Subgroups — #11990281

Order 1280.

528 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#64767	C₂	1	2	yes	maximal
3	#10600184	C₂	1	1	yes	maximal, characteristic
4	C₄×C₄⋊#168	C₂	1	1	yes	maximal, characteristic
5	#10598125	C₂	1	2	yes	maximal
6	C₂×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₅×Q₈	—	2	1	no	—
8	C₅×#391	—	2	1	no	—
9	C₅×#389	—	2	1	no	—
10	C₅×#389	C₄	1	2	yes	—
11	C₅×#384	C₂²	1	1	yes	characteristic
12	C₄×C₅×C₄⋊C₄	—	2	1	no	—
13	C₄×C₅×C₄⋊C₄	C₄	1	2	yes	—
14	C₄³×C₅	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊#168	C₂²	1	2	yes	—
16	C₂×C₄⋊#168	C₂²	1	2	yes	—
17	C₂×#3935	—	2	1	no	—
18	C₂×C₄×#168	—	2	1	no	—
19	#64767	C₄	1	8	yes	complemented
20	#64744	—	2	1	no	—
21	#64744	—	2	1	no	—
22	#8237	—	5	1	no	maximal, Sylow
23	C₂²×C₅×Q₈	—	2	1	no	—
24	C₅×C₄⋊Q₈	—	2	2	no	—
25	C₅×C₄⋊Q₈	C₂×C₄	1	4	yes	—
26	C₄×C₅×Q₈	—	4	4	no	—
27	C₂×C₅×C₄⋊C₄	—	2	1	no	—
28	C₂×C₅×C₄⋊C₄	—	2	2	no	—
29	C₂×C₅×C₄⋊C₄	—	2	1	no	—
30	C₂×C₅×C₄⋊C₄	—	2	1	no	—
31	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
32	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₅	—	2	2	no	—
34	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
41	C₂²×#168	—	2	2	no	—
42	C₂²×#168	—	2	2	no	—
43	C₅×#95	—	2	2	no	—
44	C₅×#95	—	2	2	no	—
45	C₄⋊#168	—	2	2	no	—
46	C₄⋊#168	—	2	2	no	—
47	C₄⋊#168	C₂×C₄	1	4	yes	—
48	C₄⋊#168	C₂×C₄	1	4	yes	—
49	#3935	—	4	2	no	—
50	C₄×#168	—	4	2	no	—
51	C₄×C₄⋊Q₈	D₅	1	1	yes	characteristic
52	#1767	—	5	1	no	—
53	C₄×C₄⋊C₈	—	5	1	no	—
54	C₂×#332	—	5	2	no	—
55	#1296	—	5	2	no	—
56	C₂×C₅×Q₈	—	2	4	no	—
57	C₂×C₅×Q₈	—	4	2	no	—
58	C₂×C₅×Q₈	—	4	1	no	—
59	C₂×C₅×Q₈	—	4	1	no	—
60	C₂²×C₄×C₅	—	2	2	no	—
61	C₂²×C₄×C₅	—	2	2	no	—
62	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
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	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
69	C₅×C₄⋊C₄	—	2	2	no	—
70	C₅×C₄⋊C₄	—	2	4	no	—
71	C₅×C₄⋊C₄	—	4	2	no	—
72	C₅×C₄⋊C₄	—	4	4	no	—
73	C₅×C₄⋊C₄	—	4	2	no	—
74	C₅×C₄⋊C₄	—	4	2	no	—
75	C₅×C₄⋊C₄	C₄²	1	4	yes	—
76	C₄²×C₅	—	2	2	no	—
77	C₄²×C₅	—	2	2	no	—
78	C₄²×C₅	—	2	2	no	—
79	C₄²×C₅	—	2	1	no	—
80	C₄²×C₅	—	2	1	no	—
81	C₄²×C₅	—	4	1	no	—
82	C₄²×C₅	—	4	4	no	—
83	C₄²×C₅	SD₁₆	1	2	yes	—
84	C₄²×C₅	Q₁₆	1	2	yes	—
85	C₄²×C₅	C₂²×C₄	1	2	yes	—
86	C₄²×C₅	#41	1	2	yes	—
87	C₂×#168	—	2	4	no	—
88	C₂×#168	—	2	4	no	—
89	C₂×#168	—	4	2	no	—
90	C₂×#168	—	4	2	no	—
91	C₂×#168	—	4	2	no	—
92	C₂×#168	—	4	2	no	—
93	C₂×C₄⋊Q₈	C₂×D₅	1	1	yes	characteristic
94	C₂×C₄×Q₈	—	2	1	no	—
95	C₂×C₄⋊C₈	—	5	2	no	—
96	C₂×C₄⋊C₈	—	5	2	no	—
97	C₂×#97	—	10	1	no	—
98	C₂×C₄×C₈	—	10	1	no	—
99	#391	—	2	1	no	—
100	#389	—	2	1	no	—
101	#389	Dic₅	1	2	yes	—
102	#384	C₂×D₅	1	1	yes	characteristic
103	C₄×C₄⋊C₄	—	2	1	no	—
104	C₄×C₄⋊C₄	Dic₅	1	2	yes	—
105	C₄³	C₂×D₅	1	1	yes	characteristic
106	#336	—	10	1	no	—
107	#336	—	10	1	no	—
108	#332	—	5	8	no	—
109	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₅×Q₈	—	4	4	no	—
111	C₅×Q₈	—	4	4	no	—
112	C₂×C₄×C₅	—	2	1	no	—
