Subgroups — #24919818

Order 10624.

388 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	#19128406	C₂	1	1	yes	maximal, characteristic, complemented
2	#19128351	C₂	1	1	yes	maximal, characteristic
3	#19128098	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128033	C₂	1	1	yes	maximal, characteristic, complemented
5	#19128032	C₂	1	1	yes	maximal, characteristic, complemented
6	#19128018	C₂	1	1	yes	maximal, characteristic, complemented
7	#19127989	C₂	1	1	yes	maximal, characteristic, complemented
8	#19127985	C₂	1	1	yes	maximal, characteristic, complemented
9	#19127929	C₂	1	1	yes	maximal, characteristic, complemented
10	#19127928	C₂	1	1	yes	maximal, characteristic, complemented
11	#19127792	C₂	1	1	yes	maximal, characteristic, complemented
12	(C₂²⋊D₈₃)⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#14581723	C₂	1	1	yes	maximal, characteristic, complemented
14	#106×D₈₃	C₂	1	1	yes	maximal, characteristic, complemented
15	#481×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13050589	—	2	1	no	—
17	C₂×#13050578	C₂²	1	1	yes	characteristic
18	C₂²×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₈₃	C₂²	1	1	yes	characteristic
20	#14581791	C₂²	1	1	yes	characteristic
21	#14581791	C₂²	1	1	yes	characteristic
22	#14581782	—	2	1	no	—
23	#14581774	—	2	1	no	—
24	#14581773	—	2	1	no	—
25	#14581772	C₂²	1	1	yes	characteristic
26	#14581771	—	2	1	no	—
27	#14581771	C₂²	1	1	yes	characteristic
28	#14581770	—	2	1	no	—
29	#14581769	—	2	1	no	—
30	#14581763	C₂²	1	1	yes	characteristic
31	#14581757	C₂²	1	1	yes	characteristic
32	D₄×Dic₈₃	C₂²	1	1	yes	characteristic
33	C₄×C₂²⋊D₈₃	—	2	1	no	—
34	C₄×C₂²⋊D₈₃	C₂²	1	1	yes	characteristic
35	C₂×C₄⋊Dic₈₃	C₂²	1	1	yes	characteristic
36	C₄⋊C₄×D₈₃	C₂²	1	1	yes	characteristic
37	C₄⋊C₄×D₈₃	C₂²	1	1	yes	characteristic
38	#14581724	C₂²	1	1	yes	characteristic
39	#14581724	C₂²	1	1	yes	characteristic
40	#14581723	C₂²	1	1	yes	characteristic
41	#14581723	C₂²	1	1	yes	characteristic
42	#14581723	C₂²	1	1	yes	characteristic
43	#14581723	C₂²	1	1	yes	characteristic
44	#14581723	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	#14581715	C₂²	1	1	yes	characteristic
49	#14581713	C₂²	1	1	yes	characteristic
50	(C₂×D₈₃)⋊C₈	C₂²	1	1	yes	characteristic
51	C₈×Dic₈₃	C₂²	1	1	yes	characteristic
52	#121×C₈₃	C₂²	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₈₃	C₂²	1	1	yes	characteristic
54	#106×C₈₃	C₂²	1	1	yes	characteristic
55	#106×C₈₃	C₂²	1	1	yes	characteristic
56	#102×C₈₃	C₂²	1	1	yes	characteristic
57	#102×C₈₃	C₂²	1	1	yes	characteristic
58	C₂²⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
59	#13050589	—	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×C₂²⋊D₈₃	—	2	1	no	—
