Subgroups — #13107189

Order 1600.

992 conjugacy classes of subgroups.

The subgroup lattice has 333 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅²×#123	C₂	1	1	yes	maximal, characteristic, complemented
2	#11717257	C₂	1	1	yes	maximal, characteristic
3	D₄×#1039	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#87808	C₂	1	1	yes	maximal, characteristic, complemented
5	#11717246	C₂	1	1	yes	maximal, characteristic, complemented
6	#11717218	C₂	1	1	yes	maximal, characteristic, complemented
7	#11717218	C₂	1	1	yes	maximal, characteristic, complemented
8	#11717217	C₂	1	1	yes	maximal, characteristic, complemented
9	#11717212	C₂	1	1	yes	maximal, characteristic, complemented
10	#11717206	C₂	1	1	yes	maximal, characteristic, complemented
11	#11717205	C₂	1	1	yes	maximal, characteristic, complemented
12	#11717203	C₂	1	1	yes	maximal, characteristic, complemented
13	#11717201	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂²⋊C₄×C₅⋊D₅	C₂	1	1	yes	maximal, characteristic, complemented
15	#11717198	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂³×C₅⋊D₅	C₂²	1	1	yes	characteristic
17	C₂×C₅²×D₄	C₂²	1	1	yes	characteristic
18	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊(C₅⋊D₅)	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊(C₅⋊D₅)	C₂²	1	1	yes	characteristic
21	C₂²×#1039	C₂²	1	1	yes	characteristic
22	C₂²×#1039	C₂²	1	1	yes	characteristic, complemented
23	C₂×C₄×C₅⋊D₅	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₅⋊D₅	C₂²	1	1	yes	characteristic
25	C₂×#6045	C₂²	1	1	yes	characteristic, complemented
26	C₅²×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
27	C₅²×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
28	C₅²×C₂²⋊C₄	C₂²	1	1	yes	characteristic
29	C₅²×C₂²⋊C₄	C₂²	1	1	yes	characteristic, complemented
30	C₅²×C₂²⋊C₄	C₂²	1	1	yes	characteristic, complemented
31	C₂²⋊#1039	C₂²	1	1	yes	characteristic
32	C₂²⋊#1039	C₂²	1	1	yes	characteristic
33	C₂²⋊#1039	C₂²	1	1	yes	characteristic, complemented
34	#87808	C₂²	1	1	yes	characteristic
35	#87808	C₂²	1	1	yes	characteristic
36	#87808	C₂²	1	1	yes	characteristic
37	#87808	C₂²	1	1	yes	characteristic
38	#87808	C₂²	1	1	yes	characteristic
39	#87808	C₂²	1	1	yes	characteristic
40	#87808	C₂²	1	1	yes	characteristic
41	#87808	C₂²	1	1	yes	characteristic
42	C₄⋊#1039	C₂²	1	1	yes	characteristic, complemented
43	C₄⋊#1039	C₂²	1	1	yes	characteristic, complemented
44	C₄⋊#1039	C₂²	1	1	yes	characteristic, complemented
45	C₄⋊#1039	C₂²	1	1	yes	characteristic, complemented
46	#87806	C₂²	1	1	yes	characteristic, complemented
47	#87806	C₂²	1	1	yes	characteristic, complemented
48	C₄×#1039	C₂²	1	1	yes	characteristic, complemented
