Subgroups — #35559306

Order 21248.

566 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24917107	C₂	1	1	yes	maximal, characteristic
2	#24916071	C₂	1	1	yes	maximal, characteristic, complemented
3	#24909676	C₂	1	8	yes	maximal, complemented
4	#24909237	C₂	1	1	yes	maximal, characteristic, complemented
5	#24909236	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#19127514	C₂	1	1	yes	maximal, characteristic, complemented
7	#24909083	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#361×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
9	#19128467	—	2	1	no	—
10	C₂×#14581798	C₂²	1	1	yes	characteristic
11	#19128268	—	2	1	no	—
12	C₂×#14581732	C₂²	1	1	yes	characteristic
13	#19127951	C₂²	1	4	yes	—
14	#19127934	—	2	2	no	—
15	#19127749	—	2	1	no	—
16	#19127748	C₂²	1	4	yes	complemented
17	C₂×#14581728	C₂²	1	1	yes	characteristic
18	#108×D₈₃	—	2	2	no	—
19	#19127739	C₂²	1	1	yes	characteristic
20	#19127706	—	2	1	no	—
21	#19127525	C₂²	1	4	yes	complemented
22	#19127524	C₂²	1	4	yes	complemented
23	#19127514	C₂²	1	4	yes	—
24	#19127482	C₂²	1	4	yes	complemented
25	C₂²×D₈×C₈₃	C₂²	1	1	yes	characteristic
26	C₂×M₃₂×C₈₃	C₂²	1	1	yes	characteristic
27	C₂×#108×C₈₃	C₂²	1	1	yes	characteristic
28	#361×C₈₃	C₂²	1	4	yes	—
29	C₂×#13050590	D₄	1	1	yes	characteristic
30	#14581800	—	4	2	no	—
31	#14581798	—	2	1	no	—
32	#14581798	—	2	1	no	—
33	#14581798	—	2	1	no	—
34	#14581798	—	2	1	no	—
35	#14581798	—	4	2	no	—
36	C₂×#13050574	—	2	1	no	—
37	C₂×#13050574	D₄	1	1	yes	characteristic
38	C₂×#13050573	C₂³	1	2	yes	—
39	M₁₆×D₈₃	—	2	4	no	—
40	C₂×C₈×D₈₃	D₄	1	2	yes	—
41	#14581788	—	4	1	no	—
42	#14581779	—	4	1	no	—
43	#14581763	—	2	4	no	—
44	#14581761	—	4	2	no	—
45	#14581751	—	2	1	no	—
46	#14581732	—	2	1	no	—
47	#14581732	C₂³	1	2	yes	—
48	C₂×C₈₃⋊C₁₆	—	2	1	no	—
49	#14581728	—	2	1	no	—
50	#14581728	C₂³	1	2	yes	—
51	#14581727	—	2	4	no	—
52	#14581726	—	2	1	no	—
53	#14581724	C₂³	1	2	yes	—
54	#14581713	—	2	4	no	—
55	C₂²×D₄×C₈₃	—	2	1	no	—
56	C₂×D₈×C₈₃	—	2	1	no	—
57	C₂×D₈×C₈₃	—	2	1	no	—
58	C₂×D₈×C₈₃	—	2	1	no	—
59	C₂×D₈×C₈₃	—	2	1	no	—
60	C₂×D₈×C₈₃	—	2	1	no	—
61	C₂×D₈×C₈₃	C₂³	1	2	yes	—
62	C₂×M₁₆×C₈₃	—	2	1	no	—
63	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
64	M₃₂×C₈₃	—	2	1	no	—
65	M₃₂×C₈₃	C₂³	1	2	yes	—
66	C₂×C₁₆×C₈₃	—	2	1	no	—
67	#108×C₈₃	—	2	1	no	—
68	#108×C₈₃	C₂³	1	2	yes	—
69	#13050590	—	2	1	no	—
70	#13050590	—	2	1	no	—
71	#13050590	—	2	1	no	—
72	#13050590	—	2	1	no	—
73	C₂×Dic₁₆₆	SD₁₆	1	1	yes	characteristic
74	C₂×D₃₃₂	SD₁₆	1	1	yes	characteristic
75	C₂×C₂²⋊D₈₃	—	4	4	no	—
76	C₂×C₂²⋊D₈₃	SD₁₆	1	2	yes	—
77	C₂×C₄×D₈₃	C₂×D₄	1	2	yes	—
78	#13050574	—	2	1	no	—
79	#13050574	—	2	1	no	—
80	#13050574	—	2	2	no	—
81	#13050574	—	4	1	no	—
82	#13050573	—	2	2	no	—
83	#13050573	—	2	2	no	—
