Subgroups — C₂×#10618210 #13007212

Order 1280.

9502 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#534	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#4083	C₂	1	2	yes	maximal, complemented
3	C₂×#66132	C₂	1	2	yes	maximal, complemented
4	#10618210	C₂	1	32	yes	maximal, direct product factor
5	C₂×#65992	C₂	1	8	yes	maximal, complemented
6	C₂×#65951	C₂	1	4	yes	maximal, complemented
7	C₂×#65940	C₂	1	4	yes	maximal, complemented
8	C₂×#65925	C₂	1	8	yes	maximal, complemented
9	C₂²×D₅×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#65817	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂⁵×D₅	C₂²	1	1	yes	characteristic
12	C₂³×C₅×D₄	C₂²	1	2	yes	—
13	C₂³×C₂²⋊D₅	C₂²	1	4	yes	complemented
14	C₂²×#711	C₂²	1	4	yes	complemented
15	C₂³×C₄×D₅	C₂²	1	2	yes	complemented
16	C₅×#534	C₂²	1	16	yes	complemented
17	C₂×C₅×#123	C₂²	1	4	yes	complemented
18	C₂×C₅×#121	C₂²	1	4	yes	complemented
19	C₂×C₅×#120	C₂²	1	4	yes	complemented
20	C₂²×C₅×C₂²⋊C₄	C₂²	1	1	yes	characteristic
21	C₂²×C₂²⋊Dic₅	C₂²	1	2	yes	—
22	C₂²×C₂²⋊Dic₅	C₂²	1	1	yes	characteristic
23	C₂²×C₂²⋊Dic₅	C₂²	1	4	yes	complemented
24	C₂×#4085	C₂²	1	8	yes	complemented
25	C₂×#4084	C₂²	1	4	yes	complemented
26	C₂×#4083	C₂²	1	8	yes	—
27	C₂×#4083	C₂²	1	4	yes	complemented
28	C₂×#4083	C₂²	1	4	yes	complemented
29	C₂×#4083	C₂²	1	32	yes	complemented
30	#66132	C₂²	1	32	yes	complemented
31	C₂×D₄×Dic₅	C₂²	1	8	yes	complemented
32	C₂×D₄×Dic₅	C₂²	1	8	yes	complemented
33	C₂²×#685	C₂²	1	4	yes	complemented
34	C₂²×#685	C₂²	1	2	yes	complemented
35	C₂×#4072	C₂²	1	4	yes	complemented
36	C₂×#4070	C₂²	1	4	yes	complemented
37	#65992	C₂²	1	128	yes	complemented
38	#65951	C₂²	1	64	yes	complemented
39	#65940	C₂²	1	64	yes	complemented
40	#65925	C₂²	1	128	yes	complemented
41	C₂×#4044	C₂²	1	8	yes	complemented
42	C₂×#4043	C₂²	1	4	yes	complemented
43	C₂×#4038	C₂²	1	4	yes	complemented
44	C₂×#4032	C₂²	1	8	yes	complemented
45	C₂×#4031	C₂²	1	8	yes	complemented
46	C₂×#4029	C₂²	1	8	yes	complemented
47	C₂×#4027	C₂²	1	8	yes	complemented
48	C₂×D₅×C₂²⋊C₄	C₂²	1	8	yes	complemented
49	C₂×D₅×C₂²⋊C₄	C₂²	1	16	yes	complemented
50	#65817	C₂²	1	16	yes	complemented
51	C₂×#4024	C₂²	1	8	yes	complemented
52	C₂×#3490	—	5	1	no	maximal, Sylow
53	C₂⁵×C₅	C₂³	1	2	yes	—
54	C₂⁵×C₅	C₂³	1	1	yes	characteristic
55	C₂⁴×D₅	—	2	8	no	—
56	C₂⁴×D₅	—	4	8	no	—
57	C₂⁴×D₅	C₂³	1	1	yes	characteristic
58	C₂⁴×D₅	C₂³	1	4	yes	—
59	C₂⁴×D₅	C₂³	1	8	yes	—
60	C₂⁴×D₅	C₂³	1	1	yes	characteristic
61	C₂²×C₅×D₄	—	2	16	no	—
