Subgroups — C₂×C₃×D₄×D₂₀ #14482416

Order 1920.

5072 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×D₄×D₅	C₂	1	2	yes	maximal, complemented
2	C₂³×C₃×D₂₀	C₂	1	2	yes	maximal, complemented
3	C₂×C₃×C₄×C₅×D₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×#4076	C₂	1	2	yes	maximal, complemented
5	C₃×D₄×D₂₀	C₂	1	16	yes	maximal, direct product factor
6	C₂×C₃×#4041	C₂	1	2	yes	maximal, complemented
7	C₂×C₃×#4028	C₂	1	4	yes	maximal, complemented
8	C₂×C₃×#4020	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₃×C₄×D₂₀	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×D₄×D₂₀	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₂⁴×C₃×D₅	C₂²	1	4	yes	complemented
12	C₂²×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
13	C₂³×C₃×C₄×C₅	C₂²	1	2	yes	complemented
14	C₂²×C₃×C₂²⋊D₅	C₂²	1	4	yes	complemented
15	C₂×C₃×D₄×D₅	—	2	16	no	—
16	C₂×C₃×D₄×D₅	C₂²	1	16	yes	complemented
17	C₂²×C₃×D₂₀	—	2	16	no	—
18	C₂²×C₃×D₂₀	C₂²	1	1	yes	characteristic
19	C₂²×C₃×D₂₀	C₂²	1	16	yes	complemented
20	C₂²×C₃×D₂₀	C₂²	1	2	yes	complemented
21	C₂²×C₃×D₂₀	C₂²	1	4	yes	complemented
22	C₂²×C₃×D₂₀	C₂²	1	2	yes	complemented
23	C₂²×C₃×D₂₀	C₂²	1	2	yes	complemented
24	C₂²×C₃×C₄×D₅	C₂²	1	2	yes	complemented
25	C₃×C₄×C₅×D₄	C₂²	1	8	yes	complemented
26	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
27	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	complemented
28	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic, complemented
29	C₃×#4076	C₂²	1	16	yes	complemented
30	C₂×C₃×#685	C₂²	1	4	yes	complemented
31	C₂×C₃×#685	C₂²	1	2	yes	complemented
32	C₂×C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic, complemented
33	C₃×#4041	C₂²	1	16	yes	complemented
34	C₃×#4028	C₂²	1	32	yes	complemented
35	C₃×#4020	C₂²	1	8	yes	complemented
36	C₃×C₄×D₂₀	C₂²	1	8	yes	complemented
37	C₂×C₃×D₄²	—	5	1	no	maximal, Hall
38	C₂²×D₄×D₅	C₂×C₃	1	2	yes	complemented
39	C₂³×D₂₀	C₂×C₃	1	2	yes	complemented
40	C₂×C₄×C₅×D₄	C₂×C₃	1	1	yes	characteristic, complemented
41	C₂×#4076	C₂×C₃	1	2	yes	complemented
42	D₄×D₂₀	C₂×C₃	1	16	yes	direct product factor
43	C₂×#4041	C₂×C₃	1	2	yes	complemented
44	C₂×#4028	C₂×C₃	1	4	yes	complemented
45	C₂×#4020	C₂×C₃	1	1	yes	characteristic, complemented
46	C₂×C₄×D₂₀	C₂×C₃	1	1	yes	characteristic, complemented
47	C₂⁴×C₃×C₅	C₂³	1	2	yes	—
48	C₂³×C₃×D₅	—	2	16	no	—
49	C₂³×C₃×D₅	—	2	16	no	—
50	C₂³×C₃×D₅	—	4	8	no	—
51	C₂³×C₃×D₅	C₂³	1	4	yes	—
52	C₂³×C₃×D₅	C₂³	1	2	yes	—
53	C₂³×C₃×D₅	C₂³	1	16	yes	complemented
54	C₂×C₃×C₅×D₄	D₄	1	8	yes	complemented
55	C₂×C₃×C₅×D₄	C₂³	1	4	yes	—
56	C₂²×C₃×C₄×C₅	—	2	8	no	—
57	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
58	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
59	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
60	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
61	C₂²×C₃×C₄×C₅	C₂³	1	8	yes	complemented
62	C₂×C₃×C₂²⋊D₅	—	2	16	no	—
63	C₂×C₃×C₂²⋊D₅	C₂³	1	16	yes	complemented
64	C₂²×C₃×Dic₅	C₂³	1	2	yes	—
65	C₃×D₄×D₅	—	2	64	no	—
