Subgroups — #35320341

Order 21248.

450 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24909219	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄⋊(C₈₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#19127485	C₂	1	2	yes	maximal
4	#24905552	C₂	1	2	yes	maximal, complemented
5	C₄×#131×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14581732	C₂²	1	2	yes	—
7	C₂²×C₈₃⋊C₁₆	C₂²	1	2	yes	—
8	C₄⋊(C₈₃⋊C₁₆)	—	2	1	no	—
9	C₄⋊(C₈₃⋊C₁₆)	—	2	1	no	—
10	C₄⋊(C₈₃⋊C₁₆)	—	2	1	no	—
11	C₄⋊(C₈₃⋊C₁₆)	—	2	1	no	—
12	#19127485	—	2	4	no	—
13	C₂×#131×C₈₃	C₄	1	1	yes	characteristic
14	C₂×#131×C₈₃	C₂²	1	1	yes	characteristic
15	C₄×#41×C₈₃	C₄	1	1	yes	characteristic
16	#405×C₈₃	—	2	1	no	—
17	#405×C₈₃	—	2	1	no	—
18	#405×C₈₃	C₄	1	1	yes	characteristic
19	#405×C₈₃	C₂²	1	1	yes	characteristic
20	C₄×M₁₆×C₈₃	—	2	1	no	—
21	C₄×M₁₆×C₈₃	C₄	1	1	yes	characteristic
22	C₂×C₄×C₈×C₈₃	—	2	1	no	—
23	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
24	#14581732	—	2	2	no	—
25	#14581732	—	2	2	no	—
26	C₂×C₈₃⋊C₁₆	—	2	2	no	—
27	C₂×C₈₃⋊C₁₆	—	2	2	no	—
28	C₂×C₈₃⋊C₁₆	—	2	2	no	—
29	C₂×C₈₃⋊C₁₆	—	2	2	no	—
30	C₂×#41×C₈₃	C₂×C₄	1	1	yes	characteristic
31	#131×C₈₃	—	2	2	no	—
32	#131×C₈₃	—	2	2	no	—
33	#131×C₈₃	—	2	1	no	—
34	#131×C₈₃	—	2	1	no	—
35	#131×C₈₃	D₄	1	2	yes	—
36	#131×C₈₃	Q₈	1	2	yes	—
37	C₂×M₁₆×C₈₃	—	2	1	no	—
38	C₂×M₁₆×C₈₃	—	2	1	no	—
39	C₂×M₁₆×C₈₃	C₂×C₄	1	1	yes	characteristic
40	C₂×M₁₆×C₈₃	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₈₃	—	2	1	no	—
42	C₂²×C₈×C₈₃	—	2	1	no	—
43	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
45	C₄×Q₈×C₈₃	C₈	1	1	yes	characteristic
46	C₄×Q₈×C₈₃	C₈	1	1	yes	characteristic
47	C₄×D₄×C₈₃	—	2	1	no	—
48	C₄×D₄×C₈₃	—	2	1	no	—
49	C₄×D₄×C₈₃	C₈	1	1	yes	characteristic
50	C₄×D₄×C₈₃	C₈	1	1	yes	characteristic
51	#117×C₈₃	—	2	1	no	—
52	#117×C₈₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²×C₈₃	—	2	1	no	—
54	C₂×C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
55	#97×C₈₃	—	2	1	no	—
56	#97×C₈₃	—	2	1	no	—
57	#97×C₈₃	—	2	1	no	—
58	C₄×C₈×C₈₃	—	2	1	no	—
59	C₄×C₈×C₈₃	—	2	1	no	—
60	C₄×C₈×C₈₃	—	2	1	no	—
61	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
62	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
63	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
64	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
65	C₈₃⋊C₁₆	—	4	2	no	—
