Subgroups — #35375086

Order 21248.

592 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24918031	C₂	1	1	yes	maximal, characteristic
2	#24916613	C₂	1	2	yes	maximal, complemented
3	#24906362	C₂	1	2	yes	maximal, complemented
4	#24906359	C₂	1	1	yes	maximal, characteristic, complemented
5	#24906348	C₂	1	1	yes	maximal, characteristic, complemented
6	#24906275	C₂	1	2	yes	maximal, complemented
7	C₈×Dic₃₃₂	C₂	1	1	yes	maximal, characteristic, complemented
8	#24906220	C₂	1	1	yes	maximal, characteristic, complemented
9	#24905925	C₂	1	2	yes	maximal, complemented
10	#24905745	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₈×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
12	#19128233	C₂²	1	2	yes	—
13	#19128102	C₄	1	1	yes	characteristic
14	#19128084	C₄	1	1	yes	characteristic
15	#19128054	C₄	1	2	yes	—
16	#19127943	C₄	1	1	yes	characteristic
17	#19127934	C₄	1	1	yes	characteristic
18	#19127929	C₄	1	2	yes	—
19	#19127828	C₂²	1	2	yes	—
20	C₈×Dic₁₆₆	C₂²	1	2	yes	—
21	C₄×Dic₃₃₂	C₂²	1	1	yes	characteristic
22	C₄×#13050578	C₂²	1	2	yes	—
23	#19127779	C₂²	1	2	yes	—
24	#19127776	—	2	2	no	—
25	#19127772	—	2	2	no	—
26	#19127759	—	2	2	no	—
27	#19127752	C₂²	1	1	yes	characteristic
28	#19127752	C₂²	1	1	yes	characteristic
29	#19127727	—	2	2	no	—
30	#19127723	C₂²	1	2	yes	—
31	C₄×C₈×D₈₃	C₂²	1	1	yes	characteristic
32	#19127454	C₂²	1	1	yes	characteristic
33	#19127453	C₂²	1	2	yes	—
34	#19127433	C₂²	1	2	yes	—
35	#19127431	C₂²	1	2	yes	—
36	D₄⋊(C₈₃⋊C₈)	C₂²	1	2	yes	—
37	#19127423	C₂²	1	1	yes	characteristic
38	#19127423	C₂²	1	1	yes	characteristic
39	C₈×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
40	C₄×D₈×C₈₃	C₂²	1	1	yes	characteristic
41	C₈×D₄×C₈₃	C₂²	1	2	yes	—
42	#335×C₈₃	C₂²	1	1	yes	characteristic
43	D₄⋊C₈×C₈₃	C₂²	1	2	yes	—
44	C₈²×C₈₃	C₂²	1	1	yes	characteristic
45	C₂×Dic₃₃₂	C₂×C₄	1	1	yes	characteristic
46	C₂×#13050578	C₂×C₄	1	2	yes	—
47	C₂×#13050574	—	2	2	no	—
48	C₂×#13050573	—	2	2	no	—
49	C₂×C₈×D₈₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₈×D₈₃	C₂×C₄	1	1	yes	characteristic
51	#14581785	—	2	2	no	—
52	#14581783	C₂×C₄	1	2	yes	—
53	#14581779	C₂×C₄	1	2	yes	—
54	#14581774	—	2	2	no	—
55	#14581772	C₂×C₄	1	2	yes	—
56	#14581763	C₂×C₄	1	2	yes	—
57	C₄×Dic₁₆₆	C₂³	1	2	yes	—
58	C₄×C₂²⋊D₈₃	—	2	2	no	—
59	#14581751	—	2	2	no	—
60	C₄²×D₈₃	C₂³	1	1	yes	characteristic
61	#14581726	C₂×C₄	1	1	yes	characteristic
62	#14581723	C₂×C₄	1	2	yes	—
63	C₄⋊(C₈₃⋊C₈)	C₂³	1	2	yes	—
64	Dic₈₃⋊C₈	—	2	2	no	—
65	Dic₈₃⋊C₈	—	2	2	no	—
66	Dic₈₃⋊C₈	—	2	1	no	—
67	Dic₈₃⋊C₈	—	2	2	no	—