64	C₂×C₂²⋊D₈₃	D₄	1	1	yes	characteristic
65	C₂²×Dic₈₃	—	2	1	no	—
66	C₂²×Dic₈₃	—	2	1	no	—
67	C₂×C₄×D₈₃	—	2	1	no	—
68	C₂×C₄×D₈₃	—	2	1	no	—
69	C₂×C₄×D₈₃	C₂³	1	1	yes	characteristic
70	#13050578	—	2	2	no	—
71	C₂×C₈₃⋊C₈	—	2	2	no	—
72	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
73	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
74	C₈×D₈₃	—	2	2	no	—
75	C₄⋊Dic₈₃	—	2	1	no	—
76	C₄⋊Dic₈₃	—	2	1	no	—
77	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
78	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
79	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
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	Dic₈₃⋊C₄	—	2	1	no	—
86	Dic₈₃⋊C₄	D₄	1	1	yes	characteristic
87	C₂²⋊Dic₈₃	—	2	1	no	—
88	C₂²⋊Dic₈₃	—	2	1	no	—
89	C₂²⋊Dic₈₃	—	2	1	no	—
90	C₂²⋊Dic₈₃	D₄	1	1	yes	characteristic
91	(C₂×D₈₃)⋊C₄	—	2	1	no	—
92	(C₂×D₈₃)⋊C₄	D₄	1	1	yes	characteristic
93	C₄×Dic₈₃	—	2	1	no	—
94	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
95	C₂×D₄×C₈₃	—	2	1	no	—
96	C₂×D₄×C₈₃	C₂³	1	1	yes	characteristic
97	C₂²×C₄×C₈₃	—	2	1	no	—
98	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
99	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
100	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
101	C₄⋊C₄×C₈₃	—	2	1	no	—
102	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
103	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
104	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
105	C₂²⋊C₄×C₈₃	—	2	1	no	—
106	C₂³×C₈₃	—	2	1	no	—
107	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
108	C₂²×D₈₃	C₂×D₄	1	1	yes	characteristic
109	D₄×C₈₃	—	2	1	no	—
110	D₄×C₈₃	—	2	1	no	—
111	D₄×C₈₃	—	4	2	no	—
112	C₂×C₄×C₈₃	—	2	1	no	—
113	C₂×C₄×C₈₃	—	2	1	no	—
114	C₂×C₄×C₈₃	—	2	1	no	—
115	C₂×C₄×C₈₃	—	2	1	no	—
116	C₂×C₄×C₈₃	—	2	1	no	—
117	C₂×C₄×C₈₃	—	8	1	no	—
118	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
120	C₂²⋊D₈₃	—	4	2	no	—
121	C₂²⋊D₈₃	SD₁₆	1	4	yes	complemented
122	C₂×Dic₈₃	—	2	1	no	—
123	C₂×Dic₈₃	—	2	1	no	—
124	C₂×Dic₈₃	—	2	1	no	—
125	C₂×Dic₈₃	—	2	1	no	—
126	C₂×Dic₈₃	—	4	2	no	—
127	C₂×Dic₈₃	—	4	2	no	—
128	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
129	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
130	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
131	C₄×D₈₃	—	2	1	no	—
132	C₄×D₈₃	—	2	1	no	—
133	C₄×D₈₃	—	4	2	no	—
134	C₄×D₈₃	—	8	1	no	—
135	Dic₁₆₆	—	2	1	no	—
136	Dic₁₆₆	—	2	1	no	—
137	Dic₁₆₆	—	4	2	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	#41	1	2	yes	complemented
143	C₂²×C₈₃	—	2	1	no	—
144	C₂²×C₈₃	—	2	1	no	—
145	C₂²×C₈₃	—	4	1	no	—
146	C₂²×C₈₃	—	4	1	no	—
147	C₂²×C₈₃	—	4	1	no	—
