Subgroups — #35559315

Order 21248.

534 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24917119	C₂	1	1	yes	maximal, characteristic
2	#24916071	C₂	1	1	yes	maximal, characteristic
3	#24909684	C₂	1	8	yes	maximal, complemented
4	#24909235	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#19127518	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#19127516	C₂	1	1	yes	maximal, characteristic, complemented
7	#24909083	C₂	1	1	yes	maximal, characteristic
8	C₈₃×#2155	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×Dic₃₃₂	C₂²	1	1	yes	characteristic
10	#19128467	—	2	1	no	—
11	C₂×#14581798	C₂²	1	1	yes	characteristic
12	#19128277	—	2	1	no	—
13	C₂×#14581732	C₂²	1	1	yes	characteristic
14	#19127960	C₂²	1	4	yes	—
15	#19127943	—	2	2	no	—
16	#19127749	—	2	1	no	—
17	#19127748	C₂²	1	4	yes	—
18	C₂×#14581728	C₂²	1	1	yes	characteristic
19	#108×D₈₃	—	2	2	no	—
20	#19127709	—	2	1	no	—
21	#19127523	C₂²	1	4	yes	—
22	#19127518	C₂²	1	4	yes	—
23	#19127516	C₂²	1	4	yes	—
24	#19127482	C₂²	1	4	yes	—
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	—
29	C₂×#13050590	D₄	1	1	yes	characteristic
30	C₂²×Dic₁₆₆	—	2	1	no	—
31	C₂×Dic₃₃₂	—	2	1	no	—
32	C₂×Dic₃₃₂	—	2	1	no	—
33	C₂×Dic₃₃₂	—	2	1	no	—
34	C₂×Dic₃₃₂	—	2	1	no	—
35	C₂×Dic₃₃₂	—	2	1	no	—
36	C₂×Dic₃₃₂	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	#14581792	—	4	1	no	—
49	#14581780	—	4	1	no	—
50	#14581776	—	4	2	no	—
51	#14581772	—	2	4	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	#14581715	—	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	#13050590	—	2	1	no	—
70	#13050590	—	2	1	no	—
71	#13050590	—	2	1	no	—
72	#13050590	—	2	1	no	—
73	C₂×Dic₁₆₆	—	2	2	no	—
74	C₂×Dic₁₆₆	—	4	1	no	—
75	C₂×Dic₁₆₆	—	4	1	no	—
76	C₂×Dic₁₆₆	—	4	1	no	—
77	C₂×Dic₁₆₆	—	4	1	no	—
78	C₂×Dic₁₆₆	—	4	1	no	—
79	C₂×Dic₁₆₆	SD₁₆	1	1	yes	characteristic
80	C₂×D₃₃₂	SD₁₆	1	1	yes	characteristic
81	C₂×C₂²⋊D₈₃	SD₁₆	1	2	yes	—
82	C₂²×Dic₈₃	—	4	1	no	—
83	C₂×C₄×D₈₃	C₂×D₄	1	2	yes	—
84	Dic₃₃₂	—	2	2	no	—
85	Dic₃₃₂	—	2	2	no	—
86	Dic₃₃₂	—	4	1	no	—
87	Dic₃₃₂	—	4	1	no	—
88	#13050574	—	2	1	no	—
89	#13050574	—	2	1	no	—
90	#13050574	—	2	2	no	—
91	#13050574	—	4	1	no	—
92	#13050573	—	2	2	no	—
93	#13050573	—	2	2	no	—
94	#13050573	—	4	4	no	—
95	C₂×C₈₃⋊C₈	—	4	1	no	—
96	C₂×C₈₃⋊C₈	C₂×D₄	1	2	yes	—
