Subgroups — #12017529

Order 1280.

522 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#495	C₂	1	1	yes	maximal, characteristic
2	C₂×#64784	C₂	1	2	yes	maximal, complemented
3	C₂×#64782	C₂	1	2	yes	maximal
4	#10600558	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#64681	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₅×Q₁₆	C₂²	1	1	yes	characteristic
7	C₂²×C₅×SD₁₆	—	2	1	no	—
8	C₂²×C₅×D₈	C₂²	1	1	yes	characteristic
9	C₂×C₅×#124	—	2	1	no	—
10	C₅×#495	—	2	2	no	—
11	C₅×#495	C₄	1	4	yes	complemented
12	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₂×#3950	—	2	1	no	—
14	C₂×#3950	C₂²	1	2	yes	—
15	C₂²×#672	C₂²	1	2	yes	—
16	#65224	—	2	1	no	—
17	#64784	—	2	4	no	—
18	#64782	—	2	4	no	—
19	#64681	—	2	2	no	—
20	#13205	—	5	1	no	maximal, Sylow
21	C₂²×C₅×Q₈	—	2	1	no	—
22	C₂²×C₅×D₄	—	2	1	no	—
23	C₂×C₅×Q₁₆	—	2	2	no	—
24	C₂×C₅×Q₁₆	—	2	1	no	—
25	C₂×C₅×Q₁₆	—	2	2	no	—
26	C₂×C₅×Q₁₆	C₂×C₄	1	2	yes	—
27	C₂×C₅×SD₁₆	—	2	4	no	—
28	C₂×C₅×SD₁₆	—	4	2	no	—
29	C₂×C₅×SD₁₆	—	4	2	no	—
30	C₂×C₅×D₈	—	2	2	no	—
31	C₂×C₅×D₈	—	2	1	no	—
32	C₂×C₅×D₈	—	2	2	no	—
33	C₂×C₅×D₈	C₂×C₄	1	2	yes	—
34	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
35	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
36	C₅×#124	—	2	4	no	—
37	C₅×#124	—	4	2	no	—
38	C₂×C₅×C₂²⋊C₄	—	4	2	no	—
39	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
40	C₂×C₄⋊Dic₅	—	2	2	no	—
41	C₂×C₄⋊Dic₅	—	4	2	no	—
42	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
43	C₄×C₅×C₈	D₄	1	2	yes	—
44	#3950	—	2	2	no	—
45	#3950	—	2	2	no	—
46	#3950	—	4	2	no	—
47	C₂×#672	—	2	2	no	—
48	C₂×#672	—	2	2	no	—
49	C₂×#672	D₄	1	4	yes	—
50	C₂×#495	D₅	1	1	yes	characteristic
51	C₂×#358	—	5	2	no	—
52	C₂×#357	—	5	2	no	—
53	C₂×#346	—	5	1	no	—
54	#1845	—	5	1	no	—
55	C₂⁴×C₅	—	4	1	no	—
56	C₂×C₅×Q₈	—	2	2	no	—
57	C₂×C₅×Q₈	—	4	1	no	—
58	C₂×C₅×Q₈	—	4	1	no	—
59	C₂×C₅×Q₈	—	4	1	no	—
60	C₂×C₅×Q₈	—	4	1	no	—
61	C₂×C₅×Q₈	—	4	1	no	—
62	C₂×C₅×D₄	—	2	2	no	—
63	C₂×C₅×D₄	—	4	1	no	—
64	C₂×C₅×D₄	—	4	1	no	—
65	C₂×C₅×D₄	—	4	1	no	—
66	C₂×C₅×D₄	—	4	1	no	—
67	C₂×C₅×D₄	—	4	1	no	—
68	C₂²×C₄×C₅	—	4	1	no	—
69	C₂²×C₄×C₅	D₈	1	2	yes	—
70	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
71	C₂²×Dic₅	—	4	2	no	—
72	C₅×Q₁₆	—	2	4	no	—
73	C₅×Q₁₆	—	4	2	no	—
74	C₅×SD₁₆	—	4	8	no	—
75	C₅×D₈	—	2	4	no	—
76	C₅×D₈	—	4	2	no	—
77	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
78	C₂×C₅×C₈	D₈	1	2	yes	—
79	C₂×C₅×C₈	SD₁₆	1	2	yes	—
80	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
82	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
83	C₂×C₅×C₈	#41	1	2	yes	—
84	C₅×C₂²⋊C₄	—	4	8	no	—
85	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
86	C₄²×C₅	SD₁₆	1	2	yes	—
87	C₄⋊Dic₅	—	4	2	no	—
88	C₄⋊Dic₅	—	4	2	no	—
89	C₄⋊Dic₅	—	8	4	no	—
90	#672	—	2	4	no	—
91	C₂²×Q₁₆	C₂×D₅	1	1	yes	characteristic
92	C₂²×SD₁₆	—	2	1	no	—
93	C₂²×D₈	C₂×D₅	1	1	yes	characteristic
94	C₂×#124	—	2	1	no	—
95	C₂²×C₁₆	—	5	2	no	—
