Subgroups — #24918908

Order 10624.

346 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128309	C₂	1	1	yes	maximal, characteristic, complemented
2	#19128307	C₂	1	1	yes	maximal, characteristic
3	#19128005	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128004	C₂	1	1	yes	maximal, characteristic, complemented
5	#19128000	C₂	1	1	yes	maximal, characteristic, complemented
6	#19127950	C₂	1	1	yes	maximal, characteristic, complemented
7	#19127948	C₂	1	1	yes	maximal, characteristic, complemented
8	#19127933	C₂	1	1	yes	maximal, characteristic, complemented
9	#19127931	C₂	1	1	yes	maximal, characteristic, complemented
10	#19127905	C₂	1	1	yes	maximal, characteristic, complemented
11	#19127905	C₂	1	1	yes	maximal, characteristic, complemented
12	#19127902	C₂	1	1	yes	maximal, characteristic, complemented
13	#19127901	C₂	1	1	yes	maximal, characteristic, complemented
14	#19127726	C₂	1	1	yes	maximal, characteristic, complemented
15	#479×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13050579	C₂²	1	1	yes	characteristic
17	C₂×#13050579	C₂²	1	1	yes	characteristic
18	C₂×#13050577	C₂²	1	1	yes	characteristic
19	C₂×#13050577	C₂²	1	1	yes	characteristic
20	C₂×#13050573	C₂²	1	1	yes	characteristic
21	C₂×C₈×D₈₃	C₂²	1	1	yes	characteristic
22	#14581790	C₂²	1	1	yes	characteristic
23	#14581789	C₂²	1	1	yes	characteristic
24	#14581787	—	2	1	no	—
25	#14581786	—	2	1	no	—
26	#14581784	—	2	1	no	—
27	#14581783	C₂²	1	1	yes	characteristic
28	#14581782	—	2	1	no	—
29	#14581777	—	2	1	no	—
30	#14581777	—	2	1	no	—
31	#14581772	C₂²	1	1	yes	characteristic
32	#14581768	C₂²	1	1	yes	characteristic
33	#14581767	—	2	1	no	—
34	#14581767	—	2	1	no	—
35	#14581767	C₂²	1	1	yes	characteristic
36	#14581767	C₂²	1	1	yes	characteristic
37	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
38	#14581754	C₂²	1	1	yes	characteristic
39	#14581754	C₂²	1	1	yes	characteristic
40	#14581749	C₂²	1	1	yes	characteristic
41	#14581725	C₂²	1	1	yes	characteristic
42	#14581723	C₂²	1	1	yes	characteristic
43	C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
44	Dic₈₃⋊C₈	C₂²	1	1	yes	characteristic
45	Dic₈₃⋊C₈	C₂²	1	1	yes	characteristic
46	#14581715	C₂²	1	1	yes	characteristic
47	#14581715	C₂²	1	1	yes	characteristic
48	#14581714	C₂²	1	1	yes	characteristic
49	#14581714	C₂²	1	1	yes	characteristic
50	#14581712	C₂²	1	1	yes	characteristic
51	#14581712	C₂²	1	1	yes	characteristic
52	#125×C₈₃	C₂²	1	1	yes	characteristic
53	C₄×Q₈×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	#103×C₈₃	C₂²	1	1	yes	characteristic
58	#103×C₈₃	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
60	C₂×D₃₃₂	—	2	1	no	—
61	C₂×D₃₃₂	—	2	1	no	—
62	C₂×D₃₃₂	—	2	1	no	—
63	C₂×D₃₃₂	—	2	1	no	—
64	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
