Subgroups — C₅×C₈×C₄⋊Dic₃ #14296232

Order 1920.

592 conjugacy classes of subgroups.

The subgroup lattice has 276 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×C₈×Dic₃	C₂	1	2	yes	maximal
3	C₄×C₅×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
5	C₅×#4550	C₂	1	2	yes	maximal
6	C₅×C₈×C₄⋊C₄	—	3	1	no	maximal, Hall
7	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₂×C₅×C₄⋊Dic₃	C₄	1	2	yes	—
11	C₂×C₄×C₅×Dic₃	C₂²	1	2	yes	—
12	C₂²×C₅×C₃⋊C₈	C₂²	1	2	yes	—
13	C₃×C₄×C₅×C₈	C₄	1	4	yes	complemented
14	C₅×C₃⋊#95	C₄	1	2	yes	—
15	C₅×C₈×Dic₃	—	2	4	no	—
16	C₅×C₄⋊(C₃⋊C₈)	C₄	1	4	yes	complemented
17	C₈×C₄⋊Dic₃	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₂×C₅×C₄⋊C₈	—	3	1	no	—
19	C₂×C₄×C₅×C₈	—	3	2	no	—
20	C₂×C₄×C₅×C₈	D₃	1	1	yes	characteristic
21	C₄×C₅×C₄⋊C₄	—	3	1	no	—
22	C₅×#336	—	3	2	no	—
23	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
26	C₂²×C₅×Dic₃	C₂×C₄	1	4	yes	—
27	C₂×C₃×C₅×C₈	—	2	1	no	—
28	C₂×C₃×C₅×C₈	—	2	1	no	—
29	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
30	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
31	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
32	C₂×C₃×C₅×C₈	Q₈	1	2	yes	—
33	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
34	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
35	C₅×C₄⋊Dic₃	C₈	1	8	yes	direct product factor
36	C₄×C₅×Dic₃	—	2	2	no	—
37	C₄×C₅×Dic₃	—	2	2	no	—
38	C₂×C₅×C₃⋊C₈	—	2	2	no	—
39	C₂×C₅×C₃⋊C₈	—	2	2	no	—
40	C₂×C₅×C₃⋊C₈	C₂×C₄	1	4	yes	—
41	C₂×C₃×C₄×C₈	C₂×C₅	1	1	yes	characteristic
42	C₂×C₈×Dic₃	C₂×C₅	1	2	yes	—
43	C₄×C₄⋊Dic₃	C₂×C₅	1	1	yes	characteristic
44	C₂×C₄⋊(C₃⋊C₈)	C₂×C₅	1	1	yes	characteristic
45	#4550	C₂×C₅	1	2	yes	—
46	C₂²×C₅×C₈	—	3	2	no	—
47	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
48	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
49	C₂×C₅×C₄⋊C₄	—	3	2	no	—
50	C₂×C₄²×C₅	—	3	2	no	—
51	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
52	C₅×C₄⋊C₈	—	3	4	no	—
53	C₄×C₅×C₈	—	6	4	no	—
54	C₄×C₅×C₈	Dic₃	1	4	yes	complemented
55	C₅×#95	—	3	2	no	—
56	C₈×C₄⋊C₄	—	3	1	no	Sylow
57	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₃×C₄×C₅	—	2	2	no	—
59	C₂×C₃×C₄×C₅	C₄²	1	2	yes	—
60	C₂×C₃×C₄×C₅	C₄²	1	2	yes	—
61	C₂×C₃×C₄×C₅	C₂×C₈	1	4	yes	—
62	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
65	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
66	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
67	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
68	C₂×C₅×Dic₃	—	2	4	no	—
69	C₂×C₅×Dic₃	—	2	4	no	—
70	C₂×C₅×Dic₃	C₂×C₈	1	8	yes	—
71	C₃×C₅×C₈	—	2	2	no	—
72	C₃×C₅×C₈	C₄⋊C₄	1	4	yes	complemented
73	C₅×C₃⋊C₈	—	2	4	no	—
74	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
75	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
76	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
77	C₂×C₄⋊Dic₃	C₄×C₅	1	2	yes	—
78	C₂×C₄×Dic₃	C₂²×C₅	1	2	yes	—
79	C₂²×C₃⋊C₈	C₂²×C₅	1	2	yes	—
80	C₃×C₄×C₈	C₄×C₅	1	4	yes	complemented
81	C₃⋊#95	C₄×C₅	1	2	yes	—
82	C₈×Dic₃	—	2	4	no	—
83	C₄⋊(C₃⋊C₈)	C₄×C₅	1	4	yes	complemented
84	C₂²×C₄×C₅	—	3	4	no	—
85	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
86	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
87	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
88	C₂×C₅×C₈	—	2	1	no	—
89	C₂×C₅×C₈	—	2	1	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	Dic₆	1	2	yes	—
94	C₂×C₅×C₈	D₁₂	1	2	yes	—
95	C₂×C₅×C₈	C₂×Dic₃	1	2	yes	—