84	#13050573	—	4	4	no	—
85	C₂×C₈₃⋊C₈	—	4	1	no	—
86	C₂×C₈₃⋊C₈	C₂×D₄	1	2	yes	—
87	C₈×D₈₃	—	2	2	no	—
88	C₈×D₈₃	—	2	2	no	—
89	C₈×D₈₃	—	4	4	no	—
90	C₄⋊Dic₈₃	—	2	2	no	—
91	C₂²⋊Dic₈₃	—	4	1	no	—
92	C₂²⋊Dic₈₃	—	4	4	no	—
93	C₄×Dic₈₃	—	4	1	no	—
94	C₈₃⋊C₁₆	—	2	2	no	—
95	C₂⁴×C₈₃	—	4	1	no	—
96	C₂×D₄×C₈₃	—	2	2	no	—
97	C₂×D₄×C₈₃	—	4	1	no	—
98	C₂×D₄×C₈₃	—	4	1	no	—
99	C₂×D₄×C₈₃	—	4	1	no	—
100	C₂×D₄×C₈₃	—	4	1	no	—
101	C₂×D₄×C₈₃	—	4	1	no	—
102	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
103	D₈×C₈₃	—	2	2	no	—
104	D₈×C₈₃	—	2	2	no	—
105	D₈×C₈₃	—	4	1	no	—
106	D₈×C₈₃	—	4	1	no	—
107	M₁₆×C₈₃	—	2	2	no	—
108	M₁₆×C₈₃	—	4	1	no	—
109	C₂×C₈×C₈₃	—	4	1	no	—
110	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
112	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
113	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
114	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
115	C₂×C₈×C₈₃	C₂⁴	1	1	yes	characteristic
116	C₁₆×C₈₃	—	2	2	no	—
117	C₂³×C₈₃	—	4	2	no	—
118	C₂³×C₈₃	—	8	1	no	—
119	C₂³×C₈₃	—	8	1	no	—
120	C₂³×C₈₃	—	16	2	no	—
121	C₂³×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
122	C₂²×D₈₃	C₂×SD₁₆	1	2	yes	—
123	D₄×C₈₃	—	4	2	no	—
124	D₄×C₈₃	—	4	2	no	—
125	D₄×C₈₃	—	8	1	no	—
126	D₄×C₈₃	—	8	1	no	—
127	C₂×C₄×C₈₃	#123	1	1	yes	characteristic
128	C₂×C₄×C₈₃	#123	1	1	yes	characteristic
129	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
130	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
131	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
132	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
133	C₂²⋊D₈₃	—	2	2	no	—
134	C₂²⋊D₈₃	—	2	2	no	—
135	C₂²⋊D₈₃	—	16	4	no	—
136	C₂×Dic₈₃	—	4	2	no	—
137	C₂×Dic₈₃	C₂×SD₁₆	1	2	yes	—
138	D₃₃₂	—	2	1	no	—
139	D₃₃₂	—	2	1	no	—
140	C₄×D₈₃	—	2	2	no	—
141	C₄×D₈₃	—	2	2	no	—
142	Dic₁₆₆	—	2	1	no	—
143	Dic₁₆₆	—	2	1	no	—
144	C₈×C₈₃	—	4	2	no	—
145	C₈×C₈₃	#123	1	1	yes	characteristic
146	C₈×C₈₃	#123	1	1	yes	characteristic
147	C₈×C₈₃	C₂×#41	1	1	yes	characteristic
148	C₈×C₈₃	C₂×#41	1	1	yes	characteristic
149	C₈₃⋊C₈	—	2	2	no	—
150	C₈₃⋊C₈	—	4	2	no	—
151	C₂²×C₈₃	—	4	1	no	—
152	C₂²×C₈₃	—	8	4	no	—
153	C₂²×C₈₃	—	8	2	no	—
154	C₂²×C₈₃	—	16	2	no	—
155	C₂²×C₈₃	—	16	2	no	—
156	C₂²×C₈₃	#481	1	1	yes	characteristic
157	C₂²×C₈₃	#481	1	1	yes	characteristic
158	C₂²×C₈₃	C₂²×SD₁₆	1	1	yes	characteristic
159	C₂×D₈₃	—	2	2	no	—
160	C₂×D₈₃	—	4	2	no	—
161	C₄×C₈₃	#481	1	1	yes	characteristic
162	C₄×C₈₃	#481	1	1	yes	characteristic
163	C₄×C₈₃	C₂×#123	1	1	yes	characteristic
164	C₄×C₈₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
165	Dic₈₃	—	2	2	no	—
166	Dic₈₃	—	8	2	no	—
167	#33657	—	83	1	no	maximal, Sylow
168	C₂×C₈₃	—	2	1	no	—
169	C₂×C₈₃	—	2	1	no	—
170	C₂×C₈₃	—	2	1	no	—
171	C₂×C₈₃	—	16	2	no	—
172	C₂×C₈₃	C₂×#481	1	1	yes	characteristic
173	D₈₃	—	4	2	no	—
