Subgroups — #35422904

Order 21248.

862 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19127819	C₂	1	1	yes	maximal, characteristic
2	#24916340	C₂	1	2	yes	maximal
3	#24916060	C₂	1	2	yes	maximal, complemented
4	#24908248	C₂	1	2	yes	maximal, complemented
5	#24907690	C₂	1	4	yes	maximal, complemented
6	#24907623	C₂	1	1	yes	maximal, characteristic, complemented
7	#24907524	C₂	1	2	yes	maximal, complemented
8	C₈₃×#1772	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14581814	C₂²	1	1	yes	characteristic
10	C₂×#14581798	C₂²	1	2	yes	—
11	#19128181	C₂²	1	4	yes	—
12	#19127819	C₄	1	8	yes	complemented
13	#19127803	—	2	4	no	—
14	C₄×Dic₃₃₂	—	2	2	no	—
15	C₄×D₆₆₄	—	2	2	no	—
16	C₂×#14581728	C₂²	1	1	yes	characteristic
17	#19127744	C₂²	1	4	yes	—
18	#19127740	—	2	4	no	—
19	#19127738	—	2	2	no	—
20	C₂×#14581726	C₂²	1	2	yes	—
21	C₂×#14581726	C₂²	1	2	yes	—
22	#107×D₈₃	—	2	2	no	—
23	C₂×#14581720	C₂²	1	2	yes	—
24	#19127703	C₂²	1	4	yes	—
25	#19127571	C₂²	1	2	yes	—
26	#19127469	C₂²	1	4	yes	—
27	C₂×#108×C₈₃	C₂²	1	1	yes	characteristic
28	#427×C₈₃	C₂²	1	1	yes	characteristic
29	#405×C₈₃	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
31	#339×C₈₃	C₂²	1	2	yes	—
32	C₂×#13050590	C₂³	1	2	yes	—
33	#14581814	C₂×C₄	1	4	yes	—
34	#14581814	Q₈	1	4	yes	—
35	C₂×Dic₃₃₂	D₄	1	1	yes	characteristic
36	C₂×#13050579	D₄	1	2	yes	—
37	C₂×D₆₆₄	D₄	1	1	yes	characteristic
38	#14581798	—	2	4	no	—
39	#14581798	C₂×C₄	1	8	yes	complemented
40	C₂×#13050574	C₂³	1	2	yes	—
41	C₂×#13050573	—	2	2	no	—
42	C₂×C₈×D₈₃	—	2	2	no	—
43	C₄×Dic₁₆₆	—	2	4	no	—
44	C₄×D₃₃₂	—	2	4	no	—
45	(C₂²⋊D₈₃)⋊C₄	—	4	4	no	—
46	#14581750	—	2	4	no	—
47	C₂×C₄⋊Dic₈₃	—	4	2	no	—
48	C₂×C₄⋊Dic₈₃	C₂³	1	4	yes	—
49	C₄⋊C₄×D₈₃	—	2	4	no	—
50	#14581728	C₂×C₄	1	4	yes	complemented
51	#14581726	—	2	2	no	—
52	#14581726	—	2	2	no	—
53	#14581726	—	2	2	no	—
54	#14581726	—	2	2	no	—
55	#14581725	—	2	4	no	—
56	#14581723	—	2	4	no	—
57	#14581720	C₂×C₄	1	8	yes	complemented
58	#14581717	—	2	4	no	—
59	#14581714	—	2	4	no	—
60	C₂×M₁₆×C₈₃	C₂³	1	2	yes	—
61	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
62	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
63	#117×C₈₃	C₂³	1	2	yes	—
