Subgroups — C₄×C₅×Dic₃⋊C₈ #14296094

Order 1920.

632 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	C₂×C₃×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×Dic₃⋊C₈	C₂	1	2	yes	maximal
3	C₄²×C₅×Dic₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₅×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₅×#4550	C₂	1	2	yes	maximal
6	C₄×C₅×C₄⋊C₈	—	3	1	no	maximal, Hall
7	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
8	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₅×Dic₃	C₄	1	2	yes	—
10	C₂×C₄×C₅×Dic₃	C₄	1	2	yes	—
11	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
13	C₂²×C₅×C₃⋊C₈	C₂²	1	2	yes	—
14	C₃×C₄×C₅×C₈	—	2	2	no	—
15	C₅×Dic₃⋊C₈	C₄	1	8	yes	direct product factor
16	C₄×C₅×C₃⋊C₈	—	2	2	no	—
17	C₄×Dic₃⋊C₈	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₂×C₅×C₄⋊C₈	—	3	2	no	—
19	C₂×C₄×C₅×C₈	—	3	1	no	—
20	C₂×C₄×C₅×C₈	D₃	1	1	yes	characteristic
21	C₄³×C₅	—	3	1	no	—
22	C₅×#336	—	3	2	no	—
23	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
24	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
25	C₂²×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₅×Dic₃	C₂×C₄	1	2	yes	—
27	C₂²×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₃×C₅×C₈	—	2	2	no	—
29	C₂×C₃×C₅×C₈	—	2	2	no	—
30	C₂×C₃×C₅×C₈	C₂×C₄	1	4	yes	—
31	C₃×C₄²×C₅	D₄	1	2	yes	—
32	C₃×C₄²×C₅	Q₈	1	2	yes	—
33	C₄×C₅×Dic₃	—	2	4	no	—
34	C₄×C₅×Dic₃	—	2	2	no	—
35	C₄×C₅×Dic₃	C₈	1	8	yes	complemented
36	C₄×C₅×Dic₃	C₂×C₄	1	4	yes	—
37	C₂×C₅×C₃⋊C₈	—	2	2	no	—
38	C₂×C₅×C₃⋊C₈	—	2	2	no	—
39	C₂×C₅×C₃⋊C₈	C₂×C₄	1	4	yes	—
40	C₂×C₃×C₄×C₈	C₂×C₅	1	1	yes	characteristic
41	C₂×Dic₃⋊C₈	C₂×C₅	1	2	yes	—
42	C₄²×Dic₃	C₂×C₅	1	1	yes	characteristic
43	C₂×C₄×C₃⋊C₈	C₂×C₅	1	1	yes	characteristic
44	#4550	C₂×C₅	1	2	yes	—
45	C₂²×C₅×C₈	—	3	2	no	—
46	C₂²×C₅×C₈	C₂×D₃	1	2	yes	—
47	C₂×C₄²×C₅	—	3	2	no	—
48	C₂×C₄²×C₅	—	3	1	no	—
49	C₂×C₄²×C₅	—	3	2	no	—
50	C₂×C₄²×C₅	—	3	1	no	—
51	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
52	C₅×C₄⋊C₈	—	3	8	no	—
53	C₄×C₅×C₈	—	2	2	no	—
54	C₄×C₅×C₈	—	6	2	no	—
55	C₄×C₄⋊C₈	—	3	1	no	Sylow
56	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₃×C₄×C₅	C₄⋊C₄	1	4	yes	—
58	C₂×C₃×C₄×C₅	C₂×C₈	1	4	yes	—
59	C₂×C₃×C₄×C₅	M₁₆	1	4	yes	—
60	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
61	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
62	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
63	C₂×C₃×C₄×C₅	#41	1	2	yes	—
64	C₂×C₅×Dic₃	—	2	2	no	—
65	C₂×C₅×Dic₃	—	2	2	no	—
66	C₂×C₅×Dic₃	—	2	2	no	—
67	C₂×C₅×Dic₃	C₄²	1	2	yes	—
68	C₂×C₅×Dic₃	C₄²	1	2	yes	—
69	C₂×C₅×Dic₃	C₂×C₈	1	4	yes	—
70	C₂×C₅×Dic₃	C₂×C₈	1	4	yes	—
71	C₃×C₅×C₈	—	2	4	no	—
72	C₅×C₃⋊C₈	—	2	4	no	—
73	C₂²×C₃×C₈	C₂²×C₅	1	2	yes	—
74	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
75	C₂×C₄×Dic₃	C₄×C₅	1	2	yes	—
76	C₂×C₄×Dic₃	C₄×C₅	1	2	yes	—
77	C₂×C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
78	C₂×C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
79	C₂²×C₃⋊C₈	C₂²×C₅	1	2	yes	—
80	C₃×C₄×C₈	—	2	2	no	—
81	Dic₃⋊C₈	C₄×C₅	1	8	yes	direct product factor
82	C₄×C₃⋊C₈	—	2	2	no	—
83	C₂²×C₄×C₅	—	3	1	no	—
84	C₂²×C₄×C₅	—	3	1	no	—
85	C₂²×C₄×C₅	—	3	2	no	—
86	C₂²×C₄×C₅	C₄×D₃	1	2	yes	—
