Subgroups — #11998547

Order 1280.

654 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#493	C₂	1	1	yes	maximal, characteristic
2	#10602554	C₂	1	2	yes	maximal, complemented
3	C₂×#64764	C₂	1	2	yes	maximal, complemented
4	C₈×C₄⋊Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#64678	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₅×D₈	—	2	1	no	—
7	C₂²×C₅×D₈	—	2	1	no	—
8	C₂×C₅×#127	C₂²	1	2	yes	—
9	C₅×#493	—	2	2	no	—
10	C₅×#493	C₄	1	4	yes	complemented
11	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₂×#3964	—	2	4	no	—
13	C₂×C₈×Dic₅	—	2	1	no	—
14	C₄×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊#168	C₂²	1	2	yes	—
16	C₂×C₄⋊#168	C₂²	1	1	yes	characteristic
17	#64770	—	2	1	no	—
18	#64764	—	2	4	no	—
19	#64678	—	2	2	no	—
20	#9734	—	5	1	no	maximal, Sylow
21	C₂²×C₅×D₄	—	2	2	no	—
22	C₂²×C₅×D₄	—	2	2	no	—
23	C₂²×C₅×D₄	—	2	2	no	—
24	C₂×C₅×D₈	—	2	4	no	—
25	C₂×C₅×D₈	—	2	4	no	—
26	C₂×C₅×D₈	—	4	2	no	—
27	C₂×C₅×D₈	—	4	2	no	—
28	C₂×C₅×D₈	—	4	2	no	—
29	C₂×C₅×D₈	—	4	2	no	—
30	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
31	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
32	C₅×#127	—	2	2	no	—
33	C₅×#127	—	2	2	no	—
34	C₅×#127	—	2	2	no	—
35	C₅×#127	C₂×C₄	1	4	yes	—
36	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
37	C₂×C₄⋊Dic₅	D₄	1	2	yes	—
38	C₂×C₄×Dic₅	—	2	1	no	—
39	C₂²×#168	—	2	1	no	—
40	C₂²×#168	—	2	2	no	—
41	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
42	C₄×C₅×C₈	D₄	1	2	yes	—
43	#3964	—	4	8	no	—
44	#3963	—	2	1	no	—
45	C₈×Dic₅	—	4	2	no	—
46	C₄⋊#168	—	2	1	no	—
47	C₄⋊#168	—	2	2	no	—
48	C₄⋊#168	—	2	2	no	—
49	C₄⋊#168	D₄	1	2	yes	—
50	C₂×#493	D₅	1	1	yes	characteristic
51	#1982	—	5	2	no	—
52	C₈×C₄⋊C₄	—	5	1	no	—
53	C₂×#335	—	5	1	no	—
54	C₂×#331	—	5	2	no	—
55	C₂⁴×C₅	—	4	2	no	—
56	C₂×C₅×D₄	—	2	4	no	—
57	C₂×C₅×D₄	—	2	4	no	—
58	C₂×C₅×D₄	—	2	4	no	—
59	C₂×C₅×D₄	—	4	2	no	—
60	C₂×C₅×D₄	—	4	2	no	—
61	C₂×C₅×D₄	—	4	2	no	—
62	C₂×C₅×D₄	—	4	2	no	—
63	C₂×C₅×D₄	—	4	4	no	—
64	C₂×C₅×D₄	—	4	2	no	—
65	C₂×C₅×D₄	—	4	4	no	—
66	C₂×C₅×D₄	—	4	2	no	—
67	C₂×C₅×D₄	—	4	2	no	—
68	C₂×C₅×D₄	—	8	4	no	—
69	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₅	#41	1	1	yes	characteristic
72	C₂²×Dic₅	—	2	2	no	—
73	C₅×D₈	—	4	8	no	—
74	C₅×D₈	—	4	8	no	—
75	C₂×C₅×C₈	—	2	1	no	—
76	C₂×C₅×C₈	—	2	1	no	—
77	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
