Subgroups — C₃×C₅×#2151 #14275185

Order 1920.

320 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#405	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×C₄⋊C₁₆	C₂	1	4	yes	maximal, complemented
5	C₅×#2151	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂×C₃×C₅×M₁₆	C₄	1	1	yes	characteristic
7	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₃×C₅×#117	C₄	1	1	yes	characteristic
9	C₃×C₅×M₃₂	—	2	2	no	—
10	C₂×C₃×C₅×C₁₆	—	2	2	no	—
11	C₂×C₃×C₅×C₁₆	C₂²	1	2	yes	—
12	C₂×C₃×C₅×C₁₆	C₂²	1	2	yes	—
13	C₃×C₅×#97	C₄	1	2	yes	—
14	C₃×C₄×C₅×C₈	C₂²	1	2	yes	—
15	C₃×#2151	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₂×C₅×M₃₂	C₂×C₃	1	1	yes	characteristic
17	C₂²×C₅×C₁₆	C₂×C₃	1	1	yes	characteristic
18	C₅×#405	C₂×C₃	1	1	yes	characteristic
19	C₅×C₄⋊C₁₆	C₂×C₃	1	4	yes	complemented
20	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
21	C₃×C₅×M₁₆	—	2	2	no	—
22	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
23	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
25	C₂×C₃×C₅×C₈	Q₈	1	2	yes	—
26	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
27	C₃×C₅×C₄⋊C₄	C₈	1	2	yes	—
28	C₃×C₅×C₂²⋊C₄	C₈	1	2	yes	—
29	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
30	C₃×C₅×C₁₆	—	2	2	no	—
31	C₃×C₅×C₁₆	—	2	2	no	—
32	C₂×C₃×M₃₂	C₂×C₅	1	1	yes	characteristic
33	C₂²×C₃×C₁₆	C₂×C₅	1	1	yes	characteristic
34	C₃×#405	C₂×C₅	1	1	yes	characteristic
35	C₃×C₄⋊C₁₆	C₂×C₅	1	4	yes	complemented
36	C₂×C₅×M₁₆	C₃×C₄	1	1	yes	characteristic
37	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
38	C₅×#117	C₃×C₄	1	1	yes	characteristic
39	C₅×M₃₂	—	2	2	no	—
40	C₂×C₅×C₁₆	—	2	2	no	—
41	C₂×C₅×C₁₆	C₂²×C₃	1	2	yes	—
42	C₂×C₅×C₁₆	C₂²×C₃	1	2	yes	—
43	C₅×#97	C₃×C₄	1	2	yes	—
44	C₄×C₅×C₈	C₂²×C₃	1	2	yes	—
45	#2151	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
46	C₂³×C₃×C₅	C₂×C₈	1	1	yes	characteristic
47	C₂×C₃×C₄×C₅	C₄⋊C₄	1	2	yes	—
48	C₂×C₃×C₄×C₅	C₂×C₈	1	4	yes	—
49	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
50	C₂×C₃×C₄×C₅	M₁₆	1	2	yes	—
51	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
52	C₃×C₅×C₈	—	2	2	no	—
53	C₃×C₅×C₈	C₄⋊C₄	1	2	yes	—
54	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
55	C₃×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₃×M₁₆	C₄×C₅	1	1	yes	characteristic
57	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
58	C₃×#117	C₄×C₅	1	1	yes	characteristic
59	C₃×M₃₂	—	2	2	no	—
60	C₂×C₃×C₁₆	—	2	2	no	—
61	C₂×C₃×C₁₆	C₂²×C₅	1	2	yes	—
62	C₂×C₃×C₁₆	C₂²×C₅	1	2	yes	—
63	C₃×#97	C₄×C₅	1	2	yes	—
64	C₃×C₄×C₈	C₂²×C₅	1	2	yes	—
65	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
