Subgroups — #35309451

Order 21248.

510 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24908269	C₂	1	1	yes	maximal, characteristic
2	#24908268	C₂	1	1	yes	maximal, characteristic
3	#24908268	C₂	1	1	yes	maximal, characteristic
4	#24907622	C₂	1	1	yes	maximal, characteristic, complemented
5	#24907619	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈₃⋊#1273	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₃×#1834	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14581789	C₂²	1	1	yes	characteristic
9	C₂×#14581725	—	2	2	no	—
10	C₂×#14581723	—	2	2	no	—
11	C₂×#14581714	—	2	2	no	—
12	C₂×#14581712	—	2	2	no	—
13	C₂×#14581712	—	2	2	no	—
14	C₂×C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
16	#19127571	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
18	#384×C₈₃	C₂²	1	1	yes	characteristic
19	#384×C₈₃	C₂²	1	1	yes	characteristic
20	#340×C₈₃	—	2	2	no	—
21	C₂²×D₃₃₂	—	2	1	no	—
22	C₂²×D₃₃₂	—	2	1	no	—
23	C₂²×D₃₃₂	—	2	1	no	—
24	C₂²×C₈₃⋊C₈	D₄	1	2	yes	—
25	C₂²×C₈₃⋊C₈	D₄	1	2	yes	—
26	#14581789	—	2	1	no	—
27	#14581789	—	2	1	no	—
28	#14581789	—	2	1	no	—
29	#14581789	Q₈	1	2	yes	—
30	C₂×C₄⋊Dic₈₃	—	2	1	no	—
31	C₂×C₄⋊Dic₈₃	—	2	1	no	—
32	C₂×C₄⋊Dic₈₃	—	2	1	no	—
33	#14581725	—	4	4	no	—
34	#14581723	—	4	4	no	—
35	#14581714	—	4	4	no	—
36	#14581712	—	4	4	no	—
37	#14581712	—	4	4	no	—
38	C₄×C₈₃⋊C₈	—	2	1	no	—
39	C₄×C₈₃⋊C₈	—	2	1	no	—
40	C₄×C₈₃⋊C₈	—	2	1	no	—
41	C₄×C₈₃⋊C₈	—	2	1	no	—
42	C₂²×C₈×C₈₃	D₄	1	2	yes	—
43	C₂×C₄⋊C₄×C₈₃	—	2	1	no	—
44	C₂×C₄⋊C₄×C₈₃	—	2	1	no	—
45	C₂×C₄⋊C₄×C₈₃	—	2	1	no	—
46	C₂×C₄⋊C₄×C₈₃	—	2	1	no	—
47	C₂×C₄⋊C₄×C₈₃	—	2	1	no	—
48	C₂×C₄⋊C₄×C₈₃	—	2	1	no	—
49	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
50	C₄×C₈×C₈₃	—	2	1	no	—
51	C₄×C₈×C₈₃	—	2	1	no	—
52	C₂³×D₈₃	—	4	1	no	—
53	C₂×D₃₃₂	—	2	2	no	—
54	C₂×D₃₃₂	—	2	2	no	—
55	C₂×D₃₃₂	—	2	2	no	—
56	C₂×D₃₃₂	—	4	2	no	—
57	C₂×D₃₃₂	—	4	2	no	—
58	C₂×D₃₃₂	—	4	2	no	—
59	C₂×D₃₃₂	—	4	2	no	—
60	C₂×D₃₃₂	—	4	2	no	—
61	C₂×D₃₃₂	—	4	2	no	—
62	C₂×D₃₃₂	—	4	1	no	—
63	C₂×D₃₃₂	—	4	1	no	—
64	C₂×D₃₃₂	—	4	1	no	—
65	C₂²×Dic₈₃	—	4	1	no	—
66	C₂×C₈₃⋊C₈	—	2	2	no	—
67	C₂×C₈₃⋊C₈	—	2	2	no	—
68	C₂×C₈₃⋊C₈	—	2	2	no	—
69	C₂×C₈₃⋊C₈	—	2	2	no	—
70	C₂×C₈₃⋊C₈	D₈	1	4	yes	—
71	C₂×C₈₃⋊C₈	SD₁₆	1	4	yes	—
72	C₄⋊Dic₈₃	—	4	2	no	—
73	C₄⋊Dic₈₃	—	4	2	no	—
74	C₄⋊Dic₈₃	—	4	2	no	—
75	C₂²×C₄×C₈₃	—	4	1	no	—
76	C₂²×C₄×C₈₃	—	4	1	no	—
77	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₈₃	—	2	2	no	—
81	C₂×C₈×C₈₃	—	2	2	no	—
82	C₂×C₈×C₈₃	—	2	2	no	—
83	C₄⋊C₄×C₈₃	—	4	2	no	—
84	C₄⋊C₄×C₈₃	—	4	2	no	—
85	C₄⋊C₄×C₈₃	—	4	2	no	—
86	C₄⋊C₄×C₈₃	—	4	2	no	—
