Subgroups — #35562109

Order 21248.

484 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24920473	C₂	1	1	yes	maximal, characteristic
2	#24920356	C₂	1	1	yes	maximal, characteristic, complemented
3	#24909696	C₂	1	2	yes	maximal, complemented
4	#24909688	C₂	1	1	yes	maximal, characteristic
5	#24909619	C₂	1	2	yes	maximal, complemented
6	#24909528	C₂	1	2	yes	maximal, complemented
7	#24909504	C₂	1	1	yes	maximal, characteristic
8	#24909365	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#14581738	C₂	1	1	yes	maximal, characteristic, complemented
10	#24909314	C₂	1	2	yes	maximal, complemented
11	C₈₃×#2225	C₂	1	1	yes	maximal, characteristic, complemented
12	#19128436	—	2	1	no	—
13	#19128426	—	2	1	no	—
14	C₂×#14581738	C₂²	1	1	yes	characteristic
15	C₂×#14581737	C₂²	1	2	yes	—
16	#19128102	C₂²	1	2	yes	—
17	#19128086	C₂²	1	2	yes	—
18	#19128078	C₂²	1	1	yes	characteristic
19	#19128007	—	2	2	no	—
20	#19127983	—	2	1	no	—
21	#19127919	—	2	2	no	—
22	#19127901	—	2	1	no	—
23	C₈×Dic₁₆₆	C₂²	1	1	yes	characteristic
24	C₄×Dic₃₃₂	C₂²	1	1	yes	characteristic
25	#19127795	C₂²	1	2	yes	—
26	#19127783	C₂²	1	2	yes	—
27	#19127526	C₂²	1	1	yes	characteristic
28	Dic₈₃⋊C₁₆	C₂²	1	2	yes	—
29	#19127505	C₂²	1	1	yes	characteristic
30	#19127504	C₂²	1	1	yes	characteristic
31	#19127503	C₂²	1	2	yes	—
32	#19127501	C₂²	1	2	yes	—
33	#19127501	C₂²	1	2	yes	—
34	C₄×C₈₃⋊C₁₆	C₂²	1	1	yes	characteristic
35	C₁₆×Dic₈₃	C₂²	1	2	yes	—
36	#500×C₈₃	C₂²	1	1	yes	characteristic
37	C₄×Q₁₆×C₈₃	C₂²	1	1	yes	characteristic
38	#367×C₈₃	C₂²	1	2	yes	—
39	C₄⋊C₁₆×C₈₃	C₂²	1	1	yes	characteristic
40	#358×C₈₃	C₂²	1	2	yes	—
41	C₂×Dic₃₃₂	C₂³	1	1	yes	characteristic
42	C₂×#13050577	—	2	2	no	—
43	C₂×D₆₆₄	—	2	1	no	—
44	C₂×D₆₆₄	C₂³	1	2	yes	—
45	#14581790	—	4	1	no	—
46	#14581789	—	2	1	no	—
47	#14581783	—	2	2	no	—
48	#14581782	—	4	2	no	—
49	#14581781	—	2	2	no	—
50	#14581780	—	4	1	no	—
51	#14581777	—	4	2	no	—
52	Q₈×Dic₈₃	—	4	1	no	—
53	C₄×Dic₁₆₆	—	2	1	no	—
54	C₄×Dic₁₆₆	D₄	1	1	yes	characteristic
55	#14581757	—	2	2	no	—
56	C₄×D₃₃₂	—	4	1	no	—
57	#14581754	—	2	2	no	—
58	#14581738	—	2	2	no	—
59	#14581737	—	2	4	no	—
60	C₂×C₈₃⋊C₁₆	C₂³	1	1	yes	characteristic
61	C₂×C₈₃⋊C₁₆	C₂³	1	1	yes	characteristic
62	#14581726	C₂³	1	1	yes	characteristic
63	#14581723	—	2	2	no	—
64	C₄⋊(C₈₃⋊C₈)	D₄	1	1	yes	characteristic
65	Dic₈₃⋊C₈	D₄	1	2	yes	—
66	#14581717	—	2	1	no	—
67	#14581715	—	2	2	no	—
68	#14581712	—	2	2	no	—
69	C₈×Dic₈₃	C₂³	1	2	yes	—
70	C₂×Q₁₆×C₈₃	C₂³	1	1	yes	characteristic
71	C₄⋊Q₈×C₈₃	—	2	1	no	—
72	C₄×Q₈×C₈₃	—	2	1	no	—
