Subgroups — #24917438

Order 10624.

398 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128280	C₂	1	1	yes	maximal, characteristic
2	#19128181	C₂	1	1	yes	maximal, characteristic, complemented
3	#19128045	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	#19128037	C₂	1	1	yes	maximal, characteristic, complemented
7	#19128023	C₂	1	1	yes	maximal, characteristic, complemented
8	#19128013	C₂	1	1	yes	maximal, characteristic, complemented
9	#19128012	C₂	1	1	yes	maximal, characteristic, complemented
10	#19127740	C₂	1	1	yes	maximal, characteristic, complemented
11	#107×D₈₃	C₂	1	1	yes	maximal, characteristic, complemented
12	#19127699	C₂	1	1	yes	maximal, characteristic, complemented
13	#19127694	C₂	1	1	yes	maximal, characteristic, complemented
14	#19127666	C₂	1	1	yes	maximal, characteristic, complemented
15	#482×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13050590	C₂²	1	1	yes	characteristic
17	C₂×#13050589	C₂²	1	1	yes	characteristic
18	C₂×#13050573	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₈₃	C₂²	1	1	yes	characteristic
20	#14581772	C₂²	1	1	yes	characteristic
21	#14581771	C₂²	1	1	yes	characteristic
22	#14581769	—	2	1	no	—
23	#14581765	—	2	1	no	—
24	#14581764	—	2	1	no	—
25	#14581763	—	2	1	no	—
26	#14581763	C₂²	1	1	yes	characteristic
27	C₄×Dic₁₆₆	—	2	1	no	—
28	C₄×D₃₃₂	—	2	1	no	—
29	(C₂²⋊D₈₃)⋊C₄	—	2	1	no	—
30	(C₂²⋊D₈₃)⋊C₄	—	2	1	no	—
31	#14581750	C₂²	1	1	yes	characteristic
32	#14581750	C₂²	1	1	yes	characteristic
33	C₂×C₄⋊Dic₈₃	C₂²	1	1	yes	characteristic
34	C₄⋊C₄×D₈₃	C₂²	1	1	yes	characteristic
35	C₄⋊C₄×D₈₃	C₂²	1	1	yes	characteristic
36	C₄⋊C₄×D₈₃	C₂²	1	1	yes	characteristic
37	#14581726	C₂²	1	1	yes	characteristic
38	#14581726	C₂²	1	1	yes	characteristic
39	#14581725	C₂²	1	1	yes	characteristic
40	#14581725	C₂²	1	1	yes	characteristic
41	#14581725	C₂²	1	1	yes	characteristic
42	#14581723	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	#14581713	C₂²	1	1	yes	characteristic
48	#14581713	C₂²	1	1	yes	characteristic
49	C₂²⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
50	(C₂×D₈₃)⋊C₈	C₂²	1	1	yes	characteristic
51	(C₂×D₈₃)⋊C₈	C₂²	1	1	yes	characteristic
52	#121×C₈₃	C₂²	1	1	yes	characteristic
53	#117×C₈₃	C₂²	1	1	yes	characteristic
54	#107×C₈₃	C₂²	1	1	yes	characteristic
55	#106×C₈₃	C₂²	1	1	yes	characteristic
56	#102×C₈₃	C₂²	1	1	yes	characteristic
57	#102×C₈₃	C₂²	1	1	yes	characteristic
58	C₂²⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
59	#13050590	—	2	2	no	—
60	#13050590	—	2	2	no	—
61	#13050589	—	2	2	no	—
62	#13050589	—	2	2	no	—
63	C₂×Dic₁₆₆	D₄	1	1	yes	characteristic
64	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
65	C₂×D₃₃₂	D₄	1	1	yes	characteristic
66	C₂×C₂²⋊D₈₃	—	2	1	no	—
67	C₂×C₂²⋊D₈₃	—	2	1	no	—
68	C₂×C₂²⋊D₈₃	D₄	1	1	yes	characteristic