96	C₂×C₅×C₈	C₂×Dic₃	1	2	yes	—
97	C₅×C₄⋊C₄	—	3	8	no	—
98	C₄²×C₅	—	6	2	no	—
99	C₄²×C₅	—	6	2	no	—
100	C₄²×C₅	C₄×D₃	1	2	yes	—
101	C₄²×C₅	C₂×Dic₃	1	2	yes	—
102	C₂×C₄⋊C₈	—	3	1	no	—
103	C₂×C₄×C₈	—	3	2	no	—
104	C₂×C₄×C₈	C₅×D₃	1	1	yes	characteristic
105	C₄×C₄⋊C₄	—	3	1	no	—
106	#336	—	3	2	no	—
107	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
108	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
109	C₂²×C₃×C₅	C₄×Q₈	1	1	yes	characteristic
110	C₂²×C₃×C₅	C₂²×C₈	1	2	yes	—
111	C₂²×C₃×C₅	#131	1	2	yes	—
112	C₃×C₄×C₅	—	2	2	no	—
113	C₃×C₄×C₅	C₄×C₈	1	4	yes	complemented
114	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₃×C₄×C₅	#117	1	1	yes	characteristic
116	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
117	C₃×C₄×C₅	C₄×Q₈	1	1	yes	characteristic
118	C₅×Dic₃	—	2	8	no	—
119	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
120	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
121	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
122	C₂²×Dic₃	C₂×C₄×C₅	1	4	yes	—
123	C₂×C₃×C₈	—	2	1	no	—
124	C₂×C₃×C₈	—	2	1	no	—
125	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
126	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
127	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
128	C₂×C₃×C₈	C₅×Q₈	1	2	yes	—
129	C₃×C₄²	C₂×C₄×C₅	1	2	yes	—
130	C₃×C₄²	C₂×C₄×C₅	1	2	yes	—
131	C₄⋊Dic₃	C₅×C₈	1	8	yes	direct product factor
132	C₄×Dic₃	—	2	2	no	—
133	C₄×Dic₃	—	2	2	no	—
134	C₂×C₃⋊C₈	—	2	2	no	—
135	C₂×C₃⋊C₈	—	2	2	no	—
136	C₂×C₃⋊C₈	C₂×C₄×C₅	1	4	yes	—
137	C₂³×C₅	C₂×C₄×D₃	1	1	yes	characteristic
138	C₂×C₄×C₅	—	2	2	no	—
139	C₂×C₄×C₅	—	3	8	no	—
140	C₂×C₄×C₅	—	6	4	no	—
141	C₂×C₄×C₅	—	6	4	no	—
142	C₂×C₄×C₅	D₃×C₈	1	4	yes	—
143	C₂×C₄×C₅	C₄×Dic₃	1	2	yes	—
144	C₂×C₄×C₅	C₄×Dic₃	1	2	yes	—
145	C₂×C₄×C₅	C₂×Dic₆	1	1	yes	characteristic
146	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
147	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
148	C₂×C₄×C₅	#235	1	1	yes	characteristic
149	C₂×C₄×C₅	#235	1	1	yes	characteristic
150	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
151	C₅×C₈	—	2	2	no	—
152	C₅×C₈	—	6	4	no	—
153	C₅×C₈	C₄⋊Dic₃	1	4	yes	direct product factor
154	C₂²×C₈	—	3	2	no	—
155	C₂²×C₈	C₂×C₅×D₃	1	1	yes	characteristic
156	C₂²×C₈	C₂×C₅×D₃	1	1	yes	characteristic
157	C₂×C₄⋊C₄	—	3	2	no	—
158	C₂×C₄²	—	3	2	no	—
159	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
160	C₄⋊C₈	—	3	4	no	—
161	C₄×C₈	—	6	4	no	—
162	C₄×C₈	C₅×Dic₃	1	4	yes	complemented
163	#95	—	3	2	no	—
164	C₂×C₃×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
165	C₂×C₃×C₅	C₂×C₄×C₈	1	1	yes	characteristic
166	C₂×C₃×C₅	#405	1	1	yes	characteristic
167	C₂×C₃×C₅	C₈×D₄	1	2	yes	—
168	C₂×C₃×C₅	C₈×Q₈	1	2	yes	—
169	C₂³×C₃	C₂²×C₄×C₅	1	1	yes	characteristic
170	C₂×C₃×C₄	—	2	2	no	—
171	C₂×C₃×C₄	C₄²×C₅	1	2	yes	—
172	C₂×C₃×C₄	C₄²×C₅	1	2	yes	—
173	C₂×C₃×C₄	C₂×C₅×C₈	1	4	yes	—
174	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
175	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
176	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
177	C₂×C₃×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
178	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
179	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
180	C₂×Dic₃	—	2	4	no	—
181	C₂×Dic₃	—	2	4	no	—
182	C₂×Dic₃	C₂×C₅×C₈	1	8	yes	—
183	C₃×C₈	—	2	2	no	—
184	C₃×C₈	C₅×C₄⋊C₄	1	4	yes	complemented
185	C₃⋊C₈	—	2	4	no	—
186	C₂²×C₅	C₄×Dic₆	1	1	yes	characteristic
