Subgroups — #24918903

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	#19128311	C₂	1	1	yes	maximal, characteristic, complemented
2	#19128304	C₂	1	1	yes	maximal, characteristic
3	#19128011	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128006	C₂	1	1	yes	maximal, characteristic, complemented
5	#19128003	C₂	1	1	yes	maximal, characteristic, complemented
6	#19128000	C₂	1	1	yes	maximal, characteristic, complemented
7	#19127949	C₂	1	1	yes	maximal, characteristic, complemented
8	#19127945	C₂	1	1	yes	maximal, characteristic, complemented
9	#19127930	C₂	1	1	yes	maximal, characteristic, complemented
10	#19127905	C₂	1	1	yes	maximal, characteristic, complemented
11	#19127904	C₂	1	1	yes	maximal, characteristic, complemented
12	#19127900	C₂	1	1	yes	maximal, characteristic, complemented
13	#19127899	C₂	1	1	yes	maximal, characteristic, complemented
14	#19127727	C₂	1	1	yes	maximal, characteristic, complemented
15	#465×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13050579	C₂²	1	1	yes	characteristic
17	C₂×#13050578	C₂²	1	1	yes	characteristic
18	C₂×#13050577	C₂²	1	1	yes	characteristic
19	C₂×#13050574	C₂²	1	1	yes	characteristic
20	#14581790	C₂²	1	1	yes	characteristic
21	#14581788	C₂²	1	1	yes	characteristic
22	#14581785	—	2	1	no	—
23	#14581785	—	2	1	no	—
24	#14581783	—	2	1	no	—
25	#14581783	C₂²	1	1	yes	characteristic
26	#14581783	C₂²	1	1	yes	characteristic
27	#14581782	—	2	1	no	—
28	#14581776	—	2	1	no	—
29	#14581774	—	2	1	no	—
30	#14581771	C₂²	1	1	yes	characteristic
31	#14581766	C₂²	1	1	yes	characteristic
32	#14581764	—	2	1	no	—
33	#14581763	—	2	1	no	—
34	Q₈×Dic₈₃	C₂²	1	1	yes	characteristic
35	#14581757	C₂²	1	1	yes	characteristic
36	D₄×Dic₈₃	C₂²	1	1	yes	characteristic
37	#14581754	C₂²	1	1	yes	characteristic
38	#14581751	C₂²	1	1	yes	characteristic
39	#14581725	C₂²	1	1	yes	characteristic
40	#14581725	C₂²	1	1	yes	characteristic
41	#14581724	C₂²	1	1	yes	characteristic
42	Dic₈₃⋊C₈	C₂²	1	1	yes	characteristic
43	Dic₈₃⋊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	#14581716	C₂²	1	1	yes	characteristic
48	#14581715	C₂²	1	1	yes	characteristic
49	#14581714	C₂²	1	1	yes	characteristic
50	#14581713	C₂²	1	1	yes	characteristic
51	#14581712	C₂²	1	1	yes	characteristic
52	C₂×SD₁₆×C₈₃	C₂²	1	1	yes	characteristic
53	C₂²×C₈×C₈₃	C₂²	1	1	yes	characteristic
54	C₂²⋊Q₈×C₈₃	C₂²	1	1	yes	characteristic
55	#121×C₈₃	C₂²	1	1	yes	characteristic
56	#106×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	C₂³	1	1	yes	characteristic
60	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
61	C₂×C₂²⋊D₈₃	—	2	1	no	—
62	C₂×C₂²⋊D₈₃	—	2	1	no	—
63	C₂×C₂²⋊D₈₃	—	2	1	no	—
64	C₂²×Dic₈₃	—	2	1	no	—
65	C₂×C₄×D₈₃	—	2	1	no	—
66	#13050579	—	4	1	no	—
67	#13050578	—	4	1	no	—
68	#13050577	—	4	1	no	—
69	#13050574	—	4	1	no	—
70	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
71	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
72	C₄⋊Dic₈₃	—	2	1	no	—
73	C₄⋊Dic₈₃	—	2	1	no	—
74	C₄⋊Dic₈₃	—	2	1	no	—
75	C₄⋊Dic₈₃	D₄	1	1	yes	characteristic
