Subgroups — #35557895

Order 21248.

506 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24916392	C₂	1	1	yes	maximal, characteristic
2	#24909371	C₂	1	1	yes	maximal, characteristic, complemented
3	#24909369	C₂	1	2	yes	maximal, complemented
4	Q₈×C₈₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	#24909365	C₂	1	2	yes	maximal, complemented
6	#24909365	C₂	1	1	yes	maximal, characteristic, complemented
7	#24909220	C₂	1	2	yes	maximal
8	#24909217	C₂	1	1	yes	maximal, characteristic
9	#24909061	C₂	1	2	yes	maximal
10	C₄×C₁₆×D₈₃	C₂	1	1	yes	maximal, characteristic
11	C₈₃×#2183	C₂	1	1	yes	maximal, characteristic, complemented
12	#19128255	C₄	1	1	yes	characteristic
13	C₂×#14581731	C₂²	1	2	yes	—
14	C₂×C₁₆×D₈₃	C₂²	1	1	yes	characteristic
15	C₂×C₁₆×D₈₃	C₂²	1	1	yes	characteristic
16	#19127832	C₄	1	1	yes	characteristic
17	#19127830	C₄	1	2	yes	—
18	#19127828	C₄	1	1	yes	characteristic
19	C₈×Dic₁₆₆	C₂²	1	2	yes	—
20	C₈×Dic₁₆₆	C₂²	1	1	yes	characteristic
21	#19127726	C₄	1	2	yes	—
22	#19127723	C₄	1	1	yes	characteristic
23	#19127655	C₂²	1	2	yes	—
24	C₄×C₈×D₈₃	C₂²	1	1	yes	characteristic
25	C₄⋊(C₈₃⋊C₁₆)	C₂²	1	2	yes	—
26	C₄⋊(C₈₃⋊C₁₆)	C₂²	1	1	yes	characteristic
27	Dic₈₃⋊C₁₆	C₂²	1	2	yes	—
28	Dic₈₃⋊C₁₆	C₂²	1	2	yes	—
29	Dic₈₃⋊C₁₆	C₂²	1	2	yes	—
30	#19127476	C₂²	1	2	yes	—
31	C₄×C₈₃⋊C₁₆	C₂²	1	2	yes	—
32	C₄×C₈₃⋊C₁₆	C₂²	1	1	yes	characteristic
33	C₁₆×Dic₈₃	C₂²	1	2	yes	—
34	C₁₆×Dic₈₃	C₂²	1	1	yes	characteristic
35	C₁₆×Dic₈₃	C₂²	1	1	yes	characteristic
36	C₈×Q₈×C₈₃	C₂²	1	1	yes	characteristic
37	C₄⋊C₁₆×C₈₃	C₂²	1	1	yes	characteristic
38	C₄⋊C₁₆×C₈₃	C₂²	1	2	yes	—
39	#346×C₈₃	C₂²	1	2	yes	—
40	C₄×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
41	C₂×#13050573	C₂×C₄	1	2	yes	—
42	C₂×C₈×D₈₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₈×D₈₃	C₂³	1	1	yes	characteristic
44	#14581791	C₈	1	1	yes	characteristic
45	#14581790	C₈	1	1	yes	characteristic
46	#14581782	C₈	1	2	yes	—
47	#14581772	C₈	1	1	yes	characteristic
48	#14581770	C₈	1	2	yes	—
49	#14581768	C₈	1	1	yes	characteristic
50	C₄×Dic₁₆₆	C₂×C₄	1	1	yes	characteristic
51	C₄×Dic₁₆₆	C₂×C₄	1	2	yes	—
52	#14581749	C₂×C₄	1	2	yes	—
53	C₄²×D₈₃	C₂×C₄	1	1	yes	characteristic
54	#14581731	—	2	4	no	—
