Subgroups — #24920519

Order 10624.

362 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128433	C₂	1	1	yes	maximal, characteristic
2	#19128429	C₂	1	1	yes	maximal, characteristic, complemented
3	#19128096	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128094	C₂	1	1	yes	maximal, characteristic, complemented
5	#19128072	C₂	1	1	yes	maximal, characteristic
6	#19127926	C₂	1	1	yes	maximal, characteristic, complemented
7	#19127909	C₂	1	1	yes	maximal, characteristic, complemented
8	#19127907	C₂	1	1	yes	maximal, characteristic, complemented
9	#19127904	C₂	1	1	yes	maximal, characteristic, complemented
10	#19127900	C₂	1	1	yes	maximal, characteristic
11	#19127830	C₂	1	1	yes	maximal, characteristic, complemented
12	#19127799	C₂	1	1	yes	maximal, characteristic, complemented
13	#19127798	C₂	1	1	yes	maximal, characteristic, complemented
14	#19127793	C₂	1	1	yes	maximal, characteristic, complemented
15	#464×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13050580	C₂²	1	1	yes	characteristic
17	C₂×#13050580	C₂²	1	1	yes	characteristic
18	C₂×#13050579	C₂²	1	1	yes	characteristic
19	C₂×#13050579	C₂²	1	1	yes	characteristic
20	C₂×#13050578	C₂²	1	1	yes	characteristic
21	C₂×#13050578	C₂²	1	1	yes	characteristic
22	C₂×#13050577	C₂²	1	1	yes	characteristic
23	#14581792	C₂²	1	1	yes	characteristic
24	#14581790	C₂²	1	1	yes	characteristic
25	#14581790	C₂²	1	1	yes	characteristic
26	#14581788	C₂²	1	1	yes	characteristic
27	#14581784	—	2	1	no	—
28	#14581782	—	2	1	no	—
29	#14581782	—	2	1	no	—
30	#14581781	C₂²	1	1	yes	characteristic
31	#14581778	—	2	1	no	—
32	#14581776	—	2	1	no	—
33	#14581774	—	2	1	no	—
34	Q₈×Dic₈₃	C₂²	1	1	yes	characteristic
35	C₄×Dic₁₆₆	—	2	1	no	—
36	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
37	#14581757	C₂²	1	1	yes	characteristic
38	#14581757	C₂²	1	1	yes	characteristic
39	C₄×D₃₃₂	C₂²	1	1	yes	characteristic
40	D₄×Dic₈₃	C₂²	1	1	yes	characteristic
41	(C₂²⋊D₈₃)⋊C₄	—	2	1	no	—
42	#14581725	C₂²	1	1	yes	characteristic
43	Dic₈₃⋊C₈	C₂²	1	1	yes	characteristic
44	Dic₈₃⋊C₈	C₂²	1	1	yes	characteristic
45	#14581717	C₂²	1	1	yes	characteristic
46	#14581717	C₂²	1	1	yes	characteristic
47	#14581716	C₂²	1	1	yes	characteristic
48	#14581712	C₂²	1	1	yes	characteristic
49	#14581703	C₂²	1	1	yes	characteristic
50	#14581702	C₂²	1	1	yes	characteristic
51	C₈×Dic₈₃	C₂²	1	1	yes	characteristic
52	C₂×Q₁₆×C₈₃	C₂²	1	1	yes	characteristic
53	C₂×SD₁₆×C₈₃	C₂²	1	1	yes	characteristic
54	#124×C₈₃	C₂²	1	1	yes	characteristic
55	C₄×Q₈×C₈₃	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
57	#103×C₈₃	C₂²	1	1	yes	characteristic
58	#102×C₈₃	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₆₆	D₄	1	1	yes	characteristic
60	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
61	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
62	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
63	C₂×C₂²⋊D₈₃	—	2	1	no	—
