Subgroups — C₂²×C₃×#4086 #14479962

Order 1920.

6000 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₅×D₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₃×C₂²⋊D₅	C₂	1	4	yes	maximal, complemented
3	C₂³×C₃×D₂₀	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×#4086	C₂	1	24	yes	maximal, direct product factor
5	C₂²×C₃×C₄×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#4086	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂⁵×C₃×C₅	C₂²	1	2	yes	—
8	C₂⁴×C₃×D₅	C₂²	1	2	yes	—
9	C₂²×C₃×C₅×D₄	—	2	8	no	—
10	C₂²×C₃×C₅×D₄	C₂²	1	12	yes	complemented
11	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
12	C₂²×C₃×C₂²⋊D₅	—	2	32	no	—
13	C₂²×C₃×C₂²⋊D₅	C₂²	1	48	yes	complemented
14	C₂³×C₃×Dic₅	C₂²	1	2	yes	—
15	C₂²×C₃×D₂₀	—	2	8	no	—
16	C₂²×C₃×D₂₀	C₂²	1	12	yes	complemented
17	C₃×#4086	C₂²	1	64	yes	direct product factor
18	C₂×C₃×C₄×Dic₅	C₂²	1	12	yes	complemented
19	C₂²×C₃×#127	—	5	1	no	maximal, Hall
20	C₂³×C₅×D₄	C₂×C₃	1	1	yes	characteristic, complemented
21	C₂³×C₂²⋊D₅	C₂×C₃	1	4	yes	complemented
22	C₂³×D₂₀	C₂×C₃	1	1	yes	characteristic, complemented
23	C₂×#4086	C₂×C₃	1	24	yes	direct product factor
24	C₂²×C₄×Dic₅	C₂×C₃	1	1	yes	characteristic, complemented
25	C₂⁴×C₃×C₅	—	2	16	no	—
26	C₂⁴×C₃×C₅	—	2	8	no	—
27	C₂⁴×C₃×C₅	C₂³	1	12	yes	—
28	C₂⁴×C₃×C₅	C₂³	1	1	yes	characteristic
29	C₂³×C₃×D₅	—	2	8	no	—
30	C₂³×C₃×D₅	—	2	16	no	—
31	C₂³×C₃×D₅	C₂³	1	12	yes	—
32	C₂×C₃×C₅×D₄	—	2	48	no	—
33	C₂×C₃×C₅×D₄	C₂³	1	16	yes	complemented
34	C₂²×C₃×C₄×C₅	D₄	1	8	yes	complemented
35	C₂²×C₃×C₄×C₅	C₂³	1	6	yes	—
36	C₂×C₃×C₂²⋊D₅	—	2	192	no	—
37	C₂×C₃×C₂²⋊D₅	C₂³	1	64	yes	complemented
38	C₂²×C₃×Dic₅	D₄	1	16	yes	complemented
39	C₂²×C₃×Dic₅	C₂³	1	12	yes	—
40	C₂×C₃×D₂₀	—	2	48	no	—
41	C₂×C₃×D₂₀	C₂³	1	16	yes	complemented
42	C₃×C₄×Dic₅	C₂³	1	16	yes	complemented
43	C₂³×C₃×D₄	—	5	1	no	—
44	C₂³×C₃×D₄	—	5	4	no	—
45	C₂³×C₃×D₄	D₅	1	1	yes	characteristic, complemented
46	C₂×C₃×#127	—	5	24	no	—
47	C₂²×C₃×C₄²	—	5	1	no	—
48	C₂⁵×C₅	C₂²×C₃	1	2	yes	—
