Subgroups — #35388251

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	#24917264	C₂	1	2	yes	maximal
2	C₂×C₈×Dic₁₆₆	C₂	1	1	yes	maximal, characteristic
3	#24906918	C₂	1	4	yes	maximal, complemented
4	#24906864	C₂	1	2	yes	maximal, complemented
5	#24906830	C₂	1	2	yes	maximal, complemented
6	#24906222	C₂	1	1	yes	maximal, characteristic
7	#24906153	C₂	1	2	yes	maximal
8	C₈₃×#3170	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
10	#19128005	—	2	4	no	—
11	#19127993	C₂²	1	4	yes	—
12	#19127983	C₂²	1	4	yes	—
13	C₈×Dic₁₆₆	—	2	1	no	—
14	C₈×Dic₁₆₆	—	2	1	no	—
15	C₈×Dic₁₆₆	C₂²	1	2	yes	—
16	C₈×Dic₁₆₆	C₂²	1	2	yes	—
17	#19127744	—	2	2	no	—
18	#19127725	C₂²	1	2	yes	—
19	C₂×C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
20	C₂×Dic₈₃⋊C₈	—	2	1	no	—
21	#19127709	—	2	2	no	—
22	C₂×C₈×Dic₈₃	—	2	1	no	—
23	#19127454	C₂²	1	2	yes	—
24	#19127454	C₂²	1	2	yes	—
25	#19127450	C₂²	1	4	yes	—
26	#19127447	C₂²	1	4	yes	—
27	#533×C₈₃	C₂²	1	2	yes	—
28	#500×C₈₃	C₂²	1	1	yes	characteristic
29	#494×C₈₃	C₂²	1	1	yes	characteristic
30	#487×C₈₃	C₂²	1	2	yes	—
31	#467×C₈₃	—	2	1	no	—
32	#467×C₈₃	—	2	1	no	—
33	C₄×Q₁₆×C₈₃	—	2	1	no	—
34	C₄×Q₁₆×C₈₃	—	2	1	no	—
35	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
36	C₂²×Dic₁₆₆	D₄	1	1	yes	characteristic
37	C₂×#13050574	—	2	2	no	—
38	C₂²×C₈₃⋊C₈	—	2	1	no	—
39	C₄×Dic₁₆₆	—	2	1	no	—
40	C₄×Dic₁₆₆	—	2	1	no	—
41	C₄×Dic₁₆₆	D₄	1	2	yes	—
42	C₄×Dic₁₆₆	C₂³	1	2	yes	—
43	C₂×C₄⋊Dic₈₃	D₄	1	1	yes	characteristic
44	C₂×Dic₈₃⋊C₄	—	2	1	no	—
45	C₂×C₄×Dic₈₃	—	2	1	no	—
46	#14581725	—	2	4	no	—
47	#14581723	—	2	4	no	—
48	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
49	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
50	C₄⋊(C₈₃⋊C₈)	C₂³	1	2	yes	—
51	C₄⋊(C₈₃⋊C₈)	C₂³	1	4	yes	—
52	Dic₈₃⋊C₈	—	2	2	no	—
53	Dic₈₃⋊C₈	—	2	2	no	—
54	#14581715	—	2	4	no	—
55	#14581715	—	2	4	no	—
56	C₈×Dic₈₃	—	2	2	no	—
57	C₈×Dic₈₃	—	2	2	no	—
58	C₂×Q₁₆×C₈₃	—	2	1	no	—
59	C₂×Q₁₆×C₈₃	—	2	1	no	—
60	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
61	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
62	C₄⋊Q₈×C₈₃	C₂³	1	2	yes	—
63	C₄⋊Q₈×C₈₃	C₂³	1	2	yes	—
64	#125×C₈₃	—	2	2	no	—
65	C₂²⋊Q₈×C₈₃	—	2	2	no	—
66	C₂²⋊Q₈×C₈₃	—	2	2	no	—
67	C₄×Q₈×C₈₃	—	2	2	no	—
68	C₄×Q₈×C₈₃	—	2	2	no	—
69	#117×C₈₃	—	2	2	no	—
70	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
71	#107×C₈₃	—	2	1	no	—