66	C₈₃⋊C₁₆	—	4	2	no	—
67	#41×C₈₃	—	2	1	no	—
68	#41×C₈₃	—	2	1	no	—
69	#41×C₈₃	C₄⋊C₄	1	2	yes	—
70	#41×C₈₃	M₁₆	1	2	yes	—
71	C₂×Q₈×C₈₃	C₂×C₈	1	1	yes	characteristic
72	C₂×D₄×C₈₃	—	2	1	no	—
73	C₂×D₄×C₈₃	C₂×C₈	1	1	yes	characteristic
74	C₂²×C₄×C₈₃	—	2	1	no	—
75	C₂²×C₄×C₈₃	—	2	1	no	—
76	C₂²×C₄×C₈₃	—	2	1	no	—
77	C₂²×C₄×C₈₃	M₁₆	1	1	yes	characteristic
78	C₂²×C₄×C₈₃	M₁₆	1	1	yes	characteristic
79	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
80	M₁₆×C₈₃	—	2	4	no	—
81	M₁₆×C₈₃	—	2	1	no	—
82	M₁₆×C₈₃	—	2	1	no	—
83	M₁₆×C₈₃	—	2	2	no	—
84	M₁₆×C₈₃	—	2	2	no	—
85	M₁₆×C₈₃	C₄⋊C₄	1	2	yes	—
86	M₁₆×C₈₃	M₁₆	1	2	yes	—
87	C₂×C₈×C₈₃	—	2	4	no	—
88	C₂×C₈×C₈₃	—	2	1	no	—
89	C₂×C₈×C₈₃	—	2	1	no	—
90	C₂×C₈×C₈₃	—	2	1	no	—
91	C₂×C₈×C₈₃	—	2	1	no	—
92	C₂×C₈×C₈₃	—	2	2	no	—
93	C₂×C₈×C₈₃	—	2	2	no	—
94	C₂×C₈×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₈×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
100	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
101	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
102	C₄⋊C₄×C₈₃	—	2	1	no	—
103	C₄⋊C₄×C₈₃	—	2	1	no	—
104	C₄⋊C₄×C₈₃	C₂×C₈	1	1	yes	characteristic
105	C₄⋊C₄×C₈₃	C₂×C₈	1	1	yes	characteristic
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	1	no	—
110	C₄²×C₈₃	—	2	1	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×C₈	1	1	yes	characteristic
115	C₄²×C₈₃	C₂×C₈	1	1	yes	characteristic
116	C₂³×C₈₃	—	2	1	no	—
117	C₂³×C₈₃	C₂×M₁₆	1	1	yes	characteristic
118	Q₈×C₈₃	—	2	1	no	—
119	Q₈×C₈₃	C₄⋊C₈	1	2	yes	—
120	D₄×C₈₃	—	2	2	no	—
121	D₄×C₈₃	—	2	2	no	—
122	D₄×C₈₃	—	2	1	no	—
123	D₄×C₈₃	C₄⋊C₈	1	2	yes	—
124	C₂×C₄×C₈₃	—	2	1	no	—
125	C₂×C₄×C₈₃	—	2	2	no	—
126	C₂×C₄×C₈₃	—	2	2	no	—
127	C₂×C₄×C₈₃	—	2	2	no	—
128	C₂×C₄×C₈₃	—	2	1	no	—
129	C₂×C₄×C₈₃	—	2	1	no	—
130	C₂×C₄×C₈₃	—	4	1	no	—
131	C₂×C₄×C₈₃	—	4	2	no	—
132	C₂×C₄×C₈₃	—	4	2	no	—
133	C₂×C₄×C₈₃	C₂²⋊C₈	1	1	yes	characteristic
134	C₂×C₄×C₈₃	C₂²⋊C₈	1	1	yes	characteristic
135	C₂×C₄×C₈₃	C₂²⋊C₈	1	1	yes	characteristic
136	C₂×C₄×C₈₃	C₂²⋊C₈	1	1	yes	characteristic
137	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
138	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
139	C₂×C₄×C₈₃	#117	1	1	yes	characteristic
140	C₂×C₄×C₈₃	C₂²×C₈	1	1	yes	characteristic
141	C₂×C₄×C₈₃	C₂×M₁₆	1	1	yes	characteristic
142	C₂×C₄×C₈₃	C₂×M₁₆	1	1	yes	characteristic
143	C₈×C₈₃	—	2	2	no	—
144	C₈×C₈₃	—	2	1	no	—
