Subgroups — D₅²×H(5,𝔽₂) #72221785

Order 3200.

14018 conjugacy classes of subgroups.

The subgroup lattice has 551 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	H(5,𝔽₂)×C₅⋊D₅	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×D₅×H(5,𝔽₂)	C₂	1	2	yes	maximal, complemented
3	#13107838	C₂	1	6	yes	maximal, complemented
4	D₅×#4149	C₂	1	12	yes	maximal, complemented
5	D₅²×#41	C₂	1	6	yes	maximal, complemented
6	#13107821	C₂	1	9	yes	maximal, complemented
7	D₅×#4144	C₂	1	18	yes	maximal, complemented
8	C₂×D₄×D₅²	C₂	1	9	yes	maximal, complemented
9	C₂³×D₅²	C₂²	1	6	yes	complemented
10	C₅²×H(5,𝔽₂)	C₂²	1	1	yes	characteristic, complemented
11	#11717522	C₂²	1	6	yes	complemented
12	#41×C₅⋊D₅	C₂²	1	6	yes	complemented
13	#11717517	C₂²	1	9	yes	complemented
14	C₂×D₄×C₅⋊D₅	C₂²	1	9	yes	complemented
15	C₅×#4149	C₂²	1	12	yes	complemented
16	C₅×D₅×#41	C₂²	1	12	yes	complemented
17	C₅×#4144	C₂²	1	18	yes	complemented
18	C₂×C₅×D₄×D₅	C₂²	1	18	yes	complemented
19	#11717495	C₂²	1	36	yes	complemented
20	C₂×#87886	C₂²	1	18	yes	complemented
21	C₂×D₅×C₂²⋊D₅	C₂²	1	36	yes	complemented
22	#11717485	C₂²	1	12	yes	complemented
23	#11717484	C₂²	1	6	yes	complemented
24	D₅×#713	C₂²	1	12	yes	complemented
25	Q₈×D₅²	C₂²	1	2	yes	complemented
26	#11717479	C₂²	1	72	yes	complemented
27	#11717478	C₂²	1	36	yes	complemented
28	#11717477	C₂²	1	18	yes	complemented
29	D₅×#711	C₂²	1	36	yes	complemented
30	D₄×D₅²	C₂²	1	18	yes	complemented
31	D₄×D₅²	C₂²	1	72	yes	complemented
32	D₄×D₅²	C₂²	1	36	yes	complemented
33	#11717473	C₂²	1	18	yes	complemented
34	#11717472	C₂²	1	36	yes	complemented
35	#11717471	C₂²	1	18	yes	complemented
36	D₅×#709	C₂²	1	36	yes	complemented
37	C₂×#87877	C₂²	1	9	yes	complemented
38	C₂×D₅×D₂₀	C₂²	1	18	yes	complemented
39	C₂×C₄×D₅²	C₂²	1	9	yes	complemented
40	C₂×D₅×H(5,𝔽₂)	—	5	2	no	maximal
41	C₂³×C₅⋊D₅	C₂³	1	6	yes	complemented
42	C₂³×C₅×D₅	C₂³	1	12	yes	complemented
43	C₂²×D₅²	—	4	24	no	—
44	C₂²×D₅²	C₂³	1	9	yes	—
45	C₂²×D₅²	C₂³	1	36	yes	complemented
46	C₂²×D₅²	C₂³	1	18	yes	complemented
47	C₅²×#41	C₂³	1	6	yes	complemented
48	C₂×C₅²×D₄	C₂³	1	9	yes	complemented
49	C₂×C₂²⋊(C₅⋊D₅)	C₂³	1	18	yes	complemented
50	#87904	C₂³	1	6	yes	complemented
51	Q₈×C₅⋊D₅	C₂³	1	2	yes	complemented
52	#87902	C₂³	1	18	yes	complemented
53	D₄×C₅⋊D₅	C₂³	1	18	yes	complemented
54	D₄×C₅⋊D₅	C₂³	1	36	yes	complemented
55	#87900	C₂³	1	18	yes	complemented
56	C₂×C₄⋊(C₅⋊D₅)	C₂³	1	9	yes	complemented
57	C₂×C₄×C₅⋊D₅	C₂³	1	9	yes	complemented
58	C₂×C₅×C₂²⋊D₅	C₂³	1	36	yes	complemented
59	C₅×#713	C₂³	1	12	yes	complemented
60	C₅×Q₈×D₅	C₂³	1	4	yes	complemented
