Subgroups — #35404967

Order 21248.

810 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24910072	C₂	1	2	yes	maximal
2	#24907627	C₂	1	1	yes	maximal, characteristic, complemented
3	#24907620	C₂	1	1	yes	maximal, characteristic, complemented
4	#24907597	C₂	1	2	yes	maximal, complemented
5	#24907496	C₂	1	2	yes	maximal
6	#24907488	C₂	1	2	yes	maximal
7	#24907360	C₂	1	2	yes	maximal, complemented
8	C₄×#14581724	C₂	1	2	yes	maximal, complemented
9	C₂×#405×C₈₃	C₂	1	1	yes	maximal, characteristic
10	C₂×#14581726	C₂²	1	2	yes	—
11	C₂×#14581724	C₂²	1	2	yes	—
12	C₂×#14581724	C₂²	1	2	yes	—
13	C₂×#14581724	C₂²	1	2	yes	—
14	#19127594	—	2	4	no	—
15	#19127587	—	2	4	no	—
16	#19127571	C₂²	1	2	yes	—
17	#19127567	C₂²	1	2	yes	—
18	#19127561	C₂²	1	4	yes	—
19	C₄×C₄⋊Dic₈₃	C₂²	1	4	yes	—
20	#19127464	C₂²	1	4	yes	—
21	#19127464	C₂²	1	4	yes	—
22	C₂²×M₁₆×C₈₃	C₂²	1	1	yes	characteristic
23	C₂³×C₈×C₈₃	C₂²	1	1	yes	characteristic
24	C₂×#117×C₈₃	C₂²	1	1	yes	characteristic
25	#405×C₈₃	C₄	1	8	yes	complemented
26	C₂×#97×C₈₃	C₂²	1	1	yes	characteristic
27	C₂×#97×C₈₃	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
30	C₂×C₄⋊Dic₈₃	—	2	4	no	—
31	C₂×C₄⋊Dic₈₃	C₂³	1	4	yes	—
32	C₂×C₄⋊Dic₈₃	C₂³	1	4	yes	—
33	C₂×C₂²⋊Dic₈₃	—	2	4	no	—
34	C₂×C₄×Dic₈₃	—	2	4	no	—
35	#14581726	—	2	4	no	—
36	#14581724	—	2	4	no	—
37	#14581724	—	2	4	no	—
38	#14581724	—	2	4	no	—
39	C₈₃⋊#95	—	2	4	no	—
40	C₈₃⋊#95	—	2	4	no	—
41	C₈₃⋊#95	—	2	4	no	—
42	C₂³×C₄×C₈₃	C₂³	1	1	yes	characteristic
43	C₂×M₁₆×C₈₃	—	2	2	no	—
44	C₂×M₁₆×C₈₃	—	2	2	no	—
45	C₂×M₁₆×C₈₃	C₂×C₄	1	4	yes	—
46	C₂²×C₈×C₈₃	—	2	2	no	—
47	C₂²×C₈×C₈₃	—	2	2	no	—
48	C₂²×C₈×C₈₃	C₂×C₄	1	4	yes	—
49	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
53	#117×C₈₃	C₂×C₄	1	4	yes	—
54	#117×C₈₃	Q₈	1	4	yes	—
55	C₂×C₄⋊C₄×C₈₃	D₄	1	2	yes	—
56	C₂×C₂²⋊C₄×C₈₃	D₄	1	2	yes	—
57	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
59	#97×C₈₃	C₂×C₄	1	4	yes	—
60	#97×C₈₃	C₂×C₄	1	4	yes	—
61	C₄×C₈×C₈₃	C₂×C₄	1	4	yes	—
62	C₄×C₈×C₈₃	C₂×C₄	1	4	yes	—
63	C₂²×Dic₈₃	—	2	4	no	—
64	C₂²×Dic₈₃	—	2	4	no	—
65	C₄⋊Dic₈₃	—	2	4	no	—
66	C₄⋊Dic₈₃	—	2	4	no	—
67	C₄⋊Dic₈₃	—	2	4	no	—
68	C₄⋊Dic₈₃	—	2	4	no	—
69	C₄⋊Dic₈₃	—	8	4	no	—
70	C₂²⋊Dic₈₃	—	8	4	no	—
71	C₄×Dic₈₃	—	8	4	no	—
72	C₂⁴×C₈₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₈₃	—	2	1	no	—
74	C₂²×C₄×C₈₃	—	2	1	no	—
75	C₂²×C₄×C₈₃	—	2	1	no	—
76	C₂²×C₄×C₈₃	—	2	1	no	—
77	C₂²×C₄×C₈₃	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₈₃	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₈₃	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₈₃	C₂×Q₈	1	2	yes	—
