Subgroups — #35375121

Order 21248.

658 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24918022	C₂	1	1	yes	maximal, characteristic
2	#24916397	C₂	1	1	yes	maximal, characteristic, complemented
3	#24916176	C₂	1	1	yes	maximal, characteristic
4	#24906308	C₂	1	4	yes	maximal, complemented
5	#24906301	C₂	1	4	yes	maximal, complemented
6	#24906219	C₂	1	1	yes	maximal, characteristic, complemented
7	#24905919	C₂	1	1	yes	maximal, characteristic, complemented
8	#24905915	C₂	1	1	yes	maximal, characteristic
9	C₂×#322×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14581798	—	2	1	no	—
11	#19128258	C₂²	1	1	yes	characteristic
12	#19128251	C₂²	1	1	yes	characteristic
13	#19128099	C₄	1	1	yes	characteristic
14	#19128074	C₄	1	1	yes	characteristic
15	#19128052	C₄	1	1	yes	characteristic
16	#19128048	C₄	1	1	yes	characteristic
17	#19127930	C₄	1	4	yes	—
18	#19127831	—	2	1	no	—
19	#19127831	C₂²	1	2	yes	—
20	C₈×Dic₁₆₆	—	2	1	no	—
21	C₈×Dic₁₆₆	C₂²	1	2	yes	—
22	C₄×#13050579	C₂²	1	4	yes	—
23	#19127780	—	2	1	no	—
24	#19127780	C₂²	1	2	yes	—
25	C₈×D₃₃₂	—	2	1	no	—
26	C₈×D₃₃₂	C₂²	1	2	yes	—
27	#19127727	—	2	1	no	—
28	#19127725	C₂²	1	1	yes	characteristic
29	#19127725	C₂²	1	1	yes	characteristic
30	#19127667	—	2	1	no	—
31	#19127656	—	2	1	no	—
32	#19127452	C₂²	1	4	yes	—
33	#19127433	C₂²	1	4	yes	—
34	#19127430	C₂²	1	4	yes	—
35	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
37	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
38	#322×C₈₃	C₂²	1	2	yes	—
39	#322×C₈₃	C₂²	1	2	yes	—
40	C₂×#13050590	—	2	1	no	—
41	C₂×#13050579	C₂×C₄	1	4	yes	—
42	#14581798	—	4	4	no	—
43	C₂×#13050574	—	2	1	no	—
44	C₂×#13050574	—	2	1	no	—
45	C₂×#13050573	—	2	2	no	—
46	C₂×C₈×D₈₃	—	2	2	no	—
47	#14581792	C₂×C₄	1	1	yes	characteristic
48	#14581792	C₂×C₄	1	1	yes	characteristic
49	#14581792	C₂×C₄	1	1	yes	characteristic
50	#14581789	C₂×C₄	1	1	yes	characteristic
51	#14581784	—	2	1	no	—
52	#14581780	—	2	1	no	—
53	#14581771	—	2	1	no	—
54	#14581771	C₂×C₄	1	2	yes	—
55	#14581766	—	2	1	no	—
56	#14581766	C₂×C₄	1	2	yes	—
57	C₄×Dic₁₆₆	—	2	1	no	—
58	C₄×Dic₁₆₆	C₂³	1	2	yes	—
59	C₄×D₃₃₂	—	2	1	no	—
60	C₄×D₃₃₂	C₂³	1	2	yes	—
61	#14581751	—	2	1	no	—
62	#14581724	C₂×C₄	1	4	yes	—
63	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
64	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
65	C₄⋊(C₈₃⋊C₈)	C₂³	1	2	yes	—
66	C₄⋊(C₈₃⋊C₈)	C₂³	1	2	yes	—
67	Dic₈₃⋊C₈	—	2	2	no	—
68	Dic₈₃⋊C₈	—	2	2	no	—
69	#14581717	C₂×C₄	1	4	yes	—
70	#14581714	C₂×C₄	1	4	yes	—
71	(C₂×D₈₃)⋊C₈	—	2	2	no	—
72	(C₂×D₈₃)⋊C₈	—	2	2	no	—
