Subgroups — #13106057

Order 1600.

526 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11717250	C₂	1	1	yes	maximal, characteristic
2	C₅×#4072	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#4070	C₂	1	1	yes	maximal, characteristic, complemented
4	#11717093	C₂	1	2	yes	maximal, complemented
5	#11717071	C₂	1	2	yes	maximal, complemented
6	#11717036	C₂	1	1	yes	maximal, characteristic, complemented
7	#11717013	C₂	1	1	yes	maximal, characteristic, complemented
8	#11717009	C₂	1	2	yes	maximal, complemented
9	#11716982	C₂	1	1	yes	maximal, characteristic, complemented
10	#11716966	C₂	1	2	yes	maximal, complemented
11	Dic₅×Dic₁₀	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊(C₅⋊D₅)	C₂²	1	2	yes	—
14	C₂×C₄⋊(C₅⋊D₅)	C₂²	1	1	yes	characteristic
15	C₂×C₅×Dic₁₀	C₂²	1	1	yes	characteristic
16	C₂×#6039	C₂²	1	2	yes	—
17	C₂×#6036	C₂²	1	2	yes	—
18	#87808	C₂²	1	2	yes	—
19	C₄⋊#1039	C₂²	1	1	yes	characteristic
20	C₅×C₂²⋊Dic₅	C₂²	1	2	yes	—
21	C₅×C₂²⋊Dic₅	C₂²	1	1	yes	characteristic
22	C₅×C₂²⋊Dic₅	C₂²	1	1	yes	characteristic
23	C₅×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
24	C₅×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
25	C₅×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
26	C₅×#683	C₂²	1	2	yes	—
27	C₄×C₅×Dic₅	C₂²	1	1	yes	characteristic
28	C₄×C₅×Dic₅	C₂²	1	1	yes	characteristic
29	#87780	C₂²	1	2	yes	—
30	#87780	C₂²	1	2	yes	—
31	#87779	C₂²	1	2	yes	—
32	#87779	C₂²	1	2	yes	—
33	#87779	C₂²	1	2	yes	—
34	Dic₅²	C₂²	1	2	yes	—
35	#65977	—	5	1	no	maximal
36	#65868	—	5	1	no	maximal
37	C₂³×C₅²	C₂³	1	1	yes	characteristic
38	C₂²×C₅⋊D₅	C₂³	1	2	yes	—
39	C₂×C₄×C₅²	—	2	2	no	—
40	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
41	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
42	C₂²⋊(C₅⋊D₅)	—	4	2	no	—
43	C₂×#1039	C₂³	1	2	yes	—
44	C₄⋊(C₅⋊D₅)	—	2	2	no	—
45	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
46	C₂×C₅×Dic₅	C₂³	1	2	yes	—
47	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
48	C₂×C₅×Dic₅	C₂³	1	2	yes	—
49	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
50	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
51	C₅×Dic₁₀	—	2	2	no	—
52	#6039	—	4	2	no	—
53	#6036	—	4	2	no	—
54	C₅×C₂²⋊Q₈	—	5	1	no	—
55	C₅×#117	—	5	1	no	—
56	#4093	—	5	1	no	—
57	Q₈×Dic₅	—	5	1	no	—
58	#4076	—	5	1	no	—
59	#4076	—	5	1	no	—
60	#4076	—	10	2	no	—
61	#4072	D₅	1	1	yes	characteristic, complemented
62	#4070	D₅	1	1	yes	characteristic, complemented
63	#4044	—	5	2	no	—
64	#4044	—	5	2	no	—
65	#4043	—	5	2	no	—
66	#4042	—	5	2	no	—
67	#4039	—	5	1	no	—
68	#4034	—	5	1	no	—
