Subgroups — #14297878

Order 1920.

672 conjugacy classes of subgroups.

The subgroup lattice has 272 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	#11751313	C₂	1	2	yes	maximal
3	C₂×#91101	C₂	1	2	yes	maximal, complemented
4	#11746372	C₂	1	2	yes	maximal, complemented
5	#10602763	—	3	1	no	maximal, Hall
6	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
7	C₂²×C₃×C₅×C₈	—	2	4	no	—
8	C₂²×C₃×C₅×C₈	C₄	1	4	yes	complemented
9	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
10	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₂×#6719	—	2	2	no	—
12	C₂×C₄⋊Dic₁₅	C₂²	1	4	yes	—
13	#91115	—	2	2	no	—
14	#91101	—	2	4	no	—
15	#71970	—	5	1	no	maximal, Hall
16	C₂³×C₅×C₈	D₃	1	1	yes	characteristic
17	#65501	—	3	2	no	—
18	C₂×#3949	—	3	2	no	—
19	#64771	—	3	2	no	—
20	C₂⁴×C₃×C₅	D₄	1	1	yes	characteristic
21	C₂²×C₃×C₄×C₅	—	2	2	no	—
22	C₂²×C₃×C₄×C₅	—	2	2	no	—
23	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
24	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
25	C₂²×C₃×C₄×C₅	Q₈	1	2	yes	—
26	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
27	C₂²×Dic₁₅	—	2	4	no	—
28	C₂×C₃×C₅×C₈	—	2	2	no	—
29	C₂×C₃×C₅×C₈	—	2	8	no	—
30	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
31	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
32	C₂×C₃×C₅×C₈	D₄	1	4	yes	—
33	#6719	—	4	4	no	—
34	C₄⋊Dic₁₅	—	2	4	no	—
35	C₄⋊Dic₁₅	—	2	4	no	—
36	C₂³×C₃×C₈	D₅	1	1	yes	characteristic
37	#5212	—	5	2	no	—
38	C₂×#817	—	5	2	no	—
39	#4551	—	5	2	no	—
40	C₂³×C₄×C₅	C₂×D₃	1	1	yes	characteristic
41	C₂²×C₅×C₈	—	2	4	no	—
42	C₂²×C₅×C₈	Dic₃	1	4	yes	complemented
43	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
44	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
45	C₂×C₂²⋊Dic₅	—	6	2	no	—
46	C₂×C₄⋊Dic₅	—	3	4	no	—
47	#3963	—	6	2	no	—
48	#3949	—	6	4	no	—
49	#1807	—	15	1	no	Sylow
50	C₂³×C₃×C₅	—	2	2	no	—
51	C₂³×C₃×C₅	—	2	2	no	—
52	C₂³×C₃×C₅	C₄⋊C₄	1	2	yes	—
53	C₂³×C₃×C₅	SD₁₆	1	4	yes	—
54	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
55	C₂×C₃×C₄×C₅	—	2	2	no	—
56	C₂×C₃×C₄×C₅	—	2	2	no	—
57	C₂×C₃×C₄×C₅	—	2	2	no	—
58	C₂×C₃×C₄×C₅	—	2	4	no	—
59	C₂×C₃×C₄×C₅	C₄⋊C₄	1	2	yes	—
60	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
62	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
63	C₂×C₃×C₄×C₅	#41	1	2	yes	—
64	C₂×Dic₁₅	—	4	4	no	—
65	C₂×Dic₁₅	—	4	4	no	—
66	C₂×Dic₁₅	—	4	4	no	—
67	C₃×C₅×C₈	—	2	2	no	—
