Subgroups — C₃×#66002 #11755679

Order 960.

620 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	C₂×C₃×#713	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#711	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×#124	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#4090	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#4086	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#4030	C₂	1	4	yes	maximal, complemented
7	C₃×#4027	C₂	1	4	yes	maximal, complemented
8	C₃×#4021	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×C₄²×D₅	C₂	1	1	yes	maximal, characteristic, complemented
10	#66002	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	4	yes	—
14	C₂×C₃×C₂²⋊D₅	C₂²	1	2	yes	—
15	C₂²×C₃×Dic₅	C₂²	1	2	yes	—
16	C₃×#713	—	2	4	no	—
17	C₃×#711	—	2	4	no	—
18	C₂×C₃×D₂₀	C₂²	1	1	yes	characteristic
19	C₂×C₃×D₂₀	C₂²	1	2	yes	—
20	C₂×C₃×C₄×D₅	C₂²	1	2	yes	—
21	C₂×C₃×C₄×D₅	C₂²	1	2	yes	—
22	C₂×C₃×C₄×D₅	C₂²	1	1	yes	characteristic
23	C₂×C₃×Dic₁₀	C₂²	1	1	yes	characteristic
24	C₃×C₅×C₂²⋊C₄	C₂²	1	4	yes	—
25	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
26	C₃×#685	C₂²	1	4	yes	—
27	C₃×#683	C₂²	1	4	yes	—
28	C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
29	C₃×C₄×Dic₅	C₂²	1	2	yes	—
30	C₃×#532	—	5	1	no	maximal, Hall
31	C₂×#713	C₂×C₃	1	1	yes	characteristic, complemented
32	C₂×#711	C₂×C₃	1	1	yes	characteristic, complemented
33	C₅×#124	C₂×C₃	1	1	yes	characteristic, complemented
34	#4090	C₂×C₃	1	1	yes	characteristic, complemented
35	#4086	C₂×C₃	1	1	yes	characteristic, complemented
36	#4030	C₂×C₃	1	4	yes	complemented
37	#4027	C₂×C₃	1	4	yes	complemented
38	#4021	C₂×C₃	1	1	yes	characteristic, complemented
39	C₄²×D₅	C₂×C₃	1	1	yes	characteristic, complemented
40	C₂³×C₃×C₅	C₂³	1	2	yes	—
41	C₂²×C₃×D₅	C₂³	1	2	yes	—
42	C₂²×C₃×D₅	C₂³	1	1	yes	characteristic
43	C₃×C₅×Q₈	—	2	2	no	—
44	C₃×C₅×D₄	—	2	2	no	—
45	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
46	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
47	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
48	C₃×C₂²⋊D₅	—	2	4	no	—
49	C₃×C₂²⋊D₅	—	2	8	no	—
50	C₂×C₃×Dic₅	—	2	4	no	—
51	C₂×C₃×Dic₅	C₂³	1	2	yes	—
52	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
53	C₂×C₃×Dic₅	C₂³	1	2	yes	—
54	C₃×D₂₀	—	2	4	no	—
55	C₃×D₂₀	—	2	2	no	—
56	C₃×C₄×D₅	—	2	4	no	—
57	C₃×C₄×D₅	—	2	4	no	—
58	C₃×C₄×D₅	D₄	1	4	yes	complemented
59	C₃×Dic₁₀	—	2	2	no	—
