Subgroups — C₂²×D₄×D₂₀ #13006211

Order 1280.

20038 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₄×D₅	C₂	1	2	yes	maximal, complemented
2	C₂⁴×D₂₀	C₂	1	2	yes	maximal, complemented
3	C₂²×C₄×C₅×D₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#4076	C₂	1	2	yes	maximal, complemented
5	C₂×D₄×D₂₀	C₂	1	48	yes	maximal, direct product factor
6	C₂²×#4041	C₂	1	2	yes	maximal, complemented
7	C₂²×#4028	C₂	1	4	yes	maximal, complemented
8	C₂²×#4020	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₄×D₂₀	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂⁵×D₅	C₂²	1	4	yes	complemented
11	C₂³×C₅×D₄	C₂²	1	1	yes	characteristic
12	C₂⁴×C₄×C₅	C₂²	1	2	yes	complemented
13	C₂³×C₂²⋊D₅	C₂²	1	4	yes	complemented
14	C₂²×D₄×D₅	—	2	32	no	—
15	C₂²×D₄×D₅	C₂²	1	48	yes	complemented
16	C₂³×D₂₀	—	2	32	no	—
17	C₂³×D₂₀	C₂²	1	1	yes	characteristic
18	C₂³×D₂₀	C₂²	1	48	yes	complemented
19	C₂³×D₂₀	C₂²	1	4	yes	complemented
20	C₂³×D₂₀	C₂²	1	2	yes	complemented
21	C₂³×D₂₀	C₂²	1	2	yes	complemented
22	C₂³×D₂₀	C₂²	1	2	yes	complemented
23	C₂³×C₄×D₅	C₂²	1	2	yes	complemented
24	C₂×C₄×C₅×D₄	C₂²	1	24	yes	complemented
25	C₂²×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
26	C₂²×C₅×C₂²⋊C₄	C₂²	1	2	yes	complemented
27	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic, complemented
28	C₂×#4076	C₂²	1	48	yes	complemented
29	C₂²×#685	C₂²	1	4	yes	complemented
30	C₂²×#685	C₂²	1	2	yes	complemented
31	C₂²×C₄⋊Dic₅	C₂²	1	1	yes	characteristic, complemented
32	D₄×D₂₀	C₂²	1	256	yes	direct product factor
33	C₂×#4041	C₂²	1	48	yes	complemented
34	C₂×#4028	C₂²	1	96	yes	complemented
35	C₂×#4020	C₂²	1	24	yes	complemented
36	C₂×C₄×D₂₀	C₂²	1	24	yes	complemented
37	C₂²×D₄²	—	5	1	no	maximal, Sylow
38	C₂⁵×C₅	C₂³	1	2	yes	—
39	C₂⁴×D₅	—	2	32	no	—
40	C₂⁴×D₅	—	2	32	no	—
41	C₂⁴×D₅	—	4	16	no	—
42	C₂⁴×D₅	C₂³	1	4	yes	—
43	C₂⁴×D₅	C₂³	1	2	yes	—
44	C₂⁴×D₅	C₂³	1	48	yes	complemented
45	C₂²×C₅×D₄	D₄	1	16	yes	complemented
46	C₂²×C₅×D₄	C₂³	1	12	yes	—
47	C₂³×C₄×C₅	—	2	16	no	—
48	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
49	C₂³×C₄×C₅	C₂³	1	2	yes	—
