Subgroups — #35295137

Order 21248.

516 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24916424	C₂	1	1	yes	maximal, characteristic
2	C₄×#14581720	C₂	1	1	yes	maximal, characteristic
3	C₄²×C₈₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#24905843	C₂	1	1	yes	maximal, characteristic
5	#24905753	C₂	1	1	yes	maximal, characteristic, complemented
6	#24905299	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₃×#1276	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#13050590	C₂²	1	1	yes	characteristic
9	#19127828	C₄	1	2	yes	complemented
10	#19127778	C₄	1	2	yes	complemented
11	#19127769	—	2	2	no	—
12	#19127752	—	2	1	no	—
13	C₂×#14581720	C₄	1	2	yes	—
14	C₄×#13050574	C₂²	1	1	yes	characteristic
15	C₄×#13050573	—	2	1	no	—
16	C₂×C₄×C₈₃⋊C₈	—	2	2	no	—
17	C₂×C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
19	#19127461	C₄	1	2	yes	—
20	#19127431	C₄	1	2	yes	complemented
21	#19127428	C₄	1	2	yes	complemented
22	#19127423	—	2	1	no	—
23	#19127422	—	2	1	no	—
24	#431×C₈₃	C₂²	1	1	yes	characteristic
25	#431×C₈₃	C₂²	1	1	yes	characteristic
26	C₄³×C₈₃	C₂²	1	1	yes	characteristic
27	C₂×#13050590	C₂×C₄	1	1	yes	characteristic
28	C₂×#13050574	C₂×C₄	1	2	yes	—
29	C₂×#13050573	—	2	2	no	—
30	C₂²×C₈₃⋊C₈	—	2	1	no	—
31	C₂²×C₈₃⋊C₈	D₄	1	2	yes	—
32	C₄×Dic₁₆₆	C₂×C₄	1	1	yes	characteristic
33	C₄×Dic₁₆₆	C₂×C₄	1	1	yes	characteristic
34	C₄×D₃₃₂	C₂×C₄	1	1	yes	characteristic
35	C₄×D₃₃₂	C₂×C₄	1	1	yes	characteristic
36	C₄×C₂²⋊D₈₃	—	2	1	no	—
37	C₄×C₂²⋊D₈₃	—	2	1	no	—
38	#14581751	C₂×C₄	1	1	yes	characteristic
39	#14581749	—	2	1	no	—
40	C₄²×D₈₃	—	2	1	no	—
41	Dic₈₃⋊C₈	—	2	2	no	—
42	#14581720	—	2	4	no	—
43	(C₂×D₈₃)⋊C₈	—	2	2	no	—
44	#14581703	—	2	1	no	—
45	#14581702	—	2	2	no	—
46	C₄×C₈₃⋊C₈	—	2	8	no	—
47	C₄×C₈₃⋊C₈	—	2	1	no	—
48	C₄×C₈₃⋊C₈	—	2	1	no	—
49	C₄×C₈₃⋊C₈	C₂×C₄	1	2	yes	—
50	C₄×C₈₃⋊C₈	C₂×C₄	1	2	yes	—
51	C₄×C₈₃⋊C₈	D₄	1	1	yes	characteristic
52	C₄×C₈₃⋊C₈	D₄	1	1	yes	characteristic
53	C₂×C₄²×C₈₃	—	2	1	no	—
54	C₂×C₄²×C₈₃	—	2	1	no	—
55	C₂×C₄²×C₈₃	C₂×C₄	1	2	yes	—
56	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
57	C₄⋊C₈×C₈₃	C₂×C₄	1	2	yes	—
58	C₄⋊C₈×C₈₃	C₂×C₄	1	2	yes	—
59	C₂²⋊C₈×C₈₃	—	2	1	no	—
60	C₂²⋊C₈×C₈₃	—	2	1	no	—
61	C₄×C₈×C₈₃	—	2	1	no	—
62	C₄×C₈×C₈₃	—	2	1	no	—
63	#13050590	—	2	2	no	—
64	#13050590	—	2	2	no	—
65	C₂×Dic₁₆₆	C₄²	1	1	yes	characteristic
66	C₂×D₃₃₂	C₄²	1	1	yes	characteristic
67	C₂×C₂²⋊D₈₃	—	2	1	no	—
68	C₂×C₄×D₈₃	—	2	1	no	—
69	C₂×C₄×D₈₃	—	2	1	no	—
70	C₂×C₄×D₈₃	—	2	1	no	—
71	#13050574	—	2	2	no	—
72	#13050574	—	2	2	no	—
73	#13050573	—	4	4	no	—
74	C₂×C₈₃⋊C₈	—	2	4	no	—
