Subgroups — #35557877

Order 21248.

668 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19128137	C₂	1	1	yes	maximal, characteristic
2	#24916381	C₂	1	1	yes	maximal, characteristic, complemented
3	#24909359	C₂	1	8	yes	maximal, complemented
4	#24909235	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#19127514	C₂	1	1	yes	maximal, characteristic, complemented
6	#24909198	C₂	1	1	yes	maximal, characteristic, complemented
7	#24909068	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈₃×#2948	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14581814	C₂²	1	1	yes	characteristic
10	#19128467	—	2	1	no	—
11	#19128137	C₄	1	4	yes	complemented
12	#19128137	C₂²	1	4	yes	—
13	C₂×#14581736	C₄	1	2	yes	complemented
14	C₂×#14581733	C₄	1	2	yes	complemented
15	C₂×#14581732	C₂²	1	1	yes	characteristic
16	C₂×#14581732	C₂²	1	1	yes	characteristic
17	#19127824	C₂²	1	4	yes	complemented
18	#19127819	—	2	2	no	—
19	C₂×#14581728	C₂²	1	1	yes	characteristic
20	#19127719	—	2	1	no	—
21	#19127523	C₂²	1	4	yes	complemented
22	#19127514	C₂²	1	4	yes	complemented
23	#19127510	C₂²	1	4	yes	complemented
24	#19127479	C₂²	1	4	yes	complemented
25	C₂²×D₈×C₈₃	C₂²	1	1	yes	characteristic
26	#518×C₈₃	—	2	1	no	—
27	#442×C₈₃	C₂²	1	4	yes	complemented
28	C₄×D₈×C₈₃	—	2	2	no	—
29	#427×C₈₃	C₂²	1	1	yes	characteristic
30	#418×C₈₃	—	2	1	no	—
31	#405×C₈₃	C₂²	1	1	yes	characteristic
32	C₂×#13050590	—	2	1	no	—
33	#14581814	—	2	1	no	—
34	#14581814	—	2	1	no	—
35	#14581814	C₂×C₄	1	2	yes	complemented
36	#14581814	C₂³	1	2	yes	—
37	C₂×Dic₃₃₂	C₂×C₄	1	1	yes	characteristic, complemented
38	C₂×#13050579	—	2	1	no	—
39	C₂×D₆₆₄	C₂×C₄	1	1	yes	characteristic, complemented
40	#14581800	—	4	2	no	—
41	#14581798	—	2	4	no	—
42	C₂×#13050574	—	2	1	no	—
43	M₁₆×D₈₃	—	4	2	no	—
44	#14581736	C₂×C₄	1	8	yes	complemented
45	#14581733	C₂×C₄	1	8	yes	complemented
46	#14581732	—	2	1	no	—
47	#14581732	C₂×C₄	1	2	yes	complemented
48	#14581732	C₂³	1	2	yes	—
49	#14581732	C₂³	1	2	yes	—
50	C₂×C₈₃⋊C₁₆	—	2	1	no	—
51	C₂×C₈₃⋊C₁₆	C₂×C₄	1	2	yes	complemented
52	#14581728	—	2	1	no	—
53	#14581728	C₂³	1	2	yes	—
54	#14581720	—	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	C₂×C₄	1	2	yes	—
60	C₂×D₈×C₈₃	C₂×C₄	1	2	yes	—
61	C₂×D₈×C₈₃	C₂³	1	2	yes	—
62	C₂×M₁₆×C₈₃	D₄	1	1	yes	characteristic
63	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
64	C₄×D₄×C₈₃	—	2	4	no	—
65	C₄×D₄×C₈₃	—	4	2	no	—
