Subgroups — #24920256

Order 10624.

384 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128434	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×Dic₁₆₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#19128061	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128059	C₂	1	1	yes	maximal, characteristic, complemented
5	#19128046	C₂	1	1	yes	maximal, characteristic, complemented
6	#19127991	C₂	1	1	yes	maximal, characteristic, complemented
7	#19127990	C₂	1	1	yes	maximal, characteristic, complemented
8	#19127985	C₂	1	1	yes	maximal, characteristic, complemented
9	#19127916	C₂	1	1	yes	maximal, characteristic, complemented
10	#19127914	C₂	1	1	yes	maximal, characteristic, complemented
11	#19127894	C₂	1	1	yes	maximal, characteristic
12	#19127829	C₂	1	1	yes	maximal, characteristic, complemented
13	#19127810	C₂	1	1	yes	maximal, characteristic, complemented
14	#19127792	C₂	1	1	yes	maximal, characteristic, complemented
15	#442×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×Dic₁₆₆	—	2	1	no	—
17	C₂×#13050578	C₂²	1	1	yes	characteristic
18	C₂×#13050575	C₂²	1	1	yes	characteristic
19	#14581792	C₂²	1	1	yes	characteristic
20	#14581791	C₂²	1	1	yes	characteristic
21	#14581791	C₂²	1	1	yes	characteristic
22	#14581788	C₂²	1	1	yes	characteristic
23	#14581780	C₂²	1	1	yes	characteristic
24	#14581779	C₂²	1	1	yes	characteristic
25	#14581778	—	2	1	no	—
26	#14581775	—	2	1	no	—
27	#14581774	—	2	1	no	—
28	#14581771	—	2	1	no	—
29	#14581769	—	2	1	no	—
30	#14581769	—	2	1	no	—
31	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
32	#14581757	C₂²	1	1	yes	characteristic
33	D₄×Dic₈₃	C₂²	1	1	yes	characteristic
34	D₄×Dic₈₃	C₂²	1	1	yes	characteristic
35	#14581754	C₂²	1	1	yes	characteristic
36	C₄×C₂²⋊D₈₃	—	2	1	no	—
37	#14581723	C₂²	1	1	yes	characteristic
38	#14581723	C₂²	1	1	yes	characteristic
39	C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
40	Dic₈₃⋊C₈	C₂²	1	1	yes	characteristic
41	Dic₈₃⋊C₈	C₂²	1	1	yes	characteristic
42	#14581717	C₂²	1	1	yes	characteristic
43	#14581717	C₂²	1	1	yes	characteristic
44	#14581715	C₂²	1	1	yes	characteristic
45	#14581714	C₂²	1	1	yes	characteristic
46	#14581714	C₂²	1	1	yes	characteristic
47	#14581713	C₂²	1	1	yes	characteristic
48	#14581713	C₂²	1	1	yes	characteristic
49	#14581712	C₂²	1	1	yes	characteristic
50	#14581702	C₂²	1	1	yes	characteristic
51	C₈×Dic₈₃	C₂²	1	1	yes	characteristic
52	C₂×D₈×C₈₃	C₂²	1	1	yes	characteristic
53	C₄×D₄×C₈₃	C₂²	1	1	yes	characteristic
54	C₄×D₄×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	#97×C₈₃	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₆₆	—	2	1	no	—
60	C₂×Dic₁₆₆	—	2	1	no	—
61	C₂×Dic₁₆₆	—	2	1	no	—
62	C₂×Dic₁₆₆	—	4	4	no	—
