Subgroups — #35392107

Order 21248.

706 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24918050	C₂	1	1	yes	maximal, characteristic
2	#24918048	C₂	1	2	yes	maximal
3	#24918047	C₂	1	1	yes	maximal, characteristic
4	#24917972	C₂	1	2	yes	maximal
5	#24907080	C₂	1	1	yes	maximal, characteristic, complemented
6	#24907079	C₂	1	1	yes	maximal, characteristic, complemented
7	#24907071	C₂	1	2	yes	maximal, complemented
8	#24907060	C₂	1	1	yes	maximal, characteristic, complemented
9	#24907058	C₂	1	2	yes	maximal, complemented
10	C₈²×D₈₃	C₂	1	1	yes	maximal, characteristic
11	C₈₃×#1707	C₂	1	1	yes	maximal, characteristic, complemented
12	#19128256	C₂²	1	2	yes	—
13	#19128256	C₂²	1	2	yes	—
14	#19128106	C₂²	1	1	yes	characteristic
15	#19128104	C₂²	1	2	yes	—
16	#19128104	C₂²	1	2	yes	—
17	#19128104	C₂²	1	2	yes	—
18	#19128101	—	2	4	no	—
19	#19128099	C₂²	1	2	yes	—
20	#19128081	C₂²	1	1	yes	characteristic
21	#19128080	C₂²	1	2	yes	—
22	#19128053	C₂²	1	2	yes	—
23	#19128052	C₂²	1	2	yes	—
24	#19128051	C₂²	1	2	yes	—
25	#19127943	—	2	2	no	—
26	#19127941	—	2	4	no	—
27	#19127939	—	2	2	no	—
28	Q₁₆×Dic₈₃	—	2	2	no	—
29	#19127795	—	2	2	no	—
30	#19127793	—	2	4	no	—
31	#19127736	—	2	4	no	—
32	C₄×C₈×D₈₃	C₂²	1	1	yes	characteristic
33	C₄×C₈×D₈₃	C₂²	1	1	yes	characteristic
34	C₄×C₈×D₈₃	C₂²	1	1	yes	characteristic
35	C₈×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
36	C₈×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
37	C₈×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
38	#500×C₈₃	C₂²	1	1	yes	characteristic
39	#498×C₈₃	C₂²	1	2	yes	—
40	#494×C₈₃	C₂²	1	1	yes	characteristic
41	#487×C₈₃	C₂²	1	2	yes	—
42	C₈²×C₈₃	C₂²	1	1	yes	characteristic
43	C₂×Dic₃₃₂	—	2	2	no	—
44	C₂×#13050580	—	2	4	no	—
45	C₂×C₈×D₈₃	D₄	1	2	yes	—
46	C₂×C₈×D₈₃	D₄	1	2	yes	—
47	C₂×C₈×D₈₃	D₄	1	2	yes	—
48	#14581792	C₂³	1	2	yes	—
49	#14581792	C₂³	1	2	yes	—
50	#14581783	—	2	2	no	—
51	#14581783	—	2	2	no	—
52	#14581772	—	2	2	no	—
53	#14581772	—	2	2	no	—
54	#14581768	—	2	2	no	—
55	#14581768	—	2	2	no	—
56	Q₈×Dic₈₃	—	2	2	no	—
57	Q₈×Dic₈₃	—	2	2	no	—
58	#14581757	—	2	2	no	—
59	#14581757	—	2	2	no	—
60	#14581750	—	2	2	no	—
61	#14581750	—	2	2	no	—
62	C₄²×D₈₃	C₂³	1	1	yes	characteristic
63	#14581726	—	2	2	no	—
64	#14581725	—	2	4	no	—
65	#14581725	—	2	4	no	—
66	#14581725	—	2	4	no	—
67	#14581724	—	2	4	no	—
68	#14581717	—	2	4	no	—
69	#14581716	—	2	4	no	—
70	#14581715	—	2	4	no	—
71	C₄×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
72	C₄×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
73	C₄×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
74	C₈×Dic₈₃	D₄	1	2	yes	—
