Subgroups — #35557397

Order 21248.

706 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19128119	C₂	1	2	yes	maximal
2	C₄×#14581798	C₂	1	1	yes	maximal, characteristic
3	#24909223	C₂	1	2	yes	maximal, complemented
4	#24909220	C₂	1	4	yes	maximal, complemented
5	#24909217	C₂	1	2	yes	maximal, complemented
6	C₂×C₄⋊(C₈₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄⋊C₁₆×D₈₃	C₂	1	2	yes	maximal, complemented
8	C₈₃×#2150	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14581798	C₄	1	1	yes	characteristic
10	C₂×#14581798	C₂²	1	1	yes	characteristic
11	C₄×#13050590	C₄	1	1	yes	characteristic
12	#19128119	—	2	8	no	—
13	C₂×#14581731	C₂²	1	4	yes	—
14	C₂²×C₈₃⋊C₁₆	C₂²	1	2	yes	—
15	C₂×C₁₆×D₈₃	C₂²	1	4	yes	—
16	#19127656	C₄	1	1	yes	characteristic
17	#19127656	C₂²	1	1	yes	characteristic
18	#19127655	C₄	1	2	yes	—
19	#19127655	C₂²	1	2	yes	—
20	C₄×#13050574	C₄	1	1	yes	characteristic
21	C₄×#13050573	C₄	1	2	yes	—
22	C₄×C₈×D₈₃	C₂²	1	2	yes	—
23	C₄⋊(C₈₃⋊C₁₆)	C₂²	1	1	yes	characteristic
24	C₄⋊(C₈₃⋊C₁₆)	C₂²	1	1	yes	characteristic
25	C₄⋊(C₈₃⋊C₁₆)	C₂²	1	2	yes	—
26	Dic₈₃⋊C₁₆	C₂²	1	4	yes	—
27	Dic₈₃⋊C₁₆	C₂²	1	4	yes	—
28	C₂×M₃₂×C₈₃	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
30	C₄⋊C₁₆×C₈₃	C₂²	1	2	yes	—
31	C₄⋊C₁₆×C₈₃	C₂²	1	2	yes	—
32	C₂×#13050590	C₂×C₄	1	1	yes	characteristic
33	#14581798	—	2	4	no	—
34	#14581798	D₄	1	4	yes	—
35	#14581798	Q₈	1	4	yes	—
36	C₂×#13050574	C₂×C₄	1	1	yes	characteristic
37	C₂×#13050574	C₂×C₄	1	1	yes	characteristic
38	C₂×#13050573	C₂×C₄	1	2	yes	—
39	C₂×#13050573	C₂×C₄	1	2	yes	—
40	C₂×C₈×D₈₃	C₂×C₄	1	2	yes	—
41	C₂×C₈×D₈₃	C₂³	1	2	yes	—
42	C₄×Dic₁₆₆	C₈	1	1	yes	characteristic
43	C₄×Dic₁₆₆	C₈	1	1	yes	characteristic
44	C₄×D₃₃₂	C₈	1	1	yes	characteristic
45	C₄×D₃₃₂	C₈	1	1	yes	characteristic
46	C₄×C₂²⋊D₈₃	C₈	1	2	yes	—
47	C₄×C₂²⋊D₈₃	C₈	1	2	yes	—
48	#14581751	C₂×C₄	1	1	yes	characteristic
49	#14581749	C₂×C₄	1	2	yes	—
50	C₄²×D₈₃	C₂×C₄	1	2	yes	—
51	#14581731	—	2	8	no	—
52	C₂×C₈₃⋊C₁₆	—	2	4	no	—
53	C₂×C₈₃⋊C₁₆	C₂³	1	2	yes	—
54	C₂×C₈₃⋊C₁₆	C₂³	1	2	yes	—
55	C₁₆×D₈₃	—	2	8	no	—
56	#14581703	C₂×C₄	1	2	yes	—
57	#14581702	C₂×C₄	1	2	yes	—
58	#14581702	C₂×C₄	1	2	yes	—
59	C₄×C₈₃⋊C₈	C₂×C₄	1	2	yes	—
60	C₈×Dic₈₃	C₂×C₄	1	2	yes	—
61	C₈×Dic₈₃	C₂³	1	2	yes	—
62	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