75	C₂×C₈₃⋊C₈	—	2	2	no	—
76	C₂×C₈₃⋊C₈	—	2	4	no	—
77	C₂×C₈₃⋊C₈	—	2	2	no	—
78	C₂×C₈₃⋊C₈	C₂²⋊C₄	1	2	yes	—
79	C₂×C₈₃⋊C₈	C₂²⋊C₄	1	2	yes	—
80	C₄⋊Dic₈₃	C₄²	1	1	yes	characteristic
81	C₄⋊Dic₈₃	C₄²	1	1	yes	characteristic
82	Dic₈₃⋊C₄	—	2	1	no	—
83	Dic₈₃⋊C₄	—	2	1	no	—
84	C₂²⋊Dic₈₃	—	2	1	no	—
85	(C₂×D₈₃)⋊C₄	—	2	1	no	—
86	(C₂×D₈₃)⋊C₄	—	2	1	no	—
87	C₄×Dic₈₃	—	2	1	no	—
88	C₄×Dic₈₃	—	2	1	no	—
89	C₄×Dic₈₃	—	2	1	no	—
90	C₂²×C₄×C₈₃	—	2	2	no	—
91	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₈₃	#41	1	1	yes	characteristic
94	C₂×C₈×C₈₃	—	2	2	no	—
95	C₂×C₈×C₈₃	—	2	2	no	—
96	C₄²×C₈₃	—	2	4	no	—
97	C₄²×C₈₃	—	2	4	no	—
98	C₄²×C₈₃	—	2	2	no	—
99	C₄²×C₈₃	M₁₆	1	4	yes	complemented
100	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
101	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
102	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₈₃	#41	1	1	yes	characteristic
104	C₂³×C₈₃	C₄×D₄	1	1	yes	characteristic
105	C₂²×D₈₃	—	2	1	no	—
106	C₂×C₄×C₈₃	—	2	4	no	—
107	C₂×C₄×C₈₃	—	2	4	no	—
108	C₂×C₄×C₈₃	—	2	4	no	—
109	C₂×C₄×C₈₃	—	2	2	no	—
110	C₂×C₄×C₈₃	—	2	2	no	—
111	C₂×C₄×C₈₃	C₂×C₄²	1	1	yes	characteristic
112	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₈₃	#117	1	1	yes	characteristic
114	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
117	C₂×C₄×C₈₃	C₂×M₁₆	1	2	yes	—
118	C₂×C₄×C₈₃	#131	1	2	yes	—
119	C₂²⋊D₈₃	—	4	2	no	—
120	C₂×Dic₈₃	—	2	1	no	—
121	C₂×Dic₈₃	—	2	1	no	—
122	C₂×Dic₈₃	—	2	1	no	—
123	D₃₃₂	—	2	1	no	—
124	D₃₃₂	—	2	1	no	—
125	C₄×D₈₃	—	4	2	no	—
126	C₄×D₈₃	—	4	2	no	—
127	C₄×D₈₃	—	4	2	no	—
128	Dic₁₆₆	—	2	1	no	—
129	Dic₁₆₆	—	2	1	no	—
130	C₈×C₈₃	—	4	2	no	—
131	C₈×C₈₃	—	4	2	no	—
132	C₈₃⋊C₈	—	2	4	no	—
133	C₈₃⋊C₈	—	2	2	no	—
134	C₈₃⋊C₈	—	4	2	no	—
135	C₈₃⋊C₈	#104	1	4	yes	complemented
136	C₂²×C₈₃	—	2	1	no	—
137	C₂²×C₈₃	—	2	1	no	—
138	C₂²×C₈₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
139	C₂²×C₈₃	#435	1	2	yes	—
140	C₂×D₈₃	—	4	1	no	—
141	C₂×D₈₃	—	4	1	no	—
142	C₂×D₈₃	—	4	1	no	—
143	C₄×C₈₃	—	2	8	no	—
144	C₄×C₈₃	—	2	1	no	—
145	C₄×C₈₃	—	2	1	no	—
146	C₄×C₈₃	C₄×M₁₆	1	1	yes	characteristic
147	C₄×C₈₃	#405	1	1	yes	characteristic
148	C₄×C₈₃	C₂×#104	1	2	yes	—
149	C₄×C₈₃	#436	1	2	yes	—
150	C₄×C₈₃	#443	1	2	yes	—
151	Dic₈₃	—	4	1	no	—
152	Dic₈₃	—	4	1	no	—
153	Dic₈₃	—	4	1	no	—
154	#8194	—	83	1	no	maximal, Sylow
155	C₂×C₈₃	—	2	2	no	—
156	C₂×C₈₃	#1753	1	1	yes	characteristic
157	C₂×C₈₃	C₄×#104	1	1	yes	characteristic
158	C₂×C₈₃	#1764	1	1	yes	characteristic
159	D₈₃	—	8	1	no	—
160	#2965	—	83	1	no	—
161	C₄×#104	—	83	1	no	—
162	C₄²×C₈	—	83	1	no	—
