Subgroups — C₃×#65985 #11755662

Order 960.

392 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	C₃×C₅×#123	C₂	1	1	yes	maximal, characteristic
2	C₃×#4082	C₂	1	1	yes	maximal, characteristic
3	C₃×#4081	C₂	1	1	yes	maximal, characteristic
4	C₃×#4070	C₂	1	2	yes	maximal, complemented
5	C₃×#4035	C₂	1	2	yes	maximal, complemented
6	C₃×#4034	C₂	1	2	yes	maximal, complemented
7	C₃×#4025	C₂	1	2	yes	maximal, complemented
8	C₃×#4025	C₂	1	2	yes	maximal, complemented
9	C₃×#4024	C₂	1	2	yes	maximal, complemented
10	#65985	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	C₂²×C₃×Dic₅	C₂²	1	1	yes	characteristic
14	C₂×C₃×Dic₁₀	C₂²	1	1	yes	characteristic
15	C₂×C₃×Dic₁₀	C₂²	1	2	yes	—
16	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
17	C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
18	C₃×C₅×C₂²⋊C₄	C₂²	1	1	yes	characteristic
19	C₃×C₂²⋊Dic₅	C₂²	1	1	yes	characteristic
20	C₃×C₂²⋊Dic₅	C₂²	1	2	yes	—
21	C₃×C₂²⋊Dic₅	C₂²	1	2	yes	—
22	C₃×C₂²⋊Dic₅	C₂²	1	2	yes	—
23	C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
24	C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
25	C₃×#683	C₂²	1	2	yes	—
26	C₃×#683	C₂²	1	2	yes	—
27	C₃×#683	C₂²	1	2	yes	—
28	C₃×#683	C₂²	1	2	yes	—
29	C₃×#683	C₂²	1	2	yes	—
30	C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
31	C₃×C₄×Dic₅	C₂²	1	2	yes	—
32	C₃×#563	—	5	1	no	maximal, Hall
33	C₅×#123	C₂×C₃	1	1	yes	characteristic
34	#4082	C₂×C₃	1	1	yes	characteristic
35	#4081	C₂×C₃	1	1	yes	characteristic
36	#4070	C₂×C₃	1	2	yes	complemented
37	#4035	C₂×C₃	1	2	yes	complemented
38	#4034	C₂×C₃	1	2	yes	complemented
39	#4025	C₂×C₃	1	2	yes	complemented
40	#4025	C₂×C₃	1	2	yes	complemented
41	#4024	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	C₃×C₅×D₄	—	4	1	no	—
45	C₂×C₃×C₄×C₅	—	4	1	no	—
46	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
47	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
48	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
49	C₂×C₃×Dic₅	—	4	1	no	—
50	C₂×C₃×Dic₅	C₂³	1	2	yes	—
51	C₂×C₃×Dic₅	C₂³	1	2	yes	—
52	C₂×C₃×Dic₅	C₂³	1	2	yes	—
53	C₂×C₃×Dic₅	C₂³	1	2	yes	—
54	C₃×Dic₁₀	—	4	1	no	—
55	C₃×Dic₁₀	—	4	2	no	—
56	C₃×C₄⋊Q₈	—	5	2	no	—
57	C₃×#126	—	5	2	no	—
58	C₃×#126	—	5	2	no	—
59	C₃×#125	—	5	2	no	—
60	C₃×#124	—	5	1	no	—
61	C₃×#123	—	5	1	no	—
62	C₃×#123	D₅	1	1	yes	characteristic
63	C₃×C₂²⋊Q₈	—	5	2	no	—
64	C₃×C₂²⋊Q₈	—	5	2	no	—
65	C₂×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
66	C₂²×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
67	C₂²×Dic₅	C₂²×C₃	1	1	yes	characteristic
68	C₂×Dic₁₀	C₂²×C₃	1	2	yes	—
69	C₂×Dic₁₀	C₂²×C₃	1	1	yes	characteristic
70	C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
71	C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
72	C₅×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
73	C₂²⋊Dic₅	C₂²×C₃	1	2	yes	—
74	C₂²⋊Dic₅	C₂²×C₃	1	1	yes	characteristic