78	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
79	C₂×C₅×C₈	D₈	1	2	yes	—
80	C₂×C₅×C₈	SD₁₆	1	2	yes	—
81	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₅	#41	1	1	yes	characteristic
85	C₄⋊Dic₅	—	2	2	no	—
86	C₄⋊Dic₅	D₈	1	4	yes	complemented
87	C₄×Dic₅	—	4	1	no	—
88	C₄×Dic₅	—	4	1	no	—
89	C₂×#168	—	2	2	no	—
90	C₂×#168	—	4	1	no	—
91	C₂×#168	—	4	1	no	—
92	C₂×#168	—	4	2	no	—
93	C₂×#168	—	4	4	no	—
94	C₂²×D₈	—	2	1	no	—
95	C₂²×D₈	—	2	1	no	—
96	C₂×#127	C₂×D₅	1	2	yes	—
97	#493	—	2	2	no	—
98	#493	Dic₅	1	4	yes	complemented
99	C₂×C₄⋊C₈	—	5	1	no	—
100	C₂×C₄⋊C₈	—	5	2	no	—
101	C₂×#102	—	10	4	no	—
102	C₂×C₄×C₈	—	10	1	no	—
103	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
104	C₄×C₄⋊C₄	—	5	1	no	—
105	#336	—	10	1	no	—
106	#335	—	10	2	no	—
107	#331	—	10	4	no	—
108	C₂³×C₅	—	4	4	no	—
109	C₂³×C₅	—	8	2	no	—
110	C₂³×C₅	—	8	2	no	—
111	C₂³×C₅	—	8	4	no	—
112	C₂³×C₅	—	8	4	no	—
113	C₂³×C₅	#121	1	1	yes	characteristic
114	C₅×D₄	—	4	4	no	—
115	C₅×D₄	—	4	4	no	—
116	C₅×D₄	—	4	8	no	—
117	C₅×D₄	—	8	4	no	—
118	C₅×D₄	—	8	4	no	—
119	C₅×D₄	—	8	2	no	—
120	C₅×D₄	—	8	2	no	—
121	C₅×D₄	—	8	2	no	—
122	C₅×D₄	—	8	2	no	—
123	C₂×C₄×C₅	—	2	2	no	—
124	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
126	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
127	C₂×C₄×C₅	#121	1	1	yes	characteristic
128	C₂×C₄×C₅	#124	1	1	yes	characteristic
129	C₂×C₄×C₅	#127	1	1	yes	characteristic
130	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
131	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
132	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
133	C₂×C₄×C₅	#135	1	2	yes	—
134	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
135	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
136	C₂×Dic₅	—	2	4	no	—
137	C₂×Dic₅	—	4	2	no	—
138	C₂×Dic₅	—	4	2	no	—
139	C₅×C₈	—	2	2	no	—
140	C₅×C₈	#102	1	4	yes	complemented
141	#168	—	4	2	no	—
142	#168	—	8	2	no	—
143	C₂²×D₄	—	2	2	no	—
144	C₂²×D₄	—	2	2	no	—
145	C₂²×D₄	—	2	2	no	—
146	C₂×D₈	—	2	4	no	—
147	C₂×D₈	—	2	4	no	—
148	C₂×D₈	—	4	2	no	—
149	C₂×D₈	—	4	2	no	—
150	C₂×D₈	—	4	2	no	—
151	C₂×D₈	—	4	2	no	—
152	C₂²×C₈	—	10	1	no	—
153	C₂²×C₈	—	10	2	no	—
154	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
155	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
156	#127	—	2	2	no	—
157	#127	—	2	2	no	—
158	#127	—	2	2	no	—
159	#127	C₂×Dic₅	1	4	yes	—
160	C₂×C₄⋊C₄	—	5	2	no	—
161	C₂×C₄²	—	10	1	no	—