49	C₄×#1039	C₂²	1	1	yes	characteristic, complemented
50	C₄×#1039	C₂²	1	1	yes	characteristic, complemented
51	#65992	—	5	6	no	maximal
52	C₂³×C₅²	C₂³	1	1	yes	characteristic
53	C₂³×C₅²	C₂³	1	1	yes	characteristic
54	C₂²×C₅⋊D₅	—	2	2	no	—
55	C₂²×C₅⋊D₅	—	2	2	no	—
56	C₂²×C₅⋊D₅	—	4	1	no	—
57	C₂²×C₅⋊D₅	C₂³	1	1	yes	characteristic
58	C₂²×C₅⋊D₅	C₂³	1	1	yes	characteristic
59	C₅²×D₄	—	2	2	no	—
60	C₂×C₄×C₅²	—	2	2	no	—
61	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
62	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
63	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
64	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
65	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
66	C₂²⋊(C₅⋊D₅)	—	2	2	no	—
67	C₂²⋊(C₅⋊D₅)	—	4	1	no	—
68	C₂×#1039	—	4	1	no	—
69	C₂×#1039	—	4	1	no	—
70	C₂×#1039	C₂³	1	1	yes	characteristic
71	C₂×#1039	C₂³	1	1	yes	characteristic
72	C₂×#1039	C₂³	1	1	yes	characteristic
73	C₂×#1039	C₂³	1	1	yes	characteristic
74	C₂×#1039	C₂³	1	1	yes	characteristic
75	C₂×#1039	C₂³	1	1	yes	characteristic
76	C₄×C₅⋊D₅	—	2	2	no	—
77	C₄×C₅⋊D₅	—	4	1	no	—
78	#6045	—	4	1	no	—
79	C₅×#123	D₅	1	6	yes	complemented
80	#4083	—	5	6	no	—
81	D₄×Dic₅	—	5	6	no	—
82	C₂×#685	—	5	6	no	—
83	#4072	—	5	6	no	—
84	#4044	—	5	6	no	—
85	#4044	—	5	6	no	—
86	#4043	—	5	6	no	—
87	#4038	—	5	6	no	—
88	#4032	—	5	6	no	—
89	#4031	—	5	6	no	—
90	#4029	—	5	6	no	—
91	#4027	—	5	6	no	—
92	D₅×C₂²⋊C₄	—	5	6	no	—
93	#4024	—	5	6	no	—
94	C₂²×C₅²	—	2	1	no	—
95	C₂²×C₅²	—	2	1	no	—
96	C₂²×C₅²	—	2	1	no	—
97	C₂²×C₅²	—	2	1	no	—
98	C₂²×C₅²	—	2	1	no	—
99	C₂²×C₅²	#41	1	2	yes	—
100	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
101	C₂×C₅⋊D₅	—	2	1	no	—
102	C₂×C₅⋊D₅	—	2	1	no	—
103	C₂×C₅⋊D₅	—	2	1	no	—
104	C₂×C₅⋊D₅	—	2	1	no	—
105	C₂×C₅⋊D₅	—	2	1	no	—
106	C₂×C₅⋊D₅	—	4	4	no	—
107	C₂×C₅⋊D₅	#41	1	2	yes	—
108	C₄×C₅²	—	2	1	no	—
109	C₄×C₅²	—	2	1	no	—
110	C₄×C₅²	—	2	1	no	—
111	C₄×C₅²	—	2	1	no	—
112	C₄×C₅²	—	2	1	no	—
113	#1039	—	2	1	no	—
114	#1039	—	2	1	no	—
115	#1039	—	2	1	no	—
116	#1039	—	2	1	no	—
117	#1039	—	2	1	no	—
118	#1039	—	2	1	no	—
119	C₂³×D₅	—	5	6	no	—
120	C₂×C₅×D₄	C₂×D₅	1	6	yes	—
121	C₂²×C₄×C₅	C₂×D₅	1	6	yes	—
122	C₂×C₂²⋊D₅	—	5	6	no	—
123	C₂×C₂²⋊D₅	—	5	6	no	—
124	C₂²×Dic₅	—	5	6	no	—
125	C₂²×Dic₅	—	5	6	no	—
126	C₂×C₄×D₅	—	5	6	no	—
127	C₂×C₄×D₅	—	5	6	no	—
128	C₂×Dic₁₀	—	5	6	no	—
129	C₅×C₄⋊C₄	C₂×D₅	1	6	yes	complemented
130	C₅×C₄⋊C₄	C₂×D₅	1	6	yes	complemented
131	C₅×C₂²⋊C₄	C₂×D₅	1	6	yes	—
132	C₅×C₂²⋊C₄	C₂×D₅	1	6	yes	complemented
133	C₅×C₂²⋊C₄	C₂×D₅	1	6	yes	complemented
134	C₂²⋊Dic₅	—	5	6	no	—
135	C₂²⋊Dic₅	—	5	6	no	—
136	C₂²⋊Dic₅	—	5	6	no	—
137	#685	—	5	6	no	—
138	#685	—	5	6	no	—