82	C₂²×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
83	M₁₆×C₈₃	—	2	8	no	—
84	C₂×C₈×C₈₃	—	2	1	no	—
85	C₂×C₈×C₈₃	—	2	1	no	—
86	C₂×C₈×C₈₃	—	2	1	no	—
87	C₂×C₈×C₈₃	—	2	1	no	—
88	C₂×C₈×C₈₃	—	2	8	no	—
89	C₂×C₈×C₈₃	C₂²×C₄	1	2	yes	—
90	C₂×C₈×C₈₃	C₂²×C₄	1	2	yes	—
91	C₂×C₈×C₈₃	C₂²×C₄	1	2	yes	—
92	C₂×C₈×C₈₃	C₂²×C₄	1	2	yes	—
93	C₂×C₈×C₈₃	#41	1	2	yes	—
94	C₂×C₈×C₈₃	#41	1	2	yes	—
95	C₂×C₈×C₈₃	#41	1	2	yes	—
96	C₂×C₈×C₈₃	#41	1	2	yes	—
97	C₄⋊C₄×C₈₃	C₄⋊C₄	1	8	yes	—
98	C₂²⋊C₄×C₈₃	C₄⋊C₄	1	8	yes	—
99	C₄²×C₈₃	C₂²×C₄	1	2	yes	—
100	C₄²×C₈₃	C₂²×C₄	1	2	yes	—
101	C₄²×C₈₃	C₂×Q₈	1	2	yes	—
102	C₄²×C₈₃	C₂×Q₈	1	2	yes	—
103	C₂³×C₈₃	—	2	1	no	—
104	C₂³×C₈₃	—	2	1	no	—
105	C₂³×C₈₃	—	2	4	no	—
106	C₂³×C₈₃	C₂×C₄⋊C₄	1	2	yes	—
107	C₂³×C₈₃	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₈₃	—	2	1	no	—
109	C₂×C₄×C₈₃	—	2	1	no	—
110	C₂×C₄×C₈₃	—	2	4	no	—
111	C₂×C₄×C₈₃	—	2	4	no	—
112	C₂×C₄×C₈₃	—	2	2	no	—
113	C₂×C₄×C₈₃	—	2	2	no	—
114	C₂×C₄×C₈₃	—	2	2	no	—
115	C₂×C₄×C₈₃	—	2	2	no	—
116	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	4	yes	—
117	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	4	yes	—
118	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	2	yes	—
119	C₂×C₄×C₈₃	C₂³×C₄	1	1	yes	characteristic
120	C₂×C₄×C₈₃	C₂²×Q₈	1	1	yes	characteristic
121	C₂×C₄×C₈₃	C₂×#41	1	1	yes	characteristic
122	C₂×C₄×C₈₃	C₂×#41	1	1	yes	characteristic
123	C₂×C₄×C₈₃	C₂×#41	1	1	yes	characteristic
124	C₂×C₄×C₈₃	C₂×#41	1	1	yes	characteristic
125	C₂×Dic₈₃	—	4	4	no	—
126	C₂×Dic₈₃	—	4	4	no	—
127	C₂×Dic₈₃	—	4	4	no	—
128	C₂×Dic₈₃	—	4	4	no	—
129	C₂×Dic₈₃	—	4	4	no	—
130	C₂×Dic₈₃	—	4	4	no	—
131	C₈×C₈₃	—	2	2	no	—
132	C₈×C₈₃	—	2	2	no	—
133	C₈×C₈₃	#117	1	4	yes	—
134	C₈×C₈₃	#117	1	4	yes	—
135	C₂²×C₈₃	—	2	4	no	—
136	C₂²×C₈₃	—	2	2	no	—
137	C₂²×C₈₃	—	4	2	no	—
138	C₂²×C₈₃	—	4	2	no	—
139	C₂²×C₈₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
140	C₂²×C₈₃	#532	1	1	yes	characteristic
141	C₂²×C₈₃	#532	1	1	yes	characteristic
142	C₂²×C₈₃	#577	1	4	yes	—
143	C₄×C₈₃	—	2	4	no	—
144	C₄×C₈₃	—	2	4	no	—
145	C₄×C₈₃	—	2	2	no	—
146	C₄×C₈₃	C₂×#117	1	1	yes	characteristic
147	C₄×C₈₃	C₂×#117	1	1	yes	characteristic
148	C₄×C₈₃	#533	1	1	yes	characteristic
149	C₄×C₈₃	#533	1	1	yes	characteristic
150	Dic₈₃	—	8	4	no	—
151	Dic₈₃	—	8	4	no	—
152	#20820	—	83	1	no	maximal, Sylow
153	C₂×C₈₃	—	4	2	no	—
154	C₂×C₈₃	#2285	1	1	yes	characteristic
155	C₂×C₈₃	#2912	1	2	yes	—
156	C₂×C₈₃	#3173	1	2	yes	—
157	C₂×C₈₃	#3174	1	2	yes	—
158	C₂×#405	D₈₃	1	1	yes	characteristic
159	#2314	—	83	2	no	—
160	#1846	—	83	1	no	—
161	#1844	—	83	1	no	—
162	#1836	—	83	2	no	—
163	#1809	—	83	2	no	—
