Subgroups — C₄×C₅×D₄² #11956876

Order 1280.

4804 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×D₄²	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×C₅×D₄	C₂	1	4	yes	maximal, complemented
3	C₅×#2366	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#2361	C₂	1	4	yes	maximal, complemented
5	C₅×#2358	C₂	1	4	yes	maximal, complemented
6	C₅×D₄×C₄⋊C₄	C₂	1	2	yes	maximal, complemented
7	C₅×D₄×C₂²⋊C₄	C₂	1	4	yes	maximal, complemented
8	C₄×C₅×#127	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×C₅×#121	C₂	1	4	yes	maximal, complemented
10	C₄×C₅×#120	C₂	1	4	yes	maximal, complemented
11	C₄²×C₅×D₄	C₂	1	2	yes	maximal, complemented
12	C₂³×C₅×D₄	C₂²	1	4	yes	—
13	C₂⁴×C₄×C₅	C₂²	1	4	yes	complemented
14	C₅×D₄²	C₄	1	16	yes	direct product factor
15	C₂×C₅×#127	C₂²	1	1	yes	characteristic
16	C₂×C₅×#121	C₂²	1	4	yes	—
17	C₂×C₅×#120	C₂²	1	4	yes	—
18	C₂×C₄×C₅×D₄	—	2	32	no	—
19	C₂×C₄×C₅×D₄	C₂²	1	2	yes	—
20	C₂×C₄×C₅×D₄	C₂²	1	4	yes	—
21	C₂×C₄×C₅×D₄	C₂²	1	2	yes	—
22	C₂×C₄×C₅×D₄	C₂²	1	2	yes	—
23	C₂×C₄×C₅×D₄	C₂²	1	8	yes	complemented
24	C₂×C₄×C₅×D₄	C₂²	1	4	yes	complemented
25	C₂×C₄×C₅×D₄	C₂²	1	8	yes	complemented
26	C₂×C₄×C₅×D₄	C₂²	1	4	yes	complemented
27	C₂×C₄×C₅×D₄	C₂²	1	2	yes	complemented
28	C₂²×C₅×C₄⋊C₄	C₂²	1	4	yes	complemented
29	C₂²×C₅×C₂²⋊C₄	C₂²	1	8	yes	complemented
30	C₂²×C₄²×C₅	C₂²	1	4	yes	complemented
31	C₅×#390	C₂²	1	4	yes	complemented
32	C₅×#390	C₂²	1	4	yes	complemented
33	C₅×#390	C₂²	1	2	yes	complemented
34	C₅×#389	C₂²	1	4	yes	complemented
35	C₅×#388	C₂²	1	8	yes	complemented
36	C₅×#386	C₂²	1	8	yes	complemented
37	C₅×#386	C₂²	1	4	yes	complemented
38	C₅×#386	C₂²	1	8	yes	complemented
39	C₅×#385	C₂²	1	8	yes	complemented
40	C₅×#385	C₂²	1	4	yes	complemented
41	C₅×#384	C₂²	1	1	yes	characteristic, complemented
42	C₅×#380	C₂²	1	4	yes	complemented
43	C₅×#379	C₂²	1	4	yes	complemented
44	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	complemented
45	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	complemented
46	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	complemented
47	C₄×C₅×C₂²⋊C₄	C₂²	1	4	yes	complemented
48	C₄×C₅×C₂²⋊C₄	C₂²	1	4	yes	complemented
49	C₄×C₅×C₂²⋊C₄	C₂²	1	4	yes	complemented
50	C₄×C₅×C₂²⋊C₄	C₂²	1	4	yes	complemented
51	C₄³×C₅	C₂²	1	1	yes	characteristic, complemented
52	C₄×D₄²	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
53	C₂⁵×C₅	C₂³	1	4	yes	—
54	C₂²×C₅×D₄	—	2	16	no	—
55	C₂²×C₅×D₄	—	2	16	no	—
56	C₂²×C₅×D₄	C₂×C₄	1	32	yes	complemented
57	C₂²×C₅×D₄	C₂³	1	8	yes	—
58	C₂²×C₅×D₄	C₂³	1	4	yes	—
59	C₂²×C₅×D₄	C₂³	1	2	yes	—
60	C₂²×C₅×D₄	C₂³	1	2	yes	—
61	C₂³×C₄×C₅	—	2	32	no	—
62	C₂³×C₄×C₅	—	4	8	no	—
63	C₂³×C₄×C₅	C₂³	1	4	yes	—
64	C₂³×C₄×C₅	C₂³	1	4	yes	—
