Subgroups — C₃×C₅×#949 #13087960

Order 1440.

312 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₅×#41	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×C₅×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×#215	C₂	1	2	yes	maximal, complemented
4	C₃×C₅×#213	C₂	1	2	yes	maximal, complemented
5	C₃×C₅×#208	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×C₅×AGL(1,ℤ/8)	—	3	1	no	maximal
7	C₅×#949	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂²×C₃×C₅×D₃	—	2	1	no	—
9	C₃²×C₅×Q₈	C₂²	1	1	yes	characteristic, complemented
10	C₃²×C₅×D₄	—	2	1	no	—
11	C₃²×C₅×D₄	C₂²	1	1	yes	characteristic, complemented
12	C₂×C₃²×C₄×C₅	—	2	1	no	—
13	C₂×C₃²×C₄×C₅	C₂²	1	1	yes	characteristic
14	C₃×C₅×D₁₂	—	2	1	no	—
15	C₃×C₅×D₁₂	C₂²	1	2	yes	—
16	C₃×C₅×C₃⋊C₈	C₂²	1	2	yes	complemented
17	C₃×#949	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₃×C₅×#41	—	2	1	no	—
19	C₃×C₅×#41	D₃	1	1	yes	characteristic, complemented
20	C₃×C₅×#41	C₂×C₃	1	1	yes	characteristic, complemented
21	C₂×C₃×C₅×D₄	—	3	1	no	—
22	C₂×C₅×D₁₂	C₂×C₃	1	1	yes	characteristic, complemented
23	C₃×C₅×SD₁₆	—	3	2	no	—
24	C₃×C₅×D₈	—	3	2	no	—
25	C₃×C₅×M₁₆	—	3	1	no	—
26	C₅×#215	C₂×C₃	1	2	yes	complemented
27	C₅×#213	C₂×C₃	1	2	yes	complemented
28	C₅×#208	C₂×C₃	1	1	yes	characteristic, complemented
29	C₂²×C₃²×C₅	—	2	1	no	—
30	C₂²×C₃²×C₅	D₄	1	1	yes	characteristic
31	C₂×C₃×C₅×D₃	—	2	2	no	—
32	C₂×C₃×C₅×D₃	—	4	2	no	—
33	C₃²×C₄×C₅	—	2	1	no	—
34	C₃²×C₄×C₅	D₄	1	1	yes	characteristic
35	C₃²×C₄×C₅	C₂³	1	1	yes	characteristic
36	C₅×AGL(1,ℤ/8)	—	3	1	no	Hall
37	C₃²×#41	C₂×C₅	1	1	yes	characteristic, complemented
38	C₂×C₃×D₁₂	C₂×C₅	1	1	yes	characteristic, complemented
39	C₃×#215	C₂×C₅	1	2	yes	complemented
40	C₃×#213	C₂×C₅	1	2	yes	complemented
41	C₃×#208	C₂×C₅	1	1	yes	characteristic, complemented
42	C₂³×C₃×C₅	—	6	1	no	—
43	C₂²×C₅×D₃	—	2	1	no	—
44	C₃×C₅×Q₈	—	2	1	no	—
45	C₃×C₅×Q₈	C₂×D₃	1	1	yes	characteristic, complemented
46	C₃×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic, complemented
47	C₃×C₅×D₄	—	2	2	no	—
48	C₃×C₅×D₄	—	2	1	no	—
49	C₃×C₅×D₄	—	2	1	no	—
50	C₃×C₅×D₄	—	2	1	no	—
51	C₃×C₅×D₄	—	3	2	no	—
52	C₃×C₅×D₄	—	6	1	no	—