69	C₂×C₂²⋊D₈₃	D₄	1	1	yes	characteristic
70	C₂²×Dic₈₃	—	2	1	no	—
71	C₂²×Dic₈₃	—	2	1	no	—
72	C₂×C₄×D₈₃	—	2	1	no	—
73	C₂×C₄×D₈₃	—	2	1	no	—
74	C₂×C₄×D₈₃	—	2	1	no	—
75	C₂×C₄×D₈₃	C₂³	1	1	yes	characteristic
76	C₂×C₄×D₈₃	C₂³	1	1	yes	characteristic
77	#13050573	—	2	2	no	—
78	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
79	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
80	C₈×D₈₃	—	2	2	no	—
81	C₄⋊Dic₈₃	—	2	1	no	—
82	C₄⋊Dic₈₃	—	2	1	no	—
83	C₄⋊Dic₈₃	C₂³	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	C₂²⋊Dic₈₃	—	2	1	no	—
91	C₂²⋊Dic₈₃	—	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	C₄×Dic₈₃	—	2	1	no	—
96	C₄×Dic₈₃	—	2	1	no	—
97	C₂×D₄×C₈₃	—	2	1	no	—
98	C₂×D₄×C₈₃	C₂³	1	1	yes	characteristic
99	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
100	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
101	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
103	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
104	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
105	C₂²⋊C₄×C₈₃	—	2	1	no	—
106	C₂²⋊C₄×C₈₃	—	2	1	no	—
107	C₄²×C₈₃	—	2	1	no	—
108	C₂³×C₈₃	—	2	1	no	—
109	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
110	C₂²×D₈₃	C₂×D₄	1	1	yes	characteristic
111	C₂²×D₈₃	C₂×D₄	1	1	yes	characteristic
112	D₄×C₈₃	—	2	1	no	—
113	D₄×C₈₃	—	2	1	no	—
114	D₄×C₈₃	—	8	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	—	2	1	no	—
119	C₂×C₄×C₈₃	—	2	1	no	—
120	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
122	C₂²⋊D₈₃	—	2	2	no	—
123	C₂²⋊D₈₃	—	2	2	no	—
124	C₂²⋊D₈₃	—	4	2	no	—
125	C₂²⋊D₈₃	—	8	1	no	—
126	C₂×Dic₈₃	—	2	1	no	—
127	C₂×Dic₈₃	—	2	1	no	—
128	C₂×Dic₈₃	—	2	1	no	—
129	C₂×Dic₈₃	—	2	1	no	—
130	C₂×Dic₈₃	—	4	2	no	—
131	C₂×Dic₈₃	—	8	1	no	—
132	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
133	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
134	D₃₃₂	—	2	2	no	—
135	C₄×D₈₃	—	2	1	no	—
136	C₄×D₈₃	—	2	1	no	—
137	C₄×D₈₃	—	8	1	no	—
138	C₄×D₈₃	—	8	1	no	—
139	C₄×D₈₃	—	8	1	no	—
140	C₄×D₈₃	#41	1	2	yes	—
141	C₄×D₈₃	#41	1	2	yes	—
142	Dic₁₆₆	—	2	2	no	—
143	Dic₁₆₆	—	2	1	no	—
144	Dic₁₆₆	—	2	1	no	—
145	C₈×C₈₃	—	2	1	no	—
146	C₈×C₈₃	—	2	1	no	—
147	C₈₃⋊C₈	—	2	1	no	—
148	C₈₃⋊C₈	—	2	1	no	—
149	C₂²×C₈₃	—	2	1	no	—
150	C₂²×C₈₃	—	2	1	no	—
151	C₂²×C₈₃	—	2	1	no	—
152	C₂²×C₈₃	—	4	1	no	—
153	C₂²×C₈₃	—	4	1	no	—
154	C₂²×C₈₃	—	4	1	no	—
155	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
156	C₂×D₈₃	—	2	1	no	—
157	C₂×D₈₃	—	2	1	no	—
158	C₂×D₈₃	—	2	1	no	—
159	C₂×D₈₃	—	2	1	no	—
160	C₂×D₈₃	—	2	1	no	—
161	C₂×D₈₃	#123	1	2	yes	—
162	C₄×C₈₃	—	2	1	no	—
163	C₄×C₈₃	—	4	1	no	—
164	C₄×C₈₃	—	4	1	no	—
165	C₄×C₈₃	—	4	1	no	—
166	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
167	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