69	#4031	—	5	2	no	—
70	#4031	—	5	2	no	—
71	#4029	—	5	2	no	—
72	#4029	—	5	2	no	—
73	#4021	—	5	1	no	—
74	C₄×D₂₀	—	5	1	no	—
75	#4015	—	5	1	no	—
76	C₄×Dic₁₀	—	5	1	no	—
77	C₂²×C₅²	—	2	1	no	—
78	C₂²×C₅²	—	2	1	no	—
79	C₂²×C₅²	—	2	1	no	—
80	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
81	C₂×C₅⋊D₅	—	2	2	no	—
82	C₂×C₅⋊D₅	—	2	2	no	—
83	C₂×C₅⋊D₅	—	2	2	no	—
84	C₄×C₅²	—	2	1	no	—
85	C₄×C₅²	#41	1	2	yes	—
86	#1039	—	2	2	no	—
87	C₅×Dic₅	—	2	2	no	—
88	C₅×Dic₅	—	2	1	no	—
89	C₅×Dic₅	—	2	2	no	—
90	C₅×Dic₅	—	2	1	no	—
91	C₅×Dic₅	—	2	1	no	—
92	C₅×Dic₅	—	2	1	no	—
93	C₂×C₅×Q₈	—	5	1	no	—
94	C₂²×C₄×C₅	—	2	2	no	—
95	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
96	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
97	C₂×C₂²⋊D₅	—	5	2	no	—
98	C₂×C₂²⋊D₅	—	5	2	no	—
99	C₂×C₂²⋊D₅	—	10	4	no	—
100	C₂×D₂₀	—	5	1	no	—
101	C₂×D₂₀	—	5	1	no	—
102	C₂×D₂₀	—	10	2	no	—
103	C₂×C₄×D₅	—	5	2	no	—
104	C₂×C₄×D₅	—	5	2	no	—
105	C₂×Dic₁₀	—	5	2	no	—
106	C₂×Dic₁₀	—	5	2	no	—
107	C₂×Dic₁₀	C₂×D₅	1	1	yes	characteristic
108	C₅×C₄⋊C₄	—	5	1	no	—
109	C₅×C₄⋊C₄	—	5	1	no	—
110	C₅×C₄⋊C₄	—	5	1	no	—
111	C₅×C₄⋊C₄	—	5	2	no	—
112	C₅×C₂²⋊C₄	—	5	1	no	—
113	C₅×C₂²⋊C₄	—	5	1	no	—
114	C₅×C₂²⋊C₄	—	5	2	no	—
115	C₄²×C₅	—	5	1	no	—
116	C₄²×C₅	—	5	1	no	—
117	C₂²⋊Dic₅	C₂×D₅	1	1	yes	characteristic
118	C₂²⋊Dic₅	C₂×D₅	1	2	yes	—
119	C₂²⋊Dic₅	C₂×D₅	1	1	yes	characteristic
120	#685	—	5	2	no	—
121	#685	—	5	2	no	—
122	#685	—	5	2	no	—
123	#685	—	5	2	no	—
124	#685	—	5	2	no	—
125	#685	—	5	2	no	—
126	#685	—	5	2	no	—
127	#685	—	5	2	no	—
128	#685	—	10	4	no	—
129	C₄⋊Dic₅	—	5	1	no	—
130	C₄⋊Dic₅	—	5	1	no	—
131	C₄⋊Dic₅	—	5	2	no	—
132	C₄⋊Dic₅	—	5	2	no	—
133	C₄⋊Dic₅	—	10	2	no	—
134	C₄⋊Dic₅	C₂×D₅	1	1	yes	characteristic
135	C₄⋊Dic₅	C₂×D₅	1	1	yes	characteristic
136	C₄⋊Dic₅	C₂×D₅	1	1	yes	characteristic
137	#683	—	5	2	no	—
138	#683	—	5	2	no	—
139	#683	C₂×D₅	1	2	yes	—
140	C₄×Dic₅	—	5	2	no	—
141	C₄×Dic₅	—	5	2	no	—
142	C₄×Dic₅	C₂×D₅	1	1	yes	characteristic
143	C₄×Dic₅	C₂×D₅	1	1	yes	characteristic
144	#542	—	25	1	no	Sylow
145	C₂×C₅²	—	4	1	no	—
146	C₂×C₅²	C₂×#41	1	1	yes	characteristic
147	C₂×C₅²	H(5,𝔽₂)	1	1	yes	characteristic
148	C₂×C₅²	2₋¹⁺⁴	1	1	yes	characteristic
149	C₅⋊D₅	—	4	2	no	—
150	C₂³×C₅	—	2	2	no	—
151	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
152	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
153	C₂²×D₅	—	5	2	no	—
154	C₂²×D₅	—	5	2	no	—
155	C₂²×D₅	—	10	4	no	—
156	C₅×Q₈	—	10	2	no	—
157	C₂×C₄×C₅	—	2	2	no	—
158	C₂×C₄×C₅	—	2	2	no	—
159	C₂×C₄×C₅	—	2	2	no	—
160	C₂×C₄×C₅	—	2	2	no	—
161	C₂×C₄×C₅	—	4	4	no	—
162	C₂×C₄×C₅	—	5	1	no	—