76	C₄⋊Dic₈₃	D₄	1	1	yes	characteristic
77	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
78	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
79	Dic₈₃⋊C₄	—	2	1	no	—
80	Dic₈₃⋊C₄	—	2	1	no	—
81	Dic₈₃⋊C₄	—	2	1	no	—
82	Dic₈₃⋊C₄	—	2	1	no	—
83	Dic₈₃⋊C₄	—	2	1	no	—
84	Dic₈₃⋊C₄	—	2	1	no	—
85	C₂²⋊Dic₈₃	—	2	1	no	—
86	C₂²⋊Dic₈₃	—	2	1	no	—
87	C₂²⋊Dic₈₃	D₄	1	1	yes	characteristic
88	C₂²⋊Dic₈₃	D₄	1	1	yes	characteristic
89	(C₂×D₈₃)⋊C₄	—	2	1	no	—
90	(C₂×D₈₃)⋊C₄	—	2	1	no	—
91	C₄×Dic₈₃	—	2	1	no	—
92	C₄×Dic₈₃	—	2	1	no	—
93	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
94	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
95	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
96	C₂×D₄×C₈₃	—	2	1	no	—
97	C₂×D₄×C₈₃	C₂³	1	1	yes	characteristic
98	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
99	SD₁₆×C₈₃	—	4	1	no	—
100	C₂×C₈×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	—	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	C₂²⋊C₄×C₈₃	—	2	1	no	—
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	—	8	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	—	2	1	no	—
120	C₂×C₄×C₈₃	—	2	1	no	—
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	C₂²⋊D₈₃	—	8	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	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
134	D₃₃₂	—	2	1	no	—
135	D₃₃₂	—	2	1	no	—
136	C₄×D₈₃	—	8	1	no	—
137	Dic₁₆₆	—	2	1	no	—
138	Dic₁₆₆	—	2	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	—	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	—	4	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	—	2	1	no	—
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	#537	1	1	yes	characteristic
169	C₂×C₈₃	#577	1	1	yes	characteristic
170	C₂×C₈₃	#577	1	1	yes	characteristic
171	D₈₃	—	8	1	no	—
172	#3260	—	83	1	no	maximal, Sylow
173	C₈₃	#3260	1	1	yes	characteristic, complemented, Sylow
174	#541	—	83	1	no	—
175	#540	—	83	1	no	—
176	#479	—	83	1	no	—
177	#479	—	83	1	no	—
178	#478	—	83	1	no	—
179	#478	—	83	1	no	—
180	#468	—	83	1	no	—
181	#468	—	83	1	no	—
182	#465	—	83	1	no	—
183	#465	D₈₃	1	1	yes	characteristic, complemented
184	#461	—	83	1	no	—
185	#461	—	83	1	no	—
186	#460	—	83	1	no	—
187	#460	—	83	1	no	—
188	#424	—	83	1	no	—
189	C₂×SD₁₆	—	83	1	no	—
190	C₂×SD₁₆	—	83	1	no	—
191	C₂×SD₁₆	—	83	1	no	—
192	C₂×SD₁₆	C₂×D₈₃	1	1	yes	characteristic
193	C₂×M₁₆	—	83	1	no	—
194	C₂²×C₈	C₂×D₈₃	1	1	yes	characteristic
195	C₄⋊Q₈	—	83	1	no	—
196	#127	—	83	1	no	—
197	#126	—	166	1	no	—
198	#126	—	166	1	no	—
199	#125	—	83	1	no	—
200	#125	—	83	1	no	—
201	#125	—	166	1	no	—
202	#125	—	166	1	no	—
203	#123	—	166	1	no	—
204	#123	—	166	1	no	—
205	C₂²⋊Q₈	—	83	1	no	—
206	C₂²⋊Q₈	C₂×D₈₃	1	1	yes	characteristic
207	#121	—	83	1	no	—
208	#121	—	166	1	no	—
209	#121	—	166	1	no	—
210	#121	C₂×D₈₃	1	1	yes	characteristic
211	C₄×Q₈	—	83	1	no	—
212	C₄×Q₈	—	83	1	no	—
213	C₄×D₄	—	83	1	no	—
214	C₄×D₄	—	83	1	no	—
215	#117	—	83	1	no	—
216	#107	—	83	1	no	—
217	#107	—	83	1	no	—
218	#106	—	83	1	no	—