62	C₂²×C₅×D₄	C₂³	1	4	yes	—
63	C₂²×C₅×D₄	C₂³	1	4	yes	—
64	C₂²×C₅×D₄	C₂³	1	16	yes	—
65	C₂²×C₅×D₄	C₂³	1	4	yes	complemented
66	C₂³×C₄×C₅	C₂³	1	2	yes	—
67	C₂³×C₄×C₅	C₂³	1	2	yes	—
68	C₂²×C₂²⋊D₅	—	4	16	no	—
69	C₂²×C₂²⋊D₅	C₂³	1	8	yes	—
70	C₂²×C₂²⋊D₅	C₂³	1	8	yes	—
71	C₂²×C₂²⋊D₅	C₂³	1	32	yes	complemented
72	C₂³×Dic₅	C₂³	1	4	yes	—
73	C₂³×Dic₅	C₂³	1	2	yes	—
74	C₂³×Dic₅	C₂³	1	8	yes	complemented
75	C₂×#711	—	4	16	no	—
76	C₂×#711	C₂³	1	32	yes	complemented
77	C₂²×C₄×D₅	—	4	8	no	—
78	C₂²×C₄×D₅	C₂³	1	4	yes	—
79	C₂²×C₄×D₅	C₂³	1	4	yes	—
80	C₂²×C₄×D₅	C₂³	1	16	yes	complemented
81	C₂²×Dic₁₀	C₂³	1	4	yes	complemented
82	C₅×#123	C₂³	1	32	yes	complemented
83	C₅×#121	C₂³	1	32	yes	complemented
84	C₅×#120	C₂³	1	32	yes	complemented
85	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	complemented
86	C₂×C₅×C₂²⋊C₄	C₂³	1	4	yes	—
87	C₂×C₅×C₂²⋊C₄	C₂³	1	8	yes	—
88	C₂×C₅×C₂²⋊C₄	C₂³	1	8	yes	complemented
89	C₂×C₂²⋊Dic₅	C₂³	1	4	yes	—
90	C₂×C₂²⋊Dic₅	C₂³	1	16	yes	—
91	C₂×C₂²⋊Dic₅	C₂³	1	4	yes	—
92	C₂×C₂²⋊Dic₅	C₂³	1	8	yes	—
93	C₂×C₂²⋊Dic₅	C₂³	1	4	yes	—
94	C₂×C₂²⋊Dic₅	C₂³	1	32	yes	complemented
95	C₂×C₂²⋊Dic₅	C₂³	1	8	yes	complemented
96	#4085	C₂³	1	64	yes	complemented
97	#4084	C₂³	1	32	yes	complemented
98	#4083	C₂³	1	64	yes	—
99	#4083	C₂³	1	32	yes	complemented
100	#4083	C₂³	1	32	yes	complemented
101	D₄×Dic₅	C₂³	1	64	yes	complemented
102	D₄×Dic₅	C₂³	1	64	yes	complemented
103	C₂×#685	C₂³	1	8	yes	—
104	C₂×#685	C₂³	1	8	yes	—
105	C₂×#685	C₂³	1	8	yes	—
106	C₂×#685	C₂³	1	16	yes	complemented
107	C₂×#685	C₂³	1	32	yes	complemented
108	#4072	C₂³	1	32	yes	complemented
109	C₂×C₄⋊Dic₅	C₂³	1	4	yes	complemented
110	C₂×C₄⋊Dic₅	C₂³	1	8	yes	complemented
111	#4070	C₂³	1	32	yes	complemented
112	C₂×#683	C₂³	1	8	yes	complemented
113	C₂×C₄×Dic₅	C₂³	1	8	yes	complemented
114	#4044	C₂³	1	64	yes	complemented
115	#4043	C₂³	1	32	yes	complemented
116	#4038	C₂³	1	32	yes	complemented
117	#4032	C₂³	1	64	yes	complemented
118	#4031	C₂³	1	64	yes	complemented
119	#4029	C₂³	1	64	yes	complemented
120	#4027	C₂³	1	64	yes	complemented
121	D₅×C₂²⋊C₄	C₂³	1	64	yes	complemented
122	#4024	C₂³	1	64	yes	complemented
123	#3490	—	5	32	no	—
124	C₂×#551	—	5	8	no	—
125	C₂×#546	—	5	8	no	—
126	C₂×#543	—	5	1	no	—
127	C₂×#538	—	5	4	no	—
128	C₂×#534	D₅	1	1	yes	characteristic, complemented
129	C₂×#525	—	5	4	no	—
130	C₂²×#120	—	5	2	no	—
131	C₂×#518	—	5	2	no	—
132	C₂³×C₂²⋊C₄	—	5	1	no	—
133	C₂⁴×C₅	—	2	8	no	—
134	C₂⁴×C₅	—	4	4	no	—
135	C₂⁴×C₅	—	4	8	no	—
