Subgroups — C₃×C₅×#2236 #14275270

Order 1920.

280 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	C₃×C₅×#467	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×#107	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#366	C₂	1	2	yes	maximal, complemented
4	C₃×C₅×#358	C₂	1	2	yes	maximal, complemented
5	C₃×C₅×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₅×#2236	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×Q₁₆	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₃×C₅×C₂²⋊Q₈	—	2	1	no	—
10	C₂×C₃×C₅×C₄⋊C₄	—	2	1	no	—
11	C₂×C₃×C₅×C₁₆	C₂²	1	2	yes	—
12	C₃×C₅×#107	—	2	1	no	—
13	C₃×C₅×#107	C₂²	1	1	yes	characteristic
14	C₃×C₅×#107	C₂²	1	2	yes	—
15	C₃×C₅×#103	—	2	1	no	—
16	C₃×#2236	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₅×#467	C₂×C₃	1	1	yes	characteristic
18	C₂×C₅×#107	C₂×C₃	1	1	yes	characteristic
19	C₅×#366	C₂×C₃	1	2	yes	complemented
20	C₅×#358	C₂×C₃	1	2	yes	complemented
21	C₅×C₂²⋊C₁₆	C₂×C₃	1	1	yes	characteristic
22	C₂×C₃×C₅×Q₈	—	2	1	no	—
23	C₂²×C₃×C₄×C₅	—	4	1	no	—
24	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
25	C₃×C₅×Q₁₆	—	2	2	no	—
26	C₂×C₃×C₅×C₈	—	2	2	no	—
27	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
28	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
29	C₃×C₅×C₄⋊C₄	—	2	2	no	—
30	C₃×C₅×C₄⋊C₄	—	2	1	no	—
31	C₃×C₅×C₄⋊C₄	—	4	1	no	—
32	C₃×C₅×C₄⋊C₄	—	4	1	no	—
33	C₃×C₅×C₂²⋊C₄	—	4	1	no	—
34	C₃×C₅×C₁₆	—	2	2	no	—
35	C₃×#467	C₂×C₅	1	1	yes	characteristic
36	C₂×C₃×#107	C₂×C₅	1	1	yes	characteristic
37	C₃×#366	C₂×C₅	1	2	yes	complemented
38	C₃×#358	C₂×C₅	1	2	yes	complemented
39	C₃×C₂²⋊C₁₆	C₂×C₅	1	1	yes	characteristic
40	C₂×C₅×Q₁₆	C₂²×C₃	1	1	yes	characteristic
41	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
42	C₅×C₂²⋊Q₈	—	2	1	no	—
43	C₂×C₅×C₄⋊C₄	—	2	1	no	—
44	C₂×C₅×C₁₆	C₂²×C₃	1	2	yes	—
45	C₅×#107	—	2	1	no	—
46	C₅×#107	C₂²×C₃	1	2	yes	—
47	C₅×#107	C₂²×C₃	1	1	yes	characteristic
48	C₅×#103	—	2	1	no	—
49	#2236	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
50	C₂³×C₃×C₅	D₈	1	1	yes	characteristic
51	C₃×C₅×Q₈	—	4	1	no	—
52	C₃×C₅×Q₈	—	4	1	no	—
53	C₂×C₃×C₄×C₅	—	2	1	no	—
54	C₂×C₃×C₄×C₅	—	2	1	no	—
55	C₂×C₃×C₄×C₅	—	4	1	no	—
56	C₂×C₃×C₄×C₅	—	4	1	no	—
57	C₂×C₃×C₄×C₅	—	4	2	no	—
58	C₂×C₃×C₄×C₅	—	8	2	no	—
59	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
60	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
61	C₃×C₅×C₈	—	2	1	no	—
62	C₃×C₅×C₈	#41	1	2	yes	—
63	C₂×C₃×Q₁₆	C₂²×C₅	1	1	yes	characteristic
64	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
65	C₃×C₂²⋊Q₈	—	2	1	no	—
66	C₂×C₃×C₄⋊C₄	—	2	1	no	—
67	C₂×C₃×C₁₆	C₂²×C₅	1	2	yes	—
68	C₃×#107	—	2	1	no	—
69	C₃×#107	C₂²×C₅	1	1	yes	characteristic
70	C₃×#107	C₂²×C₅	1	2	yes	—
71	C₃×#103	—	2	1	no	—
72	C₂×C₅×Q₈	—	2	1	no	—
73	C₂²×C₄×C₅	—	4	1	no	—
74	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