66	C₂×C₃×D₂₀	—	2	16	no	—
67	C₂×C₃×D₂₀	—	2	8	no	—
68	C₂×C₃×D₂₀	—	2	8	no	—
69	C₂×C₃×D₂₀	—	2	64	no	—
70	C₂×C₃×D₂₀	—	4	4	no	—
71	C₂×C₃×D₂₀	D₄	1	8	yes	direct product factor
72	C₂×C₃×D₂₀	C₂³	1	4	yes	—
73	C₂×C₃×D₂₀	C₂³	1	8	yes	complemented
74	C₂×C₃×D₂₀	C₂³	1	16	yes	complemented
75	C₂×C₃×D₂₀	C₂³	1	8	yes	complemented
76	C₂×C₃×D₂₀	C₂³	1	8	yes	complemented
77	C₂×C₃×C₄×D₅	—	2	8	no	—
78	C₂×C₃×C₄×D₅	C₂³	1	8	yes	complemented
79	C₃×C₅×C₄⋊C₄	C₂³	1	4	yes	complemented
80	C₃×C₅×C₂²⋊C₄	C₂³	1	8	yes	complemented
81	C₃×C₄²×C₅	C₂³	1	4	yes	complemented
82	C₃×#685	C₂³	1	8	yes	complemented
83	C₃×#685	C₂³	1	16	yes	complemented
84	C₃×C₄⋊Dic₅	C₂³	1	4	yes	complemented
85	C₂³×C₃×D₄	—	5	2	no	—
86	C₂³×C₃×D₄	—	5	2	no	—
87	C₃×D₄²	—	5	16	no	—
88	C₂×C₃×#127	—	5	1	no	—
89	C₂×C₃×#121	—	5	2	no	—
90	C₂×C₃×#121	—	5	2	no	—
91	C₂×C₃×#120	—	5	4	no	—
92	C₂×C₃×C₄×D₄	—	5	1	no	—
93	C₂×C₃×C₄×D₄	D₅	1	1	yes	characteristic, complemented
94	C₂⁴×D₅	C₂²×C₃	1	4	yes	complemented
95	C₂²×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
96	C₂³×C₄×C₅	C₂²×C₃	1	2	yes	complemented
97	C₂²×C₂²⋊D₅	C₂²×C₃	1	4	yes	complemented
98	C₂×D₄×D₅	—	2	16	no	—
99	C₂×D₄×D₅	C₂²×C₃	1	16	yes	complemented
100	C₂²×D₂₀	—	2	16	no	—
101	C₂²×D₂₀	C₂²×C₃	1	1	yes	characteristic
102	C₂²×D₂₀	C₂²×C₃	1	16	yes	complemented
103	C₂²×D₂₀	C₂²×C₃	1	2	yes	complemented
104	C₂²×D₂₀	C₂²×C₃	1	2	yes	complemented
105	C₂²×D₂₀	C₂²×C₃	1	4	yes	complemented
106	C₂²×D₂₀	C₂²×C₃	1	2	yes	complemented
107	C₂²×C₄×D₅	C₂²×C₃	1	2	yes	complemented
108	C₄×C₅×D₄	C₂²×C₃	1	8	yes	complemented
109	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic, complemented
110	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	complemented
111	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic, complemented
112	#4076	C₂²×C₃	1	16	yes	complemented
113	C₂×#685	C₂²×C₃	1	2	yes	complemented
114	C₂×#685	C₂²×C₃	1	4	yes	complemented
115	C₂×C₄⋊Dic₅	C₂²×C₃	1	1	yes	characteristic, complemented
116	#4041	C₂²×C₃	1	16	yes	complemented
117	#4028	C₂²×C₃	1	32	yes	complemented
118	#4020	C₂²×C₃	1	8	yes	complemented
119	C₄×D₂₀	C₂²×C₃	1	8	yes	complemented
120	C₂×D₄²	—	5	1	no	Sylow
121	C₂³×C₃×C₅	—	2	4	no	—
122	C₂³×C₃×C₅	—	2	4	no	—
123	C₂³×C₃×C₅	C₂×D₄	1	8	yes	complemented
124	C₂³×C₃×C₅	C₂⁴	1	4	yes	—
125	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
126	C₂²×C₃×D₅	—	2	4	no	—
127	C₂²×C₃×D₅	—	2	4	no	—
128	C₂²×C₃×D₅	—	2	8	no	—
129	C₂²×C₃×D₅	—	2	8	no	—
130	C₂²×C₃×D₅	—	2	32	no	—
131	C₂²×C₃×D₅	—	2	8	no	—
132	C₂²×C₃×D₅	—	2	32	no	—
133	C₂²×C₃×D₅	—	4	16	no	—
134	C₂²×C₃×D₅	—	4	64	no	—
135	C₂²×C₃×D₅	C₂×D₄	1	8	yes	complemented
136	C₂²×C₃×D₅	C₂⁴	1	8	yes	—
137	C₂²×C₃×D₅	C₂⁴	1	4	yes	—
138	C₃×C₅×D₄	C₂×D₄	1	16	yes	complemented
139	C₂×C₃×C₄×C₅	—	2	4	no	—
140	C₂×C₃×C₄×C₅	—	2	2	no	—
141	C₂×C₃×C₄×C₅	—	2	16	no	—
142	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	complemented
143	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	complemented
144	C₂×C₃×C₄×C₅	C₂⁴	1	2	yes	—
145	C₂×C₃×C₄×C₅	C₂⁴	1	4	yes	—