136	C₂⁴×C₅	#41	1	8	yes	complemented
137	C₂⁴×C₅	C₂⁴	1	4	yes	—
138	C₂⁴×C₅	C₂⁴	1	2	yes	—
139	C₂⁴×C₅	C₂⁴	1	1	yes	characteristic
140	C₂⁴×C₅	C₂⁴	1	8	yes	—
141	C₂⁴×C₅	C₂⁴	1	4	yes	—
142	C₂³×D₅	—	2	4	no	—
143	C₂³×D₅	—	2	8	no	—
144	C₂³×D₅	—	2	16	no	—
145	C₂³×D₅	—	2	4	no	—
146	C₂³×D₅	—	2	32	no	—
147	C₂³×D₅	—	2	16	no	—
148	C₂³×D₅	—	4	32	no	—
149	C₂³×D₅	—	4	8	no	—
150	C₂³×D₅	—	4	4	no	—
151	C₂³×D₅	—	4	64	no	—
152	C₂³×D₅	C₂⁴	1	4	yes	—
153	C₂³×D₅	C₂⁴	1	2	yes	—
154	C₂³×D₅	C₂⁴	1	16	yes	—
155	C₂³×D₅	C₂⁴	1	2	yes	—
156	C₂³×D₅	C₂⁴	1	4	yes	—
157	C₂×C₅×D₄	—	2	16	no	—
158	C₂×C₅×D₄	—	2	16	no	—
159	C₂×C₅×D₄	—	2	64	no	—
160	C₂×C₅×D₄	—	4	8	no	—
161	C₂×C₅×D₄	C₂⁴	1	16	yes	—
162	C₂×C₅×D₄	C₂⁴	1	16	yes	—
163	C₂×C₅×D₄	C₂⁴	1	16	yes	complemented
164	C₂²×C₄×C₅	—	2	8	no	—
165	C₂²×C₄×C₅	—	4	4	no	—
166	C₂²×C₄×C₅	C₂⁴	1	4	yes	—
167	C₂²×C₄×C₅	C₂⁴	1	4	yes	—
168	C₂²×C₄×C₅	C₂⁴	1	8	yes	—
169	C₂²×C₄×C₅	C₂⁴	1	8	yes	—
170	C₂×C₂²⋊D₅	—	2	32	no	—
171	C₂×C₂²⋊D₅	—	4	16	no	—
172	C₂×C₂²⋊D₅	—	4	64	no	—
173	C₂×C₂²⋊D₅	C₂⁴	1	32	yes	—
174	C₂×C₂²⋊D₅	C₂⁴	1	32	yes	—
175	C₂²×Dic₅	—	4	4	no	—
176	C₂²×Dic₅	—	4	16	no	—
177	C₂²×Dic₅	—	4	8	no	—
178	C₂²×Dic₅	C₂⁴	1	4	yes	—
179	C₂²×Dic₅	C₂⁴	1	8	yes	—
180	C₂²×Dic₅	C₂⁴	1	8	yes	—
181	C₂²×Dic₅	C₂⁴	1	16	yes	—
182	C₂²×Dic₅	C₂⁴	1	32	yes	complemented
183	#711	—	4	64	no	—
184	C₂×C₄×D₅	—	2	16	no	—
185	C₂×C₄×D₅	—	4	32	no	—
186	C₂×C₄×D₅	—	4	8	no	—
187	C₂×C₄×D₅	C₂⁴	1	16	yes	—
188	C₂×C₄×D₅	C₂⁴	1	16	yes	—
189	C₂×Dic₁₀	—	4	8	no	—
190	C₂×Dic₁₀	C₂⁴	1	16	yes	complemented
191	C₅×C₄⋊C₄	C₂⁴	1	16	yes	complemented
192	C₅×C₂²⋊C₄	C₂⁴	1	16	yes	—
193	C₅×C₂²⋊C₄	C₂⁴	1	32	yes	complemented
194	C₂²⋊Dic₅	C₂⁴	1	16	yes	—
195	C₂²⋊Dic₅	C₂⁴	1	16	yes	—
196	C₂²⋊Dic₅	C₂⁴	1	16	yes	—
197	C₂²⋊Dic₅	C₂⁴	1	32	yes	complemented
198	#685	C₂⁴	1	32	yes	—
199	#685	C₂⁴	1	32	yes	—
200	#685	C₂⁴	1	32	yes	—
201	C₄⋊Dic₅	C₂⁴	1	16	yes	complemented
202	C₄⋊Dic₅	C₂⁴	1	32	yes	complemented
203	#683	C₂⁴	1	32	yes	complemented
204	C₄×Dic₅	C₂⁴	1	32	yes	complemented
205	C₂⁶	—	5	1	no	—
206	C₂²×#41	—	5	4	no	—
207	C₂³×D₄	—	5	4	no	—
208	C₂³×D₄	C₂×D₅	1	2	yes	—
209	C₂⁴×C₄	—	5	2	no	—
210	#551	—	5	128	no	—
211	#546	—	5	128	no	—
212	#543	—	5	16	no	—
213	#538	—	5	64	no	—
214	#534	C₂×D₅	1	16	yes	complemented
215	C₂×#126	—	5	8	no	—
216	C₂×#124	—	5	8	no	—
217	#525	—	5	64	no	—
218	C₂×#123	—	5	8	no	—
219	C₂×#123	—	5	8	no	—
220	C₂×#123	C₂×D₅	1	4	yes	complemented
221	C₂×C₂²⋊Q₈	—	5	4	no	—
222	C₂×C₂²⋊Q₈	—	5	4	no	—
223	C₂×C₂²⋊Q₈	—	5	8	no	—
224	C₂×#121	—	5	4	no	—
225	C₂×#121	—	5	8	no	—
