×C₂²⋊D₅	—	2	8	no	—
45	C₂×C₃×Dic₅	C₂³	1	2	yes	—
46	C₂×C₃×Dic₅	C₂³	1	4	yes	—
47	C₃×D₂₀	D₄	1	4	yes	complemented
48	C₃×C₄×D₅	—	2	8	no	—
49	C₃×C₄×D₅	—	2	4	no	—
50	C₃×Dic₁₀	—	2	4	no	—
51	C₂×C₃×#41	—	5	2	no	—
52	C₃×C₄⋊Q₈	—	5	1	no	—
53	C₃×#123	—	5	4	no	—
54	C₃×C₂²⋊Q₈	—	5	2	no	—
55	C₃×#121	—	5	2	no	—
56	C₃×C₄×D₄	—	5	1	no	—
57	C₃×C₄×D₄	D₅	1	1	yes	characteristic, complemented
58	C₂×C₃×C₄⋊C₄	—	5	2	no	—
59	C₂×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
60	C₂²×C₄×C₅	C₂²×C₃	1	2	yes	—
61	C₂×C₂²⋊D₅	C₂²×C₃	1	4	yes	—
62	#709	—	2	8	no	—
63	C₂×D₂₀	C₂²×C₃	1	1	yes	characteristic
64	C₂×C₄×D₅	C₂²×C₃	1	4	yes	—
65	C₂×C₄×D₅	C₂²×C₃	1	2	yes	—
66	C₂×Dic₁₀	C₂²×C₃	1	2	yes	—
67	C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
68	C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
69	C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
70	C₂²⋊Dic₅	C₂²×C₃	1	4	yes	—
71	#685	C₂²×C₃	1	2	yes	—
72	C₄⋊Dic₅	C₂²×C₃	1	2	yes	—
73	C₄⋊Dic₅	C₂²×C₃	1	2	yes	—
74	C₄⋊Dic₅	C₂²×C₃	1	1	yes	characteristic
75	#683	C₂²×C₃	1	4	yes	—
76	#547	—	5	1	no	Sylow
77	C₂²×C₃×C₅	—	2	2	no	—
78	C₂²×C₃×C₅	—	2	2	no	—
79	C₂²×C₃×C₅	—	2	2	no	—
80	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
81	C₂×C₃×D₅	—	2	2	no	—
82	C₂×C₃×D₅	—	2	2	no	—
83	C₂×C₃×D₅	C₂×D₄	1	4	yes	—
84	C₃×C₄×C₅	—	2	2	no	—
85	C₃×C₄×C₅	—	2	1	no	—
86	C₃×C₄×C₅	C₂×D₄	1	2	yes	—
87	C₃×C₄×C₅	#41	1	2	yes	complemented
88	C₃×Dic₅	—	2	4	no	—
89	C₃×Dic₅	—	2	2	no	—
90	C₃×#41	—	10	8	no	—
91	C₂×C₃×Q₈	—	5	2	no	—
92	C₂×C₃×D₄	—	5	4	no	—
93	C₂×C₃×D₄	—	5	1	no	—
94	C₂×C₃×D₄	C₂×D₅	1	1	yes	characteristic
95	C₂²×C₃×C₄	—	5	2	no	—
96	C₂²×C₃×C₄	—	5	4	no	—
97	C₂²×C₃×C₄	C₂×D₅	1	2	yes	—
98	C₃×C₄⋊C₄	—	5	4	no	—
99	C₃×C₄⋊C₄	—	5	2	no	—
100	C₃×C₄⋊C₄	—	5	1	no	—
101	C₃×C₄⋊C₄	—	5	2	no	—
102	C₃×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
103	C₃×C₂²⋊C₄	—	5	4	no	—
104	C₃×C₂²⋊C₄	—	5	2	no	—
105	C₃×C₂²⋊C₄	C₂×D₅	1	2	yes	—
106	C₃×C₄²	C₂×D₅	1	1	yes	characteristic
107	C₂³×C₅	C₂³×C₃	1	2	yes	—
108	C₂²×D₅	C₂³×C₃	1	2	yes	—
109	C₅×D₄	—	2	2	no	—
110	C₂×C₄×C₅	—	2	4	no	—
111	C₂×C₄×C₅	C₂³×C₃	1	2	yes	—
112	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
113	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
114	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
115	C₂²⋊D₅	—	2	8	no	—
116	C₂×Dic₅	C₂³×C₃	1	2	yes	—
117	C₂×Dic₅	C₂³×C₃	1	4	yes	—
118	D₂₀	C₃×D₄	1	4	yes	complemented
119	C₄×D₅	—	2	8	no	—
120	C₄×D₅	—	2	4	no	—
