Subgroups — #35295458

Order 21248.

436 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24916439	C₂	1	1	yes	maximal, characteristic
2	#24907150	C₂	1	1	yes	maximal, characteristic, complemented
3	#24906051	C₂	1	1	yes	maximal, characteristic
4	#24905753	C₂	1	1	yes	maximal, characteristic, complemented
5	#24905363	C₂	1	1	yes	maximal, characteristic, complemented
6	#24905356	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₃×#1500	C₂	1	1	yes	maximal, characteristic
8	#19127832	C₄	1	1	yes	characteristic, complemented
9	Q₈×C₈₃⋊C₈	C₄	1	1	yes	characteristic, complemented
10	#19127779	C₄	1	1	yes	characteristic, complemented
11	#19127773	—	2	1	no	—
12	D₄×C₈₃⋊C₈	—	2	1	no	—
13	D₄×C₈₃⋊C₈	C₄	1	1	yes	characteristic, complemented
14	#19127762	—	2	1	no	—
15	#19127758	C₂²	1	1	yes	characteristic
16	#19127657	—	2	1	no	—
17	C₄×#13050574	C₂²	1	1	yes	characteristic
18	C₂×#14581703	—	2	1	no	—
19	C₂×#14581703	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈₃⋊C₈	—	2	1	no	—
21	C₂×C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
22	#19127443	C₄	1	2	yes	complemented
23	#19127441	C₄	1	2	yes	complemented
24	#19127423	—	2	1	no	—
25	#19127422	—	2	1	no	—
26	#529×C₈₃	C₂²	1	1	yes	characteristic
27	#431×C₈₃	C₂²	1	1	yes	characteristic
28	#376×C₈₃	C₂²	1	1	yes	characteristic
29	#328×C₈₃	C₄	1	2	yes	—
30	#327×C₈₃	C₄	1	2	yes	—
31	C₂×#13050574	—	2	1	no	—
32	C₂×#13050574	—	2	1	no	—
33	C₂²×C₈₃⋊C₈	—	2	1	no	—
34	C₂²×C₈₃⋊C₈	—	2	1	no	—
35	C₂²×C₈₃⋊C₈	D₄	1	1	yes	characteristic
36	C₂²×C₈₃⋊C₈	D₄	1	1	yes	characteristic
37	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
38	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
39	C₄⋊(C₈₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
40	C₄⋊(C₈₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
41	C₂²⋊(C₈₃⋊C₈)	—	2	1	no	—
42	C₂²⋊(C₈₃⋊C₈)	—	2	1	no	—
43	C₂²⋊(C₈₃⋊C₈)	—	2	1	no	—
44	#14581703	—	2	4	no	—
45	#14581703	—	2	1	no	—
46	#14581703	—	2	1	no	—
47	#14581703	—	2	1	no	—
48	#14581703	C₂×C₄	1	2	yes	—
49	#14581703	C₂×C₄	1	2	yes	—
50	C₄×C₈₃⋊C₈	—	2	4	no	—
51	C₄×C₈₃⋊C₈	—	2	1	no	—
52	C₄×C₈₃⋊C₈	—	2	1	no	—
53	C₄×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
55	C₄×C₈₃⋊C₈	D₄	1	1	yes	characteristic
56	C₄×C₈₃⋊C₈	D₄	1	1	yes	characteristic
57	#126×C₈₃	—	2	1	no	—
58	#125×C₈₃	C₂×C₄	1	1	yes	characteristic
59	#124×C₈₃	C₂×C₄	1	1	yes	characteristic
60	#123×C₈₃	—	2	1	no	—
61	C₂²⋊Q₈×C₈₃	C₂×C₄	1	1	yes	characteristic
62	#121×C₈₃	C₂×C₄	1	1	yes	characteristic
63	C₄×Q₈×C₈₃	C₂×C₄	1	1	yes	characteristic
64	C₄×D₄×C₈₃	—	2	1	no	—
65	C₄×D₄×C₈₃	C₂×C₄	1	1	yes	characteristic
66	#117×C₈₃	—	2	1	no	—
67	C₂×C₄²×C₈₃	—	2	1	no	—
68	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
69	C₄⋊C₈×C₈₃	—	2	1	no	—
70	C₂²⋊C₈×C₈₃	C₂×C₄	1	2	yes	—
71	C₄×C₈×C₈₃	—	2	1	no	—
72	#95×C₈₃	—	2	1	no	—
