Subgroups — #12087239

Order 1280.

858 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3169	C₂	1	1	yes	maximal, characteristic
2	#10615019	C₂	1	2	yes	maximal, complemented
3	#10615001	C₂	1	2	yes	maximal, complemented
4	C₂×#64971	C₂	1	1	yes	maximal, characteristic, complemented
5	#10602155	C₂	1	2	yes	maximal, complemented
6	#10602067	C₂	1	2	yes	maximal, complemented
7	#10601988	C₂	1	4	yes	maximal, complemented
8	C₈×#681	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#532	C₂²	1	2	yes	—
10	C₂×#4097	—	2	4	no	—
11	C₂×#4015	C₂²	1	2	yes	—
12	C₅×#500	C₂²	1	1	yes	characteristic
13	C₅×#493	C₂²	1	1	yes	characteristic
14	C₅×#486	C₂²	1	2	yes	—
15	C₅×#466	—	2	2	no	—
16	C₄×C₅×D₈	—	2	2	no	—
17	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
18	C₂×#3950	—	2	2	no	—
19	#65381	C₂²	1	4	yes	—
20	#65367	C₂²	1	4	yes	—
21	#65365	C₂²	1	4	yes	—
22	#65359	C₂²	1	4	yes	—
23	#65349	—	2	4	no	—
24	#65322	—	2	4	no	—
25	#65296	—	2	4	no	—
26	#65214	C₂²	1	1	yes	characteristic
27	C₄×#681	C₂²	1	1	yes	characteristic
28	#64971	—	2	2	no	—
29	#64971	C₂²	1	2	yes	—
30	#64971	C₂²	1	2	yes	—
31	#64674	C₂²	1	2	yes	—
32	C₈×#168	C₂²	1	2	yes	—
33	#19630	—	5	1	no	maximal, Sylow
34	C₂×C₅×#41	—	2	2	no	—
35	C₂²×Dic₁₀	—	2	2	no	—
36	C₂×C₅×D₈	—	2	2	no	—
37	C₂²×C₅×C₈	D₄	1	2	yes	—
38	C₅×C₄⋊Q₈	C₂³	1	2	yes	—
39	C₅×#127	C₂³	1	2	yes	—
40	C₅×#124	—	2	2	no	—
41	C₅×#121	—	2	2	no	—
42	C₅×#121	—	2	2	no	—
43	C₄×C₅×D₄	—	2	2	no	—
44	C₄×C₅×D₄	—	2	2	no	—
45	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
46	#4097	—	4	16	no	—
47	C₂×#689	—	2	8	no	—
48	C₂×#687	—	2	8	no	—
49	C₂×C₄⋊Dic₅	—	2	2	no	—
50	C₂×C₄⋊Dic₅	—	2	2	no	—
51	C₂×#681	D₄	1	2	yes	—
52	#4015	—	2	2	no	—
53	#4015	—	2	2	no	—
54	#4015	—	2	2	no	—
55	#4015	C₂³	1	4	yes	—
56	C₅×#107	—	2	2	no	—
57	C₅×#102	—	2	4	no	—
58	C₄×C₅×C₈	D₄	1	2	yes	—
59	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
60	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
61	#3964	—	2	8	no	—
62	C₂²⋊#168	D₄	1	2	yes	—
63	#3950	—	2	4	no	—
64	#3950	—	2	4	no	—
65	#3939	—	2	8	no	—
66	C₄⋊#168	D₄	1	2	yes	—
67	#3935	D₄	1	2	yes	—
68	C₄×#168	C₂³	1	2	yes	—
69	C₄×#168	C₂³	1	2	yes	—
70	#3226	—	5	2	no	—
71	#3169	D₅	1	1	yes	characteristic
72	C₂×#500	—	5	1	no	—
73	#3152	—	5	2	no	—
74	#1722	—	5	4	no	—
75	#1713	—	5	2	no	—
76	#1706	—	5	2	no	—
77	C₈×M₁₆	—	5	1	no	—
78	C₅×#41	—	4	8	no	—
79	C₂×C₅×Q₈	—	2	2	no	—
80	C₂×C₅×D₄	—	2	2	no	—
81	C₂×C₅×D₄	—	2	2	no	—
82	C₂×C₅×D₄	—	2	2	no	—
83	C₂×C₅×D₄	—	4	2	no	—
84	C₂²×C₄×C₅	—	4	2	no	—
85	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
88	C₂²×Dic₅	—	4	2	no	—
89	C₂×Dic₁₀	—	2	4	no	—
90	C₂×Dic₁₀	—	4	8	no	—
91	C₂×Dic₁₀	—	4	2	no	—
92	C₅×D₈	—	4	4	no	—
93	C₂×C₅×C₈	—	2	4	no	—
94	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
95	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
96	C₅×C₄⋊C₄	—	2	2	no	—
97	C₅×C₄⋊C₄	—	2	2	no	—
98	C₅×C₂²⋊C₄	—	4	2	no	—
99	C₅×C₂²⋊C₄	—	4	2	no	—
100	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₅	C₂⁴	1	1	yes	characteristic
104	#689	—	4	16	no	—
105	#687	—	4	16	no	—
106	C₄⋊Dic₅	—	2	4	no	—
