Subgroups — #35388255

Order 21248.

602 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24918049	C₂	1	1	yes	maximal, characteristic
2	#24917268	C₂	1	2	yes	maximal
3	#24906921	C₂	1	4	yes	maximal, complemented
4	#24906867	C₂	1	2	yes	maximal, complemented
5	#24906833	C₂	1	2	yes	maximal, complemented
6	#24906222	C₂	1	1	yes	maximal, characteristic
7	#24906151	C₂	1	2	yes	maximal
8	C₂×C₈×Q₈×C₈₃	C₂	1	1	yes	maximal, characteristic
9	#19128258	C₂²	1	2	yes	—
10	#19128106	C₂²	1	1	yes	characteristic
11	#19128081	C₂²	1	1	yes	characteristic
12	#19128052	C₂²	1	2	yes	—
13	#19128008	—	2	4	no	—
14	#19127996	C₂²	1	4	yes	—
15	#19127986	C₂²	1	4	yes	—
16	#19127942	—	2	1	no	—
17	#19127942	—	2	1	no	—
18	C₄×Dic₃₃₂	—	2	1	no	—
19	C₄×Dic₃₃₂	—	2	1	no	—
20	#19127741	—	2	2	no	—
21	#19127725	C₂²	1	2	yes	—
22	#19127710	—	2	2	no	—
23	#19127454	C₂²	1	2	yes	—
24	#19127454	C₂²	1	2	yes	—
25	#19127449	C₂²	1	4	yes	—
26	#19127448	C₂²	1	4	yes	—
27	C₂×C₄×Q₈×C₈₃	C₂²	1	1	yes	characteristic
28	C₈×Q₈×C₈₃	—	2	1	no	—
29	C₈×Q₈×C₈₃	—	2	1	no	—
30	C₈×Q₈×C₈₃	C₂²	1	2	yes	—
31	C₈×Q₈×C₈₃	C₂²	1	2	yes	—
32	C₂×C₄⋊C₈×C₈₃	—	2	1	no	—
33	C₂×C₄⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
34	C₂×C₄×C₈×C₈₃	—	2	1	no	—
35	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
36	C₂×Dic₃₃₂	—	2	1	no	—
37	C₂×Dic₃₃₂	—	2	1	no	—
38	C₂×#13050574	—	2	2	no	—
39	#14581792	C₂³	1	2	yes	—
40	#14581792	C₂³	1	2	yes	—
41	#14581784	—	2	2	no	—
42	#14581771	—	2	2	no	—
43	#14581771	—	2	2	no	—
44	C₄×Dic₁₆₆	—	2	2	no	—
45	C₄×Dic₁₆₆	—	2	2	no	—
46	#14581751	—	2	2	no	—
47	#14581726	—	2	1	no	—
48	#14581726	—	2	1	no	—
49	#14581725	—	2	4	no	—
50	#14581723	—	2	4	no	—
51	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
52	C₄⋊(C₈₃⋊C₈)	C₂³	1	4	yes	—
53	#14581717	—	2	2	no	—
54	#14581717	—	2	2	no	—
55	#14581716	—	2	4	no	—
56	#14581716	—	2	4	no	—
57	C₂²×Q₈×C₈₃	D₄	1	1	yes	characteristic
58	C₂²×C₈×C₈₃	—	2	1	no	—
59	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
60	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
61	C₄×Q₈×C₈₃	—	2	1	no	—
62	C₄×Q₈×C₈₃	—	2	1	no	—
63	C₄×Q₈×C₈₃	D₄	1	2	yes	—
64	C₄×Q₈×C₈₃	C₂³	1	2	yes	—
65	C₂×C₄⋊C₄×C₈₃	—	2	1	no	—
66	C₂×C₄⋊C₄×C₈₃	D₄	1	1	yes	characteristic
67	C₂×C₄²×C₈₃	—	2	1	no	—
68	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
69	C₄⋊C₈×C₈₃	—	2	2	no	—
70	C₄⋊C₈×C₈₃	—	2	2	no	—
71	C₄⋊C₈×C₈₃	—	2	1	no	—
72	C₄⋊C₈×C₈₃	C₂³	1	2	yes	—
73	C₄×C₈×C₈₃	—	2	2	no	—
74	C₄×C₈×C₈₃	—	2	2	no	—
75	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
76	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
77	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
79	C₂×Dic₁₆₆	—	2	2	no	—
80	C₂×Dic₁₆₆	—	2	2	no	—