75	C₈×Dic₈₃	D₄	1	2	yes	—
76	C₈×Dic₈₃	D₄	1	2	yes	—
77	C₂×Q₁₆×C₈₃	—	2	2	no	—
78	C₄⋊Q₈×C₈₃	C₂³	1	2	yes	—
79	C₄⋊Q₈×C₈₃	C₂³	1	2	yes	—
80	#107×C₈₃	—	2	2	no	—
81	#106×C₈₃	—	2	4	no	—
82	#103×C₈₃	—	2	4	no	—
83	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
84	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
85	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
86	C₂×Dic₁₆₆	—	2	2	no	—
87	C₂×Dic₁₆₆	—	2	2	no	—
88	C₂×C₄×D₈₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₄×D₈₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄×D₈₃	C₂×D₄	1	1	yes	characteristic
91	Dic₃₃₂	—	4	4	no	—
92	#13050580	—	4	8	no	—
93	C₂×C₈₃⋊C₈	C₂×D₄	1	2	yes	—
94	C₂×C₈₃⋊C₈	C₂×D₄	1	2	yes	—
95	C₂×C₈₃⋊C₈	C₂×D₄	1	2	yes	—
96	C₈×D₈₃	—	2	4	no	—
97	C₈×D₈₃	—	2	4	no	—
98	C₈×D₈₃	Q₁₆	1	8	yes	complemented
99	C₄⋊Dic₈₃	—	2	2	no	—
100	C₄⋊Dic₈₃	—	2	2	no	—
101	C₄⋊Dic₈₃	—	2	2	no	—
102	C₄⋊Dic₈₃	—	2	2	no	—
103	C₄⋊Dic₈₃	—	4	2	no	—
104	C₄⋊Dic₈₃	—	4	2	no	—
105	Dic₈₃⋊C₄	—	4	2	no	—
106	Dic₈₃⋊C₄	—	4	2	no	—
107	(C₂×D₈₃)⋊C₄	—	4	2	no	—
108	(C₂×D₈₃)⋊C₄	—	4	2	no	—
109	C₄×Dic₈₃	—	4	2	no	—
110	C₄×Dic₈₃	—	4	2	no	—
111	C₄×Dic₈₃	C₂×D₄	1	1	yes	characteristic
112	C₄×Dic₈₃	C₂×D₄	1	1	yes	characteristic
113	C₄×Dic₈₃	C₂×D₄	1	1	yes	characteristic
114	C₂×Q₈×C₈₃	—	2	2	no	—
115	C₂×Q₈×C₈₃	—	2	2	no	—
116	Q₁₆×C₈₃	—	4	4	no	—
117	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
118	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
119	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
120	C₄⋊C₄×C₈₃	—	2	2	no	—
121	C₄⋊C₄×C₈₃	—	2	2	no	—
122	C₄⋊C₄×C₈₃	—	2	2	no	—
123	C₄⋊C₄×C₈₃	—	2	2	no	—
124	C₄²×C₈₃	C₂⁴	1	1	yes	characteristic
125	C₂²×D₈₃	#127	1	1	yes	characteristic
126	Q₈×C₈₃	—	4	4	no	—
127	Q₈×C₈₃	—	4	4	no	—
128	C₂×C₄×C₈₃	—	4	2	no	—
129	C₂×C₄×C₈₃	—	4	2	no	—
130	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
133	C₂×Dic₈₃	—	4	2	no	—
134	C₂×Dic₈₃	—	4	2	no	—
135	C₂×Dic₈₃	#127	1	1	yes	characteristic
136	C₂×Dic₈₃	#127	1	1	yes	characteristic
137	C₂×Dic₈₃	#127	1	1	yes	characteristic
138	C₄×D₈₃	—	2	2	no	—
139	C₄×D₈₃	—	2	2	no	—
140	C₄×D₈₃	C₂×Q₁₆	1	4	yes	—
141	Dic₁₆₆	—	4	4	no	—
142	Dic₁₆₆	—	4	4	no	—
143	C₈×C₈₃	—	2	2	no	—
144	C₈×C₈₃	C₂×Q₁₆	1	4	yes	—
145	C₈×C₈₃	#135	1	4	yes	—
146	C₈₃⋊C₈	—	2	2	no	—
147	C₈₃⋊C₈	C₂×Q₁₆	1	4	yes	—
148	C₈₃⋊C₈	#135	1	4	yes	—
149	C₂²×C₈₃	C₂×#127	1	1	yes	characteristic
150	C₂×D₈₃	—	2	1	no	—
151	C₂×D₈₃	—	2	1	no	—
152	C₂×D₈₃	#494	1	2	yes	—
153	C₄×C₈₃	—	8	2	no	—
154	C₄×C₈₃	—	8	2	no	—
155	C₄×C₈₃	C₂²×Q₁₆	1	2	yes	—
156	C₄×C₈₃	C₂×#135	1	2	yes	—
157	C₄×C₈₃	C₂×#135	1	2	yes	—
158	Dic₈₃	—	2	1	no	—
159	Dic₈₃	—	2	1	no	—
160	Dic₈₃	—	8	2	no	—
