Subgroups — #24919596

Order 10624.

392 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128392	C₂	1	1	yes	maximal, characteristic, complemented
2	#19128371	C₂	1	1	yes	maximal, characteristic
3	#19128065	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128023	C₂	1	1	yes	maximal, characteristic, complemented
5	#19128018	C₂	1	1	yes	maximal, characteristic, complemented
6	#19128009	C₂	1	1	yes	maximal, characteristic, complemented
7	#19127944	C₂	1	1	yes	maximal, characteristic, complemented
8	#19127927	C₂	1	1	yes	maximal, characteristic, complemented
9	#19127860	C₂	1	1	yes	maximal, characteristic, complemented
10	#19127860	C₂	1	1	yes	maximal, characteristic, complemented
11	#19127806	C₂	1	1	yes	maximal, characteristic, complemented
12	#19127771	C₂	1	1	yes	maximal, characteristic, complemented
13	#19127699	C₂	1	1	yes	maximal, characteristic, complemented
14	#19127694	C₂	1	1	yes	maximal, characteristic, complemented
15	#460×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13050589	C₂²	1	1	yes	characteristic
17	C₂×#13050589	C₂²	1	1	yes	characteristic
18	C₂×#13050580	C₂²	1	1	yes	characteristic
19	C₂×#13050573	C₂²	1	1	yes	characteristic
20	C₂×C₈×D₈₃	C₂²	1	1	yes	characteristic
21	#14581788	—	2	1	no	—
22	#14581786	—	2	1	no	—
23	#14581784	C₂²	1	1	yes	characteristic
24	#14581782	—	2	1	no	—
25	#14581777	—	2	1	no	—
26	#14581768	C₂²	1	1	yes	characteristic
27	#14581768	C₂²	1	1	yes	characteristic
28	#14581764	—	2	1	no	—
29	#14581764	—	2	1	no	—
30	#14581763	C₂²	1	1	yes	characteristic
31	#14581763	C₂²	1	1	yes	characteristic
32	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
33	C₄×D₃₃₂	C₂²	1	1	yes	characteristic
34	D₄×Dic₈₃	—	2	1	no	—
35	#14581750	C₂²	1	1	yes	characteristic
36	#14581749	—	2	1	no	—
37	C₄⋊C₄×D₈₃	C₂²	1	1	yes	characteristic
38	#14581726	C₂²	1	1	yes	characteristic
39	#14581724	C₂²	1	1	yes	characteristic
40	#14581723	C₂²	1	1	yes	characteristic
41	#14581717	C₂²	1	1	yes	characteristic
42	#14581717	C₂²	1	1	yes	characteristic
43	#14581716	C₂²	1	1	yes	characteristic
44	#14581715	C₂²	1	1	yes	characteristic
45	#14581713	C₂²	1	1	yes	characteristic
46	#14581713	C₂²	1	1	yes	characteristic
47	#14581713	C₂²	1	1	yes	characteristic
48	#14581713	C₂²	1	1	yes	characteristic
49	(C₂×D₈₃)⋊C₈	C₂²	1	1	yes	characteristic
50	(C₂×D₈₃)⋊C₈	C₂²	1	1	yes	characteristic
51	#14581703	C₂²	1	1	yes	characteristic
52	C₂×SD₁₆×C₈₃	C₂²	1	1	yes	characteristic
53	C₂×SD₁₆×C₈₃	C₂²	1	1	yes	characteristic
54	#127×C₈₃	C₂²	1	1	yes	characteristic
55	C₄×Q₈×C₈₃	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
57	#102×C₈₃	C₂²	1	1	yes	characteristic
58	#102×C₈₃	C₂²	1	1	yes	characteristic
59	#13050589	—	2	2	no	—
60	#13050589	—	2	2	no	—
61	#13050589	—	4	1	no	—
62	#13050589	—	4	1	no	—
63	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
