Subgroups — #24916933

Order 10624.

484 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14581824	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14581821	C₂	1	1	yes	maximal, characteristic, complemented
3	#19128250	C₂	1	1	yes	maximal, characteristic
4	C₄×Q₈×D₈₃	C₂	1	1	yes	maximal, characteristic, complemented
5	#19127924	C₂	1	1	yes	maximal, characteristic, complemented
6	#19127923	C₂	1	1	yes	maximal, characteristic, complemented
7	#19127920	C₂	1	1	yes	maximal, characteristic, complemented
8	#19127910	C₂	1	1	yes	maximal, characteristic, complemented
9	#19127909	C₂	1	1	yes	maximal, characteristic, complemented
10	#19127900	C₂	1	1	yes	maximal, characteristic, complemented
11	#19127899	C₂	1	1	yes	maximal, characteristic, complemented
12	#19127723	C₂	1	1	yes	maximal, characteristic, complemented
13	#19127701	C₂	1	1	yes	maximal, characteristic, complemented
14	#19127699	C₂	1	1	yes	maximal, characteristic, complemented
15	#464×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13050591	C₂²	1	1	yes	characteristic
17	C₂×#13050589	C₂²	1	1	yes	characteristic
18	C₂×Q₈×D₈₃	C₂²	1	1	yes	characteristic
19	#14581824	—	2	4	no	—
20	#14581821	—	2	4	no	—
21	C₂×Dic₃₃₂	C₂²	1	1	yes	characteristic
22	C₂×#13050580	C₂²	1	1	yes	characteristic
23	C₂×#13050580	C₂²	1	1	yes	characteristic
24	C₂×#13050579	C₂²	1	1	yes	characteristic
25	C₂×#13050578	C₂²	1	1	yes	characteristic
26	C₂×#13050577	C₂²	1	1	yes	characteristic
27	C₂×#13050573	C₂²	1	1	yes	characteristic
28	C₂×#13050573	C₂²	1	1	yes	characteristic
29	#14581790	C₂²	1	1	yes	characteristic
30	#14581788	C₂²	1	1	yes	characteristic
31	#14581780	C₂²	1	1	yes	characteristic
32	#14581765	—	2	1	no	—
33	#14581765	—	2	1	no	—
34	Q₈×Dic₈₃	C₂²	1	1	yes	characteristic
35	C₄×Dic₁₆₆	—	2	1	no	—
36	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
37	#14581757	—	2	1	no	—
38	#14581757	C₂²	1	1	yes	characteristic
39	(C₂²⋊D₈₃)⋊C₄	—	2	1	no	—
40	(C₂²⋊D₈₃)⋊C₄	—	2	1	no	—
41	C₄⋊C₄×D₈₃	—	2	1	no	—
42	C₄⋊C₄×D₈₃	C₂²	1	1	yes	characteristic
43	C₄²×D₈₃	—	2	1	no	—
44	C₄²×D₈₃	C₂²	1	1	yes	characteristic
45	C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
46	Dic₈₃⋊C₈	C₂²	1	1	yes	characteristic
47	Dic₈₃⋊C₈	C₂²	1	1	yes	characteristic
48	#14581717	C₂²	1	1	yes	characteristic
49	#14581716	C₂²	1	1	yes	characteristic
50	#14581715	C₂²	1	1	yes	characteristic
51	#14581714	C₂²	1	1	yes	characteristic
52	#14581713	C₂²	1	1	yes	characteristic
53	#14581712	C₂²	1	1	yes	characteristic
54	C₂×Q₁₆×C₈₃	C₂²	1	1	yes	characteristic
55	C₂×SD₁₆×C₈₃	C₂²	1	1	yes	characteristic
56	#124×C₈₃	C₂²	1	1	yes	characteristic