81	Dic₃₃₂	—	4	2	no	—
82	Dic₃₃₂	—	4	2	no	—
83	#13050574	—	4	4	no	—
84	C₂×C₈₃⋊C₈	—	2	4	no	—
85	C₄⋊Dic₈₃	—	2	2	no	—
86	C₄⋊Dic₈₃	—	2	2	no	—
87	C₄⋊Dic₈₃	—	2	2	no	—
88	C₄⋊Dic₈₃	—	2	2	no	—
89	Dic₈₃⋊C₄	—	4	2	no	—
90	Dic₈₃⋊C₄	—	4	2	no	—
91	C₂²⋊Dic₈₃	—	4	2	no	—
92	C₄×Dic₈₃	—	4	2	no	—
93	C₂×Q₈×C₈₃	—	2	1	no	—
94	C₂×Q₈×C₈₃	—	2	2	no	—
95	C₂×Q₈×C₈₃	Q₁₆	1	4	yes	complemented
96	C₂×Q₈×C₈₃	C₂×D₄	1	2	yes	—
97	C₂²×C₄×C₈₃	—	2	1	no	—
98	C₂²×C₄×C₈₃	—	2	1	no	—
99	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₈₃	#41	1	1	yes	characteristic
102	C₂×C₈×C₈₃	—	2	2	no	—
103	C₂×C₈×C₈₃	—	2	2	no	—
104	C₂×C₈×C₈₃	—	4	1	no	—
105	C₂×C₈×C₈₃	—	4	2	no	—
106	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
110	C₄⋊C₄×C₈₃	—	2	2	no	—
111	C₄⋊C₄×C₈₃	—	2	1	no	—
112	C₄⋊C₄×C₈₃	—	2	2	no	—
113	C₄⋊C₄×C₈₃	C₂×D₄	1	2	yes	—
114	C₄²×C₈₃	—	2	2	no	—
115	C₄²×C₈₃	—	2	2	no	—
116	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₈₃	#41	1	1	yes	characteristic
119	C₄²×C₈₃	C₂⁴	1	1	yes	characteristic
120	C₂³×C₈₃	#121	1	1	yes	characteristic
121	Q₈×C₈₃	—	2	2	no	—
122	Q₈×C₈₃	—	2	4	no	—
123	Q₈×C₈₃	C₂×Q₁₆	1	4	yes	—
124	C₂×C₄×C₈₃	—	2	2	no	—
125	C₂×C₄×C₈₃	—	2	4	no	—
126	C₂×C₄×C₈₃	—	2	1	no	—
127	C₂×C₄×C₈₃	—	2	1	no	—
128	C₂×C₄×C₈₃	—	2	2	no	—
129	C₂×C₄×C₈₃	—	4	1	no	—
130	C₂×C₄×C₈₃	—	4	2	no	—
131	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
132	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
133	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
134	C₂×C₄×C₈₃	C₂×Q₁₆	1	2	yes	—
135	C₂×C₄×C₈₃	#135	1	2	yes	—
136	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄×C₈₃	C₂×#41	1	1	yes	characteristic
139	C₂×Dic₈₃	—	4	2	no	—
140	C₂×Dic₈₃	—	4	2	no	—
141	Dic₁₆₆	—	4	2	no	—
142	Dic₁₆₆	—	4	2	no	—
143	Dic₁₆₆	—	8	2	no	—
144	C₈×C₈₃	—	2	1	no	—
145	C₈×C₈₃	—	2	1	no	—
146	C₈×C₈₃	—	2	1	no	—
147	C₈×C₈₃	—	4	2	no	—
148	C₈×C₈₃	#137	1	2	yes	—
149	C₈₃⋊C₈	—	4	4	no	—
150	C₂²×C₈₃	—	2	1	no	—
151	C₂²×C₈₃	—	2	1	no	—
152	C₂²×C₈₃	#467	1	2	yes	—
153	C₂²×C₈₃	C₂×#121	1	1	yes	characteristic
154	C₄×C₈₃	—	2	1	no	—
155	C₄×C₈₃	—	2	4	no	—
156	C₄×C₈₃	—	2	1	no	—
157	C₄×C₈₃	—	2	1	no	—
158	C₄×C₈₃	—	4	2	no	—
159	C₄×C₈₃	#467	1	2	yes	—
160	C₄×C₈₃	C₂²×Q₁₆	1	1	yes	characteristic
161	C₄×C₈₃	C₂×#135	1	1	yes	characteristic
162	C₄×C₈₃	C₂×#137	1	1	yes	characteristic
163	C₄×C₈₃	#575	1	1	yes	characteristic
164	Dic₈₃	—	8	2	no	—
165	Dic₈₃	—	8	2	no	—
166	#19262	—	83	1	no	maximal, Sylow
167	C₂×C₈₃	—	2	2	no	—
168	C₂×C₈₃	#3037	1	1	yes	characteristic
169	C₂×C₈₃	C₂×#467	1	1	yes	characteristic
170	C₂×C₈₃	#3067	1	1	yes	characteristic
171	#3170	—	83	1	no	—
172	#3078	—	83	2	no	—
173	C₂×C₈×Q₈	D₈₃	1	1	yes	characteristic
174	#1684	—	83	4	no	—
175	#1676	—	83	2	no	—
176	#1670	—	83	2	no	—