63	C₂×Dic₁₆₆	D₄	1	1	yes	characteristic
64	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
65	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
66	C₂×C₂²⋊D₈₃	—	2	1	no	—
67	C₂×C₂²⋊D₈₃	—	2	1	no	—
68	C₂²×Dic₈₃	—	2	1	no	—
69	C₂²×Dic₈₃	—	2	1	no	—
70	C₂×C₄×D₈₃	—	2	1	no	—
71	#13050578	—	2	2	no	—
72	#13050575	—	2	2	no	—
73	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
74	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
75	C₄⋊Dic₈₃	—	2	1	no	—
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	D₄	1	1	yes	characteristic
83	Dic₈₃⋊C₄	D₄	1	1	yes	characteristic
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	—	2	1	no	—
91	C₄×Dic₈₃	—	2	1	no	—
92	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
93	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
94	C₂×D₄×C₈₃	C₂³	1	1	yes	characteristic
95	C₂×D₄×C₈₃	C₂³	1	1	yes	characteristic
96	C₂²×C₄×C₈₃	—	2	1	no	—
97	C₂²×C₄×C₈₃	—	2	1	no	—
98	D₈×C₈₃	—	2	2	no	—
99	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
100	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
101	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
103	C₂²⋊C₄×C₈₃	—	2	1	no	—
104	C₂²⋊C₄×C₈₃	—	2	1	no	—
105	C₄²×C₈₃	C₂³	1	1	yes	characteristic
106	C₂³×C₈₃	—	2	1	no	—
107	C₂³×C₈₃	—	2	1	no	—
108	C₂²×D₈₃	—	2	1	no	—
109	D₄×C₈₃	—	2	1	no	—
110	D₄×C₈₃	—	2	1	no	—
111	D₄×C₈₃	—	2	1	no	—
112	D₄×C₈₃	#41	1	2	yes	—
113	C₂×C₄×C₈₃	—	2	1	no	—
114	C₂×C₄×C₈₃	—	2	1	no	—
115	C₂×C₄×C₈₃	—	4	2	no	—
116	C₂×C₄×C₈₃	—	8	1	no	—
117	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
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	—	8	1	no	—
122	C₂×Dic₈₃	—	2	1	no	—
123	C₂×Dic₈₃	—	2	1	no	—
124	C₂×Dic₈₃	—	2	1	no	—
125	C₂×Dic₈₃	—	4	2	no	—
126	C₂×Dic₈₃	—	4	2	no	—
127	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
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	D₃₃₂	—	2	1	no	—
132	D₃₃₂	—	2	1	no	—
133	C₄×D₈₃	—	8	1	no	—
134	Dic₁₆₆	—	2	2	no	—
135	Dic₁₆₆	—	2	1	no	—
136	Dic₁₆₆	—	2	1	no	—
137	Dic₁₆₆	—	4	2	no	—
138	Dic₁₆₆	—	4	2	no	—
139	Dic₁₆₆	—	4	2	no	—
140	C₈×C₈₃	—	2	1	no	—
141	C₈×C₈₃	—	2	1	no	—
142	C₈₃⋊C₈	—	2	1	no	—
143	C₈₃⋊C₈	—	2	1	no	—
144	C₂²×C₈₃	—	2	2	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	—	4	1	no	—
150	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
151	C₂×D₈₃	—	4	1	no	—
152	C₂×D₈₃	—	4	1	no	—
153	C₂×D₈₃	—	4	1	no	—
154	C₄×C₈₃	—	2	1	no	—
155	C₄×C₈₃	—	2	1	no	—
156	C₄×C₈₃	—	4	1	no	—
157	C₄×C₈₃	—	4	1	no	—
158	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
159	C₄×C₈₃	2₋¹⁺⁴	1	1	yes	characteristic
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	Dic₈₃	AGL(1,ℤ/8)	1	2	yes	—
167	C₂×C₈₃	—	4	2	no	—
168	C₂×C₈₃	—	8	1	no	—