64	C₂×D₃₃₂	D₄	1	1	yes	characteristic
65	C₂×C₂²⋊D₈₃	—	2	1	no	—
66	C₂×C₂²⋊D₈₃	—	2	1	no	—
67	C₂×C₂²⋊D₈₃	—	2	1	no	—
68	C₂×C₂²⋊D₈₃	—	2	1	no	—
69	C₂²×Dic₈₃	—	2	1	no	—
70	C₂²×Dic₈₃	—	2	1	no	—
71	C₂×C₄×D₈₃	—	2	1	no	—
72	C₂×C₄×D₈₃	C₂³	1	1	yes	characteristic
73	C₂×C₄×D₈₃	C₂³	1	1	yes	characteristic
74	#13050580	—	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	C₈×D₈₃	—	4	1	no	—
79	C₄⋊Dic₈₃	D₄	1	1	yes	characteristic
80	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
81	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
82	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
83	Dic₈₃⋊C₄	—	2	1	no	—
84	Dic₈₃⋊C₄	—	2	1	no	—
85	Dic₈₃⋊C₄	—	2	1	no	—
86	Dic₈₃⋊C₄	—	2	1	no	—
87	Dic₈₃⋊C₄	—	2	1	no	—
88	Dic₈₃⋊C₄	—	2	1	no	—
89	C₂²⋊Dic₈₃	—	2	1	no	—
90	C₂²⋊Dic₈₃	—	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	(C₂×D₈₃)⋊C₄	D₄	1	1	yes	characteristic
96	C₄×Dic₈₃	—	2	1	no	—
97	C₄×Dic₈₃	—	2	1	no	—
98	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
99	C₂×D₄×C₈₃	—	2	1	no	—
100	C₂×D₄×C₈₃	—	2	1	no	—
101	C₂×D₄×C₈₃	C₂³	1	1	yes	characteristic
102	C₂×D₄×C₈₃	C₂³	1	1	yes	characteristic
103	SD₁₆×C₈₃	—	4	1	no	—
104	SD₁₆×C₈₃	—	4	1	no	—
105	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
106	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
107	C₄⋊C₄×C₈₃	—	2	1	no	—
108	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
109	C₄²×C₈₃	—	2	1	no	—
110	C₄²×C₈₃	C₂³	1	1	yes	characteristic
111	C₂³×C₈₃	—	2	1	no	—
112	C₂³×C₈₃	—	2	1	no	—
113	C₂²×D₈₃	C₂×D₄	1	1	yes	characteristic
114	C₂²×D₈₃	C₂×D₄	1	1	yes	characteristic
115	Q₈×C₈₃	—	2	1	no	—
116	Q₈×C₈₃	—	2	1	no	—
117	D₄×C₈₃	—	2	1	no	—
118	D₄×C₈₃	—	2	1	no	—
119	D₄×C₈₃	—	2	1	no	—
120	D₄×C₈₃	—	2	1	no	—
121	D₄×C₈₃	—	4	2	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	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
127	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
128	C₂²⋊D₈₃	—	4	2	no	—
129	C₂²⋊D₈₃	—	4	2	no	—
130	C₂²⋊D₈₃	—	8	1	no	—
131	C₂²⋊D₈₃	—	8	1	no	—
132	C₂×Dic₈₃	—	2	1	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	—	4	2	no	—
138	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
139	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
140	D₃₃₂	—	4	1	no	—
141	C₄×D₈₃	—	2	1	no	—
142	C₄×D₈₃	—	2	1	no	—
143	C₄×D₈₃	—	2	1	no	—
144	C₄×D₈₃	—	2	1	no	—
145	C₄×D₈₃	—	4	2	no	—
146	Dic₁₆₆	—	2	1	no	—
147	Dic₁₆₆	#41	1	2	yes	—
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	—	4	1	no	—
153	C₂²×C₈₃	—	4	1	no	—
154	C₂²×C₈₃	—	4	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	—	2	1	no	—
165	C₄×C₈₃	—	2	1	no	—
166	C₄×C₈₃	—	2	1	no	—
167	C₄×C₈₃	—	4	1	no	—
168	C₄×C₈₃	—	4	1	no	—
169	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
170	C₄×C₈₃	H(5,𝔽₂)	1	1	yes	characteristic
171	Dic₈₃	—	2	2	no	—
172	Dic₈₃	—	2	1	no	—
173	Dic₈₃	—	2	1	no	—
174	Dic₈₃	—	4	1	no	—
175	Dic₈₃	—	4	1	no	—
