Subgroups — #35392340

Order 21248.

858 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24918585	C₂	1	2	yes	maximal, complemented
2	#24918043	C₂	1	1	yes	maximal, characteristic
3	#500×D₈₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#24917921	C₂	1	2	yes	maximal, complemented
5	#24907135	C₂	1	2	yes	maximal, complemented
6	#24907133	C₂	1	2	yes	maximal, complemented
7	#24907080	C₂	1	1	yes	maximal, characteristic, complemented
8	#24907056	C₂	1	1	yes	maximal, characteristic, complemented
9	#24907055	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×#13050573	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₃×#1713	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14581824	—	2	4	no	—
13	#19128252	C₂²	1	2	yes	—
14	C₄⋊Q₈×D₈₃	C₂²	1	2	yes	—
15	#19128106	C₂²	1	1	yes	characteristic
16	#19128105	—	2	2	no	—
17	#19128104	C₂²	1	2	yes	—
18	#19128104	C₂²	1	2	yes	—
19	#19128094	—	2	4	no	—
20	#19128080	C₂²	1	2	yes	—
21	#19128078	C₂²	1	1	yes	characteristic
22	#19128075	C₂²	1	2	yes	—
23	#19128074	C₂²	1	2	yes	—
24	#19128049	C₂²	1	2	yes	—
25	#19128049	C₂²	1	2	yes	—
26	#19128048	C₂²	1	2	yes	—
27	#19128016	—	2	4	no	—
28	#19127980	—	2	4	no	—
29	#19127938	—	2	2	no	—
30	#19127783	—	2	2	no	—
31	#19127752	C₂²	1	1	yes	characteristic
32	#107×D₈₃	—	2	2	no	—
33	C₄×#13050573	C₂²	1	1	yes	characteristic
34	C₄×C₈×D₈₃	C₂²	1	1	yes	characteristic
35	#19127423	C₂²	1	1	yes	characteristic
36	#19127422	C₂²	1	1	yes	characteristic
37	C₈×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
38	#500×C₈₃	C₂²	1	1	yes	characteristic
39	#500×C₈₃	C₂²	1	1	yes	characteristic
40	#500×C₈₃	C₂²	1	2	yes	—
41	#494×C₈₃	C₂²	1	2	yes	—
42	C₈²×C₈₃	C₂²	1	1	yes	characteristic
43	C₂×#13050591	—	2	2	no	—
44	C₂×Q₈×D₈₃	—	2	2	no	—
45	#14581824	—	4	16	no	—
46	C₂×#13050580	—	2	4	no	—
47	C₂×#13050579	—	2	4	no	—
48	C₂×#13050577	—	2	4	no	—
49	C₂×D₆₆₄	—	2	2	no	—
50	C₂×#13050573	D₄	1	2	yes	—
51	C₂×C₈×D₈₃	D₄	1	2	yes	—
52	#14581792	C₂³	1	2	yes	—
53	#14581791	—	2	2	no	—
54	#14581791	—	2	2	no	—
55	#14581790	—	2	2	no	—
56	#14581789	C₂³	1	2	yes	—
57	#14581781	—	2	2	no	—
58	#14581767	—	2	2	no	—
59	#14581767	—	2	2	no	—
60	#14581754	—	2	2	no	—
61	#14581754	—	2	2	no	—
62	C₄⋊C₄×D₈₃	—	2	2	no	—
63	C₄⋊C₄×D₈₃	—	2	2	no	—
