Subgroups — #12252523

Order 1280.

682 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#65196	C₂	1	2	yes	maximal, complemented
2	#10610189	C₂	1	1	yes	maximal, characteristic
3	C₄×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
4	#10602225	C₂	1	2	yes	maximal, complemented
5	C₂×C₄⋊#672	C₂	1	1	yes	maximal, characteristic, complemented
6	#10599868	C₂	1	4	yes	maximal, complemented
7	#10599848	C₂	1	2	yes	maximal, complemented
8	C₁₆×D₂₀	C₂	1	2	yes	maximal, complemented
9	C₂×#4047	C₂²	1	2	yes	—
10	#65811	C₄	1	1	yes	characteristic
11	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
12	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
14	#65399	C₄	1	2	yes	—
15	C₂²×#672	C₂²	1	2	yes	—
16	#65213	C₄	1	1	yes	characteristic
17	#65196	—	2	8	no	—
18	C₂×#3930	C₂²	1	4	yes	—
19	C₂×D₅×C₁₆	C₂²	1	4	yes	—
20	#64977	C₄	1	2	yes	—
21	#64977	C₄	1	2	yes	—
22	C₈×D₂₀	C₂²	1	2	yes	—
23	C₈×D₂₀	C₂²	1	2	yes	—
24	C₅×#346	C₂²	1	2	yes	—
25	C₄×C₅×C₁₆	C₂²	1	2	yes	—
26	#64727	C₂²	1	4	yes	—
27	#64727	C₂²	1	4	yes	—
28	C₄⋊#672	C₂²	1	1	yes	characteristic
29	C₄⋊#672	C₂²	1	1	yes	characteristic
30	C₄⋊#672	C₂²	1	2	yes	—
31	#33438	—	5	1	no	maximal, Sylow
32	C₂×#709	C₂×C₄	1	2	yes	—
33	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
36	#4076	C₈	1	2	yes	—
37	#4076	C₈	1	2	yes	—
38	C₂×#681	C₂×C₄	1	2	yes	—
39	#4047	—	2	4	no	—
40	#4047	—	2	4	no	—
41	C₂×#676	C₂×C₄	1	4	yes	—
42	C₂×C₈×D₅	C₂³	1	4	yes	—
43	#4021	C₈	1	2	yes	—
44	#4020	C₈	1	1	yes	characteristic
45	C₄×D₂₀	C₂×C₄	1	2	yes	—
46	C₄×D₂₀	C₂×C₄	1	2	yes	—
47	#4015	C₈	1	1	yes	characteristic
48	C₅×M₃₂	—	2	2	no	—
49	C₅×M₃₂	D₄	1	4	yes	complemented
50	C₂×C₅×C₁₆	C₂³	1	2	yes	—
51	C₂×C₅×C₁₆	C₂³	1	2	yes	—
52	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
53	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
54	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
55	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
56	#3952	C₂×C₄	1	4	yes	—
57	#3952	C₂×C₄	1	4	yes	—
58	C₂×#672	—	2	4	no	—
59	C₂×#672	C₂³	1	2	yes	—
60	C₂×#672	C₂³	1	2	yes	—
61	C₄⋊#168	C₂×C₄	1	2	yes	—
62	C₄⋊#168	C₂×C₄	1	2	yes	—
63	#3930	—	2	8	no	—
64	D₅×C₁₆	—	2	8	no	—
65	C₂×#504	—	5	2	no	—
66	#2931	—	5	1	no	—
67	#2169	—	5	2	no	—
68	#2168	—	5	4	no	—
69	D₄×C₁₆	—	5	2	no	—
70	C₂×C₄⋊C₁₆	—	5	1	no	—
71	#2113	—	5	2	no	—
72	C₄×M₃₂	D₅	1	1	yes	characteristic, complemented
73	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
74	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
75	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₂²⋊D₅	C₂×C₈	1	4	yes	—
77	#709	—	2	4	no	—
78	#709	—	2	4	no	—
79	C₂×D₂₀	C₂×C₈	1	2	yes	—
80	C₂×D₂₀	C₂×C₈	1	2	yes	—
81	C₂×D₂₀	C₂×C₈	1	2	yes	—
82	C₂×C₄×D₅	C₂²×C₄	1	4	yes	—
83	C₂×Dic₁₀	C₂×C₈	1	2	yes	—
84	C₂×C₅×C₈	—	2	2	no	—
85	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
89	C₂×C₅×C₈	#41	1	2	yes	—
90	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
91	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
92	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
93	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
94	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
95	#685	C₂×C₈	1	4	yes	—
96	#685	C₂×C₈	1	4	yes	—
97	C₄⋊Dic₅	C₂×C₈	1	2	yes	—
98	C₄⋊Dic₅	C₂×C₈	1	2	yes	—
99	#681	—	2	4	no	—
100	C₂×#168	C₂²×C₄	1	4	yes	—
101	#676	—	2	8	no	—
