Subgroups — #12017524

Order 1280.

514 conjugacy classes of subgroups.

The subgroup lattice has 268 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	#10600559	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄×#672	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	C₂²	1	2	yes	—
14	C₂×#3949	—	2	1	no	—
15	C₂²×#672	C₂²	1	2	yes	—
16	#65225	—	2	1	no	—
17	#64784	—	2	4	no	—
18	#64782	—	2	4	no	—
19	C₄×#672	—	2	2	no	—
20	#13204	—	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	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
42	C₄×C₅×C₈	D₄	1	2	yes	—
43	#3963	—	4	2	no	—
44	#3950	—	2	2	no	—
45	#3950	—	2	2	no	—
46	#3949	—	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×C₄×C₁₆	—	5	1	no	—
54	#1846	—	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	#672	—	2	4	no	—
90	C₂²×Q₁₆	C₂×D₅	1	1	yes	characteristic
91	C₂²×SD₁₆	—	2	1	no	—
92	C₂²×D₈	C₂×D₅	1	1	yes	characteristic
93	C₂×#124	—	2	1	no	—
94	C₂²×C₁₆	—	5	2	no	—
95	#495	—	2	2	no	—
96	#495	Dic₅	1	4	yes	complemented
97	C₂×#107	—	5	2	no	—
98	C₂×#106	—	10	1	no	—
99	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
100	#382	—	10	1	no	—
101	#358	—	10	4	no	—
102	#357	—	10	4	no	—
103	C₄×C₁₆	—	10	2	no	—
104	C₂³×C₅	—	4	2	no	—
105	C₂³×C₅	—	8	1	no	—
106	C₂³×C₅	—	8	1	no	—
107	C₂³×C₅	—	8	4	no	—
108	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
109	C₅×Q₈	—	4	2	no	—
110	C₅×Q₈	—	4	2	no	—
111	C₅×Q₈	—	8	1	no	—
112	C₅×Q₈	—	8	1	no	—
113	C₅×D₄	—	4	2	no	—
114	C₅×D₄	—	4	2	no	—
115	C₅×D₄	—	8	1	no	—
116	C₅×D₄	—	8	1	no	—
117	C₂×C₄×C₅	—	2	4	no	—
118	C₂×C₄×C₅	—	4	2	no	—
119	C₂×C₄×C₅	—	8	1	no	—
120	C₂×C₄×C₅	—	8	1	no	—
121	C₂×C₄×C₅	#102	1	4	yes	—
122	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₅	#124	1	1	yes	characteristic
124	C₂×C₄×C₅	#127	1	1	yes	characteristic
125	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
127	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
128	C₂×Dic₅	—	8	2	no	—
129	C₂×Dic₅	—	8	4	no	—
130	C₅×C₈	#102	1	4	yes	—
131	C₅×C₈	C₄×D₄	1	2	yes	—
132	C₅×C₈	#121	1	2	yes	—
133	C₂²×Q₈	—	2	1	no	—
134	C₂²×D₄	—	2	1	no	—
135	C₂×Q₁₆	—	2	2	no	—
136	C₂×Q₁₆	—	2	2	no	—
137	C₂×Q₁₆	—	2	1	no	—
138	C₂×Q₁₆	C₂×Dic₅	1	2	yes	—
139	C₂×SD₁₆	—	2	4	no	—
140	C₂×SD₁₆	—	4	2	no	—
141	C₂×SD₁₆	—	4	2	no	—
142	C₂×D₈	—	2	2	no	—
143	C₂×D₈	—	2	2	no	—
144	C₂×D₈	—	2	1	no	—
145	C₂×D₈	C₂×Dic₅	1	2	yes	—
146	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
147	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
148	#124	—	2	4	no	—
149	#124	—	4	2	no	—
150	C₂×C₄⋊C₄	—	10	2	no	—
151	C₂×C₂²⋊C₄	—	4	2	no	—
152	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
153	C₂×C₁₆	—	5	4	no	—
154	C₂×C₁₆	—	10	2	no	—
155	C₂×C₁₆	—	10	2	no	—
156	#107	—	10	2	no	—
157	#107	—	10	2	no	—
158	#106	—	20	2	no	—
159	C₄×C₈	C₂×Dic₅	1	2	yes	—
160	C₄×C₈	C₂²⋊D₅	1	2	yes	—
161	#95	—	20	2	no	—
162	C₂²×C₅	—	8	4	no	—
163	C₂²×C₅	—	8	2	no	—
164	C₂²×C₅	—	16	2	no	—
165	C₂²×C₅	—	16	2	no	—
166	C₂²×C₅	C₂×#102	1	1	yes	characteristic
167	C₂²×C₅	#486	1	1	yes	characteristic
168	C₂²×C₅	#493	1	1	yes	characteristic