65	C₂³×C₄×C₅	C₂³	1	8	yes	—
66	C₂³×C₄×C₅	C₂³	1	4	yes	—
67	C₂³×C₄×C₅	C₂³	1	4	yes	—
68	C₂³×C₄×C₅	C₂³	1	4	yes	—
69	C₅×#127	C₂×C₄	1	8	yes	complemented
70	C₅×#121	C₂×C₄	1	32	yes	complemented
71	C₅×#120	C₂×C₄	1	32	yes	complemented
72	C₄×C₅×D₄	—	2	16	no	—
73	C₄×C₅×D₄	—	2	8	no	—
74	C₄×C₅×D₄	—	2	32	no	—
75	C₄×C₅×D₄	—	2	16	no	—
76	C₄×C₅×D₄	—	2	32	no	—
77	C₄×C₅×D₄	—	2	16	no	—
78	C₄×C₅×D₄	—	4	4	no	—
79	C₄×C₅×D₄	C₂×C₄	1	16	yes	complemented
80	C₄×C₅×D₄	D₄	1	16	yes	direct product factor
81	C₂×C₅×C₄⋊C₄	—	2	16	no	—
82	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
83	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
84	C₂×C₅×C₄⋊C₄	C₂³	1	8	yes	—
85	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
86	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
87	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
88	C₂×C₅×C₂²⋊C₄	—	2	32	no	—
89	C₂×C₅×C₂²⋊C₄	C₂³	1	4	yes	—
90	C₂×C₅×C₂²⋊C₄	C₂³	1	8	yes	—
91	C₂×C₅×C₂²⋊C₄	C₂³	1	8	yes	—
92	C₂×C₅×C₂²⋊C₄	C₂³	1	8	yes	—
93	C₂×C₅×C₂²⋊C₄	C₂³	1	4	yes	—
94	C₂×C₅×C₂²⋊C₄	C₂³	1	4	yes	—
95	C₂×C₅×C₂²⋊C₄	C₂³	1	4	yes	—
96	C₂×C₅×C₂²⋊C₄	C₂³	1	4	yes	—
97	C₂×C₅×C₂²⋊C₄	C₂³	1	4	yes	—
98	C₂×C₄²×C₅	—	2	16	no	—
99	C₂×C₄²×C₅	C₂³	1	2	yes	—
100	C₂×C₄²×C₅	C₂³	1	4	yes	—
101	C₂×C₄²×C₅	C₂³	1	4	yes	—
102	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
103	C₂×C₄²×C₅	C₂³	1	4	yes	—
104	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
105	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
106	C₂×C₄²×C₅	C₂³	1	2	yes	—
107	C₅×#95	C₂³	1	4	yes	—
108	C₅×#95	C₂³	1	4	yes	—
109	C₅×#95	C₂³	1	4	yes	—
110	C₅×#95	C₂³	1	8	yes	—
111	C₂×D₄²	C₂×C₅	1	1	yes	characteristic
112	C₂²×C₄×D₄	C₂×C₅	1	4	yes	complemented
113	#2366	C₂×C₅	1	1	yes	characteristic, complemented
114	#2361	C₂×C₅	1	4	yes	complemented
115	#2358	C₂×C₅	1	4	yes	complemented
116	D₄×C₄⋊C₄	C₂×C₅	1	2	yes	complemented
117	D₄×C₂²⋊C₄	C₂×C₅	1	4	yes	complemented
118	C₄×#127	C₂×C₅	1	1	yes	characteristic, complemented
119	C₄×#121	C₂×C₅	1	4	yes	complemented
120	C₄×#120	C₂×C₅	1	4	yes	complemented
121	C₄²×D₄	C₂×C₅	1	2	yes	complemented
122	C₂⁴×C₅	—	2	16	no	—
123	C₂⁴×C₅	—	2	16	no	—
124	C₂⁴×C₅	—	4	4	no	—
125	C₂⁴×C₅	—	4	4	no	—
126	C₂⁴×C₅	C₂²×C₄	1	16	yes	complemented
127	C₂⁴×C₅	C₂⁴	1	4	yes	—
128	C₂⁴×C₅	C₂⁴	1	4	yes	—
129	C₂×C₅×D₄	—	2	16	no	—
130	C₂×C₅×D₄	—	2	16	no	—
131	C₂×C₅×D₄	—	2	128	no	—
132	C₂×C₅×D₄	—	2	8	no	—
133	C₂×C₅×D₄	—	2	8	no	—
134	C₂×C₅×D₄	—	4	2	no	—
135	C₂×C₅×D₄	—	4	2	no	—
136	C₂×C₅×D₄	C₂²×C₄	1	8	yes	—
137	C₂×C₅×D₄	C₂²×C₄	1	32	yes	complemented
138	C₂×C₅×D₄	C₂²×C₄	1	16	yes	complemented
139	C₂×C₅×D₄	C₂²×C₄	1	8	yes	complemented
140	C₂×C₅×D₄	C₂×D₄	1	8	yes	—
141	C₂×C₅×D₄	#41	1	8	yes	—
142	C₂²×C₄×C₅	—	2	8	no	—
143	C₂²×C₄×C₅	—	2	8	no	—
