Subgroups — #12119062

Order 1280.

766 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	C₂×C₄×C₅×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	#10614194	C₂	1	2	yes	maximal
3	#10602859	C₂	1	2	yes	maximal
4	#10602420	C₂	1	2	yes	maximal
5	Dic₅×#107	C₂	1	2	yes	maximal
6	#10601016	C₂	1	4	yes	maximal
7	#10600559	C₂	1	1	yes	maximal, characteristic
8	C₈×C₄⋊Dic₅	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₅×Q₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₅×Q₈	C₂²	1	2	yes	—
11	C₄×C₅×Q₁₆	C₄	1	8	yes	complemented
12	C₂×C₅×#107	C₂²	1	1	yes	characteristic
13	C₂×C₅×#103	C₂²	1	2	yes	—
14	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	#65519	C₂²	1	4	yes	—
16	C₂×#3967	C₂²	1	4	yes	—
17	C₂×#3950	C₂²	1	2	yes	—
18	C₂×#3949	C₂²	1	2	yes	—
19	C₂×C₈×Dic₅	C₂²	1	2	yes	—
20	#65270	—	2	4	no	—
21	C₄⋊C₄×Dic₅	C₂²	1	4	yes	—
22	#65226	—	2	2	no	—
23	#65225	C₂²	1	2	yes	—
24	C₄×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
25	#65221	—	2	2	no	—
26	C₂×C₄⋊#168	C₂²	1	1	yes	characteristic
27	#64770	C₂²	1	2	yes	—
28	#64749	C₂²	1	4	yes	—
29	#22809	—	5	1	no	maximal, Sylow
30	C₂²×C₅×Q₈	C₂³	1	2	yes	—
31	C₂×C₅×Q₁₆	—	2	2	no	—
32	C₂×C₅×Q₁₆	—	2	2	no	—
33	C₂×C₅×Q₁₆	C₂×C₄	1	4	yes	—
34	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
35	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
36	C₄×C₅×Q₈	—	2	4	no	—
37	C₄×C₅×Q₈	C₂×C₄	1	8	yes	—
38	C₂×C₅×C₄⋊C₄	—	2	2	no	—
39	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
40	C₂×C₄²×C₅	—	2	2	no	—
41	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
42	C₂×C₄⋊Dic₅	D₄	1	2	yes	—
43	C₂×C₄⋊Dic₅	C₂³	1	4	yes	—
44	C₂×C₄×Dic₅	—	2	4	no	—
45	C₂×C₄×Dic₅	C₂³	1	2	yes	—
46	C₂²×#168	C₂³	1	2	yes	—
47	C₅×#107	C₂×C₄	1	4	yes	—
48	C₅×#103	C₂×C₄	1	8	yes	—
49	C₄×C₅×C₈	C₂×C₄	1	4	yes	—
50	#3967	—	4	4	no	—
51	#3963	—	2	2	no	—
52	#3963	—	2	2	no	—
53	#3963	—	2	4	no	—
54	#3963	—	2	4	no	—
55	#3963	—	2	4	no	—
56	#3963	D₄	1	1	yes	characteristic
57	#3963	D₄	1	1	yes	characteristic
58	#3950	—	2	4	no	—
59	#3949	—	4	2	no	—
60	C₈×Dic₅	—	2	4	no	—
61	C₄⋊#168	—	2	2	no	—
62	C₂×C₄×Q₁₆	D₅	1	1	yes	characteristic
63	#2356	—	5	2	no	—
64	#1969	—	5	2	no	—
65	#1927	—	5	4	no	—
66	#1846	—	5	1	no	—
67	#1805	—	5	2	no	—
68	C₄×#107	—	5	2	no	—
69	C₈×C₄⋊C₄	—	5	1	no	—
70	C₂×C₅×Q₈	—	2	2	no	—
71	C₂×C₅×Q₈	—	2	2	no	—
72	C₂×C₅×Q₈	—	2	2	no	—