64	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
65	#108×C₈₃	—	2	2	no	—
66	#107×C₈₃	D₄	1	1	yes	characteristic
67	#107×C₈₃	D₄	1	1	yes	characteristic
68	#106×C₈₃	D₄	1	2	yes	—
69	#97×C₈₃	—	2	2	no	—
70	C₄×C₈×C₈₃	—	2	2	no	—
71	C₄×C₈×C₈₃	C₂×C₄	1	4	yes	complemented
72	#13050590	—	2	2	no	—
73	#13050590	—	2	2	no	—
74	#13050590	C₂²×C₄	1	4	yes	—
75	#13050590	C₂×Q₈	1	4	yes	—
76	C₂×Dic₁₆₆	C₂×D₄	1	2	yes	—
77	C₂×D₃₃₂	C₂×D₄	1	2	yes	—
78	C₂×C₂²⋊D₈₃	—	2	2	no	—
79	C₂²×Dic₈₃	—	2	4	no	—
80	C₂×C₄×D₈₃	—	2	2	no	—
81	C₂×C₄×D₈₃	—	4	4	no	—
82	Dic₃₃₂	C₄⋊C₄	1	4	yes	complemented
83	#13050579	C₄⋊C₄	1	8	yes	complemented
84	D₆₆₄	C₄⋊C₄	1	4	yes	complemented
85	#13050574	—	2	2	no	—
86	#13050574	C₂²×C₄	1	4	yes	—
87	#13050573	—	2	8	no	—
88	C₂×C₈₃⋊C₈	—	2	2	no	—
89	C₈×D₈₃	—	2	8	no	—
90	C₄⋊Dic₈₃	—	2	4	no	—
91	C₄⋊Dic₈₃	—	2	4	no	—
92	C₄⋊Dic₈₃	—	8	4	no	—
93	Dic₈₃⋊C₄	—	4	4	no	—
94	(C₂×D₈₃)⋊C₄	—	4	4	no	—
95	C₄×Dic₈₃	—	4	4	no	—
96	C₂²×C₄×C₈₃	—	2	1	no	—
97	C₂²×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
98	M₁₆×C₈₃	—	2	2	no	—
99	M₁₆×C₈₃	—	2	2	no	—
100	C₂×C₈×C₈₃	—	2	2	no	—
101	C₂×C₈×C₈₃	C₂²×C₄	1	2	yes	—
102	C₂×C₈×C₈₃	C₂²×C₄	1	2	yes	—
103	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₈₃	C₂×Q₈	1	2	yes	—
108	C₂×C₈×C₈₃	#41	1	2	yes	—
109	C₄⋊C₄×C₈₃	C₂×D₄	1	2	yes	—
110	C₄⋊C₄×C₈₃	C₂×D₄	1	2	yes	—
111	C₂²⋊C₄×C₈₃	—	2	2	no	—
112	C₄²×C₈₃	—	2	2	no	—
113	C₄²×C₈₃	—	2	1	no	—
114	C₄²×C₈₃	C₂²×C₄	1	2	yes	—
115	C₂³×C₈₃	H(5,𝔽₂)	1	1	yes	characteristic
116	C₂²×D₈₃	—	2	2	no	—
117	C₂×C₄×C₈₃	—	2	2	no	—
118	C₂×C₄×C₈₃	—	2	2	no	—
119	C₂×C₄×C₈₃	—	2	2	no	—
120	C₂×C₄×C₈₃	—	2	2	no	—
121	C₂×C₄×C₈₃	—	2	2	no	—
122	C₂×C₄×C₈₃	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₈₃	C₂²×Q₈	1	1	yes	characteristic
126	C₂×C₄×C₈₃	C₂×#41	1	1	yes	characteristic
127	C₂×C₄×C₈₃	2₋¹⁺⁴	1	1	yes	characteristic
128	C₂²⋊D₈₃	—	2	4	no	—
129	C₂²⋊D₈₃	—	2	4	no	—
130	C₂×Dic₈₃	—	2	2	no	—
131	C₂×Dic₈₃	—	4	4	no	—
132	C₂×Dic₈₃	—	8	4	no	—
133	D₃₃₂	—	2	2	no	—
134	D₃₃₂	C₂×C₄⋊C₄	1	4	yes	—
135	C₄×D₈₃	—	2	4	no	—
136	C₄×D₈₃	—	2	4	no	—
137	C₄×D₈₃	—	16	4	no	—
138	Dic₁₆₆	—	2	2	no	—
139	Dic₁₆₆	C₂×C₄⋊C₄	1	4	yes	—
140	C₈×C₈₃	—	4	2	no	—
141	C₈×C₈₃	C₂×C₄⋊C₄	1	2	yes	—
