Subgroups — #12251944

Order 1280.

834 conjugacy classes of subgroups.

The subgroup lattice has 334 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	1	yes	maximal, characteristic
2	C₂×#65190	C₂	1	1	yes	maximal, characteristic
3	#10610280	C₂	1	1	yes	maximal, characteristic
4	C₅×#2108	C₂	1	1	yes	maximal, characteristic, complemented
5	Dic₅×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₁₆×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
7	#10600674	C₂	1	1	yes	maximal, characteristic, complemented
8	#10599865	C₂	1	2	yes	maximal, complemented
9	#10599864	C₂	1	2	yes	maximal, complemented
10	#10599844	C₂	1	2	yes	maximal, complemented
11	C₄×D₅×C₁₆	C₂	1	2	yes	maximal, complemented
12	C₂×#4053	C₄	1	1	yes	characteristic
13	C₂×#4047	C₂²	1	1	yes	characteristic
14	#65856	C₄	1	1	yes	characteristic
15	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
16	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
17	C₅×#405	C₂²	1	1	yes	characteristic
18	M₁₆×Dic₅	C₄	1	1	yes	characteristic
19	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
20	C₂×#3944	C₂²	1	1	yes	characteristic
21	C₂²×#672	C₂²	1	1	yes	characteristic
22	#65286	C₄	1	1	yes	characteristic
23	#65196	C₄	1	8	yes	complemented
24	#65190	C₄	1	8	yes	complemented
25	C₂×#3930	C₂²	1	2	yes	—
26	C₂×#3930	C₂²	1	2	yes	—
27	C₂×D₅×C₁₆	C₂²	1	2	yes	—
28	C₂×D₅×C₁₆	C₂²	1	2	yes	—
29	#64997	C₄	1	2	yes	—
30	#64996	C₄	1	2	yes	—
31	#64978	C₂²	1	2	yes	—
32	C₄×C₈×D₅	C₂²	1	2	yes	—
33	C₅×#346	C₂²	1	2	yes	—
34	C₄×C₅×C₁₆	C₂²	1	2	yes	—
35	#64722	C₂²	1	2	yes	—
36	C₁₆×Dic₅	C₂²	1	1	yes	characteristic
37	C₁₆×Dic₅	C₂²	1	2	yes	—
38	C₁₆×Dic₅	C₂²	1	1	yes	characteristic
39	C₁₆×Dic₅	C₂²	1	1	yes	characteristic
40	C₁₆×Dic₅	C₂²	1	1	yes	characteristic
41	#64681	C₂²	1	2	yes	—
42	C₄×#672	C₂²	1	2	yes	—
43	C₄×#504	—	5	1	no	maximal, Sylow
44	C₂×#709	C₂×C₄	1	1	yes	characteristic
45	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
47	C₅×#117	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
49	C₂×#681	C₂×C₄	1	1	yes	characteristic
50	C₂²×#168	C₂×C₄	1	1	yes	characteristic
51	#4053	C₈	1	8	yes	—
52	#4047	C₂×C₄	1	4	yes	—
53	#4047	C₂×C₄	1	4	yes	—
54	C₂×#676	C₂×C₄	1	2	yes	—
55	C₂×#676	C₂×C₄	1	2	yes	—
56	C₂×C₈×D₅	C₂×C₄	1	2	yes	—
57	C₂×C₈×D₅	C₂³	1	2	yes	—
58	#4039	C₈	1	2	yes	—
