Subgroups — #24919739

Order 10624.

374 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128422	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₂²⋊D₈₃	C₂	1	1	yes	maximal, characteristic
3	#19128050	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128043	C₂	1	1	yes	maximal, characteristic, complemented
5	#19128041	C₂	1	1	yes	maximal, characteristic, complemented
6	#19127984	C₂	1	1	yes	maximal, characteristic, complemented
7	#19127957	C₂	1	1	yes	maximal, characteristic, complemented
8	#19127956	C₂	1	1	yes	maximal, characteristic, complemented
9	#19127909	C₂	1	1	yes	maximal, characteristic, complemented
10	#19127855	C₂	1	1	yes	maximal, characteristic, complemented
11	Q₁₆×Dic₈₃	C₂	1	1	yes	maximal, characteristic, complemented
12	#19127776	C₂	1	1	yes	maximal, characteristic
13	#19127710	C₂	1	1	yes	maximal, characteristic, complemented
14	#19127708	C₂	1	1	yes	maximal, characteristic, complemented
15	#451×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₈₃	C₂²	1	1	yes	characteristic
17	C₂×#13050580	C₂²	1	1	yes	characteristic
18	C₂×#13050574	C₂²	1	1	yes	characteristic
19	C₂×#13050573	C₂²	1	1	yes	characteristic
20	#14581791	C₂²	1	1	yes	characteristic
21	#14581790	—	2	1	no	—
22	#14581790	C₂²	1	1	yes	characteristic
23	#14581787	—	2	1	no	—
24	#14581785	—	2	1	no	—
25	#14581778	—	2	1	no	—
26	#14581771	—	2	1	no	—
27	#14581771	C₂²	1	1	yes	characteristic
28	#14581768	—	2	1	no	—
29	#14581768	C₂²	1	1	yes	characteristic
30	#14581768	C₂²	1	1	yes	characteristic
31	Q₈×Dic₈₃	C₂²	1	1	yes	characteristic
32	Q₈×Dic₈₃	C₂²	1	1	yes	characteristic
33	#14581757	—	2	1	no	—
34	C₄×C₂²⋊D₈₃	—	2	1	no	—
35	C₄×C₂²⋊D₈₃	C₂²	1	1	yes	characteristic
36	#14581751	C₂²	1	1	yes	characteristic
37	#14581750	C₂²	1	1	yes	characteristic
38	#14581726	C₂²	1	1	yes	characteristic
39	#14581724	C₂²	1	1	yes	characteristic
40	#14581723	C₂²	1	1	yes	characteristic
41	Dic₈₃⋊C₈	C₂²	1	1	yes	characteristic
42	#14581717	C₂²	1	1	yes	characteristic
43	#14581717	C₂²	1	1	yes	characteristic
44	#14581716	C₂²	1	1	yes	characteristic
45	#14581716	C₂²	1	1	yes	characteristic
46	#14581716	C₂²	1	1	yes	characteristic
47	#14581716	C₂²	1	1	yes	characteristic
48	#14581716	C₂²	1	1	yes	characteristic
49	#14581715	C₂²	1	1	yes	characteristic
50	(C₂×D₈₃)⋊C₈	C₂²	1	1	yes	characteristic
51	C₈×Dic₈₃	C₂²	1	1	yes	characteristic
52	C₂²×Q₈×C₈₃	C₂²	1	1	yes	characteristic
53	C₂×Q₁₆×C₈₃	C₂²	1	1	yes	characteristic
54	C₂×Q₁₆×C₈₃	C₂²	1	1	yes	characteristic
55	C₂²⋊Q₈×C₈₃	C₂²	1	1	yes	characteristic
56	#103×C₈₃	C₂²	1	1	yes	characteristic
57	#103×C₈₃	C₂²	1	1	yes	characteristic
58	C₂²⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₆₆	—	2	1	no	—
60	C₂×Dic₁₆₆	—	2	1	no	—
61	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
62	Q₈×D₈₃	—	2	2	no	—
63	Q₈×D₈₃	—	4	1	no	—
64	C₂×C₂²⋊D₈₃	D₄	1	1	yes	characteristic
65	C₂×C₄×D₈₃	—	2	1	no	—
66	C₂×C₄×D₈₃	C₂³	1	1	yes	characteristic
