Subgroups — #12014332

Order 1280.

590 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₈×D₅	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#346	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#64727	C₂	1	2	yes	maximal
4	C₂×#64681	C₂	1	1	yes	maximal, characteristic, complemented
5	#10598295	C₂	1	2	yes	maximal, complemented
6	C₂²×C₈×D₅	C₄	1	1	yes	characteristic
7	C₂²×C₈×D₅	C₂²	1	1	yes	characteristic
8	C₂×C₄²×D₅	C₄	1	1	yes	characteristic
9	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
10	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₅	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
13	C₂²×#672	C₂²	1	2	yes	—
14	C₂×C₄×#168	C₄	1	1	yes	characteristic
15	C₄×C₈×D₅	—	2	2	no	—
16	C₄×C₈×D₅	—	2	2	no	—
17	C₅×#346	—	2	2	no	—
18	#64727	C₄	1	8	yes	complemented
19	#64681	—	2	2	no	—
20	#12599	—	5	1	no	maximal, Sylow
21	C₂²×C₄×D₅	C₈	1	2	yes	—
22	C₂²×C₄×D₅	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
25	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×Dic₅	C₈	1	2	yes	—
27	C₂×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
28	C₂²×#168	C₂×C₄	1	1	yes	characteristic
29	C₂²×#168	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×D₅	—	2	2	no	—
31	C₂×C₈×D₅	—	2	2	no	—
32	C₂×C₈×D₅	—	2	1	no	—
33	C₂×C₈×D₅	—	2	1	no	—
34	C₂×C₈×D₅	—	2	2	no	—
35	C₂×C₈×D₅	C₈	1	4	yes	—
36	C₂×C₈×D₅	C₂×C₄	1	4	yes	—
37	C₄²×D₅	—	2	2	no	—
38	C₄²×D₅	—	2	2	no	—
39	C₂×C₅×C₁₆	—	2	2	no	—
40	C₂×C₅×C₁₆	—	2	2	no	—
41	C₂×C₅×C₁₆	C₂×C₄	1	4	yes	complemented
42	C₄×C₅×C₈	D₄	1	2	yes	—
43	C₄×C₅×C₈	D₄	1	2	yes	—
44	C₈×Dic₅	—	2	2	no	—
45	C₈×Dic₅	—	2	1	no	—
46	C₈×Dic₅	—	2	1	no	—
47	C₂×#672	—	2	2	no	—
48	C₂×#672	—	2	2	no	—
49	C₂×#672	C₂×C₄	1	4	yes	complemented
50	C₄×#168	—	2	2	no	—
51	C₂²×C₄×C₈	—	5	1	no	—
52	C₂×C₂²⋊C₁₆	—	5	2	no	—
53	C₂×#346	—	5	1	no	—
54	C₂×#346	D₅	1	1	yes	characteristic, complemented
55	#1374	—	5	2	no	—
56	C₂³×D₅	C₂×C₈	1	1	yes	characteristic
57	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
58	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
59	C₂²×Dic₅	C₂×C₈	1	1	yes	characteristic
60	C₂²×Dic₅	C₂×C₈	1	2	yes	—
61	C₂×C₄×D₅	—	2	4	no	—
62	C₂×C₄×D₅	—	2	4	no	—
63	C₂×C₄×D₅	—	2	1	no	—
64	C₂×C₄×D₅	—	2	1	no	—
65	C₂×C₄×D₅	—	2	1	no	—
66	C₂×C₄×D₅	—	2	1	no	—
67	C₂×C₄×D₅	—	2	4	no	—
68	C₂×C₄×D₅	C₄²	1	2	yes	—
69	C₂×C₄×D₅	C₂×C₈	1	2	yes	—
70	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
71	C₂×C₅×C₈	C₂×C₈	1	2	yes	—
72	C₂×C₅×C₈	M₁₆	1	2	yes	—
73	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
74	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
75	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
76	C₂×C₅×C₈	#41	1	2	yes	—
77	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
78	C₄²×C₅	M₁₆	1	2	yes	—
79	C₄×Dic₅	—	2	4	no	—
80	C₄×Dic₅	—	2	1	no	—
81	C₄×Dic₅	—	2	1	no	—
82	C₂×#168	—	2	1	no	—
83	C₂×#168	—	2	1	no	—
84	C₂×#168	—	2	1	no	—
85	C₂×#168	—	2	1	no	—
86	C₂×#168	C₄²	1	2	yes	—
87	C₂×#168	C₂×C₈	1	2	yes	—
88	C₈×D₅	—	2	8	no	—
89	C₈×D₅	—	2	4	no	—
90	C₈×D₅	—	2	4	no	—
91	C₅×C₁₆	—	2	4	no	—
92	#672	—	2	4	no	—
93	C₂³×C₈	—	5	1	no	—
94	C₂³×C₈	—	5	1	no	—
