Subgroups — C₃×C₈×#713 #14418894

Order 1920.

912 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×#713	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×C₈×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×Q₈×#168	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#65123	C₂	1	3	yes	maximal, complemented
5	C₃×#65122	C₂	1	3	yes	maximal, complemented
6	C₃×C₈×D₂₀	C₂	1	3	yes	maximal, complemented
7	C₃×C₄×C₈×D₅	C₂	1	3	yes	maximal, complemented
8	C₈×#713	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₃×#713	C₄	1	1	yes	characteristic
10	C₃×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
11	C₃×Q₈×Dic₅	C₄	1	1	yes	characteristic
12	C₂×C₃×C₈×D₅	C₂²	1	3	yes	—
13	C₂×C₃×C₈×D₅	C₂²	1	3	yes	—
14	C₃×#4039	C₄	1	3	yes	—
15	C₃×#4038	C₄	1	3	yes	—
16	C₃×C₄×D₂₀	C₂²	1	3	yes	—
17	C₃×C₄²×D₅	C₂²	1	3	yes	—
18	C₃×C₅×C₄⋊C₈	C₂²	1	3	yes	—
19	C₃×C₄×C₅×C₈	C₂²	1	3	yes	—
20	C₃×#3952	C₂²	1	6	yes	—
21	C₃×C₈×Dic₅	C₂²	1	3	yes	—
22	C₃×C₈×Dic₅	C₂²	1	1	yes	characteristic
23	C₃×C₄⋊#168	C₂²	1	3	yes	—
24	C₃×C₄×#168	C₂²	1	3	yes	—
25	C₃×C₈×#41	—	5	1	no	maximal, Hall
26	C₄×#713	C₂×C₃	1	1	yes	characteristic
27	C₅×C₈×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
28	Q₈×#168	C₂×C₃	1	1	yes	characteristic, complemented
29	#65123	C₂×C₃	1	3	yes	complemented
30	#65122	C₂×C₃	1	3	yes	complemented
31	C₈×D₂₀	C₂×C₃	1	3	yes	complemented
32	C₄×C₈×D₅	C₂×C₃	1	3	yes	complemented
33	C₂×C₃×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
34	C₃×#713	C₈	1	8	yes	direct product factor
35	C₂×C₃×D₂₀	C₂×C₄	1	3	yes	—
36	C₂×C₃×C₄×D₅	C₂×C₄	1	3	yes	—
37	C₂×C₃×C₄×D₅	C₂×C₄	1	3	yes	—
38	C₂×C₃×C₄×D₅	C₂³	1	3	yes	—
39	C₂×C₃×C₅×C₈	C₂³	1	3	yes	—
40	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
41	C₃×C₅×C₄⋊C₄	C₂×C₄	1	3	yes	—
42	C₃×C₄²×C₅	C₂³	1	3	yes	—
43	C₃×#685	C₂×C₄	1	6	yes	—
44	C₃×C₄⋊Dic₅	C₂×C₄	1	3	yes	—
45	C₃×C₄×Dic₅	C₂×C₄	1	3	yes	—
46	C₃×C₄×Dic₅	C₂×C₄	1	3	yes	—
47	C₃×C₄×Dic₅	C₂³	1	1	yes	characteristic
48	C₂×C₃×#168	C₂³	1	3	yes	—
49	C₂×C₃×#168	C₂³	1	1	yes	characteristic
50	C₃×C₈×D₅	—	2	6	no	—
51	C₃×C₈×D₅	—	2	6	no	—
52	C₃×C₄×#41	—	5	1	no	—
53	C₃×C₈×Q₈	—	5	1	no	—
54	C₃×C₈×Q₈	D₅	1	1	yes	characteristic, complemented
55	C₃×C₈×D₄	—	5	3	no	—
56	C₃×C₈×D₄	—	5	3	no	—
57	C₃×#431	—	5	3	no	—
58	C₂×C₃×C₄×C₈	—	5	3	no	—