97	C₈×D₈₃	—	2	2	no	—
98	C₈×D₈₃	—	2	2	no	—
99	C₈×D₈₃	—	4	4	no	—
100	C₄⋊Dic₈₃	—	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	C₂×SD₁₆	1	2	yes	—
119	C₂×C₄×C₈₃	—	4	2	no	—
120	C₂×C₄×C₈₃	#123	1	1	yes	characteristic
121	C₂×C₄×C₈₃	#123	1	1	yes	characteristic
122	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
123	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
124	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
125	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
126	C₂²⋊D₈₃	—	2	2	no	—
127	C₂²⋊D₈₃	—	2	2	no	—
128	C₂×Dic₈₃	—	4	2	no	—
129	C₂×Dic₈₃	—	8	1	no	—
130	C₂×Dic₈₃	—	8	1	no	—
131	C₂×Dic₈₃	C₂×SD₁₆	1	2	yes	—
132	D₃₃₂	—	2	1	no	—
133	D₃₃₂	—	2	1	no	—
134	C₄×D₈₃	—	2	2	no	—
135	C₄×D₈₃	—	2	2	no	—
136	Dic₁₆₆	—	2	1	no	—
137	Dic₁₆₆	—	2	1	no	—
138	Dic₁₆₆	—	4	2	no	—
139	Dic₁₆₆	—	4	2	no	—
140	Dic₁₆₆	—	8	1	no	—
141	Dic₁₆₆	—	8	1	no	—
142	C₈×C₈₃	—	4	2	no	—
143	C₈×C₈₃	#123	1	1	yes	characteristic
144	C₈×C₈₃	#123	1	1	yes	characteristic
145	C₈×C₈₃	C₂×#41	1	1	yes	characteristic
146	C₈×C₈₃	C₂×#41	1	1	yes	characteristic
147	C₈₃⋊C₈	—	2	2	no	—
148	C₈₃⋊C₈	—	4	2	no	—
149	C₂²×C₈₃	—	4	1	no	—
150	C₂²×C₈₃	#481	1	1	yes	characteristic
151	C₂²×C₈₃	#481	1	1	yes	characteristic
152	C₂²×C₈₃	C₂²×SD₁₆	1	1	yes	characteristic
153	C₂×D₈₃	—	2	2	no	—
154	C₂×D₈₃	—	4	2	no	—
155	C₄×C₈₃	—	8	2	no	—
156	C₄×C₈₃	#481	1	1	yes	characteristic
157	C₄×C₈₃	#481	1	1	yes	characteristic
158	C₄×C₈₃	C₂×#123	1	1	yes	characteristic
159	C₄×C₈₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
160	Dic₈₃	—	2	2	no	—
161	Dic₈₃	—	8	2	no	—
162	#33658	—	83	1	no	maximal, Sylow
163	C₂×C₈₃	—	2	1	no	—
164	C₂×C₈₃	—	2	1	no	—
165	C₂×C₈₃	—	2	1	no	—
166	C₂×C₈₃	C₂×#481	1	1	yes	characteristic
167	D₈₃	—	4	2	no	—
168	#3059	—	83	1	no	—
169	#2916	—	83	1	no	—
170	#2239	—	83	8	no	—
171	#2155	—	83	1	no	—
172	#2155	D₈₃	1	1	yes	characteristic, complemented
173	C₂×#362	—	83	1	no	—
174	C₂×#362	—	83	1	no	—
175	#2115	—	83	1	no	—
176	C₈₃	#33658	1	1	yes	characteristic, complemented, Sylow
177	C₂²×Q₁₆	—	83	1	no	—
178	#568	—	166	1	no	—
179	C₂×#131	—	83	1	no	—
180	#536	—	166	1	no	—
181	C₂×M₃₂	—	83	1	no	—
182	C₂×M₃₂	C₂×D₈₃	1	1	yes	characteristic
183	#470	—	83	4	no	—
184	#467	—	166	2	no	—
185	#430	—	83	4	no	—
186	#430	—	166	1	no	—