219	#106	C₂×D₈₃	1	1	yes	characteristic
220	C₄⋊C₈	—	83	1	no	—
221	C₄⋊C₈	—	83	1	no	—
222	C₄⋊C₈	—	83	1	no	—
223	C₄⋊C₈	—	83	1	no	—
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	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	—	166	1	no	—
238	C₂×D₄	—	166	1	no	—
239	C₂×D₄	C₂²×D₈₃	1	1	yes	characteristic
240	C₂²×C₄	—	166	1	no	—
241	C₂²×C₄	—	166	1	no	—
242	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
243	SD₁₆	—	4	1	no	—
244	SD₁₆	—	332	1	no	—
245	SD₁₆	—	332	1	no	—
246	SD₁₆	—	332	1	no	—
247	M₁₆	—	332	1	no	—
248	C₂×C₈	—	4	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	—	2	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	—	166	1	no	—
265	C₄⋊C₄	—	166	1	no	—
266	C₄⋊C₄	—	166	1	no	—
267	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
268	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
269	C₂²⋊C₄	—	2	1	no	—
270	C₂²⋊C₄	—	2	1	no	—
271	C₂²⋊C₄	—	83	1	no	—
272	C₂²⋊C₄	—	83	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	—	83	1	no	—
279	C₄²	—	166	1	no	—
280	C₄²	—	166	1	no	—
281	C₂³	—	2	1	no	—
282	C₂³	—	166	1	no	—
283	C₂³	D₄×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	—	8	1	no	—
291	D₄	—	664	1	no	—
292	D₄	—	664	1	no	—
293	D₄	—	664	1	no	—
294	D₄	—	166	1	no	—
295	D₄	—	166	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	83	1	no	—
302	C₂×C₄	—	83	1	no	—
303	C₂×C₄	—	83	1	no	—
304	C₂×C₄	—	83	1	no	—
305	C₂×C₄	—	664	1	no	—
306	C₂×C₄	—	664	1	no	—
307	C₂×C₄	—	166	1	no	—
308	C₂×C₄	—	166	1	no	—
309	C₂×C₄	—	166	1	no	—
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	—	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	C₂×D₄×D₈₃	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	4	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	—	166	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	—	4	1	no	—
340	C₂	—	8	1	no	—
341	C₂	—	664	1	no	—
342	C₂	#19128278	1	1	yes	characteristic
343	C₂	#19128507	1	1	yes	characteristic
344	C₂	#19128508	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₃
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#465×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	#19128304	C₂
hypercenter	#465	D₈₃
hypocenter	C₈₃	#3260
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #465

Complements

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

N	H ≅ G/N	type
1. #19128311	C₂	semidirect
3. #19128011	C₂	semidirect
4. #19128006	C₂	semidirect
5. #19128003	C₂	semidirect
6. #19128000	C₂	semidirect
7. #19127949	C₂	semidirect
8. #19127945	C₂	semidirect
9. #19127930	C₂	semidirect
10. #19127905	C₂	semidirect
11. #19127904	C₂	semidirect
12. #19127900	C₂	semidirect
13. #19127899	C₂	semidirect
14. #19127727	C₂	semidirect
15. #465×C₈₃	C₂	semidirect
173. C₈₃	#3260	semidirect
183. #465	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128311	2
#19128304	2
#19128011	2
#19128006	2
#19128003	2
#19128000	2
#19127949	2
#19127945	2
#19127930	2
#19127905	2
#19127904	2
#19127900	2
#19127899	2
#19127727	2
#465×C₈₃	2
#3260	83