176	Dic₈₃	—	4	1	no	—
177	C₂×C₈₃	—	8	1	no	—
178	C₂×C₈₃	—	8	1	no	—
179	C₂×C₈₃	#546	1	1	yes	characteristic
180	C₂×C₈₃	#575	1	1	yes	characteristic
181	C₂×C₈₃	#577	1	1	yes	characteristic
182	D₈₃	—	4	1	no	—
183	D₈₃	—	4	1	no	—
184	#3317	—	83	1	no	maximal, Sylow
185	C₈₃	#3317	1	1	yes	characteristic, complemented, Sylow
186	#552	—	83	1	no	—
187	#550	—	83	1	no	—
188	#490	—	83	1	no	—
189	#482	—	83	1	no	—
190	#481	—	83	1	no	—
191	#479	—	83	1	no	—
192	#468	—	83	1	no	—
193	#465	—	83	1	no	—
194	#460	D₈₃	1	1	yes	characteristic, complemented
195	#452	—	83	1	no	—
196	#452	—	83	1	no	—
197	#441	—	83	1	no	—
198	#435	—	83	1	no	—
199	#417	—	83	1	no	—
200	#416	—	83	1	no	—
201	C₂×#41	—	83	1	no	—
202	C₂×#41	—	83	1	no	—
203	C₂×Q₁₆	—	83	1	no	—
204	C₂×SD₁₆	C₂×D₈₃	1	1	yes	characteristic
205	C₂×SD₁₆	C₂×D₈₃	1	1	yes	characteristic
206	C₂×M₁₆	—	83	1	no	—
207	C₂²×C₈	—	83	1	no	—
208	#127	—	166	1	no	—
209	#127	C₂×D₈₃	1	1	yes	characteristic
210	#126	—	166	1	no	—
211	#125	—	83	1	no	—
212	#125	—	166	1	no	—
213	#123	—	166	1	no	—
214	C₂²⋊Q₈	—	83	1	no	—
215	C₂²⋊Q₈	—	83	1	no	—
216	#121	—	83	1	no	—
217	#121	—	83	1	no	—
218	#121	—	166	1	no	—
219	#121	—	166	1	no	—
220	C₄×Q₈	—	83	1	no	—
221	C₄×Q₈	C₂×D₈₃	1	1	yes	characteristic
222	C₄×D₄	—	83	1	no	—
223	C₄×D₄	—	166	1	no	—
224	#117	—	83	1	no	—
225	#117	—	166	1	no	—
226	C₂×C₄⋊C₄	—	83	1	no	—
227	#107	—	83	1	no	—
228	#106	—	83	1	no	—
229	#106	—	83	1	no	—
230	C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
231	#103	—	83	1	no	—
232	#103	—	83	1	no	—
233	#103	—	83	1	no	—
234	#103	—	83	1	no	—
235	#102	—	83	1	no	—
236	#102	—	83	1	no	—
237	#102	—	83	1	no	—
238	#102	—	83	1	no	—
239	#102	C₂×D₈₃	1	1	yes	characteristic
240	#102	C₂×D₈₃	1	1	yes	characteristic
241	C₂²⋊C₈	—	83	1	no	—
242	C₂²⋊C₈	—	83	1	no	—
243	#97	—	83	1	no	—
244	#41	—	332	1	no	—
245	#41	—	332	1	no	—
246	#41	—	166	2	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	—	2	1	no	—
252	C₂×D₄	—	83	1	no	—
253	C₂×D₄	—	166	1	no	—
254	C₂×D₄	—	166	1	no	—
255	C₂×D₄	—	166	1	no	—
256	C₂×D₄	—	166	1	no	—
257	C₂×D₄	C₂²×D₈₃	1	1	yes	characteristic
258	C₂×D₄	C₂²×D₈₃	1	1	yes	characteristic
259	C₂²×C₄	—	83	1	no	—
260	C₂²×C₄	—	83	1	no	—
261	C₂²×C₄	—	166	1	no	—
262	C₂²×C₄	—	166	1	no	—
263	C₂²×C₄	—	166	1	no	—
264	Q₁₆	—	166	2	no	—
265	SD₁₆	—	4	1	no	—
266	SD₁₆	—	4	1	no	—
267	M₁₆	—	332	1	no	—
268	C₂×C₈	—	332	1	no	—
269	C₂×C₈	—	83	1	no	—
270	C₂×C₈	—	83	1	no	—
271	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
272	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
273	C₄⋊C₄	—	2	1	no	—
274	C₄⋊C₄	—	83	1	no	—
275	C₄⋊C₄	—	83	1	no	—
276	C₄⋊C₄	—	83	1	no	—
277	C₄⋊C₄	—	83	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	—	166	1	no	—
283	C₄⋊C₄	—	166	1	no	—
284	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
285	C₂²⋊C₄	—	83	1	no	—
286	C₂²⋊C₄	—	83	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	C₄²	—	2	1	no	—
