Subgroups — #13107149

Order 1600.

880 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅²×#121	C₂	1	1	yes	maximal, characteristic, complemented
2	#11717259	C₂	1	1	yes	maximal, characteristic, complemented
3	#11717259	C₂	1	1	yes	maximal, characteristic, complemented
4	#11717258	C₂	1	1	yes	maximal, characteristic, complemented
5	#11717256	C₂	1	1	yes	maximal, characteristic, complemented
6	#11717255	C₂	1	1	yes	maximal, characteristic, complemented
7	#11717255	C₂	1	1	yes	maximal, characteristic, complemented
8	#11717249	C₂	1	1	yes	maximal, characteristic, complemented
9	#11717244	C₂	1	1	yes	maximal, characteristic, complemented
10	#11717216	C₂	1	1	yes	maximal, characteristic, complemented
11	#11717209	C₂	1	1	yes	maximal, characteristic, complemented
12	#11717205	C₂	1	1	yes	maximal, characteristic, complemented
13	#11717203	C₂	1	1	yes	maximal, characteristic, complemented
14	#11717198	C₂	1	1	yes	maximal, characteristic, complemented
15	#11717198	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₅²×D₄	C₂²	1	1	yes	characteristic
17	C₂×C₅²×D₄	C₂²	1	1	yes	characteristic
18	C₂×C₅²×D₄	C₂²	1	1	yes	characteristic
19	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊(C₅⋊D₅)	C₂²	1	1	yes	characteristic
21	C₂×C₂²⋊(C₅⋊D₅)	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊(C₅⋊D₅)	C₂²	1	1	yes	characteristic
23	C₂²×#1039	C₂²	1	1	yes	characteristic
24	C₂²×#1039	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₅⋊D₅	C₂²	1	1	yes	characteristic
26	C₂×#6045	C₂²	1	1	yes	characteristic
27	C₂×#6045	C₂²	1	1	yes	characteristic
28	C₅²×C₄⋊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	C₂²⋊#1039	C₂²	1	1	yes	characteristic
32	C₂²⋊#1039	C₂²	1	1	yes	characteristic
33	C₂²⋊#1039	C₂²	1	1	yes	characteristic
34	C₂²⋊#1039	C₂²	1	1	yes	characteristic
35	C₂²⋊#1039	C₂²	1	1	yes	characteristic
36	C₂²⋊#1039	C₂²	1	1	yes	characteristic
37	C₂²⋊#1039	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	C₄⋊#1039	C₂²	1	1	yes	characteristic
42	C₄⋊#1039	C₂²	1	1	yes	characteristic
43	#87806	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	#65955	—	5	6	no	maximal
52	C₂³×C₅²	C₂³	1	1	yes	characteristic
53	C₂³×C₅²	C₂³	1	1	yes	characteristic
54	C₂³×C₅²	C₂³	1	1	yes	characteristic
55	C₂²×C₅⋊D₅	C₂³	1	1	yes	characteristic
56	C₅²×D₄	—	4	1	no	—
57	C₅²×D₄	—	4	1	no	—
58	C₅²×D₄	—	4	1	no	—
59	C₂×C₄×C₅²	—	4	1	no	—
60	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
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	C₂²⋊(C₅⋊D₅)	—	4	1	no	—
65	C₂²⋊(C₅⋊D₅)	—	4	1	no	—
66	C₂²⋊(C₅⋊D₅)	—	4	1	no	—
67	C₂×#1039	—	4	1	no	—
68	C₂×#1039	—	4	1	no	—
69	C₂×#1039	C₂³	1	1	yes	characteristic
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	—	4	1	no	—
77	#6045	—	4	1	no	—
78	#6045	—	4	1	no	—
79	C₅×#121	D₅	1	6	yes	complemented
80	#4085	—	5	6	no	—
81	#4085	—	5	6	no	—
82	#4084	—	5	6	no	—
83	#4082	—	5	6	no	—
84	#4081	—	5	6	no	—
85	#4081	—	5	6	no	—
86	#4075	—	5	6	no	—
87	#4070	—	5	6	no	—
88	#4042	—	5	6	no	—
89	#4035	—	5	6	no	—
90	#4031	—	5	6	no	—
91	#4029	—	5	6	no	—