87	C₄⋊C₄×C₈₃	—	4	2	no	—
88	C₄⋊C₄×C₈₃	—	4	2	no	—
89	C₄²×C₈₃	C₂×Q₈	1	1	yes	characteristic
90	C₄²×C₈₃	#41	1	1	yes	characteristic
91	C₄²×C₈₃	#41	1	1	yes	characteristic
92	C₄²×C₈₃	#41	1	1	yes	characteristic
93	C₂³×C₈₃	#127	1	1	yes	characteristic
94	C₂²×D₈₃	—	4	2	no	—
95	C₂²×D₈₃	—	8	1	no	—
96	C₂²×D₈₃	—	8	1	no	—
97	C₂²×D₈₃	—	8	1	no	—
98	C₂²×D₈₃	—	8	1	no	—
99	C₂²×D₈₃	—	8	1	no	—
100	C₂²×D₈₃	—	8	1	no	—
101	C₂×C₄×C₈₃	—	8	1	no	—
102	C₂×C₄×C₈₃	—	8	1	no	—
103	C₂×C₄×C₈₃	—	8	1	no	—
104	C₂×C₄×C₈₃	—	8	1	no	—
105	C₂×C₄×C₈₃	—	8	1	no	—
106	C₂×C₄×C₈₃	—	8	1	no	—
107	C₂×C₄×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
108	C₂×C₄×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
109	C₂×C₄×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
110	C₂×C₄×C₈₃	#123	1	1	yes	characteristic
111	C₂×C₄×C₈₃	#123	1	1	yes	characteristic
112	C₂×C₄×C₈₃	#123	1	1	yes	characteristic
113	C₂×C₄×C₈₃	C₂×D₈	1	2	yes	—
114	C₂×C₄×C₈₃	C₂×SD₁₆	1	2	yes	—
115	C₂×C₄×C₈₃	#135	1	2	yes	—
116	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	2	yes	—
117	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	2	yes	—
118	C₂×C₄×C₈₃	#137	1	2	yes	—
119	C₂×Dic₈₃	—	8	1	no	—
120	C₂×Dic₈₃	—	8	1	no	—
121	C₂×Dic₈₃	—	8	1	no	—
122	D₃₃₂	—	4	4	no	—
123	D₃₃₂	—	4	4	no	—
124	D₃₃₂	—	4	4	no	—
125	D₃₃₂	—	8	2	no	—
126	D₃₃₂	—	8	2	no	—
127	D₃₃₂	—	8	2	no	—
128	C₈×C₈₃	—	4	2	no	—
129	C₈₃⋊C₈	—	2	4	no	—
130	C₈₃⋊C₈	—	2	4	no	—
131	C₂²×C₈₃	#399	1	1	yes	characteristic
132	C₂²×C₈₃	#492	1	1	yes	characteristic
133	C₂²×C₈₃	#493	1	1	yes	characteristic
134	C₂²×C₈₃	#495	1	1	yes	characteristic
135	C₂²×C₈₃	#496	1	1	yes	characteristic
136	C₂²×C₈₃	#496	1	1	yes	characteristic
137	C₂²×C₈₃	#497	1	1	yes	characteristic
138	C₂×D₈₃	—	8	2	no	—
139	C₂×D₈₃	—	8	2	no	—
140	C₂×D₈₃	—	8	2	no	—
141	C₂×D₈₃	—	16	1	no	—
142	C₂×D₈₃	—	16	1	no	—
143	C₂×D₈₃	—	16	1	no	—
144	C₂×D₈₃	—	16	1	no	—
145	C₄×C₈₃	—	16	1	no	—
146	C₄×C₈₃	—	16	1	no	—
147	C₄×C₈₃	#474	1	2	yes	—
148	C₄×C₈₃	#476	1	2	yes	—
149	C₄×C₈₃	#478	1	2	yes	—
150	C₄×C₈₃	#480	1	2	yes	—
151	C₄×C₈₃	#481	1	2	yes	—
152	C₄×C₈₃	#482	1	2	yes	—
153	Dic₈₃	—	16	1	no	—
154	#10672	—	83	1	no	maximal, Sylow
155	C₂×C₈₃	#1706	1	1	yes	characteristic
156	C₂×C₈₃	#1714	1	1	yes	characteristic
157	C₂×C₈₃	#1720	1	1	yes	characteristic
158	C₂×C₈₃	#1721	1	1	yes	characteristic
159	C₂×C₈₃	#2092	1	1	yes	characteristic
160	C₂×C₈₃	#2093	1	1	yes	characteristic
161	C₂×C₈₃	#2094	1	1	yes	characteristic
162	D₈₃	—	16	2	no	—
163	#1957	—	83	1	no	—
164	#1957	—	83	1	no	—
165	#1957	—	83	1	no	—
166	#1845	—	83	1	no	—
167	#1844	—	83	1	no	—