163	#1476	—	83	1	no	—
164	#1451	—	83	1	no	—
165	#1278	—	83	1	no	—
166	#1276	D₈₃	1	1	yes	characteristic, complemented
167	C₈₃	#8194	1	1	yes	characteristic, complemented, Sylow
168	C₄×#41	—	83	1	no	—
169	#446	—	83	2	no	—
170	#436	—	83	2	no	—
171	#435	—	166	2	no	—
172	#431	—	166	1	no	—
173	#431	C₂×D₈₃	1	1	yes	characteristic
174	#431	C₂×D₈₃	1	1	yes	characteristic
175	C₂×#104	—	83	2	no	—
176	C₄×M₁₆	—	83	1	no	—
177	C₄×M₁₆	—	166	1	no	—
178	C₂×C₄×C₈	—	83	1	no	—
179	C₂×C₄×C₈	—	83	1	no	—
180	C₂×C₄×C₈	—	166	2	no	—
181	C₄³	C₂×D₈₃	1	1	yes	characteristic
182	#339	—	83	2	no	—
183	Q₈⋊C₈	—	83	2	no	—
184	D₄⋊C₈	—	83	2	no	—
185	#322	—	166	1	no	—
186	#322	—	166	1	no	—
187	C₂×#41	—	83	1	no	—
188	C₂×M₁₆	—	83	2	no	—
189	C₂×M₁₆	—	166	2	no	—
190	C₂²×C₈	—	83	2	no	—
191	C₂²×C₈	—	166	1	no	—
192	C₄×Q₈	—	83	1	no	—
193	C₄×Q₈	—	83	1	no	—
194	C₄×D₄	—	83	1	no	—
195	C₄×D₄	—	83	1	no	—
196	C₄×D₄	—	166	1	no	—
197	C₄×D₄	—	166	1	no	—
198	#117	—	83	1	no	—
199	#117	—	166	1	no	—
200	C₂×C₄²	—	2	1	no	—
201	C₂×C₄²	—	2	1	no	—
202	C₂×C₄²	—	166	1	no	—
203	C₂×C₄²	C₄×D₈₃	1	2	yes	—
204	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
205	C₄⋊C₈	—	166	2	no	—
206	C₄⋊C₈	C₄×D₈₃	1	2	yes	—
207	C₄⋊C₈	C₄×D₈₃	1	2	yes	—
208	#104	—	166	4	no	—
209	C₂²⋊C₈	—	2	1	no	—
210	C₂²⋊C₈	—	2	1	no	—
211	C₂²⋊C₈	—	166	2	no	—
212	#97	—	166	2	no	—
213	#97	—	166	1	no	—
214	C₄×C₈	—	2	1	no	—
215	C₄×C₈	—	2	1	no	—
216	C₄×C₈	—	83	2	no	—
217	C₄×C₈	—	83	2	no	—
218	C₄×C₈	—	83	1	no	—
219	C₄×C₈	—	83	1	no	—
220	C₄×C₈	—	166	8	no	—
221	C₄×C₈	—	166	1	no	—
222	C₄×C₈	—	166	1	no	—
223	#41	—	166	2	no	—
224	#41	—	166	2	no	—
225	C₂×Q₈	—	83	1	no	—
226	C₂×D₄	—	83	1	no	—
227	C₂×D₄	—	166	1	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	166	1	no	—
230	C₂²×C₄	—	166	1	no	—
231	C₂²×C₄	—	166	1	no	—
232	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
233	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
234	C₂²×C₄	#13050589	1	1	yes	characteristic
235	M₁₆	—	332	4	no	—
236	M₁₆	—	166	2	no	—
237	M₁₆	—	166	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	83	2	no	—
241	C₂×C₈	—	83	2	no	—
242	C₂×C₈	—	166	2	no	—
243	C₂×C₈	—	166	4	no	—
244	C₂×C₈	—	166	4	no	—
245	C₂×C₈	—	166	2	no	—
246	C₄⋊C₄	—	83	1	no	—
247	C₄⋊C₄	—	83	1	no	—
248	C₄⋊C₄	—	166	1	no	—
249	C₄⋊C₄	—	166	1	no	—
250	C₂²⋊C₄	—	166	1	no	—
251	C₂²⋊C₄	—	166	1	no	—
252	C₂²⋊C₄	—	166	1	no	—
253	C₄²	—	2	4	no	—
254	C₄²	—	2	4	no	—
255	C₄²	—	2	2	no	—
256	C₄²	—	166	1	no	—
257	C₄²	—	166	1	no	—
258	C₄²	—	166	1	no	—
259	C₄²	#13050573	1	4	yes	complemented
260	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
