Subgroups — #35408840

Order 21248.

738 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24916114	C₂	1	2	yes	maximal, complemented
2	#24908357	C₂	1	1	yes	maximal, characteristic
3	#24908352	C₂	1	1	yes	maximal, characteristic
4	#24908217	C₂	1	1	yes	maximal, characteristic, complemented
5	#24908212	C₂	1	1	yes	maximal, characteristic, complemented
6	#24908176	C₂	1	2	yes	maximal, complemented
7	#24908169	C₂	1	2	yes	maximal, complemented
8	#24907369	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#14581719	C₂	1	2	yes	maximal, complemented
10	#97×Dic₈₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₃×#2943	C₂	1	1	yes	maximal, characteristic
12	#19127762	—	2	4	no	—
13	#19127758	—	2	4	no	—
14	C₂×#14581728	C₂²	1	1	yes	characteristic
15	C₂×#14581728	C₂²	1	1	yes	characteristic
16	#19127728	C₂²	1	2	yes	—
17	#19127728	C₂²	1	2	yes	—
18	C₂×#14581719	C₂²	1	2	yes	—
19	C₂×#14581719	C₂²	1	2	yes	—
20	#19127668	C₂²	1	2	yes	—
21	#19127668	C₂²	1	2	yes	—
22	C₄×#13050574	C₂²	1	2	yes	—
23	C₂×#14581703	C₂²	1	1	yes	characteristic
24	C₂×#14581702	C₂²	1	1	yes	characteristic
25	C₂×#14581702	C₂²	1	1	yes	characteristic
26	C₂×C₈×Dic₈₃	C₂²	1	1	yes	characteristic
27	C₂×C₈×Dic₈₃	C₂²	1	1	yes	characteristic
28	C₄²×Dic₈₃	C₂²	1	1	yes	characteristic
29	#19127466	C₂²	1	2	yes	—
30	#19127460	C₂²	1	2	yes	—
31	#19127460	C₂²	1	2	yes	—
32	C₂×#135×C₈₃	C₂²	1	1	yes	characteristic
33	C₄×#41×C₈₃	C₂²	1	2	yes	—
34	#442×C₈₃	C₄	1	2	yes	complemented
35	#441×C₈₃	C₄	1	2	yes	complemented
36	#440×C₈₃	C₄	1	4	yes	complemented
37	#427×C₈₃	C₂²	1	1	yes	characteristic
38	#416×C₈₃	C₂²	1	2	yes	—
39	C₂×#97×C₈₃	C₂²	1	1	yes	characteristic
40	C₂×#13050574	C₂³	1	2	yes	—
41	C₂×#13050574	C₂³	1	2	yes	—
42	C₂²×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
43	C₂²×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
44	C₂×C₄×Dic₈₃	—	2	2	no	—
45	C₂×C₄×Dic₈₃	C₂³	1	1	yes	characteristic
46	C₂×C₄×Dic₈₃	C₂³	1	1	yes	characteristic
47	#14581728	—	2	2	no	—
48	#14581728	—	2	2	no	—
49	C₄⋊(C₈₃⋊C₈)	—	2	4	no	—
50	C₄⋊(C₈₃⋊C₈)	—	2	4	no	—
51	#14581719	—	2	4	no	—
52	#14581719	—	2	4	no	—
53	C₂²⋊(C₈₃⋊C₈)	—	2	4	no	—
54	C₂²⋊(C₈₃⋊C₈)	—	2	4	no	—
55	#14581703	—	2	2	no	—
56	#14581703	D₄	1	4	yes	—
57	#14581702	—	2	2	no	—
58	#14581702	—	2	2	no	—
59	C₄×C₈₃⋊C₈	—	2	2	no	—
60	C₈×Dic₈₃	—	2	2	no	—
61	C₈×Dic₈₃	—	2	2	no	—
62	C₂×#41×C₈₃	C₂³	1	2	yes	—
63	#135×C₈₃	—	2	2	no	—
