Subgroups — #12007573

Order 1280.

634 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#3950	C₂	1	1	yes	maximal, characteristic
2	C₅×#1729	C₂	1	1	yes	maximal, characteristic
3	C₂×#64774	C₂	1	2	yes	maximal
4	C₂×#64772	C₂	1	1	yes	maximal, characteristic
5	#10602374	C₂	1	2	yes	maximal
6	C₂²×C₅×M₁₆	C₂²	1	1	yes	characteristic
7	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₅×#117	C₂²	1	1	yes	characteristic
9	C₂²×C₄⋊Dic₅	C₂²	1	2	yes	—
10	C₂²×#681	C₂²	1	2	yes	—
11	C₂×#3950	—	2	4	no	—
12	C₂×#3950	—	2	2	no	—
13	C₅×#336	—	2	2	no	—
14	#64774	—	2	4	no	—
15	#64772	C₄	1	4	yes	complemented
16	#64749	—	2	4	no	—
17	#11270	—	5	1	no	maximal, Sylow
18	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
19	C₂³×Dic₅	—	2	2	no	—
20	C₂×C₅×M₁₆	—	2	1	no	—
21	C₂×C₅×M₁₆	—	2	2	no	—
22	C₂×C₅×M₁₆	—	2	2	no	—
23	C₂×C₅×M₁₆	Q₈	1	2	yes	—
24	C₂²×C₅×C₈	—	2	2	no	—
25	C₂²×C₅×C₈	—	2	2	no	—
26	C₂²×C₅×C₈	—	2	1	no	—
27	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
28	C₂²×C₅×C₈	D₄	1	2	yes	—
29	C₂²×C₅×C₈	Q₈	1	2	yes	—
30	C₅×#117	—	2	2	no	—
31	C₅×#117	—	2	2	no	—
32	C₂×C₅×C₄⋊C₄	D₄	1	2	yes	—
33	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
34	C₂×C₄²×C₅	—	2	1	no	—
35	C₂×C₄⋊Dic₅	—	2	2	no	—
36	C₂×C₄⋊Dic₅	—	2	2	no	—
37	C₂×C₄⋊Dic₅	—	2	2	no	—
38	C₂×C₄⋊Dic₅	—	2	2	no	—
39	C₂×C₄⋊Dic₅	—	2	2	no	—
40	C₂×C₄⋊Dic₅	D₄	1	4	yes	—
41	C₂×#681	—	2	2	no	—
42	C₂×#681	—	2	2	no	—
43	C₂×#681	—	2	2	no	—
44	C₂×#681	—	2	2	no	—
45	C₂×#681	—	2	2	no	—
46	C₂×#681	C₂×C₄	1	4	yes	—
47	C₂²×#168	—	2	2	no	—
48	#3950	—	2	8	no	—
49	C₂²×#107	—	5	1	no	—
50	C₂×#343	—	5	2	no	—
51	C₂×#341	—	5	1	no	—
52	#1736	—	5	2	no	—
53	#1729	D₅	1	1	yes	characteristic
54	C₂⁴×C₅	#41	1	1	yes	characteristic
55	C₂²×C₄×C₅	—	2	2	no	—
56	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
59	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
61	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₄×C₅	#41	1	1	yes	characteristic
63	C₂²×C₄×C₅	#41	1	2	yes	—
64	C₂²×C₄×C₅	#41	1	2	yes	—
65	C₂²×C₄×C₅	#41	1	2	yes	—
66	C₂²×Dic₅	—	2	4	no	—
67	C₂²×Dic₅	—	4	2	no	—
68	C₂²×Dic₅	—	4	2	no	—
69	C₂²×Dic₅	—	8	4	no	—
70	C₅×M₁₆	—	2	4	no	—
71	C₅×M₁₆	—	4	2	no	—
72	C₂×C₅×C₈	—	2	4	no	—
73	C₂×C₅×C₈	—	2	2	no	—
74	C₂×C₅×C₈	—	2	4	no	—
75	C₂×C₅×C₈	—	2	2	no	—
76	C₂×C₅×C₈	—	2	2	no	—
77	C₂×C₅×C₈	—	4	1	no	—
