Subgroups — C₈×#6719 #14295144

Order 1920.

768 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₅×C₈	C₂	1	1	yes	maximal, characteristic
2	C₄×#6719	C₂	1	1	yes	maximal, characteristic
3	C₂×#91114	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈×Dic₁₅	C₂	1	2	yes	maximal, complemented
5	#11746371	C₂	1	2	yes	maximal, complemented
6	C₈×C₂²⋊Dic₅	—	3	1	no	maximal, Hall
7	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₅×C₈	—	2	4	no	—
9	C₂²×C₃×C₅×C₈	C₄	1	4	yes	complemented
10	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₂×#6719	C₄	1	2	yes	—
13	C₂×C₄×Dic₁₅	C₂²	1	2	yes	—
14	C₂²×C₃⋊#168	C₂²	1	2	yes	—
15	#91115	C₄	1	2	yes	—
16	#91114	C₄	1	4	yes	complemented
17	C₈×Dic₁₅	—	2	4	no	—
18	C₈×#217	—	5	1	no	maximal, Hall
19	C₂³×C₅×C₈	D₃	1	1	yes	characteristic
20	C₄×C₂²⋊Dic₅	—	3	1	no	—
21	C₂×C₂²⋊#168	—	3	1	no	—
22	C₂×C₈×Dic₅	—	3	2	no	—
23	#64770	—	3	2	no	—
24	C₂⁴×C₃×C₅	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₃×C₄×C₅	—	2	2	no	—
26	C₂²×C₃×C₄×C₅	—	2	2	no	—
27	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
29	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
30	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
31	C₂²×Dic₁₅	C₂×C₄	1	4	yes	—
32	C₂×C₃×C₅×C₈	—	2	2	no	—
33	C₂×C₃×C₅×C₈	—	2	8	no	—
34	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
35	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
36	C₂×C₃×C₅×C₈	D₄	1	4	yes	—
37	#6719	C₈	1	8	yes	direct product factor
38	C₄×Dic₁₅	—	2	2	no	—
39	C₄×Dic₁₅	—	2	2	no	—
40	C₂×C₃⋊#168	—	2	2	no	—
41	C₂×C₃⋊#168	—	2	2	no	—
42	C₂×C₃⋊#168	C₂×C₄	1	4	yes	—
43	C₂³×C₃×C₈	D₅	1	1	yes	characteristic
44	C₄×#217	—	5	1	no	—
45	C₂×#830	—	5	1	no	—
46	C₂×C₈×Dic₃	—	5	2	no	—
47	#4550	—	5	2	no	—
48	C₂³×C₄×C₅	C₂×D₃	1	1	yes	characteristic
49	C₂²×C₅×C₈	—	2	4	no	—
50	C₂²×C₅×C₈	Dic₃	1	4	yes	complemented
51	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
52	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
53	C₂×C₂²⋊Dic₅	—	3	2	no	—
54	C₂×C₄×Dic₅	—	3	2	no	—
55	C₂²×#168	—	3	2	no	—
56	#3963	—	3	2	no	—
57	C₂²⋊#168	—	3	4	no	—
58	C₈×Dic₅	—	6	4	no	—
59	C₈×C₂²⋊C₄	—	15	1	no	Sylow
60	C₂³×C₃×C₅	—	2	2	no	—
61	C₂³×C₃×C₅	—	2	2	no	—
62	C₂³×C₃×C₅	C₄²	1	2	yes	—
63	C₂³×C₃×C₅	C₂×C₈	1	4	yes	—
64	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₃×C₄×C₅	—	2	2	no	—
66	C₂×C₃×C₄×C₅	—	2	2	no	—
67	C₂×C₃×C₄×C₅	—	2	2	no	—
68	C₂×C₃×C₄×C₅	—	2	4	no	—
69	C₂×C₃×C₄×C₅	C₄²	1	2	yes	—
70	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
73	C₂×C₃×C₄×C₅	#41	1	2	yes	—
74	C₂×Dic₁₅	—	2	4	no	—
75	C₂×Dic₁₅	—	2	4	no	—
76	C₂×Dic₁₅	C₂×C₈	1	8	yes	complemented
77	C₃×C₅×C₈	—	2	2	no	—
78	C₃×C₅×C₈	C₂²⋊C₄	1	4	yes	complemented
79	C₃⋊#168	—	2	4	no	—