66	C₅×M₁₆	—	2	2	no	—
67	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
68	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
69	C₂×C₅×C₈	C₃×D₄	1	2	yes	—
70	C₂×C₅×C₈	C₃×Q₈	1	2	yes	—
71	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
72	C₅×C₄⋊C₄	C₃×C₈	1	2	yes	—
73	C₅×C₂²⋊C₄	C₃×C₈	1	2	yes	—
74	C₄²×C₅	C₂×C₃×C₄	1	2	yes	—
75	C₅×C₁₆	—	2	2	no	—
76	C₅×C₁₆	—	2	2	no	—
77	C₂×M₃₂	C₂×C₃×C₅	1	1	yes	characteristic
78	C₂²×C₁₆	C₂×C₃×C₅	1	1	yes	characteristic
79	#405	C₂×C₃×C₅	1	1	yes	characteristic
80	C₄⋊C₁₆	C₂×C₃×C₅	1	4	yes	complemented
81	C₂²×C₃×C₅	—	2	2	no	—
82	C₂²×C₃×C₅	C₄⋊C₈	1	2	yes	—
83	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
84	C₃×C₄×C₅	—	2	4	no	—
85	C₃×C₄×C₅	C₄⋊C₈	1	2	yes	—
86	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
88	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
89	C₃×M₁₆	—	2	2	no	—
90	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
91	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
92	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
93	C₂×C₃×C₈	C₅×Q₈	1	2	yes	—
94	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
95	C₃×C₄⋊C₄	C₅×C₈	1	2	yes	—
96	C₃×C₂²⋊C₄	C₅×C₈	1	2	yes	—
97	C₃×C₄²	C₂×C₄×C₅	1	2	yes	—
98	C₃×C₁₆	—	2	2	no	—
99	C₃×C₁₆	—	2	2	no	—
100	C₂³×C₅	C₂×C₃×C₈	1	1	yes	characteristic
101	C₂×C₄×C₅	C₃×C₄⋊C₄	1	2	yes	—
102	C₂×C₄×C₅	C₂×C₃×C₈	1	4	yes	—
103	C₂×C₄×C₅	C₂×C₃×C₈	1	1	yes	characteristic
104	C₂×C₄×C₅	C₃×M₁₆	1	2	yes	—
105	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
106	C₅×C₈	—	2	2	no	—
107	C₅×C₈	C₃×C₄⋊C₄	1	2	yes	—
108	C₅×C₈	C₂×C₃×D₄	1	1	yes	characteristic
109	C₅×C₈	C₂×C₃×Q₈	1	1	yes	characteristic
110	C₂×M₁₆	C₃×C₄×C₅	1	1	yes	characteristic
111	C₂²×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
112	#117	C₃×C₄×C₅	1	1	yes	characteristic
113	M₃₂	—	2	2	no	—
114	C₂×C₁₆	—	2	2	no	—
115	C₂×C₁₆	C₂²×C₃×C₅	1	2	yes	—
116	C₂×C₁₆	C₂²×C₃×C₅	1	2	yes	—
117	#97	C₃×C₄×C₅	1	2	yes	—
118	C₄×C₈	C₂²×C₃×C₅	1	2	yes	—
119	C₂×C₃×C₅	—	2	2	no	—
120	C₂×C₃×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
121	C₂×C₃×C₅	#504	1	2	yes	—
122	C₂³×C₃	C₂×C₅×C₈	1	1	yes	characteristic
123	C₂×C₃×C₄	C₅×C₄⋊C₄	1	2	yes	—
124	C₂×C₃×C₄	C₂×C₅×C₈	1	4	yes	—
125	C₂×C₃×C₄	C₂×C₅×C₈	1	1	yes	characteristic
126	C₂×C₃×C₄	C₅×M₁₆	1	2	yes	—
127	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
128	C₃×C₈	—	2	2	no	—
129	C₃×C₈	C₅×C₄⋊C₄	1	2	yes	—
130	C₃×C₈	C₂×C₅×D₄	1	1	yes	characteristic
131	C₃×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
132	C₂²×C₅	—	2	2	no	—
133	C₂²×C₅	C₃×C₄⋊C₈	1	2	yes	—
134	C₂²×C₅	C₂²×C₃×C₈	1	1	yes	characteristic
