Subgroups — #35559759

Order 21248.

578 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24917974	C₂	1	1	yes	maximal, characteristic
2	#494×D₈₃	C₂	1	1	yes	maximal, characteristic
3	#24909707	C₂	1	4	yes	maximal, complemented
4	#24909692	C₂	1	1	yes	maximal, characteristic, complemented
5	#24909691	C₂	1	1	yes	maximal, characteristic, complemented
6	#24909690	C₂	1	1	yes	maximal, characteristic, complemented
7	#24909181	C₂	1	4	yes	maximal
8	C₄×#14581731	C₂	1	1	yes	maximal, characteristic
9	C₈₃×#2241	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Q₁₆×D₈₃	—	2	1	no	—
11	C₂×Q₁₆×D₈₃	C₂²	1	2	yes	—
12	#19128256	—	2	1	no	—
13	C₄⋊Q₈×D₈₃	—	2	1	no	—
14	C₂×#14581731	C₂²	1	2	yes	—
15	#19128106	C₂²	1	1	yes	characteristic
16	#19128102	—	2	1	no	—
17	#19128102	C₂²	1	2	yes	—
18	#19128097	—	2	1	no	—
19	#19128081	C₂²	1	1	yes	characteristic
20	#19128080	—	2	1	no	—
21	#19128079	—	2	1	no	—
22	#19128079	C₂²	1	2	yes	—
23	#19127738	—	2	1	no	—
24	#19127738	C₂²	1	2	yes	—
25	C₄×C₈×D₈₃	C₂²	1	1	yes	characteristic
26	#19127506	C₂²	1	4	yes	—
27	#19127505	C₂²	1	4	yes	—
28	#19127504	C₂²	1	4	yes	—
29	#19127476	C₂²	1	2	yes	—
30	C₄×C₈₃⋊C₁₆	C₂²	1	1	yes	characteristic
31	#500×C₈₃	C₂²	1	1	yes	characteristic
32	#494×C₈₃	C₂²	1	1	yes	characteristic
33	#358×C₈₃	C₂²	1	4	yes	—
34	C₄×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
35	C₂×Q₈×D₈₃	—	2	2	no	—
36	C₂×Dic₃₃₂	—	2	1	no	—
37	C₂×Dic₃₃₂	C₂³	1	2	yes	—
38	C₂×#13050580	—	2	2	no	—
39	C₂×#13050580	—	2	2	no	—
40	Q₁₆×D₈₃	—	2	8	no	—
41	Q₁₆×D₈₃	—	4	4	no	—
42	C₂×C₈×D₈₃	D₄	1	1	yes	characteristic
43	C₂×C₈×D₈₃	C₂³	1	1	yes	characteristic
44	#14581792	—	2	1	no	—
45	#14581792	—	2	1	no	—
46	#14581791	—	4	1	no	—
47	#14581791	—	4	1	no	—
48	#14581790	—	2	2	no	—
49	#14581783	—	2	2	no	—
50	#14581772	—	4	1	no	—
51	#14581768	—	4	1	no	—
52	Q₈×Dic₈₃	—	4	1	no	—
53	#14581757	—	4	1	no	—
54	#14581750	—	2	2	no	—
55	C₄⋊C₄×D₈₃	—	4	1	no	—
56	C₄⋊C₄×D₈₃	—	4	1	no	—
57	C₄²×D₈₃	D₄	1	1	yes	characteristic
58	#14581731	—	2	4	no	—
59	C₂×C₈₃⋊C₁₆	C₂³	1	2	yes	—
60	#14581726	—	2	1	no	—
61	#14581726	C₂³	1	2	yes	—
62	#14581723	—	2	2	no	—
63	#14581723	—	2	2	no	—
64	C₄×C₈₃⋊C₈	D₄	1	1	yes	characteristic
65	C₈×Dic₈₃	D₄	1	1	yes	characteristic
66	C₈×Dic₈₃	C₂³	1	1	yes	characteristic
67	C₂×Q₁₆×C₈₃	—	2	1	no	—
68	C₂×Q₁₆×C₈₃	C₂³	1	2	yes	—
69	C₄⋊Q₈×C₈₃	—	2	1	no	—
70	C₄⋊Q₈×C₈₃	—	2	1	no	—
71	C₂×C₁₆×C₈₃	C₂³	1	2	yes	—
72	#107×C₈₃	—	2	1	no	—
73	#107×C₈₃	C₂³	1	2	yes	—
74	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
75	C₂×Dic₁₆₆	—	2	2	no	—