61	C₅×#711	C₂³	1	36	yes	complemented
62	C₅×D₄×D₅	C₂³	1	36	yes	complemented
63	C₅×D₄×D₅	C₂³	1	72	yes	complemented
64	C₅×#709	C₂³	1	36	yes	complemented
65	C₂×C₅×D₂₀	C₂³	1	18	yes	complemented
66	C₂×C₄×C₅×D₅	C₂³	1	18	yes	complemented
67	#87886	C₂³	1	36	yes	complemented
68	#87886	C₂³	1	72	yes	complemented
69	D₅×C₂²⋊D₅	C₂³	1	72	yes	complemented
70	D₅×C₂²⋊D₅	C₂³	1	144	yes	complemented
71	C₂×#6038	C₂³	1	36	yes	complemented
72	C₂×#6037	C₂³	1	18	yes	complemented
73	C₂×#6036	C₂³	1	9	yes	complemented
74	#87880	C₂³	1	36	yes	complemented
75	#87879	C₂³	1	72	yes	complemented
76	C₂×D₅×Dic₅	C₂³	1	18	yes	complemented
77	#87877	C₂³	1	18	yes	complemented
78	#87877	C₂³	1	36	yes	complemented
79	D₅×D₂₀	C₂³	1	36	yes	complemented
80	D₅×D₂₀	C₂³	1	72	yes	complemented
81	C₄×D₅²	C₂³	1	6	yes	—
82	C₄×D₅²	C₂³	1	36	yes	complemented
83	C₄×D₅²	C₂³	1	18	yes	complemented
84	#87874	C₂³	1	36	yes	complemented
85	#87873	C₂³	1	36	yes	complemented
86	#87872	C₂³	1	6	yes	complemented
87	#87871	C₂³	1	36	yes	complemented
88	#87870	C₂³	1	36	yes	complemented
89	D₅×Dic₁₀	C₂³	1	12	yes	complemented
90	C₂×C₅×H(5,𝔽₂)	—	5	2	no	—
91	D₅×H(5,𝔽₂)	—	5	2	no	—
92	D₅×H(5,𝔽₂)	—	5	12	no	—
93	D₅×H(5,𝔽₂)	—	5	12	no	—
94	D₅×H(5,𝔽₂)	—	5	18	no	—
95	D₅×H(5,𝔽₂)	—	5	18	no	—
96	D₅×H(5,𝔽₂)	—	10	2	no	—
97	D₅×H(5,𝔽₂)	D₅	1	2	yes	direct product factor
98	C₂×#4149	—	5	12	no	—
99	C₂×D₅×#41	—	5	12	no	—
100	C₂×#4144	—	5	18	no	—
101	C₂²×D₄×D₅	—	5	18	no	—
102	C₂³×C₅²	C₂⁴	1	6	yes	complemented
103	C₂²×C₅⋊D₅	—	4	12	no	—
104	C₂²×C₅⋊D₅	C₂⁴	1	9	yes	—
105	C₂²×C₅⋊D₅	C₂⁴	1	18	yes	complemented
106	C₂²×C₅×D₅	—	4	24	no	—
107	C₂²×C₅×D₅	C₂⁴	1	18	yes	—
108	C₂²×C₅×D₅	C₂⁴	1	36	yes	complemented
109	C₂×D₅²	—	2	36	no	—
110	C₂×D₅²	—	4	72	no	—
111	C₂×D₅²	—	4	36	no	—
112	C₂×D₅²	C₂⁴	1	18	yes	—
113	C₂×D₅²	C₂⁴	1	9	yes	—
114	C₂×D₅²	C₂⁴	1	1	yes	characteristic
115	C₂×D₅²	C₂⁴	1	36	yes	complemented
116	C₅²×Q₈	C₂⁴	1	2	yes	complemented
117	C₅²×D₄	C₂⁴	1	18	yes	complemented
118	C₂×C₄×C₅²	C₂⁴	1	9	yes	complemented
119	C₂²⋊(C₅⋊D₅)	C₂⁴	1	36	yes	complemented
120	C₂×#1039	C₂⁴	1	9	yes	complemented
121	C₄⋊(C₅⋊D₅)	C₂⁴	1	18	yes	complemented
122	C₄×C₅⋊D₅	C₂⁴	1	6	yes	—
123	C₄×C₅⋊D₅	C₂⁴	1	18	yes	complemented
124	#6045	C₂⁴	1	6	yes	complemented
125	C₅×C₂²⋊D₅	C₂⁴	1	72	yes	complemented
126	C₂×C₅×Dic₅	C₂⁴	1	18	yes	complemented
127	C₅×D₂₀	C₂⁴	1	36	yes	complemented
128	C₄×C₅×D₅	C₂⁴	1	12	yes	—
129	C₄×C₅×D₅	C₂⁴	1	36	yes	complemented
130	C₅×Dic₁₀	C₂⁴	1	12	yes	complemented
131	#6039	C₂⁴	1	12	yes	complemented
132	#6038	C₂⁴	1	72	yes	complemented
133	#6037	C₂⁴	1	36	yes	complemented
134	#6036	C₂⁴	1	6	yes	—
