Subgroups — #35375284

Order 21248.

576 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24918033	C₂	1	1	yes	maximal, characteristic
2	#24916614	C₂	1	2	yes	maximal, complemented
3	#24906336	C₂	1	2	yes	maximal, complemented
4	#24906329	C₂	1	2	yes	maximal, complemented
5	#24906308	C₂	1	2	yes	maximal, complemented
6	#24906301	C₂	1	2	yes	maximal, complemented
7	#24906219	C₂	1	1	yes	maximal, characteristic, complemented
8	#24905929	C₂	1	2	yes	maximal, complemented
9	C₈₃×#1452	C₂	1	1	yes	maximal, characteristic, complemented
10	#19128235	C₂²	1	2	yes	—
11	#19128103	C₄	1	1	yes	characteristic
12	#19128085	C₄	1	1	yes	characteristic
13	#19128056	C₄	1	2	yes	—
14	#19127942	C₄	1	1	yes	characteristic
15	#19127937	C₄	1	1	yes	characteristic
16	#19127930	C₄	1	2	yes	—
17	#19127831	C₂²	1	2	yes	—
18	C₈×Dic₁₆₆	C₂²	1	2	yes	—
19	C₄×Dic₃₃₂	C₂²	1	1	yes	characteristic
20	C₄×#13050579	C₂²	1	2	yes	—
21	C₄×D₆₆₄	C₂²	1	1	yes	characteristic
22	#19127780	C₂²	1	2	yes	—
23	#19127776	—	2	2	no	—
24	#19127775	—	2	2	no	—
25	C₈×D₃₃₂	C₂²	1	2	yes	—
26	#19127761	—	2	2	no	—
27	#19127726	—	2	2	no	—
28	#19127725	C₂²	1	2	yes	—
29	#19127452	C₂²	1	1	yes	characteristic
30	#19127452	C₂²	1	1	yes	characteristic
31	#19127452	C₂²	1	2	yes	—
32	#19127433	C₂²	1	2	yes	—
33	#19127433	C₂²	1	2	yes	—
34	#19127430	C₂²	1	2	yes	—
35	#19127430	C₂²	1	2	yes	—
36	#431×C₈₃	C₂²	1	1	yes	characteristic
37	#431×C₈₃	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
39	#322×C₈₃	C₂²	1	2	yes	—
40	#322×C₈₃	C₂²	1	2	yes	—
41	C₂×Dic₃₃₂	C₂×C₄	1	1	yes	characteristic
42	C₂×#13050579	C₂×C₄	1	2	yes	—
43	C₂×D₆₆₄	C₂×C₄	1	1	yes	characteristic
44	C₂×#13050574	—	2	2	no	—
45	C₂×#13050573	—	2	2	no	—
46	C₂×C₈×D₈₃	—	2	2	no	—
47	#14581786	—	2	2	no	—
48	#14581784	C₂×C₄	1	2	yes	—
49	#14581780	C₂×C₄	1	2	yes	—
50	#14581775	—	2	2	no	—
51	#14581771	C₂×C₄	1	2	yes	—
52	#14581766	C₂×C₄	1	2	yes	—
53	C₄×Dic₁₆₆	C₂³	1	2	yes	—
54	C₄×D₃₃₂	C₂³	1	2	yes	—
55	C₄×C₂²⋊D₈₃	—	2	2	no	—
56	#14581749	—	2	2	no	—
57	#14581726	C₂×C₄	1	1	yes	characteristic
58	#14581726	C₂×C₄	1	1	yes	characteristic
59	#14581724	C₂×C₄	1	2	yes	—
60	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
61	C₄⋊(C₈₃⋊C₈)	C₂³	1	2	yes	—
62	C₄⋊(C₈₃⋊C₈)	C₂³	1	2	yes	—
63	Dic₈₃⋊C₈	—	2	2	no	—
64	Dic₈₃⋊C₈	—	2	2	no	—
65	#14581717	C₂×C₄	1	2	yes	—
66	#14581717	C₂×C₄	1	2	yes	—
67	#14581714	C₂×C₄	1	2	yes	—
68	#14581714	C₂×C₄	1	2	yes	—
69	(C₂×D₈₃)⋊C₈	—	2	2	no	—
70	(C₂×D₈₃)⋊C₈	—	2	2	no	—
71	#14581702	—	2	2	no	—
72	C₈×Dic₈₃	—	2	2	no	—
73	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
74	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
75	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
76	C₄⋊C₈×C₈₃	—	2	1	no	—
77	C₄⋊C₈×C₈₃	D₄	1	2	yes	—
78	C₂²⋊C₈×C₈₃	—	2	1	no	—
79	C₂²⋊C₈×C₈₃	D₄	1	1	yes	characteristic
80	C₂²⋊C₈×C₈₃	D₄	1	1	yes	characteristic
81	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
82	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