67	#13050580	—	4	1	no	—
68	#13050574	—	4	1	no	—
69	#13050573	—	4	1	no	—
70	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
71	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
72	C₄⋊Dic₈₃	—	2	1	no	—
73	C₄⋊Dic₈₃	—	2	1	no	—
74	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
75	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
76	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
77	Dic₈₃⋊C₄	—	2	1	no	—
78	Dic₈₃⋊C₄	—	2	1	no	—
79	Dic₈₃⋊C₄	—	2	1	no	—
80	Dic₈₃⋊C₄	—	2	1	no	—
81	Dic₈₃⋊C₄	—	2	1	no	—
82	Dic₈₃⋊C₄	—	2	1	no	—
83	Dic₈₃⋊C₄	D₄	1	1	yes	characteristic
84	C₂²⋊Dic₈₃	—	2	1	no	—
85	C₂²⋊Dic₈₃	—	2	1	no	—
86	C₂²⋊Dic₈₃	D₄	1	1	yes	characteristic
87	(C₂×D₈₃)⋊C₄	—	2	1	no	—
88	(C₂×D₈₃)⋊C₄	—	2	1	no	—
89	(C₂×D₈₃)⋊C₄	—	2	1	no	—
90	(C₂×D₈₃)⋊C₄	D₄	1	1	yes	characteristic
91	C₄×Dic₈₃	—	2	1	no	—
92	C₄×Dic₈₃	—	2	1	no	—
93	C₄×Dic₈₃	—	2	1	no	—
94	C₄×Dic₈₃	—	2	1	no	—
95	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
96	C₂×Q₈×C₈₃	—	2	1	no	—
97	C₂×Q₈×C₈₃	—	2	2	no	—
98	C₂×Q₈×C₈₃	—	4	1	no	—
99	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
100	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
101	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
102	C₂²×C₄×C₈₃	—	2	1	no	—
103	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
104	Q₁₆×C₈₃	—	2	2	no	—
105	Q₁₆×C₈₃	—	4	1	no	—
106	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
107	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
108	C₄⋊C₄×C₈₃	—	2	1	no	—
109	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
110	C₂²⋊C₄×C₈₃	—	2	1	no	—
111	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
112	C₂²×D₈₃	C₂×D₄	1	1	yes	characteristic
113	Q₈×C₈₃	—	2	1	no	—
114	Q₈×C₈₃	—	2	1	no	—
115	Q₈×C₈₃	—	2	1	no	—
116	Q₈×C₈₃	—	2	1	no	—
117	Q₈×C₈₃	—	2	1	no	—
118	Q₈×C₈₃	—	4	2	no	—
119	Q₈×C₈₃	#41	1	2	yes	—
120	C₂×C₄×C₈₃	—	2	1	no	—
121	C₂×C₄×C₈₃	—	2	1	no	—
122	C₂×C₄×C₈₃	—	2	1	no	—
123	C₂×C₄×C₈₃	—	2	1	no	—
124	C₂×C₄×C₈₃	—	2	1	no	—
125	C₂×C₄×C₈₃	—	2	1	no	—
126	C₂×C₄×C₈₃	—	4	2	no	—
127	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
128	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
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	—	2	1	no	—
134	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
135	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
136	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
137	C₄×D₈₃	—	2	1	no	—
138	C₄×D₈₃	—	2	1	no	—
139	C₄×D₈₃	—	4	2	no	—
140	Dic₁₆₆	—	2	1	no	—
141	Dic₁₆₆	—	2	1	no	—
142	Dic₁₆₆	—	4	2	no	—
143	Dic₁₆₆	—	4	2	no	—
144	C₈×C₈₃	—	2	1	no	—
145	C₈×C₈₃	—	2	1	no	—
146	C₈₃⋊C₈	—	2	1	no	—
147	C₈₃⋊C₈	—	2	1	no	—
148	C₂²×C₈₃	—	2	1	no	—
149	C₂²×C₈₃	—	2	1	no	—
150	C₂²×C₈₃	—	2	1	no	—
151	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
152	C₂×D₈₃	—	2	1	no	—
153	C₂×D₈₃	—	2	1	no	—
154	C₂×D₈₃	—	2	1	no	—
155	C₄×C₈₃	—	2	2	no	—
156	C₄×C₈₃	—	2	1	no	—
157	C₄×C₈₃	—	4	1	no	—
158	C₄×C₈₃	—	4	1	no	—
