Subgroups — #24918921

Order 10624.

346 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	#19128320	C₂	1	1	yes	maximal, characteristic, complemented
2	#19128316	C₂	1	1	yes	maximal, characteristic, complemented
3	#19127997	C₂	1	1	yes	maximal, characteristic, complemented
4	#19127994	C₂	1	1	yes	maximal, characteristic, complemented
5	#19127981	C₂	1	1	yes	maximal, characteristic, complemented
6	#19127979	C₂	1	1	yes	maximal, characteristic, complemented
7	#19127946	C₂	1	1	yes	maximal, characteristic, complemented
8	#19127944	C₂	1	1	yes	maximal, characteristic, complemented
9	#19127941	C₂	1	1	yes	maximal, characteristic, complemented
10	#19127934	C₂	1	1	yes	maximal, characteristic
11	#19127921	C₂	1	1	yes	maximal, characteristic, complemented
12	#19127920	C₂	1	1	yes	maximal, characteristic, complemented
13	#19127914	C₂	1	1	yes	maximal, characteristic, complemented
14	#19127726	C₂	1	1	yes	maximal, characteristic, complemented
15	#463×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×#13050573	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₈₃	C₂²	1	1	yes	characteristic
20	#14581792	C₂²	1	1	yes	characteristic
21	#14581791	C₂²	1	1	yes	characteristic
22	#14581785	—	2	1	no	—
23	#14581785	—	2	1	no	—
24	#14581779	C₂²	1	1	yes	characteristic
25	#14581778	—	2	1	no	—
26	#14581778	—	2	1	no	—
27	#14581776	—	2	1	no	—
28	#14581775	—	2	1	no	—
29	#14581772	C₂²	1	1	yes	characteristic
30	#14581771	—	2	1	no	—
31	#14581769	—	2	1	no	—
32	#14581768	C₂²	1	1	yes	characteristic
33	#14581763	C₂²	1	1	yes	characteristic
34	#14581763	C₂²	1	1	yes	characteristic
35	Q₈×Dic₈₃	C₂²	1	1	yes	characteristic
36	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
37	D₄×Dic₈₃	C₂²	1	1	yes	characteristic
38	#14581749	C₂²	1	1	yes	characteristic
39	#14581726	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	C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
44	Dic₈₃⋊C₈	C₂²	1	1	yes	characteristic
45	Dic₈₃⋊C₈	C₂²	1	1	yes	characteristic
46	#14581717	C₂²	1	1	yes	characteristic
47	#14581717	C₂²	1	1	yes	characteristic
48	#14581716	C₂²	1	1	yes	characteristic
49	#14581713	C₂²	1	1	yes	characteristic
50	#14581713	C₂²	1	1	yes	characteristic
51	#14581713	C₂²	1	1	yes	characteristic
52	C₂×SD₁₆×C₈₃	C₂²	1	1	yes	characteristic
53	C₂×D₈×C₈₃	C₂²	1	1	yes	characteristic
54	#124×C₈₃	C₂²	1	1	yes	characteristic
55	C₄×D₄×C₈₃	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
57	#103×C₈₃	C₂²	1	1	yes	characteristic
58	#102×C₈₃	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₆₆	—	2	1	no	—
60	C₂×Dic₁₆₆	—	2	1	no	—
61	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₈₃	—	2	1	no	—
63	C₂×C₂²⋊D₈₃	—	2	1	no	—
64	C₂²×Dic₈₃	—	2	1	no	—
65	C₂×C₄×D₈₃	C₂³	1	1	yes	characteristic
