Subgroups — #12008524

Order 1280.

454 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×C₈×D₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊#672	C₂	1	2	yes	maximal, complemented
3	#10598200	C₂	1	2	yes	maximal, complemented
4	#10598164	C₂	1	2	yes	maximal, complemented
5	C₂³×C₅×C₈	C₄	1	1	yes	characteristic
6	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₅×D₄	C₄	1	1	yes	characteristic
8	C₅×C₈×D₄	—	2	2	no	—
9	C₅×C₈×D₄	—	2	2	no	—
10	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₅×C₂²⋊C₈	C₄	1	1	yes	characteristic
12	C₂×C₅×C₂²⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₅×C₈	C₄	1	1	yes	characteristic
14	C₂²×#672	—	2	2	no	—
15	C₂²⋊#672	—	2	2	no	—
16	C₂²⋊#672	C₂²	1	4	yes	—
17	#11465	—	5	1	no	maximal, Sylow
18	C₂²×C₅×D₄	C₈	1	1	yes	characteristic
19	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
20	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₅×C₈	—	2	1	no	—
22	C₂²×C₅×C₈	—	2	1	no	—
23	C₂²×C₅×C₈	—	4	2	no	—
24	C₂²×C₅×C₈	—	4	2	no	—
25	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₅×C₈	D₄	1	2	yes	—
29	C₂²×C₅×C₈	D₄	1	2	yes	—
30	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
31	C₄×C₅×D₄	—	2	2	no	—
32	C₄×C₅×D₄	—	2	2	no	—
33	C₂×C₅×C₄⋊C₄	C₈	1	1	yes	characteristic
34	C₂×C₅×C₂²⋊C₄	C₈	1	1	yes	characteristic
35	C₂×C₅×C₂²⋊C₄	C₈	1	1	yes	characteristic
36	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
37	C₅×C₄⋊C₈	—	2	2	no	—
38	C₅×C₂²⋊C₈	—	2	2	no	—
39	C₅×C₂²⋊C₈	—	2	2	no	—
40	C₄×C₅×C₈	—	2	1	no	—
41	C₄×C₅×C₈	—	2	1	no	—
42	C₂×#672	—	2	4	no	—
43	C₂×#672	—	4	4	no	—
44	C₂×C₈×D₄	D₅	1	1	yes	characteristic
45	C₂×C₂²⋊C₁₆	—	5	2	no	—
46	#1322	—	5	2	no	—
47	#1314	—	5	2	no	—
48	C₂⁴×C₅	C₂×C₈	1	1	yes	characteristic
49	C₂⁴×C₅	C₂×C₈	1	1	yes	characteristic
50	C₂×C₅×D₄	—	2	2	no	—
51	C₂×C₅×D₄	—	2	2	no	—
52	C₂×C₅×D₄	C₁₆	1	4	yes	complemented
53	C₂²×C₄×C₅	—	2	1	no	—
54	C₂²×C₄×C₅	—	2	1	no	—
55	C₂²×C₄×C₅	—	4	1	no	—
56	C₂²×C₄×C₅	—	4	1	no	—
57	C₂²×C₄×C₅	—	4	1	no	—
58	C₂²×C₄×C₅	—	4	1	no	—
59	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
60	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
61	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
62	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
63	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
64	C₂²×C₄×C₅	M₁₆	1	2	yes	—
65	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₅×C₈	—	2	2	no	—
67	C₂×C₅×C₈	—	2	4	no	—
68	C₂×C₅×C₈	—	2	1	no	—
69	C₂×C₅×C₈	—	2	1	no	—
70	C₂×C₅×C₈	—	4	2	no	—
71	C₂×C₅×C₈	—	4	1	no	—
72	C₂×C₅×C₈	—	4	2	no	—
73	C₂×C₅×C₈	—	4	2	no	—
74	C₂×C₅×C₈	—	4	1	no	—
75	C₂×C₅×C₈	—	4	1	no	—
76	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
79	C₅×C₄⋊C₄	—	2	2	no	—
80	C₅×C₂²⋊C₄	—	2	2	no	—
81	C₅×C₂²⋊C₄	—	2	2	no	—
82	C₄²×C₅	—	2	1	no	—
83	C₄²×C₅	—	2	1	no	—
84	#672	—	4	4	no	—
85	C₂³×C₈	Dic₅	1	1	yes	characteristic
86	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
87	C₂×C₄×D₄	Dic₅	1	1	yes	characteristic
88	C₂²×C₁₆	—	10	2	no	—
89	C₈×D₄	—	2	2	no	—
90	C₈×D₄	—	2	2	no	—
91	C₂×C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