226	C₂×#121	C₂×D₅	1	4	yes	complemented
227	C₂×#120	—	5	8	no	—
228	C₂×#120	—	5	4	no	—
229	C₂×#120	—	5	4	no	—
230	C₂×#120	—	5	32	no	—
231	C₂×#120	C₂×D₅	1	4	yes	complemented
232	#518	—	5	32	no	—
233	C₂×C₄×D₄	—	5	8	no	—
234	C₂×C₄×D₄	—	5	8	no	—
235	C₂×C₄×D₄	—	5	8	no	—
236	C₂×#117	—	5	4	no	—
237	C₂×#117	—	5	4	no	—
238	C₂²×C₂²⋊C₄	—	5	1	no	—
239	C₂²×C₂²⋊C₄	—	5	2	no	—
240	C₂²×C₂²⋊C₄	—	5	4	no	—
241	C₂²×C₂²⋊C₄	—	5	2	no	—
242	C₂²×C₂²⋊C₄	—	5	4	no	—
243	C₂²×C₂²⋊C₄	—	5	8	no	—
244	C₂²×C₂²⋊C₄	—	5	16	no	—
245	C₂²×C₂²⋊C₄	C₂×D₅	1	1	yes	characteristic
246	C₂³×C₅	—	2	4	no	—
247	C₂³×C₅	—	2	8	no	—
248	C₂³×C₅	—	2	8	no	—
249	C₂³×C₅	—	2	16	no	—
250	C₂³×C₅	—	2	16	no	—
251	C₂³×C₅	—	4	16	no	—
252	C₂³×C₅	—	4	8	no	—
253	C₂³×C₅	—	4	16	no	—
254	C₂³×C₅	—	8	16	no	—
255	C₂³×C₅	C₂×#41	1	8	yes	—
256	C₂³×C₅	C₂×#41	1	4	yes	—
257	C₂³×C₅	C₂×#41	1	16	yes	complemented
258	C₂³×C₅	C₂⁵	1	8	yes	—
259	C₂³×C₅	C₂⁵	1	4	yes	—
260	C₂³×C₅	C₂⁵	1	2	yes	—
261	C₂³×C₅	C₂⁵	1	1	yes	characteristic
262	C₂²×D₅	—	2	32	no	—
263	C₂²×D₅	—	2	8	no	—
264	C₂²×D₅	—	2	4	no	—
265	C₂²×D₅	—	2	4	no	—
266	C₂²×D₅	—	2	16	no	—
267	C₂²×D₅	—	2	32	no	—
268	C₂²×D₅	—	2	8	no	—
269	C₂²×D₅	—	2	4	no	—
270	C₂²×D₅	—	2	8	no	—
271	C₂²×D₅	—	2	32	no	—
272	C₂²×D₅	—	4	16	no	—
273	C₂²×D₅	—	4	8	no	—
274	C₂²×D₅	—	4	64	no	—
275	C₂²×D₅	—	4	128	no	—
276	C₂²×D₅	—	4	16	no	—
277	C₂²×D₅	H(5,𝔽₂)	1	8	yes	—
278	C₂²×D₅	C₂⁵	1	4	yes	—
279	C₂²×D₅	C₂⁵	1	4	yes	—
280	C₅×D₄	—	2	32	no	—
281	C₅×D₄	—	2	32	no	—
282	C₅×D₄	—	4	16	no	—
283	C₂×C₄×C₅	—	2	16	no	—
284	C₂×C₄×C₅	—	2	8	no	—
285	C₂×C₄×C₅	—	2	8	no	—
286	C₂×C₄×C₅	—	4	8	no	—
287	C₂×C₄×C₅	C₂⁵	1	8	yes	—
288	C₂×C₄×C₅	C₂⁵	1	8	yes	—
289	C₂²⋊D₅	—	2	64	no	—
290	C₂²⋊D₅	—	4	32	no	—
291	C₂×Dic₅	—	2	16	no	—
292	C₂×Dic₅	—	2	8	no	—
293	C₂×Dic₅	—	4	8	no	—
294	C₂×Dic₅	—	4	32	no	—
295	C₂×Dic₅	—	4	16	no	—
296	C₂×Dic₅	C₂⁵	1	8	yes	—
297	C₂×Dic₅	C₂⁵	1	16	yes	—
298	C₄×D₅	—	2	32	no	—
299	C₄×D₅	—	4	16	no	—
300	Dic₁₀	—	4	16	no	—
301	C₂⁵	—	5	1	no	—
302	C₂⁵	—	5	4	no	—
303	C₂⁵	—	5	8	no	—
304	C₂⁵	—	5	1	no	—
305	C₂⁵	—	10	8	no	—
306	C₂⁵	—	20	8	no	—
307	C₂⁵	C₂²×D₅	1	2	yes	—
308	C₂⁵	C₂²×D₅	1	1	yes	characteristic
309	C₂×#41	—	5	32	no	—
310	C₂×#41	—	20	16	no	—
311	C₂²×Q₈	—	5	4	no	—
312	C₂²×D₄	—	2	16	no	—
313	C₂²×D₄	—	5	8	no	—
314	C₂²×D₄	—	5	8	no	—
315	C₂²×D₄	—	5	32	no	—
316	C₂²×D₄	—	20	16	no	—
317	C₂²×D₄	C₂²×D₅	1	4	yes	—
318	C₂²×D₄	C₂²×D₅	1	4	yes	—
319	C₂²×D₄	C₂²×D₅	1	16	yes	—
320	C₂²×D₄	C₂²×D₅	1	4	yes	complemented
321	C₂³×C₄	—	5	2	no	—
322	C₂³×C₄	—	5	4	no	—