55	C₂×C₈₃⋊C₁₆	C₂³	1	1	yes	characteristic
56	C₂×C₈₃⋊C₁₆	C₂³	1	2	yes	—
57	C₂×C₈₃⋊C₁₆	C₂³	1	1	yes	characteristic
58	C₁₆×D₈₃	—	2	2	no	—
59	C₁₆×D₈₃	Q₈	1	4	yes	complemented
60	C₄⋊(C₈₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
61	C₄⋊(C₈₃⋊C₈)	C₂×C₄	1	2	yes	—
62	Dic₈₃⋊C₈	C₂×C₄	1	2	yes	—
63	Dic₈₃⋊C₈	C₂×C₄	1	2	yes	—
64	Dic₈₃⋊C₈	C₂×C₄	1	2	yes	—
65	#14581703	C₂×C₄	1	2	yes	—
66	#14581702	C₂×C₄	1	2	yes	—
67	C₄×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
68	C₈×Dic₈₃	C₂×C₄	1	1	yes	characteristic
69	C₈×Dic₈₃	C₂³	1	1	yes	characteristic
70	C₈×Dic₈₃	C₂³	1	2	yes	—
71	C₄×Q₈×C₈₃	C₂×C₄	1	1	yes	characteristic
72	C₂×C₁₆×C₈₃	C₂³	1	1	yes	characteristic
73	C₂×C₁₆×C₈₃	C₂³	1	2	yes	—
74	C₂×C₁₆×C₈₃	C₂³	1	1	yes	characteristic
75	C₄⋊C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
76	C₄⋊C₈×C₈₃	C₂×C₄	1	2	yes	—
77	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₈₃	C₂³	1	2	yes	—
79	C₂×Dic₁₆₆	C₂×C₈	1	2	yes	—
80	C₂×Dic₁₆₆	C₂×C₈	1	1	yes	characteristic
81	C₂×C₄×D₈₃	C₂×C₈	1	1	yes	characteristic
82	C₂×C₄×D₈₃	C₂×C₈	1	1	yes	characteristic
83	C₂×C₄×D₈₃	C₂²×C₄	1	1	yes	characteristic
84	#13050573	—	2	4	no	—
85	C₂×C₈₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈₃⋊C₈	C₂²×C₄	1	2	yes	—
87	C₂×C₈₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
88	C₈×D₈₃	—	2	2	no	—
89	C₈×D₈₃	C₂×Q₈	1	2	yes	—
90	C₈×D₈₃	#41	1	2	yes	—
91	C₄⋊Dic₈₃	C₂×C₈	1	1	yes	characteristic
92	C₄⋊Dic₈₃	C₂×C₈	1	2	yes	—
93	Dic₈₃⋊C₄	C₂×C₈	1	2	yes	—
94	Dic₈₃⋊C₄	C₂×C₈	1	2	yes	—
95	Dic₈₃⋊C₄	C₂×C₈	1	2	yes	—
96	Dic₈₃⋊C₄	C₂×C₈	1	2	yes	—
97	Dic₈₃⋊C₄	C₂×C₈	1	2	yes	—
98	(C₂×D₈₃)⋊C₄	C₂×C₈	1	2	yes	—
99	(C₂×D₈₃)⋊C₄	C₂×C₈	1	2	yes	—
100	C₄×Dic₈₃	C₂×C₈	1	1	yes	characteristic
101	C₄×Dic₈₃	C₂×C₈	1	1	yes	characteristic
102	C₄×Dic₈₃	C₂²×C₄	1	1	yes	characteristic
103	C₄×Dic₈₃	C₂²×C₄	1	2	yes	—
104	C₈₃⋊C₁₆	—	2	1	no	—
105	C₈₃⋊C₁₆	—	2	2	no	—
106	C₈₃⋊C₁₆	C₂×Q₈	1	2	yes	—
107	C₂×Q₈×C₈₃	C₂×C₈	1	1	yes	characteristic
108	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₈₃	C₂²×C₄	1	2	yes	—
110	C₂×C₈×C₈₃	C₂⁴	1	1	yes	characteristic
111	C₄⋊C₄×C₈₃	C₂×C₈	1	2	yes	—
112	C₄⋊C₄×C₈₃	C₂×C₈	1	1	yes	characteristic
113	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
114	C₄²×C₈₃	C₂²×C₄	1	2	yes	—
