Subgroups — #24918186

Order 10624.

370 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128235	C₂	1	1	yes	maximal, characteristic, complemented
2	#19128100	C₂	1	1	yes	maximal, characteristic, complemented
3	#19128083	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128058	C₂	1	1	yes	maximal, characteristic, complemented
5	#19128055	C₂	1	1	yes	maximal, characteristic, complemented
6	#19127957	C₂	1	1	yes	maximal, characteristic, complemented
7	#19127952	C₂	1	1	yes	maximal, characteristic
8	#19127949	C₂	1	1	yes	maximal, characteristic, complemented
9	#19127945	C₂	1	1	yes	maximal, characteristic, complemented
10	#19127811	C₂	1	1	yes	maximal, characteristic, complemented
11	#19127806	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	C₄×#13050574	C₂	1	1	yes	maximal, characteristic, complemented
15	#529×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13050580	C₂²	1	1	yes	characteristic
17	C₂×#13050578	C₂²	1	1	yes	characteristic
18	C₂×#13050577	C₂²	1	1	yes	characteristic
19	C₂×#13050575	C₂²	1	1	yes	characteristic
20	C₂×#13050574	C₂²	1	1	yes	characteristic
21	C₂×#13050574	C₂²	1	1	yes	characteristic
22	#14581786	—	2	1	no	—
23	#14581784	C₂²	1	1	yes	characteristic
24	#14581780	C₂²	1	1	yes	characteristic
25	#14581775	—	2	1	no	—
26	#14581771	C₂²	1	1	yes	characteristic
27	#14581766	C₂²	1	1	yes	characteristic
28	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
29	C₄×D₃₃₂	C₂²	1	1	yes	characteristic
30	C₄×C₂²⋊D₈₃	—	2	1	no	—
31	#14581749	—	2	1	no	—
32	#14581725	C₂²	1	1	yes	characteristic
33	#14581725	C₂²	1	1	yes	characteristic
34	#14581723	C₂²	1	1	yes	characteristic
35	#14581723	C₂²	1	1	yes	characteristic
36	#14581716	C₂²	1	1	yes	characteristic
37	#14581716	C₂²	1	1	yes	characteristic
38	#14581715	C₂²	1	1	yes	characteristic
39	#14581715	C₂²	1	1	yes	characteristic
40	#14581713	C₂²	1	1	yes	characteristic
41	#14581713	C₂²	1	1	yes	characteristic
42	#14581712	C₂²	1	1	yes	characteristic
43	#14581712	C₂²	1	1	yes	characteristic
44	#14581703	C₂²	1	1	yes	characteristic
45	#14581703	C₂²	1	1	yes	characteristic
46	C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
47	C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
48	#126×C₈₃	—	2	1	no	—
49	#125×C₈₃	C₂²	1	1	yes	characteristic
50	#124×C₈₃	C₂²	1	1	yes	characteristic
51	#123×C₈₃	—	2	1	no	—
52	C₂²⋊Q₈×C₈₃	C₂²	1	1	yes	characteristic
53	#121×C₈₃	C₂²	1	1	yes	characteristic
54	C₄×Q₈×C₈₃	C₂²	1	1	yes	characteristic
55	C₄×D₄×C₈₃	—	2	1	no	—
56	C₄×D₄×C₈₃	C₂²	1	1	yes	characteristic
57	#117×C₈₃	—	2	1	no	—
58	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
60	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
61	C₂×C₂²⋊D₈₃	—	2	1	no	—
62	C₂×C₄×D₈₃	—	2	1	no	—
63	#13050580	—	2	2	no	—
64	#13050578	—	2	2	no	—
65	#13050577	—	2	2	no	—
66	#13050575	—	2	2	no	—
67	#13050574	—	2	2	no	—
68	#13050574	—	2	2	no	—