164	#1807	—	83	2	no	—
165	#1776	—	83	2	no	—
166	C₄×#106	—	83	2	no	—
167	C₈₃	#20820	1	1	yes	characteristic, complemented, Sylow
168	C₂²×M₁₆	C₂×D₈₃	1	1	yes	characteristic
169	C₂³×C₈	C₂×D₈₃	1	1	yes	characteristic
170	C₂×#117	C₂×D₈₃	1	1	yes	characteristic
171	C₂×#107	—	83	2	no	—
172	C₂×#106	—	83	2	no	—
173	C₂×#106	—	83	2	no	—
174	C₂×#106	—	83	2	no	—
175	#405	Dic₈₃	1	8	yes	complemented
176	C₂×#97	C₂×D₈₃	1	1	yes	characteristic
177	C₂×#97	C₂×D₈₃	1	1	yes	characteristic
178	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
179	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
180	#388	—	166	4	no	—
181	#387	—	166	4	no	—
182	#384	—	83	2	no	—
183	#382	—	83	2	no	—
184	#380	—	83	4	no	—
185	C₄×C₄⋊C₄	—	83	4	no	—
186	#340	—	83	4	no	—
187	#340	—	83	4	no	—
188	C₂³×C₄	C₂²×D₈₃	1	1	yes	characteristic
189	C₂×M₁₆	—	2	2	no	—
190	C₂×M₁₆	—	2	2	no	—
191	C₂×M₁₆	C₂×Dic₈₃	1	4	yes	—
192	C₂²×C₈	—	2	2	no	—
193	C₂²×C₈	—	2	2	no	—
194	C₂²×C₈	C₂×Dic₈₃	1	4	yes	—
195	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
196	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
197	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
198	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
199	#117	Dic₁₆₆	1	4	yes	—
200	#117	C₂×Dic₈₃	1	4	yes	—
201	C₂×C₄⋊C₄	—	83	4	no	—
202	C₂×C₄⋊C₄	—	83	4	no	—
203	C₂×C₄⋊C₄	—	166	4	no	—
204	C₂×C₄⋊C₄	D₃₃₂	1	2	yes	—
205	C₂×C₂²⋊C₄	—	166	4	no	—
206	C₂×C₂²⋊C₄	D₃₃₂	1	2	yes	—
207	C₂×C₄²	—	166	4	no	—
208	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
209	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
210	#107	—	166	4	no	—
211	#106	—	166	4	no	—
212	#106	—	166	4	no	—
213	#106	—	166	4	no	—
214	#97	C₂×Dic₈₃	1	4	yes	—
215	#97	C₂×Dic₈₃	1	4	yes	—
216	C₄×C₈	C₂×Dic₈₃	1	4	yes	—
217	C₄×C₈	C₂×Dic₈₃	1	4	yes	—
218	#95	—	166	4	no	—
219	#95	—	166	4	no	—
220	#95	—	166	4	no	—
221	C₂⁴	C₂×D₃₃₂	1	1	yes	characteristic
222	C₂²×C₄	—	2	1	no	—
223	C₂²×C₄	—	2	1	no	—
224	C₂²×C₄	—	2	1	no	—
225	C₂²×C₄	—	2	1	no	—
226	C₂²×C₄	—	166	4	no	—
227	C₂²×C₄	—	166	4	no	—
228	C₂²×C₄	C₂²×Dic₈₃	1	2	yes	—
229	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
230	C₂²×C₄	C₂×D₃₃₂	1	2	yes	—
231	C₂²×C₄	C₂×D₃₃₂	1	2	yes	—
232	C₂²×C₄	C₂×Dic₁₆₆	1	2	yes	—
233	C₂²×C₄	C₂³×D₈₃	1	1	yes	characteristic
234	M₁₆	—	2	8	no	—
235	C₂×C₈	—	2	1	no	—
236	C₂×C₈	—	2	1	no	—
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	2	8	no	—
240	C₂×C₈	C₂²×Dic₈₃	1	2	yes	—
241	C₂×C₈	C₂²×Dic₈₃	1	2	yes	—
242	C₂×C₈	C₂²×Dic₈₃	1	2	yes	—
243	C₂×C₈	C₂²×Dic₈₃	1	2	yes	—
244	C₂×C₈	#13050590	1	2	yes	—
245	C₂×C₈	#13050590	1	2	yes	—
246	C₂×C₈	#13050590	1	2	yes	—