63	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
64	C₂×C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
65	M₃₂×C₈₃	—	2	4	no	—
66	C₂×C₁₆×C₈₃	C₂³	1	4	yes	—
67	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
68	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
69	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
70	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
71	#13050590	C₄⋊C₄	1	4	yes	—
72	#13050590	M₁₆	1	4	yes	—
73	C₂×Dic₁₆₆	C₂×C₈	1	1	yes	characteristic
74	C₂×D₃₃₂	C₂×C₈	1	1	yes	characteristic
75	C₂×C₂²⋊D₈₃	C₂×C₈	1	2	yes	—
76	C₂×C₄×D₈₃	C₂×C₈	1	2	yes	—
77	C₂×C₄×D₈₃	C₂×C₈	1	2	yes	—
78	C₂×C₄×D₈₃	C₂²×C₄	1	2	yes	—
79	#13050574	—	2	2	no	—
80	#13050574	C₄⋊C₄	1	4	yes	—
81	#13050573	—	2	4	no	—
82	#13050573	C₄⋊C₄	1	8	yes	—
83	C₂×C₈₃⋊C₈	C₂²×C₄	1	2	yes	—
84	C₂×C₈₃⋊C₈	C₂²×C₄	1	2	yes	—
85	C₈×D₈₃	—	2	4	no	—
86	C₈×D₈₃	C₂×D₄	1	4	yes	—
87	C₈×D₈₃	C₂×Q₈	1	4	yes	—
88	C₄⋊Dic₈₃	C₂×C₈	1	1	yes	characteristic
89	C₄⋊Dic₈₃	C₂×C₈	1	1	yes	characteristic
90	Dic₈₃⋊C₄	C₂×C₈	1	2	yes	—
91	Dic₈₃⋊C₄	C₂×C₈	1	2	yes	—
92	C₂²⋊Dic₈₃	C₂×C₈	1	2	yes	—
93	(C₂×D₈₃)⋊C₄	C₂×C₈	1	2	yes	—
94	(C₂×D₈₃)⋊C₄	C₂×C₈	1	2	yes	—
95	C₄×Dic₈₃	C₂×C₈	1	2	yes	—
96	C₄×Dic₈₃	C₂×C₈	1	2	yes	—
97	C₄×Dic₈₃	C₂²×C₄	1	2	yes	—
98	C₈₃⋊C₁₆	—	2	2	no	—
99	C₈₃⋊C₁₆	—	2	2	no	—
100	C₂²×C₄×C₈₃	C₂×C₈	1	1	yes	characteristic
101	C₂²×C₄×C₈₃	C₂×C₈	1	1	yes	characteristic
102	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₈×C₈₃	—	2	2	no	—
104	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
108	C₂×C₈×C₈₃	C₂×Q₈	1	2	yes	—
109	C₂×C₈×C₈₃	C₂⁴	1	1	yes	characteristic
110	C₄²×C₈₃	C₂×C₈	1	1	yes	characteristic
111	C₄²×C₈₃	C₂×C₈	1	1	yes	characteristic
112	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
113	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
114	C₁₆×C₈₃	—	2	4	no	—
115	C₂³×C₈₃	C₂²×C₈	1	1	yes	characteristic
116	C₂²×D₈₃	C₂²×C₈	1	2	yes	—
117	C₂×C₄×C₈₃	—	2	2	no	—
118	C₂×C₄×C₈₃	—	4	1	no	—
119	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	2	yes	—
120	C₂×C₄×C₈₃	C₂²×C₈	1	1	yes	characteristic
121	C₂×C₄×C₈₃	C₂²×C₈	1	1	yes	characteristic
122	C₂×C₄×C₈₃	C₂²×C₈	1	1	yes	characteristic
123	C₂×C₄×C₈₃	C₂²×C₈	1	1	yes	characteristic
124	C₂×C₄×C₈₃	C₂²×C₈	1	1	yes	characteristic