144	C₂²×C₄×C₅	—	2	8	no	—
145	C₂²×C₄×C₅	—	2	8	no	—
146	C₂²×C₄×C₅	—	2	16	no	—
147	C₂²×C₄×C₅	—	2	8	no	—
148	C₂²×C₄×C₅	—	2	8	no	—
149	C₂²×C₄×C₅	—	2	16	no	—
150	C₂²×C₄×C₅	—	2	16	no	—
151	C₂²×C₄×C₅	—	2	16	no	—
152	C₂²×C₄×C₅	—	2	8	no	—
153	C₂²×C₄×C₅	—	4	4	no	—
154	C₂²×C₄×C₅	—	4	8	no	—
155	C₂²×C₄×C₅	—	4	4	no	—
156	C₂²×C₄×C₅	—	4	64	no	—
157	C₂²×C₄×C₅	C₂²×C₄	1	16	yes	complemented
158	C₂²×C₄×C₅	C₂×D₄	1	16	yes	complemented
159	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
160	C₂²×C₄×C₅	C₂⁴	1	4	yes	—
161	C₂²×C₄×C₅	C₂⁴	1	4	yes	—
162	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
163	C₂²×C₄×C₅	C₂⁴	1	4	yes	—
164	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
165	C₂²×C₄×C₅	C₂⁴	1	4	yes	—
166	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
167	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
168	C₅×C₄⋊C₄	—	2	16	no	—
169	C₅×C₄⋊C₄	—	2	8	no	—
170	C₅×C₄⋊C₄	—	2	4	no	—
171	C₅×C₄⋊C₄	—	4	2	no	—
172	C₅×C₄⋊C₄	C₂²×C₄	1	8	yes	complemented
173	C₅×C₄⋊C₄	C₂×D₄	1	8	yes	complemented
174	C₅×C₂²⋊C₄	—	2	16	no	—
175	C₅×C₂²⋊C₄	—	2	8	no	—
176	C₅×C₂²⋊C₄	—	2	16	no	—
177	C₅×C₂²⋊C₄	—	2	16	no	—
178	C₅×C₂²⋊C₄	—	4	4	no	—
179	C₅×C₂²⋊C₄	—	4	4	no	—
180	C₅×C₂²⋊C₄	C₂²×C₄	1	16	yes	complemented
181	C₅×C₂²⋊C₄	C₂²×C₄	1	16	yes	complemented
182	C₅×C₂²⋊C₄	C₂×D₄	1	16	yes	complemented
183	C₄²×C₅	—	2	8	no	—
184	C₄²×C₅	—	2	8	no	—
185	C₄²×C₅	—	2	8	no	—
186	C₄²×C₅	—	2	4	no	—
187	C₄²×C₅	—	2	2	no	—
188	C₄²×C₅	—	4	1	no	—
189	C₄²×C₅	C₂²×C₄	1	4	yes	complemented
190	C₄²×C₅	C₂×D₄	1	8	yes	complemented
191	C₂³×D₄	C₂²×C₅	1	4	yes	—
192	C₂⁴×C₄	C₂²×C₅	1	4	yes	complemented
193	D₄²	C₄×C₅	1	16	yes	direct product factor
194	C₂×#127	C₂²×C₅	1	1	yes	characteristic
195	C₂×#121	C₂²×C₅	1	4	yes	—
196	C₂×#120	C₂²×C₅	1	4	yes	—
197	C₂×C₄×D₄	—	2	32	no	—
198	C₂×C₄×D₄	C₂²×C₅	1	2	yes	—
199	C₂×C₄×D₄	C₂²×C₅	1	4	yes	—
200	C₂×C₄×D₄	C₂²×C₅	1	2	yes	—
201	C₂×C₄×D₄	C₂²×C₅	1	2	yes	—
202	C₂×C₄×D₄	C₂²×C₅	1	4	yes	complemented
203	C₂×C₄×D₄	C₂²×C₅	1	8	yes	complemented
204	C₂×C₄×D₄	C₂²×C₅	1	4	yes	complemented
205	C₂×C₄×D₄	C₂²×C₅	1	8	yes	complemented
206	C₂×C₄×D₄	C₂²×C₅	1	2	yes	complemented
207	C₂²×C₄⋊C₄	C₂²×C₅	1	4	yes	complemented
208	C₂²×C₂²⋊C₄	C₂²×C₅	1	8	yes	complemented
209	C₂²×C₄²	C₂²×C₅	1	4	yes	complemented
210	#390	C₂²×C₅	1	4	yes	complemented
211	#390	C₂²×C₅	1	2	yes	complemented
212	#390	C₂²×C₅	1	4	yes	complemented
213	#389	C₂²×C₅	1	4	yes	complemented
214	#388	C₂²×C₅	1	8	yes	complemented
215	#386	C₂²×C₅	1	8	yes	complemented
216	#386	C₂²×C₅	1	8	yes	complemented
217	#386	C₂²×C₅	1	4	yes	complemented
218	#385	C₂²×C₅	1	8	yes	complemented
219	#385	C₂²×C₅	1	4	yes	complemented
220	#384	C₂²×C₅	1	1	yes	characteristic, complemented
221	#380	C₂²×C₅	1	4	yes	complemented