135	#6036	C₂⁴	1	18	yes	complemented
136	D₅×Dic₅	C₂⁴	1	12	yes	—
137	D₅×Dic₅	C₂⁴	1	36	yes	complemented
138	C₂⁴×D₅	—	5	12	no	—
139	C₅×H(5,𝔽₂)	—	2	2	no	—
140	C₅×H(5,𝔽₂)	—	5	12	no	—
141	C₅×H(5,𝔽₂)	—	5	18	no	—
142	C₅×H(5,𝔽₂)	C₂×D₅	1	2	yes	complemented
143	C₂×C₅×#41	—	5	12	no	—
144	C₂²×C₅×D₄	—	5	18	no	—
145	C₂²×C₂²⋊D₅	—	5	36	no	—
146	#4149	—	5	12	no	—
147	#4149	—	5	24	no	—
148	#4149	—	5	72	no	—
149	#4149	—	5	36	no	—
150	#4149	—	5	36	no	—
151	#4149	—	10	12	no	—
152	#4149	C₂×D₅	1	12	yes	complemented
153	D₅×#41	—	5	12	no	—
154	D₅×#41	—	5	12	no	—
155	D₅×#41	—	5	12	no	—
156	D₅×#41	—	5	36	no	—
157	D₅×#41	—	5	36	no	—
158	D₅×#41	—	5	36	no	—
159	D₅×#41	—	5	36	no	—
160	D₅×#41	—	10	12	no	—
161	D₅×#41	C₂×D₅	1	12	yes	complemented
162	C₂×#713	—	5	12	no	—
163	C₂×Q₈×D₅	—	5	4	no	—
164	#4144	—	5	18	no	—
165	#4144	—	5	72	no	—
166	#4144	—	5	72	no	—
167	#4144	—	5	72	no	—
168	#4144	—	5	36	no	—
169	#4144	—	10	18	no	—
170	#4144	C₂×D₅	1	18	yes	complemented
171	C₂×#711	—	5	36	no	—
172	C₂×D₄×D₅	—	5	36	no	—
173	C₂×D₄×D₅	—	5	18	no	—
174	C₂×D₄×D₅	—	5	36	no	—
175	C₂×D₄×D₅	—	5	36	no	—
176	C₂×D₄×D₅	—	5	72	no	—
177	C₂×D₄×D₅	—	5	72	no	—
178	C₂×D₄×D₅	—	5	72	no	—
179	C₂×D₄×D₅	—	5	18	no	—
180	C₂×D₄×D₅	—	5	18	no	—
181	C₂×D₄×D₅	—	10	18	no	—
182	C₂×D₄×D₅	C₂×D₅	1	18	yes	complemented
183	C₂×#709	—	5	36	no	—
184	C₂²×D₂₀	—	5	18	no	—
185	C₂²×C₄×D₅	—	5	18	no	—
186	C₂²×H(5,𝔽₂)	—	25	1	no	Sylow
187	C₂²×C₅²	—	4	6	no	—
188	C₂²×C₅²	C₂⁵	1	9	yes	—
189	C₂×C₅⋊D₅	—	2	18	no	—
190	C₂×C₅⋊D₅	—	4	18	no	—
191	C₂×C₅⋊D₅	C₂⁵	1	9	yes	—
192	C₂×C₅⋊D₅	C₂⁵	1	1	yes	characteristic
193	C₂×C₅×D₅	—	2	36	no	—
194	C₂×C₅×D₅	—	4	36	no	—
195	C₂×C₅×D₅	C₂⁵	1	18	yes	—
196	C₂×C₅×D₅	C₂⁵	1	2	yes	—
197	D₅²	—	2	36	no	—
198	D₅²	—	2	18	no	—
199	D₅²	—	4	36	no	—
200	D₅²	H(5,𝔽₂)	1	4	yes	direct product factor
201	C₄×C₅²	C₂⁵	1	6	yes	—
202	#1039	C₂⁵	1	6	yes	—
203	C₅×Dic₅	C₂⁵	1	12	yes	—
204	C₂⁴×C₅	—	5	12	no	—
205	C₂³×D₅	—	5	12	no	—
206	C₂³×D₅	—	5	18	no	—
207	C₂³×D₅	—	5	36	no	—
208	C₂³×D₅	—	5	36	no	—
209	C₂³×D₅	—	5	36	no	—
210	C₂³×D₅	—	10	12	no	—
211	C₂³×D₅	—	20	48	no	—
212	C₂³×D₅	C₂²×D₅	1	12	yes	complemented
213	C₅×#41	—	2	12	no	—
214	C₅×#41	—	5	12	no	—
215	C₅×#41	—	5	36	no	—
216	C₅×#41	—	5	36	no	—
217	C₅×#41	C₂²×D₅	1	12	yes	complemented
218	C₂×C₅×Q₈	—	5	4	no	—
219	C₂×C₅×D₄	—	2	18	no	—
220	C₂×C₅×D₄	—	5	36	no	—
221	C₂×C₅×D₄	—	5	36	no	—
222	C₂×C₅×D₄	—	5	72	no	—
223	C₂×C₅×D₄	—	5	18	no	—
224	C₂×C₅×D₄	C₂²×D₅	1	18	yes	complemented
225	C₂²×C₄×C₅	—	5	18	no	—
226	C₂×C₂²⋊D₅	—	5	36	no	—
227	C₂×C₂²⋊D₅	—	5	36	no	—
228	C₂×C₂²⋊D₅	—	5	72	no	—
229	C₂×C₂²⋊D₅	—	5	144	no	—
230	C₂×C₂²⋊D₅	—	5	36	no	—