57	C₄×Q₈×C₈₃	C₂²	1	1	yes	characteristic
58	C₄⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
59	#103×C₈₃	C₂²	1	1	yes	characteristic
60	#102×C₈₃	C₂²	1	1	yes	characteristic
61	#13050591	—	2	2	no	—
62	#13050591	—	2	2	no	—
63	#13050589	—	2	2	no	—
64	#13050589	—	2	2	no	—
65	C₂×Dic₁₆₆	—	2	1	no	—
66	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
67	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
68	Q₈×D₈₃	—	2	2	no	—
69	Q₈×D₈₃	D₄	1	4	yes	complemented
70	C₂×D₃₃₂	—	2	1	no	—
71	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
72	C₂×C₂²⋊D₈₃	—	2	1	no	—
73	C₂×C₂²⋊D₈₃	—	2	1	no	—
74	C₂×C₂²⋊D₈₃	—	2	1	no	—
75	C₂²×Dic₈₃	—	2	1	no	—
76	C₂×C₄×D₈₃	—	2	1	no	—
77	C₂×C₄×D₈₃	—	2	1	no	—
78	C₂×C₄×D₈₃	—	2	1	no	—
79	C₂×C₄×D₈₃	D₄	1	1	yes	characteristic
80	C₂×C₄×D₈₃	D₄	1	1	yes	characteristic
81	C₂×C₄×D₈₃	C₂³	1	1	yes	characteristic
82	Dic₃₃₂	—	2	2	no	—
83	#13050580	—	2	2	no	—
84	#13050580	—	2	2	no	—
85	#13050579	—	2	2	no	—
86	#13050578	—	2	2	no	—
87	#13050577	—	2	2	no	—
88	#13050573	—	2	2	no	—
89	#13050573	—	2	2	no	—
90	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
91	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
92	C₄⋊Dic₈₃	—	2	1	no	—
93	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
94	Dic₈₃⋊C₄	—	2	1	no	—
95	Dic₈₃⋊C₄	—	2	1	no	—
96	Dic₈₃⋊C₄	—	2	1	no	—
97	Dic₈₃⋊C₄	—	2	1	no	—
98	Dic₈₃⋊C₄	—	2	1	no	—
99	(C₂×D₈₃)⋊C₄	—	2	1	no	—
100	(C₂×D₈₃)⋊C₄	—	2	1	no	—
101	C₄×Dic₈₃	—	2	1	no	—
102	C₄×Dic₈₃	—	2	1	no	—
103	C₄×Dic₈₃	—	2	1	no	—
104	C₄×Dic₈₃	D₄	1	1	yes	characteristic
105	C₄×Dic₈₃	D₄	1	1	yes	characteristic
106	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
107	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
108	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
109	C₂×D₄×C₈₃	C₂³	1	1	yes	characteristic
110	Q₁₆×C₈₃	—	2	2	no	—
111	SD₁₆×C₈₃	—	2	2	no	—
112	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
113	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
114	C₄⋊C₄×C₈₃	—	2	1	no	—
115	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
116	C₂²⋊C₄×C₈₃	—	2	1	no	—
117	C₂²⋊C₄×C₈₃	—	2	1	no	—
118	C₄²×C₈₃	—	2	1	no	—
119	C₄²×C₈₃	C₂³	1	1	yes	characteristic
120	C₂³×C₈₃	—	2	1	no	—
121	C₂²×D₈₃	—	2	1	no	—
122	C₂²×D₈₃	C₂×D₄	1	1	yes	characteristic
123	Q₈×C₈₃	—	2	1	no	—
124	Q₈×C₈₃	—	2	1	no	—
125	Q₈×C₈₃	—	2	1	no	—
126	Q₈×C₈₃	C₂×D₄	1	2	yes	—
127	D₄×C₈₃	—	2	1	no	—
128	D₄×C₈₃	—	2	1	no	—
129	C₂×C₄×C₈₃	—	2	1	no	—
130	C₂×C₄×C₈₃	—	2	1	no	—
131	C₂×C₄×C₈₃	—	2	1	no	—
132	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
133	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
