Subgroups — #35320361

Order 21248.

440 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24915885	C₂	1	1	yes	maximal, characteristic
2	C₂×#19127521	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#19127521	C₂	1	1	yes	maximal, characteristic, complemented
4	#24909092	C₂	1	1	yes	maximal, characteristic
5	#24909090	C₂	1	1	yes	maximal, characteristic
6	#24905549	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₃×#1378	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14581800	C₄	1	1	yes	characteristic
9	C₂×#14581798	C₂²	1	1	yes	characteristic
10	#19128169	C₄	1	1	yes	characteristic
11	C₂×#14581732	C₂²	1	1	yes	characteristic
12	C₂×#14581732	C₂²	1	1	yes	characteristic
13	#19127658	—	2	1	no	—
14	#19127657	C₄	1	1	yes	characteristic
15	#19127655	—	2	1	no	—
16	#19127653	—	2	1	no	—
17	M₁₆×Dic₈₃	C₄	1	1	yes	characteristic
18	C₂×C₈×Dic₈₃	C₂²	1	1	yes	characteristic
19	C₄×C₈×D₈₃	—	2	1	no	—
20	#19127521	—	2	1	no	—
21	#19127521	—	2	1	no	—
22	#19127521	—	2	1	no	—
23	#19127521	—	2	1	no	—
24	#19127486	—	2	2	no	—
25	#19127484	—	2	2	no	—
26	C₂×M₃₂×C₈₃	C₂²	1	1	yes	characteristic
27	C₂×M₃₂×C₈₃	C₂²	1	1	yes	characteristic
28	#405×C₈₃	C₂²	1	1	yes	characteristic
29	C₂×#13050590	C₂×C₄	1	1	yes	characteristic
30	#14581800	—	2	2	no	—
31	#14581800	C₈	1	4	yes	—
32	#14581798	—	2	1	no	—
33	#14581798	—	2	1	no	—
34	#14581798	D₄	1	2	yes	—
35	#14581798	Q₈	1	2	yes	—
36	C₂×#13050574	C₂×C₄	1	1	yes	characteristic
37	C₂×#13050573	—	2	1	no	—
38	C₂×#13050573	—	2	1	no	—
39	C₂²×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
40	C₂×C₈×D₈₃	—	2	1	no	—
41	C₂×C₈×D₈₃	—	2	1	no	—
42	#14581757	—	2	1	no	—
43	#14581754	—	2	1	no	—
44	(C₂²⋊D₈₃)⋊C₄	—	2	2	no	—
45	#14581749	—	2	1	no	—
46	C₂×C₄×Dic₈₃	C₂×C₄	1	1	yes	characteristic
47	C₄²×D₈₃	—	2	1	no	—
48	#14581732	—	2	1	no	—
49	#14581732	—	2	1	no	—
50	#14581732	—	2	1	no	—
51	#14581732	—	2	1	no	—
52	C₂×C₈₃⋊C₁₆	—	2	1	no	—
53	C₂×C₈₃⋊C₁₆	—	2	1	no	—
54	#14581703	—	2	1	no	—
55	#14581702	—	2	1	no	—
56	C₄×C₈₃⋊C₈	—	2	1	no	—
57	C₈×Dic₈₃	—	2	1	no	—
58	C₈×Dic₈₃	D₄	1	1	yes	characteristic
59	C₈×Dic₈₃	D₄	1	1	yes	characteristic
60	C₈×Dic₈₃	D₄	1	1	yes	characteristic
61	C₈×Dic₈₃	D₄	1	1	yes	characteristic
62	C₂×M₁₆×C₈₃	C₂×C₄	1	1	yes	characteristic
63	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
64	#117×C₈₃	C₂×C₄	1	1	yes	characteristic
65	M₃₂×C₈₃	—	2	1	no	—
66	M₃₂×C₈₃	—	2	1	no	—
67	M₃₂×C₈₃	—	2	1	no	—
68	M₃₂×C₈₃	—	2	1	no	—
69	C₂×C₁₆×C₈₃	—	2	1	no	—
70	C₂×C₁₆×C₈₃	—	2	1	no	—
71	#97×C₈₃	—	2	1	no	—
72	C₄×C₈×C₈₃	—	2	1	no	—
73	#13050590	—	2	1	no	—
74	#13050590	—	2	1	no	—
75	#13050590	C₄⋊C₄	1	2	yes	—
76	#13050590	C₂×C₈	1	2	yes	—
77	C₂×Dic₁₆₆	M₁₆	1	1	yes	characteristic
78	C₂×D₃₃₂	M₁₆	1	1	yes	characteristic