73	#14581702	—	2	2	no	—
74	C₈×Dic₈₃	—	2	2	no	—
75	C₂²×C₈×C₈₃	—	2	1	no	—
76	C₂²×C₈×C₈₃	C₂×C₄	1	2	yes	—
77	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
78	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
79	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
80	C₄×C₈×C₈₃	—	2	1	no	—
81	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
82	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
83	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
84	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
85	C₄×C₈×C₈₃	C₂³	1	2	yes	—
86	C₄×C₈×C₈₃	C₂³	1	2	yes	—
87	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
88	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
89	#13050590	—	4	2	no	—
90	#13050590	—	4	2	no	—
91	C₂×Dic₁₆₆	—	2	1	no	—
92	C₂×Dic₁₆₆	—	2	1	no	—
93	C₂×Dic₁₆₆	C₂²×C₄	1	2	yes	—
94	C₂×D₃₃₂	—	2	1	no	—
95	C₂×D₃₃₂	—	2	1	no	—
96	C₂×D₃₃₂	C₂²×C₄	1	2	yes	—
97	C₂×C₂²⋊D₈₃	—	2	2	no	—
98	C₂×C₄×D₈₃	—	2	2	no	—
99	#13050579	—	4	4	no	—
100	#13050574	—	4	2	no	—
101	#13050574	—	4	2	no	—
102	#13050573	—	4	4	no	—
103	C₂×C₈₃⋊C₈	—	2	2	no	—
104	C₂×C₈₃⋊C₈	—	2	2	no	—
105	C₈×D₈₃	—	4	4	no	—
106	C₄⋊Dic₈₃	—	2	1	no	—
107	C₄⋊Dic₈₃	—	2	1	no	—
108	C₄⋊Dic₈₃	—	2	1	no	—
109	C₄⋊Dic₈₃	—	2	1	no	—
110	C₄⋊Dic₈₃	C₂²×C₄	1	2	yes	—
111	C₄⋊Dic₈₃	C₂²×C₄	1	2	yes	—
112	Dic₈₃⋊C₄	—	2	2	no	—
113	Dic₈₃⋊C₄	—	2	2	no	—
114	C₂²⋊Dic₈₃	—	2	2	no	—
115	(C₂×D₈₃)⋊C₄	—	2	2	no	—
116	(C₂×D₈₃)⋊C₄	—	2	2	no	—
117	C₄×Dic₈₃	—	2	2	no	—
118	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
119	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₄×C₈₃	#41	1	1	yes	characteristic
121	C₂×C₈×C₈₃	—	2	2	no	—
122	C₂×C₈×C₈₃	—	2	2	no	—
123	C₂×C₈×C₈₃	—	4	2	no	—
124	C₂×C₈×C₈₃	—	4	2	no	—
125	C₂×C₈×C₈₃	SD₁₆	1	4	yes	complemented
126	C₂×C₈×C₈₃	C₂²×C₄	1	2	yes	—
127	C₂×C₈×C₈₃	C₂²×C₄	1	2	yes	—
128	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
129	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
130	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
131	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
132	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
133	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
134	C₄²×C₈₃	#41	1	1	yes	characteristic
135	C₄²×C₈₃	C₂⁴	1	1	yes	characteristic
136	C₂³×C₈₃	C₄×D₄	1	1	yes	characteristic
137	C₂²×D₈₃	—	2	2	no	—
138	C₂×C₄×C₈₃	—	2	2	no	—
139	C₂×C₄×C₈₃	—	2	2	no	—
140	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
141	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
