Subgroups — C₃×C₅×#2222 #14275256

Order 1920.

280 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₅×#504	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#473	C₂	1	2	yes	maximal
4	C₃×C₅×#368	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₅×#362	C₂	1	2	yes	maximal, complemented
6	C₅×#2222	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₃×C₅×#137	—	2	2	no	—
8	C₂×C₃×C₅×Q₁₆	C₂²	1	2	yes	—
9	C₃×C₅×#131	C₂²	1	1	yes	characteristic
10	C₃×C₅×Q₃₂	—	2	1	no	—
11	C₃×C₅×Q₃₂	—	2	1	no	—
12	C₃×C₅×M₃₂	—	2	1	no	—
13	C₃×C₅×M₃₂	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₅×C₁₆	—	2	1	no	—
15	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic
16	C₃×C₅×#108	C₂²	1	2	yes	—
17	C₃×C₅×#101	—	2	2	no	—
18	C₃×#2222	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₂×C₅×Q₃₂	C₂×C₃	1	1	yes	characteristic, complemented
20	C₅×#504	C₂×C₃	1	1	yes	characteristic
21	C₅×#473	C₂×C₃	1	2	yes	—
22	C₅×#368	C₂×C₃	1	1	yes	characteristic, complemented
23	C₅×#362	C₂×C₃	1	2	yes	complemented
24	C₃×C₅×#41	D₄	1	1	yes	characteristic
25	C₂×C₃×C₅×Q₈	—	2	2	no	—
26	C₃×C₅×Q₁₆	—	2	2	no	—
27	C₃×C₅×Q₁₆	—	2	2	no	—
28	C₃×C₅×Q₁₆	—	4	2	no	—
29	C₃×C₅×SD₁₆	—	4	2	no	—
30	C₃×C₅×M₁₆	—	2	1	no	—
31	C₃×C₅×M₁₆	—	2	2	no	—
32	C₃×C₅×M₁₆	D₄	1	1	yes	characteristic
33	C₂×C₃×C₅×C₈	—	2	1	no	—
34	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
35	C₃×C₅×C₁₆	—	2	1	no	—
36	C₃×C₅×C₁₆	D₄	1	1	yes	characteristic
37	C₃×C₅×C₁₆	D₄	1	1	yes	characteristic
38	C₂×C₃×Q₃₂	C₂×C₅	1	1	yes	characteristic, complemented
39	C₃×#504	C₂×C₅	1	1	yes	characteristic
40	C₃×#473	C₂×C₅	1	2	yes	—
41	C₃×#368	C₂×C₅	1	1	yes	characteristic, complemented
42	C₃×#362	C₂×C₅	1	2	yes	complemented
43	C₅×#137	—	2	2	no	—
44	C₂×C₅×Q₁₆	C₂²×C₃	1	2	yes	—
45	C₅×#131	C₂²×C₃	1	1	yes	characteristic
46	C₅×Q₃₂	—	2	1	no	—
47	C₅×Q₃₂	—	2	1	no	—
48	C₅×M₃₂	—	2	1	no	—
49	C₅×M₃₂	C₂²×C₃	1	1	yes	characteristic
50	C₂×C₅×C₁₆	—	2	1	no	—
51	C₂×C₅×C₁₆	C₂²×C₃	1	1	yes	characteristic
52	C₅×#108	C₂²×C₃	1	2	yes	—
53	C₅×#101	—	2	2	no	—
54	#2222	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
55	C₃×C₅×Q₈	—	4	2	no	—
56	C₃×C₅×Q₈	—	4	2	no	—
57	C₃×C₅×Q₈	D₈	1	1	yes	characteristic
58	C₃×C₅×D₄	—	2	1	no	—
59	C₃×C₅×D₄	D₈	1	1	yes	characteristic
60	C₂×C₃×C₄×C₅	—	2	1	no	—
61	C₂×C₃×C₄×C₅	—	4	2	no	—
62	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
63	C₃×C₅×C₈	—	2	1	no	—
64	C₃×C₅×C₈	—	4	2	no	—
65	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
66	C₃×C₅×C₈	#41	1	1	yes	characteristic
67	C₃×#137	—	2	2	no	—
68	C₂×C₃×Q₁₆	C₂²×C₅	1	2	yes	—
69	C₃×#131	C₂²×C₅	1	1	yes	characteristic
70	C₃×Q₃₂	—	2	1	no	—
71	C₃×Q₃₂	—	2	1	no	—
72	C₃×M₃₂	—	2	1	no	—
73	C₃×M₃₂	C₂²×C₅	1	1	yes	characteristic
