Subgroups — C₃×#66079 #11755756

Order 960.

516 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	C₂×C₃×#709	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#681	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#4053	C₂	1	4	yes	maximal, complemented
5	C₃×D₅×M₁₆	C₂	1	4	yes	maximal, complemented
6	C₃×#4047	C₂	1	4	yes	maximal, complemented
7	#66079	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂²×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₂²⋊D₅	C₄	1	2	yes	—
10	C₃×#709	C₄	1	2	yes	—
11	C₃×#709	C₄	1	2	yes	—
12	C₃×#709	C₂²	1	2	yes	—
13	C₃×#709	C₂²	1	2	yes	—
14	C₂×C₃×D₂₀	C₄	1	1	yes	characteristic
15	C₂×C₃×C₄×D₅	C₂²	1	2	yes	—
16	C₂×C₃×Dic₁₀	C₄	1	1	yes	characteristic
17	C₃×C₅×M₁₆	C₂²	1	2	yes	—
18	C₃×C₅×M₁₆	C₂²	1	2	yes	complemented
19	C₂×C₃×C₅×C₈	C₂²	1	2	yes	complemented
20	C₃×#681	C₂²	1	2	yes	—
21	C₃×#681	C₂²	1	2	yes	complemented
22	C₂×C₃×#168	C₂²	1	2	yes	complemented
23	C₃×#676	C₂²	1	8	yes	complemented
24	C₃×C₈×D₅	C₂²	1	8	yes	complemented
25	C₃×#568	—	5	1	no	maximal, Hall
26	C₂×#709	C₂×C₃	1	1	yes	characteristic
27	C₂×C₅×M₁₆	C₂×C₃	1	1	yes	characteristic, complemented
28	C₂×#681	C₂×C₃	1	1	yes	characteristic, complemented
29	#4053	C₂×C₃	1	4	yes	complemented
30	D₅×M₁₆	C₂×C₃	1	4	yes	complemented
31	#4047	C₂×C₃	1	4	yes	complemented
32	C₂³×C₃×C₅	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₃×D₅	C₂×C₄	1	2	yes	—
34	C₂×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
35	C₂×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
36	C₂×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
37	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
38	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
39	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
40	C₃×C₂²⋊D₅	C₂×C₄	1	4	yes	—
41	C₃×C₂²⋊D₅	C₂×C₄	1	4	yes	—
42	C₂×C₃×Dic₅	C₂×C₄	1	2	yes	—
43	C₃×D₂₀	C₂×C₄	1	2	yes	—
44	C₃×D₂₀	C₂×C₄	1	2	yes	—
45	C₃×C₄×D₅	C₂×C₄	1	4	yes	—
46	C₃×C₄×D₅	C₂³	1	4	yes	—
47	C₃×Dic₁₀	C₂×C₄	1	2	yes	—
48	C₃×Dic₁₀	C₂×C₄	1	2	yes	—
49	C₃×C₅×C₈	C₂³	1	4	yes	—
50	C₃×#168	C₂³	1	4	yes	—
51	C₂×C₃×#41	—	5	1	no	—
52	C₃×#131	—	5	4	no	—
53	C₃×#131	—	5	4	no	—
54	C₂×C₃×M₁₆	—	5	1	no	—
55	C₂×C₃×M₁₆	—	5	4	no	—
56	C₂×C₃×M₁₆	D₅	1	1	yes	characteristic, complemented
57	C₂²×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
58	C₂×C₂²⋊D₅	C₃×C₄	1	2	yes	—
59	#709	C₃×C₄	1	2	yes	—
60	#709	C₃×C₄	1	2	yes	—
61	#709	C₂²×C₃	1	2	yes	—
62	#709	C₂²×C₃	1	2	yes	—
63	C₂×D₂₀	C₃×C₄	1	1	yes	characteristic