96	#495	—	2	2	no	—
97	#495	Dic₅	1	4	yes	complemented
98	C₂×#107	—	5	2	no	—
99	C₂×#107	—	10	1	no	—
100	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
101	#384	—	10	1	no	—
102	#358	—	10	4	no	—
103	#357	—	10	4	no	—
104	#346	—	10	2	no	—
105	C₂³×C₅	—	4	2	no	—
106	C₂³×C₅	—	8	1	no	—
107	C₂³×C₅	—	8	1	no	—
108	C₂³×C₅	—	8	2	no	—
109	C₂³×C₅	—	8	2	no	—
110	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
111	C₅×Q₈	—	4	2	no	—
112	C₅×Q₈	—	4	2	no	—
113	C₅×Q₈	—	8	1	no	—
114	C₅×Q₈	—	8	1	no	—
115	C₅×D₄	—	4	2	no	—
116	C₅×D₄	—	4	2	no	—
117	C₅×D₄	—	8	1	no	—
118	C₅×D₄	—	8	1	no	—
119	C₂×C₄×C₅	—	2	2	no	—
120	C₂×C₄×C₅	—	2	2	no	—
121	C₂×C₄×C₅	—	4	2	no	—
122	C₂×C₄×C₅	—	8	1	no	—
123	C₂×C₄×C₅	—	8	1	no	—
124	C₂×C₄×C₅	#102	1	4	yes	—
125	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₅	#124	1	1	yes	characteristic
127	C₂×C₄×C₅	#127	1	1	yes	characteristic
128	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
130	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
131	C₂×Dic₅	—	8	2	no	—
132	C₂×Dic₅	—	8	4	no	—
133	C₅×C₈	#102	1	4	yes	—
134	C₅×C₈	C₄×D₄	1	2	yes	—
135	C₅×C₈	#121	1	2	yes	—
136	C₂²×Q₈	—	2	1	no	—
137	C₂²×D₄	—	2	1	no	—
138	C₂×Q₁₆	—	2	2	no	—
139	C₂×Q₁₆	—	2	2	no	—
140	C₂×Q₁₆	—	2	1	no	—
141	C₂×Q₁₆	C₂×Dic₅	1	2	yes	—
142	C₂×SD₁₆	—	2	4	no	—
143	C₂×SD₁₆	—	4	2	no	—
144	C₂×SD₁₆	—	4	2	no	—
145	C₂×D₈	—	2	2	no	—
146	C₂×D₈	—	2	2	no	—
147	C₂×D₈	—	2	1	no	—
148	C₂×D₈	C₂×Dic₅	1	2	yes	—
149	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
150	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
151	#124	—	2	4	no	—
152	#124	—	4	2	no	—
153	C₂×C₄⋊C₄	—	10	2	no	—
154	C₂×C₄⋊C₄	—	20	2	no	—
155	C₂×C₂²⋊C₄	—	4	2	no	—
156	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
157	C₂×C₁₆	—	5	4	no	—
158	C₂×C₁₆	—	10	2	no	—
159	C₂×C₁₆	—	10	2	no	—
160	#107	—	10	2	no	—
161	#107	—	10	2	no	—
162	#107	—	20	2	no	—
163	C₄×C₈	C₂×Dic₅	1	2	yes	—
164	C₄×C₈	C₂²⋊D₅	1	2	yes	—
165	C₂²×C₅	—	8	4	no	—
166	C₂²×C₅	—	8	2	no	—
167	C₂²×C₅	—	16	2	no	—
168	C₂²×C₅	—	16	1	no	—
169	C₂²×C₅	—	16	1	no	—
170	C₂²×C₅	C₂×#102	1	1	yes	characteristic
171	C₂²×C₅	#486	1	1	yes	characteristic
172	C₂²×C₅	#493	1	1	yes	characteristic
173	C₂²×C₅	#509	1	2	yes	—
174	C₂²×C₅	#510	1	2	yes	—
175	C₄×C₅	—	2	4	no	—
176	C₄×C₅	—	8	2	no	—
177	C₄×C₅	#390	1	1	yes	characteristic
178	C₄×C₅	C₂×#102	1	1	yes	characteristic
179	C₄×C₅	#486	1	1	yes	characteristic
180	C₄×C₅	#492	1	1	yes	characteristic
181	Dic₅	—	16	2	no	—
182	C₂⁴	—	4	1	no	—
183	C₂×Q₈	—	2	2	no	—
184	C₂×Q₈	—	4	1	no	—
185	C₂×Q₈	—	4	1	no	—
186	C₂×Q₈	—	4	1	no	—
187	C₂×Q₈	—	4	1	no	—
188	C₂×Q₈	—	4	1	no	—
189	C₂×D₄	—	2	2	no	—
190	C₂×D₄	—	4	1	no	—
191	C₂×D₄	—	4	1	no	—
192	C₂×D₄	—	4	1	no	—
193	C₂×D₄	—	4	1	no	—
194	C₂×D₄	—	4	1	no	—
195	C₂²×C₄	—	4	1	no	—
196	C₂²×C₄	—	20	2	no	—
197	C₂²×C₄	#686	1	2	yes	—
198	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
199	Q₁₆	—	2	4	no	—
200	Q₁₆	—	4	2	no	—