87	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
88	C₂×C₅×C₈	—	2	2	no	—
89	C₂×C₅×C₈	—	2	2	no	—
90	C₂×C₅×C₈	—	3	4	no	—
91	C₂×C₅×C₈	—	6	2	no	—
92	C₂×C₅×C₈	—	6	2	no	—
93	C₂×C₅×C₈	C₄×D₃	1	4	yes	—
94	C₄²×C₅	—	3	8	no	—
95	C₄²×C₅	—	3	4	no	—
96	C₄²×C₅	—	6	4	no	—
97	C₄²×C₅	—	6	2	no	—
98	C₄²×C₅	Dic₆	1	2	yes	—
99	C₄²×C₅	#67	1	2	yes	—
100	C₂×C₄⋊C₈	—	3	2	no	—
101	C₂×C₄×C₈	—	3	1	no	—
102	C₂×C₄×C₈	C₅×D₃	1	1	yes	characteristic
103	C₄³	—	3	1	no	—
104	#336	—	3	2	no	—
105	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
106	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂²×C₃×C₅	#117	1	1	yes	characteristic
108	C₂²×C₃×C₅	C₂²×C₈	1	2	yes	—
109	C₂²×C₃×C₅	C₂×M₁₆	1	2	yes	—
110	C₃×C₄×C₅	C₄⋊C₈	1	8	yes	complemented
111	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
112	C₃×C₄×C₅	C₄×Q₈	1	2	yes	—
113	C₅×Dic₃	—	2	2	no	—
114	C₅×Dic₃	—	2	4	no	—
115	C₅×Dic₃	C₄×C₈	1	4	yes	complemented
116	C₂²×C₃×C₄	C₂×C₄×C₅	1	2	yes	—
117	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
118	C₂²×Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
119	C₂²×Dic₃	C₂×C₄×C₅	1	2	yes	—
120	C₂²×Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
121	C₂×C₃×C₈	—	2	2	no	—
122	C₂×C₃×C₈	—	2	2	no	—
123	C₂×C₃×C₈	C₂×C₄×C₅	1	4	yes	—
124	C₃×C₄²	C₅×D₄	1	2	yes	—
125	C₃×C₄²	C₅×Q₈	1	2	yes	—
126	C₄×Dic₃	—	2	4	no	—
127	C₄×Dic₃	—	2	2	no	—
128	C₄×Dic₃	C₅×C₈	1	8	yes	complemented
129	C₄×Dic₃	C₂×C₄×C₅	1	4	yes	—
130	C₂×C₃⋊C₈	—	2	2	no	—
131	C₂×C₃⋊C₈	—	2	2	no	—
132	C₂×C₃⋊C₈	C₂×C₄×C₅	1	4	yes	—
133	C₂³×C₅	C₂×C₄×D₃	1	1	yes	characteristic
134	C₂×C₄×C₅	—	3	2	no	—
135	C₂×C₄×C₅	—	3	4	no	—
136	C₂×C₄×C₅	—	3	2	no	—
137	C₂×C₄×C₅	—	3	4	no	—
138	C₂×C₄×C₅	—	6	2	no	—
139	C₂×C₄×C₅	—	6	2	no	—
140	C₂×C₄×C₅	—	6	2	no	—
141	C₂×C₄×C₅	D₃×C₈	1	4	yes	—
142	C₂×C₄×C₅	ΓL(1,𝔽₂₅)	1	4	yes	—
143	C₂×C₄×C₅	Dic₃⋊C₄	1	4	yes	—
144	C₂×C₄×C₅	C₂×Dic₆	1	1	yes	characteristic
145	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
146	C₂×C₄×C₅	#235	1	2	yes	—
147	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
148	C₅×C₈	—	2	4	no	—
149	C₅×C₈	—	6	4	no	—
150	C₂²×C₈	—	3	2	no	—
151	C₂²×C₈	C₂×C₅×D₃	1	2	yes	—
152	C₂×C₄²	—	3	2	no	—
153	C₂×C₄²	—	3	1	no	—
154	C₂×C₄²	—	3	2	no	—
155	C₂×C₄²	—	3	1	no	—
156	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
157	C₄⋊C₈	—	3	8	no	—
158	C₄×C₈	—	2	2	no	—
159	C₄×C₈	—	6	2	no	—
160	C₂×C₃×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
161	C₂×C₃×C₅	C₂×C₄×C₈	1	1	yes	characteristic
162	C₂×C₃×C₅	C₄×M₁₆	1	1	yes	characteristic
163	C₂×C₃×C₅	C₂×C₄⋊C₈	1	2	yes	—
164	C₂×C₃×C₅	#431	1	2	yes	—
165	C₂³×C₃	C₂²×C₄×C₅	1	1	yes	characteristic
166	C₂×C₃×C₄	C₅×C₄⋊C₄	1	4	yes	—
167	C₂×C₃×C₄	C₂×C₅×C₈	1	4	yes	—
168	C₂×C₃×C₄	C₅×M₁₆	1	4	yes	—
169	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
170	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
171	C₂×C₃×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
172	C₂×C₃×C₄	C₅×#41	1	2	yes	—
173	C₂×Dic₃	—	2	2	no	—
174	C₂×Dic₃	—	2	2	no	—
175	C₂×Dic₃	—	2	2	no	—
176	C₂×Dic₃	C₄²×C₅	1	2	yes	—
177	C₂×Dic₃	C₄²×C₅	1	2	yes	—
178	C₂×Dic₃	C₂×C₅×C₈	1	4	yes	—
179	C₂×Dic₃	C₂×C₅×C₈	1	4	yes	—
180	C₃×C₈	—	2	4	no	—
181	C₃⋊C₈	—	2	4	no	—