146	C₂×C₃×C₄×C₅	C₂⁴	1	2	yes	—
147	C₂×C₃×C₄×C₅	C₂⁴	1	2	yes	—
148	C₃×C₂²⋊D₅	—	2	32	no	—
149	C₂×C₃×Dic₅	—	2	4	no	—
150	C₂×C₃×Dic₅	C₂⁴	1	4	yes	—
151	C₃×D₂₀	—	2	32	no	—
152	C₃×D₂₀	—	2	16	no	—
153	C₃×D₂₀	—	2	16	no	—
154	C₃×D₂₀	—	4	8	no	—
155	C₃×D₂₀	C₂×D₄	1	16	yes	direct product factor
156	C₃×C₄×D₅	—	2	16	no	—
157	C₂⁵×C₃	—	5	4	no	—
158	C₂²×C₃×D₄	—	5	4	no	—
159	C₂²×C₃×D₄	—	5	16	no	—
160	C₂²×C₃×D₄	—	5	1	no	—
161	C₂²×C₃×D₄	—	5	4	no	—
162	C₂²×C₃×D₄	—	5	2	no	—
163	C₂²×C₃×D₄	—	5	2	no	—
164	C₂²×C₃×D₄	—	5	2	no	—
165	C₂²×C₃×D₄	—	5	16	no	—
166	C₂²×C₃×D₄	—	10	16	no	—
167	C₂²×C₃×D₄	—	10	16	no	—
168	C₂²×C₃×D₄	C₂×D₅	1	1	yes	characteristic
169	C₂³×C₃×C₄	—	5	2	no	—
170	C₂³×C₃×C₄	C₂×D₅	1	2	yes	complemented
171	C₃×#127	—	5	8	no	—
172	C₃×#121	—	5	16	no	—
173	C₃×#121	—	5	16	no	—
174	C₃×#120	—	5	32	no	—
175	C₃×C₄×D₄	—	5	8	no	—
176	C₃×C₄×D₄	C₂×D₅	1	8	yes	complemented
177	C₂×C₃×C₄⋊C₄	—	5	1	no	—
178	C₂×C₃×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic, complemented
179	C₂×C₃×C₂²⋊C₄	—	5	2	no	—
180	C₂×C₃×C₂²⋊C₄	—	5	4	no	—
181	C₂×C₃×C₂²⋊C₄	C₂×D₅	1	2	yes	complemented
182	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic, complemented
183	C₂⁴×C₅	C₂³×C₃	1	2	yes	—
184	C₂³×D₅	—	2	16	no	—
185	C₂³×D₅	—	2	16	no	—
186	C₂³×D₅	—	4	8	no	—
187	C₂³×D₅	C₂³×C₃	1	4	yes	—
188	C₂³×D₅	C₂³×C₃	1	2	yes	—
189	C₂³×D₅	C₂³×C₃	1	16	yes	complemented
190	C₂×C₅×D₄	C₃×D₄	1	8	yes	complemented
191	C₂×C₅×D₄	C₂³×C₃	1	4	yes	—
192	C₂²×C₄×C₅	—	2	8	no	—
193	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
194	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
195	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
196	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
197	C₂²×C₄×C₅	C₂³×C₃	1	8	yes	complemented
198	C₂×C₂²⋊D₅	—	2	16	no	—
199	C₂×C₂²⋊D₅	C₂³×C₃	1	16	yes	complemented
200	C₂²×Dic₅	C₂³×C₃	1	2	yes	—
201	D₄×D₅	—	2	64	no	—
202	C₂×D₂₀	—	2	16	no	—
203	C₂×D₂₀	—	2	8	no	—
204	C₂×D₂₀	—	2	8	no	—
205	C₂×D₂₀	—	2	64	no	—
206	C₂×D₂₀	—	4	4	no	—
207	C₂×D₂₀	C₃×D₄	1	8	yes	direct product factor
208	C₂×D₂₀	C₂³×C₃	1	4	yes	—
209	C₂×D₂₀	C₂³×C₃	1	8	yes	complemented
210	C₂×D₂₀	C₂³×C₃	1	16	yes	complemented
211	C₂×D₂₀	C₂³×C₃	1	8	yes	complemented
212	C₂×D₂₀	C₂³×C₃	1	8	yes	complemented
213	C₂×C₄×D₅	—	2	8	no	—
214	C₂×C₄×D₅	C₂³×C₃	1	8	yes	complemented
215	C₅×C₄⋊C₄	C₂³×C₃	1	4	yes	complemented
216	C₅×C₂²⋊C₄	C₂³×C₃	1	8	yes	complemented
217	C₄²×C₅	C₂³×C₃	1	4	yes	complemented
218	#685	C₂³×C₃	1	8	yes	complemented
219	#685	C₂³×C₃	1	16	yes	complemented
220	C₄⋊Dic₅	C₂³×C₃	1	4	yes	complemented
221	C₂³×D₄	—	5	2	no	—
222	C₂³×D₄	—	5	2	no	—
223	D₄²	—	5	16	no	—
224	C₂×#127	—	5	1	no	—
225	C₂×#121	—	5	2	no	—
226	C₂×#121	—	5	2	no	—
227	C₂×#120	—	5	4	no	—
228	C₂×C₄×D₄	—	5	1	no	—
229	C₂×C₄×D₄	C₃×D₅	1	1	yes	characteristic, complemented
230	C₂²×C₃×C₅	—	2	16	no	—
231	C₂²×C₃×C₅	—	2	4	no	—
232	C₂²×C₃×C₅	—	2	4	no	—
233	C₂²×C₃×C₅	C₂²×D₄	1	2	yes	—
234	C₂²×C₃×C₅	C₂²×D₄	1	2	yes	—
235	C₂²×C₃×C₅	C₂²×D₄	1	8	yes	complemented
236	C₂²×C₃×C₅	H(5,𝔽₂)	1	2	yes	—
