Subgroups — #12087254

Order 1280.

750 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#494	C₂	1	1	yes	maximal, characteristic
2	#10614955	C₂	1	2	yes	maximal
3	#10614938	C₂	1	2	yes	maximal
4	#10610206	C₂	1	1	yes	maximal, characteristic
5	#10602095	C₂	1	4	yes	maximal, complemented
6	#10602091	C₂	1	2	yes	maximal, complemented
7	#10602089	C₂	1	2	yes	maximal, complemented
8	C₈×#681	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₅×Q₁₆	—	2	2	no	—
10	C₂×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
11	#65804	C₂²	1	2	yes	—
12	C₅×#494	—	2	2	no	—
13	C₅×#494	C₂²	1	2	yes	—
14	C₅×#494	C₂²	1	2	yes	—
15	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
16	#65405	—	2	2	no	—
17	#65382	C₂²	1	4	yes	—
18	#65380	—	2	4	no	—
19	#65369	—	2	4	no	—
20	#65368	C₂²	1	4	yes	—
21	#65367	C₂²	1	4	yes	—
22	#65367	C₂²	1	4	yes	—
23	#65259	—	2	4	no	—
24	#65253	—	2	4	no	—
25	#65214	C₂²	1	1	yes	characteristic
26	C₄×#681	C₂²	1	1	yes	characteristic
27	#64988	C₂²	1	1	yes	characteristic
28	C₄×Dic₂₀	—	2	2	no	—
29	#64971	C₂²	1	1	yes	characteristic
30	#64970	C₂²	1	2	yes	—
31	#64674	C₂²	1	2	yes	—
32	C₈×#168	C₂²	1	2	yes	—
33	#19632	—	5	1	no	maximal, Sylow
34	C₂²×C₅×Q₈	—	2	2	no	—
35	C₂×C₅×Q₁₆	—	2	4	no	—
36	C₂×C₅×Q₁₆	—	2	4	no	—
37	C₂×C₅×Q₁₆	—	4	8	no	—
38	C₂²×C₅×C₈	D₄	1	2	yes	—
39	C₅×C₄⋊Q₈	—	2	2	no	—
40	C₅×C₄⋊Q₈	—	2	2	no	—
41	C₅×C₄⋊Q₈	—	2	2	no	—
42	C₅×C₄⋊Q₈	C₂³	1	4	yes	—
43	C₂×C₅×C₄⋊C₄	—	2	2	no	—
44	C₂×C₅×C₄⋊C₄	—	2	2	no	—
45	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
46	#4072	—	2	2	no	—
47	#4070	—	2	2	no	—
48	#4070	—	2	2	no	—
49	C₂×#681	D₄	1	2	yes	—
50	C₂×Dic₂₀	—	2	2	no	—
51	#4016	—	2	2	no	—
52	#4015	C₂³	1	2	yes	—
53	#4015	C₂³	1	2	yes	—
54	C₄×Dic₁₀	—	2	2	no	—
55	C₄×Dic₁₀	—	2	2	no	—
56	C₄×C₅×C₈	D₄	1	2	yes	—
57	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
58	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
59	C₂²⋊#168	D₄	1	2	yes	—
60	#3950	—	2	2	no	—
61	#3948	—	2	4	no	—
62	#3942	—	2	8	no	—
63	#3940	—	2	8	no	—
64	#3939	—	2	8	no	—
65	#3939	—	2	8	no	—
66	C₄⋊#168	D₄	1	2	yes	—
67	#3935	D₄	1	2	yes	—
68	C₄×#168	C₂³	1	2	yes	—
69	C₄×#168	C₂³	1	2	yes	—
70	#3230	—	5	2	no	—
71	#3170	—	5	1	no	—
72	#3165	—	5	2	no	—
73	C₂×#494	D₅	1	1	yes	characteristic
74	#1723	—	5	4	no	—
75	#1713	—	5	2	no	—
76	#1707	—	5	2	no	—
77	C₈×M₁₆	—	5	1	no	—
78	C₂×C₅×Q₈	—	2	4	no	—
79	C₂×C₅×Q₈	—	4	8	no	—
80	C₂×C₅×Q₈	—	4	2	no	—
81	C₂²×C₄×C₅	—	4	2	no	—
82	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
85	C₂×Dic₁₀	—	2	2	no	—
86	C₂×Dic₁₀	—	2	2	no	—
87	C₅×Q₁₆	—	4	8	no	—
88	C₅×Q₁₆	—	4	8	no	—
89	C₂×C₅×C₈	—	2	4	no	—
90	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
91	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
92	C₅×C₄⋊C₄	—	2	4	no	—
93	C₅×C₄⋊C₄	—	2	4	no	—
94	C₅×C₄⋊C₄	—	4	2	no	—
95	C₅×C₄⋊C₄	—	4	2	no	—
