Subgroups — #24919674

Order 10624.

402 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	#19128393	C₂	1	1	yes	maximal, characteristic, complemented
2	#19128348	C₂	1	1	yes	maximal, characteristic
3	#19128054	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128044	C₂	1	1	yes	maximal, characteristic, complemented
5	#19128017	C₂	1	1	yes	maximal, characteristic, complemented
6	#19128004	C₂	1	1	yes	maximal, characteristic, complemented
7	#19127958	C₂	1	1	yes	maximal, characteristic, complemented
8	#19127921	C₂	1	1	yes	maximal, characteristic, complemented
9	#19127861	C₂	1	1	yes	maximal, characteristic, complemented
10	#19127850	C₂	1	1	yes	maximal, characteristic, complemented
11	#19127786	C₂	1	1	yes	maximal, characteristic, complemented
12	#19127777	C₂	1	1	yes	maximal, characteristic, complemented
13	#19127704	C₂	1	1	yes	maximal, characteristic, complemented
14	#19127703	C₂	1	1	yes	maximal, characteristic, complemented
15	#468×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×#13050580	C₂²	1	1	yes	characteristic, complemented
19	C₂×#13050577	C₂²	1	1	yes	characteristic
20	C₂×#13050573	C₂²	1	1	yes	characteristic
21	#14581787	—	2	1	no	—
22	#14581783	—	2	1	no	—
23	#14581783	C₂²	1	1	yes	characteristic, complemented
24	#14581779	C₂²	1	1	yes	characteristic
25	#14581776	—	2	1	no	—
26	#14581774	—	2	1	no	—
27	#14581772	C₂²	1	1	yes	characteristic
28	#14581768	C₂²	1	1	yes	characteristic
29	#14581763	—	2	1	no	—
30	#14581760	—	2	1	no	—
31	Q₈×Dic₈₃	C₂²	1	1	yes	characteristic, complemented
32	#14581754	C₂²	1	1	yes	characteristic
33	C₄×C₂²⋊D₈₃	C₂²	1	1	yes	characteristic
34	#14581750	C₂²	1	1	yes	characteristic
35	#14581748	—	2	1	no	—
36	C₂×C₄⋊Dic₈₃	C₂²	1	1	yes	characteristic, complemented
37	C₂²⋊C₄×D₈₃	—	2	1	no	—
38	#14581723	C₂²	1	1	yes	characteristic, complemented
39	#14581723	C₂²	1	1	yes	characteristic, complemented
40	Dic₈₃⋊C₈	C₂²	1	1	yes	characteristic, complemented
41	#14581717	C₂²	1	1	yes	characteristic, complemented
42	#14581717	C₂²	1	1	yes	characteristic, complemented
43	#14581715	C₂²	1	1	yes	characteristic, complemented
44	#14581714	C₂²	1	1	yes	characteristic
45	#14581714	C₂²	1	1	yes	characteristic
46	#14581713	C₂²	1	1	yes	characteristic
47	#14581713	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
51	C₈×Dic₈₃	C₂²	1	1	yes	characteristic, complemented
52	C₂×SD₁₆×C₈₃	C₂²	1	1	yes	characteristic
53	C₂×M₁₆×C₈₃	C₂²	1	1	yes	characteristic, complemented
54	C₂²⋊Q₈×C₈₃	C₂²	1	1	yes	characteristic, complemented
55	#121×C₈₃	C₂²	1	1	yes	characteristic
56	#107×C₈₃	C₂²	1	1	yes	characteristic, complemented
57	#103×C₈₃	C₂²	1	1	yes	characteristic, complemented
58	#102×C₈₃	C₂²	1	1	yes	characteristic
59	C₂³×D₈₃	—	2	1	no	—
60	#13050590	—	2	2	no	—
61	#13050590	—	4	1	no	—
62	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
63	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
64	C₂×C₂²⋊D₈₃	—	2	1	no	—
65	C₂×C₂²⋊D₈₃	—	2	1	no	—
66	C₂×C₂²⋊D₈₃	D₄	1	1	yes	characteristic
67	D₄×D₈₃	—	2	2	no	—
68	D₄×D₈₃	—	4	1	no	—
69	C₂²×Dic₈₃	—	2	1	no	—
70	C₂×C₄×D₈₃	—	2	1	no	—