237	C₂²×C₃×C₅	C₂⁵	1	1	yes	characteristic
238	C₂×C₃×D₅	—	2	16	no	—
239	C₂×C₃×D₅	—	2	4	no	—
240	C₂×C₃×D₅	—	2	4	no	—
241	C₂×C₃×D₅	—	2	8	no	—
242	C₂×C₃×D₅	—	2	8	no	—
243	C₂×C₃×D₅	—	2	8	no	—
244	C₂×C₃×D₅	—	4	64	no	—
245	C₂×C₃×D₅	—	4	16	no	—
246	C₂×C₃×D₅	C₂²×D₄	1	8	yes	complemented
247	C₃×C₄×C₅	—	2	4	no	—
248	C₃×C₄×C₅	—	2	2	no	—
249	C₃×C₄×C₅	C₂²×D₄	1	4	yes	complemented
250	C₃×C₄×C₅	C₂²×D₄	1	4	yes	complemented
251	C₃×Dic₅	—	2	4	no	—
252	C₂⁴×C₃	—	5	2	no	—
253	C₂⁴×C₃	—	5	4	no	—
254	C₂⁴×C₃	—	5	16	no	—
255	C₂⁴×C₃	—	10	16	no	—
256	C₂⁴×C₃	—	10	16	no	—
257	C₂⁴×C₃	—	20	8	no	—
258	C₂⁴×C₃	C₂²×D₅	1	2	yes	—
259	C₂×C₃×D₄	—	5	16	no	—
260	C₂×C₃×D₄	—	5	8	no	—
261	C₂×C₃×D₄	—	5	4	no	—
262	C₂×C₃×D₄	—	5	16	no	—
263	C₂×C₃×D₄	—	5	8	no	—
264	C₂×C₃×D₄	—	5	8	no	—
265	C₂×C₃×D₄	—	5	8	no	—
266	C₂×C₃×D₄	—	10	64	no	—
267	C₂×C₃×D₄	—	10	16	no	—
268	C₂×C₃×D₄	—	10	16	no	—
269	C₂×C₃×D₄	—	10	64	no	—
270	C₂×C₃×D₄	—	10	8	no	—
271	C₂×C₃×D₄	—	10	8	no	—
272	C₂×C₃×D₄	—	20	4	no	—
273	C₂×C₃×D₄	D₂₀	1	8	yes	direct product factor
274	C₂×C₃×D₄	C₂²×D₅	1	4	yes	—
275	C₂²×C₃×C₄	—	2	8	no	—
276	C₂²×C₃×C₄	—	5	2	no	—
277	C₂²×C₃×C₄	—	5	8	no	—
278	C₂²×C₃×C₄	—	10	8	no	—
279	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
280	C₂²×C₃×C₄	C₂²×D₅	1	2	yes	—
281	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
282	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
283	C₂²×C₃×C₄	C₂²×D₅	1	8	yes	complemented
284	C₃×C₄⋊C₄	—	5	4	no	—
285	C₃×C₄⋊C₄	C₂²×D₅	1	4	yes	complemented
286	C₃×C₂²⋊C₄	—	5	8	no	—
287	C₃×C₂²⋊C₄	—	5	16	no	—
288	C₃×C₂²⋊C₄	C₂²×D₅	1	8	yes	complemented
289	C₃×C₄²	C₂²×D₅	1	4	yes	complemented
290	C₂³×C₅	—	2	4	no	—
291	C₂³×C₅	—	2	4	no	—
292	C₂³×C₅	C₂×C₃×D₄	1	8	yes	complemented
293	C₂³×C₅	C₂⁴×C₃	1	4	yes	—
294	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
295	C₂²×D₅	—	2	4	no	—
296	C₂²×D₅	—	2	4	no	—
297	C₂²×D₅	—	2	8	no	—
298	C₂²×D₅	—	2	8	no	—
299	C₂²×D₅	—	2	32	no	—
300	C₂²×D₅	—	2	8	no	—
301	C₂²×D₅	—	2	32	no	—
302	C₂²×D₅	—	4	16	no	—
303	C₂²×D₅	—	4	64	no	—
304	C₂²×D₅	C₂×C₃×D₄	1	8	yes	complemented
305	C₂²×D₅	C₂⁴×C₃	1	8	yes	—
306	C₂²×D₅	C₂⁴×C₃	1	4	yes	—
307	C₅×D₄	C₂×C₃×D₄	1	16	yes	complemented
308	C₂×C₄×C₅	—	2	4	no	—
309	C₂×C₄×C₅	—	2	2	no	—
310	C₂×C₄×C₅	—	2	16	no	—
311	C₂×C₄×C₅	C₂×C₃×D₄	1	4	yes	complemented
312	C₂×C₄×C₅	C₂×C₃×D₄	1	4	yes	complemented
313	C₂×C₄×C₅	C₂⁴×C₃	1	2	yes	—
314	C₂×C₄×C₅	C₂⁴×C₃	1	4	yes	—
315	C₂×C₄×C₅	C₂⁴×C₃	1	2	yes	—
316	C₂×C₄×C₅	C₂⁴×C₃	1	2	yes	—
317	C₂²⋊D₅	—	2	32	no	—
318	C₂×Dic₅	—	2	4	no	—
319	C₂×Dic₅	C₂⁴×C₃	1	4	yes	—
320	D₂₀	—	2	32	no	—
321	D₂₀	—	2	16	no	—
322	D₂₀	—	2	16	no	—
323	D₂₀	—	4	8	no	—
324	D₂₀	C₂×C₃×D₄	1	16	yes	direct product factor
325	C₄×D₅	—	2	16	no	—
326	C₂⁵	—	5	4	no	—
327	C₂²×D₄	—	5	4	no	—
328	C₂²×D₄	—	5	16	no	—
329	C₂²×D₄	—	5	1	no	—
330	C₂²×D₄	—	5	4	no	—
331	C₂²×D₄	—	5	2	no	—
332	C₂²×D₄	—	5	2	no	—
333	C₂²×D₄	—	5	2	no	—
334	C₂²×D₄	—	5	16	no	—
335	C₂²×D₄	—	10	16	no	—
336	C₂²×D₄	—	10	16	no	—
337	C₂²×D₄	C₂×C₃×D₅	1	1	yes	characteristic
338	C₂³×C₄	—	5	2	no	—