50	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
51	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
52	C₂³×C₄×C₅	C₂³	1	24	yes	complemented
53	C₂²×C₂²⋊D₅	—	2	32	no	—
54	C₂²×C₂²⋊D₅	C₂³	1	48	yes	complemented
55	C₂³×Dic₅	C₂³	1	2	yes	—
56	C₂×D₄×D₅	—	2	384	no	—
57	C₂×D₄×D₅	C₂³	1	128	yes	complemented
58	C₂²×D₂₀	—	2	32	no	—
59	C₂²×D₂₀	—	2	16	no	—
60	C₂²×D₂₀	—	2	16	no	—
61	C₂²×D₂₀	—	2	384	no	—
62	C₂²×D₂₀	—	4	8	no	—
63	C₂²×D₂₀	D₄	1	16	yes	direct product factor
64	C₂²×D₂₀	C₂³	1	12	yes	—
65	C₂²×D₂₀	C₂³	1	128	yes	complemented
66	C₂²×D₂₀	C₂³	1	24	yes	complemented
67	C₂²×D₂₀	C₂³	1	24	yes	complemented
68	C₂²×D₂₀	C₂³	1	48	yes	complemented
69	C₂²×D₂₀	C₂³	1	24	yes	complemented
70	C₂²×C₄×D₅	—	2	16	no	—
71	C₂²×C₄×D₅	C₂³	1	24	yes	complemented
72	C₄×C₅×D₄	C₂³	1	64	yes	complemented
73	C₂×C₅×C₄⋊C₄	C₂³	1	12	yes	complemented
74	C₂×C₅×C₂²⋊C₄	C₂³	1	24	yes	complemented
75	C₂×C₄²×C₅	C₂³	1	12	yes	complemented
76	#4076	C₂³	1	128	yes	complemented
77	C₂×#685	C₂³	1	24	yes	complemented
78	C₂×#685	C₂³	1	48	yes	complemented
79	C₂×C₄⋊Dic₅	C₂³	1	12	yes	complemented
80	#4041	C₂³	1	128	yes	complemented
81	#4028	C₂³	1	256	yes	complemented
82	#4020	C₂³	1	64	yes	complemented
83	C₄×D₂₀	C₂³	1	64	yes	complemented
84	C₂⁴×D₄	—	5	2	no	—
85	C₂⁴×D₄	—	5	2	no	—
86	C₂×D₄²	—	5	48	no	—
87	C₂²×#127	—	5	1	no	—
88	C₂²×#121	—	5	2	no	—
89	C₂²×#121	—	5	2	no	—
90	C₂²×#120	—	5	4	no	—
91	C₂²×C₄×D₄	—	5	1	no	—
92	C₂²×C₄×D₄	D₅	1	1	yes	characteristic, complemented
93	C₂⁴×C₅	—	2	8	no	—
94	C₂⁴×C₅	—	2	8	no	—
95	C₂⁴×C₅	C₂×D₄	1	16	yes	complemented
96	C₂⁴×C₅	C₂⁴	1	12	yes	—
97	C₂⁴×C₅	C₂⁴	1	1	yes	characteristic
98	C₂³×D₅	—	2	8	no	—
99	C₂³×D₅	—	2	8	no	—
100	C₂³×D₅	—	2	16	no	—
101	C₂³×D₅	—	2	16	no	—
102	C₂³×D₅	—	2	192	no	—
103	C₂³×D₅	—	2	16	no	—
104	C₂³×D₅	—	2	192	no	—
105	C₂³×D₅	—	4	96	no	—
106	C₂³×D₅	—	4	256	no	—
107	C₂³×D₅	C₂×D₄	1	16	yes	complemented
108	C₂³×D₅	C₂⁴	1	24	yes	—
109	C₂³×D₅	C₂⁴	1	12	yes	—
110	C₂³×D₅	C₂⁴	1	64	yes	complemented
111	C₂×C₅×D₄	C₂×D₄	1	96	yes	complemented
112	C₂×C₅×D₄	C₂⁴	1	16	yes	—
113	C₂²×C₄×C₅	—	2	8	no	—
114	C₂²×C₄×C₅	—	2	4	no	—
115	C₂²×C₄×C₅	—	2	96	no	—
