Subgroups — #24919959

Order 10624.

360 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128414	C₂	1	1	yes	maximal, characteristic, complemented
2	#19128395	C₂	1	1	yes	maximal, characteristic, complemented
3	#19128108	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128028	C₂	1	1	yes	maximal, characteristic, complemented
5	#19128025	C₂	1	1	yes	maximal, characteristic, complemented
6	#19128025	C₂	1	1	yes	maximal, characteristic, complemented
7	#19128016	C₂	1	1	yes	maximal, characteristic, complemented
8	#19128004	C₂	1	1	yes	maximal, characteristic, complemented
9	#19127980	C₂	1	1	yes	maximal, characteristic, complemented
10	#19127955	C₂	1	1	yes	maximal, characteristic, complemented
11	#19127937	C₂	1	1	yes	maximal, characteristic, complemented
12	#19127812	C₂	1	1	yes	maximal, characteristic, complemented
13	#19127772	C₂	1	1	yes	maximal, characteristic, complemented
14	#19127740	C₂	1	1	yes	maximal, characteristic, complemented
15	#427×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×D₆₆₄	C₂²	1	1	yes	characteristic
17	C₂×#13050573	C₂²	1	1	yes	characteristic
18	#14581791	C₂²	1	1	yes	characteristic
19	#14581786	—	2	1	no	—
20	#14581783	C₂²	1	1	yes	characteristic
21	#14581780	—	2	1	no	—
22	#14581778	—	2	1	no	—
23	#14581774	—	2	1	no	—
24	#14581767	C₂²	1	1	yes	characteristic
25	#14581767	C₂²	1	1	yes	characteristic
26	#14581766	C₂²	1	1	yes	characteristic
27	#14581766	C₂²	1	1	yes	characteristic
28	#14581765	—	2	1	no	—
29	#14581765	—	2	1	no	—
30	C₄×D₃₃₂	—	2	1	no	—
31	#14581754	C₂²	1	1	yes	characteristic
32	#14581754	C₂²	1	1	yes	characteristic
33	C₄×C₂²⋊D₈₃	C₂²	1	1	yes	characteristic
34	#14581750	C₂²	1	1	yes	characteristic
35	#14581749	—	2	1	no	—
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	#14581723	C₂²	1	1	yes	characteristic
41	#14581723	C₂²	1	1	yes	characteristic
42	Dic₈₃⋊C₈	C₂²	1	1	yes	characteristic
43	#14581714	C₂²	1	1	yes	characteristic
44	#14581714	C₂²	1	1	yes	characteristic
45	#14581712	C₂²	1	1	yes	characteristic
46	#14581712	C₂²	1	1	yes	characteristic
47	#14581712	C₂²	1	1	yes	characteristic
48	#14581712	C₂²	1	1	yes	characteristic
49	C₂²⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
50	(C₂×D₈₃)⋊C₈	C₂²	1	1	yes	characteristic
51	#14581702	C₂²	1	1	yes	characteristic
52	C₂²×C₈×C₈₃	C₂²	1	1	yes	characteristic
53	#117×C₈₃	C₂²	1	1	yes	characteristic
54	#117×C₈₃	C₂²	1	1	yes	characteristic
55	#107×C₈₃	C₂²	1	1	yes	characteristic
56	#107×C₈₃	C₂²	1	1	yes	characteristic
57	#106×C₈₃	C₂²	1	1	yes	characteristic
58	#106×C₈₃	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₆₆	—	2	1	no	—
60	C₂×D₃₃₂	—	2	1	no	—
61	C₂×D₃₃₂	—	2	1	no	—
62	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
63	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
64	C₂×C₂²⋊D₈₃	—	2	1	no	—
65	C₂×C₂²⋊D₈₃	—	2	1	no	—
66	C₂×C₂²⋊D₈₃	D₄	1	1	yes	characteristic
67	C₂×C₄×D₈₃	—	2	1	no	—
68	C₂×C₄×D₈₃	—	2	1	no	—
69	C₂×C₄×D₈₃	—	2	1	no	—
70	C₂×C₄×D₈₃	C₂³	1	1	yes	characteristic
71	D₆₆₄	—	2	2	no	—
72	#13050573	—	2	2	no	—
73	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
74	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
75	C₄⋊Dic₈₃	—	2	1	no	—
76	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
77	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
78	Dic₈₃⋊C₄	—	2	1	no	—
79	Dic₈₃⋊C₄	—	2	1	no	—
80	Dic₈₃⋊C₄	—	2	1	no	—
81	Dic₈₃⋊C₄	D₄	1	1	yes	characteristic
82	C₂²⋊Dic₈₃	D₄	1	1	yes	characteristic
83	(C₂×D₈₃)⋊C₄	—	2	1	no	—
84	(C₂×D₈₃)⋊C₄	—	2	1	no	—
85	(C₂×D₈₃)⋊C₄	—	2	1	no	—
86	(C₂×D₈₃)⋊C₄	—	2	1	no	—
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	C₂³	1	1	yes	characteristic
