Subgroups — #24919919

Order 10624.

344 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128406	C₂	1	1	yes	maximal, characteristic
2	#19128390	C₂	1	1	yes	maximal, characteristic
3	#19128109	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128042	C₂	1	1	yes	maximal, characteristic, complemented
5	#19128038	C₂	1	1	yes	maximal, characteristic, complemented
6	#19128033	C₂	1	1	yes	maximal, characteristic, complemented
7	#19128031	C₂	1	1	yes	maximal, characteristic, complemented
8	#19128007	C₂	1	1	yes	maximal, characteristic, complemented
9	#19127985	C₂	1	1	yes	maximal, characteristic, complemented
10	#19127959	C₂	1	1	yes	maximal, characteristic, complemented
11	#19127943	C₂	1	1	yes	maximal, characteristic, complemented
12	#19127807	C₂	1	1	yes	maximal, characteristic, complemented
13	#19127770	C₂	1	1	yes	maximal, characteristic
14	#106×D₈₃	C₂	1	1	yes	maximal, characteristic, complemented
15	#428×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Dic₃₃₂	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₈₃	C₂²	1	1	yes	characteristic
18	#14581791	C₂²	1	1	yes	characteristic
19	#14581785	—	2	1	no	—
20	#14581784	—	2	1	no	—
21	#14581783	C₂²	1	1	yes	characteristic
22	#14581782	—	2	1	no	—
23	#14581774	—	2	1	no	—
24	#14581772	C₂²	1	1	yes	characteristic
25	#14581772	C₂²	1	1	yes	characteristic
26	#14581771	—	2	1	no	—
27	#14581771	C₂²	1	1	yes	characteristic
28	#14581771	C₂²	1	1	yes	characteristic
29	#14581770	—	2	1	no	—
30	#14581757	C₂²	1	1	yes	characteristic
31	#14581757	C₂²	1	1	yes	characteristic
32	C₄×C₂²⋊D₈₃	—	2	1	no	—
33	C₄×C₂²⋊D₈₃	C₂²	1	1	yes	characteristic
34	#14581749	—	2	1	no	—
35	C₄⋊C₄×D₈₃	C₂²	1	1	yes	characteristic
36	C₄⋊C₄×D₈₃	C₂²	1	1	yes	characteristic
37	#14581725	C₂²	1	1	yes	characteristic
38	#14581724	C₂²	1	1	yes	characteristic
39	#14581723	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	#14581717	C₂²	1	1	yes	characteristic
44	#14581717	C₂²	1	1	yes	characteristic
45	#14581715	C₂²	1	1	yes	characteristic
46	#14581715	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	#14581702	C₂²	1	1	yes	characteristic
52	C₂×M₁₆×C₈₃	C₂²	1	1	yes	characteristic
53	#117×C₈₃	C₂²	1	1	yes	characteristic
54	C₂×C₄⋊C₄×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×Dic₁₆₆	C₂³	1	1	yes	characteristic
61	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₈₃	D₄	1	1	yes	characteristic
63	C₂×C₄×D₈₃	—	2	1	no	—
64	C₂×C₄×D₈₃	—	2	1	no	—
65	C₂×C₄×D₈₃	C₂³	1	1	yes	characteristic
66	Dic₃₃₂	—	2	2	no	—
67	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
68	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
69	C₈×D₈₃	—	2	2	no	—