116	C₂²×C₄×C₅	C₂×D₄	1	8	yes	complemented
117	C₂²×C₄×C₅	C₂×D₄	1	8	yes	complemented
118	C₂²×C₄×C₅	C₂⁴	1	6	yes	—
119	C₂²×C₄×C₅	C₂⁴	1	12	yes	—
120	C₂²×C₄×C₅	C₂⁴	1	6	yes	—
121	C₂²×C₄×C₅	C₂⁴	1	6	yes	—
122	C₂²×C₄×C₅	C₂⁴	1	32	yes	complemented
123	C₂×C₂²⋊D₅	—	2	192	no	—
124	C₂×C₂²⋊D₅	C₂⁴	1	64	yes	complemented
125	C₂²×Dic₅	—	2	8	no	—
126	C₂²×Dic₅	C₂⁴	1	12	yes	—
127	D₄×D₅	—	2	512	no	—
128	C₂×D₂₀	—	2	192	no	—
129	C₂×D₂₀	—	2	96	no	—
130	C₂×D₂₀	—	2	96	no	—
131	C₂×D₂₀	—	2	512	no	—
132	C₂×D₂₀	—	4	48	no	—
133	C₂×D₂₀	C₂×D₄	1	96	yes	direct product factor
134	C₂×D₂₀	C₂⁴	1	16	yes	—
135	C₂×D₂₀	C₂⁴	1	32	yes	complemented
136	C₂×D₂₀	C₂⁴	1	64	yes	complemented
137	C₂×D₂₀	C₂⁴	1	32	yes	complemented
138	C₂×D₂₀	C₂⁴	1	32	yes	complemented
139	C₂×C₄×D₅	—	2	96	no	—
140	C₂×C₄×D₅	C₂⁴	1	32	yes	complemented
141	C₅×C₄⋊C₄	C₂⁴	1	16	yes	complemented
142	C₅×C₂²⋊C₄	C₂⁴	1	32	yes	complemented
143	C₄²×C₅	C₂⁴	1	16	yes	complemented
144	#685	C₂⁴	1	32	yes	complemented
145	#685	C₂⁴	1	64	yes	complemented
146	C₄⋊Dic₅	C₂⁴	1	16	yes	complemented
147	C₂⁶	—	5	4	no	—
148	C₂³×D₄	—	5	4	no	—
149	C₂³×D₄	—	5	48	no	—
150	C₂³×D₄	—	5	1	no	—
151	C₂³×D₄	—	5	4	no	—
152	C₂³×D₄	—	5	2	no	—
153	C₂³×D₄	—	5	2	no	—
154	C₂³×D₄	—	5	2	no	—
155	C₂³×D₄	—	5	48	no	—
156	C₂³×D₄	—	10	32	no	—
157	C₂³×D₄	—	10	32	no	—
158	C₂³×D₄	C₂×D₅	1	1	yes	characteristic
159	C₂⁴×C₄	—	5	2	no	—
160	C₂⁴×C₄	C₂×D₅	1	2	yes	complemented
161	D₄²	—	5	256	no	—
162	C₂×#127	—	5	24	no	—
163	C₂×#121	—	5	48	no	—
164	C₂×#121	—	5	48	no	—
165	C₂×#120	—	5	96	no	—
166	C₂×C₄×D₄	—	5	24	no	—
167	C₂×C₄×D₄	C₂×D₅	1	24	yes	complemented
168	C₂²×C₄⋊C₄	—	5	1	no	—
169	C₂²×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic, complemented
170	C₂²×C₂²⋊C₄	—	5	2	no	—
171	C₂²×C₂²⋊C₄	—	5	4	no	—
172	C₂²×C₂²⋊C₄	C₂×D₅	1	2	yes	complemented
173	C₂²×C₄²	C₂×D₅	1	1	yes	characteristic, complemented
174	C₂³×C₅	—	2	64	no	—
175	C₂³×C₅	—	2	24	no	—
176	C₂³×C₅	—	2	24	no	—
177	C₂³×C₅	C₂²×D₄	1	4	yes	—
178	C₂³×C₅	C₂²×D₄	1	4	yes	—
179	C₂³×C₅	C₂²×D₄	1	48	yes	complemented
180	C₂³×C₅	H(5,𝔽₂)	1	4	yes	—
181	C₂³×C₅	C₂⁵	1	8	yes	—
182	C₂³×C₅	C₂⁵	1	3	yes	—