113	C₂×C₄×C₅	—	2	1	no	—
114	C₂×C₄×C₅	—	2	1	no	—
115	C₂×C₄×C₅	—	2	1	no	—
116	C₂×C₄×C₅	—	2	1	no	—
117	C₂×C₄×C₅	—	2	1	no	—
118	C₂×C₄×C₅	—	2	1	no	—
119	C₂×C₄×C₅	—	2	1	no	—
120	C₂×C₄×C₅	—	2	4	no	—
121	C₂×C₄×C₅	—	2	1	no	—
122	C₂×C₄×C₅	—	2	1	no	—
123	C₂×C₄×C₅	—	2	1	no	—
124	C₂×C₄×C₅	—	4	2	no	—
125	C₂×C₄×C₅	—	4	2	no	—
126	C₂×C₄×C₅	—	4	2	no	—
127	C₂×C₄×C₅	—	4	2	no	—
128	C₂×C₄×C₅	—	4	2	no	—
129	C₂×C₄×C₅	#101	1	2	yes	—
130	C₂×C₄×C₅	#103	1	2	yes	—
131	C₂×C₄×C₅	#103	1	2	yes	—
132	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
133	C₂×C₄×C₅	#117	1	1	yes	characteristic
134	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
135	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
136	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
137	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
138	C₂×C₄×C₅	#135	1	2	yes	—
139	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
140	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
141	C₂×C₄×C₅	#137	1	1	yes	characteristic
142	#168	—	4	4	no	—
143	#168	—	4	4	no	—
144	C₂²×Q₈	—	2	1	no	—
145	C₂²×C₈	—	10	2	no	—
146	C₂²×C₈	—	10	2	no	—
147	C₄⋊Q₈	—	2	2	no	—
148	C₄⋊Q₈	C₄×D₅	1	4	yes	—
149	C₄×Q₈	—	4	4	no	—
150	C₂×C₄⋊C₄	—	2	1	no	—
151	C₂×C₄⋊C₄	—	2	2	no	—
152	C₂×C₄⋊C₄	—	2	1	no	—
153	C₂×C₄⋊C₄	—	2	1	no	—
154	C₂×C₄⋊C₄	C₂×Dic₅	1	2	yes	—
155	C₂×C₄⋊C₄	C₂×Dic₅	1	2	yes	—
156	C₂×C₄²	—	2	2	no	—
157	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
158	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
159	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
160	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
161	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
162	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
163	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
164	C₄⋊C₈	—	5	4	no	—
165	C₄⋊C₈	—	5	4	no	—
166	C₄⋊C₈	—	10	2	no	—
167	C₄⋊C₈	—	10	2	no	—
168	#97	—	20	2	no	—
169	C₄×C₈	—	20	2	no	—
170	#95	—	2	2	no	—
171	#95	—	2	2	no	—
172	C₂²×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
173	C₂²×C₅	C₂×#101	1	1	yes	characteristic
174	C₂²×C₅	#413	1	1	yes	characteristic
175	C₂²×C₅	C₂×#103	1	1	yes	characteristic
176	C₂²×C₅	#416	1	1	yes	characteristic
177	C₂²×C₅	#417	1	1	yes	characteristic
178	C₂²×C₅	#418	1	1	yes	characteristic
179	C₄×C₅	—	2	2	no	—
180	C₄×C₅	—	2	2	no	—
181	C₄×C₅	—	2	2	no	—
182	C₄×C₅	—	2	1	no	—
183	C₄×C₅	—	2	1	no	—
184	C₄×C₅	—	4	1	no	—
185	C₄×C₅	—	4	4	no	—
186	C₄×C₅	—	8	2	no	—
187	C₄×C₅	C₄×SD₁₆	1	2	yes	—
188	C₄×C₅	C₄×Q₁₆	1	2	yes	—
189	C₄×C₅	#440	1	2	yes	—
190	C₄×C₅	#442	1	2	yes	—
191	C₂×Q₈	—	2	4	no	—
192	C₂×Q₈	—	4	1	no	—
193	C₂×Q₈	—	4	2	no	—
194	C₂×Q₈	—	4	1	no	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
198	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
199	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
200	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
201	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
