Subgroups — #35559309

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	#24917112	C₂	1	1	yes	maximal, characteristic
2	#24916071	C₂	1	1	yes	maximal, characteristic, complemented
3	#24909679	C₂	1	8	yes	maximal, complemented
4	#24909235	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#19127515	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#19127513	C₂	1	1	yes	maximal, characteristic, complemented
7	#24909083	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈₃×#2155	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₆₆₄	C₂²	1	1	yes	characteristic
10	#19128467	—	2	1	no	—
11	C₂×#14581798	C₂²	1	1	yes	characteristic
12	#19128270	—	2	1	no	—
13	C₂×#14581732	C₂²	1	1	yes	characteristic
14	#19127955	C₂²	1	4	yes	—
15	#19127938	—	2	2	no	—
16	#19127749	—	2	1	no	—
17	#19127748	C₂²	1	4	yes	complemented
18	C₂×#14581728	C₂²	1	1	yes	characteristic
19	#108×D₈₃	—	2	2	no	—
20	#19127705	—	2	1	no	—
21	#19127523	C₂²	1	4	yes	complemented
22	#19127515	C₂²	1	4	yes	—
23	#19127513	C₂²	1	4	yes	—
24	#19127482	C₂²	1	4	yes	complemented
25	C₂×M₃₂×C₈₃	C₂²	1	1	yes	characteristic
26	C₂×#108×C₈₃	C₂²	1	1	yes	characteristic
27	#427×C₈₃	C₂²	1	1	yes	characteristic
28	#365×C₈₃	C₂²	1	4	yes	complemented
29	C₂×#13050590	D₄	1	1	yes	characteristic
30	C₂²×D₃₃₂	—	2	1	no	—
31	C₂×D₆₆₄	—	2	1	no	—
32	C₂×D₆₆₄	—	2	1	no	—
33	C₂×D₆₆₄	—	2	1	no	—
34	C₂×D₆₆₄	—	2	1	no	—
35	C₂×D₆₆₄	—	2	1	no	—
36	C₂×D₆₆₄	C₂³	1	2	yes	—
37	#14581800	—	4	2	no	—
38	#14581798	—	2	1	no	—
39	#14581798	—	2	1	no	—
40	#14581798	—	2	1	no	—
41	#14581798	—	2	1	no	—
42	#14581798	—	4	2	no	—
43	C₂×#13050574	—	2	1	no	—
44	C₂×#13050574	D₄	1	1	yes	characteristic
45	C₂×#13050573	C₂³	1	2	yes	—
46	M₁₆×D₈₃	—	2	4	no	—
47	C₂×C₈×D₈₃	D₄	1	2	yes	—
48	#14581789	—	4	1	no	—
49	#14581780	—	4	1	no	—
50	#14581767	—	2	4	no	—
51	#14581762	—	4	2	no	—
52	#14581732	—	2	1	no	—
53	#14581732	C₂³	1	2	yes	—
54	C₂×C₈₃⋊C₁₆	—	2	1	no	—
55	#14581728	—	2	1	no	—
56	#14581728	C₂³	1	2	yes	—
57	#14581727	—	2	4	no	—
58	#14581712	—	2	4	no	—
59	C₂×M₁₆×C₈₃	—	2	1	no	—
60	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
61	#117×C₈₃	—	2	1	no	—
62	M₃₂×C₈₃	—	2	1	no	—
63	M₃₂×C₈₃	C₂³	1	2	yes	—
64	C₂×C₁₆×C₈₃	—	2	1	no	—
65	#108×C₈₃	—	2	1	no	—
66	#108×C₈₃	C₂³	1	2	yes	—
67	#107×C₈₃	—	2	1	no	—
68	#106×C₈₃	C₂³	1	2	yes	—
69	C₂³×D₈₃	—	4	1	no	—
70	#13050590	—	2	1	no	—
71	#13050590	—	2	1	no	—
72	#13050590	—	2	1	no	—
73	#13050590	—	2	1	no	—
74	C₂×Dic₁₆₆	SD₁₆	1	1	yes	characteristic
75	C₂×D₃₃₂	—	2	2	no	—
76	C₂×D₃₃₂	—	4	1	no	—
77	C₂×D₃₃₂	—	4	1	no	—
78	C₂×D₃₃₂	—	4	1	no	—
79	C₂×D₃₃₂	—	4	1	no	—
80	C₂×D₃₃₂	—	4	1	no	—
81	C₂×D₃₃₂	—	4	4	no	—
82	C₂×D₃₃₂	SD₁₆	1	1	yes	characteristic
83	C₂×C₂²⋊D₈₃	SD₁₆	1	2	yes	—
84	C₂×C₄×D₈₃	C₂×D₄	1	2	yes	—
85	D₆₆₄	—	2	2	no	—
86	D₆₆₄	—	2	2	no	—
87	D₆₆₄	—	4	1	no	—
88	D₆₆₄	—	4	1	no	—
89	#13050574	—	2	1	no	—
90	#13050574	—	2	1	no	—
91	#13050574	—	2	2	no	—
92	#13050574	—	4	1	no	—
93	#13050573	—	2	2	no	—
94	#13050573	—	2	2	no	—
95	#13050573	—	4	4	no	—
96	C₂×C₈₃⋊C₈	—	4	1	no	—
97	C₂×C₈₃⋊C₈	C₂×D₄	1	2	yes	—
98	C₈×D₈₃	—	2	2	no	—
99	C₈×D₈₃	—	2	2	no	—
100	C₈×D₈₃	—	4	4	no	—