183	C₂²×D₅	—	2	64	no	—
184	C₂²×D₅	—	2	24	no	—
185	C₂²×D₅	—	2	24	no	—
186	C₂²×D₅	—	2	48	no	—
187	C₂²×D₅	—	2	48	no	—
188	C₂²×D₅	—	2	128	no	—
189	C₂²×D₅	—	2	48	no	—
190	C₂²×D₅	—	2	128	no	—
191	C₂²×D₅	—	4	64	no	—
192	C₂²×D₅	—	4	768	no	—
193	C₂²×D₅	—	4	64	no	—
194	C₂²×D₅	C₂²×D₄	1	48	yes	complemented
195	C₂²×D₅	C₂⁵	1	16	yes	—
196	C₂²×D₅	C₂⁵	1	8	yes	—
197	C₅×D₄	C₂²×D₄	1	64	yes	complemented
198	C₂×C₄×C₅	—	2	24	no	—
199	C₂×C₄×C₅	—	2	12	no	—
200	C₂×C₄×C₅	—	2	64	no	—
201	C₂×C₄×C₅	C₂²×D₄	1	24	yes	complemented
202	C₂×C₄×C₅	C₂²×D₄	1	24	yes	complemented
203	C₂×C₄×C₅	C₂⁵	1	4	yes	—
204	C₂×C₄×C₅	C₂⁵	1	8	yes	—
205	C₂×C₄×C₅	C₂⁵	1	4	yes	—
206	C₂×C₄×C₅	C₂⁵	1	4	yes	—
207	C₂²⋊D₅	—	2	128	no	—
208	C₂×Dic₅	—	2	24	no	—
209	C₂×Dic₅	C₂⁵	1	8	yes	—
210	D₂₀	—	2	128	no	—
211	D₂₀	—	2	64	no	—
212	D₂₀	—	2	64	no	—
213	D₂₀	—	4	32	no	—
214	D₂₀	C₂²×D₄	1	64	yes	direct product factor
215	C₄×D₅	—	2	64	no	—
216	C₂⁵	—	5	2	no	—
217	C₂⁵	—	5	4	no	—
218	C₂⁵	—	5	48	no	—
219	C₂⁵	—	10	32	no	—
220	C₂⁵	—	10	32	no	—
221	C₂⁵	—	20	16	no	—
222	C₂⁵	C₂²×D₅	1	2	yes	—
223	C₂²×D₄	—	5	48	no	—
224	C₂²×D₄	—	5	128	no	—
225	C₂²×D₄	—	5	16	no	—
226	C₂²×D₄	—	5	12	no	—
227	C₂²×D₄	—	5	48	no	—
228	C₂²×D₄	—	5	24	no	—
229	C₂²×D₄	—	5	24	no	—
230	C₂²×D₄	—	5	24	no	—
231	C₂²×D₄	—	5	128	no	—
232	C₂²×D₄	—	10	384	no	—
233	C₂²×D₄	—	10	32	no	—
234	C₂²×D₄	—	10	32	no	—
235	C₂²×D₄	—	10	384	no	—
236	C₂²×D₄	—	10	16	no	—
237	C₂²×D₄	—	10	16	no	—
238	C₂²×D₄	—	20	8	no	—
239	C₂²×D₄	D₂₀	1	16	yes	direct product factor
240	C₂²×D₄	C₂²×D₅	1	12	yes	—
241	C₂³×C₄	—	2	16	no	—
242	C₂³×C₄	—	5	2	no	—
243	C₂³×C₄	—	5	24	no	—
244	C₂³×C₄	—	10	16	no	—
245	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
246	C₂³×C₄	C₂²×D₅	1	2	yes	—
247	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
248	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
249	C₂³×C₄	C₂²×D₅	1	24	yes	complemented
250	#127	—	5	64	no	—
251	#121	—	5	128	no	—
252	#121	—	5	128	no	—
253	#120	—	5	256	no	—
254	C₄×D₄	—	5	64	no	—
255	C₄×D₄	C₂²×D₅	1	64	yes	complemented
256	C₂×C₄⋊C₄	—	5	12	no	—
257	C₂×C₄⋊C₄	C₂²×D₅	1	12	yes	complemented
