Subgroups — #35295447

Order 21248.

392 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24916432	C₂	1	1	yes	maximal, characteristic
2	#24907215	C₂	1	1	yes	maximal, characteristic, complemented
3	#24906052	C₂	1	1	yes	maximal, characteristic
4	#24905947	C₂	1	1	yes	maximal, characteristic
5	#24905750	C₂	1	1	yes	maximal, characteristic, complemented
6	#24905361	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₃×#1300	C₂	1	1	yes	maximal, characteristic, complemented
8	#19128235	C₂²	1	1	yes	characteristic
9	#19127828	C₄	1	1	yes	characteristic, complemented
10	#19127825	C₄	1	1	yes	characteristic, complemented
11	#19127778	C₄	1	1	yes	characteristic, complemented
12	#19127769	—	2	1	no	—
13	#19127766	C₄	1	1	yes	characteristic, complemented
14	(C₂²⋊D₈₃)⋊C₈	—	2	1	no	—
15	#19127763	—	2	1	no	—
16	#19127754	C₂²	1	1	yes	characteristic
17	#19127658	—	2	1	no	—
18	C₂×#14581703	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
20	#19127456	—	2	1	no	—
21	#19127456	—	2	1	no	—
22	#19127442	C₄	1	2	yes	complemented
23	#19127440	C₄	1	2	yes	complemented
24	#19127437	C₄	1	2	yes	—
25	#19127434	C₄	1	2	yes	—
26	#19127423	—	2	1	no	—
27	#19127423	—	2	1	no	—
28	#432×C₈₃	C₂²	1	1	yes	characteristic
29	C₄×M₁₆×C₈₃	C₂²	1	1	yes	characteristic
30	#376×C₈₃	C₂²	1	1	yes	characteristic
31	C₂×#13050573	—	2	1	no	—
32	C₂²×C₈₃⋊C₈	—	2	1	no	—
33	C₂²×C₈₃⋊C₈	D₄	1	1	yes	characteristic
34	C₂²×C₈₃⋊C₈	D₄	1	1	yes	characteristic
35	C₂×C₈×D₈₃	—	2	1	no	—
36	#14581786	—	2	1	no	—
37	#14581784	C₂×C₄	1	1	yes	characteristic
38	#14581780	C₂×C₄	1	1	yes	characteristic
39	#14581775	—	2	1	no	—
40	#14581771	C₂×C₄	1	1	yes	characteristic
41	#14581766	C₂×C₄	1	1	yes	characteristic
42	C₄×Dic₁₆₆	C₂×C₄	1	1	yes	characteristic
43	C₄×D₃₃₂	C₂×C₄	1	1	yes	characteristic
44	C₄×C₂²⋊D₈₃	—	2	1	no	—
45	#14581749	—	2	1	no	—
46	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
47	Dic₈₃⋊C₈	—	2	1	no	—
48	Dic₈₃⋊C₈	—	2	1	no	—
49	C₂²⋊(C₈₃⋊C₈)	C₂×C₄	1	2	yes	—
50	(C₂×D₈₃)⋊C₈	—	2	1	no	—
51	(C₂×D₈₃)⋊C₈	—	2	1	no	—
52	#14581703	—	2	1	no	—
53	#14581703	C₂×C₄	1	2	yes	—
54	#14581702	—	2	1	no	—
55	C₄×C₈₃⋊C₈	—	2	1	no	—
56	C₄×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
57	C₄×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
58	C₄×C₈₃⋊C₈	D₄	1	1	yes	characteristic
59	C₄×C₈₃⋊C₈	D₄	1	1	yes	characteristic
60	C₈×Dic₈₃	—	2	1	no	—
61	C₂×M₁₆×C₈₃	—	2	1	no	—
62	C₂×C₄²×C₈₃	—	2	1	no	—
63	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
64	C₄⋊C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
65	C₄⋊C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
66	C₂²⋊C₈×C₈₃	—	2	1	no	—
67	#97×C₈₃	—	2	1	no	—
68	#97×C₈₃	C₂×C₄	1	2	yes	—
69	C₄×C₈×C₈₃	—	2	1	no	—