64	C₄²×D₈₃	C₂³	1	1	yes	characteristic
65	#14581726	—	2	2	no	—
66	#14581725	—	2	4	no	—
67	#14581725	—	2	4	no	—
68	Dic₈₃⋊C₈	D₄	1	2	yes	—
69	#14581714	—	2	4	no	—
70	#14581712	—	2	4	no	—
71	#14581712	—	2	4	no	—
72	(C₂×D₈₃)⋊C₈	D₄	1	2	yes	—
73	#14581702	D₄	1	2	yes	—
74	C₄×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
75	C₄×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
76	C₄×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
77	C₈×Dic₈₃	D₄	1	2	yes	—
78	C₂×Q₁₆×C₈₃	—	2	4	no	—
79	C₄⋊Q₈×C₈₃	C₂³	1	2	yes	—
80	C₄⋊Q₈×C₈₃	C₂³	1	2	yes	—
81	#107×C₈₃	—	2	2	no	—
82	#107×C₈₃	—	2	2	no	—
83	#107×C₈₃	—	2	4	no	—
84	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
85	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
86	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
87	#13050591	—	4	8	no	—
88	C₂×Dic₁₆₆	—	2	2	no	—
89	C₂×Dic₁₆₆	—	4	2	no	—
90	Q₈×D₈₃	—	4	8	no	—
91	C₂×D₃₃₂	—	2	2	no	—
92	C₂×D₃₃₂	—	2	2	no	—
93	C₂×D₃₃₂	—	2	2	no	—
94	C₂×D₃₃₂	—	4	2	no	—
95	C₂×C₄×D₈₃	—	4	2	no	—
96	C₂×C₄×D₈₃	—	4	2	no	—
97	C₂×C₄×D₈₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×D₈₃	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×D₈₃	C₂×D₄	1	1	yes	characteristic
100	#13050580	—	4	8	no	—
101	#13050579	—	4	8	no	—
102	#13050577	—	4	8	no	—
103	D₆₆₄	—	4	4	no	—
104	#13050573	D₈	1	8	yes	complemented
105	C₂×C₈₃⋊C₈	C₂×D₄	1	2	yes	—
106	C₂×C₈₃⋊C₈	C₂×D₄	1	2	yes	—
107	C₂×C₈₃⋊C₈	C₂×D₄	1	2	yes	—
108	C₈×D₈₃	—	2	4	no	—
109	C₄⋊Dic₈₃	—	2	2	no	—
110	C₄⋊Dic₈₃	—	2	2	no	—
111	Dic₈₃⋊C₄	—	4	2	no	—
112	Dic₈₃⋊C₄	—	4	2	no	—
113	(C₂×D₈₃)⋊C₄	—	4	2	no	—
114	(C₂×D₈₃)⋊C₄	—	4	2	no	—
115	C₄×Dic₈₃	C₂×D₄	1	1	yes	characteristic
116	C₄×Dic₈₃	C₂×D₄	1	1	yes	characteristic
117	C₄×Dic₈₃	C₂×D₄	1	1	yes	characteristic
118	C₂×Q₈×C₈₃	—	2	2	no	—
119	C₂×Q₈×C₈₃	—	2	2	no	—
120	Q₁₆×C₈₃	—	4	8	no	—
121	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
122	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
123	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
124	C₄⋊C₄×C₈₃	—	2	2	no	—
125	C₄⋊C₄×C₈₃	—	2	2	no	—
126	C₄⋊C₄×C₈₃	—	2	2	no	—
127	C₄⋊C₄×C₈₃	—	2	2	no	—
128	C₄²×C₈₃	C₂⁴	1	1	yes	characteristic
129	C₂²×D₈₃	—	4	2	no	—
130	C₂²×D₈₃	#127	1	1	yes	characteristic
131	Q₈×C₈₃	—	4	4	no	—
132	Q₈×C₈₃	—	4	4	no	—
133	C₂×C₄×C₈₃	—	4	2	no	—