201	SD₁₆	—	4	8	no	—
202	D₈	—	2	4	no	—
203	D₈	—	4	2	no	—
204	C₂×C₈	#686	1	2	yes	—
205	C₂×C₈	#687	1	2	yes	—
206	C₂×C₈	C₂²⋊Dic₅	1	2	yes	—
207	C₂×C₈	D₄×D₅	1	2	yes	—
208	C₂×C₈	#711	1	2	yes	—
209	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
210	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
211	C₄⋊C₄	—	20	2	no	—
212	C₄⋊C₄	—	20	2	no	—
213	C₄⋊C₄	—	40	4	no	—
214	C₂²⋊C₄	—	4	8	no	—
215	C₄²	#687	1	2	yes	—
216	C₄²	C₂²⋊Dic₅	1	2	yes	—
217	C₁₆	—	10	4	no	—
218	C₂×C₅	—	16	2	no	—
219	C₂×C₅	#1957	1	1	yes	characteristic
220	C₂×C₅	#2139	1	2	yes	—
221	C₂×C₅	#2243	1	2	yes	—
222	C₂×C₅	#2250	1	1	yes	characteristic
223	C₂×C₅	#2251	1	1	yes	characteristic
224	C₂³	—	4	2	no	—
225	C₂³	—	8	1	no	—
226	C₂³	—	8	1	no	—
227	C₂³	—	8	2	no	—
228	C₂³	—	8	2	no	—
229	C₂³	C₂×#686	1	1	yes	characteristic
230	Q₈	—	4	2	no	—
231	Q₈	—	4	2	no	—
232	Q₈	—	8	1	no	—
233	Q₈	—	8	1	no	—
234	D₄	—	4	2	no	—
235	D₄	—	4	2	no	—
236	D₄	—	8	1	no	—
237	D₄	—	8	1	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	8	1	no	—
242	C₂×C₄	—	8	1	no	—
243	C₂×C₄	—	40	2	no	—
244	C₂×C₄	—	40	4	no	—
245	C₂×C₄	#3964	1	4	yes	—
246	C₂×C₄	C₂×#686	1	1	yes	characteristic
247	C₂×C₄	C₂×#687	1	1	yes	characteristic
248	C₂×C₄	C₂×#687	1	1	yes	characteristic
249	C₂×C₄	#4082	1	1	yes	characteristic
250	C₂×C₄	#4086	1	1	yes	characteristic
251	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
252	C₈	#3964	1	4	yes	—
253	C₈	D₄×Dic₅	1	2	yes	—
254	C₈	#4084	1	2	yes	—
255	C₅	#13205	1	1	yes	characteristic, complemented, Sylow
256	C₂²	—	8	2	no	—
257	C₂²	—	8	4	no	—
258	C₂²	—	16	2	no	—
259	C₂²	—	16	1	no	—
260	C₂²	—	16	1	no	—
261	C₂²	#65359	1	1	yes	characteristic
262	C₂²	#65362	1	1	yes	characteristic
263	C₂²	#65482	1	2	yes	—
264	C₂²	#65484	1	2	yes	—
265	C₂²	C₂×#3964	1	1	yes	characteristic
266	C₄	—	2	4	no	—
267	C₄	—	8	2	no	—
268	C₄	—	80	2	no	—
269	C₄	#65359	1	1	yes	characteristic
270	C₄	#65365	1	1	yes	characteristic
271	C₄	C₂×#3964	1	1	yes	characteristic
272	C₄	#65510	1	1	yes	characteristic
273	C₂	—	16	2	no	—
274	C₂	#10602112	1	2	yes	—
275	C₂	#10602114	1	1	yes	characteristic
276	C₂	#10602115	1	1	yes	characteristic
277	C₂	#10602557	1	1	yes	characteristic
278	C₂	#10602871	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#686
commutator	C₂×C₅×C₈	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₂×C₅×#495	C₂
socle	C₂³×C₅	C₂×D₈
prime radical	C₂²×C₅×D₈	C₂²
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	C₂×#495	D₅
hypocenter	C₅	#13205
LCS	G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#495

Complements

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

N	H ≅ G/N	type
2. C₂×#64784	C₂	semidirect
4. #10600558	C₂	semidirect
5. C₂×#64681	C₂	semidirect
11. C₅×#495	C₄	semidirect
97. #495	Dic₅	semidirect
255. C₅	#13205	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#495	2
C₂×#64784	2
C₂×#64782	2
#10600558	2
C₂×#64681	2
#13205	5