59	#4033	C₈	1	2	yes	—
60	#4027	C₈	1	4	yes	—
61	#4018	C₂×C₄	1	2	yes	—
62	C₄²×D₅	C₂×C₄	1	2	yes	—
63	C₅×M₃₂	C₂×C₄	1	4	yes	complemented
64	C₂×C₅×C₁₆	C₂×C₄	1	4	yes	complemented
65	C₂×C₅×C₁₆	C₂³	1	2	yes	—
66	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
67	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
68	C₅×#97	C₂×C₄	1	2	yes	—
69	C₄×C₅×C₈	C₂³	1	2	yes	—
70	#3947	C₂×C₄	1	2	yes	—
71	C₈×Dic₅	C₂×C₄	1	1	yes	characteristic
72	C₈×Dic₅	C₂×C₄	1	1	yes	characteristic
73	C₈×Dic₅	C₂×C₄	1	2	yes	—
74	C₈×Dic₅	C₂³	1	1	yes	characteristic
75	C₈×Dic₅	C₂³	1	1	yes	characteristic
76	#3944	C₂×C₄	1	4	yes	complemented
77	C₂×#672	C₂×C₄	1	4	yes	complemented
78	C₂×#672	C₂³	1	1	yes	characteristic
79	C₂×#672	C₂³	1	1	yes	characteristic
80	C₂×#672	C₂³	1	2	yes	—
81	#3935	C₂×C₄	1	2	yes	—
82	C₄×#168	C₂×C₄	1	2	yes	—
83	#3930	C₂×C₄	1	8	yes	complemented
84	#3930	C₂×C₄	1	8	yes	complemented
85	D₅×C₁₆	C₂×C₄	1	8	yes	complemented
86	D₅×C₁₆	C₂×C₄	1	8	yes	complemented
87	C₂×#504	—	5	1	no	—
88	C₂×#504	—	5	1	no	—
89	C₄×#131	—	5	1	no	—
90	#2108	—	5	1	no	—
91	#2108	—	5	2	no	—
92	#2108	—	5	2	no	—
93	#2108	D₅	1	1	yes	characteristic, complemented
94	C₄×M₃₂	—	5	1	no	—
95	C₄×M₃₂	—	5	2	no	—
96	C₂×C₄×C₁₆	—	5	1	no	—
97	C₂×C₄×C₁₆	—	5	2	no	—
98	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₂²⋊D₅	C₂×C₈	1	2	yes	—
100	C₂²×Dic₅	C₂×C₈	1	2	yes	—
101	#709	C₄²	1	4	yes	—
102	#709	C₂×C₈	1	4	yes	—
103	C₂×D₂₀	C₂×C₈	1	1	yes	characteristic
104	C₂×C₄×D₅	C₂×C₈	1	4	yes	—
105	C₂×C₄×D₅	C₂²×C₄	1	2	yes	—
106	C₂×Dic₁₀	C₂×C₈	1	1	yes	characteristic
107	C₅×M₁₆	C₂×C₈	1	4	yes	—
108	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
110	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
111	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
112	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
113	C₅×C₄⋊C₄	C₂×C₈	1	2	yes	—
114	C₅×C₂²⋊C₄	C₂×C₈	1	2	yes	—
115	C₄²×C₅	C₂²×C₄	1	2	yes	—
116	#685	C₂×C₈	1	4	yes	—
117	#683	C₂×C₈	1	4	yes	—
118	C₄×Dic₅	C₂×C₈	1	4	yes	—
119	C₄×Dic₅	C₂×C₈	1	2	yes	—
120	C₄×Dic₅	C₂²×C₄	1	1	yes	characteristic
121	C₄×Dic₅	C₂²×C₄	1	1	yes	characteristic
122	#681	C₄²	1	4	yes	—
123	C₂×#168	C₂×C₈	1	4	yes	—
124	C₂×#168	C₂²×C₄	1	2	yes	—
125	C₂×#168	C₂²×C₄	1	1	yes	characteristic
126	C₂×#168	C₂²×C₄	1	1	yes	characteristic
127	#676	C₄²	1	8	yes	—
128	#676	C₂×C₈	1	8	yes	—
