Subgroups — #24917620

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	#19128169	C₂	1	1	yes	maximal, characteristic
3	#19128042	C₂	1	1	yes	maximal, characteristic, complemented
4	#19127995	C₂	1	1	yes	maximal, characteristic, complemented
5	#19127990	C₂	1	1	yes	maximal, characteristic, complemented
6	#19127910	C₂	1	1	yes	maximal, characteristic, complemented
7	#19127905	C₂	1	1	yes	maximal, characteristic, complemented
8	#19127865	C₂	1	1	yes	maximal, characteristic
9	#19127861	C₂	1	1	yes	maximal, characteristic
10	#19127807	C₂	1	1	yes	maximal, characteristic, complemented
11	#19127805	C₂	1	1	yes	maximal, characteristic, complemented
12	#19127804	C₂	1	1	yes	maximal, characteristic, complemented
13	#19127800	C₂	1	1	yes	maximal, characteristic, complemented
14	#19127755	C₂	1	1	yes	maximal, characteristic, complemented
15	#485×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13050590	C₂²	1	1	yes	characteristic
17	C₂×Dic₃₃₂	C₂²	1	1	yes	characteristic
18	C₂×#13050580	C₂²	1	1	yes	characteristic
19	C₂×#13050579	C₂²	1	1	yes	characteristic
20	C₂×#13050577	C₂²	1	1	yes	characteristic
21	#14581791	C₂²	1	1	yes	characteristic
22	#14581790	C₂²	1	1	yes	characteristic
23	#14581782	—	2	1	no	—
24	#14581771	C₂²	1	1	yes	characteristic
25	#14581769	—	2	1	no	—
26	Q₈×Dic₈₃	C₂²	1	1	yes	characteristic
27	#14581757	—	2	1	no	—
28	#14581757	C₂²	1	1	yes	characteristic
29	#14581757	C₂²	1	1	yes	characteristic
30	#14581757	C₂²	1	1	yes	characteristic
31	#14581754	C₂²	1	1	yes	characteristic
32	#14581754	C₂²	1	1	yes	characteristic
33	(C₂²⋊D₈₃)⋊C₄	—	2	1	no	—
34	(C₂²⋊D₈₃)⋊C₄	—	2	1	no	—
35	#14581749	—	2	1	no	—
36	#14581748	—	2	1	no	—
37	C₂×C₄×Dic₈₃	C₂²	1	1	yes	characteristic
38	C₄²×D₈₃	—	2	1	no	—
39	#14581725	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	#14581715	C₂²	1	1	yes	characteristic
45	#14581715	C₂²	1	1	yes	characteristic
46	#14581715	C₂²	1	1	yes	characteristic
47	#14581714	C₂²	1	1	yes	characteristic
48	#14581712	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	C₂²⋊Q₈×C₈₃	C₂²	1	1	yes	characteristic
53	#117×C₈₃	C₂²	1	1	yes	characteristic
54	#107×C₈₃	C₂²	1	1	yes	characteristic
55	#106×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	#13050590	—	2	2	no	—
60	#13050590	—	2	2	no	—
61	C₂×Dic₁₆₆	—	2	1	no	—
62	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
63	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
64	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
65	C₂×C₂²⋊D₈₃	—	2	1	no	—
66	C₂²×Dic₈₃	D₄	1	1	yes	characteristic
67	C₂²×Dic₈₃	D₄	1	1	yes	characteristic
68	C₂×C₄×D₈₃	—	2	1	no	—
69	C₂×C₄×D₈₃	—	2	1	no	—
70	C₂×C₄×D₈₃	—	2	1	no	—
71	Dic₃₃₂	—	2	2	no	—
72	#13050580	—	2	2	no	—
73	#13050579	—	2	2	no	—
74	#13050577	—	2	2	no	—
75	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
76	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
77	C₄⋊Dic₈₃	—	2	1	no	—
78	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
84	Dic₈₃⋊C₄	—	2	1	no	—
85	Dic₈₃⋊C₄	—	2	1	no	—
86	C₂²⋊Dic₈₃	—	2	1	no	—
87	(C₂×D₈₃)⋊C₄	—	2	1	no	—
88	(C₂×D₈₃)⋊C₄	—	2	1	no	—
89	C₄×Dic₈₃	—	2	1	no	—
90	C₄×Dic₈₃	—	2	1	no	—