134	C₂×C₄×C₈₃	—	4	2	no	—
135	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
138	C₂×Dic₈₃	—	4	2	no	—
139	C₂×Dic₈₃	#127	1	1	yes	characteristic
140	C₂×Dic₈₃	#127	1	1	yes	characteristic
141	C₂×Dic₈₃	#127	1	1	yes	characteristic
142	D₃₃₂	—	4	4	no	—
143	D₃₃₂	—	4	4	no	—
144	D₃₃₂	—	4	4	no	—
145	D₃₃₂	—	8	4	no	—
146	C₄×D₈₃	—	2	2	no	—
147	C₄×D₈₃	—	2	2	no	—
148	C₄×D₈₃	—	8	4	no	—
149	C₄×D₈₃	—	8	4	no	—
150	C₄×D₈₃	C₂×D₈	1	4	yes	—
151	Dic₁₆₆	—	4	4	no	—
152	Dic₁₆₆	—	8	4	no	—
153	C₈×C₈₃	—	2	2	no	—
154	C₈×C₈₃	C₂×D₈	1	4	yes	—
155	C₈×C₈₃	#137	1	4	yes	complemented
156	C₈₃⋊C₈	—	2	2	no	—
157	C₈₃⋊C₈	—	2	2	no	—
158	C₈₃⋊C₈	C₂×D₈	1	4	yes	—
159	C₂²×C₈₃	C₂×#127	1	1	yes	characteristic
160	C₂×D₈₃	—	2	1	no	—
161	C₂×D₈₃	—	2	1	no	—
162	C₂×D₈₃	—	8	2	no	—
163	C₂×D₈₃	—	8	2	no	—
164	C₂×D₈₃	—	8	2	no	—
165	C₂×D₈₃	#493	1	2	yes	—
166	C₄×C₈₃	—	8	2	no	—
167	C₄×C₈₃	—	8	2	no	—
168	C₄×C₈₃	C₂²×D₈	1	2	yes	—
169	C₄×C₈₃	C₂×#137	1	2	yes	—
170	C₄×C₈₃	#578	1	2	yes	—
171	Dic₈₃	—	2	1	no	—
172	Dic₈₃	—	2	1	no	—
173	Dic₈₃	—	8	2	no	—
174	Dic₈₃	#493	1	2	yes	—
175	#19630	—	83	1	no	maximal, Sylow
176	C₂×C₈₃	C₂×#493	1	1	yes	characteristic
177	C₂×C₈₃	#3152	1	1	yes	characteristic
178	C₂×C₈₃	#3156	1	1	yes	characteristic
179	D₈₃	—	2	2	no	—
180	D₈₃	—	16	2	no	—
181	#3226	—	83	2	no	—
182	#3169	—	83	1	no	—
183	C₂×#500	—	83	1	no	—
184	#3152	—	83	2	no	—
185	#1722	—	83	2	no	—
186	#1722	—	83	2	no	—
187	#1713	—	83	1	no	—
188	#1713	D₈₃	1	1	yes	characteristic, complemented
189	#1706	—	83	1	no	—
190	#1706	—	83	1	no	—
191	C₈×M₁₆	—	83	1	no	—
192	C₈₃	#19630	1	1	yes	characteristic, complemented, Sylow
193	C₂×#137	—	166	4	no	—
194	#532	—	83	2	no	—
195	C₂×C₄⋊Q₈	—	83	2	no	—
196	#500	—	83	2	no	—
197	#500	—	83	2	no	—
198	#500	—	83	1	no	—
199	#500	—	166	2	no	—
200	#500	C₂×D₈₃	1	1	yes	characteristic
201	#500	C₂×D₈₃	1	1	yes	characteristic
202	#500	C₂×D₈₃	1	2	yes	—
203	#497	—	166	4	no	—
204	#494	—	83	2	no	—
205	#494	C₂×D₈₃	1	2	yes	—
206	#493	—	83	1	no	—
207	#492	—	83	2	no	—
208	#492	—	83	2	no	—
209	#486	—	83	2	no	—
210	#486	—	83	2	no	—
211	#486	—	83	2	no	—
212	#480	—	166	4	no	—
213	#474	—	166	4	no	—
214	#466	—	166	2	no	—
215	C₄×D₈	—	166	2	no	—
216	#431	—	83	1	no	—
217	C₂×#107	—	166	2	no	—
218	C₄×M₁₆	—	83	1	no	—
219	C₂×C₄×C₈	—	83	1	no	—