174	#3058	—	83	1	no	—
175	#2916	—	83	1	no	—
176	#2238	—	83	8	no	—
177	#2155	—	83	1	no	—
178	#2155	—	83	1	no	—
179	C₂×#361	—	83	1	no	—
180	C₂×#361	D₈₃	1	1	yes	characteristic, complemented
181	#2115	—	83	1	no	—
182	C₈₃	#33657	1	1	yes	characteristic, complemented, Sylow
183	C₂²×D₈	C₂×D₈₃	1	1	yes	characteristic
184	#568	—	166	1	no	—
185	C₂×#131	—	83	1	no	—
186	#535	—	166	1	no	—
187	C₂×M₃₂	—	83	1	no	—
188	C₂×M₃₂	C₂×D₈₃	1	1	yes	characteristic
189	#469	—	83	4	no	—
190	#466	—	166	2	no	—
191	#430	—	83	4	no	—
192	#430	—	166	1	no	—
193	C₂×#108	—	83	1	no	—
194	C₂×#108	—	166	2	no	—
195	C₂×#108	C₂×D₈₃	1	1	yes	characteristic
196	#427	—	83	1	no	—
197	#418	—	166	1	no	—
198	#365	—	83	4	no	—
199	#365	—	83	4	no	—
200	#361	—	83	4	no	—
201	#361	C₂×D₈₃	1	4	yes	—
202	#349	—	83	4	no	—
203	C₂×#41	—	83	1	no	—
204	C₂²×D₄	—	2	1	no	—
205	C₂×D₈	—	2	1	no	—
206	C₂×D₈	—	2	1	no	—
207	C₂×D₈	—	2	1	no	—
208	C₂×D₈	—	2	1	no	—
209	C₂×D₈	—	2	1	no	—
210	C₂×D₈	C₂²×D₈₃	1	2	yes	—
211	#131	—	332	2	no	—
212	#131	—	332	2	no	—
213	#131	—	166	1	no	—
214	#131	—	166	1	no	—
215	#131	—	166	1	no	—
216	#131	—	166	1	no	—
217	C₂×M₁₆	—	2	1	no	—
218	C₂×M₁₆	—	83	1	no	—
219	C₂×M₁₆	—	83	2	no	—
220	C₂×M₁₆	—	166	4	no	—
221	C₂×M₁₆	—	166	1	no	—
222	C₂²×C₈	—	83	2	no	—
223	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
224	#127	—	332	1	no	—
225	#124	—	332	1	no	—
226	#121	—	166	4	no	—
227	#120	—	332	2	no	—
228	#117	—	166	1	no	—
229	M₃₂	—	2	1	no	—
230	M₃₂	—	83	2	no	—
231	M₃₂	—	166	1	no	—
232	M₃₂	C₂²×D₈₃	1	2	yes	—
233	C₂×C₁₆	—	2	1	no	—
234	C₂×C₁₆	—	166	1	no	—
235	#108	—	2	1	no	—
236	#108	—	83	2	no	—
237	#108	—	166	4	no	—
238	#108	—	166	1	no	—
239	#108	C₂²×D₈₃	1	2	yes	—
240	#107	—	166	1	no	—
241	#106	—	83	2	no	—
242	#102	—	166	4	no	—
243	C₂⁴	—	4	1	no	—
244	#41	—	166	1	no	—
245	#41	—	166	1	no	—
246	#41	—	166	1	no	—
247	#41	—	166	1	no	—
248	C₂×Q₈	—	83	1	no	—
249	C₂×D₄	—	2	2	no	—
250	C₂×D₄	—	4	1	no	—
251	C₂×D₄	—	4	1	no	—
252	C₂×D₄	—	4	1	no	—
253	C₂×D₄	—	4	1	no	—
254	C₂×D₄	—	4	1	no	—
255	C₂×D₄	—	332	4	no	—
256	C₂×D₄	—	83	2	no	—
257	C₂×D₄	—	83	1	no	—
258	C₂²×C₄	—	83	2	no	—
259	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
260	D₈	—	2	2	no	—
261	D₈	—	2	2	no	—
262	D₈	—	4	1	no	—
263	D₈	—	4	1	no	—
264	M₁₆	—	2	2	no	—
265	M₁₆	—	4	1	no	—
266	M₁₆	—	332	4	no	—
267	M₁₆	—	332	1	no	—
268	M₁₆	—	166	2	no	—
269	M₁₆	—	166	1	no	—
270	M₁₆	—	166	1	no	—
271	M₁₆	—	166	2	no	—
272	M₁₆	—	166	2	no	—
273	C₂×C₈	—	4	1	no	—
274	C₂×C₈	—	332	4	no	—
275	C₂×C₈	—	332	1	no	—
276	C₂×C₈	—	83	2	no	—
277	C₂×C₈	—	166	2	no	—
278	C₂×C₈	—	166	2	no	—
279	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
280	C₂×C₈	#13050589	1	1	yes	characteristic
281	C₂×C₈	#13050589	1	1	yes	characteristic
282	C₂×C₈	#13050590	1	1	yes	characteristic