73	C₂×C₁₆×C₈₃	C₂³	1	2	yes	—
74	#107×C₈₃	—	2	1	no	—
75	#107×C₈₃	C₂³	1	2	yes	—
76	#107×C₈₃	C₂³	1	1	yes	characteristic
77	#103×C₈₃	—	2	1	no	—
78	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
79	C₂×Dic₁₆₆	—	2	1	no	—
80	C₂×Dic₁₆₆	D₈	1	1	yes	characteristic
81	C₂×D₃₃₂	—	2	2	no	—
82	C₂×D₃₃₂	—	4	1	no	—
83	C₂×D₃₃₂	—	4	1	no	—
84	C₂×C₄×D₈₃	—	4	2	no	—
85	Dic₃₃₂	—	2	1	no	—
86	Dic₃₃₂	#41	1	2	yes	—
87	#13050577	—	8	2	no	—
88	D₆₆₄	—	2	2	no	—
89	D₆₆₄	—	2	2	no	—
90	D₆₆₄	—	8	1	no	—
91	C₂×C₈₃⋊C₈	C₂×D₄	1	2	yes	—
92	C₄⋊Dic₈₃	—	2	1	no	—
93	C₄⋊Dic₈₃	—	4	2	no	—
94	C₄⋊Dic₈₃	D₈	1	1	yes	characteristic
95	Dic₈₃⋊C₄	—	4	1	no	—
96	Dic₈₃⋊C₄	—	4	2	no	—
97	Dic₈₃⋊C₄	—	4	2	no	—
98	Dic₈₃⋊C₄	D₈	1	2	yes	—
99	(C₂×D₈₃)⋊C₄	—	4	2	no	—
100	(C₂×D₈₃)⋊C₄	—	4	2	no	—
101	(C₂×D₈₃)⋊C₄	—	4	2	no	—
102	C₄×Dic₈₃	—	4	1	no	—
103	C₄×Dic₈₃	—	4	2	no	—
104	C₄×Dic₈₃	C₂×D₄	1	2	yes	—
105	C₈₃⋊C₁₆	—	2	1	no	—
106	C₈₃⋊C₁₆	—	2	1	no	—
107	C₂×Q₈×C₈₃	—	2	1	no	—
108	C₂×Q₈×C₈₃	—	4	1	no	—
109	Q₁₆×C₈₃	—	2	1	no	—
110	Q₁₆×C₈₃	—	2	1	no	—
111	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₈₃	C₂⁴	1	1	yes	characteristic
113	C₄⋊C₄×C₈₃	—	2	2	no	—
114	C₄⋊C₄×C₈₃	—	2	1	no	—
115	C₄⋊C₄×C₈₃	—	4	1	no	—
116	C₄⋊C₄×C₈₃	—	4	1	no	—
117	C₄²×C₈₃	—	4	1	no	—
118	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
119	C₁₆×C₈₃	—	2	2	no	—
120	C₂²×D₈₃	—	4	2	no	—
121	Q₈×C₈₃	—	4	1	no	—
122	Q₈×C₈₃	—	4	1	no	—
123	Q₈×C₈₃	—	8	2	no	—
124	C₂×C₄×C₈₃	—	4	1	no	—
125	C₂×C₄×C₈₃	—	4	1	no	—
126	C₂×C₄×C₈₃	—	4	2	no	—
127	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
130	C₂×Dic₈₃	—	4	1	no	—
131	C₂×Dic₈₃	—	4	1	no	—
132	C₂×Dic₈₃	C₂×D₈	1	2	yes	—
133	C₂×Dic₈₃	C₂×D₈	1	2	yes	—
134	D₃₃₂	—	4	2	no	—
135	D₃₃₂	—	4	2	no	—
136	D₃₃₂	—	8	2	no	—
137	D₃₃₂	—	16	1	no	—
138	C₄×D₈₃	—	16	2	no	—
139	Dic₁₆₆	—	2	2	no	—
140	Dic₁₆₆	—	2	2	no	—
141	Dic₁₆₆	—	4	1	no	—
142	C₈×C₈₃	—	2	1	no	—
143	C₈×C₈₃	C₂×#41	1	1	yes	characteristic
144	C₈×C₈₃	2₋¹⁺⁴	1	1	yes	characteristic
145	C₈₃⋊C₈	—	2	2	no	—
146	C₂²×C₈₃	C₂²×D₈	1	1	yes	characteristic
147	C₂×D₈₃	—	8	2	no	—
148	C₂×D₈₃	—	8	2	no	—
149	C₂×D₈₃	—	8	2	no	—
150	C₄×C₈₃	—	2	1	no	—
151	C₄×C₈₃	—	8	1	no	—
152	C₄×C₈₃	—	8	2	no	—
153	C₄×C₈₃	—	8	1	no	—
154	C₄×C₈₃	#508	1	2	yes	—
155	C₄×C₈₃	#548	1	1	yes	characteristic
156	C₄×C₈₃	#577	1	1	yes	characteristic
157	Dic₈₃	—	2	2	no	—
158	Dic₈₃	—	2	2	no	—
159	Dic₈₃	—	4	2	no	—
160	Dic₈₃	—	8	1	no	—
161	#33932	—	83	1	no	maximal, Sylow
162	C₂×C₈₃	#3358	1	1	yes	characteristic
163	C₂×C₈₃	C₂×#508	1	1	yes	characteristic
164	C₂×C₈₃	#3416	1	1	yes	characteristic
165	D₈₃	—	16	2	no	—