73	C₂×C₅×Q₈	—	2	2	no	—
74	C₂×C₅×Q₈	—	2	2	no	—
75	C₂×C₅×Q₈	C₂²×C₄	1	4	yes	—
76	C₂²×C₄×C₅	—	2	2	no	—
77	C₂²×C₄×C₅	—	2	2	no	—
78	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
81	C₂²×Dic₅	—	2	4	no	—
82	C₂²×Dic₅	C₂×D₄	1	2	yes	—
83	C₂²×Dic₅	C₂×D₄	1	2	yes	—
84	C₅×Q₁₆	—	2	8	no	—
85	C₂×C₅×C₈	—	2	1	no	—
86	C₂×C₅×C₈	—	2	1	no	—
87	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
88	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
89	C₂×C₅×C₈	#41	1	2	yes	—
90	C₂×C₅×C₈	#41	1	2	yes	—
91	C₅×C₄⋊C₄	—	2	8	no	—
92	C₅×C₄⋊C₄	—	2	2	no	—
93	C₅×C₄⋊C₄	C₂²×C₄	1	4	yes	—
94	C₄²×C₅	—	2	8	no	—
95	C₄²×C₅	C₂²×C₄	1	2	yes	—
96	C₄²×C₅	#41	1	2	yes	—
97	C₄⋊Dic₅	—	2	4	no	—
98	C₄⋊Dic₅	—	2	4	no	—
99	C₄⋊Dic₅	—	2	4	no	—
100	C₄×Dic₅	—	2	2	no	—
101	C₄×Dic₅	—	2	2	no	—
102	C₄×Dic₅	—	4	8	no	—
103	C₂×#168	—	2	2	no	—
104	C₂×#168	—	2	2	no	—
105	C₂×#168	—	2	2	no	—
106	C₂²×Q₁₆	C₂×D₅	1	1	yes	characteristic
107	C₂×C₄×Q₈	C₂×D₅	1	2	yes	—
108	C₄×Q₁₆	Dic₅	1	8	yes	complemented
109	C₂×#107	—	5	2	no	—
110	C₂×#107	C₂×D₅	1	1	yes	characteristic
111	C₂×#106	—	5	2	no	—
112	C₂×C₄⋊C₈	—	5	1	no	—
113	C₂×#103	—	5	4	no	—
114	C₂×#103	C₂×D₅	1	2	yes	—
115	C₂×C₄×C₈	—	5	2	no	—
116	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
117	#391	—	5	4	no	—
118	#387	—	10	4	no	—
119	#383	—	10	2	no	—
120	#382	—	5	2	no	—
121	C₄×C₄⋊C₄	—	5	1	no	—
122	C₄×C₄⋊C₄	—	5	4	no	—
123	#376	—	10	2	no	—
124	#340	—	5	4	no	—
125	#336	—	5	2	no	—
126	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
127	C₅×Q₈	—	2	4	no	—
128	C₅×Q₈	—	2	4	no	—
129	C₅×Q₈	—	4	2	no	—
130	C₅×Q₈	—	4	2	no	—
131	C₂×C₄×C₅	—	2	2	no	—
132	C₂×C₄×C₅	—	2	2	no	—
133	C₂×C₄×C₅	—	2	4	no	—
134	C₂×C₄×C₅	—	2	4	no	—
135	C₂×C₄×C₅	—	4	2	no	—
136	C₂×C₄×C₅	—	4	2	no	—
137	C₂×C₄×C₅	—	4	4	no	—
138	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
139	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
140	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
141	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
142	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
143	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
144	C₂×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
145	C₂×C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
146	C₂×Dic₅	—	2	2	no	—
147	C₂×Dic₅	—	2	2	no	—
148	C₂×Dic₅	—	2	2	no	—
149	C₂×Dic₅	—	2	2	no	—
150	C₂×Dic₅	—	2	2	no	—
