Subgroups — C₂×#10617768 #13007184

Order 1280.

14438 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#4144	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×D₄×D₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#518	C₂	1	1	yes	maximal, characteristic, complemented
4	#10617768	C₂	1	32	yes	maximal, direct product factor
5	C₂×#65888	C₂	1	8	yes	maximal, complemented
6	C₂×D₄×D₂₀	C₂	1	8	yes	maximal, complemented
7	C₂×#65859	C₂	1	2	yes	maximal, complemented
8	C₂×#65858	C₂	1	2	yes	maximal, complemented
9	C₂×#65827	C₂	1	4	yes	maximal, complemented
10	C₂²×#4028	C₂	1	4	yes	maximal, complemented
11	C₂⁵×D₅	C₂²	1	2	yes	complemented
12	C₂³×C₅×D₄	C₂²	1	1	yes	characteristic
13	C₂³×C₂²⋊D₅	C₂²	1	2	yes	complemented
14	C₂³×C₂²⋊D₅	C₂²	1	4	yes	complemented
15	C₂×#4144	—	2	16	no	—
16	C₂×#4144	C₂²	1	16	yes	complemented
17	C₂²×#711	C₂²	1	4	yes	complemented
18	C₂²×D₄×D₅	—	2	16	no	—
19	C₂²×D₄×D₅	C₂²	1	8	yes	complemented
20	C₂²×D₄×D₅	C₂²	1	16	yes	complemented
21	C₂²×D₄×D₅	C₂²	1	4	yes	complemented
22	C₂²×#709	C₂²	1	2	yes	complemented
23	C₂³×D₂₀	C₂²	1	1	yes	characteristic, complemented
24	C₂³×D₂₀	C₂²	1	8	yes	complemented
25	C₂³×C₄×D₅	C₂²	1	1	yes	characteristic, complemented
26	C₅×#518	C₂²	1	16	yes	complemented
27	C₂×C₄×C₅×D₄	C₂²	1	8	yes	complemented
28	C₂×C₅×#117	C₂²	1	2	yes	complemented
29	C₂²×C₅×C₂²⋊C₄	C₂²	1	4	yes	complemented
30	C₂×#4076	C₂²	1	16	yes	complemented
31	C₂²×#685	C₂²	1	8	yes	complemented
32	#65888	C₂²	1	128	yes	complemented
33	D₄×D₂₀	C₂²	1	128	yes	complemented
34	#65859	C₂²	1	32	yes	complemented
35	#65858	C₂²	1	32	yes	complemented
36	C₂×#4043	C₂²	1	4	yes	complemented
37	C₂×#4041	C₂²	1	4	yes	complemented
38	C₂×#4041	C₂²	1	8	yes	complemented
39	#65827	C₂²	1	64	yes	complemented
40	C₂×#4032	C₂²	1	4	yes	complemented
41	C₂×#4032	C₂²	1	8	yes	complemented
42	C₂×#4028	C₂²	1	4	yes	complemented
43	C₂×#4028	C₂²	1	8	yes	complemented
44	C₂×#4028	C₂²	1	16	yes	complemented
45	C₂×#4028	C₂²	1	8	yes	complemented
46	C₂×#4028	C₂²	1	64	yes	complemented
47	C₂×#4021	C₂²	1	4	yes	complemented
48	C₂×#4020	C₂²	1	4	yes	complemented
49	C₂×C₄×D₂₀	C₂²	1	8	yes	complemented
50	C₂×#3484	—	5	1	no	maximal, Sylow
51	C₂⁵×C₅	C₂³	1	2	yes	—
52	C₂⁴×D₅	—	2	16	no	—
53	C₂⁴×D₅	—	4	16	no	—
54	C₂⁴×D₅	C₂³	1	1	yes	characteristic
55	C₂⁴×D₅	C₂³	1	8	yes	—
56	C₂⁴×D₅	C₂³	1	2	yes	—
57	C₂⁴×D₅	C₂³	1	16	yes	complemented
58	C₂⁴×D₅	C₂³	1	4	yes	complemented
59	C₂²×C₅×D₄	D₄	1	16	yes	complemented
60	C₂²×C₅×D₄	C₂³	1	4	yes	—
61	C₂²×C₅×D₄	C₂³	1	8	yes	—
62	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
63	C₂³×C₄×C₅	C₂³	1	8	yes	complemented
64	C₂²×C₂²⋊D₅	—	2	16	no	—