169	C₂×C₈₃	#548	1	1	yes	characteristic
170	C₂×C₈₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
171	C₂×C₈₃	#577	1	1	yes	characteristic
172	D₈₃	—	8	1	no	—
173	#3373	—	83	1	no	maximal, Sylow
174	C₈₃	#3373	1	1	yes	characteristic, complemented, Sylow
175	#559	—	83	1	no	—
176	D₄×Q₈	—	83	1	no	—
177	#489	—	83	1	no	—
178	#489	—	83	1	no	—
179	#486	—	83	1	no	—
180	#476	—	83	1	no	—
181	#476	—	83	1	no	—
182	#475	—	83	1	no	—
183	#463	—	83	1	no	—
184	#463	—	83	1	no	—
185	#459	—	83	1	no	—
186	#446	—	83	1	no	—
187	#442	—	83	1	no	—
188	#442	D₈₃	1	1	yes	characteristic, complemented
189	C₄×SD₁₆	—	83	1	no	—
190	C₂²×Q₈	—	166	1	no	—
191	C₂×SD₁₆	—	83	1	no	—
192	C₂×D₈	—	83	1	no	—
193	C₂×D₈	C₂×D₈₃	1	1	yes	characteristic
194	C₄⋊Q₈	—	83	1	no	—
195	C₄⋊Q₈	—	83	1	no	—
196	C₄⋊Q₈	—	83	1	no	—
197	#127	—	83	1	no	—
198	#124	—	83	1	no	—
199	#124	—	83	1	no	—
200	#124	—	166	1	no	—
201	#123	—	166	1	no	—
202	#123	—	166	1	no	—
203	C₂²⋊Q₈	—	166	1	no	—
204	C₂²⋊Q₈	—	166	1	no	—
205	C₂²⋊Q₈	—	166	1	no	—
206	C₄×Q₈	—	83	1	no	—
207	C₄×Q₈	—	83	1	no	—
208	C₄×D₄	—	83	1	no	—
209	C₄×D₄	—	83	1	no	—
210	C₄×D₄	—	83	1	no	—
211	C₄×D₄	—	166	1	no	—
212	C₄×D₄	C₂×D₈₃	1	1	yes	characteristic
213	C₄×D₄	C₂×D₈₃	1	1	yes	characteristic
214	#106	—	83	1	no	—
215	#106	—	83	1	no	—
216	#106	C₂×D₈₃	1	1	yes	characteristic
217	C₄⋊C₈	—	83	1	no	—
218	C₄⋊C₈	—	83	1	no	—
219	C₄⋊C₈	—	83	1	no	—
220	#103	—	83	1	no	—
221	#103	—	83	1	no	—
222	#103	—	83	1	no	—
223	#102	—	83	1	no	—
224	#102	—	83	1	no	—
225	#102	—	83	1	no	—
226	#102	—	83	1	no	—
227	#102	—	83	1	no	—
228	#102	C₂×D₈₃	1	1	yes	characteristic
229	#102	C₂×D₈₃	1	1	yes	characteristic
230	#97	—	83	1	no	—
231	#97	C₂×D₈₃	1	1	yes	characteristic
232	C₄×C₈	—	83	1	no	—
233	C₂×Q₈	—	332	4	no	—
234	C₂×Q₈	—	83	1	no	—
235	C₂×Q₈	—	83	1	no	—
236	C₂×Q₈	—	166	1	no	—
237	C₂×Q₈	—	166	1	no	—
238	C₂×Q₈	—	166	1	no	—
239	C₂×D₄	—	83	1	no	—
240	C₂×D₄	—	166	1	no	—
241	C₂×D₄	—	166	1	no	—
242	C₂×D₄	C₂²×D₈₃	1	1	yes	characteristic
243	C₂×D₄	C₂²×D₈₃	1	1	yes	characteristic
244	C₂²×C₄	—	2	1	no	—
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	166	1	no	—
247	C₂²×C₄	—	166	1	no	—
248	C₂²×C₄	—	166	1	no	—
249	SD₁₆	—	166	2	no	—
250	D₈	—	2	2	no	—
251	D₈	—	166	2	no	—
252	C₂×C₈	—	83	1	no	—
253	C₂×C₈	—	83	1	no	—
254	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
255	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
256	C₄⋊C₄	—	83	1	no	—
257	C₄⋊C₄	—	83	1	no	—
258	C₄⋊C₄	—	83	1	no	—
259	C₄⋊C₄	—	83	1	no	—
260	C₄⋊C₄	—	83	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	C₂²×D₈₃	1	1	yes	characteristic
266	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
267	C₂²⋊C₄	—	2	1	no	—
268	C₂²⋊C₄	—	2	1	no	—