70	C₄⋊Dic₈₃	—	2	1	no	—
71	C₄⋊Dic₈₃	—	2	1	no	—
72	C₄⋊Dic₈₃	—	2	1	no	—
73	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
74	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
75	Dic₈₃⋊C₄	—	2	1	no	—
76	Dic₈₃⋊C₄	—	2	1	no	—
77	Dic₈₃⋊C₄	—	2	1	no	—
78	Dic₈₃⋊C₄	—	2	1	no	—
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	D₄	1	1	yes	characteristic
84	C₂²⋊Dic₈₃	—	2	1	no	—
85	C₂²⋊Dic₈₃	—	2	1	no	—
86	C₂²⋊Dic₈₃	D₄	1	1	yes	characteristic
87	(C₂×D₈₃)⋊C₄	—	2	1	no	—
88	(C₂×D₈₃)⋊C₄	—	2	1	no	—
89	(C₂×D₈₃)⋊C₄	D₄	1	1	yes	characteristic
90	C₄×Dic₈₃	—	2	1	no	—
91	C₄×Dic₈₃	—	2	1	no	—
92	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
93	C₂²×C₄×C₈₃	—	2	1	no	—
94	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
95	M₁₆×C₈₃	—	2	2	no	—
96	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
97	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
98	C₄⋊C₄×C₈₃	—	2	1	no	—
99	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
100	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
101	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
103	C₂²⋊C₄×C₈₃	—	2	1	no	—
104	C₄²×C₈₃	—	2	1	no	—
105	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
106	C₂²×D₈₃	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×C₈₃	—	2	1	no	—
108	C₂×C₄×C₈₃	—	2	1	no	—
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	—	8	1	no	—
114	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
116	C₂²⋊D₈₃	—	2	2	no	—
117	C₂×Dic₈₃	—	2	1	no	—
118	C₂×Dic₈₃	—	2	1	no	—
119	C₂×Dic₈₃	—	2	1	no	—
120	C₂×Dic₈₃	—	2	1	no	—
121	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
122	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
123	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
124	C₄×D₈₃	—	2	1	no	—
125	C₄×D₈₃	—	2	1	no	—
126	C₄×D₈₃	—	4	2	no	—
127	C₄×D₈₃	—	8	1	no	—
128	Dic₁₆₆	—	2	1	no	—
129	Dic₁₆₆	—	2	1	no	—
130	Dic₁₆₆	—	2	1	no	—
131	Dic₁₆₆	—	2	1	no	—
132	Dic₁₆₆	—	8	1	no	—
133	C₈×C₈₃	—	2	1	no	—
134	C₈×C₈₃	#41	1	2	yes	—
135	C₈₃⋊C₈	—	2	1	no	—
136	C₈₃⋊C₈	—	2	1	no	—
137	C₂²×C₈₃	—	2	1	no	—
138	C₂²×C₈₃	—	2	1	no	—
139	C₂²×C₈₃	—	2	1	no	—
140	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
141	C₂×D₈₃	—	2	1	no	—
142	C₂×D₈₃	—	2	1	no	—
143	C₂×D₈₃	#137	1	2	yes	—
144	C₄×C₈₃	—	2	1	no	—
145	C₄×C₈₃	—	4	1	no	—
146	C₄×C₈₃	—	4	1	no	—
147	C₄×C₈₃	—	4	1	no	—
148	C₄×C₈₃	—	4	1	no	—
149	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
150	C₄×C₈₃	2₋¹⁺⁴	1	1	yes	characteristic
151	Dic₈₃	—	2	1	no	—
152	Dic₈₃	—	2	1	no	—
153	Dic₈₃	—	2	1	no	—
154	Dic₈₃	—	4	1	no	—
155	Dic₈₃	—	4	1	no	—
156	Dic₈₃	—	4	1	no	—
157	Dic₈₃	—	4	1	no	—
158	C₂×C₈₃	—	4	1	no	—
159	C₂×C₈₃	#547	1	1	yes	characteristic