222	#379	C₂²×C₅	1	4	yes	complemented
223	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	complemented
224	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	complemented
225	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	complemented
226	C₄×C₂²⋊C₄	C₂²×C₅	1	4	yes	complemented
227	C₄×C₂²⋊C₄	C₂²×C₅	1	4	yes	complemented
228	C₄×C₂²⋊C₄	C₂²×C₅	1	4	yes	complemented
229	C₄×C₂²⋊C₄	C₂²×C₅	1	4	yes	complemented
230	C₄³	C₂²×C₅	1	1	yes	characteristic, complemented
231	C₂³×C₅	—	2	4	no	—
232	C₂³×C₅	—	2	4	no	—
233	C₂³×C₅	—	2	4	no	—
234	C₂³×C₅	—	2	4	no	—
235	C₂³×C₅	—	2	64	no	—
236	C₂³×C₅	—	2	8	no	—
237	C₂³×C₅	—	2	8	no	—
238	C₂³×C₅	—	2	4	no	—
239	C₂³×C₅	—	2	4	no	—
240	C₂³×C₅	—	4	16	no	—
241	C₂³×C₅	—	4	16	no	—
242	C₂³×C₅	—	4	32	no	—
243	C₂³×C₅	—	4	16	no	—
244	C₂³×C₅	C₂³×C₄	1	8	yes	—
245	C₂³×C₅	C₂³×C₄	1	8	yes	—
246	C₂³×C₅	C₂²×D₄	1	8	yes	—
247	C₂³×C₅	C₂×#41	1	8	yes	—
248	C₂³×C₅	C₂⁵	1	1	yes	characteristic
249	C₅×D₄	—	2	64	no	—
250	C₅×D₄	—	2	32	no	—
251	C₅×D₄	—	4	8	no	—
252	C₅×D₄	C₄×D₄	1	32	yes	direct product factor
253	C₂×C₄×C₅	—	2	4	no	—
254	C₂×C₄×C₅	—	2	4	no	—
255	C₂×C₄×C₅	—	2	2	no	—
256	C₂×C₄×C₅	—	2	4	no	—
257	C₂×C₄×C₅	—	2	1	no	—
258	C₂×C₄×C₅	—	2	4	no	—
259	C₂×C₄×C₅	—	2	2	no	—
260	C₂×C₄×C₅	—	2	1	no	—
261	C₂×C₄×C₅	—	2	32	no	—
262	C₂×C₄×C₅	—	2	32	no	—
263	C₂×C₄×C₅	—	2	4	no	—
264	C₂×C₄×C₅	—	2	8	no	—
265	C₂×C₄×C₅	—	2	2	no	—
266	C₂×C₄×C₅	—	2	4	no	—
267	C₂×C₄×C₅	—	2	1	no	—
268	C₂×C₄×C₅	—	2	4	no	—
269	C₂×C₄×C₅	—	2	4	no	—
270	C₂×C₄×C₅	—	4	16	no	—
271	C₂×C₄×C₅	—	4	16	no	—
272	C₂×C₄×C₅	—	4	32	no	—
273	C₂×C₄×C₅	—	4	16	no	—
274	C₂×C₄×C₅	C₂³×C₄	1	2	yes	—
275	C₂×C₄×C₅	C₂³×C₄	1	8	yes	—
276	C₂×C₄×C₅	C₂³×C₄	1	4	yes	—
277	C₂×C₄×C₅	C₂²×D₄	1	4	yes	—
278	C₂×C₄×C₅	C₂²×D₄	1	8	yes	—
279	C₂×C₄×C₅	C₂²×D₄	1	4	yes	—
280	C₂×C₄×C₅	C₂²×D₄	1	4	yes	—
281	C₂×C₄×C₅	C₂×#41	1	4	yes	—
282	C₂×C₄×C₅	H(5,𝔽₂)	1	2	yes	—
283	C₂⁵	C₂³×C₅	1	4	yes	—
284	C₂²×D₄	—	2	16	no	—
285	C₂²×D₄	—	2	16	no	—
286	C₂²×D₄	C₂×C₄×C₅	1	32	yes	complemented
287	C₂²×D₄	C₂³×C₅	1	8	yes	—
288	C₂²×D₄	C₂³×C₅	1	4	yes	—
289	C₂²×D₄	C₂³×C₅	1	2	yes	—
290	C₂²×D₄	C₂³×C₅	1	2	yes	—
291	C₂³×C₄	—	2	32	no	—
292	C₂³×C₄	—	4	8	no	—
293	C₂³×C₄	C₂³×C₅	1	4	yes	—
294	C₂³×C₄	C₂³×C₅	1	4	yes	—
295	C₂³×C₄	C₂³×C₅	1	8	yes	—
296	C₂³×C₄	C₂³×C₅	1	4	yes	—
297	C₂³×C₄	C₂³×C₅	1	4	yes	—
298	C₂³×C₄	C₂³×C₅	1	4	yes	—
299	#127	C₂×C₄×C₅	1	8	yes	complemented
300	#121	C₂×C₄×C₅	1	32	yes	complemented
301	#120	C₂×C₄×C₅	1	32	yes	complemented
302	C₄×D₄	—	2	16	no	—
303	C₄×D₄	—	2	8	no	—
304	C₄×D₄	—	2	32	no	—
305	C₄×D₄	—	2	16	no	—
306	C₄×D₄	—	2	32	no	—
307	C₄×D₄	—	2	16	no	—
308	C₄×D₄	—	4	4	no	—
309	C₄×D₄	C₂×C₄×C₅	1	16	yes	complemented
310	C₄×D₄	C₅×D₄	1	16	yes	direct product factor
311	C₂×C₄⋊C₄	—	2	16	no	—
