Subgroups — #24917638

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	#19128238	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₈×D₈₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#19128035	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128021	C₂	1	1	yes	maximal, characteristic
5	#19128011	C₂	1	1	yes	maximal, characteristic, complemented
6	#19128010	C₂	1	1	yes	maximal, characteristic, complemented
7	#19127926	C₂	1	1	yes	maximal, characteristic, complemented
8	#19127925	C₂	1	1	yes	maximal, characteristic, complemented
9	#19127855	C₂	1	1	yes	maximal, characteristic, complemented
10	#19127841	C₂	1	1	yes	maximal, characteristic
11	#19127806	C₂	1	1	yes	maximal, characteristic, complemented
12	#19127803	C₂	1	1	yes	maximal, characteristic, complemented
13	#19127799	C₂	1	1	yes	maximal, characteristic, complemented
14	#19127756	C₂	1	1	yes	maximal, characteristic, complemented
15	#441×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₈₃	C₂²	1	1	yes	characteristic
17	C₂×#13050580	C₂²	1	1	yes	characteristic
18	C₂×#13050579	C₂²	1	1	yes	characteristic
19	C₂×#13050577	C₂²	1	1	yes	characteristic
20	#14581787	—	2	1	no	—
21	#14581783	C₂²	1	1	yes	characteristic
22	#14581781	C₂²	1	1	yes	characteristic
23	#14581777	—	2	1	no	—
24	#14581768	C₂²	1	1	yes	characteristic
25	#14581767	C₂²	1	1	yes	characteristic
26	Q₈×Dic₈₃	C₂²	1	1	yes	characteristic
27	C₄×Dic₁₆₆	—	2	1	no	—
28	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
29	#14581757	—	2	1	no	—
30	#14581757	C₂²	1	1	yes	characteristic
31	C₄×D₃₃₂	—	2	1	no	—
32	C₄×D₃₃₂	C₂²	1	1	yes	characteristic
33	#14581749	—	2	1	no	—
34	C₄⋊C₄×D₈₃	—	2	1	no	—
35	C₄⋊C₄×D₈₃	C₂²	1	1	yes	characteristic
36	C₄²×D₈₃	—	2	1	no	—
37	C₄²×D₈₃	C₂²	1	1	yes	characteristic
38	#14581726	C₂²	1	1	yes	characteristic
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	#14581716	C₂²	1	1	yes	characteristic
45	#14581716	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	(C₂×D₈₃)⋊C₈	C₂²	1	1	yes	characteristic
50	(C₂×D₈₃)⋊C₈	C₂²	1	1	yes	characteristic
51	#14581703	C₂²	1	1	yes	characteristic
52	C₂×Q₁₆×C₈₃	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₈₃	C₂²	1	1	yes	characteristic
54	C₄×Q₈×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	#97×C₈₃	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₆₆	—	2	1	no	—
60	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
61	Q₈×D₈₃	—	2	2	no	—
62	Q₈×D₈₃	—	2	2	no	—
63	C₂×D₃₃₂	—	2	1	no	—
64	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
65	C₂×C₄×D₈₃	—	2	1	no	—
66	C₂×C₄×D₈₃	—	2	1	no	—
67	C₂×C₄×D₈₃	—	2	1	no	—
68	C₂×C₄×D₈₃	D₄	1	1	yes	characteristic
69	C₂×C₄×D₈₃	D₄	1	1	yes	characteristic
70	C₂×C₄×D₈₃	C₂³	1	1	yes	characteristic
71	#13050580	—	2	2	no	—
72	#13050579	—	2	2	no	—
73	#13050577	—	2	2	no	—
74	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
75	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
76	C₄⋊Dic₈₃	—	2	1	no	—
77	C₄⋊Dic₈₃	—	2	1	no	—
78	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
79	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
