Subgroups — #12253874

Order 1280.

642 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#508	C₂	1	1	yes	maximal, characteristic, complemented
2	#10615659	C₂	1	2	yes	maximal, complemented
3	C₂×#65190	C₂	1	1	yes	maximal, characteristic, complemented
4	#10602874	C₂	1	2	yes	maximal, complemented
5	#10602300	C₂	1	2	yes	maximal, complemented
6	#10602284	C₂	1	4	yes	maximal, complemented
7	#10602265	C₂	1	2	yes	maximal, complemented
8	C₂×#64726	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₅×#135	C₂²	1	2	yes	—
10	#66156	—	2	2	no	—
11	C₂×#4047	C₂²	1	1	yes	characteristic
12	C₅×#508	—	2	1	no	—
13	C₅×#508	—	2	1	no	—
14	C₅×#508	—	2	1	no	—
15	C₅×#508	—	2	1	no	—
16	C₂×C₅×Q₃₂	C₂²	1	1	yes	characteristic, complemented
17	C₂×C₅×SD₃₂	C₂²	1	2	yes	complemented
18	C₂×C₅×D₁₆	C₂²	1	1	yes	characteristic, complemented
19	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
20	#65526	—	2	2	no	—
21	#65481	C₂²	1	4	yes	—
22	#65466	—	2	4	no	—
23	#65449	C₂²	1	4	yes	—
24	C₂×#3951	C₂²	1	2	yes	—
25	C₂×#3944	C₂²	1	1	yes	characteristic
26	#65190	—	2	1	no	—
27	#65190	—	2	1	no	—
28	#65190	C₂²	1	2	yes	—
29	#65190	C₂²	1	2	yes	—
30	C₂×#3930	—	2	1	no	—
31	C₂×D₅×C₁₆	—	2	1	no	—
32	#64785	C₂²	1	4	yes	complemented
33	#64783	C₂²	1	4	yes	complemented
34	#64726	C₂²	1	2	yes	complemented
35	#64726	C₂²	1	2	yes	complemented
36	#33617	—	5	1	no	maximal, Sylow
37	C₂×C₅×#41	—	2	2	no	—
38	C₂×#709	D₄	1	1	yes	characteristic
39	C₅×#135	—	2	2	no	—
40	C₅×#135	—	2	2	no	—
41	C₅×#135	—	2	2	no	—
42	C₅×#135	—	2	2	no	—
43	C₂×C₅×Q₁₆	C₂³	1	2	yes	—
44	C₂×C₅×SD₁₆	—	2	2	no	—
45	C₂×C₅×D₈	C₂³	1	2	yes	—
46	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
47	#4097	—	4	2	no	—
48	#4096	—	4	4	no	—
49	#4095	—	4	2	no	—
50	#4088	—	2	4	no	—
51	#4078	—	2	4	no	—
52	C₂×#681	—	2	2	no	—
53	C₂×#681	D₄	1	1	yes	characteristic
54	#4047	—	2	1	no	—
55	#4047	—	2	1	no	—
56	#4047	D₄	1	2	yes	—
57	#4047	C₂³	1	2	yes	—
58	C₂×#676	—	2	1	no	—
59	C₂×C₈×D₅	—	2	1	no	—
60	C₅×Q₃₂	—	2	1	no	—
61	C₅×Q₃₂	—	2	1	no	—
62	C₅×SD₃₂	—	2	2	no	—
63	C₅×SD₃₂	—	2	2	no	—
64	C₅×D₁₆	—	2	1	no	—
65	C₅×D₁₆	—	2	1	no	—
66	C₂×C₅×C₁₆	D₄	1	1	yes	characteristic
67	C₂×C₅×C₁₆	D₄	1	1	yes	characteristic
68	C₂×C₅×C₁₆	D₄	1	1	yes	characteristic
69	C₂×C₅×C₁₆	D₄	1	1	yes	characteristic
70	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