68	Dic₈₃⋊C₈	D₄	1	1	yes	characteristic
69	Dic₈₃⋊C₈	D₄	1	1	yes	characteristic
70	#14581717	C₂×C₄	1	2	yes	—
71	#14581715	C₂×C₄	1	2	yes	—
72	#14581713	C₂×C₄	1	2	yes	—
73	C₂²⋊(C₈₃⋊C₈)	—	2	2	no	—
74	(C₂×D₈₃)⋊C₈	—	2	1	no	—
75	(C₂×D₈₃)⋊C₈	D₄	1	1	yes	characteristic
76	(C₂×D₈₃)⋊C₈	D₄	1	1	yes	characteristic
77	#14581702	—	2	2	no	—
78	C₄×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
79	C₄×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
80	C₄×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
81	C₈×Dic₈₃	—	2	2	no	—
82	C₈×Dic₈₃	C₂×C₄	1	1	yes	characteristic
83	C₈×Dic₈₃	C₂×C₄	1	1	yes	characteristic
84	C₂×D₈×C₈₃	C₂×C₄	1	1	yes	characteristic
85	C₂²×C₈×C₈₃	—	2	2	no	—
86	C₄×D₄×C₈₃	C₂³	1	2	yes	—
87	#107×C₈₃	C₂×C₄	1	1	yes	characteristic
88	C₄⋊C₈×C₈₃	C₂³	1	2	yes	—
89	#102×C₈₃	C₂×C₄	1	2	yes	—
90	C₂²⋊C₈×C₈₃	—	2	2	no	—
91	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
92	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
94	C₂×Dic₁₆₆	C₂²×C₄	1	2	yes	—
95	C₂×C₂²⋊D₈₃	—	2	2	no	—
96	C₂×C₄×D₈₃	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₄×D₈₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×D₈₃	#41	1	1	yes	characteristic
99	Dic₃₃₂	—	2	2	no	—
100	#13050578	—	2	4	no	—
101	#13050574	—	8	2	no	—
102	#13050573	—	8	2	no	—
103	C₂×C₈₃⋊C₈	—	2	2	no	—
104	C₂×C₈₃⋊C₈	—	2	1	no	—
105	C₂×C₈₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₈₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₈₃⋊C₈	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈₃⋊C₈	C₂×D₄	1	1	yes	characteristic
109	C₈×D₈₃	—	2	2	no	—
110	C₈×D₈₃	—	2	2	no	—
111	C₄⋊Dic₈₃	—	2	2	no	—
112	C₄⋊Dic₈₃	—	2	2	no	—
113	C₄⋊Dic₈₃	C₂²×C₄	1	2	yes	—
114	Dic₈₃⋊C₄	—	2	2	no	—
115	Dic₈₃⋊C₄	—	2	2	no	—
116	C₂²⋊Dic₈₃	—	2	2	no	—
117	C₂²⋊Dic₈₃	—	2	2	no	—
118	C₂²⋊Dic₈₃	—	2	2	no	—
119	(C₂×D₈₃)⋊C₄	—	2	2	no	—
120	C₄×Dic₈₃	—	2	2	no	—
121	C₄×Dic₈₃	C₂²×C₄	1	1	yes	characteristic
122	C₄×Dic₈₃	C₂×D₄	1	1	yes	characteristic
123	C₄×Dic₈₃	#41	1	1	yes	characteristic
124	C₂×D₄×C₈₃	C₂²×C₄	1	2	yes	—
125	C₂²×C₄×C₈₃	—	2	2	no	—
126	D₈×C₈₃	—	2	2	no	—
127	C₂×C₈×C₈₃	—	2	2	no	—
128	C₂×C₈×C₈₃	—	2	1	no	—
129	C₂×C₈×C₈₃	—	8	2	no	—
130	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
131	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
132	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
133	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
134	C₄⋊C₄×C₈₃	C₂²×C₄	1	2	yes	—
135	C₂²⋊C₄×C₈₃	—	2	2	no	—
136	C₄²×C₈₃	C₂⁴	1	1	yes	characteristic
137	C₂³×C₈₃	—	2	2	no	—
138	C₂²×D₈₃	C₄×D₄	1	1	yes	characteristic
