Subgroups — #12119116

Order 1280.

754 conjugacy classes of subgroups.

The subgroup lattice has 290 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	#10602860	C₂	1	2	yes	maximal
4	#10602420	C₂	1	2	yes	maximal
5	#10601398	C₂	1	2	yes	maximal
6	#10601017	C₂	1	4	yes	maximal
7	#10600559	C₂	1	1	yes	maximal, characteristic
8	#10600550	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×#3947	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	#22815	—	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	2	yes	—
57	#3950	—	4	2	no	—
58	#3949	—	2	4	no	—
59	#3947	—	2	4	no	—
60	C₄⋊#168	—	2	2	no	—
61	C₂×C₄×Q₁₆	D₅	1	1	yes	characteristic
62	#2356	—	5	2	no	—
63	#1971	—	5	2	no	—
64	#1928	—	5	4	no	—
65	#1846	—	5	1	no	—
66	#1805	—	5	2	no	—
67	#1776	—	5	2	no	—
68	#1770	—	5	1	no	—
69	C₂×C₅×Q₈	—	2	2	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	C₂²×C₄	1	4	yes	—
75	C₂²×C₄×C₅	—	2	2	no	—
76	C₂²×C₄×C₅	—	2	2	no	—
77	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
80	C₂²×Dic₅	—	2	4	no	—
81	C₂²×Dic₅	C₂×D₄	1	4	yes	—
82	C₅×Q₁₆	—	2	8	no	—
83	C₂×C₅×C₈	—	2	1	no	—
84	C₂×C₅×C₈	—	2	1	no	—
85	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
86	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
87	C₂×C₅×C₈	#41	1	2	yes	—
88	C₂×C₅×C₈	#41	1	2	yes	—
89	C₅×C₄⋊C₄	—	2	8	no	—
90	C₅×C₄⋊C₄	—	2	2	no	—
91	C₅×C₄⋊C₄	C₂²×C₄	1	4	yes	—
92	C₄²×C₅	—	2	8	no	—
93	C₄²×C₅	C₂²×C₄	1	2	yes	—
94	C₄²×C₅	#41	1	2	yes	—
95	C₄⋊Dic₅	—	2	4	no	—
96	C₄⋊Dic₅	—	2	4	no	—
97	C₄⋊Dic₅	—	4	2	no	—
98	C₄×Dic₅	—	2	2	no	—
99	C₄×Dic₅	—	2	2	no	—
100	C₄×Dic₅	—	4	8	no	—
101	C₂×#168	—	2	2	no	—
102	C₂×#168	—	2	2	no	—
103	C₂×#168	—	2	2	no	—
104	C₂²×Q₁₆	C₂×D₅	1	1	yes	characteristic
105	C₂×C₄×Q₈	C₂×D₅	1	2	yes	—
106	C₄×Q₁₆	Dic₅	1	8	yes	complemented
107	C₂×#107	—	5	2	no	—
108	C₂×#107	C₂×D₅	1	1	yes	characteristic
109	C₂×#106	—	5	2	no	—
110	C₂×C₄⋊C₈	—	5	1	no	—
111	C₂×#103	—	5	4	no	—
112	C₂×#103	C₂×D₅	1	2	yes	—
113	C₂×#97	—	5	2	no	—
114	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
115	#391	—	5	4	no	—
116	#387	—	10	4	no	—
117	#383	—	10	2	no	—
118	#382	—	5	2	no	—
119	C₄×C₄⋊C₄	—	5	1	no	—
120	C₄×C₄⋊C₄	—	5	4	no	—
121	#376	—	10	2	no	—
122	#340	—	5	4	no	—
123	#336	—	5	2	no	—
124	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
125	C₅×Q₈	—	2	4	no	—
126	C₅×Q₈	—	2	4	no	—
127	C₅×Q₈	—	4	2	no	—
128	C₅×Q₈	—	4	2	no	—
129	C₂×C₄×C₅	—	2	2	no	—
130	C₂×C₄×C₅	—	2	2	no	—