53	C₃×C₅×D₄	C₂×D₃	1	1	yes	characteristic, complemented
54	C₃×C₅×D₄	C₂²×C₃	1	1	yes	characteristic, complemented
55	C₂×C₃×C₄×C₅	—	2	2	no	—
56	C₂×C₃×C₄×C₅	—	2	1	no	—
57	C₂×C₃×C₄×C₅	—	2	1	no	—
58	C₂×C₃×C₄×C₅	—	2	1	no	—
59	C₂×C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic
60	C₂×C₃×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
61	C₅×D₁₂	—	2	1	no	—
62	C₅×D₁₂	C₂²×C₃	1	2	yes	—
63	C₃×C₅×C₈	—	3	2	no	—
64	C₅×C₃⋊C₈	C₂²×C₃	1	2	yes	complemented
65	C₃×AGL(1,ℤ/8)	—	3	1	no	—
66	#949	C₃×C₅	1	1	yes	characteristic, direct product factor
67	C₂×C₃²×C₅	—	2	1	no	—
68	C₂×C₃²×C₅	—	4	1	no	—
69	C₂×C₃²×C₅	C₂×D₄	1	1	yes	characteristic
70	C₃×C₅×D₃	—	4	2	no	—
71	C₅×#41	C₃×D₃	1	1	yes	characteristic, complemented
72	C₂×C₅×D₄	—	3	1	no	—
73	C₅×SD₁₆	—	3	2	no	—
74	C₅×D₈	—	3	2	no	—
75	C₅×M₁₆	—	3	1	no	—
76	C₂²×C₃×D₃	—	2	1	no	—
77	C₃²×Q₈	C₂²×C₅	1	1	yes	characteristic, complemented
78	C₃²×D₄	—	2	1	no	—
79	C₃²×D₄	C₂²×C₅	1	1	yes	characteristic, complemented
80	C₂×C₃²×C₄	—	2	1	no	—
81	C₂×C₃²×C₄	C₂²×C₅	1	1	yes	characteristic
82	C₃×D₁₂	—	2	1	no	—
83	C₃×D₁₂	C₂²×C₅	1	2	yes	—
84	C₃×C₃⋊C₈	C₂²×C₅	1	2	yes	complemented
85	C₂²×C₃×C₅	—	2	2	no	—
86	C₂²×C₃×C₅	—	2	1	no	—
87	C₂²×C₃×C₅	—	2	1	no	—
88	C₂²×C₃×C₅	—	2	1	no	—
89	C₂²×C₃×C₅	—	6	2	no	—
90	C₂²×C₃×C₅	—	12	2	no	—
91	C₂²×C₃×C₅	#67	1	1	yes	characteristic
92	C₂²×C₃×C₅	C₃×D₄	1	1	yes	characteristic
93	C₂×C₅×D₃	—	2	2	no	—
94	C₂×C₅×D₃	—	4	2	no	—
95	C₃×C₄×C₅	—	2	2	no	—
96	C₃×C₄×C₅	—	2	1	no	—
97	C₃×C₄×C₅	—	2	1	no	—
98	C₃×C₄×C₅	—	2	1	no	—
99	C₃×C₄×C₅	—	2	1	no	—
100	C₃×C₄×C₅	#67	1	1	yes	characteristic
101	C₃×C₄×C₅	C₃×D₄	1	1	yes	characteristic
102	C₃×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
103	C₃×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
104	C₃×#41	—	2	1	no	—
105	C₃×#41	C₅×D₃	1	1	yes	characteristic, complemented
106	C₃×#41	C₂×C₃×C₅	1	1	yes	characteristic, complemented
107	C₂×C₃×D₄	—	3	1	no	—
108	C₂×D₁₂	C₂×C₃×C₅	1	1	yes	characteristic, complemented
109	C₃×SD₁₆	—	3	2	no	—
110	C₃×D₈	—	3	2	no	—
111	C₃×M₁₆	—	3	1	no	—
112	#215	C₂×C₃×C₅	1	2	yes	complemented
113	#213	C₂×C₃×C₅	1	2	yes	complemented
114	#208	C₂×C₃×C₅	1	1	yes	characteristic, complemented
