Subgroups — #35304277

Order 21248.

516 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	#24908441	C₂	1	1	yes	maximal, characteristic, complemented
2	#24908292	C₂	1	1	yes	maximal, characteristic
3	C₄×C₈×Dic₈₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#24906219	C₂	1	1	yes	maximal, characteristic, complemented
5	#24905918	C₂	1	1	yes	maximal, characteristic, complemented
6	#24905914	C₂	1	1	yes	maximal, characteristic
7	C₈₃×#3167	C₂	1	1	yes	maximal, characteristic
8	#19127725	C₂²	1	1	yes	characteristic
9	#19127725	C₂²	1	1	yes	characteristic
10	C₂×#14581719	—	2	2	no	—
11	C₂×C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₈×Dic₈₃	—	2	2	no	—
13	C₄²×Dic₈₃	C₂²	1	1	yes	characteristic
14	#19127460	—	2	2	no	—
15	#19127452	—	2	2	no	—
16	Q₈⋊(C₈₃⋊C₈)	—	2	2	no	—
17	D₄⋊(C₈₃⋊C₈)	—	2	2	no	—
18	#533×C₈₃	C₂²	1	1	yes	characteristic
19	#532×C₈₃	C₂²	1	1	yes	characteristic
20	#498×C₈₃	C₄	1	1	yes	characteristic, complemented
21	#492×C₈₃	C₄	1	1	yes	characteristic, complemented
22	#487×C₈₃	C₄	1	1	yes	characteristic, complemented
23	#486×C₈₃	C₄	1	1	yes	characteristic, complemented
24	#465×C₈₃	—	2	2	no	—
25	C₄×SD₁₆×C₈₃	—	2	2	no	—
26	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
27	C₂×#13050574	—	2	1	no	—
28	C₂×#13050574	—	2	1	no	—
29	C₂²×C₈₃⋊C₈	—	2	1	no	—
30	C₂×C₄×Dic₈₃	—	2	1	no	—
31	C₂×C₄×Dic₈₃	—	2	1	no	—
32	C₂×C₄×Dic₈₃	D₄	1	2	yes	—
33	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
34	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
35	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
36	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
37	#14581719	—	4	4	no	—
38	C₄×C₈₃⋊C₈	—	2	1	no	—
39	C₄×C₈₃⋊C₈	D₄	1	2	yes	—
40	C₈×Dic₈₃	—	2	8	no	—
41	C₂×#41×C₈₃	—	2	1	no	—
42	C₂×SD₁₆×C₈₃	—	2	2	no	—
43	C₂²×C₈×C₈₃	D₄	1	2	yes	—
44	C₄⋊Q₈×C₈₃	C₂×C₄	1	1	yes	characteristic
45	C₄⋊Q₈×C₈₃	C₂×C₄	1	1	yes	characteristic
46	C₄⋊Q₈×C₈₃	C₂×C₄	1	1	yes	characteristic
47	#127×C₈₃	C₂×C₄	1	1	yes	characteristic
48	#125×C₈₃	—	2	1	no	—
49	#124×C₈₃	—	2	1	no	—
50	C₂²⋊Q₈×C₈₃	—	2	1	no	—
51	C₂²⋊Q₈×C₈₃	—	2	1	no	—
52	#121×C₈₃	—	2	1	no	—
53	#121×C₈₃	—	2	1	no	—
54	C₄×Q₈×C₈₃	—	2	1	no	—
55	C₄×Q₈×C₈₃	—	2	1	no	—
56	C₄×D₄×C₈₃	—	2	1	no	—
57	C₄×D₄×C₈₃	—	2	1	no	—
58	#117×C₈₃	—	2	1	no	—
59	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
60	#106×C₈₃	—	2	2	no	—
61	#103×C₈₃	—	2	2	no	—
62	#102×C₈₃	—	2	2	no	—
63	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
65	C₄×C₈×C₈₃	D₄	1	2	yes	—
66	C₂²×Dic₈₃	—	2	2	no	—
67	#13050574	—	4	2	no	—
68	#13050574	—	4	2	no	—
69	C₂×C₈₃⋊C₈	—	2	4	no	—
70	C₂×C₈₃⋊C₈	—	2	2	no	—
71	C₂×C₈₃⋊C₈	—	4	1	no	—
72	C₂×C₈₃⋊C₈	—	4	1	no	—
73	C₄×Dic₈₃	—	2	4	no	—
74	C₄×Dic₈₃	—	2	4	no	—
75	C₄×Dic₈₃	—	2	2	no	—