107	C₄⋊Dic₅	—	2	4	no	—
108	C₄⋊Dic₅	—	4	2	no	—
109	C₄⋊Dic₅	—	4	2	no	—
110	#681	D₈	1	8	yes	complemented
111	C₂×#168	C₂×D₄	1	4	yes	—
112	C₂×#168	C₂×D₄	1	4	yes	—
113	C₂×#137	—	10	4	no	—
114	#532	C₂×D₅	1	2	yes	—
115	C₂×C₄⋊Q₈	—	5	2	no	—
116	#500	—	5	4	no	—
117	#500	—	5	2	no	—
118	#500	—	5	2	no	—
119	#500	—	10	2	no	—
120	#500	C₂×D₅	1	1	yes	characteristic
121	#497	—	10	4	no	—
122	#494	—	5	4	no	—
123	#493	C₂×D₅	1	1	yes	characteristic
124	#492	—	5	4	no	—
125	#486	—	5	4	no	—
126	#486	C₂×D₅	1	2	yes	—
127	#480	—	10	4	no	—
128	#474	—	10	4	no	—
129	#466	—	2	2	no	—
130	C₄×D₈	—	2	2	no	—
131	#431	—	5	1	no	—
132	C₂×#107	—	10	2	no	—
133	C₄×M₁₆	—	5	1	no	—
134	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
135	#322	—	5	2	no	—
136	C₈²	—	5	2	no	—
137	C₂³×C₅	—	4	2	no	—
138	C₂³×C₅	#127	1	1	yes	characteristic
139	C₅×Q₈	—	4	4	no	—
140	C₅×D₄	—	4	4	no	—
141	C₅×D₄	—	4	4	no	—
142	C₅×D₄	—	4	4	no	—
143	C₅×D₄	—	8	4	no	—
144	C₂×C₄×C₅	—	2	2	no	—
145	C₂×C₄×C₅	—	2	2	no	—
146	C₂×C₄×C₅	—	4	2	no	—
147	C₂×C₄×C₅	—	8	4	no	—
148	C₂×C₄×C₅	#127	1	1	yes	characteristic
149	C₂×C₄×C₅	#127	1	1	yes	characteristic
150	C₂×C₄×C₅	#127	1	1	yes	characteristic
151	C₂×C₄×C₅	C₂×D₈	1	4	yes	—
152	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
153	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
154	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
155	C₂×Dic₅	—	4	4	no	—
156	C₂×Dic₅	—	8	4	no	—
157	Dic₁₀	—	4	8	no	—
158	Dic₁₀	—	8	4	no	—
159	C₅×C₈	—	2	2	no	—
160	C₅×C₈	#137	1	4	yes	complemented
161	#168	—	2	4	no	—
162	#168	C₂×D₈	1	8	yes	—
163	C₂×#41	—	2	2	no	—
164	C₂²×Q₈	—	10	2	no	—
165	#137	—	20	16	no	—
166	C₂×Q₁₆	—	10	8	no	—
167	C₂×SD₁₆	—	10	8	no	—
168	C₂×D₈	—	2	2	no	—
169	C₂×M₁₆	—	5	2	no	—
170	C₂²×C₈	C₂²⋊D₅	1	2	yes	—
171	C₄⋊Q₈	—	5	4	no	—
172	C₄⋊Q₈	—	10	2	no	—
173	C₄⋊Q₈	—	10	2	no	—
174	C₄⋊Q₈	—	10	2	no	—
175	C₄⋊Q₈	C₂²×D₅	1	2	yes	—
176	#127	C₂²×D₅	1	2	yes	—
177	#124	—	2	2	no	—
178	#121	—	2	2	no	—
179	#121	—	2	2	no	—
180	C₄×D₄	—	2	2	no	—
181	C₄×D₄	—	2	2	no	—
182	C₂×C₄⋊C₄	—	10	2	no	—
183	C₂×C₄⋊C₄	—	10	2	no	—
184	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
185	#107	—	2	2	no	—
186	#107	—	10	8	no	—
187	#107	—	10	4	no	—
188	#107	—	10	4	no	—
189	C₄⋊C₈	—	5	2	no	—
190	#102	—	2	4	no	—
191	#102	—	10	8	no	—
192	C₂²⋊C₈	—	5	2	no	—
193	#97	—	5	2	no	—
194	C₄×C₈	—	5	2	no	—
195	C₄×C₈	—	5	2	no	—
196	C₄×C₈	C₂²⋊D₅	1	2	yes	—
197	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
198	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
199	C₂²×C₅	—	2	1	no	—
200	C₂²×C₅	—	2	1	no	—
201	C₂²×C₅	—	8	2	no	—
202	C₂²×C₅	—	8	2	no	—
203	C₂²×C₅	—	8	2	no	—
204	C₂²×C₅	#493	1	2	yes	—
205	C₂²×C₅	C₂×#127	1	1	yes	characteristic
206	C₄×C₅	—	2	1	no	—
207	C₄×C₅	—	2	1	no	—
208	C₄×C₅	—	8	2	no	—
209	C₄×C₅	#493	1	2	yes	—
210	C₄×C₅	C₂²×D₈	1	2	yes	—
211	C₄×C₅	C₂×#137	1	2	yes	—
212	C₄×C₅	#578	1	2	yes	—
213	Dic₅	—	8	4	no	—
214	#41	—	4	8	no	—
215	C₂×Q₈	—	2	2	no	—
216	C₂×Q₈	—	10	4	no	—
217	C₂×Q₈	—	20	2	no	—
218	C₂×Q₈	—	20	8	no	—
219	C₂×D₄	—	2	2	no	—
220	C₂×D₄	—	2	2	no	—
221	C₂×D₄	—	2	2	no	—
