Subgroups — #11989776

Order 1280.

678 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	C₂²×C₄×#168	C₂	1	1	yes	maximal, characteristic
2	C₄×C₅×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₄×C₂²⋊#168	C₂	1	1	yes	maximal, characteristic
4	C₂×#64674	C₂	1	2	yes	maximal, complemented
5	#10598144	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
7	C₂³×#168	C₄	1	2	yes	—
8	C₂×C₅×C₂²⋊C₈	C₄	1	2	yes	—
9	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
10	C₂×C₂²⋊#168	C₄	1	2	yes	—
11	C₂×C₄×#168	—	2	4	no	—
12	C₂×C₄×#168	C₄	1	2	yes	—
13	C₂×C₄×#168	C₂²	1	1	yes	characteristic
14	C₂×C₄×#168	C₂²	1	2	yes	—
15	C₂×C₄×#168	C₂²	1	1	yes	characteristic
16	C₅×#336	C₄	1	2	yes	—
17	#64744	C₄	1	2	yes	—
18	#64674	—	2	4	no	—
19	#8145	—	5	1	no	maximal, Sylow
20	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
24	C₂×C₄²×C₅	—	2	2	no	—
25	C₂×C₄²×C₅	—	2	2	no	—
26	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
30	C₂²×#168	—	2	4	no	—
31	C₂²×#168	—	2	4	no	—
32	C₂²×#168	—	2	4	no	—
33	C₂²×#168	C₂×C₄	1	2	yes	—
34	C₂²×#168	C₂×C₄	1	4	yes	—
35	C₂²×#168	C₂×C₄	1	2	yes	—
36	C₅×C₂²⋊C₈	—	2	4	no	—
37	C₄×C₅×C₈	—	2	2	no	—
38	C₄×C₅×C₈	—	2	2	no	—
39	C₂²⋊#168	C₈	1	8	yes	complemented
40	C₄×#168	—	2	2	no	—
41	C₄×#168	—	2	2	no	—
42	C₄×#168	—	2	2	no	—
43	C₄×#168	—	2	4	no	—
44	C₄×#168	D₄	1	4	yes	—
45	C₂²×C₄×C₈	—	5	1	no	—
46	C₄×C₂²⋊C₈	—	5	1	no	—
47	C₄×C₂²⋊C₈	D₅	1	1	yes	characteristic
48	C₂×#322	—	5	2	no	—
49	#1273	—	5	2	no	—
50	C₂⁴×C₅	C₄²	1	1	yes	characteristic
51	C₂²×C₄×C₅	—	2	1	no	—
52	C₂²×C₄×C₅	—	2	1	no	—
53	C₂²×C₄×C₅	—	2	2	no	—
54	C₂²×C₄×C₅	—	2	2	no	—
55	C₂²×C₄×C₅	—	2	2	no	—
56	C₂²×C₄×C₅	—	2	2	no	—
57	C₂²×C₄×C₅	—	2	2	no	—
58	C₂²×C₄×C₅	C₄²	1	1	yes	characteristic
59	C₂²×C₄×C₅	C₄²	1	1	yes	characteristic
60	C₂²×C₄×C₅	C₄²	1	1	yes	characteristic
61	C₂²×C₄×C₅	C₂×C₈	1	4	yes	—
62	C₂²×C₄×C₅	M₁₆	1	4	yes	—
63	C₂²×C₄×C₅	M₁₆	1	4	yes	—
64	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₅×C₈	—	2	4	no	—
68	C₂×C₅×C₈	—	2	4	no	—
69	C₂×C₅×C₈	—	2	4	no	—
70	C₄²×C₅	—	2	1	no	—
71	C₄²×C₅	—	2	1	no	—
72	C₄²×C₅	—	2	2	no	—
73	C₄²×C₅	—	2	2	no	—
74	C₄²×C₅	C₂×D₄	1	2	yes	—
75	C₄²×C₅	#41	1	2	yes	—
76	C₂×#168	—	2	16	no	—
77	C₂×#168	—	2	2	no	—
78	C₂×#168	—	2	4	no	—
79	C₂×#168	—	2	2	no	—
80	C₂×#168	—	2	2	no	—
81	C₂×#168	—	2	2	no	—
82	C₂×#168	—	2	4	no	—
83	C₂×#168	C₂²⋊C₄	1	4	yes	—
84	C₂×#168	C₂×C₈	1	8	yes	complemented
85	C₂×#168	M₁₆	1	4	yes	—
86	C₂³×C₈	—	5	2	no	—
87	C₂²×C₄²	C₂×D₅	1	1	yes	characteristic
88	C₂×C₂²⋊C₈	—	5	2	no	—
89	C₂×C₂²⋊C₈	Dic₅	1	2	yes	—
90	C₂×C₄×C₈	—	5	1	no	—
91	C₂×C₄×C₈	—	5	2	no	—
92	C₂×C₄×C₈	—	5	1	no	—
93	C₂×C₄×C₈	—	5	2	no	—
94	C₂×C₄×C₈	—	10	4	no	—
95	C₂×C₄×C₈	C₂×D₅	1	2	yes	—
96	#336	—	5	2	no	—
97	#336	Dic₅	1	2	yes	—
98	#322	—	10	4	no	—
99	C₂³×C₅	—	2	1	no	—
100	C₂³×C₅	—	2	1	no	—
101	C₂³×C₅	—	2	1	no	—
102	C₂³×C₅	—	2	1	no	—
103	C₂³×C₅	C₄×C₈	1	2	yes	—
104	C₂³×C₅	#97	1	2	yes	—
105	C₂³×C₅	#97	1	2	yes	—
106	C₂³×C₅	C₂×C₄²	1	1	yes	characteristic
107	C₂×C₄×C₅	—	2	1	no	—
108	C₂×C₄×C₅	—	2	1	no	—
