Subgroups — #13107236

Order 1600.

872 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅²×#126	C₂	1	1	yes	maximal, characteristic, complemented
2	#11717218	C₂	1	1	yes	maximal, characteristic, complemented
3	#11717217	C₂	1	1	yes	maximal, characteristic, complemented
4	#11717216	C₂	1	1	yes	maximal, characteristic, complemented
5	#11717214	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄⋊C₄×C₅⋊D₅	C₂	1	1	yes	maximal, characteristic, complemented
7	#11717210	C₂	1	1	yes	maximal, characteristic, complemented
8	#11717206	C₂	1	1	yes	maximal, characteristic, complemented
9	#11717204	C₂	1	1	yes	maximal, characteristic
10	#11717203	C₂	1	1	yes	maximal, characteristic, complemented
11	#11717203	C₂	1	1	yes	maximal, characteristic, complemented
12	#11717201	C₂	1	1	yes	maximal, characteristic, complemented
13	#11717198	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×C₄⋊(C₅⋊D₅)	C₂	1	1	yes	maximal, characteristic, complemented
15	#11717190	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₂²⋊(C₅⋊D₅)	C₂²	1	1	yes	characteristic
17	C₂×C₂²⋊(C₅⋊D₅)	C₂²	1	1	yes	characteristic
18	C₂²×#1039	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊(C₅⋊D₅)	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₅⋊D₅	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₅⋊D₅	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₅⋊D₅	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₅⋊D₅	C₂²	1	1	yes	characteristic
24	C₂×#6045	C₂²	1	1	yes	characteristic
25	C₅²×C₄⋊C₄	C₂²	1	1	yes	characteristic
26	C₅²×C₄⋊C₄	C₂²	1	1	yes	characteristic
27	C₅²×C₄⋊C₄	C₂²	1	1	yes	characteristic
28	C₅²×C₂²⋊C₄	C₂²	1	1	yes	characteristic
29	C₅²×C₂²⋊C₄	C₂²	1	1	yes	characteristic
30	C₅²×C₂²⋊C₄	C₂²	1	1	yes	characteristic
31	C₄²×C₅²	C₂²	1	1	yes	characteristic
32	C₂²⋊#1039	C₂²	1	1	yes	characteristic
33	#87808	C₂²	1	1	yes	characteristic
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	C₄⋊#1039	C₂²	1	1	yes	characteristic
40	C₄⋊#1039	C₂²	1	1	yes	characteristic
41	C₄⋊#1039	C₂²	1	1	yes	characteristic
42	C₄⋊#1039	C₂²	1	1	yes	characteristic
43	C₄⋊#1039	C₂²	1	1	yes	characteristic
44	#87806	C₂²	1	1	yes	characteristic
45	#87806	C₂²	1	1	yes	characteristic
46	#87806	C₂²	1	1	yes	characteristic
47	#87806	C₂²	1	1	yes	characteristic
48	#87806	C₂²	1	1	yes	characteristic
49	#87806	C₂²	1	1	yes	characteristic
50	C₄×#1039	C₂²	1	1	yes	characteristic
51	#66041	—	5	6	no	maximal
52	C₂³×C₅²	C₂³	1	1	yes	characteristic
53	C₂²×C₅⋊D₅	C₂³	1	1	yes	characteristic
54	C₂²×C₅⋊D₅	C₂³	1	1	yes	characteristic
55	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
56	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