66	#13050580	—	4	1	no	—
67	#13050578	—	4	1	no	—
68	#13050573	—	4	1	no	—
69	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
70	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
71	C₈×D₈₃	—	4	1	no	—
72	C₄⋊Dic₈₃	—	2	1	no	—
73	C₄⋊Dic₈₃	—	2	1	no	—
74	C₄⋊Dic₈₃	—	2	1	no	—
75	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
76	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
77	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
78	Dic₈₃⋊C₄	—	2	1	no	—
79	Dic₈₃⋊C₄	—	2	1	no	—
80	Dic₈₃⋊C₄	—	2	1	no	—
81	Dic₈₃⋊C₄	D₄	1	1	yes	characteristic
82	Dic₈₃⋊C₄	D₄	1	1	yes	characteristic
83	C₂²⋊Dic₈₃	—	2	1	no	—
84	C₂²⋊Dic₈₃	—	2	1	no	—
85	C₂²⋊Dic₈₃	—	2	1	no	—
86	C₂²⋊Dic₈₃	—	2	1	no	—
87	C₂²⋊Dic₈₃	—	2	1	no	—
88	(C₂×D₈₃)⋊C₄	—	2	1	no	—
89	(C₂×D₈₃)⋊C₄	—	2	1	no	—
90	(C₂×D₈₃)⋊C₄	D₄	1	1	yes	characteristic
91	(C₂×D₈₃)⋊C₄	D₄	1	1	yes	characteristic
92	C₄×Dic₈₃	—	2	1	no	—
93	C₄×Dic₈₃	—	2	1	no	—
94	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
95	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
96	C₂×D₄×C₈₃	C₂³	1	1	yes	characteristic
97	C₂×D₄×C₈₃	C₂³	1	1	yes	characteristic
98	C₂²×C₄×C₈₃	—	2	1	no	—
99	SD₁₆×C₈₃	—	4	1	no	—
100	D₈×C₈₃	—	4	1	no	—
101	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
102	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
103	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
104	C₂²⋊C₄×C₈₃	—	2	1	no	—
105	C₂²⋊C₄×C₈₃	—	2	1	no	—
106	C₂²⋊C₄×C₈₃	—	2	1	no	—
107	C₄²×C₈₃	C₂³	1	1	yes	characteristic
108	C₂³×C₈₃	—	2	1	no	—
109	C₂³×C₈₃	—	2	1	no	—
110	C₂²×D₈₃	C₂×D₄	1	1	yes	characteristic
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	—	2	1	no	—
116	D₄×C₈₃	—	2	1	no	—
117	C₂×C₄×C₈₃	—	2	1	no	—
118	C₂×C₄×C₈₃	—	2	1	no	—
119	C₂×C₄×C₈₃	—	8	1	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	—	8	1	no	—
124	C₂²⋊D₈₃	—	8	1	no	—
125	C₂×Dic₈₃	—	2	1	no	—
126	C₂×Dic₈₃	—	2	1	no	—
127	C₂×Dic₈₃	—	2	1	no	—
128	C₂×Dic₈₃	—	2	1	no	—
129	C₂×Dic₈₃	—	8	1	no	—
130	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
131	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
132	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
133	C₄×D₈₃	—	2	1	no	—
134	C₄×D₈₃	—	2	1	no	—
135	Dic₁₆₆	—	2	1	no	—
136	Dic₁₆₆	—	2	1	no	—
137	Dic₁₆₆	—	8	1	no	—
138	Dic₁₆₆	—	8	1	no	—
139	C₈×C₈₃	—	2	1	no	—
140	C₈×C₈₃	—	2	1	no	—
141	C₈₃⋊C₈	—	2	1	no	—
142	C₈₃⋊C₈	—	2	1	no	—
143	C₂²×C₈₃	—	4	1	no	—
144	C₂²×C₈₃	—	4	1	no	—
145	C₂²×C₈₃	—	4	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	—	2	1	no	—
151	C₂×D₈₃	—	2	1	no	—
152	C₂×D₈₃	—	2	1	no	—
153	C₄×C₈₃	—	2	1	no	—
154	C₄×C₈₃	—	2	1	no	—
155	C₄×C₈₃	—	4	1	no	—
156	C₄×C₈₃	—	4	1	no	—
157	C₄×C₈₃	H(5,𝔽₂)	1	1	yes	characteristic
158	C₄×C₈₃	2₋¹⁺⁴	1	1	yes	characteristic
159	Dic₈₃	—	2	1	no	—
160	Dic₈₃	—	2	1	no	—