79	C₂×C₂²⋊D₈₃	—	2	1	no	—
80	C₂²×Dic₈₃	—	2	1	no	—
81	C₂×C₄×D₈₃	—	2	1	no	—
82	C₂×C₄×D₈₃	—	2	2	no	—
83	#13050574	—	2	1	no	—
84	#13050574	C₄⋊C₄	1	2	yes	—
85	#13050573	—	2	2	no	—
86	#13050573	—	2	2	no	—
87	#13050573	—	2	4	no	—
88	C₂×C₈₃⋊C₈	—	2	1	no	—
89	C₂×C₈₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₈₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₈₃⋊C₈	C₂×C₈	1	2	yes	—
92	C₂×C₈₃⋊C₈	M₁₆	1	2	yes	—
93	C₈×D₈₃	—	2	2	no	—
94	C₈×D₈₃	—	2	2	no	—
95	C₈×D₈₃	—	2	4	no	—
96	Dic₈₃⋊C₄	—	2	2	no	—
97	(C₂×D₈₃)⋊C₄	—	2	2	no	—
98	C₄×Dic₈₃	—	2	2	no	—
99	C₄×Dic₈₃	—	2	1	no	—
100	C₄×Dic₈₃	C₂²⋊C₄	1	1	yes	characteristic
101	C₄×Dic₈₃	C₂²⋊C₄	1	1	yes	characteristic
102	C₈₃⋊C₁₆	—	4	1	no	—
103	C₈₃⋊C₁₆	—	4	1	no	—
104	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
105	M₁₆×C₈₃	—	2	1	no	—
106	M₁₆×C₈₃	C₂×C₈	1	2	yes	—
107	C₂×C₈×C₈₃	—	2	1	no	—
108	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
112	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
113	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
114	C₄⋊C₄×C₈₃	—	2	1	no	—
115	C₂²⋊C₄×C₈₃	M₁₆	1	2	yes	—
116	C₄²×C₈₃	—	2	1	no	—
117	C₁₆×C₈₃	—	4	1	no	—
118	C₁₆×C₈₃	—	4	1	no	—
119	C₂³×C₈₃	C₂×M₁₆	1	1	yes	characteristic
120	C₂²×D₈₃	—	2	1	no	—
121	C₂×C₄×C₈₃	—	2	2	no	—
122	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₈₃	#117	1	1	yes	characteristic
125	C₂×C₄×C₈₃	C₂²×C₈	1	1	yes	characteristic
126	C₂×C₄×C₈₃	C₂×M₁₆	1	1	yes	characteristic
127	C₂×C₄×C₈₃	C₂×M₁₆	1	1	yes	characteristic
128	C₂²⋊D₈₃	—	2	2	no	—
129	C₂²⋊D₈₃	—	2	2	no	—
130	C₂×Dic₈₃	—	2	1	no	—
131	C₂×Dic₈₃	—	2	1	no	—
132	C₂×Dic₈₃	—	2	1	no	—
133	C₂×Dic₈₃	—	4	2	no	—
134	D₃₃₂	—	2	1	no	—
135	D₃₃₂	C₄⋊C₈	1	2	yes	—
136	C₄×D₈₃	—	2	2	no	—
137	C₄×D₈₃	—	2	2	no	—
138	C₄×D₈₃	—	4	4	no	—
139	Dic₁₆₆	—	2	1	no	—
140	Dic₁₆₆	C₄⋊C₈	1	2	yes	—
141	C₈×C₈₃	—	2	2	no	—
142	C₈×C₈₃	#121	1	1	yes	characteristic
143	C₈×C₈₃	#121	1	1	yes	characteristic
144	C₈×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
145	C₈×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
146	C₈₃⋊C₈	—	2	2	no	—
147	C₈₃⋊C₈	C₂²⋊C₈	1	2	yes	—
148	C₈₃⋊C₈	C₂²⋊C₈	1	2	yes	—
149	C₂²×C₈₃	—	4	1	no	—
150	C₂²×C₈₃	#407	1	1	yes	characteristic
151	C₂²×C₈₃	#423	1	1	yes	characteristic
152	C₂²×C₈₃	#431	1	1	yes	characteristic
153	C₂×D₈₃	—	2	2	no	—
154	C₂×D₈₃	—	4	2	no	—
155	C₄×C₈₃	—	4	2	no	—
156	C₄×C₈₃	#380	1	1	yes	characteristic
157	C₄×C₈₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
158	C₄×C₈₃	C₂×C₄⋊C₈	1	1	yes	characteristic
159	C₄×C₈₃	#432	1	1	yes	characteristic
160	Dic₈₃	—	2	2	no	—
161	Dic₈₃	—	2	2	no	—
162	Dic₈₃	—	4	1	no	—
163	#12792	—	83	1	no	maximal, Sylow
164	C₂×C₈₃	—	2	1	no	—
165	C₂×C₈₃	—	2	1	no	—
166	C₂×C₈₃	—	2	1	no	—