59	C₂×#713	C₃×C₄	1	1	yes	characteristic
60	C₄×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
61	Q₈×Dic₅	C₃×C₄	1	1	yes	characteristic
62	C₂×C₈×D₅	C₂²×C₃	1	3	yes	—
63	C₂×C₈×D₅	C₂²×C₃	1	3	yes	—
64	#4039	C₃×C₄	1	3	yes	—
65	#4038	C₃×C₄	1	3	yes	—
66	C₄×D₂₀	C₂²×C₃	1	3	yes	—
67	C₄²×D₅	C₂²×C₃	1	3	yes	—
68	C₅×C₄⋊C₈	C₂²×C₃	1	3	yes	—
69	C₄×C₅×C₈	C₂²×C₃	1	3	yes	—
70	#3952	C₂²×C₃	1	6	yes	—
71	C₈×Dic₅	C₂²×C₃	1	3	yes	—
72	C₈×Dic₅	C₂²×C₃	1	1	yes	characteristic
73	C₄⋊#168	C₂²×C₃	1	3	yes	—
74	C₄×#168	C₂²×C₃	1	3	yes	—
75	C₈×#41	—	5	1	no	Sylow
76	C₂²×C₃×D₅	C₂²×C₄	1	3	yes	—
77	C₃×C₅×Q₈	C₂×C₈	1	4	yes	complemented
78	C₂×C₃×C₄×C₅	C₂²×C₄	1	3	yes	—
79	C₂×C₃×C₄×C₅	C₂²×C₄	1	3	yes	—
80	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
81	C₂×C₃×Dic₅	C₂²×C₄	1	3	yes	—
82	C₂×C₃×Dic₅	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₃×Dic₅	C₂²×C₄	1	1	yes	characteristic
84	C₃×D₂₀	C₂×C₈	1	12	yes	complemented
85	C₃×C₄×D₅	—	2	3	no	—
86	C₃×C₄×D₅	—	2	3	no	—
87	C₃×C₄×D₅	—	2	6	no	—
88	C₃×C₄×D₅	C₂×C₈	1	12	yes	complemented
89	C₃×C₅×C₈	—	2	3	no	—
90	C₃×C₅×C₈	#41	1	2	yes	complemented
91	C₃×#168	—	2	3	no	—
92	C₃×#168	#41	1	2	yes	complemented
93	C₂×C₃×#41	—	5	1	no	—
94	C₂²×C₃×C₈	—	5	3	no	—
95	C₂²×C₃×C₈	—	5	3	no	—
96	C₃×C₄×Q₈	—	5	1	no	—
97	C₃×C₄×Q₈	C₂×D₅	1	1	yes	characteristic
98	C₃×C₄×D₄	—	5	3	no	—
99	C₃×C₄×D₄	—	5	3	no	—
100	C₃×#117	—	5	3	no	—
101	C₂×C₃×C₄²	—	5	3	no	—
102	C₃×C₄⋊C₈	—	5	3	no	—
103	C₃×C₄⋊C₈	C₂×D₅	1	3	yes	—
104	C₃×C₂²⋊C₈	—	5	6	no	—
105	C₃×C₄×C₈	—	5	3	no	—
106	C₃×C₄×C₈	—	5	1	no	—
107	C₃×C₄×C₈	—	5	3	no	—
108	C₃×C₄×C₈	C₂×D₅	1	3	yes	—
109	C₂×C₅×Q₈	C₂×C₃×C₄	1	1	yes	characteristic
110	#713	C₃×C₈	1	8	yes	direct product factor
111	C₂×D₂₀	C₂×C₃×C₄	1	3	yes	—
112	C₂×C₄×D₅	C₂×C₃×C₄	1	3	yes	—
113	C₂×C₄×D₅	C₂×C₃×C₄	1	3	yes	—
114	C₂×C₄×D₅	C₂³×C₃	1	3	yes	—
115	C₂×C₅×C₈	C₂³×C₃	1	3	yes	—
116	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
117	C₅×C₄⋊C₄	C₂×C₃×C₄	1	3	yes	—
118	C₄²×C₅	C₂³×C₃	1	3	yes	—
119	#685	C₂×C₃×C₄	1	6	yes	—
120	C₄⋊Dic₅	C₂×C₃×C₄	1	3	yes	—
121	C₄×Dic₅	C₂×C₃×C₄	1	3	yes	—
122	C₄×Dic₅	C₂×C₃×C₄	1	3	yes	—
123	C₄×Dic₅	C₂³×C₃	1	1	yes	characteristic
124	C₂×#168	C₂³×C₃	1	3	yes	—
125	C₂×#168	C₂³×C₃	1	1	yes	characteristic
126	C₈×D₅	—	2	6	no	—
127	C₈×D₅	—	2	6	no	—
128	C₄×#41	—	5	1	no	—
129	C₈×Q₈	—	5	1	no	—
