Subgroups — C₃×C₈×#681 #14280367

Order 1920.

448 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
2	C₃×#65214	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄×#681	C₂	1	1	yes	maximal, characteristic
4	C₃×#64674	C₂	1	2	yes	maximal, complemented
5	C₃×C₈×#168	C₂	1	2	yes	maximal, complemented
6	C₈×#681	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×C₅×C₈	C₄	1	2	yes	—
8	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
9	C₂×C₃×#681	C₄	1	2	yes	—
10	C₃×C₄×C₅×C₈	C₄	1	2	yes	—
11	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₃×C₂²⋊#168	C₄	1	2	yes	—
14	C₃×C₄⋊#168	C₄	1	2	yes	—
15	C₃×#3935	C₄	1	2	yes	—
16	C₃×C₄×#168	C₂²	1	2	yes	—
17	C₃×C₄×#168	C₂²	1	2	yes	—
18	C₃×C₈×M₁₆	—	5	1	no	maximal, Hall
19	C₂×C₄×C₅×C₈	C₂×C₃	1	1	yes	characteristic
20	#65214	C₂×C₃	1	1	yes	characteristic
21	C₄×#681	C₂×C₃	1	1	yes	characteristic
22	#64674	C₂×C₃	1	2	yes	complemented
23	C₈×#168	C₂×C₃	1	2	yes	complemented
24	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃×C₅×C₈	—	2	4	no	—
28	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
29	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
30	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
31	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
32	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
33	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
34	C₃×#681	C₈	1	8	yes	direct product factor
35	C₂×C₃×#168	C₂×C₄	1	4	yes	—
36	C₂×C₃×#168	C₂×C₄	1	4	yes	—
37	C₃×#431	—	5	1	no	—
38	C₃×C₄×M₁₆	—	5	1	no	—
39	C₂×C₃×C₄×C₈	D₅	1	1	yes	characteristic
40	C₃×#322	—	5	2	no	—
41	C₃×C₈²	—	5	2	no	—
42	C₂²×C₅×C₈	C₃×C₄	1	2	yes	—
43	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
44	C₂×#681	C₃×C₄	1	2	yes	—
45	C₄×C₅×C₈	C₃×C₄	1	2	yes	—
46	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
47	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
48	C₂²⋊#168	C₃×C₄	1	2	yes	—
49	C₄⋊#168	C₃×C₄	1	2	yes	—
50	#3935	C₃×C₄	1	2	yes	—
51	C₄×#168	C₂²×C₃	1	2	yes	—
52	C₄×#168	C₂²×C₃	1	2	yes	—
53	C₈×M₁₆	—	5	1	no	Sylow
54	C₂³×C₃×C₅	C₄²	1	1	yes	characteristic
55	C₂×C₃×C₄×C₅	—	2	2	no	—
56	C₂×C₃×C₄×C₅	—	2	2	no	—
57	C₂×C₃×C₄×C₅	C₄²	1	1	yes	characteristic
58	C₂×C₃×C₄×C₅	C₄²	1	1	yes	characteristic
59	C₂×C₃×C₄×C₅	C₄²	1	1	yes	characteristic
60	C₂×C₃×C₄×C₅	C₂×C₈	1	4	yes	—
61	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
64	C₃×C₅×C₈	—	2	2	no	—
65	C₃×C₅×C₈	M₁₆	1	4	yes	complemented
66	C₃×#168	—	2	4	no	—
67	C₃×#168	C₂×C₈	1	8	yes	complemented
68	C₂×C₃×M₁₆	—	5	2	no	—
69	C₂²×C₃×C₈	Dic₅	1	2	yes	—
70	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
71	C₃×C₄⋊C₈	—	5	2	no	—
72	C₃×C₂²⋊C₈	—	5	2	no	—
73	C₃×#97	—	5	2	no	—
74	C₃×C₄×C₈	—	5	2	no	—
75	C₃×C₄×C₈	—	5	2	no	—
76	C₃×C₄×C₈	Dic₅	1	2	yes	—
77	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
78	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
79	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
82	C₂×C₅×C₈	—	2	4	no	—
83	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
84	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
85	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
86	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