102	C₈×D₅	—	2	4	no	—
103	C₈×D₅	—	2	4	no	—
104	C₅×C₁₆	—	2	2	no	—
105	C₅×C₁₆	C₂×D₄	1	4	yes	—
106	#672	—	2	2	no	—
107	#672	—	2	2	no	—
108	C₂×#131	—	5	2	no	—
109	#532	—	5	1	no	—
110	#504	—	10	8	no	—
111	C₂×M₃₂	—	5	4	no	—
112	C₂×M₃₂	C₂×D₅	1	1	yes	characteristic
113	C₂×M₃₂	C₂×D₅	1	1	yes	characteristic
114	C₂²×C₁₆	—	5	2	no	—
115	C₂²×C₁₆	—	5	4	no	—
116	#436	—	5	2	no	—
117	#436	—	5	2	no	—
118	C₈×D₄	—	5	2	no	—
119	C₈×D₄	—	5	2	no	—
120	#423	—	5	1	no	—
121	#408	—	5	2	no	—
122	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
123	C₄⋊C₁₆	—	5	1	no	—
124	C₄⋊C₁₆	—	5	2	no	—
125	C₄⋊C₁₆	—	5	1	no	—
126	C₂²⋊C₁₆	—	5	4	no	—
127	C₂²⋊C₁₆	—	5	4	no	—
128	#346	C₂×D₅	1	2	yes	—
129	C₄×C₁₆	C₂×D₅	1	2	yes	—
130	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
131	C₂²×D₅	C₂²×C₈	1	4	yes	—
132	C₂×C₄×C₅	—	2	2	no	—
133	C₂×C₄×C₅	—	4	1	no	—
134	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
135	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
136	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
137	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
138	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
139	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
140	C₂×C₄×C₅	#131	1	2	yes	—
141	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
142	C₂²⋊D₅	—	2	8	no	—
143	C₂×Dic₅	C₂²×C₈	1	4	yes	—
144	D₂₀	—	2	4	no	—
145	D₂₀	—	2	4	no	—
146	D₂₀	—	4	2	no	—
147	C₄×D₅	—	2	4	no	—
148	C₄×D₅	—	2	4	no	—
149	Dic₁₀	—	2	4	no	—
150	C₅×C₈	—	2	1	no	—
151	C₅×C₈	—	2	1	no	—
152	C₅×C₈	C₄×D₄	1	2	yes	—
153	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
154	C₅×C₈	C₂×#41	1	1	yes	characteristic
155	#168	—	2	4	no	—
156	C₂×#41	—	5	2	no	—
157	#131	—	10	4	no	—
158	#131	—	10	4	no	—
159	C₂×M₁₆	—	5	2	no	—
160	C₂×M₁₆	—	5	4	no	—
161	C₂²×C₈	—	5	4	no	—
162	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
163	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
164	C₄⋊Q₈	—	5	1	no	—
165	#127	—	5	1	no	—
166	#124	—	5	2	no	—
167	#121	—	5	2	no	—
168	#121	—	5	2	no	—
169	C₄×D₄	—	5	2	no	—
170	C₄×D₄	—	5	2	no	—
171	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
172	M₃₂	—	2	2	no	—
173	M₃₂	—	10	8	no	—
174	M₃₂	D₂₀	1	4	yes	complemented
175	C₂×C₁₆	—	5	2	no	—
176	C₂×C₁₆	—	5	2	no	—
177	C₂×C₁₆	—	10	4	no	—
178	C₂×C₁₆	—	10	8	no	—
179	C₂×C₁₆	C₂²×D₅	1	2	yes	—
180	C₂×C₁₆	C₂²×D₅	1	2	yes	—
181	C₄⋊C₈	—	5	2	no	—
182	C₄⋊C₈	—	5	2	no	—
183	C₂²⋊C₈	—	5	4	no	—
184	C₂²⋊C₈	—	5	4	no	—
185	C₄×C₈	C₄×D₅	1	1	yes	characteristic
186	C₄×C₈	C₄×D₅	1	1	yes	characteristic
187	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
188	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
189	C₂²×C₅	—	2	1	no	—
190	C₂²×C₅	—	2	1	no	—
191	C₂²×C₅	C₈×D₄	1	2	yes	—
192	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
193	C₂×D₅	—	2	4	no	—
194	C₂×D₅	—	2	4	no	—
195	C₂×D₅	—	2	4	no	—
196	C₄×C₅	—	2	1	no	—
197	C₄×C₅	—	2	1	no	—
198	C₄×C₅	—	2	1	no	—
199	C₄×C₅	C₈×D₄	1	2	yes	—
200	C₄×C₅	#504	1	2	yes	complemented
201	C₄×C₅	C₂×C₄×D₄	1	1	yes	characteristic
202	C₄×C₅	C₂×#131	1	1	yes	characteristic
203	Dic₅	—	2	4	no	—
204	#41	—	10	4	no	—
205	#41	—	10	4	no	—
206	C₂×Q₈	—	5	2	no	—
207	C₂×D₄	—	5	4	no	—
208	C₂×D₄	—	5	2	no	—
209	C₂×D₄	—	5	2	no	—
210	C₂×D₄	—	5	2	no	—
211	C₂²×C₄	—	5	4	no	—
212	C₂²×C₄	C₈×D₅	1	1	yes	characteristic
213	C₂²×C₄	C₈×D₅	1	1	yes	characteristic
