Subgroups — #24920536

Order 10624.

368 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128434	C₂	1	1	yes	maximal, characteristic, complemented
2	#19128425	C₂	1	1	yes	maximal, characteristic, complemented
3	#19128093	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128087	C₂	1	1	yes	maximal, characteristic
5	#19128082	C₂	1	1	yes	maximal, characteristic, complemented
6	#19127926	C₂	1	1	yes	maximal, characteristic, complemented
7	#19127921	C₂	1	1	yes	maximal, characteristic, complemented
8	#19127919	C₂	1	1	yes	maximal, characteristic, complemented
9	#19127917	C₂	1	1	yes	maximal, characteristic, complemented
10	#19127899	C₂	1	1	yes	maximal, characteristic, complemented
11	#19127830	C₂	1	1	yes	maximal, characteristic, complemented
12	#19127811	C₂	1	1	yes	maximal, characteristic, complemented
13	#19127805	C₂	1	1	yes	maximal, characteristic, complemented
14	#19127786	C₂	1	1	yes	maximal, characteristic, complemented
15	#461×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13050580	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	C₂×#13050577	C₂²	1	1	yes	characteristic
21	C₂×D₆₆₄	C₂²	1	1	yes	characteristic
22	C₂×#13050575	C₂²	1	1	yes	characteristic
23	#14581788	C₂²	1	1	yes	characteristic
24	#14581787	—	2	1	no	—
25	#14581787	—	2	1	no	—
26	#14581781	C₂²	1	1	yes	characteristic
27	#14581781	C₂²	1	1	yes	characteristic
28	#14581780	C₂²	1	1	yes	characteristic
29	#14581779	C₂²	1	1	yes	characteristic
30	#14581778	—	2	1	no	—
31	#14581775	—	2	1	no	—
32	#14581774	—	2	1	no	—
33	#14581772	—	2	1	no	—
34	Q₈×Dic₈₃	C₂²	1	1	yes	characteristic
35	Q₈×Dic₈₃	C₂²	1	1	yes	characteristic
36	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
37	#14581757	C₂²	1	1	yes	characteristic
38	C₄×D₃₃₂	C₂²	1	1	yes	characteristic
39	#14581754	C₂²	1	1	yes	characteristic
40	C₄×C₂²⋊D₈₃	—	2	1	no	—
41	#14581750	—	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	#14581714	C₂²	1	1	yes	characteristic
47	#14581713	C₂²	1	1	yes	characteristic
48	#14581712	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×SD₁₆×C₈₃	C₂²	1	1	yes	characteristic
53	C₂×SD₁₆×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	C₄⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
57	#103×C₈₃	C₂²	1	1	yes	characteristic
58	#103×C₈₃	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₆₆	D₄	1	1	yes	characteristic
60	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
61	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
62	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
63	C₂×C₂²⋊D₈₃	—	2	1	no	—
64	C₂×C₂²⋊D₈₃	—	2	1	no	—
65	C₂×C₄×D₈₃	—	2	1	no	—
66	C₂×C₄×D₈₃	—	2	1	no	—
67	#13050580	—	2	2	no	—
68	#13050580	—	4	1	no	—
69	#13050579	—	4	1	no	—