49	C₂⁴×D₅	C₂²×C₃	1	2	yes	—
50	C₂²×C₅×D₄	—	2	8	no	—
51	C₂²×C₅×D₄	C₂²×C₃	1	12	yes	complemented
52	C₂³×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
53	C₂²×C₂²⋊D₅	—	2	32	no	—
54	C₂²×C₂²⋊D₅	C₂²×C₃	1	48	yes	complemented
55	C₂³×Dic₅	C₂²×C₃	1	2	yes	—
56	C₂²×D₂₀	—	2	8	no	—
57	C₂²×D₂₀	C₂²×C₃	1	12	yes	complemented
58	#4086	C₂²×C₃	1	64	yes	direct product factor
59	C₂×C₄×Dic₅	C₂²×C₃	1	12	yes	complemented
60	C₂²×#127	—	5	1	no	Sylow
61	C₂³×C₃×C₅	—	2	48	no	—
62	C₂³×C₃×C₅	—	2	24	no	—
63	C₂³×C₃×C₅	—	4	32	no	—
64	C₂³×C₃×C₅	C₂×D₄	1	8	yes	—
65	C₂³×C₃×C₅	C₂×D₄	1	4	yes	—
66	C₂³×C₃×C₅	C₂⁴	1	8	yes	—
67	C₂³×C₃×C₅	C₂⁴	1	3	yes	—
68	C₂²×C₃×D₅	—	2	24	no	—
69	C₂²×C₃×D₅	—	2	48	no	—
70	C₂²×C₃×D₅	—	4	32	no	—
71	C₂²×C₃×D₅	C₂⁴	1	8	yes	—
72	C₃×C₅×D₄	—	2	32	no	—
73	C₂×C₃×C₄×C₅	C₂×D₄	1	24	yes	complemented
74	C₂×C₃×C₄×C₅	C₂⁴	1	4	yes	—
75	C₃×C₂²⋊D₅	—	2	128	no	—
76	C₂×C₃×Dic₅	C₂×D₄	1	48	yes	complemented
77	C₂×C₃×Dic₅	C₂⁴	1	8	yes	—
78	C₃×D₂₀	—	2	32	no	—
79	C₂⁵×C₃	—	5	2	no	—
80	C₂⁵×C₃	C₂×D₅	1	2	yes	—
81	C₂²×C₃×D₄	—	2	8	no	—
82	C₂²×C₃×D₄	—	5	48	no	—
83	C₂²×C₃×D₄	—	5	12	no	—
84	C₂²×C₃×D₄	—	10	32	no	—
85	C₂²×C₃×D₄	—	10	8	no	—
86	C₂²×C₃×D₄	C₂×D₅	1	12	yes	complemented
87	C₂³×C₃×C₄	—	5	2	no	—
88	C₂³×C₃×C₄	C₂×D₅	1	1	yes	characteristic
89	C₃×#127	—	5	64	no	—
90	C₂×C₃×C₄²	—	5	12	no	—
91	C₂⁴×C₅	—	2	16	no	—
92	C₂⁴×C₅	—	2	8	no	—
93	C₂⁴×C₅	C₂³×C₃	1	12	yes	—
94	C₂⁴×C₅	C₂³×C₃	1	1	yes	characteristic
95	C₂³×D₅	—	2	8	no	—
96	C₂³×D₅	—	2	16	no	—
97	C₂³×D₅	C₂³×C₃	1	12	yes	—
98	C₂×C₅×D₄	—	2	48	no	—
99	C₂×C₅×D₄	C₂³×C₃	1	16	yes	complemented
100	C₂²×C₄×C₅	C₃×D₄	1	8	yes	complemented
101	C₂²×C₄×C₅	C₂³×C₃	1	6	yes	—
102	C₂×C₂²⋊D₅	—	2	192	no	—
103	C₂×C₂²⋊D₅	C₂³×C₃	1	64	yes	complemented
104	C₂²×Dic₅	C₃×D₄	1	16	yes	complemented
105	C₂²×Dic₅	C₂³×C₃	1	12	yes	—
106	C₂×D₂₀	—	2	48	no	—
107	C₂×D₂₀	C₂³×C₃	1	16	yes	complemented
108	C₄×Dic₅	C₂³×C₃	1	16	yes	complemented
109	C₂³×D₄	—	5	4	no	—
110	C₂³×D₄	—	5	1	no	—
111	C₂³×D₄	C₃×D₅	1	1	yes	characteristic, complemented
