Subgroups — C₃×#66105 #11755782

Order 960.

496 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#4150	C₂	1	2	yes	maximal, complemented
2	C₃×C₅×#135	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#4097	C₂	1	2	yes	maximal, complemented
4	C₃×D₅×Q₁₆	C₂	1	2	yes	maximal, complemented
5	C₃×#4061	C₂	1	4	yes	maximal, complemented
6	C₃×#4058	C₂	1	2	yes	maximal, complemented
7	C₂×C₃×Dic₂₀	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#4047	C₂	1	1	yes	maximal, characteristic, complemented
9	#66105	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₃×C₅×#41	C₂²	1	2	yes	—
11	C₃×Q₈×D₅	C₂²	1	4	yes	—
12	C₃×#711	—	2	4	no	—
13	C₃×#711	C₂²	1	4	yes	—
14	C₃×#709	—	2	2	no	—
15	C₃×#709	C₂²	1	1	yes	characteristic
16	C₂×C₃×Dic₁₀	—	2	2	no	—
17	C₂×C₃×Dic₁₀	C₂²	1	2	yes	—
18	C₃×C₅×Q₁₆	C₂²	1	1	yes	characteristic
19	C₃×C₅×SD₁₆	C₂²	1	2	yes	—
20	C₃×C₅×D₈	C₂²	1	1	yes	characteristic
21	C₂×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
22	C₃×#689	C₂²	1	4	yes	—
23	C₃×#687	C₂²	1	4	yes	—
24	C₃×#681	C₂²	1	1	yes	characteristic
25	C₃×Dic₂₀	C₂²	1	2	yes	—
26	C₃×Dic₂₀	C₂²	1	2	yes	—
27	C₃×#676	C₂²	1	2	yes	—
28	C₃×C₈×D₅	C₂²	1	2	yes	—
29	C₃×#578	—	5	1	no	maximal, Hall
30	#4150	C₂×C₃	1	2	yes	complemented
31	C₅×#135	C₂×C₃	1	1	yes	characteristic, complemented
32	#4097	C₂×C₃	1	2	yes	complemented
33	D₅×Q₁₆	C₂×C₃	1	2	yes	complemented
34	#4061	C₂×C₃	1	4	yes	complemented
35	#4058	C₂×C₃	1	2	yes	complemented
36	C₂×Dic₂₀	C₂×C₃	1	1	yes	characteristic, complemented
37	#4047	C₂×C₃	1	1	yes	characteristic, complemented
38	C₃×C₅×Q₈	C₂³	1	2	yes	—
39	C₃×C₅×D₄	—	2	2	no	—
40	C₃×C₅×D₄	C₂³	1	2	yes	—
41	C₂×C₃×C₄×C₅	—	2	2	no	—
42	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
43	C₃×C₂²⋊D₅	—	2	4	no	—
44	C₃×C₂²⋊D₅	D₄	1	2	yes	—
45	C₂×C₃×Dic₅	—	2	2	no	—
46	C₂×C₃×Dic₅	—	2	4	no	—
47	C₃×D₂₀	D₄	1	1	yes	characteristic
48	C₃×C₄×D₅	—	2	4	no	—
49	C₃×C₄×D₅	C₂³	1	2	yes	—
50	C₃×Dic₁₀	—	2	2	no	—
51	C₃×Dic₁₀	—	2	4	no	—
52	C₃×Dic₁₀	D₄	1	1	yes	characteristic
53	C₃×Dic₁₀	C₂³	1	4	yes	—
54	C₃×C₅×C₈	C₂³	1	1	yes	characteristic
55	C₃×C₅×C₈	C₂³	1	1	yes	characteristic
56	C₃×#168	C₂³	1	2	yes	—
57	C₃×2₋¹⁺⁴	—	5	2	no	—
58	C₃×#137	—	5	2	no	—
59	C₃×#137	—	5	4	no	—
60	C₃×#135	—	5	2	no	—
61	C₃×#135	D₅	1	1	yes	characteristic, complemented
62	C₂×C₃×Q₁₆	—	5	2	no	—
63	C₂×C₃×Q₁₆	—	5	1	no	—
64	C₃×#131	—	5	1	no	—
65	C₅×#41	C₂²×C₃	1	2	yes	—
66	Q₈×D₅	C₂²×C₃	1	4	yes	—
67	#711	—	2	4	no	—
68	#711	C₂²×C₃	1	4	yes	—
69	#709	—	2	2	no	—
70	#709	C₂²×C₃	1	1	yes	characteristic
71	C₂×Dic₁₀	—	2	2	no	—
72	C₂×Dic₁₀	C₂²×C₃	1	2	yes	—
