Subgroups — C₃×#65887 #11755564

Order 960.

624 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×C₃×#709	C₂	1	2	yes	maximal, complemented
2	C₃×C₄×C₅×D₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#4086	C₂	1	2	yes	maximal, complemented
4	C₃×#4076	C₂	1	2	yes	maximal, complemented
5	C₃×#4039	C₂	1	2	yes	maximal, complemented
6	C₃×#4030	C₂	1	4	yes	maximal, complemented
7	C₃×#4020	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×C₄×Dic₁₀	C₂	1	1	yes	maximal, characteristic, complemented
9	#65887	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	C₂×C₃×C₂²⋊D₅	C₂²	1	4	yes	—
14	C₃×#709	—	2	8	no	—
15	C₂×C₃×D₂₀	C₂²	1	2	yes	—
16	C₂×C₃×D₂₀	C₂²	1	2	yes	—
17	C₂×C₃×D₂₀	C₂²	1	2	yes	—
18	C₂×C₃×C₄×D₅	C₂²	1	4	yes	—
19	C₂×C₃×Dic₁₀	C₂²	1	1	yes	characteristic
20	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
22	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
23	C₃×#685	C₂²	1	4	yes	—
24	C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
25	C₃×#683	C₂²	1	2	yes	—
26	C₃×C₄×Dic₅	C₂²	1	2	yes	—
27	C₃×#550	—	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	#4086	C₂×C₃	1	2	yes	complemented
31	#4076	C₂×C₃	1	2	yes	complemented
32	#4039	C₂×C₃	1	2	yes	complemented
33	#4030	C₂×C₃	1	4	yes	complemented
34	#4020	C₂×C₃	1	1	yes	characteristic, complemented
35	C₄×Dic₁₀	C₂×C₃	1	1	yes	characteristic, complemented
36	C₂³×C₃×C₅	C₂³	1	2	yes	—
37	C₂²×C₃×D₅	C₂³	1	4	yes	—
38	C₃×C₅×D₄	—	2	2	no	—
39	C₂×C₃×C₄×C₅	—	2	4	no	—
40	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
41	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
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	C₃×C₂²⋊D₅	—	4	4	no	—
46	C₂×C₃×Dic₅	C₂³	1	2	yes	—
47	C₂×C₃×Dic₅	C₂³	1	2	yes	—
48	C₃×D₂₀	—	2	4	no	—
49	C₃×D₂₀	—	2	4	no	—
50	C₃×D₂₀	—	4	2	no	—
51	C₃×C₄×D₅	—	2	8	no	—
52	C₃×Dic₁₀	D₄	1	4	yes	complemented
53	C₂×C₃×#41	—	5	2	no	—
54	C₃×#127	—	5	2	no	—
55	C₃×#127	—	5	1	no	—
56	C₃×#121	—	5	2	no	—
57	C₃×#121	—	5	4	no	—
58	C₃×C₄×Q₈	—	5	1	no	—
59	C₃×C₄×D₄	—	5	2	no	—
60	C₃×C₄×D₄	D₅	1	1	yes	characteristic, complemented
61	C₂×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
62	C₂²×C₄×C₅	C₂²×C₃	1	2	yes	—
63	C₂×C₂²⋊D₅	C₂²×C₃	1	4	yes	—
64	C₂×C₂²⋊D₅	C₂²×C₃	1	4	yes	—
65	#709	—	2	8	no	—
66	C₂×D₂₀	C₂²×C₃	1	2	yes	—
67	C₂×D₂₀	C₂²×C₃	1	2	yes	—
68	C₂×D₂₀	C₂²×C₃	1	2	yes	—
69	C₂×C₄×D₅	C₂²×C₃	1	4	yes	—
70	C₂×Dic₁₀	C₂²×C₃	1	1	yes	characteristic
71	C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
72	C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
73	C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
74	#685	C₂²×C₃	1	4	yes	—
75	C₄⋊Dic₅	C₂²×C₃	1	1	yes	characteristic
76	#683	C₂²×C₃	1	2	yes	—
77	C₄×Dic₅	C₂²×C₃	1	2	yes	—
78	#550	—	5	1	no	Sylow
79	C₂²×C₃×C₅	—	2	2	no	—
80	C₂²×C₃×C₅	—	2	2	no	—
81	C₂²×C₃×C₅	—	2	2	no	—
82	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
83	C₂×C₃×D₅	—	2	4	no	—
84	C₂×C₃×D₅	—	2	4	no	—