70	#13050577	—	2	2	no	—
71	#13050577	—	4	1	no	—
72	D₆₆₄	—	4	1	no	—
73	#13050575	—	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	D₄	1	1	yes	characteristic
79	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
80	Dic₈₃⋊C₄	—	2	1	no	—
81	Dic₈₃⋊C₄	D₄	1	1	yes	characteristic
82	Dic₈₃⋊C₄	D₄	1	1	yes	characteristic
83	C₂²⋊Dic₈₃	—	2	1	no	—
84	C₂²⋊Dic₈₃	—	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	(C₂×D₈₃)⋊C₄	—	2	1	no	—
91	(C₂×D₈₃)⋊C₄	—	2	1	no	—
92	(C₂×D₈₃)⋊C₄	—	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	C₂²×C₄×C₈₃	—	2	1	no	—
102	SD₁₆×C₈₃	—	4	1	no	—
103	SD₁₆×C₈₃	—	4	1	no	—
104	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
105	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
106	C₄⋊C₄×C₈₃	—	2	1	no	—
107	C₄⋊C₄×C₈₃	—	2	1	no	—
108	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
109	C₂²⋊C₄×C₈₃	—	2	1	no	—
110	C₄²×C₈₃	C₂³	1	1	yes	characteristic
111	C₂³×C₈₃	—	2	1	no	—
112	C₂²×D₈₃	—	2	1	no	—
113	C₂²×D₈₃	—	2	1	no	—
114	Q₈×C₈₃	—	2	1	no	—
115	Q₈×C₈₃	—	2	1	no	—
116	Q₈×C₈₃	—	2	1	no	—
117	Q₈×C₈₃	—	2	1	no	—
118	D₄×C₈₃	—	2	1	no	—
119	D₄×C₈₃	—	2	1	no	—
120	C₂×C₄×C₈₃	—	2	1	no	—
121	C₂×C₄×C₈₃	—	2	1	no	—
122	C₂×C₄×C₈₃	—	2	1	no	—
123	C₂×C₄×C₈₃	—	8	1	no	—
124	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
127	C₂²⋊D₈₃	—	4	2	no	—
128	C₂²⋊D₈₃	—	8	1	no	—
129	C₂×Dic₈₃	—	2	1	no	—
130	C₂×Dic₈₃	—	2	1	no	—
131	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
132	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
133	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
134	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
135	D₃₃₂	—	2	1	no	—
136	D₃₃₂	—	2	1	no	—
137	D₃₃₂	—	2	1	no	—
138	D₃₃₂	—	2	1	no	—
139	C₄×D₈₃	—	4	2	no	—
140	C₄×D₈₃	—	8	1	no	—
141	Dic₁₆₆	—	2	1	no	—
142	Dic₁₆₆	—	2	1	no	—
143	Dic₁₆₆	—	4	1	no	—
144	C₈×C₈₃	—	2	1	no	—
145	C₈×C₈₃	—	2	1	no	—
146	C₈₃⋊C₈	—	2	1	no	—
147	C₈₃⋊C₈	#41	1	2	yes	—
148	C₂²×C₈₃	—	4	1	no	—
149	C₂²×C₈₃	—	4	1	no	—
150	C₂²×C₈₃	—	4	1	no	—
151	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
152	C₂×D₈₃	—	4	1	no	—
153	C₂×D₈₃	—	4	1	no	—
154	C₂×D₈₃	—	4	1	no	—
155	C₂×D₈₃	—	4	1	no	—
156	C₂×D₈₃	—	4	1	no	—
157	C₂×D₈₃	—	4	1	no	—
158	C₄×C₈₃	—	2	1	no	—
159	C₄×C₈₃	—	2	1	no	—
160	C₄×C₈₃	—	4	1	no	—
161	C₄×C₈₃	—	4	1	no	—
162	C₄×C₈₃	—	4	1	no	—
163	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
164	C₄×C₈₃	H(5,𝔽₂)	1	1	yes	characteristic
165	Dic₈₃	—	2	1	no	—
166	Dic₈₃	—	2	1	no	—
167	Dic₈₃	—	2	1	no	—
168	Dic₈₃	—	2	1	no	—
169	Dic₈₃	—	4	1	no	—
170	Dic₈₃	—	4	1	no	—
171	C₂×C₈₃	—	8	1	no	—
172	C₂×C₈₃	#550	1	1	yes	characteristic
173	C₂×C₈₃	#575	1	1	yes	characteristic
174	C₂×C₈₃	#577	1	1	yes	characteristic
175	D₈₃	—	8	1	no	—
176	D₈₃	—	8	1	no	—
177	#3401	—	83	1	no	maximal, Sylow