162	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
163	C₄⋊C₈	—	5	2	no	—
164	C₄⋊C₈	—	10	2	no	—
165	C₄⋊C₈	—	10	1	no	—
166	C₄⋊C₈	—	10	2	no	—
167	#102	—	20	8	no	—
168	C₄×C₈	—	20	2	no	—
169	C₄×C₈	C₂×Dic₅	1	2	yes	—
170	C₄×C₈	C₂²⋊D₅	1	2	yes	—
171	#95	—	10	1	no	—
172	C₂²×C₅	—	8	4	no	—
173	C₂²×C₅	—	8	4	no	—
174	C₂²×C₅	—	8	4	no	—
175	C₂²×C₅	—	16	4	no	—
176	C₂²×C₅	—	16	4	no	—
177	C₂²×C₅	#390	1	1	yes	characteristic
178	C₂²×C₅	#459	1	1	yes	characteristic
179	C₂²×C₅	#460	1	1	yes	characteristic
180	C₂²×C₅	#466	1	2	yes	—
181	C₂²×C₅	#472	1	2	yes	—
182	C₄×C₅	—	2	1	no	—
183	C₄×C₅	—	2	1	no	—
184	C₄×C₅	C₂×#102	1	1	yes	characteristic
185	C₄×C₅	#418	1	1	yes	characteristic
186	C₄×C₅	C₄×D₈	1	2	yes	—
187	C₄×C₅	#459	1	1	yes	characteristic
188	C₄×C₅	#460	1	1	yes	characteristic
189	C₄×C₅	#493	1	1	yes	characteristic
190	C₄×C₅	#495	1	1	yes	characteristic
191	Dic₅	—	4	4	no	—
192	C₂⁴	—	4	2	no	—
193	C₂×D₄	—	2	4	no	—
194	C₂×D₄	—	2	4	no	—
195	C₂×D₄	—	2	4	no	—
196	C₂×D₄	—	4	4	no	—
197	C₂×D₄	—	4	4	no	—
198	C₂×D₄	—	4	2	no	—
199	C₂×D₄	—	4	2	no	—
200	C₂×D₄	—	4	2	no	—
201	C₂×D₄	—	4	2	no	—
202	C₂×D₄	—	4	2	no	—
203	C₂×D₄	—	4	2	no	—
204	C₂×D₄	—	4	2	no	—
205	C₂×D₄	—	8	4	no	—
206	C₂²×C₄	—	10	2	no	—
207	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
208	C₂²×C₄	#711	1	1	yes	characteristic
209	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
210	D₈	—	4	8	no	—
211	D₈	—	4	8	no	—
212	C₂×C₈	—	2	1	no	—
213	C₂×C₈	—	2	1	no	—
214	C₂×C₈	—	10	2	no	—
215	C₂×C₈	—	20	1	no	—
216	C₂×C₈	—	20	1	no	—
217	C₂×C₈	—	20	4	no	—
218	C₂×C₈	—	20	2	no	—
219	C₂×C₈	#686	1	2	yes	—
220	C₂×C₈	#687	1	2	yes	—
221	C₂×C₈	C₂²⋊Dic₅	1	2	yes	—
222	C₂×C₈	C₂²⋊Dic₅	1	2	yes	—
223	C₄⋊C₄	—	5	4	no	—
224	C₄⋊C₄	—	10	2	no	—
225	C₄²	—	20	1	no	—
226	C₄²	—	20	1	no	—
227	C₄²	D₄×D₅	1	1	yes	characteristic
228	C₄²	#711	1	1	yes	characteristic
229	C₄²	C₂²×Dic₅	1	1	yes	characteristic
230	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
231	C₂×C₅	—	16	4	no	—
232	C₂×C₅	#1667	1	2	yes	—
233	C₂×C₅	#1668	1	2	yes	—
234	C₂×C₅	#1970	1	1	yes	characteristic
235	C₂×C₅	#1978	1	1	yes	characteristic
236	C₂×C₅	#1982	1	1	yes	characteristic
237	C₂³	—	4	4	no	—
238	C₂³	—	8	2	no	—
239	C₂³	—	8	2	no	—
240	C₂³	—	8	4	no	—
241	C₂³	—	8	4	no	—
242	C₂³	#4084	1	1	yes	characteristic
243	D₄	—	4	8	no	—
244	D₄	—	4	4	no	—
245	D₄	—	4	4	no	—
246	D₄	—	8	4	no	—
247	D₄	—	8	4	no	—