64	C₂×C₂²⋊D₈₃	—	2	1	no	—
65	C₂²×Dic₈₃	—	2	1	no	—
66	C₂×C₄×D₈₃	—	2	1	no	—
67	#13050580	—	2	2	no	—
68	#13050580	—	4	1	no	—
69	#13050579	—	4	1	no	—
70	#13050579	—	4	1	no	—
71	#13050578	—	2	2	no	—
72	#13050578	—	4	1	no	—
73	#13050577	—	2	2	no	—
74	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
75	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
76	C₄⋊Dic₈₃	—	2	1	no	—
77	C₄⋊Dic₈₃	—	2	1	no	—
78	C₄⋊Dic₈₃	—	2	1	no	—
79	C₄⋊Dic₈₃	D₄	1	1	yes	characteristic
80	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
81	Dic₈₃⋊C₄	—	2	1	no	—
82	Dic₈₃⋊C₄	—	2	1	no	—
83	Dic₈₃⋊C₄	—	2	1	no	—
84	Dic₈₃⋊C₄	—	2	1	no	—
85	Dic₈₃⋊C₄	—	2	1	no	—
86	Dic₈₃⋊C₄	—	2	1	no	—
87	Dic₈₃⋊C₄	D₄	1	1	yes	characteristic
88	Dic₈₃⋊C₄	D₄	1	1	yes	characteristic
89	C₂²⋊Dic₈₃	—	2	1	no	—
90	(C₂×D₈₃)⋊C₄	—	2	1	no	—
91	(C₂×D₈₃)⋊C₄	—	2	1	no	—
92	(C₂×D₈₃)⋊C₄	—	2	1	no	—
93	C₄×Dic₈₃	—	2	1	no	—
94	C₄×Dic₈₃	—	2	1	no	—
95	C₄×Dic₈₃	—	2	1	no	—
96	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
97	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
98	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
99	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
100	C₂×D₄×C₈₃	C₂³	1	1	yes	characteristic
101	Q₁₆×C₈₃	—	4	1	no	—
102	SD₁₆×C₈₃	—	4	1	no	—
103	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
104	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
105	C₄⋊C₄×C₈₃	—	2	1	no	—
106	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
107	C₂²⋊C₄×C₈₃	—	2	1	no	—
108	C₂²⋊C₄×C₈₃	—	2	1	no	—
109	C₄²×C₈₃	—	2	1	no	—
110	C₄²×C₈₃	C₂³	1	1	yes	characteristic
111	C₂³×C₈₃	—	2	1	no	—
112	C₂²×D₈₃	—	2	1	no	—
113	Q₈×C₈₃	—	2	1	no	—
114	Q₈×C₈₃	—	2	1	no	—
115	Q₈×C₈₃	—	2	1	no	—
116	Q₈×C₈₃	—	2	1	no	—
117	D₄×C₈₃	—	2	1	no	—
118	D₄×C₈₃	—	2	1	no	—
119	C₂×C₄×C₈₃	—	2	1	no	—
120	C₂×C₄×C₈₃	—	2	1	no	—
121	C₂×C₄×C₈₃	—	2	1	no	—
122	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
125	C₂²⋊D₈₃	—	4	2	no	—
126	C₂²⋊D₈₃	—	8	1	no	—
127	C₂×Dic₈₃	—	2	1	no	—
128	C₂×Dic₈₃	—	2	1	no	—
129	C₂×Dic₈₃	—	2	1	no	—
130	C₂×Dic₈₃	—	8	1	no	—
131	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
132	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
133	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
134	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
135	D₃₃₂	—	2	1	no	—
136	D₃₃₂	—	2	1	no	—
137	C₄×D₈₃	—	8	1	no	—
138	Dic₁₆₆	—	2	2	no	—
139	Dic₁₆₆	—	2	1	no	—
140	Dic₁₆₆	—	2	1	no	—
141	Dic₁₆₆	—	2	1	no	—
142	Dic₁₆₆	—	2	1	no	—
143	C₈×C₈₃	—	2	1	no	—
144	C₈×C₈₃	—	2	1	no	—
145	C₈₃⋊C₈	—	2	1	no	—
146	C₈₃⋊C₈	#41	1	2	yes	—
147	C₂²×C₈₃	—	4	1	no	—
148	C₂²×C₈₃	—	4	1	no	—
149	C₂²×C₈₃	—	4	1	no	—
150	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
151	C₂×D₈₃	—	4	1	no	—
152	C₂×D₈₃	—	4	1	no	—
153	C₂×D₈₃	—	4	1	no	—
154	C₄×C₈₃	—	2	1	no	—
155	C₄×C₈₃	—	2	1	no	—
156	C₄×C₈₃	—	4	1	no	—
157	C₄×C₈₃	—	4	1	no	—
158	C₄×C₈₃	—	4	1	no	—