101	(C₂×D₈₃)⋊C₄	—	4	4	no	—
102	C₈₃⋊C₁₆	—	2	2	no	—
103	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
104	M₁₆×C₈₃	—	2	2	no	—
105	M₁₆×C₈₃	—	4	1	no	—
106	C₂×C₈×C₈₃	—	4	1	no	—
107	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
109	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
110	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
111	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
112	C₂×C₈×C₈₃	C₂⁴	1	1	yes	characteristic
113	C₄⋊C₄×C₈₃	—	2	2	no	—
114	C₂²⋊C₄×C₈₃	—	4	1	no	—
115	C₄²×C₈₃	—	4	1	no	—
116	C₁₆×C₈₃	—	2	2	no	—
117	C₂³×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
118	C₂²×D₈₃	—	4	2	no	—
119	C₂²×D₈₃	—	8	1	no	—
120	C₂²×D₈₃	—	8	1	no	—
121	C₂²×D₈₃	—	16	2	no	—
122	C₂²×D₈₃	C₂×SD₁₆	1	2	yes	—
123	C₂×C₄×C₈₃	—	4	2	no	—
124	C₂×C₄×C₈₃	#123	1	1	yes	characteristic
125	C₂×C₄×C₈₃	#123	1	1	yes	characteristic
126	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
127	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
128	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
129	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
130	C₂²⋊D₈₃	—	2	2	no	—
131	C₂²⋊D₈₃	—	2	2	no	—
132	C₂×Dic₈₃	C₂×SD₁₆	1	2	yes	—
133	D₃₃₂	—	2	1	no	—
134	D₃₃₂	—	2	1	no	—
135	D₃₃₂	—	4	2	no	—
136	D₃₃₂	—	4	2	no	—
137	D₃₃₂	—	8	1	no	—
138	D₃₃₂	—	8	1	no	—
139	D₃₃₂	—	16	4	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	#481	1	1	yes	characteristic
153	C₂²×C₈₃	#481	1	1	yes	characteristic
154	C₂²×C₈₃	C₂²×SD₁₆	1	1	yes	characteristic
155	C₂×D₈₃	—	2	2	no	—
156	C₂×D₈₃	—	4	2	no	—
157	C₂×D₈₃	—	8	4	no	—
158	C₂×D₈₃	—	8	2	no	—
159	C₂×D₈₃	—	16	2	no	—
160	C₂×D₈₃	—	16	2	no	—
161	C₄×C₈₃	—	8	2	no	—
162	C₄×C₈₃	#481	1	1	yes	characteristic
163	C₄×C₈₃	#481	1	1	yes	characteristic
164	C₄×C₈₃	C₂×#123	1	1	yes	characteristic
165	C₄×C₈₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
166	Dic₈₃	—	2	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	C₂×#481	1	1	yes	characteristic
172	D₈₃	—	4	2	no	—
173	D₈₃	—	16	2	no	—
174	#3058	—	83	1	no	—
175	#2916	—	83	1	no	—
176	#2238	—	83	8	no	—
177	#2155	—	83	1	no	—
178	#2155	D₈₃	1	1	yes	characteristic, complemented
179	C₂×#361	—	83	1	no	—
180	C₂×#361	—	83	1	no	—
181	#2115	—	83	1	no	—
182	C₈₃	#33657	1	1	yes	characteristic, complemented, Sylow
183	C₂²×D₈	—	83	1	no	—
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	C₂×D₈₃	1	1	yes	characteristic
197	#418	—	166	1	no	—
198	#365	—	83	4	no	—
199	#365	C₂×D₈₃	1	4	yes	complemented
200	#361	—	83	4	no	—
201	#361	—	83	4	no	—
202	#349	—	83	4	no	—
203	C₂×#41	—	83	1	no	—
204	C₂²×D₄	—	166	1	no	—
205	C₂×D₈	—	83	2	no	—
206	C₂×D₈	—	166	1	no	—
207	C₂×D₈	—	166	1	no	—
208	C₂×D₈	—	166	1	no	—
209	C₂×D₈	—	166	1	no	—
210	C₂×D₈	—	166	1	no	—
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	—	2	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	—	2	1	no	—
241	#106	C₂²×D₈₃	1	2	yes	—
242	#102	—	166	4	no	—
243	C₂⁴	—	332	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	—	332	4	no	—
250	C₂×D₄	—	332	1	no	—
251	C₂×D₄	—	332	1	no	—
252	C₂×D₄	—	332	1	no	—
253	C₂×D₄	—	332	1	no	—
254	C₂×D₄	—	332	1	no	—