66	#117×C₈₃	—	2	1	no	—
67	#117×C₈₃	D₄	1	1	yes	characteristic
68	C₂×C₂²⋊C₄×C₈₃	—	4	2	no	—
69	#107×C₈₃	—	2	1	no	—
70	#106×C₈₃	C₂³	1	2	yes	—
71	#102×C₈₃	—	2	4	no	—
72	#97×C₈₃	C₂³	1	2	yes	—
73	C₄×C₈×C₈₃	D₄	1	2	yes	—
74	#13050590	—	2	2	no	—
75	#13050590	—	2	2	no	—
76	#13050590	—	4	1	no	—
77	#13050590	—	4	1	no	—
78	C₂×Dic₁₆₆	—	2	1	no	—
79	C₂×D₃₃₂	—	2	1	no	—
80	C₂×C₂²⋊D₈₃	—	4	1	no	—
81	C₂×C₄×D₈₃	—	4	1	no	—
82	Dic₃₃₂	—	2	1	no	—
83	Dic₃₃₂	C₂²×C₄	1	2	yes	—
84	#13050579	—	2	2	no	—
85	#13050579	—	2	2	no	—
86	D₆₆₄	—	2	1	no	—
87	D₆₆₄	C₂²×C₄	1	2	yes	—
88	#13050574	—	2	2	no	—
89	#13050574	—	4	1	no	—
90	#13050573	—	4	4	no	—
91	C₂×C₈₃⋊C₈	—	4	1	no	—
92	C₈×D₈₃	—	4	4	no	—
93	C₈₃⋊C₁₆	—	2	2	no	—
94	C₈₃⋊C₁₆	C₂²×C₄	1	4	yes	—
95	C₂⁴×C₈₃	—	4	1	no	—
96	C₂×D₄×C₈₃	—	2	2	no	—
97	C₂×D₄×C₈₃	—	2	2	no	—
98	C₂×D₄×C₈₃	—	2	2	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	—	4	4	no	—
103	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
104	D₈×C₈₃	—	2	2	no	—
105	D₈×C₈₃	—	2	2	no	—
106	D₈×C₈₃	—	2	2	no	—
107	D₈×C₈₃	C₂²×C₄	1	4	yes	—
108	M₁₆×C₈₃	—	2	1	no	—
109	M₁₆×C₈₃	—	2	1	no	—
110	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
113	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
115	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
116	C₂×C₈×C₈₃	C₂⁴	1	1	yes	characteristic
117	C₄⋊C₄×C₈₃	—	2	1	no	—
118	C₄⋊C₄×C₈₃	—	2	2	no	—
119	C₂²⋊C₄×C₈₃	—	2	1	no	—
120	C₂²⋊C₄×C₈₃	—	4	4	no	—
121	C₂²⋊C₄×C₈₃	—	4	1	no	—
122	C₄²×C₈₃	—	4	1	no	—
123	C₄²×C₈₃	C₂×D₄	1	2	yes	—
124	C₂³×C₈₃	—	4	2	no	—
125	C₂³×C₈₃	—	4	2	no	—
126	C₂³×C₈₃	—	4	2	no	—
127	C₂³×C₈₃	—	16	2	no	—
128	C₂³×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
129	C₂²×D₈₃	—	4	1	no	—
130	D₄×C₈₃	—	2	4	no	—
131	D₄×C₈₃	—	4	2	no	—
132	D₄×C₈₃	—	4	2	no	—
133	D₄×C₈₃	—	4	2	no	—
134	C₂×C₄×C₈₃	—	2	2	no	—
135	C₂×C₄×C₈₃	—	4	2	no	—
136	C₂×C₄×C₈₃	—	16	4	no	—
137	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
138	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
139	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
140	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
141	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
142	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
143	C₂²⋊D₈₃	—	4	2	no	—
144	C₂²⋊D₈₃	—	4	2	no	—
145	C₂×Dic₈₃	—	4	1	no	—
146	D₃₃₂	—	2	2	no	—
147	D₃₃₂	—	4	1	no	—
148	C₄×D₈₃	—	4	2	no	—
149	C₄×D₈₃	—	4	2	no	—
150	Dic₁₆₆	—	2	2	no	—
