Subgroups — C₃×C₅×#2161 #14275195

Order 1920.

280 conjugacy classes of subgroups.

The subgroup lattice has 152 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	C₂×C₃×C₅×#108	C₂	1	2	yes	maximal
3	C₃×C₅×#368	C₂	1	4	yes	maximal, complemented
4	C₅×#2161	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂×C₃×C₅×M₁₆	—	2	2	no	—
6	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₃×C₅×M₃₂	C₄	1	4	yes	—
8	C₂×C₃×C₅×C₁₆	C₂²	1	2	yes	—
9	C₃×C₅×#108	—	2	2	no	—
10	C₃×C₅×#108	C₂²	1	4	yes	—
11	C₃×#2161	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
12	C₂×C₅×M₃₂	C₂×C₃	1	1	yes	characteristic
13	C₂×C₅×#108	C₂×C₃	1	2	yes	—
14	C₅×#368	C₂×C₃	1	4	yes	complemented
15	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
16	C₃×C₅×M₁₆	—	2	4	no	—
17	C₃×C₅×M₁₆	—	4	2	no	—
18	C₂×C₃×C₅×C₈	—	4	2	no	—
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×C₁₆	C₂×C₄	1	4	yes	—
24	C₂×C₃×M₃₂	C₂×C₅	1	1	yes	characteristic
25	C₂×C₃×#108	C₂×C₅	1	2	yes	—
26	C₃×#368	C₂×C₅	1	4	yes	complemented
27	C₂×C₅×M₁₆	—	2	2	no	—
28	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
29	C₅×M₃₂	C₃×C₄	1	4	yes	—
30	C₂×C₅×C₁₆	C₂²×C₃	1	2	yes	—
31	C₅×#108	—	2	2	no	—
32	C₅×#108	C₂²×C₃	1	4	yes	—
33	#2161	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
34	C₂³×C₃×C₅	Q₁₆	1	1	yes	characteristic
35	C₂×C₃×C₄×C₅	C₄⋊C₄	1	2	yes	—
36	C₂×C₃×C₄×C₅	D₈	1	2	yes	—
37	C₂×C₃×C₄×C₅	Q₁₆	1	1	yes	characteristic
38	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
39	C₃×C₅×C₈	—	4	4	no	—
40	C₃×C₅×C₈	C₄⋊C₄	1	2	yes	—
41	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
42	C₃×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₃×M₁₆	—	2	2	no	—
44	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
45	C₃×M₃₂	C₄×C₅	1	4	yes	—
46	C₂×C₃×C₁₆	C₂²×C₅	1	2	yes	—
47	C₃×#108	—	2	2	no	—
48	C₃×#108	C₂²×C₅	1	4	yes	—
49	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
50	C₅×M₁₆	—	2	4	no	—
51	C₅×M₁₆	—	4	2	no	—
52	C₂×C₅×C₈	—	4	2	no	—
53	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
54	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
55	C₂×C₅×C₈	C₃×Q₈	1	2	yes	—
56	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
57	C₅×C₁₆	C₂×C₃×C₄	1	4	yes	—
58	C₂×M₃₂	C₂×C₃×C₅	1	1	yes	characteristic
59	C₂×#108	C₂×C₃×C₅	1	2	yes	—
60	#368	C₂×C₃×C₅	1	4	yes	complemented
61	C₂²×C₃×C₅	—	4	1	no	—
62	C₂²×C₃×C₅	#107	1	2	yes	—
63	C₂²×C₃×C₅	C₂×Q₁₆	1	1	yes	characteristic
64	C₃×C₄×C₅	#107	1	2	yes	—
65	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
67	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
68	C₃×M₁₆	—	2	4	no	—
69	C₃×M₁₆	—	4	2	no	—
70	C₂×C₃×C₈	—	4	2	no	—
71	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
72	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
73	C₂×C₃×C₈	C₅×Q₈	1	2	yes	—
74	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
75	C₃×C₁₆	C₂×C₄×C₅	1	4	yes	—
76	C₂³×C₅	C₃×Q₁₆	1	1	yes	characteristic
77	C₂×C₄×C₅	C₃×C₄⋊C₄	1	2	yes	—
78	C₂×C₄×C₅	C₃×D₈	1	2	yes	—
79	C₂×C₄×C₅	C₃×Q₁₆	1	1	yes	characteristic
80	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
81	C₅×C₈	—	4	4	no	—
82	C₅×C₈	C₃×C₄⋊C₄	1	2	yes	—
83	C₅×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
84	C₅×C₈	C₂×C₃×Q₈	1	1	yes	characteristic
85	C₂×M₁₆	—	2	2	no	—