161	Dic₈₃	—	8	2	no	—
162	Dic₈₃	#494	1	2	yes	—
163	#19604	—	83	1	no	maximal, Sylow
164	C₂×C₈₃	C₂×#494	1	1	yes	characteristic
165	C₂×C₈₃	#3147	1	1	yes	characteristic
166	C₂×C₈₃	#3147	1	1	yes	characteristic
167	D₈₃	—	2	2	no	—
168	#3170	—	83	1	no	—
169	#3170	—	83	1	no	—
170	#3170	—	83	2	no	—
171	#3160	—	83	2	no	—
172	#1713	—	83	1	no	—
173	#1713	—	83	1	no	—
174	#1710	—	83	2	no	—
175	#1707	—	83	2	no	—
176	#1707	—	83	1	no	—
177	#1707	D₈₃	1	1	yes	characteristic, complemented
178	C₂×C₈²	—	83	1	no	—
179	C₈₃	#19604	1	1	yes	characteristic, complemented, Sylow
180	#533	—	83	2	no	—
181	#533	—	83	2	no	—
182	#500	—	83	2	no	—
183	#500	—	83	2	no	—
184	#500	—	83	2	no	—
185	#500	—	83	1	no	—
186	#500	C₂×D₈₃	1	1	yes	characteristic
187	#499	—	166	4	no	—
188	#498	—	83	2	no	—
189	#498	C₂×D₈₃	1	2	yes	—
190	#494	—	83	2	no	—
191	#494	—	83	1	no	—
192	#494	C₂×D₈₃	1	1	yes	characteristic
193	#487	—	83	2	no	—
194	#487	—	83	2	no	—
195	#487	—	83	2	no	—
196	#487	C₂×D₈₃	1	2	yes	—
197	#467	—	166	4	no	—
198	#467	—	166	2	no	—
199	#467	—	166	2	no	—
200	C₄×Q₁₆	—	166	4	no	—
201	C₄×Q₁₆	—	166	2	no	—
202	C₄×Q₁₆	—	166	2	no	—
203	#427	—	166	4	no	—
204	C₂×C₄×C₈	—	83	1	no	—
205	C₂×C₄×C₈	—	83	1	no	—
206	C₂×C₄×C₈	—	83	1	no	—
207	C₈²	—	83	1	no	—
208	C₈²	—	83	1	no	—
209	C₈²	—	83	1	no	—
210	C₈²	C₂×D₈₃	1	1	yes	characteristic
211	C₂×Q₁₆	—	2	2	no	—
212	C₂×Q₁₆	—	166	4	no	—
213	C₂×Q₁₆	—	166	2	no	—
214	C₂²×C₈	—	83	2	no	—
215	C₂²×C₈	—	83	2	no	—
216	C₂²×C₈	—	83	2	no	—
217	C₄⋊Q₈	—	83	2	no	—
218	C₄⋊Q₈	—	83	2	no	—
219	C₄⋊Q₈	C₂²×D₈₃	1	2	yes	—
220	C₄⋊Q₈	C₂²×D₈₃	1	2	yes	—
221	#125	—	166	2	no	—
222	#125	—	166	2	no	—
223	C₂²⋊Q₈	—	166	2	no	—
224	C₂²⋊Q₈	—	166	2	no	—
225	C₂²⋊Q₈	—	166	2	no	—
226	C₂²⋊Q₈	—	166	2	no	—
227	C₄×Q₈	—	166	2	no	—
228	C₄×Q₈	—	166	2	no	—
229	C₄×Q₈	—	166	2	no	—
230	C₄×Q₈	—	166	2	no	—
231	#117	—	166	2	no	—
232	#117	—	166	2	no	—
233	C₂×C₄²	—	83	1	no	—
234	#107	—	2	2	no	—
235	#107	—	166	4	no	—
236	#107	—	166	4	no	—
237	#107	—	166	4	no	—
238	#107	—	166	2	no	—
239	#106	—	2	4	no	—
240	#106	—	166	4	no	—
241	#103	—	2	4	no	—
242	#103	—	166	4	no	—
243	#103	—	166	4	no	—
244	#103	—	166	4	no	—
245	C₄×C₈	—	83	2	no	—
246	C₄×C₈	—	83	2	no	—
247	C₄×C₈	—	83	2	no	—
248	C₄×C₈	—	83	1	no	—
249	C₄×C₈	—	83	1	no	—
250	C₄×C₈	—	83	1	no	—
251	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
252	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
253	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
254	C₂×Q₈	—	2	2	no	—
255	C₂×Q₈	—	2	2	no	—
256	C₂×Q₈	—	166	2	no	—
257	C₂×Q₈	—	166	2	no	—
258	C₂²×C₄	—	83	1	no	—
259	C₂²×C₄	—	83	1	no	—
260	C₂²×C₄	—	83	1	no	—
