Subgroups — C₃×C₅×#2227 #14275261

Order 1920.

260 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#500	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×C₅×Q₁₆	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#366	C₂	1	2	yes	maximal
4	C₃×C₅×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₃×C₅×#358	C₂	1	2	yes	maximal
6	C₅×#2227	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×Q₁₆	C₂²	1	1	yes	characteristic
8	C₃×C₅×C₄⋊Q₈	—	2	1	no	—
9	C₃×C₄×C₅×Q₈	—	2	1	no	—
10	C₂×C₃×C₅×C₁₆	C₂²	1	2	yes	—
11	C₃×C₅×#107	—	2	1	no	—
12	C₃×C₅×#107	C₂²	1	2	yes	—
13	C₃×C₅×#107	C₂²	1	1	yes	characteristic
14	C₃×C₅×#103	—	2	1	no	—
15	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
16	C₃×#2227	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₅×#500	C₂×C₃	1	1	yes	characteristic
18	C₄×C₅×Q₁₆	C₂×C₃	1	1	yes	characteristic
19	C₅×#366	C₂×C₃	1	2	yes	—
20	C₅×C₄⋊C₁₆	C₂×C₃	1	1	yes	characteristic
21	C₅×#358	C₂×C₃	1	2	yes	—
22	C₂×C₃×C₅×Q₈	—	2	1	no	—
23	C₂×C₃×C₅×Q₈	—	4	1	no	—
24	C₃×C₅×Q₁₆	—	2	1	no	—
25	C₃×C₅×Q₁₆	Q₈	1	2	yes	complemented
26	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
27	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
28	C₃×C₅×C₄⋊C₄	—	2	1	no	—
29	C₃×C₅×C₄⋊C₄	—	2	2	no	—
30	C₃×C₅×C₄⋊C₄	—	4	1	no	—
31	C₃×C₅×C₄⋊C₄	—	4	1	no	—
32	C₃×C₄²×C₅	—	4	1	no	—
33	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
34	C₃×C₅×C₁₆	—	2	2	no	—
35	C₃×#500	C₂×C₅	1	1	yes	characteristic
36	C₃×C₄×Q₁₆	C₂×C₅	1	1	yes	characteristic
37	C₃×#366	C₂×C₅	1	2	yes	—
38	C₃×C₄⋊C₁₆	C₂×C₅	1	1	yes	characteristic
39	C₃×#358	C₂×C₅	1	2	yes	—
40	C₂×C₅×Q₁₆	C₂²×C₃	1	1	yes	characteristic
41	C₅×C₄⋊Q₈	—	2	1	no	—
42	C₄×C₅×Q₈	—	2	1	no	—
43	C₂×C₅×C₁₆	C₂²×C₃	1	2	yes	—
44	C₅×#107	—	2	1	no	—
45	C₅×#107	C₂²×C₃	1	2	yes	—
46	C₅×#107	C₂²×C₃	1	1	yes	characteristic
47	C₅×#103	—	2	1	no	—
48	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
49	#2227	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
50	C₃×C₅×Q₈	—	2	2	no	—
51	C₃×C₅×Q₈	—	4	1	no	—
52	C₃×C₅×Q₈	—	8	2	no	—
53	C₂×C₃×C₄×C₅	—	4	1	no	—
54	C₂×C₃×C₄×C₅	—	4	1	no	—
55	C₂×C₃×C₄×C₅	—	4	2	no	—
56	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
57	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
58	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
59	C₃×C₅×C₈	—	2	1	no	—
60	C₃×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
61	C₃×C₅×C₈	#41	1	1	yes	characteristic
62	C₂×C₃×Q₁₆	C₂²×C₅	1	1	yes	characteristic
63	C₃×C₄⋊Q₈	—	2	1	no	—
64	C₃×C₄×Q₈	—	2	1	no	—
65	C₂×C₃×C₁₆	C₂²×C₅	1	2	yes	—
66	C₃×#107	—	2	1	no	—
67	C₃×#107	C₂²×C₅	1	1	yes	characteristic
68	C₃×#107	C₂²×C₅	1	2	yes	—
69	C₃×#103	—	2	1	no	—
70	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
71	C₂×C₅×Q₈	—	2	1	no	—
