Subgroups — #24917416

Order 10624.

418 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128267	C₂	1	1	yes	maximal, characteristic
2	#19128181	C₂	1	1	yes	maximal, characteristic, complemented
3	#19128027	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128025	C₂	1	1	yes	maximal, characteristic, complemented
5	#19128024	C₂	1	1	yes	maximal, characteristic, complemented
6	#19128020	C₂	1	1	yes	maximal, characteristic, complemented
7	#19128018	C₂	1	1	yes	maximal, characteristic, complemented
8	#19128016	C₂	1	1	yes	maximal, characteristic, complemented
9	#19128012	C₂	1	1	yes	maximal, characteristic, complemented
10	#19127742	C₂	1	1	yes	maximal, characteristic, complemented
11	#19127736	C₂	1	1	yes	maximal, characteristic, complemented
12	#19127704	C₂	1	1	yes	maximal, characteristic, complemented
13	#102×D₈₃	C₂	1	1	yes	maximal, characteristic, complemented
14	#19127666	C₂	1	1	yes	maximal, characteristic, complemented
15	#482×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13050590	C₂²	1	1	yes	characteristic
17	C₂×D₄×D₈₃	C₂²	1	1	yes	characteristic
18	C₂×#13050573	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₈₃	C₂²	1	1	yes	characteristic
20	#14581788	—	2	1	no	—
21	#14581778	—	2	1	no	—
22	#14581767	C₂²	1	1	yes	characteristic
23	#14581766	C₂²	1	1	yes	characteristic
24	#14581763	C₂²	1	1	yes	characteristic
25	#14581762	—	2	1	no	—
26	#14581760	—	2	1	no	—
27	C₄×Dic₁₆₆	—	2	1	no	—
28	C₄×D₃₃₂	—	2	1	no	—
29	(C₂²⋊D₈₃)⋊C₄	—	2	1	no	—
30	(C₂²⋊D₈₃)⋊C₄	—	2	1	no	—
31	#14581750	C₂²	1	1	yes	characteristic
32	#14581750	C₂²	1	1	yes	characteristic
33	#14581750	C₂²	1	1	yes	characteristic, complemented
34	C₂×C₄⋊Dic₈₃	C₂²	1	1	yes	characteristic, complemented
35	C₄⋊C₄×D₈₃	C₂²	1	1	yes	characteristic
36	C₄⋊C₄×D₈₃	C₂²	1	1	yes	characteristic, complemented
37	#14581725	C₂²	1	1	yes	characteristic, complemented
38	#14581725	C₂²	1	1	yes	characteristic, complemented
39	#14581725	C₂²	1	1	yes	characteristic, complemented
40	#14581724	C₂²	1	1	yes	characteristic, complemented
41	#14581724	C₂²	1	1	yes	characteristic, complemented
42	#14581723	C₂²	1	1	yes	characteristic, complemented
43	#14581714	C₂²	1	1	yes	characteristic
44	#14581714	C₂²	1	1	yes	characteristic
45	#14581713	C₂²	1	1	yes	characteristic
46	#14581713	C₂²	1	1	yes	characteristic
47	#14581712	C₂²	1	1	yes	characteristic
48	#14581712	C₂²	1	1	yes	characteristic
49	C₂²⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic, complemented
50	(C₂×D₈₃)⋊C₈	C₂²	1	1	yes	characteristic, complemented
51	(C₂×D₈₃)⋊C₈	C₂²	1	1	yes	characteristic, complemented
52	#121×C₈₃	C₂²	1	1	yes	characteristic