166	#3396	—	83	1	no	—
167	#3383	—	83	1	no	—
168	#2242	—	83	2	no	—
169	#2240	—	83	1	no	—
170	#2228	—	83	2	no	—
171	#2225	D₈₃	1	1	yes	characteristic, complemented
172	#2211	—	83	2	no	—
173	#2207	—	83	1	no	—
174	Q₈×C₁₆	—	83	1	no	—
175	C₄×Q₃₂	—	83	1	no	—
176	C₄×SD₃₂	—	83	2	no	—
177	C₈₃	#33932	1	1	yes	characteristic, complemented, Sylow
178	#559	—	166	1	no	—
179	#557	—	166	1	no	—
180	C₂×Q₃₂	—	83	1	no	—
181	C₂×SD₃₂	—	83	2	no	—
182	#500	C₂×D₈₃	1	1	yes	characteristic
183	#499	—	83	2	no	—
184	#495	—	83	2	no	—
185	#493	—	83	1	no	—
186	#479	—	166	2	no	—
187	#475	—	166	1	no	—
188	#463	—	166	2	no	—
189	#460	—	166	1	no	—
190	C₈×Q₈	—	83	1	no	—
191	C₄×Q₁₆	—	83	2	no	—
192	C₄×Q₁₆	—	83	1	no	—
193	C₄×Q₁₆	C₂×D₈₃	1	1	yes	characteristic
194	C₄×D₈	—	83	2	no	—
195	#367	C₂×D₈₃	1	2	yes	—
196	#366	—	83	1	no	—
197	C₄⋊C₁₆	—	83	2	no	—
198	C₄⋊C₁₆	C₂×D₈₃	1	1	yes	characteristic
199	#358	—	83	1	no	—
200	#358	—	83	1	no	—
201	#358	C₂×D₈₃	1	2	yes	—
202	#357	—	83	2	no	—
203	#357	—	83	2	no	—
204	#357	—	83	2	no	—
205	C₄×C₁₆	—	83	2	no	—
206	C₄×C₁₆	—	83	1	no	—
207	C₂×Q₁₆	—	83	1	no	—
208	C₂×Q₁₆	C₂²×D₈₃	1	1	yes	characteristic
209	C₂×SD₁₆	—	166	2	no	—
210	C₂×D₈	—	83	2	no	—
211	C₂×D₈	—	166	1	no	—
212	C₄⋊Q₈	—	2	1	no	—
213	C₄⋊Q₈	—	332	1	no	—
214	#127	—	166	1	no	—
215	#125	—	332	2	no	—
216	#125	—	166	2	no	—
217	#124	—	332	1	no	—
218	#124	—	166	2	no	—
219	#123	—	332	2	no	—
220	C₄×Q₈	—	2	1	no	—
221	C₄×Q₈	—	332	1	no	—
222	C₄×Q₈	—	83	1	no	—
223	C₄×Q₈	—	166	2	no	—
224	C₄×Q₈	—	166	1	no	—
225	C₄×D₄	—	332	1	no	—
226	C₄×D₄	—	166	2	no	—
227	Q₃₂	—	166	2	no	—
228	SD₃₂	—	166	4	no	—
229	C₂×C₁₆	—	83	1	no	—
230	C₂×C₁₆	—	83	1	no	—
231	C₂×C₁₆	C₂²×D₈₃	1	2	yes	—
232	#107	—	2	1	no	—
233	#107	—	83	1	no	—
234	#107	C₂²×D₈₃	1	2	yes	—
235	#107	C₂²×D₈₃	1	1	yes	characteristic
236	#106	—	166	2	no	—
237	C₄⋊C₈	—	83	2	no	—
238	C₄⋊C₈	—	83	1	no	—
239	#103	—	2	1	no	—
240	#103	—	166	2	no	—
241	#103	—	166	1	no	—
242	#102	—	166	2	no	—
243	C₄×C₈	—	83	2	no	—
244	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
245	C₂×Q₈	—	2	1	no	—
246	C₂×Q₈	—	4	1	no	—
247	C₂×Q₈	—	83	1	no	—
248	C₂×Q₈	—	166	1	no	—
249	C₂×D₄	—	332	1	no	—
250	C₂×D₄	—	332	1	no	—
251	C₂×D₄	—	166	2	no	—
252	C₂²×C₄	—	332	2	no	—
253	Q₁₆	—	2	1	no	—
254	Q₁₆	—	2	1	no	—
255	Q₁₆	—	83	2	no	—
256	Q₁₆	—	166	1	no	—
257	SD₁₆	—	664	2	no	—
258	D₈	—	664	1	no	—
259	D₈	—	166	2	no	—
260	D₈	—	166	2	no	—
261	C₂×C₈	—	83	2	no	—
262	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
263	C₂×C₈	C₂³×D₈₃	1	1	yes	characteristic
264	C₄⋊C₄	—	2	1	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	4	1	no	—