248	D₄	—	8	2	no	—
249	D₄	—	8	2	no	—
250	D₄	—	8	2	no	—
251	D₄	—	8	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	10	4	no	—
254	C₂×C₄	—	20	2	no	—
255	C₂×C₄	—	20	2	no	—
256	C₂×C₄	D₅×D₈	1	2	yes	—
257	C₂×C₄	#4058	1	2	yes	—
258	C₂×C₄	C₂×#686	1	1	yes	characteristic
259	C₂×C₄	#4078	1	1	yes	characteristic
260	C₂×C₄	#4078	1	1	yes	characteristic
261	C₂×C₄	C₂×#687	1	1	yes	characteristic
262	C₂×C₄	D₄×Dic₅	1	1	yes	characteristic
263	C₂×C₄	D₄×Dic₅	1	1	yes	characteristic
264	C₂×C₄	#4082	1	1	yes	characteristic
265	C₂×C₄	#4084	1	1	yes	characteristic
266	C₂×C₄	#4086	1	1	yes	characteristic
267	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
268	C₈	—	2	2	no	—
269	C₈	—	20	2	no	—
270	C₈	—	40	2	no	—
271	C₈	#3964	1	4	yes	complemented
272	C₅	#9734	1	1	yes	characteristic, complemented, Sylow
273	C₂²	—	8	4	no	—
274	C₂²	—	8	4	no	—
275	C₂²	—	8	4	no	—
276	C₂²	—	16	4	no	—
277	C₂²	—	16	4	no	—
278	C₂²	#65361	1	1	yes	characteristic
279	C₂²	#65364	1	1	yes	characteristic
280	C₂²	#65446	1	2	yes	—
281	C₂²	#65449	1	2	yes	—
282	C₂²	#65510	1	1	yes	characteristic
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	20	4	no	—
286	C₄	#65361	1	1	yes	characteristic
287	C₄	#65364	1	1	yes	characteristic
288	C₄	D₈×Dic₅	1	2	yes	—
289	C₄	#65440	1	1	yes	characteristic
290	C₄	#65444	1	1	yes	characteristic
291	C₄	C₂×#3964	1	1	yes	characteristic
292	C₄	#65504	1	1	yes	characteristic
293	C₂	—	16	4	no	—
294	C₂	#10602075	1	2	yes	—
295	C₂	#10602079	1	2	yes	—
296	C₂	#10602554	1	1	yes	characteristic
297	C₂	#10602831	1	1	yes	characteristic
298	C₂	#10602834	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#4084
commutator	C₄²×C₅	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅
Fitting, prime radical	C₂×C₅×#493	C₂
socle	C₂³×C₅	#121
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	C₂×#493	D₅
hypocenter	C₅	#9734
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#493

Complements

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

N	H ≅ G/N	type
2. #10602554	C₂	semidirect
3. C₂×#64764	C₂	semidirect
4. C₈×C₄⋊Dic₅	C₂	semidirect
5. C₂×#64678	C₂	semidirect
10. C₅×#493	C₄	semidirect
86. C₄⋊Dic₅	D₈	semidirect
98. #493	Dic₅	semidirect
140. C₅×C₈	#102	semidirect
271. C₈	#3964	semidirect
272. C₅	#9734	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#493	2
#10602554	2
C₂×#64764	2
C₈×C₄⋊Dic₅	2
C₂×#64678	2
#9734	5