131	C₂×C₄×C₅	—	2	4	no	—
132	C₂×C₄×C₅	—	2	4	no	—
133	C₂×C₄×C₅	—	4	2	no	—
134	C₂×C₄×C₅	—	4	2	no	—
135	C₂×C₄×C₅	—	4	4	no	—
136	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
137	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
138	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
139	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
140	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
141	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
142	C₂×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
143	C₂×C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
144	C₂×Dic₅	—	2	4	no	—
145	C₂×Dic₅	—	2	4	no	—
146	C₂×Dic₅	—	2	4	no	—
147	C₂×Dic₅	—	4	4	no	—
148	C₂×Dic₅	—	4	4	no	—
149	C₂×Dic₅	—	4	4	no	—
150	C₅×C₈	—	2	2	no	—
151	C₅×C₈	#117	1	4	yes	—
152	#168	—	4	2	no	—
153	C₂²×Q₈	C₂²×D₅	1	2	yes	—
154	C₂×Q₁₆	—	2	2	no	—
155	C₂×Q₁₆	—	2	2	no	—
156	C₂×Q₁₆	C₂×Dic₅	1	4	yes	—
157	C₂²×C₈	—	5	2	no	—
158	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
159	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
160	C₄×Q₈	—	2	4	no	—
161	C₄×Q₈	C₂×Dic₅	1	8	yes	—
162	C₂×C₄⋊C₄	—	2	2	no	—
163	C₂×C₄⋊C₄	—	5	4	no	—
164	C₂×C₄⋊C₄	—	5	2	no	—
165	C₂×C₄⋊C₄	C₂²×D₅	1	2	yes	—
166	C₂×C₄²	—	2	2	no	—
167	C₂×C₄²	—	5	2	no	—
168	C₂×C₄²	—	10	4	no	—
169	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
170	#107	—	20	2	no	—
171	#107	C₂×Dic₅	1	4	yes	—
172	#106	—	10	4	no	—
173	C₄⋊C₈	—	10	2	no	—
174	#103	—	20	4	no	—
175	#103	C₂×Dic₅	1	8	yes	—
176	#97	—	10	4	no	—
177	C₄×C₈	C₂×Dic₅	1	4	yes	—
178	#95	—	5	2	no	—
179	#95	—	10	4	no	—
180	#95	—	10	4	no	—
181	#95	—	10	4	no	—
182	#95	—	10	2	no	—
183	#95	—	10	2	no	—
184	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
185	C₂²×C₅	#547	1	1	yes	characteristic
186	C₂²×C₅	#550	1	1	yes	characteristic
187	C₂²×C₅	#575	1	2	yes	—
188	C₂²×C₅	#578	1	2	yes	—
189	C₄×C₅	—	2	2	no	—
190	C₄×C₅	—	2	2	no	—
191	C₄×C₅	—	4	4	no	—
192	C₄×C₅	—	4	4	no	—
193	C₄×C₅	C₂×#117	1	1	yes	characteristic
194	C₄×C₅	#520	1	1	yes	characteristic
195	C₄×C₅	#555	1	1	yes	characteristic
196	C₄×C₅	#556	1	1	yes	characteristic
197	Dic₅	—	4	4	no	—
198	Dic₅	—	8	4	no	—
199	C₂×Q₈	—	2	2	no	—
200	C₂×Q₈	—	2	2	no	—
201	C₂×Q₈	—	2	2	no	—
202	C₂×Q₈	—	2	2	no	—
203	C₂×Q₈	—	2	2	no	—
204	C₂×Q₈	C₂²×Dic₅	1	4	yes	—
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	—	5	4	no	—
208	C₂²×C₄	—	10	4	no	—
209	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