187	C₂²×C₅	C₄×D₁₂	1	1	yes	characteristic
188	C₂²×C₅	C₂×D₃×C₈	1	2	yes	—
189	C₂²×C₅	#901	1	2	yes	—
190	C₂²×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
191	C₄×C₅	—	2	2	no	—
192	C₄×C₅	—	6	8	no	—
193	C₄×C₅	C₈×Dic₃	1	4	yes	complemented
194	C₄×C₅	C₄×Dic₆	1	1	yes	characteristic
195	C₄×C₅	C₄×D₁₂	1	1	yes	characteristic
196	C₄×C₅	C₂×C₄⋊Dic₃	1	1	yes	characteristic
197	C₄×C₅	#926	1	1	yes	characteristic
198	C₂²×C₄	—	3	4	no	—
199	C₂²×C₄	C₄×C₅×D₃	1	1	yes	characteristic
200	C₂²×C₄	C₄×C₅×D₃	1	1	yes	characteristic
201	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
202	C₂×C₈	—	2	1	no	—
203	C₂×C₈	—	2	1	no	—
204	C₂×C₈	—	3	4	no	—
205	C₂×C₈	—	6	2	no	—
206	C₂×C₈	—	6	2	no	—
207	C₂×C₈	C₅×Dic₆	1	2	yes	—
208	C₂×C₈	C₅×D₁₂	1	2	yes	—
209	C₂×C₈	C₂×C₅×Dic₃	1	2	yes	—
210	C₂×C₈	C₂×C₅×Dic₃	1	2	yes	—
211	C₄⋊C₄	—	3	8	no	—
212	C₄²	—	6	2	no	—
213	C₄²	—	6	2	no	—
214	C₄²	C₄×C₅×D₃	1	2	yes	—
215	C₄²	C₂×C₅×Dic₃	1	2	yes	—
216	C₃×C₅	C₈×C₄⋊C₄	1	1	yes	characteristic, complemented, Hall
217	C₂²×C₃	C₂×C₄²×C₅	1	1	yes	characteristic
218	C₂²×C₃	C₄×C₅×D₄	1	1	yes	characteristic
219	C₂²×C₃	C₄×C₅×Q₈	1	1	yes	characteristic
220	C₂²×C₃	C₂²×C₅×C₈	1	2	yes	—
221	C₂²×C₃	C₅×#131	1	2	yes	—
222	C₃×C₄	—	2	2	no	—
223	C₃×C₄	C₄×C₅×C₈	1	4	yes	complemented
224	C₃×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
225	C₃×C₄	C₅×#117	1	1	yes	characteristic
226	C₃×C₄	C₄×C₅×D₄	1	1	yes	characteristic
227	C₃×C₄	C₄×C₅×Q₈	1	1	yes	characteristic
228	Dic₃	—	2	8	no	—
229	C₂×C₅	C₈×Dic₆	1	2	yes	—
230	C₂×C₅	C₈×D₁₂	1	2	yes	—
231	C₂×C₅	C₄×C₄⋊Dic₃	1	1	yes	characteristic
232	C₂×C₅	C₂×C₈×Dic₃	1	1	yes	characteristic
233	C₂×C₅	#5101	1	1	yes	characteristic
234	C₂³	C₂×C₄×C₅×D₃	1	1	yes	characteristic
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	3	8	no	—
237	C₂×C₄	—	6	4	no	—
238	C₂×C₄	—	6	4	no	—
239	C₂×C₄	C₅×D₃×C₈	1	4	yes	—
240	C₂×C₄	C₄×C₅×Dic₃	1	2	yes	—
241	C₂×C₄	C₄×C₅×Dic₃	1	2	yes	—
242	C₂×C₄	C₂×C₅×Dic₆	1	1	yes	characteristic
243	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
244	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
245	C₂×C₄	C₅×#235	1	1	yes	characteristic
246	C₂×C₄	C₅×#235	1	1	yes	characteristic
247	C₂×C₄	C₂²×C₅×Dic₃	1	1	yes	characteristic
248	C₈	—	2	2	no	—
249	C₈	—	6	4	no	—
250	C₈	C₅×C₄⋊Dic₃	1	4	yes	direct product factor
251	C₂×C₃	C₄×C₅×C₄⋊C₄	1	1	yes	characteristic
252	C₂×C₃	C₂×C₄×C₅×C₈	1	1	yes	characteristic
253	C₂×C₃	C₅×#405	1	1	yes	characteristic
254	C₂×C₃	C₅×C₈×D₄	1	2	yes	—
255	C₂×C₃	C₅×C₈×Q₈	1	2	yes	—
256	C₅	C₈×C₄⋊Dic₃	1	1	yes	characteristic, direct product factor, Sylow
257	C₂²	C₄×C₅×Dic₆	1	1	yes	characteristic
258	C₂²	C₄×C₅×D₁₂	1	1	yes	characteristic
259	C₂²	C₂×C₅×D₃×C₈	1	2	yes	—
260	C₂²	C₅×#901	1	2	yes	—
261	C₂²	C₂×C₄×C₅×Dic₃	1	1	yes	characteristic
262	C₄	—	2	2	no	—
263	C₄	—	6	8	no	—
264	C₄	C₅×C₈×Dic₃	1	4	yes	complemented
265	C₄	C₄×C₅×Dic₆	1	1	yes	characteristic
266	C₄	C₄×C₅×D₁₂	1	1	yes	characteristic
267	C₄	C₂×C₅×C₄⋊Dic₃	1	1	yes	characteristic
268	C₄	C₅×#926	1	1	yes	characteristic
269	C₃	C₅×C₈×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
270	C₂	C₅×C₈×Dic₆	1	2	yes	—
271	C₂	C₅×C₈×D₁₂	1	2	yes	—
272	C₂	C₄×C₅×C₄⋊Dic₃	1	1	yes	characteristic
273	C₂	C₂×C₅×C₈×Dic₃	1	1	yes	characteristic
274	C₂	C₅×#5101	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅×C₈	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	C₅×C₈×C₄⋊C₄
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅×C₈ ◁ C₂×C₄×C₅×C₈

