Subgroups — #24917482

Order 10624.

396 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	#19128237	C₂	1	1	yes	maximal, characteristic, complemented
2	#41×Dic₈₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#19128039	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128026	C₂	1	1	yes	maximal, characteristic, complemented
5	#19128006	C₂	1	1	yes	maximal, characteristic, complemented
6	#19127999	C₂	1	1	yes	maximal, characteristic, complemented
7	#19127925	C₂	1	1	yes	maximal, characteristic, complemented
8	#19127918	C₂	1	1	yes	maximal, characteristic, complemented
9	#19127846	C₂	1	1	yes	maximal, characteristic
10	Q₁₆×Dic₈₃	C₂	1	1	yes	maximal, characteristic, complemented
11	SD₁₆×Dic₈₃	C₂	1	1	yes	maximal, characteristic, complemented
12	#19127786	C₂	1	1	yes	maximal, characteristic, complemented
13	#19127781	C₂	1	1	yes	maximal, characteristic, complemented
14	#19127751	C₂	1	1	yes	maximal, characteristic, complemented
15	#452×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13050577	C₂²	1	1	yes	characteristic
17	C₂×#13050575	C₂²	1	1	yes	characteristic
18	#14581787	—	2	1	no	—
19	#14581783	C₂²	1	1	yes	characteristic
20	#14581781	C₂²	1	1	yes	characteristic
21	#14581776	—	2	1	no	—
22	#14581766	C₂²	1	1	yes	characteristic
23	Q₈×Dic₈₃	C₂²	1	1	yes	characteristic
24	Q₈×Dic₈₃	C₂²	1	1	yes	characteristic
25	Q₈×Dic₈₃	C₂²	1	1	yes	characteristic
26	D₄×Dic₈₃	—	2	1	no	—
27	D₄×Dic₈₃	—	2	1	no	—
28	D₄×Dic₈₃	C₂²	1	1	yes	characteristic
29	D₄×Dic₈₃	C₂²	1	1	yes	characteristic
30	#14581754	C₂²	1	1	yes	characteristic
31	(C₂²⋊D₈₃)⋊C₄	—	2	1	no	—
32	#14581751	—	2	1	no	—
33	#14581751	C₂²	1	1	yes	characteristic
34	#14581750	—	2	1	no	—
35	C₂×C₄×Dic₈₃	—	2	1	no	—
36	C₂×C₄×Dic₈₃	C₂²	1	1	yes	characteristic
37	#14581726	C₂²	1	1	yes	characteristic
38	#14581725	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	Dic₈₃⋊C₈	C₂²	1	1	yes	characteristic
43	#14581716	C₂²	1	1	yes	characteristic
44	#14581716	C₂²	1	1	yes	characteristic
45	#14581716	C₂²	1	1	yes	characteristic
46	#14581714	C₂²	1	1	yes	characteristic
47	#14581714	C₂²	1	1	yes	characteristic
48	#14581713	C₂²	1	1	yes	characteristic
49	C₂²⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
50	C₈×Dic₈₃	C₂²	1	1	yes	characteristic
51	C₈×Dic₈₃	C₂²	1	1	yes	characteristic
52	C₂×#41×C₈₃	C₂²	1	1	yes	characteristic
53	C₂×Q₁₆×C₈₃	C₂²	1	1	yes	characteristic
54	C₂×SD₁₆×C₈₃	C₂²	1	1	yes	characteristic
55	C₂²⋊Q₈×C₈₃	C₂²	1	1	yes	characteristic
56	#103×C₈₃	C₂²	1	1	yes	characteristic
57	#102×C₈₃	C₂²	1	1	yes	characteristic
58	C₂²⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
59	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
60	C₂×C₂²⋊D₈₃	—	2	1	no	—
61	C₂²×Dic₈₃	—	2	1	no	—
62	C₂²×Dic₈₃	—	2	1	no	—
63	C₂²×Dic₈₃	D₄	1	1	yes	characteristic