163	C₂×C₄×C₅	—	5	2	no	—
164	C₂×C₄×C₅	—	5	1	no	—
165	C₂×C₄×C₅	—	5	2	no	—
166	C₂×C₄×C₅	—	5	1	no	—
167	C₂×C₄×C₅	—	5	1	no	—
168	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
169	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
170	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
171	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
172	C₂²⋊D₅	—	20	2	no	—
173	C₂²⋊D₅	—	20	2	no	—
174	C₂²⋊D₅	—	20	8	no	—
175	C₂×Dic₅	—	5	2	no	—
176	C₂×Dic₅	—	5	2	no	—
177	C₂×Dic₅	—	10	4	no	—
178	C₂×Dic₅	C₂²×D₅	1	1	yes	characteristic
179	C₂×Dic₅	C₂²×D₅	1	2	yes	—
180	C₂×Dic₅	C₂²×D₅	1	1	yes	characteristic
181	C₂×Dic₅	C₂²×D₅	1	1	yes	characteristic
182	C₂×Dic₅	C₂²×D₅	1	1	yes	characteristic
183	C₂×Dic₅	C₂²×D₅	1	2	yes	—
184	D₂₀	—	10	2	no	—
185	D₂₀	—	10	2	no	—
186	D₂₀	—	20	4	no	—
187	C₄×D₅	—	20	2	no	—
188	C₄×D₅	—	20	2	no	—
189	Dic₁₀	—	2	2	no	—
190	Dic₁₀	—	20	2	no	—
191	Dic₁₀	—	20	2	no	—
192	C₄⋊Q₈	—	25	1	no	—
193	#126	—	25	2	no	—
194	#124	—	25	1	no	—
195	#124	—	25	2	no	—
196	#123	—	25	2	no	—
197	C₂²⋊Q₈	—	5	1	no	—
198	C₂²⋊Q₈	—	25	2	no	—
199	#121	—	25	1	no	—
200	C₄×Q₈	—	25	1	no	—
201	C₄×D₄	—	25	1	no	—
202	#117	—	5	1	no	—
203	C₅²	#542	1	1	yes	characteristic, complemented, Sylow
204	C₂²×C₅	—	2	4	no	—
205	C₂²×C₅	—	2	1	no	—
206	C₂²×C₅	—	2	1	no	—
207	C₂²×C₅	—	2	1	no	—
208	C₂²×C₅	—	2	1	no	—
209	C₂²×C₅	—	2	1	no	—
210	C₂²×C₅	—	2	1	no	—
211	C₂²×C₅	—	2	2	no	—
212	C₂²×C₅	—	4	2	no	—
213	C₂²×C₅	—	4	2	no	—
214	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
215	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
216	C₂×D₅	—	10	2	no	—
217	C₂×D₅	—	10	2	no	—
218	C₂×D₅	—	10	2	no	—
219	C₂×D₅	—	10	2	no	—
220	C₂×D₅	—	10	2	no	—
221	C₂×D₅	—	10	2	no	—
222	C₂×D₅	—	20	4	no	—
223	C₂×D₅	—	20	4	no	—
224	C₂×D₅	—	20	4	no	—
225	C₄×C₅	—	2	1	no	—
226	C₄×C₅	—	2	1	no	—
227	C₄×C₅	—	2	4	no	—
228	C₄×C₅	—	4	2	no	—
229	C₄×C₅	—	10	2	no	—
230	C₄×C₅	—	10	1	no	—
231	C₄×C₅	—	10	1	no	—
232	C₄×C₅	—	10	1	no	—
233	C₄×C₅	—	10	2	no	—
234	C₄×C₅	—	10	1	no	—
235	C₄×C₅	#709	1	2	yes	—
236	C₄×C₅	#711	1	2	yes	—
237	Dic₅	—	2	2	no	—
238	Dic₅	—	2	1	no	—
239	Dic₅	—	2	2	no	—
240	Dic₅	—	2	1	no	—
241	Dic₅	—	2	1	no	—
242	Dic₅	—	2	1	no	—
243	Dic₅	—	10	2	no	—
244	Dic₅	—	10	2	no	—
245	Dic₅	—	20	4	no	—
246	C₂×Q₈	—	5	1	no	—
247	C₂×Q₈	—	25	2	no	—
248	C₂×D₄	—	25	2	no	—
249	C₂×D₄	—	25	1	no	—
250	C₂²×C₄	—	25	2	no	—
251	C₂²×C₄	D₅²	1	1	yes	characteristic
252	C₄⋊C₄	—	5	1	no	—
253	C₄⋊C₄	—	5	1	no	—
254	C₄⋊C₄	—	5	1	no	—
255	C₄⋊C₄	—	5	2	no	—
256	C₄⋊C₄	—	25	2	no	—
257	C₄⋊C₄	—	25	2	no	—
258	C₄⋊C₄	—	25	1	no	—
259	C₂²⋊C₄	—	5	1	no	—
260	C₂²⋊C₄	—	5	1	no	—
