Subgroups — #11997883

Order 1280.

404 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#64677	C₂	1	2	yes	maximal
3	#10600187	C₂	1	1	yes	maximal, characteristic
4	C₄×C₄⋊#168	C₂	1	1	yes	maximal, characteristic
5	#10598145	C₂	1	2	yes	maximal
6	C₂×C₅×C₄⋊C₈	C₄	1	1	yes	characteristic
7	C₂×C₅×C₄⋊C₈	C₄	1	1	yes	characteristic
8	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
9	C₄³×C₅	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊#168	—	2	1	no	—
11	C₂×C₄⋊#168	—	2	1	no	—
12	C₂×C₄⋊#168	C₂²	1	2	yes	—
13	C₂×C₄⋊#168	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊#168	C₂²	1	1	yes	characteristic
15	C₂×C₄×#168	—	2	1	no	—
16	C₅×#336	C₄	1	2	yes	—
17	#64744	—	2	1	no	—
18	#64677	—	2	4	no	—
19	#9602	—	5	1	no	maximal, Sylow
20	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
21	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
26	C₂×C₄²×C₅	Q₈	1	1	yes	characteristic
27	C₂×C₄²×C₅	Q₈	1	1	yes	characteristic
28	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
29	C₂²×#168	—	2	1	no	—
30	C₂²×#168	—	2	1	no	—
31	C₂²×#168	—	2	2	no	—
32	C₅×C₄⋊C₈	—	2	2	no	—
33	C₅×C₄⋊C₈	C₈	1	4	yes	complemented
34	C₄×C₅×C₈	—	2	2	no	—
35	C₄×C₅×C₈	—	2	2	no	—
36	C₄⋊#168	—	2	1	no	—
37	C₄⋊#168	—	2	1	no	—
38	C₄⋊#168	—	2	2	no	—
39	C₄⋊#168	—	2	2	no	—
40	C₄⋊#168	—	4	2	no	—
41	C₄⋊#168	—	4	2	no	—
42	C₄⋊#168	D₄	1	2	yes	—
43	C₄⋊#168	Q₈	1	2	yes	—
44	C₄×#168	—	4	2	no	—
45	#1842	—	5	1	no	—
46	C₄×C₄⋊C₈	—	5	1	no	—
47	C₄×C₄⋊C₈	D₅	1	1	yes	characteristic
48	C₂×#334	—	5	2	no	—
49	#1277	—	5	2	no	—
50	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₅×C₈	—	2	2	no	—
58	C₂×C₅×C₈	—	2	2	no	—
59	C₂×C₅×C₈	—	2	2	no	—
60	C₂×C₅×C₈	—	2	2	no	—
61	C₂×C₅×C₈	—	2	2	no	—
62	C₂×C₅×C₈	C₂×C₈	1	4	yes	—
63	C₄²×C₅	—	2	2	no	—
64	C₄²×C₅	—	2	1	no	—
65	C₄²×C₅	—	2	1	no	—
66	C₄²×C₅	—	2	1	no	—
67	C₄²×C₅	—	2	1	no	—
68	C₄²×C₅	—	4	1	no	—
69	C₄²×C₅	C₂×C₈	1	2	yes	—
70	C₄²×C₅	M₁₆	1	2	yes	—
71	C₄²×C₅	SD₁₆	1	2	yes	—
72	C₄²×C₅	SD₁₆	1	2	yes	—
73	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
75	C₄²×C₅	#41	1	1	yes	characteristic
76	C₄²×C₅	#41	1	1	yes	characteristic
77	C₂×#168	—	2	2	no	—
78	C₂×#168	—	2	2	no	—
79	C₂×#168	—	4	1	no	—
80	C₂×#168	—	4	1	no	—
81	C₂×#168	—	4	1	no	—
82	C₂×#168	—	4	1	no	—
83	C₂×#168	—	4	2	no	—
84	C₂×#168	—	4	2	no	—
85	C₂×#168	—	4	2	no	—
86	C₂×C₄⋊C₈	—	5	1	no	—
87	C₂×C₄⋊C₈	—	5	1	no	—
88	C₂×C₄⋊C₈	—	5	2	no	—
89	C₂×C₄⋊C₈	—	10	1	no	—
90	C₂×C₄⋊C₈	—	10	1	no	—
91	C₂×C₄⋊C₈	Dic₅	1	1	yes	characteristic
92	C₂×C₄⋊C₈	Dic₅	1	1	yes	characteristic
93	C₂×C₄×C₈	—	10	1	no	—
94	C₂×C₄×C₈	C₂×D₅	1	2	yes	—
95	C₄³	C₂×D₅	1	1	yes	characteristic
96	#336	—	10	1	no	—
97	#336	Dic₅	1	2	yes	—
98	#334	—	10	4	no	—
99	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₅	—	2	1	no	—
101	C₂×C₄×C₅	—	2	1	no	—
102	C₂×C₄×C₅	—	2	1	no	—
103	C₂×C₄×C₅	—	2	1	no	—
104	C₂×C₄×C₅	—	2	1	no	—
105	C₂×C₄×C₅	—	2	1	no	—
106	C₂×C₄×C₅	—	2	1	no	—
107	C₂×C₄×C₅	—	2	1	no	—
108	C₂×C₄×C₅	—	2	1	no	—
109	C₂×C₄×C₅	C₄⋊C₈	1	2	yes	—
110	C₂×C₄×C₅	C₄⋊C₈	1	2	yes	—
111	C₂×C₄×C₅	#106	1	2	yes	—
112	C₂×C₄×C₅	#106	1	2	yes	—