91	C₄×Dic₈₃	—	2	1	no	—
92	C₄×Dic₈₃	—	2	1	no	—
93	C₄×Dic₈₃	D₄	1	1	yes	characteristic
94	C₄×Dic₈₃	D₄	1	1	yes	characteristic
95	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
96	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
97	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
98	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
99	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
100	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
101	C₄⋊C₄×C₈₃	—	2	1	no	—
102	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
103	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
104	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
105	C₂²⋊C₄×C₈₃	—	2	1	no	—
106	C₂²⋊C₄×C₈₃	—	2	1	no	—
107	C₄²×C₈₃	—	2	1	no	—
108	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
109	C₂²×D₈₃	—	2	1	no	—
110	Q₈×C₈₃	—	2	1	no	—
111	Q₈×C₈₃	—	2	1	no	—
112	C₂×C₄×C₈₃	—	2	1	no	—
113	C₂×C₄×C₈₃	—	2	1	no	—
114	C₂×C₄×C₈₃	—	2	1	no	—
115	C₂×C₄×C₈₃	—	2	1	no	—
116	C₂×C₄×C₈₃	—	2	1	no	—
117	C₂×C₄×C₈₃	—	2	1	no	—
118	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
120	C₂²⋊D₈₃	—	2	4	no	—
121	C₂×Dic₈₃	—	2	1	no	—
122	C₂×Dic₈₃	—	2	2	no	—
123	C₂×Dic₈₃	—	2	1	no	—
124	C₂×Dic₈₃	—	2	1	no	—
125	C₂×Dic₈₃	—	4	1	no	—
126	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
127	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
128	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
129	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
130	D₃₃₂	—	2	1	no	—
131	D₃₃₂	#41	1	2	yes	—
132	C₄×D₈₃	—	2	4	no	—
133	C₄×D₈₃	—	4	2	no	—
134	C₄×D₈₃	—	4	2	no	—
135	Dic₁₆₆	—	2	1	no	—
136	Dic₁₆₆	—	2	1	no	—
137	Dic₁₆₆	—	2	1	no	—
138	Dic₁₆₆	—	4	2	no	—
139	Dic₁₆₆	#41	1	2	yes	—
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	1	no	—
146	C₂²×C₈₃	—	2	1	no	—
147	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
148	C₂×D₈₃	—	2	2	no	—
149	C₂×D₈₃	—	4	1	no	—
150	C₂×D₈₃	—	4	1	no	—
151	C₄×C₈₃	—	2	1	no	—
152	C₄×C₈₃	—	4	1	no	—
153	C₄×C₈₃	—	4	1	no	—
154	C₄×C₈₃	—	4	1	no	—
155	C₄×C₈₃	—	4	1	no	—
156	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
157	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
158	Dic₈₃	—	2	2	no	—
159	Dic₈₃	—	2	1	no	—
160	Dic₈₃	—	2	1	no	—
161	Dic₈₃	—	2	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	#525	1	1	yes	characteristic
167	C₂×C₈₃	C₂×#137	1	1	yes	characteristic
168	C₂×C₈₃	#575	1	1	yes	characteristic
169	D₈₃	—	4	2	no	—
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	D₈₃	1	1	yes	characteristic, complemented
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	—	83	1	no	—
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	—	83	1	no	—
188	C₂×Q₁₆	—	83	1	no	—
189	C₂×Q₁₆	—	83	1	no	—
190	C₂×SD₁₆	—	83	1	no	—
191	C₂×SD₁₆	—	83	1	no	—
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	—	166	1	no	—
208	#117	—	166	1	no	—
209	#117	C₂×D₈₃	1	1	yes	characteristic
210	C₂×C₄²	—	83	1	no	—
211	C₂×C₄²	—	166	1	no	—
212	#107	—	83	1	no	—
213	#107	C₂×D₈₃	1	1	yes	characteristic
214	#106	—	83	1	no	—
215	#106	C₂×D₈₃	1	1	yes	characteristic