151	C₂×Dic₅	—	4	4	no	—
152	C₂×Dic₅	—	4	4	no	—
153	C₂×Dic₅	—	4	4	no	—
154	C₂×Dic₅	#135	1	4	yes	—
155	C₅×C₈	—	2	2	no	—
156	C₅×C₈	#117	1	4	yes	—
157	#168	—	4	2	no	—
158	C₂²×Q₈	C₂²×D₅	1	2	yes	—
159	C₂×Q₁₆	—	2	2	no	—
160	C₂×Q₁₆	—	2	2	no	—
161	C₂×Q₁₆	C₂×Dic₅	1	4	yes	—
162	C₂²×C₈	—	5	2	no	—
163	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
164	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
165	C₄×Q₈	—	2	4	no	—
166	C₄×Q₈	C₂×Dic₅	1	8	yes	—
167	C₂×C₄⋊C₄	—	2	2	no	—
168	C₂×C₄⋊C₄	—	5	2	no	—
169	C₂×C₄⋊C₄	—	5	4	no	—
170	C₂×C₄⋊C₄	C₂²×D₅	1	2	yes	—
171	C₂×C₄²	—	2	2	no	—
172	C₂×C₄²	—	5	2	no	—
173	C₂×C₄²	—	10	4	no	—
174	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
175	#107	—	10	4	no	—
176	#107	C₂×Dic₅	1	4	yes	—
177	#106	—	20	2	no	—
178	C₄⋊C₈	—	10	2	no	—
179	#103	—	20	4	no	—
180	#103	C₂×Dic₅	1	8	yes	—
181	C₄×C₈	—	10	4	no	—
182	C₄×C₈	C₂×Dic₅	1	4	yes	—
183	#95	—	5	1	no	—
184	#95	—	5	1	no	—
185	#95	—	10	4	no	—
186	#95	—	10	4	no	—
187	#95	—	10	4	no	—
188	#95	—	10	2	no	—
189	#95	—	10	2	no	—
190	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
191	C₂²×C₅	#547	1	1	yes	characteristic
192	C₂²×C₅	#550	1	1	yes	characteristic
193	C₂²×C₅	C₂×#135	1	2	yes	—
194	C₂²×C₅	#578	1	2	yes	—
195	C₄×C₅	—	2	2	no	—
196	C₄×C₅	—	2	2	no	—
197	C₄×C₅	—	4	4	no	—
198	C₄×C₅	—	4	4	no	—
199	C₄×C₅	C₂×#117	1	1	yes	characteristic
200	C₄×C₅	#520	1	1	yes	characteristic
201	C₄×C₅	#555	1	1	yes	characteristic
202	C₄×C₅	#556	1	1	yes	characteristic
203	Dic₅	—	2	4	no	—
204	Dic₅	—	4	2	no	—
205	Dic₅	—	8	4	no	—
206	C₂×Q₈	—	2	2	no	—
207	C₂×Q₈	—	2	2	no	—
208	C₂×Q₈	—	2	2	no	—
209	C₂×Q₈	—	2	2	no	—
210	C₂×Q₈	—	2	2	no	—
211	C₂×Q₈	C₂²×Dic₅	1	4	yes	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	5	2	no	—
215	C₂²×C₄	—	5	2	no	—
216	C₂²×C₄	—	10	4	no	—
217	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
218	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
219	C₂²×C₄	C₂³×D₅	1	1	yes	characteristic
220	Q₁₆	—	2	8	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	10	2	no	—
224	C₂×C₈	—	10	2	no	—
225	C₂×C₈	—	10	2	no	—
226	C₂×C₈	#709	1	2	yes	—
227	C₂×C₈	#709	1	2	yes	—
228	C₂×C₈	C₂²×Dic₅	1	2	yes	—
229	C₂×C₈	C₂²×Dic₅	1	2	yes	—
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	—	2	8	no	—
232	C₄⋊C₄	—	10	4	no	—
233	C₄⋊C₄	—	10	4	no	—
234	C₄⋊C₄	—	10	4	no	—
235	C₄⋊C₄	C₂²×Dic₅	1	4	yes	—