115	C₃²×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Hall
116	C₂³×C₅	—	6	1	no	—
117	C₅×Q₈	C₂×C₃×D₃	1	1	yes	characteristic, complemented
118	C₅×D₄	—	2	1	no	—
119	C₅×D₄	—	3	2	no	—
120	C₅×D₄	—	6	1	no	—
121	C₅×D₄	C₂×C₃×D₃	1	1	yes	characteristic, complemented
122	C₂×C₄×C₅	—	2	1	no	—
123	C₂×C₄×C₅	C₂×C₃×D₃	1	1	yes	characteristic
124	C₅×C₈	—	3	2	no	—
125	C₂²×C₃²	—	2	1	no	—
126	C₂²×C₃²	C₅×D₄	1	1	yes	characteristic
127	C₂×C₃×D₃	—	2	2	no	—
128	C₂×C₃×D₃	—	4	2	no	—
129	C₃²×C₄	—	2	1	no	—
130	C₃²×C₄	C₅×D₄	1	1	yes	characteristic
131	C₃²×C₄	C₂³×C₅	1	1	yes	characteristic
132	AGL(1,ℤ/8)	—	3	1	no	Sylow
133	C₂×C₃×C₅	—	2	1	no	—
134	C₂×C₃×C₅	—	2	1	no	—
135	C₂×C₃×C₅	—	2	1	no	—
136	C₂×C₃×C₅	—	4	1	no	—
137	C₂×C₃×C₅	—	4	2	no	—
138	C₂×C₃×C₅	—	4	1	no	—
139	C₂×C₃×C₅	—	4	1	no	—
140	C₂×C₃×C₅	—	12	2	no	—
141	C₂×C₃×C₅	C₂×#67	1	1	yes	characteristic
142	C₂×C₃×C₅	C₂×C₃×D₄	1	1	yes	characteristic
143	C₅×D₃	—	4	2	no	—
144	C₂³×C₃	—	6	1	no	—
145	C₂²×D₃	—	2	1	no	—
146	C₃×Q₈	—	2	1	no	—
147	C₃×Q₈	C₂×C₅×D₃	1	1	yes	characteristic, complemented
148	C₃×Q₈	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
149	C₃×D₄	—	2	2	no	—
150	C₃×D₄	—	2	1	no	—
151	C₃×D₄	—	2	1	no	—
152	C₃×D₄	—	2	1	no	—
153	C₃×D₄	—	3	2	no	—
154	C₃×D₄	—	6	1	no	—
155	C₃×D₄	C₂×C₅×D₃	1	1	yes	characteristic, complemented
156	C₃×D₄	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
157	C₂×C₃×C₄	—	2	2	no	—
158	C₂×C₃×C₄	—	2	1	no	—
159	C₂×C₃×C₄	—	2	1	no	—
160	C₂×C₃×C₄	—	2	1	no	—
161	C₂×C₃×C₄	C₂×C₅×D₃	1	1	yes	characteristic
162	C₂×C₃×C₄	C₂²×C₃×C₅	1	1	yes	characteristic
163	D₁₂	—	2	1	no	—
164	D₁₂	C₂²×C₃×C₅	1	2	yes	—
165	C₃×C₈	—	3	2	no	—
166	C₃⋊C₈	C₂²×C₃×C₅	1	2	yes	complemented
167	C₂²×C₅	—	2	1	no	—
168	C₂²×C₅	—	6	2	no	—
169	C₂²×C₅	—	12	2	no	—
170	C₂²×C₅	C₃×#67	1	1	yes	characteristic
171	C₄×C₅	—	2	1	no	—
172	C₄×C₅	C₃×#67	1	1	yes	characteristic
173	C₄×C₅	C₂²×C₃×D₃	1	1	yes	characteristic
174	C₂×C₃²	—	2	1	no	—
175	C₂×C₃²	—	4	1	no	—
176	C₂×C₃²	C₂×C₅×D₄	1	1	yes	characteristic
177	C₃×D₃	—	4	2	no	—
178	#41	C₃×C₅×D₃	1	1	yes	characteristic, complemented
179	C₂×D₄	—	3	1	no	—
180	SD₁₆	—	3	2	no	—
181	D₈	—	3	2	no	—
182	M₁₆	—	3	1	no	—
183	C₃×C₅	—	2	1	no	—