71	C₂×C₄×D₈₃	—	2	1	no	—
72	C₂×C₄×D₈₃	C₂³	1	1	yes	characteristic
73	#13050580	—	4	1	no	—
74	#13050577	—	2	2	no	—
75	#13050573	—	4	1	no	—
76	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
77	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
78	C₄⋊Dic₈₃	—	2	1	no	—
79	C₄⋊Dic₈₃	—	2	1	no	—
80	C₄⋊Dic₈₃	—	2	1	no	—
81	C₄⋊Dic₈₃	—	2	1	no	—
82	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
83	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
84	Dic₈₃⋊C₄	—	2	1	no	—
85	Dic₈₃⋊C₄	—	2	1	no	—
86	Dic₈₃⋊C₄	D₄	1	1	yes	characteristic
87	C₂²⋊Dic₈₃	—	2	1	no	—
88	C₂²⋊Dic₈₃	—	2	1	no	—
89	C₂²⋊Dic₈₃	D₄	1	1	yes	characteristic
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	(C₂×D₈₃)⋊C₄	D₄	1	1	yes	characteristic
95	C₄×Dic₈₃	—	2	1	no	—
96	C₄×Dic₈₃	—	2	1	no	—
97	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
98	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
99	C₂×D₄×C₈₃	—	2	1	no	—
100	C₂×D₄×C₈₃	C₂³	1	1	yes	characteristic
101	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
102	SD₁₆×C₈₃	—	4	1	no	—
103	M₁₆×C₈₃	—	4	1	no	—
104	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
105	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
106	C₄⋊C₄×C₈₃	—	2	1	no	—
107	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
108	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
109	C₂²⋊C₄×C₈₃	—	2	1	no	—
110	C₂²⋊C₄×C₈₃	—	2	1	no	—
111	C₂³×C₈₃	—	2	1	no	—
112	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
113	C₂²×D₈₃	—	2	1	no	—
114	C₂²×D₈₃	—	4	2	no	—
115	C₂²×D₈₃	—	4	2	no	—
116	C₂²×D₈₃	—	8	1	no	—
117	C₂²×D₈₃	C₂×D₄	1	1	yes	characteristic
118	Q₈×C₈₃	—	2	1	no	—
119	Q₈×C₈₃	—	2	1	no	—
120	D₄×C₈₃	—	2	1	no	—
121	D₄×C₈₃	—	2	1	no	—
122	D₄×C₈₃	—	8	1	no	—
123	C₂×C₄×C₈₃	—	2	1	no	—
124	C₂×C₄×C₈₃	—	2	1	no	—
125	C₂×C₄×C₈₃	—	2	1	no	—
126	C₂×C₄×C₈₃	—	2	1	no	—
127	C₂×C₄×C₈₃	—	2	1	no	—
128	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
129	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
130	C₂²⋊D₈₃	—	4	1	no	—
131	C₂²⋊D₈₃	—	4	2	no	—
132	C₂²⋊D₈₃	—	4	2	no	—
133	C₂×Dic₈₃	—	2	1	no	—
134	C₂×Dic₈₃	—	2	1	no	—
135	C₂×Dic₈₃	—	2	1	no	—
136	C₂×Dic₈₃	—	4	2	no	—
137	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
138	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
139	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
140	D₃₃₂	—	2	1	no	—
141	D₃₃₂	#41	1	2	yes	—
142	C₄×D₈₃	—	2	1	no	—
143	C₄×D₈₃	—	2	1	no	—
144	C₄×D₈₃	—	4	2	no	—
145	C₄×D₈₃	—	8	1	no	—
146	Dic₁₆₆	—	2	1	no	—
147	Dic₁₆₆	—	2	1	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	2	no	—
161	C₂×D₈₃	—	2	1	no	—
162	C₂×D₈₃	—	2	1	no	—
163	C₂×D₈₃	—	4	1	no	—
164	C₂×D₈₃	—	4	1	no	—
165	C₂×D₈₃	—	8	4	no	—
166	C₄×C₈₃	—	2	1	no	—
167	C₄×C₈₃	—	4	1	no	—
168	C₄×C₈₃	—	4	1	no	—
169	C₄×C₈₃	—	4	1	no	—
170	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
171	C₄×C₈₃	H(5,𝔽₂)	1	1	yes	characteristic
172	Dic₈₃	—	2	1	no	—
173	Dic₈₃	—	2	1	no	—
174	Dic₈₃	—	2	1	no	—
175	Dic₈₃	—	4	1	no	—
176	Dic₈₃	—	4	1	no	—
177	Dic₈₃	—	4	1	no	—
178	C₂×C₈₃	—	4	1	no	—
179	C₂×C₈₃	—	8	1	no	—
180	C₂×C₈₃	#546	1	1	yes	characteristic