220	#322	—	83	1	no	—
221	#322	—	83	1	no	—
222	C₈²	—	83	1	no	—
223	C₈²	C₂×D₈₃	1	1	yes	characteristic
224	C₂×#41	—	166	2	no	—
225	C₂²×Q₈	—	166	2	no	—
226	#137	—	332	16	no	—
227	C₂×Q₁₆	—	2	4	no	—
228	C₂×Q₁₆	—	166	4	no	—
229	C₂×SD₁₆	—	166	4	no	—
230	C₂×SD₁₆	—	166	4	no	—
231	C₂×D₈	—	166	2	no	—
232	C₂×M₁₆	—	83	2	no	—
233	C₂²×C₈	—	83	2	no	—
234	C₄⋊Q₈	—	83	2	no	—
235	C₄⋊Q₈	—	166	2	no	—
236	C₄⋊Q₈	—	166	2	no	—
237	C₄⋊Q₈	—	166	2	no	—
238	C₄⋊Q₈	C₂²×D₈₃	1	2	yes	—
239	C₄⋊Q₈	C₂²×D₈₃	1	2	yes	—
240	#127	—	83	2	no	—
241	#124	—	166	2	no	—
242	#121	—	166	2	no	—
243	#121	—	166	2	no	—
244	C₄×D₄	—	166	2	no	—
245	C₄×D₄	—	166	2	no	—
246	C₂×C₄⋊C₄	—	166	2	no	—
247	C₂×C₄⋊C₄	—	166	2	no	—
248	C₂×C₄²	—	83	1	no	—
249	#107	—	2	2	no	—
250	#107	—	2	2	no	—
251	#107	—	2	4	no	—
252	#107	—	166	4	no	—
253	#107	—	166	4	no	—
254	#107	—	166	2	no	—
255	C₄⋊C₈	—	83	2	no	—
256	#102	—	166	4	no	—
257	#102	—	166	4	no	—
258	#102	—	166	4	no	—
259	C₂²⋊C₈	—	83	2	no	—
260	#97	—	83	2	no	—
261	C₄×C₈	—	83	2	no	—
262	C₄×C₈	—	83	1	no	—
263	C₄×C₈	—	83	1	no	—
264	C₄×C₈	—	83	1	no	—
265	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
266	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
267	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
268	#41	—	332	8	no	—
269	C₂×Q₈	—	2	2	no	—
270	C₂×Q₈	—	2	2	no	—
271	C₂×Q₈	—	332	2	no	—
272	C₂×Q₈	—	332	8	no	—
273	C₂×Q₈	—	166	2	no	—
274	C₂×D₄	—	332	2	no	—
275	C₂×D₄	—	166	2	no	—
276	C₂×D₄	—	166	2	no	—
277	C₂×D₄	—	166	2	no	—
278	C₂²×C₄	—	332	2	no	—
279	C₂²×C₄	—	332	2	no	—
280	C₂²×C₄	—	83	1	no	—
281	C₂²×C₄	—	83	1	no	—
282	C₂²×C₄	—	83	1	no	—
283	Q₁₆	—	4	8	no	—
284	Q₁₆	—	332	8	no	—
285	SD₁₆	—	332	8	no	—
286	SD₁₆	—	332	8	no	—
287	D₈	—	332	4	no	—
288	M₁₆	—	83	8	no	—
289	C₂×C₈	—	83	2	no	—
290	C₂×C₈	—	83	2	no	—
291	C₂×C₈	—	83	2	no	—
292	C₂×C₈	—	166	4	no	—
293	C₂×C₈	D₄×D₈₃	1	2	yes	—
294	C₂×C₈	D₄×D₈₃	1	2	yes	—
295	C₂×C₈	D₄×D₈₃	1	2	yes	—
296	C₄⋊C₄	—	2	2	no	—
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	—	2	2	no	—
300	C₄⋊C₄	—	332	2	no	—
301	C₄⋊C₄	—	332	2	no	—
302	C₄⋊C₄	—	166	2	no	—
303	C₄⋊C₄	—	166	2	no	—
304	C₂²⋊C₄	—	332	2	no	—
305	C₂²⋊C₄	—	332	2	no	—
306	C₄²	—	83	1	no	—
307	C₄²	—	83	1	no	—
308	C₄²	—	83	1	no	—
309	C₄²	C₂³×D₈₃	1	1	yes	characteristic
310	C₂³	—	332	2	no	—
311	C₂³	—	83	1	no	—