121	Dic₁₀	—	2	4	no	—
122	C₂×#41	—	5	2	no	—
123	C₄⋊Q₈	—	5	1	no	—
124	#123	—	5	4	no	—
125	C₂²⋊Q₈	—	5	2	no	—
126	#121	—	5	2	no	—
127	C₄×D₄	—	5	1	no	—
128	C₄×D₄	C₃×D₅	1	1	yes	characteristic, complemented
129	C₂×C₄⋊C₄	—	5	2	no	—
130	C₂×C₃×C₅	—	4	2	no	—
131	C₂×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
133	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
134	C₃×D₅	—	2	4	no	—
135	C₂³×C₃	—	5	2	no	—
136	C₂³×C₃	C₂²×D₅	1	2	yes	—
137	C₃×Q₈	—	10	4	no	—
138	C₃×D₄	—	2	2	no	—
139	C₃×D₄	—	5	4	no	—
140	C₃×D₄	—	10	8	no	—
141	C₂×C₃×C₄	—	2	4	no	—
142	C₂×C₃×C₄	—	5	2	no	—
143	C₂×C₃×C₄	—	5	4	no	—
144	C₂×C₃×C₄	—	10	8	no	—
145	C₂×C₃×C₄	—	10	4	no	—
146	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
147	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
148	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
149	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
150	C₂²×C₅	—	2	2	no	—
151	C₂²×C₅	—	2	2	no	—
152	C₂²×C₅	—	2	2	no	—
153	C₂²×C₅	C₂⁴×C₃	1	1	yes	characteristic
154	C₂×D₅	—	2	2	no	—
155	C₂×D₅	—	2	2	no	—
156	C₂×D₅	C₂×C₃×D₄	1	4	yes	—
157	C₄×C₅	—	2	2	no	—
158	C₄×C₅	—	2	1	no	—
159	C₄×C₅	C₂×C₃×D₄	1	2	yes	—
160	C₄×C₅	C₃×#41	1	2	yes	complemented
161	Dic₅	—	2	4	no	—
162	Dic₅	—	2	2	no	—
163	#41	—	10	8	no	—
164	C₂×Q₈	—	5	2	no	—
165	C₂×D₄	—	5	4	no	—
166	C₂×D₄	—	5	1	no	—
167	C₂×D₄	C₂×C₃×D₅	1	1	yes	characteristic
168	C₂²×C₄	—	5	2	no	—
169	C₂²×C₄	—	5	4	no	—
170	C₂²×C₄	C₂×C₃×D₅	1	2	yes	—
171	C₄⋊C₄	—	5	2	no	—
172	C₄⋊C₄	—	5	4	no	—
173	C₄⋊C₄	—	5	1	no	—
174	C₄⋊C₄	—	5	2	no	—
175	C₄⋊C₄	C₂×C₃×D₅	1	1	yes	characteristic
176	C₂²⋊C₄	—	5	4	no	—
177	C₂²⋊C₄	—	5	2	no	—
178	C₂²⋊C₄	C₂×C₃×D₅	1	2	yes	—
179	C₄²	C₂×C₃×D₅	1	1	yes	characteristic
180	C₃×C₅	#547	1	1	yes	characteristic, complemented, Hall
181	C₂²×C₃	—	2	2	no	—
182	C₂²×C₃	—	2	2	no	—
183	C₂²×C₃	—	2	2	no	—
184	C₂²×C₃	—	5	4	no	—
185	C₂²×C₃	—	10	2	no	—
186	C₂²×C₃	—	10	2	no	—
187	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
188	C₃×C₄	—	2	2	no	—
189	C₃×C₄	—	2	1	no	—
190	C₃×C₄	—	10	2	no	—
191	C₃×C₄	—	10	4	no	—
192	C₃×C₄	#709	1	2	yes	complemented
193	C₃×C₄	D₄×D₅	1	2	yes	—
194	C₂×C₅	—	4	2	no	—
195	C₂×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
196	C₂×C₅	C₂×C₃×#41	1	1	yes	characteristic
197	C₂×C₅	C₃×2₋¹⁺⁴	1	1	yes	characteristic
198	D₅	—	2	4	no	—
199	C₂³	—	5	2	no	—
200	C₂³	C₂²×C₃×D₅	1	2	yes	—
201	Q₈	—	10	4	no	—
202	D₄	—	2	2	no	—
203	D₄	—	5	4	no	—