65	C₂²×C₂²⋊D₅	—	2	32	no	—
66	C₂²×C₂²⋊D₅	C₂³	1	4	yes	—
67	C₂²×C₂²⋊D₅	C₂³	1	8	yes	—
68	C₂²×C₂²⋊D₅	C₂³	1	16	yes	complemented
69	C₂²×C₂²⋊D₅	C₂³	1	8	yes	complemented
70	C₂²×C₂²⋊D₅	C₂³	1	32	yes	complemented
71	C₂³×Dic₅	C₂³	1	1	yes	characteristic
72	C₂³×Dic₅	C₂³	1	4	yes	complemented
73	#4144	—	2	128	no	—
74	C₂×#711	—	2	32	no	—
75	C₂×#711	C₂³	1	32	yes	complemented
76	C₂×D₄×D₅	—	2	64	no	—
77	C₂×D₄×D₅	—	2	128	no	—
78	C₂×D₄×D₅	—	2	32	no	—
79	C₂×D₄×D₅	C₂³	1	64	yes	complemented
80	C₂×D₄×D₅	C₂³	1	32	yes	complemented
81	C₂×#709	—	2	16	no	—
82	C₂×#709	C₂³	1	16	yes	complemented
83	C₂²×D₂₀	—	2	8	no	—
84	C₂²×D₂₀	—	4	32	no	—
85	C₂²×D₂₀	C₂³	1	8	yes	—
86	C₂²×D₂₀	C₂³	1	4	yes	—
87	C₂²×D₂₀	C₂³	1	8	yes	complemented
88	C₂²×D₂₀	C₂³	1	2	yes	complemented
89	C₂²×D₂₀	C₂³	1	64	yes	complemented
90	C₂²×D₂₀	C₂³	1	16	yes	complemented
91	C₂²×C₄×D₅	—	2	8	no	—
92	C₂²×C₄×D₅	C₂³	1	4	yes	—
93	C₂²×C₄×D₅	C₂³	1	8	yes	complemented
94	C₂²×C₄×D₅	C₂³	1	4	yes	complemented
95	C₂²×Dic₁₀	C₂³	1	2	yes	complemented
96	C₄×C₅×D₄	C₂³	1	64	yes	complemented
97	C₅×#117	C₂³	1	16	yes	complemented
98	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	complemented
99	C₂×C₅×C₂²⋊C₄	C₂³	1	4	yes	—
100	C₂×C₅×C₂²⋊C₄	C₂³	1	32	yes	complemented
101	C₂×C₅×C₂²⋊C₄	C₂³	1	8	yes	complemented
102	C₂×C₄²×C₅	C₂³	1	4	yes	complemented
103	#4076	C₂³	1	128	yes	complemented
104	C₂×#685	C₂³	1	8	yes	—
105	C₂×#685	C₂³	1	64	yes	complemented
106	C₂×#685	C₂³	1	8	yes	complemented
107	C₂×#685	C₂³	1	16	yes	complemented
108	C₂×C₄⋊Dic₅	C₂³	1	8	yes	complemented
109	#4043	C₂³	1	32	yes	complemented
110	#4041	C₂³	1	32	yes	complemented
111	#4041	C₂³	1	64	yes	complemented
112	#4032	C₂³	1	32	yes	complemented
113	#4032	C₂³	1	64	yes	complemented
114	#4028	C₂³	1	32	yes	complemented
115	#4028	C₂³	1	128	yes	complemented
116	#4028	C₂³	1	64	yes	complemented
117	#4028	C₂³	1	64	yes	complemented
118	#4021	C₂³	1	32	yes	complemented
119	#4020	C₂³	1	32	yes	complemented
120	C₄×D₂₀	C₂³	1	64	yes	complemented
121	C₂²×H(5,𝔽₂)	—	5	1	no	—
122	C₂⁴×D₄	—	5	1	no	—
123	#3484	—	5	32	no	—
124	C₂×#546	—	5	8	no	—
125	C₂×D₄²	—	5	8	no	—
126	C₂×#535	—	5	2	no	—
127	C₂×#534	—	5	4	no	—
128	C₂×#525	—	5	2	no	—
129	C₂²×#120	—	5	4	no	—
130	C₂×#518	D₅	1	1	yes	characteristic, complemented
131	C₂⁴×C₅	—	4	8	no	—
132	C₂⁴×C₅	C₂×D₄	1	16	yes	complemented
133	C₂⁴×C₅	C₂⁴	1	1	yes	characteristic
134	C₂⁴×C₅	C₂⁴	1	4	yes	—
135	C₂⁴×C₅	C₂⁴	1	8	yes	—
136	C₂³×D₅	—	2	4	no	—
137	C₂³×D₅	—	2	8	no	—
138	C₂³×D₅	—	2	8	no	—
139	C₂³×D₅	—	2	32	no	—
140	C₂³×D₅	—	2	64	no	—
141	C₂³×D₅	—	2	16	no	—
142	C₂³×D₅	—	2	32	no	—
143	C₂³×D₅	—	4	64	no	—
144	C₂³×D₅	—	4	16	no	—
145	C₂³×D₅	—	4	8	no	—
