Subgroups — #11987751

Order 1280.

432 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10605038	C₂	1	1	yes	maximal, characteristic
2	C₅×#1744	C₂	1	1	yes	maximal, characteristic
3	#10600184	C₂	1	1	yes	maximal, characteristic
4	#10597968	C₂	1	4	yes	maximal
5	C₂×#4016	—	2	1	no	—
6	C₂×C₄×C₅×C₈	—	2	1	no	—
7	C₄³×C₅	C₂²	1	1	yes	characteristic
8	#65225	—	2	1	no	—
9	#65224	C₄	1	1	yes	characteristic
10	#65224	C₄	1	1	yes	characteristic
11	#65217	—	2	1	no	—
12	#65217	—	2	1	no	—
13	#65217	C₄	1	2	yes	—
14	#65217	C₂²	1	2	yes	—
15	C₂×C₄⋊#168	—	2	1	no	—
16	C₂×#3935	—	2	1	no	—
17	C₅×#336	—	2	1	no	—
18	C₅×#336	C₂²	1	2	yes	—
19	#64744	C₂²	1	2	yes	—
20	#7680	—	5	1	no	maximal, Sylow
21	C₂²×C₅×C₈	—	2	2	no	—
22	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
26	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊Dic₅	—	2	1	no	—
30	C₂×C₄⋊Dic₅	—	2	1	no	—
31	C₂×C₄⋊Dic₅	—	2	1	no	—
32	C₂×C₄⋊Dic₅	—	2	1	no	—
33	C₂×C₄⋊Dic₅	C₂×C₄	1	2	yes	—
34	C₂×C₄⋊Dic₅	C₂×C₄	1	2	yes	—
35	C₂×#683	—	2	2	no	—
36	C₂×#683	—	2	2	no	—
37	C₂²×#168	—	2	2	no	—
38	#4016	—	4	2	no	—
39	#4016	—	4	2	no	—
40	C₄×C₅×C₈	—	2	4	no	—
41	#3963	—	2	2	no	—
42	#3963	—	2	2	no	—
43	C₄⋊#168	—	2	4	no	—
44	#3935	—	4	2	no	—
45	#2859	—	5	1	no	—
46	#1767	—	5	1	no	—
47	#1744	D₅	1	1	yes	characteristic
48	#1271	—	5	4	no	—
49	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
56	C₂²×Dic₅	—	2	2	no	—
57	C₂²×Dic₅	—	2	2	no	—
58	C₂×C₅×C₈	—	2	4	no	—
59	C₂×C₅×C₈	—	4	2	no	—
60	C₂×C₅×C₈	—	4	2	no	—
61	C₄²×C₅	—	2	2	no	—
62	C₄²×C₅	—	2	2	no	—
63	C₄²×C₅	—	2	2	no	—
64	C₄²×C₅	—	2	2	no	—
65	C₄²×C₅	—	2	2	no	—
66	C₄²×C₅	—	4	1	no	—
67	C₄²×C₅	D₈	1	2	yes	—
68	C₄²×C₅	SD₁₆	1	2	yes	—
69	C₄⋊Dic₅	—	2	4	no	—
70	C₄⋊Dic₅	—	4	2	no	—
71	C₄⋊Dic₅	—	4	2	no	—
72	C₄⋊Dic₅	—	4	2	no	—
73	C₄⋊Dic₅	—	4	2	no	—
74	C₄⋊Dic₅	—	4	2	no	—
75	#683	—	4	4	no	—
76	#683	—	4	2	no	—
77	#683	—	4	2	no	—
78	C₂×#168	—	2	4	no	—
79	C₂×#168	—	4	2	no	—
80	C₂×#168	—	4	2	no	—
81	C₂×#125	—	10	1	no	—
82	C₂×C₄⋊C₈	—	10	1	no	—
83	C₂×#97	—	10	1	no	—
84	C₂×C₄×C₈	—	2	1	no	—
85	#389	—	5	2	no	—
86	#389	—	5	2	no	—
87	#389	—	10	1	no	—
88	#389	—	10	1	no	—
89	#384	—	5	1	no	—
90	#384	—	5	1	no	—
91	#382	—	10	1	no	—
92	C₄³	C₂×D₅	1	1	yes	characteristic
93	#336	—	2	1	no	—
94	#336	—	5	2	no	—
95	#336	C₂×D₅	1	2	yes	—
96	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₅	—	2	1	no	—
98	C₂×C₄×C₅	—	2	1	no	—
99	C₂×C₄×C₅	—	2	1	no	—
100	C₂×C₄×C₅	—	2	1	no	—
101	C₂×C₄×C₅	—	2	1	no	—
102	C₂×C₄×C₅	—	2	1	no	—
103	C₂×C₄×C₅	—	2	1	no	—
104	C₂×C₄×C₅	—	2	1	no	—