68	C₃×C₅×C₈	C₂²⋊C₄	1	4	yes	complemented
69	C₂³×C₃×C₄	C₂×D₅	1	1	yes	characteristic
70	C₂²×C₃×C₈	—	2	4	no	—
71	C₂²×C₃×C₈	Dic₅	1	4	yes	complemented
72	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
73	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
74	C₂×#217	—	10	2	no	—
75	C₂×C₄⋊Dic₃	—	5	4	no	—
76	C₃⋊#95	—	10	2	no	—
77	#817	—	10	4	no	—
78	C₂⁴×C₅	D₁₂	1	1	yes	characteristic
79	C₂²×C₄×C₅	—	2	2	no	—
80	C₂²×C₄×C₅	—	2	2	no	—
81	C₂²×C₄×C₅	Dic₆	1	2	yes	—
82	C₂²×C₄×C₅	D₁₂	1	1	yes	characteristic
83	C₂²×C₄×C₅	C₂×Dic₃	1	2	yes	—
84	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
85	C₂²×Dic₅	—	6	4	no	—
86	C₂×C₅×C₈	—	2	2	no	—
87	C₂×C₅×C₈	—	2	8	no	—
88	C₂×C₅×C₈	C₂×Dic₃	1	2	yes	—
89	C₂×C₅×C₈	C₂×Dic₃	1	2	yes	—
90	C₂×C₅×C₈	#67	1	4	yes	—
91	C₂²⋊Dic₅	—	12	4	no	—
92	C₄⋊Dic₅	—	6	4	no	—
93	C₄⋊Dic₅	—	6	4	no	—
94	C₂³×C₈	D₁₅	1	1	yes	characteristic
95	C₂×#106	—	15	2	no	—
96	#380	—	15	2	no	—
97	#340	—	15	2	no	—
98	C₂²×C₃×C₅	—	2	2	no	—
99	C₂²×C₃×C₅	—	2	2	no	—
100	C₂²×C₃×C₅	—	2	8	no	—
101	C₂²×C₃×C₅	#106	1	4	yes	complemented
102	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂²×C₃×C₅	#121	1	2	yes	—
104	C₂²×C₃×C₅	C₂×SD₁₆	1	2	yes	—
105	C₂²×C₃×C₅	#135	1	2	yes	—
106	C₃×C₄×C₅	—	2	2	no	—
107	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₃×C₄×C₅	#117	1	1	yes	characteristic
109	C₃×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
110	Dic₁₅	—	8	4	no	—
111	C₂⁴×C₃	D₂₀	1	1	yes	characteristic
112	C₂²×C₃×C₄	—	2	2	no	—
113	C₂²×C₃×C₄	—	2	2	no	—
114	C₂²×C₃×C₄	Dic₁₀	1	2	yes	—
115	C₂²×C₃×C₄	D₂₀	1	1	yes	characteristic
116	C₂²×C₃×C₄	C₂×Dic₅	1	2	yes	—
117	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
118	C₂²×Dic₃	—	10	4	no	—
119	C₂×C₃×C₈	—	2	2	no	—
120	C₂×C₃×C₈	—	2	8	no	—
121	C₂×C₃×C₈	C₂×Dic₅	1	2	yes	—
122	C₂×C₃×C₈	C₂×Dic₅	1	2	yes	—
123	C₂×C₃×C₈	C₂²⋊D₅	1	4	yes	—
124	#217	—	20	4	no	—
125	C₄⋊Dic₃	—	10	4	no	—
126	C₄⋊Dic₃	—	10	4	no	—
127	C₂³×C₅	—	2	2	no	—
128	C₂³×C₅	—	2	2	no	—
129	C₂³×C₅	#204	1	4	yes	—
130	C₂³×C₅	C₄⋊Dic₃	1	2	yes	—
131	C₂³×C₅	C₂×D₁₂	1	1	yes	characteristic
132	C₂×C₄×C₅	—	2	2	no	—
133	C₂×C₄×C₅	—	2	4	no	—
134	C₂×C₄×C₅	—	2	2	no	—
135	C₂×C₄×C₅	—	2	2	no	—
136	C₂×C₄×C₅	C₄⋊Dic₃	1	2	yes	—
137	C₂×C₄×C₅	C₂×Dic₆	1	1	yes	characteristic
138	C₂×C₄×C₅	#235	1	2	yes	—
139	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
140	C₂×C₄×C₅	C₂×#67	1	2	yes	—
141	C₂×Dic₅	—	12	4	no	—
142	C₂×Dic₅	—	12	4	no	—
143	C₂×Dic₅	—	12	4	no	—
