Subgroups — #24919519

Order 10624.

376 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128392	C₂	1	1	yes	maximal, characteristic
2	Q₈×D₃₃₂	C₂	1	1	yes	maximal, characteristic
3	#19128070	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128045	C₂	1	1	yes	maximal, characteristic, complemented
5	#19128033	C₂	1	1	yes	maximal, characteristic, complemented
6	#19128008	C₂	1	1	yes	maximal, characteristic, complemented
7	#19127958	C₂	1	1	yes	maximal, characteristic, complemented
8	#19127941	C₂	1	1	yes	maximal, characteristic, complemented
9	#19127911	C₂	1	1	yes	maximal, characteristic, complemented
10	#19127855	C₂	1	1	yes	maximal, characteristic, complemented
11	#19127855	C₂	1	1	yes	maximal, characteristic, complemented
12	#19127774	C₂	1	1	yes	maximal, characteristic
13	#19127708	C₂	1	1	yes	maximal, characteristic, complemented
14	#103×D₈₃	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×Q₈×D₈₃	C₂²	1	1	yes	characteristic
18	C₂×#13050580	C₂²	1	1	yes	characteristic
19	C₂×#13050580	C₂²	1	1	yes	characteristic
20	C₂×#13050573	C₂²	1	1	yes	characteristic
21	C₂×C₈×D₈₃	C₂²	1	1	yes	characteristic
22	#14581792	—	2	1	no	—
23	#14581792	C₂²	1	1	yes	characteristic
24	#14581786	—	2	1	no	—
25	#14581784	C₂²	1	1	yes	characteristic
26	#14581782	—	2	1	no	—
27	#14581777	—	2	1	no	—
28	#14581772	—	2	1	no	—
29	#14581772	—	2	1	no	—
30	#14581772	C₂²	1	1	yes	characteristic
31	#14581772	C₂²	1	1	yes	characteristic
32	#14581768	C₂²	1	1	yes	characteristic
33	#14581768	C₂²	1	1	yes	characteristic
34	C₄×D₃₃₂	—	2	1	no	—
35	C₄×D₃₃₂	C₂²	1	1	yes	characteristic
36	#14581750	C₂²	1	1	yes	characteristic
37	#14581749	—	2	1	no	—
38	C₄⋊C₄×D₈₃	C₂²	1	1	yes	characteristic
39	#14581725	C₂²	1	1	yes	characteristic
40	#14581723	C₂²	1	1	yes	characteristic
41	C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
42	#14581717	C₂²	1	1	yes	characteristic
43	#14581717	C₂²	1	1	yes	characteristic
44	#14581717	C₂²	1	1	yes	characteristic
45	#14581717	C₂²	1	1	yes	characteristic
46	#14581716	C₂²	1	1	yes	characteristic
47	#14581716	C₂²	1	1	yes	characteristic
48	#14581715	C₂²	1	1	yes	characteristic
49	#14581715	C₂²	1	1	yes	characteristic
50	(C₂×D₈₃)⋊C₈	C₂²	1	1	yes	characteristic
51	(C₂×D₈₃)⋊C₈	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	—	2	1	no	—
61	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
62	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
63	Q₈×D₈₃	—	2	2	no	—
64	Q₈×D₈₃	—	2	2	no	—
65	Q₈×D₈₃	—	4	1	no	—
66	Q₈×D₈₃	—	4	1	no	—
67	C₂×D₃₃₂	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	C₂×C₄×D₈₃	C₂³	1	1	yes	characteristic
72	C₂×C₄×D₈₃	C₂³	1	1	yes	characteristic
73	#13050580	—	4	1	no	—
74	#13050580	—	4	1	no	—
75	#13050573	—	4	1	no	—
76	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
77	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
78	C₈×D₈₃	—	4	1	no	—
79	C₄⋊Dic₈₃	—	2	1	no	—
80	C₄⋊Dic₈₃	—	2	1	no	—
81	C₄⋊Dic₈₃	—	2	1	no	—
82	C₄⋊Dic₈₃	—	2	1	no	—
83	C₄⋊Dic₈₃	D₄	1	1	yes	characteristic
84	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
85	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
86	Dic₈₃⋊C₄	—	2	1	no	—
87	Dic₈₃⋊C₄	—	2	1	no	—
88	Dic₈₃⋊C₄	—	2	1	no	—
89	Dic₈₃⋊C₄	—	2	1	no	—
90	Dic₈₃⋊C₄	—	2	1	no	—
91	(C₂×D₈₃)⋊C₄	—	2	1	no	—
92	(C₂×D₈₃)⋊C₄	—	2	1	no	—
