Subgroups — #24918178

Order 10624.

402 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128255	C₂	1	1	yes	maximal, characteristic, complemented
2	#19128248	C₂	1	1	yes	maximal, characteristic
3	#19128105	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128072	C₂	1	1	yes	maximal, characteristic, complemented
5	#19128050	C₂	1	1	yes	maximal, characteristic, complemented
6	#19128046	C₂	1	1	yes	maximal, characteristic, complemented
7	#19127950	C₂	1	1	yes	maximal, characteristic, complemented
8	#19127948	C₂	1	1	yes	maximal, characteristic, complemented
9	#19127946	C₂	1	1	yes	maximal, characteristic, complemented
10	#19127944	C₂	1	1	yes	maximal, characteristic, complemented
11	#19127803	C₂	1	1	yes	maximal, characteristic, complemented
12	#19127801	C₂	1	1	yes	maximal, characteristic, complemented
13	#19127799	C₂	1	1	yes	maximal, characteristic, complemented
14	#19127653	C₂	1	1	yes	maximal, characteristic, complemented
15	#440×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13050590	—	2	1	no	—
17	C₂×#13050579	C₂²	1	1	yes	characteristic
18	C₂×#13050578	C₂²	1	1	yes	characteristic
19	C₂×#13050577	C₂²	1	1	yes	characteristic
20	C₂×#13050573	C₂²	1	1	yes	characteristic
21	C₂×#13050573	C₂²	1	1	yes	characteristic
22	#14581791	C₂²	1	1	yes	characteristic
23	#14581791	C₂²	1	1	yes	characteristic
24	#14581790	C₂²	1	1	yes	characteristic
25	#14581788	C₂²	1	1	yes	characteristic
26	#14581782	—	2	1	no	—
27	#14581778	—	2	1	no	—
28	#14581772	C₂²	1	1	yes	characteristic
29	#14581770	—	2	1	no	—
30	#14581768	C₂²	1	1	yes	characteristic
31	#14581767	C₂²	1	1	yes	characteristic
32	#14581765	—	2	1	no	—
33	#14581763	C₂²	1	1	yes	characteristic
34	C₄×Dic₁₆₆	—	2	1	no	—
35	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
36	C₄×D₃₃₂	C₂²	1	1	yes	characteristic
37	C₄×C₂²⋊D₈₃	—	2	1	no	—
38	#14581749	—	2	1	no	—
39	C₄²×D₈₃	C₂²	1	1	yes	characteristic
40	#14581726	C₂²	1	1	yes	characteristic
41	#14581725	C₂²	1	1	yes	characteristic
42	#14581725	C₂²	1	1	yes	characteristic
43	#14581717	C₂²	1	1	yes	characteristic
44	#14581716	C₂²	1	1	yes	characteristic
45	#14581715	C₂²	1	1	yes	characteristic
46	#14581714	C₂²	1	1	yes	characteristic
47	#14581713	C₂²	1	1	yes	characteristic
48	#14581712	C₂²	1	1	yes	characteristic
49	#14581703	C₂²	1	1	yes	characteristic
50	C₈×Dic₈₃	C₂²	1	1	yes	characteristic
51	C₈×Dic₈₃	C₂²	1	1	yes	characteristic
52	C₂×SD₁₆×C₈₃	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₈₃	C₂²	1	1	yes	characteristic
54	C₄×D₄×C₈₃	C₂²	1	1	yes	characteristic
55	#107×C₈₃	C₂²	1	1	yes	characteristic
56	#103×C₈₃	C₂²	1	1	yes	characteristic
57	#102×C₈₃	C₂²	1	1	yes	characteristic
58	#97×C₈₃	C₂²	1	1	yes	characteristic
59	#13050590	—	4	4	no	—
60	C₂×Dic₁₆₆	—	2	1	no	—
61	C₂×Dic₁₆₆	—	2	1	no	—
62	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
63	C₂×D₃₃₂	—	2	1	no	—
64	C₂×D₃₃₂	C₂³	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×C₄×D₈₃	—	2	1	no	—
69	C₂×C₄×D₈₃	D₄	1	1	yes	characteristic