76	C₄×Dic₈₃	SD₁₆	1	4	yes	complemented
77	#41×C₈₃	—	4	2	no	—
78	#41×C₈₃	—	4	2	no	—
79	C₂×Q₈×C₈₃	—	2	1	no	—
80	C₂×Q₈×C₈₃	—	2	1	no	—
81	C₂×Q₈×C₈₃	—	2	1	no	—
82	C₂×D₄×C₈₃	—	2	1	no	—
83	C₂×D₄×C₈₃	—	2	1	no	—
84	C₂×D₄×C₈₃	—	2	1	no	—
85	C₂×D₄×C₈₃	—	4	1	no	—
86	C₂²×C₄×C₈₃	—	4	1	no	—
87	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₈₃	#41	1	1	yes	characteristic
90	SD₁₆×C₈₃	—	4	4	no	—
91	C₂×C₈×C₈₃	—	2	4	no	—
92	C₂×C₈×C₈₃	C₂²⋊C₄	1	2	yes	—
93	C₂×C₈×C₈₃	C₂²⋊C₄	1	2	yes	—
94	C₄⋊C₄×C₈₃	—	2	1	no	—
95	C₄⋊C₄×C₈₃	—	2	1	no	—
96	C₄⋊C₄×C₈₃	—	2	1	no	—
97	C₄⋊C₄×C₈₃	—	2	1	no	—
98	C₄⋊C₄×C₈₃	—	2	1	no	—
99	C₄⋊C₄×C₈₃	—	2	1	no	—
100	C₄⋊C₄×C₈₃	—	4	1	no	—
101	C₄⋊C₄×C₈₃	—	4	1	no	—
102	C₂²⋊C₄×C₈₃	—	4	1	no	—
103	C₂²⋊C₄×C₈₃	—	4	1	no	—
104	C₂²⋊C₄×C₈₃	—	4	1	no	—
105	C₄²×C₈₃	—	4	1	no	—
106	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
107	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₈₃	#41	1	1	yes	characteristic
110	C₂³×C₈₃	—	4	1	no	—
111	C₂³×C₈₃	#121	1	1	yes	characteristic
112	Q₈×C₈₃	—	4	2	no	—
113	Q₈×C₈₃	—	4	2	no	—
114	Q₈×C₈₃	—	4	2	no	—
115	D₄×C₈₃	—	4	2	no	—
116	D₄×C₈₃	—	4	2	no	—
117	D₄×C₈₃	—	4	2	no	—
118	D₄×C₈₃	—	8	2	no	—
119	C₂×C₄×C₈₃	—	2	2	no	—
120	C₂×C₄×C₈₃	—	2	2	no	—
121	C₂×C₄×C₈₃	—	4	1	no	—
122	C₂×C₄×C₈₃	—	4	1	no	—
123	C₂×C₄×C₈₃	—	4	1	no	—
124	C₂×C₄×C₈₃	—	8	1	no	—
125	C₂×C₄×C₈₃	—	8	1	no	—
126	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
128	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
130	C₂×C₄×C₈₃	#124	1	1	yes	characteristic
131	C₂×C₄×C₈₃	#127	1	1	yes	characteristic
132	C₂×C₄×C₈₃	C₂×SD₁₆	1	2	yes	—
133	C₂×C₄×C₈₃	#135	1	2	yes	—
134	C₂×Dic₈₃	—	2	4	no	—
135	C₂×Dic₈₃	—	2	4	no	—
136	C₂×Dic₈₃	—	2	4	no	—
137	C₈×C₈₃	—	2	2	no	—
138	C₈×C₈₃	#104	1	4	yes	complemented
139	C₈₃⋊C₈	—	2	4	no	—
140	C₈₃⋊C₈	—	8	1	no	—
141	C₈₃⋊C₈	—	8	1	no	—
142	C₂²×C₈₃	—	2	1	no	—
143	C₂²×C₈₃	—	2	1	no	—
144	C₂²×C₈₃	—	8	1	no	—
145	C₂²×C₈₃	—	8	1	no	—
146	C₂²×C₈₃	—	8	1	no	—
147	C₂²×C₈₃	#390	1	1	yes	characteristic
148	C₂²×C₈₃	#465	1	2	yes	—
149	C₄×C₈₃	—	2	1	no	—
150	C₄×C₈₃	—	2	1	no	—
151	C₄×C₈₃	—	8	1	no	—
152	C₄×C₈₃	—	8	1	no	—
153	C₄×C₈₃	—	8	1	no	—
154	C₄×C₈₃	C₂×#104	1	2	yes	—
155	C₄×C₈₃	C₄×SD₁₆	1	2	yes	—
156	C₄×C₈₃	#443	1	2	yes	—
157	C₄×C₈₃	#492	1	1	yes	characteristic
158	C₄×C₈₃	#495	1	1	yes	characteristic
159	Dic₈₃	—	2	8	no	—
160	#9843	—	83	1	no	maximal, Sylow
161	C₂×C₈₃	—	2	2	no	—
162	C₂×C₈₃	—	16	1	no	—
163	C₂×C₈₃	#1962	1	1	yes	characteristic
164	C₂×C₈₃	#1968	1	1	yes	characteristic
165	C₂×C₈₃	#1974	1	1	yes	characteristic
166	#3167	D₈₃	1	1	yes	characteristic
167	#1994	—	83	1	no	—
168	#1962	—	83	1	no	—
169	C₄²×C₈	—	83	1	no	—
170	#1566	—	83	1	no	—
171	#1493	—	83	1	no	—
172	#1492	—	83	1	no	—
173	C₈₃	#9843	1	1	yes	characteristic, complemented, Sylow