65	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
66	C₂×C₄×D₈₃	—	2	1	no	—
67	C₂×C₄×D₈₃	—	2	1	no	—
68	C₂×C₄×D₈₃	C₂³	1	1	yes	characteristic
69	#13050579	—	4	1	no	—
70	#13050579	—	4	1	no	—
71	#13050577	—	4	1	no	—
72	#13050577	—	4	1	no	—
73	#13050573	—	4	1	no	—
74	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
75	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
76	C₈×D₈₃	—	4	1	no	—
77	C₄⋊Dic₈₃	—	2	1	no	—
78	C₄⋊Dic₈₃	—	2	1	no	—
79	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
80	Dic₈₃⋊C₄	—	2	1	no	—
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	D₄	1	1	yes	characteristic
86	Dic₈₃⋊C₄	D₄	1	1	yes	characteristic
87	(C₂×D₈₃)⋊C₄	—	2	1	no	—
88	(C₂×D₈₃)⋊C₄	—	2	1	no	—
89	(C₂×D₈₃)⋊C₄	—	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	(C₂×D₈₃)⋊C₄	D₄	1	1	yes	characteristic
94	(C₂×D₈₃)⋊C₄	D₄	1	1	yes	characteristic
95	C₄×Dic₈₃	—	2	1	no	—
96	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
97	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
98	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
99	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
100	C₄⋊C₄×C₈₃	—	2	1	no	—
101	C₄⋊C₄×C₈₃	—	2	1	no	—
102	C₄⋊C₄×C₈₃	—	2	1	no	—
103	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
104	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
105	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
106	C₄²×C₈₃	—	2	1	no	—
107	C₄²×C₈₃	C₂³	1	1	yes	characteristic
108	C₂²×D₈₃	—	2	1	no	—
109	C₂²×D₈₃	—	2	1	no	—
110	C₂²×D₈₃	C₂×D₄	1	1	yes	characteristic
111	Q₈×C₈₃	—	2	1	no	—
112	Q₈×C₈₃	—	2	1	no	—
113	C₂×C₄×C₈₃	—	2	1	no	—
114	C₂×C₄×C₈₃	—	2	1	no	—
115	C₂×C₄×C₈₃	—	2	1	no	—
116	C₂×C₄×C₈₃	—	2	1	no	—
117	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
120	C₂×Dic₈₃	—	2	1	no	—
121	C₂×Dic₈₃	—	2	1	no	—
122	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
123	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
124	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
125	D₃₃₂	—	2	1	no	—
126	D₃₃₂	—	2	1	no	—
127	D₃₃₂	—	2	1	no	—
128	D₃₃₂	—	2	1	no	—
129	D₃₃₂	—	8	1	no	—
130	D₃₃₂	—	8	1	no	—
131	D₃₃₂	—	8	1	no	—
132	D₃₃₂	—	8	1	no	—
133	C₄×D₈₃	—	2	1	no	—
134	C₄×D₈₃	—	2	1	no	—
135	C₄×D₈₃	—	8	1	no	—
136	C₄×D₈₃	—	8	1	no	—
137	Dic₁₆₆	—	2	1	no	—
138	Dic₁₆₆	—	2	1	no	—
139	C₈×C₈₃	—	2	1	no	—
140	C₈×C₈₃	—	2	1	no	—
141	C₈₃⋊C₈	—	2	1	no	—
142	C₈₃⋊C₈	—	2	1	no	—
143	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
144	C₂×D₈₃	—	2	1	no	—
145	C₂×D₈₃	—	2	1	no	—
146	C₂×D₈₃	—	2	1	no	—
147	C₂×D₈₃	—	4	1	no	—
148	C₂×D₈₃	—	4	1	no	—
149	C₂×D₈₃	—	4	1	no	—
150	C₂×D₈₃	—	4	1	no	—
151	C₂×D₈₃	—	4	1	no	—
152	C₂×D₈₃	—	4	1	no	—
153	C₄×C₈₃	—	2	1	no	—
154	C₄×C₈₃	—	2	1	no	—
155	C₄×C₈₃	—	4	1	no	—
156	C₄×C₈₃	—	4	1	no	—
157	C₄×C₈₃	—	4	1	no	—
158	C₄×C₈₃	—	4	1	no	—