204	D₄	—	10	8	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	5	2	no	—
207	C₂×C₄	—	5	4	no	—
208	C₂×C₄	—	10	8	no	—
209	C₂×C₄	—	10	4	no	—
210	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
211	C₂×C₄	C₂²×C₃×D₅	1	2	yes	—
212	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
213	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
214	C₂×C₃	—	4	2	no	—
215	C₂×C₃	—	10	4	no	—
216	C₂×C₃	C₂×#709	1	1	yes	characteristic
217	C₂×C₃	C₂×D₄×D₅	1	1	yes	characteristic
218	C₂×C₃	#4150	1	1	yes	characteristic
219	C₅	C₃×#547	1	1	yes	characteristic, complemented, Sylow
220	C₂²	—	2	2	no	—
221	C₂²	—	2	2	no	—
222	C₂²	—	2	2	no	—
223	C₂²	—	5	4	no	—
224	C₂²	—	10	2	no	—
225	C₂²	—	10	2	no	—
226	C₂²	C₂³×C₃×D₅	1	1	yes	characteristic
227	C₄	—	2	2	no	—
228	C₄	—	2	1	no	—
229	C₄	—	10	2	no	—
230	C₄	—	10	4	no	—
231	C₄	C₃×#709	1	2	yes	complemented
232	C₄	C₃×D₄×D₅	1	2	yes	—
233	C₃	#65885	1	1	yes	characteristic, direct product factor, Sylow
234	C₂	—	4	2	no	—
235	C₂	—	10	4	no	—
236	C₂	C₂×C₃×#709	1	1	yes	characteristic
237	C₂	C₂×C₃×D₄×D₅	1	1	yes	characteristic
238	C₂	C₃×#4150	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂³×D₅
commutator, cosocle	C₂²×C₅	C₂⁴×C₃
Frattini	C₂²	C₂³×C₃×D₅
Fitting	C₃×C₄×C₅×D₄	C₂
socle	C₂²×C₃×C₅	C₂⁴
hypercenter	C₃×C₄×D₄	D₅
hypocenter	C₅	C₃×#547
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#709	C₂	semidirect
2. C₃×C₄×C₅×D₄	C₂	semidirect
3. C₃×#4084	C₂	semidirect
4. C₃×#4070	C₂	semidirect
5. C₃×D₅×C₄⋊C₄	C₂	semidirect
6. C₃×#4029	C₂	semidirect
7. C₃×C₄×D₂₀	C₂	semidirect
8. C₃×#4015	C₂	semidirect
9. #65885	C₃	direct (N × H)
28. C₂×#709	C₂×C₃	semidirect
29. C₄×C₅×D₄	C₂×C₃	semidirect
30. #4084	C₂×C₃	semidirect
31. #4070	C₂×C₃	semidirect
32. D₅×C₄⋊C₄	C₂×C₃	semidirect
33. #4029	C₂×C₃	semidirect
34. C₄×D₂₀	C₂×C₃	semidirect
35. #4015	C₂×C₃	semidirect
47. C₃×D₂₀	D₄	semidirect
57. C₃×C₄×D₄	D₅	semidirect
87. C₃×C₄×C₅	#41	semidirect
118. D₂₀	C₃×D₄	semidirect
128. C₄×D₄	C₃×D₅	semidirect
160. C₄×C₅	C₃×#41	semidirect
180. C₃×C₅	#547	semidirect
192. C₃×C₄	#709	semidirect
219. C₅	C₃×#547	semidirect
231. C₄	C₃×#709	semidirect
233. C₃	#65885	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×#709	2
C₃×C₄×C₅×D₄	2
C₃×#4084	2
C₃×#4070	2
C₃×D₅×C₄⋊C₄	2
C₃×#4029	2
C₃×C₄×D₂₀	2
C₃×#4015	2
#65885	3
C₃×#547	5

Subgroups — C3×#65885 #11755562

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#65885 #11755562