Subgroups — #11990276

Order 1280.

528 conjugacy classes of subgroups.

The subgroup lattice has 304 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	#8236	—	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	2	no	—
27	C₄×C₅×Q₈	—	4	2	no	—
28	C₂×C₅×C₄⋊C₄	—	2	1	no	—
29	C₂×C₅×C₄⋊C₄	—	2	2	no	—
30	C₂×C₅×C₄⋊C₄	—	2	1	no	—
31	C₂×C₅×C₄⋊C₄	—	2	1	no	—
32	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
33	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₅	—	2	2	no	—
35	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₅	C₂×C₄	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	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
42	C₂²×#168	—	2	2	no	—
43	C₂²×#168	—	2	2	no	—
44	C₅×#95	—	2	2	no	—
45	C₅×#95	—	2	2	no	—
46	C₄⋊#168	—	2	2	no	—
47	C₄⋊#168	—	2	2	no	—
48	C₄⋊#168	C₂×C₄	1	4	yes	—
49	C₄⋊#168	C₂×C₄	1	4	yes	—
50	#3935	—	4	2	no	—
51	C₄×#168	—	4	2	no	—
52	C₄×C₄⋊Q₈	D₅	1	1	yes	characteristic
53	#1767	—	5	1	no	—
54	C₄×C₄⋊C₈	—	5	1	no	—
55	C₂×#332	—	5	2	no	—
56	#1296	—	5	2	no	—
57	C₂×C₅×Q₈	—	2	4	no	—
58	C₂×C₅×Q₈	—	4	1	no	—
59	C₂×C₅×Q₈	—	4	1	no	—
60	C₂×C₅×Q₈	—	4	2	no	—
61	C₂²×C₄×C₅	—	2	2	no	—
62	C₂²×C₄×C₅	—	2	2	no	—
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	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
70	C₅×C₄⋊C₄	—	2	2	no	—
71	C₅×C₄⋊C₄	—	2	4	no	—
72	C₅×C₄⋊C₄	—	4	2	no	—
73	C₅×C₄⋊C₄	—	4	4	no	—
74	C₅×C₄⋊C₄	—	4	1	no	—
75	C₅×C₄⋊C₄	—	4	1	no	—
76	C₅×C₄⋊C₄	—	4	1	no	—
77	C₅×C₄⋊C₄	—	4	1	no	—
78	C₅×C₄⋊C₄	C₄²	1	4	yes	—
79	C₄²×C₅	—	2	2	no	—
80	C₄²×C₅	—	2	2	no	—
81	C₄²×C₅	—	2	1	no	—
82	C₄²×C₅	—	2	1	no	—
83	C₄²×C₅	—	2	2	no	—
84	C₄²×C₅	—	4	1	no	—
85	C₄²×C₅	—	4	4	no	—
86	C₄²×C₅	D₈	1	2	yes	—
87	C₄²×C₅	SD₁₆	1	2	yes	—
88	C₄²×C₅	C₂²×C₄	1	2	yes	—
89	C₄²×C₅	#41	1	2	yes	—
90	C₂×#168	—	2	4	no	—
91	C₂×#168	—	2	4	no	—
92	C₂×#168	—	4	2	no	—
93	C₂×#168	—	4	2	no	—
94	C₂×#168	—	4	2	no	—
95	C₂×#168	—	4	2	no	—
96	C₂×C₄⋊Q₈	C₂×D₅	1	1	yes	characteristic
97	C₂×C₄×Q₈	—	2	1	no	—
98	C₂×C₄⋊C₈	—	5	2	no	—
99	C₂×C₄⋊C₈	—	5	2	no	—
100	C₂×#97	—	10	1	no	—
101	C₂×C₄×C₈	—	10	1	no	—
102	#391	—	2	1	no	—
103	#389	—	2	1	no	—
104	#389	Dic₅	1	2	yes	—
105	#384	C₂×D₅	1	1	yes	characteristic
106	C₄×C₄⋊C₄	—	2	1	no	—
107	C₄×C₄⋊C₄	Dic₅	1	2	yes	—
108	C₄³	C₂×D₅	1	1	yes	characteristic
109	#336	—	10	1	no	—
110	#336	—	10	1	no	—
111	#332	—	5	8	no	—
112	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₅×Q₈	—	4	4	no	—
114	C₅×Q₈	—	4	4	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	1	no	—
121	C₂×C₄×C₅	—	2	1	no	—
122	C₂×C₄×C₅	—	2	1	no	—
123	C₂×C₄×C₅	—	2	4	no	—
124	C₂×C₄×C₅	—	2	1	no	—
125	C₂×C₄×C₅	—	2	1	no	—
126	C₂×C₄×C₅	—	2	1	no	—
127	C₂×C₄×C₅	—	4	2	no	—
