Subgroups — #12007584

Order 1280.

574 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#3951	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#341	C₂	1	1	yes	maximal, characteristic
3	C₂×#64779	C₂	1	2	yes	maximal
4	#10602791	C₂	1	2	yes	maximal
5	#10602758	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₅×M₁₆	C₂²	1	2	yes	—
7	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂×#4072	C₂²	1	1	yes	characteristic
9	C₂²×#681	C₂²	1	1	yes	characteristic
10	C₂²×#681	C₂²	1	2	yes	—
11	C₂×#3951	—	2	4	no	—
12	C₂×#3951	—	2	2	no	—
13	C₅×#341	C₄	1	4	yes	complemented
14	#64779	—	2	4	no	—
15	#64772	—	2	4	no	—
16	#64770	—	2	2	no	—
17	#11272	—	5	1	no	maximal, Sylow
18	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
19	C₂×C₅×M₁₆	—	2	2	no	—
20	C₂×C₅×M₁₆	—	2	2	no	—
21	C₂×C₅×M₁₆	—	2	2	no	—
22	C₂×C₅×M₁₆	—	2	2	no	—
23	C₂×C₅×M₁₆	—	2	2	no	—
24	C₂×C₅×M₁₆	C₂×C₄	1	4	yes	—
25	C₂²×C₅×C₈	—	2	2	no	—
26	C₂²×C₅×C₈	—	2	2	no	—
27	C₂²×C₅×C₈	—	2	2	no	—
28	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
29	C₂²×C₅×C₈	D₄	1	2	yes	—
30	C₂²×C₅×C₈	Q₈	1	2	yes	—
31	C₂×C₂²⋊Dic₅	Q₈	1	2	yes	—
32	#4072	—	2	2	no	—
33	#4072	—	2	2	no	—
34	C₂×C₄⋊Dic₅	—	2	1	no	—
35	C₂×C₄×Dic₅	—	2	1	no	—
36	C₂×#681	—	2	2	no	—
37	C₂×#681	—	2	1	no	—
38	C₂×#681	—	2	2	no	—
39	C₂×#681	—	2	2	no	—
40	C₂×#681	—	2	2	no	—
41	C₂×#681	—	2	2	no	—
42	C₂×#681	—	2	2	no	—
43	C₂×#681	—	2	2	no	—
44	C₂×#681	D₄	1	2	yes	—
45	C₂×#681	D₄	1	4	yes	—
46	C₂²×#168	—	2	1	no	—
47	C₂²×#168	—	2	2	no	—
48	#3951	—	2	8	no	—
49	C₂²×#108	—	5	1	no	—
50	C₂×#344	—	5	2	no	—
51	#1738	—	5	2	no	—
52	C₂×#341	D₅	1	1	yes	characteristic
53	#1729	—	5	1	no	—
54	C₂⁴×C₅	C₂×Q₈	1	1	yes	characteristic
55	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
58	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
60	C₂²×C₄×C₅	#41	1	2	yes	—
61	C₂²×C₄×C₅	#41	1	2	yes	—
62	C₂²×C₄×C₅	#41	1	2	yes	—
63	C₂²×C₄×C₅	#41	1	1	yes	characteristic
64	C₂²×C₄×C₅	#41	1	1	yes	characteristic
65	C₂²×Dic₅	—	2	2	no	—
66	C₅×M₁₆	—	2	4	no	—
67	C₅×M₁₆	—	2	4	no	—
68	C₅×M₁₆	—	4	2	no	—
69	C₅×M₁₆	—	4	2	no	—
70	C₂×C₅×C₈	—	2	4	no	—
71	C₂×C₅×C₈	—	2	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	—	4	2	no	—
76	C₂×C₅×C₈	—	4	2	no	—
77	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
78	C₂²⋊Dic₅	—	2	4	no	—
79	C₄⋊Dic₅	—	4	2	no	—
80	C₄×Dic₅	—	4	1	no	—
81	C₄×Dic₅	—	4	1	no	—
82	#681	—	2	4	no	—
83	#681	—	2	4	no	—
84	#681	—	2	4	no	—
85	#681	—	4	2	no	—
86	#681	—	4	2	no	—
87	#681	—	4	2	no	—
88	C₂×#168	—	2	2	no	—
89	C₂×#168	—	2	4	no	—
90	C₂×#168	—	4	2	no	—
91	C₂×#168	—	4	1	no	—
92	C₂×#168	—	4	2	no	—
93	C₂×#168	—	4	1	no	—
94	C₂²×M₁₆	—	5	2	no	—
95	C₂²×M₁₆	—	5	1	no	—
96	C₂²×M₁₆	C₂×D₅	1	2	yes	—
97	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
98	C₂×#117	—	5	1	no	—
99	C₂×#108	—	10	4	no	—
100	C₂×#108	—	10	2	no	—
101	#344	—	10	4	no	—
102	#341	—	10	4	no	—