134	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
135	C₂²⋊D₈₃	—	4	2	no	—
136	C₂²⋊D₈₃	—	4	2	no	—
137	C₂²⋊D₈₃	—	4	2	no	—
138	C₂×Dic₈₃	—	2	1	no	—
139	C₂×Dic₈₃	—	2	1	no	—
140	C₂×Dic₈₃	—	2	1	no	—
141	C₂×Dic₈₃	—	4	2	no	—
142	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
143	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
144	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
145	D₃₃₂	—	2	1	no	—
146	D₃₃₂	—	2	1	no	—
147	D₃₃₂	—	4	2	no	—
148	C₄×D₈₃	—	2	4	no	—
149	C₄×D₈₃	—	2	2	no	—
150	C₄×D₈₃	—	4	1	no	—
151	C₄×D₈₃	—	4	2	no	—
152	C₄×D₈₃	—	4	2	no	—
153	C₄×D₈₃	C₂×D₄	1	2	yes	—
154	C₄×D₈₃	#41	1	2	yes	—
155	Dic₁₆₆	—	2	1	no	—
156	Dic₁₆₆	—	2	4	no	—
157	Dic₁₆₆	—	2	1	no	—
158	Dic₁₆₆	—	2	1	no	—
159	Dic₁₆₆	C₂×D₄	1	2	yes	—
160	C₈×C₈₃	—	2	1	no	—
161	C₈×C₈₃	—	2	1	no	—
162	C₈₃⋊C₈	—	2	1	no	—
163	C₈₃⋊C₈	—	2	1	no	—
164	C₂²×C₈₃	—	4	1	no	—
165	C₂²×C₈₃	—	4	1	no	—
166	C₂²×C₈₃	—	4	1	no	—
167	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
168	C₂×D₈₃	—	2	1	no	—
169	C₂×D₈₃	—	2	1	no	—
170	C₂×D₈₃	—	4	1	no	—
171	C₂×D₈₃	—	4	1	no	—
172	C₂×D₈₃	—	4	1	no	—
173	C₂×D₈₃	#121	1	2	yes	—
174	C₄×C₈₃	—	2	2	no	—
175	C₄×C₈₃	—	2	1	no	—
176	C₄×C₈₃	—	2	1	no	—
177	C₄×C₈₃	—	4	1	no	—
178	C₄×C₈₃	—	4	1	no	—
179	C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
180	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
181	Dic₈₃	—	2	2	no	—
182	Dic₈₃	—	2	1	no	—
183	Dic₈₃	—	4	1	no	—
184	Dic₈₃	—	4	1	no	—
185	Dic₈₃	#121	1	2	yes	—
186	Dic₈₃	#137	1	2	yes	complemented
187	C₂×C₈₃	—	8	1	no	—
188	C₂×C₈₃	C₂×#121	1	1	yes	characteristic
189	C₂×C₈₃	C₂×#137	1	1	yes	characteristic
190	C₂×C₈₃	#575	1	1	yes	characteristic
191	D₈₃	—	2	2	no	—
192	D₈₃	—	8	1	no	—
193	#3037	—	83	1	no	maximal, Sylow
194	C₈₃	#3037	1	1	yes	characteristic, complemented, Sylow
195	C₂×#137	—	83	1	no	—
196	C₂×#137	—	83	1	no	—
197	#532	—	83	1	no	—
198	C₂×C₄×Q₈	—	83	1	no	—
199	#464	—	83	1	no	—
200	#464	—	83	1	no	—
201	#464	—	83	1	no	—
202	#464	D₈₃	1	1	yes	characteristic, complemented
203	#462	—	83	1	no	—
204	#462	—	83	1	no	—
205	#460	—	83	1	no	—
206	#460	—	83	1	no	—
207	#423	—	83	1	no	—
208	#417	—	83	1	no	—
209	#417	—	83	1	no	—
210	C₂×#41	—	83	1	no	—
211	C₂×#41	—	83	1	no	—
212	C₂²×Q₈	—	83	1	no	—
213	#137	—	166	4	no	—
214	#137	—	166	4	no	—
215	C₂×Q₁₆	—	83	1	no	—
216	C₂×Q₁₆	—	83	1	no	—
217	C₂×Q₁₆	—	83	1	no	—
218	C₂×Q₁₆	C₂×D₈₃	1	1	yes	characteristic
219	C₂×SD₁₆	—	83	1	no	—
220	C₂×SD₁₆	—	83	1	no	—
221	C₂×SD₁₆	—	83	1	no	—
222	C₂×SD₁₆	C₂×D₈₃	1	1	yes	characteristic
223	C₂×M₁₆	—	83	1	no	—
224	C₂×M₁₆	—	83	1	no	—
225	C₄⋊Q₈	—	83	1	no	—
