Subgroups — #35557507

Order 21248.

600 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19127656	C₂	1	1	yes	maximal, characteristic
2	#24909267	C₂	1	4	yes	maximal, complemented
3	#24909263	C₂	1	2	yes	maximal, complemented
4	#24909259	C₂	1	2	yes	maximal, complemented
5	#24909221	C₂	1	2	yes	maximal, complemented
6	#24909074	C₂	1	1	yes	maximal, characteristic
7	#24909059	C₂	1	2	yes	maximal, complemented
8	C₂×C₂²⋊C₁₆×C₈₃	C₂	1	1	yes	maximal, characteristic
9	C₂²×#13050574	C₄	1	1	yes	characteristic
10	C₂×#14581751	C₄	1	1	yes	characteristic
11	C₂×#14581732	C₂²	1	2	yes	—
12	#19127656	C₄	1	2	yes	—
13	#19127656	C₄	1	1	yes	characteristic
14	#19127656	C₄	1	1	yes	characteristic
15	#19127656	C₂²	1	1	yes	characteristic
16	#19127656	C₂²	1	1	yes	characteristic
17	#19127656	C₂²	1	2	yes	—
18	C₂×#14581702	C₄	1	2	yes	—
19	C₂×C₈×Dic₈₃	C₂²	1	2	yes	—
20	Dic₈₃⋊C₁₆	C₂²	1	4	yes	complemented
21	Dic₈₃⋊C₁₆	C₂²	1	4	yes	complemented
22	C₂²⋊(C₈₃⋊C₁₆)	C₂²	1	2	yes	—
23	C₂²⋊(C₈₃⋊C₁₆)	C₂²	1	2	yes	—
24	#19127476	C₂²	1	4	yes	complemented
25	C₁₆×Dic₈₃	C₂²	1	4	yes	complemented
26	C₂³×C₈×C₈₃	C₂²	1	1	yes	characteristic
27	C₂²×C₁₆×C₈₃	C₂²	1	2	yes	—
28	C₂²⋊C₁₆×C₈₃	C₂²	1	1	yes	characteristic
29	C₂²⋊C₁₆×C₈₃	C₂²	1	1	yes	characteristic
30	C₂²⋊C₁₆×C₈₃	C₂²	1	2	yes	—
31	C₂×#13050574	—	2	4	no	—
32	C₂×#13050574	C₂×C₄	1	2	yes	—
33	C₂×#13050574	C₂×C₄	1	1	yes	characteristic
34	C₂×#13050574	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈₃⋊C₈	C₂×C₄	1	2	yes	—
36	#14581751	C₈	1	2	yes	—
37	#14581751	C₈	1	1	yes	characteristic
38	#14581751	C₈	1	1	yes	characteristic
39	#14581751	C₂×C₄	1	2	yes	—
40	#14581751	C₂×C₄	1	1	yes	characteristic
41	#14581751	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄⋊Dic₈₃	C₈	1	1	yes	characteristic
43	C₂×C₄⋊Dic₈₃	C₈	1	1	yes	characteristic
44	C₂×C₂²⋊Dic₈₃	C₈	1	2	yes	—
45	C₂×C₄×Dic₈₃	C₂×C₄	1	2	yes	—
46	#14581732	—	2	4	no	—
47	C₂×C₈₃⋊C₁₆	C₂³	1	4	yes	—
48	#14581702	C₂×C₄	1	2	yes	—
49	#14581702	C₂×C₄	1	2	yes	—
50	#14581702	C₂×C₄	1	2	yes	—
51	#14581702	C₂×C₄	1	2	yes	—
52	C₈×Dic₈₃	C₂×C₄	1	2	yes	—
53	C₈×Dic₈₃	C₂×C₄	1	2	yes	—
54	C₈×Dic₈₃	C₂³	1	2	yes	—
55	C₈×Dic₈₃	C₂³	1	2	yes	—
56	C₂³×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
57	C₂²×C₈×C₈₃	—	2	4	no	—
58	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
59	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
60	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
61	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
62	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
63	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
64	C₂×C₁₆×C₈₃	—	2	4	no	—
65	C₂×C₁₆×C₈₃	C₂³	1	2	yes	—
66	C₂×C₁₆×C₈₃	C₂³	1	2	yes	—
67	C₂²×Dic₈₃	C₂×C₈	1	2	yes	—
