Subgroups — C₃×C₈×#676 #14280358

Order 1920.

448 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×#676	C₂	1	1	yes	maximal, characteristic
2	C₃×#64974	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄×C₈×D₅	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×C₈²	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#64675	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#64674	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×C₈×#168	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈×#676	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₃×#676	C₄	1	2	yes	—
10	C₂×C₃×C₈×D₅	C₄	1	2	yes	—
11	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
12	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
14	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₃×#3952	C₄	1	2	yes	—
16	C₃×#3947	C₄	1	2	yes	—
17	C₃×#3946	C₄	1	2	yes	—
18	C₃×C₈×Dic₅	C₄	1	2	yes	—
19	C₃×C₄×#168	C₂²	1	1	yes	characteristic
20	C₃×C₄×#168	C₂²	1	1	yes	characteristic
21	C₃×C₄×#168	C₂²	1	1	yes	characteristic
22	C₃×C₈×M₁₆	—	5	1	no	maximal, Hall
23	C₄×#676	C₂×C₃	1	1	yes	characteristic
24	#64974	C₂×C₃	1	1	yes	characteristic
25	C₄×C₈×D₅	C₂×C₃	1	1	yes	characteristic
26	C₅×C₈²	C₂×C₃	1	1	yes	characteristic, complemented
27	#64675	C₂×C₃	1	1	yes	characteristic, complemented
28	#64674	C₂×C₃	1	1	yes	characteristic, complemented
29	C₈×#168	C₂×C₃	1	1	yes	characteristic, complemented
30	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
34	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
35	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
36	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
37	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
38	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
39	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
40	C₂×C₃×#168	C₂×C₄	1	2	yes	—
41	C₂×C₃×#168	C₂×C₄	1	2	yes	—
42	C₂×C₃×#168	C₂×C₄	1	2	yes	—
43	C₃×#676	C₈	1	8	yes	direct product factor
44	C₃×C₈×D₅	—	2	4	no	—
45	C₃×#431	—	5	1	no	—
46	C₃×C₄×M₁₆	—	5	1	no	—
47	C₂×C₃×C₄×C₈	—	5	1	no	—
48	C₃×#322	—	5	1	no	—
49	C₃×#322	—	5	1	no	—
50	C₃×C₈²	—	5	1	no	—
51	C₃×C₈²	D₅	1	1	yes	characteristic, complemented
52	C₂×#676	C₃×C₄	1	2	yes	—
53	C₂×C₈×D₅	C₃×C₄	1	2	yes	—
54	C₄²×D₅	C₂²×C₃	1	1	yes	characteristic
55	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
56	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
57	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
58	#3952	C₃×C₄	1	2	yes	—
59	#3947	C₃×C₄	1	2	yes	—
60	#3946	C₃×C₄	1	2	yes	—
61	C₈×Dic₅	C₃×C₄	1	2	yes	—
62	C₄×#168	C₂²×C₃	1	1	yes	characteristic
63	C₄×#168	C₂²×C₃	1	1	yes	characteristic
64	C₄×#168	C₂²×C₃	1	1	yes	characteristic
65	C₈×M₁₆	—	5	1	no	Sylow
66	C₂²×C₃×D₅	C₄²	1	1	yes	characteristic
67	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₃×Dic₅	C₄²	1	1	yes	characteristic
71	C₂×C₃×Dic₅	C₄²	1	1	yes	characteristic
72	C₂×C₃×Dic₅	C₄²	1	1	yes	characteristic
73	C₃×C₄×D₅	—	2	2	no	—
74	C₃×C₄×D₅	—	2	2	no	—
75	C₃×C₄×D₅	C₂×C₈	1	4	yes	—
76	C₃×C₅×C₈	—	2	2	no	—
77	C₃×C₅×C₈	C₂×C₈	1	4	yes	complemented
78	C₃×C₅×C₈	M₁₆	1	4	yes	complemented
79	C₃×#168	—	2	2	no	—