64	C₂²×Dic₈₃	D₄	1	1	yes	characteristic
65	C₂×C₄×D₈₃	—	2	1	no	—
66	#13050577	—	2	2	no	—
67	#13050575	—	2	2	no	—
68	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
69	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
70	C₄⋊Dic₈₃	—	2	1	no	—
71	C₄⋊Dic₈₃	—	2	1	no	—
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	C₂²⋊Dic₈₃	—	2	1	no	—
79	C₂²⋊Dic₈₃	—	2	1	no	—
80	C₂²⋊Dic₈₃	—	2	1	no	—
81	(C₂×D₈₃)⋊C₄	—	2	1	no	—
82	(C₂×D₈₃)⋊C₄	—	2	1	no	—
83	(C₂×D₈₃)⋊C₄	—	2	1	no	—
84	(C₂×D₈₃)⋊C₄	—	2	1	no	—
85	C₄×Dic₈₃	—	2	1	no	—
86	C₄×Dic₈₃	—	2	1	no	—
87	C₄×Dic₈₃	—	2	1	no	—
88	C₄×Dic₈₃	—	2	1	no	—
89	C₄×Dic₈₃	D₄	1	1	yes	characteristic
90	C₄×Dic₈₃	D₄	1	1	yes	characteristic
91	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
92	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
93	#41×C₈₃	—	2	2	no	—
94	#41×C₈₃	—	2	2	no	—
95	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
96	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
97	C₂×D₄×C₈₃	—	2	1	no	—
98	C₂×D₄×C₈₃	C₂³	1	1	yes	characteristic
99	C₂²×C₄×C₈₃	—	2	1	no	—
100	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
101	Q₁₆×C₈₃	—	2	2	no	—
102	SD₁₆×C₈₃	—	2	2	no	—
103	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
104	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
105	C₄⋊C₄×C₈₃	—	2	1	no	—
106	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
107	C₂²⋊C₄×C₈₃	—	2	1	no	—
108	C₂³×C₈₃	—	2	1	no	—
109	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
110	C₂²×D₈₃	—	2	1	no	—
111	Q₈×C₈₃	—	2	1	no	—
112	Q₈×C₈₃	—	2	1	no	—
113	Q₈×C₈₃	—	2	1	no	—
114	Q₈×C₈₃	#41	1	2	yes	—
115	D₄×C₈₃	—	2	4	no	—
116	D₄×C₈₃	—	2	1	no	—
117	D₄×C₈₃	#41	1	2	yes	—
118	C₂×C₄×C₈₃	—	2	4	no	—
119	C₂×C₄×C₈₃	—	2	1	no	—
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	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
126	C₂²⋊D₈₃	—	4	2	no	—
127	C₂×Dic₈₃	—	2	2	no	—
128	C₂×Dic₈₃	—	2	1	no	—
129	C₂×Dic₈₃	—	2	1	no	—
130	C₂×Dic₈₃	—	2	1	no	—
131	C₂×Dic₈₃	—	4	1	no	—
132	C₂×Dic₈₃	—	4	2	no	—
133	C₂×Dic₈₃	—	4	2	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	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
138	D₃₃₂	—	2	1	no	—
139	D₃₃₂	—	2	1	no	—
140	C₄×D₈₃	—	4	2	no	—
141	C₈×C₈₃	—	2	1	no	—
142	C₈×C₈₃	—	2	1	no	—
143	C₈₃⋊C₈	—	2	1	no	—
144	C₈₃⋊C₈	—	2	1	no	—
145	C₂²×C₈₃	—	2	1	no	—
146	C₂²×C₈₃	—	2	2	no	—
147	C₂²×C₈₃	—	2	1	no	—
148	C₂²×C₈₃	—	2	1	no	—
149	C₂²×C₈₃	—	4	1	no	—
150	C₂²×C₈₃	—	4	1	no	—
151	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
152	C₂×D₈₃	—	4	1	no	—
153	C₂×D₈₃	—	4	1	no	—
154	C₂×D₈₃	—	4	1	no	—
155	C₄×C₈₃	—	2	1	no	—
156	C₄×C₈₃	—	2	2	no	—
157	C₄×C₈₃	—	4	1	no	—
158	C₄×C₈₃	—	4	1	no	—
159	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
160	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
161	Dic₈₃	—	2	1	no	—
