Subgroups — #12014206

Order 1280.

570 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×#108	C₂	1	1	yes	maximal, characteristic
2	C₂×#65395	C₂	1	1	yes	maximal, characteristic
3	C₂²×#3944	C₂	1	1	yes	maximal, characteristic
4	#10598221	C₂	1	4	yes	maximal, complemented
5	C₂²×C₅×M₁₆	—	2	1	no	—
6	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₂×#4072	—	2	1	no	—
8	C₂×C₅×#108	—	2	1	no	—
9	C₂×C₅×#108	—	2	2	no	—
10	C₂×C₅×#108	C₄	1	4	yes	complemented
11	C₂×C₅×#108	C₂²	1	2	yes	—
12	#65395	—	2	2	no	—
13	#65395	C₄	1	4	yes	—
14	C₂×#3950	—	2	1	no	—
15	C₂×#3949	C₂²	1	2	yes	—
16	C₂×#3944	—	2	2	no	—
17	C₂×#3944	—	2	2	no	—
18	C₂×#3944	C₄	1	2	yes	complemented
19	C₂×#3944	C₂²	1	2	yes	—
20	C₂²×#672	C₄	1	2	yes	complemented
21	#12563	—	5	1	no	maximal, Sylow
22	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
23	C₂×C₅×M₁₆	—	2	4	no	—
24	C₂×C₅×M₁₆	—	2	2	no	—
25	C₂×C₅×M₁₆	—	4	2	no	—
26	C₂×C₅×M₁₆	—	4	1	no	—
27	C₂²×C₅×C₈	—	4	1	no	—
28	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
31	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
32	C₂²×C₅×C₈	D₄	1	2	yes	—
33	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
34	C₂²×C₅×C₈	D₄	1	2	yes	—
35	C₂²×C₅×C₈	Q₈	1	1	yes	characteristic
36	C₂²×C₅×C₈	Q₈	1	1	yes	characteristic
37	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
38	C₂×C₂²⋊Dic₅	—	4	1	no	—
39	#4072	—	2	4	no	—
40	#4072	—	4	2	no	—
41	C₂×C₄⋊Dic₅	—	2	2	no	—
42	C₂×C₄×Dic₅	—	4	1	no	—
43	C₅×#108	—	2	4	no	—
44	C₅×#108	—	2	2	no	—
45	C₅×#108	C₂×C₄	1	4	yes	—
46	#3950	—	2	4	no	—
47	#3949	—	2	2	no	—
48	#3949	C₂×C₄	1	4	yes	—
49	#3944	—	2	4	no	—
50	#3944	—	2	2	no	—
51	#3944	—	2	2	no	—
52	C₂×#672	—	2	4	no	—
53	C₂×#672	C₂×C₄	1	4	yes	—
54	C₂²×M₃₂	—	5	1	no	—
55	C₂²×#108	D₅	1	1	yes	characteristic
56	C₂×#427	—	5	1	no	—
57	#1383	—	5	4	no	—
58	C₂⁴×C₅	SD₁₆	1	1	yes	characteristic
59	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
60	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
61	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₅	D₈	1	2	yes	—
64	C₂²×C₄×C₅	SD₁₆	1	1	yes	characteristic
65	C₂²×C₄×C₅	SD₁₆	1	1	yes	characteristic
66	C₂²×C₄×C₅	SD₁₆	1	1	yes	characteristic
67	C₂²×C₄×C₅	Q₁₆	1	2	yes	—
68	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
69	C₂²×Dic₅	—	4	2	no	—
70	C₅×M₁₆	—	2	4	no	—
71	C₅×M₁₆	—	4	4	no	—
72	C₅×M₁₆	—	4	2	no	—
73	C₂×C₅×C₈	—	4	4	no	—
74	C₂×C₅×C₈	—	4	2	no	—
75	C₂×C₅×C₈	C₄²	1	2	yes	—
76	C₂×C₅×C₈	C₄²	1	2	yes	—
77	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