75	C₅×Q₁₆	—	2	2	no	—
76	C₂×C₅×C₈	—	2	2	no	—
77	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
78	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
79	C₅×C₄⋊C₄	—	2	2	no	—
80	C₅×C₄⋊C₄	—	2	1	no	—
81	C₅×C₄⋊C₄	—	4	1	no	—
82	C₅×C₄⋊C₄	—	4	1	no	—
83	C₅×C₂²⋊C₄	—	4	1	no	—
84	C₅×C₁₆	—	2	2	no	—
85	#467	C₂×C₃×C₅	1	1	yes	characteristic
86	C₂×#107	C₂×C₃×C₅	1	1	yes	characteristic
87	#366	C₂×C₃×C₅	1	2	yes	complemented
88	#358	C₂×C₃×C₅	1	2	yes	complemented
89	C₂²⋊C₁₆	C₂×C₃×C₅	1	1	yes	characteristic
90	C₂²×C₃×C₅	—	2	1	no	—
91	C₂²×C₃×C₅	—	2	1	no	—
92	C₂²×C₃×C₅	Q₃₂	1	2	yes	—
93	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
94	C₃×C₄×C₅	—	2	1	no	—
95	C₃×C₄×C₅	—	8	1	no	—
96	C₃×C₄×C₅	—	8	1	no	—
97	C₃×C₄×C₅	—	8	2	no	—
98	C₃×C₄×C₅	#123	1	1	yes	characteristic
99	C₃×C₄×C₅	#137	1	1	yes	characteristic
100	C₂×C₃×Q₈	—	2	1	no	—
101	C₂²×C₃×C₄	—	4	1	no	—
102	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
103	C₃×Q₁₆	—	2	2	no	—
104	C₂×C₃×C₈	—	2	2	no	—
105	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
106	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
107	C₃×C₄⋊C₄	—	2	1	no	—
108	C₃×C₄⋊C₄	—	2	2	no	—
109	C₃×C₄⋊C₄	—	4	1	no	—
110	C₃×C₄⋊C₄	—	4	1	no	—
111	C₃×C₂²⋊C₄	—	4	1	no	—
112	C₃×C₁₆	—	2	2	no	—
113	C₂³×C₅	C₃×D₈	1	1	yes	characteristic
114	C₅×Q₈	—	4	1	no	—
115	C₅×Q₈	—	4	1	no	—
116	C₂×C₄×C₅	—	2	1	no	—
117	C₂×C₄×C₅	—	2	1	no	—
118	C₂×C₄×C₅	—	4	1	no	—
119	C₂×C₄×C₅	—	4	1	no	—
120	C₂×C₄×C₅	—	4	2	no	—
121	C₂×C₄×C₅	—	8	2	no	—
122	C₂×C₄×C₅	C₃×D₈	1	1	yes	characteristic
123	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
124	C₅×C₈	—	2	1	no	—
125	C₅×C₈	C₃×#41	1	2	yes	—
126	C₂×Q₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
127	C₂²×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
128	C₂²⋊Q₈	—	2	1	no	—
129	C₂×C₄⋊C₄	—	2	1	no	—
130	C₂×C₁₆	C₂²×C₃×C₅	1	2	yes	—
131	#107	—	2	1	no	—
132	#107	C₂²×C₃×C₅	1	2	yes	—
133	#107	C₂²×C₃×C₅	1	1	yes	characteristic
134	#103	—	2	1	no	—
135	C₂×C₃×C₅	—	2	2	no	—
136	C₂×C₃×C₅	#480	1	1	yes	characteristic
137	C₂×C₃×C₅	C₂×Q₃₂	1	1	yes	characteristic
138	C₂×C₃×C₅	#509	1	1	yes	characteristic
139	C₂³×C₃	C₅×D₈	1	1	yes	characteristic
140	C₃×Q₈	—	4	1	no	—
141	C₃×Q₈	—	4	1	no	—
142	C₂×C₃×C₄	—	2	1	no	—
143	C₂×C₃×C₄	—	2	1	no	—
144	C₂×C₃×C₄	—	4	1	no	—
145	C₂×C₃×C₄	—	4	1	no	—
146	C₂×C₃×C₄	—	4	2	no	—
147	C₂×C₃×C₄	—	8	2	no	—
148	C₂×C₃×C₄	C₅×D₈	1	1	yes	characteristic
149	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
150	C₃×C₈	—	2	1	no	—
151	C₃×C₈	C₅×#41	1	2	yes	—
152	C₂²×C₅	—	2	1	no	—
153	C₂²×C₅	—	2	1	no	—
154	C₂²×C₅	C₃×Q₃₂	1	2	yes	—
155	C₂²×C₅	C₂×C₃×D₈	1	1	yes	characteristic
156	C₄×C₅	—	2	1	no	—
157	C₄×C₅	—	8	1	no	—
158	C₄×C₅	—	8	2	no	—
159	C₄×C₅	—	8	1	no	—
160	C₄×C₅	C₃×#123	1	1	yes	characteristic
161	C₄×C₅	C₃×#137	1	1	yes	characteristic
162	C₂×Q₈	—	2	1	no	—
163	C₂²×C₄	—	4	1	no	—