73	#95×C₈₃	C₂×C₄	1	2	yes	—
74	#13050574	—	4	2	no	—
75	#13050574	—	4	2	no	—
76	C₂×C₈₃⋊C₈	—	2	2	no	—
77	C₂×C₈₃⋊C₈	—	2	2	no	—
78	C₂×C₈₃⋊C₈	—	2	2	no	—
79	C₂×C₈₃⋊C₈	—	2	2	no	—
80	C₂×C₈₃⋊C₈	—	2	2	no	—
81	C₂×C₈₃⋊C₈	—	2	1	no	—
82	C₂×C₈₃⋊C₈	—	2	1	no	—
83	C₂×C₈₃⋊C₈	—	2	2	no	—
84	C₂×C₈₃⋊C₈	—	4	2	no	—
85	C₂×C₈₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₈₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₈₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₈₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×Q₈×C₈₃	C₄²	1	1	yes	characteristic
90	C₂×D₄×C₈₃	—	2	1	no	—
91	C₂×D₄×C₈₃	C₄²	1	1	yes	characteristic
92	C₂²×C₄×C₈₃	—	2	2	no	—
93	C₂²×C₄×C₈₃	—	2	1	no	—
94	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
95	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₈₃	#41	1	1	yes	characteristic
97	C₂×C₈×C₈₃	—	2	2	no	—
98	C₄⋊C₄×C₈₃	—	2	1	no	—
99	C₄⋊C₄×C₈₃	—	2	1	no	—
100	C₄⋊C₄×C₈₃	—	2	1	no	—
101	C₄⋊C₄×C₈₃	—	2	1	no	—
102	C₄⋊C₄×C₈₃	C₄²	1	1	yes	characteristic
103	C₄⋊C₄×C₈₃	C₄²	1	1	yes	characteristic
104	C₂²⋊C₄×C₈₃	—	2	1	no	—
105	C₂²⋊C₄×C₈₃	—	2	1	no	—
106	C₂²⋊C₄×C₈₃	—	2	1	no	—
107	C₂²⋊C₄×C₈₃	—	2	1	no	—
108	C₂²⋊C₄×C₈₃	—	2	1	no	—
109	C₄²×C₈₃	—	2	2	no	—
110	C₄²×C₈₃	—	2	2	no	—
111	C₄²×C₈₃	—	2	1	no	—
112	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
113	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
114	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₈₃	#41	1	1	yes	characteristic
116	C₂³×C₈₃	—	2	1	no	—
117	C₂³×C₈₃	C₄×D₄	1	1	yes	characteristic
118	Q₈×C₈₃	—	4	1	no	—
119	D₄×C₈₃	—	4	1	no	—
120	D₄×C₈₃	—	4	2	no	—
121	C₂×C₄×C₈₃	—	2	2	no	—
122	C₂×C₄×C₈₃	—	2	1	no	—
123	C₂×C₄×C₈₃	—	2	1	no	—
124	C₂×C₄×C₈₃	—	2	4	no	—
125	C₂×C₄×C₈₃	—	2	1	no	—
126	C₂×C₄×C₈₃	—	2	1	no	—
127	C₂×C₄×C₈₃	—	2	1	no	—
128	C₂×C₄×C₈₃	—	4	1	no	—
129	C₂×C₄×C₈₃	—	4	2	no	—
130	C₂×C₄×C₈₃	—	4	2	no	—
131	C₂×C₄×C₈₃	—	4	1	no	—
132	C₂×C₄×C₈₃	—	4	1	no	—
133	C₂×C₄×C₈₃	C₂×C₄²	1	1	yes	characteristic
134	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₈₃	#117	1	1	yes	characteristic
136	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
137	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
138	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
139	C₈×C₈₃	—	4	2	no	—
140	C₈₃⋊C₈	—	2	4	no	—
141	C₈₃⋊C₈	—	2	1	no	—
142	C₈₃⋊C₈	—	2	1	no	—
143	C₈₃⋊C₈	—	4	2	no	—
144	C₈₃⋊C₈	—	4	1	no	—
145	C₈₃⋊C₈	—	4	1	no	—
146	C₈₃⋊C₈	#104	1	2	yes	—
147	C₈₃⋊C₈	#104	1	2	yes	—
148	C₂²×C₈₃	—	2	1	no	—
149	C₂²×C₈₃	—	2	1	no	—
150	C₂²×C₈₃	—	4	1	no	—
151	C₂²×C₈₃	—	4	1	no	—
152	C₂²×C₈₃	—	4	1	no	—
153	C₂²×C₈₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
154	C₂²×C₈₃	#444	1	2	yes	—
155	C₄×C₈₃	—	2	1	no	—
156	C₄×C₈₃	—	2	1	no	—
157	C₄×C₈₃	—	2	1	no	—
158	C₄×C₈₃	—	2	1	no	—
159	C₄×C₈₃	—	2	1	no	—
160	C₄×C₈₃	—	4	4	no	—
161	C₄×C₈₃	—	4	1	no	—
162	C₄×C₈₃	—	4	1	no	—