87	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
88	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
89	#681	C₃×C₈	1	8	yes	direct product factor
90	C₂×#168	C₂×C₃×C₄	1	4	yes	—
91	C₂×#168	C₂×C₃×C₄	1	4	yes	—
92	#431	—	5	1	no	—
93	C₄×M₁₆	—	5	1	no	—
94	C₂×C₄×C₈	C₃×D₅	1	1	yes	characteristic
95	#322	—	5	2	no	—
96	C₈²	—	5	2	no	—
97	C₂²×C₃×C₅	—	2	1	no	—
98	C₂²×C₃×C₅	—	2	1	no	—
99	C₂²×C₃×C₅	C₄×C₈	1	2	yes	—
100	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
101	C₃×C₄×C₅	—	2	1	no	—
102	C₃×C₄×C₅	—	2	1	no	—
103	C₃×C₄×C₅	C₄×C₈	1	2	yes	—
104	C₃×C₄×C₅	C₂²×C₈	1	2	yes	—
105	C₃×C₄×C₅	C₂×M₁₆	1	2	yes	—
106	C₃×C₄×C₅	#131	1	2	yes	—
107	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
108	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
109	C₂²×C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
110	C₃×M₁₆	—	5	8	no	—
111	C₂×C₃×C₈	—	2	4	no	—
112	C₂×C₃×C₈	—	5	4	no	—
113	C₂×C₃×C₈	—	5	4	no	—
114	C₂×C₃×C₈	C₂×Dic₅	1	2	yes	—
115	C₂×C₃×C₈	C₂×Dic₅	1	2	yes	—
116	C₃×C₄²	C₄×D₅	1	1	yes	characteristic
117	C₃×C₄²	C₄×D₅	1	1	yes	characteristic
118	C₃×C₄²	C₂×Dic₅	1	1	yes	characteristic
119	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
120	C₂³×C₅	C₃×C₄²	1	1	yes	characteristic
121	C₂×C₄×C₅	—	2	2	no	—
122	C₂×C₄×C₅	—	2	2	no	—
123	C₂×C₄×C₅	C₃×C₄²	1	1	yes	characteristic
124	C₂×C₄×C₅	C₃×C₄²	1	1	yes	characteristic
125	C₂×C₄×C₅	C₃×C₄²	1	1	yes	characteristic
126	C₂×C₄×C₅	C₂×C₃×C₈	1	4	yes	—
127	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
128	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
129	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
130	C₅×C₈	—	2	2	no	—
131	C₅×C₈	C₃×M₁₆	1	4	yes	complemented
132	#168	—	2	4	no	—
133	#168	C₂×C₃×C₈	1	8	yes	complemented
134	C₂×M₁₆	—	5	2	no	—
135	C₂²×C₈	C₃×Dic₅	1	2	yes	—
136	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
137	C₄⋊C₈	—	5	2	no	—
138	C₂²⋊C₈	—	5	2	no	—
139	#97	—	5	2	no	—
140	C₄×C₈	—	5	2	no	—
141	C₄×C₈	—	5	2	no	—
142	C₄×C₈	C₃×Dic₅	1	2	yes	—
143	C₄×C₈	C₂×C₃×D₅	1	1	yes	characteristic
144	C₄×C₈	C₂×C₃×D₅	1	1	yes	characteristic
145	C₂×C₃×C₅	—	2	2	no	—
146	C₂×C₃×C₅	C₂×C₄×C₈	1	1	yes	characteristic
147	C₂×C₃×C₅	C₄×M₁₆	1	1	yes	characteristic
148	C₂×C₃×C₅	#405	1	1	yes	characteristic
149	C₂³×C₃	C₄×Dic₅	1	1	yes	characteristic
150	C₂×C₃×C₄	—	2	2	no	—
151	C₂×C₃×C₄	—	2	2	no	—
152	C₂×C₃×C₄	C₈×D₅	1	4	yes	—
153	C₂×C₃×C₄	C₄×Dic₅	1	1	yes	characteristic
154	C₂×C₃×C₄	C₄×Dic₅	1	1	yes	characteristic
155	C₂×C₃×C₄	C₄×Dic₅	1	1	yes	characteristic
156	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
157	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
158	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
159	C₃×C₈	—	2	2	no	—
160	C₃×C₈	—	5	8	no	—
161	C₃×C₈	—	10	4	no	—
162	C₃×C₈	#681	1	4	yes	direct product factor
163	C₂²×C₅	—	2	1	no	—
164	C₂²×C₅	—	2	1	no	—
165	C₂²×C₅	C₃×C₄×C₈	1	2	yes	—
166	C₂²×C₅	C₂×C₃×C₄²	1	1	yes	characteristic
167	C₄×C₅	—	2	1	no	—
168	C₄×C₅	—	2	1	no	—
169	C₄×C₅	C₃×C₄×C₈	1	2	yes	—
170	C₄×C₅	C₂²×C₃×C₈	1	2	yes	—
171	C₄×C₅	C₂×C₃×M₁₆	1	2	yes	—
172	C₄×C₅	C₃×#131	1	2	yes	—
173	C₂²×C₄	C₃×C₄×D₅	1	1	yes	characteristic
174	C₂²×C₄	C₃×C₄×D₅	1	1	yes	characteristic
175	C₂²×C₄	C₂×C₃×Dic₅	1	1	yes	characteristic