164	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
165	Q₁₆	—	2	2	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
168	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
169	C₄⋊C₄	—	2	1	no	—
170	C₄⋊C₄	—	2	2	no	—
171	C₄⋊C₄	—	4	1	no	—
172	C₄⋊C₄	—	4	1	no	—
173	C₂²⋊C₄	—	4	1	no	—
174	C₁₆	—	2	2	no	—
175	C₃×C₅	#2236	1	1	yes	characteristic, direct product factor, Hall
176	C₂²×C₃	—	2	1	no	—
177	C₂²×C₃	—	2	1	no	—
178	C₂²×C₃	C₅×Q₃₂	1	2	yes	—
179	C₂²×C₃	C₂×C₅×D₈	1	1	yes	characteristic
180	C₃×C₄	—	2	1	no	—
181	C₃×C₄	—	8	1	no	—
182	C₃×C₄	—	8	1	no	—
183	C₃×C₄	—	8	2	no	—
184	C₃×C₄	C₅×#123	1	1	yes	characteristic
185	C₃×C₄	C₅×#137	1	1	yes	characteristic
186	C₂×C₅	—	2	2	no	—
187	C₂×C₅	C₃×#480	1	1	yes	characteristic
188	C₂×C₅	C₂×C₃×Q₃₂	1	1	yes	characteristic
189	C₂×C₅	C₃×#509	1	1	yes	characteristic
190	C₂³	C₃×C₅×D₈	1	1	yes	characteristic
191	Q₈	—	4	1	no	—
192	Q₈	—	4	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	4	1	no	—
196	C₂×C₄	—	4	1	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	—	8	2	no	—
199	C₂×C₄	C₃×C₅×D₈	1	1	yes	characteristic
200	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
201	C₈	—	2	1	no	—
202	C₈	C₃×C₅×#41	1	2	yes	—
203	C₂×C₃	—	2	2	no	—
204	C₂×C₃	C₅×#480	1	1	yes	characteristic
205	C₂×C₃	C₂×C₅×Q₃₂	1	1	yes	characteristic
206	C₂×C₃	C₅×#509	1	1	yes	characteristic
207	C₅	C₃×#2236	1	1	yes	characteristic, direct product factor, Sylow
208	C₂²	—	2	1	no	—
209	C₂²	—	2	1	no	—
210	C₂²	C₃×C₅×Q₃₂	1	2	yes	—
211	C₂²	C₂×C₃×C₅×D₈	1	1	yes	characteristic
212	C₄	—	2	1	no	—
213	C₄	—	8	1	no	—
214	C₄	—	8	2	no	—
215	C₄	—	8	1	no	—
216	C₄	C₃×C₅×#123	1	1	yes	characteristic
217	C₄	C₃×C₅×#137	1	1	yes	characteristic
218	C₃	C₅×#2236	1	1	yes	characteristic, direct product factor, Sylow
219	C₂	—	2	2	no	—
220	C₂	C₃×C₅×#480	1	1	yes	characteristic
221	C₂	C₂×C₃×C₅×Q₃₂	1	1	yes	characteristic
222	C₂	C₃×C₅×#509	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃×C₅	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₃×C₅
prime radical	C₂³×C₃×C₅	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₂²×C₃×C₄×C₅ ◁ C₂²×C₃×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃×C₅×#366	C₂	semidirect
4. C₃×C₅×#358	C₂	semidirect
6. C₅×#2236	C₃	direct (N × H)
16. C₃×#2236	C₅	direct (N × H)
19. C₅×#366	C₂×C₃	semidirect
20. C₅×#358	C₂×C₃	semidirect
37. C₃×#366	C₂×C₅	semidirect
38. C₃×#358	C₂×C₅	semidirect
49. #2236	C₃×C₅	direct (N × H)
87. #366	C₂×C₃×C₅	semidirect
88. #358	C₂×C₃×C₅	semidirect
175. C₃×C₅	#2236	direct (N × H)
207. C₅	C₃×#2236	direct (N × H)
218. C₃	C₅×#2236	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×#467	2
C₂×C₃×C₅×#107	2
C₃×C₅×#366	2
C₃×C₅×#358	2
C₃×C₅×C₂²⋊C₁₆	2
C₅×#2236	3
C₃×#2236	5

Subgroups — C3×C5×#2236 #14275270

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2236 #14275270