Subgroups — #24917425

Order 10624.

386 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128274	C₂	1	1	yes	maximal, characteristic
2	#19128182	C₂	1	1	yes	maximal, characteristic
3	#19128045	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128043	C₂	1	1	yes	maximal, characteristic, complemented
5	#19128040	C₂	1	1	yes	maximal, characteristic, complemented
6	#19128039	C₂	1	1	yes	maximal, characteristic, complemented
7	#19128036	C₂	1	1	yes	maximal, characteristic, complemented
8	#19128035	C₂	1	1	yes	maximal, characteristic, complemented
9	#19128033	C₂	1	1	yes	maximal, characteristic, complemented
10	#19127740	C₂	1	1	yes	maximal, characteristic, complemented
11	#19127736	C₂	1	1	yes	maximal, characteristic, complemented
12	#19127708	C₂	1	1	yes	maximal, characteristic, complemented
13	#103×D₈₃	C₂	1	1	yes	maximal, characteristic, complemented
14	#19127666	C₂	1	1	yes	maximal, characteristic
15	#483×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13050590	C₂²	1	1	yes	characteristic
17	C₂×Q₈×D₈₃	C₂²	1	1	yes	characteristic
18	C₂×#13050573	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₈₃	C₂²	1	1	yes	characteristic
20	#14581791	—	2	1	no	—
21	#14581778	—	2	1	no	—
22	#14581777	—	2	1	no	—
23	#14581774	—	2	1	no	—
24	#14581772	C₂²	1	1	yes	characteristic
25	#14581771	C₂²	1	1	yes	characteristic
26	#14581768	C₂²	1	1	yes	characteristic
27	#14581757	—	2	1	no	—
28	#14581754	—	2	1	no	—
29	C₄×C₂²⋊D₈₃	—	2	1	no	—
30	C₄×C₂²⋊D₈₃	—	2	1	no	—
31	#14581751	C₂²	1	1	yes	characteristic
32	#14581750	C₂²	1	1	yes	characteristic
33	#14581750	C₂²	1	1	yes	characteristic
34	#14581750	C₂²	1	1	yes	characteristic
35	C₄⋊C₄×D₈₃	C₂²	1	1	yes	characteristic
36	C₄⋊C₄×D₈₃	C₂²	1	1	yes	characteristic
37	#14581726	C₂²	1	1	yes	characteristic
38	#14581725	C₂²	1	1	yes	characteristic
39	#14581725	C₂²	1	1	yes	characteristic
40	#14581725	C₂²	1	1	yes	characteristic
41	#14581724	C₂²	1	1	yes	characteristic
42	#14581723	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	#14581715	C₂²	1	1	yes	characteristic
48	#14581715	C₂²	1	1	yes	characteristic
49	C₂²⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
50	(C₂×D₈₃)⋊C₈	C₂²	1	1	yes	characteristic
51	(C₂×D₈₃)⋊C₈	C₂²	1	1	yes	characteristic
52	C₂²⋊Q₈×C₈₃	C₂²	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₈₃	C₂²	1	1	yes	characteristic
54	#106×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	D₄	1	1	yes	characteristic
63	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
64	Q₈×D₈₃	—	2	2	no	—
65	Q₈×D₈₃	—	2	2	no	—
66	C₂×D₃₃₂	D₄	1	1	yes	characteristic
67	C₂×C₂²⋊D₈₃	D₄	1	1	yes	characteristic
68	C₂×C₂²⋊D₈₃	D₄	1	1	yes	characteristic
69	C₂×C₄×D₈₃	—	2	1	no	—
70	C₂×C₄×D₈₃	—	2	1	no	—
71	C₂×C₄×D₈₃	—	2	1	no	—
72	C₂×C₄×D₈₃	C₂³	1	1	yes	characteristic
73	C₂×C₄×D₈₃	C₂³	1	1	yes	characteristic
74	#13050573	—	2	2	no	—
75	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
76	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
77	C₈×D₈₃	—	2	2	no	—
78	C₄⋊Dic₈₃	—	2	1	no	—
79	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
80	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
81	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
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	C₂²⋊Dic₈₃	—	2	1	no	—
88	(C₂×D₈₃)⋊C₄	—	2	1	no	—
89	(C₂×D₈₃)⋊C₄	—	2	1	no	—
90	(C₂×D₈₃)⋊C₄	—	2	1	no	—
91	(C₂×D₈₃)⋊C₄	—	2	1	no	—
92	(C₂×D₈₃)⋊C₄	—	2	1	no	—
93	C₄×Dic₈₃	—	2	1	no	—
94	C₄×Dic₈₃	—	2	1	no	—
95	C₄×Dic₈₃	—	2	1	no	—
96	C₄×Dic₈₃	—	2	1	no	—