80	C₃×#168	—	2	2	no	—
81	C₃×#168	C₂×C₈	1	4	yes	complemented
82	C₂×C₃×M₁₆	—	5	2	no	—
83	C₂²×C₃×C₈	—	5	2	no	—
84	C₂×C₃×C₄²	—	5	1	no	—
85	C₃×C₄⋊C₈	—	5	2	no	—
86	C₃×C₂²⋊C₈	—	5	2	no	—
87	C₃×#97	—	5	2	no	—
88	C₃×C₄×C₈	—	5	2	no	—
89	C₃×C₄×C₈	—	5	1	no	—
90	C₃×C₄×C₈	—	5	1	no	—
91	C₃×C₄×C₈	—	5	1	no	—
92	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
93	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
94	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
95	C₂×C₄×D₅	C₂×C₃×C₄	1	1	yes	characteristic
96	C₂×C₄×D₅	C₂×C₃×C₄	1	1	yes	characteristic
97	C₂×C₄×D₅	C₂×C₃×C₄	1	1	yes	characteristic
98	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
99	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
100	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
101	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
102	C₄×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
103	C₄×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
104	C₄×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
105	C₂×#168	C₂×C₃×C₄	1	2	yes	—
106	C₂×#168	C₂×C₃×C₄	1	2	yes	—
107	C₂×#168	C₂×C₃×C₄	1	2	yes	—
108	#676	C₃×C₈	1	8	yes	direct product factor
109	C₈×D₅	—	2	4	no	—
110	#431	—	5	1	no	—
111	C₄×M₁₆	—	5	1	no	—
112	C₂×C₄×C₈	—	5	1	no	—
113	#322	—	5	1	no	—
114	#322	—	5	1	no	—
115	C₈²	—	5	1	no	—
116	C₈²	C₃×D₅	1	1	yes	characteristic, complemented
117	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
118	C₂×C₃×D₅	—	2	1	no	—
119	C₂×C₃×D₅	—	2	1	no	—
120	C₂×C₃×D₅	C₄×C₈	1	2	yes	—
121	C₃×C₄×C₅	C₂²×C₈	1	2	yes	—
122	C₃×C₄×C₅	C₂×M₁₆	1	2	yes	—
123	C₃×C₄×C₅	#131	1	2	yes	—
124	C₃×Dic₅	—	2	1	no	—
125	C₃×Dic₅	—	2	1	no	—
126	C₃×Dic₅	C₄×C₈	1	2	yes	—
127	C₂²×C₃×C₄	—	5	1	no	—
128	C₂²×C₃×C₄	—	5	1	no	—
129	C₂²×C₃×C₄	—	5	1	no	—
130	C₃×M₁₆	—	5	8	no	—
131	C₂×C₃×C₈	—	5	2	no	—
132	C₂×C₃×C₈	—	5	2	no	—
133	C₂×C₃×C₈	—	5	2	no	—
134	C₂×C₃×C₈	—	10	4	no	—
135	C₂×C₃×C₈	C₄×D₅	1	2	yes	—
136	C₂×C₃×C₈	C₄×D₅	1	2	yes	—
137	C₂×C₃×C₈	C₄×D₅	1	2	yes	—
138	C₃×C₄²	—	5	1	no	—
139	C₃×C₄²	—	5	1	no	—
140	C₃×C₄²	—	5	1	no	—
141	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
142	C₂²×D₅	C₃×C₄²	1	1	yes	characteristic
143	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
144	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
145	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
146	C₂×Dic₅	C₃×C₄²	1	1	yes	characteristic
147	C₂×Dic₅	C₃×C₄²	1	1	yes	characteristic
148	C₂×Dic₅	C₃×C₄²	1	1	yes	characteristic
149	C₄×D₅	—	2	2	no	—
150	C₄×D₅	—	2	2	no	—
151	C₄×D₅	C₂×C₃×C₈	1	4	yes	—
152	C₅×C₈	—	2	2	no	—
153	C₅×C₈	C₂×C₃×C₈	1	4	yes	complemented
154	C₅×C₈	C₃×M₁₆	1	4	yes	complemented
155	#168	—	2	2	no	—
156	#168	—	2	2	no	—
157	#168	C₂×C₃×C₈	1	4	yes	complemented
158	C₂×M₁₆	—	5	2	no	—
159	C₂²×C₈	—	5	2	no	—
160	C₂×C₄²	—	5	1	no	—
161	C₄⋊C₈	—	5	2	no	—
162	C₂²⋊C₈	—	5	2	no	—
163	#97	—	5	2	no	—
164	C₄×C₈	—	5	2	no	—
165	C₄×C₈	—	5	1	no	—
166	C₄×C₈	—	5	1	no	—
167	C₄×C₈	—	5	1	no	—
168	C₄×C₈	C₂×C₃×D₅	1	1	yes	characteristic
169	C₄×C₈	C₂×C₃×D₅	1	1	yes	characteristic
170	C₄×C₈	C₂×C₃×D₅	1	1	yes	characteristic
171	C₂×C₃×C₅	C₂×C₄×C₈	1	1	yes	characteristic
172	C₂×C₃×C₅	C₄×M₁₆	1	1	yes	characteristic
