Subgroups — #24916988

Order 10624.

458 conjugacy classes of subgroups.

The subgroup lattice has 372 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
2	C₂×Q₁₆×D₈₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#19128276	C₂	1	1	yes	maximal, characteristic
4	#19128177	C₂	1	1	yes	maximal, characteristic, complemented
5	#19127926	C₂	1	1	yes	maximal, characteristic, complemented
6	#19127925	C₂	1	1	yes	maximal, characteristic, complemented
7	#19127924	C₂	1	1	yes	maximal, characteristic, complemented
8	#19127923	C₂	1	1	yes	maximal, characteristic, complemented
9	#19127910	C₂	1	1	yes	maximal, characteristic, complemented
10	#19127909	C₂	1	1	yes	maximal, characteristic, complemented
11	#19127908	C₂	1	1	yes	maximal, characteristic, complemented
12	#19127726	C₂	1	1	yes	maximal, characteristic, complemented
13	#19127708	C₂	1	1	yes	maximal, characteristic, complemented
14	#19127696	C₂	1	1	yes	maximal, characteristic
15	#462×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13050591	C₂²	1	1	yes	characteristic
17	C₂×Q₈×D₈₃	C₂²	1	1	yes	characteristic
18	C₂×Q₈×D₈₃	C₂²	1	1	yes	characteristic
19	#14581824	—	2	4	no	—
20	C₂×Dic₃₃₂	C₂²	1	1	yes	characteristic
21	C₂×#13050580	C₂²	1	1	yes	characteristic
22	C₂×#13050580	C₂²	1	1	yes	characteristic
23	C₂×#13050580	C₂²	1	1	yes	characteristic
24	Q₁₆×D₈₃	—	2	4	no	—
25	C₂×#13050579	C₂²	1	1	yes	characteristic
26	C₂×#13050577	C₂²	1	1	yes	characteristic
27	C₂×#13050573	C₂²	1	1	yes	characteristic
28	C₂×C₈×D₈₃	C₂²	1	1	yes	characteristic
29	#14581790	C₂²	1	1	yes	characteristic
30	#14581781	C₂²	1	1	yes	characteristic
31	#14581780	C₂²	1	1	yes	characteristic
32	#14581777	—	2	1	no	—
33	#14581777	—	2	1	no	—
34	#14581770	—	2	1	no	—
35	#14581770	—	2	1	no	—
36	Q₈×Dic₈₃	C₂²	1	1	yes	characteristic
37	C₄×Dic₁₆₆	—	2	1	no	—
38	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
39	#14581757	—	2	1	no	—
40	#14581757	C₂²	1	1	yes	characteristic
41	#14581750	—	2	1	no	—
42	#14581750	C₂²	1	1	yes	characteristic
43	#14581749	—	2	1	no	—
44	#14581749	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	#14581716	C₂²	1	1	yes	characteristic
51	#14581715	C₂²	1	1	yes	characteristic
52	#14581714	C₂²	1	1	yes	characteristic
53	#14581712	C₂²	1	1	yes	characteristic
54	C₂×Q₁₆×C₈₃	C₂²	1	1	yes	characteristic
55	C₂×Q₁₆×C₈₃	C₂²	1	1	yes	characteristic
56	C₄⋊Q₈×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	#103×C₈₃	C₂²	1	1	yes	characteristic
61	#13050591	—	2	2	no	—
62	#13050591	—	2	2	no	—
63	C₂×Dic₁₆₆	—	2	1	no	—
64	C₂×Dic₁₆₆	—	2	1	no	—
65	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
66	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
67	Q₈×D₈₃	—	2	2	no	—
68	Q₈×D₈₃	—	2	2	no	—
69	Q₈×D₈₃	—	2	2	no	—
70	Q₈×D₈₃	D₄	1	4	yes	—
71	C₂×D₃₃₂	—	2	1	no	—
72	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
73	C₂×C₄×D₈₃	—	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	Dic₃₃₂	—	2	2	no	—