76	C₂×Dic₁₆₆	—	4	1	no	—
77	C₂×Dic₁₆₆	—	4	2	no	—
78	Q₈×D₈₃	—	4	4	no	—
79	Q₈×D₈₃	—	4	4	no	—
80	C₂×C₄×D₈₃	—	4	2	no	—
81	C₂×C₄×D₈₃	D₈	1	1	yes	characteristic
82	C₂×C₄×D₈₃	D₈	1	1	yes	characteristic
83	C₂×C₄×D₈₃	C₂×D₄	1	1	yes	characteristic
84	Dic₃₃₂	—	2	4	no	—
85	Dic₃₃₂	—	4	2	no	—
86	#13050580	—	4	4	no	—
87	#13050580	—	4	4	no	—
88	C₂×C₈₃⋊C₈	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈₃⋊C₈	C₂×D₄	1	1	yes	characteristic
90	C₈×D₈₃	SD₁₆	1	4	yes	—
91	C₈×D₈₃	#41	1	4	yes	—
92	C₄⋊Dic₈₃	—	2	2	no	—
93	C₄⋊Dic₈₃	—	4	1	no	—
94	C₄⋊Dic₈₃	—	4	1	no	—
95	C₄⋊Dic₈₃	—	4	1	no	—
96	C₄⋊Dic₈₃	—	4	2	no	—
97	Dic₈₃⋊C₄	—	4	2	no	—
98	Dic₈₃⋊C₄	—	4	2	no	—
99	Dic₈₃⋊C₄	—	4	2	no	—
100	(C₂×D₈₃)⋊C₄	—	4	2	no	—
101	C₄×Dic₈₃	—	4	2	no	—
102	C₄×Dic₈₃	D₈	1	1	yes	characteristic
103	C₄×Dic₈₃	D₈	1	1	yes	characteristic
104	C₄×Dic₈₃	C₂×D₄	1	1	yes	characteristic
105	C₈₃⋊C₁₆	—	2	2	no	—
106	C₂×Q₈×C₈₃	—	2	2	no	—
107	C₂×Q₈×C₈₃	—	4	1	no	—
108	Q₁₆×C₈₃	—	2	4	no	—
109	Q₁₆×C₈₃	—	4	2	no	—
110	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₈₃	C₂⁴	1	1	yes	characteristic
112	C₄⋊C₄×C₈₃	—	2	2	no	—
113	C₄⋊C₄×C₈₃	—	4	1	no	—
114	C₄⋊C₄×C₈₃	—	4	1	no	—
115	C₄⋊C₄×C₈₃	—	4	1	no	—
116	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
117	C₁₆×C₈₃	—	2	2	no	—
118	C₂²×D₈₃	C₂×D₈	1	1	yes	characteristic
119	Q₈×C₈₃	—	4	2	no	—
120	Q₈×C₈₃	—	4	2	no	—
121	Q₈×C₈₃	—	8	2	no	—
122	C₂×C₄×C₈₃	—	4	2	no	—
123	C₂×C₄×C₈₃	—	4	2	no	—
124	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
127	C₂×Dic₈₃	—	4	2	no	—
128	C₂×Dic₈₃	—	4	2	no	—
129	C₂×Dic₈₃	C₂×D₈	1	1	yes	characteristic
130	C₂×Dic₈₃	C₂×D₈	1	1	yes	characteristic
131	C₂×Dic₈₃	C₂×D₈	1	1	yes	characteristic
132	C₄×D₈₃	—	2	2	no	—
133	C₄×D₈₃	—	2	2	no	—
134	C₄×D₈₃	—	8	4	no	—
135	C₄×D₈₃	#124	1	2	yes	—
136	C₄×D₈₃	C₂×SD₁₆	1	2	yes	—
137	Dic₁₆₆	—	4	2	no	—
138	Dic₁₆₆	—	4	2	no	—
139	Dic₁₆₆	—	8	2	no	—
140	Dic₁₆₆	—	8	4	no	—
141	C₈×C₈₃	C₂×SD₁₆	1	2	yes	—
142	C₈×C₈₃	C₂×#41	1	2	yes	—
143	C₈₃⋊C₈	#124	1	2	yes	—
144	C₈₃⋊C₈	C₂×SD₁₆	1	2	yes	—
145	C₂²×C₈₃	C₂²×D₈	1	1	yes	characteristic
146	C₂×D₈₃	—	2	1	no	—
147	C₂×D₈₃	—	2	1	no	—
148	C₂×D₈₃	#486	1	2	yes	—
149	C₄×C₈₃	—	8	2	no	—
150	C₄×C₈₃	—	8	2	no	—
151	C₄×C₈₃	#510	1	2	yes	—
152	C₄×C₈₃	#510	1	2	yes	—
153	C₄×C₈₃	C₂×#124	1	1	yes	characteristic
154	C₄×C₈₃	C₂²×SD₁₆	1	1	yes	characteristic
155	Dic₈₃	—	2	1	no	—
156	Dic₈₃	—	2	1	no	—
157	Dic₈₃	—	8	2	no	—
158	Dic₈₃	—	8	2	no	—
159	Dic₈₃	#486	1	2	yes	—
160	#33705	—	83	1	no	maximal, Sylow
161	C₂×C₈₃	C₂×#486	1	1	yes	characteristic
162	C₂×C₈₃	C₂×#510	1	1	yes	characteristic
