Subgroups — C₃×C₅×#468 #11751443

Order 960.

240 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×C₅×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×#121	C₂	1	1	yes	maximal, characteristic
5	C₃×C₅×#107	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×C₅×#103	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×C₅×#102	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#468	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₅×D₄	—	2	1	no	—
11	C₂×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
12	C₂²×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
13	C₃×C₅×SD₁₆	—	2	2	no	—
14	C₃×C₅×M₁₆	—	2	2	no	—
15	C₂×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
17	C₃×C₅×C₄⋊C₄	—	2	1	no	—
18	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
21	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
22	C₃×#468	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
23	C₂×C₅×SD₁₆	C₂×C₃	1	1	yes	characteristic, complemented
24	C₂×C₅×M₁₆	C₂×C₃	1	1	yes	characteristic, complemented
25	C₅×C₂²⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
26	C₅×#121	C₂×C₃	1	1	yes	characteristic
27	C₅×#107	C₂×C₃	1	1	yes	characteristic, complemented
28	C₅×#103	C₂×C₃	1	1	yes	characteristic, complemented
29	C₅×#102	C₂×C₃	1	1	yes	characteristic, complemented
30	C₂³×C₃×C₅	—	2	1	no	—
31	C₂³×C₃×C₅	D₄	1	1	yes	characteristic
32	C₃×C₅×Q₈	—	2	1	no	—
33	C₃×C₅×Q₈	—	2	1	no	—
34	C₃×C₅×D₄	—	2	1	no	—
35	C₃×C₅×D₄	—	2	1	no	—
36	C₃×C₅×D₄	—	4	2	no	—
37	C₂×C₃×C₄×C₅	—	2	1	no	—
38	C₂×C₃×C₄×C₅	—	2	1	no	—
39	C₂×C₃×C₄×C₅	—	2	1	no	—
40	C₂×C₃×C₄×C₅	—	2	1	no	—
41	C₂×C₃×C₄×C₅	—	2	1	no	—
42	C₂×C₃×C₄×C₅	D₄	1	1	yes	characteristic
43	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
44	C₃×C₅×C₈	—	2	1	no	—
45	C₃×C₅×C₈	D₄	1	2	yes	complemented
46	C₂×C₃×SD₁₆	C₂×C₅	1	1	yes	characteristic, complemented
47	C₂×C₃×M₁₆	C₂×C₅	1	1	yes	characteristic, complemented
48	C₃×C₂²⋊Q₈	C₂×C₅	1	1	yes	characteristic, complemented
49	C₃×#121	C₂×C₅	1	1	yes	characteristic
50	C₃×#107	C₂×C₅	1	1	yes	characteristic, complemented
51	C₃×#103	C₂×C₅	1	1	yes	characteristic, complemented
52	C₃×#102	C₂×C₅	1	1	yes	characteristic, complemented
53	C₂×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
54	C₂×C₅×D₄	—	2	1	no	—
55	C₂×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
56	C₂²×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
57	C₅×SD₁₆	—	2	2	no	—
58	C₅×M₁₆	—	2	2	no	—
59	C₂×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
60	C₂×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
61	C₅×C₄⋊C₄	—	2	1	no	—
62	C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
63	C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
64	C₅×C₂²⋊C₄	—	2	1	no	—
65	C₅×C₂²⋊C₄	—	2	1	no	—
66	#468	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
67	C₂²×C₃×C₅	—	2	1	no	—
68	C₂²×C₃×C₅	—	2	1	no	—
69	C₂²×C₃×C₅	—	2	1	no	—
70	C₂²×C₃×C₅	—	4	1	no	—
71	C₂²×C₃×C₅	—	4	1	no	—
72	C₂²×C₃×C₅	—	4	1	no	—
73	C₂²×C₃×C₅	C₂×D₄	1	1	yes	characteristic
74	C₃×C₄×C₅	—	2	1	no	—
75	C₃×C₄×C₅	—	4	1	no	—
76	C₃×C₄×C₅	—	4	1	no	—
77	C₃×C₄×C₅	—	4	1	no	—
78	C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