323	C₂³×C₄	—	5	8	no	—
324	C₂³×C₄	—	5	4	no	—
325	C₂³×C₄	—	5	4	no	—
326	C₂³×C₄	—	5	16	no	—
327	C₂³×C₄	—	20	8	no	—
328	C₂³×C₄	C₂²×D₅	1	2	yes	—
329	C₂³×C₄	C₂²×D₅	1	2	yes	—
330	#126	—	5	64	no	—
331	#124	—	5	64	no	—
332	#123	—	5	64	no	—
333	#123	—	5	64	no	—
334	#123	C₂²×D₅	1	32	yes	complemented
335	C₂²⋊Q₈	—	5	32	no	—
336	C₂²⋊Q₈	—	5	32	no	—
337	C₂²⋊Q₈	—	5	64	no	—
338	#121	—	5	32	no	—
339	#121	—	5	64	no	—
340	#121	C₂²×D₅	1	32	yes	complemented
341	#120	—	5	32	no	—
342	#120	—	5	32	no	—
343	#120	—	5	64	no	—
344	#120	C₂²×D₅	1	32	yes	complemented
345	C₄×D₄	—	5	64	no	—
346	C₄×D₄	—	5	64	no	—
347	C₄×D₄	—	5	64	no	—
348	#117	—	5	32	no	—
349	#117	—	5	32	no	—
350	C₂×C₄⋊C₄	—	5	8	no	—
351	C₂×C₄⋊C₄	—	5	4	no	—
352	C₂×C₄⋊C₄	—	5	8	no	—
353	C₂×C₄⋊C₄	C₂²×D₅	1	4	yes	complemented
354	C₂×C₂²⋊C₄	—	5	4	no	—
355	C₂×C₂²⋊C₄	—	5	8	no	—
356	C₂×C₂²⋊C₄	—	5	4	no	—
357	C₂×C₂²⋊C₄	—	5	4	no	—
358	C₂×C₂²⋊C₄	—	5	16	no	—
359	C₂×C₂²⋊C₄	—	5	8	no	—
360	C₂×C₂²⋊C₄	—	5	32	no	—
361	C₂×C₂²⋊C₄	—	5	8	no	—
362	C₂×C₂²⋊C₄	—	5	16	no	—
363	C₂×C₂²⋊C₄	—	5	8	no	—
364	C₂×C₂²⋊C₄	—	5	8	no	—
365	C₂×C₂²⋊C₄	—	5	32	no	—
366	C₂×C₂²⋊C₄	—	5	64	no	—
367	C₂×C₂²⋊C₄	C₂²×D₅	1	4	yes	—
368	C₂×C₂²⋊C₄	C₂²×D₅	1	8	yes	—
369	C₂×C₂²⋊C₄	C₂²×D₅	1	8	yes	complemented
370	C₂×C₄²	—	5	8	no	—
371	C₂²×C₅	—	2	4	no	—
372	C₂²×C₅	—	2	8	no	—
373	C₂²×C₅	—	2	8	no	—
374	C₂²×C₅	—	2	16	no	—
375	C₂²×C₅	—	2	8	no	—
376	C₂²×C₅	—	2	16	no	—
377	C₂²×C₅	—	4	16	no	—
378	C₂²×C₅	—	4	16	no	—
379	C₂²×C₅	—	8	16	no	—
380	C₂²×C₅	C₂²×#41	1	8	yes	—
381	C₂²×C₅	C₂²×#41	1	4	yes	—
382	C₂²×C₅	C₂²×#41	1	2	yes	—
383	C₂²×C₅	C₂×H(5,𝔽₂)	1	4	yes	—
384	C₂²×C₅	C₂⁶	1	1	yes	characteristic
385	C₂×D₅	—	2	32	no	—
386	C₂×D₅	—	2	4	no	—
387	C₂×D₅	—	2	4	no	—
388	C₂×D₅	—	2	16	no	—
389	C₂×D₅	—	2	4	no	—
390	C₂×D₅	—	2	8	no	—
391	C₂×D₅	—	4	64	no	—
392	C₂×D₅	—	4	16	no	—
393	C₂×D₅	—	4	8	no	—
394	C₂×D₅	C₂×H(5,𝔽₂)	1	8	yes	—
395	C₄×C₅	—	2	8	no	—
396	C₄×C₅	—	2	8	no	—
397	Dic₅	—	2	16	no	—
398	Dic₅	—	2	8	no	—
399	C₂⁴	—	2	8	no	—
400	C₂⁴	—	4	4	no	—
401	C₂⁴	—	4	8	no	—
402	C₂⁴	—	5	2	no	—
403	C₂⁴	—	5	4	no	—
404	C₂⁴	—	5	2	no	—
405	C₂⁴	—	5	16	no	—
406	C₂⁴	—	5	4	no	—
407	C₂⁴	—	10	8	no	—
408	C₂⁴	—	10	16	no	—
409	C₂⁴	—	10	4	no	—
410	C₂⁴	—	10	16	no	—
411	C₂⁴	—	10	4	no	—
412	C₂⁴	—	10	32	no	—
413	C₂⁴	—	20	4	no	—
414	C₂⁴	—	20	8	no	—
415	C₂⁴	—	20	32	no	—
416	C₂⁴	—	20	64	no	—
417	C₂⁴	#711	1	8	yes	complemented
418	C₂⁴	C₂³×D₅	1	4	yes	—
419	C₂⁴	C₂³×D₅	1	2	yes	—
420	C₂⁴	C₂³×D₅	1	1	yes	characteristic
421	C₂⁴	C₂³×D₅	1	8	yes	—