293	C₄²	—	166	1	no	—
294	C₄²	—	166	1	no	—
295	C₄²	C₂²×D₈₃	1	1	yes	characteristic
296	C₂³	—	2	1	no	—
297	C₂³	—	2	1	no	—
298	C₂³	—	83	1	no	—
299	C₂³	—	83	1	no	—
300	Q₈	—	2	1	no	—
301	Q₈	—	2	1	no	—
302	Q₈	—	83	2	no	—
303	Q₈	—	166	1	no	—
304	D₄	—	2	1	no	—
305	D₄	—	2	1	no	—
306	D₄	—	2	1	no	—
307	D₄	—	2	1	no	—
308	D₄	—	4	2	no	—
309	D₄	—	8	1	no	—
310	D₄	—	332	1	no	—
311	D₄	—	332	2	no	—
312	D₄	—	332	2	no	—
313	D₄	—	664	1	no	—
314	D₄	—	664	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	332	2	no	—
318	C₂×C₄	—	332	2	no	—
319	C₂×C₄	—	332	2	no	—
320	C₂×C₄	—	83	1	no	—
321	C₂×C₄	—	83	1	no	—
322	C₂×C₄	—	166	1	no	—
323	C₂×C₄	—	166	1	no	—
324	C₂×C₄	—	166	1	no	—
325	C₂×C₄	—	166	1	no	—
326	C₂×C₄	—	166	1	no	—
327	C₂×C₄	—	166	1	no	—
328	C₂×C₄	—	166	1	no	—
329	C₂×C₄	—	166	1	no	—
330	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
331	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
332	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
333	C₈	—	2	1	no	—
334	C₈	—	2	1	no	—
335	C₈	—	166	1	no	—
336	C₈	—	166	1	no	—
337	C₂²	—	4	1	no	—
338	C₂²	—	4	1	no	—
339	C₂²	—	4	1	no	—
340	C₂²	—	4	1	no	—
341	C₂²	—	4	1	no	—
342	C₂²	—	4	1	no	—
343	C₂²	—	166	1	no	—
344	C₂²	—	166	1	no	—
345	C₂²	—	166	1	no	—
346	C₂²	—	166	1	no	—
347	C₂²	—	166	1	no	—
348	C₂²	—	166	1	no	—
349	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	4	1	no	—
353	C₄	—	4	1	no	—
354	C₄	—	332	1	no	—
355	C₄	—	332	1	no	—
356	C₄	—	332	1	no	—
357	C₄	—	166	1	no	—
358	C₄	—	166	1	no	—
359	C₄	—	166	2	no	—
360	C₄	#41×D₈₃	1	1	yes	characteristic
361	C₄	#14581837	1	1	yes	characteristic
362	C₂	—	8	1	no	—
363	C₂	—	8	1	no	—
364	C₂	—	332	1	no	—
365	C₂	—	332	1	no	—
366	C₂	#19128348	1	1	yes	characteristic
367	C₂	#19128497	1	1	yes	characteristic
368	C₂	#19128507	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₃
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#460×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	#19128371	C₂
hypercenter	#460	D₈₃
hypocenter	C₈₃	#3317
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #460

Complements

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

N	H ≅ G/N	type
1. #19128392	C₂	semidirect
3. #19128065	C₂	semidirect
4. #19128023	C₂	semidirect
5. #19128018	C₂	semidirect
6. #19128009	C₂	semidirect
7. #19127944	C₂	semidirect
8. #19127927	C₂	semidirect
9. #19127860	C₂	semidirect
10. #19127860	C₂	semidirect
11. #19127806	C₂	semidirect
12. #19127771	C₂	semidirect
13. #19127699	C₂	semidirect
14. #19127694	C₂	semidirect
15. #460×C₈₃	C₂	semidirect
185. C₈₃	#3317	semidirect
194. #460	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128392	2
#19128371	2
#19128065	2
#19128023	2
#19128018	2
#19128009	2
#19127944	2
#19127927	2
#19127860	2
#19127806	2
#19127771	2
#19127699	2
#19127694	2
#460×C₈₃	2
#3317	83