92	C₂×C₂²⋊C₈	Dic₅	1	1	yes	characteristic
93	C₂×C₂²⋊C₈	C₂×D₅	1	1	yes	characteristic
94	C₂×C₄×C₈	Dic₅	1	1	yes	characteristic
95	C₂²⋊C₁₆	—	5	4	no	—
96	C₂²⋊C₁₆	—	10	2	no	—
97	C₂³×C₅	—	2	1	no	—
98	C₂³×C₅	—	2	1	no	—
99	C₂³×C₅	—	4	1	no	—
100	C₂³×C₅	—	4	1	no	—
101	C₂³×C₅	—	4	1	no	—
102	C₂³×C₅	—	4	1	no	—
103	C₂³×C₅	C₂²⋊C₈	1	2	yes	—
104	C₂³×C₅	C₂×C₁₆	1	2	yes	—
105	C₂³×C₅	C₂×C₁₆	1	2	yes	complemented
106	C₂³×C₅	M₃₂	1	2	yes	—
107	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
108	C₅×D₄	—	2	8	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	2	no	—
113	C₂×C₄×C₅	—	2	1	no	—
114	C₂×C₄×C₅	—	2	2	no	—
115	C₂×C₄×C₅	—	2	1	no	—
116	C₂×C₄×C₅	—	2	1	no	—
117	C₂×C₄×C₅	—	4	1	no	—
118	C₂×C₄×C₅	—	4	2	no	—
119	C₂×C₄×C₅	—	4	2	no	—
120	C₂×C₄×C₅	—	4	1	no	—
121	C₂×C₄×C₅	—	4	1	no	—
122	C₂×C₄×C₅	—	4	1	no	—
123	C₂×C₄×C₅	—	4	1	no	—
124	C₂×C₄×C₅	C₂²⋊C₈	1	2	yes	—
125	C₂×C₄×C₅	C₂×C₁₆	1	2	yes	complemented
126	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
128	C₅×C₈	—	2	2	no	—
129	C₅×C₈	—	2	1	no	—
130	C₅×C₈	—	4	1	no	—
131	C₅×C₈	—	4	1	no	—
132	C₅×C₈	#99	1	2	yes	—
133	C₂²×D₄	#168	1	1	yes	characteristic
134	C₂³×C₄	C₂×Dic₅	1	1	yes	characteristic
135	C₂³×C₄	C₂×Dic₅	1	1	yes	characteristic
136	C₂²×C₈	—	2	1	no	—
137	C₂²×C₈	—	2	1	no	—
138	C₂²×C₈	—	4	2	no	—
139	C₂²×C₈	—	4	2	no	—
140	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
141	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
142	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
143	C₂²×C₈	C₂²⋊D₅	1	2	yes	—
144	C₂²×C₈	C₂²⋊D₅	1	2	yes	—
145	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
146	C₄×D₄	—	2	2	no	—
147	C₄×D₄	—	2	2	no	—
148	C₂×C₄⋊C₄	#168	1	1	yes	characteristic
149	C₂×C₂²⋊C₄	#168	1	1	yes	characteristic
150	C₂×C₂²⋊C₄	#168	1	1	yes	characteristic
151	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
152	C₂×C₁₆	—	10	4	no	—
153	C₂×C₁₆	—	20	4	no	—
154	C₄⋊C₈	—	2	2	no	—
155	C₂²⋊C₈	—	2	2	no	—
156	C₂²⋊C₈	—	2	2	no	—
157	C₄×C₈	—	2	1	no	—
158	C₄×C₈	—	2	1	no	—
159	C₂²×C₅	—	2	1	no	—
160	C₂²×C₅	—	2	2	no	—
161	C₂²×C₅	—	2	1	no	—
162	C₂²×C₅	—	2	2	no	—
163	C₂²×C₅	—	2	2	no	—
164	C₂²×C₅	—	4	2	no	—
165	C₂²×C₅	—	4	2	no	—
166	C₂²×C₅	—	4	2	no	—
167	C₂²×C₅	—	4	2	no	—
168	C₂²×C₅	—	4	1	no	—
169	C₂²×C₅	—	4	1	no	—
170	C₂²×C₅	C₂²⋊C₁₆	1	4	yes	—
171	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
172	C₂²×C₅	C₂²×C₁₆	1	1	yes	characteristic
173	C₂²×C₅	C₂×M₃₂	1	1	yes	characteristic
174	C₄×C₅	—	2	2	no	—
175	C₄×C₅	—	2	1	no	—
176	C₄×C₅	—	2	2	no	—
177	C₄×C₅	—	4	1	no	—
178	C₄×C₅	—	4	1	no	—
179	C₄×C₅	C₂×#99	1	1	yes	characteristic
180	C₄×C₅	#413	1	1	yes	characteristic
181	C₂⁴	C₂×#168	1	1	yes	characteristic
182	C₂⁴	C₂×#168	1	1	yes	characteristic
183	C₂×D₄	—	2	2	no	—
184	C₂×D₄	—	2	2	no	—
185	C₂×D₄	#672	1	4	yes	complemented
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	4	1	no	—
189	C₂²×C₄	—	4	1	no	—
190	C₂²×C₄	—	4	1	no	—
191	C₂²×C₄	—	4	1	no	—
192	C₂²×C₄	C₂×#168	1	1	yes	characteristic
193	C₂²×C₄	C₂×#168	1	1	yes	characteristic
194	C₂²×C₄	C₂×#168	1	1	yes	characteristic
195	C₂²×C₄	C₂×#168	1	1	yes	characteristic
196	C₂²×C₄	#681	1	2	yes	—
