Subgroups — C₃×C₅×#3162 #14276196

Order 1920.

696 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	C₃×C₅×#533	C₂	1	2	yes	maximal
2	C₃×C₅×#501	C₂	1	8	yes	maximal, complemented
3	C₃×C₅×#427	C₂	1	4	yes	maximal
4	C₂×C₃×C₅×#97	C₂	1	1	yes	maximal, characteristic
5	C₅×#3162	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
7	C₃×C₅×C₄⋊Q₈	C₂²	1	4	yes	—
8	C₃×C₅×#125	C₂²	1	4	yes	—
9	C₃×C₅×C₂²⋊Q₈	—	2	4	no	—
10	C₃×C₄×C₅×Q₈	—	2	4	no	—
11	C₃×C₅×#117	C₂²	1	4	yes	—
12	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
13	C₃×C₅×#107	C₂²	1	8	yes	—
14	C₃×C₅×#106	C₂²	1	8	yes	—
15	C₃×C₅×#97	C₂²	1	4	yes	—
16	C₃×#3162	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₅×#533	C₂×C₃	1	2	yes	—
18	C₅×#501	C₂×C₃	1	8	yes	complemented
19	C₅×#427	C₂×C₃	1	4	yes	—
20	C₂×C₅×#97	C₂×C₃	1	1	yes	characteristic
21	C₂×C₃×C₅×Q₈	—	2	4	no	—
22	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
23	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
24	C₂×C₃×C₅×C₈	Q₈	1	8	yes	complemented
25	C₂×C₃×C₅×C₈	C₂³	1	4	yes	—
26	C₃×C₅×C₄⋊C₄	—	2	8	no	—
27	C₃×C₅×C₄⋊C₄	—	2	4	no	—
28	C₃×C₅×C₄⋊C₄	C₂³	1	8	yes	—
29	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
30	C₃×C₄²×C₅	—	2	4	no	—
31	C₃×C₄²×C₅	D₄	1	2	yes	—
32	C₃×C₄²×C₅	C₂³	1	2	yes	—
33	C₃×#533	C₂×C₅	1	2	yes	—
34	C₃×#501	C₂×C₅	1	8	yes	complemented
35	C₃×#427	C₂×C₅	1	4	yes	—
36	C₂×C₃×#97	C₂×C₅	1	1	yes	characteristic
37	C₂²×C₅×C₈	C₂²×C₃	1	2	yes	—
38	C₅×C₄⋊Q₈	C₂²×C₃	1	4	yes	—
39	C₅×#125	C₂²×C₃	1	4	yes	—
40	C₅×C₂²⋊Q₈	—	2	4	no	—
41	C₄×C₅×Q₈	—	2	4	no	—
42	C₅×#117	C₂²×C₃	1	4	yes	—
43	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
44	C₅×#107	C₂²×C₃	1	8	yes	—
45	C₅×#106	C₂²×C₃	1	8	yes	—
46	C₅×#97	C₂²×C₃	1	4	yes	—
47	#3162	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
48	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
49	C₃×C₅×Q₈	—	4	8	no	—
50	C₂×C₃×C₄×C₅	—	2	8	no	—
51	C₂×C₃×C₄×C₅	—	4	2	no	—
52	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
53	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
54	C₂×C₃×C₄×C₅	C₂×Q₈	1	4	yes	—
55	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
56	C₃×C₅×C₈	C₂×Q₈	1	8	yes	—
57	C₂²×C₃×C₈	C₂²×C₅	1	2	yes	—
58	C₃×C₄⋊Q₈	C₂²×C₅	1	4	yes	—
59	C₃×#125	C₂²×C₅	1	4	yes	—
60	C₃×C₂²⋊Q₈	—	2	4	no	—
61	C₃×C₄×Q₈	—	2	4	no	—
62	C₃×#117	C₂²×C₅	1	4	yes	—
63	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
64	C₃×#107	C₂²×C₅	1	8	yes	—
65	C₃×#106	C₂²×C₅	1	8	yes	—
66	C₃×#97	C₂²×C₅	1	4	yes	—
67	C₂×C₅×Q₈	—	2	4	no	—
