Subgroups — #12016668

Order 1280.

526 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#467	C₂	1	1	yes	maximal, characteristic
2	C₂³×#672	C₂	1	1	yes	maximal, characteristic
3	C₂×#64784	C₂	1	2	yes	maximal, complemented
4	#10602764	C₂	1	1	yes	maximal, characteristic
5	#10598213	C₂	1	2	yes	maximal, complemented
6	C₂²×C₅×Q₁₆	C₂²	1	1	yes	characteristic
7	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₅×C₂²⋊Q₈	—	2	1	no	—
9	C₅×#467	—	2	2	no	—
10	C₅×#467	C₄	1	4	yes	complemented
11	C₂×C₅×#107	C₂²	1	1	yes	characteristic
12	C₂×C₅×#103	—	2	1	no	—
13	#65501	—	2	1	no	—
14	C₂×#3950	C₂²	1	2	yes	—
15	C₂²×#672	—	2	4	no	—
16	C₂²×#672	—	2	2	no	—
17	C₂²×#672	C₂²	1	2	yes	—
18	#64784	—	2	4	no	—
19	#64771	—	2	1	no	—
20	#13060	—	5	1	no	maximal, Sylow
21	C₂²×C₅×Q₈	—	2	1	no	—
22	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
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	C₂²×C₅×C₈	—	2	2	no	—
28	C₂²×C₅×C₈	—	2	2	no	—
29	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
30	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
31	C₂²×C₅×C₈	D₄	1	2	yes	—
32	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
33	C₅×C₂²⋊Q₈	—	2	4	no	—
34	C₅×C₂²⋊Q₈	—	4	2	no	—
35	C₂×C₅×C₄⋊C₄	—	2	1	no	—
36	C₂×C₅×C₄⋊C₄	—	4	1	no	—
37	C₂×C₅×C₂²⋊C₄	—	4	1	no	—
38	C₂×C₂²⋊Dic₅	—	4	1	no	—
39	C₂×C₄⋊Dic₅	—	2	2	no	—
40	C₅×#107	—	2	1	no	—
41	C₅×#107	C₂×C₄	1	2	yes	—
42	C₅×#103	—	2	4	no	—
43	#3963	—	4	1	no	—
44	#3950	—	2	2	no	—
45	#3950	—	2	2	no	—
46	C₂×#672	—	2	8	no	—
47	C₂×#672	—	2	2	no	—
48	C₂×#672	—	2	2	no	—
49	C₂×#672	D₄	1	4	yes	—
50	C₂³×C₁₆	—	5	1	no	—
51	C₂×#467	D₅	1	1	yes	characteristic
52	C₂×#358	—	5	2	no	—
53	#1808	—	5	1	no	—
54	#1380	—	5	2	no	—
55	C₂⁴×C₅	D₈	1	1	yes	characteristic
56	C₂×C₅×Q₈	—	2	2	no	—
57	C₂×C₅×Q₈	—	4	1	no	—
58	C₂×C₅×Q₈	—	4	2	no	—
59	C₂×C₅×Q₈	—	4	2	no	—
60	C₂²×C₄×C₅	—	2	2	no	—
61	C₂²×C₄×C₅	—	2	2	no	—
62	C₂²×C₄×C₅	—	4	1	no	—
63	C₂²×C₄×C₅	—	4	1	no	—
64	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
66	C₂²×C₄×C₅	SD₁₆	1	2	yes	—
67	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
68	C₂²×Dic₅	—	4	2	no	—
69	C₅×Q₁₆	—	2	4	no	—
70	C₅×Q₁₆	—	4	1	no	—
71	C₅×Q₁₆	—	4	1	no	—
72	C₂×C₅×C₈	—	2	2	no	—
73	C₂×C₅×C₈	—	2	4	no	—
74	C₂×C₅×C₈	—	2	2	no	—
75	C₂×C₅×C₈	—	2	2	no	—
76	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
77	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
79	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
80	C₂×C₅×C₈	#41	1	2	yes	—
81	C₅×C₄⋊C₄	—	2	2	no	—
82	C₅×C₄⋊C₄	—	4	4	no	—
83	C₅×C₄⋊C₄	—	4	1	no	—
84	C₅×C₂²⋊C₄	—	4	4	no	—
85	C₂²⋊Dic₅	—	8	2	no	—
86	C₄⋊Dic₅	—	4	2	no	—
87	C₄⋊Dic₅	—	4	2	no	—
88	#672	—	2	4	no	—
89	C₂²×Q₁₆	C₂×D₅	1	1	yes	characteristic
90	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
91	C₂×C₂²⋊Q₈	—	2	1	no	—
92	C₂²×C₁₆	—	5	2	no	—