75	C₂²⋊Dic₅	C₂²×C₃	1	2	yes	—
76	C₂²⋊Dic₅	C₂²×C₃	1	2	yes	—
77	C₄⋊Dic₅	C₂²×C₃	1	2	yes	—
78	C₄⋊Dic₅	C₂²×C₃	1	2	yes	—
79	#683	C₂²×C₃	1	2	yes	—
80	#683	C₂²×C₃	1	2	yes	—
81	#683	C₂²×C₃	1	2	yes	—
82	#683	C₂²×C₃	1	2	yes	—
83	#683	C₂²×C₃	1	2	yes	—
84	C₄×Dic₅	C₂²×C₃	1	2	yes	—
85	C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic
86	#563	—	5	1	no	Sylow
87	C₂²×C₃×C₅	—	2	1	no	—
88	C₂²×C₃×C₅	—	2	1	no	—
89	C₂²×C₃×C₅	—	2	2	no	—
90	C₂²×C₃×C₅	—	2	2	no	—
91	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
92	C₃×C₄×C₅	—	2	2	no	—
93	C₃×C₄×C₅	—	2	1	no	—
94	C₃×C₄×C₅	—	2	2	no	—
95	C₃×Dic₅	—	2	2	no	—
96	C₃×Dic₅	—	2	2	no	—
97	C₃×Dic₅	—	2	2	no	—
98	C₃×Dic₅	—	2	2	no	—
99	C₂×C₃×Q₈	—	5	2	no	—
100	C₂×C₃×Q₈	—	5	1	no	—
101	C₂×C₃×D₄	C₂×D₅	1	1	yes	characteristic
102	C₂²×C₃×C₄	—	5	1	no	—
103	C₂²×C₃×C₄	C₂×D₅	1	1	yes	characteristic
104	C₃×C₄⋊C₄	—	5	2	no	—
105	C₃×C₄⋊C₄	—	5	2	no	—
106	C₃×C₄⋊C₄	—	5	2	no	—
107	C₃×C₄⋊C₄	—	5	2	no	—
108	C₃×C₄⋊C₄	—	5	2	no	—
109	C₃×C₄⋊C₄	—	5	2	no	—
110	C₃×C₄⋊C₄	—	5	2	no	—
111	C₃×C₄⋊C₄	C₂×D₅	1	2	yes	—
112	C₃×C₂²⋊C₄	—	5	2	no	—
113	C₃×C₂²⋊C₄	—	5	2	no	—
114	C₃×C₂²⋊C₄	—	5	2	no	—
115	C₃×C₂²⋊C₄	—	5	1	no	—
116	C₃×C₂²⋊C₄	C₂×D₅	1	2	yes	—
117	C₃×C₂²⋊C₄	C₂×D₅	1	1	yes	characteristic
118	C₃×C₄²	—	5	1	no	—
119	C₃×C₄²	—	5	2	no	—
120	C₂³×C₅	C₂³×C₃	1	1	yes	characteristic
121	C₂³×C₅	C₂³×C₃	1	1	yes	characteristic
122	C₅×D₄	—	4	1	no	—
123	C₂×C₄×C₅	—	4	1	no	—
124	C₂×C₄×C₅	C₂³×C₃	1	2	yes	—
125	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
126	C₂×C₄×C₅	C₂³×C₃	1	2	yes	—
127	C₂×Dic₅	—	4	1	no	—
128	C₂×Dic₅	C₂³×C₃	1	2	yes	—
129	C₂×Dic₅	C₂³×C₃	1	2	yes	—
130	C₂×Dic₅	C₂³×C₃	1	2	yes	—
131	C₂×Dic₅	C₂³×C₃	1	2	yes	—
132	Dic₁₀	—	4	1	no	—
133	Dic₁₀	—	4	2	no	—
134	C₄⋊Q₈	—	5	2	no	—
135	#126	—	5	2	no	—
136	#126	—	5	2	no	—
137	#125	—	5	2	no	—
138	#124	—	5	1	no	—
139	#123	—	5	1	no	—
140	#123	C₃×D₅	1	1	yes	characteristic
141	C₂²⋊Q₈	—	5	2	no	—
142	C₂²⋊Q₈	—	5	2	no	—
143	C₂×C₃×C₅	—	4	1	no	—
144	C₂×C₃×C₅	—	4	1	no	—
145	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
146	C₂×C₃×C₅	2₋¹⁺⁴	1	2	yes	—
147	C₂³×C₃	C₂²×D₅	1	1	yes	characteristic
148	C₂³×C₃	C₂²×D₅	1	1	yes	characteristic
149	C₃×Q₈	—	20	2	no	—
150	C₃×Q₈	—	20	1	no	—
151	C₃×D₄	—	4	1	no	—
152	C₂×C₃×C₄	—	4	1	no	—
153	C₂×C₃×C₄	—	5	2	no	—
154	C₂×C₃×C₄	—	5	2	no	—
155	C₂×C₃×C₄	—	5	2	no	—
156	C₂×C₃×C₄	—	5	2	no	—
157	C₂×C₃×C₄	—	20	1	no	—
158	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
159	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
160	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
161	C₂²×C₅	—	2	1	no	—
162	C₂²×C₅	—	2	1	no	—
163	C₂²×C₅	—	2	2	no	—
164	C₂²×C₅	—	2	2	no	—
165	C₂²×C₅	C₂⁴×C₃	1	1	yes	characteristic
166	C₄×C₅	—	2	2	no	—
167	C₄×C₅	—	2	1	no	—
168	C₄×C₅	—	2	2	no	—
169	Dic₅	—	2	2	no	—
170	Dic₅	—	2	2	no	—
171	Dic₅	—	2	2	no	—
172	Dic₅	—	2	2	no	—
173	C₂×Q₈	—	5	2	no	—
174	C₂×Q₈	—	5	1	no	—