72	#107×C₈₃	—	2	1	no	—
73	#103×C₈₃	—	2	2	no	—
74	#103×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	1	no	—
80	C₂×Dic₁₆₆	—	2	2	no	—
81	C₂×Dic₁₆₆	Q₁₆	1	4	yes	complemented
82	C₂×Dic₁₆₆	C₂×D₄	1	2	yes	—
83	C₂²×Dic₈₃	—	2	1	no	—
84	C₂²×Dic₈₃	—	2	1	no	—
85	#13050574	—	4	4	no	—
86	C₂×C₈₃⋊C₈	—	2	2	no	—
87	C₂×C₈₃⋊C₈	—	2	4	no	—
88	C₂×C₈₃⋊C₈	—	4	2	no	—
89	C₄⋊Dic₈₃	—	2	1	no	—
90	C₄⋊Dic₈₃	C₂×D₄	1	2	yes	—
91	Dic₈₃⋊C₄	—	2	2	no	—
92	Dic₈₃⋊C₄	—	2	2	no	—
93	C₄×Dic₈₃	—	2	2	no	—
94	C₄×Dic₈₃	—	2	2	no	—
95	C₂×Q₈×C₈₃	—	2	2	no	—
96	C₂×Q₈×C₈₃	—	2	2	no	—
97	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
99	C₂²×C₄×C₈₃	#41	1	1	yes	characteristic
100	Q₁₆×C₈₃	—	4	2	no	—
101	Q₁₆×C₈₃	—	4	2	no	—
102	C₂×C₈×C₈₃	—	2	2	no	—
103	C₂×C₈×C₈₃	—	4	1	no	—
104	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
108	C₄⋊C₄×C₈₃	—	2	2	no	—
109	C₄⋊C₄×C₈₃	—	2	2	no	—
110	C₄⋊C₄×C₈₃	—	2	2	no	—
111	C₄⋊C₄×C₈₃	—	2	2	no	—
112	C₄⋊C₄×C₈₃	—	4	2	no	—
113	C₄⋊C₄×C₈₃	—	4	2	no	—
114	C₂²⋊C₄×C₈₃	—	4	2	no	—
115	C₄²×C₈₃	—	4	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	—	4	2	no	—
122	Q₈×C₈₃	—	4	2	no	—
123	Q₈×C₈₃	—	8	2	no	—
124	C₂×C₄×C₈₃	—	2	1	no	—
125	C₂×C₄×C₈₃	—	2	1	no	—
126	C₂×C₄×C₈₃	—	4	1	no	—
127	C₂×C₄×C₈₃	—	4	2	no	—
128	C₂×C₄×C₈₃	—	4	2	no	—
129	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
130	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
131	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
132	C₂×C₄×C₈₃	C₂×Q₁₆	1	2	yes	—
133	C₂×C₄×C₈₃	#135	1	2	yes	—
134	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄×C₈₃	C₂×#41	1	1	yes	characteristic
137	C₂×Dic₈₃	—	2	2	no	—
138	C₂×Dic₈₃	—	2	4	no	—
139	C₂×Dic₈₃	—	2	2	no	—
140	C₂×Dic₈₃	—	4	2	no	—
141	Dic₁₆₆	—	2	2	no	—
142	Dic₁₆₆	—	2	4	no	—
143	Dic₁₆₆	C₂×Q₁₆	1	4	yes	—
144	C₈×C₈₃	—	2	1	no	—
145	C₈×C₈₃	—	2	1	no	—
146	C₈×C₈₃	—	2	1	no	—
147	C₈×C₈₃	#137	1	2	yes	—
148	C₈₃⋊C₈	—	4	4	no	—
149	C₈₃⋊C₈	—	4	2	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	1	no	—
156	C₄×C₈₃	—	2	1	no	—
157	C₄×C₈₃	—	8	2	no	—
158	C₄×C₈₃	—	8	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	—	2	4	no	—
165	Dic₈₃	—	4	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	D₈₃	1	1	yes	characteristic
172	#3078	—	83	2	no	—
173	C₂×C₈×Q₈	—	83	1	no	—
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	C₂×D₈₃	1	2	yes	—
181	C₂×C₄×Q₈	—	83	1	no	—