83	C₄×C₈×C₈₃	C₂³	1	2	yes	—
84	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
85	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
86	C₄×C₈×C₈₃	C₂³	1	2	yes	—
87	C₂×Dic₁₆₆	C₂²×C₄	1	2	yes	—
88	C₂×D₃₃₂	C₂²×C₄	1	2	yes	—
89	C₂×C₂²⋊D₈₃	—	2	2	no	—
90	C₂×C₄×D₈₃	—	2	2	no	—
91	Dic₃₃₂	—	4	1	no	—
92	#13050579	—	4	2	no	—
93	D₆₆₄	—	4	1	no	—
94	#13050574	—	8	2	no	—
95	#13050573	—	8	2	no	—
96	C₂×C₈₃⋊C₈	—	2	2	no	—
97	C₂×C₈₃⋊C₈	—	2	2	no	—
98	C₈×D₈₃	—	8	2	no	—
99	C₄⋊Dic₈₃	C₂²×C₄	1	2	yes	—
100	C₄⋊Dic₈₃	C₂²×C₄	1	2	yes	—
101	Dic₈₃⋊C₄	—	2	2	no	—
102	Dic₈₃⋊C₄	—	2	2	no	—
103	Dic₈₃⋊C₄	—	2	2	no	—
104	Dic₈₃⋊C₄	—	2	2	no	—
105	C₂²⋊Dic₈₃	—	2	2	no	—
106	(C₂×D₈₃)⋊C₄	—	2	2	no	—
107	(C₂×D₈₃)⋊C₄	—	2	2	no	—
108	(C₂×D₈₃)⋊C₄	—	2	2	no	—
109	(C₂×D₈₃)⋊C₄	—	2	2	no	—
110	C₄×Dic₈₃	—	2	2	no	—
111	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
112	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₈₃	#41	1	1	yes	characteristic
114	C₂×C₈×C₈₃	—	2	2	no	—
115	C₂×C₈×C₈₃	—	4	1	no	—
116	C₂×C₈×C₈₃	—	4	1	no	—
117	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
119	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
120	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
122	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
123	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
124	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
125	C₄²×C₈₃	#41	1	1	yes	characteristic
126	C₄²×C₈₃	C₂⁴	1	1	yes	characteristic
127	C₂³×C₈₃	C₄×D₄	1	1	yes	characteristic
128	C₂²×D₈₃	—	2	2	no	—
129	C₂×C₄×C₈₃	—	2	1	no	—
130	C₂×C₄×C₈₃	—	2	1	no	—
131	C₂×C₄×C₈₃	—	2	2	no	—
132	C₂×C₄×C₈₃	—	4	1	no	—
133	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
134	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
135	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
136	C₂×C₄×C₈₃	C₂³×C₄	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	C₂²⋊D₈₃	—	8	2	no	—
140	C₂×Dic₈₃	—	2	2	no	—
141	C₂×Dic₈₃	—	2	2	no	—
142	C₂×Dic₈₃	—	2	2	no	—
143	D₃₃₂	—	2	2	no	—
144	D₃₃₂	—	2	2	no	—
145	C₄×D₈₃	—	4	2	no	—
146	C₄×D₈₃	—	4	2	no	—
147	Dic₁₆₆	—	2	2	no	—
148	Dic₁₆₆	—	2	2	no	—
149	C₈×C₈₃	—	2	1	no	—
150	C₈×C₈₃	—	2	1	no	—
151	C₈×C₈₃	—	2	1	no	—
152	C₈×C₈₃	—	2	1	no	—
153	C₈×C₈₃	—	2	2	no	—
154	C₈×C₈₃	—	2	2	no	—
155	C₈×C₈₃	—	4	2	no	—
156	C₈₃⋊C₈	—	4	2	no	—
157	C₈₃⋊C₈	—	4	2	no	—
158	C₂²×C₈₃	—	2	1	no	—
159	C₂²×C₈₃	—	2	1	no	—
160	C₂²×C₈₃	—	2	1	no	—
161	C₂²×C₈₃	C₂×C₄×D₄	1	1	yes	characteristic
162	C₂×D₈₃	—	4	2	no	—
163	C₂×D₈₃	—	4	2	no	—
164	C₂×D₈₃	—	4	2	no	—
165	C₄×C₈₃	—	2	1	no	—
166	C₄×C₈₃	—	2	1	no	—
167	C₄×C₈₃	—	2	1	no	—
168	C₄×C₈₃	—	2	1	no	—
169	C₄×C₈₃	#568	1	1	yes	characteristic
170	C₄×C₈₃	#568	1	1	yes	characteristic
171	C₄×C₈₃	#577	1	2	yes	—
172	Dic₈₃	—	4	2	no	—
173	Dic₈₃	—	4	2	no	—
174	Dic₈₃	—	4	2	no	—
175	#17993	—	83	1	no	maximal, Sylow
176	C₂×C₈₃	—	4	1	no	—
177	C₂×C₈₃	#2932	1	1	yes	characteristic
178	C₂×C₈₃	#2952	1	1	yes	characteristic
179	C₂×C₈₃	#2957	1	1	yes	characteristic
180	D₈₃	—	8	2	no	—