142	C₈×C₈₃	C₂×C₄⋊C₄	1	2	yes	—
143	C₈×C₈₃	C₄×D₄	1	2	yes	—
144	C₈×C₈₃	C₄×D₄	1	2	yes	—
145	C₈₃⋊C₈	—	2	4	no	—
146	C₂²×C₈₃	—	2	1	no	—
147	C₂²×C₈₃	—	2	1	no	—
148	C₂²×C₈₃	#520	1	1	yes	characteristic
149	C₂²×C₈₃	D₄²	1	1	yes	characteristic
150	C₂²×C₈₃	#554	1	1	yes	characteristic
151	C₂×D₈₃	—	2	4	no	—
152	C₂×D₈₃	—	4	2	no	—
153	C₂×D₈₃	—	4	2	no	—
154	C₄×C₈₃	—	2	4	no	—
155	C₄×C₈₃	—	4	2	no	—
156	C₄×C₈₃	—	4	2	no	—
157	C₄×C₈₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
158	C₄×C₈₃	C₂×C₄×D₄	1	1	yes	characteristic
159	C₄×C₈₃	#547	1	1	yes	characteristic
160	C₄×C₈₃	D₄×Q₈	1	1	yes	characteristic
161	Dic₈₃	—	2	4	no	—
162	Dic₈₃	—	8	4	no	—
163	#22692	—	83	1	no	maximal, Sylow
164	C₂×C₈₃	—	2	1	no	—
165	C₂×C₈₃	—	2	1	no	—
166	C₂×C₈₃	D₄×C₄⋊C₄	1	1	yes	characteristic
167	C₂×C₈₃	#3292	1	1	yes	characteristic
168	C₂×C₈₃	#3293	1	1	yes	characteristic
169	D₈₃	—	4	4	no	—
170	C₂×#443	—	83	1	no	—
171	#2951	—	83	2	no	—
172	#2913	—	83	2	no	—
173	#1952	—	83	2	no	—
174	#1859	—	83	4	no	—
175	#1845	—	83	1	no	—
176	#1816	—	83	2	no	—
177	#1772	D₈₃	1	1	yes	characteristic, complemented
178	C₈₃	#22692	1	1	yes	characteristic, complemented, Sylow
179	C₂×#135	—	83	1	no	—
180	C₂×#131	—	83	2	no	—
181	#520	—	83	4	no	—
182	#443	—	83	8	no	—
183	#440	—	166	4	no	—
184	C₄×Q₁₆	—	166	2	no	—
185	C₄×D₈	—	166	2	no	—
186	C₂×#108	—	83	1	no	—
187	C₂×#108	C₂×D₈₃	1	1	yes	characteristic
188	#428	—	83	4	no	—
189	#428	—	166	4	no	—
190	#427	—	166	2	no	—
191	#427	C₂×D₈₃	1	1	yes	characteristic
192	C₂×#107	—	83	2	no	—
193	C₂×#107	—	83	2	no	—
194	C₂×#107	—	166	2	no	—
195	C₂×#104	—	83	2	no	—
196	#417	—	83	4	no	—
197	#405	C₂×D₈₃	1	2	yes	—
198	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
199	#384	—	83	2	no	—
200	#343	—	83	4	no	—
201	#339	C₂×D₈₃	1	2	yes	—
202	C₂×#41	—	83	2	no	—
203	#135	—	83	4	no	—
204	#135	—	83	4	no	—
205	C₂×Q₁₆	—	83	1	no	—
206	C₂×SD₁₆	—	83	2	no	—
207	C₂×D₈	—	83	1	no	—
208	#131	—	83	8	no	—
209	#131	—	166	4	no	—
210	C₂×M₁₆	—	83	2	no	—
211	C₂×M₁₆	—	166	2	no	—
212	C₂×M₁₆	C₂²×D₈₃	1	2	yes	—
213	C₂²×C₈	—	166	2	no	—
214	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
215	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