93	C₂²×C₁₆	—	10	4	no	—
94	C₂²×C₁₆	—	10	2	no	—
95	#467	—	2	2	no	—
96	#467	Dic₅	1	4	yes	complemented
97	C₂×#107	—	5	2	no	—
98	C₂×#107	C₂×D₅	1	1	yes	characteristic
99	C₂×#103	—	2	1	no	—
100	#380	—	10	1	no	—
101	#358	—	10	4	no	—
102	#340	—	10	1	no	—
103	C₂³×C₅	—	2	2	no	—
104	C₂³×C₅	—	2	2	no	—
105	C₂³×C₅	#102	1	2	yes	—
106	C₂³×C₅	SD₃₂	1	2	yes	—
107	C₂³×C₅	Q₃₂	1	2	yes	—
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	2	no	—
112	C₂×C₄×C₅	—	2	2	no	—
113	C₂×C₄×C₅	—	2	2	no	—
114	C₂×C₄×C₅	—	2	2	no	—
115	C₂×C₄×C₅	—	2	4	no	—
116	C₂×C₄×C₅	—	4	2	no	—
117	C₂×C₄×C₅	—	4	2	no	—
118	C₂×C₄×C₅	—	8	2	no	—
119	C₂×C₄×C₅	—	8	2	no	—
120	C₂×C₄×C₅	#102	1	2	yes	—
121	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₅	#121	1	2	yes	—
123	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
124	C₂×C₄×C₅	#135	1	2	yes	—
125	C₂×Dic₅	—	8	2	no	—
126	C₂×Dic₅	—	8	2	no	—
127	C₂×Dic₅	—	8	2	no	—
128	C₅×C₈	—	2	2	no	—
129	C₅×C₈	C₄×D₄	1	2	yes	—
130	C₅×C₈	#120	1	1	yes	characteristic
131	C₅×C₈	#123	1	1	yes	characteristic
132	C₂²×Q₈	—	2	1	no	—
133	C₂³×C₄	C₂²⋊D₅	1	1	yes	characteristic
134	C₂×Q₁₆	—	2	2	no	—
135	C₂×Q₁₆	—	2	2	no	—
136	C₂×Q₁₆	—	2	1	no	—
137	C₂×Q₁₆	C₂×Dic₅	1	2	yes	—
138	C₂²×C₈	—	2	2	no	—
139	C₂²×C₈	—	2	2	no	—
140	C₂²×C₈	C₂×Dic₅	1	2	yes	—
141	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
142	C₂²×C₈	C₂²⋊D₅	1	2	yes	—
143	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
144	C₂²⋊Q₈	—	2	4	no	—
145	C₂²⋊Q₈	—	4	2	no	—
146	C₂×C₄⋊C₄	—	2	1	no	—
147	C₂×C₄⋊C₄	—	4	1	no	—
148	C₂×C₄⋊C₄	—	10	2	no	—
149	C₂×C₂²⋊C₄	—	4	1	no	—
150	C₂×C₂²⋊C₄	—	20	1	no	—
151	C₂×C₁₆	—	5	4	no	—
152	C₂×C₁₆	—	10	8	no	—
153	C₂×C₁₆	—	10	2	no	—
154	C₂×C₁₆	—	10	2	no	—
155	#107	—	2	1	no	—
156	#107	—	10	2	no	—
157	#107	—	10	2	no	—
158	#107	C₂×Dic₅	1	2	yes	—
159	#103	—	2	4	no	—
160	#95	—	20	1	no	—
161	C₂²×C₅	—	2	2	no	—
162	C₂²×C₅	—	2	2	no	—
163	C₂²×C₅	—	2	4	no	—
164	C₂²×C₅	—	2	4	no	—
165	C₂²×C₅	#358	1	4	yes	complemented
166	C₂²×C₅	C₂×#102	1	1	yes	characteristic
167	C₂²×C₅	#466	1	2	yes	—
168	C₂²×C₅	C₂×SD₃₂	1	1	yes	characteristic
169	C₂²×C₅	C₂×Q₃₂	1	1	yes	characteristic
170	C₂²×C₅	#508	1	2	yes	—
171	C₄×C₅	—	2	2	no	—
172	C₄×C₅	—	8	2	no	—
173	C₄×C₅	—	8	2	no	—
174	C₄×C₅	#386	1	1	yes	characteristic
175	C₄×C₅	#416	1	1	yes	characteristic
176	C₄×C₅	#465	1	2	yes	—
177	Dic₅	—	16	2	no	—
178	C₂⁴	#686	1	1	yes	characteristic
179	C₂×Q₈	—	2	2	no	—
180	C₂×Q₈	—	4	1	no	—
181	C₂×Q₈	—	4	2	no	—
182	C₂×Q₈	—	4	2	no	—
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	—	2	2	no	—
185	C₂²×C₄	—	4	1	no	—
186	C₂²×C₄	—	4	1	no	—
187	C₂²×C₄	—	20	2	no	—
188	C₂²×C₄	#686	1	1	yes	characteristic
189	C₂²×C₄	#687	1	2	yes	—
190	C₂²×C₄	C₂²⋊Dic₅	1	2	yes	—
191	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
192	Q₁₆	—	2	4	no	—
193	Q₁₆	—	4	1	no	—