159	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
160	C₄×C₈₃	H(5,𝔽₂)	1	1	yes	characteristic
161	Dic₈₃	—	2	1	no	—
162	Dic₈₃	—	2	1	no	—
163	Dic₈₃	—	2	1	no	—
164	Dic₈₃	—	4	1	no	—
165	Dic₈₃	—	4	1	no	—
166	Dic₈₃	—	4	1	no	—
167	Dic₈₃	#137	1	2	yes	—
168	C₂×C₈₃	—	8	1	no	—
169	C₂×C₈₃	#550	1	1	yes	characteristic
170	C₂×C₈₃	C₂×#137	1	1	yes	characteristic
171	C₂×C₈₃	#577	1	1	yes	characteristic
172	D₈₃	—	8	1	no	—
173	#3400	—	83	1	no	maximal, Sylow
174	C₈₃	#3400	1	1	yes	characteristic, complemented, Sylow
175	#559	—	83	1	no	—
176	#557	—	83	1	no	—
177	#497	—	83	1	no	—
178	#497	—	83	1	no	—
179	#492	—	83	1	no	—
180	#464	—	83	1	no	—
181	#464	D₈₃	1	1	yes	characteristic, complemented
182	#462	—	83	1	no	—
183	#461	—	83	1	no	—
184	#461	—	83	1	no	—
185	#460	—	83	1	no	—
186	#446	—	83	1	no	—
187	#440	—	83	1	no	—
188	#440	—	83	1	no	—
189	C₄×Q₁₆	—	83	1	no	—
190	C₂×Q₁₆	—	83	1	no	—
191	C₂×Q₁₆	—	83	1	no	—
192	C₂×Q₁₆	C₂×D₈₃	1	1	yes	characteristic
193	C₂×SD₁₆	—	83	1	no	—
194	C₂×SD₁₆	—	83	1	no	—
195	C₂×SD₁₆	—	83	1	no	—
196	C₂×SD₁₆	—	83	1	no	—
197	C₂×SD₁₆	—	83	1	no	—
198	C₂×SD₁₆	C₂×D₈₃	1	1	yes	characteristic
199	C₄⋊Q₈	—	83	1	no	—
200	C₄⋊Q₈	—	83	1	no	—
201	C₄⋊Q₈	—	83	1	no	—
202	#127	—	83	1	no	—
203	#125	—	166	1	no	—
204	#125	—	166	1	no	—
205	#125	—	166	1	no	—
206	#124	—	83	1	no	—
207	#124	—	166	1	no	—
208	#124	C₂×D₈₃	1	1	yes	characteristic
209	#123	—	166	1	no	—
210	#123	—	166	1	no	—
211	C₄×Q₈	—	83	1	no	—
212	C₄×Q₈	—	83	1	no	—
213	C₄×Q₈	—	83	1	no	—
214	C₄×Q₈	—	83	1	no	—
215	C₄×Q₈	—	166	1	no	—
216	C₄×Q₈	C₂×D₈₃	1	1	yes	characteristic
217	C₄×D₄	—	83	1	no	—
218	C₄×D₄	—	83	1	no	—
219	C₄×D₄	—	166	1	no	—
220	#107	—	83	1	no	—
221	C₄⋊C₈	—	83	1	no	—
222	C₄⋊C₈	—	83	1	no	—
223	C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
224	#103	—	83	1	no	—
225	#103	—	83	1	no	—
226	#103	—	83	1	no	—
227	#103	C₂×D₈₃	1	1	yes	characteristic
228	#102	—	83	1	no	—
229	#102	C₂×D₈₃	1	1	yes	characteristic
230	#97	—	83	1	no	—
231	#97	—	83	1	no	—
232	C₄×C₈	—	83	1	no	—
233	C₂×Q₈	—	83	1	no	—
234	C₂×Q₈	—	83	1	no	—
235	C₂×Q₈	—	83	1	no	—
236	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
237	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
238	C₂×D₄	—	83	1	no	—
239	C₂×D₄	—	166	1	no	—
240	C₂×D₄	—	166	1	no	—
241	C₂×D₄	C₂²×D₈₃	1	1	yes	characteristic
242	C₂²×C₄	—	166	1	no	—
243	C₂²×C₄	—	166	1	no	—
244	Q₁₆	—	4	1	no	—
245	Q₁₆	—	332	1	no	—
246	Q₁₆	—	166	2	no	—
247	SD₁₆	—	4	1	no	—
248	SD₁₆	—	332	1	no	—
249	SD₁₆	—	332	1	no	—
250	SD₁₆	—	332	1	no	—
251	SD₁₆	—	166	2	no	—
252	SD₁₆	—	166	2	no	—
253	C₂×C₈	—	83	1	no	—
254	C₂×C₈	—	83	1	no	—
255	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
256	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
257	C₄⋊C₄	—	2	1	no	—
258	C₄⋊C₄	—	83	1	no	—
259	C₄⋊C₄	—	83	1	no	—
260	C₄⋊C₄	—	83	1	no	—
261	C₄⋊C₄	—	83	1	no	—
262	C₄⋊C₄	—	166	1	no	—
263	C₄⋊C₄	—	166	1	no	—
264	C₄⋊C₄	—	166	1	no	—
265	C₄⋊C₄	—	166	1	no	—
