Subgroups — #24919910

Order 10624.

376 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128389	C₂	1	1	yes	maximal, characteristic, complemented
2	#19128356	C₂	1	1	yes	maximal, characteristic
3	#19128111	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128038	C₂	1	1	yes	maximal, characteristic, complemented
5	#19128033	C₂	1	1	yes	maximal, characteristic, complemented
6	#19128023	C₂	1	1	yes	maximal, characteristic, complemented
7	#19128013	C₂	1	1	yes	maximal, characteristic, complemented
8	#19127997	C₂	1	1	yes	maximal, characteristic, complemented
9	#19127946	C₂	1	1	yes	maximal, characteristic, complemented
10	#19127929	C₂	1	1	yes	maximal, characteristic, complemented
11	#19127801	C₂	1	1	yes	maximal, characteristic, complemented
12	#19127771	C₂	1	1	yes	maximal, characteristic, complemented
13	#19127740	C₂	1	1	yes	maximal, characteristic, complemented
14	#107×D₈₃	C₂	1	1	yes	maximal, characteristic, complemented
15	#478×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13050590	—	2	1	no	—
17	C₂×#13050578	C₂²	1	1	yes	characteristic
18	C₂×#13050573	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₈₃	C₂²	1	1	yes	characteristic
20	#14581792	C₂²	1	1	yes	characteristic
21	#14581787	—	2	1	no	—
22	#14581783	—	2	1	no	—
23	#14581777	—	2	1	no	—
24	#14581774	—	2	1	no	—
25	#14581774	—	2	1	no	—
26	#14581772	C₂²	1	1	yes	characteristic
27	#14581772	C₂²	1	1	yes	characteristic
28	#14581771	—	2	1	no	—
29	#14581763	C₂²	1	1	yes	characteristic
30	#14581763	C₂²	1	1	yes	characteristic
31	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
32	C₄×D₃₃₂	C₂²	1	1	yes	characteristic
33	#14581750	—	2	1	no	—
34	#14581750	C₂²	1	1	yes	characteristic
35	C₄⋊C₄×D₈₃	C₂²	1	1	yes	characteristic
36	C₄⋊C₄×D₈₃	C₂²	1	1	yes	characteristic
37	C₄⋊C₄×D₈₃	C₂²	1	1	yes	characteristic
38	#14581726	C₂²	1	1	yes	characteristic
39	#14581726	C₂²	1	1	yes	characteristic
40	#14581725	C₂²	1	1	yes	characteristic
41	#14581725	C₂²	1	1	yes	characteristic
42	#14581725	C₂²	1	1	yes	characteristic
43	#14581723	C₂²	1	1	yes	characteristic
44	#14581723	C₂²	1	1	yes	characteristic
45	#14581717	C₂²	1	1	yes	characteristic
46	#14581717	C₂²	1	1	yes	characteristic
47	#14581715	C₂²	1	1	yes	characteristic
48	#14581713	C₂²	1	1	yes	characteristic
49	(C₂×D₈₃)⋊C₈	C₂²	1	1	yes	characteristic
50	(C₂×D₈₃)⋊C₈	C₂²	1	1	yes	characteristic
51	#14581703	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	C₂³	1	1	yes	characteristic
62	C₂×D₃₃₂	D₄	1	1	yes	characteristic
63	C₂×C₂²⋊D₈₃	—	2	1	no	—
64	C₂×C₂²⋊D₈₃	—	2	1	no	—
65	C₂×C₄×D₈₃	—	2	1	no	—
66	C₂×C₄×D₈₃	—	2	1	no	—
67	C₂×C₄×D₈₃	—	2	1	no	—
68	C₂×C₄×D₈₃	C₂³	1	1	yes	characteristic