339	C₂³×C₄	C₂×C₃×D₅	1	2	yes	complemented
340	#127	—	5	8	no	—
341	#121	—	5	16	no	—
342	#121	—	5	16	no	—
343	#120	—	5	32	no	—
344	C₄×D₄	—	5	8	no	—
345	C₄×D₄	C₂×C₃×D₅	1	8	yes	complemented
346	C₂×C₄⋊C₄	—	5	1	no	—
347	C₂×C₄⋊C₄	C₂×C₃×D₅	1	1	yes	characteristic, complemented
348	C₂×C₂²⋊C₄	—	5	2	no	—
349	C₂×C₂²⋊C₄	—	5	4	no	—
350	C₂×C₂²⋊C₄	C₂×C₃×D₅	1	2	yes	complemented
351	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic, complemented
352	C₂×C₃×C₅	—	2	8	no	—
353	C₂×C₃×C₅	D₄²	1	4	yes	complemented
354	C₂×C₃×C₅	C₂³×D₄	1	1	yes	characteristic
355	C₂×C₃×C₅	C₂³×D₄	1	1	yes	characteristic
356	C₂×C₃×C₅	C₂×H(5,𝔽₂)	1	1	yes	characteristic
357	C₃×D₅	—	2	8	no	—
358	C₃×D₅	—	4	8	no	—
359	C₂³×C₃	—	2	4	no	—
360	C₂³×C₃	—	2	4	no	—
361	C₂³×C₃	—	5	8	no	—
362	C₂³×C₃	—	5	4	no	—
363	C₂³×C₃	—	5	8	no	—
364	C₂³×C₃	—	10	32	no	—
365	C₂³×C₃	—	10	4	no	—
366	C₂³×C₃	—	10	4	no	—
367	C₂³×C₃	—	10	8	no	—
368	C₂³×C₃	—	10	32	no	—
369	C₂³×C₃	—	10	8	no	—
370	C₂³×C₃	—	10	8	no	—
371	C₂³×C₃	—	20	16	no	—
372	C₂³×C₃	—	20	64	no	—
373	C₂³×C₃	C₂×D₂₀	1	8	yes	complemented
374	C₂³×C₃	C₂³×D₅	1	4	yes	—
375	C₂³×C₃	C₂³×D₅	1	1	yes	characteristic
376	C₃×D₄	—	5	16	no	—
377	C₃×D₄	—	10	32	no	—
378	C₃×D₄	—	10	32	no	—
379	C₃×D₄	—	10	16	no	—
380	C₃×D₄	—	10	16	no	—
381	C₃×D₄	—	20	8	no	—
382	C₃×D₄	C₂×D₂₀	1	16	yes	direct product factor
383	C₂×C₃×C₄	—	2	4	no	—
384	C₂×C₃×C₄	—	2	2	no	—
385	C₂×C₃×C₄	—	2	16	no	—
386	C₂×C₃×C₄	—	5	4	no	—
387	C₂×C₃×C₄	—	10	4	no	—
388	C₂×C₃×C₄	—	10	16	no	—
389	C₂×C₃×C₄	C₂×D₂₀	1	4	yes	complemented
390	C₂×C₃×C₄	D₄×D₅	1	4	yes	complemented
391	C₂×C₃×C₄	C₂³×D₅	1	2	yes	—
392	C₂×C₃×C₄	C₂³×D₅	1	4	yes	—
393	C₂×C₃×C₄	C₂³×D₅	1	2	yes	—
394	C₂×C₃×C₄	C₂³×D₅	1	2	yes	—
395	C₂²×C₅	—	2	4	no	—
396	C₂²×C₅	—	2	4	no	—
397	C₂²×C₅	—	2	16	no	—
398	C₂²×C₅	C₂²×C₃×D₄	1	2	yes	—
399	C₂²×C₅	C₂²×C₃×D₄	1	2	yes	—
400	C₂²×C₅	C₂²×C₃×D₄	1	8	yes	complemented
401	C₂²×C₅	C₃×H(5,𝔽₂)	1	2	yes	—
402	C₂²×C₅	C₂⁵×C₃	1	1	yes	characteristic
403	C₂×D₅	—	2	4	no	—
404	C₂×D₅	—	2	4	no	—
405	C₂×D₅	—	2	16	no	—
406	C₂×D₅	—	2	8	no	—
407	C₂×D₅	—	2	8	no	—
408	C₂×D₅	—	2	8	no	—
409	C₂×D₅	—	4	64	no	—
410	C₂×D₅	—	4	16	no	—
411	C₂×D₅	C₂²×C₃×D₄	1	8	yes	complemented
412	C₄×C₅	—	2	4	no	—
413	C₄×C₅	—	2	2	no	—
414	C₄×C₅	C₂²×C₃×D₄	1	4	yes	complemented
415	C₄×C₅	C₂²×C₃×D₄	1	4	yes	complemented
416	Dic₅	—	2	4	no	—
417	C₂⁴	—	5	2	no	—
418	C₂⁴	—	5	4	no	—
419	C₂⁴	—	5	16	no	—
420	C₂⁴	—	10	16	no	—
421	C₂⁴	—	10	16	no	—
422	C₂⁴	—	20	8	no	—
423	C₂⁴	C₂²×C₃×D₅	1	2	yes	—
424	C₂×D₄	—	5	16	no	—
425	C₂×D₄	—	5	8	no	—
426	C₂×D₄	—	5	4	no	—
427	C₂×D₄	—	5	16	no	—
428	C₂×D₄	—	5	8	no	—
429	C₂×D₄	—	5	8	no	—
430	C₂×D₄	—	5	8	no	—
431	C₂×D₄	—	10	64	no	—
432	C₂×D₄	—	10	16	no	—
433	C₂×D₄	—	10	16	no	—
434	C₂×D₄	—	10	64	no	—
435	C₂×D₄	—	10	8	no	—
436	C₂×D₄	—	10	8	no	—
437	C₂×D₄	—	20	4	no	—
438	C₂×D₄	C₃×D₂₀	1	8	yes	direct product factor
439	C₂×D₄	C₂²×C₃×D₅	1	4	yes	—
440	C₂²×C₄	—	2	8	no	—
441	C₂²×C₄	—	5	2	no	—
442	C₂²×C₄	—	5	8	no	—
443	C₂²×C₄	—	10	8	no	—
444	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