70	#95×C₈₃	—	2	1	no	—
71	#95×C₈₃	C₂×C₄	1	2	yes	—
72	C₂×Dic₁₆₆	C₄²	1	1	yes	characteristic
73	C₂×D₃₃₂	C₄²	1	1	yes	characteristic
74	C₂×C₂²⋊D₈₃	—	2	1	no	—
75	C₂×C₄×D₈₃	—	2	1	no	—
76	#13050573	—	4	2	no	—
77	C₂×C₈₃⋊C₈	—	2	2	no	—
78	C₂×C₈₃⋊C₈	—	2	2	no	—
79	C₂×C₈₃⋊C₈	—	4	1	no	—
80	C₂×C₈₃⋊C₈	—	4	1	no	—
81	C₂×C₈₃⋊C₈	—	4	1	no	—
82	C₂×C₈₃⋊C₈	—	4	1	no	—
83	C₂×C₈₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₈₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₈₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₈₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
87	C₈×D₈₃	—	4	2	no	—
88	C₄⋊Dic₈₃	C₄²	1	1	yes	characteristic
89	C₄⋊Dic₈₃	C₄²	1	1	yes	characteristic
90	Dic₈₃⋊C₄	—	2	1	no	—
91	Dic₈₃⋊C₄	—	2	1	no	—
92	Dic₈₃⋊C₄	—	2	1	no	—
93	Dic₈₃⋊C₄	—	2	1	no	—
94	C₂²⋊Dic₈₃	—	2	1	no	—
95	(C₂×D₈₃)⋊C₄	—	2	1	no	—
96	(C₂×D₈₃)⋊C₄	—	2	1	no	—
97	(C₂×D₈₃)⋊C₄	—	2	1	no	—
98	(C₂×D₈₃)⋊C₄	—	2	1	no	—
99	C₄×Dic₈₃	—	2	1	no	—
100	C₂²×C₄×C₈₃	—	2	2	no	—
101	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
102	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₈₃	#41	1	1	yes	characteristic
104	M₁₆×C₈₃	—	4	2	no	—
105	C₂×C₈×C₈₃	—	2	1	no	—
106	C₂×C₈×C₈₃	—	2	1	no	—
107	C₂×C₈×C₈₃	—	2	2	no	—
108	C₄²×C₈₃	—	4	1	no	—
109	C₄²×C₈₃	—	4	1	no	—
110	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
111	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
112	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₈₃	#41	1	1	yes	characteristic
114	C₂³×C₈₃	C₄×D₄	1	1	yes	characteristic
115	C₂²×D₈₃	—	2	1	no	—
116	C₂×C₄×C₈₃	—	2	2	no	—
117	C₂×C₄×C₈₃	—	2	1	no	—
118	C₂×C₄×C₈₃	—	2	1	no	—
119	C₂×C₄×C₈₃	—	4	1	no	—
120	C₂×C₄×C₈₃	—	4	2	no	—
121	C₂×C₄×C₈₃	—	4	2	no	—
122	C₂×C₄×C₈₃	—	4	2	no	—
123	C₂×C₄×C₈₃	C₂×C₄²	1	1	yes	characteristic
124	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₈₃	#117	1	1	yes	characteristic
126	C₂×C₄×C₈₃	C₄×D₄	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	C₂²⋊D₈₃	—	4	2	no	—
130	C₂×Dic₈₃	—	2	1	no	—
131	C₂×Dic₈₃	—	2	1	no	—
132	C₂×Dic₈₃	—	2	1	no	—
133	D₃₃₂	—	4	1	no	—
134	C₄×D₈₃	—	4	1	no	—
135	C₄×D₈₃	—	4	1	no	—
136	Dic₁₆₆	—	4	1	no	—
137	C₈×C₈₃	—	4	2	no	—
138	C₈×C₈₃	—	4	1	no	—
139	C₈×C₈₃	—	4	1	no	—
140	C₈₃⋊C₈	—	2	1	no	—
141	C₈₃⋊C₈	—	2	1	no	—
142	C₈₃⋊C₈	—	2	1	no	—
143	C₈₃⋊C₈	—	2	1	no	—
144	C₈₃⋊C₈	—	4	2	no	—
145	C₈₃⋊C₈	—	4	2	no	—
146	C₂²×C₈₃	—	2	1	no	—
147	C₂²×C₈₃	—	2	1	no	—
148	C₂²×C₈₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
149	C₂²×C₈₃	#444	1	2	yes	—
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	—	2	1	no	—
156	C₄×C₈₃	—	2	1	no	—
157	C₄×C₈₃	—	2	1	no	—
158	C₄×C₈₃	—	8	2	no	—
159	C₄×C₈₃	#421	1	1	yes	characteristic
160	C₄×C₈₃	#421	1	1	yes	characteristic
161	Dic₈₃	—	4	1	no	—