69	C₂×C₄×D₈₃	C₂³	1	1	yes	characteristic
70	#13050578	—	2	2	no	—
71	#13050573	—	2	2	no	—
72	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
73	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
74	C₈×D₈₃	—	2	2	no	—
75	C₄⋊Dic₈₃	—	2	1	no	—
76	C₄⋊Dic₈₃	—	2	1	no	—
77	C₄⋊Dic₈₃	—	2	1	no	—
78	C₄⋊Dic₈₃	D₄	1	1	yes	characteristic
79	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
80	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
81	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
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	C₂²⋊Dic₈₃	—	2	1	no	—
87	C₂²⋊Dic₈₃	—	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	D₄	1	1	yes	characteristic
92	(C₂×D₈₃)⋊C₄	D₄	1	1	yes	characteristic
93	C₄×Dic₈₃	—	2	1	no	—
94	C₄×Dic₈₃	—	2	1	no	—
95	C₂×D₄×C₈₃	C₂³	1	1	yes	characteristic
96	C₂²×C₄×C₈₃	—	2	1	no	—
97	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
98	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
99	C₄⋊C₄×C₈₃	—	2	1	no	—
100	C₄⋊C₄×C₈₃	—	2	1	no	—
101	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
103	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
104	C₂²⋊C₄×C₈₃	—	2	1	no	—
105	C₄²×C₈₃	C₂³	1	1	yes	characteristic
106	C₂³×C₈₃	—	2	1	no	—
107	C₂²×D₈₃	C₂×D₄	1	1	yes	characteristic
108	C₂²×D₈₃	C₂×D₄	1	1	yes	characteristic
109	D₄×C₈₃	—	2	1	no	—
110	D₄×C₈₃	—	2	1	no	—
111	C₂×C₄×C₈₃	—	2	1	no	—
112	C₂×C₄×C₈₃	—	2	1	no	—
113	C₂×C₄×C₈₃	—	2	1	no	—
114	C₂×C₄×C₈₃	—	4	2	no	—
115	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
118	C₂²⋊D₈₃	—	4	2	no	—
119	C₂²⋊D₈₃	—	4	2	no	—
120	C₂×Dic₈₃	—	2	1	no	—
121	C₂×Dic₈₃	—	2	1	no	—
122	C₂×Dic₈₃	—	2	1	no	—
123	C₂×Dic₈₃	—	2	1	no	—
124	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
125	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
126	D₃₃₂	—	2	2	no	—
127	C₄×D₈₃	—	2	1	no	—
128	C₄×D₈₃	—	2	1	no	—
129	C₄×D₈₃	—	2	1	no	—
130	C₄×D₈₃	—	2	1	no	—
131	C₄×D₈₃	—	4	2	no	—
132	C₄×D₈₃	—	4	2	no	—
133	C₄×D₈₃	—	4	2	no	—
134	Dic₁₆₆	—	2	1	no	—
135	Dic₁₆₆	—	2	1	no	—
136	Dic₁₆₆	—	4	2	no	—
137	C₈×C₈₃	—	2	1	no	—
138	C₈×C₈₃	—	2	1	no	—
139	C₈₃⋊C₈	—	2	1	no	—
140	C₈₃⋊C₈	#41	1	2	yes	complemented
141	C₂²×C₈₃	—	4	1	no	—
142	C₂²×C₈₃	—	4	1	no	—
143	C₂²×C₈₃	—	4	1	no	—
144	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
145	C₂×D₈₃	—	2	1	no	—
146	C₂×D₈₃	—	2	1	no	—
147	C₂×D₈₃	—	2	1	no	—
148	C₂×D₈₃	—	2	1	no	—
149	C₂×D₈₃	—	2	1	no	—
150	C₂×D₈₃	AGL(1,ℤ/8)	1	2	yes	—
151	C₄×C₈₃	—	2	1	no	—
152	C₄×C₈₃	—	2	1	no	—
153	C₄×C₈₃	—	4	1	no	—
154	C₄×C₈₃	—	4	1	no	—
155	C₄×C₈₃	—	4	1	no	—
156	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
157	C₄×C₈₃	2₋¹⁺⁴	1	1	yes	characteristic
158	Dic₈₃	—	2	1	no	—
159	Dic₈₃	—	2	1	no	—
