Subgroups — #24917618

Order 10624.

386 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128253	C₂	1	1	yes	maximal, characteristic, complemented
2	#41×Dic₈₃	C₂	1	1	yes	maximal, characteristic
3	#19128043	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128039	C₂	1	1	yes	maximal, characteristic, complemented
5	#19127994	C₂	1	1	yes	maximal, characteristic, complemented
6	#19127989	C₂	1	1	yes	maximal, characteristic, complemented
7	#19127909	C₂	1	1	yes	maximal, characteristic, complemented
8	#19127904	C₂	1	1	yes	maximal, characteristic, complemented
9	#19127863	C₂	1	1	yes	maximal, characteristic
10	#19127808	C₂	1	1	yes	maximal, characteristic, complemented
11	#19127805	C₂	1	1	yes	maximal, characteristic, complemented
12	#19127802	C₂	1	1	yes	maximal, characteristic, complemented
13	#19127798	C₂	1	1	yes	maximal, characteristic, complemented
14	#19127755	C₂	1	1	yes	maximal, characteristic, complemented
15	#452×C₈₃	C₂	1	1	yes	maximal, characteristic
16	C₂×#13050580	C₂²	1	1	yes	characteristic
17	C₂×#13050578	C₂²	1	1	yes	characteristic
18	#14581791	C₂²	1	1	yes	characteristic
19	#14581790	C₂²	1	1	yes	characteristic
20	#14581782	—	2	1	no	—
21	#14581771	C₂²	1	1	yes	characteristic
22	#14581769	—	2	1	no	—
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	#14581757	—	2	1	no	—
27	#14581757	C₂²	1	1	yes	characteristic
28	D₄×Dic₈₃	—	2	1	no	—
29	D₄×Dic₈₃	—	2	1	no	—
30	D₄×Dic₈₃	C₂²	1	1	yes	characteristic
31	D₄×Dic₈₃	C₂²	1	1	yes	characteristic
32	#14581751	—	2	1	no	—
33	#14581751	C₂²	1	1	yes	characteristic
34	#14581748	—	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	#14581717	C₂²	1	1	yes	characteristic
44	#14581717	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	#14581713	C₂²	1	1	yes	characteristic
49	C₂²⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
50	#14581702	C₂²	1	1	yes	characteristic
51	#14581702	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×Dic₁₆₆	—	2	1	no	—
60	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
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	#13050580	—	2	2	no	—
66	#13050578	—	2	2	no	—
67	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
68	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
69	C₄⋊Dic₈₃	—	2	1	no	—
70	C₄⋊Dic₈₃	—	2	1	no	—
71	C₄⋊Dic₈₃	—	2	1	no	—
72	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
73	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
74	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
75	Dic₈₃⋊C₄	—	2	1	no	—
76	Dic₈₃⋊C₄	—	2	1	no	—
77	Dic₈₃⋊C₄	—	2	1	no	—
78	Dic₈₃⋊C₄	—	2	1	no	—
79	Dic₈₃⋊C₄	—	2	1	no	—
80	C₂²⋊Dic₈₃	—	2	1	no	—
81	C₂²⋊Dic₈₃	—	2	1	no	—
82	C₂²⋊Dic₈₃	—	2	1	no	—
83	C₂²⋊Dic₈₃	—	2	1	no	—
84	C₄×Dic₈₃	—	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	Q₈×C₈₃	—	2	1	no	—
111	Q₈×C₈₃	—	2	1	no	—
112	Q₈×C₈₃	—	2	1	no	—
113	Q₈×C₈₃	#41	1	2	yes	—
114	D₄×C₈₃	—	2	4	no	—
115	D₄×C₈₃	—	2	1	no	—
116	D₄×C₈₃	#41	1	2	yes	—
117	C₂×C₄×C₈₃	—	2	4	no	—
118	C₂×C₄×C₈₃	—	2	1	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	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
125	C₂×Dic₈₃	—	2	2	no	—
126	C₂×Dic₈₃	—	2	1	no	—