69	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
70	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
71	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
72	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
73	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
74	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
75	Dic₈₃⋊C₄	—	2	1	no	—
76	Dic₈₃⋊C₄	—	2	1	no	—
77	Dic₈₃⋊C₄	—	2	1	no	—
78	Dic₈₃⋊C₄	—	2	1	no	—
79	C₂²⋊Dic₈₃	—	2	1	no	—
80	(C₂×D₈₃)⋊C₄	—	2	1	no	—
81	(C₂×D₈₃)⋊C₄	—	2	1	no	—
82	(C₂×D₈₃)⋊C₄	—	2	1	no	—
83	(C₂×D₈₃)⋊C₄	—	2	1	no	—
84	C₄×Dic₈₃	—	2	1	no	—
85	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
86	C₂×D₄×C₈₃	—	2	1	no	—
87	C₂×D₄×C₈₃	C₂³	1	1	yes	characteristic
88	C₂²×C₄×C₈₃	—	2	1	no	—
89	C₂²×C₄×C₈₃	D₄	1	1	yes	characteristic
90	C₂²×C₄×C₈₃	D₄	1	1	yes	characteristic
91	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
92	C₄⋊C₄×C₈₃	—	2	1	no	—
93	C₄⋊C₄×C₈₃	—	2	1	no	—
94	C₄⋊C₄×C₈₃	—	2	1	no	—
95	C₄⋊C₄×C₈₃	—	2	1	no	—
96	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
97	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
98	C₂²⋊C₄×C₈₃	—	2	1	no	—
99	C₂²⋊C₄×C₈₃	—	2	1	no	—
100	C₂²⋊C₄×C₈₃	—	2	1	no	—
101	C₂²⋊C₄×C₈₃	—	2	1	no	—
102	C₂²⋊C₄×C₈₃	—	2	1	no	—
103	C₄²×C₈₃	—	2	1	no	—
104	C₄²×C₈₃	D₄	1	1	yes	characteristic
105	C₄²×C₈₃	D₄	1	1	yes	characteristic
106	C₄²×C₈₃	C₂³	1	1	yes	characteristic
107	C₄²×C₈₃	C₂³	1	1	yes	characteristic
108	C₂³×C₈₃	—	2	1	no	—
109	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
110	C₂²×D₈₃	—	2	1	no	—
111	Q₈×C₈₃	—	2	1	no	—
112	Q₈×C₈₃	—	2	1	no	—
113	D₄×C₈₃	—	2	1	no	—
114	D₄×C₈₃	—	2	1	no	—
115	D₄×C₈₃	—	4	2	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	—	2	1	no	—
120	C₂×C₄×C₈₃	—	2	1	no	—
121	C₂×C₄×C₈₃	—	4	1	no	—
122	C₂×C₄×C₈₃	—	4	1	no	—
123	C₂×C₄×C₈₃	—	4	2	no	—
124	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
127	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
128	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
129	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
130	C₂²⋊D₈₃	—	4	2	no	—
131	C₂×Dic₈₃	—	2	1	no	—
132	C₂×Dic₈₃	—	2	1	no	—
133	C₂×Dic₈₃	—	2	1	no	—
134	D₃₃₂	—	2	1	no	—
135	D₃₃₂	—	2	1	no	—
136	C₄×D₈₃	—	4	2	no	—
137	Dic₁₆₆	—	2	1	no	—
138	Dic₁₆₆	—	2	1	no	—
139	C₈₃⋊C₈	—	2	1	no	—
140	C₈₃⋊C₈	—	2	1	no	—
141	C₈₃⋊C₈	#41	1	2	yes	—
142	C₈₃⋊C₈	#41	1	2	yes	—
143	C₂²×C₈₃	—	2	1	no	—
144	C₂²×C₈₃	—	2	1	no	—
145	C₂²×C₈₃	—	2	1	no	—
146	C₂²×C₈₃	—	4	1	no	—
147	C₂²×C₈₃	—	4	1	no	—
148	C₂²×C₈₃	—	4	1	no	—
149	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
150	C₂×D₈₃	—	4	1	no	—
151	C₂×D₈₃	—	4	1	no	—
152	C₂×D₈₃	—	4	1	no	—
153	C₄×C₈₃	—	2	1	no	—
154	C₄×C₈₃	—	2	1	no	—
155	C₄×C₈₃	—	2	1	no	—
156	C₄×C₈₃	—	2	1	no	—
157	C₄×C₈₃	—	2	1	no	—
158	C₄×C₈₃	—	4	1	no	—
159	C₄×C₈₃	—	4	1	no	—
160	C₄×C₈₃	—	4	1	no	—
161	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
162	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