422	C₂⁴	C₂³×D₅	1	4	yes	—
423	#41	—	20	64	no	—
424	C₂×Q₈	—	5	16	no	—
425	C₂×Q₈	—	20	8	no	—
426	C₂×D₄	—	2	16	no	—
427	C₂×D₄	—	2	16	no	—
428	C₂×D₄	—	2	64	no	—
429	C₂×D₄	—	4	8	no	—
430	C₂×D₄	—	5	32	no	—
431	C₂×D₄	—	5	32	no	—
432	C₂×D₄	—	10	32	no	—
433	C₂×D₄	—	20	16	no	—
434	C₂×D₄	—	20	64	no	—
435	C₂×D₄	C₂³×D₅	1	16	yes	—
436	C₂×D₄	C₂³×D₅	1	16	yes	—
437	C₂×D₄	C₂³×D₅	1	16	yes	complemented
438	C₂²×C₄	—	2	8	no	—
439	C₂²×C₄	—	4	4	no	—
440	C₂²×C₄	—	5	4	no	—
441	C₂²×C₄	—	5	8	no	—
442	C₂²×C₄	—	5	8	no	—
443	C₂²×C₄	—	5	16	no	—
444	C₂²×C₄	—	5	32	no	—
445	C₂²×C₄	—	5	16	no	—
446	C₂²×C₄	—	5	16	no	—
447	C₂²×C₄	—	10	16	no	—
448	C₂²×C₄	—	20	8	no	—
449	C₂²×C₄	—	20	32	no	—
450	C₂²×C₄	—	20	4	no	—
451	C₂²×C₄	—	20	8	no	—
452	C₂²×C₄	—	20	16	no	—
453	C₂²×C₄	C₂³×D₅	1	8	yes	—
454	C₂²×C₄	C₂³×D₅	1	4	yes	—
455	C₂²×C₄	C₂³×D₅	1	4	yes	—
456	C₂²×C₄	C₂³×D₅	1	8	yes	—
457	C₄⋊C₄	—	5	32	no	—
458	C₄⋊C₄	—	5	16	no	—
459	C₄⋊C₄	—	5	32	no	—
460	C₄⋊C₄	C₂³×D₅	1	16	yes	complemented
461	C₂²⋊C₄	—	5	16	no	—
462	C₂²⋊C₄	—	5	16	no	—
463	C₂²⋊C₄	—	5	16	no	—
464	C₂²⋊C₄	—	5	32	no	—
465	C₂²⋊C₄	—	5	32	no	—
466	C₂²⋊C₄	—	5	32	no	—
467	C₂²⋊C₄	—	5	32	no	—
468	C₂²⋊C₄	C₂³×D₅	1	16	yes	—
469	C₂²⋊C₄	C₂³×D₅	1	32	yes	complemented
470	C₄²	—	5	32	no	—
471	C₂×C₅	—	2	4	no	—
472	C₂×C₅	—	2	8	no	—
473	C₂×C₅	—	4	8	no	—
474	C₂×C₅	#3490	1	4	yes	complemented
475	C₂×C₅	C₂³×#41	1	1	yes	characteristic
476	C₂×C₅	C₂²×H(5,𝔽₂)	1	2	yes	—
477	D₅	—	2	8	no	—
478	D₅	—	4	4	no	—
479	C₂³	—	2	4	no	—
480	C₂³	—	2	8	no	—
481	C₂³	—	2	8	no	—
482	C₂³	—	2	16	no	—
483	C₂³	—	2	16	no	—
484	C₂³	—	4	16	no	—
485	C₂³	—	4	8	no	—
486	C₂³	—	4	16	no	—
487	C₂³	—	5	8	no	—
488	C₂³	—	5	4	no	—
489	C₂³	—	5	4	no	—
490	C₂³	—	8	16	no	—
491	C₂³	—	10	16	no	—
492	C₂³	—	10	32	no	—
493	C₂³	—	10	32	no	—
494	C₂³	—	10	8	no	—
495	C₂³	—	10	4	no	—
496	C₂³	—	10	4	no	—
497	C₂³	—	10	32	no	—
498	C₂³	—	10	8	no	—
499	C₂³	—	10	4	no	—
500	C₂³	—	10	8	no	—
501	C₂³	—	20	8	no	—
502	C₂³	—	20	16	no	—
503	C₂³	—	20	64	no	—
504	C₂³	—	20	16	no	—
505	C₂³	—	20	128	no	—
506	C₂³	C₂×#711	1	8	yes	—
507	C₂³	C₂×#711	1	4	yes	—
508	C₂³	C₂×#711	1	16	yes	complemented
509	C₂³	C₂⁴×D₅	1	8	yes	—
510	C₂³	C₂⁴×D₅	1	4	yes	—
511	C₂³	C₂⁴×D₅	1	2	yes	—
512	C₂³	C₂⁴×D₅	1	1	yes	characteristic
513	Q₈	—	20	16	no	—
514	D₄	—	2	32	no	—
515	D₄	—	2	32	no	—
516	D₄	—	4	16	no	—
517	D₄	—	10	64	no	—
518	D₄	—	20	32	no	—
519	C₂×C₄	—	2	16	no	—
520	C₂×C₄	—	2	8	no	—
521	C₂×C₄	—	2	8	no	—
522	C₂×C₄	—	4	8	no	—
523	C₂×C₄	—	5	8	no	—
524	C₂×C₄	—	5	16	no	—
525	C₂×C₄	—	10	32	no	—