80	C₂³×C₃×C₄	C₂×D₅	1	1	yes	characteristic
81	C₂²×C₃×C₈	—	2	4	no	—
82	C₂²×C₃×C₈	Dic₅	1	4	yes	complemented
83	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
84	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
85	C₂×#217	—	5	2	no	—
86	C₂×C₄×Dic₃	—	5	2	no	—
87	C₂²×C₃⋊C₈	—	5	2	no	—
88	C₃⋊#95	—	5	2	no	—
89	#830	—	5	4	no	—
90	C₈×Dic₃	—	10	4	no	—
91	C₂⁴×C₅	C₄×D₃	1	1	yes	characteristic
92	C₂²×C₄×C₅	—	2	2	no	—
93	C₂²×C₄×C₅	—	2	2	no	—
94	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
95	C₂²×C₄×C₅	C₄×D₃	1	2	yes	—
96	C₂²×C₄×C₅	C₂×Dic₃	1	2	yes	—
97	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
98	C₂²×Dic₅	—	3	4	no	—
99	C₂×C₅×C₈	—	2	2	no	—
100	C₂×C₅×C₈	—	2	8	no	—
101	C₂×C₅×C₈	C₂×Dic₃	1	2	yes	—
102	C₂×C₅×C₈	C₂×Dic₃	1	2	yes	—
103	C₂×C₅×C₈	#67	1	4	yes	—
104	C₂²⋊Dic₅	—	3	8	no	—
105	C₄×Dic₅	—	6	2	no	—
106	C₄×Dic₅	—	6	2	no	—
107	C₂×#168	—	3	4	no	—
108	C₂×#168	—	6	2	no	—
109	C₂×#168	—	6	2	no	—
110	C₂³×C₈	D₁₅	1	1	yes	characteristic
111	C₂×C₂²⋊C₈	—	15	1	no	—
112	C₂×C₄×C₈	—	15	2	no	—
113	C₄×C₂²⋊C₄	—	15	1	no	—
114	#336	—	15	2	no	—
115	C₂²×C₃×C₅	—	2	2	no	—
116	C₂²×C₃×C₅	—	2	2	no	—
117	C₂²×C₃×C₅	—	2	8	no	—
118	C₂²×C₃×C₅	C₄×C₈	1	4	yes	complemented
119	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
120	C₂²×C₃×C₅	C₄×D₄	1	2	yes	—
121	C₂²×C₃×C₅	C₂²×C₈	1	2	yes	—
122	C₂²×C₃×C₅	#131	1	2	yes	—
123	C₃×C₄×C₅	—	2	2	no	—
124	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₃×C₄×C₅	#117	1	1	yes	characteristic
126	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
127	Dic₁₅	—	2	8	no	—
128	C₂⁴×C₃	C₄×D₅	1	1	yes	characteristic
129	C₂²×C₃×C₄	—	2	2	no	—
130	C₂²×C₃×C₄	—	2	2	no	—
131	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
132	C₂²×C₃×C₄	C₄×D₅	1	2	yes	—
133	C₂²×C₃×C₄	C₂×Dic₅	1	2	yes	—
134	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
135	C₂²×Dic₃	—	5	4	no	—
136	C₂×C₃×C₈	—	2	2	no	—
137	C₂×C₃×C₈	—	2	8	no	—
138	C₂×C₃×C₈	C₂×Dic₅	1	2	yes	—
139	C₂×C₃×C₈	C₂×Dic₅	1	2	yes	—
140	C₂×C₃×C₈	C₂²⋊D₅	1	4	yes	—
141	#217	—	5	8	no	—
142	C₄×Dic₃	—	10	2	no	—
143	C₄×Dic₃	—	10	2	no	—
144	C₂×C₃⋊C₈	—	5	4	no	—
145	C₂×C₃⋊C₈	—	10	2	no	—
146	C₂×C₃⋊C₈	—	10	2	no	—
147	C₂³×C₅	—	2	2	no	—
148	C₂³×C₅	—	2	2	no	—
149	C₂³×C₅	D₃×C₈	1	4	yes	—
150	C₂³×C₅	C₄×Dic₃	1	2	yes	—
151	C₂³×C₅	C₂×C₄×D₃	1	1	yes	characteristic
152	C₂×C₄×C₅	—	2	2	no	—
153	C₂×C₄×C₅	—	2	2	no	—
154	C₂×C₄×C₅	—	2	4	no	—
155	C₂×C₄×C₅	—	2	2	no	—
156	C₂×C₄×C₅	C₄×Dic₃	1	2	yes	—
157	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
158	C₂×C₄×C₅	#235	1	2	yes	—
159	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
160	C₂×C₄×C₅	C₂×#67	1	2	yes	—
161	C₂×Dic₅	—	3	8	no	—
162	C₂×Dic₅	—	6	4	no	—
163	C₂×Dic₅	—	6	4	no	—
164	C₅×C₈	—	2	2	no	—