72	C₂×C₅×Q₈	—	4	1	no	—
73	C₅×Q₁₆	—	2	1	no	—
74	C₅×Q₁₆	C₃×Q₈	1	2	yes	complemented
75	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
76	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
77	C₅×C₄⋊C₄	—	2	1	no	—
78	C₅×C₄⋊C₄	—	2	2	no	—
79	C₅×C₄⋊C₄	—	4	1	no	—
80	C₅×C₄⋊C₄	—	4	1	no	—
81	C₄²×C₅	—	4	1	no	—
82	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
83	C₅×C₁₆	—	2	2	no	—
84	#500	C₂×C₃×C₅	1	1	yes	characteristic
85	C₄×Q₁₆	C₂×C₃×C₅	1	1	yes	characteristic
86	#366	C₂×C₃×C₅	1	2	yes	—
87	C₄⋊C₁₆	C₂×C₃×C₅	1	1	yes	characteristic
88	#358	C₂×C₃×C₅	1	2	yes	—
89	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
90	C₃×C₄×C₅	—	2	1	no	—
91	C₃×C₄×C₅	—	4	2	no	—
92	C₃×C₄×C₅	—	8	1	no	—
93	C₃×C₄×C₅	—	8	2	no	—
94	C₃×C₄×C₅	Q₃₂	1	2	yes	—
95	C₃×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
96	C₃×C₄×C₅	#137	1	1	yes	characteristic
97	C₂×C₃×Q₈	—	2	1	no	—
98	C₂×C₃×Q₈	—	4	1	no	—
99	C₃×Q₁₆	—	2	1	no	—
100	C₃×Q₁₆	C₅×Q₈	1	2	yes	complemented
101	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
102	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
103	C₃×C₄⋊C₄	—	2	1	no	—
104	C₃×C₄⋊C₄	—	2	2	no	—
105	C₃×C₄⋊C₄	—	4	1	no	—
106	C₃×C₄⋊C₄	—	4	1	no	—
107	C₃×C₄²	—	4	1	no	—
108	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
109	C₃×C₁₆	—	2	2	no	—
110	C₅×Q₈	—	2	2	no	—
111	C₅×Q₈	—	4	1	no	—
112	C₅×Q₈	—	8	2	no	—
113	C₂×C₄×C₅	—	4	1	no	—
114	C₂×C₄×C₅	—	4	1	no	—
115	C₂×C₄×C₅	—	4	2	no	—
116	C₂×C₄×C₅	C₃×D₈	1	1	yes	characteristic
117	C₂×C₄×C₅	C₃×D₈	1	1	yes	characteristic
118	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
119	C₅×C₈	—	2	1	no	—
120	C₅×C₈	C₂×C₃×Q₈	1	1	yes	characteristic
121	C₅×C₈	C₃×#41	1	1	yes	characteristic
122	C₂×Q₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
123	C₄⋊Q₈	—	2	1	no	—
124	C₄×Q₈	—	2	1	no	—
125	C₂×C₁₆	C₂²×C₃×C₅	1	2	yes	—
126	#107	—	2	1	no	—
127	#107	C₂²×C₃×C₅	1	2	yes	—
128	#107	C₂²×C₃×C₅	1	1	yes	characteristic
129	#103	—	2	1	no	—
130	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
131	C₂×C₃×C₅	#474	1	1	yes	characteristic
132	C₂×C₃×C₅	C₂×Q₃₂	1	1	yes	characteristic
133	C₂×C₃×C₅	#509	1	1	yes	characteristic
134	C₃×Q₈	—	2	2	no	—
135	C₃×Q₈	—	4	1	no	—
136	C₃×Q₈	—	8	2	no	—
137	C₂×C₃×C₄	—	4	1	no	—
138	C₂×C₃×C₄	—	4	1	no	—
139	C₂×C₃×C₄	—	4	2	no	—
140	C₂×C₃×C₄	C₅×D₈	1	1	yes	characteristic
141	C₂×C₃×C₄	C₅×D₈	1	1	yes	characteristic
142	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
143	C₃×C₈	—	2	1	no	—
144	C₃×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
145	C₃×C₈	C₅×#41	1	1	yes	characteristic
146	C₂²×C₅	C₂×C₃×D₈	1	1	yes	characteristic
147	C₄×C₅	—	2	1	no	—
148	C₄×C₅	—	4	2	no	—
149	C₄×C₅	—	8	1	no	—
150	C₄×C₅	—	8	2	no	—
151	C₄×C₅	C₃×Q₃₂	1	2	yes	—