112	C₂×#127	—	5	24	no	—
113	C₂²×C₄²	—	5	1	no	—
114	C₂²×C₃×C₅	—	2	16	no	—
115	C₂²×C₃×C₅	—	2	8	no	—
116	C₂²×C₃×C₅	—	4	48	no	—
117	C₂²×C₃×C₅	#127	1	16	yes	complemented
118	C₂²×C₃×C₅	C₂²×D₄	1	12	yes	—
119	C₂²×C₃×C₅	C₂²×D₄	1	6	yes	—
120	C₂²×C₃×C₅	C₂⁵	1	1	yes	characteristic
121	C₂×C₃×D₅	—	2	8	no	—
122	C₂×C₃×D₅	—	2	16	no	—
123	C₂×C₃×D₅	—	4	48	no	—
124	C₃×C₄×C₅	C₂²×D₄	1	8	yes	complemented
125	C₃×Dic₅	C₂²×D₄	1	16	yes	complemented
126	C₂⁴×C₃	—	2	16	no	—
127	C₂⁴×C₃	—	2	8	no	—
128	C₂⁴×C₃	—	5	12	no	—
129	C₂⁴×C₃	—	10	16	no	—
130	C₂⁴×C₃	—	10	8	no	—
131	C₂⁴×C₃	C₂²×D₅	1	12	yes	—
132	C₂⁴×C₃	C₂²×D₅	1	1	yes	characteristic
133	C₂×C₃×D₄	—	2	48	no	—
134	C₂×C₃×D₄	—	5	64	no	—
135	C₂×C₃×D₄	—	5	16	no	—
136	C₂×C₃×D₄	—	10	192	no	—
137	C₂×C₃×D₄	—	10	48	no	—
138	C₂×C₃×D₄	C₂²×D₅	1	16	yes	complemented
139	C₂²×C₃×C₄	—	5	16	no	—
140	C₂²×C₃×C₄	—	5	12	no	—
141	C₂²×C₃×C₄	C₂²⋊D₅	1	8	yes	complemented
142	C₂²×C₃×C₄	C₂²×D₅	1	6	yes	—
143	C₃×C₄²	—	5	16	no	—
144	C₂³×C₅	—	2	48	no	—
145	C₂³×C₅	—	2	24	no	—
146	C₂³×C₅	—	4	32	no	—
147	C₂³×C₅	C₂×C₃×D₄	1	8	yes	—
148	C₂³×C₅	C₂×C₃×D₄	1	4	yes	—
149	C₂³×C₅	C₂⁴×C₃	1	8	yes	—
150	C₂³×C₅	C₂⁴×C₃	1	3	yes	—
151	C₂²×D₅	—	2	24	no	—
152	C₂²×D₅	—	2	48	no	—
153	C₂²×D₅	—	4	32	no	—
154	C₂²×D₅	C₂⁴×C₃	1	8	yes	—
155	C₅×D₄	—	2	32	no	—
156	C₂×C₄×C₅	C₂×C₃×D₄	1	24	yes	complemented
157	C₂×C₄×C₅	C₂⁴×C₃	1	4	yes	—
158	C₂²⋊D₅	—	2	128	no	—
159	C₂×Dic₅	C₂×C₃×D₄	1	48	yes	complemented
160	C₂×Dic₅	C₂⁴×C₃	1	8	yes	—
161	D₂₀	—	2	32	no	—
162	C₂⁵	—	5	2	no	—
163	C₂⁵	C₂×C₃×D₅	1	2	yes	—
164	C₂²×D₄	—	2	8	no	—
165	C₂²×D₄	—	5	48	no	—
166	C₂²×D₄	—	5	12	no	—
167	C₂²×D₄	—	10	32	no	—
168	C₂²×D₄	—	10	8	no	—
169	C₂²×D₄	C₂×C₃×D₅	1	12	yes	complemented
170	C₂³×C₄	—	5	2	no	—
171	C₂³×C₄	C₂×C₃×D₅	1	1	yes	characteristic
172	#127	—	5	64	no	—
173	C₂×C₄²	—	5	12	no	—
174	C₂×C₃×C₅	—	4	8	no	—
175	C₂×C₃×C₅	C₂×#127	1	12	yes	complemented
176	C₂×C₃×C₅	C₂³×D₄	1	2	yes	—
177	C₂×C₃×C₅	C₂³×D₄	1	1	yes	characteristic
178	C₃×D₅	—	4	8	no	—
179	C₂³×C₃	—	2	48	no	—