176	M₁₆	—	5	8	no	—
177	C₂×C₈	—	2	4	no	—
178	C₂×C₈	—	5	4	no	—
179	C₂×C₈	—	5	4	no	—
180	C₂×C₈	C₂×C₃×Dic₅	1	2	yes	—
181	C₂×C₈	C₂×C₃×Dic₅	1	2	yes	—
182	C₄²	C₃×C₄×D₅	1	1	yes	characteristic
183	C₄²	C₃×C₄×D₅	1	1	yes	characteristic
184	C₄²	C₂×C₃×Dic₅	1	1	yes	characteristic
185	C₄²	C₂²×C₃×D₅	1	1	yes	characteristic
186	C₃×C₅	C₈×M₁₆	1	1	yes	characteristic, complemented, Hall
187	C₂²×C₃	—	2	1	no	—
188	C₂²×C₃	—	2	1	no	—
189	C₂²×C₃	C₈×Dic₅	1	2	yes	—
190	C₂²×C₃	C₂×C₄×Dic₅	1	1	yes	characteristic
191	C₃×C₄	—	2	1	no	—
192	C₃×C₄	—	2	1	no	—
193	C₃×C₄	C₈×Dic₅	1	2	yes	—
194	C₃×C₄	C₂×C₈×D₅	1	2	yes	—
195	C₃×C₄	#4047	1	2	yes	—
196	C₃×C₄	C₂×#681	1	2	yes	—
197	C₂×C₅	—	2	2	no	—
198	C₂×C₅	C₂×C₃×C₄×C₈	1	1	yes	characteristic
199	C₂×C₅	C₃×C₄×M₁₆	1	1	yes	characteristic
200	C₂×C₅	C₃×#405	1	1	yes	characteristic
201	C₂³	C₃×C₄×Dic₅	1	1	yes	characteristic
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	C₃×C₈×D₅	1	4	yes	—
205	C₂×C₄	C₃×C₄×Dic₅	1	1	yes	characteristic
206	C₂×C₄	C₃×C₄×Dic₅	1	1	yes	characteristic
207	C₂×C₄	C₃×C₄×Dic₅	1	1	yes	characteristic
208	C₂×C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
209	C₂×C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
210	C₂×C₄	C₂²×C₃×Dic₅	1	1	yes	characteristic
211	C₈	—	2	2	no	—
212	C₈	—	5	8	no	—
213	C₈	—	10	4	no	—
214	C₈	C₃×#681	1	4	yes	direct product factor
215	C₂×C₃	—	2	2	no	—
216	C₂×C₃	C₄×#681	1	1	yes	characteristic
217	C₂×C₃	C₂×C₈×Dic₅	1	1	yes	characteristic
218	C₂×C₃	#65390	1	1	yes	characteristic
219	C₅	C₃×C₈×M₁₆	1	1	yes	characteristic, complemented, Sylow
220	C₂²	—	2	1	no	—
221	C₂²	—	2	1	no	—
222	C₂²	C₃×C₈×Dic₅	1	2	yes	—
223	C₂²	C₂×C₃×C₄×Dic₅	1	1	yes	characteristic
224	C₄	—	2	1	no	—
225	C₄	—	2	1	no	—
226	C₄	C₃×C₈×Dic₅	1	2	yes	—
227	C₄	C₂×C₃×C₈×D₅	1	2	yes	—
228	C₄	C₃×#4047	1	2	yes	—
229	C₄	C₂×C₃×#681	1	2	yes	—
230	C₃	C₈×#681	1	1	yes	characteristic, direct product factor, Sylow
231	C₂	—	2	2	no	—
232	C₂	C₃×C₄×#681	1	1	yes	characteristic
233	C₂	C₂×C₃×C₈×Dic₅	1	1	yes	characteristic
234	C₂	C₃×#65390	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄×C₈	C₂×D₅
commutator	C₂×C₅	C₂×C₃×C₄×C₈
Frattini	C₄²	C₂²×C₃×D₅
Fitting	C₂×C₃×C₄×C₅×C₈	C₂
socle	C₂²×C₃×C₅	C₂×C₄²
prime radical	C₂³×C₃×C₅	C₄²
cosocle	C₄²×C₅	C₂³×C₃
hypercenter	C₂×C₃×C₄×C₈	D₅
hypocenter	C₅	C₃×C₈×M₁₆
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₃×C₄×C₈ ◁ C₂×C₃×C₄×C₈

Complements

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

N	H ≅ G/N	type
4. C₃×#64674	C₂	semidirect
5. C₃×C₈×#168	C₂	semidirect
6. C₈×#681	C₃	direct (N × H)
22. #64674	C₂×C₃	semidirect
23. C₈×#168	C₂×C₃	semidirect
34. C₃×#681	C₈	direct (N × H)
65. C₃×C₅×C₈	M₁₆	semidirect
67. C₃×#168	C₂×C₈	semidirect
89. #681	C₃×C₈	direct (N × H)
131. C₅×C₈	C₃×M₁₆	semidirect
133. #168	C₂×C₃×C₈	semidirect
162. C₃×C₈	#681	direct (N × H)
186. C₃×C₅	C₈×M₁₆	semidirect
214. C₈	C₃×#681	direct (N × H)
219. C₅	C₃×C₈×M₁₆	semidirect
230. C₃	C₈×#681	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×C₄×C₅×C₈	2
C₃×#65214	2
C₃×C₄×#681	2
C₃×#64674	2
C₃×C₈×#168	2
C₈×#681	3
C₃×C₈×M₁₆	5

Subgroups — C3×C8×#681 #14280367

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₈×#681 #14280367