Subgroups — #35392089

Order 21248.

750 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24918049	C₂	1	1	yes	maximal, characteristic, complemented
2	#24918042	C₂	1	1	yes	maximal, characteristic
3	#24918022	C₂	1	2	yes	maximal, complemented
4	#24917786	C₂	1	2	yes	maximal, complemented
5	#24907075	C₂	1	2	yes	maximal, complemented
6	#24907073	C₂	1	1	yes	maximal, characteristic, complemented
7	#24907069	C₂	1	1	yes	maximal, characteristic, complemented
8	#24907064	C₂	1	2	yes	maximal, complemented
9	#24907060	C₂	1	1	yes	maximal, characteristic, complemented
10	#24907055	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₈²×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
12	#19128258	C₂²	1	2	yes	—
13	#19128251	C₂²	1	2	yes	—
14	#19128106	C₂²	1	1	yes	characteristic
15	#19128106	C₂²	1	1	yes	characteristic
16	#19128099	C₂²	1	2	yes	—
17	#19128081	C₂²	1	1	yes	characteristic
18	#19128078	C₂²	1	1	yes	characteristic
19	#19128074	C₂²	1	2	yes	—
20	#19128056	—	2	4	no	—
21	#19128052	C₂²	1	2	yes	—
22	#19128052	C₂²	1	2	yes	—
23	#19128052	C₂²	1	2	yes	—
24	#19128052	C₂²	1	2	yes	—
25	#19128048	C₂²	1	2	yes	—
26	#19128048	C₂²	1	2	yes	—
27	#19128048	C₂²	1	2	yes	—
28	#19128048	C₂²	1	2	yes	—
29	#19127942	—	2	2	no	—
30	#19127937	—	2	2	no	—
31	#19127930	—	2	4	no	—
32	C₄×Dic₃₃₂	—	2	2	no	—
33	C₄×#13050579	—	2	4	no	—
34	C₄×D₆₆₄	—	2	2	no	—
35	#19127698	—	2	4	no	—
36	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
37	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
39	C₈²×C₈₃	C₂²	1	1	yes	characteristic
40	C₈²×C₈₃	C₂²	1	1	yes	characteristic
41	C₈²×C₈₃	C₂²	1	1	yes	characteristic
42	C₈²×C₈₃	C₂²	1	1	yes	characteristic
43	C₂×#13050590	—	2	2	no	—
44	C₂×Dic₃₃₂	—	2	2	no	—
45	C₂×#13050579	—	2	4	no	—
46	C₂×D₆₆₄	—	2	2	no	—
47	#14581792	C₂³	1	2	yes	—
48	#14581792	C₂³	1	2	yes	—
49	#14581792	C₂³	1	2	yes	—
50	#14581789	C₂³	1	2	yes	—
51	#14581784	—	2	2	no	—
52	#14581780	—	2	2	no	—
53	#14581771	—	2	2	no	—
54	#14581771	—	2	2	no	—
55	#14581766	—	2	2	no	—
56	#14581766	—	2	2	no	—
57	C₄×Dic₁₆₆	—	2	2	no	—
58	C₄×Dic₁₆₆	—	2	2	no	—
59	C₄×D₃₃₂	—	2	2	no	—
60	C₄×D₃₃₂	—	2	2	no	—
61	#14581751	—	2	2	no	—
62	#14581726	—	2	2	no	—
63	#14581726	—	2	2	no	—
64	#14581724	—	2	4	no	—
65	#14581717	—	2	4	no	—
66	#14581717	—	2	4	no	—
67	#14581717	—	2	4	no	—
68	#14581717	—	2	4	no	—
69	#14581714	—	2	4	no	—
70	#14581714	—	2	4	no	—
71	#14581714	—	2	4	no	—
72	#14581714	—	2	4	no	—
73	C₂²×C₈×C₈₃	D₄	1	2	yes	—
74	C₂²×C₈×C₈₃	D₄	1	2	yes	—
75	C₂²×C₈×C₈₃	D₄	1	2	yes	—
76	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
77	C₄×C₈×C₈₃	D₄	1	2	yes	—
78	C₄×C₈×C₈₃	D₄	1	2	yes	—
79	C₄×C₈×C₈₃	D₄	1	2	yes	—
80	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
81	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
82	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
83	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
