Subgroups — #24920497

Order 10624.

350 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128429	C₂	1	1	yes	maximal, characteristic, complemented
2	#19128420	C₂	1	1	yes	maximal, characteristic
3	#19128096	C₂	1	1	yes	maximal, characteristic
4	#19128092	C₂	1	1	yes	maximal, characteristic
5	#19128079	C₂	1	1	yes	maximal, characteristic, complemented
6	#19127922	C₂	1	1	yes	maximal, characteristic
7	#19127912	C₂	1	1	yes	maximal, characteristic, complemented
8	#19127910	C₂	1	1	yes	maximal, characteristic, complemented
9	#19127909	C₂	1	1	yes	maximal, characteristic, complemented
10	#19127904	C₂	1	1	yes	maximal, characteristic
11	#19127830	C₂	1	1	yes	maximal, characteristic, complemented
12	#19127806	C₂	1	1	yes	maximal, characteristic, complemented
13	#19127805	C₂	1	1	yes	maximal, characteristic, complemented
14	#19127792	C₂	1	1	yes	maximal, characteristic
15	#464×C₈₃	C₂	1	1	yes	maximal, characteristic
16	C₂×Dic₃₃₂	C₂²	1	1	yes	characteristic
17	C₂×Dic₃₃₂	C₂²	1	1	yes	characteristic
18	C₂×#13050580	C₂²	1	1	yes	characteristic
19	C₂×#13050580	C₂²	1	1	yes	characteristic
20	C₂×#13050580	C₂²	1	1	yes	characteristic
21	C₂×#13050578	C₂²	1	1	yes	characteristic
22	C₂×#13050578	C₂²	1	1	yes	characteristic
23	#14581792	C₂²	1	1	yes	characteristic
24	#14581790	C₂²	1	1	yes	characteristic
25	#14581790	C₂²	1	1	yes	characteristic
26	#14581790	C₂²	1	1	yes	characteristic
27	#14581785	—	2	1	no	—
28	#14581785	—	2	1	no	—
29	#14581784	—	2	1	no	—
30	#14581782	—	2	1	no	—
31	#14581782	—	2	1	no	—
32	#14581779	C₂²	1	1	yes	characteristic
33	#14581769	—	2	1	no	—
34	Q₈×Dic₈₃	C₂²	1	1	yes	characteristic
35	C₄×Dic₁₆₆	—	2	1	no	—
36	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
37	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
38	#14581757	C₂²	1	1	yes	characteristic
39	#14581757	C₂²	1	1	yes	characteristic
40	D₄×Dic₈₃	C₂²	1	1	yes	characteristic
41	#14581748	—	2	1	no	—
42	#14581723	C₂²	1	1	yes	characteristic
43	Dic₈₃⋊C₈	C₂²	1	1	yes	characteristic
44	Dic₈₃⋊C₈	C₂²	1	1	yes	characteristic
45	#14581717	C₂²	1	1	yes	characteristic
46	#14581717	C₂²	1	1	yes	characteristic
47	#14581716	C₂²	1	1	yes	characteristic
48	#14581715	C₂²	1	1	yes	characteristic
49	#14581703	C₂²	1	1	yes	characteristic
50	#14581702	C₂²	1	1	yes	characteristic
51	C₈×Dic₈₃	C₂²	1	1	yes	characteristic
52	C₂×Q₁₆×C₈₃	C₂²	1	1	yes	characteristic
53	C₂×SD₁₆×C₈₃	C₂²	1	1	yes	characteristic
54	#124×C₈₃	C₂²	1	1	yes	characteristic
55	C₄×Q₈×C₈₃	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
57	#103×C₈₃	C₂²	1	1	yes	characteristic
58	#102×C₈₃	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₆₆	D₄	1	1	yes	characteristic
60	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
61	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
62	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
63	C₂²×Dic₈₃	—	2	1	no	—
64	Dic₃₃₂	—	4	1	no	—
65	Dic₃₃₂	—	4	1	no	—
66	#13050580	—	2	2	no	—
67	#13050580	—	2	2	no	—
68	#13050580	—	4	1	no	—
69	#13050578	—	2	2	no	—
70	#13050578	—	4	1	no	—
71	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
72	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
73	C₄⋊Dic₈₃	—	2	1	no	—
74	C₄⋊Dic₈₃	—	2	1	no	—
75	C₄⋊Dic₈₃	—	2	1	no	—
76	C₄⋊Dic₈₃	D₄	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	—	2	1	no	—
82	Dic₈₃⋊C₄	—	2	1	no	—
83	Dic₈₃⋊C₄	—	2	1	no	—
84	Dic₈₃⋊C₄	—	2	1	no	—
85	Dic₈₃⋊C₄	—	2	1	no	—
86	Dic₈₃⋊C₄	—	2	1	no	—
87	Dic₈₃⋊C₄	D₄	1	1	yes	characteristic