261	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
262	C₄²	D₄×D₈₃	1	1	yes	characteristic
263	C₄²	#13050591	1	1	yes	characteristic
264	C₂³	—	166	1	no	—
265	C₂³	(C₂²⋊D₈₃)⋊C₄	1	1	yes	characteristic
266	Q₈	—	166	1	no	—
267	Q₈	—	166	1	no	—
268	D₄	—	332	2	no	—
269	D₄	—	166	1	no	—
270	D₄	—	166	1	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	332	2	no	—
277	C₂×C₄	—	332	2	no	—
278	C₂×C₄	—	332	2	no	—
279	C₂×C₄	—	166	1	no	—
280	C₂×C₄	—	166	1	no	—
281	C₂×C₄	—	166	1	no	—
282	C₂×C₄	C₄²×D₈₃	1	1	yes	characteristic
283	C₂×C₄	C₂²⋊C₄×D₈₃	1	1	yes	characteristic
284	C₂×C₄	#14581750	1	1	yes	characteristic
285	C₂×C₄	(C₂²⋊D₈₃)⋊C₄	1	1	yes	characteristic
286	C₂×C₄	#14581754	1	1	yes	characteristic
287	C₂×C₄	#14581754	1	1	yes	characteristic
288	C₂×C₄	C₂×#13050573	1	2	yes	—
289	C₂×C₄	#14581800	1	2	yes	—
290	C₈	—	4	2	no	—
291	C₈	—	4	2	no	—
292	C₈	—	332	2	no	—
293	C₈	—	83	4	no	—
294	C₈	—	166	4	no	—
295	C₈	—	166	2	no	—
296	C₂²	—	2	1	no	—
297	C₂²	—	2	1	no	—
298	C₂²	—	332	1	no	—
299	C₂²	—	332	1	no	—
300	C₂²	—	332	1	no	—
301	C₂²	#19127548	1	1	yes	characteristic
302	C₂²	#19127769	1	2	yes	—
303	C₄	—	2	8	no	—
304	C₄	—	2	1	no	—
305	C₄	—	2	1	no	—
306	C₄	—	332	1	no	—
307	C₄	—	332	1	no	—
308	C₄	—	332	1	no	—
309	C₄	C₄×#13050573	1	1	yes	characteristic
310	C₄	#19127658	1	1	yes	characteristic
311	C₄	#104×D₈₃	1	2	yes	—
312	C₄	#19127778	1	2	yes	—
313	C₄	#19127815	1	2	yes	—
314	C₂	—	2	2	no	—
315	C₂	—	664	1	no	—
316	C₂	#24907243	1	1	yes	characteristic
317	C₂	#24907267	1	1	yes	characteristic
318	C₂	#24907297	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₈₃
commutator	C₂×C₄×C₈₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₈₃×#1276	C₂
socle	C₂²×C₈₃	C₄×C₂²⋊C₄
prime radical	C₂×#13050590	C₂×C₄
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	#1276	D₈₃
hypocenter	C₈₃	#8194
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ #1276

Complements

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

N	H ≅ G/N	type
3. C₄²×C₈₃⋊C₈	C₂	semidirect
5. #24905753	C₂	semidirect
6. #24905299	C₂	semidirect
7. C₈₃×#1276	C₂	semidirect
9. #19127828	C₄	semidirect
10. #19127778	C₄	semidirect
20. #19127431	C₄	semidirect
21. #19127428	C₄	semidirect
99. C₄²×C₈₃	M₁₆	semidirect
135. C₈₃⋊C₈	#104	semidirect
166. #1276	D₈₃	semidirect
167. C₈₃	#8194	semidirect
259. C₄²	#13050573	semidirect

Maximal subgroups

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

subgroup	index
#24916424	2
C₄×#14581720	2
C₄²×C₈₃⋊C₈	2
#24905843	2
#24905753	2
#24905299	2
C₈₃×#1276	2
#8194	83