146	C₂³×D₅	—	4	16	no	—
147	C₂³×D₅	—	4	128	no	—
148	C₂³×D₅	C₂⁴	1	4	yes	—
149	C₂³×D₅	C₂⁴	1	4	yes	—
150	C₂³×D₅	C₂⁴	1	4	yes	—
151	C₂³×D₅	C₂⁴	1	32	yes	—
152	C₂³×D₅	C₂⁴	1	2	yes	—
153	C₂³×D₅	C₂⁴	1	8	yes	—
154	C₂³×D₅	C₂⁴	1	16	yes	complemented
155	C₂×C₅×D₄	C₂×D₄	1	32	yes	complemented
156	C₂×C₅×D₄	C₂×D₄	1	64	yes	complemented
157	C₂×C₅×D₄	C₂⁴	1	16	yes	—
158	C₂²×C₄×C₅	—	4	16	no	—
159	C₂²×C₄×C₅	C₂×D₄	1	8	yes	complemented
160	C₂²×C₄×C₅	C₂⁴	1	8	yes	—
161	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
162	C₂²×C₄×C₅	C₂⁴	1	4	yes	—
163	C₂²×C₄×C₅	C₂⁴	1	32	yes	complemented
164	C₂×C₂²⋊D₅	—	2	16	no	—
165	C₂×C₂²⋊D₅	—	2	32	no	—
166	C₂×C₂²⋊D₅	—	2	32	no	—
167	C₂×C₂²⋊D₅	—	2	64	no	—
168	C₂×C₂²⋊D₅	—	2	128	no	—
169	C₂×C₂²⋊D₅	C₂⁴	1	16	yes	—
170	C₂×C₂²⋊D₅	C₂⁴	1	32	yes	—
171	C₂×C₂²⋊D₅	C₂⁴	1	32	yes	complemented
172	C₂²×Dic₅	—	2	4	no	—
173	C₂²×Dic₅	—	2	16	no	—
174	C₂²×Dic₅	C₂⁴	1	4	yes	—
175	C₂²×Dic₅	C₂⁴	1	2	yes	—
176	C₂²×Dic₅	C₂⁴	1	4	yes	—
177	C₂²×Dic₅	C₂⁴	1	16	yes	complemented
178	#711	—	2	128	no	—
179	D₄×D₅	—	2	256	no	—
180	D₄×D₅	—	2	128	no	—
181	#709	—	2	64	no	—
182	C₂×D₂₀	—	2	32	no	—
183	C₂×D₂₀	—	2	16	no	—
184	C₂×D₂₀	—	2	8	no	—
185	C₂×D₂₀	—	2	32	no	—
186	C₂×D₂₀	—	4	128	no	—
187	C₂×D₂₀	—	4	32	no	—
188	C₂×D₂₀	C₂⁴	1	32	yes	—
189	C₂×D₂₀	C₂⁴	1	16	yes	—
190	C₂×D₂₀	C₂⁴	1	8	yes	complemented
191	C₂×D₂₀	C₂⁴	1	64	yes	complemented
192	C₂×C₄×D₅	—	2	16	no	—
193	C₂×C₄×D₅	—	2	16	no	—
194	C₂×C₄×D₅	—	2	32	no	—
195	C₂×C₄×D₅	C₂⁴	1	16	yes	—
196	C₂×C₄×D₅	C₂⁴	1	16	yes	complemented
197	C₂×Dic₁₀	—	2	8	no	—
198	C₂×Dic₁₀	C₂⁴	1	8	yes	complemented
199	C₅×C₄⋊C₄	C₂⁴	1	16	yes	complemented
200	C₅×C₂²⋊C₄	C₂⁴	1	16	yes	—
201	C₅×C₂²⋊C₄	C₂⁴	1	32	yes	complemented
202	C₄²×C₅	C₂⁴	1	16	yes	complemented
203	#685	C₂⁴	1	32	yes	—
204	#685	C₂⁴	1	32	yes	complemented
205	#685	C₂⁴	1	64	yes	complemented
206	C₄⋊Dic₅	C₂⁴	1	32	yes	complemented
207	C₂⁶	—	5	2	no	—
208	C₂×H(5,𝔽₂)	—	5	16	no	—
209	C₂×H(5,𝔽₂)	—	10	16	no	—
210	C₂²×#41	—	5	4	no	—
211	C₂²×#41	—	5	2	no	—
212	C₂³×D₄	—	5	2	no	—
213	C₂³×D₄	—	5	4	no	—
214	C₂³×D₄	—	5	8	no	—
215	C₂³×D₄	—	5	16	no	—
216	C₂³×D₄	—	5	4	no	—
217	C₂³×D₄	—	5	1	no	—
218	C₂³×D₄	—	5	8	no	—
219	C₂³×D₄	—	10	16	no	—
220	C₂³×D₄	C₂×D₅	1	1	yes	characteristic
221	C₂⁴×C₄	—	5	1	no	—
222	#546	—	5	128	no	—
223	D₄²	—	5	128	no	—
224	#535	—	5	32	no	—
225	#534	—	5	64	no	—
226	C₂×#127	—	5	4	no	—
227	C₂×#124	—	5	4	no	—
228	#525	—	5	32	no	—
229	C₂×#123	—	5	4	no	—
230	C₂×#123	—	5	8	no	—
231	C₂×C₂²⋊Q₈	—	5	4	no	—
232	C₂×#121	—	5	16	no	—
233	C₂×#121	—	5	4	no	—
234	C₂×#121	—	5	8	no	—
235	C₂×#120	—	5	4	no	—
236	C₂×#120	—	5	8	no	—
237	C₂×#120	—	5	16	no	—
238	C₂×#120	—	5	64	no	—