84	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
85	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
86	#13050590	—	4	8	no	—
87	C₂×Dic₁₆₆	—	2	2	no	—
88	C₂×Dic₁₆₆	—	2	2	no	—
89	C₂×Dic₁₆₆	—	2	2	no	—
90	C₂×D₃₃₂	—	2	2	no	—
91	C₂×D₃₃₂	—	2	2	no	—
92	C₂×D₃₃₂	—	2	2	no	—
93	C₂×C₂²⋊D₈₃	—	4	2	no	—
94	C₂×C₄×D₈₃	—	4	2	no	—
95	Dic₃₃₂	—	4	4	no	—
96	#13050579	—	4	8	no	—
97	D₆₆₄	—	4	4	no	—
98	C₄⋊Dic₈₃	—	2	2	no	—
99	C₄⋊Dic₈₃	—	2	2	no	—
100	C₄⋊Dic₈₃	—	2	2	no	—
101	C₄⋊Dic₈₃	—	2	2	no	—
102	C₄⋊Dic₈₃	—	2	2	no	—
103	C₄⋊Dic₈₃	—	2	2	no	—
104	Dic₈₃⋊C₄	—	4	2	no	—
105	Dic₈₃⋊C₄	—	4	2	no	—
106	C₂²⋊Dic₈₃	—	4	2	no	—
107	(C₂×D₈₃)⋊C₄	—	4	2	no	—
108	(C₂×D₈₃)⋊C₄	—	4	2	no	—
109	C₄×Dic₈₃	—	4	2	no	—
110	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
113	C₂×C₈×C₈₃	—	2	4	no	—
114	C₂×C₈×C₈₃	—	2	4	no	—
115	C₂×C₈×C₈₃	—	2	4	no	—
116	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
117	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
118	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
119	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
120	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
121	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
122	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
123	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
124	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
125	C₄²×C₈₃	C₂⁴	1	1	yes	characteristic
126	C₂³×C₈₃	#127	1	1	yes	characteristic
127	C₂²×D₈₃	—	4	2	no	—
128	C₂×C₄×C₈₃	—	2	2	no	—
129	C₂×C₄×C₈₃	—	2	2	no	—
130	C₂×C₄×C₈₃	#127	1	1	yes	characteristic
131	C₂×C₄×C₈₃	#127	1	1	yes	characteristic
132	C₂×C₄×C₈₃	#127	1	1	yes	characteristic
133	C₂×C₄×C₈₃	#135	1	4	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	C₂²×D₄	1	1	yes	characteristic
137	C₂²⋊D₈₃	—	8	4	no	—
138	C₂×Dic₈₃	—	4	2	no	—
139	C₂×Dic₈₃	—	4	2	no	—
140	C₂×Dic₈₃	—	4	2	no	—
141	D₃₃₂	—	4	4	no	—
142	D₃₃₂	—	4	4	no	—
143	D₃₃₂	—	4	4	no	—
144	C₄×D₈₃	—	8	4	no	—
145	Dic₁₆₆	—	4	4	no	—
146	Dic₁₆₆	—	4	4	no	—
147	Dic₁₆₆	—	4	4	no	—
148	C₈×C₈₃	—	2	2	no	—
149	C₈×C₈₃	—	2	2	no	—
150	C₈×C₈₃	—	2	2	no	—
151	C₈×C₈₃	—	2	2	no	—
152	C₈×C₈₃	#135	1	4	yes	—
153	C₈×C₈₃	#135	1	4	yes	—
154	C₂²×C₈₃	—	2	1	no	—
155	C₂²×C₈₃	—	2	1	no	—
156	C₂²×C₈₃	#495	1	2	yes	—
157	C₂²×C₈₃	C₂×#127	1	1	yes	characteristic
158	C₂×D₈₃	—	8	2	no	—
159	C₂×D₈₃	—	8	2	no	—
160	C₂×D₈₃	—	8	2	no	—
161	C₄×C₈₃	—	2	1	no	—
162	C₄×C₈₃	—	2	1	no	—
163	C₄×C₈₃	#495	1	2	yes	—
164	C₄×C₈₃	C₂×#135	1	2	yes	—
165	C₄×C₈₃	C₂×#135	1	2	yes	—
166	C₄×C₈₃	C₂×#135	1	2	yes	—
167	Dic₈₃	—	8	2	no	—
168	Dic₈₃	—	8	2	no	—
169	Dic₈₃	—	8	2	no	—
170	#19600	—	83	1	no	maximal, Sylow
171	C₂×C₈₃	—	2	2	no	—
172	C₂×C₈₃	C₂×#495	1	1	yes	characteristic
173	C₂×C₈₃	#3147	1	1	yes	characteristic
174	C₂×C₈₃	#3147	1	1	yes	characteristic
175	D₈₃	—	16	2	no	—
176	#3170	—	83	1	no	—
177	#3169	—	83	1	no	—
178	#3167	—	83	2	no	—