163	C₂×C₈₃	C₂×#510	1	1	yes	characteristic
164	D₈₃	—	2	2	no	—
165	#3160	—	83	1	no	—
166	C₂×#494	—	83	1	no	—
167	#2245	—	83	4	no	—
168	#2241	—	83	1	no	—
169	#2241	—	83	1	no	—
170	#2241	—	83	1	no	—
171	#2241	D₈₃	1	1	yes	characteristic, complemented
172	#2141	—	83	4	no	—
173	C₄×M₃₂	—	83	1	no	—
174	C₈₃	#33705	1	1	yes	characteristic, complemented, Sylow
175	C₂²×Q₁₆	—	83	2	no	—
176	C₂²×Q₁₆	—	166	1	no	—
177	#533	—	166	1	no	—
178	C₂×C₄⋊Q₈	—	166	1	no	—
179	C₂×M₃₂	—	83	2	no	—
180	#500	—	83	1	no	—
181	#500	C₂×D₈₃	1	1	yes	characteristic
182	#499	—	83	2	no	—
183	#499	—	166	1	no	—
184	#498	—	166	1	no	—
185	#494	—	83	2	no	—
186	#494	—	83	1	no	—
187	#494	—	166	1	no	—
188	#494	—	166	1	no	—
189	#494	C₂×D₈₃	1	1	yes	characteristic
190	#427	—	83	2	no	—
191	#427	—	166	1	no	—
192	C₂×C₄×C₈	—	83	1	no	—
193	#358	—	83	4	no	—
194	#358	—	83	4	no	—
195	#358	—	83	4	no	—
196	#358	C₂×D₈₃	1	4	yes	—
197	#346	—	83	2	no	—
198	C₄×C₁₆	—	83	1	no	—
199	C₄×C₁₆	C₂×D₈₃	1	1	yes	characteristic
200	C₂²×Q₈	—	166	2	no	—
201	C₂×Q₁₆	—	2	1	no	—
202	C₂×Q₁₆	—	332	4	no	—
203	C₂×Q₁₆	—	83	2	no	—
204	C₂×Q₁₆	—	166	8	no	—
205	C₂×Q₁₆	—	166	2	no	—
206	C₂×Q₁₆	—	166	2	no	—
207	C₂×Q₁₆	—	166	1	no	—
208	C₂×Q₁₆	C₂²×D₈₃	1	2	yes	—
209	C₂²×C₈	—	83	1	no	—
210	C₂²×C₈	—	83	1	no	—
211	C₄⋊Q₈	—	2	1	no	—
212	C₄⋊Q₈	—	2	1	no	—
213	C₄⋊Q₈	—	332	1	no	—
214	C₄⋊Q₈	—	332	1	no	—
215	C₄⋊Q₈	—	166	2	no	—
216	C₄⋊Q₈	—	166	1	no	—
217	C₄⋊Q₈	—	166	1	no	—
218	#125	—	166	2	no	—
219	C₂²⋊Q₈	—	332	1	no	—
220	C₂²⋊Q₈	—	332	1	no	—
221	C₄×Q₈	—	332	1	no	—
222	C₄×Q₈	—	332	1	no	—
223	#117	—	166	2	no	—
224	C₂×C₄⋊C₄	—	332	1	no	—
225	C₂×C₄⋊C₄	—	332	1	no	—
226	C₂×C₄²	—	83	1	no	—
227	M₃₂	—	166	4	no	—
228	C₂×C₁₆	—	83	2	no	—
229	C₂×C₁₆	C₂²×D₈₃	1	2	yes	—
230	#107	—	2	1	no	—
231	#107	—	83	2	no	—
232	#107	—	166	1	no	—
233	#107	C₂²×D₈₃	1	2	yes	—
234	#106	—	166	2	no	—
235	#106	—	166	2	no	—
236	C₄×C₈	—	83	1	no	—
237	C₄×C₈	—	83	1	no	—
238	C₄×C₈	—	83	1	no	—
239	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
240	C₂×Q₈	—	2	2	no	—
241	C₂×Q₈	—	4	1	no	—
242	C₂×Q₈	—	332	2	no	—
243	C₂×Q₈	—	332	1	no	—
244	C₂×Q₈	—	332	4	no	—
245	C₂×Q₈	—	332	4	no	—
246	C₂×Q₈	—	166	2	no	—
247	C₂²×C₄	—	332	2	no	—
248	C₂²×C₄	—	83	1	no	—
249	C₂²×C₄	—	83	1	no	—
250	C₂²×C₄	—	83	1	no	—
251	Q₁₆	—	2	4	no	—
252	Q₁₆	—	4	2	no	—
253	Q₁₆	—	332	4	no	—
254	Q₁₆	—	332	4	no	—
255	Q₁₆	—	332	2	no	—
256	Q₁₆	—	166	4	no	—
257	C₂×C₈	—	83	1	no	—
258	C₂×C₈	—	83	1	no	—
259	C₂×C₈	—	83	4	no	—
260	C₂×C₈	—	83	4	no	—
261	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