148	C₂²×C₈₃	C₂×SD₁₆	1	2	yes	—
149	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
150	C₂×D₈₃	—	2	1	no	—
151	C₂×D₈₃	—	2	1	no	—
152	C₂×D₈₃	C₂×SD₁₆	1	2	yes	—
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	C₂×#41	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	—	4	1	no	—
162	Dic₈₃	—	4	1	no	—
163	Dic₈₃	—	4	1	no	—
164	Dic₈₃	—	4	1	no	—
165	Dic₈₃	C₂×SD₁₆	1	2	yes	—
166	C₂×C₈₃	—	2	2	no	—
167	C₂×C₈₃	—	8	1	no	—
168	C₂×C₈₃	#547	1	1	yes	characteristic
169	C₂×C₈₃	C₂²×SD₁₆	1	1	yes	characteristic
170	C₂×C₈₃	#577	1	1	yes	characteristic
171	D₈₃	—	2	2	no	—
172	#3335	—	83	1	no	maximal, Sylow
173	C₈₃	#3335	1	1	yes	characteristic, complemented, Sylow
174	#554	—	83	1	no	—
175	#547	—	83	1	no	—
176	#498	—	83	1	no	—
177	#483	—	83	1	no	—
178	#483	—	83	1	no	—
179	#481	—	83	1	no	—
180	#481	D₈₃	1	1	yes	characteristic, complemented
181	#476	—	83	1	no	—
182	#475	—	83	1	no	—
183	#465	—	83	1	no	—
184	#465	—	83	1	no	—
185	C₄×SD₁₆	—	83	1	no	—
186	C₈×D₄	—	83	1	no	—
187	C₂×#106	—	83	1	no	—
188	C₂×#106	—	83	1	no	—
189	C₂×#41	—	166	1	no	—
190	C₂×SD₁₆	—	83	1	no	—
191	C₂²×C₈	—	83	1	no	—
192	C₂²×C₈	—	83	1	no	—
193	C₄⋊Q₈	—	83	1	no	—
194	C₄⋊Q₈	—	83	1	no	—
195	#125	—	166	1	no	—
196	#123	—	166	1	no	—
197	#123	—	166	1	no	—
198	C₂²⋊Q₈	—	83	1	no	—
199	C₂²⋊Q₈	—	83	1	no	—
200	C₂²⋊Q₈	—	166	1	no	—
201	C₂²⋊Q₈	—	166	1	no	—
202	C₂²⋊Q₈	—	166	1	no	—
203	#121	—	83	1	no	—
204	#121	C₂×D₈₃	1	1	yes	characteristic
205	C₄×Q₈	—	83	1	no	—
206	C₄×D₄	—	83	1	no	—
207	C₄×D₄	—	83	1	no	—
208	C₄×D₄	—	166	1	no	—
209	C₂×C₄⋊C₄	—	83	1	no	—
210	C₂×C₄⋊C₄	—	83	1	no	—
211	C₂×C₄⋊C₄	—	83	1	no	—
212	C₂×C₄⋊C₄	C₂×D₈₃	1	1	yes	characteristic
213	#106	—	83	1	no	—
214	#106	—	83	1	no	—
215	#106	—	83	1	no	—
216	#106	—	83	1	no	—
217	#106	—	83	1	no	—
218	#106	—	83	1	no	—
219	#106	—	83	1	no	—
220	#106	C₂×D₈₃	1	1	yes	characteristic
221	#106	C₂×D₈₃	1	1	yes	characteristic
222	C₄⋊C₈	—	83	1	no	—
223	#103	—	83	1	no	—
224	#103	—	83	1	no	—
225	#103	—	83	1	no	—
226	#102	—	83	1	no	—
227	#102	C₂×D₈₃	1	1	yes	characteristic
228	#102	C₂×D₈₃	1	1	yes	characteristic
229	C₂²⋊C₈	—	83	1	no	—
230	C₂²⋊C₈	C₂×D₈₃	1	1	yes	characteristic
231	C₄×C₈	—	83	1	no	—
232	#41	—	332	4	no	—
233	C₂×Q₈	—	83	1	no	—
234	C₂×Q₈	—	166	1	no	—
235	C₂×Q₈	—	166	1	no	—
236	C₂×D₄	—	2	1	no	—
237	C₂×D₄	—	83	1	no	—
238	C₂×D₄	—	166	1	no	—
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	C₂²×C₄	—	166	1	no	—
244	C₂²×C₄	—	166	1	no	—
245	C₂²×C₄	—	166	1	no	—
246	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
247	SD₁₆	—	166	2	no	—
248	C₂×C₈	—	83	1	no	—
249	C₂×C₈	—	83	1	no	—
250	C₂×C₈	—	166	2	no	—
251	C₂×C₈	—	166	2	no	—
252	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