231	C₂×C₂²⋊D₅	—	10	36	no	—
232	C₂×C₂²⋊D₅	C₂²×D₅	1	36	yes	complemented
233	C₂²×Dic₅	—	5	18	no	—
234	#713	—	5	12	no	—
235	#713	—	5	36	no	—
236	#713	—	5	36	no	—
237	#713	—	10	12	no	—
238	#713	C₂²×D₅	1	12	yes	complemented
239	Q₈×D₅	—	5	4	no	—
240	Q₈×D₅	—	5	12	no	—
241	Q₈×D₅	—	5	12	no	—
242	Q₈×D₅	—	10	4	no	—
243	Q₈×D₅	C₂²×D₅	1	4	yes	complemented
244	#711	—	5	36	no	—
245	#711	—	5	72	no	—
246	#711	—	5	72	no	—
247	#711	—	5	36	no	—
248	#711	—	5	36	no	—
249	#711	—	10	36	no	—
250	#711	C₂²×D₅	1	36	yes	complemented
251	D₄×D₅	—	5	72	no	—
252	D₄×D₅	—	5	144	no	—
253	D₄×D₅	—	5	36	no	—
254	D₄×D₅	—	5	72	no	—
255	D₄×D₅	—	5	72	no	—
256	D₄×D₅	—	5	144	no	—
257	D₄×D₅	—	5	36	no	—
258	D₄×D₅	—	5	72	no	—
259	D₄×D₅	—	5	36	no	—
260	D₄×D₅	—	5	72	no	—
261	D₄×D₅	—	10	36	no	—
262	D₄×D₅	—	10	72	no	—
263	D₄×D₅	C₂²×D₅	1	36	yes	complemented
264	D₄×D₅	C₂²×D₅	1	72	yes	complemented
265	#709	—	5	36	no	—
266	#709	—	5	72	no	—
267	#709	—	5	36	no	—
268	#709	—	5	72	no	—
269	#709	—	5	36	no	—
270	#709	—	10	36	no	—
271	#709	C₂²×D₅	1	36	yes	complemented
272	C₂×D₂₀	—	5	36	no	—
273	C₂×D₂₀	—	5	18	no	—
274	C₂×D₂₀	—	5	36	no	—
275	C₂×D₂₀	—	5	72	no	—
276	C₂×D₂₀	—	10	18	no	—
277	C₂×D₂₀	C₂²×D₅	1	18	yes	complemented
278	C₂×C₄×D₅	—	5	18	no	—
279	C₂×C₄×D₅	—	5	18	no	—
280	C₂×C₄×D₅	—	5	18	no	—
281	C₂×C₄×D₅	—	5	12	no	—
282	C₂×C₄×D₅	—	5	36	no	—
283	C₂×C₄×D₅	—	5	36	no	—
284	C₂×C₄×D₅	—	5	36	no	—
285	C₂×C₄×D₅	—	10	18	no	—
286	C₂×C₄×D₅	C₂²×D₅	1	18	yes	complemented
287	C₂×Dic₁₀	—	5	12	no	—
288	C₂×H(5,𝔽₂)	—	5	2	no	—
289	C₂×H(5,𝔽₂)	—	25	18	no	—
290	C₂×H(5,𝔽₂)	—	25	12	no	—
291	C₂×H(5,𝔽₂)	—	25	6	no	—
292	C₂×H(5,𝔽₂)	—	25	9	no	—
293	C₂×H(5,𝔽₂)	—	25	1	no	—
294	C₂²×#41	—	25	6	no	—
295	C₂³×D₄	—	25	9	no	—
296	C₂×C₅²	—	2	9	no	—
297	C₂×C₅²	C₂⁶	1	1	yes	characteristic
298	C₅⋊D₅	—	2	9	no	—
299	C₅⋊D₅	C₂×H(5,𝔽₂)	1	2	yes	complemented
300	C₅×D₅	—	2	18	no	—
301	C₅×D₅	C₂×H(5,𝔽₂)	1	4	yes	complemented
302	C₂³×C₅	—	2	12	no	—
303	C₂³×C₅	—	5	18	no	—
304	C₂³×C₅	—	5	36	no	—
305	C₂³×C₅	—	20	24	no	—
306	C₂³×C₅	C₂³×D₅	1	12	yes	complemented
307	C₂²×D₅	—	4	24	no	—
308	C₂²×D₅	—	5	2	no	—
309	C₂²×D₅	—	5	18	no	—
310	C₂²×D₅	—	5	36	no	—
311	C₂²×D₅	—	5	18	no	—
312	C₂²×D₅	—	5	18	no	—
313	C₂²×D₅	—	5	18	no	—
314	C₂²×D₅	—	5	72	no	—
315	C₂²×D₅	—	10	18	no	—
316	C₂²×D₅	—	10	36	no	—
317	C₂²×D₅	—	10	72	no	—
318	C₂²×D₅	—	20	72	no	—
319	C₂²×D₅	—	20	72	no	—
320	C₂²×D₅	—	20	72	no	—
321	C₂²×D₅	—	20	24	no	—
322	C₂²×D₅	—	40	24	no	—
323	C₂²×D₅	C₂³×D₅	1	18	yes	—
324	C₂²×D₅	C₂³×D₅	1	36	yes	complemented
325	C₅×Q₈	—	2	4	no	—
326	C₅×Q₈	—	5	12	no	—
327	C₅×Q₈	C₂³×D₅	1	4	yes	complemented