97	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
98	C₂²×C₄×C₈₃	—	2	1	no	—
99	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
100	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
101	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₈₃	—	2	1	no	—
103	C₄⋊C₄×C₈₃	—	2	1	no	—
104	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
105	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
106	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
107	C₂²⋊C₄×C₈₃	—	2	1	no	—
108	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
109	C₂²×D₈₃	C₂×D₄	1	1	yes	characteristic
110	C₂²×D₈₃	C₂×D₄	1	1	yes	characteristic
111	Q₈×C₈₃	—	2	1	no	—
112	Q₈×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	—	2	1	no	—
119	C₂×C₄×C₈₃	—	8	1	no	—
120	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
122	C₂²⋊D₈₃	—	2	2	no	—
123	C₂²⋊D₈₃	—	2	2	no	—
124	C₂×Dic₈₃	—	2	1	no	—
125	C₂×Dic₈₃	—	2	1	no	—
126	C₂×Dic₈₃	—	2	1	no	—
127	C₂×Dic₈₃	—	2	1	no	—
128	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
129	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
130	D₃₃₂	—	2	2	no	—
131	C₄×D₈₃	—	2	1	no	—
132	C₄×D₈₃	—	2	1	no	—
133	C₄×D₈₃	—	4	2	no	—
134	C₄×D₈₃	—	8	1	no	—
135	C₄×D₈₃	—	8	1	no	—
136	C₄×D₈₃	#41	1	2	yes	—
137	C₄×D₈₃	#41	1	2	yes	—
138	Dic₁₆₆	—	2	2	no	—
139	Dic₁₆₆	—	2	1	no	—
140	Dic₁₆₆	—	2	1	no	—
141	Dic₁₆₆	—	4	2	no	—
142	C₈×C₈₃	—	2	1	no	—
143	C₈×C₈₃	—	2	1	no	—
144	C₈₃⋊C₈	—	2	1	no	—
145	C₈₃⋊C₈	—	2	1	no	—
146	C₂²×C₈₃	—	2	1	no	—
147	C₂²×C₈₃	—	2	1	no	—
148	C₂²×C₈₃	—	2	1	no	—
149	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
150	C₂×D₈₃	—	2	1	no	—
151	C₂×D₈₃	—	2	1	no	—
152	C₂×D₈₃	—	2	1	no	—
153	C₂×D₈₃	—	2	1	no	—
154	C₂×D₈₃	—	2	1	no	—
155	C₂×D₈₃	#123	1	2	yes	—
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	—	4	1	no	—
161	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
162	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
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	#123	1	2	yes	—
169	C₂×C₈₃	—	4	1	no	—
170	C₂×C₈₃	C₂×#123	1	1	yes	characteristic
171	C₂×C₈₃	#577	1	1	yes	characteristic
172	C₂×C₈₃	#578	1	1	yes	characteristic
173	D₈₃	—	2	2	no	—
174	D₈₃	—	4	1	no	—
175	#3097	—	83	1	no	maximal, Sylow
176	C₈₃	#3097	1	1	yes	characteristic, complemented, Sylow
177	#536	—	83	1	no	—
178	#520	—	83	1	no	—
179	#485	—	83	1	no	—
180	#485	—	83	1	no	—
181	#485	—	83	1	no	—
182	#485	—	83	1	no	—
183	#484	—	83	1	no	—
184	#484	—	83	1	no	—
185	#483	—	83	1	no	—
186	#483	D₈₃	1	1	yes	characteristic, complemented
187	#428	—	83	1	no	—
188	#427	—	83	1	no	—
189	#419	—	83	1	no	—
190	C₂×#103	—	83	1	no	—
191	#408	—	83	1	no	—
192	C₂×#41	—	83	1	no	—
193	C₂²×Q₈	—	83	1	no	—
194	C₂×M₁₆	—	83	1	no	—
195	C₂²×C₈	—	83	1	no	—
196	C₄⋊Q₈	—	166	1	no	—
197	#124	—	166	1	no	—
198	#123	—	166	1	no	—
199	#123	—	166	1	no	—
200	C₂²⋊Q₈	—	83	1	no	—
201	C₂²⋊Q₈	—	83	1	no	—
202	C₂²⋊Q₈	—	83	1	no	—
203	C₂²⋊Q₈	C₂×D₈₃	1	1	yes	characteristic
204	C₄×Q₈	—	166	1	no	—
205	C₄×D₄	—	166	1	no	—
206	C₄×D₄	—	166	1	no	—
207	C₄×D₄	—	166	1	no	—
208	#117	—	83	1	no	—
209	#117	—	83	1	no	—
210	#117	—	83	1	no	—
211	#117	—	83	1	no	—
212	C₂×C₄⋊C₄	—	83	1	no	—
213	C₂×C₄⋊C₄	—	83	1	no	—
214	C₂×C₄⋊C₄	C₂×D₈₃	1	1	yes	characteristic
215	#107	—	83	1	no	—
216	#107	—	83	1	no	—
217	#107	—	83	1	no	—
218	#107	—	83	1	no	—
219	#106	—	83	1	no	—
220	#106	—	83	1	no	—
221	#106	C₂×D₈₃	1	1	yes	characteristic
222	#106	C₂×D₈₃	1	1	yes	characteristic
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	—	83	1	no	—
233	C₂²⋊C₈	—	83	1	no	—
234	C₂²⋊C₈	C₂×D₈₃	1	1	yes	characteristic