180	C₂³×C₃	—	2	24	no	—
181	C₂³×C₃	—	4	32	no	—
182	C₂³×C₃	—	5	8	no	—
183	C₂³×C₃	—	10	48	no	—
184	C₂³×C₃	—	10	24	no	—
185	C₂³×C₃	—	20	32	no	—
186	C₂³×C₃	D₄×D₅	1	8	yes	—
187	C₂³×C₃	C₂×C₂²⋊D₅	1	4	yes	—
188	C₂³×C₃	C₂³×D₅	1	8	yes	—
189	C₂³×C₃	C₂³×D₅	1	3	yes	—
190	C₃×D₄	—	2	32	no	—
191	C₃×D₄	—	10	128	no	—
192	C₃×D₄	—	10	32	no	—
193	C₂×C₃×C₄	—	5	48	no	—
194	C₂×C₃×C₄	—	5	8	no	—
195	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	24	yes	complemented
196	C₂×C₃×C₄	C₂³×D₅	1	4	yes	—
197	C₂²×C₅	—	2	16	no	—
198	C₂²×C₅	—	2	8	no	—
199	C₂²×C₅	—	4	48	no	—
200	C₂²×C₅	C₃×#127	1	16	yes	complemented
201	C₂²×C₅	C₂²×C₃×D₄	1	12	yes	—
202	C₂²×C₅	C₂²×C₃×D₄	1	6	yes	—
203	C₂²×C₅	C₂⁵×C₃	1	1	yes	characteristic
204	C₂×D₅	—	2	8	no	—
205	C₂×D₅	—	2	16	no	—
206	C₂×D₅	—	4	48	no	—
207	C₄×C₅	C₂²×C₃×D₄	1	8	yes	complemented
208	Dic₅	C₂²×C₃×D₄	1	16	yes	complemented
209	C₂⁴	—	2	16	no	—
210	C₂⁴	—	2	8	no	—
211	C₂⁴	—	5	12	no	—
212	C₂⁴	—	10	16	no	—
213	C₂⁴	—	10	8	no	—
214	C₂⁴	C₂²×C₃×D₅	1	12	yes	—
215	C₂⁴	C₂²×C₃×D₅	1	1	yes	characteristic
216	C₂×D₄	—	2	48	no	—
217	C₂×D₄	—	5	64	no	—
218	C₂×D₄	—	5	16	no	—
219	C₂×D₄	—	10	192	no	—
220	C₂×D₄	—	10	48	no	—
221	C₂×D₄	C₂²×C₃×D₅	1	16	yes	complemented
222	C₂²×C₄	—	5	16	no	—
223	C₂²×C₄	—	5	12	no	—
224	C₂²×C₄	C₃×C₂²⋊D₅	1	8	yes	complemented
225	C₂²×C₄	C₂²×C₃×D₅	1	6	yes	—
226	C₄²	—	5	16	no	—
227	C₃×C₅	C₂²×#127	1	1	yes	characteristic, complemented, Hall
228	C₂²×C₃	—	2	16	no	—
229	C₂²×C₃	—	2	8	no	—
230	C₂²×C₃	—	4	48	no	—
231	C₂²×C₃	—	10	16	no	—
232	C₂²×C₃	—	10	8	no	—
233	C₂²×C₃	—	20	48	no	—
234	C₂²×C₃	#4086	1	16	yes	direct product factor
235	C₂²×C₃	C₂×D₄×D₅	1	12	yes	—
236	C₂²×C₃	C₂²×C₂²⋊D₅	1	6	yes	—
237	C₂²×C₃	C₂⁴×D₅	1	1	yes	characteristic
238	C₃×C₄	—	5	16	no	—
239	C₃×C₄	C₂²×C₂²⋊D₅	1	8	yes	complemented
240	C₂×C₅	—	4	8	no	—
241	C₂×C₅	C₂×C₃×#127	1	12	yes	complemented
242	C₂×C₅	C₂³×C₃×D₄	1	2	yes	—
243	C₂×C₅	C₂³×C₃×D₄	1	1	yes	characteristic
244	D₅	—	4	8	no	—
245	C₂³	—	2	48	no	—
246	C₂³	—	2	24	no	—
