Subgroups — #11989998

Order 1280.

420 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	#10610250	C₂	1	1	yes	maximal, characteristic
2	#10601547	C₂	1	1	yes	maximal, characteristic
3	#10601533	C₂	1	1	yes	maximal, characteristic, complemented
4	#10600527	C₂	1	1	yes	maximal, characteristic
5	C₂×#64675	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1278	C₂	1	1	yes	maximal, characteristic, complemented
7	#10598158	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#709	C₂²	1	1	yes	characteristic
9	C₅×#431	C₂²	1	1	yes	characteristic
10	C₄×C₅×M₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	#65418	C₄	1	2	yes	—
13	#65408	C₄	1	2	yes	—
14	#65214	C₂²	1	1	yes	characteristic
15	C₂×C₄×#168	C₂²	1	1	yes	characteristic
16	C₂×C₄×#168	C₂²	1	1	yes	characteristic
17	#64998	—	2	2	no	—
18	#64974	—	2	1	no	—
19	#64974	—	2	1	no	—
20	#64704	C₄	1	2	yes	—
21	#64703	C₄	1	2	yes	—
22	#64679	C₄	1	2	yes	—
23	#64676	C₄	1	2	yes	—
24	#64675	—	2	2	no	—
25	#64674	—	2	1	no	—
26	C₈×#168	—	2	1	no	—
27	#8189	—	5	1	no	maximal, Sylow
28	C₂×#709	C₂×C₄	1	1	yes	characteristic
29	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
30	C₂²×C₅×C₈	—	2	1	no	—
31	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
32	C₄×C₂²⋊D₅	—	2	1	no	—
33	C₄×C₂²⋊D₅	—	2	1	no	—
34	#4072	C₂×C₄	1	1	yes	characteristic
35	C₂²×#168	—	2	1	no	—
36	C₂²×#168	C₂×C₄	1	2	yes	—
37	C₄×D₂₀	C₂×C₄	1	1	yes	characteristic
38	C₄×D₂₀	C₂×C₄	1	1	yes	characteristic
39	#4018	—	2	1	no	—
40	C₄²×D₅	—	2	1	no	—
41	C₄×Dic₁₀	C₂×C₄	1	1	yes	characteristic
42	C₄×Dic₁₀	C₂×C₄	1	1	yes	characteristic
43	C₅×C₄⋊C₈	C₂×C₄	1	2	yes	—
44	C₅×C₂²⋊C₈	—	2	1	no	—
45	C₅×#97	—	2	1	no	—
46	C₄×C₅×C₈	—	2	1	no	—
47	C₄×C₅×C₈	—	2	1	no	—
48	C₄×C₅×C₈	—	2	1	no	—
49	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
50	C₂²⋊#168	—	2	1	no	—
51	#3952	—	2	2	no	—
52	#3952	—	2	2	no	—
53	#3946	—	2	2	no	—
54	#3946	—	2	2	no	—
55	C₄⋊#168	C₂×C₄	1	2	yes	—
56	C₄×#168	—	2	1	no	—
57	C₄×#168	—	2	1	no	—
58	C₄×#168	C₂×C₄	1	2	yes	—
59	C₄×#168	D₄	1	1	yes	characteristic
60	C₄×#168	D₄	1	1	yes	characteristic
61	C₄×#168	D₄	1	2	yes	—
62	#2922	—	5	1	no	—
63	#1476	—	5	1	no	—
64	#1476	—	5	1	no	—
65	#1450	—	5	1	no	—
66	C₂×#322	—	5	1	no	—
67	#1278	—	5	1	no	—
68	#1278	D₅	1	1	yes	characteristic, complemented
69	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅	#41	1	1	yes	characteristic
71	C₂²×C₄×C₅	#41	1	1	yes	characteristic
72	C₂×C₂²⋊D₅	—	2	1	no	—
73	#709	—	2	2	no	—
74	#709	—	2	2	no	—
75	C₂×D₂₀	C₄²	1	1	yes	characteristic
76	C₂×C₄×D₅	—	2	1	no	—
77	C₂×C₄×D₅	—	2	1	no	—
78	C₂×C₄×D₅	—	2	1	no	—
79	C₂×Dic₁₀	C₄²	1	1	yes	characteristic
80	C₅×M₁₆	—	2	2	no	—
81	C₅×M₁₆	—	2	2	no	—
82	C₂×C₅×C₈	—	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	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
90	C₂²⋊Dic₅	—	2	1	no	—
91	#685	—	2	1	no	—
92	#685	—	2	1	no	—
93	C₄⋊Dic₅	C₄²	1	1	yes	characteristic
94	C₄⋊Dic₅	C₄²	1	1	yes	characteristic
95	#683	—	2	1	no	—
96	#683	—	2	1	no	—
97	C₄×Dic₅	—	2	1	no	—