526	C₂×C₄	—	10	8	no	—
527	C₂×C₄	—	10	16	no	—
528	C₂×C₄	—	20	16	no	—
529	C₂×C₄	—	20	8	no	—
530	C₂×C₄	—	20	16	no	—
531	C₂×C₄	—	20	32	no	—
532	C₂×C₄	C₂⁴×D₅	1	8	yes	—
533	C₂×C₄	C₂⁴×D₅	1	8	yes	—
534	C₅	C₂×#3490	1	1	yes	characteristic, complemented, Sylow
535	C₂²	—	2	4	no	—
536	C₂²	—	2	8	no	—
537	C₂²	—	2	16	no	—
538	C₂²	—	2	8	no	—
539	C₂²	—	2	8	no	—
540	C₂²	—	2	16	no	—
541	C₂²	—	4	16	no	—
542	C₂²	—	4	16	no	—
543	C₂²	—	5	8	no	—
544	C₂²	—	8	16	no	—
545	C₂²	—	10	32	no	—
546	C₂²	—	10	4	no	—
547	C₂²	—	10	4	no	—
548	C₂²	—	10	16	no	—
549	C₂²	—	10	4	no	—
550	C₂²	—	10	8	no	—
551	C₂²	—	20	64	no	—
552	C₂²	—	20	16	no	—
553	C₂²	—	20	8	no	—
554	C₂²	C₂²×#711	1	8	yes	—
555	C₂²	C₂²×#711	1	4	yes	—
556	C₂²	C₂²×#711	1	2	yes	—
557	C₂²	D₅×H(5,𝔽₂)	1	4	yes	—
558	C₂²	C₂⁵×D₅	1	1	yes	characteristic
559	C₄	—	2	8	no	—
560	C₄	—	2	8	no	—
561	C₄	—	10	8	no	—
562	C₄	—	10	16	no	—
563	C₂	—	2	8	no	—
564	C₂	—	2	4	no	—
565	C₂	—	4	8	no	—
566	C₂	—	10	8	no	—
567	C₂	—	20	4	no	—
568	C₂	#10618210	1	4	yes	direct product factor
569	C₂	C₂³×#711	1	1	yes	characteristic
570	C₂	C₂×D₅×H(5,𝔽₂)	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂⁴×D₅
commutator, cosocle	C₂²×C₅	C₂⁶
Frattini	C₂²	C₂⁵×D₅
Fitting	C₂×C₅×#534	C₂
socle	C₂³×C₅	C₂⁵
hypercenter	C₂×#534	D₅
hypocenter	C₅	C₂×#3490
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×#534

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×#534	C₂	semidirect
2. C₂²×#4083	C₂	semidirect
3. C₂×#66132	C₂	semidirect
4. #10618210	C₂	direct (N × H)
5. C₂×#65992	C₂	semidirect
6. C₂×#65951	C₂	semidirect
7. C₂×#65940	C₂	semidirect
8. C₂×#65925	C₂	semidirect
9. C₂²×D₅×C₂²⋊C₄	C₂	semidirect
10. C₂×#65817	C₂	semidirect
13. C₂³×C₂²⋊D₅	C₂²	semidirect
14. C₂²×#711	C₂²	semidirect
15. C₂³×C₄×D₅	C₂²	semidirect
16. C₅×#534	C₂²	semidirect
17. C₂×C₅×#123	C₂²	semidirect
18. C₂×C₅×#121	C₂²	semidirect
19. C₂×C₅×#120	C₂²	semidirect
23. C₂²×C₂²⋊Dic₅	C₂²	semidirect
24. C₂×#4085	C₂²	semidirect
25. C₂×#4084	C₂²	semidirect
27. C₂×#4083	C₂²	semidirect
28. C₂×#4083	C₂²	semidirect
29. C₂×#4083	C₂²	semidirect
30. #66132	C₂²	semidirect
31. C₂×D₄×Dic₅	C₂²	semidirect
32. C₂×D₄×Dic₅	C₂²	semidirect
33. C₂²×#685	C₂²	semidirect
34. C₂²×#685	C₂²	semidirect
35. C₂×#4072	C₂²	semidirect
36. C₂×#4070	C₂²	semidirect
37. #65992	C₂²	semidirect
38. #65951	C₂²	semidirect
39. #65940	C₂²	semidirect
40. #65925	C₂²	semidirect
41. C₂×#4044	C₂²	semidirect
42. C₂×#4043	C₂²	semidirect
43. C₂×#4038	C₂²	semidirect
