Subgroups — #24919918

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	#19128408	C₂	1	1	yes	maximal, characteristic
2	#19128111	C₂	1	1	yes	maximal, characteristic, complemented
3	#19128042	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128034	C₂	1	1	yes	maximal, characteristic, complemented
5	#19128031	C₂	1	1	yes	maximal, characteristic, complemented
6	#19128029	C₂	1	1	yes	maximal, characteristic, complemented
7	#19128005	C₂	1	1	yes	maximal, characteristic, complemented
8	#19127986	C₂	1	1	yes	maximal, characteristic, complemented
9	#19127957	C₂	1	1	yes	maximal, characteristic, complemented
10	#19127940	C₂	1	1	yes	maximal, characteristic, complemented
11	#19127806	C₂	1	1	yes	maximal, characteristic, complemented
12	#19127773	C₂	1	1	yes	maximal, characteristic
13	#19127744	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×#14581723	C₂	1	1	yes	maximal, characteristic, complemented
15	#552×C₈₃	C₂	1	1	yes	maximal, characteristic
16	C₂×#13050580	C₂²	1	1	yes	characteristic
17	C₂×#13050574	C₂²	1	1	yes	characteristic
18	C₂²×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
19	#14581792	C₂²	1	1	yes	characteristic
20	#14581783	—	2	1	no	—
21	#14581771	C₂²	1	1	yes	characteristic
22	#14581771	C₂²	1	1	yes	characteristic
23	#14581769	—	2	1	no	—
24	#14581769	—	2	1	no	—
25	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
26	D₄×Dic₈₃	—	2	1	no	—
27	C₂×C₄⋊Dic₈₃	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊Dic₈₃	C₂²	1	1	yes	characteristic
29	#14581725	C₂²	1	1	yes	characteristic
30	#14581725	C₂²	1	1	yes	characteristic
31	#14581725	C₂²	1	1	yes	characteristic
32	#14581723	C₂²	1	1	yes	characteristic
33	#14581723	C₂²	1	1	yes	characteristic
34	#14581723	C₂²	1	1	yes	characteristic
35	#14581723	C₂²	1	1	yes	characteristic
36	#14581723	C₂²	1	1	yes	characteristic
37	#14581723	C₂²	1	1	yes	characteristic
38	C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
39	#14581716	C₂²	1	1	yes	characteristic
40	#14581716	C₂²	1	1	yes	characteristic
41	#14581716	C₂²	1	1	yes	characteristic
42	#14581715	C₂²	1	1	yes	characteristic
43	#14581715	C₂²	1	1	yes	characteristic
44	#14581715	C₂²	1	1	yes	characteristic
45	C₂²⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
46	C₂²⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
47	#14581703	C₂²	1	1	yes	characteristic
48	#126×C₈₃	—	2	1	no	—
49	#125×C₈₃	—	2	1	no	—
50	#125×C₈₃	C₂²	1	1	yes	characteristic
51	#123×C₈₃	—	2	1	no	—
52	C₂²⋊Q₈×C₈₃	C₂²	1	1	yes	characteristic
53	C₂²⋊Q₈×C₈₃	C₂²	1	1	yes	characteristic
54	C₄×Q₈×C₈₃	C₂²	1	1	yes	characteristic
55	C₄×D₄×C₈₃	C₂²	1	1	yes	characteristic
56	#117×C₈₃	—	2	1	no	—
57	#117×C₈₃	C₂²	1	1	yes	characteristic
58	C₂×C₄⋊C₄×C₈₃	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₆₆	—	2	1	no	—
60	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
61	C₂²×Dic₈₃	—	2	1	no	—
62	C₂²×Dic₈₃	—	2	1	no	—
63	#13050580	—	2	2	no	—
64	#13050574	—	2	2	no	—
65	C₂×C₈₃⋊C₈	—	2	2	no	—
66	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
67	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
68	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
69	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
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	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
76	Dic₈₃⋊C₄	—	2	1	no	—
77	Dic₈₃⋊C₄	—	2	1	no	—
78	Dic₈₃⋊C₄	—	2	1	no	—
79	C₂²⋊Dic₈₃	—	2	1	no	—