266	C₄⋊C₄	—	166	1	no	—
267	C₄⋊C₄	—	166	1	no	—
268	C₄⋊C₄	—	166	1	no	—
269	C₄⋊C₄	—	166	1	no	—
270	C₄⋊C₄	—	166	1	no	—
271	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
272	C₂²⋊C₄	—	2	1	no	—
273	C₂²⋊C₄	—	2	1	no	—
274	C₂²⋊C₄	—	166	1	no	—
275	C₂²⋊C₄	—	166	1	no	—
276	C₂²⋊C₄	—	166	1	no	—
277	C₂²⋊C₄	—	166	1	no	—
278	C₄²	—	2	1	no	—
279	C₄²	—	83	1	no	—
280	C₄²	—	83	1	no	—
281	C₄²	—	166	1	no	—
282	C₄²	—	166	1	no	—
283	C₄²	—	166	1	no	—
284	C₄²	C₂²×D₈₃	1	1	yes	characteristic
285	C₂³	—	2	1	no	—
286	C₂³	—	166	1	no	—
287	Q₈	—	2	1	no	—
288	Q₈	—	2	1	no	—
289	Q₈	—	2	1	no	—
290	Q₈	—	2	1	no	—
291	Q₈	—	166	2	no	—
292	Q₈	—	166	1	no	—
293	Q₈	—	166	1	no	—
294	Q₈	—	166	1	no	—
295	Q₈	—	166	1	no	—
296	D₄	—	2	1	no	—
297	D₄	—	2	1	no	—
298	D₄	—	332	2	no	—
299	D₄	—	664	1	no	—
300	D₄	—	166	1	no	—
301	D₄	—	166	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	83	1	no	—
306	C₂×C₄	—	83	1	no	—
307	C₂×C₄	—	83	1	no	—
308	C₂×C₄	—	83	1	no	—
309	C₂×C₄	—	664	1	no	—
310	C₂×C₄	—	664	1	no	—
311	C₂×C₄	—	166	1	no	—
312	C₂×C₄	—	166	1	no	—
313	C₂×C₄	—	166	1	no	—
314	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
315	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
316	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
317	C₈	—	2	1	no	—
318	C₈	—	2	1	no	—
319	C₈	—	83	2	no	—
320	C₈	—	166	1	no	—
321	C₂²	—	4	1	no	—
322	C₂²	—	4	1	no	—
323	C₂²	—	4	1	no	—
324	C₂²	—	332	1	no	—
325	C₂²	—	332	1	no	—
326	C₂²	—	332	1	no	—
327	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	4	1	no	—
331	C₄	—	4	1	no	—
332	C₄	—	4	1	no	—
333	C₄	—	332	1	no	—
334	C₄	—	332	1	no	—
335	C₄	—	332	1	no	—
336	C₄	—	83	2	no	—
337	C₄	—	166	1	no	—
338	C₄	—	166	1	no	—
339	C₄	—	166	1	no	—
340	C₄	#41×D₈₃	1	1	yes	characteristic
341	C₄	#14581837	1	1	yes	characteristic
342	C₂	—	8	1	no	—
343	C₂	—	664	1	no	—
344	C₂	#19128369	1	1	yes	characteristic
345	C₂	#137×D₈₃	1	1	yes	characteristic
346	C₂	#19128508	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₃
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#464×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	#14581788	C₂²
hypercenter	#464	D₈₃
hypocenter	C₈₃	#3400
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #464

Complements

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

N	H ≅ G/N	type
2. #19128429	C₂	semidirect
3. #19128096	C₂	semidirect
4. #19128094	C₂	semidirect
6. #19127926	C₂	semidirect
7. #19127909	C₂	semidirect
8. #19127907	C₂	semidirect
9. #19127904	C₂	semidirect
11. #19127830	C₂	semidirect
12. #19127799	C₂	semidirect
13. #19127798	C₂	semidirect
14. #19127793	C₂	semidirect
15. #464×C₈₃	C₂	semidirect
174. C₈₃	#3400	semidirect
181. #464	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128433	2
#19128429	2
#19128096	2
#19128094	2
#19128072	2
#19127926	2
#19127909	2
#19127907	2
#19127904	2
#19127900	2
#19127830	2
#19127799	2
#19127798	2
#19127793	2
#464×C₈₃	2
#3400	83