93	(C₂×D₈₃)⋊C₄	—	2	1	no	—
94	(C₂×D₈₃)⋊C₄	—	2	1	no	—
95	(C₂×D₈₃)⋊C₄	—	2	1	no	—
96	(C₂×D₈₃)⋊C₄	D₄	1	1	yes	characteristic
97	(C₂×D₈₃)⋊C₄	D₄	1	1	yes	characteristic
98	C₄×Dic₈₃	—	2	1	no	—
99	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
100	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
101	Q₁₆×C₈₃	—	2	2	no	—
102	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
103	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
104	C₄⋊C₄×C₈₃	—	2	1	no	—
105	C₄⋊C₄×C₈₃	—	2	1	no	—
106	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
107	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
108	C₄²×C₈₃	—	2	1	no	—
109	C₄²×C₈₃	—	2	1	no	—
110	C₄²×C₈₃	C₂³	1	1	yes	characteristic
111	C₂²×D₈₃	C₂×D₄	1	1	yes	characteristic
112	C₂²×D₈₃	C₂×D₄	1	1	yes	characteristic
113	Q₈×C₈₃	—	2	1	no	—
114	Q₈×C₈₃	—	2	1	no	—
115	Q₈×C₈₃	—	2	1	no	—
116	Q₈×C₈₃	#41	1	2	yes	—
117	C₂×C₄×C₈₃	—	2	1	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	C₂×D₄	1	1	yes	characteristic
122	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
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	—	2	1	no	—
134	C₄×D₈₃	—	2	1	no	—
135	C₄×D₈₃	—	4	2	no	—
136	C₄×D₈₃	—	4	2	no	—
137	C₄×D₈₃	—	4	2	no	—
138	Dic₁₆₆	—	2	1	no	—
139	Dic₁₆₆	—	2	1	no	—
140	Dic₁₆₆	—	2	1	no	—
141	Dic₁₆₆	—	2	1	no	—
142	Dic₁₆₆	—	4	2	no	—
143	Dic₁₆₆	—	4	2	no	—
144	C₈×C₈₃	—	2	1	no	—
145	C₈×C₈₃	—	2	1	no	—
146	C₈₃⋊C₈	—	2	1	no	—
147	C₈₃⋊C₈	—	2	1	no	—
148	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
149	C₂×D₈₃	—	2	1	no	—
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	#137	1	2	yes	—
155	C₄×C₈₃	—	2	2	no	—
156	C₄×C₈₃	—	2	1	no	—
157	C₄×C₈₃	—	2	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	H(5,𝔽₂)	1	1	yes	characteristic
163	Dic₈₃	—	2	1	no	—
164	Dic₈₃	—	2	1	no	—
165	Dic₈₃	—	4	1	no	—
166	Dic₈₃	—	4	1	no	—
167	Dic₈₃	—	4	1	no	—
168	Dic₈₃	—	4	1	no	—
169	C₂×C₈₃	#546	1	1	yes	characteristic
170	C₂×C₈₃	C₂×#137	1	1	yes	characteristic
171	C₂×C₈₃	#578	1	1	yes	characteristic
172	D₈₃	—	2	2	no	—
173	D₈₃	—	4	1	no	—
174	#3312	—	83	1	no	maximal, Sylow
175	C₈₃	#3312	1	1	yes	characteristic, complemented, Sylow
176	#552	—	83	1	no	—
177	D₄×Q₈	—	83	1	no	—
178	#491	—	83	1	no	—
179	#485	—	83	1	no	—
180	#483	—	83	1	no	—
181	#479	—	83	1	no	—
182	#470	—	83	1	no	—
183	#467	—	83	1	no	—
184	#462	—	83	1	no	—
185	#451	—	83	1	no	—
186	#451	—	83	1	no	—
187	#441	D₈₃	1	1	yes	characteristic, complemented
188	#435	—	83	1	no	—
189	#419	—	83	1	no	—
190	C₂×#103	—	83	1	no	—
191	C₂²×Q₈	—	83	1	no	—
192	C₂²×Q₈	—	83	1	no	—
193	C₂×Q₁₆	—	83	1	no	—
194	C₂×Q₁₆	—	83	1	no	—
195	C₂×Q₁₆	C₂×D₈₃	1	1	yes	characteristic
196	C₂×M₁₆	—	83	1	no	—
197	C₂²×C₈	—	83	1	no	—
198	C₄⋊Q₈	—	83	1	no	—
199	C₄⋊Q₈	—	166	1	no	—
200	#126	—	166	1	no	—
201	#125	—	83	1	no	—
202	#125	—	166	1	no	—
203	#123	—	166	1	no	—
204	C₂²⋊Q₈	—	83	1	no	—
205	C₂²⋊Q₈	—	83	1	no	—
206	C₂²⋊Q₈	—	83	1	no	—
207	C₂²⋊Q₈	—	83	1	no	—
208	C₂²⋊Q₈	—	166	1	no	—
209	C₂²⋊Q₈	—	166	1	no	—
210	C₄×Q₈	C₂×D₈₃	1	1	yes	characteristic
211	C₄×Q₈	C₂×D₈₃	1	1	yes	characteristic
212	C₄×D₄	—	83	1	no	—
213	C₄×D₄	—	166	1	no	—
214	#117	—	83	1	no	—
215	#117	—	166	1	no	—
216	C₂×C₄⋊C₄	—	83	1	no	—
217	#107	—	83	1	no	—
218	#106	—	83	1	no	—
219	#106	C₂×D₈₃	1	1	yes	characteristic
220	C₄⋊C₈	—	83	1	no	—
221	#103	—	83	1	no	—
