Subgroups — C₃×#65998 #11755675

Order 960.

472 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	C₃×C₅×#123	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#4086	C₂	1	1	yes	maximal, characteristic
3	C₃×#4081	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₄×C₂²⋊D₅	C₂	1	2	yes	maximal, complemented
5	C₃×#4040	C₂	1	2	yes	maximal, complemented
6	C₃×#4033	C₂	1	2	yes	maximal, complemented
7	C₃×#4031	C₂	1	2	yes	maximal, complemented
8	C₃×#4031	C₂	1	2	yes	maximal, complemented
9	C₃×#4027	C₂	1	2	yes	maximal, complemented
10	#65998	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₂×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
12	C₂²×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₂²⋊D₅	C₂²	1	2	yes	—
14	C₂×C₃×C₂²⋊D₅	C₂²	1	2	yes	—
15	C₂²×C₃×Dic₅	C₂²	1	1	yes	characteristic
16	C₂×C₃×D₂₀	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₄×D₅	C₂²	1	2	yes	—
18	C₂×C₃×Dic₁₀	C₂²	1	2	yes	—
19	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
20	C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
21	C₃×C₅×C₂²⋊C₄	C₂²	1	1	yes	characteristic
22	C₃×C₂²⋊Dic₅	C₂²	1	1	yes	characteristic
23	C₃×C₂²⋊Dic₅	C₂²	1	2	yes	—
24	C₃×#685	C₂²	1	2	yes	—
25	C₃×#685	C₂²	1	2	yes	—
26	C₃×#685	C₂²	1	2	yes	—
27	C₃×#683	C₂²	1	2	yes	—
28	C₃×#683	C₂²	1	2	yes	—
29	C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
30	C₃×C₄×Dic₅	C₂²	1	2	yes	—
31	C₃×C₄×Dic₅	C₂²	1	2	yes	—
32	C₃×#559	—	5	1	no	maximal, Hall
33	C₅×#123	C₂×C₃	1	1	yes	characteristic, complemented
34	#4086	C₂×C₃	1	1	yes	characteristic
35	#4081	C₂×C₃	1	1	yes	characteristic, complemented
36	C₄×C₂²⋊D₅	C₂×C₃	1	2	yes	complemented
37	#4040	C₂×C₃	1	2	yes	complemented
38	#4033	C₂×C₃	1	2	yes	complemented
39	#4031	C₂×C₃	1	2	yes	complemented
40	#4031	C₂×C₃	1	2	yes	complemented
41	#4027	C₂×C₃	1	2	yes	complemented
42	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
43	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
44	C₂²×C₃×D₅	C₂³	1	2	yes	—
45	C₃×C₅×D₄	—	4	1	no	—
46	C₂×C₃×C₄×C₅	—	4	1	no	—
47	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
48	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
49	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
50	C₃×C₂²⋊D₅	—	2	4	no	—
51	C₃×C₂²⋊D₅	—	2	4	no	—
52	C₂×C₃×Dic₅	—	4	1	no	—
53	C₂×C₃×Dic₅	C₂³	1	2	yes	—
54	C₂×C₃×Dic₅	C₂³	1	2	yes	—
55	C₂×C₃×Dic₅	C₂³	1	2	yes	—