115	C₁₆×C₈₃	—	2	1	no	—
116	C₁₆×C₈₃	—	2	2	no	—
117	C₁₆×C₈₃	C₂×Q₈	1	2	yes	—
118	C₂²×D₈₃	C₂²×C₈	1	1	yes	characteristic
119	Q₈×C₈₃	—	2	2	no	—
120	C₂×C₄×C₈₃	C₂²×C₈	1	1	yes	characteristic
121	C₂×C₄×C₈₃	C₂²×C₈	1	2	yes	—
122	C₂×C₄×C₈₃	C₂²×C₈	1	2	yes	—
123	C₂×C₄×C₈₃	C₂²×C₈	1	1	yes	characteristic
124	C₂×C₄×C₈₃	C₂³×C₄	1	1	yes	characteristic
125	C₂×Dic₈₃	C₂²×C₈	1	1	yes	characteristic
126	C₂×Dic₈₃	C₂²×C₈	1	2	yes	—
127	C₂×Dic₈₃	C₂²×C₈	1	2	yes	—
128	C₂×Dic₈₃	C₂²×C₈	1	1	yes	characteristic
129	C₂×Dic₈₃	C₂²×C₈	1	1	yes	characteristic
130	C₄×D₈₃	—	2	2	no	—
131	C₄×D₈₃	—	2	2	no	—
132	C₄×D₈₃	C₄×Q₈	1	2	yes	—
133	C₄×D₈₃	#131	1	2	yes	—
134	Dic₁₆₆	—	2	2	no	—
135	Dic₁₆₆	—	2	4	no	—
136	C₈×C₈₃	—	2	1	no	—
137	C₈×C₈₃	—	2	2	no	—
138	C₈×C₈₃	C₂²×Q₈	1	1	yes	characteristic
139	C₈×C₈₃	C₂×#41	1	1	yes	characteristic
140	C₈₃⋊C₈	—	2	1	no	—
141	C₈₃⋊C₈	—	2	2	no	—
142	C₈₃⋊C₈	C₄×Q₈	1	2	yes	—
143	C₂²×C₈₃	C₂³×C₈	1	1	yes	characteristic
144	C₂×D₈₃	—	2	1	no	—
145	C₂×D₈₃	—	2	1	no	—
146	C₂×D₈₃	C₈×Q₈	1	2	yes	—
147	C₄×C₈₃	—	2	2	no	—
148	C₄×C₈₃	—	2	1	no	—
149	C₄×C₈₃	—	2	2	no	—
150	C₄×C₈₃	#504	1	2	yes	—
151	C₄×C₈₃	C₂×C₄×Q₈	1	1	yes	characteristic
152	C₄×C₈₃	C₂×#131	1	1	yes	characteristic
153	Dic₈₃	—	2	2	no	—
154	Dic₈₃	—	2	2	no	—
155	Dic₈₃	—	2	1	no	—
156	Dic₈₃	C₈×Q₈	1	2	yes	—
157	Dic₈₃	#504	1	2	yes	—
158	#33474	—	83	1	no	maximal, Sylow
159	C₂×C₈₃	C₂×C₈×Q₈	1	1	yes	characteristic
160	C₂×C₈₃	C₂×#504	1	1	yes	characteristic
161	C₂×C₈₃	C₂×#504	1	1	yes	characteristic
162	D₈₃	—	2	2	no	—
163	#2960	—	83	1	no	—
164	#2183	—	83	2	no	—
165	#2183	—	83	1	no	—
166	#2183	D₈₃	1	1	yes	characteristic, complemented
167	Q₈×C₁₆	—	83	1	no	—
168	Q₈×C₁₆	—	83	2	no	—
169	Q₈×C₁₆	—	83	1	no	—
170	#2151	—	83	2	no	—
171	#2150	—	83	1	no	—
172	#2108	—	83	2	no	—
173	C₂×C₄×C₁₆	—	83	1	no	—
174	C₈₃	#33474	1	1	yes	characteristic, complemented, Sylow
175	#533	—	83	1	no	—
176	C₂×M₃₂	—	83	2	no	—
177	C₂²×C₁₆	—	83	1	no	—
178	C₂²×C₁₆	—	83	1	no	—
179	#446	—	83	1	no	—
180	#446	—	83	1	no	—
181	#446	—	83	2	no	—
182	C₈×Q₈	—	83	2	no	—
183	C₈×Q₈	—	83	1	no	—
184	C₈×Q₈	C₂×D₈₃	1	1	yes	characteristic
185	#424	—	83	2	no	—
186	#423	—	83	1	no	—
187	#405	—	83	2	no	—
188	C₂×C₄×C₈	—	83	1	no	—
189	C₄⋊C₁₆	—	83	2	no	—