93	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
94	C₂×C₈×C₈₃	—	2	2	no	—
95	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
96	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
97	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
98	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
99	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
100	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
101	C₂²⋊C₄×C₈₃	—	2	1	no	—
102	C₂²⋊C₄×C₈₃	—	2	1	no	—
103	C₄²×C₈₃	—	2	1	no	—
104	C₄²×C₈₃	—	2	1	no	—
105	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
106	C₂²×D₈₃	—	2	1	no	—
107	C₂²×D₈₃	—	2	1	no	—
108	C₂²×D₈₃	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×C₈₃	—	2	1	no	—
110	C₂×C₄×C₈₃	—	2	1	no	—
111	C₂×C₄×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	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
117	C₂²⋊D₈₃	—	4	1	no	—
118	C₂²⋊D₈₃	—	8	1	no	—
119	C₂²⋊D₈₃	—	8	1	no	—
120	C₂×Dic₈₃	—	2	1	no	—
121	C₂×Dic₈₃	—	2	1	no	—
122	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
123	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
124	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
125	D₃₃₂	—	2	1	no	—
126	D₃₃₂	—	2	1	no	—
127	D₃₃₂	—	2	1	no	—
128	D₃₃₂	—	2	1	no	—
129	D₃₃₂	—	4	2	no	—
130	D₃₃₂	—	8	1	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	Dic₁₆₆	—	8	1	no	—
137	C₈×C₈₃	—	2	1	no	—
138	C₈×C₈₃	#41	1	2	yes	—
139	C₈₃⋊C₈	—	2	1	no	—
140	C₈₃⋊C₈	—	2	1	no	—
141	C₂²×C₈₃	—	2	1	no	—
142	C₂²×C₈₃	—	2	1	no	—
143	C₂²×C₈₃	—	2	1	no	—
144	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
145	C₂×D₈₃	—	2	1	no	—
146	C₂×D₈₃	—	2	1	no	—
147	C₂×D₈₃	—	2	1	no	—
148	C₂×D₈₃	—	4	1	no	—
149	C₂×D₈₃	—	4	1	no	—
150	C₂×D₈₃	—	4	1	no	—
151	C₂×D₈₃	—	4	1	no	—
152	C₂×D₈₃	—	4	1	no	—
153	C₂×D₈₃	—	4	1	no	—
154	C₄×C₈₃	—	2	1	no	—
155	C₄×C₈₃	—	4	1	no	—
156	C₄×C₈₃	—	4	1	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	2₋¹⁺⁴	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	C₂×C₈₃	—	4	1	no	—
167	C₂×C₈₃	#547	1	1	yes	characteristic
168	C₂×C₈₃	#575	1	1	yes	characteristic
169	C₂×C₈₃	#576	1	1	yes	characteristic
170	D₈₃	—	4	1	no	—
171	D₈₃	—	8	1	no	—
172	D₈₃	—	8	1	no	—
173	#3347	—	83	1	no	maximal, Sylow
174	C₈₃	#3347	1	1	yes	characteristic, complemented, Sylow
175	#555	—	83	1	no	—
176	#553	—	83	1	no	—
177	#501	—	83	1	no	—
178	#482	—	83	1	no	—
179	#482	—	83	1	no	—
180	#482	—	83	1	no	—
181	#480	—	83	1	no	—
182	#478	—	83	1	no	—
183	#474	—	83	1	no	—
184	#469	—	83	1	no	—
185	#466	—	83	1	no	—
186	#442	—	83	1	no	—
187	#435	—	83	1	no	—
188	#428	—	83	1	no	—
189	#427	D₈₃	1	1	yes	characteristic, complemented
190	C₂×D₈	—	83	1	no	—
191	C₂×M₁₆	—	83	1	no	—
192	C₂²×C₈	C₂×D₈₃	1	1	yes	characteristic
193	C₄⋊Q₈	—	83	1	no	—
194	#126	—	166	1	no	—
195	#125	—	83	1	no	—
196	#124	—	166	1	no	—
197	#124	—	166	1	no	—
198	#123	—	166	1	no	—
199	#121	—	83	1	no	—
200	#121	—	83	1	no	—
201	#121	—	83	1	no	—
202	#121	—	83	1	no	—
203	#121	—	166	1	no	—
204	#121	—	166	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	#117	—	83	1	no	—
210	#117	—	166	1	no	—
211	#117	C₂×D₈₃	1	1	yes	characteristic
212	#117	C₂×D₈₃	1	1	yes	characteristic
213	C₂×C₄⋊C₄	—	83	1	no	—
214	#107	—	83	1	no	—
215	#107	—	83	1	no	—
216	#107	—	83	1	no	—
217	#107	C₂×D₈₃	1	1	yes	characteristic
218	#107	C₂×D₈₃	1	1	yes	characteristic
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	C₄⋊C₈	—	83	1	no	—
224	#102	—	83	1	no	—
225	#102	—	83	1	no	—
226	#102	—	83	1	no	—
227	#102	—	83	1	no	—
228	#102	—	83	1	no	—
229	#102	—	83	1	no	—
230	C₂²⋊C₈	—	83	1	no	—