161	Dic₈₃	—	2	1	no	—
162	Dic₈₃	—	4	1	no	—
163	Dic₈₃	—	4	1	no	—
164	Dic₈₃	—	4	1	no	—
165	Dic₈₃	—	4	1	no	—
166	C₂×C₈₃	—	8	1	no	—
167	C₂×C₈₃	—	8	1	no	—
168	C₂×C₈₃	#537	1	1	yes	characteristic
169	C₂×C₈₃	#576	1	1	yes	characteristic
170	C₂×C₈₃	#578	1	1	yes	characteristic
171	D₈₃	—	4	1	no	—
172	#3262	—	83	1	no	maximal, Sylow
173	C₈₃	#3262	1	1	yes	characteristic, complemented, Sylow
174	#542	—	83	1	no	—
175	#542	—	83	1	no	—
176	#477	—	83	1	no	—
177	#477	—	83	1	no	—
178	#474	—	83	1	no	—
179	#474	—	83	1	no	—
180	#468	—	83	1	no	—
181	#468	—	83	1	no	—
182	#467	—	83	1	no	—
183	#466	—	83	1	no	—
184	#464	—	83	1	no	—
185	#464	—	83	1	no	—
186	#463	—	83	1	no	—
187	#463	D₈₃	1	1	yes	characteristic, complemented
188	#424	—	83	1	no	—
189	C₂×Q₁₆	—	83	1	no	—
190	C₂×SD₁₆	—	83	1	no	—
191	C₂×SD₁₆	C₂×D₈₃	1	1	yes	characteristic
192	C₂×D₈	C₂×D₈₃	1	1	yes	characteristic
193	C₂×M₁₆	—	83	1	no	—
194	C₂²×C₈	—	83	1	no	—
195	C₄⋊Q₈	—	83	1	no	—
196	C₄⋊Q₈	—	83	1	no	—
197	#126	—	166	1	no	—
198	#126	—	166	1	no	—
199	#124	—	83	1	no	—
200	#124	—	166	1	no	—
201	#124	—	166	1	no	—
202	#124	C₂×D₈₃	1	1	yes	characteristic
203	#123	—	166	1	no	—
204	#123	—	166	1	no	—
205	C₂²⋊Q₈	—	83	1	no	—
206	C₂²⋊Q₈	—	83	1	no	—
207	C₂²⋊Q₈	—	166	1	no	—
208	C₂²⋊Q₈	—	166	1	no	—
209	#121	—	83	1	no	—
210	#121	—	83	1	no	—
211	C₄×Q₈	—	83	1	no	—
212	C₄×Q₈	—	83	1	no	—
213	C₄×D₄	—	83	1	no	—
214	C₄×D₄	C₂×D₈₃	1	1	yes	characteristic
215	#117	—	83	1	no	—
216	#107	—	83	1	no	—
217	#107	—	83	1	no	—
218	#107	—	83	1	no	—
219	#107	—	83	1	no	—
220	C₄⋊C₈	—	83	1	no	—
221	C₄⋊C₈	—	83	1	no	—
222	C₄⋊C₈	—	83	1	no	—
223	C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
224	#103	—	83	1	no	—
225	#103	—	83	1	no	—
226	#103	—	83	1	no	—
227	#103	C₂×D₈₃	1	1	yes	characteristic
228	#102	—	83	1	no	—
229	#102	—	83	1	no	—
230	#102	—	83	1	no	—
231	#102	C₂×D₈₃	1	1	yes	characteristic
232	C₂×Q₈	—	83	1	no	—
233	C₂×Q₈	—	166	1	no	—
234	C₂×Q₈	—	166	1	no	—
235	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
236	C₂×D₄	—	166	1	no	—
237	C₂×D₄	—	166	1	no	—
238	C₂×D₄	C₂²×D₈₃	1	1	yes	characteristic
239	C₂×D₄	C₂²×D₈₃	1	1	yes	characteristic
240	C₂²×C₄	—	2	1	no	—
241	C₂²×C₄	—	83	1	no	—
242	C₂²×C₄	—	166	1	no	—
243	Q₁₆	—	332	1	no	—
244	SD₁₆	—	4	1	no	—
245	SD₁₆	—	332	1	no	—
246	D₈	—	4	1	no	—
247	M₁₆	—	332	1	no	—
248	C₂×C₈	—	332	1	no	—
249	C₂×C₈	—	83	1	no	—
250	C₂×C₈	—	83	1	no	—
251	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
252	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
253	C₄⋊C₄	—	83	1	no	—
254	C₄⋊C₄	—	83	1	no	—
255	C₄⋊C₄	—	83	1	no	—
256	C₄⋊C₄	—	83	1	no	—
257	C₄⋊C₄	—	83	1	no	—
258	C₄⋊C₄	—	166	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	—	166	1	no	—
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	—	83	1	no	—