73	C₅×Q₁₆	C₂²×C₃	1	1	yes	characteristic
74	C₅×SD₁₆	C₂²×C₃	1	2	yes	—
75	C₅×D₈	C₂²×C₃	1	1	yes	characteristic
76	C₂×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
77	#689	C₂²×C₃	1	4	yes	—
78	#687	C₂²×C₃	1	4	yes	—
79	#681	C₂²×C₃	1	1	yes	characteristic
80	Dic₂₀	C₂²×C₃	1	2	yes	—
81	Dic₂₀	C₂²×C₃	1	2	yes	—
82	#676	C₂²×C₃	1	2	yes	—
83	C₈×D₅	C₂²×C₃	1	2	yes	—
84	#578	—	5	1	no	Sylow
85	C₂²×C₃×C₅	—	2	2	no	—
86	C₂²×C₃×C₅	C₂×D₄	1	1	yes	characteristic
87	C₂×C₃×D₅	C₂×D₄	1	2	yes	—
88	C₃×C₄×C₅	—	2	2	no	—
89	C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
90	C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
91	C₃×Dic₅	—	2	4	no	—
92	C₃×Dic₅	C₂×D₄	1	2	yes	—
93	C₃×#41	—	5	1	no	—
94	C₃×#41	—	5	4	no	—
95	C₃×#41	—	10	2	no	—
96	C₃×#41	—	10	4	no	—
97	C₃×#41	C₂×D₅	1	2	yes	—
98	C₂×C₃×Q₈	—	5	4	no	—
99	C₂×C₃×Q₈	—	5	2	no	—
100	C₂×C₃×Q₈	—	10	2	no	—
101	C₃×Q₁₆	—	5	4	no	—
102	C₃×Q₁₆	—	5	2	no	—
103	C₃×Q₁₆	—	5	2	no	—
104	C₃×Q₁₆	C₂×D₅	1	1	yes	characteristic
105	C₃×SD₁₆	—	5	4	no	—
106	C₃×SD₁₆	C₂×D₅	1	2	yes	—
107	C₃×D₈	C₂×D₅	1	1	yes	characteristic
108	C₃×M₁₆	—	5	1	no	—
109	C₃×M₁₆	—	5	2	no	—
110	C₂×C₃×C₈	—	5	2	no	—
111	C₂×C₃×C₈	C₂×D₅	1	1	yes	characteristic
112	C₅×Q₈	C₂³×C₃	1	2	yes	—
113	C₅×D₄	—	2	2	no	—
114	C₅×D₄	C₂³×C₃	1	2	yes	—
115	C₂×C₄×C₅	—	2	2	no	—
116	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
117	C₂²⋊D₅	—	2	4	no	—
118	C₂²⋊D₅	C₃×D₄	1	2	yes	—
119	C₂×Dic₅	—	2	2	no	—
120	C₂×Dic₅	—	2	4	no	—
121	D₂₀	C₃×D₄	1	1	yes	characteristic
122	C₄×D₅	—	2	4	no	—
123	C₄×D₅	C₂³×C₃	1	2	yes	—
124	Dic₁₀	—	2	2	no	—
125	Dic₁₀	—	2	4	no	—
126	Dic₁₀	C₃×D₄	1	1	yes	characteristic
127	Dic₁₀	C₂³×C₃	1	4	yes	—
128	C₅×C₈	C₂³×C₃	1	1	yes	characteristic
129	C₅×C₈	C₂³×C₃	1	1	yes	characteristic
130	#168	C₂³×C₃	1	2	yes	—
131	2₋¹⁺⁴	—	5	2	no	—
132	#137	—	5	2	no	—
133	#137	—	5	4	no	—
134	#135	—	5	2	no	—
135	#135	C₃×D₅	1	1	yes	characteristic, complemented
136	C₂×Q₁₆	—	5	2	no	—
137	C₂×Q₁₆	—	5	1	no	—
138	#131	—	5	1	no	—
139	C₂×C₃×C₅	—	2	1	no	—
140	C₂×C₃×C₅	—	4	2	no	—
141	C₂×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
142	C₃×D₅	—	2	2	no	—
143	C₃×Q₈	—	5	1	no	—
144	C₃×Q₈	—	5	4	no	—
145	C₃×Q₈	—	10	4	no	—
146	C₃×Q₈	—	10	2	no	—
147	C₃×Q₈	C₂²×D₅	1	2	yes	—
148	C₃×D₄	—	2	2	no	—
149	C₃×D₄	—	5	2	no	—
150	C₃×D₄	—	5	1	no	—
151	C₃×D₄	—	10	4	no	—
152	C₃×D₄	C₂²×D₅	1	2	yes	—
153	C₂×C₃×C₄	—	2	2	no	—
154	C₂×C₃×C₄	—	5	2	no	—
155	C₂×C₃×C₄	—	10	4	no	—
156	C₂×C₃×C₄	—	10	4	no	—
157	C₂×C₃×C₄	—	10	2	no	—
158	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
159	C₃×C₈	—	5	2	no	—
