Subgroups — C₃×#66009 #11755686

Order 960.

540 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×Q₈×D₅	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#711	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#124	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#4090	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#4082	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#4029	C₂	1	4	yes	maximal, complemented
7	C₃×#4024	C₂	1	4	yes	maximal, complemented
8	C₃×#4018	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#4015	C₂	1	1	yes	maximal, characteristic, complemented
10	#66009	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₂×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₂²⋊D₅	C₂²	1	2	yes	—
14	C₂²×C₃×Dic₅	C₂²	1	2	yes	—
15	C₃×Q₈×D₅	—	2	4	no	—
16	C₃×#711	—	2	4	no	—
17	C₂×C₃×C₄×D₅	C₂²	1	2	yes	—
18	C₂×C₃×C₄×D₅	C₂²	1	1	yes	characteristic
19	C₂×C₃×Dic₁₀	C₂²	1	1	yes	characteristic
20	C₂×C₃×Dic₁₀	C₂²	1	1	yes	characteristic
21	C₂×C₃×Dic₁₀	C₂²	1	2	yes	—
22	C₃×C₅×C₂²⋊C₄	C₂²	1	4	yes	—
23	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
24	C₃×C₂²⋊Dic₅	C₂²	1	4	yes	—
25	C₃×#685	C₂²	1	2	yes	—
26	C₃×C₄⋊Dic₅	C₂²	1	4	yes	—
27	C₃×#683	C₂²	1	4	yes	—
28	C₃×#683	C₂²	1	2	yes	—
29	C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
30	C₃×#536	—	5	1	no	maximal, Hall
31	C₂×Q₈×D₅	C₂×C₃	1	1	yes	characteristic, complemented
32	C₂×#711	C₂×C₃	1	1	yes	characteristic
33	C₅×#124	C₂×C₃	1	1	yes	characteristic, complemented
34	#4090	C₂×C₃	1	1	yes	characteristic, complemented
35	#4082	C₂×C₃	1	1	yes	characteristic, complemented
36	#4029	C₂×C₃	1	4	yes	complemented
37	#4024	C₂×C₃	1	4	yes	complemented
38	#4018	C₂×C₃	1	1	yes	characteristic, complemented
39	#4015	C₂×C₃	1	1	yes	characteristic, complemented
40	C₂³×C₃×C₅	C₂³	1	2	yes	—
41	C₂²×C₃×D₅	C₂³	1	1	yes	characteristic
42	C₃×C₅×Q₈	—	2	2	no	—
43	C₃×C₅×D₄	—	2	2	no	—
44	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
45	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
46	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
47	C₃×C₂²⋊D₅	—	2	4	no	—
48	C₂×C₃×Dic₅	—	2	4	no	—
49	C₂×C₃×Dic₅	C₂³	1	2	yes	—
50	C₂×C₃×Dic₅	C₂³	1	2	yes	—
51	C₂×C₃×Dic₅	C₂³	1	2	yes	—
52	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
53	C₃×C₄×D₅	—	2	4	no	—
54	C₃×C₄×D₅	D₄	1	4	yes	—