178	C₈₃	#3401	1	1	yes	characteristic, complemented, Sylow
179	#559	—	83	1	no	—
180	#556	—	83	1	no	—
181	#497	—	83	1	no	—
182	#496	—	83	1	no	—
183	#495	—	83	1	no	—
184	#464	—	83	1	no	—
185	#464	—	83	1	no	—
186	#463	—	83	1	no	—
187	#463	—	83	1	no	—
188	#461	D₈₃	1	1	yes	characteristic, complemented
189	#460	—	83	1	no	—
190	#446	—	83	1	no	—
191	#442	—	83	1	no	—
192	#441	—	83	1	no	—
193	C₄×SD₁₆	—	83	1	no	—
194	C₂×Q₁₆	—	83	1	no	—
195	C₂×Q₁₆	—	83	1	no	—
196	C₂×SD₁₆	—	83	1	no	—
197	C₂×SD₁₆	—	83	1	no	—
198	C₂×SD₁₆	—	83	1	no	—
199	C₂×SD₁₆	C₂×D₈₃	1	1	yes	characteristic
200	C₂×SD₁₆	C₂×D₈₃	1	1	yes	characteristic
201	C₂×D₈	—	83	1	no	—
202	C₂×D₈	—	83	1	no	—
203	C₄⋊Q₈	C₂×D₈₃	1	1	yes	characteristic
204	#127	—	83	1	no	—
205	#126	—	166	1	no	—
206	#126	—	166	1	no	—
207	#124	—	83	1	no	—
208	#124	—	83	1	no	—
209	#124	—	83	1	no	—
210	#124	—	83	1	no	—
211	#124	—	166	1	no	—
212	#123	—	166	1	no	—
213	#123	—	166	1	no	—
214	C₂²⋊Q₈	—	166	1	no	—
215	C₄×Q₈	—	83	1	no	—
216	C₄×Q₈	—	83	1	no	—
217	C₄×Q₈	—	83	1	no	—
218	C₄×Q₈	—	83	1	no	—
219	C₄×D₄	—	83	1	no	—
220	C₄×D₄	—	83	1	no	—
221	C₄×D₄	—	166	1	no	—
222	C₄×D₄	C₂×D₈₃	1	1	yes	characteristic
223	#117	—	166	1	no	—
224	#106	—	83	1	no	—
225	C₄⋊C₈	—	83	1	no	—
226	C₄⋊C₈	—	83	1	no	—
227	C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
228	#103	—	83	1	no	—
229	#103	C₂×D₈₃	1	1	yes	characteristic
230	#103	C₂×D₈₃	1	1	yes	characteristic
231	#102	—	83	1	no	—
232	#102	—	83	1	no	—
233	#102	—	83	1	no	—
234	#97	—	83	1	no	—
235	#97	—	83	1	no	—
236	C₄×C₈	—	83	1	no	—
237	C₂×Q₈	—	83	1	no	—
238	C₂×Q₈	—	83	1	no	—
239	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
240	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
241	C₂×D₄	—	83	1	no	—
242	C₂×D₄	—	83	1	no	—
243	C₂×D₄	—	166	1	no	—
244	C₂×D₄	—	166	1	no	—
245	C₂×D₄	C₂²×D₈₃	1	1	yes	characteristic
246	C₂²×C₄	—	2	1	no	—
247	C₂²×C₄	—	166	1	no	—
248	C₂²×C₄	—	166	1	no	—
249	Q₁₆	—	332	1	no	—
250	Q₁₆	—	166	2	no	—
251	SD₁₆	—	4	1	no	—
252	SD₁₆	—	4	1	no	—
253	SD₁₆	—	332	1	no	—
254	SD₁₆	—	332	1	no	—
255	SD₁₆	—	166	2	no	—
256	D₈	—	332	1	no	—
257	D₈	—	166	2	no	—
258	C₂×C₈	—	83	1	no	—
259	C₂×C₈	—	83	1	no	—
260	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
261	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
262	C₄⋊C₄	—	2	1	no	—
263	C₄⋊C₄	—	2	1	no	—
264	C₄⋊C₄	—	83	1	no	—
265	C₄⋊C₄	—	83	1	no	—
266	C₄⋊C₄	—	83	1	no	—
267	C₄⋊C₄	—	83	1	no	—
268	C₄⋊C₄	—	166	1	no	—
269	C₄⋊C₄	—	166	1	no	—
270	C₄⋊C₄	—	166	1	no	—
271	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
272	C₂²⋊C₄	—	2	1	no	—
273	C₂²⋊C₄	—	166	1	no	—
274	C₂²⋊C₄	—	166	1	no	—
275	C₂²⋊C₄	—	166	1	no	—
276	C₂²⋊C₄	—	166	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	C₂²⋊C₄	—	166	1	no	—