56	C₃×D₂₀	—	4	1	no	—
57	C₃×C₄×D₅	—	4	2	no	—
58	C₃×Dic₁₀	—	2	4	no	—
59	C₃×#127	—	5	1	no	—
60	C₃×#124	—	5	2	no	—
61	C₃×#124	—	5	2	no	—
62	C₃×#123	—	5	1	no	—
63	C₃×#123	—	5	2	no	—
64	C₃×#123	D₅	1	1	yes	characteristic, complemented
65	C₃×C₄×Q₈	—	5	2	no	—
66	C₃×C₄×D₄	—	5	2	no	—
67	C₃×C₄×D₄	—	5	2	no	—
68	C₂×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
69	C₂²×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
70	C₂×C₂²⋊D₅	C₂²×C₃	1	2	yes	—
71	C₂×C₂²⋊D₅	C₂²×C₃	1	2	yes	—
72	C₂²×Dic₅	C₂²×C₃	1	1	yes	characteristic
73	C₂×D₂₀	C₂²×C₃	1	1	yes	characteristic
74	C₂×C₄×D₅	C₂²×C₃	1	2	yes	—
75	C₂×Dic₁₀	C₂²×C₃	1	2	yes	—
76	C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
77	C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
78	C₅×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
79	C₂²⋊Dic₅	C₂²×C₃	1	1	yes	characteristic
80	C₂²⋊Dic₅	C₂²×C₃	1	2	yes	—
81	#685	C₂²×C₃	1	2	yes	—
82	#685	C₂²×C₃	1	2	yes	—
83	#685	C₂²×C₃	1	2	yes	—
84	#683	C₂²×C₃	1	2	yes	—
85	#683	C₂²×C₃	1	2	yes	—
86	C₄×Dic₅	C₂²×C₃	1	2	yes	—
87	C₄×Dic₅	C₂²×C₃	1	2	yes	—
88	C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic
89	#559	—	5	1	no	Sylow
90	C₂²×C₃×C₅	—	2	2	no	—
91	C₂²×C₃×C₅	—	2	2	no	—
92	C₂²×C₃×C₅	—	2	1	no	—
93	C₂²×C₃×C₅	—	2	1	no	—
94	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
95	C₂×C₃×D₅	—	2	2	no	—
96	C₂×C₃×D₅	—	2	2	no	—
97	C₂×C₃×D₅	—	2	2	no	—
98	C₃×C₄×C₅	—	2	2	no	—
99	C₃×C₄×C₅	—	2	2	no	—
100	C₃×C₄×C₅	—	2	1	no	—
101	C₃×Dic₅	—	2	2	no	—
102	C₃×Dic₅	—	2	2	no	—
103	C₃×Dic₅	#41	1	4	yes	—
104	C₂×C₃×Q₈	—	5	2	no	—
105	C₂×C₃×D₄	—	5	2	no	—
106	C₂×C₃×D₄	—	5	2	no	—
107	C₂×C₃×D₄	—	5	1	no	—
108	C₂×C₃×D₄	C₂×D₅	1	1	yes	characteristic
109	C₂²×C₃×C₄	—	5	1	no	—
110	C₂²×C₃×C₄	—	5	2	no	—
111	C₂²×C₃×C₄	C₂×D₅	1	1	yes	characteristic
112	C₃×C₄⋊C₄	—	5	2	no	—
113	C₃×C₄⋊C₄	—	5	2	no	—
114	C₃×C₄⋊C₄	C₂×D₅	1	2	yes	—
115	C₃×C₂²⋊C₄	—	5	2	no	—
116	C₃×C₂²⋊C₄	—	5	1	no	—
117	C₃×C₂²⋊C₄	—	5	2	no	—
118	C₃×C₂²⋊C₄	—	5	2	no	—
119	C₃×C₂²⋊C₄	—	5	2	no	—
120	C₃×C₂²⋊C₄	C₂×D₅	1	2	yes	—
121	C₃×C₂²⋊C₄	C₂×D₅	1	1	yes	characteristic
122	C₃×C₄²	—	5	2	no	—
123	C₃×C₄²	—	5	2	no	—
124	C₃×C₄²	—	5	1	no	—
125	C₂³×C₅	C₂³×C₃	1	1	yes	characteristic
126	C₂³×C₅	C₂³×C₃	1	1	yes	characteristic
127	C₂²×D₅	C₂³×C₃	1	2	yes	—
128	C₅×D₄	—	4	1	no	—
129	C₂×C₄×C₅	—	4	1	no	—
130	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
131	C₂×C₄×C₅	C₂³×C₃	1	2	yes	—