187	C₂×#108	—	83	1	no	—
188	C₂×#108	—	166	2	no	—
189	C₂×#108	C₂×D₈₃	1	1	yes	characteristic
190	#427	C₂×D₈₃	1	1	yes	characteristic
191	#419	—	166	1	no	—
192	#365	—	83	4	no	—
193	#365	C₂×D₈₃	1	4	yes	—
194	#362	—	83	4	no	—
195	#362	—	83	4	no	—
196	#349	—	83	4	no	—
197	C₂×#41	—	83	1	no	—
198	C₂²×Q₈	—	166	1	no	—
199	C₂×Q₁₆	—	83	2	no	—
200	C₂×Q₁₆	—	166	1	no	—
201	C₂×Q₁₆	—	166	1	no	—
202	C₂×Q₁₆	—	166	1	no	—
203	C₂×Q₁₆	—	166	1	no	—
204	C₂×Q₁₆	—	166	1	no	—
205	#131	—	332	2	no	—
206	#131	—	332	2	no	—
207	#131	—	166	1	no	—
208	#131	—	166	1	no	—
209	#131	—	166	1	no	—
210	#131	—	166	1	no	—
211	C₂×M₁₆	—	2	1	no	—
212	C₂×M₁₆	—	83	1	no	—
213	C₂×M₁₆	—	83	2	no	—
214	C₂×M₁₆	—	166	4	no	—
215	C₂×M₁₆	—	166	1	no	—
216	C₂²×C₈	—	83	2	no	—
217	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
218	C₄⋊Q₈	—	332	1	no	—
219	#124	—	332	1	no	—
220	#123	—	332	2	no	—
221	C₂²⋊Q₈	—	166	4	no	—
222	#117	—	2	1	no	—
223	M₃₂	—	2	1	no	—
224	M₃₂	—	83	2	no	—
225	M₃₂	—	166	1	no	—
226	M₃₂	C₂²×D₈₃	1	2	yes	—
227	C₂×C₁₆	—	2	1	no	—
228	C₂×C₁₆	—	166	1	no	—
229	#108	—	2	1	no	—
230	#108	—	83	2	no	—
231	#108	—	166	4	no	—
232	#108	—	166	1	no	—
233	#108	C₂²×D₈₃	1	2	yes	—
234	#107	—	2	1	no	—
235	#106	C₂²×D₈₃	1	2	yes	—
236	#103	—	166	4	no	—
237	#41	—	166	1	no	—
238	#41	—	166	1	no	—
239	#41	—	166	1	no	—
240	#41	—	166	1	no	—
241	C₂×Q₈	—	332	1	no	—
242	C₂×Q₈	—	332	1	no	—
243	C₂×Q₈	—	332	1	no	—
244	C₂×Q₈	—	332	1	no	—
245	C₂×Q₈	—	332	1	no	—
246	C₂×Q₈	—	83	1	no	—
247	C₂×Q₈	—	166	2	no	—
248	C₂×D₄	—	83	2	no	—
249	C₂×D₄	—	83	1	no	—
250	C₂²×C₄	—	332	1	no	—
251	C₂²×C₄	—	83	2	no	—
252	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
253	Q₁₆	—	332	1	no	—
254	Q₁₆	—	332	1	no	—
255	Q₁₆	—	166	2	no	—
256	Q₁₆	—	166	2	no	—
257	M₁₆	—	2	2	no	—
258	M₁₆	—	4	1	no	—
259	M₁₆	—	332	4	no	—
260	M₁₆	—	332	1	no	—
261	M₁₆	—	166	2	no	—
262	M₁₆	—	166	1	no	—
263	M₁₆	—	166	1	no	—
264	M₁₆	—	166	2	no	—
265	M₁₆	—	166	2	no	—
266	C₂×C₈	—	4	1	no	—
267	C₂×C₈	—	332	4	no	—
268	C₂×C₈	—	332	1	no	—
269	C₂×C₈	—	83	2	no	—
270	C₂×C₈	—	166	2	no	—
271	C₂×C₈	—	166	2	no	—
272	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
273	C₂×C₈	#13050590	1	1	yes	characteristic
274	C₂×C₈	#13050590	1	1	yes	characteristic
275	C₂×C₈	#13050591	1	1	yes	characteristic