182	#500	C₂×D₈₃	1	1	yes	characteristic
183	#494	C₂×D₈₃	1	1	yes	characteristic
184	#487	C₂×D₈₃	1	2	yes	—
185	#479	—	166	4	no	—
186	#477	—	83	4	no	—
187	#475	—	83	4	no	—
188	#467	—	2	1	no	—
189	#467	—	2	1	no	—
190	C₈×Q₈	—	83	2	no	—
191	C₈×Q₈	—	83	2	no	—
192	C₈×Q₈	—	166	1	no	—
193	C₈×Q₈	—	166	1	no	—
194	C₄×Q₁₆	—	2	1	no	—
195	C₄×Q₁₆	—	2	1	no	—
196	#428	—	166	2	no	—
197	#423	—	83	2	no	—
198	C₂×C₄⋊C₈	—	83	1	no	—
199	C₂×C₄⋊C₈	—	166	1	no	—
200	#419	—	166	2	no	—
201	C₂×C₄×C₈	—	166	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	—	83	1	no	—
208	C₂×Q₁₆	—	2	1	no	—
209	C₂×Q₁₆	—	2	1	no	—
210	C₂×M₁₆	—	166	2	no	—
211	C₂²×C₈	—	166	1	no	—
212	C₂²×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
213	C₂²×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
214	C₄⋊Q₈	C₂²×D₈₃	1	2	yes	—
215	C₄⋊Q₈	C₂²×D₈₃	1	2	yes	—
216	#125	—	2	2	no	—
217	C₂²⋊Q₈	—	2	2	no	—
218	C₂²⋊Q₈	—	2	2	no	—
219	C₄×Q₈	—	2	2	no	—
220	C₄×Q₈	—	2	2	no	—
221	C₄×Q₈	—	83	2	no	—
222	C₄×Q₈	—	83	2	no	—
223	C₄×Q₈	—	166	1	no	—
224	C₄×Q₈	—	166	1	no	—
225	#117	—	2	2	no	—
226	C₂×C₄⋊C₄	—	83	1	no	—
227	C₂×C₄⋊C₄	—	166	1	no	—
228	C₂×C₄²	—	166	1	no	—
229	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
230	#107	—	2	1	no	—
231	#107	—	2	1	no	—
232	#107	—	166	4	no	—
233	#106	—	166	4	no	—
234	C₄⋊C₈	—	83	2	no	—
235	C₄⋊C₈	—	83	4	no	—
236	C₄⋊C₈	—	166	2	no	—
237	C₄⋊C₈	—	166	2	no	—
238	C₄⋊C₈	—	166	2	no	—
239	C₄⋊C₈	—	166	1	no	—
240	#103	—	2	2	no	—
241	#103	—	2	2	no	—
242	#103	—	166	4	no	—
243	#103	—	166	4	no	—
244	C₄×C₈	—	166	2	no	—
245	C₄×C₈	—	166	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	2	no	—
251	C₂×Q₈	—	2	2	no	—
252	C₂×Q₈	—	83	4	no	—
253	C₂×Q₈	—	83	2	no	—
254	C₂×Q₈	—	166	1	no	—
255	C₂×Q₈	—	166	2	no	—
256	C₂²×C₄	—	166	1	no	—
257	C₂²×C₄	—	166	1	no	—
258	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
259	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
260	C₂²×C₄	#13050589	1	1	yes	characteristic
261	Q₁₆	—	4	2	no	—
262	Q₁₆	—	4	2	no	—
263	M₁₆	—	332	4	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	4	1	no	—
266	C₂×C₈	—	332	2	no	—
267	C₂×C₈	—	166	4	no	—
268	C₂×C₈	—	166	2	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	2	no	—
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	—	4	2	no	—
278	C₄⋊C₄	—	4	2	no	—
279	C₄⋊C₄	—	83	2	no	—
280	C₄⋊C₄	—	166	1	no	—
281	C₄⋊C₄	—	166	2	no	—