68	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
69	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
70	C₂×C₅×C₈	C₃×Q₈	1	8	yes	complemented
71	C₂×C₅×C₈	C₂³×C₃	1	4	yes	—
72	C₅×C₄⋊C₄	—	2	8	no	—
73	C₅×C₄⋊C₄	—	2	4	no	—
74	C₅×C₄⋊C₄	C₂³×C₃	1	8	yes	—
75	C₅×C₂²⋊C₄	—	2	4	no	—
76	C₄²×C₅	—	2	4	no	—
77	C₄²×C₅	C₃×D₄	1	2	yes	—
78	C₄²×C₅	C₂³×C₃	1	2	yes	—
79	#533	C₂×C₃×C₅	1	2	yes	—
80	#501	C₂×C₃×C₅	1	8	yes	complemented
81	#427	C₂×C₃×C₅	1	4	yes	—
82	C₂×#97	C₂×C₃×C₅	1	1	yes	characteristic
83	C₂²×C₃×C₅	—	2	2	no	—
84	C₂²×C₃×C₅	C₄⋊Q₈	1	2	yes	—
85	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
86	C₃×C₄×C₅	—	2	4	no	—
87	C₃×C₄×C₅	—	4	8	no	—
88	C₃×C₄×C₅	C₄⋊Q₈	1	2	yes	—
89	C₃×C₄×C₅	C₂²×Q₈	1	2	yes	—
90	C₂×C₃×Q₈	—	2	4	no	—
91	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
92	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
93	C₂×C₃×C₈	C₅×Q₈	1	8	yes	complemented
94	C₂×C₃×C₈	C₂³×C₅	1	4	yes	—
95	C₃×C₄⋊C₄	—	2	8	no	—
96	C₃×C₄⋊C₄	—	2	4	no	—
97	C₃×C₄⋊C₄	C₂³×C₅	1	8	yes	—
98	C₃×C₂²⋊C₄	—	2	4	no	—
99	C₃×C₄²	—	2	4	no	—
100	C₃×C₄²	C₅×D₄	1	2	yes	—
101	C₃×C₄²	C₂³×C₅	1	2	yes	—
102	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
103	C₅×Q₈	—	4	8	no	—
104	C₂×C₄×C₅	—	2	8	no	—
105	C₂×C₄×C₅	—	4	2	no	—
106	C₂×C₄×C₅	C₂×C₃×D₄	1	4	yes	—
107	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
108	C₂×C₄×C₅	C₂×C₃×Q₈	1	4	yes	—
109	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
110	C₅×C₈	C₂×C₃×Q₈	1	8	yes	—
111	C₂²×C₈	C₂²×C₃×C₅	1	2	yes	—
112	C₄⋊Q₈	C₂²×C₃×C₅	1	4	yes	—
113	#125	C₂²×C₃×C₅	1	4	yes	—
114	C₂²⋊Q₈	—	2	4	no	—
115	C₄×Q₈	—	2	4	no	—
116	#117	C₂²×C₃×C₅	1	4	yes	—
117	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
118	#107	C₂²×C₃×C₅	1	8	yes	—
119	#106	C₂²×C₃×C₅	1	8	yes	—
120	#97	C₂²×C₃×C₅	1	4	yes	—
121	C₂×C₃×C₅	—	2	2	no	—
122	C₂×C₃×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
123	C₂×C₃×C₅	#575	1	2	yes	—
124	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
125	C₃×Q₈	—	4	8	no	—
126	C₂×C₃×C₄	—	2	8	no	—
127	C₂×C₃×C₄	—	4	2	no	—
128	C₂×C₃×C₄	C₂×C₅×D₄	1	4	yes	—
129	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
130	C₂×C₃×C₄	C₂×C₅×Q₈	1	4	yes	—
131	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
132	C₃×C₈	C₂×C₅×Q₈	1	8	yes	—
133	C₂²×C₅	—	2	2	no	—
134	C₂²×C₅	C₃×C₄⋊Q₈	1	2	yes	—
135	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
136	C₄×C₅	—	2	4	no	—
137	C₄×C₅	—	4	8	no	—
138	C₄×C₅	C₃×C₄⋊Q₈	1	2	yes	—
139	C₄×C₅	C₂²×C₃×Q₈	1	2	yes	—
140	C₂×Q₈	—	2	4	no	—