167	C₂×C₈₃	#1797	1	1	yes	characteristic
168	D₈₃	—	4	2	no	—
169	C₄×#131	—	83	1	no	—
170	C₂×#364	—	83	1	no	—
171	C₂×#364	—	83	1	no	—
172	#2117	—	83	1	no	—
173	#2117	—	83	1	no	—
174	#1378	—	83	1	no	—
175	#1378	D₈₃	1	1	yes	characteristic, complemented
176	C₈₃	#12792	1	1	yes	characteristic, complemented, Sylow
177	C₂×#131	—	83	1	no	—
178	C₂×#131	—	83	1	no	—
179	C₄×#41	—	83	1	no	—
180	C₂×M₃₂	—	83	1	no	—
181	C₂×M₃₂	—	83	1	no	—
182	C₂×M₃₂	C₂×D₈₃	1	1	yes	characteristic
183	C₂×M₃₂	C₂×D₈₃	1	1	yes	characteristic
184	#405	—	83	1	no	—
185	#405	—	166	1	no	—
186	#405	—	166	1	no	—
187	#405	C₂×D₈₃	1	1	yes	characteristic
188	C₄×M₁₆	—	83	1	no	—
189	C₄×M₁₆	—	166	1	no	—
190	C₂×C₄×C₈	—	83	1	no	—
191	C₂×C₄×C₈	—	166	1	no	—
192	#364	—	166	1	no	—
193	#364	—	166	1	no	—
194	#364	—	166	1	no	—
195	#364	—	166	1	no	—
196	#350	—	166	2	no	—
197	#350	—	166	2	no	—
198	C₂×#41	—	83	1	no	—
199	#131	—	83	4	no	—
200	#131	—	83	2	no	—
201	#131	—	83	2	no	—
202	#131	—	166	2	no	—
203	#131	—	166	1	no	—
204	#131	—	166	1	no	—
205	C₂×M₁₆	—	83	1	no	—
206	C₂×M₁₆	—	166	1	no	—
207	C₂×M₁₆	—	166	1	no	—
208	C₂×M₁₆	C₄×D₈₃	1	1	yes	characteristic
209	C₂²×C₈	—	83	1	no	—
210	C₂²×C₈	—	166	1	no	—
211	C₂²×C₈	—	166	1	no	—
212	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
213	C₄×Q₈	—	166	1	no	—
214	C₄×D₄	—	166	1	no	—
215	C₄×D₄	—	166	2	no	—
216	#117	—	166	1	no	—
217	#117	C₄×D₈₃	1	1	yes	characteristic
218	C₂×C₄²	—	83	1	no	—
219	C₂×C₄²	—	166	1	no	—
220	M₃₂	—	2	1	no	—
221	M₃₂	—	2	1	no	—
222	M₃₂	—	2	1	no	—
223	M₃₂	—	2	1	no	—
224	M₃₂	—	166	1	no	—
225	M₃₂	—	166	1	no	—
226	M₃₂	—	166	1	no	—
227	M₃₂	—	166	1	no	—
228	C₂×C₁₆	—	2	1	no	—
229	C₂×C₁₆	—	2	1	no	—
230	C₂×C₁₆	—	166	1	no	—
231	C₂×C₁₆	—	166	1	no	—
232	#97	—	2	1	no	—
233	#97	—	166	1	no	—
234	#97	—	166	1	no	—
235	C₄×C₈	—	2	1	no	—
236	C₄×C₈	—	83	1	no	—
237	C₄×C₈	—	83	1	no	—
238	C₄×C₈	—	83	1	no	—
239	C₄×C₈	—	83	1	no	—
240	C₄×C₈	—	166	1	no	—
241	C₄×C₈	—	166	1	no	—
242	#41	—	83	2	no	—
243	#41	—	83	2	no	—
244	#41	—	166	1	no	—
245	#41	—	166	1	no	—
246	C₂×Q₈	—	83	1	no	—
247	C₂×D₄	—	83	1	no	—
248	C₂×D₄	—	166	1	no	—
249	C₂²×C₄	—	166	2	no	—
250	C₂²×C₄	—	166	1	no	—
251	C₂²×C₄	—	166	1	no	—
252	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
253	M₁₆	—	2	1	no	—
254	M₁₆	—	83	2	no	—
255	M₁₆	—	166	4	no	—
256	M₁₆	—	166	1	no	—
257	M₁₆	—	166	2	no	—
258	M₁₆	—	166	2	no	—
259	M₁₆	C₈×D₈₃	1	2	yes	—
260	C₂×C₈	—	2	1	no	—
261	C₂×C₈	—	83	2	no	—
262	C₂×C₈	—	83	2	no	—
263	C₂×C₈	—	83	1	no	—
264	C₂×C₈	—	83	1	no	—
265	C₂×C₈	—	166	4	no	—
266	C₂×C₈	—	166	1	no	—
267	C₂×C₈	—	166	2	no	—
268	C₂×C₈	—	166	2	no	—
269	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
270	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