60	C₂×C₃×#41	—	5	1	no	—
61	C₂×C₃×#41	—	5	1	no	—
62	C₃×C₄⋊Q₈	—	5	1	no	—
63	C₃×#127	—	5	1	no	—
64	C₃×#124	—	5	1	no	—
65	C₃×#124	D₅	1	1	yes	characteristic, complemented
66	C₃×#121	—	5	4	no	—
67	C₃×C₄×D₄	—	5	4	no	—
68	C₂×C₃×C₄²	—	5	1	no	—
69	C₂×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
70	C₂×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
71	C₂×C₂²⋊D₅	C₂²×C₃	1	2	yes	—
72	C₂×C₂²⋊D₅	C₂²×C₃	1	4	yes	—
73	C₂²×Dic₅	C₂²×C₃	1	2	yes	—
74	#713	—	2	4	no	—
75	#711	—	2	4	no	—
76	C₂×D₂₀	C₂²×C₃	1	1	yes	characteristic
77	C₂×D₂₀	C₂²×C₃	1	2	yes	—
78	C₂×C₄×D₅	C₂²×C₃	1	2	yes	—
79	C₂×C₄×D₅	C₂²×C₃	1	2	yes	—
80	C₂×C₄×D₅	C₂²×C₃	1	1	yes	characteristic
81	C₂×Dic₁₀	C₂²×C₃	1	1	yes	characteristic
82	C₅×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
83	C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
84	#685	C₂²×C₃	1	4	yes	—
85	#683	C₂²×C₃	1	4	yes	—
86	C₄×Dic₅	C₂²×C₃	1	2	yes	—
87	C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic
88	#532	—	5	1	no	Sylow
89	C₂²×C₃×C₅	—	2	2	no	—
90	C₂²×C₃×C₅	—	2	4	no	—
91	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
92	C₂×C₃×D₅	—	2	2	no	—
93	C₂×C₃×D₅	—	2	2	no	—
94	C₂×C₃×D₅	—	2	4	no	—
95	C₂×C₃×D₅	C₂×D₄	1	2	yes	—
96	C₃×C₄×C₅	—	2	2	no	—
97	C₃×C₄×C₅	—	2	2	no	—
98	C₃×C₄×C₅	C₂×D₄	1	2	yes	—
99	C₃×Dic₅	—	2	2	no	—
100	C₃×Dic₅	C₂×D₄	1	2	yes	—
101	C₃×Dic₅	#41	1	4	yes	complemented
102	C₃×#41	—	10	4	no	—
103	C₃×#41	—	10	4	no	—
104	C₂×C₃×Q₈	—	5	1	no	—
105	C₂×C₃×Q₈	C₂×D₅	1	1	yes	characteristic
106	C₂×C₃×D₄	—	5	4	no	—
107	C₂×C₃×D₄	—	5	2	no	—
108	C₂×C₃×D₄	—	5	2	no	—
109	C₂×C₃×D₄	—	5	1	no	—
110	C₂×C₃×D₄	C₂×D₅	1	1	yes	characteristic
111	C₂²×C₃×C₄	—	5	1	no	—
112	C₂²×C₃×C₄	—	5	2	no	—
113	C₂²×C₃×C₄	—	5	2	no	—
114	C₂²×C₃×C₄	—	5	2	no	—
115	C₃×C₄⋊C₄	—	5	4	no	—
116	C₃×C₂²⋊C₄	—	5	4	no	—
117	C₃×C₂²⋊C₄	C₂×D₅	1	4	yes	—
118	C₃×C₄²	—	5	2	no	—
119	C₃×C₄²	—	5	1	no	—
120	C₃×C₄²	C₂×D₅	1	1	yes	characteristic
121	C₂³×C₅	C₂³×C₃	1	2	yes	—
122	C₂²×D₅	C₂³×C₃	1	2	yes	—
123	C₂²×D₅	C₂³×C₃	1	1	yes	characteristic
124	C₅×Q₈	—	2	2	no	—
125	C₅×D₄	—	2	2	no	—
126	C₂×C₄×C₅	C₂³×C₃	1	2	yes	—
127	C₂×C₄×C₅	C₂³×C₃	1	2	yes	—
128	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
129	C₂²⋊D₅	—	2	4	no	—
130	C₂²⋊D₅	—	2	8	no	—
131	C₂×Dic₅	—	2	4	no	—
132	C₂×Dic₅	C₂³×C₃	1	2	yes	—
133	C₂×Dic₅	C₂³×C₃	1	1	yes	characteristic
134	C₂×Dic₅	C₂³×C₃	1	2	yes	—
135	D₂₀	—	2	4	no	—
136	D₂₀	—	2	2	no	—
137	C₄×D₅	—	2	4	no	—