85	C₂×C₃×D₅	—	2	4	no	—
86	C₃×C₄×C₅	—	2	2	no	—
87	C₃×C₄×C₅	—	2	1	no	—
88	C₃×C₄×C₅	C₂×D₄	1	2	yes	—
89	C₃×C₄×C₅	#41	1	2	yes	complemented
90	C₃×Dic₅	—	2	2	no	—
91	C₃×Dic₅	C₂×D₄	1	4	yes	—
92	C₃×#41	—	10	8	no	—
93	C₂×C₃×Q₈	—	5	1	no	—
94	C₂×C₃×D₄	—	5	4	no	—
95	C₂×C₃×D₄	—	5	4	no	—
96	C₂×C₃×D₄	—	5	2	no	—
97	C₂×C₃×D₄	—	5	2	no	—
98	C₂×C₃×D₄	—	5	2	no	—
99	C₂×C₃×D₄	C₂×D₅	1	1	yes	characteristic
100	C₂²×C₃×C₄	—	5	4	no	—
101	C₂²×C₃×C₄	C₂×D₅	1	2	yes	—
102	C₃×C₄⋊C₄	—	5	2	no	—
103	C₃×C₄⋊C₄	—	5	1	no	—
104	C₃×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
105	C₃×C₂²⋊C₄	—	5	4	no	—
106	C₃×C₂²⋊C₄	C₂×D₅	1	2	yes	—
107	C₃×C₄²	—	5	2	no	—
108	C₃×C₄²	C₂×D₅	1	1	yes	characteristic
109	C₂³×C₅	C₂³×C₃	1	2	yes	—
110	C₂²×D₅	C₂³×C₃	1	4	yes	—
111	C₅×D₄	—	2	2	no	—
112	C₂×C₄×C₅	—	2	4	no	—
113	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
114	C₂×C₄×C₅	C₂³×C₃	1	2	yes	—
115	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
116	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
117	C₂²⋊D₅	—	2	8	no	—
118	C₂²⋊D₅	—	4	4	no	—
119	C₂×Dic₅	C₂³×C₃	1	2	yes	—
120	C₂×Dic₅	C₂³×C₃	1	2	yes	—
121	D₂₀	—	2	4	no	—
122	D₂₀	—	2	4	no	—
123	D₂₀	—	4	2	no	—
124	C₄×D₅	—	2	8	no	—
125	Dic₁₀	C₃×D₄	1	4	yes	complemented
126	C₂×#41	—	5	2	no	—
127	#127	—	5	2	no	—
128	#127	—	5	1	no	—
129	#121	—	5	2	no	—
130	#121	—	5	4	no	—
131	C₄×Q₈	—	5	1	no	—
132	C₄×D₄	—	5	2	no	—
133	C₄×D₄	C₃×D₅	1	1	yes	characteristic, complemented
134	C₂×C₃×C₅	—	4	2	no	—
135	C₂×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
137	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
138	C₃×D₅	—	4	4	no	—
139	C₂³×C₃	—	5	4	no	—
140	C₂³×C₃	C₂²×D₅	1	2	yes	—
141	C₃×Q₈	—	5	4	no	—
142	C₃×D₄	—	2	2	no	—
143	C₃×D₄	—	10	8	no	—
144	C₃×D₄	—	10	4	no	—
145	C₃×D₄	—	10	4	no	—
146	C₃×D₄	—	20	2	no	—
147	C₃×D₄	—	20	4	no	—
148	C₂×C₃×C₄	—	2	4	no	—
149	C₂×C₃×C₄	—	5	2	no	—
150	C₂×C₃×C₄	—	5	2	no	—
151	C₂×C₃×C₄	—	10	8	no	—
152	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
153	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
154	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
155	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
156	C₂²×C₅	—	2	2	no	—
157	C₂²×C₅	—	2	2	no	—
158	C₂²×C₅	—	2	2	no	—
159	C₂²×C₅	C₂⁴×C₃	1	1	yes	characteristic
160	C₂×D₅	—	2	4	no	—
161	C₂×D₅	—	2	4	no	—
162	C₂×D₅	—	2	4	no	—
163	C₄×C₅	—	2	2	no	—
164	C₄×C₅	—	2	1	no	—
165	C₄×C₅	C₂×C₃×D₄	1	2	yes	—
166	C₄×C₅	C₃×#41	1	2	yes	complemented
167	Dic₅	—	2	2	no	—
168	Dic₅	C₂×C₃×D₄	1	4	yes	—
169	#41	—	10	8	no	—
170	C₂×Q₈	—	5	1	no	—
171	C₂×D₄	—	5	4	no	—
172	C₂×D₄	—	5	4	no	—
173	C₂×D₄	—	5	2	no	—
174	C₂×D₄	—	5	2	no	—
175	C₂×D₄	—	5	2	no	—
176	C₂×D₄	C₂×C₃×D₅	1	1	yes	characteristic
177	C₂²×C₄	—	5	4	no	—
178	C₂²×C₄	C₂×C₃×D₅	1	2	yes	—
179	C₄⋊C₄	—	5	2	no	—
180	C₄⋊C₄	—	5	1	no	—
181	C₄⋊C₄	C₂×C₃×D₅	1	1	yes	characteristic
182	C₂²⋊C₄	—	5	4	no	—
183	C₂²⋊C₄	C₂×C₃×D₅	1	2	yes	—
184	C₄²	—	5	2	no	—
185	C₄²	C₂×C₃×D₅	1	1	yes	characteristic