239	C₂×#120	—	5	8	no	—
240	#518	C₂×D₅	1	16	yes	complemented
241	C₂×C₄×D₄	—	5	8	no	—
242	C₂×C₄×D₄	C₂×D₅	1	8	yes	complemented
243	C₂×#117	C₂×D₅	1	2	yes	complemented
244	C₂²×C₂²⋊C₄	—	5	8	no	—
245	C₂²×C₂²⋊C₄	C₂×D₅	1	4	yes	complemented
246	C₂³×C₅	—	2	16	no	—
247	C₂³×C₅	—	2	4	no	—
248	C₂³×C₅	—	4	32	no	—
249	C₂³×C₅	—	4	16	no	—
250	C₂³×C₅	C₂²×D₄	1	4	yes	—
251	C₂³×C₅	C₂²×D₄	1	16	yes	—
252	C₂³×C₅	C₂²×D₄	1	32	yes	complemented
253	C₂³×C₅	C₂⁵	1	2	yes	—
254	C₂³×C₅	C₂⁵	1	8	yes	—
255	C₂³×C₅	C₂⁵	1	1	yes	characteristic
256	C₂²×D₅	—	2	32	no	—
257	C₂²×D₅	—	2	8	no	—
258	C₂²×D₅	—	2	4	no	—
259	C₂²×D₅	—	2	4	no	—
260	C₂²×D₅	—	2	16	no	—
261	C₂²×D₅	—	2	8	no	—
262	C₂²×D₅	—	2	16	no	—
263	C₂²×D₅	—	2	8	no	—
264	C₂²×D₅	—	2	64	no	—
265	C₂²×D₅	—	2	16	no	—
266	C₂²×D₅	—	2	16	no	—
267	C₂²×D₅	—	2	32	no	—
268	C₂²×D₅	—	2	64	no	—
269	C₂²×D₅	—	4	32	no	—
270	C₂²×D₅	—	4	16	no	—
271	C₂²×D₅	—	4	32	no	—
272	C₂²×D₅	—	4	128	no	—
273	C₂²×D₅	—	4	256	no	—
274	C₂²×D₅	—	4	32	no	—
275	C₂²×D₅	—	8	32	no	—
276	C₂²×D₅	H(5,𝔽₂)	1	8	yes	complemented
277	C₂²×D₅	C₂⁵	1	8	yes	—
278	C₂²×D₅	C₂⁵	1	8	yes	—
279	C₂²×D₅	C₂⁵	1	4	yes	—
280	C₅×D₄	C₂²×D₄	1	64	yes	complemented
281	C₂×C₄×C₅	—	2	16	no	—
282	C₂×C₄×C₅	—	4	32	no	—
283	C₂×C₄×C₅	C₂²×D₄	1	8	yes	—
284	C₂×C₄×C₅	C₂²×D₄	1	16	yes	complemented
285	C₂×C₄×C₅	C₂⁵	1	16	yes	—
286	C₂×C₄×C₅	C₂⁵	1	4	yes	—
287	C₂²⋊D₅	—	2	32	no	—
288	C₂²⋊D₅	—	2	64	no	—
289	C₂²⋊D₅	—	2	64	no	—
290	C₂×Dic₅	—	2	8	no	—
291	C₂×Dic₅	—	2	8	no	—
292	C₂×Dic₅	—	2	4	no	—
293	C₂×Dic₅	—	2	32	no	—
294	C₂×Dic₅	C₂⁵	1	4	yes	—
295	C₂×Dic₅	C₂⁵	1	8	yes	—
296	D₂₀	—	2	64	no	—
297	D₂₀	—	2	32	no	—
298	D₂₀	—	2	16	no	—
299	D₂₀	—	4	64	no	—
300	C₄×D₅	—	2	32	no	—
301	C₄×D₅	—	2	32	no	—
302	Dic₁₀	—	2	16	no	—
303	C₂⁵	—	5	1	no	—
304	C₂⁵	—	5	8	no	—
305	C₂⁵	—	5	2	no	—
306	C₂⁵	—	5	16	no	—
307	C₂⁵	—	5	4	no	—
308	C₂⁵	—	10	16	no	—
309	C₂⁵	—	20	16	no	—
310	C₂⁵	C₂²×D₅	1	2	yes	—
311	H(5,𝔽₂)	—	10	128	no	—
312	C₂×#41	—	5	32	no	—
313	C₂×#41	—	5	16	no	—
314	C₂×#41	—	10	32	no	—
315	C₂×#41	—	10	16	no	—
316	C₂²×Q₈	—	5	2	no	—
317	C₂²×D₄	—	5	4	no	—
318	C₂²×D₄	—	5	8	no	—
319	C₂²×D₄	—	5	32	no	—
320	C₂²×D₄	—	5	8	no	—
321	C₂²×D₄	—	5	16	no	—
322	C₂²×D₄	—	5	32	no	—
323	C₂²×D₄	—	5	64	no	—
324	C₂²×D₄	—	5	8	no	—
325	C₂²×D₄	—	5	4	no	—
326	C₂²×D₄	—	5	2	no	—
327	C₂²×D₄	—	5	8	no	—
328	C₂²×D₄	—	5	16	no	—
329	C₂²×D₄	—	5	64	no	—
330	C₂²×D₄	—	10	16	no	—
331	C₂²×D₄	—	10	32	no	—
332	C₂²×D₄	—	10	64	no	—
333	C₂²×D₄	—	10	128	no	—
334	C₂²×D₄	—	10	32	no	—
335	C₂²×D₄	—	10	8	no	—
336	C₂²×D₄	—	20	32	no	—
337	C₂²×D₄	D₂₀	1	16	yes	complemented
338	C₂²×D₄	C₂²×D₅	1	4	yes	—
339	C₂²×D₄	C₂²×D₅	1	8	yes	—
340	C₂³×C₄	—	5	4	no	—
341	C₂³×C₄	—	5	1	no	—
