Subgroups — C₃×C₅×C₈×Q₁₆ #14274611

Order 1920.

328 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	C₃×C₅×C₈×Q₈	C₂	1	2	yes	maximal
2	C₃×C₄×C₅×Q₁₆	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#335	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×Q₈⋊C₈	C₂	1	2	yes	maximal
5	C₃×C₅×C₈²	C₂	1	1	yes	maximal, characteristic
6	C₅×C₈×Q₁₆	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×Q₁₆	C₄	1	1	yes	characteristic
8	C₃×C₄×C₅×Q₈	C₂²	1	2	yes	—
9	C₃×C₅×#107	C₄	1	1	yes	characteristic
10	C₃×C₅×C₄⋊C₈	—	2	2	no	—
11	C₃×C₅×C₄⋊C₈	C₂²	1	2	yes	—
12	C₃×C₅×#103	C₄	1	2	yes	—
13	C₃×C₄×C₅×C₈	—	2	2	no	—
14	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
16	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
17	C₃×C₈×Q₁₆	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₅×C₈×Q₈	C₂×C₃	1	2	yes	—
19	C₄×C₅×Q₁₆	C₂×C₃	1	1	yes	characteristic
20	C₅×#335	C₂×C₃	1	1	yes	characteristic
21	C₅×Q₈⋊C₈	C₂×C₃	1	2	yes	—
22	C₅×C₈²	C₂×C₃	1	1	yes	characteristic
23	C₂×C₃×C₅×Q₈	C₂×C₄	1	2	yes	—
24	C₃×C₅×Q₁₆	C₈	1	4	yes	direct product factor
25	C₂×C₃×C₅×C₈	—	2	1	no	—
26	C₂×C₃×C₅×C₈	—	2	2	no	—
27	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
30	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
31	C₃×C₅×C₄⋊C₄	—	2	2	no	—
32	C₃×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
33	C₃×C₄²×C₅	—	2	2	no	—
34	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
35	C₃×C₈×Q₈	C₂×C₅	1	2	yes	—
36	C₃×C₄×Q₁₆	C₂×C₅	1	1	yes	characteristic
37	C₃×#335	C₂×C₅	1	1	yes	characteristic
38	C₃×Q₈⋊C₈	C₂×C₅	1	2	yes	—
39	C₃×C₈²	C₂×C₅	1	1	yes	characteristic
40	C₂×C₅×Q₁₆	C₃×C₄	1	1	yes	characteristic
41	C₄×C₅×Q₈	C₂²×C₃	1	2	yes	—
42	C₅×#107	C₃×C₄	1	1	yes	characteristic
43	C₅×C₄⋊C₈	—	2	2	no	—
44	C₅×C₄⋊C₈	C₂²×C₃	1	2	yes	—
45	C₅×#103	C₃×C₄	1	2	yes	—
46	C₄×C₅×C₈	—	2	2	no	—
47	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
48	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
49	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
50	C₈×Q₁₆	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
51	C₃×C₅×Q₈	—	2	2	no	—
52	C₃×C₅×Q₈	C₂×C₈	1	4	yes	—
53	C₂×C₃×C₄×C₅	—	2	2	no	—
54	C₂×C₃×C₄×C₅	—	2	2	no	—
55	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
57	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
58	C₃×C₅×C₈	—	2	2	no	—
59	C₃×C₅×C₈	—	2	1	no	—
60	C₃×C₅×C₈	—	2	1	no	—
61	C₃×C₅×C₈	—	4	2	no	—
62	C₃×C₅×C₈	C₂×C₈	1	2	yes	—
63	C₃×C₅×C₈	Q₁₆	1	2	yes	direct product factor
64	C₂×C₃×Q₁₆	C₄×C₅	1	1	yes	characteristic
65	C₃×C₄×Q₈	C₂²×C₅	1	2	yes	—
66	C₃×#107	C₄×C₅	1	1	yes	characteristic
67	C₃×C₄⋊C₈	—	2	2	no	—
68	C₃×C₄⋊C₈	C₂²×C₅	1	2	yes	—
69	C₃×#103	C₄×C₅	1	2	yes	—
70	C₃×C₄×C₈	—	2	2	no	—
71	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
72	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
73	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
74	C₂×C₅×Q₈	C₂×C₃×C₄	1	2	yes	—
75	C₅×Q₁₆	C₃×C₈	1	4	yes	direct product factor
76	C₂×C₅×C₈	—	2	1	no	—
77	C₂×C₅×C₈	—	2	2	no	—
78	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