55	C₃×Dic₁₀	—	2	2	no	—
56	C₃×Dic₁₀	—	2	2	no	—
57	C₃×Dic₁₀	—	2	4	no	—
58	C₂×C₃×#41	—	5	1	no	—
59	C₂²×C₃×Q₈	—	5	1	no	—
60	C₃×C₄⋊Q₈	—	5	1	no	—
61	C₃×C₄⋊Q₈	—	5	1	no	—
62	C₃×#124	—	5	1	no	—
63	C₃×#124	D₅	1	1	yes	characteristic, complemented
64	C₃×#123	—	5	4	no	—
65	C₃×C₂²⋊Q₈	—	5	4	no	—
66	C₃×#117	—	5	1	no	—
67	C₂×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
68	C₂×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
69	C₂×C₂²⋊D₅	C₂²×C₃	1	2	yes	—
70	C₂²×Dic₅	C₂²×C₃	1	2	yes	—
71	Q₈×D₅	—	2	4	no	—
72	#711	—	2	4	no	—
73	C₂×C₄×D₅	C₂²×C₃	1	2	yes	—
74	C₂×C₄×D₅	C₂²×C₃	1	1	yes	characteristic
75	C₂×Dic₁₀	C₂²×C₃	1	1	yes	characteristic
76	C₂×Dic₁₀	C₂²×C₃	1	1	yes	characteristic
77	C₂×Dic₁₀	C₂²×C₃	1	2	yes	—
78	C₅×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
79	C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
80	C₂²⋊Dic₅	C₂²×C₃	1	4	yes	—
81	#685	C₂²×C₃	1	2	yes	—
82	C₄⋊Dic₅	C₂²×C₃	1	4	yes	—
83	#683	C₂²×C₃	1	2	yes	—
84	#683	C₂²×C₃	1	4	yes	—
85	C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic
86	#536	—	5	1	no	Sylow
87	C₂²×C₃×C₅	—	2	2	no	—
88	C₂²×C₃×C₅	—	2	4	no	—
89	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
90	C₂×C₃×D₅	—	2	2	no	—
91	C₂×C₃×D₅	C₂×D₄	1	2	yes	—
92	C₃×C₄×C₅	—	2	2	no	—
93	C₃×C₄×C₅	—	2	2	no	—
94	C₃×C₄×C₅	C₂×D₄	1	2	yes	—
95	C₃×Dic₅	—	2	2	no	—
96	C₃×Dic₅	—	2	2	no	—
97	C₃×Dic₅	—	2	2	no	—
98	C₃×Dic₅	C₂×D₄	1	2	yes	—
99	C₃×#41	—	10	4	no	—
100	C₂×C₃×Q₈	—	5	2	no	—
101	C₂×C₃×Q₈	—	5	1	no	—
102	C₂×C₃×Q₈	—	5	1	no	—
103	C₂×C₃×Q₈	—	10	4	no	—
104	C₂×C₃×Q₈	C₂×D₅	1	1	yes	characteristic
105	C₂×C₃×D₄	—	5	2	no	—
106	C₂×C₃×D₄	C₂×D₅	1	1	yes	characteristic
107	C₂²×C₃×C₄	—	5	1	no	—
108	C₂²×C₃×C₄	—	5	2	no	—
109	C₂²×C₃×C₄	—	5	2	no	—
110	C₃×C₄⋊C₄	—	5	2	no	—
111	C₃×C₄⋊C₄	—	5	4	no	—
112	C₃×C₄⋊C₄	—	5	4	no	—
113	C₃×C₂²⋊C₄	—	5	4	no	—
114	C₃×C₂²⋊C₄	—	5	2	no	—
115	C₃×C₂²⋊C₄	C₂×D₅	1	4	yes	—
116	C₃×C₄²	—	5	1	no	—
117	C₃×C₄²	C₂×D₅	1	1	yes	characteristic
118	C₂³×C₅	C₂³×C₃	1	2	yes	—
119	C₂²×D₅	C₂³×C₃	1	1	yes	characteristic
120	C₅×Q₈	—	2	2	no	—
121	C₅×D₄	—	2	2	no	—
122	C₂×C₄×C₅	C₂³×C₃	1	2	yes	—
123	C₂×C₄×C₅	C₂³×C₃	1	2	yes	—
124	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
125	C₂²⋊D₅	—	2	4	no	—
126	C₂×Dic₅	—	2	4	no	—
127	C₂×Dic₅	C₂³×C₃	1	2	yes	—
128	C₂×Dic₅	C₂³×C₃	1	2	yes	—
129	C₂×Dic₅	C₂³×C₃	1	2	yes	—