68	C₂²×Dic₈₃	C₂×C₈	1	2	yes	—
69	#13050574	—	2	2	no	—
70	#13050574	—	2	4	no	—
71	#13050574	M₁₆	1	4	yes	—
72	C₂×C₈₃⋊C₈	—	2	4	no	—
73	C₂×C₈₃⋊C₈	C₂²×C₄	1	2	yes	—
74	C₂×C₈₃⋊C₈	C₂²×C₄	1	2	yes	—
75	C₄⋊Dic₈₃	C₂×C₈	1	2	yes	—
76	C₄⋊Dic₈₃	C₂×C₈	1	2	yes	—
77	C₄⋊Dic₈₃	C₂×C₈	1	1	yes	characteristic
78	C₄⋊Dic₈₃	C₂×C₈	1	1	yes	characteristic
79	C₄⋊Dic₈₃	C₂×C₈	1	1	yes	characteristic
80	C₄⋊Dic₈₃	C₂×C₈	1	1	yes	characteristic
81	C₂²⋊Dic₈₃	C₂×C₈	1	2	yes	—
82	C₂²⋊Dic₈₃	C₂×C₈	1	2	yes	—
83	C₂²⋊Dic₈₃	C₂×C₈	1	2	yes	—
84	C₂²⋊Dic₈₃	C₂×C₈	1	2	yes	—
85	C₄×Dic₈₃	C₂×C₈	1	2	yes	—
86	C₄×Dic₈₃	C₂×C₈	1	2	yes	—
87	C₄×Dic₈₃	C₂²×C₄	1	2	yes	—
88	C₄×Dic₈₃	C₂²×C₄	1	2	yes	—
89	C₈₃⋊C₁₆	—	2	4	no	—
90	C₂⁴×C₈₃	C₂×C₈	1	1	yes	characteristic
91	C₂²×C₄×C₈₃	—	2	2	no	—
92	C₂²×C₄×C₈₃	—	2	2	no	—
93	C₂²×C₄×C₈₃	C₂×C₈	1	1	yes	characteristic
94	C₂²×C₄×C₈₃	C₂×C₈	1	1	yes	characteristic
95	C₂²×C₄×C₈₃	C₂×C₈	1	1	yes	characteristic
96	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
98	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₈₃	—	2	2	no	—
100	C₂×C₈×C₈₃	—	2	2	no	—
101	C₂×C₈×C₈₃	—	2	2	no	—
102	C₂×C₈×C₈₃	—	2	2	no	—
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	#41	1	4	yes	—
108	C₂×C₈×C₈₃	C₂⁴	1	1	yes	characteristic
109	C₁₆×C₈₃	—	2	2	no	—
110	C₁₆×C₈₃	—	2	2	no	—
111	C₂³×C₈₃	—	2	2	no	—
112	C₂³×C₈₃	—	2	2	no	—
113	C₂³×C₈₃	—	4	1	no	—
114	C₂³×C₈₃	C₂²×C₈	1	1	yes	characteristic
115	C₂³×C₈₃	C₂²×C₈	1	1	yes	characteristic
116	C₂³×C₈₃	C₂²×C₈	1	1	yes	characteristic
117	C₂×C₄×C₈₃	—	2	2	no	—
118	C₂×C₄×C₈₃	—	2	1	no	—
119	C₂×C₄×C₈₃	—	2	1	no	—
120	C₂×C₄×C₈₃	—	2	1	no	—
121	C₂×C₄×C₈₃	—	2	1	no	—
122	C₂×C₄×C₈₃	—	2	2	no	—
123	C₂×C₄×C₈₃	—	2	2	no	—
124	C₂×C₄×C₈₃	#117	1	2	yes	—
125	C₂×C₄×C₈₃	C₂²×C₈	1	1	yes	characteristic
126	C₂×C₄×C₈₃	C₂²×C₈	1	1	yes	characteristic
127	C₂×C₄×C₈₃	C₂²×C₈	1	1	yes	characteristic
128	C₂×C₄×C₈₃	C₂×M₁₆	1	2	yes	—
129	C₂×C₄×C₈₃	C₂³×C₄	1	1	yes	characteristic
130	C₂×Dic₈₃	—	2	4	no	—
131	C₂×Dic₈₃	—	4	2	no	—
132	C₂×Dic₈₃	C₂²×C₈	1	2	yes	—
133	C₂×Dic₈₃	C₂²×C₈	1	2	yes	—
134	C₂×Dic₈₃	C₂²×C₈	1	2	yes	—
135	C₂×Dic₈₃	C₂²×C₈	1	2	yes	—
136	C₈×C₈₃	—	2	1	no	—
137	C₈×C₈₃	—	2	1	no	—
138	C₈×C₈₃	#117	1	2	yes	—
139	C₈×C₈₃	C₂×#41	1	2	yes	—
140	C₈₃⋊C₈	—	2	2	no	—
141	C₈₃⋊C₈	C₂×M₁₆	1	4	yes	—
142	C₂²×C₈₃	—	2	1	no	—
143	C₂²×C₈₃	—	2	1	no	—
144	C₂²×C₈₃	—	2	1	no	—
145	C₂²×C₈₃	—	2	1	no	—
146	C₂²×C₈₃	—	2	1	no	—
147	C₂²×C₈₃	—	2	1	no	—
148	C₂²×C₈₃	—	2	1	no	—
149	C₂²×C₈₃	—	4	4	no	—
150	C₂²×C₈₃	#431	1	2	yes	—
151	C₂²×C₈₃	#504	1	2	yes	—
152	C₂²×C₈₃	C₂³×C₈	1	1	yes	characteristic
153	C₄×C₈₃	—	2	1	no	—
154	C₄×C₈₃	—	2	1	no	—
155	C₄×C₈₃	#431	1	2	yes	—
156	C₄×C₈₃	C₂×#117	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	—	2	2	no	—
161	Dic₈₃	—	2	2	no	—