78	#13050580	—	2	2	no	—
79	#13050580	—	2	2	no	—
80	#13050580	—	2	2	no	—
81	#13050579	—	2	2	no	—
82	#13050577	—	2	2	no	—
83	#13050573	—	2	2	no	—
84	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
85	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
86	C₈×D₈₃	—	2	2	no	—
87	C₄⋊Dic₈₃	—	2	1	no	—
88	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
89	Dic₈₃⋊C₄	—	2	1	no	—
90	Dic₈₃⋊C₄	—	2	1	no	—
91	Dic₈₃⋊C₄	—	2	1	no	—
92	Dic₈₃⋊C₄	—	2	1	no	—
93	Dic₈₃⋊C₄	D₄	1	1	yes	characteristic
94	Dic₈₃⋊C₄	D₄	1	1	yes	characteristic
95	(C₂×D₈₃)⋊C₄	—	2	1	no	—
96	(C₂×D₈₃)⋊C₄	—	2	1	no	—
97	(C₂×D₈₃)⋊C₄	—	2	1	no	—
98	(C₂×D₈₃)⋊C₄	—	2	1	no	—
99	(C₂×D₈₃)⋊C₄	—	2	1	no	—
100	(C₂×D₈₃)⋊C₄	D₄	1	1	yes	characteristic
101	(C₂×D₈₃)⋊C₄	D₄	1	1	yes	characteristic
102	C₄×Dic₈₃	—	2	1	no	—
103	C₄×Dic₈₃	—	2	1	no	—
104	C₄×Dic₈₃	—	2	1	no	—
105	C₄×Dic₈₃	—	2	1	no	—
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×Q₈×C₈₃	C₂³	1	1	yes	characteristic
110	Q₁₆×C₈₃	—	2	2	no	—
111	Q₁₆×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	—	2	1	no	—
116	C₄⋊C₄×C₈₃	—	2	1	no	—
117	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
118	C₄²×C₈₃	—	2	1	no	—
119	C₄²×C₈₃	C₂³	1	1	yes	characteristic
120	C₂²×D₈₃	—	2	1	no	—
121	C₂²×D₈₃	C₂×D₄	1	1	yes	characteristic
122	Q₈×C₈₃	—	2	1	no	—
123	Q₈×C₈₃	—	2	1	no	—
124	Q₈×C₈₃	—	2	1	no	—
125	Q₈×C₈₃	—	2	1	no	—
126	Q₈×C₈₃	—	2	1	no	—
127	Q₈×C₈₃	C₂×D₄	1	2	yes	—
128	C₂×C₄×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	C₂×Dic₈₃	—	2	1	no	—
136	C₂×Dic₈₃	—	2	1	no	—
137	C₂×Dic₈₃	—	2	1	no	—
138	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
139	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
140	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
141	D₃₃₂	—	2	1	no	—
142	D₃₃₂	—	2	1	no	—
143	D₃₃₂	—	4	2	no	—
144	C₄×D₈₃	—	2	4	no	—
145	C₄×D₈₃	—	4	2	no	—
146	C₄×D₈₃	—	4	2	no	—
147	C₄×D₈₃	C₂×D₄	1	2	yes	—
148	C₄×D₈₃	#41	1	2	yes	—
149	Dic₁₆₆	—	2	1	no	—
150	Dic₁₆₆	—	2	4	no	—
151	Dic₁₆₆	—	2	1	no	—
152	Dic₁₆₆	—	2	1	no	—
153	Dic₁₆₆	—	4	2	no	—
154	Dic₁₆₆	C₂×D₄	1	2	yes	—
155	C₈×C₈₃	—	2	1	no	—
156	C₈×C₈₃	—	2	1	no	—
157	C₈₃⋊C₈	—	2	1	no	—
158	C₈₃⋊C₈	—	2	1	no	—
159	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
160	C₂×D₈₃	—	2	1	no	—
161	C₂×D₈₃	—	2	1	no	—
162	C₂×D₈₃	—	4	1	no	—
163	C₂×D₈₃	—	4	1	no	—
164	C₂×D₈₃	—	4	1	no	—
165	C₂×D₈₃	#121	1	2	yes	—
166	C₄×C₈₃	—	2	2	no	—
167	C₄×C₈₃	—	2	1	no	—
168	C₄×C₈₃	—	2	1	no	—
169	C₄×C₈₃	—	4	1	no	—
170	C₄×C₈₃	—	4	1	no	—
171	C₄×C₈₃	—	4	1	no	—
172	C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
173	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
174	Dic₈₃	—	2	2	no	—
175	Dic₈₃	—	2	1	no	—
176	Dic₈₃	—	2	1	no	—
177	Dic₈₃	—	4	1	no	—
178	Dic₈₃	—	4	1	no	—
179	Dic₈₃	#121	1	2	yes	—
180	C₂×C₈₃	C₂×#121	1	1	yes	characteristic
181	C₂×C₈₃	#578	1	1	yes	characteristic
182	C₂×C₈₃	#578	1	1	yes	characteristic
183	D₈₃	—	2	2	no	—