160	Dic₈₃	—	4	1	no	—
161	Dic₈₃	—	4	1	no	—
162	Dic₈₃	—	4	1	no	—
163	Dic₈₃	—	4	1	no	—
164	C₂×C₈₃	—	8	1	no	—
165	C₂×C₈₃	#547	1	1	yes	characteristic
166	C₂×C₈₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
167	C₂×C₈₃	#578	1	1	yes	characteristic
168	D₈₃	—	2	2	no	—
169	D₈₃	—	4	1	no	—
170	#3343	—	83	1	no	maximal, Sylow
171	C₈₃	#3343	1	1	yes	characteristic, complemented, Sylow
172	#552	—	83	1	no	—
173	#547	—	83	1	no	—
174	#501	—	83	1	no	—
175	#484	—	83	1	no	—
176	#483	—	83	1	no	—
177	#482	—	83	1	no	—
178	#480	—	83	1	no	—
179	#478	D₈₃	1	1	yes	characteristic, complemented
180	#477	—	83	1	no	—
181	#468	—	83	1	no	—
182	#465	—	83	1	no	—
183	#440	—	83	1	no	—
184	#435	—	83	1	no	—
185	#428	—	83	1	no	—
186	C₂×#107	—	83	1	no	—
187	C₂×#41	—	166	1	no	—
188	C₂×SD₁₆	—	83	1	no	—
189	C₂×M₁₆	—	83	1	no	—
190	C₂²×C₈	—	83	1	no	—
191	C₄⋊Q₈	—	83	1	no	—
192	#126	—	166	1	no	—
193	#125	—	166	1	no	—
194	#125	C₂×D₈₃	1	1	yes	characteristic
195	#123	—	166	1	no	—
196	#123	—	166	1	no	—
197	#123	—	166	1	no	—
198	C₂²⋊Q₈	—	83	1	no	—
199	C₂²⋊Q₈	—	83	1	no	—
200	C₂²⋊Q₈	—	166	1	no	—
201	#121	—	83	1	no	—
202	#121	—	83	1	no	—
203	C₄×Q₈	—	83	1	no	—
204	C₄×D₄	—	83	1	no	—
205	C₄×D₄	C₂×D₈₃	1	1	yes	characteristic
206	#117	—	83	1	no	—
207	#117	—	166	1	no	—
208	C₂×C₄⋊C₄	—	83	1	no	—
209	C₂×C₄⋊C₄	—	83	1	no	—
210	C₂×C₄⋊C₄	—	83	1	no	—
211	#107	—	83	1	no	—
212	#107	—	83	1	no	—
213	#107	—	83	1	no	—
214	#107	—	83	1	no	—
215	#107	—	83	1	no	—
216	#107	C₂×D₈₃	1	1	yes	characteristic
217	#106	—	83	1	no	—
218	#106	—	83	1	no	—
219	#106	C₂×D₈₃	1	1	yes	characteristic
220	C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
221	#103	—	83	1	no	—
222	#103	—	83	1	no	—
223	#103	—	83	1	no	—
224	#102	—	83	1	no	—
225	#102	C₂×D₈₃	1	1	yes	characteristic
226	#102	C₂×D₈₃	1	1	yes	characteristic
227	C₂²⋊C₈	—	83	1	no	—
228	C₂²⋊C₈	—	83	1	no	—
229	#97	—	83	1	no	—
230	#41	—	332	4	no	—
231	C₂×Q₈	—	83	1	no	—
232	C₂×Q₈	—	166	1	no	—
233	C₂×D₄	—	83	1	no	—
234	C₂×D₄	—	166	1	no	—
235	C₂×D₄	—	166	1	no	—
236	C₂×D₄	C₂²×D₈₃	1	1	yes	characteristic
237	C₂²×C₄	—	2	1	no	—
238	C₂²×C₄	—	83	1	no	—
239	C₂²×C₄	—	83	1	no	—
240	C₂²×C₄	—	166	1	no	—
241	C₂²×C₄	—	166	1	no	—
242	C₂²×C₄	—	166	1	no	—
243	SD₁₆	—	166	2	no	—
244	M₁₆	—	166	2	no	—
245	C₂×C₈	—	83	1	no	—
246	C₂×C₈	—	83	1	no	—
247	C₂×C₈	—	166	2	no	—
248	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
249	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
250	C₄⋊C₄	—	2	1	no	—
251	C₄⋊C₄	—	2	1	no	—
252	C₄⋊C₄	—	83	1	no	—
253	C₄⋊C₄	—	83	1	no	—
254	C₄⋊C₄	—	83	1	no	—
255	C₄⋊C₄	—	83	1	no	—
256	C₄⋊C₄	—	166	1	no	—
257	C₄⋊C₄	—	166	1	no	—
258	C₄⋊C₄	—	166	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	C₂²×D₈₃	1	1	yes	characteristic