168	#1834	D₈₃	1	1	yes	characteristic, complemented
169	#1273	—	83	1	no	—
170	C₈₃	#10672	1	1	yes	characteristic, complemented, Sylow
171	C₂×#127	—	83	1	no	—
172	C₂×#107	—	166	2	no	—
173	C₂×#106	—	166	2	no	—
174	C₂×#102	—	166	2	no	—
175	C₂×#102	—	166	2	no	—
176	C₂×#102	—	166	2	no	—
177	C₂×C₄×C₈	—	83	1	no	—
178	C₂×C₄×C₈	—	83	1	no	—
179	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
180	#384	—	83	1	no	—
181	#384	C₂×D₈₃	1	1	yes	characteristic
182	#384	C₂×D₈₃	1	1	yes	characteristic
183	#340	—	2	2	no	—
184	C₂²×D₄	—	166	1	no	—
185	C₂²×D₄	—	166	1	no	—
186	C₂²×D₄	—	166	1	no	—
187	C₂²×C₈	—	83	2	no	—
188	C₂²×C₈	—	83	2	no	—
189	C₂²×C₈	C₂²⋊D₈₃	1	2	yes	—
190	#127	—	83	2	no	—
191	#127	—	166	1	no	—
192	#127	—	166	1	no	—
193	#127	—	166	1	no	—
194	C₂×C₄⋊C₄	—	2	1	no	—
195	C₂×C₄⋊C₄	—	2	1	no	—
196	C₂×C₄⋊C₄	—	2	1	no	—
197	C₂×C₄⋊C₄	—	2	1	no	—
198	C₂×C₄⋊C₄	—	2	1	no	—
199	C₂×C₄⋊C₄	—	2	1	no	—
200	C₂×C₄⋊C₄	—	166	1	no	—
201	C₂×C₄⋊C₄	—	166	1	no	—
202	C₂×C₄⋊C₄	—	166	1	no	—
203	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
204	#107	—	332	4	no	—
205	#106	—	332	4	no	—
206	#102	—	332	4	no	—
207	#102	—	332	4	no	—
208	#102	—	332	4	no	—
209	C₄×C₈	—	2	1	no	—
210	C₄×C₈	—	2	1	no	—
211	C₄×C₈	—	166	1	no	—
212	C₄×C₈	—	166	1	no	—
213	C₄×C₈	—	166	1	no	—
214	C₄×C₈	—	166	1	no	—
215	C₂⁴	—	332	1	no	—
216	C₂×D₄	—	332	1	no	—
217	C₂×D₄	—	332	1	no	—
218	C₂×D₄	—	332	1	no	—
219	C₂×D₄	—	332	2	no	—
220	C₂×D₄	—	332	2	no	—
221	C₂×D₄	—	332	2	no	—
222	C₂×D₄	—	332	2	no	—
223	C₂×D₄	—	332	2	no	—
224	C₂×D₄	—	332	2	no	—
225	C₂×D₄	—	166	2	no	—
226	C₂×D₄	—	166	2	no	—
227	C₂×D₄	—	166	2	no	—
228	C₂²×C₄	—	4	1	no	—
229	C₂²×C₄	—	4	1	no	—
230	C₂²×C₄	—	332	1	no	—
231	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
232	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
233	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	83	4	no	—
238	C₂×C₈	—	83	4	no	—
239	C₂×C₈	—	166	2	no	—
240	C₂×C₈	—	166	2	no	—
241	C₂×C₈	—	166	2	no	—
242	C₂×C₈	—	166	2	no	—
243	C₄⋊C₄	—	4	2	no	—
244	C₄⋊C₄	—	4	2	no	—
245	C₄⋊C₄	—	4	2	no	—
246	C₄⋊C₄	—	4	2	no	—
247	C₄⋊C₄	—	4	2	no	—
248	C₄⋊C₄	—	4	2	no	—
249	C₄⋊C₄	—	332	2	no	—
250	C₄⋊C₄	—	332	2	no	—
251	C₄⋊C₄	—	332	2	no	—
252	C₄²	Q₈×D₈₃	1	1	yes	characteristic
253	C₄²	#13050590	1	1	yes	characteristic
254	C₄²	#13050590	1	1	yes	characteristic
255	C₄²	#13050591	1	1	yes	characteristic
256	C₂³	—	332	2	no	—
257	C₂³	—	664	1	no	—
258	C₂³	—	664	1	no	—
259	C₂³	—	664	1	no	—
260	C₂³	—	664	1	no	—
261	C₂³	—	664	1	no	—
262	C₂³	—	664	1	no	—
263	C₂³	#14581788	1	1	yes	characteristic