57	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
58	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
59	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
60	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
61	C₂²⋊(C₅⋊D₅)	—	2	2	no	—
62	C₂²⋊(C₅⋊D₅)	—	4	1	no	—
63	C₂×#1039	—	4	1	no	—
64	C₂×#1039	C₂³	1	1	yes	characteristic
65	C₂×#1039	C₂³	1	1	yes	characteristic
66	C₂×#1039	C₂³	1	1	yes	characteristic
67	C₂×#1039	C₂³	1	1	yes	characteristic
68	C₂×#1039	C₂³	1	1	yes	characteristic
69	C₂×#1039	C₂³	1	1	yes	characteristic
70	C₄⋊(C₅⋊D₅)	—	2	2	no	—
71	C₄×C₅⋊D₅	—	2	2	no	—
72	C₄×C₅⋊D₅	—	4	1	no	—
73	C₄×C₅⋊D₅	—	4	1	no	—
74	C₄×C₅⋊D₅	—	4	1	no	—
75	#6045	—	4	1	no	—
76	C₅×#126	D₅	1	6	yes	complemented
77	#4044	—	5	6	no	—
78	#4043	—	5	6	no	—
79	#4042	—	5	6	no	—
80	#4040	—	5	6	no	—
81	D₅×C₄⋊C₄	—	5	6	no	—
82	#4036	—	5	6	no	—
83	#4032	—	5	6	no	—
84	#4030	—	5	6	no	—
85	#4029	—	5	6	no	—
86	#4029	—	5	6	no	—
87	#4027	—	5	6	no	—
88	#4024	—	5	6	no	—
89	C₄×D₂₀	—	5	6	no	—
90	#4016	—	5	6	no	—
91	C₂²×C₅²	—	2	1	no	—
92	C₂²×C₅²	—	2	1	no	—
93	C₂²×C₅²	—	2	1	no	—
94	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
95	C₂×C₅⋊D₅	—	2	1	no	—
96	C₂×C₅⋊D₅	—	2	1	no	—
97	C₂×C₅⋊D₅	—	2	1	no	—
98	C₂×C₅⋊D₅	—	2	1	no	—
99	C₂×C₅⋊D₅	—	2	1	no	—
100	C₂×C₅⋊D₅	#41	1	2	yes	—
101	C₄×C₅²	—	2	1	no	—
102	C₄×C₅²	—	2	1	no	—
103	C₄×C₅²	—	2	1	no	—
104	C₄×C₅²	—	2	1	no	—
105	C₄×C₅²	—	2	1	no	—
106	C₄×C₅²	—	2	1	no	—
107	#1039	—	2	1	no	—
108	#1039	—	2	1	no	—
109	#1039	—	2	1	no	—
110	#1039	—	2	1	no	—
111	#1039	—	2	1	no	—
112	#1039	—	2	1	no	—
113	C₂×C₂²⋊D₅	—	5	6	no	—
114	C₂×C₂²⋊D₅	—	5	6	no	—
115	C₂²×Dic₅	—	5	6	no	—
116	C₂×D₂₀	—	5	6	no	—
117	C₂×C₄×D₅	—	5	6	no	—
118	C₂×C₄×D₅	—	5	6	no	—
119	C₂×C₄×D₅	—	5	6	no	—
120	C₂×C₄×D₅	—	5	6	no	—
121	C₂×Dic₁₀	—	5	6	no	—
122	C₅×C₄⋊C₄	C₂×D₅	1	6	yes	—
123	C₅×C₄⋊C₄	C₂×D₅	1	6	yes	—
124	C₅×C₄⋊C₄	C₂×D₅	1	6	yes	—
125	C₅×C₂²⋊C₄	C₂×D₅	1	6	yes	—
126	C₅×C₂²⋊C₄	C₂×D₅	1	6	yes	—
127	C₅×C₂²⋊C₄	C₂×D₅	1	6	yes	—
128	C₄²×C₅	C₂×D₅	1	6	yes	—
129	C₂²⋊Dic₅	—	5	6	no	—
130	#685	—	5	6	no	—
131	#685	—	5	6	no	—
132	#685	—	5	6	no	—
133	#685	—	5	6	no	—
134	#685	—	5	6	no	—
135	#685	—	5	6	no	—
136	C₄⋊Dic₅	—	5	6	no	—
137	C₄⋊Dic₅	—	5	6	no	—
138	C₄⋊Dic₅	—	5	6	no	—
139	C₄⋊Dic₅	—	5	6	no	—
140	C₄⋊Dic₅	—	5	6	no	—
141	#683	—	5	6	no	—
142	#683	—	5	6	no	—
143	#683	—	5	6	no	—
144	#683	—	5	6	no	—
145	#683	—	5	6	no	—
146	#683	—	5	6	no	—
147	C₄×Dic₅	—	5	6	no	—
148	#539	—	25	1	no	Sylow
149	C₂×C₅²	—	4	1	no	—
150	C₂×C₅²	C₂×#41	1	1	yes	characteristic
151	C₂×C₅²	H(5,𝔽₂)	1	1	yes	characteristic
152	C₂×C₅²	2₋¹⁺⁴	1	1	yes	characteristic
153	C₅⋊D₅	—	2	2	no	—
154	C₅⋊D₅	—	4	1	no	—