179	#3131	—	83	2	no	—
180	#1711	—	83	2	no	—
181	#1710	—	83	1	no	—
182	#1709	—	83	1	no	—
183	#1708	—	83	2	no	—
184	#1707	—	83	1	no	—
185	#1706	—	83	1	no	—
186	C₂×C₈²	D₈₃	1	1	yes	characteristic, complemented
187	C₈₃	#19600	1	1	yes	characteristic, complemented, Sylow
188	#533	—	83	2	no	—
189	#532	—	83	2	no	—
190	#500	—	83	1	no	—
191	#500	—	83	1	no	—
192	#498	—	83	2	no	—
193	#494	—	83	1	no	—
194	#493	—	83	1	no	—
195	#492	—	83	2	no	—
196	#488	—	166	4	no	—
197	#487	—	83	2	no	—
198	#487	—	83	2	no	—
199	#487	—	83	2	no	—
200	#487	—	83	2	no	—
201	#486	—	83	2	no	—
202	#486	—	83	2	no	—
203	#486	—	83	2	no	—
204	#486	—	83	2	no	—
205	#467	—	166	2	no	—
206	#466	—	166	2	no	—
207	#465	—	166	4	no	—
208	C₄×Q₁₆	—	166	2	no	—
209	C₄×SD₁₆	—	166	4	no	—
210	C₄×D₈	—	166	2	no	—
211	#416	—	166	4	no	—
212	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
213	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
214	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
215	C₈²	C₂×D₈₃	1	1	yes	characteristic
216	C₈²	C₂×D₈₃	1	1	yes	characteristic
217	C₈²	C₂×D₈₃	1	1	yes	characteristic
218	C₈²	C₂×D₈₃	1	1	yes	characteristic
219	C₂×#41	—	166	2	no	—
220	C₂×Q₁₆	—	166	2	no	—
221	C₂×SD₁₆	—	166	4	no	—
222	C₂×D₈	—	166	2	no	—
223	C₂²×C₈	D₃₃₂	1	2	yes	—
224	C₂²×C₈	D₃₃₂	1	2	yes	—
225	C₂²×C₈	D₃₃₂	1	2	yes	—
226	C₄⋊Q₈	—	83	2	no	—
227	C₄⋊Q₈	—	83	2	no	—
228	C₄⋊Q₈	—	83	2	no	—
229	#127	—	83	2	no	—
230	#125	—	166	2	no	—
231	#124	—	166	2	no	—
232	C₂²⋊Q₈	—	166	2	no	—
233	C₂²⋊Q₈	—	166	2	no	—
234	#121	—	166	2	no	—
235	#121	—	166	2	no	—
236	C₄×Q₈	—	166	2	no	—
237	C₄×Q₈	—	166	2	no	—
238	C₄×D₄	—	166	2	no	—
239	C₄×D₄	—	166	2	no	—
240	#117	—	166	2	no	—
241	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
242	#107	—	166	2	no	—
243	#107	—	166	2	no	—
244	#106	—	166	4	no	—
245	#103	—	166	4	no	—
246	#103	—	166	4	no	—
247	#103	—	166	4	no	—
248	#103	—	166	4	no	—
249	#102	—	166	4	no	—
250	#102	—	166	4	no	—
251	#102	—	166	4	no	—
252	#102	—	166	4	no	—
253	C₄×C₈	D₃₃₂	1	2	yes	—
254	C₄×C₈	D₃₃₂	1	2	yes	—
255	C₄×C₈	D₃₃₂	1	2	yes	—
256	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
257	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
258	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
259	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
260	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
261	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
262	#41	—	332	8	no	—
263	C₂×Q₈	—	166	2	no	—
264	C₂×Q₈	—	166	2	no	—
265	C₂×Q₈	—	166	2	no	—
266	C₂×D₄	—	332	2	no	—
267	C₂×D₄	—	166	2	no	—
268	C₂×D₄	—	166	2	no	—
269	C₂×D₄	—	166	2	no	—
270	C₂²×C₄	—	332	2	no	—
271	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
272	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
273	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
274	Q₁₆	—	332	4	no	—
275	SD₁₆	—	332	8	no	—
276	D₈	—	332	4	no	—