342	C₂³×C₄	—	5	4	no	—
343	C₂³×C₄	—	5	4	no	—
344	C₂³×C₄	—	5	8	no	—
345	C₂³×C₄	—	10	8	no	—
346	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
347	C₂³×C₄	C₂²×D₅	1	8	yes	complemented
348	#127	—	5	32	no	—
349	#124	—	5	32	no	—
350	#123	—	5	32	no	—
351	#123	—	5	64	no	—
352	C₂²⋊Q₈	—	5	32	no	—
353	#121	—	5	128	no	—
354	#121	—	5	32	no	—
355	#121	—	5	64	no	—
356	#120	—	5	32	no	—
357	#120	—	5	64	no	—
358	#120	—	5	128	no	—
359	#120	—	5	64	no	—
360	C₄×D₄	—	5	64	no	—
361	C₄×D₄	C₂²×D₅	1	64	yes	complemented
362	#117	C₂²×D₅	1	16	yes	complemented
363	C₂×C₄⋊C₄	—	5	8	no	—
364	C₂×C₄⋊C₄	C₂²×D₅	1	4	yes	complemented
365	C₂×C₂²⋊C₄	—	5	8	no	—
366	C₂×C₂²⋊C₄	—	5	16	no	—
367	C₂×C₂²⋊C₄	—	5	8	no	—
368	C₂×C₂²⋊C₄	—	5	64	no	—
369	C₂×C₂²⋊C₄	C₂²×D₅	1	4	yes	—
370	C₂×C₂²⋊C₄	C₂²×D₅	1	32	yes	complemented
371	C₂×C₂²⋊C₄	C₂²×D₅	1	8	yes	complemented
372	C₂×C₄²	C₂²×D₅	1	4	yes	complemented
373	C₂²×C₅	—	2	16	no	—
374	C₂²×C₅	—	2	4	no	—
375	C₂²×C₅	—	2	8	no	—
376	C₂²×C₅	—	2	32	no	—
377	C₂²×C₅	—	4	32	no	—
378	C₂²×C₅	C₂³×D₄	1	4	yes	—
379	C₂²×C₅	C₂³×D₄	1	16	yes	—
380	C₂²×C₅	C₂³×D₄	1	2	yes	—
381	C₂²×C₅	C₂×H(5,𝔽₂)	1	4	yes	—
382	C₂²×C₅	C₂⁶	1	1	yes	characteristic
383	C₂×D₅	—	2	32	no	—
384	C₂×D₅	—	2	4	no	—
385	C₂×D₅	—	2	4	no	—
386	C₂×D₅	—	2	16	no	—
387	C₂×D₅	—	2	8	no	—
388	C₂×D₅	—	2	8	no	—
389	C₂×D₅	—	2	16	no	—
390	C₂×D₅	—	2	16	no	—
391	C₂×D₅	—	4	128	no	—
392	C₂×D₅	—	4	32	no	—
393	C₂×D₅	—	4	16	no	—
394	C₂×D₅	—	4	16	no	—
395	C₂×D₅	—	8	32	no	—
396	C₂×D₅	C₂×H(5,𝔽₂)	1	8	yes	complemented
397	C₄×C₅	—	2	16	no	—
398	C₄×C₅	C₂³×D₄	1	8	yes	—
399	Dic₅	—	2	8	no	—
400	Dic₅	—	2	4	no	—
401	C₂⁴	—	4	8	no	—
402	C₂⁴	—	5	2	no	—
403	C₂⁴	—	5	4	no	—
404	C₂⁴	—	5	4	no	—
405	C₂⁴	—	5	4	no	—
406	C₂⁴	—	5	32	no	—
407	C₂⁴	—	5	8	no	—
408	C₂⁴	—	5	16	no	—
409	C₂⁴	—	10	8	no	—
410	C₂⁴	—	10	32	no	—
411	C₂⁴	—	10	4	no	—
412	C₂⁴	—	10	32	no	—
413	C₂⁴	—	10	8	no	—
414	C₂⁴	—	10	64	no	—
415	C₂⁴	—	10	16	no	—
416	C₂⁴	—	20	16	no	—
417	C₂⁴	—	20	8	no	—
418	C₂⁴	—	20	16	no	—
419	C₂⁴	—	20	64	no	—
420	C₂⁴	—	20	128	no	—
421	C₂⁴	C₂×D₂₀	1	16	yes	complemented
422	C₂⁴	C₂³×D₅	1	1	yes	characteristic
423	C₂⁴	C₂³×D₅	1	4	yes	—
424	C₂⁴	C₂³×D₅	1	8	yes	—
425	#41	—	10	128	no	—
426	#41	—	10	64	no	—
427	C₂×Q₈	—	5	8	no	—
428	C₂×Q₈	—	10	8	no	—
429	C₂×D₄	—	5	16	no	—
430	C₂×D₄	—	5	32	no	—
431	C₂×D₄	—	5	32	no	—
432	C₂×D₄	—	5	32	no	—
433	C₂×D₄	—	5	16	no	—
434	C₂×D₄	—	5	8	no	—
435	C₂×D₄	—	5	64	no	—
436	C₂×D₄	—	10	32	no	—
437	C₂×D₄	—	10	32	no	—
438	C₂×D₄	—	10	16	no	—
439	C₂×D₄	—	10	32	no	—
440	C₂×D₄	—	10	64	no	—
441	C₂×D₄	—	10	128	no	—
442	C₂×D₄	—	10	256	no	—
443	C₂×D₄	—	10	128	no	—
444	C₂×D₄	—	10	16	no	—
445	C₂×D₄	—	10	8	no	—
446	C₂×D₄	—	10	32	no	—