132	C₂×C₄×C₅	C₂³×C₃	1	2	yes	—
133	C₂²⋊D₅	—	2	4	no	—
134	C₂²⋊D₅	—	2	4	no	—
135	C₂×Dic₅	—	4	1	no	—
136	C₂×Dic₅	C₂³×C₃	1	2	yes	—
137	C₂×Dic₅	C₂³×C₃	1	2	yes	—
138	C₂×Dic₅	C₂³×C₃	1	2	yes	—
139	D₂₀	—	4	1	no	—
140	C₄×D₅	—	4	2	no	—
141	Dic₁₀	—	2	4	no	—
142	#127	—	5	1	no	—
143	#124	—	5	2	no	—
144	#124	—	5	2	no	—
145	#123	—	5	1	no	—
146	#123	—	5	2	no	—
147	#123	C₃×D₅	1	1	yes	characteristic, complemented
148	C₄×Q₈	—	5	2	no	—
149	C₄×D₄	—	5	2	no	—
150	C₄×D₄	—	5	2	no	—
151	C₂×C₃×C₅	—	4	1	no	—
152	C₂×C₃×C₅	—	4	1	no	—
153	C₂×C₃×C₅	C₂×#41	1	2	yes	—
154	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
155	C₃×D₅	—	4	2	no	—
156	C₂³×C₃	—	5	2	no	—
157	C₂³×C₃	C₂²×D₅	1	1	yes	characteristic
158	C₂³×C₃	C₂²×D₅	1	1	yes	characteristic
159	C₃×Q₈	—	10	4	no	—
160	C₃×D₄	—	4	1	no	—
161	C₃×D₄	—	10	4	no	—
162	C₃×D₄	—	10	4	no	—
163	C₃×D₄	—	20	1	no	—
164	C₂×C₃×C₄	—	4	1	no	—
165	C₂×C₃×C₄	—	5	2	no	—
166	C₂×C₃×C₄	—	5	2	no	—
167	C₂×C₃×C₄	—	5	2	no	—
168	C₂×C₃×C₄	—	20	2	no	—
169	C₂×C₃×C₄	—	20	1	no	—
170	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
171	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
172	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
173	C₂²×C₅	—	2	2	no	—
174	C₂²×C₅	—	2	2	no	—
175	C₂²×C₅	—	2	1	no	—
176	C₂²×C₅	—	2	1	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	2	no	—
181	C₄×C₅	—	2	2	no	—
182	C₄×C₅	—	2	2	no	—
183	C₄×C₅	—	2	1	no	—
184	Dic₅	—	2	2	no	—
185	Dic₅	—	2	2	no	—
186	Dic₅	C₃×#41	1	4	yes	—
187	C₂×Q₈	—	5	2	no	—
188	C₂×D₄	—	5	2	no	—
189	C₂×D₄	—	5	2	no	—
190	C₂×D₄	—	5	1	no	—
191	C₂×D₄	C₂×C₃×D₅	1	1	yes	characteristic
192	C₂²×C₄	—	5	1	no	—
193	C₂²×C₄	—	5	2	no	—
194	C₂²×C₄	C₂×C₃×D₅	1	1	yes	characteristic
195	C₄⋊C₄	—	5	2	no	—
196	C₄⋊C₄	—	5	2	no	—
197	C₄⋊C₄	C₂×C₃×D₅	1	2	yes	—
198	C₂²⋊C₄	—	5	2	no	—
199	C₂²⋊C₄	—	5	1	no	—
200	C₂²⋊C₄	—	5	2	no	—
201	C₂²⋊C₄	—	5	2	no	—
202	C₂²⋊C₄	—	5	2	no	—
203	C₂²⋊C₄	C₂×C₃×D₅	1	2	yes	—
204	C₂²⋊C₄	C₂×C₃×D₅	1	1	yes	characteristic
205	C₄²	—	5	2	no	—
206	C₄²	—	5	2	no	—
207	C₄²	—	5	1	no	—
208	C₃×C₅	#559	1	1	yes	characteristic, complemented, Hall
209	C₂²×C₃	—	2	1	no	—
210	C₂²×C₃	—	2	1	no	—
211	C₂²×C₃	—	2	2	no	—
212	C₂²×C₃	—	2	2	no	—
213	C₂²×C₃	—	10	2	no	—
214	C₂²×C₃	—	10	2	no	—
215	C₂²×C₃	—	10	2	no	—
216	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
217	C₃×C₄	—	2	2	no	—
218	C₃×C₄	—	2	1	no	—
219	C₃×C₄	—	2	2	no	—
220	C₃×C₄	—	5	4	no	—
221	C₃×C₄	—	10	2	no	—