165	C₅×C₈	#217	1	4	yes	complemented
166	#168	—	6	4	no	—
167	C₂³×C₄	C₂×D₁₅	1	1	yes	characteristic
168	C₂²×C₈	—	2	4	no	—
169	C₂²×C₈	—	15	2	no	—
170	C₂²×C₈	Dic₁₅	1	4	yes	complemented
171	C₂²×C₈	C₂×D₁₅	1	1	yes	characteristic
172	C₂²×C₈	C₂×D₁₅	1	1	yes	characteristic
173	C₂×C₂²⋊C₄	—	15	2	no	—
174	C₂×C₄²	—	15	2	no	—
175	C₂²⋊C₈	—	15	4	no	—
176	C₄×C₈	—	30	4	no	—
177	#95	—	15	2	no	—
178	C₂×C₃×C₅	—	2	4	no	—
179	C₂×C₃×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
180	C₂×C₃×C₅	C₂×C₄×C₈	1	1	yes	characteristic
181	C₂×C₃×C₅	#405	1	1	yes	characteristic
182	C₂×C₃×C₅	C₈×D₄	1	4	yes	—
183	C₂³×C₃	—	2	2	no	—
184	C₂³×C₃	—	2	2	no	—
185	C₂³×C₃	C₈×D₅	1	4	yes	—
186	C₂³×C₃	C₄×Dic₅	1	2	yes	—
187	C₂³×C₃	C₂×C₄×D₅	1	1	yes	characteristic
188	C₂×C₃×C₄	—	2	2	no	—
189	C₂×C₃×C₄	—	2	2	no	—
190	C₂×C₃×C₄	—	2	2	no	—
191	C₂×C₃×C₄	—	2	4	no	—
192	C₂×C₃×C₄	C₄×Dic₅	1	2	yes	—
193	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
194	C₂×C₃×C₄	#709	1	2	yes	—
195	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
196	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
197	C₂×Dic₃	—	5	8	no	—
198	C₂×Dic₃	—	10	4	no	—
199	C₂×Dic₃	—	10	4	no	—
200	C₃×C₈	—	2	2	no	—
201	C₃×C₈	C₂²⋊Dic₅	1	4	yes	complemented
202	C₃⋊C₈	—	10	4	no	—
203	C₂²×C₅	—	2	2	no	—
204	C₂²×C₅	—	2	2	no	—
205	C₂²×C₅	—	2	8	no	—
206	C₂²×C₅	C₈×Dic₃	1	4	yes	complemented
207	C₂²×C₅	C₂×D₃×C₈	1	2	yes	—
208	C₂²×C₅	#901	1	2	yes	—
209	C₂²×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
210	C₂²×C₅	C₄×#67	1	2	yes	—
211	C₄×C₅	—	2	2	no	—
212	C₄×C₅	#926	1	1	yes	characteristic
213	C₄×C₅	C₄×#67	1	2	yes	—
214	C₄×C₅	C₂×#217	1	1	yes	characteristic
215	Dic₅	—	6	8	no	—
216	C₂⁴	C₄×D₁₅	1	1	yes	characteristic
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	15	4	no	—
220	C₂²×C₄	C₄×D₁₅	1	1	yes	characteristic
221	C₂²×C₄	C₄×D₁₅	1	2	yes	—
222	C₂²×C₄	C₂×Dic₁₅	1	2	yes	—
223	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	8	no	—
226	C₂×C₈	—	15	4	no	—
227	C₂×C₈	—	30	2	no	—
228	C₂×C₈	—	30	2	no	—
229	C₂×C₈	C₂×Dic₁₅	1	2	yes	—
230	C₂×C₈	C₂×Dic₁₅	1	2	yes	—
231	C₂×C₈	#1220	1	4	yes	—
232	C₂²⋊C₄	—	15	8	no	—
233	C₄²	—	30	2	no	—
234	C₄²	—	30	2	no	—
235	C₃×C₅	C₈×C₂²⋊C₄	1	1	yes	characteristic, complemented, Hall
236	C₂²×C₃	—	2	2	no	—
237	C₂²×C₃	—	2	8	no	—
238	C₂²×C₃	—	2	2	no	—
239	C₂²×C₃	C₈×Dic₅	1	4	yes	complemented
240	C₂²×C₃	C₂×C₈×D₅	1	2	yes	—
241	C₂²×C₃	#4047	1	2	yes	—
242	C₂²×C₃	C₂×C₄×Dic₅	1	1	yes	characteristic
243	C₂²×C₃	C₄×C₂²⋊D₅	1	2	yes	—
244	C₃×C₄	—	2	2	no	—
245	C₃×C₄	#4072	1	1	yes	characteristic
246	C₃×C₄	C₄×C₂²⋊D₅	1	2	yes	—
247	C₃×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
248	Dic₃	—	10	8	no	—
249	C₂×C₅	—	2	4	no	—
250	C₂×C₅	C₂×C₈×Dic₃	1	1	yes	characteristic