139	D₄×C₈₃	—	2	2	no	—
140	D₄×C₈₃	—	2	2	no	—
141	C₂×C₄×C₈₃	—	2	2	no	—
142	C₂×C₄×C₈₃	—	4	2	no	—
143	C₂×C₄×C₈₃	—	4	2	no	—
144	C₂×C₄×C₈₃	C₂³×C₄	1	1	yes	characteristic
145	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄×C₈₃	C₂×#41	1	1	yes	characteristic
147	C₂²⋊D₈₃	—	8	2	no	—
148	C₂×Dic₈₃	—	2	2	no	—
149	C₂×Dic₈₃	—	2	2	no	—
150	C₂×Dic₈₃	C₄×D₄	1	1	yes	characteristic
151	C₂×Dic₈₃	C₄×D₄	1	1	yes	characteristic
152	C₂×Dic₈₃	C₄×D₄	1	1	yes	characteristic
153	C₄×D₈₃	—	2	1	no	—
154	C₄×D₈₃	—	2	1	no	—
155	C₄×D₈₃	—	2	1	no	—
156	C₄×D₈₃	—	2	1	no	—
157	C₄×D₈₃	—	4	1	no	—
158	Dic₁₆₆	—	2	2	no	—
159	Dic₁₆₆	—	2	2	no	—
160	C₈×C₈₃	—	2	1	no	—
161	C₈×C₈₃	—	2	1	no	—
162	C₈×C₈₃	—	2	1	no	—
163	C₈×C₈₃	—	4	1	no	—
164	C₈×C₈₃	—	4	2	no	—
165	C₈×C₈₃	#131	1	2	yes	—
166	C₈₃⋊C₈	—	2	1	no	—
167	C₈₃⋊C₈	—	2	1	no	—
168	C₈₃⋊C₈	—	2	1	no	—
169	C₈₃⋊C₈	—	4	1	no	—
170	C₈₃⋊C₈	—	4	2	no	—
171	C₈₃⋊C₈	#131	1	2	yes	—
172	C₂²×C₈₃	—	4	2	no	—
173	C₂²×C₈₃	—	4	2	no	—
174	C₂²×C₈₃	—	4	2	no	—
175	C₂²×C₈₃	C₂×C₄×D₄	1	1	yes	characteristic
176	C₂×D₈₃	—	2	1	no	—
177	C₂×D₈₃	—	2	1	no	—
178	C₂×D₈₃	—	2	1	no	—
179	C₄×C₈₃	—	2	1	no	—
180	C₄×C₈₃	—	4	2	no	—
181	C₄×C₈₃	C₂×#131	1	1	yes	characteristic
182	C₄×C₈₃	C₂×#131	1	1	yes	characteristic
183	C₄×C₈₃	#576	1	1	yes	characteristic
184	C₄×C₈₃	#578	1	1	yes	characteristic
185	Dic₈₃	—	2	1	no	—
186	Dic₈₃	—	2	1	no	—
187	Dic₈₃	—	2	1	no	—
188	Dic₈₃	—	4	2	no	—
189	Dic₈₃	—	4	2	no	—
190	#17973	—	83	1	no	maximal, Sylow
191	C₂×C₈₃	—	8	2	no	—
192	C₂×C₈₃	#2931	1	1	yes	characteristic
193	C₂×C₈₃	#2951	1	1	yes	characteristic
194	C₂×C₈₃	#2957	1	1	yes	characteristic
195	D₈₃	—	4	1	no	—
196	#3168	—	83	1	no	—
197	#2993	—	83	2	no	—
198	#1592	—	83	2	no	—
199	#1591	—	83	1	no	—
200	#1589	—	83	1	no	—
201	SD₁₆⋊C₈	—	83	2	no	—
202	C₈×Q₁₆	—	83	1	no	—
203	C₈×D₈	D₈₃	1	1	yes	characteristic, complemented
204	#1567	—	83	1	no	—
205	#1495	—	83	2	no	—
206	#1450	—	83	1	no	—
207	C₈₃	#17973	1	1	yes	characteristic, complemented, Sylow
208	#529	—	83	2	no	—
209	#499	—	83	1	no	—
210	#495	—	83	1	no	—
211	#488	—	83	2	no	—
212	#467	—	83	1	no	—
213	#466	—	83	1	no	—
214	#465	—	83	2	no	—
215	#446	—	83	2	no	—
216	C₈×Q₈	—	83	2	no	—
217	C₄×Q₁₆	—	83	1	no	—
218	C₄×SD₁₆	—	83	2	no	—
219	C₄×D₈	C₂×D₈₃	1	1	yes	characteristic
220	#436	—	83	2	no	—
221	#436	—	166	2	no	—
222	#435	—	166	2	no	—
223	C₈×D₄	C₂×D₈₃	1	2	yes	—
224	#433	—	166	2	no	—
225	#431	—	83	1	no	—
226	#431	—	83	1	no	—
227	#424	—	166	2	no	—
228	#423	—	83	2	no	—
229	C₂×C₄×C₈	—	83	1	no	—
230	#335	—	83	2	no	—
231	#335	—	83	1	no	—