141	C₂²×C₄	C₃×C₅×D₄	1	2	yes	—
142	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
143	C₂×C₈	C₃×C₅×Q₈	1	8	yes	complemented
144	C₂×C₈	C₂³×C₃×C₅	1	4	yes	—
145	C₄⋊C₄	—	2	8	no	—
146	C₄⋊C₄	—	2	4	no	—
147	C₄⋊C₄	C₂³×C₃×C₅	1	8	yes	—
148	C₂²⋊C₄	—	2	4	no	—
149	C₄²	—	2	4	no	—
150	C₄²	C₃×C₅×D₄	1	2	yes	—
151	C₄²	C₂³×C₃×C₅	1	2	yes	—
152	C₃×C₅	#3162	1	1	yes	characteristic, direct product factor, Hall
153	C₂²×C₃	—	2	2	no	—
154	C₂²×C₃	C₅×C₄⋊Q₈	1	2	yes	—
155	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
156	C₃×C₄	—	2	4	no	—
157	C₃×C₄	—	4	8	no	—
158	C₃×C₄	C₅×C₄⋊Q₈	1	2	yes	—
159	C₃×C₄	C₂²×C₅×Q₈	1	2	yes	—
160	C₂×C₅	—	2	2	no	—
161	C₂×C₅	C₂×C₃×C₄⋊Q₈	1	1	yes	characteristic
162	C₂×C₅	C₃×#575	1	2	yes	—
163	C₂³	C₂×C₃×C₅×D₄	1	1	yes	characteristic
164	Q₈	—	4	8	no	—
165	C₂×C₄	—	2	8	no	—
166	C₂×C₄	—	4	2	no	—
167	C₂×C₄	C₂×C₃×C₅×D₄	1	4	yes	—
168	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
169	C₂×C₄	C₂×C₃×C₅×Q₈	1	4	yes	—
170	C₂×C₄	C₂⁴×C₃×C₅	1	1	yes	characteristic
171	C₈	C₂×C₃×C₅×Q₈	1	8	yes	—
172	C₂×C₃	—	2	2	no	—
173	C₂×C₃	C₂×C₅×C₄⋊Q₈	1	1	yes	characteristic
174	C₂×C₃	C₅×#575	1	2	yes	—
175	C₅	C₃×#3162	1	1	yes	characteristic, direct product factor, Sylow
176	C₂²	—	2	2	no	—
177	C₂²	C₃×C₅×C₄⋊Q₈	1	2	yes	—
178	C₂²	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
179	C₄	—	2	4	no	—
180	C₄	—	4	8	no	—
181	C₄	C₃×C₅×C₄⋊Q₈	1	2	yes	—
182	C₄	C₂²×C₃×C₅×Q₈	1	2	yes	—
183	C₃	C₅×#3162	1	1	yes	characteristic, direct product factor, Sylow
184	C₂	—	2	2	no	—
185	C₂	C₂×C₃×C₅×C₄⋊Q₈	1	1	yes	characteristic
186	C₂	C₃×C₅×#575	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₄×C₅	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃×C₅
socle	C₂²×C₃×C₅	C₂²×D₄
prime radical	C₂³×C₃×C₅	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄×C₅ ◁ C₂×C₃×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#501	C₂	semidirect
5. C₅×#3162	C₃	direct (N × H)
16. C₃×#3162	C₅	direct (N × H)
18. C₅×#501	C₂×C₃	semidirect
24. C₂×C₃×C₅×C₈	Q₈	semidirect
34. C₃×#501	C₂×C₅	semidirect
47. #3162	C₃×C₅	direct (N × H)
70. C₂×C₅×C₈	C₃×Q₈	semidirect
80. #501	C₂×C₃×C₅	semidirect
93. C₂×C₃×C₈	C₅×Q₈	semidirect
143. C₂×C₈	C₃×C₅×Q₈	semidirect
152. C₃×C₅	#3162	direct (N × H)
175. C₅	C₃×#3162	direct (N × H)
183. C₃	C₅×#3162	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×#533	2
C₃×C₅×#501	2
C₃×C₅×#427	2
C₂×C₃×C₅×#97	2
C₅×#3162	3
C₃×#3162	5

Subgroups — C3×C5×#3162 #14276196

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#3162 #14276196