79	C₃×C₄×C₅	#41	1	1	yes	characteristic
80	C₂×C₃×Q₈	C₂²×C₅	1	1	yes	characteristic
81	C₂×C₃×D₄	—	2	1	no	—
82	C₂×C₃×D₄	C₂²×C₅	1	1	yes	characteristic
83	C₂²×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
84	C₃×SD₁₆	—	2	2	no	—
85	C₃×M₁₆	—	2	2	no	—
86	C₂×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
87	C₂×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
88	C₃×C₄⋊C₄	—	2	1	no	—
89	C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
90	C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
91	C₃×C₂²⋊C₄	—	2	1	no	—
92	C₃×C₂²⋊C₄	—	2	1	no	—
93	C₂³×C₅	—	2	1	no	—
94	C₂³×C₅	C₃×D₄	1	1	yes	characteristic
95	C₅×Q₈	—	2	1	no	—
96	C₅×Q₈	—	2	1	no	—
97	C₅×D₄	—	2	1	no	—
98	C₅×D₄	—	2	1	no	—
99	C₅×D₄	—	4	2	no	—
100	C₂×C₄×C₅	—	2	1	no	—
101	C₂×C₄×C₅	—	2	1	no	—
102	C₂×C₄×C₅	—	2	1	no	—
103	C₂×C₄×C₅	—	2	1	no	—
104	C₂×C₄×C₅	—	2	1	no	—
105	C₂×C₄×C₅	C₃×D₄	1	1	yes	characteristic
106	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
107	C₅×C₈	—	2	1	no	—
108	C₅×C₈	C₃×D₄	1	2	yes	complemented
109	C₂×SD₁₆	C₂×C₃×C₅	1	1	yes	characteristic, complemented
110	C₂×M₁₆	C₂×C₃×C₅	1	1	yes	characteristic, complemented
111	C₂²⋊Q₈	C₂×C₃×C₅	1	1	yes	characteristic, complemented
112	#121	C₂×C₃×C₅	1	1	yes	characteristic
113	#107	C₂×C₃×C₅	1	1	yes	characteristic, complemented
114	#103	C₂×C₃×C₅	1	1	yes	characteristic, complemented
115	#102	C₂×C₃×C₅	1	1	yes	characteristic, complemented
116	C₂×C₃×C₅	—	4	1	no	—
117	C₂×C₃×C₅	—	8	1	no	—
118	C₂×C₃×C₅	#121	1	1	yes	characteristic
119	C₂×C₃×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
120	C₂×C₃×C₅	#137	1	1	yes	characteristic
121	C₂³×C₃	—	2	1	no	—
122	C₂³×C₃	C₅×D₄	1	1	yes	characteristic
123	C₃×Q₈	—	2	1	no	—
124	C₃×Q₈	—	2	1	no	—
125	C₃×D₄	—	2	1	no	—
126	C₃×D₄	—	2	1	no	—
127	C₃×D₄	—	4	2	no	—
128	C₂×C₃×C₄	—	2	1	no	—
129	C₂×C₃×C₄	—	2	1	no	—
130	C₂×C₃×C₄	—	2	1	no	—
131	C₂×C₃×C₄	—	2	1	no	—
132	C₂×C₃×C₄	—	2	1	no	—
133	C₂×C₃×C₄	C₅×D₄	1	1	yes	characteristic
134	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
135	C₃×C₈	—	2	1	no	—
136	C₃×C₈	C₅×D₄	1	2	yes	complemented
137	C₂²×C₅	—	2	1	no	—
138	C₂²×C₅	—	2	1	no	—
139	C₂²×C₅	—	2	1	no	—
140	C₂²×C₅	—	4	1	no	—
141	C₂²×C₅	—	4	1	no	—
142	C₂²×C₅	—	4	1	no	—
143	C₂²×C₅	C₂×C₃×D₄	1	1	yes	characteristic
144	C₄×C₅	—	2	1	no	—
145	C₄×C₅	—	4	1	no	—
146	C₄×C₅	—	4	1	no	—
147	C₄×C₅	—	4	1	no	—
148	C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
149	C₄×C₅	C₃×#41	1	1	yes	characteristic
150	C₂×Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
151	C₂×D₄	—	2	1	no	—
152	C₂×D₄	C₂²×C₃×C₅	1	1	yes	characteristic
153	C₂²×C₄	C₂²×C₃×C₅	1	1	yes	characteristic
154	SD₁₆	—	2	2	no	—
155	M₁₆	—	2	2	no	—
156	C₂×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
157	C₂×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
158	C₄⋊C₄	—	2	1	no	—
159	C₄⋊C₄	C₂²×C₃×C₅	1	1	yes	characteristic
160	C₄⋊C₄	C₂²×C₃×C₅	1	1	yes	characteristic
161	C₂²⋊C₄	—	2	1	no	—
162	C₂²⋊C₄	—	2	1	no	—
163	C₃×C₅	#468	1	1	yes	characteristic, direct product factor, Hall
164	C₂²×C₃	—	2	1	no	—
165	C₂²×C₃	—	2	1	no	—
166	C₂²×C₃	—	2	1	no	—
167	C₂²×C₃	—	4	1	no	—
168	C₂²×C₃	—	4	1	no	—
169	C₂²×C₃	—	4	1	no	—