267	C₄⋊C₄	—	4	1	no	—
268	C₄⋊C₄	—	332	2	no	—
269	C₄⋊C₄	—	332	2	no	—
270	C₄⋊C₄	—	332	2	no	—
271	C₄⋊C₄	—	332	1	no	—
272	C₄⋊C₄	—	83	1	no	—
273	C₄⋊C₄	—	83	2	no	—
274	C₄⋊C₄	—	166	1	no	—
275	C₂²⋊C₄	—	332	2	no	—
276	C₂²⋊C₄	—	332	2	no	—
277	C₂²⋊C₄	—	332	2	no	—
278	C₄²	—	4	1	no	—
279	C₄²	—	332	2	no	—
280	C₄²	—	332	1	no	—
281	C₄²	—	83	2	no	—
282	C₄²	D₄×D₈₃	1	1	yes	characteristic
283	C₁₆	—	2	2	no	—
284	C₁₆	—	166	1	no	—
285	C₁₆	—	166	1	no	—
286	C₂³	—	332	2	no	—
287	Q₈	—	4	1	no	—
288	Q₈	—	4	1	no	—
289	Q₈	—	8	2	no	—
290	Q₈	—	332	1	no	—
291	Q₈	—	166	2	no	—
292	Q₈	—	166	2	no	—
293	D₄	—	1328	1	no	—
294	D₄	—	332	2	no	—
295	D₄	—	332	2	no	—
296	D₄	—	664	2	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	1328	2	no	—
301	C₂×C₄	—	332	1	no	—
302	C₂×C₄	—	332	1	no	—
303	C₂×C₄	—	83	2	no	—
304	C₂×C₄	—	83	2	no	—
305	C₂×C₄	D₈×D₈₃	1	1	yes	characteristic
306	C₂×C₄	D₈×D₈₃	1	1	yes	characteristic
307	C₂×C₄	C₂×D₄×D₈₃	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	166	2	no	—
310	C₈	#41×D₈₃	1	1	yes	characteristic
311	C₈	#14581840	1	1	yes	characteristic
312	C₂²	—	664	2	no	—
313	C₂²	—	664	2	no	—
314	C₂²	—	664	2	no	—
315	C₂²	C₂×D₈×D₈₃	1	1	yes	characteristic
316	C₄	—	2	1	no	—
317	C₄	—	8	2	no	—
318	C₄	—	8	1	no	—
319	C₄	—	8	1	no	—
320	C₄	—	332	2	no	—
321	C₄	—	664	1	no	—
322	C₄	—	166	2	no	—
323	C₄	—	166	2	no	—
324	C₄	#19128132	1	2	yes	—
325	C₄	#19128361	1	1	yes	characteristic
326	C₄	#19128510	1	1	yes	characteristic
327	C₂	—	1328	2	no	—
328	C₂	#24920097	1	1	yes	characteristic
329	C₂	C₂×#19128132	1	1	yes	characteristic
330	C₂	#24920608	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₈₃
commutator, cosocle	C₂×C₈×C₈₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₃
Fitting	C₈₃×#2225	C₂
socle	C₂²×C₈₃	C₂²×D₈
prime radical	#19128078	C₂²
hypercenter	#2225	D₈₃
hypocenter	C₈₃	#33932
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2225

Complements

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

N	H ≅ G/N	type
2. #24920356	C₂	semidirect
3. #24909696	C₂	semidirect
5. #24909619	C₂	semidirect
6. #24909528	C₂	semidirect
8. #24909365	C₂	semidirect
9. C₄×#14581738	C₂	semidirect
10. #24909314	C₂	semidirect
11. C₈₃×#2225	C₂	semidirect
171. #2225	D₈₃	semidirect
177. C₈₃	#33932	semidirect

Maximal subgroups

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

subgroup	index
#24920473	2
#24920356	2
#24909696	2
#24909688	2
#24909619	2
#24909528	2
#24909504	2
#24909365	2
C₄×#14581738	2
#24909314	2
C₈₃×#2225	2
#33932	83