163	Dic₈₃	—	4	1	no	—
164	Dic₈₃	—	4	1	no	—
165	Dic₈₃	—	4	1	no	—
166	C₂×C₈₃	—	4	1	no	—
167	C₂×C₈₃	—	8	1	no	—
168	C₂×C₈₃	#532	1	1	yes	characteristic
169	C₂×C₈₃	#575	1	1	yes	characteristic
170	C₂×C₈₃	#575	1	1	yes	characteristic
171	D₈₃	—	8	1	no	—
172	#3183	—	83	1	no	maximal, Sylow
173	C₈₃	#3183	1	1	yes	characteristic, complemented, Sylow
174	#529	—	83	1	no	—
175	#529	D₈₃	1	1	yes	characteristic, complemented
176	#499	—	83	1	no	—
177	#495	—	83	1	no	—
178	#488	—	83	1	no	—
179	#488	—	83	1	no	—
180	#470	—	83	1	no	—
181	#469	—	83	1	no	—
182	#468	—	83	1	no	—
183	#468	—	83	1	no	—
184	#442	—	83	1	no	—
185	#441	—	83	1	no	—
186	#440	—	83	1	no	—
187	#440	—	83	1	no	—
188	C₄×M₁₆	—	83	1	no	—
189	C₂×Q₁₆	—	83	1	no	—
190	C₂×SD₁₆	—	83	1	no	—
191	C₂×SD₁₆	—	83	1	no	—
192	C₂×D₈	—	83	1	no	—
193	C₂×M₁₆	—	83	1	no	—
194	C₂×M₁₆	—	83	1	no	—
195	#126	—	2	1	no	—
196	#126	—	166	1	no	—
197	#125	—	83	1	no	—
198	#125	C₂×D₈₃	1	1	yes	characteristic
199	#124	—	83	1	no	—
200	#124	C₂×D₈₃	1	1	yes	characteristic
201	#123	—	2	1	no	—
202	#123	—	166	1	no	—
203	C₂²⋊Q₈	—	83	1	no	—
204	C₂²⋊Q₈	C₂×D₈₃	1	1	yes	characteristic
205	#121	—	83	1	no	—
206	#121	C₂×D₈₃	1	1	yes	characteristic
207	C₄×Q₈	—	83	1	no	—
208	C₄×Q₈	C₂×D₈₃	1	1	yes	characteristic
209	C₄×D₄	—	2	1	no	—
210	C₄×D₄	—	83	1	no	—
211	C₄×D₄	—	166	1	no	—
212	C₄×D₄	C₂×D₈₃	1	1	yes	characteristic
213	#117	—	2	1	no	—
214	#117	—	166	1	no	—
215	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
216	#107	—	83	1	no	—
217	#107	—	83	1	no	—
218	#106	—	83	1	no	—
219	#106	—	83	1	no	—
220	#103	—	83	1	no	—
221	#103	—	83	1	no	—
222	#103	—	83	1	no	—
223	#103	—	83	1	no	—
224	#102	—	83	1	no	—
225	#102	—	83	1	no	—
226	#102	—	83	1	no	—
227	#102	—	83	1	no	—
228	#97	—	83	1	no	—
229	#97	—	83	1	no	—
230	C₄×C₈	—	83	1	no	—
231	C₄×C₈	—	83	1	no	—
232	C₂×Q₈	—	83	1	no	—
233	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
234	C₂×D₄	—	2	1	no	—
235	C₂×D₄	—	83	1	no	—
236	C₂×D₄	—	166	1	no	—
237	C₂×D₄	C₂²×D₈₃	1	1	yes	characteristic
238	C₂²×C₄	—	2	1	no	—
239	C₂²×C₄	—	166	1	no	—
240	C₂²×C₄	C₂²⋊D₈₃	1	1	yes	characteristic
241	C₂²×C₄	C₂²⋊D₈₃	1	1	yes	characteristic
242	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
243	Q₁₆	—	166	2	no	—
244	SD₁₆	—	166	2	no	—
245	SD₁₆	—	166	2	no	—
246	D₈	—	166	2	no	—
247	M₁₆	—	166	2	no	—
248	M₁₆	—	166	2	no	—
249	C₂×C₈	—	83	1	no	—
250	C₂×C₈	—	83	1	no	—
251	C₂×C₈	—	83	1	no	—
252	C₂×C₈	—	83	1	no	—
253	C₄⋊C₄	—	2	1	no	—
254	C₄⋊C₄	—	2	1	no	—
255	C₄⋊C₄	—	2	1	no	—
256	C₄⋊C₄	—	2	1	no	—
257	C₄⋊C₄	—	83	1	no	—
258	C₄⋊C₄	—	83	1	no	—
259	C₄⋊C₄	—	166	1	no	—
260	C₄⋊C₄	—	166	1	no	—
261	C₄⋊C₄	—	166	1	no	—
262	C₄⋊C₄	—	166	1	no	—
263	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
264	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
265	C₂²⋊C₄	—	2	1	no	—
266	C₂²⋊C₄	—	2	1	no	—
267	C₂²⋊C₄	—	2	1	no	—
268	C₂²⋊C₄	—	2	1	no	—
269	C₂²⋊C₄	—	2	1	no	—