181	C₂×C₈₃	#575	1	1	yes	characteristic
182	C₂×C₈₃	#577	1	1	yes	characteristic
183	D₈₃	—	4	1	no	—
184	D₈₃	—	4	2	no	—
185	#3323	—	83	1	no	maximal, Sylow
186	C₈₃	#3323	1	1	yes	characteristic, complemented, Sylow
187	#552	—	83	1	no	—
188	#546	—	83	1	no	—
189	#488	—	83	1	no	—
190	#485	—	83	1	no	—
191	#481	—	83	1	no	—
192	#478	—	83	1	no	—
193	#470	—	83	1	no	—
194	#468	D₈₃	1	1	yes	characteristic, complemented
195	#464	—	83	1	no	—
196	#452	—	83	1	no	—
197	#450	—	83	1	no	—
198	C₄×SD₁₆	—	83	1	no	—
199	#436	—	83	1	no	—
200	#418	—	83	1	no	—
201	#417	—	83	1	no	—
202	C₂×#41	—	83	1	no	—
203	C₂²×D₄	—	83	1	no	—
204	C₂×Q₁₆	—	83	1	no	—
205	C₂×SD₁₆	—	83	1	no	—
206	C₂×SD₁₆	C₂×D₈₃	1	1	yes	characteristic
207	C₂×M₁₆	—	83	1	no	—
208	C₂×M₁₆	C₂×D₈₃	1	1	yes	characteristic, complemented
209	#126	—	166	1	no	—
210	#125	—	83	1	no	—
211	#125	—	166	1	no	—
212	#124	—	83	1	no	—
213	#123	—	166	1	no	—
214	#123	—	166	1	no	—
215	C₂²⋊Q₈	—	83	1	no	—
216	C₂²⋊Q₈	—	83	1	no	—
217	C₂²⋊Q₈	C₂×D₈₃	1	1	yes	characteristic, complemented
218	#121	—	166	1	no	—
219	#121	C₂×D₈₃	1	1	yes	characteristic
220	#120	—	166	1	no	—
221	C₄×Q₈	—	83	1	no	—
222	C₄×D₄	—	83	1	no	—
223	C₄×D₄	—	83	1	no	—
224	#117	—	83	1	no	—
225	#117	—	166	1	no	—
226	C₂×C₄⋊C₄	—	83	1	no	—
227	C₂×C₂²⋊C₄	—	166	1	no	—
228	#107	C₂×D₈₃	1	1	yes	characteristic, complemented
229	#106	—	83	1	no	—
230	#106	—	83	1	no	—
231	C₄⋊C₈	—	83	1	no	—
232	#103	—	83	1	no	—
233	#103	—	83	1	no	—
234	#103	—	83	1	no	—
235	#103	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	C₂×D₈₃	1	1	yes	characteristic
242	C₂²⋊C₈	—	83	1	no	—
243	C₂²⋊C₈	—	83	1	no	—
244	C₄×C₈	—	83	1	no	—
245	C₂⁴	—	166	1	no	—
246	#41	—	332	1	no	—
247	#41	—	166	2	no	—
248	C₂×Q₈	—	83	1	no	—
249	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
250	C₂×D₄	—	2	1	no	—
251	C₂×D₄	—	332	1	no	—
252	C₂×D₄	—	83	1	no	—
253	C₂×D₄	—	83	1	no	—
254	C₂×D₄	—	166	1	no	—
255	C₂×D₄	—	166	2	no	—
256	C₂×D₄	—	166	1	no	—
257	C₂×D₄	C₂²×D₈₃	1	1	yes	characteristic
258	C₂²×C₄	—	83	1	no	—
259	C₂²×C₄	—	166	1	no	—
260	C₂²×C₄	—	166	1	no	—
261	C₂²×C₄	—	166	1	no	—
262	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
263	Q₁₆	—	332	1	no	—
264	SD₁₆	—	4	1	no	—
265	SD₁₆	—	166	2	no	—
266	M₁₆	—	4	1	no	—
267	M₁₆	—	332	1	no	—
268	C₂×C₈	—	83	1	no	—
269	C₂×C₈	—	83	1	no	—
270	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
271	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
272	C₄⋊C₄	—	2	1	no	—
273	C₄⋊C₄	—	83	1	no	—
274	C₄⋊C₄	—	83	1	no	—
275	C₄⋊C₄	—	83	1	no	—
276	C₄⋊C₄	—	166	1	no	—
277	C₄⋊C₄	—	166	1	no	—
278	C₄⋊C₄	—	166	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	C₂²⋊C₄	—	2	1	no	—
285	C₂²⋊C₄	—	2	1	no	—
286	C₂²⋊C₄	—	83	1	no	—
287	C₂²⋊C₄	—	83	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	C₂²⋊C₄	—	166	1	no	—
294	C₄²	—	83	1	no	—
295	C₄²	—	166	1	no	—
296	C₄²	—	166	1	no	—
297	C₂³	—	2	1	no	—
298	C₂³	—	332	2	no	—
299	C₂³	—	332	2	no	—
300	C₂³	—	83	1	no	—
301	C₂³	—	664	1	no	—
302	C₂³	—	166	1	no	—
303	C₂³	D₄×D₈₃	1	1	yes	characteristic
304	Q₈	—	2	1	no	—
305	Q₈	—	2	1	no	—
306	Q₈	—	166	1	no	—
307	Q₈	—	166	1	no	—