216	C₄×Q₈	—	166	4	no	—
217	C₄×D₄	—	332	4	no	—
218	C₄×D₄	—	166	4	no	—
219	#117	—	166	4	no	—
220	#117	C₂²×D₈₃	1	2	yes	—
221	C₂×C₄⋊C₄	—	332	2	no	—
222	C₂×C₄⋊C₄	—	83	4	no	—
223	C₂×C₄⋊C₄	—	166	4	no	—
224	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
225	#108	—	2	2	no	—
226	#108	—	83	4	no	—
227	#107	—	166	2	no	—
228	#107	—	166	2	no	—
229	#107	—	166	2	no	—
230	#107	—	166	2	no	—
231	#107	—	166	4	no	—
232	#107	D₃₃₂	1	1	yes	characteristic
233	#107	D₃₃₂	1	1	yes	characteristic
234	#106	—	166	4	no	—
235	#106	D₃₃₂	1	2	yes	—
236	#104	—	83	8	no	—
237	#103	—	166	4	no	—
238	#102	—	166	4	no	—
239	#97	—	2	2	no	—
240	C₄×C₈	—	2	2	no	—
241	C₄×C₈	C₄×D₈₃	1	4	yes	complemented
242	#41	—	83	4	no	—
243	#41	—	83	4	no	—
244	#41	—	166	2	no	—
245	#41	—	166	2	no	—
246	C₂×Q₈	—	83	2	no	—
247	C₂×D₄	—	83	2	no	—
248	C₂×D₄	—	166	2	no	—
249	C₂²×C₄	—	2	1	no	—
250	C₂²×C₄	—	332	4	no	—
251	C₂²×C₄	—	166	4	no	—
252	C₂²×C₄	—	166	2	no	—
253	C₂²×C₄	C₂³×D₈₃	1	1	yes	characteristic
254	Q₁₆	—	83	4	no	—
255	SD₁₆	—	83	8	no	—
256	D₈	—	83	4	no	—
257	M₁₆	—	2	2	no	—
258	M₁₆	—	2	2	no	—
259	M₁₆	—	83	4	no	—
260	M₁₆	—	166	2	no	—
261	M₁₆	—	166	8	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	166	2	no	—
264	C₂×C₈	—	166	8	no	—
265	C₂×C₈	C₂×C₄×D₈₃	1	2	yes	—
266	C₂×C₈	C₂×C₄×D₈₃	1	2	yes	—
267	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
268	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
269	C₂×C₈	C₂×D₃₃₂	1	1	yes	characteristic
270	C₂×C₈	C₂×D₃₃₂	1	1	yes	characteristic
271	C₂×C₈	Q₈×D₈₃	1	2	yes	—
272	C₂×C₈	#13050590	1	2	yes	—
273	C₄⋊C₄	—	332	4	no	—
274	C₄⋊C₄	—	664	4	no	—
275	C₄⋊C₄	—	166	4	no	—
276	C₄⋊C₄	—	166	4	no	—
277	C₄⋊C₄	C₂×D₃₃₂	1	2	yes	—
278	C₄⋊C₄	C₂×D₃₃₂	1	2	yes	—
279	C₂²⋊C₄	—	2	2	no	—
280	C₂²⋊C₄	—	332	4	no	—
281	C₄²	—	2	2	no	—
282	C₄²	—	2	1	no	—
283	C₄²	—	332	4	no	—
284	C₄²	C₂×C₄×D₈₃	1	2	yes	—
285	C₂³	—	166	2	no	—
286	C₂³	#14581838	1	1	yes	characteristic
287	Q₈	—	83	4	no	—
288	Q₈	—	166	2	no	—
289	D₄	—	83	4	no	—
290	D₄	—	166	4	no	—
291	D₄	—	166	4	no	—
292	D₄	—	166	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	1328	4	no	—
299	C₂×C₄	—	332	4	no	—
300	C₂×C₄	—	664	4	no	—
301	C₂×C₄	—	166	2	no	—
302	C₂×C₄	—	166	4	no	—