162	Dic₈₃	—	4	1	no	—
163	Dic₈₃	—	4	1	no	—
164	#8248	—	83	1	no	maximal, Sylow
165	C₂×C₈₃	—	2	2	no	—
166	C₂×C₈₃	#1757	1	1	yes	characteristic
167	C₂×C₈₃	#1759	1	1	yes	characteristic
168	C₂×C₈₃	#1765	1	1	yes	characteristic
169	D₈₃	—	8	1	no	—
170	#2966	—	83	1	no	—
171	#1746	—	83	1	no	—
172	#1524	—	83	1	no	—
173	#1500	—	83	1	no	—
174	#1451	—	83	1	no	—
175	#1302	—	83	1	no	—
176	#1300	D₈₃	1	1	yes	characteristic, complemented
177	C₈₃	#8248	1	1	yes	characteristic, complemented, Sylow
178	#529	—	83	1	no	—
179	#446	—	83	1	no	—
180	C₈×Q₈	—	83	1	no	—
181	#436	—	83	1	no	—
182	#435	—	166	1	no	—
183	C₈×D₄	—	83	1	no	—
184	C₈×D₄	—	166	1	no	—
185	#433	—	166	1	no	—
186	#432	C₂×D₈₃	1	1	yes	characteristic
187	#431	—	83	1	no	—
188	#405	—	166	1	no	—
189	C₄×M₁₆	C₂×D₈₃	1	1	yes	characteristic
190	C₂×#97	—	83	1	no	—
191	C₂×C₄×C₈	—	83	1	no	—
192	#376	C₂×D₈₃	1	1	yes	characteristic
193	#336	—	166	1	no	—
194	#336	—	166	1	no	—
195	#330	—	83	2	no	—
196	#329	—	83	2	no	—
197	#328	—	83	2	no	—
198	#327	—	83	2	no	—
199	#322	—	166	1	no	—
200	#322	—	166	1	no	—
201	C₂×M₁₆	—	2	1	no	—
202	C₂×M₁₆	—	166	1	no	—
203	C₂²×C₈	—	83	1	no	—
204	C₂²×C₈	—	83	1	no	—
205	C₂²×C₈	—	166	1	no	—
206	C₂²×C₈	—	166	1	no	—
207	#126	—	166	1	no	—
208	#125	—	83	1	no	—
209	#124	—	83	1	no	—
210	#123	—	166	1	no	—
211	C₂²⋊Q₈	—	83	1	no	—
212	#121	—	83	1	no	—
213	C₄×Q₈	—	83	1	no	—
214	C₄×D₄	—	83	1	no	—
215	C₄×D₄	—	166	1	no	—
216	#117	—	166	1	no	—
217	C₂×C₄²	—	2	1	no	—
218	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
219	C₄⋊C₈	—	166	1	no	—
220	C₄⋊C₈	—	166	1	no	—
221	C₄⋊C₈	—	166	1	no	—
222	C₄⋊C₈	C₄×D₈₃	1	1	yes	characteristic
223	C₄⋊C₈	C₄×D₈₃	1	1	yes	characteristic
224	C₂²⋊C₈	—	2	1	no	—
225	C₂²⋊C₈	—	83	2	no	—
226	C₂²⋊C₈	—	166	1	no	—
227	C₂²⋊C₈	—	166	1	no	—
228	#97	—	2	1	no	—
229	#97	—	83	2	no	—
230	#97	—	166	1	no	—
231	#97	—	166	1	no	—
232	#97	C₄×D₈₃	1	2	yes	—
233	C₄×C₈	—	2	1	no	—
234	C₄×C₈	—	83	1	no	—
235	C₄×C₈	—	83	1	no	—
236	C₄×C₈	—	83	1	no	—
237	C₄×C₈	—	83	1	no	—
238	C₄×C₈	—	166	1	no	—
239	C₄×C₈	—	166	1	no	—
240	#95	—	2	1	no	—
241	#95	C₄×D₈₃	1	2	yes	—
242	C₂×Q₈	—	83	1	no	—
243	C₂×D₄	—	83	1	no	—
244	C₂×D₄	—	166	1	no	—
245	C₂²×C₄	—	2	2	no	—
246	C₂²×C₄	—	166	1	no	—
247	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
248	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
249	C₂²×C₄	#13050589	1	1	yes	characteristic
250	M₁₆	—	4	2	no	—
251	M₁₆	—	332	2	no	—
252	C₂×C₈	—	2	1	no	—
253	C₂×C₈	—	2	1	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	332	2	no	—
256	C₂×C₈	—	332	1	no	—
257	C₂×C₈	—	332	1	no	—
258	C₂×C₈	—	332	1	no	—
259	C₂×C₈	—	332	1	no	—
260	C₂×C₈	—	83	1	no	—
261	C₂×C₈	—	83	1	no	—
262	C₂×C₈	—	83	1	no	—
263	C₂×C₈	—	83	1	no	—