159	C₄×C₈₃	H(5,𝔽₂)	1	1	yes	characteristic
160	C₄×C₈₃	2₋¹⁺⁴	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	C₂×C₈₃	#537	1	1	yes	characteristic
167	C₂×C₈₃	#577	1	1	yes	characteristic
168	C₂×C₈₃	#577	1	1	yes	characteristic
169	D₈₃	—	4	1	no	—
170	D₈₃	—	8	1	no	—
171	D₈₃	—	8	1	no	—
172	#3260	—	83	1	no	maximal, Sylow
173	C₈₃	#3260	1	1	yes	characteristic, complemented, Sylow
174	#541	—	83	1	no	—
175	#540	—	83	1	no	—
176	#479	—	83	1	no	—
177	#479	D₈₃	1	1	yes	characteristic, complemented
178	#478	—	83	1	no	—
179	#478	—	83	1	no	—
180	#468	—	83	1	no	—
181	#468	—	83	1	no	—
182	#465	—	83	1	no	—
183	#465	—	83	1	no	—
184	#461	—	83	1	no	—
185	#461	—	83	1	no	—
186	#460	—	83	1	no	—
187	#460	—	83	1	no	—
188	#424	—	83	1	no	—
189	C₂×SD₁₆	—	83	1	no	—
190	C₂×SD₁₆	—	83	1	no	—
191	C₂×SD₁₆	—	83	1	no	—
192	C₂×SD₁₆	—	83	1	no	—
193	C₂×M₁₆	—	83	1	no	—
194	C₂²×C₈	—	83	1	no	—
195	C₄⋊Q₈	—	83	1	no	—
196	#127	—	83	1	no	—
197	#126	—	166	1	no	—
198	#126	—	166	1	no	—
199	#125	—	83	1	no	—
200	#125	—	166	1	no	—
201	#125	—	166	1	no	—
202	#125	C₂×D₈₃	1	1	yes	characteristic
203	#123	—	166	1	no	—
204	#123	—	166	1	no	—
205	C₂²⋊Q₈	—	83	1	no	—
206	C₂²⋊Q₈	—	83	1	no	—
207	#121	—	83	1	no	—
208	#121	—	83	1	no	—
209	#121	—	166	1	no	—
210	#121	—	166	1	no	—
211	C₄×Q₈	—	83	1	no	—
212	C₄×Q₈	C₂×D₈₃	1	1	yes	characteristic
213	C₄×D₄	—	83	1	no	—
214	C₄×D₄	—	83	1	no	—
215	#117	—	83	1	no	—
216	#107	—	83	1	no	—
217	#107	C₂×D₈₃	1	1	yes	characteristic
218	#106	—	83	1	no	—
219	#106	C₂×D₈₃	1	1	yes	characteristic
220	C₄⋊C₈	—	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	C₂×D₈₃	1	1	yes	characteristic
227	#103	C₂×D₈₃	1	1	yes	characteristic
228	#102	—	83	1	no	—
229	#102	—	83	1	no	—
230	#102	—	83	1	no	—
231	#102	—	83	1	no	—
232	C₂×Q₈	—	83	1	no	—
233	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
234	C₂×D₄	—	83	1	no	—
235	C₂×D₄	—	83	1	no	—
236	C₂×D₄	—	166	1	no	—
237	C₂×D₄	—	166	1	no	—
238	C₂×D₄	—	166	1	no	—
239	C₂×D₄	—	166	1	no	—
240	C₂²×C₄	—	83	1	no	—
241	C₂²×C₄	—	166	1	no	—
242	C₂²×C₄	—	166	1	no	—
243	SD₁₆	—	332	1	no	—
244	SD₁₆	—	332	1	no	—
245	SD₁₆	—	332	1	no	—
246	SD₁₆	—	332	1	no	—
247	M₁₆	—	332	1	no	—
248	C₂×C₈	—	332	1	no	—
249	C₂×C₈	—	83	1	no	—
250	C₂×C₈	—	83	1	no	—
251	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
252	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
253	C₄⋊C₄	—	2	1	no	—
254	C₄⋊C₄	—	2	1	no	—
255	C₄⋊C₄	—	2	1	no	—
256	C₄⋊C₄	—	83	1	no	—
257	C₄⋊C₄	—	83	1	no	—
258	C₄⋊C₄	—	83	1	no	—
259	C₄⋊C₄	—	166	1	no	—
260	C₄⋊C₄	—	166	1	no	—
261	C₄⋊C₄	—	166	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	C₂²×D₈₃	1	1	yes	characteristic
267	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
268	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
269	C₂²⋊C₄	—	83	1	no	—
270	C₂²⋊C₄	—	83	1	no	—