105	C₂×C₄×C₅	—	2	1	no	—
106	C₂×C₄×C₅	—	2	1	no	—
107	C₂×C₄×C₅	—	2	1	no	—
108	C₂×C₄×C₅	—	2	1	no	—
109	C₂×C₄×C₅	—	2	1	no	—
110	C₂×C₄×C₅	#101	1	2	yes	—
111	C₂×C₄×C₅	#102	1	2	yes	—
112	C₂×C₄×C₅	#102	1	2	yes	—
113	C₂×C₄×C₅	#104	1	2	yes	—
114	C₂×C₄×C₅	#104	1	2	yes	—
115	C₂×C₄×C₅	#131	1	2	yes	—
116	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
117	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
118	C₂×C₄×C₅	#137	1	2	yes	—
119	C₂×Dic₅	—	4	2	no	—
120	C₂×Dic₅	—	4	2	no	—
121	C₂×Dic₅	—	4	2	no	—
122	C₂×Dic₅	—	4	2	no	—
123	C₂×Dic₅	—	4	2	no	—
124	C₂×Dic₅	—	4	2	no	—
125	C₅×C₈	—	4	4	no	—
126	#168	—	4	4	no	—
127	C₂²×C₈	—	2	2	no	—
128	C₂²×C₈	—	10	2	no	—
129	#125	—	20	2	no	—
130	#125	—	20	2	no	—
131	C₂×C₄⋊C₄	—	5	2	no	—
132	C₂×C₄⋊C₄	—	5	2	no	—
133	C₂×C₄⋊C₄	—	10	1	no	—
134	C₂×C₄⋊C₄	—	10	1	no	—
135	C₂×C₄⋊C₄	—	10	2	no	—
136	C₂×C₄⋊C₄	—	10	1	no	—
137	C₂×C₄⋊C₄	—	10	1	no	—
138	C₂×C₄⋊C₄	—	10	2	no	—
139	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
140	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
141	C₂×C₄²	D₂₀	1	1	yes	characteristic
142	C₂×C₄²	D₂₀	1	1	yes	characteristic
143	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
144	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
145	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
146	C₄⋊C₈	—	10	4	no	—
147	#97	—	20	2	no	—
148	C₄×C₈	—	2	4	no	—
149	#95	—	10	2	no	—
150	#95	—	10	2	no	—
151	C₂²×C₅	#408	1	1	yes	characteristic
152	C₂²×C₅	#410	1	1	yes	characteristic
153	C₂²×C₅	C₂×#101	1	1	yes	characteristic
154	C₂²×C₅	C₂×#102	1	1	yes	characteristic
155	C₂²×C₅	#419	1	1	yes	characteristic
156	C₂²×C₅	C₂×#104	1	1	yes	characteristic
157	C₂²×C₅	C₂×#104	1	1	yes	characteristic
158	C₄×C₅	—	2	2	no	—
159	C₄×C₅	—	2	2	no	—
160	C₄×C₅	—	2	2	no	—
161	C₄×C₅	—	2	1	no	—
162	C₄×C₅	—	2	1	no	—
163	C₄×C₅	—	2	2	no	—
164	C₄×C₅	—	4	1	no	—
165	C₄×C₅	#330	1	4	yes	—
166	Dic₅	—	8	2	no	—
167	Dic₅	—	8	2	no	—
168	C₂²×C₄	—	10	2	no	—
169	C₂²×C₄	—	10	2	no	—
170	C₂²×C₄	#685	1	1	yes	characteristic
171	C₂²×C₄	#685	1	1	yes	characteristic
172	C₂²×C₄	#685	1	1	yes	characteristic
173	C₂²×C₄	#685	1	1	yes	characteristic
174	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
175	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
176	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
177	C₂×C₈	—	2	4	no	—
178	C₂×C₈	—	4	2	no	—
179	C₂×C₈	—	4	2	no	—
180	C₂×C₈	—	10	4	no	—
181	C₂×C₈	—	20	2	no	—
182	C₂×C₈	—	20	2	no	—
183	C₄⋊C₄	—	10	4	no	—
184	C₄⋊C₄	—	20	2	no	—
185	C₄⋊C₄	—	20	2	no	—