64	C₂×C₄×D₅	C₂²×C₃	1	2	yes	—
65	C₂×Dic₁₀	C₃×C₄	1	1	yes	characteristic
66	C₅×M₁₆	C₂²×C₃	1	2	yes	—
67	C₅×M₁₆	C₂²×C₃	1	2	yes	complemented
68	C₂×C₅×C₈	C₂²×C₃	1	2	yes	complemented
69	#681	C₂²×C₃	1	2	yes	—
70	#681	C₂²×C₃	1	2	yes	complemented
71	C₂×#168	C₂²×C₃	1	2	yes	complemented
72	#676	C₂²×C₃	1	8	yes	complemented
73	C₈×D₅	C₂²×C₃	1	8	yes	complemented
74	#568	—	5	1	no	Sylow
75	C₂²×C₃×C₅	—	4	1	no	—
76	C₂²×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₃×D₅	—	4	2	no	—
80	C₂×C₃×D₅	C₂²×C₄	1	4	yes	—
81	C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
82	C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
83	C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
84	C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
85	C₃×Dic₅	C₂²×C₄	1	4	yes	—
86	C₃×#41	—	5	2	no	—
87	C₃×#41	—	5	2	no	—
88	C₃×#41	—	5	2	no	—
89	C₃×#41	—	5	2	no	—
90	C₂×C₃×Q₈	—	5	1	no	—
91	C₂×C₃×D₄	—	5	2	no	—
92	C₂×C₃×D₄	—	5	1	no	—
93	C₂²×C₃×C₄	—	5	2	no	—
94	C₂²×C₃×C₄	C₂×D₅	1	1	yes	characteristic
95	C₃×M₁₆	—	5	2	no	—
96	C₃×M₁₆	—	5	2	no	—
97	C₃×M₁₆	—	5	8	no	—
98	C₃×M₁₆	C₂×D₅	1	2	yes	—
99	C₃×M₁₆	C₂×D₅	1	2	yes	complemented
100	C₂×C₃×C₈	—	5	2	no	—
101	C₂×C₃×C₈	—	5	8	no	—
102	C₂×C₃×C₈	C₂×D₅	1	2	yes	complemented
103	C₂³×C₅	C₂×C₃×C₄	1	1	yes	characteristic
104	C₂²×D₅	C₂×C₃×C₄	1	2	yes	—
105	C₂×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
106	C₂×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
107	C₂×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
108	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
109	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
110	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
111	C₂²⋊D₅	C₂×C₃×C₄	1	4	yes	—
112	C₂²⋊D₅	C₂×C₃×C₄	1	4	yes	—
113	C₂×Dic₅	C₂×C₃×C₄	1	2	yes	—
114	D₂₀	C₂×C₃×C₄	1	2	yes	—
115	D₂₀	C₂×C₃×C₄	1	2	yes	—
116	C₄×D₅	C₂×C₃×C₄	1	4	yes	—
117	C₄×D₅	C₂³×C₃	1	4	yes	—
118	Dic₁₀	C₂×C₃×C₄	1	2	yes	—
119	Dic₁₀	C₂×C₃×C₄	1	2	yes	—
120	C₅×C₈	C₂³×C₃	1	4	yes	—
121	#168	C₂³×C₃	1	4	yes	—
122	C₂×#41	—	5	1	no	—
123	#131	—	5	4	no	—
124	#131	—	5	4	no	—
125	C₂×M₁₆	—	5	1	no	—
126	C₂×M₁₆	—	5	4	no	—
127	C₂×M₁₆	C₃×D₅	1	1	yes	characteristic, complemented
128	C₂×C₃×C₅	—	2	1	no	—
129	C₂×C₃×C₅	—	2	1	no	—
130	C₂×C₃×C₅	—	2	1	no	—
131	C₂×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
132	C₃×D₅	—	2	4	no	—
133	C₂³×C₃	—	5	2	no	—
134	C₂³×C₃	C₄×D₅	1	1	yes	characteristic
135	C₃×Q₈	—	5	2	no	—
136	C₃×Q₈	—	5	2	no	—
137	C₃×D₄	—	5	4	no	—
138	C₃×D₄	—	5	4	no	—
139	C₃×D₄	—	5	2	no	—