261	C₂²⋊C₄	—	5	2	no	—
262	C₂²⋊C₄	—	25	2	no	—
263	C₂²⋊C₄	—	25	2	no	—
264	C₂²⋊C₄	—	25	2	no	—
265	C₂²⋊C₄	—	25	2	no	—
266	C₄²	—	5	1	no	—
267	C₄²	—	5	1	no	—
268	C₄²	—	25	2	no	—
269	C₂×C₅	—	2	2	no	—
270	C₂×C₅	—	2	2	no	—
271	C₂×C₅	—	2	2	no	—
272	C₂×C₅	—	4	4	no	—
273	C₂×C₅	—	4	1	no	—
274	C₂×C₅	—	4	1	no	—
275	C₂×C₅	C₂×#709	1	1	yes	characteristic
276	C₂×C₅	C₂×#711	1	1	yes	characteristic
277	C₂×C₅	#4147	1	1	yes	characteristic
278	C₂×C₅	#4149	1	1	yes	characteristic
279	C₂×C₅	#4149	1	1	yes	characteristic
280	C₂×C₅	#4150	1	1	yes	characteristic
281	D₅	—	20	2	no	—
282	D₅	—	20	2	no	—
283	D₅	—	40	4	no	—
284	C₂³	—	25	2	no	—
285	C₂³	C₂×D₅²	1	1	yes	characteristic
286	Q₈	—	10	2	no	—
287	Q₈	—	100	2	no	—
288	D₄	—	50	2	no	—
289	D₄	—	100	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	5	1	no	—
292	C₂×C₄	—	5	1	no	—
293	C₂×C₄	—	5	1	no	—
294	C₂×C₄	—	5	1	no	—
295	C₂×C₄	—	5	2	no	—
296	C₂×C₄	—	5	2	no	—
297	C₂×C₄	—	25	2	no	—
298	C₂×C₄	—	100	2	no	—
299	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
300	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
301	C₅	—	2	2	no	—
302	C₅	#65868	1	1	yes	characteristic, complemented
303	C₅	#65977	1	1	yes	characteristic, complemented
304	C₂²	—	2	1	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	50	2	no	—
308	C₂²	—	50	2	no	—
309	C₂²	—	50	2	no	—
310	C₂²	C₂²×D₅²	1	1	yes	characteristic
311	C₄	—	2	1	no	—
312	C₄	—	10	1	no	—
313	C₄	—	10	1	no	—
314	C₄	—	10	1	no	—
315	C₄	—	10	2	no	—
316	C₄	—	10	1	no	—
317	C₄	—	10	2	no	—
318	C₄	—	50	2	no	—
319	C₄	#87879	1	2	yes	—
320	C₂	—	4	1	no	—
321	C₂	—	100	2	no	—
322	C₂	#11717462	1	1	yes	characteristic
323	C₂	#11717473	1	1	yes	characteristic
324	C₂	C₂×#87879	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₅²
commutator, socle, cosocle	C₂²×C₅²	C₂⁴
Fitting	C₂²×C₄×C₅²	C₂²
prime radical	#11717250	C₂
hypercenter	C₂²×C₄	D₅²
hypocenter	C₅²	#542
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
2. C₅×#4072	C₂	semidirect
3. C₅×#4070	C₂	semidirect
4. #11717093	C₂	semidirect
5. #11717071	C₂	semidirect
6. #11717036	C₂	semidirect
7. #11717013	C₂	semidirect
8. #11717009	C₂	semidirect
9. #11716982	C₂	semidirect
10. #11716966	C₂	semidirect
11. Dic₅×Dic₁₀	C₂	semidirect
61. #4072	D₅	semidirect
62. #4070	D₅	semidirect
203. C₅²	#542	semidirect
302. C₅	#65868	semidirect
303. C₅	#65977	semidirect

Maximal subgroups

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

subgroup	index
#11717250	2
C₅×#4072	2
C₅×#4070	2
#11717093	2
#11717071	2
#11717036	2
#11717013	2
#11717009	2
#11716982	2
#11716966	2
Dic₅×Dic₁₀	2
#65977	5
#65868	5