232	#335	C₂×D₈₃	1	1	yes	characteristic
233	Q₈⋊C₈	—	83	2	no	—
234	Q₈⋊C₈	—	83	2	no	—
235	D₄⋊C₈	—	83	2	no	—
236	D₄⋊C₈	C₂×D₈₃	1	2	yes	—
237	#322	—	83	1	no	—
238	#322	—	83	1	no	—
239	C₈²	—	83	1	no	—
240	C₈²	C₂×D₈₃	1	1	yes	characteristic
241	C₂×Q₁₆	—	83	1	no	—
242	C₂×SD₁₆	—	83	2	no	—
243	C₂×D₈	C₄×D₈₃	1	1	yes	characteristic
244	C₂×M₁₆	—	166	2	no	—
245	C₂×M₁₆	—	166	2	no	—
246	C₂²×C₈	—	2	2	no	—
247	C₂²×C₈	—	83	1	no	—
248	C₂²×C₈	—	83	1	no	—
249	#126	—	166	2	no	—
250	#125	—	83	2	no	—
251	#124	—	83	2	no	—
252	#123	—	166	2	no	—
253	C₂²⋊Q₈	—	83	2	no	—
254	#121	—	83	2	no	—
255	C₄×Q₈	—	83	2	no	—
256	C₄×D₄	—	166	2	no	—
257	C₄×D₄	C₂²×D₈₃	1	2	yes	—
258	#117	—	166	2	no	—
259	C₂×C₄²	—	83	1	no	—
260	#107	—	83	1	no	—
261	#107	C₄×D₈₃	1	1	yes	characteristic
262	#106	—	83	2	no	—
263	C₄⋊C₈	—	83	1	no	—
264	C₄⋊C₈	—	83	1	no	—
265	C₄⋊C₈	—	83	2	no	—
266	C₄⋊C₈	—	166	2	no	—
267	C₄⋊C₈	—	166	1	no	—
268	C₄⋊C₈	—	166	2	no	—
269	C₄⋊C₈	—	166	2	no	—
270	C₄⋊C₈	C₂²×D₈₃	1	2	yes	—
271	#103	—	83	2	no	—
272	#103	—	83	2	no	—
273	#102	—	83	2	no	—
274	#102	C₄×D₈₃	1	2	yes	—
275	C₂²⋊C₈	—	2	2	no	—
276	C₂²⋊C₈	—	83	1	no	—
277	C₂²⋊C₈	—	83	1	no	—
278	C₂²⋊C₈	—	166	1	no	—
279	C₂²⋊C₈	—	166	2	no	—
280	#97	—	166	2	no	—
281	C₄×C₈	—	83	1	no	—
282	C₄×C₈	—	83	1	no	—
283	C₄×C₈	—	83	1	no	—
284	C₄×C₈	—	83	1	no	—
285	C₄×C₈	—	83	1	no	—
286	C₄×C₈	—	166	2	no	—
287	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
288	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
289	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
290	C₂×Q₈	—	83	2	no	—
291	C₂×D₄	—	166	2	no	—
292	C₂×D₄	C₂×C₄×D₈₃	1	2	yes	—
293	C₂²×C₄	—	2	2	no	—
294	C₂²×C₄	—	83	1	no	—
295	C₂²×C₄	—	83	1	no	—
296	C₂²×C₄	—	83	1	no	—
297	Q₁₆	—	166	2	no	—
298	SD₁₆	—	166	4	no	—
299	D₈	—	2	2	no	—
300	M₁₆	—	664	2	no	—
301	M₁₆	—	664	2	no	—
302	C₂×C₈	—	2	2	no	—
303	C₂×C₈	—	2	1	no	—
304	C₂×C₈	—	8	2	no	—
305	C₂×C₈	—	83	1	no	—
306	C₂×C₈	—	83	1	no	—
307	C₂×C₈	—	83	1	no	—
308	C₂×C₈	—	83	1	no	—
309	C₂×C₈	—	166	2	no	—
310	C₂×C₈	—	166	2	no	—
311	C₂×C₈	—	166	1	no	—
312	C₂×C₈	—	166	2	no	—
313	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
314	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
315	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
316	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
317	C₄⋊C₄	—	83	2	no	—
318	C₄⋊C₄	—	166	2	no	—
319	C₄⋊C₄	—	166	2	no	—
320	C₄⋊C₄	—	166	2	no	—
321	C₄⋊C₄	—	166	2	no	—
322	C₄⋊C₄	C₂×C₄×D₈₃	1	2	yes	—
323	C₂²⋊C₄	—	2	2	no	—
324	C₂²⋊C₄	—	166	2	no	—
325	C₂²⋊C₄	—	166	2	no	—
326	C₂²⋊C₄	—	166	2	no	—