98	C₄×Dic₅	—	2	1	no	—
99	C₄×Dic₅	—	2	1	no	—
100	C₂×#168	—	2	2	no	—
101	C₂×#168	—	2	2	no	—
102	C₂×#168	—	2	2	no	—
103	C₂×#168	—	2	2	no	—
104	C₂×#168	—	4	2	no	—
105	C₂×#168	C₂²⋊C₄	1	2	yes	—
106	C₂×#168	C₂²⋊C₄	1	2	yes	—
107	C₄×#41	—	5	1	no	—
108	#433	—	10	2	no	—
109	#431	—	5	1	no	—
110	#431	—	10	1	no	—
111	#431	—	10	1	no	—
112	#431	C₂×D₅	1	1	yes	characteristic
113	#424	—	5	2	no	—
114	#408	—	5	2	no	—
115	C₄×M₁₆	C₂×D₅	1	1	yes	characteristic
116	C₂×C₄×C₈	—	5	1	no	—
117	C₂×C₄×C₈	—	5	1	no	—
118	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
119	Q₈⋊C₈	—	5	2	no	—
120	Q₈⋊C₈	—	5	2	no	—
121	D₄⋊C₈	—	5	2	no	—
122	D₄⋊C₈	—	5	2	no	—
123	#322	—	10	1	no	—
124	#322	—	10	2	no	—
125	C₈²	—	10	1	no	—
126	C₂³×C₅	#117	1	1	yes	characteristic
127	C₂²×D₅	—	2	1	no	—
128	C₂×C₄×C₅	—	2	2	no	—
129	C₂×C₄×C₅	—	2	2	no	—
130	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
131	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
133	C₂×C₄×C₅	C₄×D₄	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	#131	1	2	yes	—
137	C₂×C₄×C₅	#131	1	2	yes	—
138	C₂²⋊D₅	—	4	2	no	—
139	C₂×Dic₅	—	2	1	no	—
140	C₂×Dic₅	—	2	1	no	—
141	C₂×Dic₅	—	2	1	no	—
142	D₂₀	—	2	1	no	—
143	D₂₀	—	2	1	no	—
144	C₄×D₅	—	4	2	no	—
145	C₄×D₅	—	4	2	no	—
146	C₄×D₅	—	4	2	no	—
147	Dic₁₀	—	2	1	no	—
148	Dic₁₀	—	2	1	no	—
149	C₅×C₈	—	2	4	no	—
150	C₅×C₈	—	4	2	no	—
151	C₅×C₈	—	4	2	no	—
152	#168	—	2	2	no	—
153	#168	—	2	2	no	—
154	#168	—	4	2	no	—
155	#168	—	4	2	no	—
156	C₂×#41	—	5	1	no	—
157	C₂×M₁₆	C₄×D₅	1	2	yes	—
158	C₂²×C₈	—	2	1	no	—
159	C₂²×C₈	—	5	2	no	—
160	C₂²×C₈	—	10	1	no	—
161	C₄×Q₈	—	5	1	no	—
162	C₄×Q₈	—	5	1	no	—
163	C₄×D₄	—	5	1	no	—
164	C₄×D₄	—	5	1	no	—
165	C₄×D₄	—	10	1	no	—
166	C₄×D₄	—	10	1	no	—
167	#117	—	5	1	no	—
168	#117	—	10	1	no	—
169	C₂×C₄²	—	10	1	no	—
170	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
171	C₄⋊C₈	—	5	2	no	—
172	C₄⋊C₈	—	10	2	no	—
173	C₄⋊C₈	—	10	2	no	—
174	C₄⋊C₈	C₄×D₅	1	2	yes	—
175	C₂²⋊C₈	—	2	1	no	—
176	C₂²⋊C₈	—	10	2	no	—
177	C₂²⋊C₈	—	10	2	no	—
178	C₂²⋊C₈	—	10	1	no	—
179	#97	—	2	1	no	—
180	C₄×C₈	—	2	1	no	—
181	C₄×C₈	—	2	1	no	—
182	C₄×C₈	—	2	1	no	—
183	C₄×C₈	—	5	2	no	—
184	C₄×C₈	—	5	2	no	—
185	C₄×C₈	—	5	1	no	—
186	C₄×C₈	—	5	1	no	—
187	C₄×C₈	—	10	1	no	—
188	C₄×C₈	—	10	1	no	—
189	C₄×C₈	C₄×D₅	1	2	yes	—
190	C₂²×C₅	—	2	1	no	—
191	C₂²×C₅	—	2	1	no	—
192	C₂²×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
193	C₂²×C₅	#433	1	2	yes	—
194	C₂×D₅	—	4	1	no	—
195	C₂×D₅	—	4	1	no	—
196	C₂×D₅	—	4	1	no	—
197	C₄×C₅	—	2	1	no	—
198	C₄×C₅	—	2	1	no	—
199	C₄×C₅	#405	1	1	yes	characteristic
200	C₄×C₅	#405	1	1	yes	characteristic
201	C₄×C₅	#408	1	2	yes	—
202	C₄×C₅	#443	1	2	yes	—
203	C₄×C₅	#444	1	2	yes	—
204	Dic₅	—	4	1	no	—
205	Dic₅	—	4	1	no	—
206	Dic₅	—	4	1	no	—
207	#41	—	10	2	no	—
208	#41	—	10	2	no	—
209	C₂×Q₈	—	5	1	no	—
210	C₂×D₄	—	5	1	no	—
211	C₂×D₄	—	10	1	no	—
212	C₂²×C₄	—	10	1	no	—
213	C₂²×C₄	—	10	1	no	—
214	C₂²×C₄	—	10	1	no	—
215	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
216	C₂²×C₄	#711	1	1	yes	characteristic