447	C₂×D₄	—	20	32	no	—
448	C₂×D₄	—	20	128	no	—
449	C₂×D₄	C₂×D₂₀	1	32	yes	complemented
450	C₂×D₄	C₂×D₂₀	1	64	yes	complemented
451	C₂×D₄	C₂³×D₅	1	16	yes	—
452	C₂²×C₄	—	4	16	no	—
453	C₂²×C₄	—	5	16	no	—
454	C₂²×C₄	—	5	4	no	—
455	C₂²×C₄	—	5	4	no	—
456	C₂²×C₄	—	5	2	no	—
457	C₂²×C₄	—	5	16	no	—
458	C₂²×C₄	—	5	16	no	—
459	C₂²×C₄	—	10	4	no	—
460	C₂²×C₄	—	10	16	no	—
461	C₂²×C₄	—	10	16	no	—
462	C₂²×C₄	—	10	16	no	—
463	C₂²×C₄	—	10	32	no	—
464	C₂²×C₄	C₂×D₂₀	1	8	yes	complemented
465	C₂²×C₄	C₂³×D₅	1	8	yes	—
466	C₂²×C₄	C₂³×D₅	1	2	yes	—
467	C₂²×C₄	C₂³×D₅	1	4	yes	—
468	C₂²×C₄	C₂³×D₅	1	32	yes	complemented
469	C₄⋊C₄	—	5	32	no	—
470	C₄⋊C₄	C₂³×D₅	1	16	yes	complemented
471	C₂²⋊C₄	—	5	32	no	—
472	C₂²⋊C₄	—	5	64	no	—
473	C₂²⋊C₄	—	5	32	no	—
474	C₂²⋊C₄	C₂³×D₅	1	16	yes	—
475	C₂²⋊C₄	C₂³×D₅	1	32	yes	complemented
476	C₄²	C₂³×D₅	1	16	yes	complemented
477	C₂×C₅	—	2	4	no	—
478	C₂×C₅	—	2	16	no	—
479	C₂×C₅	#3484	1	4	yes	complemented
480	C₂×C₅	C₂⁴×D₄	1	1	yes	characteristic
481	C₂×C₅	C₂²×H(5,𝔽₂)	1	2	yes	—
482	D₅	—	2	8	no	—
483	D₅	—	4	8	no	—
484	D₅	—	4	8	no	—
485	C₂³	—	2	16	no	—
486	C₂³	—	2	4	no	—
487	C₂³	—	4	32	no	—
488	C₂³	—	4	16	no	—
489	C₂³	—	5	8	no	—
490	C₂³	—	5	4	no	—
491	C₂³	—	5	8	no	—
492	C₂³	—	5	8	no	—
493	C₂³	—	10	16	no	—
494	C₂³	—	10	64	no	—
495	C₂³	—	10	32	no	—
496	C₂³	—	10	8	no	—
497	C₂³	—	10	4	no	—
498	C₂³	—	10	4	no	—
499	C₂³	—	10	8	no	—
500	C₂³	—	10	64	no	—
501	C₂³	—	10	16	no	—
502	C₂³	—	10	8	no	—
503	C₂³	—	10	16	no	—
504	C₂³	—	10	16	no	—
505	C₂³	—	10	32	no	—
506	C₂³	—	20	32	no	—
507	C₂³	—	20	16	no	—
508	C₂³	—	20	32	no	—
509	C₂³	—	20	128	no	—
510	C₂³	—	20	32	no	—
511	C₂³	—	20	256	no	—
512	C₂³	—	40	32	no	—
513	C₂³	C₂²×D₂₀	1	4	yes	—
514	C₂³	C₂²×D₂₀	1	16	yes	—
515	C₂³	C₂²×D₂₀	1	32	yes	complemented
516	C₂³	C₂⁴×D₅	1	2	yes	—
517	C₂³	C₂⁴×D₅	1	8	yes	—
518	C₂³	C₂⁴×D₅	1	1	yes	characteristic
519	Q₈	—	10	16	no	—
520	D₄	—	10	64	no	—
521	D₄	—	10	64	no	—
522	D₄	—	10	32	no	—
523	D₄	—	10	64	no	—
524	D₄	—	10	32	no	—
525	D₄	—	10	16	no	—
526	D₄	—	20	64	no	—
527	D₄	C₂²×D₂₀	1	64	yes	complemented
528	C₂×C₄	—	2	16	no	—
529	C₂×C₄	—	4	32	no	—
530	C₂×C₄	—	5	8	no	—
531	C₂×C₄	—	5	4	no	—
532	C₂×C₄	—	10	8	no	—
533	C₂×C₄	—	10	4	no	—
534	C₂×C₄	—	10	8	no	—
535	C₂×C₄	—	10	32	no	—
536	C₂×C₄	—	10	32	no	—
537	C₂×C₄	—	10	32	no	—
538	C₂×C₄	C₂²×D₂₀	1	8	yes	—
539	C₂×C₄	C₂²×D₂₀	1	16	yes	complemented
540	C₂×C₄	C₂⁴×D₅	1	16	yes	—
541	C₂×C₄	C₂⁴×D₅	1	4	yes	—
542	C₅	C₂×#3484	1	1	yes	characteristic, complemented, Sylow
543	C₂²	—	2	16	no	—
544	C₂²	—	2	4	no	—
545	C₂²	—	2	8	no	—
546	C₂²	—	2	32	no	—
547	C₂²	—	4	32	no	—
548	C₂²	—	5	8	no	—
549	C₂²	—	10	32	no	—
550	C₂²	—	10	4	no	—