130	C₈×Q₈	C₃×D₅	1	1	yes	characteristic, complemented
131	C₈×D₄	—	5	3	no	—
132	C₈×D₄	—	5	3	no	—
133	#431	—	5	3	no	—
134	C₂×C₄×C₈	—	5	3	no	—
135	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
136	C₂×C₃×D₅	—	2	3	no	—
137	C₂×C₃×D₅	—	2	3	no	—
138	C₂×C₃×D₅	C₂²×C₈	1	6	yes	—
139	C₃×C₄×C₅	—	2	3	no	—
140	C₃×C₄×C₅	C₂²×C₈	1	6	yes	—
141	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
142	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
143	C₃×Dic₅	—	2	3	no	—
144	C₃×Dic₅	C₂²×C₈	1	2	yes	—
145	C₃×Dic₅	#131	1	2	yes	—
146	C₃×#41	—	5	8	no	—
147	C₂×C₃×Q₈	C₄×D₅	1	1	yes	characteristic
148	C₂×C₃×D₄	—	5	3	no	—
149	C₂²×C₃×C₄	—	5	3	no	—
150	C₂²×C₃×C₄	—	5	3	no	—
151	C₂²×C₃×C₄	—	5	3	no	—
152	C₂×C₃×C₈	—	5	1	no	—
153	C₂×C₃×C₈	—	5	3	no	—
154	C₂×C₃×C₈	—	10	6	no	—
155	C₂×C₃×C₈	—	10	6	no	—
156	C₂×C₃×C₈	C₂²×D₅	1	3	yes	—
157	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
158	C₃×C₄⋊C₄	—	5	3	no	—
159	C₃×C₄⋊C₄	C₄×D₅	1	3	yes	—
160	C₃×C₂²⋊C₄	—	5	6	no	—
161	C₃×C₄²	—	5	3	no	—
162	C₃×C₄²	—	5	3	no	—
163	C₃×C₄²	—	5	1	no	—
164	C₃×C₄²	C₂²×D₅	1	3	yes	—
165	C₂²×D₅	C₂²×C₃×C₄	1	3	yes	—
166	C₅×Q₈	C₂×C₃×C₈	1	4	yes	complemented
167	C₂×C₄×C₅	C₂²×C₃×C₄	1	3	yes	—
168	C₂×C₄×C₅	C₂²×C₃×C₄	1	3	yes	—
169	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
170	C₂×Dic₅	C₂²×C₃×C₄	1	3	yes	—
171	C₂×Dic₅	C₂²×C₃×C₄	1	1	yes	characteristic
172	C₂×Dic₅	C₂²×C₃×C₄	1	1	yes	characteristic
173	D₂₀	C₂×C₃×C₈	1	12	yes	complemented
174	C₄×D₅	—	2	3	no	—
175	C₄×D₅	—	2	3	no	—
176	C₄×D₅	—	2	6	no	—
177	C₄×D₅	C₂×C₃×C₈	1	12	yes	complemented
178	C₅×C₈	—	2	3	no	—
179	C₅×C₈	C₃×#41	1	2	yes	complemented
180	#168	—	2	3	no	—
181	#168	C₃×#41	1	2	yes	complemented
182	C₂×#41	—	5	1	no	—
183	C₂²×C₈	—	5	3	no	—
184	C₂²×C₈	—	5	3	no	—
185	C₄×Q₈	—	5	1	no	—
186	C₄×Q₈	C₂×C₃×D₅	1	1	yes	characteristic
187	C₄×D₄	—	5	3	no	—
188	C₄×D₄	—	5	3	no	—
189	#117	—	5	3	no	—
190	C₂×C₄²	—	5	3	no	—
191	C₄⋊C₈	—	5	3	no	—
192	C₄⋊C₈	C₂×C₃×D₅	1	3	yes	—
193	C₂²⋊C₈	—	5	6	no	—
194	C₄×C₈	—	5	1	no	—
195	C₄×C₈	—	5	3	no	—
196	C₄×C₈	—	5	3	no	—
197	C₄×C₈	C₂×C₃×D₅	1	3	yes	—
198	C₂×C₃×C₅	C₄×#41	1	1	yes	characteristic
199	C₂×C₃×C₅	C₂³×C₈	1	1	yes	characteristic
200	C₂×C₃×C₅	C₂×#131	1	1	yes	characteristic
201	C₃×D₅	—	2	6	no	—
202	C₂³×C₃	—	5	3	no	—
203	C₃×Q₈	C₈×D₅	1	4	yes	complemented
204	C₃×D₄	—	5	12	no	—