44. C₂×#4032	C₂²	semidirect
45. C₂×#4031	C₂²	semidirect
46. C₂×#4029	C₂²	semidirect
47. C₂×#4027	C₂²	semidirect
48. C₂×D₅×C₂²⋊C₄	C₂²	semidirect
49. C₂×D₅×C₂²⋊C₄	C₂²	semidirect
50. #65817	C₂²	semidirect
51. C₂×#4024	C₂²	semidirect
65. C₂²×C₅×D₄	C₂³	semidirect
71. C₂²×C₂²⋊D₅	C₂³	semidirect
74. C₂³×Dic₅	C₂³	semidirect
76. C₂×#711	C₂³	semidirect
80. C₂²×C₄×D₅	C₂³	semidirect
81. C₂²×Dic₁₀	C₂³	semidirect
82. C₅×#123	C₂³	semidirect
83. C₅×#121	C₂³	semidirect
84. C₅×#120	C₂³	semidirect
85. C₂×C₅×C₄⋊C₄	C₂³	semidirect
88. C₂×C₅×C₂²⋊C₄	C₂³	semidirect
94. C₂×C₂²⋊Dic₅	C₂³	semidirect
95. C₂×C₂²⋊Dic₅	C₂³	semidirect
96. #4085	C₂³	semidirect
97. #4084	C₂³	semidirect
99. #4083	C₂³	semidirect
100. #4083	C₂³	semidirect
101. D₄×Dic₅	C₂³	semidirect
102. D₄×Dic₅	C₂³	semidirect
106. C₂×#685	C₂³	semidirect
107. C₂×#685	C₂³	semidirect
108. #4072	C₂³	semidirect
109. C₂×C₄⋊Dic₅	C₂³	semidirect
110. C₂×C₄⋊Dic₅	C₂³	semidirect
111. #4070	C₂³	semidirect
112. C₂×#683	C₂³	semidirect
113. C₂×C₄×Dic₅	C₂³	semidirect
114. #4044	C₂³	semidirect
115. #4043	C₂³	semidirect
116. #4038	C₂³	semidirect
117. #4032	C₂³	semidirect
118. #4031	C₂³	semidirect
119. #4029	C₂³	semidirect
120. #4027	C₂³	semidirect
121. D₅×C₂²⋊C₄	C₂³	semidirect
122. #4024	C₂³	semidirect
128. C₂×#534	D₅	semidirect
136. C₂⁴×C₅	#41	semidirect
163. C₂×C₅×D₄	C₂⁴	semidirect
182. C₂²×Dic₅	C₂⁴	semidirect
190. C₂×Dic₁₀	C₂⁴	semidirect
191. C₅×C₄⋊C₄	C₂⁴	semidirect
193. C₅×C₂²⋊C₄	C₂⁴	semidirect
197. C₂²⋊Dic₅	C₂⁴	semidirect
201. C₄⋊Dic₅	C₂⁴	semidirect
202. C₄⋊Dic₅	C₂⁴	semidirect
203. #683	C₂⁴	semidirect
204. C₄×Dic₅	C₂⁴	semidirect
214. #534	C₂×D₅	semidirect
220. C₂×#123	C₂×D₅	semidirect
226. C₂×#121	C₂×D₅	semidirect
231. C₂×#120	C₂×D₅	semidirect
257. C₂³×C₅	C₂×#41	semidirect
320. C₂²×D₄	C₂²×D₅	semidirect
334. #123	C₂²×D₅	semidirect
340. #121	C₂²×D₅	semidirect
344. #120	C₂²×D₅	semidirect
353. C₂×C₄⋊C₄	C₂²×D₅	semidirect
369. C₂×C₂²⋊C₄	C₂²×D₅	semidirect
417. C₂⁴	#711	semidirect
437. C₂×D₄	C₂³×D₅	semidirect
460. C₄⋊C₄	C₂³×D₅	semidirect
469. C₂²⋊C₄	C₂³×D₅	semidirect
474. C₂×C₅	#3490	semidirect
508. C₂³	C₂×#711	semidirect
534. C₅	C₂×#3490	semidirect
568. C₂	#10618210	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×#534	2
C₂²×#4083	2
C₂×#66132	2
#10618210	2
C₂×#65992	2
C₂×#65951	2
C₂×#65940	2
C₂×#65925	2
C₂²×D₅×C₂²⋊C₄	2
C₂×#65817	2
C₂×#3490	5

Subgroups — C2×#10618210 #13007212

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10618210 #13007212