269	C₂²⋊C₄	—	83	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	C₂²⋊C₄	—	166	1	no	—
276	C₂²⋊C₄	—	166	1	no	—
277	C₄²	—	83	1	no	—
278	C₄²	—	166	1	no	—
279	C₄²	—	166	1	no	—
280	C₄²	C₂²×D₈₃	1	1	yes	characteristic
281	C₂³	—	2	1	no	—
282	C₂³	—	2	1	no	—
283	C₂³	—	83	1	no	—
284	Q₈	—	2	1	no	—
285	Q₈	—	2	1	no	—
286	Q₈	—	664	1	no	—
287	Q₈	—	664	1	no	—
288	Q₈	—	166	1	no	—
289	Q₈	—	166	1	no	—
290	D₄	—	2	1	no	—
291	D₄	—	2	1	no	—
292	D₄	—	2	1	no	—
293	D₄	—	2	1	no	—
294	D₄	—	664	1	no	—
295	D₄	—	664	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	8	1	no	—
299	C₂×C₄	—	83	1	no	—
300	C₂×C₄	—	83	1	no	—
301	C₂×C₄	—	83	1	no	—
302	C₂×C₄	—	664	1	no	—
303	C₂×C₄	—	166	1	no	—
304	C₂×C₄	—	166	1	no	—
305	C₂×C₄	—	166	1	no	—
306	C₂×C₄	—	166	1	no	—
307	C₂×C₄	—	166	1	no	—
308	C₂×C₄	—	166	1	no	—
309	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
310	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
311	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
312	C₈	—	2	1	no	—
313	C₈	—	2	1	no	—
314	C₈	—	166	1	no	—
315	C₈	—	166	1	no	—
316	C₂²	—	4	1	no	—
317	C₂²	—	4	1	no	—
318	C₂²	—	4	1	no	—
319	C₂²	—	4	1	no	—
320	C₂²	—	4	1	no	—
321	C₂²	—	4	1	no	—
322	C₂²	—	166	1	no	—
323	C₂²	—	166	1	no	—
324	C₂²	—	166	1	no	—
325	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	4	1	no	—
329	C₄	—	4	1	no	—
330	C₄	—	332	1	no	—
331	C₄	—	332	1	no	—
332	C₄	—	332	1	no	—
333	C₄	—	332	1	no	—
334	C₄	—	166	1	no	—
335	C₄	—	166	1	no	—
336	C₄	—	166	1	no	—
337	C₄	#14581837	1	1	yes	characteristic
338	C₄	#14581839	1	1	yes	characteristic
339	C₂	—	8	1	no	—
340	C₂	—	8	1	no	—
341	C₂	—	332	1	no	—
342	C₂	#19128280	1	1	yes	characteristic
343	C₂	#19128505	1	1	yes	characteristic
344	C₂	#19128515	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₃
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#463×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	#19127934	C₂
hypercenter	#463	D₈₃
hypocenter	C₈₃	#3262
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #463

Complements

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

N	H ≅ G/N	type
1. #19128320	C₂	semidirect
2. #19128316	C₂	semidirect
3. #19127997	C₂	semidirect
4. #19127994	C₂	semidirect
5. #19127981	C₂	semidirect
6. #19127979	C₂	semidirect
7. #19127946	C₂	semidirect
8. #19127944	C₂	semidirect
9. #19127941	C₂	semidirect
11. #19127921	C₂	semidirect
12. #19127920	C₂	semidirect
13. #19127914	C₂	semidirect
14. #19127726	C₂	semidirect
15. #463×C₈₃	C₂	semidirect
173. C₈₃	#3262	semidirect
187. #463	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128320	2
#19128316	2
#19127997	2
#19127994	2
#19127981	2
#19127979	2
#19127946	2
#19127944	2
#19127941	2
#19127934	2
#19127921	2
#19127920	2
#19127914	2
#19127726	2
#463×C₈₃	2
#3262	83