445	C₂²×C₄	C₂²×C₃×D₅	1	2	yes	—
446	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
447	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
448	C₂²×C₄	C₂²×C₃×D₅	1	8	yes	complemented
449	C₄⋊C₄	—	5	4	no	—
450	C₄⋊C₄	C₂²×C₃×D₅	1	4	yes	complemented
451	C₂²⋊C₄	—	5	8	no	—
452	C₂²⋊C₄	—	5	16	no	—
453	C₂²⋊C₄	C₂²×C₃×D₅	1	8	yes	complemented
454	C₄²	C₂²×C₃×D₅	1	4	yes	complemented
455	C₃×C₅	C₂×D₄²	1	1	yes	characteristic, complemented, Hall
456	C₂²×C₃	—	2	16	no	—
457	C₂²×C₃	—	2	4	no	—
458	C₂²×C₃	—	2	4	no	—
459	C₂²×C₃	—	5	8	no	—
460	C₂²×C₃	—	10	16	no	—
461	C₂²×C₃	—	10	4	no	—
462	C₂²×C₃	—	10	4	no	—
463	C₂²×C₃	—	10	8	no	—
464	C₂²×C₃	—	10	8	no	—
465	C₂²×C₃	—	10	8	no	—
466	C₂²×C₃	—	20	64	no	—
467	C₂²×C₃	—	20	16	no	—
468	C₂²×C₃	C₂²×D₂₀	1	2	yes	—
469	C₂²×C₃	C₂²×D₂₀	1	8	yes	complemented
470	C₂²×C₃	C₂×D₄×D₅	1	2	yes	—
471	C₂²×C₃	#4149	1	2	yes	—
472	C₂²×C₃	C₂⁴×D₅	1	1	yes	characteristic
473	C₃×C₄	—	2	4	no	—
474	C₃×C₄	—	2	2	no	—
475	C₃×C₄	—	10	4	no	—
476	C₃×C₄	C₂²×D₂₀	1	4	yes	complemented
477	C₃×C₄	C₂×D₄×D₅	1	4	yes	complemented
478	C₂×C₅	—	2	8	no	—
479	C₂×C₅	C₃×D₄²	1	4	yes	complemented
480	C₂×C₅	C₂³×C₃×D₄	1	1	yes	characteristic
481	C₂×C₅	C₂³×C₃×D₄	1	1	yes	characteristic
482	C₂×C₅	C₂×C₃×H(5,𝔽₂)	1	1	yes	characteristic
483	D₅	—	2	8	no	—
484	D₅	—	4	8	no	—
485	C₂³	—	2	4	no	—
486	C₂³	—	2	4	no	—
487	C₂³	—	5	8	no	—
488	C₂³	—	5	4	no	—
489	C₂³	—	5	8	no	—
490	C₂³	—	10	32	no	—
491	C₂³	—	10	4	no	—
492	C₂³	—	10	4	no	—
493	C₂³	—	10	8	no	—
494	C₂³	—	10	32	no	—
495	C₂³	—	10	8	no	—
496	C₂³	—	10	8	no	—
497	C₂³	—	20	16	no	—
498	C₂³	—	20	64	no	—
499	C₂³	C₂×C₃×D₂₀	1	8	yes	complemented
500	C₂³	C₂³×C₃×D₅	1	4	yes	—
501	C₂³	C₂³×C₃×D₅	1	1	yes	characteristic
502	D₄	—	5	16	no	—
503	D₄	—	10	32	no	—
504	D₄	—	10	32	no	—
505	D₄	—	10	16	no	—
506	D₄	—	10	16	no	—
507	D₄	—	20	8	no	—
508	D₄	C₂×C₃×D₂₀	1	16	yes	direct product factor
509	C₂×C₄	—	2	4	no	—
510	C₂×C₄	—	2	2	no	—
511	C₂×C₄	—	2	16	no	—
512	C₂×C₄	—	5	4	no	—
513	C₂×C₄	—	10	4	no	—
514	C₂×C₄	—	10	16	no	—
515	C₂×C₄	C₂×C₃×D₂₀	1	4	yes	complemented
516	C₂×C₄	C₃×D₄×D₅	1	4	yes	complemented
517	C₂×C₄	C₂³×C₃×D₅	1	2	yes	—
518	C₂×C₄	C₂³×C₃×D₅	1	4	yes	—
519	C₂×C₄	C₂³×C₃×D₅	1	2	yes	—
520	C₂×C₄	C₂³×C₃×D₅	1	2	yes	—
521	C₂×C₃	—	2	8	no	—
522	C₂×C₃	—	10	8	no	—
523	C₂×C₃	—	20	8	no	—
524	C₂×C₃	D₄×D₂₀	1	4	yes	direct product factor
525	C₂×C₃	C₂³×D₂₀	1	1	yes	characteristic
526	C₂×C₃	C₂²×D₄×D₅	1	1	yes	characteristic
527	C₂×C₃	C₂×#4149	1	1	yes	characteristic
528	C₅	C₂×C₃×D₄²	1	1	yes	characteristic, complemented, Sylow
529	C₂²	—	2	16	no	—
530	C₂²	—	2	4	no	—
531	C₂²	—	2	4	no	—
532	C₂²	—	5	8	no	—
533	C₂²	—	10	4	no	—
534	C₂²	—	10	16	no	—
535	C₂²	—	10	4	no	—
536	C₂²	—	10	8	no	—
537	C₂²	—	10	8	no	—
538	C₂²	—	10	8	no	—
539	C₂²	—	20	64	no	—
540	C₂²	—	20	16	no	—
541	C₂²	C₂²×C₃×D₂₀	1	2	yes	—
542	C₂²	C₂²×C₃×D₂₀	1	8	yes	complemented
543	C₂²	C₂×C₃×D₄×D₅	1	2	yes	—
544	C₂²	C₃×#4149	1	2	yes	—
545	C₂²	C₂⁴×C₃×D₅	1	1	yes	characteristic
546	C₄	—	2	4	no	—
547	C₄	—	2	2	no	—
548	C₄	—	10	4	no	—
