Subgroups — #24918598

Order 10624.

370 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128257	C₂	1	1	yes	maximal, characteristic, complemented
2	#19128248	C₂	1	1	yes	maximal, characteristic
3	#19128105	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128098	C₂	1	1	yes	maximal, characteristic, complemented
5	#19128046	C₂	1	1	yes	maximal, characteristic, complemented
6	#19128046	C₂	1	1	yes	maximal, characteristic, complemented
7	#19128028	C₂	1	1	yes	maximal, characteristic, complemented
8	#19128027	C₂	1	1	yes	maximal, characteristic, complemented
9	#19128024	C₂	1	1	yes	maximal, characteristic, complemented
10	#19128023	C₂	1	1	yes	maximal, characteristic, complemented
11	#19128002	C₂	1	1	yes	maximal, characteristic, complemented
12	#19128001	C₂	1	1	yes	maximal, characteristic, complemented
13	#19127998	C₂	1	1	yes	maximal, characteristic, complemented
14	#19127753	C₂	1	1	yes	maximal, characteristic, complemented
15	#478×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13050590	—	2	1	no	—
17	#14581791	C₂²	1	1	yes	characteristic
18	#14581791	C₂²	1	1	yes	characteristic
19	#14581791	C₂²	1	1	yes	characteristic
20	#14581788	C₂²	1	1	yes	characteristic
21	#14581784	C₂²	1	1	yes	characteristic
22	#14581778	—	2	1	no	—
23	#14581770	—	2	1	no	—
24	#14581770	—	2	1	no	—
25	#14581767	C₂²	1	1	yes	characteristic
26	#14581765	—	2	1	no	—
27	#14581763	C₂²	1	1	yes	characteristic
28	#14581757	—	2	1	no	—
29	#14581757	—	2	1	no	—
30	C₄×D₃₃₂	C₂²	1	1	yes	characteristic
31	C₄×C₂²⋊D₈₃	—	2	1	no	—
32	#14581750	C₂²	1	1	yes	characteristic
33	#14581750	C₂²	1	1	yes	characteristic
34	C₄²×D₈₃	C₂²	1	1	yes	characteristic
35	#14581726	C₂²	1	1	yes	characteristic
36	#14581725	C₂²	1	1	yes	characteristic
37	#14581725	C₂²	1	1	yes	characteristic
38	#14581724	C₂²	1	1	yes	characteristic
39	#14581723	C₂²	1	1	yes	characteristic
40	#14581723	C₂²	1	1	yes	characteristic
41	C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
42	#14581714	C₂²	1	1	yes	characteristic
43	#14581714	C₂²	1	1	yes	characteristic
44	#14581713	C₂²	1	1	yes	characteristic
45	#14581713	C₂²	1	1	yes	characteristic
46	#14581712	C₂²	1	1	yes	characteristic
47	#14581712	C₂²	1	1	yes	characteristic
48	(C₂×D₈₃)⋊C₈	C₂²	1	1	yes	characteristic
49	(C₂×D₈₃)⋊C₈	C₂²	1	1	yes	characteristic
50	C₈×Dic₈₃	C₂²	1	1	yes	characteristic
51	C₈×Dic₈₃	C₂²	1	1	yes	characteristic
52	#125×C₈₃	C₂²	1	1	yes	characteristic
53	C₄×D₄×C₈₃	C₂²	1	1	yes	characteristic
54	#107×C₈₃	C₂²	1	1	yes	characteristic
55	#106×C₈₃	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
57	#102×C₈₃	C₂²	1	1	yes	characteristic
58	#102×C₈₃	C₂²	1	1	yes	characteristic
59	#13050590	—	4	4	no	—
60	C₂×Dic₁₆₆	—	2	1	no	—
61	C₂×Dic₁₆₆	—	2	1	no	—
62	C₂×Dic₁₆₆	—	2	1	no	—
63	C₂×D₃₃₂	—	2	1	no	—
64	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