205	C₂×C₃×C₄	—	5	1	no	—
206	C₂×C₃×C₄	—	5	1	no	—
207	C₂×C₃×C₄	—	5	12	no	—
208	C₂×C₃×C₄	—	5	3	no	—
209	C₂×C₃×C₄	—	10	6	no	—
210	C₂×C₃×C₄	—	10	3	no	—
211	C₂×C₃×C₄	—	10	3	no	—
212	C₂×C₃×C₄	C₂×C₄×D₅	1	3	yes	—
213	C₂×C₃×C₄	C₂×C₄×D₅	1	3	yes	—
214	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
215	C₃×C₈	—	2	3	no	—
216	C₃×C₈	—	5	2	no	—
217	C₃×C₈	—	10	3	no	—
218	C₃×C₈	#713	1	2	yes	direct product factor
219	C₂²×C₅	C₂³×C₃×C₄	1	1	yes	characteristic
220	C₂×D₅	—	2	3	no	—
221	C₂×D₅	—	2	3	no	—
222	C₂×D₅	C₂²×C₃×C₈	1	6	yes	—
223	C₄×C₅	—	2	3	no	—
224	C₄×C₅	C₂²×C₃×C₈	1	6	yes	—
225	C₄×C₅	C₂×C₃×#41	1	1	yes	characteristic
226	C₄×C₅	C₂×C₃×#41	1	1	yes	characteristic
227	Dic₅	—	2	3	no	—
228	Dic₅	C₂²×C₃×C₈	1	2	yes	—
229	Dic₅	C₃×#131	1	2	yes	—
230	#41	—	5	8	no	—
231	C₂×Q₈	C₃×C₄×D₅	1	1	yes	characteristic
232	C₂×D₄	—	5	3	no	—
233	C₂²×C₄	—	5	3	no	—
234	C₂²×C₄	—	5	3	no	—
235	C₂²×C₄	—	5	3	no	—
236	C₂×C₈	—	5	1	no	—
237	C₂×C₈	—	5	3	no	—
238	C₂×C₈	—	10	6	no	—
239	C₂×C₈	—	10	6	no	—
240	C₂×C₈	C₂²×C₃×D₅	1	3	yes	—
241	C₂×C₈	C₂²×C₃×D₅	1	1	yes	characteristic
242	C₄⋊C₄	—	5	3	no	—
243	C₄⋊C₄	C₃×C₄×D₅	1	3	yes	—
244	C₂²⋊C₄	—	5	6	no	—
245	C₄²	—	5	3	no	—
246	C₄²	—	5	3	no	—
247	C₄²	—	5	1	no	—
248	C₄²	C₂²×C₃×D₅	1	3	yes	—
249	C₃×C₅	C₈×#41	1	1	yes	characteristic, complemented, Hall
250	C₂²×C₃	—	5	6	no	—
251	C₂²×C₃	—	10	3	no	—
252	C₂²×C₃	—	10	3	no	—
253	C₂²×C₃	C₂²×C₄×D₅	1	1	yes	characteristic
254	C₃×C₄	—	2	3	no	—
255	C₃×C₄	—	5	2	no	—
256	C₃×C₄	—	5	2	no	—
257	C₃×C₄	—	10	3	no	—
258	C₃×C₄	C₂×C₈×D₅	1	6	yes	—
259	C₃×C₄	C₂×#713	1	1	yes	characteristic
260	C₃×C₄	D₅×#41	1	1	yes	characteristic
261	C₂×C₅	C₃×C₄×#41	1	1	yes	characteristic
262	C₂×C₅	C₂³×C₃×C₈	1	1	yes	characteristic
263	C₂×C₅	C₂×C₃×#131	1	1	yes	characteristic
264	D₅	—	2	6	no	—
265	C₂³	—	5	3	no	—
266	Q₈	C₃×C₈×D₅	1	4	yes	complemented
267	D₄	—	5	12	no	—
268	C₂×C₄	—	5	1	no	—
269	C₂×C₄	—	5	1	no	—
270	C₂×C₄	—	5	12	no	—
271	C₂×C₄	—	5	3	no	—
272	C₂×C₄	—	10	6	no	—
273	C₂×C₄	—	10	3	no	—
274	C₂×C₄	—	10	3	no	—
275	C₂×C₄	C₂×C₃×C₄×D₅	1	3	yes	—
276	C₂×C₄	C₂×C₃×C₄×D₅	1	3	yes	—
277	C₂×C₄	C₂³×C₃×D₅	1	1	yes	characteristic
278	C₈	—	2	3	no	—
279	C₈	—	5	2	no	—
280	C₈	—	10	3	no	—
281	C₈	C₃×#713	1	2	yes	direct product factor
282	C₂×C₃	—	10	6	no	—
283	C₂×C₃	C₄×#713	1	1	yes	characteristic
