Subgroups — #14438924

Order 1920.

1144 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#91273	C₂	1	4	yes	maximal, complemented
2	#11752542	C₂	1	2	yes	maximal, complemented
3	#11750800	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₅×D₃×C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#64984	C₂	1	1	yes	maximal, characteristic, complemented
6	#11747971	C₂	1	4	yes	maximal, complemented
7	#11747071	C₂	1	2	yes	maximal, complemented
8	#10610198	—	3	1	no	maximal, Hall
9	C₂×#6765	C₂²	1	4	yes	—
10	C₂×Dic₆₀	C₂²	1	4	yes	—
11	#91758	C₂²	1	2	yes	—
12	C₂×C₅×D₃×C₈	C₂²	1	2	yes	—
13	C₄²×C₅×D₃	C₂²	1	1	yes	characteristic
14	C₂×C₃×D₄₀	C₂²	1	4	yes	—
15	C₃×#4020	C₂²	1	2	yes	—
16	#91456	—	2	4	no	—
17	Dic₃×D₂₀	—	2	4	no	—
18	#91364	C₂²	1	4	yes	—
19	C₂×#6657	C₂²	1	8	yes	—
20	#91273	—	2	16	no	—
21	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
22	C₅×C₈×Dic₃	C₂²	1	2	yes	—
23	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
24	#81782	—	5	1	no	maximal, Hall
25	C₂×#4050	—	3	4	no	—
26	#65811	—	3	2	no	—
27	C₂×C₄×C₅×C₈	—	3	1	no	—
28	#64988	—	3	1	no	—
29	#64985	—	3	4	no	—
30	#64984	D₃	1	1	yes	characteristic, complemented
31	#64982	—	3	2	no	—
32	C₂×Dic₃₀	C₂³	1	4	yes	—
33	C₂×C₄×C₅×D₃	D₄	1	2	yes	—
34	C₂×C₄×C₅×D₃	C₂³	1	1	yes	characteristic
35	C₂×C₃×D₂₀	—	2	4	no	—
36	C₂×C₃×D₂₀	C₂³	1	4	yes	—
37	C₂×#1201	—	2	4	no	—
38	C₂×D₅×Dic₃	—	2	4	no	—
39	#6765	—	2	16	no	—
40	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
41	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
42	C₄⋊Dic₁₅	—	2	4	no	—
43	Dic₆₀	—	2	8	no	—
44	C₄×C₅×Dic₃	D₄	1	2	yes	—
45	C₄×C₅×Dic₃	C₂³	1	1	yes	characteristic
46	C₂×C₅×C₃⋊C₈	C₂³	1	2	yes	—
47	C₅×D₃×C₈	D₄	1	8	yes	complemented
48	C₃×D₄₀	—	2	8	no	—
49	#6665	—	2	8	no	—
50	#6657	—	2	16	no	—
51	C₂×#912	—	5	4	no	—
52	#5716	—	5	2	no	—
53	C₃×#493	—	5	1	no	—
54	#5155	—	5	4	no	—
55	#5076	—	5	2	no	—
56	#4699	—	5	1	no	—
57	C₄×D₃×C₈	D₅	1	1	yes	characteristic, complemented
58	C₂×#709	—	3	4	no	—
59	C₂²×C₅×C₈	—	3	2	no	—
60	C₂×C₄²×C₅	—	3	1	no	—
61	#4076	—	6	4	no	—
62	C₂×Dic₂₀	—	3	4	no	—
63	#4050	—	6	16	no	—
64	C₂×D₄₀	C₂×D₃	1	4	yes	—
65	C₂×#677	—	3	8	no	—
66	#4021	—	3	4	no	—
67	#4020	C₂×D₃	1	2	yes	—
68	C₄×D₂₀	—	6	4	no	—
69	#4015	—	3	2	no	—
70	C₄×C₅×C₈	—	3	2	no	—
71	C₄×C₅×C₈	—	3	1	no	—
72	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