71	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
72	#3968	—	2	4	no	—
73	#3965	—	2	4	no	—
74	#3951	—	2	2	no	—
75	#3951	C₂³	1	4	yes	—
76	#3944	—	2	1	no	—
77	#3944	C₂³	1	2	yes	—
78	C₂×#672	—	2	1	no	—
79	#3930	—	2	2	no	—
80	#3930	—	2	2	no	—
81	D₅×C₁₆	—	2	2	no	—
82	D₅×C₁₆	—	2	2	no	—
83	C₂×#508	D₅	1	1	yes	characteristic, complemented
84	C₂×#504	—	5	1	no	—
85	#3067	—	5	2	no	—
86	#2223	—	5	4	no	—
87	#2222	—	5	2	no	—
88	#2221	—	5	2	no	—
89	C₂×#368	—	5	1	no	—
90	#2148	—	5	2	no	—
91	C₅×#41	—	4	2	no	—
92	C₅×#41	—	4	2	no	—
93	C₅×#41	—	4	2	no	—
94	C₅×#41	—	4	2	no	—
95	C₂×C₅×Q₈	—	2	2	no	—
96	C₂×C₅×D₄	—	2	2	no	—
97	C₂×C₅×D₄	—	4	2	no	—
98	C₂²×C₄×C₅	—	4	2	no	—
99	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
100	C₂×C₂²⋊D₅	—	2	1	no	—
101	#709	—	2	1	no	—
102	#709	—	2	1	no	—
103	#709	D₈	1	2	yes	—
104	#709	C₂×D₄	1	2	yes	—
105	C₂×D₂₀	D₈	1	1	yes	characteristic
106	C₂×C₄×D₅	—	2	1	no	—
107	C₂×Dic₁₀	D₈	1	1	yes	characteristic
108	C₅×Q₁₆	—	2	2	no	—
109	C₅×Q₁₆	—	2	2	no	—
110	C₅×SD₁₆	—	4	2	no	—
111	C₅×SD₁₆	—	4	2	no	—
112	C₅×D₈	—	2	2	no	—
113	C₅×D₈	—	2	2	no	—
114	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
115	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
116	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
117	C₂×C₅×C₈	#41	1	1	yes	characteristic
118	C₂×C₅×C₈	#41	1	1	yes	characteristic
119	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
120	#689	—	4	4	no	—
121	#688	—	4	4	no	—
122	#687	—	4	4	no	—
123	#686	—	4	4	no	—
124	#681	—	2	1	no	—
125	#681	—	2	4	no	—
126	#681	—	4	2	no	—
127	#681	C₂×D₄	1	2	yes	—
128	C₂×#168	—	2	1	no	—
129	C₂×#168	—	4	2	no	—
130	#676	—	2	2	no	—
131	#676	—	2	2	no	—
132	C₈×D₅	—	2	2	no	—
133	C₈×D₅	—	2	2	no	—
134	C₅×C₁₆	C₂×D₄	1	1	yes	characteristic
135	C₅×C₁₆	C₂×D₄	1	1	yes	characteristic
136	C₅×C₁₆	C₂×D₄	1	1	yes	characteristic
137	C₅×C₁₆	C₂×D₄	1	1	yes	characteristic
138	#672	—	2	2	no	—
139	#575	—	10	2	no	—
140	C₂×#135	C₂×D₅	1	2	yes	—
141	C₂×#131	—	5	1	no	—
142	#508	—	2	1	no	—
143	#508	—	2	1	no	—
144	#508	—	2	1	no	—
145	#508	—	2	1	no	—
146	C₂×Q₃₂	C₂×D₅	1	1	yes	characteristic, complemented
147	C₂×SD₃₂	C₂×D₅	1	2	yes	complemented
148	C₂×D₁₆	C₂×D₅	1	1	yes	characteristic, complemented
149	#504	—	5	2	no	—
150	#504	—	5	2	no	—
151	#504	—	10	1	no	—
152	#504	—	10	1	no	—