138	C₄×D₅	—	2	4	no	—
139	C₄×D₅	C₃×D₄	1	4	yes	complemented
140	Dic₁₀	—	2	2	no	—
141	C₂×#41	—	5	1	no	—
142	C₂×#41	—	5	1	no	—
143	C₄⋊Q₈	—	5	1	no	—
144	#127	—	5	1	no	—
145	#124	—	5	1	no	—
146	#124	C₃×D₅	1	1	yes	characteristic, complemented
147	#121	—	5	4	no	—
148	C₄×D₄	—	5	4	no	—
149	C₂×C₄²	—	5	1	no	—
150	C₂×C₃×C₅	—	4	2	no	—
151	C₂×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
152	C₂×C₃×C₅	C₂×#41	1	2	yes	—
153	C₃×D₅	—	2	2	no	—
154	C₃×D₅	—	4	2	no	—
155	C₂³×C₃	—	5	1	no	—
156	C₂³×C₃	—	5	2	no	—
157	C₂³×C₃	C₂²×D₅	1	2	yes	—
158	C₃×Q₈	—	2	2	no	—
159	C₃×Q₈	—	10	2	no	—
160	C₃×D₄	—	2	2	no	—
161	C₃×D₄	—	10	4	no	—
162	C₃×D₄	—	10	8	no	—
163	C₃×D₄	—	10	4	no	—
164	C₃×D₄	—	10	2	no	—
165	C₂×C₃×C₄	—	5	2	no	—
166	C₂×C₃×C₄	—	5	1	no	—
167	C₂×C₃×C₄	—	5	4	no	—
168	C₂×C₃×C₄	—	5	2	no	—
169	C₂×C₃×C₄	—	10	4	no	—
170	C₂×C₃×C₄	—	10	4	no	—
171	C₂×C₃×C₄	—	10	4	no	—
172	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
173	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
174	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
175	C₂²×C₅	—	2	2	no	—
176	C₂²×C₅	—	2	4	no	—
177	C₂²×C₅	C₂⁴×C₃	1	1	yes	characteristic
178	C₂×D₅	—	2	2	no	—
179	C₂×D₅	—	2	2	no	—
180	C₂×D₅	—	2	4	no	—
181	C₂×D₅	C₂×C₃×D₄	1	2	yes	—
182	C₄×C₅	—	2	2	no	—
183	C₄×C₅	—	2	2	no	—
184	C₄×C₅	C₂×C₃×D₄	1	2	yes	—
185	Dic₅	—	2	2	no	—
186	Dic₅	C₂×C₃×D₄	1	2	yes	—
187	Dic₅	C₃×#41	1	4	yes	complemented
188	#41	—	10	4	no	—
189	#41	—	10	4	no	—
190	C₂×Q₈	—	5	1	no	—
191	C₂×Q₈	C₂×C₃×D₅	1	1	yes	characteristic
192	C₂×D₄	—	5	4	no	—
193	C₂×D₄	—	5	2	no	—
194	C₂×D₄	—	5	2	no	—
195	C₂×D₄	—	5	1	no	—
196	C₂×D₄	C₂×C₃×D₅	1	1	yes	characteristic
197	C₂²×C₄	—	5	1	no	—
198	C₂²×C₄	—	5	2	no	—
199	C₂²×C₄	—	5	2	no	—
200	C₂²×C₄	—	5	2	no	—
201	C₄⋊C₄	—	5	4	no	—
202	C₂²⋊C₄	—	5	4	no	—
203	C₂²⋊C₄	C₂×C₃×D₅	1	4	yes	—
204	C₄²	—	5	2	no	—
205	C₄²	—	5	1	no	—
206	C₄²	C₂×C₃×D₅	1	1	yes	characteristic
207	C₃×C₅	#532	1	1	yes	characteristic, complemented, Hall
208	C₂²×C₃	—	2	2	no	—
209	C₂²×C₃	—	2	4	no	—
210	C₂²×C₃	—	5	2	no	—
211	C₂²×C₃	—	10	2	no	—
212	C₂²×C₃	—	10	2	no	—
213	C₂²×C₃	—	10	4	no	—
214	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
215	C₃×C₄	—	2	2	no	—
216	C₃×C₄	—	2	2	no	—
217	C₃×C₄	—	5	2	no	—
218	C₃×C₄	—	5	4	no	—
219	C₃×C₄	—	10	2	no	—
220	C₃×C₄	D₄×D₅	1	2	yes	—
221	C₂×C₅	—	4	2	no	—
222	C₂×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
223	C₂×C₅	C₂×C₃×#41	1	2	yes	—