181	#3168	—	83	1	no	—
182	#2993	—	83	2	no	—
183	#1587	—	83	2	no	—
184	#1586	—	83	2	no	—
185	#1583	—	83	2	no	—
186	#1582	—	83	2	no	—
187	#1566	—	83	1	no	—
188	#1495	—	83	2	no	—
189	#1452	D₈₃	1	1	yes	characteristic, complemented
190	C₈₃	#17993	1	1	yes	characteristic, complemented, Sylow
191	#529	—	83	2	no	—
192	#499	—	83	1	no	—
193	#495	—	83	1	no	—
194	#488	—	83	2	no	—
195	#467	—	83	1	no	—
196	#466	—	83	1	no	—
197	#465	—	83	2	no	—
198	#446	—	83	2	no	—
199	C₈×Q₈	—	83	2	no	—
200	C₄×Q₁₆	—	83	1	no	—
201	C₄×SD₁₆	—	83	2	no	—
202	C₄×D₈	—	83	1	no	—
203	#436	—	83	2	no	—
204	#436	—	166	2	no	—
205	#435	—	166	2	no	—
206	C₈×D₄	—	83	2	no	—
207	#433	—	166	2	no	—
208	#431	C₂×D₈₃	1	1	yes	characteristic
209	#431	C₂×D₈₃	1	1	yes	characteristic
210	#424	—	166	2	no	—
211	#423	—	83	2	no	—
212	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
213	#334	—	83	1	no	—
214	#334	—	83	1	no	—
215	#334	—	83	2	no	—
216	Q₈⋊C₈	—	83	2	no	—
217	Q₈⋊C₈	—	83	2	no	—
218	D₄⋊C₈	—	83	2	no	—
219	D₄⋊C₈	—	83	2	no	—
220	#322	C₂×D₈₃	1	2	yes	—
221	#322	C₂×D₈₃	1	2	yes	—
222	C₂×Q₁₆	—	83	1	no	—
223	C₂×SD₁₆	—	83	2	no	—
224	C₂×D₈	—	83	1	no	—
225	C₂×M₁₆	—	166	2	no	—
226	C₂×M₁₆	—	166	2	no	—
227	C₂²×C₈	—	166	2	no	—
228	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
229	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
230	#126	—	166	2	no	—
231	#125	—	83	2	no	—
232	#124	—	83	2	no	—
233	#123	—	166	2	no	—
234	C₂²⋊Q₈	—	83	2	no	—
235	#121	—	83	2	no	—
236	C₄×Q₈	—	83	2	no	—
237	C₄×D₄	—	83	2	no	—
238	C₄×D₄	—	166	2	no	—
239	#117	—	166	2	no	—
240	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
241	#107	—	83	1	no	—
242	#107	—	83	1	no	—
243	#106	—	83	2	no	—
244	C₄⋊C₈	—	2	1	no	—
245	C₄⋊C₈	—	83	2	no	—
246	C₄⋊C₈	—	83	2	no	—
247	C₄⋊C₈	—	166	2	no	—
248	C₄⋊C₈	—	166	2	no	—
249	C₄⋊C₈	—	166	2	no	—
250	C₄⋊C₈	D₃₃₂	1	2	yes	—
251	#103	—	83	2	no	—
252	#103	—	83	2	no	—
253	#102	—	83	2	no	—
254	#102	—	83	2	no	—
255	C₂²⋊C₈	—	2	1	no	—
256	C₂²⋊C₈	—	166	2	no	—
257	C₂²⋊C₈	—	166	2	no	—
258	C₂²⋊C₈	D₃₃₂	1	1	yes	characteristic
259	C₂²⋊C₈	D₃₃₂	1	1	yes	characteristic
260	#97	—	166	2	no	—
261	C₄×C₈	—	166	2	no	—
262	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
263	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
264	C₄×C₈	C₂²×D₈₃	1	2	yes	—
265	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
266	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
267	C₄×C₈	C₂²×D₈₃	1	2	yes	—
268	C₂×Q₈	—	83	2	no	—
269	C₂×D₄	—	83	2	no	—
270	C₂×D₄	—	166	2	no	—
271	C₂²×C₄	—	166	2	no	—
272	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
273	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
274	C₂²×C₄	#13050590	1	1	yes	characteristic
275	Q₁₆	—	332	1	no	—
276	SD₁₆	—	332	2	no	—
277	D₈	—	332	1	no	—
278	M₁₆	—	664	2	no	—
279	M₁₆	—	664	2	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	—	4	1	no	—
282	C₂×C₈	—	4	1	no	—
283	C₂×C₈	—	664	2	no	—
284	C₂×C₈	—	166	2	no	—
285	C₂×C₈	—	166	2	no	—