74	C₂×C₃×C₁₆	—	2	1	no	—
75	C₂×C₃×C₁₆	C₂²×C₅	1	1	yes	characteristic
76	C₃×#108	C₂²×C₅	1	2	yes	—
77	C₃×#101	—	2	2	no	—
78	C₅×#41	C₃×D₄	1	1	yes	characteristic
79	C₂×C₅×Q₈	—	2	2	no	—
80	C₅×Q₁₆	—	2	2	no	—
81	C₅×Q₁₆	—	2	2	no	—
82	C₅×Q₁₆	—	4	2	no	—
83	C₅×SD₁₆	—	4	2	no	—
84	C₅×M₁₆	—	2	1	no	—
85	C₅×M₁₆	—	2	2	no	—
86	C₅×M₁₆	C₃×D₄	1	1	yes	characteristic
87	C₂×C₅×C₈	—	2	1	no	—
88	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
89	C₅×C₁₆	—	2	1	no	—
90	C₅×C₁₆	C₃×D₄	1	1	yes	characteristic
91	C₅×C₁₆	C₃×D₄	1	1	yes	characteristic
92	C₂×Q₃₂	C₂×C₃×C₅	1	1	yes	characteristic, complemented
93	#504	C₂×C₃×C₅	1	1	yes	characteristic
94	#473	C₂×C₃×C₅	1	2	yes	—
95	#368	C₂×C₃×C₅	1	1	yes	characteristic, complemented
96	#362	C₂×C₃×C₅	1	2	yes	complemented
97	C₂²×C₃×C₅	—	2	1	no	—
98	C₂²×C₃×C₅	#135	1	1	yes	characteristic
99	C₃×C₄×C₅	—	2	1	no	—
100	C₃×C₄×C₅	—	8	2	no	—
101	C₃×C₄×C₅	#121	1	1	yes	characteristic
102	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
103	C₃×#41	C₅×D₄	1	1	yes	characteristic
104	C₂×C₃×Q₈	—	2	2	no	—
105	C₃×Q₁₆	—	2	2	no	—
106	C₃×Q₁₆	—	2	2	no	—
107	C₃×Q₁₆	—	4	2	no	—
108	C₃×SD₁₆	—	4	2	no	—
109	C₃×M₁₆	—	2	1	no	—
110	C₃×M₁₆	—	2	2	no	—
111	C₃×M₁₆	C₅×D₄	1	1	yes	characteristic
112	C₂×C₃×C₈	—	2	1	no	—
113	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
114	C₃×C₁₆	—	2	1	no	—
115	C₃×C₁₆	C₅×D₄	1	1	yes	characteristic
116	C₃×C₁₆	C₅×D₄	1	1	yes	characteristic
117	C₅×Q₈	—	4	2	no	—
118	C₅×Q₈	—	4	2	no	—
119	C₅×Q₈	C₃×D₈	1	1	yes	characteristic
120	C₅×D₄	—	2	1	no	—
121	C₅×D₄	C₃×D₈	1	1	yes	characteristic
122	C₂×C₄×C₅	—	2	1	no	—
123	C₂×C₄×C₅	—	4	2	no	—
124	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
125	C₅×C₈	—	2	1	no	—
126	C₅×C₈	—	4	2	no	—
127	C₅×C₈	C₂×C₃×D₄	1	1	yes	characteristic
128	C₅×C₈	C₃×#41	1	1	yes	characteristic
129	#137	—	2	2	no	—
130	C₂×Q₁₆	C₂²×C₃×C₅	1	2	yes	—
131	#131	C₂²×C₃×C₅	1	1	yes	characteristic
132	Q₃₂	—	2	1	no	—
133	Q₃₂	—	2	1	no	—
134	M₃₂	—	2	1	no	—
135	M₃₂	C₂²×C₃×C₅	1	1	yes	characteristic
136	C₂×C₁₆	—	2	1	no	—
137	C₂×C₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
138	#108	C₂²×C₃×C₅	1	2	yes	—
139	#101	—	2	2	no	—
140	C₂×C₃×C₅	—	2	1	no	—
141	C₂×C₃×C₅	—	4	1	no	—
142	C₂×C₃×C₅	#466	1	1	yes	characteristic
143	C₃×Q₈	—	4	2	no	—
144	C₃×Q₈	—	4	2	no	—
145	C₃×Q₈	C₅×D₈	1	1	yes	characteristic
146	C₃×D₄	—	2	1	no	—
147	C₃×D₄	C₅×D₈	1	1	yes	characteristic
148	C₂×C₃×C₄	—	2	1	no	—
149	C₂×C₃×C₄	—	4	2	no	—
150	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
151	C₃×C₈	—	2	1	no	—
152	C₃×C₈	—	4	2	no	—
153	C₃×C₈	C₂×C₅×D₄	1	1	yes	characteristic
154	C₃×C₈	C₅×#41	1	1	yes	characteristic
155	C₂²×C₅	—	2	1	no	—
156	C₂²×C₅	C₃×#135	1	1	yes	characteristic
157	C₄×C₅	—	2	1	no	—