328	C₅×D₄	—	2	36	no	—
329	C₅×D₄	—	5	72	no	—
330	C₅×D₄	—	5	36	no	—
331	C₅×D₄	C₂³×D₅	1	36	yes	complemented
332	C₂×C₄×C₅	—	2	18	no	—
333	C₂×C₄×C₅	—	5	18	no	—
334	C₂×C₄×C₅	—	5	12	no	—
335	C₂×C₄×C₅	—	5	36	no	—
336	C₂×C₄×C₅	C₂³×D₅	1	18	yes	complemented
337	C₂²⋊D₅	—	5	72	no	—
338	C₂²⋊D₅	—	5	72	no	—
339	C₂²⋊D₅	—	5	72	no	—
340	C₂²⋊D₅	—	10	72	no	—
341	C₂²⋊D₅	C₂³×D₅	1	72	yes	complemented
342	C₂×Dic₅	—	5	18	no	—
343	C₂×Dic₅	—	5	12	no	—
344	C₂×Dic₅	—	5	36	no	—
345	C₂×Dic₅	—	10	18	no	—
346	C₂×Dic₅	C₂³×D₅	1	18	yes	complemented
347	D₂₀	—	5	72	no	—
348	D₂₀	—	5	36	no	—
349	D₂₀	—	10	36	no	—
350	D₂₀	C₂³×D₅	1	36	yes	complemented
351	C₄×D₅	—	5	12	no	—
352	C₄×D₅	—	5	36	no	—
353	C₄×D₅	—	5	12	no	—
354	C₄×D₅	—	5	36	no	—
355	C₄×D₅	—	5	12	no	—
356	C₄×D₅	—	5	36	no	—
357	C₄×D₅	—	10	12	no	—
358	C₄×D₅	—	10	36	no	—
359	C₄×D₅	C₂³×D₅	1	12	yes	—
360	C₄×D₅	C₂³×D₅	1	36	yes	complemented
361	Dic₁₀	—	5	12	no	—
362	Dic₁₀	—	5	24	no	—
363	Dic₁₀	—	10	12	no	—
364	Dic₁₀	C₂³×D₅	1	12	yes	complemented
365	C₂⁵	—	25	6	no	—
366	H(5,𝔽₂)	—	5	12	no	—
367	H(5,𝔽₂)	—	5	18	no	—
368	H(5,𝔽₂)	—	25	36	no	—
369	H(5,𝔽₂)	—	25	36	no	—
370	H(5,𝔽₂)	—	25	72	no	—
371	H(5,𝔽₂)	—	25	36	no	—
372	H(5,𝔽₂)	—	25	12	no	—
373	H(5,𝔽₂)	—	25	18	no	—
374	H(5,𝔽₂)	—	25	6	no	—
375	H(5,𝔽₂)	—	25	9	no	—
376	H(5,𝔽₂)	D₅²	1	1	yes	characteristic, direct product factor
377	C₂×#41	—	5	12	no	—
378	C₂×#41	—	25	36	no	—
379	C₂×#41	—	25	36	no	—
380	C₂×#41	—	25	12	no	—
381	C₂×#41	—	25	6	no	—
382	C₂×#41	—	25	18	no	—
383	C₂×#41	—	25	18	no	—
384	C₂×#41	—	25	6	no	—
385	C₂²×Q₈	—	25	2	no	—
386	C₂²×D₄	—	5	18	no	—
387	C₂²×D₄	—	25	18	no	—
388	C₂²×D₄	—	25	18	no	—
389	C₂²×D₄	—	25	18	no	—
390	C₂²×D₄	—	25	72	no	—
391	C₂²×D₄	—	25	36	no	—
392	C₂²×D₄	—	25	9	no	—
393	C₂²×D₄	—	25	36	no	—
394	C₂²×D₄	—	25	9	no	—
395	C₂³×C₄	—	25	9	no	—
396	C₅²	C₂²×H(5,𝔽₂)	1	1	yes	characteristic, complemented, Sylow
397	C₂²×C₅	—	2	18	no	—
398	C₂²×C₅	—	4	12	no	—
399	C₂²×C₅	—	5	2	no	—
400	C₂²×C₅	—	5	18	no	—
401	C₂²×C₅	—	8	12	no	—
402	C₂²×C₅	—	10	36	no	—
403	C₂²×C₅	—	20	36	no	—
404	C₂²×C₅	C₂⁴×D₅	1	18	yes	—
405	C₂×D₅	—	2	36	no	—
406	C₂×D₅	—	4	36	no	—
407	C₂×D₅	—	5	2	no	—
408	C₂×D₅	—	5	8	no	—
409	C₂×D₅	—	5	18	no	—
410	C₂×D₅	—	10	2	no	—
411	C₂×D₅	—	10	18	no	—
412	C₂×D₅	—	10	36	no	—
413	C₂×D₅	—	10	36	no	—
414	C₂×D₅	—	10	36	no	—
415	C₂×D₅	—	10	36	no	—
416	C₂×D₅	—	20	36	no	—
417	C₂×D₅	—	20	72	no	—
418	C₂×D₅	—	20	36	no	—
419	C₂×D₅	—	40	36	no	—
420	C₂×D₅	C₂⁴×D₅	1	18	yes	—
421	C₂×D₅	C₂⁴×D₅	1	2	yes	—
422	C₄×C₅	—	2	12	no	—
423	C₄×C₅	—	5	12	no	—
424	C₄×C₅	C₂⁴×D₅	1	12	yes	—
425	Dic₅	—	5	12	no	—
426	Dic₅	—	10	12	no	—