153	C₂×M₃₂	—	5	1	no	—
154	C₂×M₃₂	—	10	1	no	—
155	C₂²×C₁₆	—	10	1	no	—
156	C₂²×C₁₆	C₂×D₅	1	1	yes	characteristic
157	#473	—	5	4	no	—
158	#472	—	5	4	no	—
159	#471	—	10	4	no	—
160	C₂×#108	—	5	2	no	—
161	#413	—	10	2	no	—
162	#368	—	5	2	no	—
163	#368	—	5	2	no	—
164	#362	—	5	4	no	—
165	#361	—	5	4	no	—
166	C₂³×C₅	—	4	2	no	—
167	C₂³×C₅	#135	1	1	yes	characteristic
168	C₂²×D₅	—	2	1	no	—
169	C₅×Q₈	—	4	2	no	—
170	C₅×Q₈	—	4	2	no	—
171	C₅×D₄	—	4	2	no	—
172	C₅×D₄	—	4	2	no	—
173	C₅×D₄	—	8	2	no	—
174	C₅×D₄	—	8	2	no	—
175	C₂×C₄×C₅	—	4	2	no	—
176	C₂×C₄×C₅	—	8	2	no	—
177	C₂×C₄×C₅	—	8	2	no	—
178	C₂×C₄×C₅	#121	1	1	yes	characteristic
179	C₂×C₄×C₅	#121	1	1	yes	characteristic
180	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
181	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
182	C₂×C₄×C₅	#135	1	1	yes	characteristic
183	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
184	C₂²⋊D₅	—	2	2	no	—
185	C₂²⋊D₅	—	2	2	no	—
186	C₂×Dic₅	—	2	1	no	—
187	D₂₀	—	2	1	no	—
188	D₂₀	C₂×D₈	1	2	yes	—
189	C₄×D₅	—	2	2	no	—
190	C₄×D₅	—	2	2	no	—
191	Dic₁₀	—	2	1	no	—
192	Dic₁₀	C₂×D₈	1	2	yes	—
193	C₅×C₈	#121	1	1	yes	characteristic
194	C₅×C₈	#121	1	1	yes	characteristic
195	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
196	C₅×C₈	C₂×#41	1	1	yes	characteristic
197	#168	—	2	2	no	—
198	#168	—	4	4	no	—
199	C₂×#41	—	2	2	no	—
200	C₂×#41	—	5	1	no	—
201	#137	—	10	4	no	—
202	#137	—	20	2	no	—
203	AGL(1,ℤ/8)	—	10	4	no	—
204	AGL(1,ℤ/8)	—	20	2	no	—
205	#135	—	2	2	no	—
206	#135	—	2	2	no	—
207	#135	—	2	2	no	—
208	#135	—	2	2	no	—
209	#135	—	20	4	no	—
210	C₂×Q₁₆	C₂²×D₅	1	2	yes	—
211	C₂×SD₁₆	—	2	2	no	—
212	C₂×D₈	C₂²×D₅	1	2	yes	—
213	#131	—	5	2	no	—
214	#131	—	5	2	no	—
215	#131	—	10	1	no	—
216	#131	—	10	1	no	—
217	C₂×M₁₆	—	5	1	no	—
218	C₂×M₁₆	—	10	1	no	—
219	C₂×M₁₆	—	10	2	no	—
220	C₂²×C₈	—	10	1	no	—
221	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
222	Q₃₂	—	2	1	no	—
223	Q₃₂	—	2	1	no	—
224	SD₃₂	—	2	2	no	—
225	SD₃₂	—	2	2	no	—
226	D₁₆	—	2	1	no	—
227	D₁₆	—	2	1	no	—
228	M₃₂	—	5	2	no	—
229	M₃₂	—	10	1	no	—
230	M₃₂	—	10	2	no	—
231	M₃₂	—	10	2	no	—
232	C₂×C₁₆	—	10	1	no	—
233	C₂×C₁₆	—	10	2	no	—
234	C₂×C₁₆	—	10	2	no	—
235	C₂×C₁₆	C₂²⋊D₅	1	1	yes	characteristic
236	C₂×C₁₆	C₂²⋊D₅	1	1	yes	characteristic
237	C₂×C₁₆	C₂²⋊D₅	1	1	yes	characteristic