140	C₃×D₄	—	5	2	no	—
141	C₂×C₃×C₄	—	5	2	no	—
142	C₂×C₃×C₄	—	5	4	no	—
143	C₂×C₃×C₄	—	5	4	no	—
144	C₂×C₃×C₄	C₄×D₅	1	1	yes	characteristic
145	C₂×C₃×C₄	C₄×D₅	1	1	yes	characteristic
146	C₂×C₃×C₄	C₄×D₅	1	1	yes	characteristic
147	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
148	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
149	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
150	C₃×C₈	—	5	4	no	—
151	C₃×C₈	C₂²×D₅	1	4	yes	—
152	C₂²×C₅	—	4	1	no	—
153	C₂²×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
154	C₂²×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
155	C₂²×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
156	C₂×D₅	—	4	2	no	—
157	C₂×D₅	C₂²×C₃×C₄	1	4	yes	—
158	C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
159	C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
160	C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
161	C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
162	Dic₅	C₂²×C₃×C₄	1	4	yes	—
163	#41	—	5	2	no	—
164	#41	—	5	2	no	—
165	#41	—	5	2	no	—
166	#41	—	5	2	no	—
167	C₂×Q₈	—	5	1	no	—
168	C₂×D₄	—	5	2	no	—
169	C₂×D₄	—	5	1	no	—
170	C₂²×C₄	—	5	2	no	—
171	C₂²×C₄	C₂×C₃×D₅	1	1	yes	characteristic
172	M₁₆	—	5	2	no	—
173	M₁₆	—	5	2	no	—
174	M₁₆	—	5	8	no	—
175	M₁₆	C₂×C₃×D₅	1	2	yes	—
176	M₁₆	C₂×C₃×D₅	1	2	yes	complemented
177	C₂×C₈	—	5	2	no	—
178	C₂×C₈	—	5	8	no	—
179	C₂×C₈	C₂×C₃×D₅	1	2	yes	complemented
180	C₃×C₅	#568	1	1	yes	characteristic, complemented, Hall
181	C₂²×C₃	—	4	1	no	—
182	C₂²×C₃	—	5	4	no	—
183	C₂²×C₃	—	20	2	no	—
184	C₂²×C₃	C₂×C₄×D₅	1	1	yes	characteristic
185	C₂²×C₃	C₂×C₄×D₅	1	1	yes	characteristic
186	C₂²×C₃	C₂×C₄×D₅	1	1	yes	characteristic
187	C₃×C₄	—	5	4	no	—
188	C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
189	C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
190	C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
191	C₃×C₄	C₂³×D₅	1	1	yes	characteristic
192	C₂×C₅	—	2	1	no	—
193	C₂×C₅	—	2	1	no	—
194	C₂×C₅	—	2	1	no	—
195	C₂×C₅	C₂³×C₃×C₄	1	1	yes	characteristic
196	D₅	—	2	4	no	—
197	C₂³	—	5	2	no	—
198	C₂³	C₃×C₄×D₅	1	1	yes	characteristic
199	Q₈	—	5	2	no	—
200	Q₈	—	5	2	no	—
201	D₄	—	5	4	no	—
202	D₄	—	5	4	no	—
203	D₄	—	5	2	no	—
204	D₄	—	5	2	no	—
205	C₂×C₄	—	5	2	no	—
206	C₂×C₄	—	5	4	no	—
207	C₂×C₄	—	5	4	no	—
208	C₂×C₄	C₃×C₄×D₅	1	1	yes	characteristic
209	C₂×C₄	C₃×C₄×D₅	1	1	yes	characteristic
210	C₂×C₄	C₃×C₄×D₅	1	1	yes	characteristic
211	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