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	#135	1	2	yes	—
168	C₂×C₈₃	—	4	1	no	—
169	C₂×C₈₃	—	4	2	no	—
170	C₂×C₈₃	#525	1	1	yes	characteristic
171	C₂×C₈₃	C₂×#135	1	1	yes	characteristic
172	C₂×C₈₃	#575	1	1	yes	characteristic
173	D₈₃	—	8	1	no	—
174	#3102	—	83	1	no	maximal, Sylow
175	C₈₃	#3102	1	1	yes	characteristic, complemented, Sylow
176	#529	—	83	1	no	—
177	C₄×#41	—	83	1	no	—
178	#485	—	83	1	no	—
179	#482	—	83	1	no	—
180	#479	—	83	1	no	—
181	#478	—	83	1	no	—
182	#464	—	83	1	no	—
183	#463	—	83	1	no	—
184	#452	D₈₃	1	1	yes	characteristic, complemented
185	#449	—	83	1	no	—
186	C₄×Q₁₆	—	83	1	no	—
187	C₄×SD₁₆	—	83	1	no	—
188	C₄×SD₁₆	—	83	1	no	—
189	C₄×D₈	—	83	1	no	—
190	#431	—	83	1	no	—
191	C₂×#41	C₂×D₈₃	1	1	yes	characteristic
192	C₂×Q₁₆	C₂×D₈₃	1	1	yes	characteristic
193	C₂×SD₁₆	—	83	1	no	—
194	C₂×SD₁₆	C₂×D₈₃	1	1	yes	characteristic
195	C₂×D₈	—	83	1	no	—
196	#126	—	166	1	no	—
197	#125	—	83	1	no	—
198	#124	—	83	1	no	—
199	#123	—	166	1	no	—
200	C₂²⋊Q₈	C₂×D₈₃	1	1	yes	characteristic
201	#121	—	83	1	no	—
202	C₄×Q₈	—	83	1	no	—
203	C₄×Q₈	—	83	1	no	—
204	C₄×Q₈	—	83	1	no	—
205	C₄×D₄	—	83	1	no	—
206	C₄×D₄	—	83	1	no	—
207	C₄×D₄	—	83	1	no	—
208	C₄×D₄	—	166	1	no	—
209	C₄×D₄	—	166	1	no	—
210	C₄×D₄	—	166	1	no	—
211	#117	—	83	1	no	—
212	#117	—	166	1	no	—
213	#117	—	166	1	no	—
214	C₂×C₄²	—	83	1	no	—
215	C₂×C₄²	—	166	1	no	—
216	#107	—	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	C₄⋊C₈	—	83	1	no	—
222	#103	—	83	1	no	—
223	#103	—	83	1	no	—
224	#103	—	83	1	no	—
225	#103	C₂×D₈₃	1	1	yes	characteristic
226	#102	—	83	1	no	—
227	#102	—	83	1	no	—
228	#102	—	83	1	no	—
229	#102	C₂×D₈₃	1	1	yes	characteristic
230	C₂²⋊C₈	—	83	1	no	—
231	C₂²⋊C₈	C₂×D₈₃	1	1	yes	characteristic
232	C₄×C₈	—	83	1	no	—
233	C₄×C₈	—	83	1	no	—
234	#41	—	2	2	no	—
235	#41	—	2	2	no	—
236	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
237	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
238	C₂×D₄	—	2	1	no	—
239	C₂×D₄	—	83	1	no	—
240	C₂×D₄	—	166	1	no	—
241	C₂×D₄	C₂²×D₈₃	1	1	yes	characteristic
242	C₂²×C₄	—	2	1	no	—
243	C₂²×C₄	—	83	1	no	—
244	C₂²×C₄	—	83	1	no	—
245	C₂²×C₄	—	166	1	no	—
246	C₂²×C₄	—	166	1	no	—
247	C₂²×C₄	—	166	1	no	—
248	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
249	Q₁₆	—	2	2	no	—
250	SD₁₆	—	2	2	no	—
251	SD₁₆	—	166	2	no	—
252	D₈	—	166	2	no	—
253	C₂×C₈	—	83	1	no	—
254	C₂×C₈	—	83	1	no	—
255	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
256	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
257	C₄⋊C₄	—	2	1	no	—
258	C₄⋊C₄	—	83	1	no	—
259	C₄⋊C₄	—	83	1	no	—
260	C₄⋊C₄	—	83	1	no	—
261	C₄⋊C₄	—	166	1	no	—
262	C₄⋊C₄	—	166	1	no	—
263	C₄⋊C₄	—	166	1	no	—
264	C₄⋊C₄	—	166	1	no	—
265	C₄⋊C₄	—	166	1	no	—
266	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
267	C₂²⋊C₄	—	2	1	no	—
268	C₂²⋊C₄	—	166	1	no	—
269	C₂²⋊C₄	—	166	1	no	—