139	#685	—	5	6	no	—
140	#685	—	5	6	no	—
141	#685	—	5	6	no	—
142	#685	—	5	6	no	—
143	#685	—	5	6	no	—
144	#685	—	5	6	no	—
145	C₄⋊Dic₅	—	5	6	no	—
146	C₄⋊Dic₅	—	5	6	no	—
147	C₄⋊Dic₅	—	5	6	no	—
148	C₄⋊Dic₅	—	5	6	no	—
149	#683	—	5	6	no	—
150	#683	—	5	6	no	—
151	C₄×Dic₅	—	5	6	no	—
152	C₄×Dic₅	—	5	6	no	—
153	C₄×Dic₅	—	5	6	no	—
154	#551	—	25	1	no	Sylow
155	C₂×C₅²	—	2	2	no	—
156	C₂×C₅²	—	4	1	no	—
157	C₂×C₅²	C₂×#41	1	1	yes	characteristic
158	C₂×C₅²	C₂×#41	1	1	yes	characteristic
159	C₂×C₅²	H(5,𝔽₂)	1	1	yes	characteristic
160	C₅⋊D₅	—	2	2	no	—
161	C₅⋊D₅	—	4	1	no	—
162	C₂³×C₅	C₂²×D₅	1	6	yes	—
163	C₂³×C₅	C₂²×D₅	1	6	yes	—
164	C₂²×D₅	—	5	6	no	—
165	C₂²×D₅	—	5	6	no	—
166	C₂²×D₅	—	10	12	no	—
167	C₂²×D₅	—	10	12	no	—
168	C₂²×D₅	—	20	6	no	—
169	C₅×D₄	—	2	12	no	—
170	C₂×C₄×C₅	—	2	12	no	—
171	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
172	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
173	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
174	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
175	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
176	C₂²⋊D₅	—	10	12	no	—
177	C₂²⋊D₅	—	20	6	no	—
178	C₂×Dic₅	—	5	6	no	—
179	C₂×Dic₅	—	5	6	no	—
180	C₂×Dic₅	—	5	6	no	—
181	C₂×Dic₅	—	5	6	no	—
182	C₂×Dic₅	—	5	6	no	—
183	C₂×Dic₅	—	5	6	no	—
184	C₂×Dic₅	—	20	6	no	—
185	C₂×Dic₅	—	20	6	no	—
186	C₄×D₅	—	10	12	no	—
187	C₄×D₅	—	20	6	no	—
188	Dic₁₀	—	20	6	no	—
189	#126	—	25	1	no	—
190	#126	—	25	1	no	—
191	#124	—	25	1	no	—
192	#123	—	25	1	no	—
193	#123	—	25	1	no	—
194	#123	C₅⋊D₅	1	1	yes	characteristic, complemented
195	C₂²⋊Q₈	—	25	1	no	—
196	C₂²⋊Q₈	—	25	1	no	—
197	#120	—	25	1	no	—
198	C₄×D₄	—	25	1	no	—
199	C₄×D₄	—	25	1	no	—
200	#117	—	25	1	no	—
201	#117	—	25	1	no	—
202	C₂×C₂²⋊C₄	—	25	1	no	—
203	C₂×C₂²⋊C₄	—	25	1	no	—
204	C₅²	#551	1	1	yes	characteristic, complemented, Sylow
205	C₂²×C₅	—	2	6	no	—
206	C₂²×C₅	—	2	6	no	—
207	C₂²×C₅	—	2	6	no	—
208	C₂²×C₅	—	2	6	no	—
209	C₂²×C₅	—	2	6	no	—
210	C₂²×C₅	#711	1	12	yes	—
211	C₂²×C₅	C₂³×D₅	1	6	yes	—
212	C₂×D₅	—	5	12	no	—
213	C₂×D₅	—	10	6	no	—
214	C₂×D₅	—	10	6	no	—
215	C₂×D₅	—	10	6	no	—
216	C₂×D₅	—	10	6	no	—
217	C₂×D₅	—	10	6	no	—
218	C₂×D₅	—	20	24	no	—
219	C₄×C₅	—	2	6	no	—
220	C₄×C₅	—	2	6	no	—
221	C₄×C₅	—	2	6	no	—
222	C₄×C₅	—	2	6	no	—
223	C₄×C₅	—	2	6	no	—
224	Dic₅	—	10	6	no	—
225	Dic₅	—	10	6	no	—
226	Dic₅	—	10	6	no	—
227	Dic₅	—	10	6	no	—
228	Dic₅	—	10	6	no	—
229	Dic₅	—	10	6	no	—
230	C₂⁴	—	25	1	no	—
231	C₂×Q₈	—	25	1	no	—
232	C₂×D₄	—	25	1	no	—
233	C₂×D₄	—	25	1	no	—
234	C₂×D₄	C₂×C₅⋊D₅	1	1	yes	characteristic
235	C₂²×C₄	—	25	1	no	—
236	C₂²×C₄	—	25	1	no	—
237	C₂²×C₄	—	25	1	no	—
238	C₂²×C₄	—	25	1	no	—
239	C₂²×C₄	C₂×C₅⋊D₅	1	1	yes	characteristic