64	#135×C₈₃	—	2	2	no	—
65	C₂×Q₁₆×C₈₃	C₂×C₄	1	1	yes	characteristic
66	C₂×SD₁₆×C₈₃	C₂×C₄	1	2	yes	—
67	C₂×D₈×C₈₃	C₂×C₄	1	1	yes	characteristic
68	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
69	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
70	C₄×Q₈×C₈₃	C₂×C₄	1	4	yes	—
71	C₄×D₄×C₈₃	—	2	4	no	—
72	C₄×D₄×C₈₃	C₂×C₄	1	4	yes	—
73	#117×C₈₃	—	2	2	no	—
74	#117×C₈₃	C₂³	1	2	yes	—
75	C₂×C₄²×C₈₃	—	2	2	no	—
76	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
77	#107×C₈₃	C₂×C₄	1	2	yes	—
78	#106×C₈₃	C₂×C₄	1	2	yes	—
79	#103×C₈₃	C₂×C₄	1	4	yes	—
80	#102×C₈₃	C₂×C₄	1	4	yes	—
81	#97×C₈₃	C₂×C₄	1	2	yes	—
82	#97×C₈₃	C₂×C₄	1	2	yes	—
83	C₂²×Dic₈₃	—	2	1	no	—
84	C₂²×Dic₈₃	—	2	1	no	—
85	#13050574	—	2	2	no	—
86	#13050574	—	2	2	no	—
87	#13050574	—	2	2	no	—
88	#13050574	—	2	2	no	—
89	C₂×C₈₃⋊C₈	—	2	1	no	—
90	C₂×C₈₃⋊C₈	—	2	1	no	—
91	C₂×C₈₃⋊C₈	—	2	1	no	—
92	C₂×C₈₃⋊C₈	—	2	1	no	—
93	C₂×C₈₃⋊C₈	—	2	2	no	—
94	C₂×C₈₃⋊C₈	—	2	2	no	—
95	C₂×C₈₃⋊C₈	C₂×D₄	1	2	yes	—
96	C₂×C₈₃⋊C₈	C₂×D₄	1	2	yes	—
97	C₄×Dic₈₃	—	2	1	no	—
98	C₄×Dic₈₃	—	2	1	no	—
99	C₄×Dic₈₃	—	4	4	no	—
100	C₄×Dic₈₃	#41	1	2	yes	—
101	C₄×Dic₈₃	#41	1	2	yes	—
102	#41×C₈₃	—	2	2	no	—
103	#41×C₈₃	—	2	2	no	—
104	#41×C₈₃	—	2	2	no	—
105	#41×C₈₃	—	2	2	no	—
106	C₂×Q₈×C₈₃	C₂²×C₄	1	2	yes	—
107	C₂×D₄×C₈₃	—	2	2	no	—
108	C₂×D₄×C₈₃	C₂²×C₄	1	2	yes	—
109	C₂²×C₄×C₈₃	—	2	2	no	—
110	C₂²×C₄×C₈₃	—	2	4	no	—
111	C₂²×C₄×C₈₃	#41	1	2	yes	—
112	C₂²×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
113	Q₁₆×C₈₃	—	2	2	no	—
114	SD₁₆×C₈₃	—	2	4	no	—
115	D₈×C₈₃	—	2	2	no	—
116	C₂×C₈×C₈₃	—	2	1	no	—
117	C₂×C₈×C₈₃	—	2	1	no	—
118	C₂×C₈×C₈₃	C₂²×C₄	1	2	yes	—
119	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
120	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₈×C₈₃	#41	1	2	yes	—
122	C₂×C₈×C₈₃	#41	1	2	yes	—
123	C₄⋊C₄×C₈₃	—	2	4	no	—
124	C₄⋊C₄×C₈₃	C₂²×C₄	1	4	yes	—
125	C₂²⋊C₄×C₈₃	—	2	2	no	—
126	C₂²⋊C₄×C₈₃	—	2	4	no	—
127	C₄²×C₈₃	—	2	4	no	—
128	C₄²×C₈₃	—	2	2	no	—
129	C₄²×C₈₃	C₂²×C₄	1	2	yes	—
130	C₄²×C₈₃	C₂×D₄	1	2	yes	—
131	C₂³×C₈₃	—	2	2	no	—
132	C₂³×C₈₃	C₂×#41	1	1	yes	characteristic
133	Q₈×C₈₃	—	2	2	no	—
134	Q₈×C₈₃	—	2	2	no	—
135	D₄×C₈₃	—	2	2	no	—
136	D₄×C₈₃	—	2	2	no	—
137	D₄×C₈₃	—	4	2	no	—
138	D₄×C₈₃	—	4	2	no	—
139	C₂×C₄×C₈₃	—	2	4	no	—
140	C₂×C₄×C₈₃	—	2	2	no	—
141	C₂×C₄×C₈₃	—	4	2	no	—
142	C₂×C₄×C₈₃	—	4	2	no	—
143	C₂×C₄×C₈₃	—	4	2	no	—
144	C₂×C₄×C₈₃	—	8	4	no	—