160	C₃×C₈	C₂²×D₅	1	1	yes	characteristic
161	C₃×C₈	C₂²×D₅	1	1	yes	characteristic
162	C₂²×C₅	—	2	2	no	—
163	C₂²×C₅	C₂×C₃×D₄	1	1	yes	characteristic
164	C₂×D₅	C₂×C₃×D₄	1	2	yes	—
165	C₄×C₅	—	2	2	no	—
166	C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
167	C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
168	Dic₅	—	2	4	no	—
169	Dic₅	C₂×C₃×D₄	1	2	yes	—
170	#41	—	5	1	no	—
171	#41	—	5	4	no	—
172	#41	—	10	2	no	—
173	#41	—	10	4	no	—
174	#41	C₂×C₃×D₅	1	2	yes	—
175	C₂×Q₈	—	5	4	no	—
176	C₂×Q₈	—	5	2	no	—
177	C₂×Q₈	—	10	2	no	—
178	Q₁₆	—	5	4	no	—
179	Q₁₆	—	5	2	no	—
180	Q₁₆	—	5	2	no	—
181	Q₁₆	C₂×C₃×D₅	1	1	yes	characteristic
182	SD₁₆	—	5	4	no	—
183	SD₁₆	C₂×C₃×D₅	1	2	yes	—
184	D₈	C₂×C₃×D₅	1	1	yes	characteristic
185	M₁₆	—	5	1	no	—
186	M₁₆	—	5	2	no	—
187	C₂×C₈	—	5	2	no	—
188	C₂×C₈	C₂×C₃×D₅	1	1	yes	characteristic
189	C₃×C₅	#578	1	1	yes	characteristic, complemented, Hall
190	C₂²×C₃	—	2	2	no	—
191	C₂²×C₃	—	5	2	no	—
192	C₂²×C₃	D₄×D₅	1	1	yes	characteristic
193	C₃×C₄	—	2	2	no	—
194	C₃×C₄	—	5	2	no	—
195	C₃×C₄	—	10	4	no	—
196	C₃×C₄	D₄×D₅	1	1	yes	characteristic
197	C₃×C₄	C₂³×D₅	1	1	yes	characteristic
198	C₂×C₅	—	2	1	no	—
199	C₂×C₅	—	4	2	no	—
200	C₂×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
201	D₅	—	2	2	no	—
202	Q₈	—	5	1	no	—
203	Q₈	—	5	4	no	—
204	Q₈	—	10	4	no	—
205	Q₈	—	10	2	no	—
206	Q₈	C₂²×C₃×D₅	1	2	yes	—
207	D₄	—	2	2	no	—
208	D₄	—	5	2	no	—
209	D₄	—	5	1	no	—
210	D₄	—	10	4	no	—
211	D₄	C₂²×C₃×D₅	1	2	yes	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	5	2	no	—
214	C₂×C₄	—	10	4	no	—
215	C₂×C₄	—	10	4	no	—
216	C₂×C₄	—	10	2	no	—
217	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
218	C₈	—	5	2	no	—
219	C₈	C₂²×C₃×D₅	1	1	yes	characteristic
220	C₈	C₂²×C₃×D₅	1	1	yes	characteristic
221	C₂×C₃	—	2	1	no	—
222	C₂×C₃	—	4	2	no	—
223	C₂×C₃	—	10	2	no	—
224	C₂×C₃	C₂×D₄×D₅	1	1	yes	characteristic
225	C₅	C₃×#578	1	1	yes	characteristic, complemented, Sylow
226	C₂²	—	2	2	no	—
227	C₂²	—	5	2	no	—
228	C₂²	C₃×D₄×D₅	1	1	yes	characteristic
229	C₄	—	2	2	no	—
230	C₄	—	5	2	no	—
231	C₄	—	10	4	no	—
232	C₄	C₃×D₄×D₅	1	1	yes	characteristic
233	C₄	C₂³×C₃×D₅	1	1	yes	characteristic
234	C₃	#66105	1	1	yes	characteristic, direct product factor, Sylow
235	C₂	—	2	1	no	—
236	C₂	—	4	2	no	—
237	C₂	—	10	2	no	—
238	C₂	C₂×C₃×D₄×D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃	C₂×D₄×D₅
commutator, cosocle	C₄×C₅	C₂⁴×C₃
Frattini	C₄	C₂³×C₃×D₅
Fitting	C₃×C₅×#135	C₂
socle	C₂×C₃×C₅	C₂²×D₄
hypercenter	C₃×#135	D₅
hypocenter	C₅	C₃×#578
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×C₄ ◁ C₃×#135

