Subgroups — #35297607

Order 21248.

434 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24916174	C₂	1	1	yes	maximal, characteristic
2	#24907625	C₂	1	1	yes	maximal, characteristic, complemented
3	#24906209	C₂	1	1	yes	maximal, characteristic, complemented
4	#24906024	C₂	1	1	yes	maximal, characteristic
5	#24906022	C₂	1	1	yes	maximal, characteristic
6	#24905323	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₃×#1286	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14581800	—	2	1	no	—
9	#19128251	C₂²	1	1	yes	characteristic
10	#19127778	—	2	1	no	—
11	#19127778	C₄	1	2	yes	complemented
12	#19127766	—	2	1	no	—
13	#19127766	C₄	1	2	yes	complemented
14	#19127754	C₂²	1	1	yes	characteristic
15	C₂×#14581725	—	2	1	no	—
16	C₂×#14581723	—	2	1	no	—
17	#19127669	—	2	1	no	—
18	C₂×C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
19	#19127567	C₂²	1	1	yes	characteristic
20	#19127453	—	2	1	no	—
21	#19127451	—	2	1	no	—
22	#19127436	—	2	2	no	—
23	#19127434	—	2	2	no	—
24	#431×C₈₃	C₂²	1	1	yes	characteristic
25	#423×C₈₃	C₂²	1	1	yes	characteristic
26	#382×C₈₃	C₂²	1	1	yes	characteristic
27	C₂×#13050590	—	2	1	no	—
28	#14581800	—	4	2	no	—
29	#14581800	—	4	2	no	—
30	C₂×#13050573	—	2	2	no	—
31	C₂²×C₈₃⋊C₈	D₄	1	1	yes	characteristic
32	C₂²×C₈₃⋊C₈	D₄	1	1	yes	characteristic
33	C₂×C₈×D₈₃	—	2	2	no	—
34	#14581792	Q₈	1	1	yes	characteristic
35	#14581789	Q₈	1	1	yes	characteristic
36	#14581780	—	2	1	no	—
37	#14581766	—	2	1	no	—
38	#14581766	D₄	1	2	yes	—
39	C₄×D₃₃₂	—	2	1	no	—
40	C₄×D₃₃₂	C₂×C₄	1	2	yes	—
41	C₂×C₄⋊Dic₈₃	—	2	1	no	—
42	#14581725	—	4	2	no	—
43	#14581723	—	4	2	no	—
44	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
45	C₂²⋊(C₈₃⋊C₈)	—	2	1	no	—
46	(C₂×D₈₃)⋊C₈	—	2	2	no	—
47	(C₂×D₈₃)⋊C₈	—	2	2	no	—
48	C₄×C₈₃⋊C₈	—	2	1	no	—
49	C₄×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈₃⋊C₈	D₄	1	1	yes	characteristic
52	C₄×C₈₃⋊C₈	D₄	1	1	yes	characteristic
53	C₈₃⋊#95	—	2	1	no	—
54	C₈₃⋊#95	—	2	1	no	—
55	C₂×M₁₆×C₈₃	—	2	1	no	—
56	C₂×C₄⋊C₄×C₈₃	—	2	1	no	—
57	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
58	C₄⋊C₈×C₈₃	—	2	1	no	—
59	C₄⋊C₈×C₈₃	—	2	1	no	—
60	C₄⋊C₈×C₈₃	C₂×C₄	1	2	yes	—
61	C₂²⋊C₈×C₈₃	—	2	1	no	—
62	C₄×C₈×C₈₃	—	2	1	no	—
63	#95×C₈₃	—	2	1	no	—
64	#95×C₈₃	—	2	1	no	—
65	#13050590	—	4	1	no	—
66	#13050590	—	4	1	no	—
67	#13050590	—	4	1	no	—
68	#13050590	—	4	1	no	—
69	C₂×Dic₁₆₆	—	2	1	no	—
70	C₂×D₃₃₂	—	2	1	no	—
71	C₂×D₃₃₂	—	2	1	no	—
72	C₂×D₃₃₂	C₄⋊C₄	1	2	yes	—
73	C₂×C₂²⋊D₈₃	—	2	2	no	—
74	C₂²×Dic₈₃	—	4	1	no	—
75	C₂×C₄×D₈₃	—	2	2	no	—
76	#13050573	—	4	4	no	—
77	C₂×C₈₃⋊C₈	—	2	1	no	—
78	C₂×C₈₃⋊C₈	—	2	1	no	—
79	C₂×C₈₃⋊C₈	—	4	1	no	—