80	C₂²⋊Dic₈₃	—	2	1	no	—
81	C₄×Dic₈₃	—	2	1	no	—
82	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
83	C₂×D₄×C₈₃	D₄	1	1	yes	characteristic
84	C₂²×C₄×C₈₃	—	2	1	no	—
85	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
86	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
87	C₄⋊C₄×C₈₃	—	2	1	no	—
88	C₄⋊C₄×C₈₃	—	2	1	no	—
89	C₄⋊C₄×C₈₃	—	2	1	no	—
90	C₄⋊C₄×C₈₃	—	2	1	no	—
91	C₄⋊C₄×C₈₃	—	2	1	no	—
92	C₄⋊C₄×C₈₃	—	2	1	no	—
93	C₄⋊C₄×C₈₃	D₄	1	1	yes	characteristic
94	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
95	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
96	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
97	C₂²⋊C₄×C₈₃	—	2	1	no	—
98	C₂²⋊C₄×C₈₃	—	2	1	no	—
99	C₂²⋊C₄×C₈₃	—	2	1	no	—
100	C₂²⋊C₄×C₈₃	D₄	1	1	yes	characteristic
101	C₂²⋊C₄×C₈₃	D₄	1	1	yes	characteristic
102	C₄²×C₈₃	—	2	1	no	—
103	C₄²×C₈₃	—	2	1	no	—
104	C₄²×C₈₃	C₂³	1	1	yes	characteristic
105	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
106	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
107	Q₈×C₈₃	—	2	1	no	—
108	Q₈×C₈₃	—	2	1	no	—
109	D₄×C₈₃	—	2	2	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	—	2	1	no	—
116	C₂×C₄×C₈₃	—	2	1	no	—
117	C₂×C₄×C₈₃	—	2	1	no	—
118	C₂×C₄×C₈₃	—	4	2	no	—
119	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
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	—	8	1	no	—
129	C₂×Dic₈₃	—	8	1	no	—
130	Dic₁₆₆	—	2	1	no	—
131	Dic₁₆₆	—	2	1	no	—
132	Dic₁₆₆	—	8	1	no	—
133	C₈₃⋊C₈	—	2	1	no	—
134	C₈₃⋊C₈	—	2	1	no	—
135	C₈₃⋊C₈	—	2	1	no	—
136	C₈₃⋊C₈	#41	1	2	yes	—
137	C₂²×C₈₃	—	2	1	no	—
138	C₂²×C₈₃	—	2	1	no	—
139	C₂²×C₈₃	—	2	1	no	—
140	C₂²×C₈₃	—	2	1	no	—
141	C₂²×C₈₃	—	2	1	no	—
142	C₂²×C₈₃	#137	1	2	yes	—
143	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
144	C₄×C₈₃	—	2	1	no	—
145	C₄×C₈₃	—	2	1	no	—
146	C₄×C₈₃	—	2	1	no	—
147	C₄×C₈₃	—	2	1	no	—
148	C₄×C₈₃	—	4	1	no	—
149	C₄×C₈₃	—	4	1	no	—
150	C₄×C₈₃	—	4	1	no	—
151	C₄×C₈₃	—	4	1	no	—
152	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
153	C₄×C₈₃	2₋¹⁺⁴	1	1	yes	characteristic
154	Dic₈₃	—	4	1	no	—
155	Dic₈₃	—	4	1	no	—
156	Dic₈₃	—	4	1	no	—
157	Dic₈₃	—	4	1	no	—
158	C₂×C₈₃	—	2	2	no	—
159	C₂×C₈₃	—	4	1	no	—
160	C₂×C₈₃	#547	1	1	yes	characteristic
161	C₂×C₈₃	C₂×#137	1	1	yes	characteristic
162	C₂×C₈₃	#577	1	1	yes	characteristic
163	#3344	—	83	1	no	maximal, Sylow
164	C₈₃	#3344	1	1	yes	characteristic, complemented, Sylow
165	#554	—	83	1	no	—
166	#552	D₈₃	1	1	yes	characteristic
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	—	83	1	no	—
179	C₂×#106	—	83	1	no	—
180	C₂×Q₁₆	—	83	1	no	—
181	C₂×M₁₆	—	83	1	no	—
182	C₂²×C₈	—	83	1	no	—
183	C₄⋊Q₈	—	83	1	no	—
184	#126	—	2	1	no	—
185	#125	—	2	1	no	—
186	#125	—	166	1	no	—
187	#125	C₂×D₈₃	1	1	yes	characteristic
188	#123	—	2	1	no	—
189	C₂²⋊Q₈	—	83	1	no	—
190	C₂²⋊Q₈	—	83	1	no	—
191	C₂²⋊Q₈	—	166	1	no	—
192	C₂²⋊Q₈	—	166	1	no	—
193	C₂²⋊Q₈	C₂×D₈₃	1	1	yes	characteristic
194	C₂²⋊Q₈	C₂×D₈₃	1	1	yes	characteristic
195	C₄×Q₈	—	83	1	no	—
196	C₄×Q₈	C₂×D₈₃	1	1	yes	characteristic
197	C₄×D₄	—	166	1	no	—
198	C₄×D₄	C₂×D₈₃	1	1	yes	characteristic
199	#117	—	2	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	—	83	1	no	—
206	#107	—	83	1	no	—
207	#106	—	83	1	no	—
208	#106	—	83	1	no	—
209	#106	—	83	1	no	—
210	#106	—	83	1	no	—
211	#106	—	83	1	no	—
212	#106	—	83	1	no	—
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	—	166	1	no	—
225	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