145	C₂×C₄×C₈₃	C₄×D₄	1	4	yes	—
146	C₂×C₄×C₈₃	C₂³×C₄	1	1	yes	characteristic
147	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₄×C₈₃	C₂×#41	1	1	yes	characteristic
149	C₂×C₄×C₈₃	C₂×#41	1	1	yes	characteristic
150	C₂×C₄×C₈₃	C₂×#41	1	1	yes	characteristic
151	C₂×C₄×C₈₃	C₂×#41	1	1	yes	characteristic
152	C₂×Dic₈₃	—	2	4	no	—
153	C₂×Dic₈₃	—	2	2	no	—
154	C₂×Dic₈₃	—	2	2	no	—
155	C₂×Dic₈₃	—	4	2	no	—
156	C₈×C₈₃	—	2	2	no	—
157	C₈×C₈₃	#117	1	2	yes	—
158	C₈×C₈₃	#117	1	2	yes	—
159	C₈₃⋊C₈	—	2	2	no	—
160	C₈₃⋊C₈	—	2	2	no	—
161	C₈₃⋊C₈	—	4	2	no	—
162	C₈₃⋊C₈	—	4	2	no	—
163	C₂²×C₈₃	—	2	2	no	—
164	C₂²×C₈₃	—	4	4	no	—
165	C₂²×C₈₃	—	4	2	no	—
166	C₂²×C₈₃	C₂×C₄×D₄	1	1	yes	characteristic
167	C₂²×C₈₃	#529	1	1	yes	characteristic
168	C₂²×C₈₃	#529	1	1	yes	characteristic
169	C₄×C₈₃	—	2	4	no	—
170	C₄×C₈₃	—	4	4	no	—
171	C₄×C₈₃	—	4	2	no	—
172	C₄×C₈₃	C₂×#117	1	1	yes	characteristic
173	C₄×C₈₃	C₄×#41	1	1	yes	characteristic
174	C₄×C₈₃	#532	1	1	yes	characteristic
175	C₄×C₈₃	#532	1	1	yes	characteristic
176	Dic₈₃	—	2	4	no	—
177	Dic₈₃	—	4	2	no	—
178	#21297	—	83	1	no	maximal, Sylow
179	C₂×C₈₃	—	2	1	no	—
180	C₂×C₈₃	—	2	1	no	—
181	C₂×C₈₃	—	8	2	no	—
182	C₂×C₈₃	#2293	1	1	yes	characteristic
183	C₂×C₈₃	#2956	1	2	yes	—
184	#2943	D₈₃	1	1	yes	characteristic
185	#2923	—	83	2	no	—
186	#1975	—	83	1	no	—
187	#1974	—	83	1	no	—
188	#1946	—	83	1	no	—
189	#1945	—	83	1	no	—
190	#1938	—	83	2	no	—
191	#1937	—	83	2	no	—
192	#1778	—	83	1	no	—
193	C₄×#104	—	83	2	no	—
194	C₄×#97	—	83	1	no	—
195	C₈₃	#21297	1	1	yes	characteristic, complemented, Sylow
196	C₂×#135	C₂×D₈₃	1	1	yes	characteristic
197	C₄×#41	C₂×D₈₃	1	2	yes	—
198	#442	Dic₈₃	1	2	yes	complemented
199	#441	Dic₈₃	1	2	yes	complemented
200	#440	Dic₈₃	1	4	yes	complemented
201	#433	—	166	4	no	—
202	#432	—	166	4	no	—
203	C₂×#108	—	83	1	no	—
204	C₂×#108	—	83	1	no	—
205	#427	C₂×D₈₃	1	1	yes	characteristic
206	#424	—	83	2	no	—
207	#424	—	83	2	no	—
208	C₂×#104	—	83	2	no	—
209	C₂×#104	—	83	2	no	—
210	#416	C₂×D₈₃	1	2	yes	—
211	#408	—	83	2	no	—
212	#408	—	83	2	no	—
213	C₄×M₁₆	—	83	2	no	—
214	C₂×#97	—	83	1	no	—
215	C₂×#97	—	83	1	no	—
216	C₂×#97	—	83	1	no	—
217	C₂×#97	C₂×D₈₃	1	1	yes	characteristic
218	C₂×C₄×C₈	—	83	1	no	—
219	C₂×C₄×C₈	—	83	1	no	—
220	C₄³	—	83	1	no	—
221	#341	—	83	2	no	—
222	#339	—	83	2	no	—
223	#339	—	83	2	no	—
224	C₂×#41	C₂²×D₈₃	1	2	yes	—
225	#135	—	2	2	no	—
226	#135	—	2	2	no	—