255	C₂×D₄	—	83	2	no	—
256	C₂×D₄	—	83	1	no	—
257	C₂×D₄	—	166	2	no	—
258	C₂²×C₄	—	83	2	no	—
259	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
260	D₈	—	332	1	no	—
261	D₈	—	332	1	no	—
262	D₈	—	166	2	no	—
263	D₈	—	166	2	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	#13050590	1	1	yes	characteristic
281	C₂×C₈	#13050590	1	1	yes	characteristic
282	C₂×C₈	#13050591	1	1	yes	characteristic
283	C₂×C₈	#13050591	1	1	yes	characteristic
284	C₂×C₈	C₂³×D₈₃	1	1	yes	characteristic
285	C₄⋊C₄	—	2	2	no	—
286	C₂²⋊C₄	—	4	1	no	—
287	C₂²⋊C₄	—	332	4	no	—
288	C₄²	—	4	1	no	—
289	C₁₆	—	2	2	no	—
290	C₁₆	—	166	2	no	—
291	C₂³	—	1328	2	no	—
292	C₂³	—	332	2	no	—
293	C₂³	—	83	2	no	—
294	C₂³	—	664	1	no	—
295	C₂³	—	664	1	no	—
296	C₂³	SD₁₆×D₈₃	1	1	yes	characteristic
297	Q₈	—	166	1	no	—
298	Q₈	—	166	1	no	—
299	D₄	—	1328	4	no	—
300	D₄	—	332	2	no	—
301	D₄	—	332	2	no	—
302	D₄	—	664	1	no	—
303	D₄	—	664	1	no	—
304	D₄	—	166	2	no	—
305	D₄	—	166	2	no	—
306	D₄	—	166	1	no	—
307	D₄	—	166	1	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	83	2	no	—
310	C₂×C₄	—	166	2	no	—
311	C₂×C₄	—	166	2	no	—
312	C₂×C₄	#14581777	1	1	yes	characteristic
313	C₂×C₄	#14581777	1	1	yes	characteristic
314	C₂×C₄	SD₁₆×D₈₃	1	1	yes	characteristic
315	C₂×C₄	#14581819	1	1	yes	characteristic
316	C₂×C₄	#14581819	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	#14581777	1	1	yes	characteristic
322	C₈	#14581777	1	1	yes	characteristic
323	C₈	C₂×#13050590	1	1	yes	characteristic
324	C₈	C₂×#13050591	1	1	yes	characteristic
325	C₂²	—	4	1	no	—
326	C₂²	—	1328	2	no	—
327	C₂²	—	1328	2	no	—
328	C₂²	—	332	2	no	—
329	C₂²	—	664	4	no	—
330	C₂²	—	664	2	no	—
331	C₂²	—	166	2	no	—
332	C₂²	#19128021	1	1	yes	characteristic
333	C₂²	#19128021	1	1	yes	characteristic
334	C₂²	C₂×SD₁₆×D₈₃	1	1	yes	characteristic
335	C₄	—	8	2	no	—
336	C₄	—	166	2	no	—
337	C₄	#19128021	1	1	yes	characteristic
338	C₄	#19128021	1	1	yes	characteristic
339	C₄	C₂×#14581777	1	1	yes	characteristic
340	C₄	C₂×#14581819	1	1	yes	characteristic
341	C₂	—	2	1	no	—
342	C₂	—	2	1	no	—
343	C₂	—	2	1	no	—
344	C₂	—	1328	2	no	—
345	C₂	—	332	2	no	—
346	C₂	C₂×#19128021	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#19128021
commutator, cosocle	C₂×C₈×C₈₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₃
Fitting	C₈₃×#2155	C₂
socle	C₂×C₈₃	C₂×#481
prime radical	#24917112	C₂
hypercenter	#2155	D₈₃
hypocenter	C₈₃	#33657
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2155

Complements

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

N	H ≅ G/N	type
2. #24916071	C₂	semidirect
3. #24909679	C₂	semidirect
4. #24909235	C₂	semidirect
5. C₂×#19127515	C₂	semidirect
6. C₂×#19127513	C₂	semidirect
7. #24909083	C₂	semidirect
8. C₈₃×#2155	C₂	semidirect
17. #19127748	C₂²	semidirect
21. #19127523	C₂²	semidirect
24. #19127482	C₂²	semidirect
28. #365×C₈₃	C₂²	semidirect
178. #2155	D₈₃	semidirect
182. C₈₃	#33657	semidirect
199. #365	C₂×D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#24917112	2
#24916071	2
#24909679	2
#24909235	2
C₂×#19127515	2
C₂×#19127513	2
#24909083	2
C₈₃×#2155	2
#33657	83