251	C₂×C₅	#5101	1	1	yes	characteristic
252	C₂×C₅	C₈×#67	1	4	yes	—
253	C₂×C₅	C₄×#217	1	1	yes	characteristic
254	C₂³	—	2	2	no	—
255	C₂³	—	2	2	no	—
256	C₂³	C₈×D₁₅	1	4	yes	—
257	C₂³	C₄×Dic₁₅	1	2	yes	—
258	C₂³	C₂×C₄×D₁₅	1	1	yes	characteristic
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	15	8	no	—
264	C₂×C₄	—	30	4	no	—
265	C₂×C₄	—	30	4	no	—
266	C₂×C₄	C₄×Dic₁₅	1	2	yes	—
267	C₂×C₄	C₂×C₄×D₁₅	1	1	yes	characteristic
268	C₂×C₄	#6817	1	2	yes	—
269	C₂×C₄	C₂²×Dic₁₅	1	1	yes	characteristic
270	C₂×C₄	C₂×#1220	1	2	yes	—
271	C₈	—	2	2	no	—
272	C₈	—	30	4	no	—
273	C₈	#6719	1	4	yes	direct product factor
274	C₂×C₃	—	2	4	no	—
275	C₂×C₃	C₂×C₈×Dic₅	1	1	yes	characteristic
276	C₂×C₃	#65390	1	1	yes	characteristic
277	C₂×C₃	C₈×C₂²⋊D₅	1	4	yes	—
278	C₂×C₃	C₄×C₂²⋊Dic₅	1	1	yes	characteristic
279	C₅	C₈×#217	1	1	yes	characteristic, complemented, Sylow
280	C₂²	—	2	2	no	—
281	C₂²	—	2	8	no	—
282	C₂²	—	2	2	no	—
283	C₂²	C₈×Dic₁₅	1	4	yes	complemented
284	C₂²	C₂×C₈×D₁₅	1	2	yes	—
285	C₂²	#91790	1	2	yes	—
286	C₂²	C₂×C₄×Dic₁₅	1	1	yes	characteristic
287	C₂²	C₄×#1220	1	2	yes	—
288	C₄	—	2	2	no	—
289	C₄	—	30	8	no	—
290	C₄	#91815	1	1	yes	characteristic
291	C₄	C₄×#1220	1	2	yes	—
292	C₄	C₂×#6719	1	1	yes	characteristic
293	C₃	C₈×C₂²⋊Dic₅	1	1	yes	characteristic, complemented, Sylow
294	C₂	—	2	4	no	—
295	C₂	C₂×C₈×Dic₁₅	1	1	yes	characteristic
296	C₂	#11751202	1	1	yes	characteristic
297	C₂	C₈×#1220	1	4	yes	—
298	C₂	C₄×#6719	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂×D₁₅
commutator	C₂×C₃×C₅	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×D₁₅
Fitting	C₂³×C₃×C₅×C₈	C₂
socle	C₂³×C₃×C₅	C₂²×C₄
prime radical	C₂⁴×C₃×C₅	C₂×C₄
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	C₂³×C₈	D₁₅
hypocenter	C₃×C₅	C₈×C₂²⋊C₄
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂²×C₈ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
4. C₂×C₈×Dic₁₅	C₂	semidirect
5. #11746371	C₂	semidirect
9. C₂²×C₃×C₅×C₈	C₄	semidirect
16. #91114	C₄	semidirect
37. #6719	C₈	direct (N × H)
50. C₂²×C₅×C₈	Dic₃	semidirect
76. C₂×Dic₁₅	C₂×C₈	semidirect
78. C₃×C₅×C₈	C₂²⋊C₄	semidirect
82. C₂²×C₃×C₈	Dic₅	semidirect
118. C₂²×C₃×C₅	C₄×C₈	semidirect
165. C₅×C₈	#217	semidirect
170. C₂²×C₈	Dic₁₅	semidirect
201. C₃×C₈	C₂²⋊Dic₅	semidirect
206. C₂²×C₅	C₈×Dic₃	semidirect
235. C₃×C₅	C₈×C₂²⋊C₄	semidirect
239. C₂²×C₃	C₈×Dic₅	semidirect
273. C₈	#6719	direct (N × H)
279. C₅	C₈×#217	semidirect
283. C₂²	C₈×Dic₁₅	semidirect
293. C₃	C₈×C₂²⋊Dic₅	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×C₃×C₅×C₈	2
C₄×#6719	2
C₂×#91114	2
C₂×C₈×Dic₁₅	2
#11746371	2
C₈×C₂²⋊Dic₅	3
C₈×#217	5

Subgroups — C8×#6719 #14295144

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×#6719 #14295144