224	D₅	—	2	2	no	—
225	D₅	—	4	2	no	—
226	C₂³	—	5	1	no	—
227	C₂³	—	5	2	no	—
228	C₂³	C₂²×C₃×D₅	1	2	yes	—
229	Q₈	—	2	2	no	—
230	Q₈	—	10	2	no	—
231	D₄	—	2	2	no	—
232	D₄	—	10	4	no	—
233	D₄	—	10	8	no	—
234	D₄	—	10	4	no	—
235	D₄	—	10	2	no	—
236	C₂×C₄	—	5	2	no	—
237	C₂×C₄	—	5	1	no	—
238	C₂×C₄	—	5	4	no	—
239	C₂×C₄	—	5	2	no	—
240	C₂×C₄	—	10	4	no	—
241	C₂×C₄	—	10	4	no	—
242	C₂×C₄	—	10	4	no	—
243	C₂×C₄	C₂²×C₃×D₅	1	2	yes	—
244	C₂×C₄	C₂²×C₃×D₅	1	2	yes	—
245	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
246	C₂×C₃	—	4	2	no	—
247	C₂×C₃	—	10	2	no	—
248	C₂×C₃	—	20	2	no	—
249	C₂×C₃	C₂×D₄×D₅	1	1	yes	characteristic
250	C₂×C₃	D₅×#41	1	2	yes	—
251	C₅	C₃×#532	1	1	yes	characteristic, complemented, Sylow
252	C₂²	—	2	2	no	—
253	C₂²	—	2	4	no	—
254	C₂²	—	5	2	no	—
255	C₂²	—	10	2	no	—
256	C₂²	—	10	2	no	—
257	C₂²	—	10	4	no	—
258	C₂²	C₂³×C₃×D₅	1	1	yes	characteristic
259	C₄	—	2	2	no	—
260	C₄	—	2	2	no	—
261	C₄	—	5	2	no	—
262	C₄	—	5	4	no	—
263	C₄	—	10	2	no	—
264	C₄	C₃×D₄×D₅	1	2	yes	—
265	C₃	#66002	1	1	yes	characteristic, direct product factor, Sylow
266	C₂	—	4	2	no	—
267	C₂	—	10	2	no	—
268	C₂	—	20	2	no	—
269	C₂	C₂×C₃×D₄×D₅	1	1	yes	characteristic
270	C₂	C₃×D₅×#41	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
hypercenter	C₃×#124	D₅
hypocenter	C₅	C₃×#532
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×#713	C₂	semidirect
2. C₂×C₃×#711	C₂	semidirect
3. C₃×C₅×#124	C₂	semidirect
4. C₃×#4090	C₂	semidirect
5. C₃×#4086	C₂	semidirect
6. C₃×#4030	C₂	semidirect
7. C₃×#4027	C₂	semidirect
8. C₃×#4021	C₂	semidirect
9. C₃×C₄²×D₅	C₂	semidirect
10. #66002	C₃	direct (N × H)
31. C₂×#713	C₂×C₃	semidirect
32. C₂×#711	C₂×C₃	semidirect
33. C₅×#124	C₂×C₃	semidirect
34. #4090	C₂×C₃	semidirect
35. #4086	C₂×C₃	semidirect
36. #4030	C₂×C₃	semidirect
37. #4027	C₂×C₃	semidirect
38. #4021	C₂×C₃	semidirect
39. C₄²×D₅	C₂×C₃	semidirect
58. C₃×C₄×D₅	D₄	semidirect
65. C₃×#124	D₅	semidirect
101. C₃×Dic₅	#41	semidirect
139. C₄×D₅	C₃×D₄	semidirect
146. #124	C₃×D₅	semidirect
187. Dic₅	C₃×#41	semidirect
207. C₃×C₅	#532	semidirect
251. C₅	C₃×#532	semidirect
265. C₃	#66002	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×#713	2
C₂×C₃×#711	2
C₃×C₅×#124	2
C₃×#4090	2
C₃×#4086	2
C₃×#4030	2
C₃×#4027	2
C₃×#4021	2
C₃×C₄²×D₅	2
#66002	3
C₃×#532	5

Subgroups — C3×#66002 #11755679

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#66002 #11755679