129	C₈×D₅	C₂×C₈	1	8	yes	—
130	C₈×D₅	C₂²×C₄	1	4	yes	—
131	C₈×D₅	C₂²×C₄	1	4	yes	—
132	C₅×C₁₆	C₂²×C₄	1	2	yes	—
133	C₅×C₁₆	C₂²×C₄	1	4	yes	—
134	C₅×C₁₆	C₂²×C₄	1	2	yes	—
135	#672	C₂²×C₄	1	2	yes	—
136	#672	C₂²×C₄	1	4	yes	—
137	#672	C₂²×C₄	1	2	yes	—
138	C₂×#131	—	5	1	no	—
139	C₂×#131	—	5	1	no	—
140	C₄×#41	—	5	1	no	—
141	#504	—	5	8	no	—
142	#504	—	5	8	no	—
143	C₂×M₃₂	—	5	1	no	—
144	C₂×M₃₂	—	5	2	no	—
145	C₂×M₃₂	—	5	2	no	—
146	C₂×M₃₂	C₂×D₅	1	1	yes	characteristic
147	C₂²×C₁₆	—	5	1	no	—
148	C₂²×C₁₆	—	5	2	no	—
149	C₂²×C₁₆	—	5	2	no	—
150	C₂²×C₁₆	C₂×D₅	1	1	yes	characteristic
151	#405	—	5	1	no	—
152	#405	—	5	2	no	—
153	#405	—	5	2	no	—
154	#405	C₂×D₅	1	1	yes	characteristic
155	C₄×M₁₆	—	5	1	no	—
156	C₄×M₁₆	—	5	2	no	—
157	C₂×C₄×C₈	—	5	1	no	—
158	C₂×C₄×C₈	—	5	2	no	—
159	#346	—	5	2	no	—
160	#346	—	5	2	no	—
161	#346	C₂×D₅	1	2	yes	—
162	C₄×C₁₆	—	5	1	no	—
163	C₄×C₁₆	—	5	2	no	—
164	C₄×C₁₆	—	5	1	no	—
165	C₄×C₁₆	—	5	1	no	—
166	C₄×C₁₆	—	5	1	no	—
167	C₄×C₁₆	—	5	2	no	—
168	C₄×C₁₆	C₂×D₅	1	2	yes	—
169	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
170	C₂²×D₅	C₂²×C₈	1	2	yes	—
171	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
172	C₂×C₄×C₅	C₂²×C₈	1	4	yes	—
173	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
174	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
175	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
176	C₂²⋊D₅	C₄×C₈	1	8	yes	—
177	C₂×Dic₅	—	2	4	no	—
178	C₂×Dic₅	C₂²×C₈	1	2	yes	—
179	C₂×Dic₅	C₂²×C₈	1	2	yes	—
180	C₂×Dic₅	C₂²×C₈	1	2	yes	—
181	D₂₀	C₄×C₈	1	4	yes	—
182	C₄×D₅	—	2	8	no	—
183	C₄×D₅	C₂×C₄²	1	4	yes	—
184	C₄×D₅	C₂²×C₈	1	4	yes	—
185	Dic₁₀	C₄×C₈	1	4	yes	—
186	C₅×C₈	C₂×C₄²	1	2	yes	—
187	C₅×C₈	C₂²×C₈	1	4	yes	—
188	C₅×C₈	C₂³×C₄	1	1	yes	characteristic
189	C₅×C₈	C₂³×C₄	1	1	yes	characteristic
190	#168	C₂×C₄²	1	4	yes	—
191	#168	C₂²×C₈	1	2	yes	—
192	#168	C₂²×C₈	1	2	yes	—
193	C₂×#41	—	5	1	no	—
194	#131	—	5	8	no	—
195	#131	—	5	4	no	—
196	#131	—	5	4	no	—
197	C₂×M₁₆	—	5	1	no	—
198	C₂×M₁₆	—	5	2	no	—
199	C₂×M₁₆	—	5	2	no	—
200	C₂×M₁₆	C₄×D₅	1	1	yes	characteristic
201	C₂²×C₈	—	5	1	no	—
202	C₂²×C₈	—	5	2	no	—
203	C₂²×C₈	—	5	2	no	—
204	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
205	C₄×Q₈	—	5	2	no	—
206	C₄×D₄	—	5	2	no	—
207	C₄×D₄	—	5	4	no	—
208	#117	—	5	2	no	—
209	#117	C₄×D₅	1	1	yes	characteristic