262	C₂×C₈	C₂³×D₈₃	1	1	yes	characteristic
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	4	1	no	—
265	C₄⋊C₄	—	4	1	no	—
266	C₄⋊C₄	—	4	1	no	—
267	C₄⋊C₄	—	332	2	no	—
268	C₄⋊C₄	—	332	2	no	—
269	C₄⋊C₄	—	332	2	no	—
270	C₄⋊C₄	—	332	2	no	—
271	C₄⋊C₄	—	332	1	no	—
272	C₄⋊C₄	—	332	1	no	—
273	C₄⋊C₄	—	332	1	no	—
274	C₄⋊C₄	—	166	2	no	—
275	C₂²⋊C₄	—	332	2	no	—
276	C₄²	—	332	2	no	—
277	C₄²	—	83	1	no	—
278	C₄²	—	83	1	no	—
279	C₄²	—	83	1	no	—
280	C₄²	D₄×D₈₃	1	1	yes	characteristic
281	C₁₆	—	2	2	no	—
282	C₁₆	—	166	2	no	—
283	C₂³	—	83	1	no	—
284	Q₈	—	4	2	no	—
285	Q₈	—	4	2	no	—
286	Q₈	—	8	2	no	—
287	Q₈	—	332	2	no	—
288	Q₈	—	332	2	no	—
289	Q₈	—	664	4	no	—
290	Q₈	—	664	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	332	2	no	—
294	C₂×C₄	—	332	2	no	—
295	C₂×C₄	—	83	1	no	—
296	C₂×C₄	—	83	2	no	—
297	C₂×C₄	—	83	2	no	—
298	C₂×C₄	—	83	1	no	—
299	C₂×C₄	—	83	1	no	—
300	C₂×C₄	—	664	4	no	—
301	C₂×C₄	—	166	2	no	—
302	C₂×C₄	—	166	2	no	—
303	C₂×C₄	D₈×D₈₃	1	1	yes	characteristic
304	C₂×C₄	D₈×D₈₃	1	1	yes	characteristic
305	C₂×C₄	C₂×D₄×D₈₃	1	1	yes	characteristic
306	C₈	—	83	2	no	—
307	C₈	—	83	2	no	—
308	C₈	SD₁₆×D₈₃	1	2	yes	—
309	C₈	#41×D₈₃	1	2	yes	—
310	C₂²	—	83	2	no	—
311	C₂²	—	166	1	no	—
312	C₂²	—	166	1	no	—
313	C₂²	C₂×D₈×D₈₃	1	1	yes	characteristic
314	C₄	—	8	2	no	—
315	C₄	—	8	2	no	—
316	C₄	—	83	2	no	—
317	C₄	—	664	2	no	—
318	C₄	—	664	2	no	—
319	C₄	—	166	1	no	—
320	C₄	—	166	1	no	—
321	C₄	#19128140	1	2	yes	—
322	C₄	#19128143	1	2	yes	—
323	C₄	#124×D₈₃	1	1	yes	characteristic
324	C₄	C₂×SD₁₆×D₈₃	1	1	yes	characteristic
325	C₂	—	166	2	no	—
326	C₂	#486×D₈₃	1	1	yes	characteristic
327	C₂	C₂×#19128140	1	1	yes	characteristic
328	C₂	C₂×#19128143	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₈₃
commutator, cosocle	C₂×C₈×C₈₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₃
Fitting	C₈₃×#2241	C₂
socle	C₂²×C₈₃	C₂²×D₈
prime radical	C₂²×D₈₃	C₂×D₈
hypercenter	#2241	D₈₃
hypocenter	C₈₃	#33705
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2241

Complements

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

N	H ≅ G/N	type
3. #24909707	C₂	semidirect
4. #24909692	C₂	semidirect
5. #24909691	C₂	semidirect
6. #24909690	C₂	semidirect
9. C₈₃×#2241	C₂	semidirect
171. #2241	D₈₃	semidirect
174. C₈₃	#33705	semidirect

Maximal subgroups

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

subgroup	index
#24917974	2
#494×D₈₃	2
#24909707	2
#24909692	2
#24909691	2
#24909690	2
#24909181	2
C₄×#14581731	2
C₈₃×#2241	2
#33705	83