277	C₂×C₈	—	2	4	no	—
278	C₂×C₈	—	2	4	no	—
279	C₂×C₈	—	2	4	no	—
280	C₂×C₈	C₂×D₃₃₂	1	2	yes	—
281	C₂×C₈	C₂×D₃₃₂	1	2	yes	—
282	C₂×C₈	C₂×D₃₃₂	1	2	yes	—
283	C₂×C₈	C₂×D₃₃₂	1	2	yes	—
284	C₂×C₈	C₂×D₃₃₂	1	2	yes	—
285	C₂×C₈	C₂×D₃₃₂	1	2	yes	—
286	C₄⋊C₄	—	332	2	no	—
287	C₄⋊C₄	—	332	2	no	—
288	C₄⋊C₄	—	166	2	no	—
289	C₄⋊C₄	—	166	2	no	—
290	C₄⋊C₄	—	166	2	no	—
291	C₄⋊C₄	—	166	2	no	—
292	C₄⋊C₄	—	166	2	no	—
293	C₄⋊C₄	—	166	2	no	—
294	C₂²⋊C₄	—	332	2	no	—
295	C₂²⋊C₄	—	332	2	no	—
296	C₂²⋊C₄	—	332	2	no	—
297	C₄²	—	332	2	no	—
298	C₄²	C₂×D₃₃₂	1	1	yes	characteristic
299	C₄²	C₂×D₃₃₂	1	1	yes	characteristic
300	C₄²	C₂×D₃₃₂	1	1	yes	characteristic
301	C₄²	C₂³×D₈₃	1	1	yes	characteristic
302	C₂³	—	332	2	no	—
303	C₂³	#14581789	1	1	yes	characteristic
304	Q₈	—	332	4	no	—
305	Q₈	—	332	4	no	—
306	Q₈	—	332	4	no	—
307	D₄	—	332	4	no	—
308	D₄	—	332	4	no	—
309	D₄	—	332	4	no	—
310	D₄	—	664	4	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	332	2	no	—
314	C₂×C₄	—	332	2	no	—
315	C₂×C₄	—	332	2	no	—
316	C₂×C₄	—	664	4	no	—
317	C₂×C₄	#14581789	1	1	yes	characteristic
318	C₂×C₄	#14581789	1	1	yes	characteristic
319	C₂×C₄	#14581789	1	1	yes	characteristic
320	C₂×C₄	#14581814	1	4	yes	—
321	C₂×C₄	C₂²×D₃₃₂	1	1	yes	characteristic
322	C₂×C₄	C₂²×D₃₃₂	1	1	yes	characteristic
323	C₂×C₄	C₂²×D₃₃₂	1	1	yes	characteristic
324	C₈	—	2	2	no	—
325	C₈	—	2	2	no	—
326	C₈	—	2	2	no	—
327	C₈	—	2	2	no	—
328	C₈	#14581814	1	4	yes	—
329	C₈	#14581814	1	4	yes	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	664	2	no	—
333	C₂²	—	664	2	no	—
334	C₂²	—	664	2	no	—
335	C₂²	#19128085	1	2	yes	—
336	C₂²	C₂×#14581789	1	1	yes	characteristic
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	664	2	no	—
340	C₄	—	664	2	no	—
341	C₄	—	664	2	no	—
342	C₄	#19128085	1	2	yes	—
343	C₄	C₂×#14581814	1	2	yes	—
344	C₄	C₂×#14581814	1	2	yes	—
345	C₄	C₂×#14581814	1	2	yes	—
346	C₂	—	2	2	no	—
347	C₂	—	1328	2	no	—
348	C₂	C₂×#19128085	1	1	yes	characteristic
349	C₂	#24917891	1	1	yes	characteristic
350	C₂	#24917891	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14581789
commutator, cosocle	C₄²×C₈₃	C₂⁴
Frattini	C₄²	C₂³×D₈₃
Fitting	C₂×C₈²×C₈₃	C₂
socle	C₂²×C₈₃	C₂×#127
prime radical	#24918042	C₂
hypercenter	C₂×C₈²	D₈₃
hypocenter	C₈₃	#19600
LCS	G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
1. #24918049	C₂	semidirect
3. #24918022	C₂	semidirect
4. #24917786	C₂	semidirect
5. #24907075	C₂	semidirect
6. #24907073	C₂	semidirect
7. #24907069	C₂	semidirect
8. #24907064	C₂	semidirect
9. #24907060	C₂	semidirect
10. #24907055	C₂	semidirect
11. C₂×C₈²×C₈₃	C₂	semidirect
186. C₂×C₈²	D₈₃	semidirect
187. C₈₃	#19600	semidirect

Maximal subgroups

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

subgroup	index
#24918049	2
#24918042	2
#24918022	2
#24917786	2
#24907075	2
#24907073	2
#24907069	2
#24907064	2
#24907060	2
#24907055	2
C₂×C₈²×C₈₃	2
#19600	83