151	Dic₁₆₆	—	4	1	no	—
152	C₈×C₈₃	—	2	2	no	—
153	C₈×C₈₃	C₄×D₄	1	1	yes	characteristic
154	C₈×C₈₃	C₄×D₄	1	1	yes	characteristic
155	C₈×C₈₃	C₂³×C₄	1	1	yes	characteristic
156	C₈×C₈₃	C₂×#41	1	1	yes	characteristic
157	C₈₃⋊C₈	—	4	2	no	—
158	C₂²×C₈₃	—	4	1	no	—
159	C₂²×C₈₃	—	4	4	no	—
160	C₂²×C₈₃	—	8	4	no	—
161	C₂²×C₈₃	—	16	2	no	—
162	C₂²×C₈₃	—	16	2	no	—
163	C₂²×C₈₃	#442	1	1	yes	characteristic
164	C₂²×C₈₃	#442	1	1	yes	characteristic
165	C₂²×C₈₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
166	C₂×D₈₃	—	4	2	no	—
167	C₂×D₈₃	—	16	1	no	—
168	C₄×C₈₃	—	4	2	no	—
169	C₄×C₈₃	—	8	2	no	—
170	C₄×C₈₃	#442	1	1	yes	characteristic
171	C₄×C₈₃	#442	1	1	yes	characteristic
172	C₄×C₈₃	C₂×C₄×D₄	1	1	yes	characteristic
173	C₄×C₈₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
174	Dic₈₃	—	4	2	no	—
175	#33471	—	83	1	no	maximal, Sylow
176	C₂×C₈₃	—	2	1	no	—
177	C₂×C₈₃	—	2	1	no	—
178	C₂×C₈₃	—	2	1	no	—
179	C₂×C₈₃	—	8	4	no	—
180	C₂×C₈₃	C₂×#442	1	1	yes	characteristic
181	D₈₃	—	8	2	no	—
182	C₂×#509	—	83	1	no	—
183	#2957	—	83	1	no	—
184	#2948	D₈₃	1	1	yes	characteristic, complemented
185	AGL(1,ℤ/16)	—	83	8	no	—
186	#2155	—	83	1	no	—
187	C₂×#361	—	83	1	no	—
188	#2145	—	83	1	no	—
189	#2110	—	83	1	no	—
190	C₈₃	#33471	1	1	yes	characteristic, complemented, Sylow
191	C₂×#135	—	83	1	no	—
192	C₂²×D₈	C₂×D₈₃	1	1	yes	characteristic
193	#568	—	166	1	no	—
194	#518	—	2	1	no	—
195	#509	—	83	4	no	—
196	#509	—	83	4	no	—
197	C₂×SD₃₂	—	83	2	no	—
198	C₂×D₁₆	—	83	2	no	—
199	C₂×M₃₂	—	83	1	no	—
200	C₂×M₃₂	—	83	1	no	—
201	#444	—	83	4	no	—
202	#443	—	166	2	no	—
203	#442	C₂×D₈₃	1	4	yes	complemented
204	C₄×D₈	—	2	2	no	—
205	C₂×#108	—	83	1	no	—
206	#427	C₂×D₈₃	1	1	yes	characteristic
207	#421	—	166	1	no	—
208	#418	—	2	1	no	—
209	#405	C₂×D₈₃	1	1	yes	characteristic
210	#365	—	83	4	no	—
211	#361	—	83	4	no	—
212	#360	—	83	4	no	—
213	#347	—	83	4	no	—
214	C₂×#41	—	166	1	no	—
215	C₂²×D₄	—	2	1	no	—
216	#135	—	83	2	no	—
217	#135	—	83	2	no	—
218	#135	—	166	1	no	—
219	#135	—	166	1	no	—
220	C₂×Q₁₆	—	83	1	no	—
221	C₂×SD₁₆	—	166	1	no	—
222	C₂×D₈	—	2	1	no	—
223	C₂×D₈	—	2	1	no	—
224	C₂×D₈	—	2	1	no	—
225	C₂×D₈	—	83	1	no	—
226	C₂×D₈	C₄×D₈₃	1	2	yes	—
227	C₂×D₈	C₄×D₈₃	1	2	yes	—
228	C₂×D₈	C₂²×D₈₃	1	2	yes	—
229	#131	—	332	2	no	—
230	#131	—	166	4	no	—
231	C₂×M₁₆	—	332	2	no	—
232	C₂×M₁₆	—	166	1	no	—
233	C₂×M₁₆	C₂²⋊D₈₃	1	1	yes	characteristic