269	C₂²⋊C₄	—	166	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	C₄²	—	83	1	no	—
277	C₄²	—	83	1	no	—
278	C₄²	—	166	1	no	—
279	C₄²	C₂²×D₈₃	1	1	yes	characteristic
280	C₂³	—	2	1	no	—
281	C₂³	—	2	1	no	—
282	C₂³	—	166	1	no	—
283	Q₈	—	332	2	no	—
284	Q₈	—	332	2	no	—
285	Q₈	—	332	2	no	—
286	Q₈	—	166	2	no	—
287	Q₈	—	166	1	no	—
288	Q₈	—	166	1	no	—
289	D₄	—	2	1	no	—
290	D₄	—	2	1	no	—
291	D₄	—	2	1	no	—
292	D₄	—	332	2	no	—
293	D₄	—	664	1	no	—
294	D₄	—	166	1	no	—
295	D₄	—	166	1	no	—
296	D₄	#13050590	1	2	yes	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	8	1	no	—
301	C₂×C₄	—	332	2	no	—
302	C₂×C₄	—	332	2	no	—
303	C₂×C₄	—	83	1	no	—
304	C₂×C₄	—	83	1	no	—
305	C₂×C₄	—	83	1	no	—
306	C₂×C₄	—	83	1	no	—
307	C₂×C₄	—	664	1	no	—
308	C₂×C₄	—	166	1	no	—
309	C₂×C₄	—	166	1	no	—
310	C₂×C₄	—	166	1	no	—
311	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
312	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
313	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
314	C₈	—	2	1	no	—
315	C₈	—	2	1	no	—
316	C₈	—	166	1	no	—
317	C₈	—	166	1	no	—
318	C₂²	—	2	2	no	—
319	C₂²	—	4	1	no	—
320	C₂²	—	4	1	no	—
321	C₂²	—	4	1	no	—
322	C₂²	—	4	1	no	—
323	C₂²	—	4	1	no	—
324	C₂²	—	332	1	no	—
325	C₂²	—	332	1	no	—
326	C₂²	—	332	1	no	—
327	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	4	1	no	—
331	C₄	—	4	1	no	—
332	C₄	—	332	1	no	—
333	C₄	—	332	1	no	—
334	C₄	—	332	1	no	—
335	C₄	—	83	2	no	—
336	C₄	—	166	1	no	—
337	C₄	—	166	1	no	—
338	C₄	—	166	1	no	—
339	C₄	C₂×#13050590	1	1	yes	characteristic
340	C₄	#14581839	1	1	yes	characteristic
341	C₂	—	4	2	no	—
342	C₂	—	8	1	no	—
343	C₂	—	664	1	no	—
344	C₂	#19128360	1	1	yes	characteristic
345	C₂	AGL(1,ℤ/8)×D₈₃	1	1	yes	characteristic
346	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	#442×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	#19127894	C₂
hypercenter	#442	D₈₃
hypocenter	C₈₃	#3373
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #442

Complements

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

N	H ≅ G/N	type
1. #19128434	C₂	semidirect
2. D₄×Dic₁₆₆	C₂	semidirect
3. #19128061	C₂	semidirect
4. #19128059	C₂	semidirect
5. #19128046	C₂	semidirect
6. #19127991	C₂	semidirect
7. #19127990	C₂	semidirect
8. #19127985	C₂	semidirect
9. #19127916	C₂	semidirect
10. #19127914	C₂	semidirect
12. #19127829	C₂	semidirect
13. #19127810	C₂	semidirect
14. #19127792	C₂	semidirect
15. #442×C₈₃	C₂	semidirect
174. C₈₃	#3373	semidirect
188. #442	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128434	2
D₄×Dic₁₆₆	2
#19128061	2
#19128059	2
#19128046	2
#19127991	2
#19127990	2
#19127985	2
#19127916	2
#19127914	2
#19127894	2
#19127829	2
#19127810	2
#19127792	2
#442×C₈₃	2
#3373	83