142	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
143	C₂×C₄×C₈₃	C₂×SD₁₆	1	2	yes	—
144	C₂×C₄×C₈₃	#135	1	2	yes	—
145	C₂×C₄×C₈₃	C₂³×C₄	1	1	yes	characteristic
146	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄×C₈₃	C₂×#41	1	1	yes	characteristic
148	C₂²⋊D₈₃	—	4	4	no	—
149	C₂×Dic₈₃	—	2	2	no	—
150	C₂×Dic₈₃	—	2	2	no	—
151	C₂×Dic₈₃	—	2	2	no	—
152	D₃₃₂	—	2	4	no	—
153	D₃₃₂	—	4	2	no	—
154	D₃₃₂	—	4	2	no	—
155	C₄×D₈₃	—	4	2	no	—
156	C₄×D₈₃	—	4	2	no	—
157	Dic₁₆₆	—	2	4	no	—
158	Dic₁₆₆	—	4	2	no	—
159	Dic₁₆₆	—	4	2	no	—
160	C₈×C₈₃	—	2	2	no	—
161	C₈×C₈₃	—	2	2	no	—
162	C₈×C₈₃	—	2	1	no	—
163	C₈×C₈₃	—	2	1	no	—
164	C₈×C₈₃	—	4	2	no	—
165	C₈×C₈₃	C₂×SD₁₆	1	2	yes	—
166	C₈×C₈₃	C₂×SD₁₆	1	2	yes	—
167	C₈₃⋊C₈	—	4	2	no	—
168	C₈₃⋊C₈	—	4	2	no	—
169	C₂²×C₈₃	—	2	1	no	—
170	C₂²×C₈₃	—	2	1	no	—
171	C₂²×C₈₃	C₄×SD₁₆	1	2	yes	—
172	C₂²×C₈₃	C₂×C₄×D₄	1	1	yes	characteristic
173	C₂×D₈₃	—	4	2	no	—
174	C₂×D₈₃	—	4	2	no	—
175	C₂×D₈₃	—	4	2	no	—
176	C₄×C₈₃	—	2	1	no	—
177	C₄×C₈₃	—	2	1	no	—
178	C₄×C₈₃	C₄×SD₁₆	1	2	yes	—
179	C₄×C₈₃	#444	1	2	yes	—
180	C₄×C₈₃	#568	1	2	yes	—
181	C₄×C₈₃	C₂²×SD₁₆	1	1	yes	characteristic
182	C₄×C₈₃	C₂×#135	1	1	yes	characteristic
183	Dic₈₃	—	4	2	no	—
184	Dic₈₃	—	4	2	no	—
185	Dic₈₃	—	4	2	no	—
186	#17976	—	83	1	no	maximal, Sylow
187	C₂×C₈₃	—	2	2	no	—
188	C₂×C₈₃	#2932	1	1	yes	characteristic
189	C₂×C₈₃	C₂×C₄×SD₁₆	1	1	yes	characteristic
190	C₂×C₈₃	C₂×#444	1	1	yes	characteristic
191	D₈₃	—	8	2	no	—
192	#3167	—	83	1	no	—
193	#2960	—	83	1	no	—
194	#2931	—	83	1	no	—
195	#1583	—	83	4	no	—
196	#1582	—	83	4	no	—
197	#1566	—	83	1	no	—
198	#1493	—	83	1	no	—
199	#1492	—	83	1	no	—
200	C₂×#322	D₈₃	1	1	yes	characteristic, complemented
201	C₈₃	#17976	1	1	yes	characteristic, complemented, Sylow
202	C₂×#131	—	166	1	no	—
203	#533	—	83	1	no	—
204	#532	—	83	1	no	—
205	#498	—	83	1	no	—
206	#492	—	83	1	no	—
207	#487	—	83	1	no	—
208	#486	—	83	1	no	—
209	#465	—	83	4	no	—
210	#446	—	83	2	no	—
211	#446	—	166	1	no	—
212	C₈×Q₈	—	83	2	no	—
213	C₈×Q₈	—	166	1	no	—
214	C₄×SD₁₆	—	83	4	no	—
215	#436	—	83	2	no	—
216	#436	—	166	1	no	—
217	C₈×D₄	—	83	2	no	—
218	C₈×D₄	—	166	1	no	—
219	#424	—	166	1	no	—
220	#423	—	83	1	no	—
221	#423	—	83	1	no	—
222	#408	—	166	1	no	—
223	#405	—	166	1	no	—
224	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
225	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
226	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
227	#334	—	83	4	no	—
228	Q₈⋊C₈	—	83	4	no	—
229	D₄⋊C₈	—	83	4	no	—
230	#322	C₂×D₈₃	1	2	yes	—
231	#322	C₂×D₈₃	1	2	yes	—
232	C₂×#41	—	166	1	no	—
