Subgroups — #11990325

Order 1280.

444 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10610189	C₂	1	1	yes	maximal, characteristic
2	C₅×C₈×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#10601566	C₂	1	1	yes	maximal, characteristic
4	#10600652	C₂	1	1	yes	maximal, characteristic
5	#10600518	C₂	1	1	yes	maximal, characteristic, complemented
6	#10598234	C₂	1	1	yes	maximal, characteristic, complemented
7	#10598160	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#4047	C₄	1	2	yes	—
9	#65811	C₂²	1	1	yes	characteristic
10	C₅×#431	C₂²	1	1	yes	characteristic
11	C₄×C₅×M₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
13	#65399	C₄	1	2	yes	—
14	#65214	C₂²	1	1	yes	characteristic
15	#65213	C₂²	1	1	yes	characteristic
16	C₄×#681	C₂²	1	1	yes	characteristic
17	#64977	—	2	1	no	—
18	#64977	—	2	1	no	—
19	C₈×D₂₀	—	2	1	no	—
20	C₈×D₂₀	—	2	1	no	—
21	C₅×#322	—	2	1	no	—
22	C₅×C₈²	—	2	1	no	—
23	#64705	C₄	1	2	yes	—
24	#64701	C₄	1	2	yes	—
25	#64684	C₄	1	4	yes	—
26	#64675	—	2	1	no	—
27	#64674	—	2	1	no	—
28	#8245	—	5	1	no	maximal, Sylow
29	C₂×#709	C₂×C₄	1	2	yes	—
30	C₂×C₅×M₁₆	—	2	1	no	—
31	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
34	#4076	—	2	1	no	—
35	#4076	—	2	1	no	—
36	C₂×#681	—	2	1	no	—
37	C₂×#681	C₂×C₄	1	2	yes	—
38	#4047	—	4	4	no	—
39	C₂×#676	—	2	2	no	—
40	C₂×C₈×D₅	—	2	2	no	—
41	#4021	C₂×C₄	1	2	yes	—
42	#4020	C₂×C₄	1	1	yes	characteristic
43	C₄×D₂₀	—	2	1	no	—
44	C₄×D₂₀	—	2	1	no	—
45	#4015	C₂×C₄	1	1	yes	characteristic
46	C₅×C₄⋊C₈	C₂×C₄	1	2	yes	—
47	C₅×C₂²⋊C₈	—	2	1	no	—
48	C₅×#97	C₂×C₄	1	2	yes	—
49	C₄×C₅×C₈	—	2	1	no	—
50	C₄×C₅×C₈	—	2	1	no	—
51	C₄×C₅×C₈	D₄	1	1	yes	characteristic
52	C₄×C₅×C₈	D₄	1	1	yes	characteristic
53	C₄×C₅×C₈	D₄	1	1	yes	characteristic
54	C₄×C₅×C₈	D₄	1	1	yes	characteristic
55	C₂²⋊#168	—	2	1	no	—
56	#3952	—	2	2	no	—
57	#3952	—	2	2	no	—
58	C₄⋊#168	—	2	1	no	—
59	C₄⋊#168	—	2	1	no	—
60	C₄⋊#168	C₂×C₄	1	2	yes	—
61	#3935	C₂×C₄	1	2	yes	—
62	C₄×#168	—	2	1	no	—
63	C₄×#168	—	2	1	no	—
64	#2931	—	5	1	no	—
65	#1543	—	5	1	no	—
66	#1525	—	5	1	no	—
67	#1451	—	5	1	no	—
68	C₈×M₁₆	D₅	1	1	yes	characteristic, complemented
69	#1299	—	5	1	no	—
70	#1299	—	5	1	no	—
71	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₅	#41	1	1	yes	characteristic
73	C₂²×C₄×C₅	#41	1	1	yes	characteristic
74	C₂×C₂²⋊D₅	—	2	2	no	—
75	#709	—	4	2	no	—
76	#709	—	4	2	no	—
77	C₂×D₂₀	—	2	1	no	—
78	C₂×D₂₀	—	2	1	no	—
79	C₂×D₂₀	C₄²	1	2	yes	—
80	C₂×C₄×D₅	—	2	2	no	—
81	C₂×Dic₁₀	C₄²	1	2	yes	—
82	C₅×M₁₆	—	2	4	no	—
83	C₂×C₅×C₈	—	2	2	no	—
84	C₂×C₅×C₈	—	2	2	no	—
85	C₂×C₅×C₈	—	2	2	no	—
86	C₂×C₅×C₈	—	4	1	no	—
87	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
91	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
95	#685	—	2	2	no	—
96	#685	—	2	2	no	—
97	C₄⋊Dic₅	—	2	1	no	—
98	C₄⋊Dic₅	—	2	1	no	—
99	#681	—	4	2	no	—
100	#681	—	4	2	no	—
101	C₂×#168	—	2	2	no	—
102	C₂×#168	—	2	2	no	—
103	C₂×#168	—	2	2	no	—
104	#676	—	4	4	no	—
105	C₈×D₅	—	4	4	no	—