551	C₂²	—	10	4	no	—
552	C₂²	—	10	16	no	—
553	C₂²	—	10	8	no	—
554	C₂²	—	10	8	no	—
555	C₂²	—	10	16	no	—
556	C₂²	—	10	16	no	—
557	C₂²	—	20	128	no	—
558	C₂²	—	20	32	no	—
559	C₂²	—	20	16	no	—
560	C₂²	—	20	16	no	—
561	C₂²	—	40	32	no	—
562	C₂²	C₂³×D₂₀	1	4	yes	—
563	C₂²	C₂³×D₂₀	1	16	yes	—
564	C₂²	C₂³×D₂₀	1	2	yes	—
565	C₂²	D₅×H(5,𝔽₂)	1	4	yes	—
566	C₂²	C₂⁵×D₅	1	1	yes	characteristic
567	C₄	—	2	16	no	—
568	C₄	—	10	4	no	—
569	C₄	—	10	8	no	—
570	C₄	C₂³×D₂₀	1	8	yes	—
571	C₂	—	2	4	no	—
572	C₂	—	2	16	no	—
573	C₂	—	10	8	no	—
574	C₂	—	20	8	no	—
575	C₂	—	20	8	no	—
576	C₂	#10617768	1	4	yes	direct product factor
577	C₂	C₂⁴×D₂₀	1	1	yes	characteristic
578	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×#518	C₂
socle	C₂³×C₅	C₂⁵
hypercenter	C₂×#518	D₅
hypocenter	C₅	C₂×#3484
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×#518

Complements

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

N	H ≅ G/N	type
1. C₂²×#4144	C₂	semidirect
2. C₂³×D₄×D₅	C₂	semidirect
3. C₂×C₅×#518	C₂	semidirect
4. #10617768	C₂	direct (N × H)
5. C₂×#65888	C₂	semidirect
6. C₂×D₄×D₂₀	C₂	semidirect
7. C₂×#65859	C₂	semidirect
8. C₂×#65858	C₂	semidirect
9. C₂×#65827	C₂	semidirect
10. C₂²×#4028	C₂	semidirect
11. C₂⁵×D₅	C₂²	semidirect
13. C₂³×C₂²⋊D₅	C₂²	semidirect
14. C₂³×C₂²⋊D₅	C₂²	semidirect
16. C₂×#4144	C₂²	semidirect
17. C₂²×#711	C₂²	semidirect
19. C₂²×D₄×D₅	C₂²	semidirect
20. C₂²×D₄×D₅	C₂²	semidirect
21. C₂²×D₄×D₅	C₂²	semidirect
22. C₂²×#709	C₂²	semidirect
23. C₂³×D₂₀	C₂²	semidirect
24. C₂³×D₂₀	C₂²	semidirect
25. C₂³×C₄×D₅	C₂²	semidirect
26. C₅×#518	C₂²	semidirect
27. C₂×C₄×C₅×D₄	C₂²	semidirect
28. C₂×C₅×#117	C₂²	semidirect
29. C₂²×C₅×C₂²⋊C₄	C₂²	semidirect
30. C₂×#4076	C₂²	semidirect
31. C₂²×#685	C₂²	semidirect
32. #65888	C₂²	semidirect
33. D₄×D₂₀	C₂²	semidirect
34. #65859	C₂²	semidirect
35. #65858	C₂²	semidirect
36. C₂×#4043	C₂²	semidirect
37. C₂×#4041	C₂²	semidirect
38. C₂×#4041	C₂²	semidirect
39. #65827	C₂²	semidirect
40. C₂×#4032	C₂²	semidirect
41. C₂×#4032	C₂²	semidirect
42. C₂×#4028	C₂²	semidirect
43. C₂×#4028	C₂²	semidirect
44. C₂×#4028	C₂²	semidirect
45. C₂×#4028	C₂²	semidirect
46. C₂×#4028	C₂²	semidirect
47. C₂×#4021	C₂²	semidirect
48. C₂×#4020	C₂²	semidirect
49. C₂×C₄×D₂₀	C₂²	semidirect
57. C₂⁴×D₅	C₂³	semidirect
58. C₂⁴×D₅	C₂³	semidirect
59. C₂²×C₅×D₄	D₄	semidirect
63. C₂³×C₄×C₅	C₂³	semidirect
68. C₂²×C₂²⋊D₅	C₂³	semidirect
69. C₂²×C₂²⋊D₅	C₂³	semidirect
70. C₂²×C₂²⋊D₅	C₂³	semidirect
72. C₂³×Dic₅	C₂³	semidirect
75. C₂×#711	C₂³	semidirect
79. C₂×D₄×D₅	C₂³	semidirect
80. C₂×D₄×D₅	C₂³	semidirect
82. C₂×#709	C₂³	semidirect
87. C₂²×D₂₀	C₂³	semidirect