80	C₂×C₈₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₈₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₈₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₈₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₈₃⋊C₈	D₈	1	2	yes	—
85	C₂×C₈₃⋊C₈	SD₁₆	1	2	yes	—
86	C₈×D₈₃	—	4	4	no	—
87	C₄⋊Dic₈₃	—	2	1	no	—
88	C₄⋊Dic₈₃	—	4	1	no	—
89	C₄⋊Dic₈₃	—	4	1	no	—
90	C₄⋊Dic₈₃	C₄⋊C₄	1	2	yes	—
91	(C₂×D₈₃)⋊C₄	—	2	2	no	—
92	(C₂×D₈₃)⋊C₄	—	2	2	no	—
93	C₂²×C₄×C₈₃	—	4	1	no	—
94	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₈₃	#41	1	1	yes	characteristic
97	M₁₆×C₈₃	—	4	1	no	—
98	M₁₆×C₈₃	—	4	1	no	—
99	C₂×C₈×C₈₃	—	2	2	no	—
100	C₂×C₈×C₈₃	—	4	1	no	—
101	C₄⋊C₄×C₈₃	—	4	1	no	—
102	C₄⋊C₄×C₈₃	—	4	1	no	—
103	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
104	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₈₃	C₂×Q₈	1	1	yes	characteristic
106	C₄²×C₈₃	#41	1	1	yes	characteristic
107	C₂³×C₈₃	#121	1	1	yes	characteristic
108	C₂²×D₈₃	—	2	2	no	—
109	C₂×C₄×C₈₃	—	4	1	no	—
110	C₂×C₄×C₈₃	—	4	1	no	—
111	C₂×C₄×C₈₃	—	8	1	no	—
112	C₂×C₄×C₈₃	—	8	1	no	—
113	C₂×C₄×C₈₃	—	8	1	no	—
114	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₈₃	#117	1	1	yes	characteristic
117	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
118	C₂×C₄×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
119	C₂×C₄×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
120	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
122	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
123	C₂×C₄×C₈₃	#137	1	1	yes	characteristic
124	C₂²⋊D₈₃	—	4	2	no	—
125	C₂²⋊D₈₃	—	4	2	no	—
126	C₂×Dic₈₃	—	2	2	no	—
127	C₂×Dic₈₃	—	8	1	no	—
128	C₂×Dic₈₃	—	8	1	no	—
129	C₂×Dic₈₃	—	8	1	no	—
130	D₃₃₂	—	4	2	no	—
131	D₃₃₂	—	4	2	no	—
132	D₃₃₂	—	4	1	no	—
133	D₃₃₂	—	4	1	no	—
134	C₄×D₈₃	—	4	2	no	—
135	C₄×D₈₃	—	4	2	no	—
136	Dic₁₆₆	—	4	1	no	—
137	Dic₁₆₆	—	4	1	no	—
138	C₈×C₈₃	—	4	2	no	—
139	C₈×C₈₃	—	8	1	no	—
140	C₈₃⋊C₈	—	2	1	no	—
141	C₈₃⋊C₈	—	2	1	no	—
142	C₈₃⋊C₈	—	8	1	no	—
143	C₈₃⋊C₈	#102	1	2	yes	—
144	C₈₃⋊C₈	#102	1	2	yes	—
145	C₂²×C₈₃	—	2	1	no	—
146	C₂²×C₈₃	—	2	1	no	—
147	C₂²×C₈₃	#380	1	1	yes	characteristic
148	C₂²×C₈₃	#459	1	1	yes	characteristic
149	C₂²×C₈₃	#461	1	1	yes	characteristic
150	C₂×D₈₃	—	4	2	no	—
151	C₂×D₈₃	—	4	2	no	—
152	C₂×D₈₃	—	4	2	no	—
153	C₄×C₈₃	—	2	1	no	—
154	C₄×C₈₃	—	2	1	no	—
155	C₄×C₈₃	—	2	1	no	—
156	C₄×C₈₃	—	2	1	no	—
157	C₄×C₈₃	—	16	1	no	—
158	C₄×C₈₃	C₂×#102	1	1	yes	characteristic
159	C₄×C₈₃	#417	1	1	yes	characteristic
160	C₄×C₈₃	#474	1	1	yes	characteristic
161	C₄×C₈₃	#476	1	1	yes	characteristic
162	Dic₈₃	—	4	2	no	—
163	Dic₈₃	—	16	1	no	—
164	#8674	—	83	1	no	maximal, Sylow
165	C₂×C₈₃	—	2	1	no	—
166	C₂×C₈₃	—	2	1	no	—
167	C₂×C₈₃	#1802	1	1	yes	characteristic
168	C₂×C₈₃	#1812	1	1	yes	characteristic
169	C₂×C₈₃	#1814	1	1	yes	characteristic