78	C₂×C₅×C₈	—	4	1	no	—
79	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
80	C₅×C₄⋊C₄	—	2	4	no	—
81	C₅×C₂²⋊C₄	—	4	2	no	—
82	C₄²×C₅	—	4	1	no	—
83	C₄²×C₅	—	4	1	no	—
84	C₄⋊Dic₅	—	2	4	no	—
85	C₄⋊Dic₅	—	2	4	no	—
86	C₄⋊Dic₅	—	4	2	no	—
87	C₄⋊Dic₅	—	4	2	no	—
88	#681	—	2	4	no	—
89	#681	—	2	4	no	—
90	#681	—	4	2	no	—
91	#681	—	4	2	no	—
92	C₂×#168	—	2	4	no	—
93	C₂×#168	—	4	2	no	—
94	C₂×#168	—	4	2	no	—
95	C₂²×M₁₆	—	5	2	no	—
96	C₂²×M₁₆	C₂×D₅	1	1	yes	characteristic
97	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
98	C₂×#117	C₂×D₅	1	1	yes	characteristic
99	C₂²×C₄⋊C₄	—	5	2	no	—
100	C₂×#107	—	10	4	no	—
101	C₂×#107	—	10	2	no	—
102	#343	—	10	4	no	—
103	#341	—	5	4	no	—
104	#340	—	10	4	no	—
105	#336	—	2	2	no	—
106	C₂³×C₅	—	2	2	no	—
107	C₂³×C₅	—	2	2	no	—
108	C₂³×C₅	C₄×Q₈	1	1	yes	characteristic
109	C₂³×C₅	#121	1	1	yes	characteristic
110	C₂³×C₅	#123	1	2	yes	—
111	C₂³×C₅	#125	1	1	yes	characteristic
112	C₂³×C₅	#126	1	2	yes	—
113	C₂×C₄×C₅	—	4	2	no	—
114	C₂×C₄×C₅	—	4	2	no	—
115	C₂×C₄×C₅	—	4	2	no	—
116	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₅	#117	1	2	yes	—
118	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₅	#120	1	1	yes	characteristic
121	C₂×C₄×C₅	#121	1	2	yes	—
122	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
123	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
124	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
125	C₂×C₄×C₅	#123	1	2	yes	—
126	C₂×C₄×C₅	#123	1	1	yes	characteristic
127	C₂×C₄×C₅	#124	1	2	yes	—
128	C₂×C₄×C₅	#124	1	1	yes	characteristic
129	C₂×C₄×C₅	#124	1	1	yes	characteristic
130	C₂×C₄×C₅	#125	1	2	yes	—
131	C₂×C₄×C₅	#125	1	2	yes	—
132	C₂×C₄×C₅	#126	1	2	yes	—
133	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
134	C₂×Dic₅	—	4	4	no	—
135	C₂×Dic₅	—	8	4	no	—
136	C₂×Dic₅	—	8	4	no	—
137	C₂×Dic₅	—	8	4	no	—
138	C₅×C₈	—	2	4	no	—
139	C₅×C₈	—	4	2	no	—
140	#168	—	4	4	no	—
141	C₂³×C₄	—	10	2	no	—
142	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
143	C₂×M₁₆	—	2	1	no	—
144	C₂×M₁₆	—	2	2	no	—
145	C₂×M₁₆	—	2	2	no	—
146	C₂×M₁₆	—	5	4	no	—
147	C₂×M₁₆	—	10	2	no	—
148	C₂×M₁₆	—	10	2	no	—
149	C₂×M₁₆	—	10	2	no	—
150	C₂×M₁₆	—	10	2	no	—
151	C₂×M₁₆	—	10	2	no	—
152	C₂×M₁₆	Dic₁₀	1	2	yes	—
153	C₂²×C₈	—	2	2	no	—
154	C₂²×C₈	—	2	2	no	—
155	C₂²×C₈	—	2	1	no	—
156	C₂²×C₈	—	10	2	no	—