145	C₈×C₈₃	—	2	2	no	—
146	C₈×C₈₃	—	2	1	no	—
147	C₈×C₈₃	#121	1	1	yes	characteristic
148	C₈×C₈₃	#121	1	1	yes	characteristic
149	C₈×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
150	C₈×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
151	C₂²×C₈₃	—	2	1	no	—
152	C₂²×C₈₃	—	2	1	no	—
153	C₂²×C₈₃	—	2	2	no	—
154	C₂²×C₈₃	—	4	1	no	—
155	C₂²×C₈₃	—	4	1	no	—
156	C₂²×C₈₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
157	C₂²×C₈₃	#423	1	1	yes	characteristic
158	C₂²×C₈₃	#432	1	1	yes	characteristic
159	C₄×C₈₃	—	2	2	no	—
160	C₄×C₈₃	—	2	1	no	—
161	C₄×C₈₃	—	2	1	no	—
162	C₄×C₈₃	—	2	1	no	—
163	C₄×C₈₃	—	4	1	no	—
164	C₄×C₈₃	—	4	1	no	—
165	C₄×C₈₃	#350	1	2	yes	complemented
166	C₄×C₈₃	#380	1	1	yes	characteristic
167	C₄×C₈₃	#407	1	1	yes	characteristic
168	C₄×C₈₃	C₂×C₄⋊C₈	1	1	yes	characteristic
169	C₄×C₈₃	#431	1	1	yes	characteristic
170	#12787	—	83	1	no	maximal, Sylow
171	C₂×C₈₃	—	2	1	no	—
172	C₂×C₈₃	—	2	1	no	—
173	C₂×C₈₃	—	4	2	no	—
174	C₂×C₈₃	#1797	1	1	yes	characteristic
175	C₂×C₈₃	C₂×#350	1	1	yes	characteristic
176	C₂×C₈₃	#2117	1	1	yes	characteristic
177	C₄×#131	D₈₃	1	1	yes	characteristic
178	#2150	—	83	1	no	—
179	C₂×C₄⋊C₁₆	—	83	1	no	—
180	C₂×#350	—	83	2	no	—
181	#1379	—	83	2	no	—
182	C₈₃	#12787	1	1	yes	characteristic, complemented, Sylow
183	C₂×#131	Dic₈₃	1	1	yes	characteristic
184	C₂×#131	C₂×D₈₃	1	1	yes	characteristic
185	C₄×#41	Dic₈₃	1	1	yes	characteristic
186	C₂×M₃₂	—	83	2	no	—
187	C₂²×C₁₆	—	83	2	no	—
188	#405	—	2	1	no	—
189	#405	—	2	1	no	—
190	#405	Dic₈₃	1	1	yes	characteristic
191	#405	C₂×D₈₃	1	1	yes	characteristic
192	C₄×M₁₆	—	2	1	no	—
193	C₄×M₁₆	Dic₈₃	1	1	yes	characteristic
194	C₂×C₄×C₈	—	2	1	no	—
195	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
196	C₄⋊C₁₆	—	166	1	no	—
197	C₄⋊C₁₆	—	166	1	no	—
198	C₄⋊C₁₆	—	166	1	no	—
199	C₄⋊C₁₆	—	166	1	no	—
200	#350	—	166	4	no	—
201	C₂×#41	C₂×Dic₈₃	1	1	yes	characteristic
202	#131	—	2	2	no	—
203	#131	—	2	2	no	—
204	#131	—	2	1	no	—
205	#131	—	2	1	no	—
206	#131	Dic₁₆₆	1	2	yes	—
207	#131	D₃₃₂	1	2	yes	—
208	C₂×M₁₆	—	2	1	no	—
209	C₂×M₁₆	—	2	1	no	—
210	C₂×M₁₆	C₂×Dic₈₃	1	1	yes	characteristic
211	C₂×M₁₆	C₂×Dic₈₃	1	1	yes	characteristic
212	C₂²×C₈	—	2	1	no	—
213	C₂²×C₈	—	2	1	no	—
214	C₂²×C₈	C₂×Dic₈₃	1	1	yes	characteristic
215	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
216	C₄×Q₈	C₈₃⋊C₈	1	1	yes	characteristic