427	Dic₅	C₂⁴×D₅	1	12	yes	—
428	C₂⁴	—	5	12	no	—
429	C₂⁴	—	25	9	no	—
430	C₂⁴	—	25	36	no	—
431	C₂⁴	—	25	18	no	—
432	C₂⁴	—	25	6	no	—
433	C₂⁴	—	100	24	no	—
434	#41	—	5	12	no	—
435	#41	—	5	36	no	—
436	#41	—	5	36	no	—
437	#41	—	25	36	no	—
438	#41	—	25	72	no	—
439	#41	—	25	36	no	—
440	#41	—	25	36	no	—
441	#41	—	25	36	no	—
442	#41	—	25	36	no	—
443	#41	—	25	6	no	—
444	#41	—	25	18	no	—
445	#41	—	25	18	no	—
446	#41	C₂×D₅²	1	6	yes	complemented
447	C₂×Q₈	—	5	4	no	—
448	C₂×Q₈	—	25	12	no	—
449	C₂×Q₈	—	25	6	no	—
450	C₂×Q₈	—	25	2	no	—
451	C₂×D₄	—	5	36	no	—
452	C₂×D₄	—	5	36	no	—
453	C₂×D₄	—	5	72	no	—
454	C₂×D₄	—	5	18	no	—
455	C₂×D₄	—	25	36	no	—
456	C₂×D₄	—	25	72	no	—
457	C₂×D₄	—	25	36	no	—
458	C₂×D₄	—	25	36	no	—
459	C₂×D₄	—	25	18	no	—
460	C₂×D₄	—	25	72	no	—
461	C₂×D₄	—	25	36	no	—
462	C₂×D₄	—	25	72	no	—
463	C₂×D₄	—	25	144	no	—
464	C₂×D₄	—	25	18	no	—
465	C₂×D₄	—	25	18	no	—
466	C₂×D₄	—	25	18	no	—
467	C₂×D₄	—	25	36	no	—
468	C₂×D₄	—	25	9	no	—
469	C₂×D₄	C₂×D₅²	1	9	yes	complemented
470	C₂²×C₄	—	5	18	no	—
471	C₂²×C₄	—	25	6	no	—
472	C₂²×C₄	—	25	9	no	—
473	C₂²×C₄	—	25	36	no	—
474	C₂²×C₄	—	25	18	no	—
475	C₂²×C₄	—	25	18	no	—
476	C₂²×C₄	—	25	9	no	—
477	C₂×C₅	—	2	18	no	—
478	C₂×C₅	—	2	2	no	—
479	C₂×C₅	—	4	18	no	—
480	C₂×C₅	—	5	4	no	—
481	C₂×C₅	—	10	18	no	—
482	C₂×C₅	C₂⁵×D₅	1	2	yes	—
483	D₅	—	2	18	no	—
484	D₅	—	5	4	no	—
485	D₅	—	10	4	no	—
486	D₅	—	10	18	no	—
487	D₅	—	20	18	no	—
488	D₅	D₅×H(5,𝔽₂)	1	4	yes	direct product factor
489	C₂³	—	5	18	no	—
490	C₂³	—	5	36	no	—
491	C₂³	—	20	24	no	—
492	C₂³	—	25	1	no	—
493	C₂³	—	25	18	no	—
494	C₂³	—	25	9	no	—
495	C₂³	—	25	36	no	—
496	C₂³	—	25	9	no	—
497	C₂³	—	25	18	no	—
498	C₂³	—	50	36	no	—
499	C₂³	—	100	12	no	—
500	C₂³	—	100	72	no	—
501	C₂³	—	100	36	no	—
502	C₂³	C₂²×D₅²	1	6	yes	complemented
503	Q₈	—	5	12	no	—
504	Q₈	—	25	12	no	—
505	Q₈	—	25	6	no	—
506	Q₈	C₂²×D₅²	1	2	yes	complemented
507	D₄	—	5	72	no	—
508	D₄	—	5	36	no	—
509	D₄	—	25	72	no	—
510	D₄	—	25	36	no	—
511	D₄	—	25	36	no	—
512	D₄	—	25	18	no	—
513	D₄	C₂²×D₅²	1	18	yes	complemented
514	C₂×C₄	—	5	18	no	—
515	C₂×C₄	—	5	12	no	—
516	C₂×C₄	—	5	36	no	—
517	C₂×C₄	—	25	6	no	—
518	C₂×C₄	—	25	18	no	—
519	C₂×C₄	—	25	12	no	—
520	C₂×C₄	—	25	36	no	—
521	C₂×C₄	—	25	9	no	—
522	C₂×C₄	—	25	6	no	—
523	C₂×C₄	—	25	18	no	—
524	C₂×C₄	C₂²×D₅²	1	9	yes	complemented
525	C₅	—	2	2	no	—
526	C₅	C₂×D₅×H(5,𝔽₂)	1	2	yes	complemented
527	C₂²	—	4	6	no	—
528	C₂²	—	5	2	no	—
529	C₂²	—	5	18	no	—
530	C₂²	—	10	36	no	—
531	C₂²	—	20	36	no	—
532	C₂²	—	25	4	no	—
533	C₂²	—	25	1	no	—
534	C₂²	—	25	9	no	—
535	C₂²	—	50	18	no	—
536	C₂²	—	50	36	no	—
537	C₂²	—	50	18	no	—