234	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
235	C₄×D₄	—	2	4	no	—
236	C₄×D₄	—	4	2	no	—
237	#117	—	2	1	no	—
238	#117	C₂²⋊D₈₃	1	1	yes	characteristic
239	C₂×C₂²⋊C₄	—	4	2	no	—
240	SD₃₂	—	83	8	no	—
241	D₁₆	—	83	8	no	—
242	M₃₂	—	83	2	no	—
243	M₃₂	—	83	2	no	—
244	M₃₂	—	83	2	no	—
245	M₃₂	—	166	1	no	—
246	C₂×C₁₆	—	83	2	no	—
247	C₂×C₁₆	—	166	1	no	—
248	#108	—	83	2	no	—
249	#108	—	166	1	no	—
250	#107	—	2	1	no	—
251	#106	C₂²×D₈₃	1	2	yes	—
252	#104	—	166	4	no	—
253	#102	—	2	4	no	—
254	#97	C₂²×D₈₃	1	2	yes	—
255	C₄×C₈	C₂²⋊D₈₃	1	2	yes	—
256	C₂⁴	—	4	1	no	—
257	#41	—	332	1	no	—
258	#41	—	332	1	no	—
259	#41	—	166	2	no	—
260	#41	—	166	2	no	—
261	C₂×Q₈	—	166	1	no	—
262	C₂×D₄	—	2	2	no	—
263	C₂×D₄	—	2	2	no	—
264	C₂×D₄	—	2	2	no	—
265	C₂×D₄	—	4	1	no	—
266	C₂×D₄	—	4	1	no	—
267	C₂×D₄	—	4	1	no	—
268	C₂×D₄	—	332	1	no	—
269	C₂×D₄	—	166	1	no	—
270	C₂²×C₄	—	4	4	no	—
271	C₂²×C₄	—	332	1	no	—
272	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
273	Q₁₆	—	83	2	no	—
274	Q₁₆	—	166	1	no	—
275	SD₁₆	—	166	2	no	—
276	SD₁₆	—	166	2	no	—
277	D₈	—	2	2	no	—
278	D₈	—	2	2	no	—
279	D₈	—	2	2	no	—
280	D₈	—	83	2	no	—
281	D₈	—	166	1	no	—
282	D₈	C₂×C₄×D₈₃	1	4	yes	—
283	M₁₆	—	2	1	no	—
284	M₁₆	—	2	1	no	—
285	M₁₆	—	332	4	no	—
286	M₁₆	—	332	1	no	—
287	M₁₆	—	166	2	no	—
288	C₂×C₈	—	332	4	no	—
289	C₂×C₈	—	332	1	no	—
290	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
291	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
292	C₂×C₈	C₂×C₂²⋊D₈₃	1	2	yes	—
293	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
294	C₂×C₈	#13050590	1	1	yes	characteristic
295	C₂×C₈	#13050590	1	1	yes	characteristic
296	C₂×C₈	C₂³×D₈₃	1	1	yes	characteristic
297	C₄⋊C₄	—	2	1	no	—
298	C₄⋊C₄	—	2	2	no	—
299	C₂²⋊C₄	—	2	1	no	—
300	C₂²⋊C₄	—	4	4	no	—
301	C₂²⋊C₄	—	4	1	no	—
302	C₄²	—	4	1	no	—
303	C₄²	C₂×C₂²⋊D₈₃	1	2	yes	—
304	C₁₆	—	83	4	no	—
305	C₁₆	—	166	2	no	—
306	C₂³	—	4	2	no	—
307	C₂³	—	4	2	no	—
308	C₂³	—	4	2	no	—
309	C₂³	—	16	2	no	—
310	C₂³	—	332	1	no	—
311	C₂³	#14581818	1	1	yes	characteristic
312	Q₈	—	332	1	no	—
313	Q₈	—	166	2	no	—
314	D₄	—	2	4	no	—
315	D₄	—	4	2	no	—
316	D₄	—	4	2	no	—
317	D₄	—	4	2	no	—
318	D₄	—	332	2	no	—
319	D₄	—	332	2	no	—
320	D₄	—	332	1	no	—
321	D₄	—	166	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	16	4	no	—
325	C₂×C₄	—	332	1	no	—
326	C₂×C₄	—	332	2	no	—
327	C₂×C₄	—	332	2	no	—
328	C₂×C₄	C₄×C₂²⋊D₈₃	1	1	yes	characteristic
329	C₂×C₄	C₄×C₂²⋊D₈₃	1	1	yes	characteristic