103	#341	Dic₅	1	4	yes	complemented
104	#336	—	10	2	no	—
105	C₂³×C₅	—	4	2	no	—
106	C₂³×C₅	C₄×Q₈	1	1	yes	characteristic
107	C₂³×C₅	C₂²⋊Q₈	1	2	yes	—
108	C₂³×C₅	C₂²⋊Q₈	1	1	yes	characteristic
109	C₂³×C₅	#125	1	2	yes	—
110	C₂³×C₅	C₄⋊Q₈	1	1	yes	characteristic
111	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₅	#117	1	2	yes	—
113	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₅	#120	1	1	yes	characteristic
116	C₂×C₄×C₅	#121	1	2	yes	—
117	C₂×C₄×C₅	#121	1	1	yes	characteristic
118	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
119	C₂×C₄×C₅	#123	1	2	yes	—
120	C₂×C₄×C₅	#123	1	1	yes	characteristic
121	C₂×C₄×C₅	#123	1	2	yes	—
122	C₂×C₄×C₅	#124	1	1	yes	characteristic
123	C₂×C₄×C₅	#124	1	1	yes	characteristic
124	C₂×C₄×C₅	#124	1	2	yes	—
125	C₂×C₄×C₅	#125	1	2	yes	—
126	C₂×C₄×C₅	#125	1	1	yes	characteristic
127	C₂×C₄×C₅	#126	1	2	yes	—
128	C₂×C₄×C₅	#126	1	2	yes	—
129	C₂×Dic₅	—	4	4	no	—
130	C₂×Dic₅	—	4	2	no	—
131	C₅×C₈	—	2	4	no	—
132	C₅×C₈	—	4	4	no	—
133	#168	—	4	4	no	—
134	#168	—	4	2	no	—
135	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
136	C₂×M₁₆	—	2	2	no	—
137	C₂×M₁₆	—	2	2	no	—
138	C₂×M₁₆	—	2	2	no	—
139	C₂×M₁₆	—	2	2	no	—
140	C₂×M₁₆	—	2	2	no	—
141	C₂×M₁₆	—	5	2	no	—
142	C₂×M₁₆	—	5	4	no	—
143	C₂×M₁₆	—	10	2	no	—
144	C₂×M₁₆	—	10	1	no	—
145	C₂×M₁₆	—	10	2	no	—
146	C₂×M₁₆	—	10	2	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	C₂×Dic₅	1	4	yes	—
152	C₂²×C₈	—	2	2	no	—
153	C₂²×C₈	—	2	2	no	—
154	C₂²×C₈	—	2	2	no	—
155	C₂²×C₈	—	10	1	no	—
156	C₂²×C₈	—	10	2	no	—
157	C₂²×C₈	Dic₁₀	1	2	yes	—
158	C₂²×C₈	D₂₀	1	2	yes	—
159	C₂²×C₈	C₂×Dic₅	1	2	yes	—
160	#117	—	10	2	no	—
161	#117	—	10	2	no	—
162	C₂×C₄⋊C₄	—	10	1	no	—
163	C₂×C₂²⋊C₄	—	5	2	no	—
164	C₂×C₄²	—	10	1	no	—
165	#108	—	10	8	no	—
166	C₂²×C₅	—	2	2	no	—
167	C₂²×C₅	—	2	2	no	—
168	C₂²×C₅	—	2	2	no	—
169	C₂²×C₅	—	4	2	no	—
170	C₂²×C₅	—	4	2	no	—
171	C₂²×C₅	#387	1	2	yes	—
172	C₂²×C₅	#389	1	1	yes	characteristic
173	C₂²×C₅	#393	1	1	yes	characteristic
174	C₂²×C₅	#398	1	2	yes	—
175	C₂²×C₅	#400	1	1	yes	characteristic
176	C₄×C₅	#382	1	1	yes	characteristic
177	C₄×C₅	#385	1	1	yes	characteristic
178	C₄×C₅	#388	1	2	yes	—
179	C₄×C₅	#394	1	1	yes	characteristic
180	C₄×C₅	#397	1	1	yes	characteristic
181	C₄×C₅	#400	1	2	yes	—
182	Dic₅	—	8	2	no	—
183	C₂⁴	Q₈×D₅	1	1	yes	characteristic
184	C₂²×C₄	—	10	2	no	—
185	C₂²×C₄	C₂×Dic₁₀	1	1	yes	characteristic
186	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
187	C₂²×C₄	#709	1	2	yes	—
188	C₂²×C₄	#709	1	2	yes	—
189	C₂²×C₄	D₄×D₅	1	2	yes	—
190	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
191	C₂²×C₄	#711	1	2	yes	—
192	C₂²×C₄	#711	1	1	yes	characteristic
193	C₂²×C₄	#713	1	1	yes	characteristic
194	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
195	M₁₆	—	2	4	no	—
196	M₁₆	—	2	4	no	—
197	M₁₆	—	4	2	no	—
198	M₁₆	—	4	2	no	—
199	M₁₆	—	10	4	no	—
200	M₁₆	—	10	4	no	—