113	C₂×C₄×C₅	#108	1	2	yes	—
114	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₅	C₄×Q₈	1	1	yes	characteristic
116	C₂×C₄×C₅	#121	1	1	yes	characteristic
117	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
118	C₂×C₄×C₅	#125	1	1	yes	characteristic
119	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
120	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
121	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
122	C₂×C₄×C₅	#131	1	1	yes	characteristic
123	C₂×C₄×C₅	#131	1	1	yes	characteristic
124	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
125	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
126	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
127	C₂×C₄×C₅	#137	1	1	yes	characteristic
128	C₅×C₈	—	2	4	no	—
129	C₅×C₈	—	4	2	no	—
130	#168	—	4	2	no	—
131	#168	—	4	2	no	—
132	#168	—	8	2	no	—
133	C₂²×C₈	—	10	1	no	—
134	C₂²×C₈	—	10	1	no	—
135	C₂²×C₈	—	10	2	no	—
136	C₂²×C₈	C₂×Dic₅	1	2	yes	—
137	C₂²×C₈	C₂×Dic₅	1	2	yes	—
138	C₂×C₄²	Dic₁₀	1	1	yes	characteristic
139	C₂×C₄²	Dic₁₀	1	1	yes	characteristic
140	C₂×C₄²	D₂₀	1	1	yes	characteristic
141	C₂×C₄²	D₂₀	1	1	yes	characteristic
142	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
143	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
144	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
145	C₄⋊C₈	—	2	2	no	—
146	C₄⋊C₈	—	5	2	no	—
147	C₄⋊C₈	—	5	2	no	—
148	C₄⋊C₈	—	10	2	no	—
149	C₄⋊C₈	—	10	1	no	—
150	C₄⋊C₈	—	10	1	no	—
151	C₄⋊C₈	—	10	2	no	—
152	C₄⋊C₈	—	20	2	no	—
153	C₄⋊C₈	—	20	2	no	—
154	C₄⋊C₈	#168	1	4	yes	complemented
155	C₄×C₈	—	2	2	no	—
156	C₄×C₈	—	2	2	no	—
157	C₄×C₈	—	20	2	no	—
158	C₂²×C₅	#389	1	1	yes	characteristic
159	C₂²×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
160	C₂²×C₅	#424	1	1	yes	characteristic
161	C₂²×C₅	C₂×#106	1	1	yes	characteristic
162	C₂²×C₅	#428	1	1	yes	characteristic
163	C₂²×C₅	C₂×#108	1	1	yes	characteristic
164	C₂²×C₅	#430	1	1	yes	characteristic
165	C₄×C₅	—	2	2	no	—
166	C₄×C₅	—	2	2	no	—
167	C₄×C₅	—	2	1	no	—
168	C₄×C₅	—	2	1	no	—
169	C₄×C₅	—	4	1	no	—
170	C₄×C₅	#334	1	4	yes	complemented
171	C₄×C₅	C₈×D₄	1	1	yes	characteristic
172	C₄×C₅	#436	1	1	yes	characteristic
173	C₄×C₅	C₈×Q₈	1	1	yes	characteristic
174	C₄×C₅	#446	1	1	yes	characteristic
175	C₄×C₅	#460	1	1	yes	characteristic
176	C₄×C₅	#461	1	1	yes	characteristic
177	C₄×C₅	#475	1	1	yes	characteristic
178	C₄×C₅	#476	1	1	yes	characteristic
179	C₂²×C₄	C₄⋊Dic₅	1	1	yes	characteristic
180	C₂²×C₄	C₄⋊Dic₅	1	1	yes	characteristic
181	C₂²×C₄	C₄⋊Dic₅	1	1	yes	characteristic
182	C₂²×C₄	C₄⋊Dic₅	1	1	yes	characteristic
183	C₂²×C₄	C₂×Dic₁₀	1	1	yes	characteristic
184	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
185	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	10	2	no	—
192	C₂×C₈	—	10	2	no	—
193	C₂×C₈	—	20	1	no	—
194	C₂×C₈	—	20	1	no	—
195	C₂×C₈	—	20	1	no	—
196	C₂×C₈	—	20	1	no	—
197	C₂×C₈	—	20	2	no	—
198	C₂×C₈	—	20	2	no	—
199	C₂×C₈	—	20	2	no	—
200	C₂×C₈	C₂×#168	1	4	yes	—
201	C₄²	—	2	2	no	—
202	C₄²	—	2	1	no	—
203	C₄²	—	2	1	no	—
204	C₄²	—	2	1	no	—
205	C₄²	—	2	1	no	—