182	C₂²×C₅	C₄²×D₃	1	1	yes	characteristic
183	C₂²×C₅	#872	1	1	yes	characteristic
184	C₂²×C₅	C₂×D₃×C₈	1	2	yes	—
185	C₂²×C₅	C₂×ΓL(1,𝔽₂₅)	1	2	yes	—
186	C₂²×C₅	C₂×Dic₃⋊C₄	1	1	yes	characteristic
187	C₄×C₅	—	3	4	no	—
188	C₄×C₅	—	6	4	no	—
189	C₄×C₅	—	6	2	no	—
190	C₄×C₅	Dic₃⋊C₈	1	8	yes	direct product factor
191	C₄×C₅	C₄×Dic₆	1	2	yes	—
192	C₄×C₅	C₄×#67	1	2	yes	—
193	C₂²×C₄	—	3	1	no	—
194	C₂²×C₄	—	3	1	no	—
195	C₂²×C₄	—	3	2	no	—
196	C₂²×C₄	C₄×C₅×D₃	1	2	yes	—
197	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	3	4	no	—
201	C₂×C₈	—	6	2	no	—
202	C₂×C₈	—	6	2	no	—
203	C₂×C₈	C₄×C₅×D₃	1	4	yes	—
204	C₄²	—	3	8	no	—
205	C₄²	—	3	4	no	—
206	C₄²	—	6	4	no	—
207	C₄²	—	6	2	no	—
208	C₄²	C₅×Dic₆	1	2	yes	—
209	C₄²	C₅×#67	1	2	yes	—
210	C₃×C₅	C₄×C₄⋊C₈	1	1	yes	characteristic, complemented, Hall
211	C₂²×C₃	C₂×C₄²×C₅	1	1	yes	characteristic
212	C₂²×C₃	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
213	C₂²×C₃	C₅×#117	1	1	yes	characteristic
214	C₂²×C₃	C₂²×C₅×C₈	1	2	yes	—
215	C₂²×C₃	C₂×C₅×M₁₆	1	2	yes	—
216	C₃×C₄	C₅×C₄⋊C₈	1	8	yes	complemented
217	C₃×C₄	C₄×C₅×D₄	1	2	yes	—
218	C₃×C₄	C₄×C₅×Q₈	1	2	yes	—
219	Dic₃	—	2	2	no	—
220	Dic₃	—	2	4	no	—
221	Dic₃	C₄×C₅×C₈	1	4	yes	complemented
222	C₂×C₅	C₄×D₃×C₈	1	1	yes	characteristic
223	C₂×C₅	#4685	1	2	yes	—
224	C₂×C₅	C₄×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
225	C₂×C₅	C₄×Dic₃⋊C₄	1	1	yes	characteristic
226	C₂×C₅	C₂×Dic₃⋊C₈	1	2	yes	—
227	C₂³	C₂×C₄×C₅×D₃	1	1	yes	characteristic
228	C₂×C₄	—	3	2	no	—
229	C₂×C₄	—	3	4	no	—
230	C₂×C₄	—	3	2	no	—
231	C₂×C₄	—	3	4	no	—
232	C₂×C₄	—	6	2	no	—
233	C₂×C₄	—	6	2	no	—
234	C₂×C₄	—	6	2	no	—
235	C₂×C₄	C₅×D₃×C₈	1	4	yes	—
236	C₂×C₄	C₅×ΓL(1,𝔽₂₅)	1	4	yes	—
237	C₂×C₄	C₅×Dic₃⋊C₄	1	4	yes	—
238	C₂×C₄	C₂×C₅×Dic₆	1	1	yes	characteristic
239	C₂×C₄	C₂×C₄×C₅×D₃	1	2	yes	—
240	C₂×C₄	C₅×#235	1	2	yes	—
241	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
242	C₈	—	2	4	no	—
243	C₈	—	6	4	no	—
244	C₂×C₃	C₄×C₅×C₄⋊C₄	1	1	yes	characteristic
245	C₂×C₃	C₂×C₄×C₅×C₈	1	1	yes	characteristic
246	C₂×C₃	C₄×C₅×M₁₆	1	1	yes	characteristic
247	C₂×C₃	C₂×C₅×C₄⋊C₈	1	2	yes	—
248	C₂×C₃	C₅×#431	1	2	yes	—
249	C₅	C₄×Dic₃⋊C₈	1	1	yes	characteristic, direct product factor, Sylow
250	C₂²	C₄²×C₅×D₃	1	1	yes	characteristic
251	C₂²	C₅×#872	1	1	yes	characteristic
252	C₂²	C₂×C₅×D₃×C₈	1	2	yes	—
253	C₂²	C₂×C₅×ΓL(1,𝔽₂₅)	1	2	yes	—
254	C₂²	C₂×C₅×Dic₃⋊C₄	1	1	yes	characteristic
255	C₄	—	3	4	no	—
256	C₄	—	6	4	no	—
257	C₄	—	6	2	no	—
258	C₄	C₅×Dic₃⋊C₈	1	8	yes	direct product factor
259	C₄	C₄×C₅×Dic₆	1	2	yes	—
260	C₄	C₄×C₅×#67	1	2	yes	—
261	C₃	C₄×C₅×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
262	C₂	C₄×C₅×D₃×C₈	1	1	yes	characteristic
263	C₂	C₅×#4685	1	2	yes	—
264	C₂	C₄×C₅×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
265	C₂	C₄×C₅×Dic₃⋊C₄	1	1	yes	characteristic
266	C₂	C₂×C₅×Dic₃⋊C₈	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²×C₅	C₂×D₃
commutator	C₂×C₃	C₂×C₄×C₅×C₈
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₂×C₃×C₄×C₅×C₈	C₂
socle, prime radical	C₂³×C₃×C₅	C₂²×C₄
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₂×C₄×C₅×C₈	D₃
hypocenter	C₃	C₄×C₅×C₄⋊C₈
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×C₈