186	C₃×C₅	#550	1	1	yes	characteristic, complemented, Hall
187	C₂²×C₃	—	2	2	no	—
188	C₂²×C₃	—	2	2	no	—
189	C₂²×C₃	—	2	2	no	—
190	C₂²×C₃	—	10	4	no	—
191	C₂²×C₃	—	10	4	no	—
192	C₂²×C₃	—	10	4	no	—
193	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
194	C₃×C₄	—	2	2	no	—
195	C₃×C₄	—	2	1	no	—
196	C₃×C₄	—	5	4	no	—
197	C₃×C₄	—	10	2	no	—
198	C₃×C₄	#709	1	2	yes	complemented
199	C₃×C₄	D₄×D₅	1	2	yes	—
200	C₂×C₅	—	4	2	no	—
201	C₂×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
202	C₂×C₅	C₂×C₃×#41	1	1	yes	characteristic
203	C₂×C₅	C₃×H(5,𝔽₂)	1	1	yes	characteristic
204	D₅	—	4	4	no	—
205	C₂³	—	5	4	no	—
206	C₂³	C₂²×C₃×D₅	1	2	yes	—
207	Q₈	—	5	4	no	—
208	D₄	—	2	2	no	—
209	D₄	—	10	8	no	—
210	D₄	—	10	4	no	—
211	D₄	—	10	4	no	—
212	D₄	—	20	2	no	—
213	D₄	—	20	4	no	—
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	5	2	no	—
216	C₂×C₄	—	5	2	no	—
217	C₂×C₄	—	10	8	no	—
218	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
219	C₂×C₄	C₂²×C₃×D₅	1	2	yes	—
220	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
221	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
222	C₂×C₃	—	4	2	no	—
223	C₂×C₃	—	20	4	no	—
224	C₂×C₃	C₂×#709	1	1	yes	characteristic
225	C₂×C₃	C₂×D₄×D₅	1	1	yes	characteristic
226	C₂×C₃	#4149	1	1	yes	characteristic
227	C₅	C₃×#550	1	1	yes	characteristic, complemented, Sylow
228	C₂²	—	2	2	no	—
229	C₂²	—	2	2	no	—
230	C₂²	—	2	2	no	—
231	C₂²	—	10	4	no	—
232	C₂²	—	10	4	no	—
233	C₂²	—	10	4	no	—
234	C₂²	C₂³×C₃×D₅	1	1	yes	characteristic
235	C₄	—	2	2	no	—
236	C₄	—	2	1	no	—
237	C₄	—	5	4	no	—
238	C₄	—	10	2	no	—
239	C₄	C₃×#709	1	2	yes	complemented
240	C₄	C₃×D₄×D₅	1	2	yes	—
241	C₃	#65887	1	1	yes	characteristic, direct product factor, Sylow
242	C₂	—	4	2	no	—
243	C₂	—	20	4	no	—
244	C₂	C₂×C₃×#709	1	1	yes	characteristic
245	C₂	C₂×C₃×D₄×D₅	1	1	yes	characteristic
246	C₂	C₃×#4149	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×#550
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×#4086	C₂	semidirect
4. C₃×#4076	C₂	semidirect
5. C₃×#4039	C₂	semidirect
6. C₃×#4030	C₂	semidirect
7. C₃×#4020	C₂	semidirect
8. C₃×C₄×Dic₁₀	C₂	semidirect
9. #65887	C₃	direct (N × H)
28. C₂×#709	C₂×C₃	semidirect
29. C₄×C₅×D₄	C₂×C₃	semidirect
30. #4086	C₂×C₃	semidirect
31. #4076	C₂×C₃	semidirect
32. #4039	C₂×C₃	semidirect
33. #4030	C₂×C₃	semidirect
34. #4020	C₂×C₃	semidirect
35. C₄×Dic₁₀	C₂×C₃	semidirect
52. C₃×Dic₁₀	D₄	semidirect
60. C₃×C₄×D₄	D₅	semidirect
89. C₃×C₄×C₅	#41	semidirect
125. Dic₁₀	C₃×D₄	semidirect
133. C₄×D₄	C₃×D₅	semidirect
166. C₄×C₅	C₃×#41	semidirect
186. C₃×C₅	#550	semidirect
198. C₃×C₄	#709	semidirect
227. C₅	C₃×#550	semidirect
239. C₄	C₃×#709	semidirect
241. C₃	#65887	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×#4086	2
C₃×#4076	2
C₃×#4039	2
C₃×#4030	2
C₃×#4020	2
C₃×C₄×Dic₁₀	2
#65887	3
C₃×#550	5

Subgroups — C3×#65887 #11755564

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#65887 #11755564