233	C₂×SD₁₆	—	83	4	no	—
234	#131	—	332	4	no	—
235	C₂×M₁₆	—	166	1	no	—
236	C₂×M₁₆	—	166	1	no	—
237	C₂×M₁₆	—	166	2	no	—
238	C₂²×C₈	—	2	1	no	—
239	C₂²×C₈	—	166	2	no	—
240	C₂²×C₈	C₄×D₈₃	1	2	yes	—
241	C₂²×C₈	D₃₃₂	1	1	yes	characteristic
242	C₂²×C₈	D₃₃₂	1	1	yes	characteristic
243	C₄⋊Q₈	—	83	1	no	—
244	C₄⋊Q₈	—	83	1	no	—
245	C₄⋊Q₈	—	83	1	no	—
246	#127	—	83	1	no	—
247	#125	—	166	1	no	—
248	#124	—	166	1	no	—
249	C₂²⋊Q₈	—	83	2	no	—
250	C₂²⋊Q₈	—	166	1	no	—
251	#121	—	83	2	no	—
252	#121	—	166	1	no	—
253	C₄×Q₈	—	83	2	no	—
254	C₄×Q₈	—	166	1	no	—
255	C₄×D₄	—	83	2	no	—
256	C₄×D₄	—	166	1	no	—
257	#117	—	166	1	no	—
258	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
259	#106	—	83	4	no	—
260	C₄⋊C₈	—	83	2	no	—
261	C₄⋊C₈	—	83	2	no	—
262	C₄⋊C₈	—	166	2	no	—
263	C₄⋊C₈	—	166	2	no	—
264	C₄⋊C₈	—	166	1	no	—
265	C₄⋊C₈	—	166	1	no	—
266	#103	—	83	4	no	—
267	#102	—	83	4	no	—
268	C₂²⋊C₈	—	166	2	no	—
269	C₂²⋊C₈	—	166	2	no	—
270	#97	—	166	2	no	—
271	C₄×C₈	—	2	1	no	—
272	C₄×C₈	—	166	2	no	—
273	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
274	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
275	C₄×C₈	D₃₃₂	1	1	yes	characteristic
276	C₄×C₈	D₃₃₂	1	1	yes	characteristic
277	C₄×C₈	C₂²×D₈₃	1	2	yes	—
278	C₄×C₈	C₂²×D₈₃	1	2	yes	—
279	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
280	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
281	#41	—	332	2	no	—
282	#41	—	332	2	no	—
283	C₂×Q₈	—	83	2	no	—
284	C₂×Q₈	—	166	1	no	—
285	C₂×Q₈	—	166	1	no	—
286	C₂×D₄	—	83	2	no	—
287	C₂×D₄	—	166	2	no	—
288	C₂×D₄	—	166	1	no	—
289	C₂×D₄	—	166	1	no	—
290	C₂²×C₄	—	166	2	no	—
291	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
292	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
293	C₂²×C₄	#13050590	1	1	yes	characteristic
294	SD₁₆	—	332	4	no	—
295	M₁₆	—	332	2	no	—
296	M₁₆	—	332	2	no	—
297	M₁₆	—	332	4	no	—
298	C₂×C₈	—	2	2	no	—
299	C₂×C₈	—	2	2	no	—
300	C₂×C₈	—	4	2	no	—
301	C₂×C₈	—	4	2	no	—
302	C₂×C₈	—	332	4	no	—
303	C₂×C₈	—	166	2	no	—
304	C₂×C₈	—	166	2	no	—
305	C₂×C₈	#13050579	1	4	yes	complemented
306	C₂×C₈	C₂×C₄×D₈₃	1	2	yes	—
307	C₂×C₈	C₂×C₄×D₈₃	1	2	yes	—
308	C₂×C₈	C₂×D₃₃₂	1	1	yes	characteristic
309	C₂×C₈	C₂×D₃₃₂	1	1	yes	characteristic
310	C₂×C₈	C₂×D₃₃₂	1	1	yes	characteristic
311	C₂×C₈	C₂×D₃₃₂	1	1	yes	characteristic
312	C₄⋊C₄	—	83	2	no	—
313	C₄⋊C₄	—	83	2	no	—
314	C₄⋊C₄	—	166	2	no	—
315	C₄⋊C₄	—	166	1	no	—
316	C₄⋊C₄	—	166	1	no	—
317	C₄⋊C₄	—	166	1	no	—
318	C₄⋊C₄	—	166	2	no	—
319	C₄⋊C₄	—	166	1	no	—
320	C₂²⋊C₄	—	166	2	no	—
321	C₂²⋊C₄	—	166	2	no	—