236	C₄²	—	2	8	no	—
237	C₄²	—	10	2	no	—
238	C₄²	—	10	2	no	—
239	C₄²	—	20	8	no	—
240	C₄²	#711	1	2	yes	—
241	C₄²	C₂²×Dic₅	1	2	yes	—
242	C₂×C₅	#2354	1	1	yes	characteristic
243	C₂×C₅	C₄×#135	1	1	yes	characteristic
244	C₂×C₅	#2950	1	1	yes	characteristic
245	C₂×C₅	#3336	1	2	yes	—
246	C₂×C₅	#3393	1	2	yes	—
247	C₂³	C₂×D₄×D₅	1	1	yes	characteristic
248	Q₈	—	2	4	no	—
249	Q₈	—	2	4	no	—
250	Q₈	—	4	2	no	—
251	Q₈	—	4	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	4	4	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	5	4	no	—
260	C₂×C₄	—	10	2	no	—
261	C₂×C₄	—	10	2	no	—
262	C₂×C₄	—	10	2	no	—
263	C₂×C₄	—	10	2	no	—
264	C₂×C₄	—	10	2	no	—
265	C₂×C₄	—	20	4	no	—
266	C₂×C₄	—	20	4	no	—
267	C₂×C₄	—	20	4	no	—
268	C₂×C₄	D₄×Dic₅	1	2	yes	—
269	C₂×C₄	D₄×Dic₅	1	2	yes	—
270	C₂×C₄	C₂×#709	1	1	yes	characteristic
271	C₂×C₄	C₂×#709	1	1	yes	characteristic
272	C₂×C₄	C₂×#711	1	1	yes	characteristic
273	C₂×C₄	#4149	1	1	yes	characteristic
274	C₂×C₄	#4150	1	1	yes	characteristic
275	C₂×C₄	C₂³×Dic₅	1	1	yes	characteristic
276	C₈	—	2	2	no	—
277	C₈	—	20	2	no	—
278	C₈	#4072	1	4	yes	—
279	C₅	#22809	1	1	yes	characteristic, complemented, Sylow
280	C₂²	#65885	1	1	yes	characteristic
281	C₂²	#65887	1	1	yes	characteristic
282	C₂²	#66100	1	2	yes	—
283	C₂²	D₅×#135	1	2	yes	—
284	C₂²	C₂×D₄×Dic₅	1	1	yes	characteristic
285	C₄	—	2	2	no	—
286	C₄	—	2	2	no	—
287	C₄	—	4	4	no	—
288	C₄	—	4	4	no	—
289	C₄	—	10	4	no	—
290	C₄	—	20	2	no	—
291	C₄	—	40	4	no	—
292	C₄	#65876	1	1	yes	characteristic
293	C₄	#65897	1	1	yes	characteristic
294	C₄	C₂×#4072	1	1	yes	characteristic
295	C₄	#66161	1	1	yes	characteristic
296	C₂	#10611441	1	2	yes	—
297	C₂	#10611472	1	2	yes	—
298	C₂	#10614068	1	1	yes	characteristic
299	C₂	#10615605	1	1	yes	characteristic
300	C₂	Dic₅×#135	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₄×D₅
commutator	C₂×C₄×C₅	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅
Fitting	C₂×C₄×C₅×Q₁₆	C₂
socle, prime radical	C₂³×C₅	C₂²×D₄
cosocle	C₂²×C₄×C₅	C₂⁴
hypercenter	C₂×C₄×Q₁₆	D₅
hypocenter	C₅	#22809
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
11. C₄×C₅×Q₁₆	C₄	semidirect
108. C₄×Q₁₆	Dic₅	semidirect
279. C₅	#22809	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₅×Q₁₆	2
#10614194	2
#10602859	2
#10602420	2
Dic₅×#107	2
#10601016	2
#10600559	2
C₈×C₄⋊Dic₅	2
#22809	5