276	C₂×C₈	#13050591	1	1	yes	characteristic
277	C₂×C₈	C₂³×D₈₃	1	1	yes	characteristic
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	—	332	4	no	—
280	C₂²⋊C₄	—	4	1	no	—
281	C₂²⋊C₄	—	332	4	no	—
282	C₄²	—	4	1	no	—
283	C₁₆	—	2	2	no	—
284	C₁₆	—	166	2	no	—
285	C₂³	—	83	2	no	—
286	C₂³	SD₁₆×D₈₃	1	1	yes	characteristic
287	Q₈	—	332	2	no	—
288	Q₈	—	332	2	no	—
289	Q₈	—	664	1	no	—
290	Q₈	—	664	1	no	—
291	Q₈	—	166	1	no	—
292	Q₈	—	166	1	no	—
293	D₄	—	166	2	no	—
294	D₄	—	166	2	no	—
295	D₄	—	166	1	no	—
296	D₄	—	166	1	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	332	2	no	—
299	C₂×C₄	—	83	2	no	—
300	C₂×C₄	—	664	1	no	—
301	C₂×C₄	—	664	1	no	—
302	C₂×C₄	—	166	2	no	—
303	C₂×C₄	—	166	2	no	—
304	C₂×C₄	#14581777	1	1	yes	characteristic
305	C₂×C₄	#14581777	1	1	yes	characteristic
306	C₂×C₄	SD₁₆×D₈₃	1	1	yes	characteristic
307	C₂×C₄	#14581819	1	1	yes	characteristic
308	C₂×C₄	#14581819	1	1	yes	characteristic
309	C₂×C₄	C₂×D₄×D₈₃	1	1	yes	characteristic
310	C₈	—	4	2	no	—
311	C₈	—	332	2	no	—
312	C₈	—	166	2	no	—
313	C₈	#14581777	1	1	yes	characteristic
314	C₈	#14581777	1	1	yes	characteristic
315	C₈	C₂×#13050590	1	1	yes	characteristic
316	C₈	C₂×#13050591	1	1	yes	characteristic
317	C₂²	—	4	1	no	—
318	C₂²	—	332	2	no	—
319	C₂²	—	166	2	no	—
320	C₂²	#19128021	1	1	yes	characteristic
321	C₂²	#19128021	1	1	yes	characteristic
322	C₂²	C₂×SD₁₆×D₈₃	1	1	yes	characteristic
323	C₄	—	8	2	no	—
324	C₄	—	664	2	no	—
325	C₄	—	166	2	no	—
326	C₄	#19128021	1	1	yes	characteristic
327	C₄	#19128021	1	1	yes	characteristic
328	C₄	C₂×#14581777	1	1	yes	characteristic
329	C₄	C₂×#14581819	1	1	yes	characteristic
330	C₂	—	2	1	no	—
331	C₂	—	2	1	no	—
332	C₂	—	2	1	no	—
333	C₂	—	332	2	no	—
334	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	C₂×#13050590	D₄
hypercenter	#2155	D₈₃
hypocenter	C₈₃	#33658
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
3. #24909684	C₂	semidirect
4. #24909235	C₂	semidirect
5. C₂×#19127518	C₂	semidirect
6. C₂×#19127516	C₂	semidirect
8. C₈₃×#2155	C₂	semidirect
172. #2155	D₈₃	semidirect
176. C₈₃	#33658	semidirect

Maximal subgroups

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

subgroup	index
#24917119	2
#24916071	2
#24909684	2
#24909235	2
C₂×#19127518	2
C₂×#19127516	2
#24909083	2
C₈₃×#2155	2
#33658	83