194	Q₁₆	—	4	1	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	2	4	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	C₂²⋊Dic₅	1	2	yes	—
200	C₂×C₈	D₄×D₅	1	2	yes	—
201	C₂×C₈	#711	1	2	yes	—
202	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
203	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
204	C₄⋊C₄	—	2	2	no	—
205	C₄⋊C₄	—	4	4	no	—
206	C₄⋊C₄	—	4	1	no	—
207	C₄⋊C₄	—	20	2	no	—
208	C₄⋊C₄	—	20	2	no	—
209	C₂²⋊C₄	—	4	4	no	—
210	C₂²⋊C₄	—	40	2	no	—
211	C₁₆	—	10	4	no	—
212	C₂×C₅	—	2	4	no	—
213	C₂×C₅	#1893	1	1	yes	characteristic
214	C₂×C₅	C₂×#358	1	1	yes	characteristic
215	C₂×C₅	#2138	1	1	yes	characteristic
216	C₂×C₅	#2216	1	2	yes	—
217	C₂×C₅	#2217	1	2	yes	—
218	C₂³	—	2	2	no	—
219	C₂³	—	2	2	no	—
220	C₂³	#3960	1	2	yes	—
221	C₂³	#3961	1	2	yes	—
222	C₂³	#3964	1	2	yes	—
223	C₂³	C₂×#686	1	1	yes	characteristic
224	Q₈	—	4	2	no	—
225	Q₈	—	4	2	no	—
226	Q₈	—	8	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	8	2	no	—
234	C₂×C₄	—	8	2	no	—
235	C₂×C₄	—	40	2	no	—
236	C₂×C₄	—	40	2	no	—
237	C₂×C₄	—	40	2	no	—
238	C₂×C₄	#3964	1	2	yes	—
239	C₂×C₄	C₂×#687	1	1	yes	characteristic
240	C₂×C₄	#4085	1	2	yes	—
241	C₂×C₄	#4096	1	2	yes	—
242	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
243	C₈	—	2	2	no	—
244	C₈	D₄×Dic₅	1	2	yes	—
245	C₈	#4081	1	1	yes	characteristic
246	C₈	#4083	1	1	yes	characteristic
247	C₅	#13060	1	1	yes	characteristic, complemented, Sylow
248	C₂²	—	2	2	no	—
249	C₂²	—	2	4	no	—
250	C₂²	—	2	4	no	—
251	C₂²	—	2	2	no	—
252	C₂²	#64784	1	4	yes	complemented
253	C₂²	#65327	1	2	yes	—
254	C₂²	C₂×#3960	1	1	yes	characteristic
255	C₂²	C₂×#3961	1	1	yes	characteristic
256	C₂²	#65483	1	2	yes	—
257	C₂²	C₂×#3964	1	1	yes	characteristic
258	C₄	—	2	2	no	—
259	C₄	—	8	2	no	—
260	C₄	—	8	2	no	—
261	C₄	—	80	2	no	—
262	C₄	#65330	1	2	yes	—
263	C₄	#65509	1	1	yes	characteristic
264	C₄	#65522	1	1	yes	characteristic
265	C₂	—	2	4	no	—
266	C₂	#10601874	1	2	yes	—
267	C₂	#10601875	1	2	yes	—
268	C₂	#10602474	1	1	yes	characteristic
269	C₂	C₂×#64784	1	1	yes	characteristic
270	C₂	#10602870	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#686
commutator	C₂×C₅×C₈	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₂×C₅×#467	C₂
socle	C₂³×C₅	C₂×D₈
prime radical	C₂⁴×C₅	D₈
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	C₂×#467	D₅
hypocenter	C₅	#13060
LCS	G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#467

Complements

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

N	H ≅ G/N	type
3. C₂×#64784	C₂	semidirect
5. #10598213	C₂	semidirect
10. C₅×#467	C₄	semidirect
96. #467	Dic₅	semidirect
165. C₂²×C₅	#358	semidirect
247. C₅	#13060	semidirect
252. C₂²	#64784	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#467	2
C₂³×#672	2
C₂×#64784	2
#10602764	2
#10598213	2
#13060	5