286	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
287	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
288	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
289	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
290	C₂×C₈	C₂×D₃₃₂	1	2	yes	—
291	C₂×C₈	C₂×D₃₃₂	1	2	yes	—
292	C₄⋊C₄	—	83	2	no	—
293	C₄⋊C₄	—	83	2	no	—
294	C₄⋊C₄	—	166	2	no	—
295	C₄⋊C₄	—	166	2	no	—
296	C₄⋊C₄	—	166	2	no	—
297	C₄⋊C₄	—	166	2	no	—
298	C₂²⋊C₄	—	166	2	no	—
299	C₂²⋊C₄	—	166	2	no	—
300	C₂²⋊C₄	—	166	2	no	—
301	C₂²⋊C₄	—	166	2	no	—
302	C₂²⋊C₄	—	166	2	no	—
303	C₄²	—	166	2	no	—
304	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
305	C₄²	C₂×D₃₃₂	1	1	yes	characteristic
306	C₄²	#13050590	1	1	yes	characteristic
307	C₄²	C₂³×D₈₃	1	1	yes	characteristic
308	C₂³	—	166	2	no	—
309	C₂³	C₄×D₃₃₂	1	1	yes	characteristic
310	Q₈	—	166	2	no	—
311	Q₈	—	166	2	no	—
312	D₄	—	664	2	no	—
313	D₄	—	166	2	no	—
314	D₄	—	166	2	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	332	2	no	—
320	C₂×C₄	—	332	2	no	—
321	C₂×C₄	—	166	2	no	—
322	C₂×C₄	—	166	2	no	—
323	C₂×C₄	—	166	2	no	—
324	C₂×C₄	C₄×D₃₃₂	1	1	yes	characteristic
325	C₂×C₄	C₄×D₃₃₂	1	1	yes	characteristic
326	C₂×C₄	C₄×D₃₃₂	1	1	yes	characteristic
327	C₂×C₄	C₂²×C₄×D₈₃	1	1	yes	characteristic
328	C₂×C₄	C₂²×D₃₃₂	1	1	yes	characteristic
329	C₂×C₄	C₂×#13050590	1	1	yes	characteristic
330	C₈	—	2	1	no	—
331	C₈	—	2	1	no	—
332	C₈	—	2	1	no	—
333	C₈	—	2	1	no	—
334	C₈	—	2	2	no	—
335	C₈	—	2	2	no	—
336	C₈	—	4	2	no	—
337	C₈	—	332	2	no	—
338	C₈	—	332	2	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	332	2	no	—
343	C₂²	—	332	2	no	—
344	C₂²	—	332	2	no	—
345	C₂²	C₂×C₄×D₃₃₂	1	1	yes	characteristic
346	C₄	—	2	1	no	—
347	C₄	—	2	1	no	—
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	—	332	2	no	—
351	C₄	—	332	2	no	—
352	C₄	—	332	2	no	—
353	C₄	#19128467	1	1	yes	characteristic
354	C₄	#19128467	1	1	yes	characteristic
355	C₄	#19128513	1	2	yes	—
356	C₂	—	4	1	no	—
357	C₂	—	664	2	no	—
358	C₂	#24916182	1	1	yes	characteristic
359	C₂	#24916348	1	1	yes	characteristic
360	C₂	#24916381	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₃₃₂
commutator	C₂×C₄×C₈₃	C₂³×C₄
Frattini	C₄²	C₂³×D₈₃
Fitting	C₈₃×#1452	C₂
socle	C₂²×C₈₃	C₂×C₄×D₄
prime radical	#19127937	C₄
cosocle	C₄²×C₈₃	C₂⁴
hypercenter	#1452	D₈₃
hypocenter	C₈₃	#17993
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1452

Complements

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

N	H ≅ G/N	type
2. #24916614	C₂	semidirect
3. #24906336	C₂	semidirect
4. #24906329	C₂	semidirect
5. #24906308	C₂	semidirect
6. #24906301	C₂	semidirect
7. #24906219	C₂	semidirect
8. #24905929	C₂	semidirect
9. C₈₃×#1452	C₂	semidirect
189. #1452	D₈₃	semidirect
190. C₈₃	#17993	semidirect

Maximal subgroups

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

subgroup	index
#24918033	2
#24916614	2
#24906336	2
#24906329	2
#24906308	2
#24906301	2
#24906219	2
#24905929	2
C₈₃×#1452	2
#17993	83