312	Q₈	—	4	4	no	—
313	Q₈	—	4	4	no	—
314	Q₈	—	332	4	no	—
315	Q₈	—	664	4	no	—
316	D₄	—	332	4	no	—
317	D₄	—	332	4	no	—
318	D₄	—	332	4	no	—
319	D₄	—	664	4	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	332	2	no	—
323	C₂×C₄	—	83	1	no	—
324	C₂×C₄	—	83	4	no	—
325	C₂×C₄	—	83	1	no	—
326	C₂×C₄	—	83	1	no	—
327	C₂×C₄	—	664	4	no	—
328	C₂×C₄	—	664	4	no	—
329	C₂×C₄	—	166	2	no	—
330	C₂×C₄	—	166	2	no	—
331	C₂×C₄	C₂×D₄×D₈₃	1	1	yes	characteristic
332	C₂×C₄	C₂×D₄×D₈₃	1	1	yes	characteristic
333	C₂×C₄	C₂×D₄×D₈₃	1	1	yes	characteristic
334	C₈	—	2	2	no	—
335	C₈	—	83	4	no	—
336	C₈	—	166	2	no	—
337	C₈	—	166	2	no	—
338	C₈	D₈×D₈₃	1	4	yes	—
339	C₈	#14581824	1	4	yes	complemented
340	C₂²	—	83	2	no	—
341	C₂²	—	664	2	no	—
342	C₂²	—	664	2	no	—
343	C₂²	—	664	2	no	—
344	C₂²	—	166	1	no	—
345	C₂²	—	166	1	no	—
346	C₂²	#127×D₈₃	1	1	yes	characteristic
347	C₄	—	8	2	no	—
348	C₄	—	8	2	no	—
349	C₄	—	83	2	no	—
350	C₄	—	664	2	no	—
351	C₄	—	166	1	no	—
352	C₄	—	166	1	no	—
353	C₄	C₂×D₈×D₈₃	1	2	yes	—
354	C₄	C₂×#14581824	1	2	yes	—
355	C₄	#19128514	1	2	yes	—
356	C₂	—	1328	2	no	—
357	C₂	—	166	2	no	—
358	C₂	#493×D₈₃	1	1	yes	characteristic
359	C₂	#24917921	1	1	yes	characteristic
360	C₂	#24917951	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#127×D₈₃
commutator, cosocle	C₄²×C₈₃	C₂⁴
Frattini	C₄²	C₂³×D₈₃
Fitting	C₈₃×#1713	C₂
socle	C₂²×C₈₃	C₂×#127
prime radical	#24918043	C₂
hypercenter	#1713	D₈₃
hypocenter	C₈₃	#19630
LCS	G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1713

Complements

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

N	H ≅ G/N	type
1. #24918585	C₂	semidirect
3. #500×D₈₃	C₂	semidirect
4. #24917921	C₂	semidirect
5. #24907135	C₂	semidirect
6. #24907133	C₂	semidirect
7. #24907080	C₂	semidirect
8. #24907056	C₂	semidirect
9. #24907055	C₂	semidirect
10. C₈×#13050573	C₂	semidirect
11. C₈₃×#1713	C₂	semidirect
104. #13050573	D₈	semidirect
155. C₈×C₈₃	#137	semidirect
188. #1713	D₈₃	semidirect
192. C₈₃	#19630	semidirect
339. C₈	#14581824	semidirect

Maximal subgroups

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

subgroup	index
#24918585	2
#24918043	2
#500×D₈₃	2
#24917921	2
#24907135	2
#24907133	2
#24907080	2
#24907056	2
#24907055	2
C₈×#13050573	2
C₈₃×#1713	2
#19630	83