92	#4024	—	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	—	2	1	no	—
100	C₂²×C₅²	—	2	1	no	—
101	C₂²×C₅²	—	2	1	no	—
102	C₂²×C₅²	—	2	1	no	—
103	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
104	C₂×C₅⋊D₅	—	2	1	no	—
105	C₂×C₅⋊D₅	—	2	1	no	—
106	C₂×C₅⋊D₅	—	2	1	no	—
107	C₄×C₅²	—	2	1	no	—
108	C₄×C₅²	—	2	1	no	—
109	C₄×C₅²	—	2	1	no	—
110	C₄×C₅²	—	2	1	no	—
111	#1039	—	2	1	no	—
112	#1039	—	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	C₂×C₅×D₄	C₂×D₅	1	6	yes	—
119	C₂×C₅×D₄	C₂×D₅	1	6	yes	—
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	C₂×C₂²⋊D₅	—	5	6	no	—
125	C₂²×Dic₅	—	5	6	no	—
126	C₂²×Dic₅	—	5	6	no	—
127	C₂×C₄×D₅	—	5	6	no	—
128	C₂×Dic₁₀	—	5	6	no	—
129	C₂×Dic₁₀	—	5	6	no	—
130	C₅×C₄⋊C₄	C₂×D₅	1	6	yes	—
131	C₅×C₂²⋊C₄	C₂×D₅	1	6	yes	—
132	C₅×C₂²⋊C₄	C₂×D₅	1	6	yes	—
133	C₂²⋊Dic₅	—	5	6	no	—
134	C₂²⋊Dic₅	—	5	6	no	—
135	C₂²⋊Dic₅	—	5	6	no	—
136	C₂²⋊Dic₅	—	5	6	no	—
137	C₂²⋊Dic₅	—	5	6	no	—
138	C₂²⋊Dic₅	—	5	6	no	—
139	C₂²⋊Dic₅	—	5	6	no	—
140	#685	—	5	6	no	—
141	#685	—	5	6	no	—
142	#685	—	5	6	no	—
143	C₄⋊Dic₅	—	5	6	no	—
144	C₄⋊Dic₅	—	5	6	no	—
145	#683	—	5	6	no	—
146	#683	—	5	6	no	—
147	#683	—	5	6	no	—
148	#683	—	5	6	no	—
149	#683	—	5	6	no	—
150	#683	—	5	6	no	—
151	#683	—	5	6	no	—
152	C₄×Dic₅	—	5	6	no	—
153	#562	—	25	1	no	Sylow
154	C₂×C₅²	—	4	1	no	—
155	C₂×C₅²	—	4	1	no	—
156	C₂×C₅²	—	4	1	no	—
157	C₂×C₅²	H(5,𝔽₂)	1	1	yes	characteristic
158	C₂×C₅²	H(5,𝔽₂)	1	1	yes	characteristic
159	C₂×C₅²	2₋¹⁺⁴	1	1	yes	characteristic
160	C₅⋊D₅	—	4	1	no	—
161	C₂³×C₅	C₂²×D₅	1	6	yes	—
162	C₂³×C₅	C₂²×D₅	1	6	yes	—
163	C₂³×C₅	C₂²×D₅	1	6	yes	—
164	C₂²×D₅	—	5	6	no	—
165	C₅×D₄	—	4	6	no	—
166	C₅×D₄	—	4	6	no	—
167	C₅×D₄	—	4	6	no	—
168	C₂×C₄×C₅	—	4	6	no	—
169	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
170	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
171	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
172	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
173	C₂²⋊D₅	—	20	6	no	—
174	C₂²⋊D₅	—	20	6	no	—
175	C₂²⋊D₅	—	20	6	no	—
176	C₂×Dic₅	—	5	6	no	—
177	C₂×Dic₅	—	5	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	—	20	6	no	—
184	C₂×Dic₅	—	20	6	no	—
185	C₄×D₅	—	20	6	no	—
186	Dic₁₀	—	20	6	no	—
187	Dic₁₀	—	20	6	no	—
188	#125	—	25	1	no	—
189	#124	—	25	1	no	—
190	#124	—	25	1	no	—
191	#123	—	25	1	no	—
192	#123	—	25	1	no	—
193	#123	—	25	1	no	—
194	#123	—	25	1	no	—
195	C₂²⋊Q₈	—	25	1	no	—
196	C₂²⋊Q₈	—	25	1	no	—
197	C₂²⋊Q₈	—	25	1	no	—
198	C₂²⋊Q₈	—	25	1	no	—
199	#121	—	25	1	no	—
200	#121	—	25	1	no	—
201	#121	—	25	1	no	—
202	#121	C₅⋊D₅	1	1	yes	characteristic, complemented
203	C₅²	#562	1	1	yes	characteristic, complemented, Sylow
204	C₂²×C₅	—	2	6	no	—
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	—	2	6	no	—
211	C₂²×C₅	—	2	6	no	—
212	C₂²×C₅	—	2	6	no	—
213	C₂²×C₅	C₂³×D₅	1	6	yes	—
214	C₂×D₅	—	10	6	no	—
215	C₂×D₅	—	10	6	no	—
216	C₂×D₅	—	10	6	no	—
217	C₄×C₅	—	2	6	no	—
218	C₄×C₅	—	2	6	no	—
219	C₄×C₅	—	2	6	no	—
220	C₄×C₅	—	2	6	no	—
221	Dic₅	—	10	6	no	—
222	Dic₅	—	10	6	no	—
223	Dic₅	—	10	6	no	—
224	Dic₅	—	10	6	no	—
225	Dic₅	—	10	6	no	—
226	Dic₅	—	10	6	no	—
227	Dic₅	—	10	6	no	—
228	C₂×Q₈	—	25	1	no	—