95	C₂²×C₄²	—	5	1	no	—
96	C₂²×C₁₆	—	5	2	no	—
97	C₂²×C₁₆	C₂×D₅	1	2	yes	—
98	C₂×C₄×C₈	—	5	1	no	—
99	C₂×C₄×C₈	—	5	1	no	—
100	C₂×C₄×C₈	—	5	1	no	—
101	C₂×C₄×C₈	—	10	2	no	—
102	C₂×C₄×C₈	—	10	2	no	—
103	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
104	C₂²⋊C₁₆	—	5	8	no	—
105	#346	—	2	2	no	—
106	#346	—	10	2	no	—
107	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
108	C₂²×D₅	—	2	1	no	—
109	C₂²×D₅	—	2	1	no	—
110	C₂²×D₅	—	2	2	no	—
111	C₂²×D₅	C₄×C₈	1	2	yes	—
112	C₂²×D₅	M₃₂	1	4	yes	—
113	C₂×C₄×C₅	—	2	2	no	—
114	C₂×C₄×C₅	—	2	2	no	—
115	C₂×C₄×C₅	C₂²⋊C₈	1	4	yes	—
116	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
117	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₅	#117	1	1	yes	characteristic
119	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
120	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
121	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
122	C₂×Dic₅	—	2	2	no	—
123	C₂×Dic₅	—	2	1	no	—
124	C₂×Dic₅	—	2	1	no	—
125	C₂×Dic₅	—	2	2	no	—
126	C₂×Dic₅	C₄×C₈	1	2	yes	—
127	C₂×Dic₅	M₃₂	1	4	yes	—
128	C₄×D₅	—	2	16	no	—
129	C₄×D₅	—	2	2	no	—
130	C₄×D₅	—	2	2	no	—
131	C₄×D₅	—	2	2	no	—
132	C₄×D₅	—	2	2	no	—
133	C₅×C₈	C₂²⋊C₈	1	4	yes	—
134	C₅×C₈	C₄×D₄	1	2	yes	—
135	C₅×C₈	C₄×D₄	1	2	yes	—
136	#168	—	2	2	no	—
137	#168	—	2	2	no	—
138	C₂³×C₄	—	5	1	no	—
139	C₂³×C₄	—	5	2	no	—
140	C₂²×C₈	—	5	1	no	—
141	C₂²×C₈	—	5	1	no	—
142	C₂²×C₈	—	5	4	no	—
143	C₂²×C₈	—	5	4	no	—
144	C₂²×C₈	—	10	2	no	—
145	C₂²×C₈	—	10	2	no	—
146	C₂²×C₈	—	10	1	no	—
147	C₂²×C₈	—	10	2	no	—
148	C₂²×C₈	—	10	1	no	—
149	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
150	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
151	C₂×C₄²	—	5	2	no	—
152	C₂×C₄²	—	5	1	no	—
153	C₂×C₄²	—	10	2	no	—
154	C₂×C₄²	—	10	2	no	—
155	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
156	C₂×C₁₆	—	2	2	no	—
157	C₂×C₁₆	—	2	2	no	—
158	C₂×C₁₆	—	5	4	no	—
159	C₂×C₁₆	—	10	2	no	—
160	C₂×C₁₆	—	10	2	no	—
161	C₂×C₁₆	C₄×D₅	1	4	yes	complemented
162	C₄×C₈	—	10	2	no	—
163	C₄×C₈	—	10	1	no	—
164	C₄×C₈	—	10	1	no	—
165	C₄×C₈	—	10	2	no	—
166	C₄×C₈	D₂₀	1	2	yes	—
167	C₄×C₈	C₂²⋊D₅	1	2	yes	—
168	C₂²×C₅	C₂×C₄×C₈	1	1	yes	characteristic
169	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
170	C₂²×C₅	#431	1	1	yes	characteristic
171	C₂²×C₅	C₂×M₃₂	1	2	yes	—
172	C₂²×C₅	C₂×M₃₂	1	2	yes	—
173	C₂×D₅	—	2	2	no	—
174	C₂×D₅	—	2	4	no	—
175	C₂×D₅	—	2	4	no	—
176	C₂×D₅	—	2	2	no	—
177	C₂×D₅	#346	1	4	yes	complemented
178	C₄×C₅	—	2	4	no	—
179	C₄×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
180	C₄×C₅	C₄×M₁₆	1	1	yes	characteristic
181	C₄×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
182	C₄×C₅	#431	1	1	yes	characteristic
183	Dic₅	—	2	4	no	—
184	Dic₅	—	2	2	no	—
185	C₂⁴	—	5	1	no	—
186	C₂²×C₄	—	5	2	no	—
187	C₂²×C₄	—	5	1	no	—
188	C₂²×C₄	—	5	2	no	—
189	C₂²×C₄	—	5	2	no	—
190	C₂²×C₄	—	10	4	no	—
191	C₂²×C₄	—	10	4	no	—
192	C₂²×C₄	—	10	4	no	—
193	C₂²×C₄	—	10	1	no	—
194	C₂²×C₄	—	10	1	no	—
195	C₂²×C₄	—	10	1	no	—
196	C₂²×C₄	—	10	1	no	—
197	C₂²×C₄	C₈×D₅	1	2	yes	—
198	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