312	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
313	C₂×C₄⋊C₄	C₂³×C₅	1	8	yes	—
314	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
315	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
316	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
317	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
318	C₂×C₂²⋊C₄	—	2	32	no	—
319	C₂×C₂²⋊C₄	C₂³×C₅	1	8	yes	—
320	C₂×C₂²⋊C₄	C₂³×C₅	1	8	yes	—
321	C₂×C₂²⋊C₄	C₂³×C₅	1	8	yes	—
322	C₂×C₂²⋊C₄	C₂³×C₅	1	4	yes	—
323	C₂×C₂²⋊C₄	C₂³×C₅	1	4	yes	—
324	C₂×C₂²⋊C₄	C₂³×C₅	1	4	yes	—
325	C₂×C₂²⋊C₄	C₂³×C₅	1	4	yes	—
326	C₂×C₂²⋊C₄	C₂³×C₅	1	4	yes	—
327	C₂×C₂²⋊C₄	C₂³×C₅	1	4	yes	—
328	C₂×C₄²	—	2	16	no	—
329	C₂×C₄²	C₂³×C₅	1	4	yes	—
330	C₂×C₄²	C₂³×C₅	1	2	yes	—
331	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
332	C₂×C₄²	C₂³×C₅	1	4	yes	—
333	C₂×C₄²	C₂³×C₅	1	4	yes	—
334	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
335	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
336	C₂×C₄²	C₂³×C₅	1	2	yes	—
337	#95	C₂³×C₅	1	4	yes	—
338	#95	C₂³×C₅	1	4	yes	—
339	#95	C₂³×C₅	1	4	yes	—
340	#95	C₂³×C₅	1	8	yes	—
341	C₂²×C₅	—	2	8	no	—
342	C₂²×C₅	—	2	8	no	—
343	C₂²×C₅	—	2	8	no	—
344	C₂²×C₅	—	2	8	no	—
345	C₂²×C₅	—	2	8	no	—
346	C₂²×C₅	—	2	8	no	—
347	C₂²×C₅	—	2	16	no	—
348	C₂²×C₅	—	2	8	no	—
349	C₂²×C₅	—	4	64	no	—
350	C₂²×C₅	—	4	4	no	—
351	C₂²×C₅	—	4	8	no	—
352	C₂²×C₅	—	4	4	no	—
353	C₂²×C₅	C₂×C₄×D₄	1	16	yes	complemented
354	C₂²×C₅	C₂⁴×C₄	1	1	yes	characteristic
355	C₂²×C₅	C₂³×D₄	1	2	yes	—
356	C₂²×C₅	C₂²×#41	1	2	yes	—
357	C₂²×C₅	C₂×H(5,𝔽₂)	1	1	yes	characteristic
358	C₂²×C₅	#585	1	1	yes	characteristic
359	C₄×C₅	—	2	8	no	—
360	C₄×C₅	—	2	2	no	—
361	C₄×C₅	—	2	8	no	—
362	C₄×C₅	—	2	4	no	—
363	C₄×C₅	—	4	4	no	—
364	C₄×C₅	—	4	4	no	—
365	C₄×C₅	—	4	1	no	—
366	C₄×C₅	C₂×C₄×D₄	1	8	yes	complemented
367	C₄×C₅	D₄²	1	4	yes	direct product factor
368	C₂⁴	—	2	16	no	—
369	C₂⁴	—	2	16	no	—
370	C₂⁴	—	4	4	no	—
371	C₂⁴	—	4	4	no	—
372	C₂⁴	C₂²×C₄×C₅	1	16	yes	complemented
373	C₂⁴	C₂⁴×C₅	1	4	yes	—
374	C₂⁴	C₂⁴×C₅	1	4	yes	—
375	C₂×D₄	—	2	16	no	—
376	C₂×D₄	—	2	16	no	—
377	C₂×D₄	—	2	128	no	—
378	C₂×D₄	—	2	8	no	—
379	C₂×D₄	—	2	8	no	—
380	C₂×D₄	—	4	2	no	—
381	C₂×D₄	—	4	2	no	—
382	C₂×D₄	C₂²×C₄×C₅	1	8	yes	—
383	C₂×D₄	C₂²×C₄×C₅	1	32	yes	complemented
384	C₂×D₄	C₂²×C₄×C₅	1	16	yes	complemented
385	C₂×D₄	C₂²×C₄×C₅	1	8	yes	complemented
386	C₂×D₄	C₂×C₅×D₄	1	8	yes	—
387	C₂×D₄	C₅×#41	1	8	yes	—
388	C₂²×C₄	—	2	8	no	—
389	C₂²×C₄	—	2	8	no	—
390	C₂²×C₄	—	2	8	no	—
391	C₂²×C₄	—	2	8	no	—
392	C₂²×C₄	—	2	16	no	—
393	C₂²×C₄	—	2	8	no	—
394	C₂²×C₄	—	2	8	no	—
395	C₂²×C₄	—	2	16	no	—
396	C₂²×C₄	—	2	16	no	—
397	C₂²×C₄	—	2	16	no	—
398	C₂²×C₄	—	2	8	no	—
399	C₂²×C₄	—	4	4	no	—
400	C₂²×C₄	—	4	8	no	—
401	C₂²×C₄	—	4	4	no	—