227	C₂×Q₁₆	C₂×Dic₈₃	1	1	yes	characteristic
228	C₂×SD₁₆	C₂×Dic₈₃	1	2	yes	—
229	C₂×D₈	C₂×Dic₈₃	1	1	yes	characteristic
230	C₂×M₁₆	—	83	2	no	—
231	C₂×M₁₆	—	83	2	no	—
232	C₂²×C₈	—	83	1	no	—
233	C₂²×C₈	—	83	1	no	—
234	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
235	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
236	C₄×Q₈	C₂×Dic₈₃	1	4	yes	—
237	C₄×D₄	—	2	4	no	—
238	C₄×D₄	C₂×Dic₈₃	1	4	yes	—
239	#117	—	2	2	no	—
240	#117	C₂²×D₈₃	1	2	yes	—
241	C₂×C₄²	—	2	2	no	—
242	C₂×C₄²	—	83	1	no	—
243	C₂×C₄²	—	83	1	no	—
244	C₂×C₄²	—	166	2	no	—
245	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
246	#108	—	166	2	no	—
247	#108	—	166	2	no	—
248	#107	C₂×Dic₈₃	1	2	yes	—
249	#106	C₂×Dic₈₃	1	2	yes	—
250	C₄⋊C₈	—	166	4	no	—
251	C₄⋊C₈	—	166	4	no	—
252	#104	—	166	4	no	—
253	#104	—	166	4	no	—
254	#103	C₂×Dic₈₃	1	4	yes	—
255	#102	C₂×Dic₈₃	1	4	yes	—
256	C₂²⋊C₈	—	166	4	no	—
257	C₂²⋊C₈	—	166	4	no	—
258	#97	—	83	4	no	—
259	#97	—	166	2	no	—
260	#97	—	166	2	no	—
261	#97	—	166	2	no	—
262	#97	C₂×Dic₈₃	1	2	yes	—
263	#97	C₂×Dic₈₃	1	2	yes	—
264	C₄×C₈	—	166	2	no	—
265	C₄×C₈	—	166	2	no	—
266	C₄×C₈	—	166	2	no	—
267	#41	—	2	2	no	—
268	#41	—	2	2	no	—
269	#41	—	2	2	no	—
270	#41	—	2	2	no	—
271	C₂×Q₈	C₂²×Dic₈₃	1	2	yes	—
272	C₂×D₄	—	2	2	no	—
273	C₂×D₄	C₂²×Dic₈₃	1	2	yes	—
274	C₂²×C₄	—	2	2	no	—
275	C₂²×C₄	—	2	4	no	—
276	C₂²×C₄	—	166	1	no	—
277	C₂²×C₄	—	166	1	no	—
278	C₂²×C₄	#13050589	1	2	yes	—
279	C₂²×C₄	C₂³×D₈₃	1	1	yes	characteristic
280	Q₁₆	—	2	2	no	—
281	SD₁₆	—	2	4	no	—
282	D₈	—	2	2	no	—
283	M₁₆	—	166	2	no	—
284	M₁₆	—	166	2	no	—
285	M₁₆	—	166	2	no	—
286	M₁₆	—	166	2	no	—
287	C₂×C₈	—	2	1	no	—
288	C₂×C₈	—	2	1	no	—
289	C₂×C₈	—	83	2	no	—
290	C₂×C₈	—	83	2	no	—
291	C₂×C₈	—	166	2	no	—
292	C₂×C₈	—	166	2	no	—
293	C₂×C₈	—	166	1	no	—
294	C₂×C₈	—	166	1	no	—
295	C₂×C₈	—	166	1	no	—
296	C₂×C₈	—	166	1	no	—
297	C₂×C₈	C₂²×Dic₈₃	1	2	yes	—
298	C₂×C₈	C₂²×Dic₈₃	1	1	yes	characteristic
299	C₂×C₈	C₂²×Dic₈₃	1	1	yes	characteristic
300	C₂×C₈	#13050590	1	2	yes	—
301	C₂×C₈	#13050590	1	2	yes	—
302	C₄⋊C₄	—	2	4	no	—
303	C₄⋊C₄	C₂²×Dic₈₃	1	4	yes	—
304	C₂²⋊C₄	—	2	2	no	—
305	C₂²⋊C₄	—	2	4	no	—
306	C₄²	—	2	2	no	—
307	C₄²	—	2	4	no	—
308	C₄²	—	332	4	no	—
309	C₄²	—	83	2	no	—
310	C₄²	—	83	2	no	—
311	C₄²	—	166	1	no	—
312	C₄²	—	166	1	no	—
313	C₄²	C₂²×Dic₈₃	1	2	yes	—
314	C₄²	D₄×D₈₃	1	2	yes	—
315	C₂³	—	2	2	no	—
316	C₂³	C₂×#13050589	1	1	yes	characteristic
317	Q₈	—	2	2	no	—