264	C₂×C₈	—	166	2	no	—
265	C₂×C₈	—	166	2	no	—
266	C₄⋊C₄	—	83	1	no	—
267	C₄⋊C₄	—	83	1	no	—
268	C₄⋊C₄	—	166	1	no	—
269	C₄⋊C₄	—	166	1	no	—
270	C₄⋊C₄	—	166	1	no	—
271	C₄⋊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	—	4	1	no	—
278	C₄²	—	4	1	no	—
279	C₄²	—	166	1	no	—
280	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
281	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
282	C₄²	D₄×D₈₃	1	1	yes	characteristic
283	C₄²	#13050591	1	1	yes	characteristic
284	C₂³	—	166	1	no	—
285	C₂³	(C₂²⋊D₈₃)⋊C₄	1	1	yes	characteristic
286	Q₈	—	332	1	no	—
287	D₄	—	332	1	no	—
288	D₄	—	332	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	332	1	no	—
297	C₂×C₄	—	332	1	no	—
298	C₂×C₄	—	166	1	no	—
299	C₂×C₄	—	166	1	no	—
300	C₂×C₄	—	166	1	no	—
301	C₂×C₄	C₄²×D₈₃	1	1	yes	characteristic
302	C₂×C₄	C₂²⋊C₄×D₈₃	1	1	yes	characteristic
303	C₂×C₄	#14581750	1	1	yes	characteristic
304	C₂×C₄	(C₂²⋊D₈₃)⋊C₄	1	1	yes	characteristic
305	C₂×C₄	#14581754	1	1	yes	characteristic
306	C₂×C₄	#14581754	1	1	yes	characteristic
307	C₈	—	4	2	no	—
308	C₈	—	4	1	no	—
309	C₈	—	4	1	no	—
310	C₈	—	332	2	no	—
311	C₈	—	332	2	no	—
312	C₈	—	166	1	no	—
313	C₈	—	166	1	no	—
314	C₈	—	166	1	no	—
315	C₈	—	166	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	332	1	no	—
319	C₂²	—	332	1	no	—
320	C₂²	—	332	1	no	—
321	C₂²	#19127548	1	1	yes	characteristic
322	C₂²	#19127820	1	2	yes	—
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	8	2	no	—
329	C₄	—	332	1	no	—
330	C₄	—	332	1	no	—
331	C₄	—	332	1	no	—
332	C₄	#19127716	1	1	yes	characteristic
333	C₄	#19127716	1	1	yes	characteristic
334	C₂	—	2	2	no	—
335	C₂	—	664	1	no	—
336	C₂	#24907262	1	1	yes	characteristic
337	C₂	#24907272	1	1	yes	characteristic
338	C₂	#24907304	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊C₄×D₈₃
commutator	C₂×C₄×C₈₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₈₃×#1300	C₂
socle	C₂²×C₈₃	C₄×C₂²⋊C₄
prime radical	#14581766	C₂×C₄
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	#1300	D₈₃
hypocenter	C₈₃	#8248
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1300

Complements

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

N	H ≅ G/N	type
2. #24907215	C₂	semidirect
5. #24905750	C₂	semidirect
6. #24905361	C₂	semidirect
7. C₈₃×#1300	C₂	semidirect
9. #19127828	C₄	semidirect
10. #19127825	C₄	semidirect
11. #19127778	C₄	semidirect
13. #19127766	C₄	semidirect
22. #19127442	C₄	semidirect
23. #19127440	C₄	semidirect
176. #1300	D₈₃	semidirect
177. C₈₃	#8248	semidirect

Maximal subgroups

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

subgroup	index
#24916432	2
#24907215	2
#24906052	2
#24905947	2
#24905750	2
#24905361	2
C₈₃×#1300	2
#8248	83