247	C₂³	—	4	32	no	—
248	C₂³	—	5	8	no	—
249	C₂³	—	10	48	no	—
250	C₂³	—	10	24	no	—
251	C₂³	—	20	32	no	—
252	C₂³	C₃×D₄×D₅	1	8	yes	—
253	C₂³	C₂×C₃×C₂²⋊D₅	1	4	yes	—
254	C₂³	C₂³×C₃×D₅	1	8	yes	—
255	C₂³	C₂³×C₃×D₅	1	3	yes	—
256	D₄	—	2	32	no	—
257	D₄	—	10	128	no	—
258	D₄	—	10	32	no	—
259	C₂×C₄	—	5	48	no	—
260	C₂×C₄	—	5	8	no	—
261	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	24	yes	complemented
262	C₂×C₄	C₂³×C₃×D₅	1	4	yes	—
263	C₂×C₃	—	4	8	no	—
264	C₂×C₃	—	20	8	no	—
265	C₂×C₃	C₂×#4086	1	12	yes	direct product factor
266	C₂×C₃	C₂²×D₄×D₅	1	2	yes	—
267	C₂×C₃	C₂³×C₂²⋊D₅	1	1	yes	characteristic
268	C₅	C₂²×C₃×#127	1	1	yes	characteristic, complemented, Sylow
269	C₂²	—	2	16	no	—
270	C₂²	—	2	8	no	—
271	C₂²	—	4	48	no	—
272	C₂²	—	10	16	no	—
273	C₂²	—	10	8	no	—
274	C₂²	—	20	48	no	—
275	C₂²	C₃×#4086	1	16	yes	direct product factor
276	C₂²	C₂×C₃×D₄×D₅	1	12	yes	—
277	C₂²	C₂²×C₃×C₂²⋊D₅	1	6	yes	—
278	C₂²	C₂⁴×C₃×D₅	1	1	yes	characteristic
279	C₄	—	5	16	no	—
280	C₄	C₂²×C₃×C₂²⋊D₅	1	8	yes	complemented
281	C₃	C₂²×#4086	1	1	yes	characteristic, direct product factor, Sylow
282	C₂	—	4	8	no	—
283	C₂	—	20	8	no	—
284	C₂	C₂×C₃×#4086	1	12	yes	direct product factor
285	C₂	C₂²×C₃×D₄×D₅	1	2	yes	—
286	C₂	C₂³×C₃×C₂²⋊D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴×C₃	C₂²×D₅
commutator, cosocle	C₂²×C₅	C₂⁵×C₃
Frattini	C₂²	C₂⁴×C₃×D₅
Fitting	C₂³×C₃×C₅×D₄	C₂
socle	C₂⁴×C₃×C₅	C₂³
hypercenter	C₂³×C₃×D₄	D₅
hypocenter	C₅	C₂²×C₃×#127
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂⁴×C₃ ◁ C₂³×C₃×D₄

Complements

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

N	H ≅ G/N	type
1. C₂³×C₃×C₅×D₄	C₂	semidirect
2. C₂³×C₃×C₂²⋊D₅	C₂	semidirect
3. C₂³×C₃×D₂₀	C₂	semidirect
4. C₂×C₃×#4086	C₂	direct (N × H)
5. C₂²×C₃×C₄×Dic₅	C₂	semidirect
6. C₂²×#4086	C₃	direct (N × H)
10. C₂²×C₃×C₅×D₄	C₂²	semidirect
13. C₂²×C₃×C₂²⋊D₅	C₂²	semidirect
16. C₂²×C₃×D₂₀	C₂²	semidirect
17. C₃×#4086	C₂²	direct (N × H)
18. C₂×C₃×C₄×Dic₅	C₂²	semidirect