271	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
272	C₂×C₈	Q₈×D₈₃	1	1	yes	characteristic
273	C₂×C₈	#13050590	1	1	yes	characteristic
274	C₂×C₈	#13050590	1	1	yes	characteristic
275	C₄⋊C₄	—	2	1	no	—
276	C₄⋊C₄	—	166	2	no	—
277	C₂²⋊C₄	—	166	2	no	—
278	C₂²⋊C₄	#13050573	1	2	yes	—
279	C₄²	—	2	1	no	—
280	C₄²	—	83	1	no	—
281	C₄²	—	83	1	no	—
282	C₄²	—	166	2	no	—
283	C₄²	—	166	1	no	—
284	C₁₆	—	4	1	no	—
285	C₁₆	—	4	1	no	—
286	C₁₆	—	332	1	no	—
287	C₁₆	—	332	1	no	—
288	C₂³	—	166	1	no	—
289	C₂³	C₂×#13050573	1	1	yes	characteristic
290	Q₈	—	83	2	no	—
291	Q₈	—	166	1	no	—
292	D₄	—	83	2	no	—
293	D₄	—	166	2	no	—
294	D₄	—	166	2	no	—
295	D₄	—	166	1	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	332	4	no	—
298	C₂×C₄	—	332	2	no	—
299	C₂×C₄	—	166	1	no	—
300	C₂×C₄	—	166	1	no	—
301	C₂×C₄	—	166	1	no	—
302	C₂×C₄	—	166	2	no	—
303	C₂×C₄	—	166	2	no	—
304	C₂×C₄	C₂²⋊C₄×D₈₃	1	1	yes	characteristic
305	C₂×C₄	C₄⋊C₄×D₈₃	1	1	yes	characteristic
306	C₂×C₄	#14581749	1	1	yes	characteristic
307	C₂×C₄	C₂×C₈×D₈₃	1	1	yes	characteristic
308	C₂×C₄	M₁₆×D₈₃	1	1	yes	characteristic
309	C₂×C₄	M₁₆×D₈₃	1	1	yes	characteristic
310	C₈	—	2	2	no	—
311	C₈	—	83	2	no	—
312	C₈	—	83	2	no	—
313	C₈	—	166	2	no	—
314	C₈	#14581765	1	1	yes	characteristic
315	C₈	#14581765	1	1	yes	characteristic
316	C₈	#14581770	1	1	yes	characteristic
317	C₈	#14581770	1	1	yes	characteristic
318	C₂²	—	4	1	no	—
319	C₂²	—	332	2	no	—
320	C₂²	—	166	2	no	—
321	C₂²	#19127663	1	1	yes	characteristic
322	C₂²	#19127723	1	1	yes	characteristic
323	C₂²	#19127752	1	1	yes	characteristic
324	C₄	—	4	2	no	—
325	C₄	—	332	1	no	—
326	C₄	—	166	2	no	—
327	C₄	—	166	2	no	—
328	C₄	#19127560	1	1	yes	characteristic
329	C₄	C₂²⋊C₈×D₈₃	1	1	yes	characteristic
330	C₄	C₄⋊C₈×D₈₃	1	1	yes	characteristic
331	C₄	#19127756	1	1	yes	characteristic
332	C₂	—	2	1	no	—
333	C₂	—	2	1	no	—
334	C₂	—	2	1	no	—
335	C₂	—	332	2	no	—
336	C₂	#24907442	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#14581765
commutator	C₂×C₄×C₈₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₈₃
Fitting	C₈₃×#1378	C₂
socle	C₂×C₈₃	#1797
prime radical	C₂×#13050590	C₂×C₄
cosocle	C₂²×C₈×C₈₃	C₂³
hypercenter	#1378	D₈₃
hypocenter	C₈₃	#12792
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #1378

Complements

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

N	H ≅ G/N	type
2. C₂×#19127521	C₂	semidirect
3. C₂×#19127521	C₂	semidirect
6. #24905549	C₂	semidirect
7. C₈₃×#1378	C₂	semidirect
175. #1378	D₈₃	semidirect
176. C₈₃	#12792	semidirect

Maximal subgroups

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

subgroup	index
#24915885	2
C₂×#19127521	2
#24909092	2
#24909090	2
#24905549	2
C₈₃×#1378	2
#12792	83