247	C₂×C₈	#13050590	1	2	yes	—
248	C₄⋊C₄	—	664	4	no	—
249	C₄⋊C₄	—	166	4	no	—
250	C₄⋊C₄	—	166	4	no	—
251	C₄⋊C₄	—	166	4	no	—
252	C₄⋊C₄	—	166	4	no	—
253	C₄⋊C₄	C₄⋊Dic₈₃	1	8	yes	—
254	C₂²⋊C₄	—	664	4	no	—
255	C₂²⋊C₄	C₄⋊Dic₈₃	1	8	yes	—
256	C₄²	—	664	4	no	—
257	C₄²	C₂²×Dic₈₃	1	2	yes	—
258	C₄²	C₂²×Dic₈₃	1	2	yes	—
259	C₄²	C₂×Dic₁₆₆	1	2	yes	—
260	C₄²	C₂×Dic₁₆₆	1	2	yes	—
261	C₂³	—	2	1	no	—
262	C₂³	—	2	1	no	—
263	C₂³	—	2	4	no	—
264	C₂³	C₂×C₄⋊Dic₈₃	1	2	yes	—
265	C₂³	C₂²×D₃₃₂	1	1	yes	characteristic
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	332	4	no	—
275	C₂×C₄	—	332	4	no	—
276	C₂×C₄	—	332	4	no	—
277	C₂×C₄	—	332	4	no	—
278	C₂×C₄	—	332	4	no	—
279	C₂×C₄	—	332	4	no	—
280	C₂×C₄	C₂×C₄⋊Dic₈₃	1	4	yes	—
281	C₂×C₄	C₂×C₄⋊Dic₈₃	1	4	yes	—
282	C₂×C₄	C₂×C₄⋊Dic₈₃	1	2	yes	—
283	C₂×C₄	C₂³×Dic₈₃	1	1	yes	characteristic
284	C₂×C₄	C₂²×Dic₁₆₆	1	1	yes	characteristic
285	C₂×C₄	C₂×#13050590	1	1	yes	characteristic
286	C₂×C₄	C₂×#13050590	1	1	yes	characteristic
287	C₂×C₄	C₂×#13050590	1	1	yes	characteristic
288	C₂×C₄	C₂×#13050590	1	1	yes	characteristic
289	C₈	—	2	2	no	—
290	C₈	—	2	2	no	—
291	C₈	#14581751	1	4	yes	—
292	C₈	#14581751	1	4	yes	—
293	C₂²	—	2	2	no	—
294	C₂²	—	2	4	no	—
295	C₂²	—	4	2	no	—
296	C₂²	—	4	2	no	—
297	C₂²	C₂²×C₄⋊Dic₈₃	1	1	yes	characteristic
298	C₂²	#19128251	1	1	yes	characteristic
299	C₂²	#19128251	1	1	yes	characteristic
300	C₂²	#19128513	1	4	yes	—
301	C₄	—	2	2	no	—
302	C₄	—	2	4	no	—
303	C₄	—	2	4	no	—
304	C₄	—	664	4	no	—
305	C₄	—	664	4	no	—
306	C₄	C₂×#14581751	1	1	yes	characteristic
307	C₄	C₂×#14581751	1	1	yes	characteristic
308	C₄	#19128258	1	1	yes	characteristic
309	C₄	#19128258	1	1	yes	characteristic
310	C₂	—	4	2	no	—
311	C₂	#24909870	1	1	yes	characteristic
312	C₂	#24916058	1	2	yes	—
313	C₂	#24918080	1	2	yes	—
314	C₂	#24918091	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₃₃₂
commutator	C₂×C₄×C₈₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₈₃
Fitting	C₂×#405×C₈₃	C₂
socle	C₂³×C₈₃	C₂²×D₄
prime radical	C₂⁴×C₈₃	C₂×D₄
cosocle	C₂²×C₄×C₈₃	C₂⁴
hypercenter	C₂×#405	D₈₃
hypocenter	C₈₃	#20820
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
2. #24907627	C₂	semidirect
3. #24907620	C₂	semidirect
4. #24907597	C₂	semidirect
7. #24907360	C₂	semidirect
8. C₄×#14581724	C₂	semidirect
25. #405×C₈₃	C₄	semidirect
167. C₈₃	#20820	semidirect
175. #405	Dic₈₃	semidirect

Maximal subgroups

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

subgroup	index
#24910072	2
#24907627	2
#24907620	2
#24907597	2
#24907496	2
#24907488	2
#24907360	2
C₄×#14581724	2
C₂×#405×C₈₃	2
#20820	83