402	C₂²×C₄	—	4	64	no	—
403	C₂²×C₄	C₂²×C₄×C₅	1	16	yes	complemented
404	C₂²×C₄	C₂×C₅×D₄	1	16	yes	complemented
405	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
406	C₂²×C₄	C₂⁴×C₅	1	4	yes	—
407	C₂²×C₄	C₂⁴×C₅	1	4	yes	—
408	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
409	C₂²×C₄	C₂⁴×C₅	1	4	yes	—
410	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
411	C₂²×C₄	C₂⁴×C₅	1	4	yes	—
412	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
413	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
414	C₄⋊C₄	—	2	16	no	—
415	C₄⋊C₄	—	2	8	no	—
416	C₄⋊C₄	—	2	4	no	—
417	C₄⋊C₄	—	4	2	no	—
418	C₄⋊C₄	C₂²×C₄×C₅	1	8	yes	complemented
419	C₄⋊C₄	C₂×C₅×D₄	1	8	yes	complemented
420	C₂²⋊C₄	—	2	16	no	—
421	C₂²⋊C₄	—	2	8	no	—
422	C₂²⋊C₄	—	2	16	no	—
423	C₂²⋊C₄	—	2	16	no	—
424	C₂²⋊C₄	—	4	4	no	—
425	C₂²⋊C₄	—	4	4	no	—
426	C₂²⋊C₄	C₂²×C₄×C₅	1	16	yes	complemented
427	C₂²⋊C₄	C₂²×C₄×C₅	1	16	yes	complemented
428	C₂²⋊C₄	C₂×C₅×D₄	1	16	yes	complemented
429	C₄²	—	2	8	no	—
430	C₄²	—	2	8	no	—
431	C₄²	—	2	8	no	—
432	C₄²	—	2	4	no	—
433	C₄²	—	2	2	no	—
434	C₄²	—	4	1	no	—
435	C₄²	C₂²×C₄×C₅	1	4	yes	complemented
436	C₄²	C₂×C₅×D₄	1	8	yes	complemented
437	C₂×C₅	—	2	16	no	—
438	C₂×C₅	—	4	8	no	—
439	C₂×C₅	C₂²×C₄×D₄	1	2	yes	—
440	C₂×C₅	C₄×H(5,𝔽₂)	1	1	yes	characteristic
441	C₂×C₅	C₂×D₄²	1	1	yes	characteristic
442	C₂×C₅	D₄×#41	1	2	yes	—
443	C₂×C₅	#3481	1	1	yes	characteristic
444	C₂³	—	2	4	no	—
445	C₂³	—	2	4	no	—
446	C₂³	—	2	4	no	—
447	C₂³	—	2	4	no	—
448	C₂³	—	2	64	no	—
449	C₂³	—	2	8	no	—
450	C₂³	—	2	8	no	—
451	C₂³	—	2	4	no	—
452	C₂³	—	2	4	no	—
453	C₂³	—	4	16	no	—
454	C₂³	—	4	16	no	—
455	C₂³	—	4	32	no	—
456	C₂³	—	4	16	no	—
457	C₂³	C₂³×C₄×C₅	1	8	yes	—
458	C₂³	C₂³×C₄×C₅	1	8	yes	—
459	C₂³	C₂²×C₅×D₄	1	8	yes	—
460	C₂³	C₂×C₅×#41	1	8	yes	—
461	C₂³	C₂⁵×C₅	1	1	yes	characteristic
462	D₄	—	2	64	no	—
463	D₄	—	2	32	no	—
464	D₄	—	4	8	no	—
465	D₄	C₄×C₅×D₄	1	32	yes	direct product factor
466	C₂×C₄	—	2	4	no	—
467	C₂×C₄	—	2	4	no	—
468	C₂×C₄	—	2	2	no	—
469	C₂×C₄	—	2	4	no	—
470	C₂×C₄	—	2	1	no	—
471	C₂×C₄	—	2	4	no	—
472	C₂×C₄	—	2	2	no	—
473	C₂×C₄	—	2	1	no	—
474	C₂×C₄	—	2	32	no	—
475	C₂×C₄	—	2	32	no	—
476	C₂×C₄	—	2	4	no	—
477	C₂×C₄	—	2	8	no	—
478	C₂×C₄	—	2	2	no	—
479	C₂×C₄	—	2	4	no	—
480	C₂×C₄	—	2	1	no	—
481	C₂×C₄	—	2	4	no	—
482	C₂×C₄	—	2	4	no	—
483	C₂×C₄	—	4	16	no	—
484	C₂×C₄	—	4	32	no	—
485	C₂×C₄	—	4	16	no	—
486	C₂×C₄	—	4	16	no	—
487	C₂×C₄	C₂³×C₄×C₅	1	2	yes	—
488	C₂×C₄	C₂³×C₄×C₅	1	8	yes	—
489	C₂×C₄	C₂³×C₄×C₅	1	4	yes	—
490	C₂×C₄	C₂²×C₅×D₄	1	4	yes	—
491	C₂×C₄	C₂²×C₅×D₄	1	8	yes	—
492	C₂×C₄	C₂²×C₅×D₄	1	4	yes	—
493	C₂×C₄	C₂²×C₅×D₄	1	4	yes	—
494	C₂×C₄	C₂×C₅×#41	1	4	yes	—
495	C₂×C₄	C₅×H(5,𝔽₂)	1	2	yes	—
496	C₅	C₄×D₄²	1	1	yes	characteristic, direct product factor, Sylow