173	C₂×C₃×C₅	#405	1	1	yes	characteristic
174	C₃×D₅	—	2	2	no	—
175	C₂³×C₃	—	5	1	no	—
176	C₂×C₃×C₄	—	5	1	no	—
177	C₂×C₃×C₄	—	5	4	no	—
178	C₂×C₃×C₄	—	5	1	no	—
179	C₂×C₃×C₄	—	5	1	no	—
180	C₂×C₃×C₄	—	10	2	no	—
181	C₂×C₃×C₄	—	10	2	no	—
182	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
183	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
184	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
185	C₃×C₈	—	2	2	no	—
186	C₃×C₈	—	5	4	no	—
187	C₃×C₈	—	10	2	no	—
188	C₃×C₈	—	10	2	no	—
189	C₃×C₈	C₈×D₅	1	4	yes	complemented
190	C₃×C₈	#676	1	4	yes	direct product factor
191	C₂²×C₅	C₂×C₃×C₄²	1	1	yes	characteristic
192	C₂×D₅	—	2	1	no	—
193	C₂×D₅	—	2	1	no	—
194	C₂×D₅	C₃×C₄×C₈	1	2	yes	—
195	C₄×C₅	C₂²×C₃×C₈	1	2	yes	—
196	C₄×C₅	C₂×C₃×M₁₆	1	2	yes	—
197	C₄×C₅	C₃×#131	1	2	yes	—
198	Dic₅	—	2	1	no	—
199	Dic₅	—	2	1	no	—
200	Dic₅	C₃×C₄×C₈	1	2	yes	—
201	C₂²×C₄	—	5	1	no	—
202	C₂²×C₄	—	5	1	no	—
203	C₂²×C₄	—	5	1	no	—
204	M₁₆	—	5	8	no	—
205	C₂×C₈	—	5	2	no	—
206	C₂×C₈	—	5	2	no	—
207	C₂×C₈	—	5	2	no	—
208	C₂×C₈	—	10	4	no	—
209	C₂×C₈	C₃×C₄×D₅	1	2	yes	—
210	C₂×C₈	C₃×C₄×D₅	1	2	yes	—
211	C₂×C₈	C₃×C₄×D₅	1	2	yes	—
212	C₄²	—	5	1	no	—
213	C₄²	—	5	1	no	—
214	C₄²	—	5	1	no	—
215	C₄²	C₂²×C₃×D₅	1	1	yes	characteristic
216	C₃×C₅	C₈×M₁₆	1	1	yes	characteristic, complemented, Hall
217	C₂²×C₃	—	5	2	no	—
218	C₂²×C₃	—	10	1	no	—
219	C₂²×C₃	—	10	1	no	—
220	C₂²×C₃	C₄²×D₅	1	1	yes	characteristic
221	C₃×C₄	—	5	2	no	—
222	C₃×C₄	—	10	1	no	—
223	C₃×C₄	—	10	1	no	—
224	C₃×C₄	C₂×C₈×D₅	1	2	yes	—
225	C₃×C₄	C₂×#676	1	2	yes	—
226	C₃×C₄	#4047	1	2	yes	—
227	C₂×C₅	C₂×C₃×C₄×C₈	1	1	yes	characteristic
228	C₂×C₅	C₃×C₄×M₁₆	1	1	yes	characteristic
229	C₂×C₅	C₃×#405	1	1	yes	characteristic
230	D₅	—	2	2	no	—
231	C₂³	—	5	1	no	—
232	C₂×C₄	—	5	1	no	—
233	C₂×C₄	—	5	4	no	—
234	C₂×C₄	—	5	1	no	—
235	C₂×C₄	—	5	1	no	—
236	C₂×C₄	—	10	2	no	—
237	C₂×C₄	—	10	2	no	—
238	C₂×C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
239	C₂×C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
240	C₂×C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
241	C₈	—	2	2	no	—
242	C₈	—	5	4	no	—
243	C₈	—	10	2	no	—
244	C₈	—	10	2	no	—
245	C₈	C₃×C₈×D₅	1	4	yes	complemented
246	C₈	C₃×#676	1	4	yes	direct product factor
247	C₂×C₃	—	10	2	no	—
248	C₂×C₃	C₄×C₈×D₅	1	1	yes	characteristic
249	C₂×C₃	C₄×#676	1	1	yes	characteristic
250	C₂×C₃	#64978	1	1	yes	characteristic
251	C₅	C₃×C₈×M₁₆	1	1	yes	characteristic, complemented, Sylow
252	C₂²	—	5	2	no	—
253	C₂²	—	10	1	no	—
254	C₂²	—	10	1	no	—
255	C₂²	C₃×C₄²×D₅	1	1	yes	characteristic
256	C₄	—	5	2	no	—
257	C₄	—	10	1	no	—
258	C₄	—	10	1	no	—
259	C₄	C₂×C₃×C₈×D₅	1	2	yes	—
260	C₄	C₂×C₃×#676	1	2	yes	—
261	C₄	C₃×#4047	1	2	yes	—
262	C₃	C₈×#676	1	1	yes	characteristic, direct product factor, Sylow
263	C₂	—	10	2	no	—
264	C₂	C₃×C₄×C₈×D₅	1	1	yes	characteristic
265	C₂	C₃×C₄×#676	1	1	yes	characteristic
266	C₂	C₃×#64978	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	C₃×C₅×C₈²	C₂
socle	C₂²×C₃×C₅	C₂×C₄²
prime radical	C₂²×C₃×D₅	C₄²
cosocle	C₄²×C₅	C₂³×C₃
hypercenter	C₃×C₈²	D₅
hypocenter	C₅	C₃×C₈×M₁₆
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₃×C₄×C₈ ◁ C₃×C₈²