163	C₄×C₈₃	—	4	1	no	—
164	C₄×C₈₃	C₂×#104	1	1	yes	characteristic
165	C₄×C₈₃	C₂×#104	1	1	yes	characteristic
166	#8249	—	83	1	no	maximal, Sylow
167	C₂×C₈₃	—	2	2	no	—
168	C₂×C₈₃	—	8	1	no	—
169	C₂×C₈₃	#1757	1	1	yes	characteristic
170	C₂×C₈₃	C₄×#104	1	1	yes	characteristic
171	C₂×C₈₃	#1765	1	1	yes	characteristic
172	#2966	—	83	1	no	—
173	#1726	—	83	1	no	—
174	#1524	—	83	1	no	—
175	#1500	D₈₃	1	1	yes	characteristic
176	#1451	—	83	1	no	—
177	#1302	—	83	1	no	—
178	#1300	—	83	1	no	—
179	C₈₃	#8249	1	1	yes	characteristic, complemented, Sylow
180	#529	C₂×D₈₃	1	1	yes	characteristic
181	#446	—	83	1	no	—
182	C₈×Q₈	—	83	1	no	—
183	#436	—	83	1	no	—
184	#435	—	166	1	no	—
185	C₈×D₄	—	83	1	no	—
186	C₈×D₄	—	166	1	no	—
187	#433	—	166	1	no	—
188	#432	—	83	1	no	—
189	#431	C₂×D₈₃	1	1	yes	characteristic
190	#405	—	166	1	no	—
191	C₄×M₁₆	—	83	1	no	—
192	C₂×#97	—	83	1	no	—
193	C₂×#97	—	166	1	no	—
194	C₂×C₄×C₈	—	83	1	no	—
195	C₂×C₄×C₈	—	166	1	no	—
196	#376	C₂×D₈₃	1	1	yes	characteristic
197	#330	—	83	2	no	—
198	#329	—	83	2	no	—
199	#328	Dic₈₃	1	2	yes	—
200	#327	Dic₈₃	1	2	yes	—
201	#322	—	166	1	no	—
202	#322	—	166	1	no	—
203	C₂×M₁₆	—	166	1	no	—
204	C₂×M₁₆	—	166	1	no	—
205	C₂²×C₈	—	83	1	no	—
206	C₂²×C₈	—	83	1	no	—
207	C₂²×C₈	—	166	1	no	—
208	C₂²×C₈	—	166	1	no	—
209	#126	—	2	1	no	—
210	#125	C₄×D₈₃	1	1	yes	characteristic
211	#124	C₄×D₈₃	1	1	yes	characteristic
212	#123	—	2	1	no	—
213	C₂²⋊Q₈	C₂×Dic₈₃	1	1	yes	characteristic
214	#121	C₂×Dic₈₃	1	1	yes	characteristic
215	C₄×Q₈	C₄×D₈₃	1	1	yes	characteristic
216	C₄×D₄	—	2	1	no	—
217	C₄×D₄	C₄×D₈₃	1	1	yes	characteristic
218	#117	—	2	1	no	—
219	C₂×C₄²	—	2	1	no	—
220	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
221	C₄⋊C₈	—	2	1	no	—
222	C₄⋊C₈	—	83	1	no	—
223	C₄⋊C₈	—	83	1	no	—
224	C₄⋊C₈	—	166	1	no	—
225	C₄⋊C₈	—	166	1	no	—
226	C₂²⋊C₈	—	166	1	no	—
227	C₂²⋊C₈	—	166	1	no	—
228	C₂²⋊C₈	—	166	1	no	—
229	C₂²⋊C₈	C₂×Dic₈₃	1	2	yes	—
230	#97	—	83	2	no	—
231	#97	—	83	2	no	—
232	#97	—	166	1	no	—
233	#97	—	166	4	no	—
234	#97	—	166	1	no	—
235	#97	—	166	1	no	—
236	C₄×C₈	—	2	1	no	—
237	C₄×C₈	—	83	1	no	—
238	C₄×C₈	—	83	1	no	—
239	C₄×C₈	—	83	1	no	—
240	C₄×C₈	—	83	1	no	—
241	C₄×C₈	—	166	1	no	—
242	C₄×C₈	—	166	4	no	—
243	C₄×C₈	—	166	1	no	—
244	#95	—	2	1	no	—
245	#95	C₂×Dic₈₃	1	2	yes	—
246	C₂×Q₈	C₄×Dic₈₃	1	1	yes	characteristic
247	C₂×D₄	—	2	1	no	—
248	C₂×D₄	C₄×Dic₈₃	1	1	yes	characteristic
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	—	2	1	no	—
251	C₂²×C₄	C₂²×Dic₈₃	1	1	yes	characteristic
252	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
253	C₂²×C₄	#13050589	1	1	yes	characteristic
254	M₁₆	—	332	2	no	—
255	M₁₆	—	332	2	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	332	2	no	—
258	C₂×C₈	—	83	1	no	—
259	C₂×C₈	—	83	1	no	—
260	C₂×C₈	—	83	1	no	—
261	C₂×C₈	—	83	1	no	—
262	C₂×C₈	—	166	1	no	—