190	C₄⋊C₁₆	—	83	2	no	—
191	C₄⋊C₁₆	—	83	2	no	—
192	C₄⋊C₁₆	—	83	1	no	—
193	C₄⋊C₁₆	—	83	2	no	—
194	C₄⋊C₁₆	C₂×D₈₃	1	1	yes	characteristic
195	C₄⋊C₁₆	C₂×D₈₃	1	2	yes	—
196	#346	—	83	2	no	—
197	#346	C₂×D₈₃	1	2	yes	—
198	C₄×C₁₆	—	83	1	no	—
199	C₄×C₁₆	—	83	1	no	—
200	C₄×C₁₆	—	83	2	no	—
201	C₄×C₁₆	—	83	1	no	—
202	C₄×C₁₆	—	83	2	no	—
203	C₄×C₁₆	C₂×D₈₃	1	1	yes	characteristic
204	C₂×M₁₆	—	83	2	no	—
205	C₂²×C₈	—	83	1	no	—
206	C₂²×C₈	—	83	1	no	—
207	C₄⋊Q₈	—	83	1	no	—
208	C₄⋊Q₈	—	83	1	no	—
209	#125	—	83	2	no	—
210	C₂²⋊Q₈	—	83	1	no	—
211	C₂²⋊Q₈	—	83	2	no	—
212	C₂²⋊Q₈	—	83	1	no	—
213	C₄×Q₈	—	83	2	no	—
214	C₄×Q₈	—	83	1	no	—
215	C₄×Q₈	C₄×D₈₃	1	1	yes	characteristic
216	#117	—	83	2	no	—
217	C₂×C₄²	—	83	1	no	—
218	M₃₂	—	166	4	no	—
219	C₂×C₁₆	—	83	1	no	—
220	C₂×C₁₆	—	83	4	no	—
221	C₂×C₁₆	—	83	1	no	—
222	C₂×C₁₆	—	83	2	no	—
223	C₂×C₁₆	—	166	2	no	—
224	C₂×C₁₆	C₂²×D₈₃	1	1	yes	characteristic
225	C₂×C₁₆	C₂²×D₈₃	1	2	yes	—
226	C₂×C₁₆	C₂²×D₈₃	1	1	yes	characteristic
227	C₄⋊C₈	—	83	2	no	—
228	C₄⋊C₈	—	83	2	no	—
229	C₄⋊C₈	—	83	2	no	—
230	C₄⋊C₈	—	83	1	no	—
231	C₄⋊C₈	—	83	2	no	—
232	C₄⋊C₈	C₄×D₈₃	1	1	yes	characteristic
233	C₄⋊C₈	C₄×D₈₃	1	2	yes	—
234	#97	—	83	2	no	—
235	#97	—	83	2	no	—
236	C₄×C₈	—	83	1	no	—
237	C₄×C₈	—	83	1	no	—
238	C₄×C₈	—	83	2	no	—
239	C₄×C₈	—	83	1	no	—
240	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
241	C₄×C₈	C₂²×D₈₃	1	2	yes	—
242	C₂×Q₈	—	83	2	no	—
243	C₂×Q₈	—	83	1	no	—
244	C₂×Q₈	C₈×D₈₃	1	1	yes	characteristic
245	C₂²×C₄	—	83	1	no	—
246	C₂²×C₄	—	83	1	no	—
247	C₂²×C₄	—	83	1	no	—
248	M₁₆	—	166	4	no	—
249	C₂×C₈	—	83	1	no	—
250	C₂×C₈	—	83	2	no	—
251	C₂×C₈	—	83	2	no	—
252	C₂×C₈	—	83	1	no	—
253	C₂×C₈	—	83	2	no	—
254	C₂×C₈	—	166	2	no	—
255	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
256	C₂×C₈	C₂×C₄×D₈₃	1	2	yes	—
257	C₂×C₈	C₂³×D₈₃	1	1	yes	characteristic
258	C₄⋊C₄	—	83	2	no	—
259	C₄⋊C₄	—	83	2	no	—
260	C₄⋊C₄	—	83	2	no	—
261	C₄⋊C₄	—	83	2	no	—
262	C₄⋊C₄	—	83	2	no	—
263	C₄⋊C₄	—	83	2	no	—
264	C₄⋊C₄	—	83	1	no	—
265	C₄⋊C₄	C₈×D₈₃	1	2	yes	—
266	C₄⋊C₄	C₈×D₈₃	1	1	yes	characteristic
267	C₂²⋊C₄	—	83	2	no	—
268	C₂²⋊C₄	—	83	2	no	—
269	C₄²	—	83	1	no	—
270	C₄²	—	83	1	no	—
271	C₄²	—	83	1	no	—
272	C₄²	—	83	2	no	—