327	C₂²⋊C₄	—	166	2	no	—
328	C₄²	—	83	1	no	—
329	C₄²	—	83	1	no	—
330	C₄²	—	83	1	no	—
331	C₄²	—	166	2	no	—
332	C₄²	C₂³×D₈₃	1	1	yes	characteristic
333	C₂³	—	2	2	no	—
334	C₂³	—	83	1	no	—
335	Q₈	—	166	2	no	—
336	Q₈	—	166	2	no	—
337	D₄	—	2	2	no	—
338	D₄	—	2	2	no	—
339	D₄	—	664	2	no	—
340	C₂×C₄	—	2	2	no	—
341	C₂×C₄	—	4	2	no	—
342	C₂×C₄	—	4	2	no	—
343	C₂×C₄	—	332	1	no	—
344	C₂×C₄	—	83	1	no	—
345	C₂×C₄	—	83	1	no	—
346	C₂×C₄	—	83	1	no	—
347	C₂×C₄	—	166	2	no	—
348	C₂×C₄	—	166	2	no	—
349	C₂×C₄	—	166	1	no	—
350	C₂×C₄	—	166	1	no	—
351	C₂×C₄	—	166	1	no	—
352	C₂×C₄	—	166	1	no	—
353	C₂×C₄	C₂²×C₄×D₈₃	1	1	yes	characteristic
354	C₂×C₄	C₂×D₄×D₈₃	1	1	yes	characteristic
355	C₂×C₄	#41×D₈₃	1	1	yes	characteristic
356	C₈	—	2	1	no	—
357	C₈	—	2	1	no	—
358	C₈	—	2	1	no	—
359	C₈	—	4	1	no	—
360	C₈	—	4	2	no	—
361	C₈	—	332	1	no	—
362	C₈	—	332	2	no	—
363	C₈	—	83	2	no	—
364	C₈	—	166	1	no	—
365	C₈	—	166	1	no	—
366	C₈	—	166	1	no	—
367	C₈	#14581798	1	2	yes	—
368	C₂²	—	4	2	no	—
369	C₂²	—	4	2	no	—
370	C₂²	—	4	2	no	—
371	C₂²	—	166	1	no	—
372	C₂²	—	166	1	no	—
373	C₂²	—	166	1	no	—
374	C₂²	C₄×D₄×D₈₃	1	1	yes	characteristic
375	C₄	—	2	1	no	—
376	C₄	—	4	2	no	—
377	C₄	—	332	2	no	—
378	C₄	—	332	2	no	—
379	C₄	—	166	1	no	—
380	C₄	—	166	1	no	—
381	C₄	—	166	1	no	—
382	C₄	#131×D₈₃	1	1	yes	characteristic
383	C₄	C₂×#14581798	1	1	yes	characteristic
384	C₄	#19128502	1	1	yes	characteristic
385	C₄	#19128515	1	1	yes	characteristic
386	C₂	—	8	2	no	—
387	C₂	—	332	1	no	—
388	C₂	#24916170	1	1	yes	characteristic
389	C₂	#24916333	1	1	yes	characteristic
390	C₂	#24916375	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₈₃
commutator	C₂×C₄×C₈₃	C₂³×C₄
Frattini	C₄²	C₂³×D₈₃
Fitting	C₈×D₈×C₈₃	C₂
socle	C₂²×C₈₃	C₂×C₄×D₄
prime radical	#19127934	C₄
cosocle	C₄²×C₈₃	C₂⁴
hypercenter	C₈×D₈	D₈₃
hypocenter	C₈₃	#17973
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈×D₈

Complements

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

N	H ≅ G/N	type
2. #24916613	C₂	semidirect
3. #24906362	C₂	semidirect
4. #24906359	C₂	semidirect
5. #24906348	C₂	semidirect
6. #24906275	C₂	semidirect
7. C₈×Dic₃₃₂	C₂	semidirect
8. #24906220	C₂	semidirect
9. #24905925	C₂	semidirect
10. #24905745	C₂	semidirect
11. C₈×D₈×C₈₃	C₂	semidirect
203. C₈×D₈	D₈₃	semidirect
207. C₈₃	#17973	semidirect

Maximal subgroups

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

subgroup	index
#24918031	2
#24916613	2
#24906362	2
#24906359	2
#24906348	2
#24906275	2
C₈×Dic₃₃₂	2
#24906220	2
#24905925	2
#24905745	2
C₈×D₈×C₈₃	2
#17973	83