253	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
254	C₄⋊C₄	—	2	1	no	—
255	C₄⋊C₄	—	83	1	no	—
256	C₄⋊C₄	—	83	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	—	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	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
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	—	166	1	no	—
279	C₂³	—	2	1	no	—
280	C₂³	—	83	1	no	—
281	C₂³	D₄×D₈₃	1	1	yes	characteristic
282	Q₈	—	332	2	no	—
283	Q₈	—	664	1	no	—
284	Q₈	—	166	1	no	—
285	Q₈	—	166	1	no	—
286	D₄	—	2	1	no	—
287	D₄	—	2	1	no	—
288	D₄	—	4	2	no	—
289	D₄	—	332	2	no	—
290	D₄	—	83	4	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	8	1	no	—
297	C₂×C₄	—	332	2	no	—
298	C₂×C₄	—	332	2	no	—
299	C₂×C₄	—	332	2	no	—
300	C₂×C₄	—	83	1	no	—
301	C₂×C₄	—	83	1	no	—
302	C₂×C₄	—	83	1	no	—
303	C₂×C₄	—	664	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	—	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	—	83	2	no	—
315	C₈	—	166	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	4	1	no	—
319	C₂²	—	4	1	no	—
320	C₂²	—	4	1	no	—
321	C₂²	—	83	2	no	—
322	C₂²	—	166	1	no	—
323	C₂²	—	166	1	no	—
324	C₂²	SD₁₆×D₈₃	1	2	yes	—
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	—	332	1	no	—
334	C₄	—	83	2	no	—
335	C₄	—	166	1	no	—
336	C₄	—	166	1	no	—
337	C₄	#41×D₈₃	1	1	yes	characteristic
338	C₄	#14581839	1	1	yes	characteristic
339	C₂	—	2	2	no	—
340	C₂	—	8	1	no	—
341	C₂	—	166	2	no	—
342	C₂	#19128349	1	1	yes	characteristic
343	C₂	C₂×SD₁₆×D₈₃	1	1	yes	characteristic
344	C₂	#19128511	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₃
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#481×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	#19128351	C₂
hypercenter	#481	D₈₃
hypocenter	C₈₃	#3335
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #481

Complements

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

N	H ≅ G/N	type
1. #19128406	C₂	semidirect
3. #19128098	C₂	semidirect
4. #19128033	C₂	semidirect
5. #19128032	C₂	semidirect
6. #19128018	C₂	semidirect
7. #19127989	C₂	semidirect
8. #19127985	C₂	semidirect
9. #19127929	C₂	semidirect
10. #19127928	C₂	semidirect
11. #19127792	C₂	semidirect
12. (C₂²⋊D₈₃)⋊C₈	C₂	semidirect
13. C₂×#14581723	C₂	semidirect
14. #106×D₈₃	C₂	semidirect
15. #481×C₈₃	C₂	semidirect
121. C₂²⋊D₈₃	SD₁₆	semidirect
142. C₈₃⋊C₈	#41	semidirect
173. C₈₃	#3335	semidirect
180. #481	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128406	2
#19128351	2
#19128098	2
#19128033	2
#19128032	2
#19128018	2
#19127989	2
#19127985	2
#19127929	2
#19127928	2
#19127792	2
(C₂²⋊D₈₃)⋊C₈	2
C₂×#14581723	2
#106×D₈₃	2
#481×C₈₃	2
#3335	83