197	C₂²×C₄	C₂²⋊Dic₅	1	2	yes	—
198	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
199	C₂×C₈	—	2	4	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	1	no	—
202	C₂×C₈	—	2	1	no	—
203	C₂×C₈	—	4	2	no	—
204	C₂×C₈	—	4	1	no	—
205	C₂×C₈	—	4	2	no	—
206	C₂×C₈	—	4	2	no	—
207	C₂×C₈	—	4	1	no	—
208	C₂×C₈	—	4	1	no	—
209	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
210	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
211	C₂×C₈	C₂×C₂²⋊D₅	1	2	yes	—
212	C₄⋊C₄	—	2	2	no	—
213	C₂²⋊C₄	—	2	2	no	—
214	C₂²⋊C₄	—	2	2	no	—
215	C₄²	—	2	1	no	—
216	C₄²	—	2	1	no	—
217	C₁₆	—	20	4	no	—
218	C₂×C₅	—	2	1	no	—
219	C₂×C₅	—	2	1	no	—
220	C₂×C₅	—	2	4	no	—
221	C₂×C₅	—	4	2	no	—
222	C₂×C₅	#1459	1	1	yes	characteristic
223	C₂×C₅	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
224	C₂×C₅	#2115	1	1	yes	characteristic
225	C₂³	—	2	1	no	—
226	C₂³	—	2	1	no	—
227	C₂³	—	4	1	no	—
228	C₂³	—	4	1	no	—
229	C₂³	—	4	1	no	—
230	C₂³	—	4	1	no	—
231	C₂³	C₂×#672	1	2	yes	—
232	C₂³	C₂×#672	1	2	yes	complemented
233	C₂³	#3944	1	2	yes	—
234	C₂³	C₂²⋊#168	1	2	yes	—
235	C₂³	C₂²×#168	1	1	yes	characteristic
236	D₄	—	2	8	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	4	1	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	4	1	no	—
251	C₂×C₄	—	4	1	no	—
252	C₂×C₄	C₂×#672	1	2	yes	complemented
253	C₂×C₄	C₂²⋊#168	1	2	yes	—
254	C₂×C₄	C₂×#681	1	1	yes	characteristic
255	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
256	C₈	—	2	2	no	—
257	C₈	—	2	1	no	—
258	C₈	—	4	1	no	—
259	C₈	—	4	1	no	—
260	C₈	#3966	1	2	yes	—
261	C₅	#11465	1	1	yes	characteristic, complemented, Sylow
262	C₂²	—	2	1	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	4	2	no	—
268	C₂²	—	4	2	no	—
269	C₂²	—	4	2	no	—
270	C₂²	—	4	2	no	—
271	C₂²	—	4	1	no	—
272	C₂²	—	4	1	no	—
273	C₂²	C₂²⋊#672	1	4	yes	—
274	C₂²	C₂²×#672	1	1	yes	characteristic
275	C₂²	C₂×#3944	1	1	yes	characteristic
276	C₂²	C₂×C₂²⋊#168	1	1	yes	characteristic
277	C₄	—	2	2	no	—
278	C₄	—	2	1	no	—
279	C₄	—	2	2	no	—
280	C₄	—	4	1	no	—
281	C₄	—	4	1	no	—
282	C₄	C₂×#3966	1	1	yes	characteristic
283	C₄	#65526	1	1	yes	characteristic
284	C₂	—	2	1	no	—
285	C₂	—	2	1	no	—
286	C₂	—	2	4	no	—
287	C₂	—	4	2	no	—
288	C₂	#10602393	1	1	yes	characteristic
289	C₂	C₂×C₂²⋊#672	1	1	yes	characteristic
290	C₂	#10602800	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²⋊Dic₅
commutator	C₂²×C₅	C₂²×C₁₆
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₂×C₅×C₈×D₄	C₂
socle	C₂²×C₅	C₂×C₂²⋊C₈
prime radical	C₂²×C₅×D₄	C₈
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	C₂×C₈×D₄	D₅
hypocenter	C₅	#11465
LCS	G ▷ C₂²×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ C₂×C₈×D₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₂²⋊#672	C₂	semidirect
3. #10598200	C₂	semidirect
4. #10598164	C₂	semidirect
52. C₂×C₅×D₄	C₁₆	semidirect
105. C₂³×C₅	C₂×C₁₆	semidirect
125. C₂×C₄×C₅	C₂×C₁₆	semidirect
185. C₂×D₄	#672	semidirect
232. C₂³	C₂×#672	semidirect
252. C₂×C₄	C₂×#672	semidirect
261. C₅	#11465	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×C₈×D₄	2
C₂×C₂²⋊#672	2
#10598200	2
#10598164	2
#11465	5