210	C₂×C₄²	—	5	1	no	—
211	C₂×C₄²	—	5	2	no	—
212	M₃₂	—	5	4	no	—
213	M₃₂	—	5	8	no	—
214	M₃₂	—	5	8	no	—
215	M₃₂	C₄×D₅	1	4	yes	complemented
216	C₂×C₁₆	—	5	4	no	—
217	C₂×C₁₆	—	5	1	no	—
218	C₂×C₁₆	—	5	1	no	—
219	C₂×C₁₆	—	5	2	no	—
220	C₂×C₁₆	—	5	8	no	—
221	C₂×C₁₆	—	5	8	no	—
222	C₂×C₁₆	C₄×D₅	1	4	yes	complemented
223	C₂×C₁₆	C₂²×D₅	1	1	yes	characteristic
224	C₂×C₁₆	C₂²×D₅	1	2	yes	—
225	C₂×C₁₆	C₂²×D₅	1	1	yes	characteristic
226	#97	—	5	2	no	—
227	#97	—	5	2	no	—
228	#97	C₄×D₅	1	2	yes	—
229	C₄×C₈	—	5	1	no	—
230	C₄×C₈	—	5	2	no	—
231	C₄×C₈	—	5	1	no	—
232	C₄×C₈	—	5	1	no	—
233	C₄×C₈	—	5	1	no	—
234	C₄×C₈	—	5	2	no	—
235	C₄×C₈	C₂²×D₅	1	2	yes	—
236	C₂²×C₅	—	2	2	no	—
237	C₂²×C₅	C₂×C₄×C₈	1	2	yes	—
238	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
239	C₂×D₅	—	2	4	no	—
240	C₂×D₅	C₂×C₄×C₈	1	4	yes	—
241	C₄×C₅	—	2	4	no	—
242	C₄×C₅	C₂×C₄×C₈	1	2	yes	—
243	C₄×C₅	C₂²×C₄²	1	1	yes	characteristic
244	C₄×C₅	C₂³×C₈	1	1	yes	characteristic
245	Dic₅	—	2	2	no	—
246	Dic₅	C₂×C₄×C₈	1	4	yes	—
247	Dic₅	#504	1	4	yes	complemented
248	#41	—	5	4	no	—
249	#41	—	5	4	no	—
250	C₂×Q₈	—	5	1	no	—
251	C₂×D₄	—	5	2	no	—
252	C₂×D₄	—	5	1	no	—
253	C₂²×C₄	—	5	2	no	—
254	C₂²×C₄	—	5	2	no	—
255	C₂²×C₄	—	5	4	no	—
256	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
257	M₁₆	—	5	4	no	—
258	M₁₆	—	5	8	no	—
259	M₁₆	—	5	8	no	—
260	M₁₆	C₈×D₅	1	4	yes	—
261	C₂×C₈	—	5	1	no	—
262	C₂×C₈	—	5	2	no	—
263	C₂×C₈	—	5	1	no	—
264	C₂×C₈	—	5	4	no	—
265	C₂×C₈	—	5	4	no	—
266	C₂×C₈	—	5	4	no	—
267	C₂×C₈	—	5	8	no	—
268	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
269	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
270	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
271	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
272	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
273	C₄⋊C₄	—	5	4	no	—
274	C₄⋊C₄	C₈×D₅	1	2	yes	—
275	C₂²⋊C₄	—	5	4	no	—
276	C₂²⋊C₄	C₈×D₅	1	2	yes	—
277	C₄²	—	5	1	no	—
278	C₄²	—	5	1	no	—
279	C₄²	—	5	4	no	—
280	C₄²	—	5	2	no	—
281	C₄²	C₂×C₄×D₅	1	2	yes	—
282	C₁₆	—	5	2	no	—
283	C₁₆	—	5	2	no	—
284	C₁₆	—	5	4	no	—
285	C₁₆	C₂×C₄×D₅	1	2	yes	—
286	C₁₆	C₂×C₄×D₅	1	4	yes	—
287	C₁₆	C₂×C₄×D₅	1	2	yes	—
288	C₂×C₅	—	2	2	no	—
289	C₂×C₅	C₂²×C₄×C₈	1	1	yes	characteristic
290	C₂×C₅	C₂×#504	1	2	yes	—
291	D₅	—	2	4	no	—
292	C₂³	—	5	2	no	—