212	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
213	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
214	C₈	—	5	4	no	—
215	C₈	C₂²×C₃×D₅	1	4	yes	—
216	C₂×C₃	—	2	1	no	—
217	C₂×C₃	—	2	1	no	—
218	C₂×C₃	—	2	1	no	—
219	C₂×C₃	—	10	4	no	—
220	C₂×C₃	C₂²×C₄×D₅	1	1	yes	characteristic
221	C₅	C₃×#568	1	1	yes	characteristic, complemented, Sylow
222	C₂²	—	4	1	no	—
223	C₂²	—	5	4	no	—
224	C₂²	—	20	2	no	—
225	C₂²	C₂×C₃×C₄×D₅	1	1	yes	characteristic
226	C₂²	C₂×C₃×C₄×D₅	1	1	yes	characteristic
227	C₂²	C₂×C₃×C₄×D₅	1	1	yes	characteristic
228	C₄	—	5	4	no	—
229	C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
230	C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
231	C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
232	C₄	C₂³×C₃×D₅	1	1	yes	characteristic
233	C₃	#66079	1	1	yes	characteristic, direct product factor, Sylow
234	C₂	—	2	1	no	—
235	C₂	—	2	1	no	—
236	C₂	—	2	1	no	—
237	C₂	—	10	4	no	—
238	C₂	C₂²×C₃×C₄×D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄	C₂³×D₅
commutator	C₂×C₅	C₂³×C₃×C₄
Frattini	C₄	C₂³×C₃×D₅
Fitting	C₂×C₃×C₅×M₁₆	C₂
socle	C₂×C₃×C₅	C₂³×C₄
prime radical	C₂×C₃×#709	C₂
cosocle	C₄×C₅	C₂⁴×C₃
hypercenter	C₂×C₃×M₁₆	D₅
hypocenter	C₅	C₃×#568
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₃×C₄ ◁ C₂×C₃×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₅×M₁₆	C₂	semidirect
3. C₂×C₃×#681	C₂	semidirect
4. C₃×#4053	C₂	semidirect
5. C₃×D₅×M₁₆	C₂	semidirect
6. C₃×#4047	C₂	semidirect
7. #66079	C₃	direct (N × H)
18. C₃×C₅×M₁₆	C₂²	semidirect
19. C₂×C₃×C₅×C₈	C₂²	semidirect
21. C₃×#681	C₂²	semidirect
22. C₂×C₃×#168	C₂²	semidirect
23. C₃×#676	C₂²	semidirect
24. C₃×C₈×D₅	C₂²	semidirect
27. C₂×C₅×M₁₆	C₂×C₃	semidirect
28. C₂×#681	C₂×C₃	semidirect
29. #4053	C₂×C₃	semidirect
30. D₅×M₁₆	C₂×C₃	semidirect
31. #4047	C₂×C₃	semidirect
56. C₂×C₃×M₁₆	D₅	semidirect
67. C₅×M₁₆	C₂²×C₃	semidirect
68. C₂×C₅×C₈	C₂²×C₃	semidirect
70. #681	C₂²×C₃	semidirect
71. C₂×#168	C₂²×C₃	semidirect
72. #676	C₂²×C₃	semidirect
73. C₈×D₅	C₂²×C₃	semidirect
99. C₃×M₁₆	C₂×D₅	semidirect
102. C₂×C₃×C₈	C₂×D₅	semidirect
127. C₂×M₁₆	C₃×D₅	semidirect
176. M₁₆	C₂×C₃×D₅	semidirect
179. C₂×C₈	C₂×C₃×D₅	semidirect
180. C₃×C₅	#568	semidirect
221. C₅	C₃×#568	semidirect
233. C₃	#66079	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×#709	2
C₂×C₃×C₅×M₁₆	2
C₂×C₃×#681	2
C₃×#4053	2
C₃×D₅×M₁₆	2
C₃×#4047	2
#66079	3
C₃×#568	5

Subgroups — C3×#66079 #11755756

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#66079 #11755756