226	#127	—	83	1	no	—
227	#124	—	83	1	no	—
228	#124	C₂×D₈₃	1	1	yes	characteristic
229	#121	—	166	1	no	—
230	#121	—	166	1	no	—
231	C₄×Q₈	—	83	1	no	—
232	C₄×Q₈	—	83	1	no	—
233	C₄×Q₈	—	83	1	no	—
234	C₄×Q₈	—	166	1	no	—
235	C₄×Q₈	—	166	1	no	—
236	C₄×Q₈	C₂×D₈₃	1	1	yes	characteristic
237	C₄×D₄	—	166	1	no	—
238	C₄×D₄	—	166	1	no	—
239	C₂×C₄⋊C₄	—	83	1	no	—
240	C₂×C₄⋊C₄	—	166	1	no	—
241	C₂×C₄²	—	83	1	no	—
242	C₂×C₄²	—	166	1	no	—
243	C₄⋊C₈	—	83	1	no	—
244	C₄⋊C₈	—	83	1	no	—
245	C₄⋊C₈	—	83	1	no	—
246	C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
247	#103	—	83	1	no	—
248	#103	—	83	1	no	—
249	#103	—	83	1	no	—
250	#103	C₂×D₈₃	1	1	yes	characteristic
251	#102	—	83	1	no	—
252	#102	—	83	1	no	—
253	#102	—	83	1	no	—
254	#102	C₂×D₈₃	1	1	yes	characteristic
255	#41	—	166	2	no	—
256	#41	—	166	2	no	—
257	#41	—	166	2	no	—
258	#41	—	166	2	no	—
259	C₂×Q₈	—	83	4	no	—
260	C₂×Q₈	—	83	1	no	—
261	C₂×Q₈	—	83	1	no	—
262	C₂×Q₈	—	166	1	no	—
263	C₂×Q₈	—	166	2	no	—
264	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
265	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
266	C₂×D₄	—	83	1	no	—
267	C₂×D₄	—	166	1	no	—
268	C₂×D₄	—	166	1	no	—
269	C₂×D₄	—	166	1	no	—
270	C₂×D₄	—	166	1	no	—
271	C₂×D₄	C₂²×D₈₃	1	1	yes	characteristic
272	C₂²×C₄	—	83	1	no	—
273	C₂²×C₄	—	83	1	no	—
274	C₂²×C₄	—	83	1	no	—
275	C₂²×C₄	—	166	1	no	—
276	C₂²×C₄	—	166	1	no	—
277	C₂²×C₄	—	166	1	no	—
278	C₂²×C₄	—	166	1	no	—
279	Q₁₆	—	2	2	no	—
280	Q₁₆	—	166	2	no	—
281	Q₁₆	—	166	2	no	—
282	Q₁₆	—	166	2	no	—
283	SD₁₆	—	2	2	no	—
284	SD₁₆	—	166	2	no	—
285	SD₁₆	—	166	2	no	—
286	SD₁₆	—	166	2	no	—
287	M₁₆	—	166	2	no	—
288	M₁₆	—	166	2	no	—
289	C₂×C₈	—	83	1	no	—
290	C₂×C₈	—	83	1	no	—
291	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
292	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
293	C₄⋊C₄	—	2	1	no	—
294	C₄⋊C₄	—	83	1	no	—
295	C₄⋊C₄	—	166	1	no	—
296	C₄⋊C₄	—	166	1	no	—
297	C₄⋊C₄	—	166	1	no	—
298	C₄⋊C₄	—	166	1	no	—
299	C₄⋊C₄	—	166	1	no	—
300	C₄⋊C₄	—	166	1	no	—
301	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
302	C₂²⋊C₄	—	2	1	no	—
303	C₂²⋊C₄	—	2	1	no	—
304	C₂²⋊C₄	—	166	1	no	—
305	C₂²⋊C₄	—	166	1	no	—
306	C₄²	—	2	1	no	—
307	C₄²	—	83	1	no	—
308	C₄²	—	83	1	no	—
309	C₄²	—	83	1	no	—
310	C₄²	—	166	1	no	—
311	C₄²	—	166	1	no	—
312	C₄²	—	166	1	no	—
313	C₄²	C₂²×D₈₃	1	1	yes	characteristic
314	C₂³	—	2	1	no	—
315	C₂³	—	83	1	no	—
316	C₂³	—	166	1	no	—
317	Q₈	—	2	1	no	—
318	Q₈	—	2	1	no	—
319	Q₈	—	2	1	no	—
320	Q₈	—	83	2	no	—
321	Q₈	—	166	4	no	—
322	Q₈	—	166	1	no	—
323	Q₈	—	166	1	no	—
324	Q₈	—	166	1	no	—
325	Q₈	D₄×D₈₃	1	2	yes	—
326	D₄	—	2	1	no	—
327	D₄	—	2	1	no	—