258	C₂×C₂²⋊C₄	—	5	24	no	—
259	C₂×C₂²⋊C₄	—	5	48	no	—
260	C₂×C₂²⋊C₄	C₂²×D₅	1	24	yes	complemented
261	C₂×C₄²	C₂²×D₅	1	12	yes	complemented
262	C₂²×C₅	—	2	8	no	—
263	C₂²×C₅	—	2	8	no	—
264	C₂²×C₅	—	2	96	no	—
265	C₂²×C₅	D₄²	1	16	yes	complemented
266	C₂²×C₅	C₂³×D₄	1	6	yes	—
267	C₂²×C₅	C₂³×D₄	1	6	yes	—
268	C₂²×C₅	C₂³×D₄	1	16	yes	complemented
269	C₂²×C₅	C₂×H(5,𝔽₂)	1	6	yes	—
270	C₂²×C₅	C₂⁶	1	1	yes	characteristic
271	C₂×D₅	—	2	8	no	—
272	C₂×D₅	—	2	8	no	—
273	C₂×D₅	—	2	96	no	—
274	C₂×D₅	—	2	16	no	—
275	C₂×D₅	—	2	16	no	—
276	C₂×D₅	—	2	16	no	—
277	C₂×D₅	—	4	256	no	—
278	C₂×D₅	—	4	96	no	—
279	C₂×D₅	C₂³×D₄	1	16	yes	complemented
280	C₄×C₅	—	2	8	no	—
281	C₄×C₅	—	2	4	no	—
282	C₄×C₅	C₂³×D₄	1	8	yes	complemented
283	C₄×C₅	C₂³×D₄	1	8	yes	complemented
284	Dic₅	—	2	8	no	—
285	C₂⁴	—	2	8	no	—
286	C₂⁴	—	2	8	no	—
287	C₂⁴	—	5	16	no	—
288	C₂⁴	—	5	12	no	—
289	C₂⁴	—	5	24	no	—
290	C₂⁴	—	5	64	no	—
291	C₂⁴	—	10	192	no	—
292	C₂⁴	—	10	8	no	—
293	C₂⁴	—	10	8	no	—
294	C₂⁴	—	10	16	no	—
295	C₂⁴	—	10	192	no	—
296	C₂⁴	—	10	16	no	—
297	C₂⁴	—	10	16	no	—
298	C₂⁴	—	20	96	no	—
299	C₂⁴	—	20	256	no	—
300	C₂⁴	C₂×D₂₀	1	16	yes	complemented
301	C₂⁴	C₂³×D₅	1	12	yes	—
302	C₂⁴	C₂³×D₅	1	1	yes	characteristic
303	C₂×D₄	—	5	64	no	—
304	C₂×D₄	—	5	96	no	—
305	C₂×D₄	—	5	16	no	—
306	C₂×D₄	—	5	64	no	—
307	C₂×D₄	—	5	32	no	—
308	C₂×D₄	—	5	32	no	—
309	C₂×D₄	—	5	32	no	—
310	C₂×D₄	—	10	512	no	—
311	C₂×D₄	—	10	192	no	—
312	C₂×D₄	—	10	192	no	—
313	C₂×D₄	—	10	512	no	—
314	C₂×D₄	—	10	96	no	—
315	C₂×D₄	—	10	96	no	—
316	C₂×D₄	—	20	48	no	—
317	C₂×D₄	C₂×D₂₀	1	96	yes	direct product factor
318	C₂×D₄	C₂³×D₅	1	16	yes	—
319	C₂²×C₄	—	2	8	no	—
320	C₂²×C₄	—	2	4	no	—
321	C₂²×C₄	—	2	96	no	—
322	C₂²×C₄	—	5	12	no	—
323	C₂²×C₄	—	5	32	no	—
324	C₂²×C₄	—	10	8	no	—
325	C₂²×C₄	—	10	96	no	—
326	C₂²×C₄	C₂×D₂₀	1	8	yes	complemented
327	C₂²×C₄	D₄×D₅	1	8	yes	complemented
328	C₂²×C₄	C₂³×D₅	1	6	yes	—
329	C₂²×C₄	C₂³×D₅	1	12	yes	—
330	C₂²×C₄	C₂³×D₅	1	6	yes	—
331	C₂²×C₄	C₂³×D₅	1	6	yes	—
332	C₂²×C₄	C₂³×D₅	1	32	yes	complemented
333	C₄⋊C₄	—	5	16	no	—
334	C₄⋊C₄	C₂³×D₅	1	16	yes	complemented
335	C₂²⋊C₄	—	5	32	no	—