240	C₄⋊C₄	—	25	1	no	—
241	C₄⋊C₄	—	25	1	no	—
242	C₄⋊C₄	—	25	1	no	—
243	C₄⋊C₄	—	25	1	no	—
244	C₄⋊C₄	—	25	1	no	—
245	C₄⋊C₄	—	25	1	no	—
246	C₄⋊C₄	C₂×C₅⋊D₅	1	1	yes	characteristic, complemented
247	C₄⋊C₄	C₂×C₅⋊D₅	1	1	yes	characteristic, complemented
248	C₂²⋊C₄	—	25	1	no	—
249	C₂²⋊C₄	—	25	1	no	—
250	C₂²⋊C₄	—	25	1	no	—
251	C₂²⋊C₄	—	25	1	no	—
252	C₂²⋊C₄	—	25	1	no	—
253	C₂²⋊C₄	—	25	1	no	—
254	C₂²⋊C₄	—	25	1	no	—
255	C₂²⋊C₄	—	25	1	no	—
256	C₂²⋊C₄	—	25	1	no	—
257	C₂²⋊C₄	—	25	1	no	—
258	C₂²⋊C₄	—	25	1	no	—
259	C₂²⋊C₄	C₂×C₅⋊D₅	1	1	yes	characteristic
260	C₂²⋊C₄	C₂×C₅⋊D₅	1	1	yes	characteristic, complemented
261	C₂²⋊C₄	C₂×C₅⋊D₅	1	1	yes	characteristic, complemented
262	C₄²	—	25	1	no	—
263	C₄²	—	25	1	no	—
264	C₄²	—	25	1	no	—
265	C₂×C₅	—	2	12	no	—
266	C₂×C₅	—	4	6	no	—
267	C₂×C₅	C₂×#711	1	6	yes	—
268	C₂×C₅	D₅×#41	1	6	yes	—
269	C₂×C₅	#4149	1	6	yes	—
270	D₅	—	10	12	no	—
271	D₅	—	20	6	no	—
272	C₂³	—	25	1	no	—
273	C₂³	—	25	1	no	—
274	C₂³	—	50	2	no	—
275	C₂³	—	50	2	no	—
276	C₂³	—	100	1	no	—
277	C₂³	C₂²×C₅⋊D₅	1	1	yes	characteristic
278	C₂³	C₂²×C₅⋊D₅	1	1	yes	characteristic
279	Q₈	—	100	1	no	—
280	D₄	—	2	2	no	—
281	D₄	—	50	2	no	—
282	D₄	—	100	1	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	25	1	no	—
285	C₂×C₄	—	25	1	no	—
286	C₂×C₄	—	25	1	no	—
287	C₂×C₄	—	25	1	no	—
288	C₂×C₄	—	25	1	no	—
289	C₂×C₄	—	25	1	no	—
290	C₂×C₄	—	50	2	no	—
291	C₂×C₄	—	100	1	no	—
292	C₂×C₄	—	100	1	no	—
293	C₂×C₄	—	100	1	no	—
294	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
295	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
296	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
297	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
298	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
299	C₅	#65992	1	6	yes	complemented
300	C₂²	—	2	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	25	2	no	—
306	C₂²	—	50	1	no	—
307	C₂²	—	50	1	no	—
308	C₂²	—	50	1	no	—
309	C₂²	—	50	1	no	—
310	C₂²	—	50	1	no	—
311	C₂²	—	100	4	no	—
312	C₂²	#87902	1	2	yes	—
313	C₂²	C₂³×C₅⋊D₅	1	1	yes	characteristic
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	2	1	no	—
319	C₄	—	50	1	no	—
320	C₄	—	50	1	no	—
321	C₄	—	50	1	no	—
322	C₄	—	50	1	no	—
323	C₄	—	50	1	no	—
324	C₄	—	50	1	no	—
325	C₂	—	2	2	no	—
326	C₂	—	4	1	no	—
327	C₂	—	50	2	no	—
328	C₂	—	100	1	no	—
329	C₂	C₂×#87902	1	1	yes	characteristic
330	C₂	#41×C₅⋊D₅	1	1	yes	characteristic
331	C₂	#11717522	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×C₅⋊D₅
commutator, socle, cosocle	C₂²×C₅²	C₂⁴
Fitting	C₅²×#123	C₂
prime radical	#11717257	C₂
hypercenter	#123	C₅⋊D₅
hypocenter	C₅²	#551
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ #123