128	C₂×C₄×C₅	—	4	2	no	—
129	C₂×C₄×C₅	—	4	2	no	—
130	C₂×C₄×C₅	—	4	2	no	—
131	C₂×C₄×C₅	—	4	2	no	—
132	C₂×C₄×C₅	#101	1	2	yes	—
133	C₂×C₄×C₅	#102	1	2	yes	—
134	C₂×C₄×C₅	#102	1	2	yes	—
135	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
136	C₂×C₄×C₅	#117	1	1	yes	characteristic
137	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
138	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
139	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
140	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
141	C₂×C₄×C₅	#135	1	2	yes	—
142	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
143	C₂×C₄×C₅	#137	1	1	yes	characteristic
144	C₂×C₄×C₅	#137	1	2	yes	—
145	#168	—	4	4	no	—
146	#168	—	4	4	no	—
147	C₂²×Q₈	—	2	1	no	—
148	C₂²×C₈	—	10	2	no	—
149	C₂²×C₈	—	10	2	no	—
150	C₄⋊Q₈	—	2	2	no	—
151	C₄⋊Q₈	C₄×D₅	1	4	yes	—
152	C₄×Q₈	—	4	2	no	—
153	C₄×Q₈	—	4	2	no	—
154	C₂×C₄⋊C₄	—	2	1	no	—
155	C₂×C₄⋊C₄	—	2	2	no	—
156	C₂×C₄⋊C₄	—	2	1	no	—
157	C₂×C₄⋊C₄	—	2	1	no	—
158	C₂×C₄⋊C₄	C₂×Dic₅	1	2	yes	—
159	C₂×C₄⋊C₄	C₂×Dic₅	1	2	yes	—
160	C₂×C₄²	—	2	2	no	—
161	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
162	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
163	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
164	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
165	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
166	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
167	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
168	C₄⋊C₈	—	5	4	no	—
169	C₄⋊C₈	—	5	4	no	—
170	C₄⋊C₈	—	10	2	no	—
171	C₄⋊C₈	—	10	2	no	—
172	#97	—	20	2	no	—
173	C₄×C₈	—	20	2	no	—
174	#95	—	2	2	no	—
175	#95	—	2	2	no	—
176	C₂²×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
177	C₂²×C₅	C₂×#101	1	1	yes	characteristic
178	C₂²×C₅	#413	1	1	yes	characteristic
179	C₂²×C₅	C₂×#102	1	1	yes	characteristic
180	C₂²×C₅	#416	1	1	yes	characteristic
181	C₂²×C₅	#417	1	1	yes	characteristic
182	C₂²×C₅	#419	1	1	yes	characteristic
183	C₄×C₅	—	2	2	no	—
184	C₄×C₅	—	2	2	no	—
185	C₄×C₅	—	2	2	no	—
186	C₄×C₅	—	2	1	no	—
187	C₄×C₅	—	2	1	no	—
188	C₄×C₅	—	4	1	no	—
189	C₄×C₅	—	4	4	no	—
190	C₄×C₅	—	8	2	no	—
191	C₄×C₅	C₄×D₈	1	2	yes	—
192	C₄×C₅	C₄×SD₁₆	1	2	yes	—
193	C₄×C₅	#440	1	2	yes	—
194	C₄×C₅	#441	1	2	yes	—
195	C₂×Q₈	—	2	4	no	—
196	C₂×Q₈	—	4	2	no	—
197	C₂×Q₈	—	4	1	no	—
198	C₂×Q₈	—	4	1	no	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
202	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
203	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
204	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
205	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