235	#41	—	166	2	no	—
236	#41	—	166	2	no	—
237	C₂×Q₈	—	83	1	no	—
238	C₂×Q₈	—	83	1	no	—
239	C₂×Q₈	—	166	1	no	—
240	C₂×Q₈	—	166	2	no	—
241	C₂×Q₈	—	166	2	no	—
242	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
243	C₂×D₄	—	83	1	no	—
244	C₂×D₄	—	83	1	no	—
245	C₂×D₄	—	83	1	no	—
246	C₂²×C₄	—	2	1	no	—
247	C₂²×C₄	—	83	1	no	—
248	C₂²×C₄	—	83	1	no	—
249	C₂²×C₄	—	166	1	no	—
250	C₂²×C₄	—	166	1	no	—
251	C₂²×C₄	—	166	1	no	—
252	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
253	M₁₆	—	166	2	no	—
254	C₂×C₈	—	83	1	no	—
255	C₂×C₈	—	83	1	no	—
256	C₂×C₈	—	166	2	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	—	2	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	C₂²×D₈₃	1	1	yes	characteristic
270	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
271	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
272	C₂²⋊C₄	—	2	1	no	—
273	C₂²⋊C₄	—	166	1	no	—
274	C₂²⋊C₄	—	166	1	no	—
275	C₂²⋊C₄	—	166	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	C₄²	—	166	1	no	—
281	C₄²	—	166	1	no	—
282	C₄²	—	166	1	no	—
283	C₄²	—	166	1	no	—
284	C₂³	—	83	1	no	—
285	C₂³	—	83	1	no	—
286	C₂³	D₄×D₈₃	1	1	yes	characteristic
287	Q₈	—	2	1	no	—
288	Q₈	—	2	1	no	—
289	Q₈	—	332	2	no	—
290	Q₈	—	166	2	no	—
291	Q₈	—	166	1	no	—
292	Q₈	—	166	1	no	—
293	D₄	—	166	2	no	—
294	D₄	—	166	2	no	—
295	D₄	—	166	2	no	—
296	C₂×C₄	—	2	1	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	—	8	1	no	—
303	C₂×C₄	—	332	2	no	—
304	C₂×C₄	—	83	1	no	—
305	C₂×C₄	—	83	2	no	—
306	C₂×C₄	—	83	2	no	—
307	C₂×C₄	—	83	1	no	—
308	C₂×C₄	—	664	1	no	—
309	C₂×C₄	—	664	1	no	—
310	C₂×C₄	—	166	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	D₄×D₈₃	1	1	yes	characteristic
317	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
318	C₈	—	2	1	no	—
319	C₈	—	2	1	no	—
320	C₈	—	166	1	no	—
321	C₈	—	166	1	no	—
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	83	2	no	—
326	C₂²	—	166	1	no	—
327	C₂²	—	166	1	no	—
328	C₂²	—	166	1	no	—
329	C₂²	—	166	1	no	—
330	C₂²	—	166	1	no	—
331	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
332	C₄	—	2	1	no	—
333	C₄	—	4	1	no	—
334	C₄	—	4	1	no	—
335	C₄	—	4	1	no	—
336	C₄	—	4	1	no	—
337	C₄	—	332	1	no	—
338	C₄	—	332	1	no	—
339	C₄	—	332	1	no	—
340	C₄	—	332	1	no	—
341	C₄	—	83	2	no	—
342	C₄	—	166	1	no	—
343	C₄	#41×D₈₃	1	1	yes	characteristic
344	C₄	#41×D₈₃	1	1	yes	characteristic
345	C₂	—	4	1	no	—
346	C₂	—	332	1	no	—
347	C₂	—	166	2	no	—
348	C₂	#123×D₈₃	1	1	yes	characteristic
349	C₂	#19128507	1	1	yes	characteristic
350	C₂	#19128517	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₃
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#483×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	C₂×#13050590	C₂²
hypercenter	#483	D₈₃
hypocenter	C₈₃	#3097
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #483

Complements

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

N	H ≅ G/N	type
3. #19128045	C₂	semidirect
4. #19128043	C₂	semidirect
5. #19128040	C₂	semidirect
6. #19128039	C₂	semidirect
7. #19128036	C₂	semidirect
8. #19128035	C₂	semidirect
9. #19128033	C₂	semidirect
10. #19127740	C₂	semidirect
11. #19127736	C₂	semidirect
12. #19127708	C₂	semidirect
13. #103×D₈₃	C₂	semidirect
15. #483×C₈₃	C₂	semidirect
176. C₈₃	#3097	semidirect
186. #483	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128274	2
#19128182	2
#19128045	2
#19128043	2
#19128040	2
#19128039	2
#19128036	2
#19128035	2
#19128033	2
#19127740	2
#19127736	2
#19127708	2
#103×D₈₃	2
#19127666	2
#483×C₈₃	2
#3097	83