206	C₄²	—	4	1	no	—
207	C₄²	#677	1	2	yes	—
208	C₄²	#677	1	2	yes	—
209	C₄²	C₂×#168	1	2	yes	—
210	C₄²	#681	1	2	yes	—
211	C₄²	D₄×D₅	1	1	yes	characteristic
212	C₄²	#711	1	1	yes	characteristic
213	C₄²	Q₈×D₅	1	1	yes	characteristic
214	C₄²	#713	1	1	yes	characteristic
215	C₂×C₅	C₂×#334	1	1	yes	characteristic
216	C₂×C₅	#1565	1	1	yes	characteristic
217	C₂×C₅	#1570	1	1	yes	characteristic
218	C₂×C₅	#1574	1	1	yes	characteristic
219	C₂×C₅	#1939	1	1	yes	characteristic
220	C₂×C₅	#1948	1	1	yes	characteristic
221	C₂×C₅	#1950	1	1	yes	characteristic
222	C₂³	C₂×C₄⋊Dic₅	1	1	yes	characteristic
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	C₄⋊#168	1	2	yes	—
233	C₂×C₄	C₄⋊#168	1	2	yes	—
234	C₂×C₄	#3949	1	2	yes	—
235	C₂×C₄	#3949	1	2	yes	—
236	C₂×C₄	#3951	1	2	yes	—
237	C₂×C₄	#4034	1	1	yes	characteristic
238	C₂×C₄	#4036	1	1	yes	characteristic
239	C₂×C₄	#4041	1	1	yes	characteristic
240	C₂×C₄	#4043	1	1	yes	characteristic
241	C₂×C₄	C₂×#677	1	1	yes	characteristic
242	C₂×C₄	C₂×#677	1	1	yes	characteristic
243	C₂×C₄	#4054	1	1	yes	characteristic
244	C₂×C₄	#4055	1	1	yes	characteristic
245	C₂×C₄	C₂²×#168	1	1	yes	characteristic
246	C₂×C₄	C₂×#681	1	1	yes	characteristic
247	C₂×C₄	D₄×Dic₅	1	1	yes	characteristic
248	C₂×C₄	Q₈×Dic₅	1	1	yes	characteristic
249	C₂×C₄	#4094	1	1	yes	characteristic
250	C₂×C₄	#4094	1	1	yes	characteristic
251	C₈	—	2	4	no	—
252	C₈	—	4	2	no	—
253	C₈	—	20	2	no	—
254	C₈	—	20	2	no	—
255	C₈	—	40	2	no	—
256	C₅	#9602	1	1	yes	characteristic, complemented, Sylow
257	C₂²	C₂×C₄⋊#168	1	1	yes	characteristic
258	C₂²	#65274	1	1	yes	characteristic
259	C₂²	#65288	1	1	yes	characteristic
260	C₂²	C₂×#3949	1	1	yes	characteristic
261	C₂²	C₂×#3951	1	1	yes	characteristic
262	C₂²	#65410	1	1	yes	characteristic
263	C₂²	#65414	1	1	yes	characteristic
264	C₄	—	2	1	no	—
265	C₄	—	2	1	no	—
266	C₄	—	2	2	no	—
267	C₄	—	2	2	no	—
268	C₄	—	4	1	no	—
269	C₄	#64677	1	4	yes	complemented
270	C₄	#65130	1	1	yes	characteristic
271	C₄	#65131	1	1	yes	characteristic
272	C₄	#65137	1	1	yes	characteristic
273	C₄	#65140	1	1	yes	characteristic
274	C₄	D₄×#168	1	1	yes	characteristic
275	C₄	#65301	1	1	yes	characteristic
276	C₄	Q₈×#168	1	1	yes	characteristic
277	C₄	#65313	1	1	yes	characteristic
278	C₂	#10600277	1	1	yes	characteristic
279	C₂	C₂×#64677	1	1	yes	characteristic
280	C₂	#10600643	1	1	yes	characteristic
281	C₂	#10601247	1	1	yes	characteristic
282	C₂	#10601267	1	1	yes	characteristic
283	C₂	#10601542	1	1	yes	characteristic
284	C₂	#10601546	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₀
commutator	C₂×C₄×C₅	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₄×C₅×C₄⋊C₈	C₂
socle, prime radical	C₂³×C₅	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	C₄×C₄⋊C₈	D₅
hypocenter	C₅	#9602
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
33. C₅×C₄⋊C₈	C₈	semidirect
154. C₄⋊C₈	#168	semidirect
170. C₄×C₅	#334	semidirect
256. C₅	#9602	semidirect
269. C₄	#64677	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₅×C₄⋊C₈	2
C₂×#64677	2
#10600187	2
C₄×C₄⋊#168	2
#10598145	2
#9602	5