202	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
203	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
204	C₂×C₈	—	10	4	no	—
205	C₂×C₈	—	10	4	no	—
206	C₂×C₈	—	20	2	no	—
207	C₂×C₈	—	20	2	no	—
208	C₂×C₈	—	20	2	no	—
209	C₂×C₈	—	20	2	no	—
210	C₄⋊C₄	—	2	2	no	—
211	C₄⋊C₄	—	2	4	no	—
212	C₄⋊C₄	—	4	2	no	—
213	C₄⋊C₄	—	4	4	no	—
214	C₄⋊C₄	—	4	2	no	—
215	C₄⋊C₄	—	4	2	no	—
216	C₄⋊C₄	C₄×Dic₅	1	4	yes	—
217	C₄²	—	2	2	no	—
218	C₄²	—	2	1	no	—
219	C₄²	—	2	2	no	—
220	C₄²	—	2	2	no	—
221	C₄²	—	2	1	no	—
222	C₄²	—	4	1	no	—
223	C₄²	—	4	4	no	—
224	C₄²	#688	1	2	yes	—
225	C₄²	#689	1	2	yes	—
226	C₄²	C₂×C₄×D₅	1	2	yes	—
227	C₄²	#709	1	2	yes	—
228	C₂×C₅	#1549	1	1	yes	characteristic
229	C₂×C₅	#1550	1	1	yes	characteristic
230	C₂×C₅	#1552	1	1	yes	characteristic
231	C₂×C₅	#1553	1	1	yes	characteristic
232	C₂×C₅	C₄×#101	1	1	yes	characteristic
233	C₂×C₅	C₄×#103	1	1	yes	characteristic
234	C₂×C₅	#1762	1	1	yes	characteristic
235	C₂³	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
236	Q₈	—	4	4	no	—
237	Q₈	—	4	4	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	#3967	1	2	yes	—
256	C₂×C₄	#3967	1	2	yes	—
257	C₂×C₄	#3968	1	2	yes	—
258	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
259	C₂×C₄	#4072	1	1	yes	characteristic
260	C₂×C₄	C₄×C₂²⋊D₅	1	2	yes	—
261	C₂×C₄	C₄×C₂²⋊D₅	1	2	yes	—
262	C₂×C₄	#4078	1	2	yes	—
263	C₂×C₄	C₂×#688	1	1	yes	characteristic
264	C₂×C₄	C₂×#689	1	1	yes	characteristic
265	C₂×C₄	#4095	1	1	yes	characteristic
266	C₂×C₄	#4096	1	2	yes	—
267	C₂×C₄	#4097	1	1	yes	characteristic
268	C₈	—	20	4	no	—
269	C₈	—	20	4	no	—
270	C₅	#8237	1	1	yes	characteristic, complemented, Sylow
271	C₂²	C₄×C₂²⋊Dic₅	1	1	yes	characteristic
272	C₂²	#65504	1	1	yes	characteristic
273	C₂²	C₂×#3967	1	1	yes	characteristic
274	C₂²	C₂×#3968	1	1	yes	characteristic
275	C₂²	#65521	1	1	yes	characteristic
276	C₂²	#65522	1	1	yes	characteristic
277	C₂²	#65526	1	1	yes	characteristic
278	C₄	—	2	2	no	—
279	C₄	—	2	2	no	—
280	C₄	—	2	2	no	—
281	C₄	—	2	1	no	—
282	C₄	—	2	1	no	—
283	C₄	—	4	1	no	—
284	C₄	—	4	4	no	—
285	C₄	—	8	2	no	—
286	C₄	#65303	1	2	yes	—
287	C₄	#65307	1	2	yes	—
288	C₄	C₄×#688	1	2	yes	—
289	C₄	C₄×#689	1	2	yes	—
290	C₂	#10602402	1	1	yes	characteristic
291	C₂	C₄×#3967	1	1	yes	characteristic
292	C₂	C₄×#3968	1	1	yes	characteristic
293	C₂	#10602568	1	1	yes	characteristic
294	C₂	#10602601	1	1	yes	characteristic
295	C₂	#10602623	1	1	yes	characteristic
296	C₂	#10602624	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₅
commutator	C₂×C₄×C₅	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₄×C₅×C₄⋊Q₈	C₂
socle, prime radical	C₂³×C₅	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	C₄×C₄⋊Q₈	D₅
hypocenter	C₅	#8237
LCS	G ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
19. #64767	C₄	semidirect
270. C₅	#8237	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⋊Q₈	2
C₂×#64767	2
#10600184	2
C₄×C₄⋊#168	2
#10598125	2
#8237	5