53	#117×C₈₃	C₂²	1	1	yes	characteristic, complemented
54	#107×C₈₃	C₂²	1	1	yes	characteristic, complemented
55	#106×C₈₃	C₂²	1	1	yes	characteristic, complemented
56	#102×C₈₃	C₂²	1	1	yes	characteristic
57	#102×C₈₃	C₂²	1	1	yes	characteristic
58	C₂²⋊C₈×C₈₃	C₂²	1	1	yes	characteristic, complemented
59	C₂³×D₈₃	—	2	1	no	—
60	#13050590	—	2	2	no	—
61	#13050590	—	2	2	no	—
62	C₂×Dic₁₆₆	D₄	1	1	yes	characteristic
63	C₂×D₃₃₂	—	2	1	no	—
64	C₂×D₃₃₂	D₄	1	1	yes	characteristic
65	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
66	C₂×C₂²⋊D₈₃	—	2	1	no	—
67	C₂×C₂²⋊D₈₃	—	2	1	no	—
68	C₂×C₂²⋊D₈₃	—	2	1	no	—
69	C₂×C₂²⋊D₈₃	D₄	1	1	yes	characteristic
70	C₂×C₂²⋊D₈₃	D₄	1	1	yes	characteristic
71	D₄×D₈₃	—	2	2	no	—
72	D₄×D₈₃	—	2	2	no	—
73	C₂²×Dic₈₃	—	2	1	no	—
74	C₂×C₄×D₈₃	—	2	1	no	—
75	C₂×C₄×D₈₃	—	2	1	no	—
76	C₂×C₄×D₈₃	C₂³	1	1	yes	characteristic
77	C₂×C₄×D₈₃	C₂³	1	1	yes	characteristic
78	#13050573	—	2	2	no	—
79	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
80	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
81	C₈×D₈₃	—	2	2	no	—
82	C₄⋊Dic₈₃	—	2	1	no	—
83	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
84	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
85	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
86	Dic₈₃⋊C₄	—	2	1	no	—
87	Dic₈₃⋊C₄	—	2	1	no	—
88	C₂²⋊Dic₈₃	—	2	1	no	—
89	(C₂×D₈₃)⋊C₄	—	2	1	no	—
90	(C₂×D₈₃)⋊C₄	—	2	1	no	—
91	(C₂×D₈₃)⋊C₄	—	2	1	no	—
92	(C₂×D₈₃)⋊C₄	—	2	1	no	—
93	(C₂×D₈₃)⋊C₄	—	2	1	no	—
94	C₄×Dic₈₃	—	2	1	no	—
95	C₄×Dic₈₃	—	2	1	no	—
96	C₄×Dic₈₃	—	2	1	no	—
97	C₂×D₄×C₈₃	—	2	1	no	—
98	C₂×D₄×C₈₃	C₂³	1	1	yes	characteristic
99	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
100	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
101	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
103	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
104	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
105	C₂²⋊C₄×C₈₃	—	2	1	no	—
106	C₂²⋊C₄×C₈₃	—	2	1	no	—
107	C₄²×C₈₃	—	2	1	no	—
108	C₂³×C₈₃	—	2	1	no	—
109	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
110	C₂²×D₈₃	—	2	1	no	—
111	C₂²×D₈₃	—	4	2	no	—
112	C₂²×D₈₃	—	8	1	no	—
113	C₂²×D₈₃	—	8	1	no	—
114	C₂²×D₈₃	C₂×D₄	1	1	yes	characteristic
115	C₂²×D₈₃	C₂×D₄	1	1	yes	characteristic
116	D₄×C₈₃	—	2	1	no	—
117	D₄×C₈₃	—	2	1	no	—
118	D₄×C₈₃	—	8	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	1	no	—
123	C₂×C₄×C₈₃	—	2	1	no	—
124	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
126	C₂²⋊D₈₃	—	2	2	no	—
127	C₂²⋊D₈₃	—	2	2	no	—
128	C₂²⋊D₈₃	—	4	2	no	—
129	C₂²⋊D₈₃	—	8	1	no	—
130	C₂²⋊D₈₃	—	8	1	no	—
131	C₂×Dic₈₃	—	2	1	no	—
132	C₂×Dic₈₃	—	2	1	no	—
133	C₂×Dic₈₃	—	2	1	no	—