184	C₃×C₅	#949	1	1	yes	characteristic, direct product factor
185	C₃×C₅	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic, complemented
186	C₂²×C₃	—	2	2	no	—
187	C₂²×C₃	—	2	1	no	—
188	C₂²×C₃	—	2	1	no	—
189	C₂²×C₃	—	2	1	no	—
190	C₂²×C₃	—	6	2	no	—
191	C₂²×C₃	—	12	2	no	—
192	C₂²×C₃	C₅×#67	1	1	yes	characteristic
193	C₂²×C₃	C₃×C₅×D₄	1	1	yes	characteristic
194	C₂×D₃	—	2	2	no	—
195	C₂×D₃	—	4	2	no	—
196	C₃×C₄	—	2	2	no	—
197	C₃×C₄	—	2	1	no	—
198	C₃×C₄	—	2	1	no	—
199	C₃×C₄	—	2	1	no	—
200	C₃×C₄	—	2	1	no	—
201	C₃×C₄	C₅×#67	1	1	yes	characteristic
202	C₃×C₄	C₃×C₅×D₄	1	1	yes	characteristic
203	C₃×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
204	C₃×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
205	C₂×C₅	—	2	1	no	—
206	C₂×C₅	—	4	1	no	—
207	C₂×C₅	—	12	2	no	—
208	C₂×C₅	C₂×C₃×#67	1	1	yes	characteristic
209	C₃²	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
210	C₂³	—	6	1	no	—
211	Q₈	C₂×C₃×C₅×D₃	1	1	yes	characteristic, complemented
212	D₄	—	2	1	no	—
213	D₄	—	3	2	no	—
214	D₄	—	6	1	no	—
215	D₄	C₂×C₃×C₅×D₃	1	1	yes	characteristic, complemented
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	C₂×C₃×C₅×D₃	1	1	yes	characteristic
218	C₈	—	3	2	no	—
219	C₂×C₃	—	2	1	no	—
220	C₂×C₃	—	2	1	no	—
221	C₂×C₃	—	2	1	no	—
222	C₂×C₃	—	4	1	no	—
223	C₂×C₃	—	4	2	no	—
224	C₂×C₃	—	4	1	no	—
225	C₂×C₃	—	4	1	no	—
226	C₂×C₃	—	12	2	no	—
227	C₂×C₃	C₂×C₅×#67	1	1	yes	characteristic
228	C₂×C₃	C₂×C₃×C₅×D₄	1	1	yes	characteristic
229	D₃	—	4	2	no	—
230	C₅	C₃×#949	1	1	yes	characteristic, direct product factor, Sylow
231	C₂²	—	2	1	no	—
232	C₂²	—	6	2	no	—
233	C₂²	—	12	2	no	—
234	C₂²	C₃×C₅×#67	1	1	yes	characteristic
235	C₄	—	2	1	no	—
236	C₄	C₃×C₅×#67	1	1	yes	characteristic
237	C₄	C₂²×C₃×C₅×D₃	1	1	yes	characteristic
238	C₃	—	2	1	no	—
239	C₃	C₅×#949	1	1	yes	characteristic, direct product factor
240	C₃	C₃×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic, complemented
241	C₂	—	2	1	no	—
242	C₂	—	4	1	no	—
243	C₂	—	12	2	no	—
244	C₂	C₂×C₃×C₅×#67	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₅	C₂×#67
commutator, cosocle	C₃×C₄	C₂³×C₃×C₅
Frattini	C₄	C₂²×C₃×C₅×D₃
Fitting	C₃²×C₅×#41	C₂
socle	C₂×C₃²×C₅	C₂×D₄