549	C₄	C₂²×C₃×D₂₀	1	4	yes	complemented
550	C₄	C₂×C₃×D₄×D₅	1	4	yes	complemented
551	C₃	C₂×D₄×D₂₀	1	1	yes	characteristic, direct product factor, Sylow
552	C₂	—	2	8	no	—
553	C₂	—	10	8	no	—
554	C₂	—	20	8	no	—
555	C₂	C₃×D₄×D₂₀	1	4	yes	direct product factor
556	C₂	C₂³×C₃×D₂₀	1	1	yes	characteristic
557	C₂	C₂²×C₃×D₄×D₅	1	1	yes	characteristic
558	C₂	C₂×C₃×#4149	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₃	C₂³×D₅
commutator, cosocle	C₂²×C₅	C₂⁵×C₃
Frattini	C₂²	C₂⁴×C₃×D₅
Fitting	C₂×C₃×C₄×C₅×D₄	C₂
socle	C₂³×C₃×C₅	C₂⁴
hypercenter	C₂×C₃×C₄×D₄	D₅
hypocenter	C₅	C₂×C₃×D₄²
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×D₄×D₅	C₂	semidirect
2. C₂³×C₃×D₂₀	C₂	semidirect
3. C₂×C₃×C₄×C₅×D₄	C₂	semidirect
4. C₂×C₃×#4076	C₂	semidirect
5. C₃×D₄×D₂₀	C₂	direct (N × H)
6. C₂×C₃×#4041	C₂	semidirect
7. C₂×C₃×#4028	C₂	semidirect
8. C₂×C₃×#4020	C₂	semidirect
9. C₂×C₃×C₄×D₂₀	C₂	semidirect
10. C₂×D₄×D₂₀	C₃	direct (N × H)
11. C₂⁴×C₃×D₅	C₂²	semidirect
13. C₂³×C₃×C₄×C₅	C₂²	semidirect
14. C₂²×C₃×C₂²⋊D₅	C₂²	semidirect
16. C₂×C₃×D₄×D₅	C₂²	semidirect
19. C₂²×C₃×D₂₀	C₂²	semidirect
20. C₂²×C₃×D₂₀	C₂²	semidirect
21. C₂²×C₃×D₂₀	C₂²	semidirect
22. C₂²×C₃×D₂₀	C₂²	semidirect
23. C₂²×C₃×D₂₀	C₂²	semidirect
24. C₂²×C₃×C₄×D₅	C₂²	semidirect
25. C₃×C₄×C₅×D₄	C₂²	semidirect
26. C₂×C₃×C₅×C₄⋊C₄	C₂²	semidirect
27. C₂×C₃×C₅×C₂²⋊C₄	C₂²	semidirect
28. C₂×C₃×C₄²×C₅	C₂²	semidirect
29. C₃×#4076	C₂²	semidirect
30. C₂×C₃×#685	C₂²	semidirect
31. C₂×C₃×#685	C₂²	semidirect
32. C₂×C₃×C₄⋊Dic₅	C₂²	semidirect
33. C₃×#4041	C₂²	semidirect
34. C₃×#4028	C₂²	semidirect
35. C₃×#4020	C₂²	semidirect
36. C₃×C₄×D₂₀	C₂²	semidirect
38. C₂²×D₄×D₅	C₂×C₃	semidirect
39. C₂³×D₂₀	C₂×C₃	semidirect
40. C₂×C₄×C₅×D₄	C₂×C₃	semidirect
41. C₂×#4076	C₂×C₃	semidirect
42. D₄×D₂₀	C₂×C₃	direct (N × H)
43. C₂×#4041	C₂×C₃	semidirect
44. C₂×#4028	C₂×C₃	semidirect
45. C₂×#4020	C₂×C₃	semidirect
46. C₂×C₄×D₂₀	C₂×C₃	semidirect
53. C₂³×C₃×D₅	C₂³	semidirect
54. C₂×C₃×C₅×D₄	D₄	semidirect
61. C₂²×C₃×C₄×C₅	C₂³	semidirect
63. C₂×C₃×C₂²⋊D₅	C₂³	semidirect
71. C₂×C₃×D₂₀	D₄	direct (N × H)
73. C₂×C₃×D₂₀	C₂³	semidirect
74. C₂×C₃×D₂₀	C₂³	semidirect
75. C₂×C₃×D₂₀	C₂³	semidirect
76. C₂×C₃×D₂₀	C₂³	semidirect
78. C₂×C₃×C₄×D₅	C₂³	semidirect
79. C₃×C₅×C₄⋊C₄	C₂³	semidirect
80. C₃×C₅×C₂²⋊C₄	C₂³	semidirect
81. C₃×C₄²×C₅	C₂³	semidirect
82. C₃×#685	C₂³	semidirect
83. C₃×#685	C₂³	semidirect
84. C₃×C₄⋊Dic₅	C₂³	semidirect
93. C₂×C₃×C₄×D₄	D₅	semidirect
94. C₂⁴×D₅	C₂²×C₃	semidirect
96. C₂³×C₄×C₅	C₂²×C₃	semidirect
97. C₂²×C₂²⋊D₅	C₂²×C₃	semidirect
99. C₂×D₄×D₅	C₂²×C₃	semidirect
102. C₂²×D₂₀	C₂²×C₃	semidirect
103. C₂²×D₂₀	C₂²×C₃	semidirect
104. C₂²×D₂₀	C₂²×C₃	semidirect
105. C₂²×D₂₀	C₂²×C₃	semidirect
106. C₂²×D₂₀	C₂²×C₃	semidirect
107. C₂²×C₄×D₅	C₂²×C₃	semidirect
108. C₄×C₅×D₄	C₂²×C₃	semidirect
109. C₂×C₅×C₄⋊C₄	C₂²×C₃	semidirect
110. C₂×C₅×C₂²⋊C₄	C₂²×C₃	semidirect
111. C₂×C₄²×C₅	C₂²×C₃	semidirect
112. #4076	C₂²×C₃	semidirect
113. C₂×#685	C₂²×C₃	semidirect
114. C₂×#685	C₂²×C₃	semidirect