284	C₂×C₃	C₂²×C₈×D₅	1	1	yes	characteristic
285	C₂×C₃	D₅×#131	1	1	yes	characteristic
286	C₅	C₃×C₈×#41	1	1	yes	characteristic, complemented, Sylow
287	C₂²	—	5	6	no	—
288	C₂²	—	10	3	no	—
289	C₂²	—	10	3	no	—
290	C₂²	C₂²×C₃×C₄×D₅	1	1	yes	characteristic
291	C₄	—	2	3	no	—
292	C₄	—	5	2	no	—
293	C₄	—	5	2	no	—
294	C₄	—	10	3	no	—
295	C₄	C₂×C₃×C₈×D₅	1	6	yes	—
296	C₄	C₂×C₃×#713	1	1	yes	characteristic
297	C₄	C₃×D₅×#41	1	1	yes	characteristic
298	C₃	C₈×#713	1	1	yes	characteristic, direct product factor, Sylow
299	C₂	—	10	6	no	—
300	C₂	C₃×C₄×#713	1	1	yes	characteristic
301	C₂	C₂²×C₃×C₈×D₅	1	1	yes	characteristic
302	C₂	C₃×D₅×#131	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₈	C₂²×D₅
commutator	C₂×C₅	C₂³×C₃×C₈
Frattini	C₂×C₄	C₂³×C₃×D₅
Fitting	C₃×C₅×C₈×Q₈	C₂
socle	C₂²×C₃×C₅	C₂³×C₄
prime radical	C₂×C₃×#713	C₄
cosocle	C₂×C₄×C₅	C₂⁴×C₃
hypercenter	C₃×C₈×Q₈	D₅
hypocenter	C₅	C₃×C₈×#41
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₈ ◁ C₃×C₈×Q₈

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×C₈×Q₈	C₂	semidirect
3. C₃×Q₈×#168	C₂	semidirect
4. C₃×#65123	C₂	semidirect
5. C₃×#65122	C₂	semidirect
6. C₃×C₈×D₂₀	C₂	semidirect
7. C₃×C₄×C₈×D₅	C₂	semidirect
8. C₈×#713	C₃	direct (N × H)
27. C₅×C₈×Q₈	C₂×C₃	semidirect
28. Q₈×#168	C₂×C₃	semidirect
29. #65123	C₂×C₃	semidirect
30. #65122	C₂×C₃	semidirect
31. C₈×D₂₀	C₂×C₃	semidirect
32. C₄×C₈×D₅	C₂×C₃	semidirect
34. C₃×#713	C₈	direct (N × H)
54. C₃×C₈×Q₈	D₅	semidirect
77. C₃×C₅×Q₈	C₂×C₈	semidirect
84. C₃×D₂₀	C₂×C₈	semidirect
88. C₃×C₄×D₅	C₂×C₈	semidirect
90. C₃×C₅×C₈	#41	semidirect
92. C₃×#168	#41	semidirect
110. #713	C₃×C₈	direct (N × H)
130. C₈×Q₈	C₃×D₅	semidirect
166. C₅×Q₈	C₂×C₃×C₈	semidirect
173. D₂₀	C₂×C₃×C₈	semidirect
177. C₄×D₅	C₂×C₃×C₈	semidirect
179. C₅×C₈	C₃×#41	semidirect
181. #168	C₃×#41	semidirect
203. C₃×Q₈	C₈×D₅	semidirect
218. C₃×C₈	#713	direct (N × H)
249. C₃×C₅	C₈×#41	semidirect
266. Q₈	C₃×C₈×D₅	semidirect
281. C₈	C₃×#713	direct (N × H)
286. C₅	C₃×C₈×#41	semidirect
298. C₃	C₈×#713	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×#713	2
C₃×C₅×C₈×Q₈	2
C₃×Q₈×#168	2
C₃×#65123	2
C₃×#65122	2
C₃×C₈×D₂₀	2
C₃×C₄×C₈×D₅	2
C₈×#713	3
C₃×C₈×#41	5

Subgroups — C3×C8×#713 #14418894

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₈×#713 #14418894