322	C₂²⋊C₄	—	166	2	no	—
323	C₄²	—	166	2	no	—
324	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
325	C₄²	C₂×D₃₃₂	1	1	yes	characteristic
326	C₄²	#13050590	1	1	yes	characteristic
327	C₄²	C₂³×D₈₃	1	1	yes	characteristic
328	C₂³	—	166	2	no	—
329	C₂³	C₄×D₃₃₂	1	1	yes	characteristic
330	Q₈	—	332	2	no	—
331	Q₈	—	332	2	no	—
332	Q₈	—	166	4	no	—
333	D₄	—	332	4	no	—
334	D₄	—	332	2	no	—
335	D₄	—	332	2	no	—
336	D₄	—	166	4	no	—
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	332	2	no	—
340	C₂×C₄	—	332	2	no	—
341	C₂×C₄	—	166	2	no	—
342	C₂×C₄	—	166	2	no	—
343	C₂×C₄	—	166	2	no	—
344	C₂×C₄	C₄×D₃₃₂	1	1	yes	characteristic
345	C₂×C₄	C₄×D₃₃₂	1	1	yes	characteristic
346	C₂×C₄	C₄×D₃₃₂	1	1	yes	characteristic
347	C₂×C₄	C₂×#13050579	1	2	yes	—
348	C₂×C₄	#14581814	1	2	yes	—
349	C₂×C₄	C₂²×C₄×D₈₃	1	1	yes	characteristic
350	C₂×C₄	C₂²×D₃₃₂	1	1	yes	characteristic
351	C₂×C₄	C₂×#13050590	1	1	yes	characteristic
352	C₈	—	2	2	no	—
353	C₈	—	2	2	no	—
354	C₈	—	2	1	no	—
355	C₈	—	2	1	no	—
356	C₈	—	4	2	no	—
357	C₈	—	332	2	no	—
358	C₈	—	332	2	no	—
359	C₈	C₂×#13050579	1	2	yes	—
360	C₈	C₂×#13050579	1	2	yes	—
361	C₂²	—	2	1	no	—
362	C₂²	—	2	1	no	—
363	C₂²	—	332	2	no	—
364	C₂²	—	332	2	no	—
365	C₂²	—	332	2	no	—
366	C₂²	C₄×#13050579	1	2	yes	—
367	C₂²	C₂×C₄×D₃₃₂	1	1	yes	characteristic
368	C₄	—	2	1	no	—
369	C₄	—	2	1	no	—
370	C₄	—	332	2	no	—
371	C₄	—	332	2	no	—
372	C₄	—	332	2	no	—
373	C₄	C₄×#13050579	1	2	yes	—
374	C₄	#19127824	1	2	yes	—
375	C₄	#19128467	1	2	yes	—
376	C₄	C₂²×#13050579	1	1	yes	characteristic
377	C₄	C₂×#14581814	1	1	yes	characteristic
378	C₂	—	2	2	no	—
379	C₂	—	664	2	no	—
380	C₂	#24916182	1	1	yes	characteristic
381	C₂	C₂×C₄×#13050579	1	1	yes	characteristic
382	C₂	C₂×#19127824	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₃₃₂
commutator	C₂×C₄×C₈₃	C₂³×C₄
Frattini	C₄²	C₂³×D₈₃
Fitting	C₂×#322×C₈₃	C₂
socle	C₂²×C₈₃	C₂×C₄×D₄
prime radical	#19128251	C₂²
cosocle	C₄²×C₈₃	C₂⁴
hypercenter	C₂×#322	D₈₃
hypocenter	C₈₃	#17976
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
2. #24916397	C₂	semidirect
4. #24906308	C₂	semidirect
5. #24906301	C₂	semidirect
6. #24906219	C₂	semidirect
7. #24905919	C₂	semidirect
9. C₂×#322×C₈₃	C₂	semidirect
125. C₂×C₈×C₈₃	SD₁₆	semidirect
200. C₂×#322	D₈₃	semidirect
201. C₈₃	#17976	semidirect
305. C₂×C₈	#13050579	semidirect

Maximal subgroups

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

subgroup	index
#24918022	2
#24916397	2
#24916176	2
#24906308	2
#24906301	2
#24906219	2
#24905919	2
#24905915	2
C₂×#322×C₈₃	2
#17976	83