134	C₂×Dic₈₃	—	8	1	no	—
135	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
136	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
137	D₃₃₂	—	2	2	no	—
138	D₃₃₂	—	2	1	no	—
139	D₃₃₂	—	2	1	no	—
140	D₃₃₂	—	8	1	no	—
141	C₄×D₈₃	—	2	1	no	—
142	C₄×D₈₃	—	2	1	no	—
143	C₄×D₈₃	—	8	1	no	—
144	C₄×D₈₃	—	8	1	no	—
145	C₄×D₈₃	#41	1	2	yes	—
146	C₄×D₈₃	#41	1	2	yes	—
147	Dic₁₆₆	—	2	2	no	—
148	C₈×C₈₃	—	2	1	no	—
149	C₈×C₈₃	—	2	1	no	—
150	C₈₃⋊C₈	—	2	1	no	—
151	C₈₃⋊C₈	—	2	1	no	—
152	C₂²×C₈₃	—	2	1	no	—
153	C₂²×C₈₃	—	2	1	no	—
154	C₂²×C₈₃	—	2	1	no	—
155	C₂²×C₈₃	—	4	1	no	—
156	C₂²×C₈₃	—	4	1	no	—
157	C₂²×C₈₃	—	4	1	no	—
158	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
159	C₂×D₈₃	—	2	1	no	—
160	C₂×D₈₃	—	2	1	no	—
161	C₂×D₈₃	—	2	1	no	—
162	C₂×D₈₃	—	2	1	no	—
163	C₂×D₈₃	—	2	1	no	—
164	C₂×D₈₃	—	4	1	no	—
165	C₂×D₈₃	—	4	1	no	—
166	C₂×D₈₃	—	4	1	no	—
167	C₂×D₈₃	—	8	4	no	—
168	C₂×D₈₃	#123	1	2	yes	—
169	C₄×C₈₃	—	2	1	no	—
170	C₄×C₈₃	—	4	1	no	—
171	C₄×C₈₃	—	4	1	no	—
172	C₄×C₈₃	—	4	1	no	—
173	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
174	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
175	Dic₈₃	—	2	1	no	—
176	Dic₈₃	—	4	1	no	—
177	Dic₈₃	—	4	1	no	—
178	Dic₈₃	—	4	1	no	—
179	Dic₈₃	#123	1	2	yes	—
180	C₂×C₈₃	—	4	1	no	—
181	C₂×C₈₃	—	8	1	no	—
182	C₂×C₈₃	C₂×#123	1	1	yes	characteristic
183	C₂×C₈₃	#576	1	1	yes	characteristic
184	C₂×C₈₃	#577	1	1	yes	characteristic
185	D₈₃	—	2	2	no	—
186	D₈₃	—	4	1	no	—
187	D₈₃	—	8	1	no	—
188	#3096	—	83	1	no	maximal, Sylow
189	C₈₃	#3096	1	1	yes	characteristic, complemented, Sylow
190	#535	—	83	1	no	—
191	#520	—	83	1	no	—
192	#482	—	83	1	no	—
193	#482	—	83	1	no	—
194	#482	—	83	1	no	—
195	#482	D₈₃	1	1	yes	characteristic, complemented
196	#481	—	83	1	no	—
197	#481	—	83	1	no	—
198	#480	—	83	1	no	—
199	#480	—	83	1	no	—
200	#428	—	83	1	no	—
201	#427	—	83	1	no	—
202	#418	—	83	1	no	—
203	C₂×#102	—	83	1	no	—
204	#408	—	83	1	no	—
205	C₂×#41	—	83	1	no	—
206	C₂²×D₄	—	83	1	no	—
207	C₂×M₁₆	—	83	1	no	—
208	C₂²×C₈	—	83	1	no	—
209	#127	—	166	1	no	—
210	#124	—	166	1	no	—
211	#121	—	83	1	no	—
212	#121	—	83	1	no	—
213	#121	—	83	1	no	—
214	#121	C₂×D₈₃	1	1	yes	characteristic
215	#120	—	166	1	no	—
216	#120	—	166	1	no	—
217	C₄×Q₈	—	166	1	no	—
218	C₄×D₄	—	166	1	no	—
219	C₄×D₄	—	166	1	no	—
220	C₄×D₄	—	166	1	no	—
221	#117	—	83	1	no	—
222	#117	—	83	1	no	—
223	#117	—	83	1	no	—
224	#117	C₂×D₈₃	1	1	yes	characteristic, complemented
225	C₂×C₄⋊C₄	—	83	1	no	—
226	C₂×C₄⋊C₄	—	83	1	no	—
227	C₂×C₄⋊C₄	—	83	1	no	—
228	#107	—	83	1	no	—
229	#107	—	83	1	no	—
230	#107	—	83	1	no	—
231	#107	C₂×D₈₃	1	1	yes	characteristic, complemented