170	D₈₃	—	8	2	no	—
171	#2931	—	83	1	no	—
172	#1846	—	83	1	no	—
173	#1564	—	83	1	no	—
174	#1517	—	83	1	no	—
175	#1517	—	83	1	no	—
176	#1286	—	83	1	no	—
177	#1286	D₈₃	1	1	yes	characteristic, complemented
178	C₈₃	#8674	1	1	yes	characteristic, complemented, Sylow
179	C₂×#131	—	166	1	no	—
180	#532	—	83	1	no	—
181	#436	—	83	2	no	—
182	#436	—	166	1	no	—
183	C₈×D₄	—	83	2	no	—
184	C₈×D₄	—	166	1	no	—
185	#431	—	83	1	no	—
186	#431	C₂×D₈₃	1	1	yes	characteristic
187	C₂×#107	—	166	1	no	—
188	C₂×#106	—	166	1	no	—
189	#423	C₂×D₈₃	1	1	yes	characteristic
190	#408	—	166	1	no	—
191	C₂×C₄×C₈	—	83	1	no	—
192	#382	—	83	1	no	—
193	#382	C₂×D₈₃	1	1	yes	characteristic
194	#335	—	166	1	no	—
195	#334	—	166	1	no	—
196	#327	—	166	2	no	—
197	#327	—	166	2	no	—
198	C₂×#41	—	166	1	no	—
199	#131	—	332	2	no	—
200	#131	—	332	2	no	—
201	C₂×M₁₆	—	2	1	no	—
202	C₂×M₁₆	—	166	2	no	—
203	C₂²×C₈	—	83	1	no	—
204	C₂²×C₈	—	83	1	no	—
205	C₂²×C₈	—	166	2	no	—
206	C₄⋊Q₈	—	83	1	no	—
207	#127	—	83	1	no	—
208	#124	—	166	1	no	—
209	#121	—	83	2	no	—
210	#121	—	166	1	no	—
211	C₄×D₄	—	83	2	no	—
212	C₄×D₄	—	166	1	no	—
213	C₂×C₄⋊C₄	—	2	1	no	—
214	C₂×C₄⋊C₄	—	166	1	no	—
215	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
216	#107	—	332	2	no	—
217	#106	—	332	2	no	—
218	C₄⋊C₈	—	2	1	no	—
219	C₄⋊C₈	—	2	1	no	—
220	C₄⋊C₈	—	166	1	no	—
221	C₄⋊C₈	C₄×D₈₃	1	2	yes	—
222	C₂²⋊C₈	—	2	1	no	—
223	C₂²⋊C₈	—	166	2	no	—
224	C₂²⋊C₈	—	166	2	no	—
225	C₂²⋊C₈	—	166	1	no	—
226	C₄×C₈	—	2	1	no	—
227	C₄×C₈	—	83	1	no	—
228	C₄×C₈	—	83	1	no	—
229	C₄×C₈	—	83	1	no	—
230	C₄×C₈	—	83	1	no	—
231	C₄×C₈	—	166	1	no	—
232	#95	—	2	1	no	—
233	#95	—	2	1	no	—
234	#95	—	166	1	no	—
235	#95	—	166	1	no	—
236	#41	—	332	1	no	—
237	#41	—	332	1	no	—
238	#41	—	332	1	no	—
239	#41	—	332	1	no	—
240	C₂×Q₈	—	166	1	no	—
241	C₂×D₄	—	83	2	no	—
242	C₂×D₄	—	166	2	no	—
243	C₂×D₄	—	166	1	no	—
244	C₂×D₄	—	166	1	no	—
245	C₂²×C₄	—	4	1	no	—
246	C₂²×C₄	—	332	1	no	—
247	C₂²×C₄	—	166	2	no	—
248	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
249	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
250	C₂²×C₄	#13050590	1	1	yes	characteristic
251	M₁₆	—	4	1	no	—
252	M₁₆	—	4	1	no	—
253	M₁₆	—	332	4	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	4	1	no	—
256	C₂×C₈	—	332	4	no	—
257	C₂×C₈	—	332	1	no	—
258	C₂×C₈	—	83	2	no	—
259	C₂×C₈	—	83	2	no	—
260	C₂×C₈	—	83	1	no	—
261	C₂×C₈	—	83	1	no	—
262	C₂×C₈	—	83	1	no	—
263	C₂×C₈	—	83	1	no	—
264	C₂×C₈	—	166	1	no	—
265	C₂×C₈	—	166	1	no	—
266	C₄⋊C₄	—	4	1	no	—
267	C₄⋊C₄	—	4	1	no	—
268	C₄⋊C₄	—	332	1	no	—
269	C₄⋊C₄	—	332	1	no	—
270	C₄⋊C₄	—	83	2	no	—
271	C₄⋊C₄	—	166	1	no	—
272	C₂²⋊C₄	—	166	2	no	—
273	C₂²⋊C₄	—	166	2	no	—
274	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