238	C₂×C₁₆	C₂²⋊D₅	1	1	yes	characteristic
239	C₂×C₁₆	C₂²×D₅	1	1	yes	characteristic
240	C₂×C₁₆	C₂²×D₅	1	1	yes	characteristic
241	#108	—	5	4	no	—
242	#108	—	10	2	no	—
243	#101	—	10	4	no	—
244	#100	—	10	4	no	—
245	C₂²×C₅	—	4	1	no	—
246	C₂²×C₅	—	8	2	no	—
247	C₂²×C₅	—	16	2	no	—
248	C₂²×C₅	#466	1	1	yes	characteristic
249	C₂²×C₅	#466	1	1	yes	characteristic
250	C₂²×C₅	C₂×#135	1	1	yes	characteristic
251	C₂×D₅	—	2	2	no	—
252	C₂×D₅	—	4	2	no	—
253	C₄×C₅	—	8	2	no	—
254	C₄×C₅	#466	1	1	yes	characteristic
255	C₄×C₅	#466	1	1	yes	characteristic
256	C₄×C₅	C₂×#121	1	1	yes	characteristic
257	C₄×C₅	C₂²×D₈	1	1	yes	characteristic
258	Dic₅	—	2	2	no	—
259	#41	—	4	2	no	—
260	#41	—	4	2	no	—
261	#41	—	4	2	no	—
262	#41	—	4	2	no	—
263	#41	—	5	2	no	—
264	#41	—	5	2	no	—
265	#41	—	10	1	no	—
266	#41	—	10	1	no	—
267	C₂×Q₈	—	2	2	no	—
268	C₂×Q₈	—	5	1	no	—
269	C₂×D₄	—	2	2	no	—
270	C₂×D₄	—	4	2	no	—
271	C₂×D₄	—	5	1	no	—
272	C₂×D₄	—	10	1	no	—
273	C₂²×C₄	—	4	2	no	—
274	C₂²×C₄	—	10	1	no	—
275	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
276	Q₁₆	—	2	2	no	—
277	Q₁₆	—	2	2	no	—
278	Q₁₆	—	20	4	no	—
279	SD₁₆	—	4	2	no	—
280	SD₁₆	—	4	2	no	—
281	SD₁₆	—	20	4	no	—
282	SD₁₆	—	20	4	no	—
283	D₈	—	2	2	no	—
284	D₈	—	2	2	no	—
285	D₈	—	20	4	no	—
286	M₁₆	—	5	2	no	—
287	M₁₆	—	10	1	no	—
288	M₁₆	—	10	2	no	—
289	M₁₆	—	10	2	no	—
290	M₁₆	—	10	4	no	—
291	M₁₆	—	20	2	no	—
292	C₂×C₈	—	10	1	no	—
293	C₂×C₈	—	10	2	no	—
294	C₂×C₈	—	10	2	no	—
295	C₂×C₈	—	20	2	no	—
296	C₂×C₈	D₄×D₅	1	1	yes	characteristic
297	C₂×C₈	#711	1	1	yes	characteristic
298	C₂×C₈	#711	1	1	yes	characteristic
299	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
300	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
301	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
302	C₁₆	—	10	2	no	—
303	C₁₆	C₂×C₂²⋊D₅	1	1	yes	characteristic
304	C₁₆	C₂×C₂²⋊D₅	1	1	yes	characteristic
305	C₁₆	C₂×C₂²⋊D₅	1	1	yes	characteristic
306	C₁₆	C₂×C₂²⋊D₅	1	1	yes	characteristic
307	C₂×C₅	—	2	1	no	—
308	C₂×C₅	—	2	1	no	—
309	C₂×C₅	—	2	1	no	—
310	C₂×C₅	—	16	2	no	—
311	C₂×C₅	C₂×#466	1	1	yes	characteristic
312	D₅	—	4	2	no	—
313	C₂³	—	4	2	no	—
314	C₂³	—	10	1	no	—
315	C₂³	#4058	1	1	yes	characteristic
316	Q₈	—	4	2	no	—
317	Q₈	—	4	2	no	—
318	Q₈	—	5	2	no	—
319	Q₈	—	10	1	no	—
320	D₄	—	4	2	no	—
321	D₄	—	4	2	no	—
322	D₄	—	5	2	no	—