70	C₂×C₄×D₈₃	D₄	1	1	yes	characteristic
71	C₂×C₄×D₈₃	C₂³	1	1	yes	characteristic
72	#13050579	—	2	2	no	—
73	#13050578	—	2	2	no	—
74	#13050577	—	2	2	no	—
75	#13050573	—	2	2	no	—
76	#13050573	—	2	2	no	—
77	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
78	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
79	C₄⋊Dic₈₃	—	2	1	no	—
80	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
81	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
82	Dic₈₃⋊C₄	—	2	1	no	—
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	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	C₄×Dic₈₃	—	2	1	no	—
94	C₄×Dic₈₃	D₄	1	1	yes	characteristic
95	C₄×Dic₈₃	D₄	1	1	yes	characteristic
96	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
97	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
98	C₂×D₄×C₈₃	C₂³	1	1	yes	characteristic
99	C₂²×C₄×C₈₃	—	2	1	no	—
100	SD₁₆×C₈₃	—	2	2	no	—
101	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
102	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
103	C₄⋊C₄×C₈₃	—	2	1	no	—
104	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
105	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
106	C₂²⋊C₄×C₈₃	—	2	1	no	—
107	C₄²×C₈₃	—	2	1	no	—
108	C₄²×C₈₃	C₂³	1	1	yes	characteristic
109	C₂³×C₈₃	—	2	1	no	—
110	C₂²×D₈₃	—	2	1	no	—
111	C₂²×D₈₃	C₂×D₄	1	1	yes	characteristic
112	Q₈×C₈₃	—	2	1	no	—
113	Q₈×C₈₃	—	2	1	no	—
114	D₄×C₈₃	—	2	1	no	—
115	D₄×C₈₃	—	2	1	no	—
116	C₂×C₄×C₈₃	—	2	1	no	—
117	C₂×C₄×C₈₃	—	2	1	no	—
118	C₂×C₄×C₈₃	—	2	1	no	—
119	C₂×C₄×C₈₃	—	4	2	no	—
120	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
122	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
123	C₂²⋊D₈₃	—	4	2	no	—
124	C₂²⋊D₈₃	—	4	2	no	—
125	C₂²⋊D₈₃	—	4	2	no	—
126	C₂×Dic₈₃	—	2	1	no	—
127	C₂×Dic₈₃	—	2	1	no	—
128	C₂×Dic₈₃	—	2	1	no	—
129	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
130	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
131	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
132	D₃₃₂	—	2	1	no	—
133	D₃₃₂	—	2	1	no	—
134	D₃₃₂	—	4	2	no	—
135	C₄×D₈₃	—	2	1	no	—
136	C₄×D₈₃	—	2	1	no	—
137	C₄×D₈₃	—	2	2	no	—
138	C₄×D₈₃	—	2	2	no	—
139	C₄×D₈₃	—	4	2	no	—
140	Dic₁₆₆	—	2	1	no	—
141	Dic₁₆₆	—	2	1	no	—
142	Dic₁₆₆	—	4	2	no	—
143	Dic₁₆₆	—	4	2	no	—
144	C₈×C₈₃	—	2	1	no	—
145	C₈×C₈₃	#41	1	2	yes	complemented
146	C₈₃⋊C₈	—	2	1	no	—
147	C₈₃⋊C₈	#41	1	2	yes	complemented
148	C₂²×C₈₃	—	4	1	no	—
149	C₂²×C₈₃	—	4	1	no	—
150	C₂²×C₈₃	—	4	1	no	—
151	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
152	C₂×D₈₃	—	2	1	no	—
153	C₂×D₈₃	—	2	1	no	—
154	C₂×D₈₃	—	2	1	no	—
155	C₂×D₈₃	—	4	1	no	—
156	C₂×D₈₃	—	4	1	no	—
157	C₂×D₈₃	—	4	1	no	—
158	C₄×C₈₃	—	2	1	no	—
159	C₄×C₈₃	—	2	1	no	—
160	C₄×C₈₃	—	4	1	no	—
161	C₄×C₈₃	—	4	1	no	—
162	C₄×C₈₃	—	4	1	no	—
163	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
164	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
165	Dic₈₃	—	2	1	no	—
166	Dic₈₃	—	4	1	no	—
167	Dic₈₃	—	4	1	no	—
168	Dic₈₃	—	4	1	no	—