Complements

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

N	H ≅ G/N	type
4. C₃×C₅×C₈²	C₂	semidirect
5. C₃×#64675	C₂	semidirect
6. C₃×#64674	C₂	semidirect
7. C₃×C₈×#168	C₂	semidirect
8. C₈×#676	C₃	direct (N × H)
26. C₅×C₈²	C₂×C₃	semidirect
27. #64675	C₂×C₃	semidirect
28. #64674	C₂×C₃	semidirect
29. C₈×#168	C₂×C₃	semidirect
43. C₃×#676	C₈	direct (N × H)
51. C₃×C₈²	D₅	semidirect
77. C₃×C₅×C₈	C₂×C₈	semidirect
78. C₃×C₅×C₈	M₁₆	semidirect
81. C₃×#168	C₂×C₈	semidirect
108. #676	C₃×C₈	direct (N × H)
116. C₈²	C₃×D₅	semidirect
153. C₅×C₈	C₂×C₃×C₈	semidirect
154. C₅×C₈	C₃×M₁₆	semidirect
157. #168	C₂×C₃×C₈	semidirect
189. C₃×C₈	C₈×D₅	semidirect
190. C₃×C₈	#676	direct (N × H)
216. C₃×C₅	C₈×M₁₆	semidirect
245. C₈	C₃×C₈×D₅	semidirect
246. C₈	C₃×#676	direct (N × H)
251. C₅	C₃×C₈×M₁₆	semidirect
262. C₃	C₈×#676	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×C₄×#676	2
C₃×#64974	2
C₃×C₄×C₈×D₅	2
C₃×C₅×C₈²	2
C₃×#64675	2
C₃×#64674	2
C₃×C₈×#168	2
C₈×#676	3
C₃×C₈×M₁₆	5

Subgroups — C3×C8×#676 #14280358

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₈×#676 #14280358