283	C₄²	—	83	1	no	—
284	C₄²	—	83	1	no	—
285	C₄²	—	166	1	no	—
286	C₄²	—	166	1	no	—
287	C₄²	—	166	1	no	—
288	C₄²	C₂²×D₈₃	1	1	yes	characteristic
289	C₂³	—	2	1	no	—
290	C₂³	—	166	1	no	—
291	C₂³	—	166	1	no	—
292	Q₈	—	2	1	no	—
293	Q₈	—	2	1	no	—
294	Q₈	—	2	1	no	—
295	Q₈	—	2	1	no	—
296	Q₈	—	332	1	no	—
297	Q₈	—	166	1	no	—
298	Q₈	—	166	1	no	—
299	D₄	—	2	1	no	—
300	D₄	—	2	1	no	—
301	D₄	—	332	2	no	—
302	D₄	—	664	1	no	—
303	D₄	—	166	1	no	—
304	D₄	—	166	1	no	—
305	D₄	—	166	1	no	—
306	D₄	—	166	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	8	1	no	—
311	C₂×C₄	—	332	2	no	—
312	C₂×C₄	—	83	1	no	—
313	C₂×C₄	—	83	1	no	—
314	C₂×C₄	—	83	1	no	—
315	C₂×C₄	—	83	1	no	—
316	C₂×C₄	—	664	1	no	—
317	C₂×C₄	—	166	1	no	—
318	C₂×C₄	—	166	1	no	—
319	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
320	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
321	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
322	C₈	—	2	1	no	—
323	C₈	—	2	1	no	—
324	C₈	—	83	2	no	—
325	C₈	—	166	1	no	—
326	C₂²	—	4	1	no	—
327	C₂²	—	4	1	no	—
328	C₂²	—	4	1	no	—
329	C₂²	—	332	1	no	—
330	C₂²	—	332	1	no	—
331	C₂²	—	332	1	no	—
332	C₂²	—	332	1	no	—
333	C₂²	—	332	1	no	—
334	C₂²	—	332	1	no	—
335	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	4	1	no	—
339	C₄	—	4	1	no	—
340	C₄	—	4	1	no	—
341	C₄	—	332	1	no	—
342	C₄	—	332	1	no	—
343	C₄	—	166	1	no	—
344	C₄	—	166	1	no	—
345	C₄	—	166	1	no	—
346	C₄	—	166	1	no	—
347	C₄	#41×D₈₃	1	1	yes	characteristic
348	C₄	#14581837	1	1	yes	characteristic
349	C₂	—	8	1	no	—
350	C₂	—	664	1	no	—
351	C₂	—	664	1	no	—
352	C₂	#19128369	1	1	yes	characteristic
353	C₂	#19128500	1	1	yes	characteristic
354	C₂	#19128507	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₃
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#461×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	#19128087	C₂
hypercenter	#461	D₈₃
hypocenter	C₈₃	#3401
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #461

Complements

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

N	H ≅ G/N	type
1. #19128434	C₂	semidirect
2. #19128425	C₂	semidirect
3. #19128093	C₂	semidirect
5. #19128082	C₂	semidirect
6. #19127926	C₂	semidirect
7. #19127921	C₂	semidirect
8. #19127919	C₂	semidirect
9. #19127917	C₂	semidirect
10. #19127899	C₂	semidirect
11. #19127830	C₂	semidirect
12. #19127811	C₂	semidirect
13. #19127805	C₂	semidirect
14. #19127786	C₂	semidirect
15. #461×C₈₃	C₂	semidirect
178. C₈₃	#3401	semidirect
188. #461	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128434	2
#19128425	2
#19128093	2
#19128087	2
#19128082	2
#19127926	2
#19127921	2
#19127919	2
#19127917	2
#19127899	2
#19127830	2
#19127811	2
#19127805	2
#19127786	2
#461×C₈₃	2
#3401	83