184	D₈₃	—	8	1	no	—
185	#3044	—	83	1	no	maximal, Sylow
186	C₈₃	#3044	1	1	yes	characteristic, complemented, Sylow
187	C₂×#137	—	83	1	no	—
188	C₂²×Q₁₆	—	83	1	no	—
189	#536	—	83	1	no	—
190	#519	—	83	1	no	—
191	#464	—	83	1	no	—
192	#464	—	83	1	no	—
193	#464	—	83	1	no	—
194	#464	—	83	1	no	—
195	#462	—	83	1	no	—
196	#462	—	83	1	no	—
197	#462	—	83	1	no	—
198	#462	D₈₃	1	1	yes	characteristic, complemented
199	#424	—	83	1	no	—
200	#419	—	83	1	no	—
201	#416	—	83	1	no	—
202	C₂×#41	—	83	1	no	—
203	C₂²×Q₈	—	83	1	no	—
204	C₂²×Q₈	—	83	1	no	—
205	#137	—	166	4	no	—
206	C₂×Q₁₆	—	83	1	no	—
207	C₂×Q₁₆	—	83	1	no	—
208	C₂×Q₁₆	—	83	1	no	—
209	C₂×Q₁₆	—	83	1	no	—
210	C₂×Q₁₆	—	166	4	no	—
211	C₂×Q₁₆	C₂×D₈₃	1	1	yes	characteristic
212	C₂×Q₁₆	C₂×D₈₃	1	1	yes	characteristic
213	C₂×SD₁₆	—	83	1	no	—
214	C₂×SD₁₆	—	83	1	no	—
215	C₂×M₁₆	—	83	1	no	—
216	C₂²×C₈	—	83	1	no	—
217	C₄⋊Q₈	—	83	1	no	—
218	C₄⋊Q₈	C₂×D₈₃	1	1	yes	characteristic
219	#124	—	83	1	no	—
220	#124	—	83	1	no	—
221	#123	—	166	1	no	—
222	#123	—	166	1	no	—
223	C₂²⋊Q₈	—	166	1	no	—
224	C₂²⋊Q₈	—	166	1	no	—
225	C₄×Q₈	—	83	1	no	—
226	C₄×Q₈	—	83	1	no	—
227	C₄×Q₈	—	83	1	no	—
228	C₄×Q₈	—	166	1	no	—
229	C₄×Q₈	—	166	1	no	—
230	C₄×Q₈	C₂×D₈₃	1	1	yes	characteristic
231	#117	—	83	1	no	—
232	#117	—	83	1	no	—
233	#117	—	166	1	no	—
234	#117	—	166	1	no	—
235	C₄⋊C₈	—	83	1	no	—
236	C₄⋊C₈	—	83	1	no	—
237	C₄⋊C₈	—	83	1	no	—
238	C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
239	#103	—	83	1	no	—
240	#103	—	83	1	no	—
241	#103	—	83	1	no	—
242	#103	—	83	1	no	—
243	#103	C₂×D₈₃	1	1	yes	characteristic
244	#103	C₂×D₈₃	1	1	yes	characteristic
245	#102	—	83	1	no	—
246	#102	—	83	1	no	—
247	#41	—	166	2	no	—
248	#41	—	166	2	no	—
249	C₂×Q₈	—	83	4	no	—
250	C₂×Q₈	—	83	1	no	—
251	C₂×Q₈	—	83	1	no	—
252	C₂×Q₈	—	166	1	no	—
253	C₂×Q₈	—	166	1	no	—
254	C₂×Q₈	—	166	2	no	—
255	C₂×Q₈	—	166	2	no	—
256	C₂×Q₈	—	166	2	no	—
257	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
258	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
259	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
260	C₂×D₄	—	83	1	no	—
261	C₂×D₄	—	166	1	no	—
262	C₂²×C₄	—	83	1	no	—
263	C₂²×C₄	—	166	1	no	—
264	C₂²×C₄	—	166	1	no	—
265	C₂²×C₄	—	166	1	no	—
266	Q₁₆	—	2	2	no	—
267	Q₁₆	—	2	2	no	—
268	Q₁₆	—	166	2	no	—
269	Q₁₆	—	166	2	no	—
270	Q₁₆	—	166	2	no	—
271	Q₁₆	—	166	2	no	—
272	SD₁₆	—	166	2	no	—
273	SD₁₆	—	166	2	no	—
274	M₁₆	—	166	2	no	—
275	C₂×C₈	—	83	1	no	—
276	C₂×C₈	—	83	1	no	—
277	C₂×C₈	—	166	2	no	—
278	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
279	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
280	C₄⋊C₄	—	2	1	no	—
281	C₄⋊C₄	—	2	1	no	—
282	C₄⋊C₄	—	2	1	no	—
283	C₄⋊C₄	—	83	1	no	—
284	C₄⋊C₄	—	83	1	no	—
285	C₄⋊C₄	—	83	1	no	—
286	C₄⋊C₄	—	166	1	no	—
287	C₄⋊C₄	—	166	1	no	—
288	C₄⋊C₄	—	166	1	no	—
289	C₄⋊C₄	—	166	1	no	—
290	C₄⋊C₄	—	166	1	no	—
291	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