Complements

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

N	H ≅ G/N	type
1. C₃×#4150	C₂	semidirect
2. C₃×C₅×#135	C₂	semidirect
3. C₃×#4097	C₂	semidirect
4. C₃×D₅×Q₁₆	C₂	semidirect
5. C₃×#4061	C₂	semidirect
6. C₃×#4058	C₂	semidirect
7. C₂×C₃×Dic₂₀	C₂	semidirect
8. C₃×#4047	C₂	semidirect
9. #66105	C₃	direct (N × H)
30. #4150	C₂×C₃	semidirect
31. C₅×#135	C₂×C₃	semidirect
32. #4097	C₂×C₃	semidirect
33. D₅×Q₁₆	C₂×C₃	semidirect
34. #4061	C₂×C₃	semidirect
35. #4058	C₂×C₃	semidirect
36. C₂×Dic₂₀	C₂×C₃	semidirect
37. #4047	C₂×C₃	semidirect
61. C₃×#135	D₅	semidirect
135. #135	C₃×D₅	semidirect
189. C₃×C₅	#578	semidirect
225. C₅	C₃×#578	semidirect
234. C₃	#66105	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×#4150	2
C₃×C₅×#135	2
C₃×#4097	2
C₃×D₅×Q₁₆	2
C₃×#4061	2
C₃×#4058	2
C₂×C₃×Dic₂₀	2
C₃×#4047	2
#66105	3
C₃×#578	5

Subgroups — C3×#66105 #11755782

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#66105 #11755782