323	D₄	—	8	2	no	—
324	D₄	—	8	2	no	—
325	D₄	—	10	2	no	—
326	D₄	—	10	2	no	—
327	D₄	—	10	1	no	—
328	C₂×C₄	—	4	2	no	—
329	C₂×C₄	—	8	2	no	—
330	C₂×C₄	—	8	2	no	—
331	C₂×C₄	—	10	1	no	—
332	C₂×C₄	—	10	2	no	—
333	C₂×C₄	—	10	2	no	—
334	C₂×C₄	D₅×D₈	1	1	yes	characteristic
335	C₂×C₄	D₅×D₈	1	1	yes	characteristic
336	C₂×C₄	#4058	1	1	yes	characteristic
337	C₂×C₄	#4084	1	1	yes	characteristic
338	C₂×C₄	#4084	1	1	yes	characteristic
339	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
340	C₈	—	10	2	no	—
341	C₈	—	20	4	no	—
342	C₈	#4084	1	1	yes	characteristic
343	C₈	#4084	1	1	yes	characteristic
344	C₈	C₂×#711	1	1	yes	characteristic
345	C₈	C₂²×C₂²⋊D₅	1	1	yes	characteristic
346	C₅	#33617	1	1	yes	characteristic, complemented, Sylow
347	C₂²	—	4	1	no	—
348	C₂²	—	8	2	no	—
349	C₂²	—	10	2	no	—
350	C₂²	—	16	2	no	—
351	C₂²	—	20	2	no	—
352	C₂²	#65446	1	1	yes	characteristic
353	C₂²	#65446	1	1	yes	characteristic
354	C₂²	C₂×#4058	1	1	yes	characteristic
355	C₄	—	8	2	no	—
356	C₄	—	10	2	no	—
357	C₄	#65446	1	1	yes	characteristic
358	C₄	#65446	1	1	yes	characteristic
359	C₄	C₂×D₅×D₈	1	1	yes	characteristic
360	C₄	C₂×#4084	1	1	yes	characteristic
361	C₂	—	2	1	no	—
362	C₂	—	2	1	no	—
363	C₂	—	2	1	no	—
364	C₂	—	16	2	no	—
365	C₂	—	20	2	no	—
366	C₂	C₂×#65446	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#65446
commutator, cosocle	C₂×C₅×C₈	C₂⁴
Frattini	C₂×C₈	C₂³×D₅
Fitting	C₂×C₅×#508	C₂
socle	C₂×C₅	C₂×#466
hypercenter	C₂×#508	D₅
hypocenter	C₅	#33617
LCS	G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#508

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×#508	C₂	semidirect
2. #10615659	C₂	semidirect
3. C₂×#65190	C₂	semidirect
4. #10602874	C₂	semidirect
5. #10602300	C₂	semidirect
6. #10602284	C₂	semidirect
7. #10602265	C₂	semidirect
8. C₂×#64726	C₂	semidirect
16. C₂×C₅×Q₃₂	C₂²	semidirect
17. C₂×C₅×SD₃₂	C₂²	semidirect
18. C₂×C₅×D₁₆	C₂²	semidirect
32. #64785	C₂²	semidirect
33. #64783	C₂²	semidirect
34. #64726	C₂²	semidirect
35. #64726	C₂²	semidirect
83. C₂×#508	D₅	semidirect
146. C₂×Q₃₂	C₂×D₅	semidirect
147. C₂×SD₃₂	C₂×D₅	semidirect
148. C₂×D₁₆	C₂×D₅	semidirect
346. C₅	#33617	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#508	2
#10615659	2
C₂×#65190	2
#10602874	2
#10602300	2
#10602284	2
#10602265	2
C₂×#64726	2
#33617	5