96	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₅	C₂⁴	1	1	yes	characteristic
100	C₂²⋊Dic₅	—	4	2	no	—
101	C₄⋊Dic₅	—	2	2	no	—
102	C₄⋊Dic₅	—	2	2	no	—
103	C₄⋊Dic₅	—	2	2	no	—
104	C₄⋊Dic₅	—	2	2	no	—
105	#683	—	4	2	no	—
106	#683	—	4	2	no	—
107	C₄×Dic₅	—	4	2	no	—
108	#681	Q₁₆	1	8	yes	complemented
109	C₂×#168	C₂×D₄	1	4	yes	—
110	C₂×#168	C₂×D₄	1	4	yes	—
111	Dic₂₀	—	4	4	no	—
112	C₂²×Q₁₆	—	2	2	no	—
113	#533	—	5	2	no	—
114	C₂×C₄⋊Q₈	C₂×D₅	1	2	yes	—
115	#501	—	10	4	no	—
116	#500	—	5	4	no	—
117	#500	—	5	4	no	—
118	#500	—	5	1	no	—
119	#498	—	5	4	no	—
120	#494	—	2	2	no	—
121	#494	—	5	1	no	—
122	#494	C₂×D₅	1	2	yes	—
123	#494	C₂×D₅	1	2	yes	—
124	#487	—	5	4	no	—
125	#487	—	5	2	no	—
126	#484	—	10	4	no	—
127	#477	—	10	4	no	—
128	#467	—	10	2	no	—
129	C₄×Q₁₆	—	10	2	no	—
130	#431	—	5	1	no	—
131	#428	—	10	4	no	—
132	C₄×M₁₆	—	5	1	no	—
133	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
134	#322	—	5	2	no	—
135	C₈²	—	5	2	no	—
136	C₂³×C₅	#127	1	1	yes	characteristic
137	C₅×Q₈	—	4	8	no	—
138	C₅×Q₈	—	8	4	no	—
139	C₂×C₄×C₅	—	2	2	no	—
140	C₂×C₄×C₅	—	2	2	no	—
141	C₂×C₄×C₅	—	4	4	no	—
142	C₂×C₄×C₅	—	8	4	no	—
143	C₂×C₄×C₅	#127	1	1	yes	characteristic
144	C₂×C₄×C₅	#127	1	1	yes	characteristic
145	C₂×C₄×C₅	#127	1	1	yes	characteristic
146	C₂×C₄×C₅	C₂×Q₁₆	1	4	yes	—
147	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
149	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
150	C₂×Dic₅	—	4	2	no	—
151	C₂×Dic₅	—	4	2	no	—
152	Dic₁₀	—	4	4	no	—
153	Dic₁₀	—	4	4	no	—
154	C₅×C₈	—	2	2	no	—
155	C₅×C₈	AGL(1,ℤ/8)	1	4	yes	—
156	#168	—	2	4	no	—
157	#168	C₂×Q₁₆	1	8	yes	complemented
158	C₂²×Q₈	—	2	2	no	—
159	C₂×Q₁₆	—	2	4	no	—
160	C₂×Q₁₆	—	2	4	no	—
161	C₂×Q₁₆	—	4	8	no	—
162	C₂×Q₁₆	—	10	2	no	—
163	C₂×M₁₆	—	5	2	no	—
164	C₂²×C₈	C₂²⋊D₅	1	2	yes	—
165	C₄⋊Q₈	—	2	2	no	—
166	C₄⋊Q₈	—	2	2	no	—
167	C₄⋊Q₈	—	2	2	no	—
168	C₄⋊Q₈	—	5	2	no	—
169	C₄⋊Q₈	—	5	2	no	—
170	C₄⋊Q₈	C₂²×D₅	1	4	yes	—
171	#125	—	10	2	no	—
172	C₂²⋊Q₈	—	10	2	no	—
173	C₂²⋊Q₈	—	10	2	no	—
174	C₄×Q₈	—	10	2	no	—
175	C₄×Q₈	—	10	2	no	—
176	#117	—	10	2	no	—
177	C₂×C₄⋊C₄	—	2	2	no	—
178	C₂×C₄⋊C₄	—	2	2	no	—
179	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
180	#107	—	10	8	no	—
181	#107	—	10	8	no	—
182	#107	—	10	2	no	—
183	#106	—	10	8	no	—
184	C₄⋊C₈	—	5	2	no	—
185	#103	—	10	8	no	—
186	#103	—	10	4	no	—
187	C₂²⋊C₈	—	5	2	no	—
188	#97	—	5	2	no	—
189	C₄×C₈	—	5	2	no	—
190	C₄×C₈	—	5	2	no	—
191	C₄×C₈	C₂²⋊D₅	1	2	yes	—
192	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
193	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
194	C₂²×C₅	—	2	1	no	—
195	C₂²×C₅	—	2	1	no	—
196	C₂²×C₅	#494	1	2	yes	—
197	C₂²×C₅	C₂×#127	1	1	yes	characteristic
198	C₄×C₅	—	2	1	no	—
199	C₄×C₅	—	2	1	no	—
200	C₄×C₅	—	8	4	no	—
201	C₄×C₅	#494	1	2	yes	—
202	C₄×C₅	C₂²×Q₁₆	1	2	yes	—
203	C₄×C₅	C₂×AGL(1,ℤ/8)	1	2	yes	—
204	C₄×C₅	#578	1	2	yes	—
205	Dic₅	—	8	2	no	—
206	Dic₅	—	8	2	no	—