159	C₄×C₈₃	—	4	1	no	—
160	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
161	C₄×C₈₃	H(5,𝔽₂)	1	1	yes	characteristic
162	Dic₈₃	—	2	1	no	—
163	Dic₈₃	—	2	1	no	—
164	Dic₈₃	—	4	1	no	—
165	Dic₈₃	—	4	1	no	—
166	Dic₈₃	—	4	1	no	—
167	Dic₈₃	—	4	1	no	—
168	Dic₈₃	#137	1	2	yes	—
169	C₂×C₈₃	—	4	1	no	—
170	C₂×C₈₃	#546	1	1	yes	characteristic
171	C₂×C₈₃	C₂×#137	1	1	yes	characteristic
172	C₂×C₈₃	#578	1	1	yes	characteristic
173	D₈₃	—	4	1	no	—
174	#3328	—	83	1	no	maximal, Sylow
175	C₈₃	#3328	1	1	yes	characteristic, complemented, Sylow
176	#556	—	83	1	no	—
177	D₄×Q₈	—	83	1	no	—
178	#487	—	83	1	no	—
179	#485	—	83	1	no	—
180	#485	—	83	1	no	—
181	#475	—	83	1	no	—
182	#470	—	83	1	no	—
183	#470	—	83	1	no	—
184	#462	—	83	1	no	—
185	#451	—	83	1	no	—
186	#451	D₈₃	1	1	yes	characteristic, complemented
187	C₄×Q₁₆	—	83	1	no	—
188	#436	—	83	1	no	—
189	#419	—	83	1	no	—
190	#419	—	83	1	no	—
191	C₂²×Q₈	—	83	1	no	—
192	C₂²×Q₈	C₂×D₈₃	1	1	yes	characteristic
193	C₂×Q₁₆	—	83	1	no	—
194	C₂×Q₁₆	C₂×D₈₃	1	1	yes	characteristic
195	C₂×Q₁₆	C₂×D₈₃	1	1	yes	characteristic
196	C₂×M₁₆	—	83	1	no	—
197	C₂×M₁₆	—	83	1	no	—
198	C₄⋊Q₈	—	83	1	no	—
199	C₄⋊Q₈	—	83	1	no	—
200	C₄⋊Q₈	—	166	1	no	—
201	#126	—	166	1	no	—
202	#126	—	166	1	no	—
203	#124	—	166	1	no	—
204	C₂²⋊Q₈	—	83	1	no	—
205	C₂²⋊Q₈	—	83	1	no	—
206	C₂²⋊Q₈	—	83	1	no	—
207	C₂²⋊Q₈	—	166	1	no	—
208	C₂²⋊Q₈	—	166	1	no	—
209	C₂²⋊Q₈	C₂×D₈₃	1	1	yes	characteristic
210	C₄×Q₈	—	83	1	no	—
211	C₄×Q₈	—	83	1	no	—
212	C₄×Q₈	—	166	1	no	—
213	C₄×D₄	—	83	1	no	—
214	C₄×D₄	—	166	1	no	—
215	#117	—	83	1	no	—
216	#117	—	83	1	no	—
217	#107	—	83	1	no	—
218	#106	—	83	1	no	—
219	#106	—	83	1	no	—
220	C₄⋊C₈	—	83	1	no	—
221	#103	—	83	1	no	—
222	#103	—	83	1	no	—
223	#103	—	83	1	no	—
224	#103	—	83	1	no	—
225	#103	—	83	1	no	—
226	#103	—	83	1	no	—
227	#103	—	83	1	no	—
228	#103	—	83	1	no	—
229	#103	C₂×D₈₃	1	1	yes	characteristic
230	#103	C₂×D₈₃	1	1	yes	characteristic
231	C₂²⋊C₈	—	83	1	no	—
232	C₂²⋊C₈	C₂×D₈₃	1	1	yes	characteristic
233	C₄×C₈	—	83	1	no	—
234	C₂×Q₈	—	2	1	no	—
235	C₂×Q₈	—	2	2	no	—
236	C₂×Q₈	—	4	1	no	—
237	C₂×Q₈	—	332	1	no	—
238	C₂×Q₈	—	83	1	no	—
239	C₂×Q₈	—	166	1	no	—
240	C₂×Q₈	—	166	1	no	—
241	C₂×Q₈	—	166	2	no	—
242	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
243	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
244	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
245	C₂×D₄	—	83	1	no	—
246	C₂²×C₄	—	2	1	no	—
247	C₂²×C₄	—	83	1	no	—
248	C₂²×C₄	—	166	1	no	—
249	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
250	Q₁₆	—	2	2	no	—
251	Q₁₆	—	4	1	no	—
252	Q₁₆	—	332	1	no	—
253	M₁₆	—	332	1	no	—
254	M₁₆	—	332	1	no	—
255	C₂×C₈	—	83	1	no	—
256	C₂×C₈	—	83	1	no	—
257	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
258	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
259	C₄⋊C₄	—	2	1	no	—
260	C₄⋊C₄	—	83	1	no	—
261	C₄⋊C₄	—	83	1	no	—
262	C₄⋊C₄	—	83	1	no	—
263	C₄⋊C₄	—	83	1	no	—
264	C₄⋊C₄	—	166	1	no	—