170	C₂²×C₃	C₂×C₅×D₄	1	1	yes	characteristic
171	C₃×C₄	—	2	1	no	—
172	C₃×C₄	—	4	1	no	—
173	C₃×C₄	—	4	1	no	—
174	C₃×C₄	—	4	1	no	—
175	C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
176	C₃×C₄	C₅×#41	1	1	yes	characteristic
177	C₂×C₅	—	4	1	no	—
178	C₂×C₅	—	8	1	no	—
179	C₂×C₅	C₃×#121	1	1	yes	characteristic
180	C₂×C₅	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
181	C₂×C₅	C₃×#137	1	1	yes	characteristic
182	C₂³	—	2	1	no	—
183	C₂³	C₃×C₅×D₄	1	1	yes	characteristic
184	Q₈	—	2	1	no	—
185	Q₈	—	2	1	no	—
186	D₄	—	2	1	no	—
187	D₄	—	2	1	no	—
188	D₄	—	4	2	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	C₃×C₅×D₄	1	1	yes	characteristic
195	C₂×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
196	C₈	—	2	1	no	—
197	C₈	C₃×C₅×D₄	1	2	yes	complemented
198	C₂×C₃	—	4	1	no	—
199	C₂×C₃	—	8	1	no	—
200	C₂×C₃	C₅×#121	1	1	yes	characteristic
201	C₂×C₃	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
202	C₂×C₃	C₅×#137	1	1	yes	characteristic
203	C₅	C₃×#468	1	1	yes	characteristic, direct product factor, Sylow
204	C₂²	—	2	1	no	—
205	C₂²	—	2	1	no	—
206	C₂²	—	2	1	no	—
207	C₂²	—	4	1	no	—
208	C₂²	—	4	1	no	—
209	C₂²	—	4	1	no	—
210	C₂²	C₂×C₃×C₅×D₄	1	1	yes	characteristic
211	C₄	—	2	1	no	—
212	C₄	—	4	1	no	—
213	C₄	—	4	1	no	—
214	C₄	—	4	1	no	—
215	C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
216	C₄	C₃×C₅×#41	1	1	yes	characteristic
217	C₃	C₅×#468	1	1	yes	characteristic, direct product factor, Sylow
218	C₂	—	4	1	no	—
219	C₂	—	8	1	no	—
220	C₂	C₃×C₅×#121	1	1	yes	characteristic
221	C₂	C₃×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
222	C₂	C₃×C₅×#137	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃×C₅	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₃×C₅
prime radical	C₃×C₅×#121	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₂²×C₃×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅×SD₁₆	C₂	semidirect
2. C₂×C₃×C₅×M₁₆	C₂	semidirect
3. C₃×C₅×C₂²⋊Q₈	C₂	semidirect
5. C₃×C₅×#107	C₂	semidirect
6. C₃×C₅×#103	C₂	semidirect
7. C₃×C₅×#102	C₂	semidirect
8. C₅×#468	C₃	direct (N × H)
22. C₃×#468	C₅	direct (N × H)
23. C₂×C₅×SD₁₆	C₂×C₃	semidirect
24. C₂×C₅×M₁₆	C₂×C₃	semidirect
25. C₅×C₂²⋊Q₈	C₂×C₃	semidirect
27. C₅×#107	C₂×C₃	semidirect
28. C₅×#103	C₂×C₃	semidirect
29. C₅×#102	C₂×C₃	semidirect
45. C₃×C₅×C₈	D₄	semidirect
46. C₂×C₃×SD₁₆	C₂×C₅	semidirect
47. C₂×C₃×M₁₆	C₂×C₅	semidirect
48. C₃×C₂²⋊Q₈	C₂×C₅	semidirect
50. C₃×#107	C₂×C₅	semidirect
51. C₃×#103	C₂×C₅	semidirect
52. C₃×#102	C₂×C₅	semidirect
66. #468	C₃×C₅	direct (N × H)
108. C₅×C₈	C₃×D₄	semidirect
109. C₂×SD₁₆	C₂×C₃×C₅	semidirect
110. C₂×M₁₆	C₂×C₃×C₅	semidirect
111. C₂²⋊Q₈	C₂×C₃×C₅	semidirect
113. #107	C₂×C₃×C₅	semidirect
114. #103	C₂×C₃×C₅	semidirect
115. #102	C₂×C₃×C₅	semidirect
136. C₃×C₈	C₅×D₄	semidirect
163. C₃×C₅	#468	direct (N × H)
197. C₈	C₃×C₅×D₄	semidirect
203. C₅	C₃×#468	direct (N × H)
217. C₃	C₅×#468	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×C₅×SD₁₆	2
C₂×C₃×C₅×M₁₆	2
C₃×C₅×C₂²⋊Q₈	2
C₃×C₅×#121	2
C₃×C₅×#107	2
C₃×C₅×#103	2
C₃×C₅×#102	2
C₅×#468	3
C₃×#468	5

Subgroups — C3×C5×#468 #11751443

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#468 #11751443