270	C₂²⋊C₄	—	166	1	no	—
271	C₂²⋊C₄	—	166	1	no	—
272	C₂²⋊C₄	—	166	1	no	—
273	C₂²⋊C₄	—	166	1	no	—
274	C₂²⋊C₄	—	166	1	no	—
275	C₄²	—	83	1	no	—
276	C₄²	—	83	1	no	—
277	C₄²	—	83	1	no	—
278	C₄²	—	83	1	no	—
279	C₄²	—	166	1	no	—
280	C₄²	—	166	1	no	—
281	C₄²	—	166	1	no	—
282	C₄²	—	166	1	no	—
283	C₂³	—	2	1	no	—
284	C₂³	—	166	1	no	—
285	C₂³	D₄×D₈₃	1	1	yes	characteristic
286	Q₈	—	2	1	no	—
287	Q₈	—	2	1	no	—
288	Q₈	—	2	1	no	—
289	Q₈	#13050589	1	2	yes	—
290	D₄	—	2	4	no	—
291	D₄	—	2	1	no	—
292	D₄	—	332	2	no	—
293	D₄	—	166	1	no	—
294	D₄	—	166	1	no	—
295	D₄	#13050589	1	2	yes	—
296	C₂×C₄	—	2	4	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	332	2	no	—
303	C₂×C₄	—	332	2	no	—
304	C₂×C₄	—	332	2	no	—
305	C₂×C₄	—	332	1	no	—
306	C₂×C₄	—	83	1	no	—
307	C₂×C₄	—	83	1	no	—
308	C₂×C₄	—	83	1	no	—
309	C₂×C₄	—	83	1	no	—
310	C₂×C₄	—	166	1	no	—
311	C₂×C₄	—	166	1	no	—
312	C₂×C₄	—	166	1	no	—
313	C₂×C₄	—	166	2	no	—
314	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
315	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
316	C₈	—	2	1	no	—
317	C₈	—	2	1	no	—
318	C₈	—	166	1	no	—
319	C₈	—	166	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	2	no	—
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	4	1	no	—
325	C₂²	—	4	1	no	—
326	C₂²	—	332	1	no	—
327	C₂²	—	332	1	no	—
328	C₂²	—	332	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	—	332	1	no	—
335	C₄	—	332	1	no	—
336	C₄	—	332	1	no	—
337	C₄	—	83	2	no	—
338	C₄	—	166	1	no	—
339	C₄	—	166	1	no	—
340	C₄	—	166	1	no	—
341	C₄	C₂×#13050589	1	1	yes	characteristic
342	C₄	C₂×#13050589	1	1	yes	characteristic
343	C₂	—	4	1	no	—
344	C₂	—	4	2	no	—
345	C₂	—	664	1	no	—
346	C₂	#19128207	1	1	yes	characteristic
347	C₂	#135×D₈₃	1	1	yes	characteristic
348	C₂	#19128500	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₃
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#452×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	#19127846	C₂
hypercenter	#452	D₈₃
hypocenter	C₈₃	#3102
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #452

Complements

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

N	H ≅ G/N	type
1. #19128237	C₂	semidirect
2. #41×Dic₈₃	C₂	semidirect
3. #19128039	C₂	semidirect
4. #19128026	C₂	semidirect
5. #19128006	C₂	semidirect
6. #19127999	C₂	semidirect
7. #19127925	C₂	semidirect
8. #19127918	C₂	semidirect
10. Q₁₆×Dic₈₃	C₂	semidirect
11. SD₁₆×Dic₈₃	C₂	semidirect
12. #19127786	C₂	semidirect
13. #19127781	C₂	semidirect
14. #19127751	C₂	semidirect
15. #452×C₈₃	C₂	semidirect
175. C₈₃	#3102	semidirect
184. #452	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128237	2
#41×Dic₈₃	2
#19128039	2
#19128026	2
#19128006	2
#19127999	2
#19127925	2
#19127918	2
#19127846	2
Q₁₆×Dic₈₃	2
SD₁₆×Dic₈₃	2
#19127786	2
#19127781	2
#19127751	2
#452×C₈₃	2
#3102	83