186	C₄⋊C₄	—	20	4	no	—
187	C₄⋊C₄	—	20	2	no	—
188	C₄⋊C₄	—	20	2	no	—
189	C₄⋊C₄	—	20	2	no	—
190	C₄⋊C₄	—	20	2	no	—
191	C₄⋊C₄	—	20	2	no	—
192	C₄²	—	2	2	no	—
193	C₄²	—	2	2	no	—
194	C₄²	—	2	2	no	—
195	C₄²	—	2	2	no	—
196	C₄²	—	2	2	no	—
197	C₄²	—	4	1	no	—
198	C₄²	#677	1	2	yes	—
199	C₄²	D₄₀	1	2	yes	—
200	C₂×C₅	#1470	1	1	yes	characteristic
201	C₂×C₅	#1485	1	1	yes	characteristic
202	C₂×C₅	#1492	1	1	yes	characteristic
203	C₂×C₅	#1502	1	1	yes	characteristic
204	C₂×C₅	#1506	1	1	yes	characteristic
205	C₂×C₅	C₂×#330	1	1	yes	characteristic
206	C₂×C₅	#1546	1	1	yes	characteristic
207	C₂³	C₂×#685	1	1	yes	characteristic
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	20	2	no	—
222	C₂×C₄	—	20	2	no	—
223	C₂×C₄	—	20	2	no	—
224	C₂×C₄	—	20	2	no	—
225	C₂×C₄	—	20	2	no	—
226	C₂×C₄	—	20	2	no	—
227	C₂×C₄	#3937	1	2	yes	—
228	C₂×C₄	#3953	1	2	yes	—
229	C₂×C₄	#3953	1	2	yes	—
230	C₂×C₄	#3956	1	2	yes	—
231	C₂×C₄	#3957	1	2	yes	—
232	C₂×C₄	#4047	1	2	yes	—
233	C₂×C₄	C₂×#677	1	1	yes	characteristic
234	C₂×C₄	C₂×D₄₀	1	1	yes	characteristic
235	C₂×C₄	#4097	1	2	yes	—
236	C₈	—	4	4	no	—
237	C₈	—	20	4	no	—
238	C₅	#7680	1	1	yes	characteristic, complemented, Sylow
239	C₂²	C₂×#3937	1	1	yes	characteristic
240	C₂²	#65293	1	1	yes	characteristic
241	C₂²	#65295	1	1	yes	characteristic
242	C₂²	#65399	1	1	yes	characteristic
243	C₂²	C₂×#3953	1	1	yes	characteristic
244	C₂²	C₂×#3956	1	1	yes	characteristic
245	C₂²	C₂×#3957	1	1	yes	characteristic
246	C₄	—	2	2	no	—
247	C₄	—	2	2	no	—
248	C₄	—	2	2	no	—
249	C₄	—	2	2	no	—
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	4	1	no	—
253	C₄	—	40	2	no	—
254	C₄	—	40	2	no	—
255	C₄	#64685	1	4	yes	—
256	C₂	#10600226	1	1	yes	characteristic
257	C₂	#10600528	1	1	yes	characteristic
258	C₂	C₂×#64685	1	1	yes	characteristic
259	C₂	#10601562	1	1	yes	characteristic
260	C₂	#10601571	1	1	yes	characteristic
261	C₂	#10601580	1	1	yes	characteristic
262	C₂	#10601582	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#685
commutator	C₂²×C₄×C₅	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#1744	C₂
socle, prime radical	C₂³×C₅	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	#1744	D₅
hypocenter	C₅	#7680
LCS	G ▷ C₂²×C₄×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1744

Complements

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

N	H ≅ G/N	type
238. C₅	#7680	semidirect

Maximal subgroups

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

subgroup	index
#10605038	2
C₅×#1744	2
#10600184	2
#10597968	2
#7680	5