210	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
211	C₂²×C₄	C₂³×D₅	1	1	yes	characteristic
212	Q₁₆	—	2	8	no	—
213	C₂×C₈	—	2	1	no	—
214	C₂×C₈	—	2	1	no	—
215	C₂×C₈	—	10	2	no	—
216	C₂×C₈	—	10	2	no	—
217	C₂×C₈	—	10	2	no	—
218	C₂×C₈	#709	1	2	yes	—
219	C₂×C₈	#709	1	2	yes	—
220	C₂×C₈	C₂²×Dic₅	1	2	yes	—
221	C₂×C₈	C₂²×Dic₅	1	2	yes	—
222	C₄⋊C₄	—	2	2	no	—
223	C₄⋊C₄	—	2	8	no	—
224	C₄⋊C₄	—	10	4	no	—
225	C₄⋊C₄	—	10	4	no	—
226	C₄⋊C₄	—	20	2	no	—
227	C₄⋊C₄	C₂²×Dic₅	1	4	yes	—
228	C₄²	—	2	8	no	—
229	C₄²	—	10	2	no	—
230	C₄²	—	10	2	no	—
231	C₄²	—	20	8	no	—
232	C₄²	#711	1	2	yes	—
233	C₄²	C₂²×Dic₅	1	2	yes	—
234	C₂×C₅	#2354	1	1	yes	characteristic
235	C₂×C₅	#2943	1	1	yes	characteristic
236	C₂×C₅	#2950	1	1	yes	characteristic
237	C₂×C₅	#3352	1	2	yes	—
238	C₂×C₅	#3402	1	2	yes	—
239	C₂³	C₂×D₄×D₅	1	1	yes	characteristic
240	Q₈	—	2	4	no	—
241	Q₈	—	2	4	no	—
242	Q₈	—	4	2	no	—
243	Q₈	—	4	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	10	4	no	—
252	C₂×C₄	—	10	4	no	—
253	C₂×C₄	—	10	4	no	—
254	C₂×C₄	—	20	4	no	—
255	C₂×C₄	—	20	4	no	—
256	C₂×C₄	—	20	4	no	—
257	C₂×C₄	D₄×Dic₅	1	2	yes	—
258	C₂×C₄	D₄×Dic₅	1	2	yes	—
259	C₂×C₄	C₂×#709	1	1	yes	characteristic
260	C₂×C₄	C₂×#709	1	1	yes	characteristic
261	C₂×C₄	C₂×#711	1	1	yes	characteristic
262	C₂×C₄	#4149	1	1	yes	characteristic
263	C₂×C₄	#4150	1	1	yes	characteristic
264	C₂×C₄	C₂³×Dic₅	1	1	yes	characteristic
265	C₈	—	2	2	no	—
266	C₈	—	20	2	no	—
267	C₈	#4072	1	4	yes	—
268	C₅	#22815	1	1	yes	characteristic, complemented, Sylow
269	C₂²	#65885	1	1	yes	characteristic
270	C₂²	#65887	1	1	yes	characteristic
271	C₂²	#66100	1	2	yes	—
272	C₂²	#66102	1	2	yes	—
273	C₂²	C₂×D₄×Dic₅	1	1	yes	characteristic
274	C₄	—	2	2	no	—
275	C₄	—	2	2	no	—
276	C₄	—	4	4	no	—
277	C₄	—	4	4	no	—
278	C₄	—	20	4	no	—
279	C₄	—	40	4	no	—
280	C₄	#65876	1	1	yes	characteristic
281	C₄	#65897	1	1	yes	characteristic
282	C₄	C₂×#4072	1	1	yes	characteristic
283	C₄	#66161	1	1	yes	characteristic
284	C₂	#10611445	1	2	yes	—
285	C₂	#10611475	1	2	yes	—
286	C₂	#10614068	1	1	yes	characteristic
287	C₂	#10615605	1	1	yes	characteristic
288	C₂	#10615628	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	#22815
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
106. C₄×Q₁₆	Dic₅	semidirect
268. C₅	#22815	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
#10602860	2
#10602420	2
#10601398	2
#10601017	2
#10600559	2
#10600550	2
#22815	5