65	C₂×C₂²⋊D₈₃	—	2	1	no	—
66	C₂×C₂²⋊D₈₃	—	2	1	no	—
67	C₂×C₂²⋊D₈₃	—	2	1	no	—
68	C₂×C₄×D₈₃	—	2	1	no	—
69	C₂×C₄×D₈₃	D₄	1	1	yes	characteristic
70	C₂×C₄×D₈₃	D₄	1	1	yes	characteristic
71	C₂×C₄×D₈₃	C₂³	1	1	yes	characteristic
72	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
73	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
74	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
75	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
76	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
77	Dic₈₃⋊C₄	—	2	1	no	—
78	Dic₈₃⋊C₄	—	2	1	no	—
79	Dic₈₃⋊C₄	—	2	1	no	—
80	Dic₈₃⋊C₄	—	2	1	no	—
81	Dic₈₃⋊C₄	—	2	1	no	—
82	C₂²⋊Dic₈₃	—	2	1	no	—
83	(C₂×D₈₃)⋊C₄	—	2	1	no	—
84	(C₂×D₈₃)⋊C₄	—	2	1	no	—
85	(C₂×D₈₃)⋊C₄	—	2	1	no	—
86	(C₂×D₈₃)⋊C₄	—	2	1	no	—
87	C₄×Dic₈₃	—	2	1	no	—
88	C₄×Dic₈₃	—	2	1	no	—
89	C₄×Dic₈₃	D₄	1	1	yes	characteristic
90	C₄×Dic₈₃	D₄	1	1	yes	characteristic
91	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
92	C₂×D₄×C₈₃	C₂³	1	1	yes	characteristic
93	C₂²×C₄×C₈₃	—	2	1	no	—
94	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
95	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
96	C₄⋊C₄×C₈₃	—	2	1	no	—
97	C₄⋊C₄×C₈₃	—	2	1	no	—
98	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
99	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
100	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
101	C₂²⋊C₄×C₈₃	—	2	1	no	—
102	C₄²×C₈₃	C₂³	1	1	yes	characteristic
103	C₂³×C₈₃	—	2	1	no	—
104	C₂²×D₈₃	—	2	1	no	—
105	C₂²×D₈₃	C₂×D₄	1	1	yes	characteristic
106	D₄×C₈₃	—	2	1	no	—
107	D₄×C₈₃	—	2	1	no	—
108	C₂×C₄×C₈₃	—	2	1	no	—
109	C₂×C₄×C₈₃	—	2	1	no	—
110	C₂×C₄×C₈₃	—	2	1	no	—
111	C₂×C₄×C₈₃	—	4	2	no	—
112	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
115	C₂²⋊D₈₃	—	4	2	no	—
116	C₂²⋊D₈₃	—	4	2	no	—
117	C₂²⋊D₈₃	—	4	2	no	—
118	C₂×Dic₈₃	—	2	1	no	—
119	C₂×Dic₈₃	—	2	1	no	—
120	C₂×Dic₈₃	—	2	1	no	—
121	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
122	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
123	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
124	D₃₃₂	—	2	1	no	—
125	D₃₃₂	—	2	1	no	—
126	D₃₃₂	—	4	2	no	—
127	C₄×D₈₃	—	2	1	no	—
128	C₄×D₈₃	—	2	1	no	—
129	C₄×D₈₃	—	2	2	no	—
130	C₄×D₈₃	—	2	2	no	—
131	C₄×D₈₃	—	4	2	no	—
132	Dic₁₆₆	—	4	2	no	—
133	Dic₁₆₆	—	4	2	no	—
134	Dic₁₆₆	—	4	2	no	—
135	C₈×C₈₃	—	2	1	no	—
136	C₈×C₈₃	—	2	1	no	—
137	C₈₃⋊C₈	—	2	1	no	—
138	C₈₃⋊C₈	—	2	1	no	—
139	C₂²×C₈₃	—	4	1	no	—
140	C₂²×C₈₃	—	4	1	no	—
141	C₂²×C₈₃	—	4	1	no	—
142	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
143	C₂×D₈₃	—	2	1	no	—
144	C₂×D₈₃	—	2	1	no	—
145	C₂×D₈₃	—	2	1	no	—
146	C₂×D₈₃	—	4	1	no	—
147	C₂×D₈₃	—	4	1	no	—
148	C₂×D₈₃	—	4	1	no	—
149	C₄×C₈₃	—	2	1	no	—
150	C₄×C₈₃	—	2	1	no	—
151	C₄×C₈₃	—	4	1	no	—
152	C₄×C₈₃	—	4	1	no	—
153	C₄×C₈₃	—	4	1	no	—
154	C₄×C₈₃	2₋¹⁺⁴	1	1	yes	characteristic
155	C₄×C₈₃	2₋¹⁺⁴	1	1	yes	characteristic
156	Dic₈₃	—	2	1	no	—