229	C₂×Q₈	—	25	1	no	—
230	C₂×D₄	—	25	1	no	—
231	C₂×D₄	—	25	1	no	—
232	C₂×D₄	—	25	1	no	—
233	C₂×D₄	C₂×C₅⋊D₅	1	1	yes	characteristic
234	C₂×D₄	C₂×C₅⋊D₅	1	1	yes	characteristic
235	C₂×D₄	C₂×C₅⋊D₅	1	1	yes	characteristic
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	—	25	1	no	—
247	C₄⋊C₄	—	25	1	no	—
248	C₄⋊C₄	—	25	1	no	—
249	C₄⋊C₄	C₂×C₅⋊D₅	1	1	yes	characteristic
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	—	25	1	no	—
260	C₂²⋊C₄	C₂×C₅⋊D₅	1	1	yes	characteristic
261	C₂²⋊C₄	C₂×C₅⋊D₅	1	1	yes	characteristic
262	C₄²	—	25	1	no	—
263	C₂×C₅	—	4	6	no	—
264	C₂×C₅	—	4	6	no	—
265	C₂×C₅	—	4	6	no	—
266	C₂×C₅	#4144	1	6	yes	—
267	C₂×C₅	#4144	1	6	yes	—
268	C₂×C₅	#4150	1	6	yes	—
269	D₅	—	20	6	no	—
270	C₂³	—	25	1	no	—
271	C₂³	C₂²×C₅⋊D₅	1	1	yes	characteristic
272	C₂³	C₂²×C₅⋊D₅	1	1	yes	characteristic
273	C₂³	C₂²×C₅⋊D₅	1	1	yes	characteristic
274	Q₈	—	100	1	no	—
275	Q₈	—	100	1	no	—
276	D₄	—	4	1	no	—
277	D₄	—	4	1	no	—
278	D₄	—	4	1	no	—
279	D₄	—	100	1	no	—
280	D₄	—	100	1	no	—
281	D₄	—	100	1	no	—
282	C₂×C₄	—	4	1	no	—
283	C₂×C₄	—	25	1	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	—	100	1	no	—
291	C₂×C₄	—	100	1	no	—
292	C₂×C₄	—	100	1	no	—
293	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
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	C₅	#65955	1	6	yes	complemented
298	C₂²	—	2	1	no	—
299	C₂²	—	2	1	no	—
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	—	2	1	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	50	1	no	—
308	C₂²	—	50	1	no	—
309	C₂²	—	50	1	no	—
310	C₂²	C₂³×C₅⋊D₅	1	1	yes	characteristic
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	50	1	no	—
316	C₄	—	50	1	no	—
317	C₄	—	50	1	no	—
318	C₄	—	50	1	no	—
319	C₄	—	50	1	no	—
320	C₄	—	50	1	no	—
321	C₄	—	50	1	no	—
322	C₂	—	4	1	no	—
323	C₂	—	4	1	no	—
324	C₂	—	4	1	no	—
325	C₂	—	100	1	no	—
326	C₂	#11717517	1	1	yes	characteristic
327	C₂	#11717517	1	1	yes	characteristic
328	C₂	#11717523	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×C₅⋊D₅
commutator, socle, cosocle	C₂²×C₅²	C₂⁴
Fitting	C₅²×#121	C₂
hypercenter	#121	C₅⋊D₅
hypocenter	C₅²	#562
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ #121

Complements

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

N	H ≅ G/N	type
1. C₅²×#121	C₂	semidirect
2. #11717259	C₂	semidirect
3. #11717259	C₂	semidirect
4. #11717258	C₂	semidirect
5. #11717256	C₂	semidirect
6. #11717255	C₂	semidirect
7. #11717255	C₂	semidirect
8. #11717249	C₂	semidirect
9. #11717244	C₂	semidirect
10. #11717216	C₂	semidirect
11. #11717209	C₂	semidirect
12. #11717205	C₂	semidirect
13. #11717203	C₂	semidirect
14. #11717198	C₂	semidirect
15. #11717198	C₂	semidirect
79. C₅×#121	D₅	semidirect
202. #121	C₅⋊D₅	semidirect
203. C₅²	#562	semidirect
297. C₅	#65955	semidirect

Maximal subgroups

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

subgroup	index
C₅²×#121	2
#11717259	2
#11717258	2
#11717256	2
#11717255	2
#11717249	2
#11717244	2
#11717216	2
#11717209	2
#11717205	2
#11717203	2
#11717198	2
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