162	#33422	—	83	1	no	maximal, Sylow
163	C₂×C₈₃	—	2	2	no	—
164	C₂×C₈₃	—	2	2	no	—
165	C₂×C₈₃	—	4	1	no	—
166	C₂×C₈₃	C₂×#431	1	1	yes	characteristic
167	C₂×C₈₃	C₂×#504	1	1	yes	characteristic
168	C₂×C₈₃	#3407	1	1	yes	characteristic
169	C₂×#405	—	83	1	no	—
170	#2164	—	83	4	no	—
171	#2163	—	83	2	no	—
172	#2162	—	83	2	no	—
173	#2151	—	83	2	no	—
174	#2112	—	83	1	no	—
175	C₂×C₂²⋊C₁₆	D₈₃	1	1	yes	characteristic
176	#2108	—	83	2	no	—
177	C₈₃	#33422	1	1	yes	characteristic, complemented, Sylow
178	C₂²×M₁₆	—	83	1	no	—
179	C₂³×C₈	C₂×D₈₃	1	1	yes	characteristic
180	C₂×#117	—	83	1	no	—
181	C₂×M₃₂	—	83	2	no	—
182	C₂²×C₁₆	C₂×D₈₃	1	2	yes	—
183	#405	—	83	1	no	—
184	#405	—	83	2	no	—
185	#405	—	83	1	no	—
186	#405	—	83	1	no	—
187	#405	—	83	1	no	—
188	#405	—	83	2	no	—
189	C₂×#97	—	83	2	no	—
190	C₂×C₄×C₈	—	83	2	no	—
191	C₄⋊C₁₆	—	83	4	no	—
192	C₄⋊C₁₆	—	83	4	no	—
193	C₂²⋊C₁₆	—	83	2	no	—
194	C₂²⋊C₁₆	—	83	2	no	—
195	C₂²⋊C₁₆	C₂×D₈₃	1	1	yes	characteristic
196	C₂²⋊C₁₆	C₂×D₈₃	1	1	yes	characteristic
197	C₂²⋊C₁₆	C₂×D₈₃	1	2	yes	—
198	#346	—	83	4	no	—
199	C₄×C₁₆	—	83	4	no	—
200	C₂³×C₄	C₄×D₈₃	1	1	yes	characteristic
201	C₂×M₁₆	—	83	1	no	—
202	C₂×M₁₆	—	83	2	no	—
203	C₂×M₁₆	—	83	1	no	—
204	C₂×M₁₆	—	166	4	no	—
205	C₂²×C₈	—	2	4	no	—
206	C₂²×C₈	—	83	2	no	—
207	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
208	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
209	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
210	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
211	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
212	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
213	#117	—	83	1	no	—
214	#117	—	83	1	no	—
215	#117	—	83	2	no	—
216	#117	—	83	2	no	—
217	#117	—	83	1	no	—
218	#117	—	83	1	no	—
219	C₂×C₄⋊C₄	—	83	1	no	—
220	C₂×C₄⋊C₄	—	83	1	no	—
221	C₂×C₂²⋊C₄	—	83	2	no	—
222	C₂×C₄²	—	83	2	no	—
223	M₃₂	—	166	4	no	—
224	C₂×C₁₆	—	2	4	no	—
225	C₂×C₁₆	—	83	4	no	—
226	C₂×C₁₆	C₂²×D₈₃	1	2	yes	—
227	C₂×C₁₆	C₂²×D₈₃	1	2	yes	—
228	#97	—	83	2	no	—
229	#97	—	83	2	no	—
230	#97	—	83	2	no	—
231	#97	—	83	2	no	—
232	C₄×C₈	—	83	2	no	—
233	C₄×C₈	—	83	2	no	—
234	C₄×C₈	—	83	2	no	—
235	C₄×C₈	—	83	2	no	—
236	C₂⁴	C₈×D₈₃	1	1	yes	characteristic
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	83	2	no	—
240	C₂²×C₄	—	83	2	no	—
241	C₂²×C₄	C₈×D₈₃	1	1	yes	characteristic
242	C₂²×C₄	C₈×D₈₃	1	1	yes	characteristic
243	C₂²×C₄	C₈×D₈₃	1	1	yes	characteristic
244	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
245	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
246	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
247	M₁₆	—	83	4	no	—
248	M₁₆	—	166	4	no	—
249	M₁₆	—	166	2	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	83	2	no	—