222	C₂×D₄	—	4	2	no	—
223	C₂²×C₄	—	4	2	no	—
224	C₂²×C₄	—	20	2	no	—
225	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
226	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
228	Q₁₆	—	20	16	no	—
229	SD₁₆	—	20	16	no	—
230	D₈	—	4	4	no	—
231	M₁₆	—	5	8	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	5	4	no	—
234	C₂×C₈	—	5	4	no	—
235	C₂×C₈	C₂×C₂²⋊D₅	1	2	yes	—
236	C₂×C₈	C₂×C₂²⋊D₅	1	2	yes	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	—	2	2	no	—
239	C₄⋊C₄	—	10	4	no	—
240	C₄⋊C₄	—	10	4	no	—
241	C₄⋊C₄	—	20	2	no	—
242	C₄⋊C₄	—	20	2	no	—
243	C₂²⋊C₄	—	4	2	no	—
244	C₂²⋊C₄	—	4	2	no	—
245	C₄²	D₄×D₅	1	1	yes	characteristic
246	C₄²	D₄×D₅	1	1	yes	characteristic
247	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
248	C₄²	C₂³×D₅	1	1	yes	characteristic
249	C₂×C₅	—	2	2	no	—
250	C₂×C₅	—	16	2	no	—
251	C₂×C₅	C₂×#493	1	1	yes	characteristic
252	C₂×C₅	#3152	1	1	yes	characteristic
253	C₂×C₅	#3156	1	1	yes	characteristic
254	C₂³	—	4	2	no	—
255	C₂³	#4086	1	1	yes	characteristic
256	Q₈	—	4	4	no	—
257	Q₈	—	20	8	no	—
258	Q₈	—	40	4	no	—
259	D₄	—	4	4	no	—
260	D₄	—	4	4	no	—
261	D₄	—	4	4	no	—
262	D₄	—	8	4	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	8	4	no	—
267	C₂×C₄	—	20	4	no	—
268	C₂×C₄	—	40	4	no	—
269	C₂×C₄	D₅×D₈	1	4	yes	—
270	C₂×C₄	#4086	1	1	yes	characteristic
271	C₂×C₄	#4086	1	1	yes	characteristic
272	C₂×C₄	#4086	1	1	yes	characteristic
273	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
274	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
275	C₂×C₄	C₂²×C₂²⋊D₅	1	1	yes	characteristic
276	C₈	—	2	2	no	—
277	C₈	—	5	8	no	—
278	C₈	—	10	4	no	—
279	C₈	#4097	1	4	yes	complemented
280	C₅	#19630	1	1	yes	characteristic, complemented, Sylow
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	8	2	no	—
284	C₂²	—	8	2	no	—
285	C₂²	—	8	2	no	—
286	C₂²	#65440	1	2	yes	—
287	C₂²	C₂×#4086	1	1	yes	characteristic
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	8	2	no	—
291	C₄	—	40	4	no	—
292	C₄	#65440	1	2	yes	—
293	C₄	C₂×D₅×D₈	1	2	yes	—
294	C₄	#66105	1	2	yes	—
295	C₄	C₂×#4097	1	2	yes	—
296	C₂	—	2	2	no	—
297	C₂	—	16	2	no	—
298	C₂	#10615019	1	1	yes	characteristic
299	C₂	C₂×#65440	1	1	yes	characteristic
300	C₂	#10615638	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#4086
commutator, cosocle	C₄²×C₅	C₂⁴
Frattini	C₄²	C₂³×D₅
Fitting, prime radical	C₅×#3169	C₂
socle	C₂²×C₅	C₂×#127
hypercenter	#3169	D₅
hypocenter	C₅	#19630
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3169

Complements

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

N	H ≅ G/N	type
2. #10615019	C₂	semidirect
3. #10615001	C₂	semidirect
4. C₂×#64971	C₂	semidirect
5. #10602155	C₂	semidirect
6. #10602067	C₂	semidirect
7. #10601988	C₂	semidirect
8. C₈×#681	C₂	semidirect
110. #681	D₈	semidirect
160. C₅×C₈	#137	semidirect
279. C₈	#4097	semidirect
280. C₅	#19630	semidirect

Maximal subgroups

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

subgroup	index
C₅×#3169	2
#10615019	2
#10615001	2
C₂×#64971	2
#10602155	2
#10602067	2
#10601988	2
C₈×#681	2
#19630	5