73	#3147	—	15	1	no	Sylow
74	C₂²×C₅×D₃	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₃×D₅	—	2	4	no	—
76	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
77	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
78	C₂×Dic₁₅	—	2	4	no	—
79	Dic₃₀	—	2	8	no	—
80	C₂×C₅×Dic₃	C₂×D₄	1	2	yes	—
81	C₂×C₅×Dic₃	C₂×D₄	1	1	yes	characteristic
82	C₄×C₅×D₃	—	2	4	no	—
83	C₄×C₅×D₃	C₂×D₄	1	4	yes	—
84	C₃×D₂₀	—	2	8	no	—
85	C₃×D₂₀	—	4	8	no	—
86	#1201	—	4	8	no	—
87	D₅×Dic₃	—	4	8	no	—
88	C₃×C₅×C₈	C₂×D₄	1	4	yes	—
89	C₅×C₃⋊C₈	C₂×D₄	1	4	yes	—
90	C₂×#237	—	5	4	no	—
91	C₂×C₃×D₈	—	5	4	no	—
92	C₃×#127	—	5	2	no	—
93	#938	—	10	4	no	—
94	#936	—	5	4	no	—
95	D₄×Dic₃	—	10	4	no	—
96	C₂×#214	—	5	8	no	—
97	#912	—	10	16	no	—
98	C₂×Dic₁₂	—	5	4	no	—
99	C₂×D₃×C₈	C₂×D₅	1	2	yes	—
100	C₄²×D₃	C₂×D₅	1	1	yes	characteristic
101	#869	—	5	2	no	—
102	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
103	C₈×Dic₃	C₂×D₅	1	2	yes	—
104	C₄×C₃⋊C₈	C₂×D₅	1	1	yes	characteristic
105	C₂²×C₄×C₅	—	3	1	no	—
106	C₂²×C₄×C₅	—	3	2	no	—
107	C₂×C₂²⋊D₅	—	6	4	no	—
108	#709	—	6	16	no	—
109	C₂×D₂₀	—	2	4	no	—
110	C₂×D₂₀	C₂²×D₃	1	4	yes	—
111	C₂×C₄×D₅	—	6	4	no	—
112	C₂×Dic₁₀	—	3	4	no	—
113	C₂×C₅×C₈	—	3	2	no	—
114	C₂×C₅×C₈	—	3	8	no	—
115	C₂×C₅×C₈	C₂²×D₃	1	2	yes	—
116	C₄²×C₅	—	3	2	no	—
117	C₄²×C₅	—	3	1	no	—
118	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
119	#685	—	6	8	no	—
120	C₄⋊Dic₅	—	6	4	no	—
121	Dic₂₀	—	6	8	no	—
122	D₄₀	—	2	8	no	—
123	#677	—	6	16	no	—
124	C₂×#135	—	15	4	no	—
125	#532	—	15	2	no	—
126	#495	—	15	4	no	—
127	#494	—	15	1	no	—
128	#493	—	5	1	no	—
129	#492	—	15	2	no	—
130	C₂×C₄×C₈	—	3	1	no	—
131	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₅×D₃	—	2	2	no	—
133	C₂×C₅×D₃	#127	1	2	yes	—
134	C₂×C₃×D₅	—	4	4	no	—
135	C₂×C₃×D₅	—	4	8	no	—
136	C₃×C₄×C₅	#135	1	4	yes	complemented
137	C₃×C₄×C₅	C₂²×D₄	1	2	yes	—
138	Dic₁₅	—	4	4	no	—
139	C₅×Dic₃	—	2	2	no	—
140	C₅×Dic₃	#127	1	2	yes	—
141	C₂×C₃×D₄	—	5	4	no	—
142	C₂×C₃×D₄	—	10	4	no	—
143	C₂×#67	—	10	4	no	—
144	C₂²×Dic₃	—	10	4	no	—
145	#237	—	10	16	no	—
146	C₂×C₄×D₃	D₂₀	1	2	yes	—
147	C₂×C₄×D₃	C₂²×D₅	1	1	yes	characteristic
148	C₂×Dic₆	—	5	4	no	—
149	C₃×D₈	—	10	8	no	—
150	C₂×C₃×C₈	C₂²×D₅	1	2	yes	—
151	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
152	#217	—	10	8	no	—
153	#214	—	10	16	no	—
154	C₄⋊Dic₃	—	10	4	no	—