144	C₅×C₈	—	2	2	no	—
145	C₅×C₈	#217	1	4	yes	complemented
146	C₂³×C₄	C₂×D₁₅	1	1	yes	characteristic
147	C₂²×C₈	—	2	4	no	—
148	C₂²×C₈	Dic₁₅	1	4	yes	complemented
149	C₂²×C₈	C₂×D₁₅	1	1	yes	characteristic
150	C₂²×C₈	C₂×D₁₅	1	1	yes	characteristic
151	C₂×C₄⋊C₄	—	15	4	no	—
152	C₂×C₂²⋊C₄	—	30	2	no	—
153	#106	—	30	4	no	—
154	#95	—	30	2	no	—
155	C₂×C₃×C₅	—	2	4	no	—
156	C₂×C₃×C₅	#380	1	1	yes	characteristic
157	C₂×C₃×C₅	C₂×#106	1	1	yes	characteristic
158	C₂×C₃×C₅	#427	1	1	yes	characteristic
159	C₂×C₃×C₅	#465	1	4	yes	—
160	C₂³×C₃	—	2	2	no	—
161	C₂³×C₃	—	2	2	no	—
162	C₂³×C₃	#677	1	4	yes	—
163	C₂³×C₃	C₄⋊Dic₅	1	2	yes	—
164	C₂³×C₃	C₂×D₂₀	1	1	yes	characteristic
165	C₂×C₃×C₄	—	2	2	no	—
166	C₂×C₃×C₄	—	2	2	no	—
167	C₂×C₃×C₄	—	2	4	no	—
168	C₂×C₃×C₄	—	2	2	no	—
169	C₂×C₃×C₄	C₄⋊Dic₅	1	2	yes	—
170	C₂×C₃×C₄	C₂×Dic₁₀	1	1	yes	characteristic
171	C₂×C₃×C₄	#709	1	2	yes	—
172	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
173	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
174	C₂×Dic₃	—	20	4	no	—
175	C₂×Dic₃	—	20	4	no	—
176	C₂×Dic₃	—	20	4	no	—
177	C₃×C₈	—	2	2	no	—
178	C₃×C₈	C₂²⋊Dic₅	1	4	yes	complemented
179	C₂²×C₅	—	2	2	no	—
180	C₂²×C₅	—	2	2	no	—
181	C₂²×C₅	—	2	8	no	—
182	C₂²×C₅	#817	1	4	yes	complemented
183	C₂²×C₅	C₂×#204	1	2	yes	—
184	C₂²×C₅	#904	1	2	yes	—
185	C₂²×C₅	C₂×C₄⋊Dic₃	1	1	yes	characteristic
186	C₂²×C₅	#930	1	2	yes	—
187	C₄×C₅	—	2	2	no	—
188	C₄×C₅	#924	1	2	yes	—
189	C₄×C₅	#926	1	1	yes	characteristic
190	C₄×C₅	C₂×#217	1	1	yes	characteristic
191	Dic₅	—	24	4	no	—
192	C₂⁴	D₆₀	1	1	yes	characteristic
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	30	4	no	—
196	C₂²×C₄	Dic₃₀	1	2	yes	—
197	C₂²×C₄	D₆₀	1	1	yes	characteristic
198	C₂²×C₄	C₂×Dic₁₅	1	2	yes	—
199	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	8	no	—
202	C₂×C₈	C₂×Dic₁₅	1	2	yes	—
203	C₂×C₈	C₂×Dic₁₅	1	2	yes	—
204	C₂×C₈	#1220	1	4	yes	—
205	C₄⋊C₄	—	30	4	no	—
206	C₄⋊C₄	—	30	4	no	—
207	C₂²⋊C₄	—	60	4	no	—
208	C₃×C₅	#1807	1	1	yes	characteristic, complemented, Hall
209	C₂²×C₃	—	2	2	no	—
210	C₂²×C₃	—	2	8	no	—
211	C₂²×C₃	—	2	2	no	—
212	C₂²×C₃	#3949	1	4	yes	complemented
213	C₂²×C₃	C₂×#677	1	2	yes	—
214	C₂²×C₃	#4050	1	2	yes	—
215	C₂²×C₃	C₂×C₄⋊Dic₅	1	1	yes	characteristic
216	C₂²×C₃	#4076	1	2	yes	—
217	C₃×C₄	—	2	2	no	—
218	C₃×C₄	#4070	1	2	yes	—
219	C₃×C₄	#4072	1	1	yes	characteristic
220	C₃×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