538	C₂²	—	100	18	no	—
539	C₂²	—	100	36	no	—
540	C₂²	C₂³×D₅²	1	9	yes	—
541	C₄	—	5	12	no	—
542	C₄	—	25	6	no	—
543	C₄	C₂³×D₅²	1	6	yes	—
544	C₂	—	2	9	no	—
545	C₂	—	5	4	no	—
546	C₂	—	10	18	no	—
547	C₂	—	25	2	no	—
548	C₂	—	50	9	no	—
549	C₂	C₂⁴×D₅²	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂⁴×D₅²
commutator, socle, cosocle	C₂×C₅²	C₂⁶
Fitting	C₅²×H(5,𝔽₂)	C₂²
hypercenter	H(5,𝔽₂)	D₅²
hypocenter	C₅²	C₂²×H(5,𝔽₂)
LCS	G ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. H(5,𝔽₂)×C₅⋊D₅	C₂	semidirect
2. C₅×D₅×H(5,𝔽₂)	C₂	semidirect
3. #13107838	C₂	semidirect
4. D₅×#4149	C₂	semidirect
5. D₅²×#41	C₂	semidirect
6. #13107821	C₂	semidirect
7. D₅×#4144	C₂	semidirect
8. C₂×D₄×D₅²	C₂	semidirect
9. C₂³×D₅²	C₂²	semidirect
10. C₅²×H(5,𝔽₂)	C₂²	semidirect
11. #11717522	C₂²	semidirect
12. #41×C₅⋊D₅	C₂²	semidirect
13. #11717517	C₂²	semidirect
14. C₂×D₄×C₅⋊D₅	C₂²	semidirect
15. C₅×#4149	C₂²	semidirect
16. C₅×D₅×#41	C₂²	semidirect
17. C₅×#4144	C₂²	semidirect
18. C₂×C₅×D₄×D₅	C₂²	semidirect
19. #11717495	C₂²	semidirect
20. C₂×#87886	C₂²	semidirect
21. C₂×D₅×C₂²⋊D₅	C₂²	semidirect
22. #11717485	C₂²	semidirect
23. #11717484	C₂²	semidirect
24. D₅×#713	C₂²	semidirect
25. Q₈×D₅²	C₂²	semidirect
26. #11717479	C₂²	semidirect
27. #11717478	C₂²	semidirect
28. #11717477	C₂²	semidirect
29. D₅×#711	C₂²	semidirect
30. D₄×D₅²	C₂²	semidirect
31. D₄×D₅²	C₂²	semidirect
32. D₄×D₅²	C₂²	semidirect
33. #11717473	C₂²	semidirect
34. #11717472	C₂²	semidirect
35. #11717471	C₂²	semidirect
36. D₅×#709	C₂²	semidirect
37. C₂×#87877	C₂²	semidirect
38. C₂×D₅×D₂₀	C₂²	semidirect
39. C₂×C₄×D₅²	C₂²	semidirect
41. C₂³×C₅⋊D₅	C₂³	semidirect
42. C₂³×C₅×D₅	C₂³	semidirect
45. C₂²×D₅²	C₂³	semidirect
46. C₂²×D₅²	C₂³	semidirect
47. C₅²×#41	C₂³	semidirect
48. C₂×C₅²×D₄	C₂³	semidirect
49. C₂×C₂²⋊(C₅⋊D₅)	C₂³	semidirect
50. #87904	C₂³	semidirect
51. Q₈×C₅⋊D₅	C₂³	semidirect
52. #87902	C₂³	semidirect
53. D₄×C₅⋊D₅	C₂³	semidirect
54. D₄×C₅⋊D₅	C₂³	semidirect
55. #87900	C₂³	semidirect
56. C₂×C₄⋊(C₅⋊D₅)	C₂³	semidirect
57. C₂×C₄×C₅⋊D₅	C₂³	semidirect
58. C₂×C₅×C₂²⋊D₅	C₂³	semidirect
59. C₅×#713	C₂³	semidirect
60. C₅×Q₈×D₅	C₂³	semidirect
61. C₅×#711	C₂³	semidirect
62. C₅×D₄×D₅	C₂³	semidirect
63. C₅×D₄×D₅	C₂³	semidirect
64. C₅×#709	C₂³	semidirect
65. C₂×C₅×D₂₀	C₂³	semidirect
66. C₂×C₄×C₅×D₅	C₂³	semidirect
67. #87886	C₂³	semidirect
68. #87886	C₂³	semidirect
69. D₅×C₂²⋊D₅	C₂³	semidirect
70. D₅×C₂²⋊D₅	C₂³	semidirect
71. C₂×#6038	C₂³	semidirect
72. C₂×#6037	C₂³	semidirect
73. C₂×#6036	C₂³	semidirect
74. #87880	C₂³	semidirect
75. #87879	C₂³	semidirect