160	C₂×C₈₃	C₂×#137	1	1	yes	characteristic
161	C₂×C₈₃	#577	1	1	yes	characteristic
162	D₈₃	—	2	2	no	—
163	#3344	—	83	1	no	maximal, Sylow
164	C₈₃	#3344	1	1	yes	characteristic, complemented, Sylow
165	#554	—	83	1	no	—
166	#552	—	83	1	no	—
167	#501	—	83	1	no	—
168	#485	—	83	1	no	—
169	#484	—	83	1	no	—
170	#483	—	83	1	no	—
171	#483	—	83	1	no	—
172	#479	—	83	1	no	—
173	#475	—	83	1	no	—
174	#470	—	83	1	no	—
175	#467	—	83	1	no	—
176	#441	—	83	1	no	—
177	#435	—	83	1	no	—
178	#428	D₈₃	1	1	yes	characteristic, complemented
179	C₂×#106	—	83	1	no	—
180	C₂×Q₁₆	—	83	1	no	—
181	C₂×M₁₆	C₂×D₈₃	1	1	yes	characteristic
182	C₂²×C₈	—	83	1	no	—
183	C₄⋊Q₈	—	83	1	no	—
184	#126	—	166	1	no	—
185	#125	—	83	1	no	—
186	#125	—	166	1	no	—
187	#125	—	166	1	no	—
188	#123	—	166	1	no	—
189	C₂²⋊Q₈	—	83	1	no	—
190	C₂²⋊Q₈	—	83	1	no	—
191	C₂²⋊Q₈	—	83	1	no	—
192	C₂²⋊Q₈	—	83	1	no	—
193	C₂²⋊Q₈	—	166	1	no	—
194	C₂²⋊Q₈	—	166	1	no	—
195	C₄×Q₈	—	83	1	no	—
196	C₄×Q₈	—	83	1	no	—
197	C₄×D₄	—	83	1	no	—
198	C₄×D₄	—	166	1	no	—
199	#117	—	166	1	no	—
200	#117	C₂×D₈₃	1	1	yes	characteristic
201	C₂×C₄⋊C₄	—	83	1	no	—
202	C₂×C₄⋊C₄	—	83	1	no	—
203	C₂×C₄⋊C₄	C₂×D₈₃	1	1	yes	characteristic
204	#107	—	83	1	no	—
205	#107	C₂×D₈₃	1	1	yes	characteristic
206	#107	C₂×D₈₃	1	1	yes	characteristic
207	#106	—	83	1	no	—
208	#106	—	83	1	no	—
209	#106	—	83	1	no	—
210	#106	—	83	1	no	—
211	#106	C₂×D₈₃	1	1	yes	characteristic
212	#106	C₂×D₈₃	1	1	yes	characteristic
213	C₄⋊C₈	—	83	1	no	—
214	#103	—	83	1	no	—
215	#103	—	83	1	no	—
216	#103	—	83	1	no	—
217	#103	—	83	1	no	—
218	#103	—	83	1	no	—
219	#103	—	83	1	no	—
220	C₂²⋊C₈	—	83	1	no	—
221	C₂²⋊C₈	—	83	1	no	—
222	#97	—	83	1	no	—
223	C₂×Q₈	—	83	1	no	—
224	C₂×Q₈	—	83	1	no	—
225	C₂×Q₈	—	166	1	no	—
226	C₂×D₄	—	83	1	no	—
227	C₂²×C₄	—	2	1	no	—
228	C₂²×C₄	—	83	1	no	—
229	C₂²×C₄	—	166	1	no	—
230	C₂²×C₄	—	166	1	no	—
231	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
232	Q₁₆	—	166	2	no	—
233	M₁₆	—	2	2	no	—
234	C₂×C₈	—	83	1	no	—
235	C₂×C₈	—	83	1	no	—
236	C₂×C₈	—	166	2	no	—
237	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
238	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
239	C₄⋊C₄	—	2	1	no	—
240	C₄⋊C₄	—	83	1	no	—
241	C₄⋊C₄	—	83	1	no	—
242	C₄⋊C₄	—	83	1	no	—
243	C₄⋊C₄	—	166	1	no	—
244	C₄⋊C₄	—	166	1	no	—
245	C₄⋊C₄	—	166	1	no	—
246	C₄⋊C₄	—	166	1	no	—
247	C₄⋊C₄	—	166	1	no	—
248	C₄⋊C₄	—	166	1	no	—
249	C₄⋊C₄	—	166	1	no	—
250	C₄⋊C₄	—	166	1	no	—
251	C₄⋊C₄	—	166	1	no	—
252	C₄⋊C₄	—	166	1	no	—
253	C₄⋊C₄	—	166	1	no	—
254	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
255	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
256	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