265	C₄⋊C₄	—	166	1	no	—
266	C₄⋊C₄	—	166	1	no	—
267	C₄⋊C₄	—	166	1	no	—
268	C₄⋊C₄	—	166	1	no	—
269	C₄⋊C₄	—	166	1	no	—
270	C₄⋊C₄	—	166	1	no	—
271	C₄⋊C₄	—	166	1	no	—
272	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
273	C₂²⋊C₄	—	2	1	no	—
274	C₂²⋊C₄	—	83	1	no	—
275	C₂²⋊C₄	—	83	1	no	—
276	C₂²⋊C₄	—	166	1	no	—
277	C₂²⋊C₄	—	166	1	no	—
278	C₂²⋊C₄	—	166	1	no	—
279	C₂²⋊C₄	—	166	1	no	—
280	C₂²⋊C₄	—	166	1	no	—
281	C₄²	—	83	1	no	—
282	C₄²	—	166	1	no	—
283	C₄²	—	166	1	no	—
284	C₄²	—	166	1	no	—
285	C₄²	—	166	1	no	—
286	C₂³	—	83	1	no	—
287	C₂³	D₄×D₈₃	1	1	yes	characteristic
288	Q₈	—	2	1	no	—
289	Q₈	—	2	1	no	—
290	Q₈	—	2	1	no	—
291	Q₈	—	2	1	no	—
292	Q₈	—	2	1	no	—
293	Q₈	—	4	2	no	—
294	Q₈	—	332	2	no	—
295	Q₈	—	332	2	no	—
296	Q₈	—	166	1	no	—
297	Q₈	—	166	1	no	—
298	Q₈	#13050589	1	2	yes	—
299	D₄	—	166	2	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	332	2	no	—
308	C₂×C₄	—	83	1	no	—
309	C₂×C₄	—	83	1	no	—
310	C₂×C₄	—	83	1	no	—
311	C₂×C₄	—	166	1	no	—
312	C₂×C₄	—	166	1	no	—
313	C₂×C₄	—	166	1	no	—
314	C₂×C₄	—	166	1	no	—
315	C₂×C₄	—	166	1	no	—
316	C₂×C₄	—	166	1	no	—
317	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
318	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
319	C₈	—	2	1	no	—
320	C₈	—	2	1	no	—
321	C₈	—	166	1	no	—
322	C₈	—	166	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	166	1	no	—
327	C₂²	—	166	1	no	—
328	C₂²	—	166	1	no	—
329	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
330	C₄	—	2	1	no	—
331	C₄	—	2	2	no	—
332	C₄	—	4	1	no	—
333	C₄	—	4	1	no	—
334	C₄	—	4	1	no	—
335	C₄	—	332	1	no	—
336	C₄	—	332	1	no	—
337	C₄	—	332	1	no	—
338	C₄	—	332	1	no	—
339	C₄	—	83	2	no	—
340	C₄	—	166	1	no	—
341	C₄	—	166	1	no	—
342	C₄	C₂×#13050589	1	1	yes	characteristic
343	C₄	#14581837	1	1	yes	characteristic
344	C₂	—	4	1	no	—
345	C₂	—	332	1	no	—
346	C₂	#19128338	1	1	yes	characteristic
347	C₂	#137×D₈₃	1	1	yes	characteristic
348	C₂	#19128514	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₃
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#451×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	C₂×C₂²⋊D₈₃	D₄
hypercenter	#451	D₈₃
hypocenter	C₈₃	#3328
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #451

Complements

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

N	H ≅ G/N	type
3. #19128050	C₂	semidirect
4. #19128043	C₂	semidirect
5. #19128041	C₂	semidirect
6. #19127984	C₂	semidirect
7. #19127957	C₂	semidirect
8. #19127956	C₂	semidirect
9. #19127909	C₂	semidirect
10. #19127855	C₂	semidirect
11. Q₁₆×Dic₈₃	C₂	semidirect
13. #19127710	C₂	semidirect
14. #19127708	C₂	semidirect
15. #451×C₈₃	C₂	semidirect
175. C₈₃	#3328	semidirect
186. #451	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128422	2
Q₈×C₂²⋊D₈₃	2
#19128050	2
#19128043	2
#19128041	2
#19127984	2
#19127957	2
#19127956	2
#19127909	2
#19127855	2
Q₁₆×Dic₈₃	2
#19127776	2
#19127710	2
#19127708	2
#451×C₈₃	2
#3328	83