152	C₄×C₅	C₃×C₂²⋊Q₈	1	1	yes	characteristic
153	C₄×C₅	C₃×#137	1	1	yes	characteristic
154	C₂×Q₈	—	2	1	no	—
155	C₂×Q₈	—	4	1	no	—
156	Q₁₆	—	2	1	no	—
157	Q₁₆	C₃×C₅×Q₈	1	2	yes	complemented
158	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
159	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
160	C₄⋊C₄	—	2	1	no	—
161	C₄⋊C₄	—	2	2	no	—
162	C₄⋊C₄	—	4	1	no	—
163	C₄⋊C₄	—	4	1	no	—
164	C₄²	—	4	1	no	—
165	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
166	C₁₆	—	2	2	no	—
167	C₃×C₅	#2227	1	1	yes	characteristic, direct product factor, Hall
168	C₂²×C₃	C₂×C₅×D₈	1	1	yes	characteristic
169	C₃×C₄	—	2	1	no	—
170	C₃×C₄	—	4	2	no	—
171	C₃×C₄	—	8	1	no	—
172	C₃×C₄	—	8	2	no	—
173	C₃×C₄	C₅×Q₃₂	1	2	yes	—
174	C₃×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
175	C₃×C₄	C₅×#137	1	1	yes	characteristic
176	C₂×C₅	C₃×#474	1	1	yes	characteristic
177	C₂×C₅	C₂×C₃×Q₃₂	1	1	yes	characteristic
178	C₂×C₅	C₃×#509	1	1	yes	characteristic
179	Q₈	—	2	2	no	—
180	Q₈	—	4	1	no	—
181	Q₈	—	8	2	no	—
182	C₂×C₄	—	4	1	no	—
183	C₂×C₄	—	4	1	no	—
184	C₂×C₄	—	4	2	no	—
185	C₂×C₄	C₃×C₅×D₈	1	1	yes	characteristic
186	C₂×C₄	C₃×C₅×D₈	1	1	yes	characteristic
187	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
188	C₈	—	2	1	no	—
189	C₈	C₂×C₃×C₅×Q₈	1	1	yes	characteristic
190	C₈	C₃×C₅×#41	1	1	yes	characteristic
191	C₂×C₃	C₅×#474	1	1	yes	characteristic
192	C₂×C₃	C₂×C₅×Q₃₂	1	1	yes	characteristic
193	C₂×C₃	C₅×#509	1	1	yes	characteristic
194	C₅	C₃×#2227	1	1	yes	characteristic, direct product factor, Sylow
195	C₂²	C₂×C₃×C₅×D₈	1	1	yes	characteristic
196	C₄	—	2	1	no	—
197	C₄	—	4	2	no	—
198	C₄	—	8	2	no	—
199	C₄	—	8	1	no	—
200	C₄	C₃×C₅×Q₃₂	1	2	yes	—
201	C₄	C₃×C₅×C₂²⋊Q₈	1	1	yes	characteristic
202	C₄	C₃×C₅×#137	1	1	yes	characteristic
203	C₃	C₅×#2227	1	1	yes	characteristic, direct product factor, Sylow
204	C₂	C₃×C₅×#474	1	1	yes	characteristic
205	C₂	C₂×C₃×C₅×Q₃₂	1	1	yes	characteristic
206	C₂	C₃×C₅×#509	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₃×C₅	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₃×C₅
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₃×C₄²×C₅ ◁ C₃×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
6. C₅×#2227	C₃	direct (N × H)
16. C₃×#2227	C₅	direct (N × H)
25. C₃×C₅×Q₁₆	Q₈	semidirect
49. #2227	C₃×C₅	direct (N × H)
74. C₅×Q₁₆	C₃×Q₈	semidirect
100. C₃×Q₁₆	C₅×Q₈	semidirect
157. Q₁₆	C₃×C₅×Q₈	semidirect
167. C₃×C₅	#2227	direct (N × H)
194. C₅	C₃×#2227	direct (N × H)
203. C₃	C₅×#2227	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×#500	2
C₃×C₄×C₅×Q₁₆	2
C₃×C₅×#366	2
C₃×C₅×C₄⋊C₁₆	2
C₃×C₅×#358	2
C₅×#2227	3
C₃×#2227	5

Subgroups — C3×C5×#2227 #14275261

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2227 #14275261