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	(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	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	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
97	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
98	Q₁₆×C₈₃	—	2	2	no	—
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	—	2	1	no	—
103	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
104	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
105	C₄²×C₈₃	—	2	1	no	—
106	C₄²×C₈₃	—	2	1	no	—
107	C₄²×C₈₃	C₂³	1	1	yes	characteristic
108	C₂²×D₈₃	—	2	1	no	—
109	C₂²×D₈₃	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	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	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
121	C₂×Dic₈₃	—	2	1	no	—
122	C₂×Dic₈₃	—	2	1	no	—
123	C₂×Dic₈₃	—	2	1	no	—
124	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
125	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
126	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
127	D₃₃₂	—	2	1	no	—
128	D₃₃₂	—	2	1	no	—
129	D₃₃₂	—	4	2	no	—
130	C₄×D₈₃	—	2	4	no	—
131	C₄×D₈₃	—	2	2	no	—
132	C₄×D₈₃	—	2	1	no	—
133	C₄×D₈₃	—	2	1	no	—
134	C₄×D₈₃	—	4	1	no	—
135	C₄×D₈₃	—	4	2	no	—
136	C₄×D₈₃	—	4	2	no	—
137	Dic₁₆₆	—	2	4	no	—
138	Dic₁₆₆	—	2	1	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	C₂²×D₄	1	1	yes	characteristic
145	C₂×D₈₃	—	2	1	no	—
146	C₂×D₈₃	—	2	1	no	—
147	C₂×D₈₃	—	4	1	no	—
148	C₂×D₈₃	—	4	1	no	—
149	C₂×D₈₃	—	4	1	no	—
150	C₂×D₈₃	#137	1	2	yes	—
151	C₄×C₈₃	—	2	2	no	—
152	C₄×C₈₃	—	2	1	no	—
153	C₄×C₈₃	—	2	1	no	—
154	C₄×C₈₃	—	4	1	no	—
155	C₄×C₈₃	—	4	1	no	—
156	C₄×C₈₃	—	4	1	no	—
157	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
158	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
159	Dic₈₃	—	2	1	no	—
160	Dic₈₃	—	2	2	no	—
161	Dic₈₃	—	2	1	no	—
162	Dic₈₃	—	2	1	no	—
163	Dic₈₃	—	4	1	no	—
164	Dic₈₃	—	4	1	no	—
165	C₂×C₈₃	#525	1	1	yes	characteristic
166	C₂×C₈₃	C₂×#137	1	1	yes	characteristic
167	C₂×C₈₃	#575	1	1	yes	characteristic
168	D₈₃	—	2	2	no	—
169	D₈₃	—	8	1	no	—
170	#3117	—	83	1	no	maximal, Sylow
171	C₈₃	#3117	1	1	yes	characteristic, complemented, Sylow
172	#529	—	83	1	no	—
173	C₂×C₄×Q₈	—	83	1	no	—
174	#484	—	83	1	no	—
175	#481	—	83	1	no	—
176	#479	—	83	1	no	—
177	#479	—	83	1	no	—
178	#464	—	83	1	no	—
179	#464	—	83	1	no	—
180	#451	—	83	1	no	—
181	#448	—	83	1	no	—
182	#441	—	83	1	no	—
183	#441	D₈₃	1	1	yes	characteristic, complemented
184	#440	—	83	1	no	—
185	#440	—	83	1	no	—
186	#432	—	83	1	no	—
187	C₂²×Q₈	—	83	1	no	—
188	C₂×Q₁₆	—	83	1	no	—
189	C₂×Q₁₆	C₂×D₈₃	1	1	yes	characteristic
190	C₂×SD₁₆	—	83	1	no	—
191	C₂×SD₁₆	—	83	1	no	—
192	#126	—	166	1	no	—
193	#125	—	83	1	no	—
194	#124	—	83	1	no	—
195	#123	—	166	1	no	—
196	C₂²⋊Q₈	—	83	1	no	—
197	#121	—	83	1	no	—
198	C₄×Q₈	—	83	1	no	—
199	C₄×Q₈	—	83	1	no	—
200	C₄×Q₈	—	83	1	no	—
201	C₄×Q₈	—	166	1	no	—
202	C₄×Q₈	—	166	1	no	—
203	C₄×Q₈	C₂×D₈₃	1	1	yes	characteristic
204	C₄×Q₈	C₂×D₈₃	1	1	yes	characteristic
205	C₄×D₄	—	83	1	no	—
206	C₄×D₄	—	166	1	no	—
207	#117	—	166	1	no	—
208	C₂×C₄⋊C₄	—	83	1	no	—