78	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
79	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
80	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
81	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
84	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
85	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
86	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
87	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
91	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
92	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
93	C₂²⋊Dic₅	—	4	4	no	—
94	C₄⋊Dic₅	—	2	4	no	—
95	C₄⋊Dic₅	—	4	2	no	—
96	C₄×Dic₅	—	4	4	no	—
97	#672	—	2	4	no	—
98	C₂²×M₁₆	—	2	1	no	—
99	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
100	C₂×#117	—	10	1	no	—
101	C₂×M₃₂	—	5	2	no	—
102	C₂×M₃₂	—	5	2	no	—
103	C₂×M₃₂	—	10	2	no	—
104	C₂×M₃₂	—	10	2	no	—
105	C₂²×C₁₆	—	5	2	no	—
106	C₂×#108	—	2	1	no	—
107	C₂×#108	—	2	2	no	—
108	C₂×#108	Dic₅	1	4	yes	complemented
109	C₂×#108	C₂×D₅	1	2	yes	—
110	#427	—	5	4	no	—
111	#427	—	10	2	no	—
112	C₂×#107	—	10	1	no	—
113	C₂×#106	—	5	2	no	—
114	C₂³×C₅	—	4	2	no	—
115	C₂³×C₅	#102	1	2	yes	—
116	C₂³×C₅	#103	1	2	yes	—
117	C₂³×C₅	#106	1	1	yes	characteristic
118	C₂³×C₅	#106	1	1	yes	characteristic
119	C₂³×C₅	C₂×SD₁₆	1	1	yes	characteristic
120	C₂×C₄×C₅	#95	1	2	yes	—
121	C₂×C₄×C₅	#95	1	2	yes	—
122	C₂×C₄×C₅	#102	1	2	yes	—
123	C₂×C₄×C₅	#102	1	2	yes	—
124	C₂×C₄×C₅	#102	1	2	yes	—
125	C₂×C₄×C₅	#103	1	2	yes	—
126	C₂×C₄×C₅	#103	1	2	yes	—
127	C₂×C₄×C₅	#103	1	2	yes	—
128	C₂×C₄×C₅	#106	1	1	yes	characteristic
129	C₂×C₄×C₅	#106	1	1	yes	characteristic
130	C₂×C₄×C₅	#107	1	2	yes	—
131	C₂×C₄×C₅	#107	1	2	yes	—
132	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
137	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
138	C₂×Dic₅	—	4	4	no	—
139	C₂×Dic₅	—	8	4	no	—
140	C₅×C₈	—	4	4	no	—
141	C₅×C₈	#95	1	2	yes	—
142	C₅×C₈	#95	1	2	yes	—
143	C₅×C₈	C₂×C₄²	1	1	yes	characteristic
144	C₅×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
145	C₅×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
146	C₅×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
147	C₂³×C₄	C₂²⋊D₅	1	1	yes	characteristic
148	C₂×M₁₆	—	2	4	no	—
149	C₂×M₁₆	—	2	2	no	—
150	C₂×M₁₆	—	4	2	no	—
151	C₂×M₁₆	—	4	1	no	—
152	C₂²×C₈	—	4	1	no	—
153	C₂²×C₈	Dic₁₀	1	1	yes	characteristic
154	C₂²×C₈	Dic₁₀	1	1	yes	characteristic
155	C₂²×C₈	C₄×D₅	1	2	yes	—
156	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
157	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
