Subgroups — C₃×C₅×#2158 #14275192

Order 1920.

304 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#427	C₂	1	2	yes	maximal
3	C₃×C₅×#367	C₂	1	2	yes	maximal, complemented
4	C₃×C₅×#366	C₂	1	2	yes	maximal, complemented
5	C₅×#2158	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₃×C₅×#117	—	2	2	no	—
8	C₂×C₃×C₅×C₁₆	C₄	1	4	yes	complemented
9	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic
11	C₃×C₅×#107	C₂²	1	4	yes	—
12	C₃×C₅×#106	—	2	2	no	—
13	C₃×#2158	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
14	C₂²×C₅×C₁₆	C₂×C₃	1	1	yes	characteristic
15	C₅×#427	C₂×C₃	1	2	yes	—
16	C₅×#367	C₂×C₃	1	2	yes	complemented
17	C₅×#366	C₂×C₃	1	2	yes	complemented
18	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
19	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
20	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
21	C₂×C₃×C₅×C₈	Q₈	1	2	yes	—
22	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
23	C₃×C₅×C₄⋊C₄	—	2	4	no	—
24	C₃×C₅×C₂²⋊C₄	—	4	2	no	—
25	C₃×C₄²×C₅	—	4	2	no	—
26	C₃×C₅×C₁₆	C₂×C₄	1	2	yes	—
27	C₃×C₅×C₁₆	C₂×C₄	1	2	yes	—
28	C₂²×C₃×C₁₆	C₂×C₅	1	1	yes	characteristic
29	C₃×#427	C₂×C₅	1	2	yes	—
30	C₃×#367	C₂×C₅	1	2	yes	complemented
31	C₃×#366	C₂×C₅	1	2	yes	complemented
32	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
33	C₅×#117	—	2	2	no	—
34	C₂×C₅×C₁₆	C₃×C₄	1	4	yes	complemented
35	C₂×C₅×C₁₆	C₂²×C₃	1	1	yes	characteristic
36	C₂×C₅×C₁₆	C₂²×C₃	1	1	yes	characteristic
37	C₅×#107	C₂²×C₃	1	4	yes	—
38	C₅×#106	—	2	2	no	—
39	#2158	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
40	C₂³×C₃×C₅	D₈	1	1	yes	characteristic
41	C₂×C₃×C₄×C₅	—	4	4	no	—
42	C₂×C₃×C₄×C₅	C₄⋊C₄	1	2	yes	—
43	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
44	C₂×C₃×C₄×C₅	Q₁₆	1	2	yes	—
45	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
46	C₃×C₅×C₈	C₄⋊C₄	1	2	yes	—
47	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
48	C₃×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
49	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
50	C₃×#117	—	2	2	no	—
51	C₂×C₃×C₁₆	C₄×C₅	1	4	yes	complemented
52	C₂×C₃×C₁₆	C₂²×C₅	1	1	yes	characteristic
53	C₂×C₃×C₁₆	C₂²×C₅	1	1	yes	characteristic
54	C₃×#107	C₂²×C₅	1	4	yes	—
55	C₃×#106	—	2	2	no	—
56	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
57	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
58	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
59	C₂×C₅×C₈	C₃×Q₈	1	2	yes	—
60	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
61	C₅×C₄⋊C₄	—	2	4	no	—
62	C₅×C₂²⋊C₄	—	4	2	no	—
63	C₄²×C₅	—	4	2	no	—
64	C₅×C₁₆	C₂×C₃×C₄	1	2	yes	—
65	C₅×C₁₆	C₂×C₃×C₄	1	2	yes	—
66	C₂²×C₁₆	C₂×C₃×C₅	1	1	yes	characteristic
67	#427	C₂×C₃×C₅	1	2	yes	—
68	#367	C₂×C₃×C₅	1	2	yes	complemented
69	#366	C₂×C₃×C₅	1	2	yes	complemented
70	C₂²×C₃×C₅	—	2	2	no	—
71	C₂²×C₃×C₅	#107	1	2	yes	—
72	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
73	C₃×C₄×C₅	—	8	4	no	—
74	C₃×C₄×C₅	#107	1	2	yes	—
75	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₃×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
77	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
78	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
79	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
80	C₂×C₃×C₈	C₅×Q₈	1	2	yes	—
81	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
82	C₃×C₄⋊C₄	—	2	4	no	—
83	C₃×C₂²⋊C₄	—	4	2	no	—
84	C₃×C₄²	—	4	2	no	—
85	C₃×C₁₆	C₂×C₄×C₅	1	2	yes	—
86	C₃×C₁₆	C₂×C₄×C₅	1	2	yes	—
87	C₂³×C₅	C₃×D₈	1	1	yes	characteristic
88	C₂×C₄×C₅	—	4	4	no	—
89	C₂×C₄×C₅	C₃×C₄⋊C₄	1	2	yes	—
90	C₂×C₄×C₅	C₃×D₈	1	1	yes	characteristic
91	C₂×C₄×C₅	C₃×Q₁₆	1	2	yes	—
92	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
93	C₅×C₈	C₃×C₄⋊C₄	1	2	yes	—
94	C₅×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
95	C₅×C₈	C₂×C₃×Q₈	1	1	yes	characteristic