201	M₁₆	—	10	4	no	—
202	M₁₆	—	20	2	no	—
203	M₁₆	—	20	2	no	—
204	M₁₆	—	20	2	no	—
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	2	4	no	—
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	—	4	2	no	—
211	C₂×C₈	—	4	2	no	—
212	C₂×C₈	—	10	4	no	—
213	C₂×C₈	—	10	2	no	—
214	C₂×C₈	—	20	1	no	—
215	C₂×C₈	—	20	1	no	—
216	C₂×C₈	—	20	2	no	—
217	C₂×C₈	—	20	2	no	—
218	C₂×C₈	C₄⋊Dic₅	1	4	yes	—
219	C₄⋊C₄	—	20	2	no	—
220	C₂²⋊C₄	—	10	4	no	—
221	C₄²	—	20	1	no	—
222	C₄²	—	20	1	no	—
223	C₂×C₅	—	2	1	no	—
224	C₂×C₅	—	2	2	no	—
225	C₂×C₅	—	2	2	no	—
226	C₂×C₅	—	2	1	no	—
227	C₂×C₅	#1442	1	1	yes	characteristic
228	C₂×C₅	#1953	1	2	yes	—
229	C₂³	—	4	2	no	—
230	C₂³	#4034	1	1	yes	characteristic
231	C₂³	#4035	1	2	yes	—
232	C₂³	#4042	1	2	yes	—
233	C₂³	#4043	1	1	yes	characteristic
234	C₂³	Q₈×Dic₅	1	1	yes	characteristic
235	C₂×C₄	—	20	4	no	—
236	C₂×C₄	—	20	2	no	—
237	C₂×C₄	#4016	1	2	yes	—
238	C₂×C₄	#4021	1	1	yes	characteristic
239	C₂×C₄	#4021	1	1	yes	characteristic
240	C₂×C₄	#4024	1	2	yes	—
241	C₂×C₄	#4025	1	2	yes	—
242	C₂×C₄	#4028	1	1	yes	characteristic
243	C₂×C₄	#4029	1	2	yes	—
244	C₂×C₄	#4030	1	2	yes	—
245	C₂×C₄	#4031	1	2	yes	—
246	C₂×C₄	#4032	1	1	yes	characteristic
247	C₂×C₄	#4036	1	1	yes	characteristic
248	C₂×C₄	#4040	1	2	yes	—
249	C₂×C₄	#4041	1	1	yes	characteristic
250	C₂×C₄	#4044	1	2	yes	—
251	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
252	C₂×C₄	#4072	1	2	yes	—
253	C₂×C₄	D₄×Dic₅	1	2	yes	—
254	C₂×C₄	D₄×Dic₅	1	1	yes	characteristic
255	C₈	—	2	4	no	—
256	C₈	—	4	4	no	—
257	C₈	—	20	4	no	—
258	C₈	—	20	2	no	—
259	C₅	#11272	1	1	yes	characteristic, complemented, Sylow
260	C₂²	—	2	2	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	4	2	no	—
264	C₂²	—	4	2	no	—
265	C₂²	#64946	1	2	yes	—
266	C₂²	#64948	1	1	yes	characteristic
267	C₂²	#64962	1	1	yes	characteristic
268	C₂²	#65270	1	2	yes	—
269	C₂²	#65274	1	1	yes	characteristic
270	C₄	—	40	2	no	—
271	C₄	#64949	1	2	yes	—
272	C₄	#64961	1	1	yes	characteristic
273	C₄	#64963	1	1	yes	characteristic
274	C₄	#65225	1	1	yes	characteristic
275	C₄	#65239	1	1	yes	characteristic
276	C₄	#65240	1	2	yes	—
277	C₂	—	2	1	no	—
278	C₂	—	2	2	no	—
279	C₂	—	2	1	no	—
280	C₂	—	2	2	no	—
281	C₂	#10600205	1	1	yes	characteristic
282	C₂	#10601500	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#4041
commutator	C₂²×C₄×C₅	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₅
Fitting	C₂×C₅×#341	C₂
socle	C₂²×C₅	#389
prime radical	C₂⁴×C₅	C₂×Q₈
cosocle	C₂³×C₄×C₅	C₂³
hypercenter	C₂×#341	D₅
hypocenter	C₅	#11272
LCS	G ▷ C₂²×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#341

Complements

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

N	H ≅ G/N	type
13. C₅×#341	C₄	semidirect
103. #341	Dic₅	semidirect
259. C₅	#11272	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#3951	2
C₂×C₅×#341	2
C₂×#64779	2
#10602791	2
#10602758	2
#11272	5