Complements

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

N	H ≅ G/N	type
15. C₅×Dic₃⋊C₈	C₄	direct (N × H)
17. C₄×Dic₃⋊C₈	C₅	direct (N × H)
35. C₄×C₅×Dic₃	C₈	semidirect
81. Dic₃⋊C₈	C₄×C₅	direct (N × H)
110. C₃×C₄×C₅	C₄⋊C₈	semidirect
115. C₅×Dic₃	C₄×C₈	semidirect
128. C₄×Dic₃	C₅×C₈	semidirect
190. C₄×C₅	Dic₃⋊C₈	direct (N × H)
210. C₃×C₅	C₄×C₄⋊C₈	semidirect
216. C₃×C₄	C₅×C₄⋊C₈	semidirect
221. Dic₃	C₄×C₅×C₈	semidirect
249. C₅	C₄×Dic₃⋊C₈	direct (N × H)
258. C₄	C₅×Dic₃⋊C₈	direct (N × H)
261. C₃	C₄×C₅×C₄⋊C₈	semidirect

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
C₂×C₅×Dic₃⋊C₈	2
C₄²×C₅×Dic₃	2
C₂×C₄×C₅×C₃⋊C₈	2
C₅×#4550	2
C₄×C₅×C₄⋊C₈	3
C₄×Dic₃⋊C₈	5

Subgroups — C4×C5×Dic3⋊C8 #14296094

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₅×Dic₃⋊C₈ #14296094