497	C₂²	—	2	8	no	—
498	C₂²	—	2	8	no	—
499	C₂²	—	2	8	no	—
500	C₂²	—	2	8	no	—
501	C₂²	—	2	8	no	—
502	C₂²	—	2	8	no	—
503	C₂²	—	2	16	no	—
504	C₂²	—	2	8	no	—
505	C₂²	—	4	64	no	—
506	C₂²	—	4	4	no	—
507	C₂²	—	4	8	no	—
508	C₂²	—	4	4	no	—
509	C₂²	C₂×C₄×C₅×D₄	1	16	yes	complemented
510	C₂²	C₂⁴×C₄×C₅	1	1	yes	characteristic
511	C₂²	C₂³×C₅×D₄	1	2	yes	—
512	C₂²	C₂²×C₅×#41	1	2	yes	—
513	C₂²	C₂×C₅×H(5,𝔽₂)	1	1	yes	characteristic
514	C₂²	C₅×#585	1	1	yes	characteristic
515	C₄	—	2	8	no	—
516	C₄	—	2	2	no	—
517	C₄	—	2	8	no	—
518	C₄	—	2	4	no	—
519	C₄	—	4	4	no	—
520	C₄	—	4	4	no	—
521	C₄	—	4	1	no	—
522	C₄	C₂×C₄×C₅×D₄	1	8	yes	complemented
523	C₄	C₅×D₄²	1	4	yes	direct product factor
524	C₂	—	2	16	no	—
525	C₂	—	4	8	no	—
526	C₂	C₂²×C₄×C₅×D₄	1	2	yes	—
527	C₂	C₄×C₅×H(5,𝔽₂)	1	1	yes	characteristic
528	C₂	C₂×C₅×D₄²	1	1	yes	characteristic
529	C₂	C₅×D₄×#41	1	2	yes	—
530	C₂	C₅×#3481	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₅
Frattini, cosocle	C₂³	C₂⁵×C₅
socle	C₂³×C₅	C₂⁵
prime radical	C₂×C₅×D₄²	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄×C₅×D₄	C₂	semidirect
3. C₅×#2366	C₂	semidirect
4. C₅×#2361	C₂	semidirect
5. C₅×#2358	C₂	semidirect
6. C₅×D₄×C₄⋊C₄	C₂	semidirect
7. C₅×D₄×C₂²⋊C₄	C₂	semidirect
8. C₄×C₅×#127	C₂	semidirect
9. C₄×C₅×#121	C₂	semidirect
10. C₄×C₅×#120	C₂	semidirect
11. C₄²×C₅×D₄	C₂	semidirect
13. C₂⁴×C₄×C₅	C₂²	semidirect
14. C₅×D₄²	C₄	direct (N × H)
23. C₂×C₄×C₅×D₄	C₂²	semidirect
24. C₂×C₄×C₅×D₄	C₂²	semidirect
25. C₂×C₄×C₅×D₄	C₂²	semidirect
26. C₂×C₄×C₅×D₄	C₂²	semidirect
27. C₂×C₄×C₅×D₄	C₂²	semidirect
28. C₂²×C₅×C₄⋊C₄	C₂²	semidirect
29. C₂²×C₅×C₂²⋊C₄	C₂²	semidirect
30. C₂²×C₄²×C₅	C₂²	semidirect
31. C₅×#390	C₂²	semidirect
32. C₅×#390	C₂²	semidirect
33. C₅×#390	C₂²	semidirect
34. C₅×#389	C₂²	semidirect
35. C₅×#388	C₂²	semidirect
36. C₅×#386	C₂²	semidirect
37. C₅×#386	C₂²	semidirect
38. C₅×#386	C₂²	semidirect
39. C₅×#385	C₂²	semidirect
40. C₅×#385	C₂²	semidirect
41. C₅×#384	C₂²	semidirect
42. C₅×#380	C₂²	semidirect
43. C₅×#379	C₂²	semidirect
44. C₄×C₅×C₄⋊C₄	C₂²	semidirect
45. C₄×C₅×C₄⋊C₄	C₂²	semidirect
46. C₄×C₅×C₄⋊C₄	C₂²	semidirect
47. C₄×C₅×C₂²⋊C₄	C₂²	semidirect
48. C₄×C₅×C₂²⋊C₄	C₂²	semidirect
49. C₄×C₅×C₂²⋊C₄	C₂²	semidirect
50. C₄×C₅×C₂²⋊C₄	C₂²	semidirect
51. C₄³×C₅	C₂²	semidirect
52. C₄×D₄²	C₅	direct (N × H)
56. C₂²×C₅×D₄	C₂×C₄	semidirect
69. C₅×#127	C₂×C₄	semidirect
70. C₅×#121	C₂×C₄	semidirect
71. C₅×#120	C₂×C₄	semidirect
79. C₄×C₅×D₄	C₂×C₄	semidirect
80. C₄×C₅×D₄	D₄	direct (N × H)
112. C₂²×C₄×D₄	C₂×C₅	semidirect
113. #2366	C₂×C₅	semidirect
114. #2361	C₂×C₅	semidirect
115. #2358	C₂×C₅	semidirect
116. D₄×C₄⋊C₄	C₂×C₅	semidirect
117. D₄×C₂²⋊C₄	C₂×C₅	semidirect