76. C₂×D₅×Dic₅	C₂³	semidirect
77. #87877	C₂³	semidirect
78. #87877	C₂³	semidirect
79. D₅×D₂₀	C₂³	semidirect
80. D₅×D₂₀	C₂³	semidirect
82. C₄×D₅²	C₂³	semidirect
83. C₄×D₅²	C₂³	semidirect
84. #87874	C₂³	semidirect
85. #87873	C₂³	semidirect
86. #87872	C₂³	semidirect
87. #87871	C₂³	semidirect
88. #87870	C₂³	semidirect
89. D₅×Dic₁₀	C₂³	semidirect
97. D₅×H(5,𝔽₂)	D₅	direct (N × H)
102. C₂³×C₅²	C₂⁴	semidirect
105. C₂²×C₅⋊D₅	C₂⁴	semidirect
108. C₂²×C₅×D₅	C₂⁴	semidirect
115. C₂×D₅²	C₂⁴	semidirect
116. C₅²×Q₈	C₂⁴	semidirect
117. C₅²×D₄	C₂⁴	semidirect
118. C₂×C₄×C₅²	C₂⁴	semidirect
119. C₂²⋊(C₅⋊D₅)	C₂⁴	semidirect
120. C₂×#1039	C₂⁴	semidirect
121. C₄⋊(C₅⋊D₅)	C₂⁴	semidirect
123. C₄×C₅⋊D₅	C₂⁴	semidirect
124. #6045	C₂⁴	semidirect
125. C₅×C₂²⋊D₅	C₂⁴	semidirect
126. C₂×C₅×Dic₅	C₂⁴	semidirect
127. C₅×D₂₀	C₂⁴	semidirect
129. C₄×C₅×D₅	C₂⁴	semidirect
130. C₅×Dic₁₀	C₂⁴	semidirect
131. #6039	C₂⁴	semidirect
132. #6038	C₂⁴	semidirect
133. #6037	C₂⁴	semidirect
135. #6036	C₂⁴	semidirect
137. D₅×Dic₅	C₂⁴	semidirect
142. C₅×H(5,𝔽₂)	C₂×D₅	semidirect
152. #4149	C₂×D₅	semidirect
161. D₅×#41	C₂×D₅	semidirect
170. #4144	C₂×D₅	semidirect
182. C₂×D₄×D₅	C₂×D₅	semidirect
200. D₅²	H(5,𝔽₂)	direct (N × H)
212. C₂³×D₅	C₂²×D₅	semidirect
217. C₅×#41	C₂²×D₅	semidirect
224. C₂×C₅×D₄	C₂²×D₅	semidirect
232. C₂×C₂²⋊D₅	C₂²×D₅	semidirect
238. #713	C₂²×D₅	semidirect
243. Q₈×D₅	C₂²×D₅	semidirect
250. #711	C₂²×D₅	semidirect
263. D₄×D₅	C₂²×D₅	semidirect
264. D₄×D₅	C₂²×D₅	semidirect
271. #709	C₂²×D₅	semidirect
277. C₂×D₂₀	C₂²×D₅	semidirect
286. C₂×C₄×D₅	C₂²×D₅	semidirect
299. C₅⋊D₅	C₂×H(5,𝔽₂)	semidirect
301. C₅×D₅	C₂×H(5,𝔽₂)	semidirect
306. C₂³×C₅	C₂³×D₅	semidirect
324. C₂²×D₅	C₂³×D₅	semidirect
327. C₅×Q₈	C₂³×D₅	semidirect
331. C₅×D₄	C₂³×D₅	semidirect
336. C₂×C₄×C₅	C₂³×D₅	semidirect
341. C₂²⋊D₅	C₂³×D₅	semidirect
346. C₂×Dic₅	C₂³×D₅	semidirect
350. D₂₀	C₂³×D₅	semidirect
360. C₄×D₅	C₂³×D₅	semidirect
364. Dic₁₀	C₂³×D₅	semidirect
376. H(5,𝔽₂)	D₅²	direct (N × H)
396. C₅²	C₂²×H(5,𝔽₂)	semidirect
446. #41	C₂×D₅²	semidirect
469. C₂×D₄	C₂×D₅²	semidirect
488. D₅	D₅×H(5,𝔽₂)	direct (N × H)
502. C₂³	C₂²×D₅²	semidirect
506. Q₈	C₂²×D₅²	semidirect
513. D₄	C₂²×D₅²	semidirect
524. C₂×C₄	C₂²×D₅²	semidirect
526. C₅	C₂×D₅×H(5,𝔽₂)	semidirect

Maximal subgroups

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

subgroup	index
H(5,𝔽₂)×C₅⋊D₅	2
C₅×D₅×H(5,𝔽₂)	2
#13107838	2
D₅×#4149	2
D₅²×#41	2
#13107821	2
D₅×#4144	2
C₂×D₄×D₅²	2
C₂×D₅×H(5,𝔽₂)	5

Subgroups — D52×H(5,𝔽2) #72221785

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅²×H(5,𝔽₂) #72221785