157	C₂²×C₈	Dic₁₀	1	2	yes	—
158	C₂²×C₈	C₄×D₅	1	2	yes	—
159	C₂²×C₈	C₂²⋊D₅	1	2	yes	—
160	#117	—	2	2	no	—
161	#117	—	2	2	no	—
162	C₂×C₄⋊C₄	—	5	4	no	—
163	C₂×C₄⋊C₄	—	10	2	no	—
164	C₂×C₄⋊C₄	—	10	2	no	—
165	C₂×C₄⋊C₄	—	10	2	no	—
166	C₂×C₄⋊C₄	—	10	2	no	—
167	C₂×C₄⋊C₄	—	10	2	no	—
168	C₂×C₄⋊C₄	D₂₀	1	2	yes	—
169	C₂×C₂²⋊C₄	—	2	1	no	—
170	C₂×C₄²	—	2	1	no	—
171	#107	—	10	8	no	—
172	C₂²×C₅	—	2	4	no	—
173	C₂²×C₅	—	2	2	no	—
174	C₂²×C₅	—	2	2	no	—
175	C₂²×C₅	—	2	2	no	—
176	C₂²×C₅	—	2	2	no	—
177	C₂²×C₅	#387	1	2	yes	—
178	C₂²×C₅	#389	1	1	yes	characteristic
179	C₂²×C₅	#394	1	1	yes	characteristic
180	C₂²×C₅	#397	1	1	yes	characteristic
181	C₂²×C₅	#400	1	2	yes	—
182	C₂²×C₅	#472	1	2	yes	—
183	C₂²×C₅	#473	1	2	yes	—
184	C₄×C₅	—	8	2	no	—
185	C₄×C₅	#382	1	1	yes	characteristic
186	C₄×C₅	#385	1	1	yes	characteristic
187	C₄×C₅	#388	1	2	yes	—
188	C₄×C₅	#393	1	1	yes	characteristic
189	C₄×C₅	#398	1	2	yes	—
190	C₄×C₅	#400	1	1	yes	characteristic
191	Dic₅	—	8	4	no	—
192	C₂⁴	#709	1	1	yes	characteristic
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	10	4	no	—
195	C₂²×C₄	—	20	2	no	—
196	C₂²×C₄	—	20	2	no	—
197	C₂²×C₄	—	40	4	no	—
198	C₂²×C₄	C₂×Dic₁₀	1	1	yes	characteristic
199	C₂²×C₄	C₂×Dic₁₀	1	1	yes	characteristic
200	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
201	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
202	C₂²×C₄	#709	1	1	yes	characteristic
203	C₂²×C₄	D₄×D₅	1	2	yes	—
204	C₂²×C₄	#711	1	2	yes	—
205	C₂²×C₄	#711	1	2	yes	—
206	C₂²×C₄	#713	1	2	yes	—
207	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
208	M₁₆	—	2	4	no	—
209	M₁₆	—	4	2	no	—
210	M₁₆	—	10	4	no	—
211	M₁₆	—	10	4	no	—
212	M₁₆	—	20	2	no	—
213	M₁₆	—	20	2	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	4	1	no	—
220	C₂×C₈	—	4	1	no	—
221	C₂×C₈	—	10	4	no	—
222	C₂×C₈	—	20	2	no	—
223	C₂×C₈	—	20	2	no	—
224	C₂×C₈	#683	1	4	yes	—
225	C₄⋊C₄	—	2	4	no	—
226	C₄⋊C₄	—	10	4	no	—
227	C₄⋊C₄	—	10	4	no	—
228	C₄⋊C₄	—	20	2	no	—
229	C₄⋊C₄	—	20	2	no	—
230	C₂²⋊C₄	—	4	2	no	—
231	C₄²	—	4	1	no	—
232	C₄²	—	4	1	no	—
233	C₂×C₅	—	2	2	no	—
234	C₂×C₅	—	2	2	no	—
235	C₂×C₅	#1442	1	1	yes	characteristic
236	C₂×C₅	#1952	1	2	yes	—
237	C₂×C₅	#2045	1	1	yes	characteristic
238	C₂×C₅	#2047	1	1	yes	characteristic
239	C₂×C₅	#2078	1	2	yes	—
240	C₂³	—	2	2	no	—
241	C₂³	—	2	2	no	—
242	C₂³	C₄×Dic₁₀	1	1	yes	characteristic