177	#1567	—	83	1	no	—
178	#1550	—	83	2	no	—
179	C₈₃	#19262	1	1	yes	characteristic, complemented, Sylow
180	#533	—	83	2	no	—
181	C₂×C₄×Q₈	C₂×D₈₃	1	1	yes	characteristic
182	#500	—	83	1	no	—
183	#494	—	83	1	no	—
184	#487	—	83	2	no	—
185	#479	—	166	4	no	—
186	#477	—	83	4	no	—
187	#475	—	83	4	no	—
188	#467	—	166	1	no	—
189	#467	—	166	1	no	—
190	C₈×Q₈	—	2	1	no	—
191	C₈×Q₈	—	2	1	no	—
192	C₈×Q₈	C₂×D₈₃	1	2	yes	—
193	C₈×Q₈	C₂×D₈₃	1	2	yes	—
194	C₄×Q₁₆	—	166	1	no	—
195	C₄×Q₁₆	—	166	1	no	—
196	#428	—	166	2	no	—
197	#423	—	83	2	no	—
198	C₂×C₄⋊C₈	—	2	1	no	—
199	C₂×C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
200	#419	—	166	2	no	—
201	C₂×C₄×C₈	—	2	1	no	—
202	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
203	#335	—	83	2	no	—
204	#335	—	83	2	no	—
205	#333	—	83	4	no	—
206	#332	—	83	4	no	—
207	C₂²×Q₈	D₃₃₂	1	1	yes	characteristic
208	C₂×Q₁₆	—	166	1	no	—
209	C₂×Q₁₆	—	166	1	no	—
210	C₂×M₁₆	—	166	2	no	—
211	C₂²×C₈	—	2	1	no	—
212	C₂²×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
213	C₂²×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
214	C₄⋊Q₈	—	83	2	no	—
215	C₄⋊Q₈	—	83	2	no	—
216	#125	—	166	2	no	—
217	C₂²⋊Q₈	—	166	2	no	—
218	C₂²⋊Q₈	—	166	2	no	—
219	C₄×Q₈	—	2	1	no	—
220	C₄×Q₈	—	2	1	no	—
221	C₄×Q₈	—	166	2	no	—
222	C₄×Q₈	—	166	2	no	—
223	C₄×Q₈	D₃₃₂	1	2	yes	—
224	C₄×Q₈	C₂²×D₈₃	1	2	yes	—
225	#117	—	166	2	no	—
226	C₂×C₄⋊C₄	—	2	1	no	—
227	C₂×C₄⋊C₄	D₃₃₂	1	1	yes	characteristic
228	C₂×C₄²	—	2	1	no	—
229	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
230	#107	—	166	1	no	—
231	#107	—	166	1	no	—
232	#107	—	166	4	no	—
233	#106	—	166	4	no	—
234	C₄⋊C₈	—	2	2	no	—
235	C₄⋊C₈	—	2	2	no	—
236	C₄⋊C₈	—	2	1	no	—
237	C₄⋊C₈	—	83	4	no	—
238	C₄⋊C₈	—	166	2	no	—
239	C₄⋊C₈	C₂²×D₈₃	1	2	yes	—
240	#103	—	166	4	no	—
241	#103	—	166	4	no	—
242	#103	—	166	2	no	—
243	#103	—	166	2	no	—
244	C₄×C₈	—	2	2	no	—
245	C₄×C₈	—	2	2	no	—
246	C₄×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
247	C₄×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
248	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
249	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
250	C₂×Q₈	—	2	1	no	—
251	C₂×Q₈	—	2	2	no	—
252	C₂×Q₈	—	166	2	no	—
253	C₂×Q₈	—	166	2	no	—
254	C₂×Q₈	Dic₃₃₂	1	4	yes	complemented
255	C₂×Q₈	C₂×D₃₃₂	1	2	yes	—
256	C₂²×C₄	—	2	1	no	—
257	C₂²×C₄	—	2	1	no	—
258	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
259	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
260	C₂²×C₄	#13050590	1	1	yes	characteristic
261	Q₁₆	—	332	2	no	—
262	Q₁₆	—	332	2	no	—
263	M₁₆	—	332	4	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	4	1	no	—
267	C₂×C₈	—	4	2	no	—
268	C₂×C₈	—	166	4	no	—