217	C₄×Q₈	C₈₃⋊C₈	1	1	yes	characteristic
218	C₄×D₄	—	2	1	no	—
219	C₄×D₄	—	2	1	no	—
220	C₄×D₄	C₈₃⋊C₈	1	1	yes	characteristic
221	C₄×D₄	C₈₃⋊C₈	1	1	yes	characteristic
222	#117	—	2	1	no	—
223	#117	C₂×Dic₈₃	1	1	yes	characteristic
224	C₂×C₄²	—	2	1	no	—
225	C₂×C₄²	C₂×Dic₈₃	1	1	yes	characteristic
226	M₃₂	—	166	2	no	—
227	M₃₂	—	166	2	no	—
228	C₂×C₁₆	—	166	2	no	—
229	C₂×C₁₆	—	166	2	no	—
230	C₂×C₁₆	—	166	2	no	—
231	C₂×C₁₆	—	166	2	no	—
232	#97	—	2	1	no	—
233	#97	—	2	1	no	—
234	#97	—	2	1	no	—
235	C₄×C₈	—	2	1	no	—
236	C₄×C₈	—	2	1	no	—
237	C₄×C₈	—	2	1	no	—
238	C₄×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
239	C₄×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
240	C₄×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
241	C₄×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
242	#41	—	2	1	no	—
243	#41	—	2	1	no	—
244	#41	C₄⋊Dic₈₃	1	2	yes	—
245	#41	#13050574	1	2	yes	—
246	C₂×Q₈	C₂×C₈₃⋊C₈	1	1	yes	characteristic
247	C₂×D₄	—	2	1	no	—
248	C₂×D₄	C₂×C₈₃⋊C₈	1	1	yes	characteristic
249	C₂²×C₄	—	2	1	no	—
250	C₂²×C₄	—	2	1	no	—
251	C₂²×C₄	—	2	1	no	—
252	C₂²×C₄	#13050574	1	1	yes	characteristic
253	C₂²×C₄	#13050574	1	1	yes	characteristic
254	C₂²×C₄	C₂²×Dic₈₃	1	1	yes	characteristic
255	M₁₆	—	2	4	no	—
256	M₁₆	—	2	1	no	—
257	M₁₆	—	2	1	no	—
258	M₁₆	—	2	2	no	—
259	M₁₆	—	2	2	no	—
260	M₁₆	C₄⋊Dic₈₃	1	2	yes	—
261	M₁₆	#13050574	1	2	yes	—
262	C₂×C₈	—	2	4	no	—
263	C₂×C₈	—	2	1	no	—
264	C₂×C₈	—	2	1	no	—
265	C₂×C₈	—	2	1	no	—
266	C₂×C₈	—	2	1	no	—
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	C₂²⋊Dic₈₃	1	1	yes	characteristic
270	C₂×C₈	C₂²⋊Dic₈₃	1	1	yes	characteristic
271	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
272	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
273	C₂×C₈	C₂×D₃₃₂	1	1	yes	characteristic
274	C₂×C₈	C₂×Dic₁₆₆	1	1	yes	characteristic
275	C₂×C₈	#13050590	1	1	yes	characteristic
276	C₂×C₈	#13050590	1	1	yes	characteristic
277	C₄⋊C₄	—	2	1	no	—
278	C₄⋊C₄	—	2	1	no	—
279	C₄⋊C₄	C₂×C₈₃⋊C₈	1	1	yes	characteristic
280	C₄⋊C₄	C₂×C₈₃⋊C₈	1	1	yes	characteristic
281	C₂²⋊C₄	—	2	1	no	—
282	C₂²⋊C₄	—	2	1	no	—
283	C₂²⋊C₄	—	2	1	no	—
284	C₄²	—	2	1	no	—
285	C₄²	—	2	1	no	—
286	C₄²	—	2	1	no	—
287	C₄²	C₂²⋊Dic₈₃	1	1	yes	characteristic
288	C₄²	C₂²⋊Dic₈₃	1	1	yes	characteristic
289	C₄²	C₂×C₈₃⋊C₈	1	1	yes	characteristic
290	C₄²	C₂×C₈₃⋊C₈	1	1	yes	characteristic
291	C₁₆	—	332	2	no	—