118. C₄×#127	C₂×C₅	semidirect
119. C₄×#121	C₂×C₅	semidirect
120. C₄×#120	C₂×C₅	semidirect
121. C₄²×D₄	C₂×C₅	semidirect
126. C₂⁴×C₅	C₂²×C₄	semidirect
137. C₂×C₅×D₄	C₂²×C₄	semidirect
138. C₂×C₅×D₄	C₂²×C₄	semidirect
139. C₂×C₅×D₄	C₂²×C₄	semidirect
157. C₂²×C₄×C₅	C₂²×C₄	semidirect
158. C₂²×C₄×C₅	C₂×D₄	semidirect
172. C₅×C₄⋊C₄	C₂²×C₄	semidirect
173. C₅×C₄⋊C₄	C₂×D₄	semidirect
180. C₅×C₂²⋊C₄	C₂²×C₄	semidirect
181. C₅×C₂²⋊C₄	C₂²×C₄	semidirect
182. C₅×C₂²⋊C₄	C₂×D₄	semidirect
189. C₄²×C₅	C₂²×C₄	semidirect
190. C₄²×C₅	C₂×D₄	semidirect
192. C₂⁴×C₄	C₂²×C₅	semidirect
193. D₄²	C₄×C₅	direct (N × H)
202. C₂×C₄×D₄	C₂²×C₅	semidirect
203. C₂×C₄×D₄	C₂²×C₅	semidirect
204. C₂×C₄×D₄	C₂²×C₅	semidirect
205. C₂×C₄×D₄	C₂²×C₅	semidirect
206. C₂×C₄×D₄	C₂²×C₅	semidirect
207. C₂²×C₄⋊C₄	C₂²×C₅	semidirect
208. C₂²×C₂²⋊C₄	C₂²×C₅	semidirect
209. C₂²×C₄²	C₂²×C₅	semidirect
210. #390	C₂²×C₅	semidirect
211. #390	C₂²×C₅	semidirect
212. #390	C₂²×C₅	semidirect
213. #389	C₂²×C₅	semidirect
214. #388	C₂²×C₅	semidirect
215. #386	C₂²×C₅	semidirect
216. #386	C₂²×C₅	semidirect
217. #386	C₂²×C₅	semidirect
218. #385	C₂²×C₅	semidirect
219. #385	C₂²×C₅	semidirect
220. #384	C₂²×C₅	semidirect
221. #380	C₂²×C₅	semidirect
222. #379	C₂²×C₅	semidirect
223. C₄×C₄⋊C₄	C₂²×C₅	semidirect
224. C₄×C₄⋊C₄	C₂²×C₅	semidirect
225. C₄×C₄⋊C₄	C₂²×C₅	semidirect
226. C₄×C₂²⋊C₄	C₂²×C₅	semidirect
227. C₄×C₂²⋊C₄	C₂²×C₅	semidirect
228. C₄×C₂²⋊C₄	C₂²×C₅	semidirect
229. C₄×C₂²⋊C₄	C₂²×C₅	semidirect
230. C₄³	C₂²×C₅	semidirect
252. C₅×D₄	C₄×D₄	direct (N × H)
286. C₂²×D₄	C₂×C₄×C₅	semidirect
299. #127	C₂×C₄×C₅	semidirect
300. #121	C₂×C₄×C₅	semidirect
301. #120	C₂×C₄×C₅	semidirect
309. C₄×D₄	C₂×C₄×C₅	semidirect
310. C₄×D₄	C₅×D₄	direct (N × H)
353. C₂²×C₅	C₂×C₄×D₄	semidirect
366. C₄×C₅	C₂×C₄×D₄	semidirect
367. C₄×C₅	D₄²	direct (N × H)
372. C₂⁴	C₂²×C₄×C₅	semidirect
383. C₂×D₄	C₂²×C₄×C₅	semidirect
384. C₂×D₄	C₂²×C₄×C₅	semidirect
385. C₂×D₄	C₂²×C₄×C₅	semidirect
403. C₂²×C₄	C₂²×C₄×C₅	semidirect
404. C₂²×C₄	C₂×C₅×D₄	semidirect
418. C₄⋊C₄	C₂²×C₄×C₅	semidirect
419. C₄⋊C₄	C₂×C₅×D₄	semidirect
426. C₂²⋊C₄	C₂²×C₄×C₅	semidirect
427. C₂²⋊C₄	C₂²×C₄×C₅	semidirect
428. C₂²⋊C₄	C₂×C₅×D₄	semidirect
435. C₄²	C₂²×C₄×C₅	semidirect
436. C₄²	C₂×C₅×D₄	semidirect
465. D₄	C₄×C₅×D₄	direct (N × H)
496. C₅	C₄×D₄²	direct (N × H)
509. C₂²	C₂×C₄×C₅×D₄	semidirect
522. C₄	C₂×C₄×C₅×D₄	semidirect
523. C₄	C₅×D₄²	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅×D₄²	2
C₂²×C₄×C₅×D₄	2
C₅×#2366	2
C₅×#2361	2
C₅×#2358	2
C₅×D₄×C₄⋊C₄	2
C₅×D₄×C₂²⋊C₄	2
C₄×C₅×#127	2
C₄×C₅×#121	2
C₄×C₅×#120	2
C₄²×C₅×D₄	2
C₄×D₄²	5

Subgroups — C4×C5×D42 #11956876

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₅×D₄² #11956876