283	C₂×C₈	#13050590	1	1	yes	characteristic
284	C₂×C₈	C₂³×D₈₃	1	1	yes	characteristic
285	C₄⋊C₄	—	166	2	no	—
286	C₂²⋊C₄	—	332	1	no	—
287	C₂²⋊C₄	—	332	4	no	—
288	C₄²	—	332	1	no	—
289	C₁₆	—	2	2	no	—
290	C₁₆	—	166	2	no	—
291	C₂³	—	4	2	no	—
292	C₂³	—	8	1	no	—
293	C₂³	—	8	1	no	—
294	C₂³	—	16	2	no	—
295	C₂³	—	83	2	no	—
296	C₂³	SD₁₆×D₈₃	1	1	yes	characteristic
297	Q₈	—	166	1	no	—
298	Q₈	—	166	1	no	—
299	D₄	—	4	2	no	—
300	D₄	—	4	2	no	—
301	D₄	—	8	1	no	—
302	D₄	—	8	1	no	—
303	D₄	—	1328	4	no	—
304	D₄	—	166	2	no	—
305	D₄	—	166	2	no	—
306	D₄	—	166	1	no	—
307	D₄	—	166	1	no	—
308	C₂×C₄	—	332	2	no	—
309	C₂×C₄	—	83	2	no	—
310	C₂×C₄	—	166	2	no	—
311	C₂×C₄	—	166	2	no	—
312	C₂×C₄	#14581774	1	1	yes	characteristic
313	C₂×C₄	#14581774	1	1	yes	characteristic
314	C₂×C₄	SD₁₆×D₈₃	1	1	yes	characteristic
315	C₂×C₄	#14581816	1	1	yes	characteristic
316	C₂×C₄	#14581816	1	1	yes	characteristic
317	C₂×C₄	C₂×D₄×D₈₃	1	1	yes	characteristic
318	C₈	—	4	2	no	—
319	C₈	—	332	2	no	—
320	C₈	—	166	2	no	—
321	C₈	#14581774	1	1	yes	characteristic
322	C₈	#14581774	1	1	yes	characteristic
323	C₈	C₂×#13050589	1	1	yes	characteristic
324	C₈	C₂×#13050590	1	1	yes	characteristic
325	C₂²	—	4	1	no	—
326	C₂²	—	8	4	no	—
327	C₂²	—	8	2	no	—
328	C₂²	—	16	2	no	—
329	C₂²	—	16	2	no	—
330	C₂²	—	332	2	no	—
331	C₂²	—	166	2	no	—
332	C₂²	#19128018	1	1	yes	characteristic
333	C₂²	#19128018	1	1	yes	characteristic
334	C₂²	C₂×SD₁₆×D₈₃	1	1	yes	characteristic
335	C₄	—	664	2	no	—
336	C₄	—	166	2	no	—
337	C₄	#19128018	1	1	yes	characteristic
338	C₄	#19128018	1	1	yes	characteristic
339	C₄	C₂×#14581774	1	1	yes	characteristic
340	C₄	C₂×#14581816	1	1	yes	characteristic
341	C₂	—	2	1	no	—
342	C₂	—	2	1	no	—
343	C₂	—	2	1	no	—
344	C₂	—	16	2	no	—
345	C₂	—	332	2	no	—
346	C₂	C₂×#19128018	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#19128018
commutator, cosocle	C₂×C₈×C₈₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₃
Fitting	C₂×#361×C₈₃	C₂
socle	C₂×C₈₃	C₂×#481
prime radical	#24917107	C₂
hypercenter	C₂×#361	D₈₃
hypocenter	C₈₃	#33657
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#361

Complements

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

N	H ≅ G/N	type
2. #24916071	C₂	semidirect
3. #24909676	C₂	semidirect
4. #24909237	C₂	semidirect
5. #24909236	C₂	semidirect
6. C₂×#19127514	C₂	semidirect
7. #24909083	C₂	semidirect
8. C₂×#361×C₈₃	C₂	semidirect
16. #19127748	C₂²	semidirect
21. #19127525	C₂²	semidirect
22. #19127524	C₂²	semidirect
24. #19127482	C₂²	semidirect
180. C₂×#361	D₈₃	semidirect
182. C₈₃	#33657	semidirect

Maximal subgroups

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

subgroup	index
#24917107	2
#24916071	2
#24909676	2
#24909237	2
#24909236	2
C₂×#19127514	2
#24909083	2
C₂×#361×C₈₃	2
#33657	83