155	C₂³×C₅	C₂²×D₅	1	6	yes	—
156	C₂²×D₅	—	5	6	no	—
157	C₂²×D₅	—	5	6	no	—
158	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
159	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
160	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
161	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
162	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
163	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
164	C₂²⋊D₅	—	10	12	no	—
165	C₂²⋊D₅	—	20	6	no	—
166	C₂×Dic₅	—	5	6	no	—
167	C₂×Dic₅	—	5	6	no	—
168	C₂×Dic₅	—	5	6	no	—
169	C₂×Dic₅	—	5	6	no	—
170	C₂×Dic₅	—	5	6	no	—
171	C₂×Dic₅	—	5	6	no	—
172	C₂×Dic₅	—	20	6	no	—
173	D₂₀	—	10	12	no	—
174	C₄×D₅	—	10	12	no	—
175	C₄×D₅	—	20	6	no	—
176	C₄×D₅	—	20	6	no	—
177	C₄×D₅	—	20	6	no	—
178	Dic₁₀	—	20	6	no	—
179	#126	—	25	1	no	—
180	#126	C₅⋊D₅	1	1	yes	characteristic, complemented
181	#125	—	25	1	no	—
182	#125	—	25	1	no	—
183	#123	—	25	1	no	—
184	#123	—	25	1	no	—
185	#123	—	25	1	no	—
186	#123	—	25	1	no	—
187	C₂²⋊Q₈	—	25	1	no	—
188	C₂²⋊Q₈	—	25	1	no	—
189	C₂²⋊Q₈	—	25	1	no	—
190	#121	—	25	1	no	—
191	C₄×D₄	—	25	1	no	—
192	C₄×D₄	—	25	1	no	—
193	C₂×C₄⋊C₄	—	25	1	no	—
194	C₅²	#539	1	1	yes	characteristic, complemented, Sylow
195	C₂²×C₅	—	2	6	no	—
196	C₂²×C₅	—	2	6	no	—
197	C₂²×C₅	—	2	6	no	—
198	C₂²×C₅	C₂³×D₅	1	6	yes	—
199	C₂×D₅	—	5	12	no	—
200	C₂×D₅	—	10	6	no	—
201	C₂×D₅	—	10	6	no	—
202	C₂×D₅	—	10	6	no	—
203	C₂×D₅	—	10	6	no	—
204	C₂×D₅	—	10	6	no	—
205	C₄×C₅	—	2	6	no	—
206	C₄×C₅	—	2	6	no	—
207	C₄×C₅	—	2	6	no	—
208	C₄×C₅	—	2	6	no	—
209	C₄×C₅	—	2	6	no	—
210	C₄×C₅	—	2	6	no	—
211	Dic₅	—	10	6	no	—
212	Dic₅	—	10	6	no	—
213	Dic₅	—	10	6	no	—
214	Dic₅	—	10	6	no	—
215	Dic₅	—	10	6	no	—
216	Dic₅	—	10	6	no	—
217	C₂×Q₈	—	25	1	no	—
218	C₂×D₄	—	25	1	no	—
219	C₂×D₄	—	25	1	no	—
220	C₂×D₄	—	25	1	no	—
221	C₂²×C₄	—	25	1	no	—
222	C₂²×C₄	—	25	1	no	—
223	C₂²×C₄	—	25	1	no	—
224	C₂²×C₄	—	25	1	no	—
225	C₂²×C₄	—	25	1	no	—
226	C₄⋊C₄	—	25	1	no	—
227	C₄⋊C₄	—	25	1	no	—
228	C₄⋊C₄	—	25	1	no	—
229	C₄⋊C₄	—	25	1	no	—
230	C₄⋊C₄	—	25	1	no	—
231	C₄⋊C₄	—	25	1	no	—
232	C₄⋊C₄	—	25	1	no	—
233	C₄⋊C₄	—	25	1	no	—
234	C₄⋊C₄	—	25	1	no	—
235	C₄⋊C₄	—	25	1	no	—
236	C₄⋊C₄	—	25	1	no	—
237	C₄⋊C₄	C₂×C₅⋊D₅	1	1	yes	characteristic
238	C₄⋊C₄	C₂×C₅⋊D₅	1	1	yes	characteristic
239	C₄⋊C₄	C₂×C₅⋊D₅	1	1	yes	characteristic
240	C₂²⋊C₄	—	25	1	no	—
241	C₂²⋊C₄	—	25	1	no	—
242	C₂²⋊C₄	—	25	1	no	—
243	C₂²⋊C₄	—	25	1	no	—
244	C₂²⋊C₄	—	25	1	no	—
245	C₂²⋊C₄	—	25	1	no	—
246	C₂²⋊C₄	—	25	1	no	—
247	C₂²⋊C₄	C₂×C₅⋊D₅	1	1	yes	characteristic
248	C₂²⋊C₄	C₂×C₅⋊D₅	1	1	yes	characteristic
249	C₂²⋊C₄	C₂×C₅⋊D₅	1	1	yes	characteristic
250	C₄²	—	25	1	no	—
251	C₄²	C₂×C₅⋊D₅	1	1	yes	characteristic
252	C₂×C₅	—	4	6	no	—