222	#103	—	83	1	no	—
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	#97	C₂×D₈₃	1	1	yes	characteristic
234	C₂×Q₈	—	332	1	no	—
235	C₂×Q₈	—	332	1	no	—
236	C₂×Q₈	—	83	1	no	—
237	C₂×Q₈	—	83	1	no	—
238	C₂×Q₈	—	166	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×Q₈	C₂²×D₈₃	1	1	yes	characteristic
244	C₂×D₄	—	83	1	no	—
245	C₂²×C₄	—	83	1	no	—
246	C₂²×C₄	—	83	1	no	—
247	C₂²×C₄	—	166	1	no	—
248	C₂²×C₄	—	166	1	no	—
249	C₂²×C₄	—	166	1	no	—
250	Q₁₆	—	2	2	no	—
251	Q₁₆	—	332	1	no	—
252	Q₁₆	—	332	1	no	—
253	M₁₆	—	332	1	no	—
254	C₂×C₈	—	332	1	no	—
255	C₂×C₈	—	83	1	no	—
256	C₂×C₈	—	83	1	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	—	166	1	no	—
270	C₄⋊C₄	—	166	1	no	—
271	C₄⋊C₄	—	166	1	no	—
272	C₄⋊C₄	—	166	1	no	—
273	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
274	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
275	C₂²⋊C₄	—	83	1	no	—
276	C₂²⋊C₄	—	83	1	no	—
277	C₂²⋊C₄	—	166	1	no	—
278	C₂²⋊C₄	—	166	1	no	—
279	C₂²⋊C₄	—	166	1	no	—
280	C₂²⋊C₄	—	166	1	no	—
281	C₂²⋊C₄	—	166	1	no	—
282	C₄²	—	2	1	no	—
283	C₄²	—	2	1	no	—
284	C₄²	—	166	1	no	—
285	C₄²	C₂²×D₈₃	1	1	yes	characteristic
286	C₂³	—	83	1	no	—
287	C₂³	—	83	1	no	—
288	Q₈	—	2	1	no	—
289	Q₈	—	2	1	no	—
290	Q₈	—	2	1	no	—
291	Q₈	—	332	2	no	—
292	Q₈	—	332	2	no	—
293	Q₈	—	166	1	no	—
294	Q₈	—	166	1	no	—
295	Q₈	—	166	1	no	—
296	Q₈	—	166	1	no	—
297	Q₈	#13050590	1	2	yes	—
298	D₄	—	166	2	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	332	2	no	—
304	C₂×C₄	—	332	2	no	—
305	C₂×C₄	—	332	2	no	—
306	C₂×C₄	—	83	1	no	—
307	C₂×C₄	—	83	1	no	—
308	C₂×C₄	—	166	1	no	—
309	C₂×C₄	—	166	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	D₄×D₈₃	1	1	yes	characteristic
318	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
319	C₈	—	2	1	no	—
320	C₈	—	2	1	no	—
321	C₈	—	166	1	no	—
322	C₈	—	166	1	no	—
323	C₂²	—	83	2	no	—
324	C₂²	—	166	1	no	—
325	C₂²	—	166	1	no	—
326	C₂²	—	166	1	no	—
327	C₂²	—	166	1	no	—
328	C₂²	—	166	1	no	—
329	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
330	C₄	—	2	2	no	—
331	C₄	—	2	1	no	—
332	C₄	—	2	1	no	—
333	C₄	—	4	1	no	—
334	C₄	—	4	1	no	—
335	C₄	—	4	1	no	—
336	C₄	—	332	1	no	—
337	C₄	—	332	1	no	—
338	C₄	—	332	1	no	—
339	C₄	—	332	1	no	—
340	C₄	—	166	1	no	—
341	C₄	—	166	1	no	—
342	C₄	C₂×#13050590	1	1	yes	characteristic
343	C₄	#14581838	1	1	yes	characteristic
344	C₂	—	332	1	no	—
345	C₂	—	166	2	no	—
346	C₂	#19128347	1	1	yes	characteristic
347	C₂	#137×D₈₃	1	1	yes	characteristic
348	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	#441×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	C₂×D₃₃₂	D₄
hypercenter	#441	D₈₃
hypocenter	C₈₃	#3312
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
3. #19128070	C₂	semidirect
4. #19128045	C₂	semidirect
5. #19128033	C₂	semidirect
6. #19128008	C₂	semidirect
7. #19127958	C₂	semidirect
8. #19127941	C₂	semidirect
9. #19127911	C₂	semidirect
10. #19127855	C₂	semidirect
11. #19127855	C₂	semidirect
13. #19127708	C₂	semidirect
14. #103×D₈₃	C₂	semidirect
15. #441×C₈₃	C₂	semidirect
175. C₈₃	#3312	semidirect
187. #441	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128392	2
Q₈×D₃₃₂	2
#19128070	2
#19128045	2
#19128033	2
#19128008	2
#19127958	2
#19127941	2
#19127911	2
#19127855	2
#19127774	2
#19127708	2
#103×D₈₃	2
#441×C₈₃	2
#3312	83