336	C₂²⋊C₄	—	5	64	no	—
337	C₂²⋊C₄	C₂³×D₅	1	32	yes	complemented
338	C₄²	C₂³×D₅	1	16	yes	complemented
339	C₂×C₅	—	2	16	no	—
340	C₂×C₅	C₂×D₄²	1	12	yes	complemented
341	C₂×C₅	C₂⁴×D₄	1	1	yes	characteristic
342	C₂×C₅	C₂⁴×D₄	1	1	yes	characteristic
343	C₂×C₅	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
344	D₅	—	2	16	no	—
345	D₅	—	4	16	no	—
346	C₂³	—	2	64	no	—
347	C₂³	—	2	24	no	—
348	C₂³	—	2	24	no	—
349	C₂³	—	5	48	no	—
350	C₂³	—	5	8	no	—
351	C₂³	—	5	16	no	—
352	C₂³	—	10	128	no	—
353	C₂³	—	10	64	no	—
354	C₂³	—	10	24	no	—
355	C₂³	—	10	24	no	—
356	C₂³	—	10	48	no	—
357	C₂³	—	10	128	no	—
358	C₂³	—	10	48	no	—
359	C₂³	—	10	48	no	—
360	C₂³	—	20	64	no	—
361	C₂³	—	20	768	no	—
362	C₂³	—	20	64	no	—
363	C₂³	C₂²×D₂₀	1	4	yes	—
364	C₂³	C₂²×D₂₀	1	48	yes	complemented
365	C₂³	C₂×D₄×D₅	1	4	yes	—
366	C₂³	#4149	1	4	yes	—
367	C₂³	C₂⁴×D₅	1	8	yes	—
368	C₂³	C₂⁴×D₅	1	3	yes	—
369	D₄	—	5	64	no	—
370	D₄	—	10	128	no	—
371	D₄	—	10	128	no	—
372	D₄	—	10	64	no	—
373	D₄	—	10	64	no	—
374	D₄	—	20	32	no	—
375	D₄	C₂²×D₂₀	1	64	yes	direct product factor
376	C₂×C₄	—	2	24	no	—
377	C₂×C₄	—	2	12	no	—
378	C₂×C₄	—	2	64	no	—
379	C₂×C₄	—	5	8	no	—
380	C₂×C₄	—	10	24	no	—
381	C₂×C₄	—	10	64	no	—
382	C₂×C₄	C₂²×D₂₀	1	24	yes	complemented
383	C₂×C₄	C₂×D₄×D₅	1	24	yes	complemented
384	C₂×C₄	C₂⁴×D₅	1	4	yes	—
385	C₂×C₄	C₂⁴×D₅	1	8	yes	—
386	C₂×C₄	C₂⁴×D₅	1	4	yes	—
387	C₂×C₄	C₂⁴×D₅	1	4	yes	—
388	C₅	C₂²×D₄²	1	1	yes	characteristic, complemented, Sylow
389	C₂²	—	2	8	no	—
390	C₂²	—	2	8	no	—
391	C₂²	—	2	96	no	—
392	C₂²	—	5	16	no	—
393	C₂²	—	10	8	no	—
394	C₂²	—	10	8	no	—
395	C₂²	—	10	96	no	—
396	C₂²	—	10	16	no	—
397	C₂²	—	10	16	no	—
398	C₂²	—	10	16	no	—
399	C₂²	—	20	256	no	—
400	C₂²	—	20	96	no	—
401	C₂²	D₄×D₂₀	1	16	yes	direct product factor
402	C₂²	C₂³×D₂₀	1	6	yes	—
403	C₂²	C₂³×D₂₀	1	16	yes	complemented
404	C₂²	C₂²×D₄×D₅	1	6	yes	—
405	C₂²	C₂×#4149	1	6	yes	—
406	C₂²	C₂⁵×D₅	1	1	yes	characteristic
407	C₄	—	2	8	no	—
408	C₄	—	2	4	no	—
409	C₄	—	10	8	no	—
410	C₄	C₂³×D₂₀	1	8	yes	complemented
411	C₄	C₂²×D₄×D₅	1	8	yes	complemented
412	C₂	—	2	16	no	—
413	C₂	—	10	16	no	—