88. C₂²×D₂₀	C₂³	semidirect
89. C₂²×D₂₀	C₂³	semidirect
90. C₂²×D₂₀	C₂³	semidirect
93. C₂²×C₄×D₅	C₂³	semidirect
94. C₂²×C₄×D₅	C₂³	semidirect
95. C₂²×Dic₁₀	C₂³	semidirect
96. C₄×C₅×D₄	C₂³	semidirect
97. C₅×#117	C₂³	semidirect
98. C₂×C₅×C₄⋊C₄	C₂³	semidirect
100. C₂×C₅×C₂²⋊C₄	C₂³	semidirect
101. C₂×C₅×C₂²⋊C₄	C₂³	semidirect
102. C₂×C₄²×C₅	C₂³	semidirect
103. #4076	C₂³	semidirect
105. C₂×#685	C₂³	semidirect
106. C₂×#685	C₂³	semidirect
107. C₂×#685	C₂³	semidirect
108. C₂×C₄⋊Dic₅	C₂³	semidirect
109. #4043	C₂³	semidirect
110. #4041	C₂³	semidirect
111. #4041	C₂³	semidirect
112. #4032	C₂³	semidirect
113. #4032	C₂³	semidirect
114. #4028	C₂³	semidirect
115. #4028	C₂³	semidirect
116. #4028	C₂³	semidirect
117. #4028	C₂³	semidirect
118. #4021	C₂³	semidirect
119. #4020	C₂³	semidirect
120. C₄×D₂₀	C₂³	semidirect
130. C₂×#518	D₅	semidirect
132. C₂⁴×C₅	C₂×D₄	semidirect
154. C₂³×D₅	C₂⁴	semidirect
155. C₂×C₅×D₄	C₂×D₄	semidirect
156. C₂×C₅×D₄	C₂×D₄	semidirect
159. C₂²×C₄×C₅	C₂×D₄	semidirect
163. C₂²×C₄×C₅	C₂⁴	semidirect
171. C₂×C₂²⋊D₅	C₂⁴	semidirect
177. C₂²×Dic₅	C₂⁴	semidirect
190. C₂×D₂₀	C₂⁴	semidirect
191. C₂×D₂₀	C₂⁴	semidirect
196. C₂×C₄×D₅	C₂⁴	semidirect
198. C₂×Dic₁₀	C₂⁴	semidirect
199. C₅×C₄⋊C₄	C₂⁴	semidirect
201. C₅×C₂²⋊C₄	C₂⁴	semidirect
202. C₄²×C₅	C₂⁴	semidirect
204. #685	C₂⁴	semidirect
205. #685	C₂⁴	semidirect
206. C₄⋊Dic₅	C₂⁴	semidirect
240. #518	C₂×D₅	semidirect
242. C₂×C₄×D₄	C₂×D₅	semidirect
243. C₂×#117	C₂×D₅	semidirect
245. C₂²×C₂²⋊C₄	C₂×D₅	semidirect
252. C₂³×C₅	C₂²×D₄	semidirect
276. C₂²×D₅	H(5,𝔽₂)	semidirect
280. C₅×D₄	C₂²×D₄	semidirect
284. C₂×C₄×C₅	C₂²×D₄	semidirect
337. C₂²×D₄	D₂₀	semidirect
347. C₂³×C₄	C₂²×D₅	semidirect
361. C₄×D₄	C₂²×D₅	semidirect
362. #117	C₂²×D₅	semidirect
364. C₂×C₄⋊C₄	C₂²×D₅	semidirect
370. C₂×C₂²⋊C₄	C₂²×D₅	semidirect
371. C₂×C₂²⋊C₄	C₂²×D₅	semidirect
372. C₂×C₄²	C₂²×D₅	semidirect
396. C₂×D₅	C₂×H(5,𝔽₂)	semidirect
421. C₂⁴	C₂×D₂₀	semidirect
449. C₂×D₄	C₂×D₂₀	semidirect
450. C₂×D₄	C₂×D₂₀	semidirect
464. C₂²×C₄	C₂×D₂₀	semidirect
468. C₂²×C₄	C₂³×D₅	semidirect
470. C₄⋊C₄	C₂³×D₅	semidirect
475. C₂²⋊C₄	C₂³×D₅	semidirect
476. C₄²	C₂³×D₅	semidirect
479. C₂×C₅	#3484	semidirect
515. C₂³	C₂²×D₂₀	semidirect
527. D₄	C₂²×D₂₀	semidirect
539. C₂×C₄	C₂²×D₂₀	semidirect
542. C₅	C₂×#3484	semidirect
576. C₂	#10617768	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×#4144	2
C₂³×D₄×D₅	2
C₂×C₅×#518	2
#10617768	2
C₂×#65888	2
C₂×D₄×D₂₀	2
C₂×#65859	2
C₂×#65858	2
C₂×#65827	2
C₂²×#4028	2
C₂×#3484	5

Subgroups — C2×#10617768 #13007184

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10617768 #13007184