130	C₂×Dic₅	C₂³×C₃	1	1	yes	characteristic
131	C₄×D₅	—	2	4	no	—
132	C₄×D₅	C₃×D₄	1	4	yes	—
133	Dic₁₀	—	2	2	no	—
134	Dic₁₀	—	2	2	no	—
135	Dic₁₀	—	2	4	no	—
136	C₂×#41	—	5	1	no	—
137	C₂²×Q₈	—	5	1	no	—
138	C₄⋊Q₈	—	5	1	no	—
139	C₄⋊Q₈	—	5	1	no	—
140	#124	—	5	1	no	—
141	#124	C₃×D₅	1	1	yes	characteristic, complemented
142	#123	—	5	4	no	—
143	C₂²⋊Q₈	—	5	4	no	—
144	#117	—	5	1	no	—
145	C₂×C₃×C₅	—	4	2	no	—
146	C₂×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₃×C₅	2₋¹⁺⁴	1	2	yes	—
148	C₃×D₅	—	2	2	no	—
149	C₂³×C₃	—	5	1	no	—
150	C₂³×C₃	C₂²×D₅	1	2	yes	—
151	C₃×Q₈	—	2	2	no	—
152	C₃×Q₈	—	10	4	no	—
153	C₃×Q₈	—	10	2	no	—
154	C₃×Q₈	—	10	2	no	—
155	C₃×D₄	—	2	2	no	—
156	C₃×D₄	—	10	4	no	—
157	C₂×C₃×C₄	—	5	1	no	—
158	C₂×C₃×C₄	—	5	4	no	—
159	C₂×C₃×C₄	—	5	2	no	—
160	C₂×C₃×C₄	—	5	2	no	—
161	C₂×C₃×C₄	—	5	2	no	—
162	C₂×C₃×C₄	—	10	4	no	—
163	C₂×C₃×C₄	—	10	4	no	—
164	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
165	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
166	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
167	C₂²×C₅	—	2	2	no	—
168	C₂²×C₅	—	2	4	no	—
169	C₂²×C₅	C₂⁴×C₃	1	1	yes	characteristic
170	C₂×D₅	—	2	2	no	—
171	C₂×D₅	C₂×C₃×D₄	1	2	yes	—
172	C₄×C₅	—	2	2	no	—
173	C₄×C₅	—	2	2	no	—
174	C₄×C₅	C₂×C₃×D₄	1	2	yes	—
175	Dic₅	—	2	2	no	—
176	Dic₅	—	2	2	no	—
177	Dic₅	—	2	2	no	—
178	Dic₅	C₂×C₃×D₄	1	2	yes	—
179	#41	—	10	4	no	—
180	C₂×Q₈	—	5	2	no	—
181	C₂×Q₈	—	5	1	no	—
182	C₂×Q₈	—	5	1	no	—
183	C₂×Q₈	—	10	4	no	—
184	C₂×Q₈	C₂×C₃×D₅	1	1	yes	characteristic
185	C₂×D₄	—	5	2	no	—
186	C₂×D₄	C₂×C₃×D₅	1	1	yes	characteristic
187	C₂²×C₄	—	5	1	no	—
188	C₂²×C₄	—	5	2	no	—
189	C₂²×C₄	—	5	2	no	—
190	C₄⋊C₄	—	5	2	no	—
191	C₄⋊C₄	—	5	4	no	—
192	C₄⋊C₄	—	5	4	no	—
193	C₂²⋊C₄	—	5	4	no	—
194	C₂²⋊C₄	—	5	2	no	—
195	C₂²⋊C₄	C₂×C₃×D₅	1	4	yes	—
196	C₄²	—	5	1	no	—
197	C₄²	C₂×C₃×D₅	1	1	yes	characteristic
198	C₃×C₅	#536	1	1	yes	characteristic, complemented, Hall
199	C₂²×C₃	—	2	2	no	—
200	C₂²×C₃	—	2	4	no	—
201	C₂²×C₃	—	5	2	no	—
202	C₂²×C₃	—	10	2	no	—
203	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
204	C₃×C₄	—	2	2	no	—
205	C₃×C₄	—	2	2	no	—
206	C₃×C₄	—	5	2	no	—
207	C₃×C₄	—	10	2	no	—
208	C₃×C₄	—	10	2	no	—
209	C₃×C₄	—	10	2	no	—
210	C₃×C₄	D₄×D₅	1	2	yes	—
211	C₂×C₅	—	4	2	no	—
212	C₂×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