hypercenter	C₃×C₅×#41	D₃
hypocenter	C₃	C₃×C₅×AGL(1,ℤ/8)
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₅ ◁ C₂×C₃×C₄×C₅ ◁ C₃×C₅×#41

Complements

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

N	H ≅ G/N	type
1. C₃²×C₅×#41	C₂	semidirect
2. C₂×C₃×C₅×D₁₂	C₂	semidirect
3. C₃×C₅×#215	C₂	semidirect
4. C₃×C₅×#213	C₂	semidirect
5. C₃×C₅×#208	C₂	semidirect
7. C₅×#949	C₃	direct (N × H)
9. C₃²×C₅×Q₈	C₂²	semidirect
11. C₃²×C₅×D₄	C₂²	semidirect
16. C₃×C₅×C₃⋊C₈	C₂²	semidirect
17. C₃×#949	C₅	direct (N × H)
19. C₃×C₅×#41	D₃	semidirect
20. C₃×C₅×#41	C₂×C₃	semidirect
22. C₂×C₅×D₁₂	C₂×C₃	semidirect
26. C₅×#215	C₂×C₃	semidirect
27. C₅×#213	C₂×C₃	semidirect
28. C₅×#208	C₂×C₃	semidirect
37. C₃²×#41	C₂×C₅	semidirect
38. C₂×C₃×D₁₂	C₂×C₅	semidirect
39. C₃×#215	C₂×C₅	semidirect
40. C₃×#213	C₂×C₅	semidirect
41. C₃×#208	C₂×C₅	semidirect
45. C₃×C₅×Q₈	C₂×D₃	semidirect
46. C₃×C₅×Q₈	C₂²×C₃	semidirect
53. C₃×C₅×D₄	C₂×D₃	semidirect
54. C₃×C₅×D₄	C₂²×C₃	semidirect
64. C₅×C₃⋊C₈	C₂²×C₃	semidirect
66. #949	C₃×C₅	direct (N × H)
71. C₅×#41	C₃×D₃	semidirect
77. C₃²×Q₈	C₂²×C₅	semidirect
79. C₃²×D₄	C₂²×C₅	semidirect
84. C₃×C₃⋊C₈	C₂²×C₅	semidirect
105. C₃×#41	C₅×D₃	semidirect
106. C₃×#41	C₂×C₃×C₅	semidirect
108. C₂×D₁₂	C₂×C₃×C₅	semidirect
112. #215	C₂×C₃×C₅	semidirect
113. #213	C₂×C₃×C₅	semidirect
114. #208	C₂×C₃×C₅	semidirect
115. C₃²×C₅	AGL(1,ℤ/8)	semidirect
117. C₅×Q₈	C₂×C₃×D₃	semidirect
121. C₅×D₄	C₂×C₃×D₃	semidirect
147. C₃×Q₈	C₂×C₅×D₃	semidirect
148. C₃×Q₈	C₂²×C₃×C₅	semidirect
155. C₃×D₄	C₂×C₅×D₃	semidirect
156. C₃×D₄	C₂²×C₃×C₅	semidirect
166. C₃⋊C₈	C₂²×C₃×C₅	semidirect
178. #41	C₃×C₅×D₃	semidirect
184. C₃×C₅	#949	direct (N × H)
185. C₃×C₅	C₃×AGL(1,ℤ/8)	semidirect
209. C₃²	C₅×AGL(1,ℤ/8)	semidirect
211. Q₈	C₂×C₃×C₅×D₃	semidirect
215. D₄	C₂×C₃×C₅×D₃	semidirect
230. C₅	C₃×#949	direct (N × H)
239. C₃	C₅×#949	direct (N × H)
240. C₃	C₃×C₅×AGL(1,ℤ/8)	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₅×#41	2
C₂×C₃×C₅×D₁₂	2
C₃×C₅×#215	2
C₃×C₅×#213	2
C₃×C₅×#208	2
C₃×C₅×AGL(1,ℤ/8)	3
C₅×#949	3
C₃×#949	5

Subgroups — C3×C5×#949 #13087960

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#949 #13087960