Subgroups — #12119108

Order 1280.

770 conjugacy classes of subgroups.

The subgroup lattice has 296 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	#10614193	C₂	1	2	yes	maximal
3	#10602860	C₂	1	2	yes	maximal
4	#10602419	C₂	1	2	yes	maximal
5	#10601398	C₂	1	2	yes	maximal
6	#10601016	C₂	1	4	yes	maximal
7	#10600557	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×#3949	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	#65225	—	2	2	no	—
23	#65224	C₂²	1	2	yes	—
24	C₄×C₄⋊Dic₅	—	2	2	no	—
25	C₄×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊#168	C₂²	1	1	yes	characteristic
27	#64770	C₂²	1	2	yes	—
28	#64749	C₂²	1	4	yes	—
29	#22814	—	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	—	2	2	no	—
43	C₂×C₄⋊Dic₅	—	2	2	no	—
44	C₂×C₄⋊Dic₅	D₄	1	2	yes	—
45	C₂×C₄⋊Dic₅	C₂³	1	4	yes	—
46	C₂×C₄×Dic₅	—	2	4	no	—
47	C₂×C₄×Dic₅	C₂³	1	2	yes	—
48	C₂²×#168	C₂³	1	2	yes	—
49	C₅×#107	C₂×C₄	1	4	yes	—
50	C₅×#103	C₂×C₄	1	8	yes	—
51	C₄×C₅×C₈	C₂×C₄	1	4	yes	—
52	#3967	—	4	4	no	—
53	#3963	—	2	4	no	—
54	#3963	—	2	4	no	—
55	#3963	—	2	4	no	—
56	#3963	D₄	1	2	yes	—
57	#3949	—	2	4	no	—
58	#3949	—	4	2	no	—
59	#3947	—	2	4	no	—
60	C₄⋊#168	—	2	2	no	—
61	C₂×C₄×Q₁₆	D₅	1	1	yes	characteristic
62	#2355	—	5	2	no	—
63	#1971	—	5	2	no	—
64	#1927	—	5	4	no	—
65	#1844	—	5	1	no	—
66	#1803	—	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	—	2	4	no	—
98	C₄⋊Dic₅	—	4	4	no	—
99	C₄⋊Dic₅	—	8	2	no	—
100	C₄×Dic₅	—	2	2	no	—
101	C₄×Dic₅	—	2	2	no	—
102	C₄×Dic₅	—	8	4	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	C₂×D₅	1	1	yes	characteristic
110	C₂×#106	—	5	2	no	—
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×#97	—	5	2	no	—
116	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
117	#391	—	5	4	no	—
118	#387	—	10	4	no	—
119	#384	—	5	2	no	—
120	#382	—	10	2	no	—
121	C₄×C₄⋊C₄	—	5	1	no	—
122	C₄×C₄⋊C₄	—	5	4	no	—
123	C₄×C₄⋊C₄	—	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	4	no	—
133	C₂×C₄×C₅	—	2	4	no	—
134	C₂×C₄×C₅	—	4	2	no	—
135	C₂×C₄×C₅	—	4	2	no	—
136	C₂×C₄×C₅	—	4	2	no	—
137	C₂×C₄×C₅	—	4	2	no	—
138	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
139	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
140	C₂×C₄×C₅	#137	1	4	yes	—
141	C₂×C₄×C₅	C₂³×C₄	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	C₂×#41	1	1	yes	characteristic
145	C₂×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
146	C₂×C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
147	C₂×Dic₅	—	2	4	no	—
148	C₂×Dic₅	—	2	4	no	—
149	C₂×Dic₅	—	2	4	no	—
150	C₂×Dic₅	—	4	4	no	—
151	C₂×Dic₅	—	4	4	no	—
152	C₂×Dic₅	—	4	4	no	—
153	C₅×C₈	—	2	2	no	—
154	C₅×C₈	#117	1	4	yes	—
155	#168	—	4	2	no	—
156	C₂²×Q₈	C₂²×D₅	1	2	yes	—
157	C₂×Q₁₆	—	2	2	no	—
158	C₂×Q₁₆	—	2	2	no	—
159	C₂×Q₁₆	C₂×Dic₅	1	4	yes	—
160	C₂²×C₈	—	5	2	no	—
161	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
162	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
163	C₄×Q₈	—	2	4	no	—
164	C₄×Q₈	C₂×Dic₅	1	8	yes	—
165	C₂×C₄⋊C₄	—	2	2	no	—
166	C₂×C₄⋊C₄	—	5	2	no	—
167	C₂×C₄⋊C₄	—	5	4	no	—
168	C₂×C₄⋊C₄	—	10	2	no	—
169	C₂×C₄⋊C₄	—	10	2	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	C₂×Dic₅	1	4	yes	—
176	#106	—	10	4	no	—
177	#106	—	20	2	no	—
178	C₄⋊C₈	—	10	2	no	—
179	#103	—	20	4	no	—
180	#103	C₂×Dic₅	1	8	yes	—
181	#97	—	10	4	no	—
182	C₄×C₈	C₂×Dic₅	1	4	yes	—