206	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
207	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
208	C₂×C₈	—	10	4	no	—
209	C₂×C₈	—	10	4	no	—
210	C₂×C₈	—	20	2	no	—
211	C₂×C₈	—	20	2	no	—
212	C₂×C₈	—	20	2	no	—
213	C₂×C₈	—	20	2	no	—
214	C₄⋊C₄	—	2	2	no	—
215	C₄⋊C₄	—	2	4	no	—
216	C₄⋊C₄	—	4	2	no	—
217	C₄⋊C₄	—	4	1	no	—
218	C₄⋊C₄	—	4	4	no	—
219	C₄⋊C₄	—	4	1	no	—
220	C₄⋊C₄	—	4	1	no	—
221	C₄⋊C₄	—	4	1	no	—
222	C₄⋊C₄	C₄×Dic₅	1	4	yes	—
223	C₄²	—	2	2	no	—
224	C₄²	—	2	1	no	—
225	C₄²	—	2	2	no	—
226	C₄²	—	2	2	no	—
227	C₄²	—	2	1	no	—
228	C₄²	—	4	1	no	—
229	C₄²	—	4	4	no	—
230	C₄²	#686	1	2	yes	—
231	C₄²	#687	1	2	yes	—
232	C₄²	C₂×C₄×D₅	1	2	yes	—
233	C₄²	#709	1	2	yes	—
234	C₂×C₅	#1546	1	1	yes	characteristic
235	C₂×C₅	#1547	1	1	yes	characteristic
236	C₂×C₅	#1552	1	1	yes	characteristic
237	C₂×C₅	#1554	1	1	yes	characteristic
238	C₂×C₅	C₄×#101	1	1	yes	characteristic
239	C₂×C₅	C₄×#102	1	1	yes	characteristic
240	C₂×C₅	#1763	1	1	yes	characteristic
241	C₂³	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
242	Q₈	—	4	4	no	—
243	Q₈	—	4	4	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	#3964	1	2	yes	—
262	C₂×C₄	#3964	1	2	yes	—
263	C₂×C₄	#3968	1	2	yes	—
264	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
265	C₂×C₄	#4072	1	1	yes	characteristic
266	C₂×C₄	C₄×C₂²⋊D₅	1	2	yes	—
267	C₂×C₄	C₄×C₂²⋊D₅	1	2	yes	—
268	C₂×C₄	C₂×#686	1	1	yes	characteristic
269	C₂×C₄	C₂×#687	1	1	yes	characteristic
270	C₂×C₄	#4088	1	2	yes	—
271	C₂×C₄	#4095	1	1	yes	characteristic
272	C₂×C₄	#4096	1	2	yes	—
273	C₂×C₄	#4097	1	1	yes	characteristic
274	C₈	—	20	4	no	—
275	C₈	—	20	4	no	—
276	C₅	#8236	1	1	yes	characteristic, complemented, Sylow
277	C₂²	C₄×C₂²⋊Dic₅	1	1	yes	characteristic
278	C₂²	C₂×#3964	1	1	yes	characteristic
279	C₂²	#65514	1	1	yes	characteristic
280	C₂²	C₂×#3968	1	1	yes	characteristic
281	C₂²	#65521	1	1	yes	characteristic
282	C₂²	#65522	1	1	yes	characteristic
283	C₂²	#65526	1	1	yes	characteristic
284	C₄	—	2	2	no	—
285	C₄	—	2	2	no	—
286	C₄	—	2	2	no	—
287	C₄	—	2	1	no	—
288	C₄	—	2	1	no	—
289	C₄	—	4	1	no	—
290	C₄	—	4	4	no	—
291	C₄	—	8	2	no	—
292	C₄	C₄×#686	1	2	yes	—
293	C₄	C₄×#687	1	2	yes	—
294	C₄	#65315	1	2	yes	—
295	C₄	#65319	1	2	yes	—
296	C₂	C₄×#3964	1	1	yes	characteristic
297	C₂	#10602424	1	1	yes	characteristic
298	C₂	C₄×#3968	1	1	yes	characteristic
299	C₂	#10602566	1	1	yes	characteristic
300	C₂	#10602599	1	1	yes	characteristic
301	C₂	#10602624	1	1	yes	characteristic
302	C₂	#10602625	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	#8236
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
276. C₅	#8236	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
#8236	5