158	C₂²×C₈	D₂₀	1	2	yes	—
159	C₂²×C₈	C₂×Dic₅	1	2	yes	—
160	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
161	C₂²×C₈	C₂²⋊D₅	1	2	yes	—
162	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
163	#117	—	10	4	no	—
164	#117	—	20	2	no	—
165	C₂×C₄⋊C₄	—	10	2	no	—
166	C₂×C₂²⋊C₄	—	20	1	no	—
167	C₂×C₄²	—	20	1	no	—
168	M₃₂	—	10	4	no	—
169	M₃₂	—	10	2	no	—
170	M₃₂	—	10	2	no	—
171	C₂×C₁₆	—	5	4	no	—
172	C₂×C₁₆	—	10	4	no	—
173	#108	—	2	4	no	—
174	#108	—	2	2	no	—
175	#108	C₂×Dic₅	1	4	yes	—
176	#107	—	10	4	no	—
177	#106	—	5	4	no	—
178	#106	—	10	2	no	—
179	C₂²×C₅	—	2	2	no	—
180	C₂²×C₅	—	2	2	no	—
181	C₂²×C₅	—	2	2	no	—
182	C₂²×C₅	—	4	2	no	—
183	C₂²×C₅	—	4	2	no	—
184	C₂²×C₅	#340	1	2	yes	—
185	C₂²×C₅	#340	1	2	yes	—
186	C₂²×C₅	C₂×#102	1	1	yes	characteristic
187	C₂²×C₅	C₂×#103	1	1	yes	characteristic
188	C₂²×C₅	C₂×#106	1	1	yes	characteristic
189	C₄×C₅	#340	1	2	yes	—
190	C₄×C₅	#340	1	2	yes	—
191	C₄×C₅	C₂×#95	1	1	yes	characteristic
192	C₄×C₅	C₂×#102	1	1	yes	characteristic
193	C₄×C₅	C₂×#103	1	1	yes	characteristic
194	C₄×C₅	C₂×#107	1	1	yes	characteristic
195	Dic₅	—	8	4	no	—
196	C₂⁴	#688	1	1	yes	characteristic
197	C₂²×C₄	—	20	2	no	—
198	C₂²×C₄	#683	1	1	yes	characteristic
199	C₂²×C₄	#683	1	1	yes	characteristic
200	C₂²×C₄	#685	1	2	yes	—
201	C₂²×C₄	#686	1	2	yes	—
202	C₂²×C₄	#687	1	1	yes	characteristic
203	C₂²×C₄	#687	1	1	yes	characteristic
204	C₂²×C₄	#688	1	1	yes	characteristic
205	C₂²×C₄	#689	1	2	yes	—
206	C₂²×C₄	C₂²⋊Dic₅	1	2	yes	—
207	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
208	M₁₆	—	2	4	no	—
209	M₁₆	—	4	4	no	—
210	M₁₆	—	4	2	no	—
211	C₂×C₈	—	4	4	no	—
212	C₂×C₈	—	4	2	no	—
213	C₂×C₈	C₄×Dic₅	1	2	yes	—
214	C₂×C₈	C₄×Dic₅	1	2	yes	—
215	C₂×C₈	#683	1	1	yes	characteristic
216	C₂×C₈	#683	1	1	yes	characteristic
217	C₂×C₈	#683	1	2	yes	—
218	C₂×C₈	#683	1	2	yes	—
219	C₂×C₈	C₄⋊Dic₅	1	2	yes	—
220	C₂×C₈	C₄⋊Dic₅	1	2	yes	—
221	C₂×C₈	#685	1	2	yes	—
222	C₂×C₈	#685	1	2	yes	—
223	C₂×C₈	#685	1	2	yes	—
224	C₂×C₈	C₂²⋊Dic₅	1	2	yes	—
225	C₂×C₈	C₂×Dic₁₀	1	1	yes	characteristic
226	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
227	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
228	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
229	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
230	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
231	C₄⋊C₄	—	10	4	no	—
232	C₄⋊C₄	—	20	2	no	—
233	C₂²⋊C₄	—	20	4	no	—
234	C₄²	—	20	4	no	—
235	C₁₆	—	10	4	no	—
236	C₂×C₅	—	2	1	no	—
237	C₂×C₅	—	2	2	no	—
238	C₂×C₅	—	2	2	no	—
239	C₂×C₅	—	2	1	no	—