125	C₂×C₄×C₈₃	C₂×M₁₆	1	2	yes	—
126	C₂×C₄×C₈₃	C₂³×C₄	1	1	yes	characteristic
127	C₂²⋊D₈₃	C₄⋊C₈	1	8	yes	—
128	C₂×Dic₈₃	C₂²×C₈	1	2	yes	—
129	C₂×Dic₈₃	C₂²×C₈	1	2	yes	—
130	C₂×Dic₈₃	C₂²×C₈	1	2	yes	—
131	D₃₃₂	C₄⋊C₈	1	4	yes	—
132	C₄×D₈₃	—	2	4	no	—
133	C₄×D₈₃	—	4	2	no	—
134	C₄×D₈₃	C₂×C₄⋊C₄	1	4	yes	—
135	C₄×D₈₃	C₂×M₁₆	1	4	yes	—
136	Dic₁₆₆	C₄⋊C₈	1	4	yes	—
137	C₈×C₈₃	—	2	1	no	—
138	C₈×C₈₃	—	2	1	no	—
139	C₈×C₈₃	C₂×C₄⋊C₄	1	2	yes	—
140	C₈×C₈₃	C₂²×D₄	1	1	yes	characteristic
141	C₈×C₈₃	C₂²×Q₈	1	1	yes	characteristic
142	C₈₃⋊C₈	—	2	2	no	—
143	C₈₃⋊C₈	C₂×C₄⋊C₄	1	4	yes	—
144	C₂²×C₈₃	—	2	1	no	—
145	C₂²×C₈₃	—	2	1	no	—
146	C₂²×C₈₃	C₂×C₄⋊C₈	1	2	yes	—
147	C₂²×C₈₃	C₂³×C₈	1	1	yes	characteristic
148	C₂×D₈₃	—	2	2	no	—
149	C₂×D₈₃	—	2	2	no	—
150	C₂×D₈₃	C₂×C₄⋊C₈	1	4	yes	—
151	C₄×C₈₃	—	2	1	no	—
152	C₄×C₈₃	—	2	1	no	—
153	C₄×C₈₃	—	2	1	no	—
154	C₄×C₈₃	C₂×C₄⋊C₈	1	2	yes	—
155	C₄×C₈₃	#504	1	2	yes	complemented
156	C₄×C₈₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
157	C₄×C₈₃	C₂²×M₁₆	1	1	yes	characteristic
158	Dic₈₃	—	2	2	no	—
159	Dic₈₃	—	2	2	no	—
160	Dic₈₃	C₂×C₄⋊C₈	1	4	yes	—
161	#33398	—	83	1	no	maximal, Sylow
162	C₂×C₈₃	—	2	2	no	—
163	C₂×C₈₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
164	C₂×C₈₃	C₂×#504	1	1	yes	characteristic
165	C₂×C₈₃	#3407	1	1	yes	characteristic
166	D₈₃	—	2	4	no	—
167	C₂×#504	—	83	2	no	—
168	C₄×#131	—	83	1	no	—
169	#2151	—	83	2	no	—
170	#2151	—	83	4	no	—
171	#2150	—	83	2	no	—
172	#2150	D₈₃	1	1	yes	characteristic, complemented
173	C₂×C₄⋊C₁₆	—	83	1	no	—
174	C₂×C₄⋊C₁₆	—	83	2	no	—
175	C₈₃	#33398	1	1	yes	characteristic, complemented, Sylow
176	C₂×#131	—	83	1	no	—
177	C₂×#131	—	83	1	no	—
178	C₄×#41	—	83	1	no	—
179	#504	—	166	8	no	—
180	C₂×M₃₂	—	83	4	no	—
181	C₂×M₃₂	C₂×D₈₃	1	2	yes	—
182	C₂²×C₁₆	—	83	2	no	—
183	C₂²×C₁₆	—	83	4	no	—
184	#405	—	83	1	no	—
185	#405	—	83	1	no	—
186	#405	—	83	2	no	—
187	#405	—	83	2	no	—
188	C₄×M₁₆	—	83	1	no	—
189	C₄×M₁₆	—	83	2	no	—
190	C₂×C₄×C₈	—	83	2	no	—
191	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
192	C₄⋊C₁₆	—	83	4	no	—
193	C₄⋊C₁₆	—	83	4	no	—
194	C₄⋊C₁₆	—	83	1	no	—
195	C₄⋊C₁₆	—	83	1	no	—
196	C₄⋊C₁₆	—	83	2	no	—
197	C₄⋊C₁₆	C₂×D₈₃	1	2	yes	—
198	C₄⋊C₁₆	C₂×D₈₃	1	2	yes	—
199	C₂×#41	—	83	1	no	—
200	#131	—	83	4	no	—
201	#131	—	83	4	no	—