109	C₂×C₄×C₅	—	2	1	no	—
110	C₂×C₄×C₅	—	2	1	no	—
111	C₂×C₄×C₅	—	2	1	no	—
112	C₂×C₄×C₅	—	2	1	no	—
113	C₂×C₄×C₅	—	2	4	no	—
114	C₂×C₄×C₅	—	2	4	no	—
115	C₂×C₄×C₅	—	2	8	no	—
116	C₂×C₄×C₅	—	2	8	no	—
117	C₂×C₄×C₅	C₄×C₈	1	2	yes	—
118	C₂×C₄×C₅	#97	1	2	yes	—
119	C₂×C₄×C₅	#97	1	2	yes	—
120	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₅	#117	1	1	yes	characteristic
122	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
123	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
124	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
125	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
126	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
127	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
128	C₂×C₄×C₅	#131	1	2	yes	—
129	C₂×C₄×C₅	#131	1	2	yes	—
130	C₅×C₈	—	4	4	no	—
131	#168	—	2	4	no	—
132	#168	—	2	4	no	—
133	#168	—	2	8	no	—
134	C₂³×C₄	C₄×D₅	1	1	yes	characteristic
135	C₂³×C₄	C₄×D₅	1	1	yes	characteristic
136	C₂³×C₄	C₂×Dic₅	1	1	yes	characteristic
137	C₂²×C₈	—	5	2	no	—
138	C₂²×C₈	—	5	2	no	—
139	C₂²×C₈	—	5	4	no	—
140	C₂²×C₈	—	10	4	no	—
141	C₂²×C₈	—	10	4	no	—
142	C₂²×C₈	—	10	4	no	—
143	C₂²×C₈	C₂×Dic₅	1	4	yes	—
144	C₂×C₄²	—	2	2	no	—
145	C₂×C₄²	—	2	2	no	—
146	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
147	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
148	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
149	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
150	C₂²⋊C₈	—	2	4	no	—
151	C₂²⋊C₈	—	5	8	no	—
152	C₄×C₈	—	2	2	no	—
153	C₄×C₈	—	2	2	no	—
154	C₄×C₈	—	5	4	no	—
155	C₄×C₈	—	10	4	no	—
156	C₄×C₈	—	10	2	no	—
157	C₄×C₈	—	10	2	no	—
158	C₄×C₈	—	10	2	no	—
159	C₂²×C₅	—	2	2	no	—
160	C₂²×C₅	—	2	2	no	—
161	C₂²×C₅	—	2	2	no	—
162	C₂²×C₅	—	2	2	no	—
163	C₂²×C₅	—	2	2	no	—
164	C₂²×C₅	—	2	2	no	—
165	C₂²×C₅	#322	1	4	yes	complemented
166	C₂²×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
167	C₂²×C₅	C₂×C₄×C₈	1	1	yes	characteristic
168	C₂²×C₅	C₂×#97	1	1	yes	characteristic
169	C₂²×C₅	C₂×#97	1	1	yes	characteristic
170	C₂²×C₅	C₄×M₁₆	1	1	yes	characteristic
171	C₂²×C₅	#405	1	1	yes	characteristic
172	C₂²×C₅	#405	1	1	yes	characteristic
173	C₄×C₅	—	2	2	no	—
174	C₄×C₅	—	2	2	no	—
175	C₄×C₅	—	2	2	no	—
176	C₄×C₅	#407	1	2	yes	—
177	C₄×C₅	#432	1	2	yes	—
178	C₄×C₅	C₈×D₄	1	4	yes	—
179	C₄×C₅	#435	1	4	yes	—
180	C₂⁴	C₄×Dic₅	1	1	yes	characteristic
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	—	2	2	no	—
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	C₈×D₅	1	4	yes	—
189	C₂²×C₄	#676	1	4	yes	—
190	C₂²×C₄	#681	1	4	yes	—
191	C₂²×C₄	C₄×Dic₅	1	1	yes	characteristic
192	C₂²×C₄	C₄×Dic₅	1	1	yes	characteristic
193	C₂²×C₄	C₄×Dic₅	1	1	yes	characteristic
194	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
195	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
196	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
197	C₂×C₈	—	2	4	no	—
198	C₂×C₈	—	2	4	no	—
199	C₂×C₈	—	2	4	no	—
200	C₂×C₈	—	5	4	no	—
201	C₂×C₈	—	5	8	no	—
202	C₂×C₈	—	5	4	no	—
203	C₂×C₈	—	10	16	no	—