264	D₄	—	332	4	no	—
265	D₄	—	332	4	no	—
266	D₄	—	332	4	no	—
267	D₄	—	664	2	no	—
268	D₄	—	664	2	no	—
269	D₄	—	664	2	no	—
270	C₂×C₄	—	8	1	no	—
271	C₂×C₄	—	8	1	no	—
272	C₂×C₄	—	8	1	no	—
273	C₂×C₄	—	8	1	no	—
274	C₂×C₄	—	8	1	no	—
275	C₂×C₄	—	8	1	no	—
276	C₂×C₄	—	664	1	no	—
277	C₂×C₄	—	664	1	no	—
278	C₂×C₄	—	664	1	no	—
279	C₂×C₄	#14581768	1	1	yes	characteristic
280	C₂×C₄	#14581770	1	1	yes	characteristic
281	C₂×C₄	#14581770	1	1	yes	characteristic
282	C₂×C₄	#14581775	1	1	yes	characteristic
283	C₂×C₄	#14581777	1	1	yes	characteristic
284	C₂×C₄	#14581777	1	1	yes	characteristic
285	C₂×C₄	D₈×D₈₃	1	2	yes	—
286	C₂×C₄	SD₁₆×D₈₃	1	2	yes	—
287	C₂×C₄	#14581812	1	2	yes	—
288	C₂×C₄	#14581817	1	2	yes	—
289	C₂×C₄	#14581819	1	2	yes	—
290	C₂×C₄	#14581824	1	2	yes	—
291	C₈	—	4	2	no	—
292	C₈	—	166	4	no	—
293	C₈	—	166	4	no	—
294	C₂²	—	1328	1	no	—
295	C₂²	—	1328	1	no	—
296	C₂²	—	1328	1	no	—
297	C₂²	—	1328	1	no	—
298	C₂²	—	664	2	no	—
299	C₂²	—	664	2	no	—
300	C₂²	—	664	2	no	—
301	C₂²	#19127636	1	1	yes	characteristic
302	C₂²	#19128072	1	1	yes	characteristic
303	C₂²	#19128076	1	1	yes	characteristic
304	C₂²	#19128083	1	1	yes	characteristic
305	C₂²	#19128087	1	1	yes	characteristic
306	C₂²	#19128091	1	1	yes	characteristic
307	C₂²	#19128094	1	1	yes	characteristic
308	C₄	—	16	1	no	—
309	C₄	—	16	1	no	—
310	C₄	—	1328	1	no	—
311	C₄	#19127980	1	2	yes	—
312	C₄	#19127990	1	2	yes	—
313	C₄	#19127998	1	2	yes	—
314	C₄	#19128016	1	2	yes	—
315	C₄	#19128021	1	2	yes	—
316	C₄	#19128025	1	2	yes	—
317	C₂	—	1328	2	no	—
318	C₂	#24907054	1	1	yes	characteristic
319	C₂	#24907087	1	1	yes	characteristic
320	C₂	#24907120	1	1	yes	characteristic
321	C₂	#24907131	1	1	yes	characteristic
322	C₂	#24908987	1	1	yes	characteristic
323	C₂	#24908988	1	1	yes	characteristic
324	C₂	#24908993	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14581788
commutator, cosocle	C₂×C₄²×C₈₃	C₂³
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₈₃×#1834	C₂
socle	C₂³×C₈₃	#127
prime radical	C₂×#14581789	C₂²
hypercenter	#1834	D₈₃
hypocenter	C₈₃	#10672
LCS	G ▷ C₂×C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1834

Complements

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

N	H ≅ G/N	type
4. #24907622	C₂	semidirect
5. #24907619	C₂	semidirect
6. C₈₃⋊#1273	C₂	semidirect
7. C₈₃×#1834	C₂	semidirect
168. #1834	D₈₃	semidirect
170. C₈₃	#10672	semidirect

Maximal subgroups

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

subgroup	index
#24908269	2
#24908268	2
#24907622	2
#24907619	2
C₈₃⋊#1273	2
C₈₃×#1834	2
#10672	83