271	C₂²⋊C₄	—	166	1	no	—
272	C₂²⋊C₄	—	166	1	no	—
273	C₂²⋊C₄	—	166	1	no	—
274	C₂²⋊C₄	—	166	1	no	—
275	C₂²⋊C₄	—	166	1	no	—
276	C₂²⋊C₄	—	166	1	no	—
277	C₄²	—	2	1	no	—
278	C₄²	—	83	1	no	—
279	C₄²	—	166	1	no	—
280	C₄²	C₂²×D₈₃	1	1	yes	characteristic
281	C₂³	—	83	1	no	—
282	C₂³	—	166	1	no	—
283	C₂³	—	166	1	no	—
284	Q₈	—	2	1	no	—
285	Q₈	—	2	1	no	—
286	Q₈	—	166	1	no	—
287	Q₈	—	166	1	no	—
288	D₄	—	664	1	no	—
289	D₄	—	664	1	no	—
290	D₄	—	664	1	no	—
291	D₄	—	664	1	no	—
292	D₄	—	166	1	no	—
293	D₄	—	166	1	no	—
294	D₄	—	166	1	no	—
295	D₄	—	166	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	83	1	no	—
301	C₂×C₄	—	83	1	no	—
302	C₂×C₄	—	83	1	no	—
303	C₂×C₄	—	664	1	no	—
304	C₂×C₄	—	664	1	no	—
305	C₂×C₄	—	166	1	no	—
306	C₂×C₄	—	166	1	no	—
307	C₂×C₄	—	166	1	no	—
308	C₂×C₄	—	166	1	no	—
309	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
310	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
311	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
312	C₈	—	2	1	no	—
313	C₈	—	2	1	no	—
314	C₈	—	166	1	no	—
315	C₈	—	166	1	no	—
316	C₂²	—	332	1	no	—
317	C₂²	—	332	1	no	—
318	C₂²	—	332	1	no	—
319	C₂²	—	332	1	no	—
320	C₂²	—	332	1	no	—
321	C₂²	—	332	1	no	—
322	C₂²	—	166	1	no	—
323	C₂²	—	166	1	no	—
324	C₂²	—	166	1	no	—
325	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	4	1	no	—
329	C₄	—	4	1	no	—
330	C₄	—	4	1	no	—
331	C₄	—	4	1	no	—
332	C₄	—	332	1	no	—
333	C₄	—	332	1	no	—
334	C₄	—	166	1	no	—
335	C₄	—	166	1	no	—
336	C₄	—	166	1	no	—
337	C₄	#14581838	1	1	yes	characteristic
338	C₄	#14581840	1	1	yes	characteristic
339	C₂	—	332	1	no	—
340	C₂	—	664	1	no	—
341	C₂	—	664	1	no	—
342	C₂	#19128281	1	1	yes	characteristic
343	C₂	#19128508	1	1	yes	characteristic
344	C₂	#19128510	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₃
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#479×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	#19128307	C₂
hypercenter	#479	D₈₃
hypocenter	C₈₃	#3260
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #479

Complements

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

N	H ≅ G/N	type
1. #19128309	C₂	semidirect
3. #19128005	C₂	semidirect
4. #19128004	C₂	semidirect
5. #19128000	C₂	semidirect
6. #19127950	C₂	semidirect
7. #19127948	C₂	semidirect
8. #19127933	C₂	semidirect
9. #19127931	C₂	semidirect
10. #19127905	C₂	semidirect
11. #19127905	C₂	semidirect
12. #19127902	C₂	semidirect
13. #19127901	C₂	semidirect
14. #19127726	C₂	semidirect
15. #479×C₈₃	C₂	semidirect
173. C₈₃	#3260	semidirect
177. #479	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128309	2
#19128307	2
#19128005	2
#19128004	2
#19128000	2
#19127950	2
#19127948	2
#19127933	2
#19127931	2
#19127905	2
#19127902	2
#19127901	2
#19127726	2
#479×C₈₃	2
#3260	83