263	C₂×C₈	—	166	1	no	—
264	C₂×C₈	—	166	2	no	—
265	C₂×C₈	—	166	2	no	—
266	C₂×C₈	—	166	2	no	—
267	C₂×C₈	—	166	2	no	—
268	C₂×C₈	—	166	2	no	—
269	C₂×C₈	—	166	2	no	—
270	C₄⋊C₄	—	2	1	no	—
271	C₄⋊C₄	—	2	1	no	—
272	C₄⋊C₄	—	2	1	no	—
273	C₄⋊C₄	—	2	1	no	—
274	C₄⋊C₄	C₄×Dic₈₃	1	1	yes	characteristic
275	C₄⋊C₄	C₄×Dic₈₃	1	1	yes	characteristic
276	C₂²⋊C₄	—	2	1	no	—
277	C₂²⋊C₄	—	2	1	no	—
278	C₂²⋊C₄	—	2	1	no	—
279	C₂²⋊C₄	—	2	1	no	—
280	C₂²⋊C₄	—	2	1	no	—
281	C₄²	—	2	2	no	—
282	C₄²	—	2	2	no	—
283	C₄²	—	2	1	no	—
284	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
285	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
286	C₄²	D₄×D₈₃	1	1	yes	characteristic
287	C₄²	#13050589	1	1	yes	characteristic
288	C₂³	—	2	1	no	—
289	C₂³	D₄×Dic₈₃	1	1	yes	characteristic
290	Q₈	—	4	1	no	—
291	D₄	—	4	1	no	—
292	D₄	—	4	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	4	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	C₂×C₄×Dic₈₃	1	1	yes	characteristic
306	C₂×C₄	C₂²⋊C₄×D₈₃	1	1	yes	characteristic
307	C₂×C₄	#14581748	1	1	yes	characteristic
308	C₂×C₄	(C₂²⋊D₈₃)⋊C₄	1	1	yes	characteristic
309	C₂×C₄	(C₂²⋊D₈₃)⋊C₄	1	1	yes	characteristic
310	C₂×C₄	D₄×Dic₈₃	1	1	yes	characteristic
311	C₈	—	4	2	no	—
312	C₈	—	332	2	no	—
313	C₈	—	332	1	no	—
314	C₈	—	332	1	no	—
315	C₈	—	83	2	no	—
316	C₈	—	83	2	no	—
317	C₈	—	166	4	no	—
318	C₈	—	166	1	no	—
319	C₈	—	166	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	4	1	no	—
323	C₂²	—	4	1	no	—
324	C₂²	—	4	1	no	—
325	C₂²	C₂²⋊C₄×Dic₈₃	1	1	yes	characteristic
326	C₂²	#19127823	1	2	yes	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	4	4	no	—
333	C₄	—	4	1	no	—
334	C₄	—	4	1	no	—
335	C₄	—	4	1	no	—
336	C₄	#104×D₈₃	1	1	yes	characteristic
337	C₄	#104×D₈₃	1	1	yes	characteristic
338	C₂	—	2	2	no	—
339	C₂	—	8	1	no	—
340	C₂	#24907265	1	1	yes	characteristic
341	C₂	#104×Dic₈₃	1	1	yes	characteristic
342	C₂	#24907310	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊C₄×D₈₃
commutator	C₂×C₄×C₈₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₈₃×#1500	C₂
socle	C₂²×C₈₃	C₄×C₂²⋊C₄
prime radical	#121×C₈₃	C₂×C₄
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	#1500	D₈₃
hypocenter	C₈₃	#8249
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1500

Complements

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

N	H ≅ G/N	type
2. #24907150	C₂	semidirect
4. #24905753	C₂	semidirect
5. #24905363	C₂	semidirect
6. #24905356	C₂	semidirect
8. #19127832	C₄	semidirect
9. Q₈×C₈₃⋊C₈	C₄	semidirect
10. #19127779	C₄	semidirect
13. D₄×C₈₃⋊C₈	C₄	semidirect
22. #19127443	C₄	semidirect
23. #19127441	C₄	semidirect
179. C₈₃	#8249	semidirect

Maximal subgroups

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

subgroup	index
#24916439	2
#24907150	2
#24906051	2
#24905753	2
#24905363	2
#24905356	2
C₈₃×#1500	2
#8249	83