155	C₄×Dic₃	D₂₀	1	2	yes	—
156	C₄×Dic₃	C₂²×D₅	1	1	yes	characteristic
157	C₂×C₃⋊C₈	C₂²×D₅	1	2	yes	—
158	Dic₁₂	—	10	8	no	—
159	D₃×C₈	D₂₀	1	8	yes	complemented
160	C₂³×C₅	—	3	1	no	—
161	C₂²×D₅	—	2	4	no	—
162	C₂×C₄×C₅	—	3	1	no	—
163	C₂×C₄×C₅	—	3	4	no	—
164	C₂×C₄×C₅	—	3	2	no	—
165	C₂×C₄×C₅	—	6	4	no	—
166	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
167	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
168	C₂²⋊D₅	—	12	8	no	—
169	C₂×Dic₅	—	6	4	no	—
170	D₂₀	—	2	8	no	—
171	D₂₀	—	4	8	no	—
172	C₄×D₅	—	12	8	no	—
173	Dic₁₀	—	6	8	no	—
174	C₅×C₈	—	3	4	no	—
175	C₅×C₈	D₃×D₄	1	4	yes	—
176	C₂×#41	—	15	4	no	—
177	#135	—	30	16	no	—
178	C₂×Q₁₆	—	15	4	no	—
179	C₂×SD₁₆	—	15	8	no	—
180	C₂×D₈	—	5	4	no	—
181	C₂²×C₈	—	3	2	no	—
182	C₄⋊Q₈	—	15	2	no	—
183	#127	—	5	2	no	—
184	#124	—	15	4	no	—
185	#121	—	30	4	no	—
186	C₄×D₄	—	30	4	no	—
187	C₂×C₄²	—	3	1	no	—
188	C₄×C₈	—	3	2	no	—
189	C₄×C₈	—	3	1	no	—
190	C₄×C₈	D₃×D₅	1	1	yes	characteristic
191	C₂×C₃×C₅	C₂×#127	1	1	yes	characteristic
192	C₂×C₃×C₅	C₂×#135	1	2	yes	—
193	C₃×D₅	—	8	4	no	—
194	C₅×D₃	—	2	2	no	—
195	C₂³×C₃	—	10	4	no	—
196	C₂²×D₃	C₂×D₂₀	1	1	yes	characteristic
197	C₃×D₄	—	10	8	no	—
198	C₃×D₄	—	20	8	no	—
199	C₂×C₃×C₄	C₂×D₂₀	1	2	yes	—
200	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
201	#67	—	20	8	no	—
202	C₂×Dic₃	—	10	4	no	—
203	C₂×Dic₃	—	20	8	no	—
204	C₂×Dic₃	C₂×D₂₀	1	2	yes	—
205	C₂×Dic₃	C₂×D₂₀	1	1	yes	characteristic
206	C₄×D₃	—	2	4	no	—
207	C₄×D₃	C₂×D₂₀	1	4	yes	—
208	Dic₆	—	10	8	no	—
209	C₃×C₈	C₂×D₂₀	1	4	yes	—
210	C₃⋊C₈	C₂×D₂₀	1	4	yes	—
211	C₂²×C₅	—	3	2	no	—
212	C₂²×C₅	—	6	2	no	—
213	C₂²×C₅	C₂×D₃×D₄	1	1	yes	characteristic
214	C₂×D₅	—	4	4	no	—
215	C₂×D₅	—	4	8	no	—
216	C₄×C₅	—	3	2	no	—
217	C₄×C₅	—	6	2	no	—
218	C₄×C₅	#912	1	4	yes	complemented
219	C₄×C₅	C₂×D₃×D₄	1	2	yes	—
220	Dic₅	—	12	4	no	—
221	#41	—	30	16	no	—
222	C₂×Q₈	—	15	4	no	—
223	C₂×D₄	—	5	4	no	—
224	C₂×D₄	—	10	4	no	—
225	C₂×D₄	—	30	4	no	—
226	C₂²×C₄	—	3	1	no	—
227	C₂²×C₄	—	3	2	no	—
228	C₂²×C₄	—	30	4	no	—
229	Q₁₆	—	30	8	no	—
230	SD₁₆	—	30	16	no	—
231	D₈	—	10	8	no	—
232	C₂×C₈	—	3	2	no	—
233	C₂×C₈	—	3	8	no	—
234	C₂×C₈	C₂×D₃×D₅	1	2	yes	—
235	C₄⋊C₄	—	30	4	no	—
236	C₂²⋊C₄	—	30	8	no	—
237	C₄²	—	3	2	no	—
238	C₄²	—	3	1	no	—
239	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
240	C₃×C₅	#3147	1	1	yes	characteristic, complemented, Hall