292	C₁₆	—	332	2	no	—
293	C₂³	—	2	1	no	—
294	C₂³	C₂×#13050574	1	1	yes	characteristic
295	Q₈	—	2	1	no	—
296	Q₈	C₄⋊(C₈₃⋊C₈)	1	2	yes	—
297	D₄	—	2	2	no	—
298	D₄	—	2	2	no	—
299	D₄	—	2	1	no	—
300	D₄	C₄⋊(C₈₃⋊C₈)	1	2	yes	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	C₂²⋊(C₈₃⋊C₈)	1	1	yes	characteristic
311	C₂×C₄	C₂²⋊(C₈₃⋊C₈)	1	1	yes	characteristic
312	C₂×C₄	C₂²⋊(C₈₃⋊C₈)	1	1	yes	characteristic
313	C₂×C₄	C₂²⋊(C₈₃⋊C₈)	1	1	yes	characteristic
314	C₂×C₄	C₂×C₂²⋊Dic₈₃	1	1	yes	characteristic
315	C₂×C₄	C₂×C₄⋊Dic₈₃	1	1	yes	characteristic
316	C₂×C₄	#14581751	1	1	yes	characteristic
317	C₂×C₄	C₂²×C₈₃⋊C₈	1	1	yes	characteristic
318	C₂×C₄	C₂×#13050574	1	1	yes	characteristic
319	C₂×C₄	C₂×#13050574	1	1	yes	characteristic
320	C₈	—	2	2	no	—
321	C₈	—	2	1	no	—
322	C₈	—	2	2	no	—
323	C₈	—	2	1	no	—
324	C₈	#14581766	1	1	yes	characteristic
325	C₈	#14581766	1	1	yes	characteristic
326	C₈	#14581771	1	1	yes	characteristic
327	C₈	#14581771	1	1	yes	characteristic
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	2	no	—
331	C₂²	—	4	1	no	—
332	C₂²	—	4	1	no	—
333	C₂²	C₂×C₂²⋊(C₈₃⋊C₈)	1	1	yes	characteristic
334	C₂²	#19127725	1	1	yes	characteristic
335	C₂²	#19127758	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	2	2	no	—
340	C₄	—	4	1	no	—
341	C₄	—	4	1	no	—
342	C₄	#19127485	1	2	yes	complemented
343	C₄	#19127561	1	1	yes	characteristic
344	C₄	#19127665	1	1	yes	characteristic
345	C₄	C₂×C₄⋊(C₈₃⋊C₈)	1	1	yes	characteristic
346	C₄	#19127754	1	1	yes	characteristic
347	C₂	—	2	1	no	—
348	C₂	—	2	1	no	—
349	C₂	—	4	2	no	—
350	C₂	#24907443	1	1	yes	characteristic
351	C₂	C₂×#19127485	1	1	yes	characteristic
352	C₂	#24909091	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₈₃
commutator	C₂×C₄×C₈₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₈₃
Fitting	C₄×#131×C₈₃	C₂
socle	C₂²×C₈₃	C₂×C₂²⋊C₈
prime radical	C₂×#41×C₈₃	C₂×C₄
cosocle	C₂²×C₈×C₈₃	C₂³
hypercenter	C₄×#131	D₈₃
hypocenter	C₈₃	#12787
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
1. #24909219	C₂	semidirect
2. C₂×C₄⋊(C₈₃⋊C₁₆)	C₂	semidirect
4. #24905552	C₂	semidirect
165. C₄×C₈₃	#350	semidirect
182. C₈₃	#12787	semidirect
342. C₄	#19127485	semidirect

Maximal subgroups

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

subgroup	index
#24909219	2
C₂×C₄⋊(C₈₃⋊C₁₆)	2
C₂×#19127485	2
#24905552	2
C₄×#131×C₈₃	2
#12787	83