257	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
258	C₂²⋊C₄	—	2	1	no	—
259	C₂²⋊C₄	—	83	1	no	—
260	C₂²⋊C₄	—	83	1	no	—
261	C₂²⋊C₄	—	166	1	no	—
262	C₂²⋊C₄	—	166	1	no	—
263	C₂²⋊C₄	—	166	1	no	—
264	C₂²⋊C₄	—	166	1	no	—
265	C₄²	—	2	1	no	—
266	C₄²	—	83	1	no	—
267	C₄²	—	166	1	no	—
268	C₄²	—	166	1	no	—
269	C₂³	—	83	1	no	—
270	C₂³	D₄×D₈₃	1	1	yes	characteristic
271	Q₈	—	664	1	no	—
272	Q₈	—	166	1	no	—
273	Q₈	—	166	1	no	—
274	Q₈	—	166	1	no	—
275	Q₈	—	166	1	no	—
276	D₄	—	166	2	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	8	1	no	—
284	C₂×C₄	—	332	2	no	—
285	C₂×C₄	—	83	1	no	—
286	C₂×C₄	—	83	1	no	—
287	C₂×C₄	—	83	1	no	—
288	C₂×C₄	—	664	1	no	—
289	C₂×C₄	—	166	1	no	—
290	C₂×C₄	—	166	1	no	—
291	C₂×C₄	—	166	1	no	—
292	C₂×C₄	—	166	1	no	—
293	C₂×C₄	—	166	1	no	—
294	C₂×C₄	—	166	1	no	—
295	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
296	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
297	C₈	—	2	1	no	—
298	C₈	—	166	1	no	—
299	C₈	—	166	1	no	—
300	C₈	#13050590	1	2	yes	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	83	2	no	—
305	C₂²	—	166	1	no	—
306	C₂²	—	166	1	no	—
307	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
308	C₄	—	2	1	no	—
309	C₄	—	4	1	no	—
310	C₄	—	4	1	no	—
311	C₄	—	4	1	no	—
312	C₄	—	4	1	no	—
313	C₄	—	332	1	no	—
314	C₄	—	332	1	no	—
315	C₄	—	332	1	no	—
316	C₄	—	332	1	no	—
317	C₄	—	166	1	no	—
318	C₄	—	166	1	no	—
319	C₄	—	166	1	no	—
320	C₄	C₂×#13050590	1	1	yes	characteristic
321	C₄	#14581840	1	1	yes	characteristic
322	C₂	—	4	1	no	—
323	C₂	—	166	2	no	—
324	C₂	#19128352	1	1	yes	characteristic
325	C₂	#137×D₈₃	1	1	yes	characteristic
326	C₂	#19128511	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₃
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#428×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	C₂×C₂²⋊D₈₃	D₄
hypercenter	#428	D₈₃
hypocenter	C₈₃	#3344
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #428

Complements

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

N	H ≅ G/N	type
3. #19128109	C₂	semidirect
4. #19128042	C₂	semidirect
5. #19128038	C₂	semidirect
6. #19128033	C₂	semidirect
7. #19128031	C₂	semidirect
8. #19128007	C₂	semidirect
9. #19127985	C₂	semidirect
10. #19127959	C₂	semidirect
11. #19127943	C₂	semidirect
12. #19127807	C₂	semidirect
14. #106×D₈₃	C₂	semidirect
15. #428×C₈₃	C₂	semidirect
164. C₈₃	#3344	semidirect
178. #428	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128406	2
#19128390	2
#19128109	2
#19128042	2
#19128038	2
#19128033	2
#19128031	2
#19128007	2
#19127985	2
#19127959	2
#19127943	2
#19127807	2
#19127770	2
#106×D₈₃	2
#428×C₈₃	2
#3344	83