264	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
265	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
266	C₂²⋊C₄	—	2	1	no	—
267	C₂²⋊C₄	—	83	1	no	—
268	C₂²⋊C₄	—	83	1	no	—
269	C₂²⋊C₄	—	166	1	no	—
270	C₂²⋊C₄	—	166	1	no	—
271	C₂²⋊C₄	—	166	1	no	—
272	C₂²⋊C₄	—	166	1	no	—
273	C₂²⋊C₄	—	166	1	no	—
274	C₄²	—	166	1	no	—
275	C₄²	—	166	1	no	—
276	C₄²	C₂²×D₈₃	1	1	yes	characteristic
277	C₂³	—	2	1	no	—
278	C₂³	—	83	1	no	—
279	C₂³	—	83	1	no	—
280	Q₈	—	332	2	no	—
281	Q₈	—	166	1	no	—
282	Q₈	—	166	1	no	—
283	D₄	—	2	1	no	—
284	D₄	—	2	1	no	—
285	D₄	—	332	2	no	—
286	D₄	—	332	2	no	—
287	D₄	—	166	2	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	332	2	no	—
293	C₂×C₄	—	332	2	no	—
294	C₂×C₄	—	332	2	no	—
295	C₂×C₄	—	83	1	no	—
296	C₂×C₄	—	83	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	—	166	1	no	—
302	C₂×C₄	—	166	1	no	—
303	C₂×C₄	—	166	1	no	—
304	C₂×C₄	—	166	1	no	—
305	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
306	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
307	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	83	2	no	—
311	C₈	—	166	1	no	—
312	C₂²	—	4	1	no	—
313	C₂²	—	4	1	no	—
314	C₂²	—	4	1	no	—
315	C₂²	—	83	2	no	—
316	C₂²	—	166	1	no	—
317	C₂²	—	166	1	no	—
318	C₂²	—	166	1	no	—
319	C₂²	—	166	1	no	—
320	C₂²	—	166	1	no	—
321	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	4	1	no	—
325	C₄	—	4	1	no	—
326	C₄	—	4	1	no	—
327	C₄	—	332	1	no	—
328	C₄	—	332	1	no	—
329	C₄	—	332	1	no	—
330	C₄	—	332	1	no	—
331	C₄	—	166	1	no	—
332	C₄	—	166	1	no	—
333	C₄	#41×D₈₃	1	1	yes	characteristic
334	C₄	#14581840	1	1	yes	characteristic
335	C₂	—	8	1	no	—
336	C₂	—	332	1	no	—
337	C₂	—	166	2	no	—
338	C₂	#19128353	1	1	yes	characteristic
339	C₂	AGL(1,ℤ/8)×D₈₃	1	1	yes	characteristic
340	C₂	#19128515	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	#19128356	C₂
hypercenter	#478	D₈₃
hypocenter	C₈₃	#3343
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. #19128389	C₂	semidirect
3. #19128111	C₂	semidirect
4. #19128038	C₂	semidirect
5. #19128033	C₂	semidirect
6. #19128023	C₂	semidirect
7. #19128013	C₂	semidirect
8. #19127997	C₂	semidirect
9. #19127946	C₂	semidirect
10. #19127929	C₂	semidirect
11. #19127801	C₂	semidirect
12. #19127771	C₂	semidirect
13. #19127740	C₂	semidirect
14. #107×D₈₃	C₂	semidirect
15. #478×C₈₃	C₂	semidirect
140. C₈₃⋊C₈	#41	semidirect
171. C₈₃	#3343	semidirect
179. #478	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128389	2
#19128356	2
#19128111	2
#19128038	2
#19128033	2
#19128023	2
#19128013	2
#19127997	2
#19127946	2
#19127929	2
#19127801	2
#19127771	2
#19127740	2
#107×D₈₃	2
#478×C₈₃	2
#3343	83