115. C₂×C₄⋊Dic₅	C₂²×C₃	semidirect
116. #4041	C₂²×C₃	semidirect
117. #4028	C₂²×C₃	semidirect
118. #4020	C₂²×C₃	semidirect
119. C₄×D₂₀	C₂²×C₃	semidirect
123. C₂³×C₃×C₅	C₂×D₄	semidirect
135. C₂²×C₃×D₅	C₂×D₄	semidirect
138. C₃×C₅×D₄	C₂×D₄	semidirect
142. C₂×C₃×C₄×C₅	C₂×D₄	semidirect
143. C₂×C₃×C₄×C₅	C₂×D₄	semidirect
155. C₃×D₂₀	C₂×D₄	direct (N × H)
170. C₂³×C₃×C₄	C₂×D₅	semidirect
176. C₃×C₄×D₄	C₂×D₅	semidirect
178. C₂×C₃×C₄⋊C₄	C₂×D₅	semidirect
181. C₂×C₃×C₂²⋊C₄	C₂×D₅	semidirect
182. C₂×C₃×C₄²	C₂×D₅	semidirect
189. C₂³×D₅	C₂³×C₃	semidirect
190. C₂×C₅×D₄	C₃×D₄	semidirect
197. C₂²×C₄×C₅	C₂³×C₃	semidirect
199. C₂×C₂²⋊D₅	C₂³×C₃	semidirect
207. C₂×D₂₀	C₃×D₄	direct (N × H)
209. C₂×D₂₀	C₂³×C₃	semidirect
210. C₂×D₂₀	C₂³×C₃	semidirect
211. C₂×D₂₀	C₂³×C₃	semidirect
212. C₂×D₂₀	C₂³×C₃	semidirect
214. C₂×C₄×D₅	C₂³×C₃	semidirect
215. C₅×C₄⋊C₄	C₂³×C₃	semidirect
216. C₅×C₂²⋊C₄	C₂³×C₃	semidirect
217. C₄²×C₅	C₂³×C₃	semidirect
218. #685	C₂³×C₃	semidirect
219. #685	C₂³×C₃	semidirect
220. C₄⋊Dic₅	C₂³×C₃	semidirect
229. C₂×C₄×D₄	C₃×D₅	semidirect
235. C₂²×C₃×C₅	C₂²×D₄	semidirect
246. C₂×C₃×D₅	C₂²×D₄	semidirect
249. C₃×C₄×C₅	C₂²×D₄	semidirect
250. C₃×C₄×C₅	C₂²×D₄	semidirect
273. C₂×C₃×D₄	D₂₀	direct (N × H)
283. C₂²×C₃×C₄	C₂²×D₅	semidirect
285. C₃×C₄⋊C₄	C₂²×D₅	semidirect
288. C₃×C₂²⋊C₄	C₂²×D₅	semidirect
289. C₃×C₄²	C₂²×D₅	semidirect
292. C₂³×C₅	C₂×C₃×D₄	semidirect
304. C₂²×D₅	C₂×C₃×D₄	semidirect
307. C₅×D₄	C₂×C₃×D₄	semidirect
311. C₂×C₄×C₅	C₂×C₃×D₄	semidirect
312. C₂×C₄×C₅	C₂×C₃×D₄	semidirect
324. D₂₀	C₂×C₃×D₄	direct (N × H)
339. C₂³×C₄	C₂×C₃×D₅	semidirect
345. C₄×D₄	C₂×C₃×D₅	semidirect
347. C₂×C₄⋊C₄	C₂×C₃×D₅	semidirect
350. C₂×C₂²⋊C₄	C₂×C₃×D₅	semidirect
351. C₂×C₄²	C₂×C₃×D₅	semidirect
353. C₂×C₃×C₅	D₄²	semidirect
373. C₂³×C₃	C₂×D₂₀	semidirect
382. C₃×D₄	C₂×D₂₀	direct (N × H)
389. C₂×C₃×C₄	C₂×D₂₀	semidirect
390. C₂×C₃×C₄	D₄×D₅	semidirect
400. C₂²×C₅	C₂²×C₃×D₄	semidirect
411. C₂×D₅	C₂²×C₃×D₄	semidirect
414. C₄×C₅	C₂²×C₃×D₄	semidirect
415. C₄×C₅	C₂²×C₃×D₄	semidirect
438. C₂×D₄	C₃×D₂₀	direct (N × H)
448. C₂²×C₄	C₂²×C₃×D₅	semidirect
450. C₄⋊C₄	C₂²×C₃×D₅	semidirect
453. C₂²⋊C₄	C₂²×C₃×D₅	semidirect
454. C₄²	C₂²×C₃×D₅	semidirect
455. C₃×C₅	C₂×D₄²	semidirect
469. C₂²×C₃	C₂²×D₂₀	semidirect
476. C₃×C₄	C₂²×D₂₀	semidirect
477. C₃×C₄	C₂×D₄×D₅	semidirect
479. C₂×C₅	C₃×D₄²	semidirect
499. C₂³	C₂×C₃×D₂₀	semidirect
508. D₄	C₂×C₃×D₂₀	direct (N × H)
515. C₂×C₄	C₂×C₃×D₂₀	semidirect
516. C₂×C₄	C₃×D₄×D₅	semidirect
524. C₂×C₃	D₄×D₂₀	direct (N × H)
528. C₅	C₂×C₃×D₄²	semidirect
542. C₂²	C₂²×C₃×D₂₀	semidirect
549. C₄	C₂²×C₃×D₂₀	semidirect
550. C₄	C₂×C₃×D₄×D₅	semidirect
551. C₃	C₂×D₄×D₂₀	direct (N × H)
555. C₂	C₃×D₄×D₂₀	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×D₄×D₅	2
C₂³×C₃×D₂₀	2
C₂×C₃×C₄×C₅×D₄	2
C₂×C₃×#4076	2
C₃×D₄×D₂₀	2
C₂×C₃×#4041	2
C₂×C₃×#4028	2
C₂×C₃×#4020	2
C₂×C₃×C₄×D₂₀	2
C₂×D₄×D₂₀	3
C₂×C₃×D₄²	5

Subgroups — C2×C3×D4×D20 #14482416

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×D₄×D₂₀ #14482416