Complements

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

N	H ≅ G/N	type
13. C₃×C₄×C₅×C₈	C₄	semidirect
16. C₅×C₄⋊(C₃⋊C₈)	C₄	semidirect
17. C₈×C₄⋊Dic₃	C₅	direct (N × H)
35. C₅×C₄⋊Dic₃	C₈	direct (N × H)
54. C₄×C₅×C₈	Dic₃	semidirect
72. C₃×C₅×C₈	C₄⋊C₄	semidirect
80. C₃×C₄×C₈	C₄×C₅	semidirect
83. C₄⋊(C₃⋊C₈)	C₄×C₅	semidirect
113. C₃×C₄×C₅	C₄×C₈	semidirect
131. C₄⋊Dic₃	C₅×C₈	direct (N × H)
153. C₅×C₈	C₄⋊Dic₃	direct (N × H)
162. C₄×C₈	C₅×Dic₃	semidirect
184. C₃×C₈	C₅×C₄⋊C₄	semidirect
193. C₄×C₅	C₈×Dic₃	semidirect
216. C₃×C₅	C₈×C₄⋊C₄	semidirect
223. C₃×C₄	C₄×C₅×C₈	semidirect
250. C₈	C₅×C₄⋊Dic₃	direct (N × H)
256. C₅	C₈×C₄⋊Dic₃	direct (N × H)
264. C₄	C₅×C₈×Dic₃	semidirect
269. C₃	C₅×C₈×C₄⋊C₄	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×C₈×Dic₃	2
C₄×C₅×C₄⋊Dic₃	2
C₂×C₅×C₄⋊(C₃⋊C₈)	2
C₅×#4550	2
C₅×C₈×C₄⋊C₄	3
C₈×C₄⋊Dic₃	5

Subgroups — C5×C8×C4⋊Dic3 #14296232

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×C₈×C₄⋊Dic₃ #14296232