253	C₂×C₅	D₅×#41	1	6	yes	—
254	C₂×C₅	#4149	1	6	yes	—
255	C₂×C₅	#4150	1	6	yes	—
256	D₅	—	10	12	no	—
257	D₅	—	20	6	no	—
258	C₂³	—	25	1	no	—
259	C₂³	—	25	1	no	—
260	C₂³	C₂²×C₅⋊D₅	1	1	yes	characteristic
261	Q₈	—	100	1	no	—
262	D₄	—	50	2	no	—
263	D₄	—	50	2	no	—
264	D₄	—	100	1	no	—
265	C₂×C₄	—	25	1	no	—
266	C₂×C₄	—	25	1	no	—
267	C₂×C₄	—	25	1	no	—
268	C₂×C₄	—	25	1	no	—
269	C₂×C₄	—	25	1	no	—
270	C₂×C₄	—	25	1	no	—
271	C₂×C₄	—	50	2	no	—
272	C₂×C₄	—	100	1	no	—
273	C₂×C₄	—	100	1	no	—
274	C₂×C₄	—	100	1	no	—
275	C₂×C₄	—	100	1	no	—
276	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
277	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
278	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
279	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
280	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
281	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
282	C₅	#66041	1	6	yes	complemented
283	C₂²	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	—	2	1	no	—
286	C₂²	—	25	2	no	—
287	C₂²	—	50	1	no	—
288	C₂²	—	50	1	no	—
289	C₂²	—	50	1	no	—
290	C₂²	—	50	1	no	—
291	C₂²	—	50	1	no	—
292	C₂²	C₂³×C₅⋊D₅	1	1	yes	characteristic
293	C₄	—	2	1	no	—
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	50	1	no	—
300	C₄	—	50	1	no	—
301	C₄	—	50	1	no	—
302	C₄	—	50	1	no	—
303	C₄	—	50	1	no	—
304	C₄	—	50	1	no	—
305	C₂	—	4	1	no	—
306	C₂	—	50	2	no	—
307	C₂	—	100	1	no	—
308	C₂	#41×C₅⋊D₅	1	1	yes	characteristic
309	C₂	#11717522	1	1	yes	characteristic
310	C₂	#11717523	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×C₅⋊D₅
commutator, socle, cosocle	C₂²×C₅²	C₂⁴
Fitting	C₅²×#126	C₂
prime radical	#11717204	C₂
hypercenter	#126	C₅⋊D₅
hypocenter	C₅²	#539
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ #126

Complements

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

N	H ≅ G/N	type
1. C₅²×#126	C₂	semidirect
2. #11717218	C₂	semidirect
3. #11717217	C₂	semidirect
4. #11717216	C₂	semidirect
5. #11717214	C₂	semidirect
6. C₄⋊C₄×C₅⋊D₅	C₂	semidirect
7. #11717210	C₂	semidirect
8. #11717206	C₂	semidirect
10. #11717203	C₂	semidirect
11. #11717203	C₂	semidirect
12. #11717201	C₂	semidirect
13. #11717198	C₂	semidirect
14. C₄×C₄⋊(C₅⋊D₅)	C₂	semidirect
15. #11717190	C₂	semidirect
76. C₅×#126	D₅	semidirect
180. #126	C₅⋊D₅	semidirect
194. C₅²	#539	semidirect
282. C₅	#66041	semidirect

Maximal subgroups

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

subgroup	index
C₅²×#126	2
#11717218	2
#11717217	2
#11717216	2
#11717214	2
C₄⋊C₄×C₅⋊D₅	2
#11717210	2
#11717206	2
#11717204	2
#11717203	2
#11717201	2
#11717198	2
C₄×C₄⋊(C₅⋊D₅)	2
#11717190	2
#66041	5