282	C₄⋊C₄	—	166	2	no	—
283	C₂²⋊C₄	—	4	2	no	—
284	C₄²	—	4	2	no	—
285	C₄²	—	166	2	no	—
286	C₄²	—	166	2	no	—
287	C₄²	D₄×D₈₃	1	1	yes	characteristic
288	C₄²	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
289	C₄²	#13050589	1	1	yes	characteristic
290	C₄²	C₂³×D₈₃	1	1	yes	characteristic
291	C₂³	#14581763	1	1	yes	characteristic
292	Q₈	—	4	2	no	—
293	Q₈	—	4	2	no	—
294	Q₈	—	8	2	no	—
295	Q₈	—	83	4	no	—
296	Q₈	—	166	4	no	—
297	Q₈	—	166	2	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	332	2	no	—
304	C₂×C₄	—	166	2	no	—
305	C₂×C₄	—	166	2	no	—
306	C₂×C₄	—	166	4	no	—
307	C₂×C₄	#14581763	1	1	yes	characteristic
308	C₂×C₄	#14581763	1	1	yes	characteristic
309	C₂×C₄	#14581763	1	1	yes	characteristic
310	C₂×C₄	Q₁₆×D₈₃	1	2	yes	—
311	C₂×C₄	#14581815	1	2	yes	—
312	C₂×C₄	C₂×D₄×D₈₃	1	1	yes	characteristic
313	C₂×C₄	C₂²×C₂²⋊D₈₃	1	1	yes	characteristic
314	C₂×C₄	C₂×#13050589	1	1	yes	characteristic
315	C₈	—	2	1	no	—
316	C₈	—	2	1	no	—
317	C₈	—	2	1	no	—
318	C₈	—	332	4	no	—
319	C₈	—	332	2	no	—
320	C₈	#14581822	1	2	yes	—
321	C₂²	—	2	1	no	—
322	C₂²	—	2	1	no	—
323	C₂²	#19127939	1	2	yes	—
324	C₂²	C₂×#14581763	1	1	yes	characteristic
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	8	2	no	—
329	C₄	—	8	2	no	—
330	C₄	—	332	2	no	—
331	C₄	—	166	4	no	—
332	C₄	#19127939	1	2	yes	—
333	C₄	C₂×Q₁₆×D₈₃	1	1	yes	characteristic
334	C₄	C₂×#14581815	1	1	yes	characteristic
335	C₄	C₂×#14581822	1	1	yes	characteristic
336	C₄	#19128499	1	1	yes	characteristic
337	C₂	—	2	2	no	—
338	C₂	#24916931	1	1	yes	characteristic
339	C₂	C₂×#19127939	1	1	yes	characteristic
340	C₂	#24917179	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14581763
commutator, cosocle	C₄²×C₈₃	C₂⁴
Frattini	C₄²	C₂³×D₈₃
Fitting	C₈₃×#3170	C₂
socle	C₂²×C₈₃	C₂×#121
prime radical	C₂³×C₈₃	#121
hypercenter	#3170	D₈₃
hypocenter	C₈₃	#19262
LCS	G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3170

Complements

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

N	H ≅ G/N	type
3. #24906918	C₂	semidirect
4. #24906864	C₂	semidirect
5. #24906830	C₂	semidirect
81. C₂×Dic₁₆₆	Q₁₆	semidirect
179. C₈₃	#19262	semidirect

Maximal subgroups

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

subgroup	index
#24917264	2
C₂×C₈×Dic₁₆₆	2
#24906918	2
#24906864	2
#24906830	2
#24906222	2
#24906153	2
C₈₃×#3170	2
#19262	83