261	Q₁₆	—	4	4	no	—
262	Q₁₆	—	332	8	no	—
263	Q₁₆	—	332	4	no	—
264	C₂×C₈	—	83	2	no	—
265	C₂×C₈	—	83	2	no	—
266	C₂×C₈	—	83	8	no	—
267	C₂×C₈	—	83	2	no	—
268	C₂×C₈	—	166	4	no	—
269	C₂×C₈	—	166	4	no	—
270	C₂×C₈	D₄×D₈₃	1	2	yes	—
271	C₂×C₈	D₄×D₈₃	1	2	yes	—
272	C₂×C₈	D₄×D₈₃	1	2	yes	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	—	332	2	no	—
278	C₄⋊C₄	—	332	2	no	—
279	C₄⋊C₄	—	332	2	no	—
280	C₄⋊C₄	—	332	2	no	—
281	C₄⋊C₄	—	166	2	no	—
282	C₄⋊C₄	—	166	2	no	—
283	C₄⋊C₄	—	166	2	no	—
284	C₄⋊C₄	—	166	2	no	—
285	C₂²⋊C₄	—	332	2	no	—
286	C₂²⋊C₄	—	332	2	no	—
287	C₄²	—	332	2	no	—
288	C₄²	—	332	2	no	—
289	C₄²	—	83	1	no	—
290	C₄²	—	83	1	no	—
291	C₄²	—	83	1	no	—
292	C₄²	C₂³×D₈₃	1	1	yes	characteristic
293	C₂³	—	83	1	no	—
294	Q₈	—	4	4	no	—
295	Q₈	—	4	4	no	—
296	Q₈	—	332	4	no	—
297	Q₈	—	332	4	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	332	2	no	—
301	C₂×C₄	—	332	2	no	—
302	C₂×C₄	—	83	1	no	—
303	C₂×C₄	—	83	4	no	—
304	C₂×C₄	—	83	1	no	—
305	C₂×C₄	—	83	1	no	—
306	C₂×C₄	—	166	2	no	—
307	C₂×C₄	—	166	2	no	—
308	C₂×C₄	C₂×D₄×D₈₃	1	1	yes	characteristic
309	C₂×C₄	C₂×D₄×D₈₃	1	1	yes	characteristic
310	C₂×C₄	C₂×D₄×D₈₃	1	1	yes	characteristic
311	C₈	—	2	2	no	—
312	C₈	—	83	4	no	—
313	C₈	—	83	4	no	—
314	C₈	—	166	2	no	—
315	C₈	Q₁₆×D₈₃	1	4	yes	—
316	C₈	#14581813	1	4	yes	—
317	C₂²	—	83	2	no	—
318	C₂²	—	166	1	no	—
319	C₂²	—	166	1	no	—
320	C₂²	#127×D₈₃	1	1	yes	characteristic
321	C₄	—	8	2	no	—
322	C₄	—	8	2	no	—
323	C₄	—	83	2	no	—
324	C₄	—	664	2	no	—
325	C₄	—	664	2	no	—
326	C₄	—	166	1	no	—
327	C₄	—	166	1	no	—
328	C₄	C₂×Q₁₆×D₈₃	1	2	yes	—
329	C₄	#135×D₈₃	1	2	yes	—
330	C₄	C₂×#14581813	1	2	yes	—
331	C₂	—	166	2	no	—
332	C₂	#494×D₈₃	1	1	yes	characteristic
333	C₂	#24917886	1	1	yes	characteristic
334	C₂	#24917888	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#127×D₈₃
commutator, cosocle	C₄²×C₈₃	C₂⁴
Frattini	C₄²	C₂³×D₈₃
Fitting	C₈₃×#1707	C₂
socle	C₂²×C₈₃	C₂×#127
prime radical	C₂²×D₈₃	#127
hypercenter	#1707	D₈₃
hypocenter	C₈₃	#19604
LCS	G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1707

Complements

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

N	H ≅ G/N	type
5. #24907080	C₂	semidirect
6. #24907079	C₂	semidirect
7. #24907071	C₂	semidirect
8. #24907060	C₂	semidirect
9. #24907058	C₂	semidirect
11. C₈₃×#1707	C₂	semidirect
98. C₈×D₈₃	Q₁₆	semidirect
177. #1707	D₈₃	semidirect
179. C₈₃	#19604	semidirect

Maximal subgroups

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

subgroup	index
#24918050	2
#24918048	2
#24918047	2
#24917972	2
#24907080	2
#24907079	2
#24907071	2
#24907060	2
#24907058	2
C₈²×D₈₃	2
C₈₃×#1707	2
#19604	83