157	Dic₈₃	—	4	1	no	—
158	Dic₈₃	—	4	1	no	—
159	Dic₈₃	—	4	1	no	—
160	Dic₈₃	#135	1	2	yes	—
161	Dic₈₃	AGL(1,ℤ/8)	1	2	yes	—
162	C₂×C₈₃	—	8	1	no	—
163	C₂×C₈₃	#536	1	1	yes	characteristic
164	C₂×C₈₃	C₂×#135	1	1	yes	characteristic
165	C₂×C₈₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
166	D₈₃	—	4	1	no	—
167	D₈₃	—	8	1	no	—
168	#3228	—	83	1	no	maximal, Sylow
169	C₈₃	#3228	1	1	yes	characteristic, complemented, Sylow
170	#533	—	83	1	no	—
171	#532	—	83	1	no	—
172	#500	—	83	1	no	—
173	#498	—	83	1	no	—
174	#486	—	83	1	no	—
175	#486	—	83	1	no	—
176	#482	—	83	1	no	—
177	#482	—	83	1	no	—
178	#482	—	83	1	no	—
179	#482	—	83	1	no	—
180	#478	—	83	1	no	—
181	#478	—	83	1	no	—
182	#478	—	83	1	no	—
183	#478	D₈₃	1	1	yes	characteristic, complemented
184	#431	—	83	1	no	—
185	C₂×#41	—	166	1	no	—
186	C₄⋊Q₈	—	83	1	no	—
187	C₄⋊Q₈	—	83	1	no	—
188	C₄⋊Q₈	—	83	1	no	—
189	#127	—	83	1	no	—
190	#125	—	83	1	no	—
191	#125	C₂×D₈₃	1	1	yes	characteristic
192	#124	—	166	1	no	—
193	C₂²⋊Q₈	—	166	1	no	—
194	C₂²⋊Q₈	—	166	1	no	—
195	#121	—	83	1	no	—
196	#121	—	83	1	no	—
197	#121	—	166	1	no	—
198	C₄×Q₈	—	166	1	no	—
199	C₄×Q₈	—	166	1	no	—
200	C₄×D₄	—	83	1	no	—
201	C₄×D₄	—	166	1	no	—
202	C₄×D₄	C₂×D₈₃	1	1	yes	characteristic
203	#117	—	83	1	no	—
204	#117	—	83	1	no	—
205	C₂×C₄²	—	83	1	no	—
206	#107	—	83	1	no	—
207	#107	—	83	1	no	—
208	#107	—	83	1	no	—
209	#107	C₂×D₈₃	1	1	yes	characteristic
210	#106	—	83	1	no	—
211	#106	—	83	1	no	—
212	#106	—	83	1	no	—
213	#106	C₂×D₈₃	1	1	yes	characteristic
214	C₄⋊C₈	—	83	1	no	—
215	C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
216	#102	—	83	1	no	—
217	#102	—	83	1	no	—
218	#102	—	83	1	no	—
219	#102	—	83	1	no	—
220	#102	—	83	1	no	—
221	#102	—	83	1	no	—
222	#102	C₂×D₈₃	1	1	yes	characteristic
223	#102	C₂×D₈₃	1	1	yes	characteristic
224	C₂²⋊C₈	—	83	1	no	—
225	C₂²⋊C₈	—	83	1	no	—
226	C₄×C₈	—	83	1	no	—
227	C₄×C₈	—	83	1	no	—
228	#41	—	332	4	no	—
229	C₂×Q₈	—	166	1	no	—
230	C₂×Q₈	—	166	1	no	—
231	C₂×Q₈	—	166	1	no	—
232	C₂×D₄	—	83	1	no	—
233	C₂×D₄	—	166	1	no	—
234	C₂×D₄	—	166	1	no	—
235	C₂×D₄	—	166	1	no	—
236	C₂×D₄	—	166	1	no	—
237	C₂×D₄	C₂²×D₈₃	1	1	yes	characteristic
238	C₂²×C₄	—	2	1	no	—
239	C₂²×C₄	—	83	1	no	—
240	C₂²×C₄	—	83	1	no	—
241	C₂²×C₄	—	83	1	no	—
242	C₂²×C₄	—	166	1	no	—
243	C₂×C₈	—	83	1	no	—
244	C₂×C₈	—	83	1	no	—
245	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
246	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
247	C₄⋊C₄	—	2	1	no	—
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	83	1	no	—
250	C₄⋊C₄	—	83	1	no	—
251	C₄⋊C₄	—	83	1	no	—
252	C₄⋊C₄	—	166	1	no	—
253	C₄⋊C₄	—	166	1	no	—
254	C₄⋊C₄	—	166	1	no	—
255	C₄⋊C₄	—	166	1	no	—
256	C₄⋊C₄	—	166	1	no	—
257	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
258	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
259	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