175	C₂×D₄	C₂×C₃×D₅	1	1	yes	characteristic
176	C₂²×C₄	—	5	1	no	—
177	C₂²×C₄	C₂×C₃×D₅	1	1	yes	characteristic
178	C₄⋊C₄	—	5	2	no	—
179	C₄⋊C₄	—	5	2	no	—
180	C₄⋊C₄	—	5	2	no	—
181	C₄⋊C₄	—	5	2	no	—
182	C₄⋊C₄	—	5	2	no	—
183	C₄⋊C₄	—	5	2	no	—
184	C₄⋊C₄	—	5	2	no	—
185	C₄⋊C₄	C₂×C₃×D₅	1	2	yes	—
186	C₂²⋊C₄	—	5	2	no	—
187	C₂²⋊C₄	—	5	2	no	—
188	C₂²⋊C₄	—	5	2	no	—
189	C₂²⋊C₄	—	5	1	no	—
190	C₂²⋊C₄	C₂×C₃×D₅	1	2	yes	—
191	C₂²⋊C₄	C₂×C₃×D₅	1	1	yes	characteristic
192	C₄²	—	5	1	no	—
193	C₄²	—	5	2	no	—
194	C₃×C₅	#563	1	1	yes	characteristic, complemented, Hall
195	C₂²×C₃	—	2	1	no	—
196	C₂²×C₃	—	2	1	no	—
197	C₂²×C₃	—	2	2	no	—
198	C₂²×C₃	—	2	2	no	—
199	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
200	C₃×C₄	—	2	2	no	—
201	C₃×C₄	—	2	1	no	—
202	C₃×C₄	—	2	2	no	—
203	C₃×C₄	—	10	2	no	—
204	C₃×C₄	—	10	2	no	—
205	C₃×C₄	—	10	2	no	—
206	C₃×C₄	—	10	2	no	—
207	C₂×C₅	—	4	1	no	—
208	C₂×C₅	—	4	1	no	—
209	C₂×C₅	C₃×H(5,𝔽₂)	1	1	yes	characteristic
210	C₂×C₅	C₃×2₋¹⁺⁴	1	2	yes	—
211	C₂³	C₂²×C₃×D₅	1	1	yes	characteristic
212	C₂³	C₂²×C₃×D₅	1	1	yes	characteristic
213	Q₈	—	20	2	no	—
214	Q₈	—	20	1	no	—
215	D₄	—	4	1	no	—
216	C₂×C₄	—	4	1	no	—
217	C₂×C₄	—	5	2	no	—
218	C₂×C₄	—	5	2	no	—
219	C₂×C₄	—	5	2	no	—
220	C₂×C₄	—	5	2	no	—
221	C₂×C₄	—	20	1	no	—
222	C₂×C₄	C₂²×C₃×D₅	1	2	yes	—
223	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
224	C₂×C₄	C₂²×C₃×D₅	1	2	yes	—
225	C₂×C₃	—	4	1	no	—
226	C₂×C₃	—	4	1	no	—
227	C₂×C₃	#4144	1	1	yes	characteristic
228	C₂×C₃	#4150	1	2	yes	—
229	C₅	C₃×#563	1	1	yes	characteristic, complemented, Sylow
230	C₂²	—	2	1	no	—
231	C₂²	—	2	1	no	—
232	C₂²	—	2	2	no	—
233	C₂²	—	2	2	no	—
234	C₂²	C₂³×C₃×D₅	1	1	yes	characteristic
235	C₄	—	2	2	no	—
236	C₄	—	2	1	no	—
237	C₄	—	2	2	no	—
238	C₄	—	10	2	no	—
239	C₄	—	10	2	no	—
240	C₄	—	10	2	no	—
241	C₄	—	10	2	no	—
242	C₃	#65985	1	1	yes	characteristic, direct product factor, Sylow
243	C₂	—	4	1	no	—
244	C₂	—	4	1	no	—
245	C₂	C₃×#4144	1	1	yes	characteristic
246	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×#123	C₂
socle	C₂²×C₃×C₅	C₂⁴
prime radical	C₂×C₃×C₅×D₄	C₂²
hypercenter	C₃×#123	D₅
hypocenter	C₅	C₃×#563
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
4. C₃×#4070	C₂	semidirect
5. C₃×#4035	C₂	semidirect
6. C₃×#4034	C₂	semidirect
7. C₃×#4025	C₂	semidirect
8. C₃×#4025	C₂	semidirect
9. C₃×#4024	C₂	semidirect
10. #65985	C₃	direct (N × H)
36. #4070	C₂×C₃	semidirect
37. #4035	C₂×C₃	semidirect
38. #4034	C₂×C₃	semidirect
39. #4025	C₂×C₃	semidirect
40. #4025	C₂×C₃	semidirect
41. #4024	C₂×C₃	semidirect
194. C₃×C₅	#563	semidirect
229. C₅	C₃×#563	semidirect
242. C₃	#65985	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×#4082	2
C₃×#4081	2
C₃×#4070	2
C₃×#4035	2
C₃×#4034	2
C₃×#4025	2
C₃×#4024	2
#65985	3
C₃×#563	5

Subgroups — C3×#65985 #11755662

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#65985 #11755662