96	C₂²×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
97	#117	—	2	2	no	—
98	C₂×C₁₆	C₃×C₄×C₅	1	4	yes	complemented
99	C₂×C₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
100	C₂×C₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
101	#107	C₂²×C₃×C₅	1	4	yes	—
102	#106	—	2	2	no	—
103	C₂×C₃×C₅	—	2	2	no	—
104	C₂×C₃×C₅	C₂×#107	1	1	yes	characteristic
105	C₂×C₃×C₅	#508	1	2	yes	—
106	C₂³×C₃	C₅×D₈	1	1	yes	characteristic
107	C₂×C₃×C₄	—	4	4	no	—
108	C₂×C₃×C₄	C₅×C₄⋊C₄	1	2	yes	—
109	C₂×C₃×C₄	C₅×D₈	1	1	yes	characteristic
110	C₂×C₃×C₄	C₅×Q₁₆	1	2	yes	—
111	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
112	C₃×C₈	C₅×C₄⋊C₄	1	2	yes	—
113	C₃×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
114	C₃×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
115	C₂²×C₅	—	2	2	no	—
116	C₂²×C₅	C₃×#107	1	2	yes	—
117	C₂²×C₅	C₂×C₃×D₈	1	1	yes	characteristic
118	C₄×C₅	—	8	4	no	—
119	C₄×C₅	C₃×#107	1	2	yes	—
120	C₄×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
121	C₄×C₅	C₂×C₃×Q₁₆	1	1	yes	characteristic
122	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
123	C₂×C₈	C₂×C₃×C₄×C₅	1	2	yes	—
124	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
125	C₂×C₈	C₃×C₅×Q₈	1	2	yes	—
126	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
127	C₄⋊C₄	—	2	4	no	—
128	C₂²⋊C₄	—	4	2	no	—
129	C₄²	—	4	2	no	—
130	C₁₆	C₂×C₃×C₄×C₅	1	2	yes	—
131	C₁₆	C₂×C₃×C₄×C₅	1	2	yes	—
132	C₃×C₅	#2158	1	1	yes	characteristic, direct product factor, Hall
133	C₂²×C₃	—	2	2	no	—
134	C₂²×C₃	C₅×#107	1	2	yes	—
135	C₂²×C₃	C₂×C₅×D₈	1	1	yes	characteristic
136	C₃×C₄	—	8	4	no	—
137	C₃×C₄	C₅×#107	1	2	yes	—
138	C₃×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
139	C₃×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
140	C₂×C₅	—	2	2	no	—
141	C₂×C₅	C₂×C₃×#107	1	1	yes	characteristic
142	C₂×C₅	C₃×#508	1	2	yes	—
143	C₂³	C₃×C₅×D₈	1	1	yes	characteristic
144	C₂×C₄	—	4	4	no	—
145	C₂×C₄	C₃×C₅×C₄⋊C₄	1	2	yes	—
146	C₂×C₄	C₃×C₅×D₈	1	1	yes	characteristic
147	C₂×C₄	C₃×C₅×Q₁₆	1	2	yes	—
148	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
149	C₈	C₃×C₅×C₄⋊C₄	1	2	yes	—
150	C₈	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
151	C₈	C₂×C₃×C₅×Q₈	1	1	yes	characteristic
152	C₂×C₃	—	2	2	no	—
153	C₂×C₃	C₂×C₅×#107	1	1	yes	characteristic
154	C₂×C₃	C₅×#508	1	2	yes	—
155	C₅	C₃×#2158	1	1	yes	characteristic, direct product factor, Sylow
156	C₂²	—	2	2	no	—
157	C₂²	C₃×C₅×#107	1	2	yes	—
158	C₂²	C₂×C₃×C₅×D₈	1	1	yes	characteristic
159	C₄	—	8	4	no	—
160	C₄	C₃×C₅×#107	1	2	yes	—
161	C₄	C₂×C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
162	C₄	C₂×C₃×C₅×Q₁₆	1	1	yes	characteristic
163	C₃	C₅×#2158	1	1	yes	characteristic, direct product factor, Sylow
164	C₂	—	2	2	no	—
165	C₂	C₂×C₃×C₅×#107	1	1	yes	characteristic
166	C₂	C₃×C₅×#508	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₄×C₅	D₈
commutator	C₈	C₂²×C₃×C₄×C₅
Frattini, cosocle	C₂×C₈	C₂³×C₃×C₅
socle	C₂²×C₃×C₅	C₂×D₈
prime radical	C₂³×C₃×C₅	D₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄×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×#367	C₂	semidirect
4. C₃×C₅×#366	C₂	semidirect
5. C₅×#2158	C₃	direct (N × H)
8. C₂×C₃×C₅×C₁₆	C₄	semidirect
13. C₃×#2158	C₅	direct (N × H)
16. C₅×#367	C₂×C₃	semidirect
17. C₅×#366	C₂×C₃	semidirect
30. C₃×#367	C₂×C₅	semidirect
31. C₃×#366	C₂×C₅	semidirect
34. C₂×C₅×C₁₆	C₃×C₄	semidirect
39. #2158	C₃×C₅	direct (N × H)
51. C₂×C₃×C₁₆	C₄×C₅	semidirect
68. #367	C₂×C₃×C₅	semidirect
69. #366	C₂×C₃×C₅	semidirect
98. C₂×C₁₆	C₃×C₄×C₅	semidirect
132. C₃×C₅	#2158	direct (N × H)
155. C₅	C₃×#2158	direct (N × H)
163. C₃	C₅×#2158	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₃×C₅×C₁₆	2
C₃×C₅×#427	2
C₃×C₅×#367	2
C₃×C₅×#366	2
C₅×#2158	3
C₃×#2158	5

Subgroups — C3×C5×#2158 #14275192

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2158 #14275192