202	#131	—	166	4	no	—
203	C₂×M₁₆	—	83	1	no	—
204	C₂×M₁₆	—	83	1	no	—
205	C₂×M₁₆	—	83	2	no	—
206	C₂×M₁₆	—	83	2	no	—
207	C₂²×C₈	—	83	2	no	—
208	C₂²×C₈	—	83	2	no	—
209	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
210	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
211	C₄×Q₈	—	83	1	no	—
212	C₄×Q₈	—	83	1	no	—
213	C₄×D₄	—	83	2	no	—
214	C₄×D₄	—	83	2	no	—
215	C₄×D₄	—	83	1	no	—
216	C₄×D₄	—	83	1	no	—
217	#117	—	83	1	no	—
218	#117	—	83	2	no	—
219	C₂×C₄²	—	83	2	no	—
220	C₂×C₄²	C₄×D₈₃	1	1	yes	characteristic
221	M₃₂	—	2	4	no	—
222	M₃₂	—	166	8	no	—
223	C₂×C₁₆	—	83	2	no	—
224	C₂×C₁₆	—	83	2	no	—
225	C₂×C₁₆	—	166	8	no	—
226	C₂×C₁₆	—	166	4	no	—
227	C₂×C₁₆	C₂²×D₈₃	1	4	yes	—
228	#97	—	83	2	no	—
229	#97	—	83	2	no	—
230	#97	—	83	2	no	—
231	C₄×C₈	—	83	2	no	—
232	C₄×C₈	—	83	2	no	—
233	C₄×C₈	—	83	2	no	—
234	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
235	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
236	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
237	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
238	#41	—	83	4	no	—
239	#41	—	83	4	no	—
240	C₂×Q₈	—	83	1	no	—
241	C₂×D₄	—	83	2	no	—
242	C₂×D₄	—	83	1	no	—
243	C₂²×C₄	—	83	2	no	—
244	C₂²×C₄	—	83	2	no	—
245	C₂²×C₄	—	83	2	no	—
246	C₂²×C₄	C₈×D₈₃	1	1	yes	characteristic
247	C₂²×C₄	C₈×D₈₃	1	1	yes	characteristic
248	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
249	M₁₆	—	83	4	no	—
250	M₁₆	—	83	8	no	—
251	M₁₆	—	166	4	no	—
252	M₁₆	—	166	2	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	83	2	no	—
255	C₂×C₈	—	83	4	no	—
256	C₂×C₈	—	83	4	no	—
257	C₂×C₈	—	83	2	no	—
258	C₂×C₈	—	166	4	no	—
259	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
260	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
261	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
262	C₂×C₈	D₄×D₈₃	1	2	yes	—
263	C₂×C₈	Q₈×D₈₃	1	2	yes	—
264	C₂×C₈	C₂³×D₈₃	1	1	yes	characteristic
265	C₄⋊C₄	—	83	2	no	—
266	C₄⋊C₄	—	83	2	no	—
267	C₄⋊C₄	—	83	1	no	—
268	C₄⋊C₄	—	83	1	no	—
269	C₂²⋊C₄	—	83	2	no	—
270	C₂²⋊C₄	—	83	2	no	—
271	C₂²⋊C₄	—	83	2	no	—
272	C₄²	—	83	2	no	—
273	C₄²	—	83	2	no	—
274	C₄²	—	83	2	no	—
275	C₄²	C₈×D₈₃	1	1	yes	characteristic
276	C₄²	C₈×D₈₃	1	1	yes	characteristic
277	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