217	C₂²×C₄	#713	1	1	yes	characteristic
218	M₁₆	—	2	2	no	—
219	M₁₆	—	2	2	no	—
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	5	2	no	—
225	C₂×C₈	—	5	2	no	—
226	C₂×C₈	—	10	2	no	—
227	C₂×C₈	—	10	2	no	—
228	C₂×C₈	—	10	2	no	—
229	C₂×C₈	—	10	2	no	—
230	C₂×C₈	—	20	2	no	—
231	C₄⋊C₄	—	5	1	no	—
232	C₄⋊C₄	—	5	1	no	—
233	C₄⋊C₄	—	10	1	no	—
234	C₄⋊C₄	—	10	1	no	—
235	C₂²⋊C₄	—	10	1	no	—
236	C₂²⋊C₄	—	10	1	no	—
237	C₂²⋊C₄	—	10	1	no	—
238	C₄²	—	10	1	no	—
239	C₄²	—	10	1	no	—
240	C₄²	—	10	1	no	—
241	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
242	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
243	C₄²	D₄×D₅	1	1	yes	characteristic
244	C₄²	D₄×D₅	1	1	yes	characteristic
245	C₂×C₅	—	2	2	no	—
246	C₂×C₅	#1750	1	1	yes	characteristic
247	C₂×C₅	#1764	1	1	yes	characteristic
248	C₂×C₅	#1765	1	1	yes	characteristic
249	D₅	—	8	1	no	—
250	C₂³	—	10	1	no	—
251	C₂³	#4038	1	1	yes	characteristic
252	Q₈	—	10	1	no	—
253	Q₈	—	10	1	no	—
254	D₄	—	10	1	no	—
255	D₄	—	10	1	no	—
256	D₄	—	20	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	10	1	no	—
260	C₂×C₄	—	10	1	no	—
261	C₂×C₄	—	10	1	no	—
262	C₂×C₄	—	20	2	no	—
263	C₂×C₄	—	20	2	no	—
264	C₂×C₄	—	20	2	no	—
265	C₂×C₄	C₄²×D₅	1	1	yes	characteristic
266	C₂×C₄	D₅×C₂²⋊C₄	1	1	yes	characteristic
267	C₂×C₄	#4027	1	1	yes	characteristic
268	C₂×C₄	#4027	1	1	yes	characteristic
269	C₂×C₄	#4039	1	1	yes	characteristic
270	C₂×C₄	#4039	1	1	yes	characteristic
271	C₂×C₄	#4047	1	2	yes	—
272	C₂×C₄	#4053	1	2	yes	—
273	C₈	—	2	4	no	—
274	C₈	—	4	2	no	—
275	C₈	—	4	2	no	—
276	C₈	—	10	2	no	—
277	C₈	—	10	2	no	—
278	C₈	—	20	2	no	—
279	C₈	—	20	2	no	—
280	C₅	#8189	1	1	yes	characteristic, complemented, Sylow
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	20	1	no	—
284	C₂²	—	20	1	no	—
285	C₂²	—	20	1	no	—
286	C₂²	#64955	1	1	yes	characteristic
287	C₂²	#65129	1	2	yes	—
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	20	1	no	—
291	C₄	—	20	1	no	—
292	C₄	—	20	1	no	—
293	C₄	#64978	1	1	yes	characteristic
294	C₄	#64997	1	1	yes	characteristic
295	C₄	#65016	1	2	yes	—
296	C₄	#65113	1	2	yes	—
297	C₄	#65183	1	2	yes	—
298	C₂	—	2	2	no	—
299	C₂	—	40	1	no	—
300	C₂	#10599394	1	1	yes	characteristic
301	C₂	#10599532	1	1	yes	characteristic
302	C₂	#10599720	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₅
commutator	C₂×C₄×C₅	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#1278	C₂
socle	C₂²×C₅	C₄×C₂²⋊C₄
prime radical	C₂×#709	C₂×C₄
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	#1278	D₅
hypocenter	C₅	#8189
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ #1278

Complements

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

N	H ≅ G/N	type
3. #10601533	C₂	semidirect
5. C₂×#64675	C₂	semidirect
6. C₅×#1278	C₂	semidirect
7. #10598158	C₂	semidirect
68. #1278	D₅	semidirect
280. C₅	#8189	semidirect

Maximal subgroups

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

subgroup	index
#10610250	2
#10601547	2
#10601533	2
#10600527	2
C₂×#64675	2
C₅×#1278	2
#10598158	2
#8189	5