414	C₂	—	20	16	no	—
415	C₂	C₂×D₄×D₂₀	1	12	yes	direct product factor
416	C₂	C₂⁴×D₂₀	1	1	yes	characteristic
417	C₂	C₂³×D₄×D₅	1	1	yes	characteristic
418	C₂	C₂²×#4149	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂³×D₅
commutator, cosocle	C₂²×C₅	C₂⁶
Frattini	C₂²	C₂⁵×D₅
Fitting	C₂²×C₄×C₅×D₄	C₂
socle	C₂⁴×C₅	C₂⁴
hypercenter	C₂²×C₄×D₄	D₅
hypocenter	C₅	C₂²×D₄²
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂³×D₄×D₅	C₂	semidirect
2. C₂⁴×D₂₀	C₂	semidirect
3. C₂²×C₄×C₅×D₄	C₂	semidirect
4. C₂²×#4076	C₂	semidirect
5. C₂×D₄×D₂₀	C₂	direct (N × H)
6. C₂²×#4041	C₂	semidirect
7. C₂²×#4028	C₂	semidirect
8. C₂²×#4020	C₂	semidirect
9. C₂²×C₄×D₂₀	C₂	semidirect
10. C₂⁵×D₅	C₂²	semidirect
12. C₂⁴×C₄×C₅	C₂²	semidirect
13. C₂³×C₂²⋊D₅	C₂²	semidirect
15. C₂²×D₄×D₅	C₂²	semidirect
18. C₂³×D₂₀	C₂²	semidirect
19. C₂³×D₂₀	C₂²	semidirect
20. C₂³×D₂₀	C₂²	semidirect
21. C₂³×D₂₀	C₂²	semidirect
22. C₂³×D₂₀	C₂²	semidirect
23. C₂³×C₄×D₅	C₂²	semidirect
24. C₂×C₄×C₅×D₄	C₂²	semidirect
25. C₂²×C₅×C₄⋊C₄	C₂²	semidirect
26. C₂²×C₅×C₂²⋊C₄	C₂²	semidirect
27. C₂²×C₄²×C₅	C₂²	semidirect
28. C₂×#4076	C₂²	semidirect
29. C₂²×#685	C₂²	semidirect
30. C₂²×#685	C₂²	semidirect
31. C₂²×C₄⋊Dic₅	C₂²	semidirect
32. D₄×D₂₀	C₂²	direct (N × H)
33. C₂×#4041	C₂²	semidirect
34. C₂×#4028	C₂²	semidirect
35. C₂×#4020	C₂²	semidirect
36. C₂×C₄×D₂₀	C₂²	semidirect
44. C₂⁴×D₅	C₂³	semidirect
45. C₂²×C₅×D₄	D₄	semidirect
52. C₂³×C₄×C₅	C₂³	semidirect
54. C₂²×C₂²⋊D₅	C₂³	semidirect
57. C₂×D₄×D₅	C₂³	semidirect
63. C₂²×D₂₀	D₄	direct (N × H)
65. C₂²×D₂₀	C₂³	semidirect
66. C₂²×D₂₀	C₂³	semidirect
67. C₂²×D₂₀	C₂³	semidirect
68. C₂²×D₂₀	C₂³	semidirect
69. C₂²×D₂₀	C₂³	semidirect
71. C₂²×C₄×D₅	C₂³	semidirect
72. C₄×C₅×D₄	C₂³	semidirect
73. C₂×C₅×C₄⋊C₄	C₂³	semidirect
74. C₂×C₅×C₂²⋊C₄	C₂³	semidirect
75. C₂×C₄²×C₅	C₂³	semidirect
76. #4076	C₂³	semidirect
77. C₂×#685	C₂³	semidirect
78. C₂×#685	C₂³	semidirect
79. C₂×C₄⋊Dic₅	C₂³	semidirect
80. #4041	C₂³	semidirect
81. #4028	C₂³	semidirect
82. #4020	C₂³	semidirect
83. C₄×D₂₀	C₂³	semidirect
92. C₂²×C₄×D₄	D₅	semidirect
95. C₂⁴×C₅	C₂×D₄	semidirect