292	C₂²⋊C₄	—	83	1	no	—
293	C₂²⋊C₄	—	83	1	no	—
294	C₂²⋊C₄	—	166	1	no	—
295	C₂²⋊C₄	—	166	1	no	—
296	C₂²⋊C₄	—	166	1	no	—
297	C₂²⋊C₄	—	166	1	no	—
298	C₂²⋊C₄	—	166	1	no	—
299	C₄²	—	2	1	no	—
300	C₄²	—	83	1	no	—
301	C₄²	—	166	1	no	—
302	C₄²	—	166	1	no	—
303	C₄²	—	166	1	no	—
304	C₄²	—	166	1	no	—
305	C₄²	C₂²×D₈₃	1	1	yes	characteristic
306	C₂³	—	83	1	no	—
307	C₂³	—	166	1	no	—
308	Q₈	—	2	1	no	—
309	Q₈	—	2	1	no	—
310	Q₈	—	2	1	no	—
311	Q₈	—	2	1	no	—
312	Q₈	—	2	1	no	—
313	Q₈	—	332	2	no	—
314	Q₈	—	83	2	no	—
315	Q₈	—	166	4	no	—
316	Q₈	—	166	1	no	—
317	Q₈	—	166	1	no	—
318	Q₈	—	166	1	no	—
319	Q₈	D₄×D₈₃	1	2	yes	—
320	D₄	—	332	2	no	—
321	D₄	—	166	1	no	—
322	D₄	—	166	1	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	2	1	no	—
326	C₂×C₄	—	2	1	no	—
327	C₂×C₄	—	332	2	no	—
328	C₂×C₄	—	332	2	no	—
329	C₂×C₄	—	83	1	no	—
330	C₂×C₄	—	83	2	no	—
331	C₂×C₄	—	83	2	no	—
332	C₂×C₄	—	83	1	no	—
333	C₂×C₄	—	83	1	no	—
334	C₂×C₄	—	166	4	no	—
335	C₂×C₄	—	166	1	no	—
336	C₂×C₄	—	166	1	no	—
337	C₂×C₄	—	166	1	no	—
338	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
339	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
340	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
341	C₈	—	2	1	no	—
342	C₈	—	2	1	no	—
343	C₈	—	166	1	no	—
344	C₈	—	166	1	no	—
345	C₂²	—	332	1	no	—
346	C₂²	—	332	1	no	—
347	C₂²	—	332	1	no	—
348	C₂²	—	83	2	no	—
349	C₂²	—	166	1	no	—
350	C₂²	—	166	1	no	—
351	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
352	C₄	—	2	2	no	—
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	4	1	no	—
356	C₄	—	4	1	no	—
357	C₄	—	4	1	no	—
358	C₄	—	332	1	no	—
359	C₄	—	332	1	no	—
360	C₄	—	83	2	no	—
361	C₄	—	166	2	no	—
362	C₄	—	166	1	no	—
363	C₄	—	166	1	no	—
364	C₄	C₂×D₄×D₈₃	1	1	yes	characteristic
365	C₄	#41×D₈₃	1	1	yes	characteristic
366	C₂	—	664	1	no	—
367	C₂	—	166	2	no	—
368	C₂	#121×D₈₃	1	1	yes	characteristic
369	C₂	#19128514	1	1	yes	characteristic
370	C₂	#19128517	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₃
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#462×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	C₂×#13050591	C₂²
hypercenter	#462	D₈₃
hypocenter	C₈₃	#3044
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #462

Complements

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

N	H ≅ G/N	type
2. C₂×Q₁₆×D₈₃	C₂	semidirect
4. #19128177	C₂	semidirect
5. #19127926	C₂	semidirect
6. #19127925	C₂	semidirect
7. #19127924	C₂	semidirect
8. #19127923	C₂	semidirect
9. #19127910	C₂	semidirect
10. #19127909	C₂	semidirect
11. #19127908	C₂	semidirect
12. #19127726	C₂	semidirect
13. #19127708	C₂	semidirect
15. #462×C₈₃	C₂	semidirect
186. C₈₃	#3044	semidirect
198. #462	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×Q₁₆×D₈₃	2
#19128276	2
#19128177	2
#19127926	2
#19127925	2
#19127924	2
#19127923	2
#19127910	2
#19127909	2
#19127908	2
#19127726	2
#19127708	2
#19127696	2
#462×C₈₃	2
#3044	83