232	#106	—	83	1	no	—
233	#106	—	83	1	no	—
234	#106	—	83	1	no	—
235	#106	C₂×D₈₃	1	1	yes	characteristic, complemented
236	#102	—	83	1	no	—
237	#102	—	83	1	no	—
238	#102	—	83	1	no	—
239	#102	—	83	1	no	—
240	#102	—	83	1	no	—
241	#102	—	83	1	no	—
242	#102	C₂×D₈₃	1	1	yes	characteristic
243	#102	C₂×D₈₃	1	1	yes	characteristic
244	C₂²⋊C₈	—	83	1	no	—
245	C₂²⋊C₈	—	83	1	no	—
246	C₂²⋊C₈	—	83	1	no	—
247	C₂²⋊C₈	C₂×D₈₃	1	1	yes	characteristic, complemented
248	C₂⁴	—	166	1	no	—
249	#41	—	166	2	no	—
250	#41	—	166	2	no	—
251	C₂×Q₈	—	83	1	no	—
252	C₂×D₄	—	2	1	no	—
253	C₂×D₄	—	83	1	no	—
254	C₂×D₄	—	83	1	no	—
255	C₂×D₄	—	83	1	no	—
256	C₂×D₄	—	83	1	no	—
257	C₂×D₄	—	166	1	no	—
258	C₂×D₄	—	166	2	no	—
259	C₂×D₄	—	166	2	no	—
260	C₂×D₄	—	166	1	no	—
261	C₂×D₄	—	166	1	no	—
262	C₂×D₄	—	166	1	no	—
263	C₂×D₄	C₂²×D₈₃	1	1	yes	characteristic
264	C₂²×C₄	—	83	1	no	—
265	C₂²×C₄	—	83	1	no	—
266	C₂²×C₄	—	166	1	no	—
267	C₂²×C₄	—	166	1	no	—
268	C₂²×C₄	—	166	1	no	—
269	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
270	M₁₆	—	166	2	no	—
271	C₂×C₈	—	83	1	no	—
272	C₂×C₈	—	83	1	no	—
273	C₂×C₈	—	166	2	no	—
274	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
275	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
276	C₄⋊C₄	—	83	1	no	—
277	C₄⋊C₄	—	83	1	no	—
278	C₄⋊C₄	—	83	1	no	—
279	C₄⋊C₄	—	166	1	no	—
280	C₄⋊C₄	—	166	1	no	—
281	C₄⋊C₄	—	166	1	no	—
282	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
283	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
284	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
285	C₂²⋊C₄	—	2	1	no	—
286	C₂²⋊C₄	—	2	1	no	—
287	C₂²⋊C₄	—	166	1	no	—
288	C₂²⋊C₄	—	166	1	no	—
289	C₂²⋊C₄	—	166	1	no	—
290	C₂²⋊C₄	—	166	1	no	—
291	C₂²⋊C₄	—	166	1	no	—
292	C₂²⋊C₄	—	166	1	no	—
293	C₄²	—	2	1	no	—
294	C₄²	—	166	1	no	—
295	C₄²	—	166	1	no	—
296	C₄²	—	166	1	no	—
297	C₂³	—	2	1	no	—
298	C₂³	—	332	2	no	—
299	C₂³	—	83	1	no	—
300	C₂³	—	83	1	no	—
301	C₂³	—	664	1	no	—
302	C₂³	—	664	1	no	—
303	C₂³	—	166	1	no	—
304	C₂³	D₄×D₈₃	1	1	yes	characteristic
305	Q₈	—	166	2	no	—
306	D₄	—	2	1	no	—
307	D₄	—	2	1	no	—
308	D₄	—	8	1	no	—
309	D₄	—	332	2	no	—
310	D₄	—	664	1	no	—
311	D₄	—	664	1	no	—
312	D₄	—	664	1	no	—
313	D₄	—	166	2	no	—
314	D₄	—	166	2	no	—
315	D₄	—	166	2	no	—
316	D₄	—	166	1	no	—
317	D₄	—	166	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	83	1	no	—
324	C₂×C₄	—	83	2	no	—
325	C₂×C₄	—	83	2	no	—
326	C₂×C₄	—	83	1	no	—
327	C₂×C₄	—	664	1	no	—
328	C₂×C₄	—	664	1	no	—
329	C₂×C₄	—	664	1	no	—
330	C₂×C₄	—	166	1	no	—
331	C₂×C₄	—	166	1	no	—
332	C₂×C₄	—	166	1	no	—
333	C₂×C₄	—	166	1	no	—
334	C₂×C₄	—	166	1	no	—
335	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