107. C₂³×D₅	C₂×D₄	semidirect
110. C₂³×D₅	C₂⁴	semidirect
111. C₂×C₅×D₄	C₂×D₄	semidirect
116. C₂²×C₄×C₅	C₂×D₄	semidirect
117. C₂²×C₄×C₅	C₂×D₄	semidirect
122. C₂²×C₄×C₅	C₂⁴	semidirect
124. C₂×C₂²⋊D₅	C₂⁴	semidirect
133. C₂×D₂₀	C₂×D₄	direct (N × H)
135. C₂×D₂₀	C₂⁴	semidirect
136. C₂×D₂₀	C₂⁴	semidirect
137. C₂×D₂₀	C₂⁴	semidirect
138. C₂×D₂₀	C₂⁴	semidirect
140. C₂×C₄×D₅	C₂⁴	semidirect
141. C₅×C₄⋊C₄	C₂⁴	semidirect
142. C₅×C₂²⋊C₄	C₂⁴	semidirect
143. C₄²×C₅	C₂⁴	semidirect
144. #685	C₂⁴	semidirect
145. #685	C₂⁴	semidirect
146. C₄⋊Dic₅	C₂⁴	semidirect
160. C₂⁴×C₄	C₂×D₅	semidirect
167. C₂×C₄×D₄	C₂×D₅	semidirect
169. C₂²×C₄⋊C₄	C₂×D₅	semidirect
172. C₂²×C₂²⋊C₄	C₂×D₅	semidirect
173. C₂²×C₄²	C₂×D₅	semidirect
179. C₂³×C₅	C₂²×D₄	semidirect
194. C₂²×D₅	C₂²×D₄	semidirect
197. C₅×D₄	C₂²×D₄	semidirect
201. C₂×C₄×C₅	C₂²×D₄	semidirect
202. C₂×C₄×C₅	C₂²×D₄	semidirect
214. D₂₀	C₂²×D₄	direct (N × H)
239. C₂²×D₄	D₂₀	direct (N × H)
249. C₂³×C₄	C₂²×D₅	semidirect
255. C₄×D₄	C₂²×D₅	semidirect
257. C₂×C₄⋊C₄	C₂²×D₅	semidirect
260. C₂×C₂²⋊C₄	C₂²×D₅	semidirect
261. C₂×C₄²	C₂²×D₅	semidirect
265. C₂²×C₅	D₄²	semidirect
268. C₂²×C₅	C₂³×D₄	semidirect
279. C₂×D₅	C₂³×D₄	semidirect
282. C₄×C₅	C₂³×D₄	semidirect
283. C₄×C₅	C₂³×D₄	semidirect
300. C₂⁴	C₂×D₂₀	semidirect
317. C₂×D₄	C₂×D₂₀	direct (N × H)
326. C₂²×C₄	C₂×D₂₀	semidirect
327. C₂²×C₄	D₄×D₅	semidirect
332. C₂²×C₄	C₂³×D₅	semidirect
334. C₄⋊C₄	C₂³×D₅	semidirect
337. C₂²⋊C₄	C₂³×D₅	semidirect
338. C₄²	C₂³×D₅	semidirect
340. C₂×C₅	C₂×D₄²	semidirect
364. C₂³	C₂²×D₂₀	semidirect
375. D₄	C₂²×D₂₀	direct (N × H)
382. C₂×C₄	C₂²×D₂₀	semidirect
383. C₂×C₄	C₂×D₄×D₅	semidirect
388. C₅	C₂²×D₄²	semidirect
401. C₂²	D₄×D₂₀	direct (N × H)
403. C₂²	C₂³×D₂₀	semidirect
410. C₄	C₂³×D₂₀	semidirect
411. C₄	C₂²×D₄×D₅	semidirect
415. C₂	C₂×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
C₂³×D₄×D₅	2
C₂⁴×D₂₀	2
C₂²×C₄×C₅×D₄	2
C₂²×#4076	2
C₂×D₄×D₂₀	2
C₂²×#4041	2
C₂²×#4028	2
C₂²×#4020	2
C₂²×C₄×D₂₀	2
C₂²×D₄²	5

Subgroups — C22×D4×D20 #13006211

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×D₄×D₂₀ #13006211