293	C₂³	C₂×C₈×D₅	1	1	yes	characteristic
294	Q₈	—	5	4	no	—
295	D₄	—	5	8	no	—
296	D₄	—	5	4	no	—
297	C₂×C₄	—	5	2	no	—
298	C₂×C₄	—	5	2	no	—
299	C₂×C₄	—	5	4	no	—
300	C₂×C₄	—	5	4	no	—
301	C₂×C₄	—	5	2	no	—
302	C₂×C₄	—	10	8	no	—
303	C₂×C₄	—	10	4	no	—
304	C₂×C₄	C₄²×D₅	1	2	yes	—
305	C₂×C₄	C₂×C₈×D₅	1	4	yes	—
306	C₂×C₄	C₂×C₈×D₅	1	2	yes	—
307	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
308	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
309	C₈	—	5	4	no	—
310	C₈	—	5	2	no	—
311	C₈	—	5	2	no	—
312	C₈	C₄²×D₅	1	2	yes	—
313	C₈	C₂×C₈×D₅	1	4	yes	—
314	C₈	C₂²×C₄×D₅	1	1	yes	characteristic
315	C₈	C₂²×C₄×D₅	1	1	yes	characteristic
316	C₅	C₄×#504	1	1	yes	characteristic, complemented, Sylow
317	C₂²	—	2	2	no	—
318	C₂²	—	5	4	no	—
319	C₂²	—	10	4	no	—
320	C₂²	C₄×C₈×D₅	1	2	yes	—
321	C₂²	C₂²×C₈×D₅	1	1	yes	characteristic
322	C₄	—	2	4	no	—
323	C₄	—	5	4	no	—
324	C₄	—	5	4	no	—
325	C₄	—	10	2	no	—
326	C₄	C₄×C₈×D₅	1	2	yes	—
327	C₄	C₂×C₄²×D₅	1	1	yes	characteristic
328	C₄	C₂²×C₈×D₅	1	1	yes	characteristic
329	C₂	—	2	2	no	—
330	C₂	—	10	4	no	—
331	C₂	C₂×C₄×C₈×D₅	1	1	yes	characteristic
332	C₂	D₅×#504	1	2	yes	—

	subgroup	G/H
center	C₂×C₁₆	C₂²×D₅
commutator	C₂×C₅	C₂²×C₄×C₈
Frattini	C₂×C₈	C₂³×D₅
Fitting	C₅×#2108	C₂
socle	C₂²×C₅	C₂³×C₈
prime radical	C₂×#709	C₂×C₄
cosocle	C₂×C₅×C₈	C₂⁴
hypercenter	#2108	D₅
hypocenter	C₅	C₄×#504
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₁₆ ◁ #2108

Complements

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

N	H ≅ G/N	type
4. C₅×#2108	C₂	semidirect
5. Dic₅×M₃₂	C₂	semidirect
6. C₂×C₁₆×Dic₅	C₂	semidirect
7. #10600674	C₂	semidirect
8. #10599865	C₂	semidirect
9. #10599864	C₂	semidirect
10. #10599844	C₂	semidirect
11. C₄×D₅×C₁₆	C₂	semidirect
23. #65196	C₄	semidirect
24. #65190	C₄	semidirect
63. C₅×M₃₂	C₂×C₄	semidirect
64. C₂×C₅×C₁₆	C₂×C₄	semidirect
76. #3944	C₂×C₄	semidirect
77. C₂×#672	C₂×C₄	semidirect
83. #3930	C₂×C₄	semidirect
84. #3930	C₂×C₄	semidirect
85. D₅×C₁₆	C₂×C₄	semidirect
86. D₅×C₁₆	C₂×C₄	semidirect
93. #2108	D₅	semidirect
215. M₃₂	C₄×D₅	semidirect
222. C₂×C₁₆	C₄×D₅	semidirect
247. Dic₅	#504	semidirect
316. C₅	C₄×#504	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
C₂×#65190	2
#10610280	2
C₅×#2108	2
Dic₅×M₃₂	2
C₂×C₁₆×Dic₅	2
#10600674	2
#10599865	2
#10599864	2
#10599844	2
C₄×D₅×C₁₆	2
C₄×#504	5