79	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
80	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
81	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
82	C₅×C₄⋊C₄	—	2	2	no	—
83	C₅×C₄⋊C₄	C₂×C₃×C₄	1	2	yes	—
84	C₄²×C₅	—	2	2	no	—
85	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
86	C₈×Q₈	C₂×C₃×C₅	1	2	yes	—
87	C₄×Q₁₆	C₂×C₃×C₅	1	1	yes	characteristic
88	#335	C₂×C₃×C₅	1	1	yes	characteristic
89	Q₈⋊C₈	C₂×C₃×C₅	1	2	yes	—
90	C₈²	C₂×C₃×C₅	1	1	yes	characteristic
91	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
92	C₃×C₄×C₅	—	2	4	no	—
93	C₃×C₄×C₅	—	2	1	no	—
94	C₃×C₄×C₅	—	4	2	no	—
95	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
96	C₃×C₄×C₅	#131	1	1	yes	characteristic
97	C₃×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
98	C₃×C₄×C₅	#135	1	1	yes	characteristic
99	C₂×C₃×Q₈	C₂×C₄×C₅	1	2	yes	—
100	C₃×Q₁₆	C₅×C₈	1	4	yes	direct product factor
101	C₂×C₃×C₈	—	2	1	no	—
102	C₂×C₃×C₈	—	2	2	no	—
103	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
104	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
105	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
106	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
107	C₃×C₄⋊C₄	—	2	2	no	—
108	C₃×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
109	C₃×C₄²	—	2	2	no	—
110	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
111	C₅×Q₈	—	2	2	no	—
112	C₅×Q₈	C₂×C₃×C₈	1	4	yes	—
113	C₂×C₄×C₅	—	2	2	no	—
114	C₂×C₄×C₅	—	2	2	no	—
115	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
116	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
117	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
118	C₅×C₈	—	2	2	no	—
119	C₅×C₈	—	2	1	no	—
120	C₅×C₈	—	2	1	no	—
121	C₅×C₈	—	4	2	no	—
122	C₅×C₈	C₂×C₃×C₈	1	2	yes	—
123	C₅×C₈	C₃×Q₁₆	1	2	yes	direct product factor
124	C₂×Q₁₆	C₃×C₄×C₅	1	1	yes	characteristic
125	C₄×Q₈	C₂²×C₃×C₅	1	2	yes	—
126	#107	C₃×C₄×C₅	1	1	yes	characteristic
127	C₄⋊C₈	—	2	2	no	—
128	C₄⋊C₈	C₂²×C₃×C₅	1	2	yes	—
129	#103	C₃×C₄×C₅	1	2	yes	—
130	C₄×C₈	—	2	2	no	—
131	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
132	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
133	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
134	C₂×C₃×C₅	C₈×D₄	1	1	yes	characteristic
135	C₂×C₃×C₅	C₄×Q₁₆	1	1	yes	characteristic
136	C₂×C₃×C₅	#443	1	1	yes	characteristic
137	C₃×Q₈	—	2	2	no	—
138	C₃×Q₈	C₂×C₅×C₈	1	4	yes	—
139	C₂×C₃×C₄	—	2	2	no	—
140	C₂×C₃×C₄	—	2	2	no	—
141	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
142	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
143	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
144	C₃×C₈	—	2	2	no	—
145	C₃×C₈	—	2	1	no	—
146	C₃×C₈	—	2	1	no	—
147	C₃×C₈	—	4	2	no	—
148	C₃×C₈	C₂×C₅×C₈	1	2	yes	—
149	C₃×C₈	C₅×Q₁₆	1	2	yes	direct product factor
150	C₂²×C₅	C₃×C₄×D₄	1	1	yes	characteristic
151	C₄×C₅	—	2	4	no	—
152	C₄×C₅	—	2	1	no	—
153	C₄×C₅	—	4	2	no	—
154	C₄×C₅	C₂²×C₃×C₈	1	1	yes	characteristic
155	C₄×C₅	C₃×#131	1	1	yes	characteristic
156	C₄×C₅	C₂×C₃×Q₁₆	1	1	yes	characteristic