318	Q₈	—	2	2	no	—
319	D₄	—	2	2	no	—
320	D₄	—	2	2	no	—
321	D₄	—	4	2	no	—
322	D₄	—	4	2	no	—
323	C₂×C₄	—	2	4	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	8	4	no	—
329	C₂×C₄	—	332	2	no	—
330	C₂×C₄	—	166	2	no	—
331	C₂×C₄	—	166	4	no	—
332	C₂×C₄	—	166	2	no	—
333	C₂×C₄	D₄×Dic₈₃	1	4	yes	—
334	C₂×C₄	C₂³×Dic₈₃	1	1	yes	characteristic
335	C₂×C₄	C₂×D₄×D₈₃	1	1	yes	characteristic
336	C₂×C₄	#41×D₈₃	1	1	yes	characteristic
337	C₂×C₄	#41×D₈₃	1	1	yes	characteristic
338	C₂×C₄	C₂×#13050590	1	1	yes	characteristic
339	C₂×C₄	C₂×#13050590	1	1	yes	characteristic
340	C₈	—	2	2	no	—
341	C₈	—	332	2	no	—
342	C₈	—	332	2	no	—
343	C₈	—	166	2	no	—
344	C₈	—	166	2	no	—
345	C₈	#14581751	1	2	yes	—
346	C₈	#14581751	1	2	yes	—
347	C₂²	—	2	2	no	—
348	C₂²	—	4	2	no	—
349	C₂²	—	4	4	no	—
350	C₂²	C₂×D₄×Dic₈₃	1	1	yes	characteristic
351	C₂²	#19128233	1	1	yes	characteristic
352	C₂²	#19128233	1	1	yes	characteristic
353	C₄	—	2	4	no	—
354	C₄	—	4	2	no	—
355	C₄	—	4	4	no	—
356	C₄	—	332	2	no	—
357	C₄	—	166	4	no	—
358	C₄	C₂×#14581751	1	1	yes	characteristic
359	C₄	#41×Dic₈₃	1	1	yes	characteristic
360	C₄	#19128248	1	1	yes	characteristic
361	C₄	#19128248	1	1	yes	characteristic
362	C₂	—	2	1	no	—
363	C₂	—	2	1	no	—
364	C₂	—	8	2	no	—
365	C₂	#24909928	1	1	yes	characteristic
366	C₂	#24916373	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₈₃
commutator	C₂×C₄×C₈₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₈₃
Fitting	C₈₃×#2943	C₂
socle	C₂²×C₈₃	C₂×C₄×D₄
prime radical	C₂×#135×C₈₃	C₂²
cosocle	C₂²×C₄×C₈₃	C₂⁴
hypercenter	#2943	D₈₃
hypocenter	C₈₃	#21297
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #2943

Complements

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

N	H ≅ G/N	type
1. #24916114	C₂	semidirect
4. #24908217	C₂	semidirect
5. #24908212	C₂	semidirect
6. #24908176	C₂	semidirect
7. #24908169	C₂	semidirect
8. #24907369	C₂	semidirect
9. C₄×#14581719	C₂	semidirect
10. #97×Dic₈₃	C₂	semidirect
34. #442×C₈₃	C₄	semidirect
35. #441×C₈₃	C₄	semidirect
36. #440×C₈₃	C₄	semidirect
195. C₈₃	#21297	semidirect
198. #442	Dic₈₃	semidirect
199. #441	Dic₈₃	semidirect
200. #440	Dic₈₃	semidirect

Maximal subgroups

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

subgroup	index
#24916114	2
#24908357	2
#24908352	2
#24908217	2
#24908212	2
#24908176	2
#24908169	2
#24907369	2
C₄×#14581719	2
#97×Dic₈₃	2
C₈₃×#2943	2
#21297	83