127	C₂×Dic₈₃	—	2	1	no	—
128	C₂×Dic₈₃	—	2	1	no	—
129	C₂×Dic₈₃	—	2	1	no	—
130	C₂×Dic₈₃	—	4	1	no	—
131	C₂×Dic₈₃	—	4	2	no	—
132	C₂×Dic₈₃	—	4	2	no	—
133	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
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	Dic₁₆₆	—	2	1	no	—
138	Dic₁₆₆	—	2	1	no	—
139	Dic₁₆₆	—	4	2	no	—
140	C₈×C₈₃	—	2	1	no	—
141	C₈×C₈₃	—	2	1	no	—
142	C₈₃⋊C₈	—	2	1	no	—
143	C₈₃⋊C₈	—	2	1	no	—
144	C₂²×C₈₃	—	2	1	no	—
145	C₂²×C₈₃	—	2	2	no	—
146	C₂²×C₈₃	—	2	1	no	—
147	C₂²×C₈₃	—	2	1	no	—
148	C₂²×C₈₃	—	4	1	no	—
149	C₂²×C₈₃	—	4	1	no	—
150	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
151	C₄×C₈₃	—	2	1	no	—
152	C₄×C₈₃	—	2	2	no	—
153	C₄×C₈₃	—	4	1	no	—
154	C₄×C₈₃	—	4	1	no	—
155	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
156	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
157	Dic₈₃	—	2	1	no	—
158	Dic₈₃	—	2	1	no	—
159	Dic₈₃	—	2	1	no	—
160	Dic₈₃	—	4	1	no	—
161	Dic₈₃	—	4	1	no	—
162	Dic₈₃	—	4	1	no	—
163	Dic₈₃	—	4	1	no	—
164	Dic₈₃	#137	1	2	yes	—
165	C₂×C₈₃	—	4	1	no	—
166	C₂×C₈₃	—	4	2	no	—
167	C₂×C₈₃	#525	1	1	yes	characteristic
168	C₂×C₈₃	C₂×#137	1	1	yes	characteristic
169	C₂×C₈₃	#575	1	1	yes	characteristic
170	#3116	—	83	1	no	maximal, Sylow
171	C₈₃	#3116	1	1	yes	characteristic, complemented, Sylow
172	#533	—	83	1	no	—
173	C₄×#41	—	83	1	no	—
174	#485	—	83	1	no	—
175	#485	—	83	1	no	—
176	#477	—	83	1	no	—
177	#476	—	83	1	no	—
178	#462	—	83	1	no	—
179	#461	—	83	1	no	—
180	#452	—	83	1	no	—
181	#452	D₈₃	1	1	yes	characteristic
182	#441	—	83	1	no	—
183	#441	—	83	1	no	—
184	#440	—	83	1	no	—
185	#440	—	83	1	no	—
186	#432	—	83	1	no	—
187	C₂×#41	C₂×D₈₃	1	1	yes	characteristic
188	C₂×Q₁₆	—	83	1	no	—
189	C₂×Q₁₆	C₂×D₈₃	1	1	yes	characteristic
190	C₂×SD₁₆	—	83	1	no	—
191	C₂×SD₁₆	C₂×D₈₃	1	1	yes	characteristic
192	C₄⋊Q₈	—	83	1	no	—
193	C₄⋊Q₈	—	83	1	no	—
194	#125	—	166	1	no	—
195	C₂²⋊Q₈	—	83	1	no	—
196	C₂²⋊Q₈	—	166	1	no	—
197	C₂²⋊Q₈	C₂×D₈₃	1	1	yes	characteristic
198	C₄×Q₈	—	83	1	no	—
199	C₄×Q₈	—	83	1	no	—
200	C₄×Q₈	—	83	1	no	—
201	C₄×Q₈	—	83	1	no	—
202	C₄×Q₈	—	166	1	no	—
203	C₄×D₄	—	83	1	no	—
204	C₄×D₄	—	83	1	no	—
205	C₄×D₄	—	166	1	no	—
206	C₄×D₄	—	166	1	no	—
207	#117	—	83	1	no	—
208	#117	—	166	1	no	—
209	#117	—	166	1	no	—
210	C₂×C₄²	—	83	1	no	—
211	C₂×C₄²	—	166	1	no	—
212	#107	—	83	1	no	—
213	#107	—	83	1	no	—
214	#106	—	83	1	no	—
215	#106	—	83	1	no	—
216	C₄⋊C₈	—	83	1	no	—
217	C₄⋊C₈	—	83	1	no	—
218	#103	—	83	1	no	—
219	#103	—	83	1	no	—
220	#103	—	83	1	no	—
221	#103	—	83	1	no	—
222	#103	—	83	1	no	—
223	#103	C₂×D₈₃	1	1	yes	characteristic
224	#102	—	83	1	no	—
225	#102	C₂×D₈₃	1	1	yes	characteristic
226	C₂²⋊C₈	—	83	1	no	—
227	C₂²⋊C₈	C₂×D₈₃	1	1	yes	characteristic
228	#97	—	83	1	no	—
229	#97	—	83	1	no	—
230	#41	—	2	2	no	—
231	#41	—	2	2	no	—
232	C₂×Q₈	—	83	1	no	—
233	C₂×Q₈	—	166	1	no	—
234	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
235	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
236	C₂×D₄	—	2	1	no	—
237	C₂×D₄	C₂²×D₈₃	1	1	yes	characteristic
238	C₂²×C₄	—	2	1	no	—
239	C₂²×C₄	—	83	1	no	—
240	C₂²×C₄	—	83	1	no	—
241	C₂²×C₄	—	166	1	no	—
242	C₂²×C₄	—	166	1	no	—
243	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