330	C₂×C₄	#14581817	1	1	yes	characteristic
331	C₂×C₄	#14581817	1	1	yes	characteristic
332	C₂×C₄	#14581818	1	1	yes	characteristic
333	C₂×C₄	C₂²×C₂²⋊D₈₃	1	1	yes	characteristic
334	C₈	—	2	2	no	—
335	C₈	—	332	2	no	—
336	C₈	C₄×C₂²⋊D₈₃	1	1	yes	characteristic
337	C₈	C₄×C₂²⋊D₈₃	1	1	yes	characteristic
338	C₈	C₂²×C₄×D₈₃	1	1	yes	characteristic
339	C₈	C₂×#13050590	1	1	yes	characteristic
340	C₂²	—	4	1	no	—
341	C₂²	—	4	4	no	—
342	C₂²	—	8	4	no	—
343	C₂²	—	16	2	no	—
344	C₂²	—	16	2	no	—
345	C₂²	—	1328	1	no	—
346	C₂²	—	332	2	no	—
347	C₂²	#19127811	1	1	yes	characteristic
348	C₂²	#19127811	1	1	yes	characteristic
349	C₂²	C₂×#14581818	1	1	yes	characteristic
350	C₄	—	4	2	no	—
351	C₄	—	8	2	no	—
352	C₄	—	332	2	no	—
353	C₄	#19127811	1	1	yes	characteristic
354	C₄	#19127811	1	1	yes	characteristic
355	C₄	C₂×C₄×C₂²⋊D₈₃	1	1	yes	characteristic
356	C₄	C₂×#14581817	1	1	yes	characteristic
357	C₂	—	2	1	no	—
358	C₂	—	2	1	no	—
359	C₂	—	2	1	no	—
360	C₂	—	8	4	no	—
361	C₂	—	664	2	no	—
362	C₂	C₂×#19127811	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#19127811
commutator	C₈×C₈₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₈₃
Fitting	C₈₃×#2948	C₂
socle	C₂×C₈₃	C₂×#442
prime radical	C₂×#19128137	C₂
cosocle	C₂×C₈×C₈₃	C₂⁴
hypercenter	#2948	D₈₃
hypocenter	C₈₃	#33471
LCS	G ▷ C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ #2948

Complements

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

N	H ≅ G/N	type
2. #24916381	C₂	semidirect
3. #24909359	C₂	semidirect
4. #24909235	C₂	semidirect
5. C₂×#19127514	C₂	semidirect
6. #24909198	C₂	semidirect
7. #24909068	C₂	semidirect
8. C₈₃×#2948	C₂	semidirect
11. #19128137	C₄	semidirect
13. C₂×#14581736	C₄	semidirect
14. C₂×#14581733	C₄	semidirect
17. #19127824	C₂²	semidirect
21. #19127523	C₂²	semidirect
22. #19127514	C₂²	semidirect
23. #19127510	C₂²	semidirect
24. #19127479	C₂²	semidirect
27. #442×C₈₃	C₂²	semidirect
35. #14581814	C₂×C₄	semidirect
37. C₂×Dic₃₃₂	C₂×C₄	semidirect
39. C₂×D₆₆₄	C₂×C₄	semidirect
44. #14581736	C₂×C₄	semidirect
45. #14581733	C₂×C₄	semidirect
47. #14581732	C₂×C₄	semidirect
51. C₂×C₈₃⋊C₁₆	C₂×C₄	semidirect
184. #2948	D₈₃	semidirect
190. C₈₃	#33471	semidirect
203. #442	C₂×D₈₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19128137	2
#24916381	2
#24909359	2
#24909235	2
C₂×#19127514	2
#24909198	2
#24909068	2
C₈₃×#2948	2
#33471	83