174	#533	C₂×D₈₃	1	1	yes	characteristic
175	#532	C₂×D₈₃	1	1	yes	characteristic
176	#498	Dic₈₃	1	1	yes	characteristic, complemented
177	#492	Dic₈₃	1	1	yes	characteristic, complemented
178	#487	Dic₈₃	1	1	yes	characteristic, complemented
179	#486	Dic₈₃	1	1	yes	characteristic, complemented
180	#465	—	2	2	no	—
181	C₄×SD₁₆	—	2	2	no	—
182	#423	—	83	1	no	—
183	#423	—	83	1	no	—
184	C₂×#104	—	166	2	no	—
185	C₂×C₄×C₈	—	83	1	no	—
186	C₂×C₄×C₈	—	166	2	no	—
187	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
188	C₄³	—	83	1	no	—
189	#339	—	166	2	no	—
190	#334	—	166	2	no	—
191	Q₈⋊C₈	—	166	2	no	—
192	D₄⋊C₈	—	166	2	no	—
193	C₂×#41	—	2	1	no	—
194	C₂×SD₁₆	—	2	2	no	—
195	C₂×M₁₆	—	166	1	no	—
196	C₂×M₁₆	—	166	1	no	—
197	C₂²×C₈	—	166	1	no	—
198	C₂²×C₈	C₂²⋊D₈₃	1	2	yes	—
199	C₄⋊Q₈	C₂×Dic₈₃	1	1	yes	characteristic
200	C₄⋊Q₈	C₂×Dic₈₃	1	1	yes	characteristic
201	C₄⋊Q₈	C₂×Dic₈₃	1	1	yes	characteristic
202	#127	C₂×Dic₈₃	1	1	yes	characteristic
203	#125	—	2	1	no	—
204	#124	—	2	1	no	—
205	C₂²⋊Q₈	—	2	1	no	—
206	C₂²⋊Q₈	—	2	1	no	—
207	#121	—	2	1	no	—
208	#121	—	2	1	no	—
209	C₄×Q₈	—	2	1	no	—
210	C₄×Q₈	—	2	1	no	—
211	C₄×D₄	—	2	1	no	—
212	C₄×D₄	—	2	1	no	—
213	#117	—	2	1	no	—
214	C₂×C₄²	—	83	2	no	—
215	C₂×C₄²	—	166	1	no	—
216	C₂×C₄²	—	166	1	no	—
217	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
218	#106	—	2	2	no	—
219	C₄⋊C₈	—	166	1	no	—
220	C₄⋊C₈	—	166	1	no	—
221	C₄⋊C₈	—	166	1	no	—
222	C₄⋊C₈	—	166	1	no	—
223	#104	—	332	4	no	—
224	#103	—	2	2	no	—
225	#102	—	2	2	no	—
226	C₄×C₈	—	83	2	no	—
227	C₄×C₈	—	166	8	no	—
228	C₄×C₈	—	166	1	no	—
229	C₄×C₈	C₂×Dic₈₃	1	1	yes	characteristic
230	C₄×C₈	C₂×Dic₈₃	1	1	yes	characteristic
231	C₄×C₈	C₂²⋊D₈₃	1	2	yes	—
232	#41	—	4	2	no	—
233	#41	—	4	2	no	—
234	C₂×Q₈	—	2	1	no	—
235	C₂×Q₈	—	2	1	no	—
236	C₂×Q₈	—	2	1	no	—
237	C₂×D₄	—	2	1	no	—
238	C₂×D₄	—	2	1	no	—
239	C₂×D₄	—	2	1	no	—
240	C₂×D₄	—	4	1	no	—
241	C₂²×C₄	—	4	1	no	—
242	C₂²×C₄	—	166	2	no	—
243	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
244	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
245	C₂²×C₄	#13050589	1	1	yes	characteristic
246	SD₁₆	—	4	4	no	—
247	M₁₆	—	332	2	no	—
248	M₁₆	—	332	2	no	—
249	C₂×C₈	—	2	4	no	—
250	C₂×C₈	—	332	1	no	—
251	C₂×C₈	—	332	1	no	—
252	C₂×C₈	—	166	4	no	—
253	C₂×C₈	—	166	2	no	—
254	C₂×C₈	C₂²⋊Dic₈₃	1	2	yes	—
255	C₂×C₈	C₂²⋊Dic₈₃	1	2	yes	—
256	C₄⋊C₄	—	2	1	no	—
257	C₄⋊C₄	—	2	1	no	—
258	C₄⋊C₄	—	2	1	no	—
259	C₄⋊C₄	—	2	1	no	—
260	C₄⋊C₄	—	2	1	no	—
261	C₄⋊C₄	—	2	1	no	—
262	C₄⋊C₄	—	4	1	no	—
263	C₄⋊C₄	—	4	1	no	—
264	C₂²⋊C₄	—	4	1	no	—
265	C₂²⋊C₄	—	4	1	no	—
266	C₂²⋊C₄	—	4	1	no	—
267	C₄²	—	4	1	no	—
268	C₄²	—	83	4	no	—
269	C₄²	—	166	4	no	—
270	C₄²	—	166	4	no	—
271	C₄²	—	166	2	no	—