269	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
270	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
271	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
272	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	—	332	2	no	—
277	C₄⋊C₄	—	332	2	no	—
278	C₄⋊C₄	—	166	2	no	—
279	C₄⋊C₄	—	166	2	no	—
280	C₄⋊C₄	—	166	2	no	—
281	C₄⋊C₄	—	166	2	no	—
282	C₄⋊C₄	C₂×D₃₃₂	1	2	yes	—
283	C₂²⋊C₄	—	332	2	no	—
284	C₄²	—	2	2	no	—
285	C₄²	—	2	2	no	—
286	C₄²	—	332	2	no	—
287	C₄²	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
288	C₄²	C₂×D₃₃₂	1	1	yes	characteristic
289	C₄²	#13050590	1	1	yes	characteristic
290	C₄²	C₂³×D₈₃	1	1	yes	characteristic
291	C₂³	#14581766	1	1	yes	characteristic
292	Q₈	—	2	4	no	—
293	Q₈	—	2	2	no	—
294	Q₈	—	332	2	no	—
295	Q₈	—	332	2	no	—
296	Q₈	—	664	2	no	—
297	Q₈	C₂×Dic₃₃₂	1	4	yes	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	4	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	332	2	no	—
306	C₂×C₄	—	332	2	no	—
307	C₂×C₄	#14581766	1	1	yes	characteristic
308	C₂×C₄	#14581766	1	1	yes	characteristic
309	C₂×C₄	#14581766	1	1	yes	characteristic
310	C₂×C₄	C₂×Dic₃₃₂	1	2	yes	—
311	C₂×C₄	#14581814	1	2	yes	—
312	C₂×C₄	C₂²×C₂²⋊D₈₃	1	1	yes	characteristic
313	C₂×C₄	C₂²×D₃₃₂	1	1	yes	characteristic
314	C₂×C₄	C₂×#13050590	1	1	yes	characteristic
315	C₈	—	2	1	no	—
316	C₈	—	2	1	no	—
317	C₈	—	2	1	no	—
318	C₈	—	4	2	no	—
319	C₈	—	332	4	no	—
320	C₈	#14581823	1	2	yes	—
321	C₂²	—	2	1	no	—
322	C₂²	—	2	1	no	—
323	C₂²	#19127942	1	2	yes	—
324	C₂²	C₂×#14581766	1	1	yes	characteristic
325	C₄	—	2	4	no	—
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	4	2	no	—
330	C₄	—	664	2	no	—
331	C₄	—	664	2	no	—
332	C₄	#19127942	1	2	yes	—
333	C₄	C₂²×Dic₃₃₂	1	1	yes	characteristic
334	C₄	C₂×#14581814	1	1	yes	characteristic
335	C₄	C₂×#14581823	1	1	yes	characteristic
336	C₄	#19128499	1	1	yes	characteristic
337	C₂	—	2	2	no	—
338	C₂	#24916935	1	1	yes	characteristic
339	C₂	C₂×#19127942	1	1	yes	characteristic
340	C₂	#24917183	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14581766
commutator, cosocle	C₄²×C₈₃	C₂⁴
Frattini	C₄²	C₂³×D₈₃
Fitting	C₂×C₈×Q₈×C₈₃	C₂
socle	C₂²×C₈₃	C₂×#121
prime radical	C₂³×C₈₃	#121
hypercenter	C₂×C₈×Q₈	D₈₃
hypocenter	C₈₃	#19262
LCS	G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
3. #24906921	C₂	semidirect
4. #24906867	C₂	semidirect
5. #24906833	C₂	semidirect
95. C₂×Q₈×C₈₃	Q₁₆	semidirect
179. C₈₃	#19262	semidirect
254. C₂×Q₈	Dic₃₃₂	semidirect

Maximal subgroups

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

subgroup	index
#24918049	2
#24917268	2
#24906921	2
#24906867	2
#24906833	2
#24906222	2
#24906151	2
C₂×C₈×Q₈×C₈₃	2
#19262	83