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	C₂×D₈₃	1	1	yes	characteristic
223	#103	C₂×D₈₃	1	1	yes	characteristic
224	#102	—	83	1	no	—
225	#102	—	83	1	no	—
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	—	166	2	no	—
231	#41	—	166	2	no	—
232	C₂×Q₈	—	83	1	no	—
233	C₂×Q₈	—	83	1	no	—
234	C₂×Q₈	—	166	1	no	—
235	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
236	C₂×D₄	—	83	1	no	—
237	C₂×D₄	—	166	1	no	—
238	C₂²×C₄	—	83	1	no	—
239	C₂²×C₄	—	83	1	no	—
240	C₂²×C₄	—	166	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	—	166	2	no	—
245	Q₁₆	—	166	2	no	—
246	SD₁₆	—	166	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	—	166	1	no	—
255	C₄⋊C₄	—	166	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	C₂²×D₈₃	1	1	yes	characteristic
263	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
264	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
265	C₂²⋊C₄	—	2	1	no	—
266	C₂²⋊C₄	—	2	1	no	—
267	C₂²⋊C₄	—	166	1	no	—
268	C₂²⋊C₄	—	166	1	no	—
269	C₂²⋊C₄	—	166	1	no	—
270	C₄²	—	2	1	no	—
271	C₄²	—	83	1	no	—
272	C₄²	—	83	1	no	—
273	C₄²	—	83	1	no	—
274	C₄²	—	83	1	no	—
275	C₄²	—	166	1	no	—
276	C₄²	—	166	1	no	—
277	C₄²	—	166	1	no	—
278	C₄²	—	166	1	no	—
279	C₂³	—	166	1	no	—
280	C₂³	D₄×D₈₃	1	1	yes	characteristic
281	Q₈	—	2	1	no	—
282	Q₈	—	2	1	no	—
283	Q₈	—	332	2	no	—
284	Q₈	—	83	2	no	—
285	Q₈	—	166	1	no	—
286	Q₈	—	166	1	no	—
287	Q₈	—	166	1	no	—
288	D₄	—	83	2	no	—
289	D₄	—	166	4	no	—
290	D₄	—	166	1	no	—
291	C₂×C₄	—	2	1	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	4	no	—
305	C₂×C₄	—	166	1	no	—
306	C₂×C₄	—	166	1	no	—
307	C₂×C₄	—	166	2	no	—
308	C₂×C₄	—	166	1	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	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	332	1	no	—
319	C₂²	—	332	1	no	—
320	C₂²	—	166	2	no	—
321	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	4	1	no	—
324	C₄	—	4	1	no	—
325	C₄	—	4	1	no	—
326	C₄	—	4	1	no	—
327	C₄	—	332	1	no	—
328	C₄	—	332	1	no	—
329	C₄	—	83	2	no	—
330	C₄	—	166	2	no	—
331	C₄	—	166	1	no	—
332	C₄	—	166	1	no	—
333	C₄	—	166	1	no	—
334	C₄	#41×D₈₃	1	1	yes	characteristic
335	C₄	#41×D₈₃	1	1	yes	characteristic
336	C₂	—	4	1	no	—
337	C₂	—	332	2	no	—
338	C₂	#19128209	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	#485×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	C₂×#13050590	C₂²
hypercenter	#485	D₈₃
hypocenter	C₈₃	#3116
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #485

Complements

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

N	H ≅ G/N	type
1. #19128253	C₂	semidirect
3. #19128042	C₂	semidirect
4. #19127995	C₂	semidirect
5. #19127990	C₂	semidirect
6. #19127910	C₂	semidirect
7. #19127905	C₂	semidirect
10. #19127807	C₂	semidirect
11. #19127805	C₂	semidirect
12. #19127804	C₂	semidirect
13. #19127800	C₂	semidirect
14. #19127755	C₂	semidirect
15. #485×C₈₃	C₂	semidirect
171. C₈₃	#3116	semidirect
175. #485	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128253	2
#19128169	2
#19128042	2
#19127995	2
#19127990	2
#19127910	2
#19127905	2
#19127865	2
#19127861	2
#19127807	2
#19127805	2
#19127804	2
#19127800	2
#19127755	2
#485×C₈₃	2
#3116	83