214	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
215	M₁₆	—	10	4	no	—
216	M₁₆	—	10	8	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	5	4	no	—
219	C₂×C₈	—	10	4	no	—
220	C₂×C₈	—	10	4	no	—
221	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
222	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
223	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
224	C₂×C₈	C₂×D₂₀	1	2	yes	—
225	C₂×C₈	#709	1	2	yes	—
226	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
227	C₄⋊C₄	—	5	2	no	—
228	C₄⋊C₄	—	5	2	no	—
229	C₂²⋊C₄	—	5	4	no	—
230	C₂²⋊C₄	—	5	4	no	—
231	C₄²	C₈×D₅	1	1	yes	characteristic
232	C₄²	C₈×D₅	1	1	yes	characteristic
233	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
234	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
235	C₁₆	—	2	2	no	—
236	C₁₆	—	10	2	no	—
237	C₁₆	—	10	2	no	—
238	C₁₆	C₂×D₂₀	1	4	yes	—
239	C₂×C₅	—	2	2	no	—
240	C₂×C₅	C₂×C₈×D₄	1	1	yes	characteristic
241	C₂×C₅	C₂×#504	1	1	yes	characteristic
242	C₂×C₅	#3407	1	1	yes	characteristic
243	D₅	—	4	4	no	—
244	C₂³	—	5	4	no	—
245	C₂³	C₂×C₈×D₅	1	1	yes	characteristic
246	Q₈	—	10	4	no	—
247	D₄	—	10	8	no	—
248	D₄	—	10	4	no	—
249	D₄	—	10	4	no	—
250	D₄	—	20	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	4	1	no	—
253	C₂×C₄	—	5	4	no	—
254	C₂×C₄	—	10	4	no	—
255	C₂×C₄	—	10	4	no	—
256	C₂×C₄	C₄×D₂₀	1	2	yes	—
257	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
258	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
259	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
260	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
261	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
262	C₂×C₄	#4047	1	2	yes	—
263	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
264	C₈	—	2	1	no	—
265	C₈	—	2	1	no	—
266	C₈	—	10	4	no	—
267	C₈	C₄×D₂₀	1	2	yes	—
268	C₈	C₂²×D₂₀	1	1	yes	characteristic
269	C₈	C₂×#709	1	1	yes	characteristic
270	C₅	#33438	1	1	yes	characteristic, complemented, Sylow
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	10	4	no	—
274	C₂²	—	10	4	no	—
275	C₂²	—	10	4	no	—
276	C₂²	C₈×D₂₀	1	2	yes	—
277	C₂²	C₂²×C₈×D₅	1	1	yes	characteristic
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	2	1	no	—
281	C₄	—	10	4	no	—
282	C₄	C₈×D₂₀	1	2	yes	—
283	C₄	#65196	1	2	yes	complemented
284	C₄	C₂×C₄×D₂₀	1	1	yes	characteristic
285	C₄	C₂×#4047	1	1	yes	characteristic
286	C₂	—	2	2	no	—
287	C₂	—	20	4	no	—
288	C₂	C₂×C₈×D₂₀	1	1	yes	characteristic
289	C₂	C₂×#65196	1	1	yes	characteristic
290	C₂	#10613738	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅
commutator, socle	C₂²×C₅	C₂³×C₈
Fitting	C₄×C₅×M₃₂	C₂
prime radical	#65811	C₄
cosocle	C₂×C₅×C₈	C₂⁴
hypercenter	C₄×M₃₂	D₅
hypocenter	C₅	#33438
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
1. C₂×#65196	C₂	semidirect
3. C₄×C₅×M₃₂	C₂	semidirect
4. #10602225	C₂	semidirect
5. C₂×C₄⋊#672	C₂	semidirect
6. #10599868	C₂	semidirect
7. #10599848	C₂	semidirect
8. C₁₆×D₂₀	C₂	semidirect
49. C₅×M₃₂	D₄	semidirect
72. C₄×M₃₂	D₅	semidirect
174. M₃₂	D₂₀	semidirect
200. C₄×C₅	#504	semidirect
270. C₅	#33438	semidirect
283. C₄	#65196	semidirect

Maximal subgroups

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

subgroup	index
C₂×#65196	2
#10610189	2
C₄×C₅×M₃₂	2
#10602225	2
C₂×C₄⋊#672	2
#10599868	2
#10599848	2
C₁₆×D₂₀	2
#33438	5