158	C₄×C₅	—	8	2	no	—
159	C₄×C₅	C₃×#121	1	1	yes	characteristic
160	C₄×C₅	C₂×C₃×D₈	1	1	yes	characteristic
161	#41	C₃×C₅×D₄	1	1	yes	characteristic
162	C₂×Q₈	—	2	2	no	—
163	Q₁₆	—	2	2	no	—
164	Q₁₆	—	2	2	no	—
165	Q₁₆	—	4	2	no	—
166	SD₁₆	—	4	2	no	—
167	M₁₆	—	2	1	no	—
168	M₁₆	—	2	2	no	—
169	M₁₆	C₃×C₅×D₄	1	1	yes	characteristic
170	C₂×C₈	—	2	1	no	—
171	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
172	C₁₆	—	2	1	no	—
173	C₁₆	C₃×C₅×D₄	1	1	yes	characteristic
174	C₁₆	C₃×C₅×D₄	1	1	yes	characteristic
175	C₃×C₅	#2222	1	1	yes	characteristic, direct product factor, Hall
176	C₂²×C₃	—	2	1	no	—
177	C₂²×C₃	C₅×#135	1	1	yes	characteristic
178	C₃×C₄	—	2	1	no	—
179	C₃×C₄	—	8	2	no	—
180	C₃×C₄	C₅×#121	1	1	yes	characteristic
181	C₃×C₄	C₂×C₅×D₈	1	1	yes	characteristic
182	C₂×C₅	—	2	1	no	—
183	C₂×C₅	—	4	1	no	—
184	C₂×C₅	C₃×#466	1	1	yes	characteristic
185	Q₈	—	4	2	no	—
186	Q₈	—	4	2	no	—
187	Q₈	C₃×C₅×D₈	1	1	yes	characteristic
188	D₄	—	2	1	no	—
189	D₄	C₃×C₅×D₈	1	1	yes	characteristic
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
193	C₈	—	2	1	no	—
194	C₈	—	4	2	no	—
195	C₈	C₂×C₃×C₅×D₄	1	1	yes	characteristic
196	C₈	C₃×C₅×#41	1	1	yes	characteristic
197	C₂×C₃	—	2	1	no	—
198	C₂×C₃	—	4	1	no	—
199	C₂×C₃	C₅×#466	1	1	yes	characteristic
200	C₅	C₃×#2222	1	1	yes	characteristic, direct product factor, Sylow
201	C₂²	—	2	1	no	—
202	C₂²	C₃×C₅×#135	1	1	yes	characteristic
203	C₄	—	2	1	no	—
204	C₄	—	8	2	no	—
205	C₄	C₃×C₅×#121	1	1	yes	characteristic
206	C₄	C₂×C₃×C₅×D₈	1	1	yes	characteristic
207	C₃	C₅×#2222	1	1	yes	characteristic, direct product factor, Sylow
208	C₂	—	2	1	no	—
209	C₂	—	4	1	no	—
210	C₂	C₃×C₅×#466	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃×C₅	#466
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₃×C₅
prime radical	C₃×C₅×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₅ ◁ C₂×C₃×C₄×C₅ ◁ C₃×C₅×#131 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅×Q₃₂	C₂	semidirect
4. C₃×C₅×#368	C₂	semidirect
5. C₃×C₅×#362	C₂	semidirect
6. C₅×#2222	C₃	direct (N × H)
18. C₃×#2222	C₅	direct (N × H)
19. C₂×C₅×Q₃₂	C₂×C₃	semidirect
22. C₅×#368	C₂×C₃	semidirect
23. C₅×#362	C₂×C₃	semidirect
38. C₂×C₃×Q₃₂	C₂×C₅	semidirect
41. C₃×#368	C₂×C₅	semidirect
42. C₃×#362	C₂×C₅	semidirect
54. #2222	C₃×C₅	direct (N × H)
92. C₂×Q₃₂	C₂×C₃×C₅	semidirect
95. #368	C₂×C₃×C₅	semidirect
96. #362	C₂×C₃×C₅	semidirect
175. C₃×C₅	#2222	direct (N × H)
200. C₅	C₃×#2222	direct (N × H)
207. C₃	C₅×#2222	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×Q₃₂	2
C₃×C₅×#504	2
C₃×C₅×#473	2
C₃×C₅×#368	2
C₃×C₅×#362	2
C₅×#2222	3
C₃×#2222	5

Subgroups — C3×C5×#2222 #14275256

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2222 #14275256