86	C₂²×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
87	M₃₂	C₃×C₄×C₅	1	4	yes	—
88	C₂×C₁₆	C₂²×C₃×C₅	1	2	yes	—
89	#108	—	2	2	no	—
90	#108	C₂²×C₃×C₅	1	4	yes	—
91	C₂×C₃×C₅	—	2	2	no	—
92	C₂×C₃×C₅	—	2	1	no	—
93	C₂×C₃×C₅	C₂×#107	1	1	yes	characteristic
94	C₂³×C₃	C₅×Q₁₆	1	1	yes	characteristic
95	C₂×C₃×C₄	C₅×C₄⋊C₄	1	2	yes	—
96	C₂×C₃×C₄	C₅×D₈	1	2	yes	—
97	C₂×C₃×C₄	C₅×Q₁₆	1	1	yes	characteristic
98	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
99	C₃×C₈	—	4	4	no	—
100	C₃×C₈	C₅×C₄⋊C₄	1	2	yes	—
101	C₃×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
102	C₃×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
103	C₂²×C₅	—	4	1	no	—
104	C₂²×C₅	C₃×#107	1	2	yes	—
105	C₂²×C₅	C₂×C₃×Q₁₆	1	1	yes	characteristic
106	C₄×C₅	C₃×#107	1	2	yes	—
107	C₄×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
108	C₄×C₅	C₂×C₃×D₈	1	1	yes	characteristic
109	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
110	M₁₆	—	2	4	no	—
111	M₁₆	—	4	2	no	—
112	C₂×C₈	—	4	2	no	—
113	C₂×C₈	C₂×C₃×C₄×C₅	1	2	yes	—
114	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
115	C₂×C₈	C₃×C₅×Q₈	1	2	yes	—
116	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
117	C₁₆	C₂×C₃×C₄×C₅	1	4	yes	—
118	C₃×C₅	#2161	1	1	yes	characteristic, direct product factor, Hall
119	C₂²×C₃	—	4	1	no	—
120	C₂²×C₃	C₅×#107	1	2	yes	—
121	C₂²×C₃	C₂×C₅×Q₁₆	1	1	yes	characteristic
122	C₃×C₄	C₅×#107	1	2	yes	—
123	C₃×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
124	C₃×C₄	C₂×C₅×D₈	1	1	yes	characteristic
125	C₂×C₅	—	2	2	no	—
126	C₂×C₅	—	2	1	no	—
127	C₂×C₅	C₂×C₃×#107	1	1	yes	characteristic
128	C₂³	C₃×C₅×Q₁₆	1	1	yes	characteristic
129	C₂×C₄	C₃×C₅×C₄⋊C₄	1	2	yes	—
130	C₂×C₄	C₃×C₅×D₈	1	2	yes	—
131	C₂×C₄	C₃×C₅×Q₁₆	1	1	yes	characteristic
132	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
133	C₈	—	4	4	no	—
134	C₈	C₃×C₅×C₄⋊C₄	1	2	yes	—
135	C₈	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
136	C₈	C₂×C₃×C₅×Q₈	1	1	yes	characteristic
137	C₂×C₃	—	2	1	no	—
138	C₂×C₃	—	2	2	no	—
139	C₂×C₃	C₂×C₅×#107	1	1	yes	characteristic
140	C₅	C₃×#2161	1	1	yes	characteristic, direct product factor, Sylow
141	C₂²	—	4	1	no	—
142	C₂²	C₃×C₅×#107	1	2	yes	—
143	C₂²	C₂×C₃×C₅×Q₁₆	1	1	yes	characteristic
144	C₄	C₃×C₅×#107	1	2	yes	—
145	C₄	C₂×C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
146	C₄	C₂×C₃×C₅×D₈	1	1	yes	characteristic
147	C₃	C₅×#2161	1	1	yes	characteristic, direct product factor, Sylow
148	C₂	—	2	1	no	—
149	C₂	—	2	2	no	—
150	C₂	C₂×C₃×C₅×#107	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄×C₅	C₂×D₈
commutator	C₈	C₂²×C₃×C₄×C₅
Frattini, cosocle	C₂×C₈	C₂³×C₃×C₅
socle	C₂×C₃×C₅	C₂×#107
prime radical	C₂³×C₃×C₅	Q₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ 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×#368	C₂	semidirect
4. C₅×#2161	C₃	direct (N × H)
11. C₃×#2161	C₅	direct (N × H)
14. C₅×#368	C₂×C₃	semidirect
26. C₃×#368	C₂×C₅	semidirect
33. #2161	C₃×C₅	direct (N × H)
60. #368	C₂×C₃×C₅	semidirect
118. C₃×C₅	#2161	direct (N × H)
140. C₅	C₃×#2161	direct (N × H)
147. C₃	C₅×#2161	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
C₂×C₃×C₅×#108	2
C₃×C₅×#368	2
C₅×#2161	3
C₃×#2161	5

Subgroups — C3×C5×#2161 #14275195

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2161 #14275195