260	C₂²⋊C₄	—	2	1	no	—
261	C₂²⋊C₄	—	166	1	no	—
262	C₂²⋊C₄	—	166	1	no	—
263	C₂²⋊C₄	—	166	1	no	—
264	C₂²⋊C₄	—	166	1	no	—
265	C₂²⋊C₄	—	166	1	no	—
266	C₄²	—	83	1	no	—
267	C₄²	—	83	1	no	—
268	C₄²	—	83	1	no	—
269	C₄²	—	166	1	no	—
270	C₄²	—	166	1	no	—
271	C₄²	C₂²×D₈₃	1	1	yes	characteristic
272	C₂³	—	2	1	no	—
273	C₂³	—	83	1	no	—
274	C₂³	—	166	1	no	—
275	Q₈	—	332	2	no	—
276	Q₈	—	332	2	no	—
277	Q₈	—	332	2	no	—
278	D₄	—	2	1	no	—
279	D₄	—	2	1	no	—
280	D₄	—	332	2	no	—
281	D₄	—	332	2	no	—
282	D₄	—	332	2	no	—
283	D₄	—	332	2	no	—
284	D₄	—	166	1	no	—
285	D₄	—	166	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	332	2	no	—
291	C₂×C₄	—	83	1	no	—
292	C₂×C₄	—	83	1	no	—
293	C₂×C₄	—	83	1	no	—
294	C₂×C₄	—	166	2	no	—
295	C₂×C₄	—	166	2	no	—
296	C₂×C₄	—	166	1	no	—
297	C₂×C₄	—	166	1	no	—
298	C₂×C₄	—	166	1	no	—
299	C₂×C₄	—	166	1	no	—
300	C₂×C₄	—	166	1	no	—
301	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
302	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
303	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
304	C₈	—	2	1	no	—
305	C₈	—	2	1	no	—
306	C₈	—	166	1	no	—
307	C₈	—	166	1	no	—
308	C₂²	—	4	1	no	—
309	C₂²	—	4	1	no	—
310	C₂²	—	4	1	no	—
311	C₂²	—	332	1	no	—
312	C₂²	—	332	1	no	—
313	C₂²	—	332	1	no	—
314	C₂²	—	166	1	no	—
315	C₂²	—	166	1	no	—
316	C₂²	—	166	1	no	—
317	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	4	1	no	—
321	C₄	—	4	1	no	—
322	C₄	—	4	1	no	—
323	C₄	—	332	1	no	—
324	C₄	—	332	1	no	—
325	C₄	—	332	1	no	—
326	C₄	—	83	2	no	—
327	C₄	—	83	2	no	—
328	C₄	—	166	1	no	—
329	C₄	#14581839	1	1	yes	characteristic
330	C₄	#14581840	1	1	yes	characteristic
331	C₂	—	8	1	no	—
332	C₂	—	332	1	no	—
333	C₂	—	664	1	no	—
334	C₂	#19128274	1	1	yes	characteristic
335	C₂	#135×D₈₃	1	1	yes	characteristic
336	C₂	AGL(1,ℤ/8)×D₈₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₃
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#478×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	#19128248	C₂
hypercenter	#478	D₈₃
hypocenter	C₈₃	#3228
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #478

Complements

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

N	H ≅ G/N	type
1. #19128257	C₂	semidirect
3. #19128105	C₂	semidirect
4. #19128098	C₂	semidirect
5. #19128046	C₂	semidirect
6. #19128046	C₂	semidirect
7. #19128028	C₂	semidirect
8. #19128027	C₂	semidirect
9. #19128024	C₂	semidirect
10. #19128023	C₂	semidirect
11. #19128002	C₂	semidirect
12. #19128001	C₂	semidirect
13. #19127998	C₂	semidirect
14. #19127753	C₂	semidirect
15. #478×C₈₃	C₂	semidirect
169. C₈₃	#3228	semidirect
183. #478	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128257	2
#19128248	2
#19128105	2
#19128098	2
#19128046	2
#19128028	2
#19128027	2
#19128024	2
#19128023	2
#19128002	2
#19128001	2
#19127998	2
#19127753	2
#478×C₈₃	2
#3228	83