135	C₄×C₅	—	2	4	no	—
136	C₄×C₅	C₃×C₄⋊C₈	1	2	yes	—
137	C₄×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
138	C₄×C₅	C₂×C₃×M₁₆	1	1	yes	characteristic
139	C₂²×C₄	C₂×C₃×C₄×C₅	1	1	yes	characteristic
140	M₁₆	—	2	2	no	—
141	C₂×C₈	C₂×C₃×C₄×C₅	1	2	yes	—
142	C₂×C₈	C₂×C₃×C₄×C₅	1	1	yes	characteristic
143	C₂×C₈	C₃×C₅×D₄	1	2	yes	—
144	C₂×C₈	C₃×C₅×Q₈	1	2	yes	—
145	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
146	C₄⋊C₄	C₃×C₅×C₈	1	2	yes	—
147	C₂²⋊C₄	C₃×C₅×C₈	1	2	yes	—
148	C₄²	C₂×C₃×C₄×C₅	1	2	yes	—
149	C₁₆	—	2	2	no	—
150	C₁₆	—	2	2	no	—
151	C₃×C₅	#2151	1	1	yes	characteristic, direct product factor, Hall
152	C₂²×C₃	—	2	2	no	—
153	C₂²×C₃	C₅×C₄⋊C₈	1	2	yes	—
154	C₂²×C₃	C₂²×C₅×C₈	1	1	yes	characteristic
155	C₃×C₄	—	2	4	no	—
156	C₃×C₄	C₅×C₄⋊C₈	1	2	yes	—
157	C₃×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
158	C₃×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
159	C₂×C₅	—	2	2	no	—
160	C₂×C₅	C₂×C₃×C₄⋊C₈	1	1	yes	characteristic
161	C₂×C₅	C₃×#504	1	2	yes	—
162	C₂³	C₂×C₃×C₅×C₈	1	1	yes	characteristic
163	C₂×C₄	C₃×C₅×C₄⋊C₄	1	2	yes	—
164	C₂×C₄	C₂×C₃×C₅×C₈	1	4	yes	—
165	C₂×C₄	C₂×C₃×C₅×C₈	1	1	yes	characteristic
166	C₂×C₄	C₃×C₅×M₁₆	1	2	yes	—
167	C₂×C₄	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
168	C₈	—	2	2	no	—
169	C₈	C₃×C₅×C₄⋊C₄	1	2	yes	—
170	C₈	C₂×C₃×C₅×D₄	1	1	yes	characteristic
171	C₈	C₂×C₃×C₅×Q₈	1	1	yes	characteristic
172	C₂×C₃	—	2	2	no	—
173	C₂×C₃	C₂×C₅×C₄⋊C₈	1	1	yes	characteristic
174	C₂×C₃	C₅×#504	1	2	yes	—
175	C₅	C₃×#2151	1	1	yes	characteristic, direct product factor, Sylow
176	C₂²	—	2	2	no	—
177	C₂²	C₃×C₅×C₄⋊C₈	1	2	yes	—
178	C₂²	C₂²×C₃×C₅×C₈	1	1	yes	characteristic
179	C₄	—	2	4	no	—
180	C₄	C₃×C₅×C₄⋊C₈	1	2	yes	—
181	C₄	C₂×C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
182	C₄	C₂×C₃×C₅×M₁₆	1	1	yes	characteristic
183	C₃	C₅×#2151	1	1	yes	characteristic, direct product factor, Sylow
184	C₂	—	2	2	no	—
185	C₂	C₂×C₃×C₅×C₄⋊C₈	1	1	yes	characteristic
186	C₂	C₃×C₅×#504	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₃×C₅×C₄⋊C₁₆	C₂	semidirect
5. C₅×#2151	C₃	direct (N × H)
15. C₃×#2151	C₅	direct (N × H)
19. C₅×C₄⋊C₁₆	C₂×C₃	semidirect
35. C₃×C₄⋊C₁₆	C₂×C₅	semidirect
45. #2151	C₃×C₅	direct (N × H)
80. C₄⋊C₁₆	C₂×C₃×C₅	semidirect
151. C₃×C₅	#2151	direct (N × H)
175. C₅	C₃×#2151	direct (N × H)
183. C₃	C₅×#2151	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×M₃₂	2
C₂²×C₃×C₅×C₁₆	2
C₃×C₅×#405	2
C₃×C₅×C₄⋊C₁₆	2
C₅×#2151	3
C₃×#2151	5

Subgroups — C3×C5×#2151 #14275185

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2151 #14275185