88	Dic₈₃⋊C₄	D₄	1	1	yes	characteristic
89	C₂²⋊Dic₈₃	—	2	1	no	—
90	C₂²⋊Dic₈₃	—	2	1	no	—
91	C₂²⋊Dic₈₃	—	2	1	no	—
92	C₄×Dic₈₃	—	2	1	no	—
93	C₄×Dic₈₃	—	2	1	no	—
94	C₄×Dic₈₃	—	2	1	no	—
95	C₄×Dic₈₃	—	2	1	no	—
96	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
97	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
98	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
99	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
100	C₂×D₄×C₈₃	C₂³	1	1	yes	characteristic
101	Q₁₆×C₈₃	—	4	1	no	—
102	SD₁₆×C₈₃	—	4	1	no	—
103	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
104	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
105	C₄⋊C₄×C₈₃	—	2	1	no	—
106	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
107	C₂²⋊C₄×C₈₃	—	2	1	no	—
108	C₂²⋊C₄×C₈₃	—	2	1	no	—
109	C₄²×C₈₃	—	2	1	no	—
110	C₄²×C₈₃	C₂³	1	1	yes	characteristic
111	C₂³×C₈₃	—	2	1	no	—
112	Q₈×C₈₃	—	2	1	no	—
113	Q₈×C₈₃	—	2	1	no	—
114	Q₈×C₈₃	—	2	1	no	—
115	Q₈×C₈₃	—	2	1	no	—
116	D₄×C₈₃	—	2	1	no	—
117	D₄×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	—	4	2	no	—
129	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
130	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
131	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
132	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
133	Dic₁₆₆	—	2	2	no	—
134	Dic₁₆₆	—	2	1	no	—
135	Dic₁₆₆	—	2	1	no	—
136	Dic₁₆₆	—	2	1	no	—
137	Dic₁₆₆	—	2	1	no	—
138	Dic₁₆₆	—	2	1	no	—
139	Dic₁₆₆	—	2	1	no	—
140	C₈×C₈₃	—	2	1	no	—
141	C₈×C₈₃	—	2	1	no	—
142	C₈₃⋊C₈	—	2	1	no	—
143	C₈₃⋊C₈	#41	1	2	yes	—
144	C₂²×C₈₃	—	4	1	no	—
145	C₂²×C₈₃	—	4	1	no	—
146	C₂²×C₈₃	—	4	1	no	—
147	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
148	C₄×C₈₃	—	2	1	no	—
149	C₄×C₈₃	—	2	1	no	—
150	C₄×C₈₃	—	4	1	no	—
151	C₄×C₈₃	—	4	1	no	—
152	C₄×C₈₃	—	4	1	no	—
153	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
154	C₄×C₈₃	H(5,𝔽₂)	1	1	yes	characteristic
155	Dic₈₃	—	2	1	no	—
156	Dic₈₃	—	2	1	no	—
157	Dic₈₃	—	2	1	no	—
158	Dic₈₃	—	4	1	no	—
159	Dic₈₃	—	4	1	no	—
160	Dic₈₃	—	4	1	no	—
161	Dic₈₃	—	4	1	no	—
162	Dic₈₃	#137	1	2	yes	—
163	C₂×C₈₃	—	8	1	no	—
164	C₂×C₈₃	#550	1	1	yes	characteristic
165	C₂×C₈₃	C₂×#137	1	1	yes	characteristic
166	C₂×C₈₃	#578	1	1	yes	characteristic
167	#3398	—	83	1	no	maximal, Sylow
168	C₈₃	#3398	1	1	yes	characteristic, complemented, Sylow
169	#557	—	83	1	no	—
170	#556	—	83	1	no	—
171	#497	—	83	1	no	—
172	#497	—	83	1	no	—
173	#494	—	83	1	no	—
174	#464	—	83	1	no	—
175	#464	D₈₃	1	1	yes	characteristic
176	#462	—	83	1	no	—
177	#462	—	83	1	no	—
178	#462	—	83	1	no	—
179	#461	—	83	1	no	—
180	#446	—	83	1	no	—
181	#441	—	83	1	no	—
182	#441	—	83	1	no	—
183	C₄×SD₁₆	—	83	1	no	—
184	C₂×Q₁₆	—	83	1	no	—
185	C₂×Q₁₆	—	83	1	no	—
186	C₂×Q₁₆	—	83	1	no	—
187	C₂×Q₁₆	—	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	C₂×SD₁₆	C₂×D₈₃	1	1	yes	characteristic
193	C₄⋊Q₈	—	83	1	no	—
194	C₄⋊Q₈	—	83	1	no	—
195	C₄⋊Q₈	—	83	1	no	—
196	C₄⋊Q₈	—	83	1	no	—
197	#126	—	166	1	no	—
198	#126	—	166	1	no	—
199	#125	—	166	1	no	—
200	#125	—	166	1	no	—
201	#125	—	166	1	no	—
202	#124	—	83	1	no	—
203	#124	C₂×D₈₃	1	1	yes	characteristic
204	C₂²⋊Q₈	—	166	1	no	—
205	C₄×Q₈	—	83	1	no	—
206	C₄×Q₈	—	83	1	no	—
207	C₄×Q₈	—	83	1	no	—
208	C₄×Q₈	—	83	1	no	—
209	C₄×Q₈	—	83	1	no	—
210	C₄×Q₈	—	166	1	no	—
211	C₄×Q₈	C₂×D₈₃	1	1	yes	characteristic
212	C₄×D₄	—	83	1	no	—
213	#117	—	166	1	no	—
214	#106	—	83	1	no	—
215	C₄⋊C₈	—	83	1	no	—
216	C₄⋊C₈	—	83	1	no	—
217	C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