273	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
274	C₄²	C₂×C₄×D₈₃	1	2	yes	—
275	C₁₆	—	2	1	no	—
276	C₁₆	—	2	2	no	—
277	C₁₆	—	83	2	no	—
278	C₁₆	—	166	1	no	—
279	C₁₆	—	166	2	no	—
280	C₁₆	Q₈×D₈₃	1	2	yes	—
281	C₂³	—	83	1	no	—
282	Q₈	—	2	2	no	—
283	Q₈	—	166	4	no	—
284	Q₈	—	166	2	no	—
285	C₂×C₄	—	83	1	no	—
286	C₂×C₄	—	83	1	no	—
287	C₂×C₄	—	83	2	no	—
288	C₂×C₄	—	83	2	no	—
289	C₂×C₄	—	83	1	no	—
290	C₂×C₄	—	83	2	no	—
291	C₂×C₄	—	83	2	no	—
292	C₂×C₄	—	166	2	no	—
293	C₂×C₄	—	166	2	no	—
294	C₂×C₄	C₂×C₈×D₈₃	1	1	yes	characteristic
295	C₂×C₄	C₂×C₈×D₈₃	1	2	yes	—
296	C₂×C₄	C₂×C₈×D₈₃	1	2	yes	—
297	C₂×C₄	C₂×C₈×D₈₃	1	1	yes	characteristic
298	C₂×C₄	C₂²×C₄×D₈₃	1	1	yes	characteristic
299	C₈	—	2	1	no	—
300	C₈	—	2	2	no	—
301	C₈	—	83	2	no	—
302	C₈	—	166	1	no	—
303	C₈	—	166	2	no	—
304	C₈	C₂×Q₈×D₈₃	1	1	yes	characteristic
305	C₈	#41×D₈₃	1	1	yes	characteristic
306	C₂²	—	83	2	no	—
307	C₂²	—	166	1	no	—
308	C₂²	—	166	1	no	—
309	C₂²	C₂²×C₈×D₈₃	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	2	no	—
312	C₄	—	2	2	no	—
313	C₄	—	83	2	no	—
314	C₄	—	83	2	no	—
315	C₄	—	166	2	no	—
316	C₄	—	166	1	no	—
317	C₄	—	166	2	no	—
318	C₄	#19128119	1	2	yes	—
319	C₄	C₄×Q₈×D₈₃	1	1	yes	characteristic
320	C₄	#131×D₈₃	1	1	yes	characteristic
321	C₂	—	166	2	no	—
322	C₂	C₈×Q₈×D₈₃	1	1	yes	characteristic
323	C₂	#504×D₈₃	1	1	yes	characteristic
324	C₂	C₂×#19128119	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₈₃
commutator, socle	C₂²×C₈₃	C₂³×C₈
Fitting	C₈₃×#2183	C₂
prime radical	C₂²×D₈₃	C₂²×C₈
cosocle	C₂×C₈×C₈₃	C₂⁴
hypercenter	#2183	D₈₃
hypocenter	C₈₃	#33474
LCS	G ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₈ ◁ #2183

Complements

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

N	H ≅ G/N	type
2. #24909371	C₂	semidirect
3. #24909369	C₂	semidirect
4. Q₈×C₈₃⋊C₁₆	C₂	semidirect
5. #24909365	C₂	semidirect
6. #24909365	C₂	semidirect
11. C₈₃×#2183	C₂	semidirect
59. C₁₆×D₈₃	Q₈	semidirect
166. #2183	D₈₃	semidirect
174. C₈₃	#33474	semidirect

Maximal subgroups

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

subgroup	index
#24916392	2
#24909371	2
#24909369	2
Q₈×C₈₃⋊C₁₆	2
#24909365	2
#24909220	2
#24909217	2
#24909061	2
C₄×C₁₆×D₈₃	2
C₈₃×#2183	2
#33474	83