209	C₂×C₄⋊C₄	—	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	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	—	83	1	no	—
228	#97	—	83	1	no	—
229	#97	C₂×D₈₃	1	1	yes	characteristic
230	C₂×Q₈	—	83	1	no	—
231	C₂×Q₈	—	166	1	no	—
232	C₂×Q₈	—	166	2	no	—
233	C₂×Q₈	—	166	2	no	—
234	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
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	—	83	1	no	—
241	C₂²×C₄	—	166	1	no	—
242	C₂²×C₄	—	166	1	no	—
243	C₂²×C₄	—	166	1	no	—
244	Q₁₆	—	2	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	—	2	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	C₂²×D₈₃	1	1	yes	characteristic
264	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
265	C₂²⋊C₄	—	166	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	—	2	1	no	—
271	C₄²	—	2	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	C₂²×D₈₃	1	1	yes	characteristic
279	C₂³	—	83	1	no	—
280	C₂³	—	166	1	no	—
281	Q₈	—	2	1	no	—
282	Q₈	—	2	1	no	—
283	Q₈	—	2	1	no	—
284	Q₈	—	83	2	no	—
285	Q₈	—	166	4	no	—
286	Q₈	—	166	1	no	—
287	Q₈	#13050590	1	2	yes	—
288	D₄	—	332	2	no	—
289	D₄	—	166	1	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	—	332	2	no	—
296	C₂×C₄	—	332	1	no	—
297	C₂×C₄	—	332	2	no	—
298	C₂×C₄	—	83	1	no	—
299	C₂×C₄	—	83	1	no	—
300	C₂×C₄	—	83	1	no	—
301	C₂×C₄	—	166	2	no	—
302	C₂×C₄	—	166	4	no	—
303	C₂×C₄	—	166	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	D₄×D₈₃	1	1	yes	characteristic
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	—	332	1	no	—
316	C₂²	—	332	1	no	—
317	C₂²	—	332	1	no	—
318	C₂²	—	83	2	no	—
319	C₂²	—	166	1	no	—
320	C₂²	—	166	1	no	—
321	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
322	C₄	—	2	2	no	—
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	4	1	no	—
326	C₄	—	4	1	no	—
327	C₄	—	4	1	no	—
328	C₄	—	332	1	no	—
329	C₄	—	332	1	no	—
330	C₄	—	166	1	no	—
331	C₄	—	166	2	no	—
332	C₄	—	166	1	no	—
333	C₄	—	166	1	no	—
334	C₄	#41×D₈₃	1	1	yes	characteristic
335	C₄	C₂×#13050590	1	1	yes	characteristic
336	C₂	—	664	1	no	—
337	C₂	—	166	2	no	—
338	C₂	#19128210	1	1	yes	characteristic
339	C₂	#137×D₈₃	1	1	yes	characteristic
340	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	#441×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	#14581767	C₂²
hypercenter	#441	D₈₃
hypocenter	C₈₃	#3117
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #441

Complements

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

N	H ≅ G/N	type
2. C₄×Q₈×D₈₃	C₂	semidirect
3. #19128035	C₂	semidirect
5. #19128011	C₂	semidirect
6. #19128010	C₂	semidirect
7. #19127926	C₂	semidirect
8. #19127925	C₂	semidirect
9. #19127855	C₂	semidirect
11. #19127806	C₂	semidirect
12. #19127803	C₂	semidirect
13. #19127799	C₂	semidirect
14. #19127756	C₂	semidirect
15. #441×C₈₃	C₂	semidirect
171. C₈₃	#3117	semidirect
183. #441	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128238	2
C₄×Q₈×D₈₃	2
#19128035	2
#19128021	2
#19128011	2
#19128010	2
#19127926	2
#19127925	2
#19127855	2
#19127841	2
#19127806	2
#19127803	2
#19127799	2
#19127756	2
#441×C₈₃	2
#3117	83