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	#102	C₂×D₈₃	1	1	yes	characteristic
224	#97	—	83	1	no	—
225	#97	—	83	1	no	—
226	C₄×C₈	—	83	1	no	—
227	C₂×Q₈	—	83	1	no	—
228	C₂×Q₈	—	83	1	no	—
229	C₂×Q₈	—	83	1	no	—
230	C₂×Q₈	—	83	1	no	—
231	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
232	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
233	C₂×D₄	C₂²×D₈₃	1	1	yes	characteristic
234	C₂²×C₄	—	166	1	no	—
235	Q₁₆	—	4	1	no	—
236	Q₁₆	—	332	1	no	—
237	Q₁₆	—	332	1	no	—
238	Q₁₆	—	332	1	no	—
239	Q₁₆	—	166	2	no	—
240	Q₁₆	—	166	2	no	—
241	SD₁₆	—	4	1	no	—
242	SD₁₆	—	332	1	no	—
243	SD₁₆	—	166	2	no	—
244	C₂×C₈	—	83	1	no	—
245	C₂×C₈	—	83	1	no	—
246	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
247	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	83	1	no	—
250	C₄⋊C₄	—	83	1	no	—
251	C₄⋊C₄	—	83	1	no	—
252	C₄⋊C₄	—	83	1	no	—
253	C₄⋊C₄	—	166	1	no	—
254	C₄⋊C₄	—	166	1	no	—
255	C₄⋊C₄	—	166	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	—	166	1	no	—
264	C₄⋊C₄	—	166	1	no	—
265	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
266	C₂²⋊C₄	—	2	1	no	—
267	C₂²⋊C₄	—	2	1	no	—
268	C₂²⋊C₄	—	166	1	no	—
269	C₂²⋊C₄	—	166	1	no	—
270	C₂²⋊C₄	—	166	1	no	—
271	C₄²	—	2	1	no	—
272	C₄²	—	83	1	no	—
273	C₄²	—	83	1	no	—
274	C₄²	—	166	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	—	2	1	no	—
280	Q₈	—	2	1	no	—
281	Q₈	—	2	1	no	—
282	Q₈	—	2	1	no	—
283	Q₈	—	2	1	no	—
284	Q₈	—	166	2	no	—
285	Q₈	—	166	1	no	—
286	Q₈	—	166	1	no	—
287	Q₈	—	166	1	no	—
288	Q₈	—	166	1	no	—
289	Q₈	—	166	1	no	—
290	Q₈	—	166	1	no	—
291	D₄	—	2	1	no	—
292	D₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	332	2	no	—
297	C₂×C₄	—	83	1	no	—
298	C₂×C₄	—	83	1	no	—
299	C₂×C₄	—	83	1	no	—
300	C₂×C₄	—	83	1	no	—
301	C₂×C₄	—	166	1	no	—
302	C₂×C₄	—	166	1	no	—
303	C₂×C₄	—	166	1	no	—
304	C₂×C₄	—	166	1	no	—
305	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
306	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
307	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	83	2	no	—
311	C₈	—	166	1	no	—
312	C₂²	—	4	1	no	—
313	C₂²	—	4	1	no	—
314	C₂²	—	4	1	no	—
315	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	4	1	no	—
319	C₄	—	4	1	no	—
320	C₄	—	4	1	no	—
321	C₄	—	332	1	no	—
322	C₄	—	332	1	no	—
323	C₄	—	332	1	no	—
324	C₄	—	332	1	no	—
325	C₄	—	83	2	no	—
326	C₄	—	166	1	no	—
327	C₄	—	166	1	no	—
328	C₄	—	166	1	no	—
329	C₄	#41×D₈₃	1	1	yes	characteristic
330	C₄	#14581837	1	1	yes	characteristic
331	C₂	—	8	1	no	—
332	C₂	#19128369	1	1	yes	characteristic
333	C₂	#137×D₈₃	1	1	yes	characteristic
334	C₂	#19128515	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₃
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#464×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	C₂×D₄×C₈₃	C₂³
hypercenter	#464	D₈₃
hypocenter	C₈₃	#3398
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #464

Complements

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

N	H ≅ G/N	type
1. #19128429	C₂	semidirect
5. #19128079	C₂	semidirect
7. #19127912	C₂	semidirect
8. #19127910	C₂	semidirect
9. #19127909	C₂	semidirect
11. #19127830	C₂	semidirect
12. #19127806	C₂	semidirect
13. #19127805	C₂	semidirect
168. C₈₃	#3398	semidirect

Maximal subgroups

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

subgroup	index
#19128429	2
#19128420	2
#19128096	2
#19128092	2
#19128079	2
#19127922	2
#19127912	2
#19127910	2
#19127909	2
#19127904	2
#19127830	2
#19127806	2
#19127805	2
#19127792	2
#464×C₈₃	2
#3398	83