20. C₂³×C₅×D₄	C₂×C₃	semidirect
21. C₂³×C₂²⋊D₅	C₂×C₃	semidirect
22. C₂³×D₂₀	C₂×C₃	semidirect
23. C₂×#4086	C₂×C₃	direct (N × H)
24. C₂²×C₄×Dic₅	C₂×C₃	semidirect
33. C₂×C₃×C₅×D₄	C₂³	semidirect
34. C₂²×C₃×C₄×C₅	D₄	semidirect
37. C₂×C₃×C₂²⋊D₅	C₂³	semidirect
38. C₂²×C₃×Dic₅	D₄	semidirect
41. C₂×C₃×D₂₀	C₂³	semidirect
42. C₃×C₄×Dic₅	C₂³	semidirect
45. C₂³×C₃×D₄	D₅	semidirect
51. C₂²×C₅×D₄	C₂²×C₃	semidirect
54. C₂²×C₂²⋊D₅	C₂²×C₃	semidirect
57. C₂²×D₂₀	C₂²×C₃	semidirect
58. #4086	C₂²×C₃	direct (N × H)
59. C₂×C₄×Dic₅	C₂²×C₃	semidirect
73. C₂×C₃×C₄×C₅	C₂×D₄	semidirect
76. C₂×C₃×Dic₅	C₂×D₄	semidirect
86. C₂²×C₃×D₄	C₂×D₅	semidirect
99. C₂×C₅×D₄	C₂³×C₃	semidirect
100. C₂²×C₄×C₅	C₃×D₄	semidirect
103. C₂×C₂²⋊D₅	C₂³×C₃	semidirect
104. C₂²×Dic₅	C₃×D₄	semidirect
107. C₂×D₂₀	C₂³×C₃	semidirect
108. C₄×Dic₅	C₂³×C₃	semidirect
111. C₂³×D₄	C₃×D₅	semidirect
117. C₂²×C₃×C₅	#127	semidirect
124. C₃×C₄×C₅	C₂²×D₄	semidirect
125. C₃×Dic₅	C₂²×D₄	semidirect
138. C₂×C₃×D₄	C₂²×D₅	semidirect
141. C₂²×C₃×C₄	C₂²⋊D₅	semidirect
156. C₂×C₄×C₅	C₂×C₃×D₄	semidirect
159. C₂×Dic₅	C₂×C₃×D₄	semidirect
169. C₂²×D₄	C₂×C₃×D₅	semidirect
175. C₂×C₃×C₅	C₂×#127	semidirect
195. C₂×C₃×C₄	C₂×C₂²⋊D₅	semidirect
200. C₂²×C₅	C₃×#127	semidirect
207. C₄×C₅	C₂²×C₃×D₄	semidirect
208. Dic₅	C₂²×C₃×D₄	semidirect
221. C₂×D₄	C₂²×C₃×D₅	semidirect
224. C₂²×C₄	C₃×C₂²⋊D₅	semidirect
227. C₃×C₅	C₂²×#127	semidirect
234. C₂²×C₃	#4086	direct (N × H)
239. C₃×C₄	C₂²×C₂²⋊D₅	semidirect
241. C₂×C₅	C₂×C₃×#127	semidirect
261. C₂×C₄	C₂×C₃×C₂²⋊D₅	semidirect
265. C₂×C₃	C₂×#4086	direct (N × H)
268. C₅	C₂²×C₃×#127	semidirect
275. C₂²	C₃×#4086	direct (N × H)
280. C₄	C₂²×C₃×C₂²⋊D₅	semidirect
281. C₃	C₂²×#4086	direct (N × H)
284. C₂	C₂×C₃×#4086	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃×C₅×D₄	2
C₂³×C₃×C₂²⋊D₅	2
C₂³×C₃×D₂₀	2
C₂×C₃×#4086	2
C₂²×C₃×C₄×Dic₅	2
C₂²×#4086	3
C₂²×C₃×#127	5

Subgroups — C22×C3×#4086 #14479962

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×#4086 #14479962