244	Q₁₆	—	2	2	no	—
245	Q₁₆	—	166	2	no	—
246	SD₁₆	—	2	2	no	—
247	SD₁₆	—	166	2	no	—
248	C₂×C₈	—	83	1	no	—
249	C₂×C₈	—	83	1	no	—
250	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
251	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
252	C₄⋊C₄	—	2	1	no	—
253	C₄⋊C₄	—	83	1	no	—
254	C₄⋊C₄	—	83	1	no	—
255	C₄⋊C₄	—	83	1	no	—
256	C₄⋊C₄	—	166	1	no	—
257	C₄⋊C₄	—	166	1	no	—
258	C₄⋊C₄	—	166	1	no	—
259	C₄⋊C₄	—	166	1	no	—
260	C₄⋊C₄	—	166	1	no	—
261	C₄⋊C₄	—	166	1	no	—
262	C₄⋊C₄	—	166	1	no	—
263	C₄⋊C₄	—	166	1	no	—
264	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
265	C₂²⋊C₄	—	2	1	no	—
266	C₂²⋊C₄	—	166	1	no	—
267	C₂²⋊C₄	—	166	1	no	—
268	C₂²⋊C₄	—	166	1	no	—
269	C₂²⋊C₄	—	166	1	no	—
270	C₄²	—	83	1	no	—
271	C₄²	—	83	1	no	—
272	C₄²	—	83	1	no	—
273	C₄²	—	83	1	no	—
274	C₄²	—	166	1	no	—
275	C₄²	—	166	1	no	—
276	C₄²	—	166	1	no	—
277	C₄²	—	166	1	no	—
278	C₄²	—	166	1	no	—
279	C₂³	—	2	1	no	—
280	C₂³	D₄×D₈₃	1	1	yes	characteristic
281	Q₈	—	2	1	no	—
282	Q₈	—	2	1	no	—
283	Q₈	—	2	1	no	—
284	Q₈	—	332	2	no	—
285	Q₈	—	166	1	no	—
286	Q₈	—	166	1	no	—
287	Q₈	#13050589	1	2	yes	—
288	D₄	—	2	4	no	—
289	D₄	—	2	1	no	—
290	D₄	#13050589	1	2	yes	—
291	C₂×C₄	—	2	4	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	332	2	no	—
298	C₂×C₄	—	332	2	no	—
299	C₂×C₄	—	332	1	no	—
300	C₂×C₄	—	83	1	no	—
301	C₂×C₄	—	83	1	no	—
302	C₂×C₄	—	83	1	no	—
303	C₂×C₄	—	83	1	no	—
304	C₂×C₄	—	166	1	no	—
305	C₂×C₄	—	166	1	no	—
306	C₂×C₄	—	166	1	no	—
307	C₂×C₄	—	166	1	no	—
308	C₂×C₄	—	166	2	no	—
309	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
310	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
311	C₈	—	2	1	no	—
312	C₈	—	2	1	no	—
313	C₈	—	166	1	no	—
314	C₈	—	166	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	2	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	4	1	no	—
320	C₂²	—	4	1	no	—
321	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	2	2	no	—
324	C₄	—	4	1	no	—
325	C₄	—	4	1	no	—
326	C₄	—	332	1	no	—
327	C₄	—	332	1	no	—
328	C₄	—	332	1	no	—
329	C₄	—	332	1	no	—
330	C₄	—	83	2	no	—
331	C₄	—	166	1	no	—
332	C₄	—	166	1	no	—
333	C₄	—	166	1	no	—
334	C₄	C₂×#13050589	1	1	yes	characteristic
335	C₄	C₂×#13050589	1	1	yes	characteristic
336	C₂	—	4	1	no	—
337	C₂	—	4	2	no	—
338	C₂	#19128207	1	1	yes	characteristic
339	C₂	#137×D₈₃	1	1	yes	characteristic
340	C₂	#19128498	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	C₂×#41×C₈₃	C₂²
hypercenter	#452	D₈₃
hypocenter	C₈₃	#3116
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. #19128253	C₂	semidirect
3. #19128043	C₂	semidirect
4. #19128039	C₂	semidirect
5. #19127994	C₂	semidirect
6. #19127989	C₂	semidirect
7. #19127909	C₂	semidirect
8. #19127904	C₂	semidirect
10. #19127808	C₂	semidirect
11. #19127805	C₂	semidirect
12. #19127802	C₂	semidirect
13. #19127798	C₂	semidirect
14. #19127755	C₂	semidirect
171. C₈₃	#3116	semidirect

Maximal subgroups

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

subgroup	index
#19128253	2
#41×Dic₈₃	2
#19128043	2
#19128039	2
#19127994	2
#19127989	2
#19127909	2
#19127904	2
#19127863	2
#19127808	2
#19127805	2
#19127802	2
#19127798	2
#19127755	2
#452×C₈₃	2
#3116	83