222	C₃×C₄	—	10	2	no	—
223	C₂×C₅	—	4	1	no	—
224	C₂×C₅	—	4	1	no	—
225	C₂×C₅	C₂×C₃×#41	1	2	yes	—
226	C₂×C₅	C₃×H(5,𝔽₂)	1	1	yes	characteristic
227	D₅	—	4	2	no	—
228	C₂³	—	5	2	no	—
229	C₂³	C₂²×C₃×D₅	1	1	yes	characteristic
230	C₂³	C₂²×C₃×D₅	1	1	yes	characteristic
231	Q₈	—	10	4	no	—
232	D₄	—	4	1	no	—
233	D₄	—	10	4	no	—
234	D₄	—	10	4	no	—
235	D₄	—	20	1	no	—
236	C₂×C₄	—	4	1	no	—
237	C₂×C₄	—	5	2	no	—
238	C₂×C₄	—	5	2	no	—
239	C₂×C₄	—	5	2	no	—
240	C₂×C₄	—	20	2	no	—
241	C₂×C₄	—	20	1	no	—
242	C₂×C₄	C₂²×C₃×D₅	1	2	yes	—
243	C₂×C₄	C₂²×C₃×D₅	1	2	yes	—
244	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
245	C₂×C₃	—	4	1	no	—
246	C₂×C₃	—	4	1	no	—
247	C₂×C₃	—	20	2	no	—
248	C₂×C₃	#4144	1	1	yes	characteristic
249	C₂×C₃	D₅×#41	1	2	yes	—
250	C₅	C₃×#559	1	1	yes	characteristic, complemented, Sylow
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	2	no	—
255	C₂²	—	10	2	no	—
256	C₂²	—	10	2	no	—
257	C₂²	—	10	2	no	—
258	C₂²	C₂³×C₃×D₅	1	1	yes	characteristic
259	C₄	—	2	2	no	—
260	C₄	—	2	1	no	—
261	C₄	—	2	2	no	—
262	C₄	—	5	4	no	—
263	C₄	—	10	2	no	—
264	C₄	—	10	2	no	—
265	C₃	#65998	1	1	yes	characteristic, direct product factor, Sylow
266	C₂	—	4	1	no	—
267	C₂	—	4	1	no	—
268	C₂	—	20	2	no	—
269	C₂	C₃×#4144	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×#123	C₂
socle	C₂²×C₃×C₅	C₂⁴
prime radical	C₃×#4086	C₂
hypercenter	C₃×#123	D₅
hypocenter	C₅	C₃×#559
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×#123

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×#123	C₂	semidirect
3. C₃×#4081	C₂	semidirect
4. C₃×C₄×C₂²⋊D₅	C₂	semidirect
5. C₃×#4040	C₂	semidirect
6. C₃×#4033	C₂	semidirect
7. C₃×#4031	C₂	semidirect
8. C₃×#4031	C₂	semidirect
9. C₃×#4027	C₂	semidirect
10. #65998	C₃	direct (N × H)
33. C₅×#123	C₂×C₃	semidirect
35. #4081	C₂×C₃	semidirect
36. C₄×C₂²⋊D₅	C₂×C₃	semidirect
37. #4040	C₂×C₃	semidirect
38. #4033	C₂×C₃	semidirect
39. #4031	C₂×C₃	semidirect
40. #4031	C₂×C₃	semidirect
41. #4027	C₂×C₃	semidirect
64. C₃×#123	D₅	semidirect
147. #123	C₃×D₅	semidirect
208. C₃×C₅	#559	semidirect
250. C₅	C₃×#559	semidirect
265. C₃	#65998	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₅×#123	2
C₃×#4086	2
C₃×#4081	2
C₃×C₄×C₂²⋊D₅	2
C₃×#4040	2
C₃×#4033	2
C₃×#4031	2
C₃×#4027	2
#65998	3
C₃×#559	5

Subgroups — C3×#65998 #11755675

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#65998 #11755675