221	Dic₃	—	40	4	no	—
222	C₂×C₅	—	2	4	no	—
223	C₂×C₅	C₂×#817	1	1	yes	characteristic
224	C₂×C₅	#5106	1	1	yes	characteristic
225	C₂×C₅	#5114	1	4	yes	—
226	C₂×C₅	#5212	1	1	yes	characteristic
227	C₂³	—	2	2	no	—
228	C₂³	—	2	2	no	—
229	C₂³	#6706	1	4	yes	—
230	C₂³	C₄⋊Dic₁₅	1	2	yes	—
231	C₂³	C₂×D₆₀	1	1	yes	characteristic
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	60	4	no	—
237	C₂×C₄	—	60	4	no	—
238	C₂×C₄	—	60	4	no	—
239	C₂×C₄	C₄⋊Dic₁₅	1	2	yes	—
240	C₂×C₄	C₂×Dic₃₀	1	1	yes	characteristic
241	C₂×C₄	#6817	1	2	yes	—
242	C₂×C₄	C₂²×Dic₁₅	1	1	yes	characteristic
243	C₂×C₄	C₂×#1220	1	2	yes	—
244	C₈	—	2	2	no	—
245	C₈	#6719	1	4	yes	complemented
246	C₂×C₃	—	2	4	no	—
247	C₂×C₃	C₂×#3949	1	1	yes	characteristic
248	C₂×C₃	#65395	1	1	yes	characteristic
249	C₂×C₃	#65403	1	4	yes	—
250	C₂×C₃	#65501	1	1	yes	characteristic
251	C₅	#71970	1	1	yes	characteristic, complemented, Sylow
252	C₂²	—	2	2	no	—
253	C₂²	—	2	8	no	—
254	C₂²	—	2	2	no	—
255	C₂²	#91101	1	4	yes	complemented
256	C₂²	C₂×#6706	1	2	yes	—
257	C₂²	#91793	1	2	yes	—
258	C₂²	C₂×C₄⋊Dic₁₅	1	1	yes	characteristic
259	C₂²	#91819	1	2	yes	—
260	C₄	—	2	2	no	—
261	C₄	—	120	4	no	—
262	C₄	#91813	1	2	yes	—
263	C₄	#91815	1	1	yes	characteristic
264	C₄	C₂×#6719	1	1	yes	characteristic
265	C₃	#10602763	1	1	yes	characteristic, complemented, Sylow
266	C₂	—	2	4	no	—
267	C₂	C₂×#91101	1	1	yes	characteristic
268	C₂	#11751207	1	1	yes	characteristic
269	C₂	#11751215	1	4	yes	—
270	C₂	#11751313	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₆₀
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅
Fitting	C₂³×C₃×C₅×C₈	C₂
socle	C₂³×C₃×C₅	C₂×D₄
prime radical	C₂⁴×C₃×C₅	D₄
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	C₂³×C₈	D₁₅
hypocenter	C₃×C₅	#1807
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
3. C₂×#91101	C₂	semidirect
4. #11746372	C₂	semidirect
8. C₂²×C₃×C₅×C₈	C₄	semidirect
42. C₂²×C₅×C₈	Dic₃	semidirect
68. C₃×C₅×C₈	C₂²⋊C₄	semidirect
71. C₂²×C₃×C₈	Dic₅	semidirect
101. C₂²×C₃×C₅	#106	semidirect
145. C₅×C₈	#217	semidirect
148. C₂²×C₈	Dic₁₅	semidirect
178. C₃×C₈	C₂²⋊Dic₅	semidirect
182. C₂²×C₅	#817	semidirect
208. C₃×C₅	#1807	semidirect
212. C₂²×C₃	#3949	semidirect
245. C₈	#6719	semidirect
251. C₅	#71970	semidirect
255. C₂²	#91101	semidirect
265. C₃	#10602763	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
#11751313	2
C₂×#91101	2
#11746372	2
#10602763	3
#71970	5