207	C₂×Q₈	—	2	4	no	—
208	C₂×Q₈	—	4	8	no	—
209	C₂×Q₈	—	4	2	no	—
210	C₂×Q₈	—	10	2	no	—
211	C₂×Q₈	—	10	2	no	—
212	C₂²×C₄	—	4	2	no	—
213	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
214	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
215	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
216	Q₁₆	—	4	8	no	—
217	Q₁₆	—	4	8	no	—
218	Q₁₆	—	20	4	no	—
219	M₁₆	—	5	8	no	—
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	5	4	no	—
222	C₂×C₈	—	5	4	no	—
223	C₂×C₈	C₂×C₂²⋊D₅	1	2	yes	—
224	C₂×C₈	C₂×C₂²⋊D₅	1	2	yes	—
225	C₄⋊C₄	—	2	4	no	—
226	C₄⋊C₄	—	2	4	no	—
227	C₄⋊C₄	—	4	2	no	—
228	C₄⋊C₄	—	4	2	no	—
229	C₄⋊C₄	—	10	2	no	—
230	C₄⋊C₄	—	10	2	no	—
231	C₄⋊C₄	—	10	2	no	—
232	C₄⋊C₄	—	10	2	no	—
233	C₄⋊C₄	—	20	2	no	—
234	C₄⋊C₄	—	20	2	no	—
235	C₂²⋊C₄	—	20	2	no	—
236	C₄²	—	20	2	no	—
237	C₄²	D₄×D₅	1	1	yes	characteristic
238	C₄²	D₄×D₅	1	1	yes	characteristic
239	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
240	C₄²	C₂³×D₅	1	1	yes	characteristic
241	C₂×C₅	—	2	2	no	—
242	C₂×C₅	C₂×#494	1	1	yes	characteristic
243	C₂×C₅	#3151	1	1	yes	characteristic
244	C₂×C₅	#3156	1	1	yes	characteristic
245	C₂³	#4086	1	1	yes	characteristic
246	Q₈	—	4	8	no	—
247	Q₈	—	8	4	no	—
248	Q₈	—	20	4	no	—
249	Q₈	—	20	4	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	—	8	4	no	—
254	C₂×C₄	—	20	2	no	—
255	C₂×C₄	—	20	2	no	—
256	C₂×C₄	D₅×Q₁₆	1	4	yes	—
257	C₂×C₄	#4086	1	1	yes	characteristic
258	C₂×C₄	#4086	1	1	yes	characteristic
259	C₂×C₄	#4086	1	1	yes	characteristic
260	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
261	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
262	C₂×C₄	C₂²×C₂²⋊D₅	1	1	yes	characteristic
263	C₈	—	2	2	no	—
264	C₈	—	5	8	no	—
265	C₈	—	10	4	no	—
266	C₈	#4078	1	4	yes	—
267	C₅	#19632	1	1	yes	characteristic, complemented, Sylow
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	#65470	1	2	yes	—
271	C₂²	C₂×#4086	1	1	yes	characteristic
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	8	4	no	—
275	C₄	—	40	2	no	—
276	C₄	—	40	2	no	—
277	C₄	#65470	1	2	yes	—
278	C₄	C₂×D₅×Q₁₆	1	2	yes	—
279	C₄	#66100	1	2	yes	—
280	C₄	C₂×#4078	1	2	yes	—
281	C₂	—	2	2	no	—
282	C₂	#10614894	1	1	yes	characteristic
283	C₂	C₂×#65470	1	1	yes	characteristic
284	C₂	#10615612	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#4086
commutator, cosocle	C₄²×C₅	C₂⁴
Frattini	C₄²	C₂³×D₅
Fitting	C₂×C₅×#494	C₂
socle	C₂²×C₅	C₂×#127
prime radical	C₂³×C₅	#127
hypercenter	C₂×#494	D₅
hypocenter	C₅	#19632
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
5. #10602095	C₂	semidirect
6. #10602091	C₂	semidirect
7. #10602089	C₂	semidirect
108. #681	Q₁₆	semidirect
157. #168	C₂×Q₁₆	semidirect
267. C₅	#19632	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#494	2
#10614955	2
#10614938	2
#10610206	2
#10602095	2
#10602091	2
#10602089	2
C₈×#681	2
#19632	5