270	C₂²⋊C₄	—	166	1	no	—
271	C₂²⋊C₄	—	166	1	no	—
272	C₂²⋊C₄	—	166	1	no	—
273	C₂²⋊C₄	—	166	1	no	—
274	C₂²⋊C₄	—	166	1	no	—
275	C₄²	—	2	1	no	—
276	C₄²	—	166	1	no	—
277	C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
278	C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
279	C₄²	C₂²×D₈₃	1	1	yes	characteristic
280	C₄²	C₂²×D₈₃	1	1	yes	characteristic
281	C₂³	—	2	1	no	—
282	C₂³	—	166	1	no	—
283	C₂³	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
284	Q₈	—	2	1	no	—
285	Q₈	—	2	1	no	—
286	Q₈	—	166	1	no	—
287	Q₈	—	166	1	no	—
288	D₄	—	2	1	no	—
289	D₄	—	2	1	no	—
290	D₄	—	4	2	no	—
291	D₄	—	332	2	no	—
292	D₄	—	166	1	no	—
293	D₄	—	166	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	332	2	no	—
303	C₂×C₄	—	166	1	no	—
304	C₂×C₄	—	166	1	no	—
305	C₂×C₄	—	166	1	no	—
306	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
307	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
308	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
309	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
310	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
311	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
312	C₈	—	83	2	no	—
313	C₈	—	83	2	no	—
314	C₈	—	166	1	no	—
315	C₈	—	166	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	4	1	no	—
320	C₂²	—	4	1	no	—
321	C₂²	—	4	1	no	—
322	C₂²	—	332	1	no	—
323	C₂²	—	332	1	no	—
324	C₂²	—	332	1	no	—
325	C₂²	C₂²×C₂²⋊D₈₃	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	4	1	no	—
332	C₄	—	4	1	no	—
333	C₄	—	4	1	no	—
334	C₄	—	332	1	no	—
335	C₄	—	332	1	no	—
336	C₄	—	332	1	no	—
337	C₄	#41×D₈₃	1	1	yes	characteristic
338	C₄	#41×D₈₃	1	1	yes	characteristic
339	C₂	—	4	1	no	—
340	C₂	—	8	1	no	—
341	C₂	—	664	1	no	—
342	C₂	#19128247	1	1	yes	characteristic
343	C₂	#19128499	1	1	yes	characteristic
344	C₂	#19128499	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₈₃
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#529×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	#19127952	C₂
hypercenter	#529	D₈₃
hypocenter	C₈₃	#3183
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #529

Complements

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

N	H ≅ G/N	type
1. #19128235	C₂	semidirect
2. #19128100	C₂	semidirect
3. #19128083	C₂	semidirect
4. #19128058	C₂	semidirect
5. #19128055	C₂	semidirect
6. #19127957	C₂	semidirect
8. #19127949	C₂	semidirect
9. #19127945	C₂	semidirect
10. #19127811	C₂	semidirect
11. #19127806	C₂	semidirect
12. #19127801	C₂	semidirect
13. #19127799	C₂	semidirect
14. C₄×#13050574	C₂	semidirect
15. #529×C₈₃	C₂	semidirect
173. C₈₃	#3183	semidirect
175. #529	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128235	2
#19128100	2
#19128083	2
#19128058	2
#19128055	2
#19127957	2
#19127952	2
#19127949	2
#19127945	2
#19127811	2
#19127806	2
#19127801	2
#19127799	2
C₄×#13050574	2
#529×C₈₃	2
#3183	83