256	C₂×C₈	—	83	2	no	—
257	C₂×C₈	—	166	4	no	—
258	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
259	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
260	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
261	C₂×C₈	#13050589	1	4	yes	—
262	C₂×C₈	C₂³×D₈₃	1	1	yes	characteristic
263	C₄⋊C₄	—	83	2	no	—
264	C₄⋊C₄	—	83	2	no	—
265	C₄⋊C₄	—	83	1	no	—
266	C₄⋊C₄	—	83	1	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	C₂²⋊C₄	—	83	2	no	—
273	C₄²	—	83	2	no	—
274	C₄²	—	83	2	no	—
275	C₄²	—	83	2	no	—
276	C₄²	—	83	2	no	—
277	C₁₆	—	2	2	no	—
278	C₁₆	—	2	2	no	—
279	C₁₆	—	166	4	no	—
280	C₂³	—	2	2	no	—
281	C₂³	—	2	2	no	—
282	C₂³	—	4	1	no	—
283	C₂³	C₂×C₈×D₈₃	1	1	yes	characteristic
284	C₂³	C₂×C₈×D₈₃	1	1	yes	characteristic
285	C₂³	C₂×C₈×D₈₃	1	1	yes	characteristic
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	332	2	no	—
294	C₂×C₄	—	83	2	no	—
295	C₂×C₄	—	83	2	no	—
296	C₂×C₄	—	83	2	no	—
297	C₂×C₄	—	83	2	no	—
298	C₂×C₄	—	166	4	no	—
299	C₂×C₄	#14581748	1	2	yes	—
300	C₂×C₄	C₂×C₈×D₈₃	1	1	yes	characteristic
301	C₂×C₄	C₂×C₈×D₈₃	1	1	yes	characteristic
302	C₂×C₄	C₂×C₈×D₈₃	1	1	yes	characteristic
303	C₂×C₄	M₁₆×D₈₃	1	2	yes	—
304	C₂×C₄	C₂²×C₄×D₈₃	1	1	yes	characteristic
305	C₈	—	2	1	no	—
306	C₈	—	2	1	no	—
307	C₈	—	83	4	no	—
308	C₈	—	166	2	no	—
309	C₈	#14581748	1	2	yes	—
310	C₈	C₂×#13050589	1	2	yes	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	4	4	no	—
319	C₂²	#19127751	1	2	yes	—
320	C₂²	#19128117	1	2	yes	—
321	C₂²	C₂²×C₈×D₈₃	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	166	2	no	—
325	C₄	—	166	2	no	—
326	C₄	—	166	2	no	—
327	C₄	—	166	2	no	—
328	C₄	#19127751	1	2	yes	—
329	C₄	C₂×#14581748	1	1	yes	characteristic
330	C₄	C₂×M₁₆×D₈₃	1	1	yes	characteristic
331	C₂	—	2	2	no	—
332	C₂	—	2	2	no	—
333	C₂	—	4	1	no	—
334	C₂	C₂×#19127751	1	1	yes	characteristic
335	C₂	C₂×#19128117	1	1	yes	characteristic
336	C₂	#24920565	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₈₃
commutator, socle	C₂²×C₈₃	C₂³×C₈
Fitting	C₂×C₂²⋊C₁₆×C₈₃	C₂
prime radical	C₂⁴×C₈₃	C₂×C₈
cosocle	C₂×C₈×C₈₃	C₂⁴
hypercenter	C₂×C₂²⋊C₁₆	D₈₃
hypocenter	C₈₃	#33422
LCS	G ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₂²⋊C₁₆

Complements

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

N	H ≅ G/N	type
2. #24909267	C₂	semidirect
3. #24909263	C₂	semidirect
4. #24909259	C₂	semidirect
5. #24909221	C₂	semidirect
7. #24909059	C₂	semidirect
20. Dic₈₃⋊C₁₆	C₂²	semidirect
21. Dic₈₃⋊C₁₆	C₂²	semidirect
24. #19127476	C₂²	semidirect
25. C₁₆×Dic₈₃	C₂²	semidirect
177. C₈₃	#33422	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19127656	2
#24909267	2
#24909263	2
#24909259	2
#24909221	2
#24909074	2
#24909059	2
C₂×C₂²⋊C₁₆×C₈₃	2
#33422	83