272	C₄²	C₂²×Dic₈₃	1	1	yes	characteristic
273	C₄²	D₄×D₈₃	1	1	yes	characteristic
274	C₄²	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
275	C₄²	#13050589	1	1	yes	characteristic
276	C₂³	—	4	1	no	—
277	C₂³	#14581763	1	1	yes	characteristic
278	Q₈	—	4	2	no	—
279	Q₈	—	4	2	no	—
280	Q₈	—	4	2	no	—
281	D₄	—	4	2	no	—
282	D₄	—	4	2	no	—
283	D₄	—	4	2	no	—
284	D₄	—	8	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	4	1	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	8	1	no	—
291	C₂×C₄	—	8	1	no	—
292	C₂×C₄	—	166	4	no	—
293	C₂×C₄	—	166	4	no	—
294	C₂×C₄	—	166	4	no	—
295	C₂×C₄	C₂×C₂²⋊Dic₈₃	1	1	yes	characteristic
296	C₂×C₄	D₄×Dic₈₃	1	1	yes	characteristic
297	C₂×C₄	D₄×Dic₈₃	1	1	yes	characteristic
298	C₂×C₄	#14581763	1	1	yes	characteristic
299	C₂×C₄	#14581779	1	1	yes	characteristic
300	C₂×C₄	#14581788	1	1	yes	characteristic
301	C₂×C₄	SD₁₆×D₈₃	1	2	yes	—
302	C₂×C₄	#14581813	1	2	yes	—
303	C₈	—	2	2	no	—
304	C₈	—	664	1	no	—
305	C₈	—	664	1	no	—
306	C₈	—	166	4	no	—
307	C₈	#14581719	1	4	yes	complemented
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	8	1	no	—
311	C₂²	—	8	1	no	—
312	C₂²	—	8	1	no	—
313	C₂²	#19127604	1	1	yes	characteristic
314	C₂²	#19127927	1	2	yes	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	8	1	no	—
318	C₄	—	8	1	no	—
319	C₄	—	8	1	no	—
320	C₄	—	166	8	no	—
321	C₄	C₂×#14581719	1	2	yes	—
322	C₄	SD₁₆×Dic₈₃	1	2	yes	—
323	C₄	#19127818	1	2	yes	—
324	C₄	#19128072	1	1	yes	characteristic
325	C₄	#19128082	1	1	yes	characteristic
326	C₂	—	2	2	no	—
327	C₂	—	16	1	no	—
328	C₂	#24908292	1	1	yes	characteristic
329	C₂	#24908319	1	1	yes	characteristic
330	C₂	#24908352	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14581788
commutator	C₄²×C₈₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₈₃×#3167	C₂
socle	C₂²×C₈₃	#390
prime radical	#532×C₈₃	C₂²
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	#3167	D₈₃
hypocenter	C₈₃	#9843
LCS	G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3167

Complements

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

N	H ≅ G/N	type
1. #24908441	C₂	semidirect
3. C₄×C₈×Dic₈₃	C₂	semidirect
4. #24906219	C₂	semidirect
5. #24905918	C₂	semidirect
20. #498×C₈₃	C₄	semidirect
21. #492×C₈₃	C₄	semidirect
22. #487×C₈₃	C₄	semidirect
23. #486×C₈₃	C₄	semidirect
76. C₄×Dic₈₃	SD₁₆	semidirect
138. C₈×C₈₃	#104	semidirect
173. C₈₃	#9843	semidirect
176. #498	Dic₈₃	semidirect
177. #492	Dic₈₃	semidirect
178. #487	Dic₈₃	semidirect
179. #486	Dic₈₃	semidirect
307. C₈	#14581719	semidirect

Maximal subgroups

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

subgroup	index
#24908441	2
#24908292	2
C₄×C₈×Dic₈₃	2
#24906219	2
#24905918	2
#24905914	2
C₈₃×#3167	2
#9843	83