231	C₂²⋊C₈	—	83	1	no	—
232	#97	—	83	1	no	—
233	C₂×Q₈	—	166	1	no	—
234	C₂×D₄	—	83	1	no	—
235	C₂×D₄	—	83	1	no	—
236	C₂×D₄	—	83	1	no	—
237	C₂×D₄	—	166	1	no	—
238	C₂×D₄	—	166	1	no	—
239	C₂×D₄	—	166	1	no	—
240	C₂×D₄	—	166	1	no	—
241	C₂²×C₄	—	83	1	no	—
242	C₂²×C₄	—	166	1	no	—
243	C₂²×C₄	—	166	1	no	—
244	C₂²×C₄	—	166	1	no	—
245	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
246	D₈	—	166	2	no	—
247	M₁₆	—	166	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	83	1	no	—
250	C₂×C₈	—	83	1	no	—
251	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
252	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
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	C₂²×D₈₃	1	1	yes	characteristic
261	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
262	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
263	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
264	C₂²⋊C₄	—	2	1	no	—
265	C₂²⋊C₄	—	2	1	no	—
266	C₂²⋊C₄	—	83	1	no	—
267	C₂²⋊C₄	—	83	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	—	2	1	no	—
276	C₄²	—	2	1	no	—
277	C₄²	—	83	1	no	—
278	C₄²	—	166	1	no	—
279	C₂³	—	83	1	no	—
280	C₂³	—	166	1	no	—
281	C₂³	—	166	1	no	—
282	C₂³	D₄×D₈₃	1	1	yes	characteristic
283	Q₈	—	664	1	no	—
284	D₄	—	332	2	no	—
285	D₄	—	332	1	no	—
286	D₄	—	664	1	no	—
287	D₄	—	664	1	no	—
288	D₄	—	664	1	no	—
289	D₄	—	166	1	no	—
290	D₄	—	166	1	no	—
291	D₄	—	166	1	no	—
292	D₄	—	166	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	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	332	2	no	—
300	C₂×C₄	—	83	1	no	—
301	C₂×C₄	—	83	1	no	—
302	C₂×C₄	—	83	1	no	—
303	C₂×C₄	—	664	1	no	—
304	C₂×C₄	—	664	1	no	—
305	C₂×C₄	—	166	1	no	—
306	C₂×C₄	—	166	1	no	—
307	C₂×C₄	—	166	1	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	—	166	1	no	—
313	C₈	—	166	1	no	—
314	C₈	#13050590	1	2	yes	—
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	—	332	1	no	—
321	C₂²	—	332	1	no	—
322	C₂²	—	332	1	no	—
323	C₂²	—	332	1	no	—
324	C₂²	—	166	1	no	—
325	C₂²	—	166	1	no	—
326	C₂²	—	166	1	no	—
327	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
328	C₄	—	2	1	no	—
329	C₄	—	4	1	no	—
330	C₄	—	4	1	no	—
331	C₄	—	4	1	no	—
332	C₄	—	4	1	no	—
333	C₄	—	332	1	no	—
334	C₄	—	332	1	no	—
335	C₄	—	166	1	no	—
336	C₄	—	166	1	no	—
337	C₄	—	166	1	no	—
338	C₄	C₂×#13050590	1	1	yes	characteristic
339	C₄	#14581840	1	1	yes	characteristic
340	C₂	—	4	1	no	—
341	C₂	—	332	1	no	—
342	C₂	—	664	1	no	—
343	C₂	—	664	1	no	—
344	C₂	#19128352	1	1	yes	characteristic
345	C₂	#19128498	1	1	yes	characteristic
346	C₂	#19128503	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₃
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#427×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
hypercenter	#427	D₈₃
hypocenter	C₈₃	#3347
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #427

Complements

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

N	H ≅ G/N	type
1. #19128414	C₂	semidirect
2. #19128395	C₂	semidirect
3. #19128108	C₂	semidirect
4. #19128028	C₂	semidirect
5. #19128025	C₂	semidirect
6. #19128025	C₂	semidirect
7. #19128016	C₂	semidirect
8. #19128004	C₂	semidirect
9. #19127980	C₂	semidirect
10. #19127955	C₂	semidirect
11. #19127937	C₂	semidirect
12. #19127812	C₂	semidirect
13. #19127772	C₂	semidirect
14. #19127740	C₂	semidirect
15. #427×C₈₃	C₂	semidirect
174. C₈₃	#3347	semidirect
189. #427	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128414	2
#19128395	2
#19128108	2
#19128028	2
#19128025	2
#19128016	2
#19128004	2
#19127980	2
#19127955	2
#19127937	2
#19127812	2
#19127772	2
#19127740	2
#427×C₈₃	2
#3347	83