243	C₂³	#4016	1	1	yes	characteristic
244	C₂³	#4025	1	2	yes	—
245	C₂³	#4040	1	2	yes	—
246	C₂³	#4076	1	1	yes	characteristic
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	20	4	no	—
251	C₂×C₄	—	40	4	no	—
252	C₂×C₄	—	40	4	no	—
253	C₂×C₄	—	40	4	no	—
254	C₂×C₄	#4015	1	1	yes	characteristic
255	C₂×C₄	C₄×D₂₀	1	1	yes	characteristic
256	C₂×C₄	#4024	1	2	yes	—
257	C₂×C₄	#4024	1	2	yes	—
258	C₂×C₄	#4027	1	2	yes	—
259	C₂×C₄	#4031	1	2	yes	—
260	C₂×C₄	#4032	1	2	yes	—
261	C₂×C₄	#4036	1	2	yes	—
262	C₂×C₄	#4036	1	2	yes	—
263	C₂×C₄	#4038	1	2	yes	—
264	C₂×C₄	#4041	1	2	yes	—
265	C₂×C₄	#4044	1	2	yes	—
266	C₂×C₄	C₂×#683	1	1	yes	characteristic
267	C₂×C₄	#4070	1	1	yes	characteristic
268	C₂×C₄	#4081	1	1	yes	characteristic
269	C₂×C₄	#4082	1	1	yes	characteristic
270	C₂×C₄	#4083	1	1	yes	characteristic
271	C₂×C₄	#4093	1	1	yes	characteristic
272	C₈	—	2	4	no	—
273	C₈	—	4	2	no	—
274	C₈	—	20	4	no	—
275	C₅	#11270	1	1	yes	characteristic, complemented, Sylow
276	C₂²	—	2	4	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	2	2	no	—
281	C₂²	#64944	1	2	yes	—
282	C₂²	#64949	1	2	yes	—
283	C₂²	#65217	1	1	yes	characteristic
284	C₂²	#65242	1	1	yes	characteristic
285	C₂²	#65280	1	1	yes	characteristic
286	C₂²	#65431	1	2	yes	—
287	C₂²	#65432	1	2	yes	—
288	C₄	—	8	2	no	—
289	C₄	—	40	4	no	—
290	C₄	#64950	1	2	yes	—
291	C₄	#64959	1	2	yes	—
292	C₄	#65235	1	1	yes	characteristic
293	C₄	#65236	1	1	yes	characteristic
294	C₄	#65266	1	1	yes	characteristic
295	C₄	#65275	1	1	yes	characteristic
296	C₂	—	2	2	no	—
297	C₂	—	2	2	no	—
298	C₂	#10600208	1	1	yes	characteristic
299	C₂	#10600687	1	2	yes	—
300	C₂	#10600787	1	2	yes	—
301	C₂	#10601333	1	1	yes	characteristic
302	C₂	#10601336	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#4076
commutator	C₂²×C₄×C₅	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₅
Fitting	C₅×#1729	C₂
socle	C₂³×C₅	#121
prime radical	C₂⁴×C₅	#41
cosocle	C₂³×C₄×C₅	C₂³
hypercenter	#1729	D₅
hypocenter	C₅	#11270
LCS	G ▷ C₂²×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ #1729

Complements

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

N	H ≅ G/N	type
15. #64772	C₄	semidirect
275. C₅	#11270	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#3950	2
C₅×#1729	2
C₂×#64774	2
C₂×#64772	2
#10602374	2
#11270	5