Complements

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

N	H ≅ G/N	type
1. C₅²×#123	C₂	semidirect
3. D₄×#1039	C₂	semidirect
4. C₂×#87808	C₂	semidirect
5. #11717246	C₂	semidirect
6. #11717218	C₂	semidirect
7. #11717218	C₂	semidirect
8. #11717217	C₂	semidirect
9. #11717212	C₂	semidirect
10. #11717206	C₂	semidirect
11. #11717205	C₂	semidirect
12. #11717203	C₂	semidirect
13. #11717201	C₂	semidirect
14. C₂²⋊C₄×C₅⋊D₅	C₂	semidirect
15. #11717198	C₂	semidirect
22. C₂²×#1039	C₂²	semidirect
25. C₂×#6045	C₂²	semidirect
26. C₅²×C₄⋊C₄	C₂²	semidirect
27. C₅²×C₄⋊C₄	C₂²	semidirect
29. C₅²×C₂²⋊C₄	C₂²	semidirect
30. C₅²×C₂²⋊C₄	C₂²	semidirect
33. C₂²⋊#1039	C₂²	semidirect
42. C₄⋊#1039	C₂²	semidirect
43. C₄⋊#1039	C₂²	semidirect
44. C₄⋊#1039	C₂²	semidirect
45. C₄⋊#1039	C₂²	semidirect
46. #87806	C₂²	semidirect
47. #87806	C₂²	semidirect
48. C₄×#1039	C₂²	semidirect
49. C₄×#1039	C₂²	semidirect
50. C₄×#1039	C₂²	semidirect
79. C₅×#123	D₅	semidirect
129. C₅×C₄⋊C₄	C₂×D₅	semidirect
130. C₅×C₄⋊C₄	C₂×D₅	semidirect
132. C₅×C₂²⋊C₄	C₂×D₅	semidirect
133. C₅×C₂²⋊C₄	C₂×D₅	semidirect
194. #123	C₅⋊D₅	semidirect
204. C₅²	#551	semidirect
246. C₄⋊C₄	C₂×C₅⋊D₅	semidirect
247. C₄⋊C₄	C₂×C₅⋊D₅	semidirect
260. C₂²⋊C₄	C₂×C₅⋊D₅	semidirect
261. C₂²⋊C₄	C₂×C₅⋊D₅	semidirect
299. C₅	#65992	semidirect

Maximal subgroups

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

subgroup	index
C₅²×#123	2
#11717257	2
D₄×#1039	2
C₂×#87808	2
#11717246	2
#11717218	2
#11717217	2
#11717212	2
#11717206	2
#11717205	2
#11717203	2
#11717201	2
C₂²⋊C₄×C₅⋊D₅	2
#11717198	2
#65992	5