157	C₄×C₅	C₃×#135	1	1	yes	characteristic
158	C₂×Q₈	C₂×C₃×C₄×C₅	1	2	yes	—
159	Q₁₆	C₃×C₅×C₈	1	4	yes	direct product factor
160	C₂×C₈	—	2	1	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	C₂×C₃×C₄×C₅	1	1	yes	characteristic
163	C₂×C₈	C₂×C₃×C₄×C₅	1	1	yes	characteristic
164	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
165	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
166	C₄⋊C₄	—	2	2	no	—
167	C₄⋊C₄	C₂×C₃×C₄×C₅	1	2	yes	—
168	C₄²	—	2	2	no	—
169	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
170	C₃×C₅	C₈×Q₁₆	1	1	yes	characteristic, direct product factor, Hall
171	C₂²×C₃	C₄×C₅×D₄	1	1	yes	characteristic
172	C₃×C₄	—	2	4	no	—
173	C₃×C₄	—	2	1	no	—
174	C₃×C₄	—	4	2	no	—
175	C₃×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
176	C₃×C₄	C₅×#131	1	1	yes	characteristic
177	C₃×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
178	C₃×C₄	C₅×#135	1	1	yes	characteristic
179	C₂×C₅	C₃×C₈×D₄	1	1	yes	characteristic
180	C₂×C₅	C₃×C₄×Q₁₆	1	1	yes	characteristic
181	C₂×C₅	C₃×#443	1	1	yes	characteristic
182	Q₈	—	2	2	no	—
183	Q₈	C₂×C₃×C₅×C₈	1	4	yes	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
187	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
188	C₂×C₄	C₃×C₅×#41	1	1	yes	characteristic
189	C₈	—	2	2	no	—
190	C₈	—	2	1	no	—
191	C₈	—	2	1	no	—
192	C₈	—	4	2	no	—
193	C₈	C₂×C₃×C₅×C₈	1	2	yes	—
194	C₈	C₃×C₅×Q₁₆	1	2	yes	direct product factor
195	C₂×C₃	C₅×C₈×D₄	1	1	yes	characteristic
196	C₂×C₃	C₄×C₅×Q₁₆	1	1	yes	characteristic
197	C₂×C₃	C₅×#443	1	1	yes	characteristic
198	C₅	C₃×C₈×Q₁₆	1	1	yes	characteristic, direct product factor, Sylow
199	C₂²	C₃×C₄×C₅×D₄	1	1	yes	characteristic
200	C₄	—	2	4	no	—
201	C₄	—	2	1	no	—
202	C₄	—	4	2	no	—
203	C₄	C₂²×C₃×C₅×C₈	1	1	yes	characteristic
204	C₄	C₃×C₅×#131	1	1	yes	characteristic
205	C₄	C₂×C₃×C₅×Q₁₆	1	1	yes	characteristic
206	C₄	C₃×C₅×#135	1	1	yes	characteristic
207	C₃	C₅×C₈×Q₁₆	1	1	yes	characteristic, direct product factor, Sylow
208	C₂	C₃×C₅×C₈×D₄	1	1	yes	characteristic
209	C₂	C₃×C₄×C₅×Q₁₆	1	1	yes	characteristic
210	C₂	C₃×C₅×#443	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₅×C₈	D₄
commutator	C₄	C₂²×C₃×C₅×C₈
Frattini, cosocle	C₄²	C₂³×C₃×C₅
socle, prime radical	C₂²×C₃×C₅	C₄×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₅×C₈ ◁ C₃×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
6. C₅×C₈×Q₁₆	C₃	direct (N × H)
17. C₃×C₈×Q₁₆	C₅	direct (N × H)
24. C₃×C₅×Q₁₆	C₈	direct (N × H)
50. C₈×Q₁₆	C₃×C₅	direct (N × H)
63. C₃×C₅×C₈	Q₁₆	direct (N × H)
75. C₅×Q₁₆	C₃×C₈	direct (N × H)
100. C₃×Q₁₆	C₅×C₈	direct (N × H)
123. C₅×C₈	C₃×Q₁₆	direct (N × H)
149. C₃×C₈	C₅×Q₁₆	direct (N × H)
159. Q₁₆	C₃×C₅×C₈	direct (N × H)
170. C₃×C₅	C₈×Q₁₆	direct (N × H)
194. C₈	C₃×C₅×Q₁₆	direct (N × H)
198. C₅	C₃×C₈×Q₁₆	direct (N × H)
207. C₃	C₅×C₈×Q₁₆	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×C₈×Q₈	2
C₃×C₄×C₅×Q₁₆	2
C₃×C₅×#335	2
C₃×C₅×Q₈⋊C₈	2
C₃×C₅×C₈²	2
C₅×C₈×Q₁₆	3
C₃×C₈×Q₁₆	5

Subgroups — C3×C5×C8×Q16 #14274611

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×C₈×Q₁₆ #14274611