204	C₂×C₈	—	10	2	no	—
205	C₂×C₈	—	10	4	no	—
206	C₂×C₈	—	10	2	no	—
207	C₂×C₈	—	10	2	no	—
208	C₂×C₈	—	10	2	no	—
209	C₂×C₈	—	10	4	no	—
210	C₄²	—	2	1	no	—
211	C₄²	—	2	1	no	—
212	C₄²	—	2	2	no	—
213	C₄²	—	2	2	no	—
214	C₄²	D₄×D₅	1	2	yes	—
215	C₄²	#711	1	2	yes	—
216	C₂×C₅	—	2	4	no	—
217	C₂×C₅	C₂×#322	1	1	yes	characteristic
218	C₂×C₅	C₈×M₁₆	1	1	yes	characteristic
219	C₂×C₅	#1451	1	1	yes	characteristic
220	C₂×C₅	#1452	1	1	yes	characteristic
221	C₂×C₅	#1749	1	1	yes	characteristic
222	C₂×C₅	C₈×C₂²⋊C₄	1	1	yes	characteristic
223	C₂×C₅	#1752	1	1	yes	characteristic
224	C₂³	—	2	1	no	—
225	C₂³	—	2	1	no	—
226	C₂³	—	2	1	no	—
227	C₂³	—	2	1	no	—
228	C₂³	#3935	1	2	yes	—
229	C₂³	C₈×Dic₅	1	2	yes	—
230	C₂³	#3947	1	2	yes	—
231	C₂³	C₂×C₄×Dic₅	1	1	yes	characteristic
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	8	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	2	8	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	#3935	1	2	yes	—
243	C₂×C₄	C₈×Dic₅	1	2	yes	—
244	C₂×C₄	#3947	1	2	yes	—
245	C₂×C₄	#4023	1	1	yes	characteristic
246	C₂×C₄	D₅×C₂²⋊C₄	1	1	yes	characteristic
247	C₂×C₄	#4027	1	2	yes	—
248	C₂×C₄	C₂×C₈×D₅	1	2	yes	—
249	C₂×C₄	C₂×#676	1	2	yes	—
250	C₂×C₄	D₅×M₁₆	1	2	yes	—
251	C₂×C₄	#4053	1	2	yes	—
252	C₂×C₄	C₂×#681	1	2	yes	—
253	C₂×C₄	D₄×Dic₅	1	2	yes	—
254	C₂×C₄	#4094	1	2	yes	—
255	C₈	—	4	4	no	—
256	C₈	—	10	8	no	—
257	C₈	—	10	4	no	—
258	C₈	—	10	4	no	—
259	C₅	#8145	1	1	yes	characteristic, complemented, Sylow
260	C₂²	—	2	2	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	2	2	no	—
266	C₂²	#64674	1	4	yes	complemented
267	C₂²	C₂×#3935	1	1	yes	characteristic
268	C₂²	C₂²⋊C₄×Dic₅	1	1	yes	characteristic
269	C₂²	#65286	1	1	yes	characteristic
270	C₂²	C₂×C₈×Dic₅	1	1	yes	characteristic
271	C₂²	C₂×#3947	1	1	yes	characteristic
272	C₂²	M₁₆×Dic₅	1	1	yes	characteristic
273	C₂²	#65411	1	1	yes	characteristic
274	C₄	—	2	2	no	—
275	C₄	—	2	2	no	—
276	C₄	—	2	2	no	—
277	C₄	#65008	1	2	yes	—
278	C₄	#65014	1	4	yes	—
279	C₄	#65015	1	2	yes	—
280	C₄	#65300	1	4	yes	—
281	C₂	—	2	4	no	—
282	C₂	#10600240	1	1	yes	characteristic
283	C₂	C₂×#64674	1	1	yes	characteristic
284	C₂	#10600640	1	1	yes	characteristic
285	C₂	#10600699	1	1	yes	characteristic
286	C₂	#10600740	1	1	yes	characteristic
287	C₂	#10601532	1	1	yes	characteristic
288	C₂	#10601535	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₅
commutator	C₂²×C₅	C₂×C₄×C₈
Fitting	C₄×C₅×C₂²⋊C₈	C₂
socle	C₂³×C₅	C₂×C₄²
prime radical	C₂⁴×C₅	C₄²
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	C₄×C₂²⋊C₈	D₅
hypocenter	C₅	#8145
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄² ◁ C₄×C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
4. C₂×#64674	C₂	semidirect
5. #10598144	C₂	semidirect
39. C₂²⋊#168	C₈	semidirect
84. C₂×#168	C₂×C₈	semidirect
165. C₂²×C₅	#322	semidirect
259. C₅	#8145	semidirect
266. C₂²	#64674	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×#168	2
C₄×C₅×C₂²⋊C₈	2
C₄×C₂²⋊#168	2
C₂×#64674	2
#10598144	2
#8145	5