240	C₂×C₅	C₂×#340	1	1	yes	characteristic
241	C₂×C₅	#2155	1	2	yes	—
242	C₂³	—	4	2	no	—
243	C₂³	#3940	1	1	yes	characteristic
244	C₂³	#3940	1	1	yes	characteristic
245	C₂³	#3941	1	2	yes	—
246	C₂³	#3967	1	2	yes	—
247	C₂³	C₂×#688	1	1	yes	characteristic
248	C₂×C₄	—	20	4	no	—
249	C₂×C₄	—	40	4	no	—
250	C₂×C₄	#3939	1	2	yes	—
251	C₂×C₄	#3939	1	2	yes	—
252	C₂×C₄	#3940	1	1	yes	characteristic
253	C₂×C₄	#3940	1	1	yes	characteristic
254	C₂×C₄	#3941	1	2	yes	—
255	C₂×C₄	#3942	1	2	yes	—
256	C₂×C₄	#3942	1	2	yes	—
257	C₂×C₄	#3963	1	2	yes	—
258	C₂×C₄	#3963	1	2	yes	—
259	C₂×C₄	#3964	1	2	yes	—
260	C₂×C₄	#3964	1	2	yes	—
261	C₂×C₄	#3967	1	2	yes	—
262	C₂×C₄	C₂×#683	1	1	yes	characteristic
263	C₂×C₄	C₂×#685	1	1	yes	characteristic
264	C₂×C₄	C₂×#686	1	1	yes	characteristic
265	C₂×C₄	C₂×#687	1	1	yes	characteristic
266	C₂×C₄	C₂×#689	1	1	yes	characteristic
267	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
268	C₈	—	4	4	no	—
269	C₈	#3963	1	2	yes	—
270	C₈	#3963	1	2	yes	—
271	C₈	C₂×C₄×Dic₅	1	1	yes	characteristic
272	C₈	C₂×#683	1	1	yes	characteristic
273	C₈	C₂×C₄⋊Dic₅	1	1	yes	characteristic
274	C₈	C₂×#685	1	1	yes	characteristic
275	C₅	#12563	1	1	yes	characteristic, complemented, Sylow
276	C₂²	—	2	2	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	4	2	no	—
280	C₂²	—	4	2	no	—
281	C₂²	#64749	1	2	yes	—
282	C₂²	#64749	1	2	yes	—
283	C₂²	C₂×#3940	1	1	yes	characteristic
284	C₂²	C₂×#3941	1	1	yes	characteristic
285	C₂²	C₂×#3967	1	1	yes	characteristic
286	C₄	—	40	4	no	—
287	C₄	#64749	1	2	yes	—
288	C₄	#64749	1	2	yes	—
289	C₄	C₂×#3939	1	1	yes	characteristic
290	C₄	C₂×#3942	1	1	yes	characteristic
291	C₄	C₂×#3963	1	1	yes	characteristic
292	C₄	C₂×#3964	1	1	yes	characteristic
293	C₂	—	2	1	no	—
294	C₂	—	2	1	no	—
295	C₂	—	2	2	no	—
296	C₂	—	2	2	no	—
297	C₂	#10601478	1	2	yes	—
298	C₂	C₂×#64749	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#3940
commutator	C₅×C₈	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₂²×C₅×#108	C₂
socle	C₂²×C₅	C₂×#106
prime radical	C₂⁴×C₅	SD₁₆
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	C₂²×#108	D₅
hypocenter	C₅	#12563
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×#108

Complements

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

N	H ≅ G/N	type
4. #10598221	C₂	semidirect
10. C₂×C₅×#108	C₄	semidirect
18. C₂×#3944	C₄	semidirect
20. C₂²×#672	C₄	semidirect
108. C₂×#108	Dic₅	semidirect
275. C₅	#12563	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₅×#108	2
C₂×#65395	2
C₂²×#3944	2
#10598221	2
#12563	5