169	Dic₈₃	AGL(1,ℤ/8)	1	2	yes	—
170	Dic₈₃	#137	1	2	yes	—
171	C₂×C₈₃	—	8	1	no	—
172	C₂×C₈₃	#532	1	1	yes	characteristic
173	C₂×C₈₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
174	C₂×C₈₃	C₂×#137	1	1	yes	characteristic
175	D₈₃	—	4	1	no	—
176	D₈₃	—	8	1	no	—
177	#3182	—	83	1	no	maximal, Sylow
178	C₈₃	#3182	1	1	yes	characteristic, complemented, Sylow
179	#533	—	83	1	no	—
180	#532	—	83	1	no	—
181	#500	—	83	1	no	—
182	#492	—	83	1	no	—
183	#487	—	83	1	no	—
184	#486	—	83	1	no	—
185	#468	—	83	1	no	—
186	#468	—	83	1	no	—
187	#468	—	83	1	no	—
188	#468	—	83	1	no	—
189	#440	—	83	1	no	—
190	#440	—	83	1	no	—
191	#440	—	83	1	no	—
192	#440	D₈₃	1	1	yes	characteristic, complemented
193	C₄×M₁₆	—	83	1	no	—
194	C₂×#41	—	166	1	no	—
195	C₂×SD₁₆	—	83	1	no	—
196	C₂×SD₁₆	—	83	1	no	—
197	C₂×SD₁₆	—	83	1	no	—
198	C₂×SD₁₆	C₂×D₈₃	1	1	yes	characteristic
199	C₂×M₁₆	—	83	1	no	—
200	C₂×M₁₆	—	83	1	no	—
201	C₄⋊Q₈	—	83	1	no	—
202	C₄⋊Q₈	—	83	1	no	—
203	C₄⋊Q₈	—	83	1	no	—
204	#127	—	83	1	no	—
205	#125	—	166	1	no	—
206	#124	—	166	1	no	—
207	C₂²⋊Q₈	—	83	1	no	—
208	C₂²⋊Q₈	—	83	1	no	—
209	C₂²⋊Q₈	—	166	1	no	—
210	#121	—	83	1	no	—
211	#121	—	83	1	no	—
212	#121	—	166	1	no	—
213	C₄×Q₈	—	83	1	no	—
214	C₄×Q₈	—	166	1	no	—
215	C₄×Q₈	C₂×D₈₃	1	1	yes	characteristic
216	C₄×D₄	—	83	1	no	—
217	C₄×D₄	—	166	1	no	—
218	C₄×D₄	C₂×D₈₃	1	1	yes	characteristic
219	#117	—	166	1	no	—
220	C₂×C₄²	—	83	1	no	—
221	#107	—	83	1	no	—
222	#107	—	83	1	no	—
223	#107	—	83	1	no	—
224	#107	C₂×D₈₃	1	1	yes	characteristic
225	#103	—	83	1	no	—
226	#103	—	83	1	no	—
227	#103	—	83	1	no	—
228	#103	C₂×D₈₃	1	1	yes	characteristic
229	#102	—	83	1	no	—
230	#102	—	83	1	no	—
231	#102	—	83	1	no	—
232	#102	C₂×D₈₃	1	1	yes	characteristic
233	#97	—	83	1	no	—
234	#97	C₂×D₈₃	1	1	yes	characteristic
235	C₄×C₈	—	83	1	no	—
236	C₄×C₈	—	83	1	no	—
237	#41	—	332	4	no	—
238	C₂×Q₈	—	83	1	no	—
239	C₂×Q₈	—	166	1	no	—
240	C₂×Q₈	—	166	1	no	—
241	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
242	C₂×D₄	—	83	1	no	—
243	C₂×D₄	—	166	1	no	—
244	C₂×D₄	—	166	1	no	—
245	C₂×D₄	—	166	1	no	—
246	C₂×D₄	—	166	1	no	—
247	C₂×D₄	C₂²×D₈₃	1	1	yes	characteristic
248	C₂²×C₄	—	2	1	no	—
249	C₂²×C₄	—	83	1	no	—
250	C₂²×C₄	—	83	1	no	—
251	C₂²×C₄	—	83	1	no	—
252	C₂²×C₄	—	166	1	no	—
253	SD₁₆	—	2	2	no	—
254	SD₁₆	—	166	2	no	—
255	SD₁₆	—	166	2	no	—
256	SD₁₆	—	166	2	no	—
257	M₁₆	—	166	2	no	—
258	M₁₆	—	166	2	no	—
259	C₂×C₈	—	83	1	no	—
260	C₂×C₈	—	83	1	no	—
261	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
262	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
263	C₄⋊C₄	—	2	1	no	—
264	C₄⋊C₄	—	83	1	no	—
265	C₄⋊C₄	—	83	1	no	—
266	C₄⋊C₄	—	166	1	no	—
267	C₄⋊C₄	—	166	1	no	—
268	C₄⋊C₄	—	166	1	no	—
269	C₄⋊C₄	—	166	1	no	—
270	C₄⋊C₄	—	166	1	no	—
271	C₄⋊C₄	—	166	1	no	—
272	C₄⋊C₄	—	166	1	no	—
273	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
274	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
275	C₂²⋊C₄	—	2	1	no	—
276	C₂²⋊C₄	—	166	1	no	—
277	C₂²⋊C₄	—	166	1	no	—