275	C₄²	D₄×D₈₃	1	1	yes	characteristic
276	C₄²	Q₈×D₈₃	1	1	yes	characteristic
277	C₄²	#13050590	1	1	yes	characteristic
278	C₂³	—	166	2	no	—
279	C₂³	#14581765	1	1	yes	characteristic
280	Q₈	—	332	1	no	—
281	Q₈	—	332	1	no	—
282	D₄	—	332	2	no	—
283	D₄	—	332	2	no	—
284	D₄	—	332	2	no	—
285	D₄	—	332	2	no	—
286	D₄	—	332	1	no	—
287	D₄	—	332	1	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	8	1	no	—
291	C₂×C₄	—	8	1	no	—
292	C₂×C₄	—	8	1	no	—
293	C₂×C₄	—	332	2	no	—
294	C₂×C₄	—	332	2	no	—
295	C₂×C₄	—	664	1	no	—
296	C₂×C₄	—	664	1	no	—
297	C₂×C₄	—	664	1	no	—
298	C₂×C₄	—	166	2	no	—
299	C₂×C₄	C₂²⋊C₄×D₈₃	1	1	yes	characteristic
300	C₂×C₄	C₄⋊C₄×D₈₃	1	1	yes	characteristic
301	C₂×C₄	#14581749	1	1	yes	characteristic
302	C₂×C₄	#14581765	1	1	yes	characteristic
303	C₂×C₄	#14581770	1	1	yes	characteristic
304	C₂×C₄	#14581770	1	1	yes	characteristic
305	C₂×C₄	D₈×D₈₃	1	1	yes	characteristic
306	C₂×C₄	SD₁₆×D₈₃	1	1	yes	characteristic
307	C₂×C₄	#14581819	1	1	yes	characteristic
308	C₂×C₄	#14581824	1	1	yes	characteristic
309	C₈	—	4	2	no	—
310	C₈	—	8	1	no	—
311	C₈	—	83	2	no	—
312	C₈	—	83	2	no	—
313	C₈	—	664	1	no	—
314	C₈	—	166	1	no	—
315	C₈	—	166	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	332	2	no	—
319	C₂²	—	332	2	no	—
320	C₂²	—	332	2	no	—
321	C₂²	#19127560	1	1	yes	characteristic
322	C₂²	#19127895	1	1	yes	characteristic
323	C₂²	#19127905	1	1	yes	characteristic
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	16	1	no	—
329	C₄	—	1328	1	no	—
330	C₄	—	332	2	no	—
331	C₄	#102×D₈₃	1	1	yes	characteristic
332	C₄	#19127701	1	1	yes	characteristic
333	C₄	#19127980	1	1	yes	characteristic
334	C₄	#19127990	1	1	yes	characteristic
335	C₂	—	2	1	no	—
336	C₂	—	2	1	no	—
337	C₂	—	664	2	no	—
338	C₂	#24907462	1	1	yes	characteristic
339	C₂	#24907508	1	1	yes	characteristic
340	C₂	#24907516	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19127560
commutator	C₄²×C₈₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₈₃×#1286	C₂
socle	C₂²×C₈₃	#380
prime radical	#19128251	C₂²
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	#1286	D₈₃
hypocenter	C₈₃	#8674
LCS	G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1286

Complements

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

N	H ≅ G/N	type
2. #24907625	C₂	semidirect
3. #24906209	C₂	semidirect
6. #24905323	C₂	semidirect
7. C₈₃×#1286	C₂	semidirect
11. #19127778	C₄	semidirect
13. #19127766	C₄	semidirect
177. #1286	D₈₃	semidirect
178. C₈₃	#8674	semidirect

Maximal subgroups

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

subgroup	index
#24916174	2
#24907625	2
#24906209	2
#24906024	2
#24906022	2
#24905323	2
C₈₃×#1286	2
#8674	83