226	C₂×D₄	C₂²⋊D₈₃	1	1	yes	characteristic
227	C₂²×C₄	—	2	1	no	—
228	C₂²×C₄	—	166	1	no	—
229	C₂²×C₄	—	166	1	no	—
230	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
231	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
232	Q₁₆	—	166	2	no	—
233	M₁₆	—	166	2	no	—
234	C₂×C₈	—	83	1	no	—
235	C₂×C₈	—	83	1	no	—
236	C₂×C₈	—	83	1	no	—
237	C₂×C₈	—	83	1	no	—
238	C₂×C₈	—	166	2	no	—
239	C₄⋊C₄	—	2	1	no	—
240	C₄⋊C₄	—	2	1	no	—
241	C₄⋊C₄	—	2	1	no	—
242	C₄⋊C₄	—	2	1	no	—
243	C₄⋊C₄	—	2	1	no	—
244	C₄⋊C₄	—	2	1	no	—
245	C₄⋊C₄	—	83	1	no	—
246	C₄⋊C₄	—	83	1	no	—
247	C₄⋊C₄	—	83	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	—	2	1	no	—
260	C₂²⋊C₄	—	2	1	no	—
261	C₂²⋊C₄	—	166	1	no	—
262	C₂²⋊C₄	—	166	1	no	—
263	C₂²⋊C₄	C₂²⋊D₈₃	1	1	yes	characteristic
264	C₂²⋊C₄	C₂²⋊D₈₃	1	1	yes	characteristic
265	C₄²	—	2	1	no	—
266	C₄²	—	2	1	no	—
267	C₄²	—	166	1	no	—
268	C₄²	C₂²×D₈₃	1	1	yes	characteristic
269	C₂³	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
270	C₂³	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
271	Q₈	—	2	1	no	—
272	Q₈	—	2	1	no	—
273	Q₈	—	664	1	no	—
274	Q₈	—	166	1	no	—
275	Q₈	—	166	1	no	—
276	D₄	—	2	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	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	664	1	no	—
287	C₂×C₄	—	664	1	no	—
288	C₂×C₄	—	166	1	no	—
289	C₂×C₄	—	166	1	no	—
290	C₂×C₄	—	166	1	no	—
291	C₂×C₄	—	166	1	no	—
292	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
293	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
294	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
295	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
296	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
297	C₈	—	83	2	no	—
298	C₈	—	166	1	no	—
299	C₈	—	166	1	no	—
300	C₈	—	166	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	2	1	no	—
306	C₂²	#14581822	1	2	yes	—
307	C₂²	C₂²×C₂²⋊D₈₃	1	1	yes	characteristic
308	C₄	—	2	1	no	—
309	C₄	—	2	1	no	—
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	4	1	no	—
313	C₄	—	4	1	no	—
314	C₄	—	4	1	no	—
315	C₄	—	4	1	no	—
316	C₄	—	332	1	no	—
317	C₄	—	332	1	no	—
318	C₄	—	332	1	no	—
319	C₄	—	332	1	no	—
320	C₄	#41×D₈₃	1	1	yes	characteristic
321	C₄	#14581839	1	1	yes	characteristic
322	C₂	—	2	2	no	—
323	C₂	—	4	1	no	—
324	C₂	#19128350	1	1	yes	characteristic
325	C₂	C₂×#14581822	1	1	yes	characteristic
326	C₂	#19128509	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₈₃
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#552×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	C₂×D₄×C₈₃	D₄
hypercenter	#552	D₈₃
hypocenter	C₈₃	#3344
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ #552

Complements

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

N	H ≅ G/N	type
2. #19128111	C₂	semidirect
3. #19128042	C₂	semidirect
4. #19128034	C₂	semidirect
5. #19128031	C₂	semidirect
6. #19128029	C₂	semidirect
7. #19128005	C₂	semidirect
8. #19127986	C₂	semidirect
9. #19127957	C₂	semidirect
10. #19127940	C₂	semidirect
11. #19127806	C₂	semidirect
13. #19127744	C₂	semidirect
14. C₂×#14581723	C₂	semidirect
164. C₈₃	#3344	semidirect

Maximal subgroups

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

subgroup	index
#19128408	2
#19128111	2
#19128042	2
#19128034	2
#19128031	2
#19128029	2
#19128005	2
#19127986	2
#19127957	2
#19127940	2
#19127806	2
#19127773	2
#19127744	2
C₂×#14581723	2
#552×C₈₃	2
#3344	83