278	C₂²⋊C₄	—	166	1	no	—
279	C₂²⋊C₄	—	166	1	no	—
280	C₂²⋊C₄	—	166	1	no	—
281	C₄²	—	2	1	no	—
282	C₄²	—	83	1	no	—
283	C₄²	—	83	1	no	—
284	C₄²	—	83	1	no	—
285	C₄²	—	166	1	no	—
286	C₄²	C₂²×D₈₃	1	1	yes	characteristic
287	C₂³	—	2	1	no	—
288	C₂³	—	83	1	no	—
289	C₂³	—	166	1	no	—
290	Q₈	—	2	1	no	—
291	Q₈	—	2	1	no	—
292	Q₈	—	332	2	no	—
293	Q₈	—	332	2	no	—
294	Q₈	—	166	1	no	—
295	Q₈	—	166	1	no	—
296	D₄	—	2	1	no	—
297	D₄	—	2	1	no	—
298	D₄	—	332	2	no	—
299	D₄	—	332	2	no	—
300	D₄	—	332	2	no	—
301	D₄	—	332	2	no	—
302	D₄	—	166	1	no	—
303	D₄	—	166	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	332	2	no	—
309	C₂×C₄	—	83	1	no	—
310	C₂×C₄	—	83	1	no	—
311	C₂×C₄	—	83	1	no	—
312	C₂×C₄	—	166	2	no	—
313	C₂×C₄	—	166	2	no	—
314	C₂×C₄	—	166	1	no	—
315	C₂×C₄	—	166	1	no	—
316	C₂×C₄	—	166	1	no	—
317	C₂×C₄	—	166	1	no	—
318	C₂×C₄	—	166	1	no	—
319	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
320	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
321	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
322	C₈	—	2	1	no	—
323	C₈	—	83	2	no	—
324	C₈	—	166	1	no	—
325	C₈	#13050590	1	2	yes	complemented
326	C₂²	—	4	1	no	—
327	C₂²	—	4	1	no	—
328	C₂²	—	4	1	no	—
329	C₂²	—	332	1	no	—
330	C₂²	—	332	1	no	—
331	C₂²	—	332	1	no	—
332	C₂²	—	166	1	no	—
333	C₂²	—	166	1	no	—
334	C₂²	—	166	1	no	—
335	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	4	1	no	—
339	C₄	—	4	1	no	—
340	C₄	—	4	1	no	—
341	C₄	—	332	1	no	—
342	C₄	—	332	1	no	—
343	C₄	—	332	1	no	—
344	C₄	—	83	2	no	—
345	C₄	—	83	2	no	—
346	C₄	—	166	1	no	—
347	C₄	#41×D₈₃	1	1	yes	characteristic
348	C₄	C₂×#13050590	1	1	yes	characteristic
349	C₂	—	8	1	no	—
350	C₂	—	332	1	no	—
351	C₂	—	664	1	no	—
352	C₂	#19128248	1	1	yes	characteristic
353	C₂	AGL(1,ℤ/8)×D₈₃	1	1	yes	characteristic
354	C₂	#137×D₈₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₃
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#440×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	#19128248	C₂
hypercenter	#440	D₈₃
hypocenter	C₈₃	#3182
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #440

Complements

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

N	H ≅ G/N	type
1. #19128255	C₂	semidirect
3. #19128105	C₂	semidirect
4. #19128072	C₂	semidirect
5. #19128050	C₂	semidirect
6. #19128046	C₂	semidirect
7. #19127950	C₂	semidirect
8. #19127948	C₂	semidirect
9. #19127946	C₂	semidirect
10. #19127944	C₂	semidirect
11. #19127803	C₂	semidirect
12. #19127801	C₂	semidirect
13. #19127799	C₂	semidirect
14. #19127653	C₂	semidirect
15. #440×C₈₃	C₂	semidirect
145. C₈×C₈₃	#41	semidirect
147. C₈₃⋊C₈	#41	semidirect
178. C₈₃	#3182	semidirect
192. #440	D₈₃	semidirect
325. C₈	#13050590	semidirect

Maximal subgroups

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

subgroup	index
#19128255	2
#19128248	2
#19128105	2
#19128072	2
#19128050	2
#19128046	2
#19127950	2
#19127948	2
#19127946	2
#19127944	2
#19127803	2
#19127801	2
#19127799	2
#19127653	2
#440×C₈₃	2
#3182	83