241	C₂²×C₃	—	20	4	no	—
242	C₂²×C₃	—	20	8	no	—
243	C₂²×C₃	C₂²×D₂₀	1	1	yes	characteristic
244	C₂×D₃	—	2	2	no	—
245	C₂×D₃	#4020	1	2	yes	—
246	C₃×C₄	#4050	1	4	yes	complemented
247	C₃×C₄	C₂²×D₂₀	1	2	yes	—
248	Dic₃	—	2	2	no	—
249	Dic₃	—	20	4	no	—
250	Dic₃	#4020	1	2	yes	—
251	C₂×C₅	—	6	2	no	—
252	C₂×C₅	D₃×#127	1	1	yes	characteristic
253	C₂×C₅	C₂×#912	1	2	yes	—
254	D₅	—	8	4	no	—
255	C₂³	—	3	1	no	—
256	C₂³	—	10	4	no	—
257	Q₈	—	30	8	no	—
258	D₄	—	10	8	no	—
259	D₄	—	20	8	no	—
260	D₄	—	60	8	no	—
261	C₂×C₄	—	3	1	no	—
262	C₂×C₄	—	3	4	no	—
263	C₂×C₄	—	3	2	no	—
264	C₂×C₄	—	6	4	no	—
265	C₂×C₄	—	30	4	no	—
266	C₂×C₄	—	60	8	no	—
267	C₂×C₄	D₃×D₂₀	1	2	yes	—
268	C₂×C₄	C₂²×D₃×D₅	1	1	yes	characteristic
269	C₈	—	3	4	no	—
270	C₈	D₃×D₂₀	1	4	yes	—
271	C₂×C₃	—	40	4	no	—
272	C₂×C₃	C₂×#4020	1	1	yes	characteristic
273	C₂×C₃	C₂×#4050	1	2	yes	—
274	D₃	—	2	2	no	—
275	C₅	#81782	1	1	yes	characteristic, complemented, Sylow
276	C₂²	—	3	2	no	—
277	C₂²	—	6	2	no	—
278	C₂²	—	20	8	no	—
279	C₂²	—	20	4	no	—
280	C₂²	C₂×D₃×D₂₀	1	1	yes	characteristic
281	C₄	—	3	2	no	—
282	C₄	—	6	2	no	—
283	C₄	—	60	4	no	—
284	C₄	#91273	1	4	yes	complemented
285	C₄	C₂×D₃×D₂₀	1	2	yes	—
286	C₃	#10610198	1	1	yes	characteristic, complemented, Sylow
287	C₂	—	6	2	no	—
288	C₂	—	40	4	no	—
289	C₂	D₃×#4020	1	1	yes	characteristic
290	C₂	C₂×#91273	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₃×D₂₀
commutator, cosocle	C₂×C₃×C₄×C₅	C₂⁴
Frattini	C₂×C₄	C₂²×D₃×D₅
Fitting	C₃×C₄×C₅×C₈	C₂²
socle	C₂²×C₃×C₅	C₂²×D₄
hypercenter	C₄×C₈	D₃×D₅
hypocenter	C₃×C₅	#3147
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
1. C₂×#91273	C₂	semidirect
2. #11752542	C₂	semidirect
3. #11750800	C₂	semidirect
4. C₄×C₅×D₃×C₈	C₂	semidirect
5. C₃×#64984	C₂	semidirect
6. #11747971	C₂	semidirect
7. #11747071	C₂	semidirect
30. #64984	D₃	semidirect
47. C₅×D₃×C₈	D₄	semidirect
57. C₄×D₃×C₈	D₅	semidirect
136. C₃×C₄×C₅	#135	semidirect
159. D₃×C₈	D₂₀	semidirect
218. C₄×C₅	#912	semidirect
240. C₃×C₅	#3147	semidirect
246. C₃×C₄	#4050	semidirect
275. C₅	#81782	semidirect
284. C₄	#91273	semidirect
286. C₃	#10610198	semidirect

Maximal subgroups

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

subgroup	index
C₂×#91273	2
#11752542	2
#11750800	2
C₄×C₅×D₃×C₈	2
C₃×#64984	2
#11747971	2
#11747071	2
#10610198	3
#81782	5