336	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
337	C₈	—	2	1	no	—
338	C₈	—	2	1	no	—
339	C₈	—	166	1	no	—
340	C₈	—	166	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	4	1	no	—
345	C₂²	—	4	1	no	—
346	C₂²	—	4	1	no	—
347	C₂²	—	332	1	no	—
348	C₂²	—	332	1	no	—
349	C₂²	—	332	1	no	—
350	C₂²	—	83	2	no	—
351	C₂²	—	664	4	no	—
352	C₂²	—	166	1	no	—
353	C₂²	—	166	1	no	—
354	C₂²	—	166	1	no	—
355	C₂²	—	166	1	no	—
356	C₂²	—	166	1	no	—
357	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
358	C₄	—	2	1	no	—
359	C₄	—	4	1	no	—
360	C₄	—	4	1	no	—
361	C₄	—	4	1	no	—
362	C₄	—	332	1	no	—
363	C₄	—	332	1	no	—
364	C₄	—	332	1	no	—
365	C₄	—	83	2	no	—
366	C₄	—	166	1	no	—
367	C₄	#41×D₈₃	1	1	yes	characteristic
368	C₄	#41×D₈₃	1	1	yes	characteristic
369	C₂	—	4	1	no	—
370	C₂	—	8	1	no	—
371	C₂	—	332	1	no	—
372	C₂	—	664	1	no	—
373	C₂	—	166	2	no	—
374	C₂	#123×D₈₃	1	1	yes	characteristic
375	C₂	#19128505	1	1	yes	characteristic
376	C₂	#19128508	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₃
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#482×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	#19128267	C₂
hypercenter	#482	D₈₃
hypocenter	C₈₃	#3096
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #482

Complements

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

N	H ≅ G/N	type
2. #19128181	C₂	semidirect
3. #19128027	C₂	semidirect
4. #19128025	C₂	semidirect
5. #19128024	C₂	semidirect
6. #19128020	C₂	semidirect
7. #19128018	C₂	semidirect
8. #19128016	C₂	semidirect
9. #19128012	C₂	semidirect
10. #19127742	C₂	semidirect
11. #19127736	C₂	semidirect
12. #19127704	C₂	semidirect
13. #102×D₈₃	C₂	semidirect
14. #19127666	C₂	semidirect
15. #482×C₈₃	C₂	semidirect
33. #14581750	C₂²	semidirect
34. C₂×C₄⋊Dic₈₃	C₂²	semidirect
36. C₄⋊C₄×D₈₃	C₂²	semidirect
37. #14581725	C₂²	semidirect
38. #14581725	C₂²	semidirect
39. #14581725	C₂²	semidirect
40. #14581724	C₂²	semidirect
41. #14581724	C₂²	semidirect
42. #14581723	C₂²	semidirect
49. C₂²⋊(C₈₃⋊C₈)	C₂²	semidirect
50. (C₂×D₈₃)⋊C₈	C₂²	semidirect
51. (C₂×D₈₃)⋊C₈	C₂²	semidirect
53. #117×C₈₃	C₂²	semidirect
54. #107×C₈₃	C₂²	semidirect
55. #106×C₈₃	C₂²	semidirect
58. C₂²⋊C₈×C₈₃	C₂²	semidirect
189. C₈₃	#3096	semidirect
195. #482	D₈₃	semidirect
224. #117	C₂×D₈₃	semidirect
231. #107	C₂×D₈₃	semidirect
235. #106	C₂×D₈₃	semidirect
247. C₂²⋊C₈	C₂×D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128267	2
#19128181	2
#19128027	2
#19128025	2
#19128024	2
#19128020	2
#19128018	2
#19128016	2
#19128012	2
#19127742	2
#19127736	2
#19127704	2
#102×D₈₃	2
#19127666	2
#482×C₈₃	2
#3096	83