183	#95	—	5	2	no	—
184	#95	—	10	4	no	—
185	#95	—	10	4	no	—
186	#95	—	10	4	no	—
187	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
188	C₂²×C₅	#547	1	1	yes	characteristic
189	C₂²×C₅	#550	1	1	yes	characteristic
190	C₂²×C₅	C₂×#137	1	2	yes	—
191	C₂²×C₅	#577	1	2	yes	—
192	C₄×C₅	—	2	2	no	—
193	C₄×C₅	—	4	4	no	—
194	C₄×C₅	—	4	4	no	—
195	C₄×C₅	#441	1	4	yes	—
196	C₄×C₅	C₂×#117	1	1	yes	characteristic
197	C₄×C₅	#520	1	1	yes	characteristic
198	C₄×C₅	#555	1	1	yes	characteristic
199	C₄×C₅	#556	1	1	yes	characteristic
200	Dic₅	—	4	4	no	—
201	Dic₅	—	8	4	no	—
202	C₂×Q₈	—	2	2	no	—
203	C₂×Q₈	—	2	2	no	—
204	C₂×Q₈	—	2	2	no	—
205	C₂×Q₈	—	2	2	no	—
206	C₂×Q₈	—	2	2	no	—
207	C₂×Q₈	C₂²×Dic₅	1	4	yes	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	5	4	no	—
211	C₂²×C₄	—	10	4	no	—
212	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
213	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
214	C₂²×C₄	C₂³×D₅	1	1	yes	characteristic
215	Q₁₆	—	2	8	no	—
216	C₂×C₈	—	2	1	no	—
217	C₂×C₈	—	2	1	no	—
218	C₂×C₈	—	10	2	no	—
219	C₂×C₈	—	10	2	no	—
220	C₂×C₈	—	10	2	no	—
221	C₂×C₈	#709	1	2	yes	—
222	C₂×C₈	#709	1	2	yes	—
223	C₂×C₈	C₂²×Dic₅	1	2	yes	—
224	C₂×C₈	C₂²×Dic₅	1	2	yes	—
225	C₄⋊C₄	—	2	2	no	—
226	C₄⋊C₄	—	2	8	no	—
227	C₄⋊C₄	—	10	4	no	—
228	C₄⋊C₄	—	10	4	no	—
229	C₄⋊C₄	—	10	4	no	—
230	C₄⋊C₄	—	20	4	no	—
231	C₄⋊C₄	—	40	2	no	—
232	C₄⋊C₄	C₂²×Dic₅	1	4	yes	—
233	C₄²	—	2	8	no	—
234	C₄²	—	10	2	no	—
235	C₄²	—	10	2	no	—
236	C₄²	—	40	4	no	—
237	C₄²	#711	1	2	yes	—
238	C₄²	C₂²×Dic₅	1	2	yes	—
239	C₂×C₅	#2354	1	1	yes	characteristic
240	C₂×C₅	C₂×#441	1	1	yes	characteristic
241	C₂×C₅	#2952	1	1	yes	characteristic
242	C₂×C₅	#3356	1	2	yes	—
243	C₂×C₅	#3400	1	2	yes	—
244	C₂³	C₂×D₄×D₅	1	1	yes	characteristic
245	Q₈	—	2	4	no	—
246	Q₈	—	2	4	no	—
247	Q₈	—	4	2	no	—
248	Q₈	—	4	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	10	4	no	—
257	C₂×C₄	—	10	4	no	—
258	C₂×C₄	—	10	4	no	—
259	C₂×C₄	—	20	4	no	—
260	C₂×C₄	—	20	4	no	—
261	C₂×C₄	—	20	4	no	—
262	C₂×C₄	#4064	1	4	yes	—
263	C₂×C₄	D₄×Dic₅	1	2	yes	—
264	C₂×C₄	D₄×Dic₅	1	2	yes	—
265	C₂×C₄	C₂×#709	1	1	yes	characteristic
266	C₂×C₄	C₂×#709	1	1	yes	characteristic
267	C₂×C₄	C₂×#711	1	1	yes	characteristic
268	C₂×C₄	#4149	1	1	yes	characteristic
269	C₂×C₄	#4150	1	1	yes	characteristic
270	C₂×C₄	C₂³×Dic₅	1	1	yes	characteristic
271	C₈	—	2	2	no	—
272	C₈	—	20	2	no	—
273	C₈	#4072	1	4	yes	—
274	C₅	#22814	1	1	yes	characteristic, complemented, Sylow
275	C₂²	#65885	1	1	yes	characteristic
276	C₂²	#65887	1	1	yes	characteristic
277	C₂²	C₂×#4064	1	2	yes	—
278	C₂²	#66104	1	2	yes	—
279	C₂²	C₂×D₄×Dic₅	1	1	yes	characteristic
280	C₄	—	2	2	no	—
281	C₄	—	4	4	no	—
282	C₄	—	4	4	no	—
283	C₄	—	20	4	no	—
284	C₄	—	40	4	no	—
285	C₄	#65473	1	4	yes	—
286	C₄	#65876	1	1	yes	characteristic
287	C₄	#65897	1	1	yes	characteristic
288	C₄	C₂×#4072	1	1	yes	characteristic
289	C₄	#66161	1	1	yes	characteristic
290	C₂	#10611444	1	2	yes	—
291	C₂	#10611476	1	2	yes	—
292	C₂	#10614068	1	1	yes	characteristic
293	C₂	C₂×#65473	1	1	yes	characteristic
294	C₂	#10615630	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	#22814
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
274. C₅	#22814	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
#10614193	2
#10602860	2
#10602419	2
#10601398	2
#10601016	2
#10600557	2
#10600550	2
#22814	5