308	D₄	—	2	1	no	—
309	D₄	—	2	1	no	—
310	D₄	—	8	1	no	—
311	D₄	—	332	2	no	—
312	D₄	—	332	1	no	—
313	D₄	—	332	2	no	—
314	D₄	—	83	2	no	—
315	D₄	—	166	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	332	2	no	—
322	C₂×C₄	—	332	2	no	—
323	C₂×C₄	—	83	1	no	—
324	C₂×C₄	—	83	1	no	—
325	C₂×C₄	—	83	1	no	—
326	C₂×C₄	—	664	1	no	—
327	C₂×C₄	—	166	1	no	—
328	C₂×C₄	—	166	1	no	—
329	C₂×C₄	—	166	1	no	—
330	C₂×C₄	—	166	1	no	—
331	C₂×C₄	—	166	1	no	—
332	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
333	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
334	C₈	—	2	1	no	—
335	C₈	—	2	1	no	—
336	C₈	—	166	1	no	—
337	C₈	—	166	1	no	—
338	C₂²	—	2	1	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	4	1	no	—
342	C₂²	—	4	1	no	—
343	C₂²	—	4	1	no	—
344	C₂²	—	332	1	no	—
345	C₂²	—	332	1	no	—
346	C₂²	—	664	4	no	—
347	C₂²	—	166	2	no	—
348	C₂²	—	166	1	no	—
349	C₂²	—	166	1	no	—
350	C₂²	—	166	1	no	—
351	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
352	C₄	—	2	1	no	—
353	C₄	—	4	1	no	—
354	C₄	—	4	1	no	—
355	C₄	—	4	1	no	—
356	C₄	—	332	1	no	—
357	C₄	—	332	1	no	—
358	C₄	—	332	1	no	—
359	C₄	—	166	1	no	—
360	C₄	—	166	1	no	—
361	C₄	—	166	1	no	—
362	C₄	#41×D₈₃	1	1	yes	characteristic
363	C₄	#14581837	1	1	yes	characteristic
364	C₂	—	4	1	no	—
365	C₂	—	8	1	no	—
366	C₂	—	332	2	no	—
367	C₂	—	332	1	no	—
368	C₂	#19128340	1	1	yes	characteristic
369	C₂	#19128500	1	1	yes	characteristic
370	C₂	#19128511	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₃
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#468×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	#19128348	C₂
hypercenter	#468	D₈₃
hypocenter	C₈₃	#3323
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #468

Complements

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

N	H ≅ G/N	type
1. #19128393	C₂	semidirect
3. #19128054	C₂	semidirect
4. #19128044	C₂	semidirect
5. #19128017	C₂	semidirect
6. #19128004	C₂	semidirect
7. #19127958	C₂	semidirect
8. #19127921	C₂	semidirect
9. #19127861	C₂	semidirect
10. #19127850	C₂	semidirect
11. #19127786	C₂	semidirect
12. #19127777	C₂	semidirect
13. #19127704	C₂	semidirect
14. #19127703	C₂	semidirect
15. #468×C₈₃	C₂	semidirect
18. C₂×#13050580	C₂²	semidirect
23. #14581783	C₂²	semidirect
31. Q₈×Dic₈₃	C₂²	semidirect
36. C₂×C₄⋊Dic₈₃	C₂²	semidirect
38. #14581723	C₂²	semidirect
39. #14581723	C₂²	semidirect
40. Dic₈₃⋊C₈	C₂²	semidirect
41. #14581717	C₂²	semidirect
42. #14581717	C₂²	semidirect
43. #14581715	C₂²	semidirect
49. C₂²⋊(C₈₃⋊C₈)	C₂²	semidirect
51. C₈×Dic₈₃	C₂²	semidirect
53. C₂×M₁₆×C₈₃	C₂²	semidirect
54. C₂²⋊Q₈×C₈₃	C₂²	semidirect
56. #107×C₈₃	C₂²	semidirect
57. #103×C₈₃	C₂²	semidirect
186. C₈₃	#3323	semidirect
194. #468	D₈₃	semidirect
208. C₂×M₁₆	C₂×D₈₃	semidirect
217. C₂²⋊Q₈	C₂×D₈₃	semidirect
228. #107	C₂×D₈₃	semidirect
235. #103	C₂×D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128393	2
#19128348	2
#19128054	2
#19128044	2
#19128017	2
#19128004	2
#19127958	2
#19127921	2
#19127861	2
#19127850	2
#19127786	2
#19127777	2
#19127704	2
#19127703	2
#468×C₈₃	2
#3323	83