213	C₂×C₅	C₃×2₋¹⁺⁴	1	2	yes	—
214	D₅	—	2	2	no	—
215	C₂³	—	5	1	no	—
216	C₂³	C₂²×C₃×D₅	1	2	yes	—
217	Q₈	—	2	2	no	—
218	Q₈	—	10	4	no	—
219	Q₈	—	10	2	no	—
220	Q₈	—	10	2	no	—
221	D₄	—	2	2	no	—
222	D₄	—	10	4	no	—
223	C₂×C₄	—	5	1	no	—
224	C₂×C₄	—	5	4	no	—
225	C₂×C₄	—	5	2	no	—
226	C₂×C₄	—	5	2	no	—
227	C₂×C₄	—	5	2	no	—
228	C₂×C₄	—	10	4	no	—
229	C₂×C₄	—	10	4	no	—
230	C₂×C₄	C₂²×C₃×D₅	1	2	yes	—
231	C₂×C₄	C₂²×C₃×D₅	1	2	yes	—
232	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
233	C₂×C₃	—	4	2	no	—
234	C₂×C₃	—	10	2	no	—
235	C₂×C₃	C₂×D₄×D₅	1	1	yes	characteristic
236	C₂×C₃	#4150	1	2	yes	—
237	C₅	C₃×#536	1	1	yes	characteristic, complemented, Sylow
238	C₂²	—	2	2	no	—
239	C₂²	—	2	4	no	—
240	C₂²	—	5	2	no	—
241	C₂²	—	10	2	no	—
242	C₂²	C₂³×C₃×D₅	1	1	yes	characteristic
243	C₄	—	2	2	no	—
244	C₄	—	2	2	no	—
245	C₄	—	5	2	no	—
246	C₄	—	10	2	no	—
247	C₄	—	10	2	no	—
248	C₄	—	10	2	no	—
249	C₄	C₃×D₄×D₅	1	2	yes	—
250	C₃	#66009	1	1	yes	characteristic, direct product factor, Sylow
251	C₂	—	4	2	no	—
252	C₂	—	10	2	no	—
253	C₂	C₂×C₃×D₄×D₅	1	1	yes	characteristic
254	C₂	C₃×#4150	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃	C₂³×D₅
commutator, cosocle	C₂²×C₅	C₂⁴×C₃
Frattini	C₂²	C₂³×C₃×D₅
Fitting	C₃×C₅×#124	C₂
socle	C₂²×C₃×C₅	C₂⁴
prime radical	C₂×C₃×#711	C₂
hypercenter	C₃×#124	D₅
hypocenter	C₅	C₃×#536
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×#124

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×Q₈×D₅	C₂	semidirect
3. C₃×C₅×#124	C₂	semidirect
4. C₃×#4090	C₂	semidirect
5. C₃×#4082	C₂	semidirect
6. C₃×#4029	C₂	semidirect
7. C₃×#4024	C₂	semidirect
8. C₃×#4018	C₂	semidirect
9. C₃×#4015	C₂	semidirect
10. #66009	C₃	direct (N × H)
31. C₂×Q₈×D₅	C₂×C₃	semidirect
33. C₅×#124	C₂×C₃	semidirect
34. #4090	C₂×C₃	semidirect
35. #4082	C₂×C₃	semidirect
36. #4029	C₂×C₃	semidirect
37. #4024	C₂×C₃	semidirect
38. #4018	C₂×C₃	semidirect
39. #4015	C₂×C₃	semidirect
63. C₃×#124	D₅	semidirect
141. #124	C₃×D₅	semidirect
198. C₃×C₅	#536	semidirect
237. C₅	C₃×#536	semidirect
250. C₃	#66009	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×Q₈×D₅	2
C₂×C₃×#711	2
C₃×C₅×#124	2
C₃×#4090	2
C₃×#4082	2
C₃×#4029	2
C₃×#4024	2
C₃×#4018	2
C₃×#4015	2
#66009	3
C₃×#536	5

Subgroups — C3×#66009 #11755686

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#66009 #11755686