303	C₂×C₄	—	166	4	no	—
304	C₂×C₄	C₂²×C₄×D₈₃	1	1	yes	characteristic
305	C₂×C₄	C₂×D₄×D₈₃	1	1	yes	characteristic
306	C₂×C₄	C₂²×D₃₃₂	1	1	yes	characteristic
307	C₂×C₄	C₂×Q₈×D₈₃	1	1	yes	characteristic
308	C₂×C₄	C₂×#13050590	1	1	yes	characteristic
309	C₂×C₄	#14581839	1	1	yes	characteristic
310	C₈	—	4	2	no	—
311	C₈	—	166	4	no	—
312	C₈	C₄⋊C₄×D₈₃	1	2	yes	—
313	C₈	C₄⋊C₄×D₈₃	1	2	yes	—
314	C₈	C₄×D₃₃₂	1	2	yes	—
315	C₈	C₄×D₃₃₂	1	2	yes	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	332	2	no	—
319	C₂²	—	332	2	no	—
320	C₂²	—	166	4	no	—
321	C₂²	#19128181	1	1	yes	characteristic
322	C₂²	D₄×D₃₃₂	1	1	yes	characteristic
323	C₂²	#19128411	1	1	yes	characteristic
324	C₄	—	2	4	no	—
325	C₄	—	4	2	no	—
326	C₄	—	4	2	no	—
327	C₄	—	664	4	no	—
328	C₄	—	166	4	no	—
329	C₄	C₂×C₄⋊C₄×D₈₃	1	1	yes	characteristic
330	C₄	C₂×C₄×D₃₃₂	1	1	yes	characteristic
331	C₄	#19128356	1	1	yes	characteristic
332	C₄	Q₈×D₃₃₂	1	1	yes	characteristic
333	C₂	—	2	1	no	—
334	C₂	—	2	1	no	—
335	C₂	—	332	4	no	—
336	C₂	C₄⋊C₄×D₃₃₂	1	1	yes	characteristic
337	C₂	#24919263	1	1	yes	characteristic
338	C₂	#24919268	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₃₃₂
commutator	C₂×C₄×C₈₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₈₃
Fitting	C₈₃×#1772	C₂
socle	C₂²×C₈₃	D₄²
prime radical	C₂×#14581814	C₂²
cosocle	C₂²×C₄×C₈₃	C₂⁴
hypercenter	#1772	D₈₃
hypocenter	C₈₃	#22692
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772

Complements

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

N	H ≅ G/N	type
3. #24916060	C₂	semidirect
4. #24908248	C₂	semidirect
5. #24907690	C₂	semidirect
6. #24907623	C₂	semidirect
7. #24907524	C₂	semidirect
8. C₈₃×#1772	C₂	semidirect
12. #19127819	C₄	semidirect
39. #14581798	C₂×C₄	semidirect
50. #14581728	C₂×C₄	semidirect
57. #14581720	C₂×C₄	semidirect
71. C₄×C₈×C₈₃	C₂×C₄	semidirect
82. Dic₃₃₂	C₄⋊C₄	semidirect
83. #13050579	C₄⋊C₄	semidirect
84. D₆₆₄	C₄⋊C₄	semidirect
177. #1772	D₈₃	semidirect
178. C₈₃	#22692	semidirect
241. C₄×C₈	C₄×D₈₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19127819	2
#24916340	2
#24916060	2
#24908248	2
#24907690	2
#24907623	2
#24907524	2
C₈₃×#1772	2
#22692	83