106	C₂×#131	—	5	2	no	—
107	#532	—	5	1	no	—
108	#436	—	10	1	no	—
109	#436	—	10	1	no	—
110	C₈×D₄	—	10	1	no	—
111	C₈×D₄	—	10	1	no	—
112	#431	—	5	1	no	—
113	#431	C₂×D₅	1	1	yes	characteristic
114	#423	—	5	1	no	—
115	#408	—	5	2	no	—
116	C₄×M₁₆	—	5	1	no	—
117	C₄×M₁₆	C₂×D₅	1	1	yes	characteristic
118	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
119	#333	—	5	2	no	—
120	#331	—	5	2	no	—
121	#329	—	5	4	no	—
122	#322	—	2	1	no	—
123	#322	—	10	1	no	—
124	#322	—	10	1	no	—
125	C₈²	—	2	1	no	—
126	C₂³×C₅	#117	1	1	yes	characteristic
127	C₂²×D₅	—	2	2	no	—
128	C₂×C₄×C₅	—	2	1	no	—
129	C₂×C₄×C₅	—	2	1	no	—
130	C₂×C₄×C₅	—	2	2	no	—
131	C₂×C₄×C₅	—	2	2	no	—
132	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
133	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
135	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
136	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
137	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
138	C₂²⋊D₅	—	4	4	no	—
139	C₂×Dic₅	—	2	2	no	—
140	D₂₀	—	4	2	no	—
141	D₂₀	—	4	2	no	—
142	D₂₀	—	4	2	no	—
143	C₄×D₅	—	4	2	no	—
144	C₄×D₅	—	4	2	no	—
145	Dic₁₀	—	4	2	no	—
146	C₅×C₈	—	2	4	no	—
147	C₅×C₈	—	2	1	no	—
148	C₅×C₈	—	2	1	no	—
149	C₅×C₈	—	2	1	no	—
150	C₅×C₈	—	4	2	no	—
151	C₅×C₈	#100	1	2	yes	—
152	#168	—	4	2	no	—
153	#168	—	4	2	no	—
154	#168	—	4	2	no	—
155	C₂×#41	—	5	2	no	—
156	#131	—	20	4	no	—
157	C₂×M₁₆	—	2	1	no	—
158	C₂×M₁₆	—	5	2	no	—
159	C₂×M₁₆	—	10	1	no	—
160	C₂×M₁₆	—	10	2	no	—
161	C₂²×C₈	—	10	2	no	—
162	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
163	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
164	C₄⋊Q₈	—	5	1	no	—
165	#127	—	5	1	no	—
166	#124	—	5	2	no	—
167	#121	—	10	1	no	—
168	#121	—	10	1	no	—
169	C₄×D₄	—	10	1	no	—
170	C₄×D₄	—	10	1	no	—
171	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
172	C₄⋊C₈	—	5	2	no	—
173	C₄⋊C₈	—	10	1	no	—
174	C₄⋊C₈	—	10	1	no	—
175	C₄⋊C₈	C₄×D₅	1	2	yes	—
176	C₂²⋊C₈	—	2	1	no	—
177	C₂²⋊C₈	—	10	1	no	—
178	C₂²⋊C₈	—	10	2	no	—
179	C₂²⋊C₈	—	10	2	no	—
180	#97	—	5	2	no	—
181	#97	C₄×D₅	1	2	yes	—
182	C₄×C₈	—	2	1	no	—
183	C₄×C₈	—	2	1	no	—
184	C₄×C₈	—	10	1	no	—
185	C₄×C₈	—	10	1	no	—
186	C₄×C₈	D₂₀	1	1	yes	characteristic
187	C₄×C₈	D₂₀	1	1	yes	characteristic
188	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
189	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
190	C₂²×C₅	—	2	1	no	—
191	C₂²×C₅	—	2	1	no	—
192	C₂²×C₅	—	2	1	no	—
193	C₂²×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
194	C₂×D₅	—	4	2	no	—
195	C₂×D₅	—	4	2	no	—
196	C₂×D₅	—	4	2	no	—
197	C₄×C₅	—	2	1	no	—
198	C₄×C₅	—	2	1	no	—
199	C₄×C₅	—	2	1	no	—
200	C₄×C₅	C₂×#100	1	1	yes	characteristic
201	C₄×C₅	#413	1	1	yes	characteristic
202	C₄×C₅	#443	1	2	yes	—
203	C₄×C₅	#444	1	2	yes	—
204	Dic₅	—	4	2	no	—
205	#41	—	20	2	no	—
206	#41	—	20	2	no	—
207	C₂×Q₈	—	5	2	no	—
208	C₂×D₄	—	5	2	no	—
209	C₂×D₄	—	10	2	no	—
210	C₂×D₄	—	10	1	no	—
211	C₂×D₄	—	10	1	no	—
212	C₂²×C₄	—	10	2	no	—