199	C₂×C₈	—	5	2	no	—
200	C₂×C₈	—	5	2	no	—
201	C₂×C₈	—	10	1	no	—
202	C₂×C₈	—	10	1	no	—
203	C₂×C₈	—	10	1	no	—
204	C₂×C₈	—	10	1	no	—
205	C₂×C₈	—	10	8	no	—
206	C₂×C₈	—	10	4	no	—
207	C₂×C₈	—	10	4	no	—
208	C₂×C₈	C₈×D₅	1	2	yes	—
209	C₂×C₈	#676	1	2	yes	—
210	C₂×C₈	#685	1	2	yes	—
211	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
212	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
213	C₂×C₈	#709	1	2	yes	—
214	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
215	C₄²	—	10	4	no	—
216	C₄²	—	10	1	no	—
217	C₄²	—	10	1	no	—
218	C₄²	#676	1	2	yes	—
219	C₄²	#685	1	2	yes	—
220	C₁₆	—	2	4	no	—
221	C₁₆	—	10	4	no	—
222	C₂×C₅	C₄×C₂²⋊C₈	1	1	yes	characteristic
223	C₂×C₅	C₂×#346	1	1	yes	characteristic
224	C₂×C₅	C₄×M₃₂	1	1	yes	characteristic
225	C₂×C₅	#2112	1	2	yes	—
226	C₂×C₅	#2163	1	2	yes	—
227	D₅	—	2	4	no	—
228	C₂³	—	5	4	no	—
229	C₂³	—	5	2	no	—
230	C₂³	—	10	1	no	—
231	C₂³	—	10	2	no	—
232	C₂³	—	10	1	no	—
233	C₂³	C₂×C₈×D₅	1	1	yes	characteristic
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	5	4	no	—
237	C₂×C₄	—	5	2	no	—
238	C₂×C₄	—	10	16	no	—
239	C₂×C₄	—	10	1	no	—
240	C₂×C₄	—	10	2	no	—
241	C₂×C₄	—	10	2	no	—
242	C₂×C₄	—	10	1	no	—
243	C₂×C₄	—	10	2	no	—
244	C₂×C₄	—	10	2	no	—
245	C₂×C₄	—	10	2	no	—
246	C₂×C₄	—	10	2	no	—
247	C₂×C₄	#3952	1	4	yes	—
248	C₂×C₄	C₄²×D₅	1	1	yes	characteristic
249	C₂×C₄	#4018	1	1	yes	characteristic
250	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
251	C₂×C₄	C₂×#676	1	1	yes	characteristic
252	C₂×C₄	C₂×#676	1	1	yes	characteristic
253	C₂×C₄	C₂×#685	1	1	yes	characteristic
254	C₈	—	10	2	no	—
255	C₈	—	10	2	no	—
256	C₈	#3952	1	4	yes	—
257	C₈	C₄×D₂₀	1	2	yes	—
258	C₈	C₄×C₂²⋊D₅	1	2	yes	—
259	C₅	#12599	1	1	yes	characteristic, complemented, Sylow
260	C₂²	—	5	4	no	—
261	C₂²	—	10	2	no	—
262	C₂²	—	10	2	no	—
263	C₂²	—	10	4	no	—
264	C₂²	—	10	4	no	—
265	C₂²	C₄×C₈×D₅	1	1	yes	characteristic
266	C₂²	#64974	1	1	yes	characteristic
267	C₂²	D₅×M₃₂	1	2	yes	—
268	C₂²	D₅×M₃₂	1	2	yes	—
269	C₂²	C₂×#3952	1	1	yes	characteristic
270	C₄	—	2	4	no	—
271	C₄	—	10	2	no	—
272	C₄	—	10	4	no	—
273	C₄	#64974	1	1	yes	characteristic
274	C₄	C₄×#676	1	1	yes	characteristic
275	C₄	C₄×#685	1	1	yes	characteristic
276	C₄	C₂×#3952	1	1	yes	characteristic
277	C₂	—	10	4	no	—
278	C₂	D₅×#346	1	1	yes	characteristic
279	C₂	#10599864	1	1	yes	characteristic
280	C₂	#10599866	1	2	yes	—
281	C₂	C₄×#3952	1	1	yes	characteristic
282	C₂	#10602224	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅
commutator	C₂²×C₅	C₂×C₄×C₈
Fitting	C₂×C₅×#346	C₂
socle	C₂³×C₅	C₂²×C₈
prime radical	C₂³×D₅	C₂×C₈
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	C₂×#346	D₅
hypocenter	C₅	#12599
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#346	C₂	semidirect
4. C₂×#64681	C₂	semidirect
5. #10598295	C₂	semidirect
18. #64727	C₄	semidirect
41. C₂×C₅×C₁₆	C₂×C₄	semidirect
49. C₂×#672	C₂×C₄	semidirect
54. C₂×#346	D₅	semidirect
161. C₂×C₁₆	C₄×D₅	semidirect
177. C₂×D₅	#346	semidirect
259. C₅	#12599	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₈×D₅	2
C₂×C₅×#346	2
C₂×#64727	2
C₂×#64681	2
#10598295	2
#12599	5