278	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
279	C₁₆	—	2	4	no	—
280	C₁₆	—	166	2	no	—
281	C₁₆	—	166	2	no	—
282	C₂³	—	83	2	no	—
283	C₂³	C₂×C₈×D₈₃	1	1	yes	characteristic
284	Q₈	—	83	4	no	—
285	D₄	—	83	8	no	—
286	D₄	—	83	4	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	—	332	2	no	—
290	C₂×C₄	—	83	2	no	—
291	C₂×C₄	—	83	4	no	—
292	C₂×C₄	—	83	4	no	—
293	C₂×C₄	—	83	2	no	—
294	C₂×C₄	—	83	2	no	—
295	C₂×C₄	—	166	4	no	—
296	C₂×C₄	C₄⋊C₄×D₈₃	1	2	yes	—
297	C₂×C₄	C₂×C₈×D₈₃	1	1	yes	characteristic
298	C₂×C₄	C₂×C₈×D₈₃	1	1	yes	characteristic
299	C₂×C₄	C₂×C₈×D₈₃	1	1	yes	characteristic
300	C₂×C₄	C₂×C₈×D₈₃	1	1	yes	characteristic
301	C₂×C₄	C₂×C₈×D₈₃	1	1	yes	characteristic
302	C₂×C₄	M₁₆×D₈₃	1	2	yes	—
303	C₂×C₄	C₂²×C₄×D₈₃	1	1	yes	characteristic
304	C₈	—	2	1	no	—
305	C₈	—	2	1	no	—
306	C₈	—	83	4	no	—
307	C₈	—	166	2	no	—
308	C₈	C₄⋊C₄×D₈₃	1	2	yes	—
309	C₈	C₂×D₄×D₈₃	1	1	yes	characteristic
310	C₈	C₂×Q₈×D₈₃	1	1	yes	characteristic
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	83	4	no	—
314	C₂²	—	166	2	no	—
315	C₂²	—	166	2	no	—
316	C₂²	C₄⋊C₈×D₈₃	1	2	yes	—
317	C₂²	C₂²×C₈×D₈₃	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	2	1	no	—
321	C₄	—	83	4	no	—
322	C₄	—	166	2	no	—
323	C₄	—	166	2	no	—
324	C₄	C₄⋊C₈×D₈₃	1	2	yes	—
325	C₄	#19128119	1	2	yes	complemented
326	C₄	C₂×C₄⋊C₄×D₈₃	1	1	yes	characteristic
327	C₄	C₂×M₁₆×D₈₃	1	1	yes	characteristic
328	C₂	—	2	2	no	—
329	C₂	—	166	4	no	—
330	C₂	C₂×C₄⋊C₈×D₈₃	1	1	yes	characteristic
331	C₂	C₂×#19128119	1	1	yes	characteristic
332	C₂	#24920565	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₈₃
commutator, socle	C₂²×C₈₃	C₂³×C₈
Fitting	C₈₃×#2150	C₂
prime radical	C₂×#13050590	C₂×C₄
cosocle	C₂×C₈×C₈₃	C₂⁴
hypercenter	#2150	D₈₃
hypocenter	C₈₃	#33398
LCS	G ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
3. #24909223	C₂	semidirect
4. #24909220	C₂	semidirect
5. #24909217	C₂	semidirect
6. C₂×C₄⋊(C₈₃⋊C₁₆)	C₂	semidirect
7. C₄⋊C₁₆×D₈₃	C₂	semidirect
8. C₈₃×#2150	C₂	semidirect
155. C₄×C₈₃	#504	semidirect
172. #2150	D₈₃	semidirect
175. C₈₃	#33398	semidirect
325. C₄	#19128119	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19128119	2
C₄×#14581798	2
#24909223	2
#24909220	2
#24909217	2
C₂×C₄⋊(C₈₃⋊C₁₆)	2
C₄⋊C₁₆×D₈₃	2
C₈₃×#2150	2
#33398	83