213	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
214	C₂²×C₄	#709	1	1	yes	characteristic
215	C₂²×C₄	#709	1	1	yes	characteristic
216	M₁₆	—	2	4	no	—
217	M₁₆	—	20	2	no	—
218	M₁₆	—	20	4	no	—
219	M₁₆	—	20	2	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	4	1	no	—
224	C₂×C₈	—	10	2	no	—
225	C₂×C₈	—	10	2	no	—
226	C₂×C₈	—	10	2	no	—
227	C₂×C₈	—	20	4	no	—
228	C₂×C₈	#685	1	1	yes	characteristic
229	C₂×C₈	#685	1	1	yes	characteristic
230	C₂×C₈	#685	1	1	yes	characteristic
231	C₂×C₈	#685	1	1	yes	characteristic
232	C₄⋊C₄	—	10	1	no	—
233	C₄⋊C₄	—	10	1	no	—
234	C₂²⋊C₄	—	10	2	no	—
235	C₂²⋊C₄	—	10	2	no	—
236	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
237	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
238	C₄²	C₂×D₂₀	1	1	yes	characteristic
239	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
240	C₂×C₅	—	4	1	no	—
241	C₂×C₅	C₄×#100	1	1	yes	characteristic
242	C₂×C₅	#1764	1	1	yes	characteristic
243	C₂×C₅	#1765	1	1	yes	characteristic
244	D₅	—	8	2	no	—
245	C₂³	—	10	2	no	—
246	C₂³	#4018	1	1	yes	characteristic
247	Q₈	—	20	2	no	—
248	D₄	—	20	2	no	—
249	D₄	—	20	4	no	—
250	D₄	—	20	2	no	—
251	D₄	—	20	2	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	10	2	no	—
257	C₂×C₄	—	20	2	no	—
258	C₂×C₄	—	20	2	no	—
259	C₂×C₄	C₄²×D₅	1	1	yes	characteristic
260	C₂×C₄	C₄×D₂₀	1	1	yes	characteristic
261	C₂×C₄	C₄×D₂₀	1	1	yes	characteristic
262	C₂×C₄	C₂×#685	1	1	yes	characteristic
263	C₂×C₄	C₄×C₂²⋊D₅	1	1	yes	characteristic
264	C₂×C₄	C₄×C₂²⋊D₅	1	1	yes	characteristic
265	C₈	—	2	4	no	—
266	C₈	—	2	1	no	—
267	C₈	—	2	1	no	—
268	C₈	—	2	1	no	—
269	C₈	—	4	2	no	—
270	C₈	—	20	2	no	—
271	C₈	—	20	2	no	—
272	C₈	—	20	2	no	—
273	C₈	#3955	1	2	yes	—
274	C₅	#8245	1	1	yes	characteristic, complemented, Sylow
275	C₂²	—	2	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	20	2	no	—
279	C₂²	—	20	2	no	—
280	C₂²	—	20	2	no	—
281	C₂²	C₄×#685	1	1	yes	characteristic
282	C₄	—	2	1	no	—
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	20	2	no	—
286	C₄	#64989	1	2	yes	—
287	C₄	C₂×#3955	1	1	yes	characteristic
288	C₄	#65421	1	1	yes	characteristic
289	C₄	#65434	1	2	yes	—
290	C₂	—	4	1	no	—
291	C₂	—	40	2	no	—
292	C₂	#10600582	1	1	yes	characteristic
293	C₂	C₄×#3955	1	1	yes	characteristic
294	C₂	#10600681	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#685
commutator	C₂×C₄×C₅	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×C₈×M₁₆	C₂
socle	C₂²×C₅	C₄×C₂²⋊C₄
prime radical	#65811	C₂²
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	C₈×M₁₆	D₅
hypocenter	C₅	#8245
LCS	G ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₅×C₈×M₁₆	C₂	semidirect
5. #10600518	C₂	semidirect
6. #10598234	C₂	semidirect
7. #10598160	C₂	semidirect
68. C₈×M₁₆	D₅	semidirect
274. C₅	#8245	semidirect

Maximal subgroups

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

subgroup	index
#10610189	2
C₅×C₈×M₁₆	2
#10601566	2
#10600652	2
#10600518	2
#10598234	2
#10598160	2
#8245	5