169	C₂²×C₅	#508	1	4	yes	—
170	C₄×C₅	—	2	4	no	—
171	C₄×C₅	—	8	2	no	—
172	C₄×C₅	#390	1	1	yes	characteristic
173	C₄×C₅	C₂×#102	1	1	yes	characteristic
174	C₄×C₅	#486	1	1	yes	characteristic
175	C₄×C₅	#492	1	1	yes	characteristic
176	Dic₅	—	16	2	no	—
177	C₂⁴	—	4	1	no	—
178	C₂×Q₈	—	2	2	no	—
179	C₂×Q₈	—	4	1	no	—
180	C₂×Q₈	—	4	1	no	—
181	C₂×Q₈	—	4	1	no	—
182	C₂×Q₈	—	4	1	no	—
183	C₂×Q₈	—	4	1	no	—
184	C₂×D₄	—	2	2	no	—
185	C₂×D₄	—	4	1	no	—
186	C₂×D₄	—	4	1	no	—
187	C₂×D₄	—	4	1	no	—
188	C₂×D₄	—	4	1	no	—
189	C₂×D₄	—	4	1	no	—
190	C₂²×C₄	—	4	1	no	—
191	C₂²×C₄	—	20	2	no	—
192	C₂²×C₄	#686	1	2	yes	—
193	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
194	Q₁₆	—	2	4	no	—
195	Q₁₆	—	4	2	no	—
196	SD₁₆	—	4	8	no	—
197	D₈	—	2	4	no	—
198	D₈	—	4	2	no	—
199	C₂×C₈	#686	1	2	yes	—
200	C₂×C₈	#687	1	2	yes	—
201	C₂×C₈	C₂²⋊Dic₅	1	2	yes	—
202	C₂×C₈	D₄×D₅	1	2	yes	—
203	C₂×C₈	#711	1	2	yes	—
204	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
205	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
206	C₄⋊C₄	—	20	2	no	—
207	C₄⋊C₄	—	20	2	no	—
208	C₂²⋊C₄	—	4	8	no	—
209	C₄²	#687	1	2	yes	—
210	C₄²	C₂²⋊Dic₅	1	2	yes	—
211	C₁₆	—	10	4	no	—
212	C₂×C₅	—	16	2	no	—
213	C₂×C₅	#1957	1	1	yes	characteristic
214	C₂×C₅	#2138	1	2	yes	—
215	C₂×C₅	#2242	1	2	yes	—
216	C₂×C₅	#2249	1	2	yes	—
217	C₂³	—	4	2	no	—
218	C₂³	—	8	1	no	—
219	C₂³	—	8	1	no	—
220	C₂³	—	8	4	no	—
221	C₂³	C₂×#686	1	1	yes	characteristic
222	Q₈	—	4	2	no	—
223	Q₈	—	4	2	no	—
224	Q₈	—	8	1	no	—
225	Q₈	—	8	1	no	—
226	D₄	—	4	2	no	—
227	D₄	—	4	2	no	—
228	D₄	—	8	1	no	—
229	D₄	—	8	1	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	8	1	no	—
233	C₂×C₄	—	8	1	no	—
234	C₂×C₄	—	40	2	no	—
235	C₂×C₄	—	40	4	no	—
236	C₂×C₄	#3964	1	4	yes	—
237	C₂×C₄	C₂×#686	1	1	yes	characteristic
238	C₂×C₄	C₂×#687	1	1	yes	characteristic
239	C₂×C₄	C₂×#687	1	1	yes	characteristic
240	C₂×C₄	#4082	1	1	yes	characteristic
241	C₂×C₄	#4086	1	1	yes	characteristic
242	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
243	C₈	#3964	1	4	yes	—
244	C₈	D₄×Dic₅	1	2	yes	—
245	C₈	#4084	1	2	yes	—
246	C₅	#13204	1	1	yes	characteristic, complemented, Sylow
247	C₂²	—	8	2	no	—
248	C₂²	—	8	4	no	—
249	C₂²	—	16	2	no	—
250	C₂²	—	16	2	no	—
251	C₂²	#65359	1	1	yes	characteristic
252	C₂²	#65362	1	1	yes	characteristic
253	C₂²	#65483	1	4	yes	—
254	C₂²	C₂×#3964	1	1	yes	characteristic
255	C₄	—	2	4	no	—
256	C₄	—	8	2	no	—
257	C₄	—	80	2	no	—
258	C₄	#65359	1	1	yes	characteristic
259	C₄	#65365	1	1	yes	characteristic
260	C₄	C₂×#3964	1	1	yes	characteristic
261	C₄	#65510	1	1	yes	characteristic
262	C₂	—	16	2	no	—
263	C₂	#10602111	1	2	yes	—
264	C₂	#10602113	1	2	yes	—
265	C₂	#10602557	1	1	yes	characteristic
266	C₂	#10602870	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	#13204
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. #10600559	C₂	semidirect
5. C₂×C₄×#672	C₂	semidirect
11. C₅×#495	C₄	semidirect
96. #495	Dic₅	semidirect
246. C₅	#13204	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
#10600559	2
C₂×C₄×#672	2
#13204	5