328	D₄	—	332	2	no	—
329	D₄	—	332	2	no	—
330	D₄	—	332	2	no	—
331	D₄	—	332	2	no	—
332	D₄	—	166	1	no	—
333	D₄	—	166	1	no	—
334	C₂×C₄	—	2	1	no	—
335	C₂×C₄	—	2	1	no	—
336	C₂×C₄	—	2	1	no	—
337	C₂×C₄	—	332	1	no	—
338	C₂×C₄	—	332	2	no	—
339	C₂×C₄	—	332	2	no	—
340	C₂×C₄	—	332	2	no	—
341	C₂×C₄	—	83	1	no	—
342	C₂×C₄	—	83	1	no	—
343	C₂×C₄	—	83	2	no	—
344	C₂×C₄	—	83	2	no	—
345	C₂×C₄	—	83	1	no	—
346	C₂×C₄	—	166	2	no	—
347	C₂×C₄	—	166	4	no	—
348	C₂×C₄	—	166	1	no	—
349	C₂×C₄	—	166	1	no	—
350	C₂×C₄	—	166	1	no	—
351	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
352	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
353	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
354	C₈	—	2	1	no	—
355	C₈	—	2	1	no	—
356	C₈	—	166	1	no	—
357	C₈	—	166	1	no	—
358	C₂²	—	4	1	no	—
359	C₂²	—	4	1	no	—
360	C₂²	—	4	1	no	—
361	C₂²	—	332	1	no	—
362	C₂²	—	332	1	no	—
363	C₂²	—	332	1	no	—
364	C₂²	—	83	2	no	—
365	C₂²	—	166	1	no	—
366	C₂²	—	166	1	no	—
367	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
368	C₄	—	2	2	no	—
369	C₄	—	2	1	no	—
370	C₄	—	2	1	no	—
371	C₄	—	4	1	no	—
372	C₄	—	4	1	no	—
373	C₄	—	332	1	no	—
374	C₄	—	332	1	no	—
375	C₄	—	83	2	no	—
376	C₄	—	83	2	no	—
377	C₄	—	166	2	no	—
378	C₄	—	166	1	no	—
379	C₄	C₂×D₄×D₈₃	1	1	yes	characteristic
380	C₄	#41×D₈₃	1	1	yes	characteristic
381	C₂	—	8	1	no	—
382	C₂	—	664	1	no	—
383	C₂	—	166	2	no	—
384	C₂	#121×D₈₃	1	1	yes	characteristic
385	C₂	#137×D₈₃	1	1	yes	characteristic
386	C₂	#19128498	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₃
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#464×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	#19128250	C₂
hypercenter	#464	D₈₃
hypocenter	C₈₃	#3037
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #464

Complements

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

N	H ≅ G/N	type
1. C₂×#14581824	C₂	semidirect
2. C₂×#14581821	C₂	semidirect
4. C₄×Q₈×D₈₃	C₂	semidirect
5. #19127924	C₂	semidirect
6. #19127923	C₂	semidirect
7. #19127920	C₂	semidirect
8. #19127910	C₂	semidirect
9. #19127909	C₂	semidirect
10. #19127900	C₂	semidirect
11. #19127899	C₂	semidirect
12. #19127723	C₂	semidirect
13. #19127701	C₂	semidirect
14. #19127699	C₂	semidirect
15. #464×C₈₃	C₂	semidirect
69. Q₈×D₈₃	D₄	semidirect
186. Dic₈₃	#137	semidirect
194. C₈₃	#3037	semidirect
202. #464	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14581824	2
C₂×#14581821	2
#19128250	2
C₄×Q₈×D₈₃	2
#19127924	2
#19127923	2
#19127920	2
#19127910	2
#19127909	2
#19127900	2
#19127899	2
#19127723	2
#19127701	2
#19127699	2
#464×C₈₃	2
#3037	83