168	Dic₈₃	—	2	1	no	—
169	Dic₈₃	—	4	1	no	—
170	Dic₈₃	—	4	1	no	—
171	Dic₈₃	—	4	1	no	—
172	Dic₈₃	—	4	1	no	—
173	Dic₈₃	#123	1	2	yes	—
174	C₂×C₈₃	—	4	1	no	—
175	C₂×C₈₃	—	8	1	no	—
176	C₂×C₈₃	C₂×#123	1	1	yes	characteristic
177	C₂×C₈₃	#576	1	1	yes	characteristic
178	C₂×C₈₃	#578	1	1	yes	characteristic
179	D₈₃	—	2	2	no	—
180	D₈₃	—	4	1	no	—
181	#3098	—	83	1	no	maximal, Sylow
182	C₈₃	#3098	1	1	yes	characteristic, complemented, Sylow
183	#537	—	83	1	no	—
184	#520	—	83	1	no	—
185	#485	—	83	1	no	—
186	#485	—	83	1	no	—
187	#484	—	83	1	no	—
188	#484	—	83	1	no	—
189	#482	—	83	1	no	—
190	#482	D₈₃	1	1	yes	characteristic, complemented
191	#480	—	83	1	no	—
192	#480	—	83	1	no	—
193	#428	—	83	1	no	—
194	C₂×#107	—	83	1	no	—
195	#417	—	83	1	no	—
196	#416	—	83	1	no	—
197	#408	—	83	1	no	—
198	C₂×#41	—	83	1	no	—
199	C₂×#41	—	83	1	no	—
200	C₂×M₁₆	—	83	1	no	—
201	C₂²×C₈	—	83	1	no	—
202	C₂²⋊Q₈	—	83	1	no	—
203	C₂²⋊Q₈	—	83	1	no	—
204	C₂²⋊Q₈	—	166	1	no	—
205	#121	—	83	1	no	—
206	#121	—	166	1	no	—
207	#121	—	166	1	no	—
208	#121	—	166	1	no	—
209	#121	C₂×D₈₃	1	1	yes	characteristic
210	C₄×Q₈	—	166	1	no	—
211	C₄×D₄	—	166	1	no	—
212	C₄×D₄	—	166	1	no	—
213	C₄×D₄	—	166	1	no	—
214	#117	—	83	1	no	—
215	#117	—	83	1	no	—
216	#117	C₂×D₈₃	1	1	yes	characteristic
217	C₂×C₄⋊C₄	—	83	1	no	—
218	C₂×C₄⋊C₄	—	83	1	no	—
219	C₂×C₄⋊C₄	—	83	1	no	—
220	C₂×C₄⋊C₄	—	83	1	no	—
221	#107	—	83	1	no	—
222	#107	—	83	1	no	—
223	#107	—	83	1	no	—
224	#107	—	83	1	no	—
225	#107	—	83	1	no	—
226	#107	C₂×D₈₃	1	1	yes	characteristic
227	#106	—	83	1	no	—
228	#106	C₂×D₈₃	1	1	yes	characteristic
229	#103	—	83	1	no	—
230	#103	—	83	1	no	—
231	#103	—	83	1	no	—
232	#103	—	83	1	no	—
233	#102	—	83	1	no	—
234	#102	—	83	1	no	—
235	#102	C₂×D₈₃	1	1	yes	characteristic
236	#102	C₂×D₈₃	1	1	yes	characteristic
237	C₂²⋊C₈	—	83	1	no	—
238	C₂²⋊C₈	—	83	1	no	—
239	C₂²⋊C₈	—	83	1	no	—
240	C₂²⋊C₈	C₂×D₈₃	1	1	yes	characteristic
241	#41	—	166	2	no	—
242	#41	—	166	2	no	—
243	#41	—	166	2	no	—
244	#41	—	166	2	no	—
245	C₂×Q₈	—	83	1	no	—
246	C₂×Q₈	—	83	1	no	—
247	C₂×D₄	—	2	1	no	—
248	C₂×D₄	—	83	1	no	—
249	C₂×D₄	—	83	1	no	—
250	C₂×D₄	—	83	1	no	—
251	C₂×D₄	—	166	1	no	—
252	C₂×D₄	—	166	1	no	—
253	C₂×D₄	C₂²×D₈₃	1	1	yes	characteristic
254	C₂²×C₄	—	83	1	no	—
255	C₂²×C₄	—	83	1	no	—
256	C₂²×C₄	—	166	1	no	—
257	C₂²×C₄	—	166	1	no	—
258	C₂²×C₄	—	166	1	no	—
259	C₂²×C₄	—	166	1	no	—
260	C₂²×C₄	—	166	1	no	—
261	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
262	M₁₆	—	166	2	no	—
263	C₂×C₈	—	83	1	no	—
264	C₂×C₈	—	83	1	no	—
265	C₂×C₈	—	166	2	no	—
266	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
267	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
268	C₄⋊C₄	—	83	1	no	—
269	C₄⋊C₄	—	83	1	no	—
270	C₄⋊C₄	—	83	1	no	—
271	C₄⋊C₄	—	166	1	no	—
272	C₄⋊C₄	—	166	1	no	—
273	C₄⋊C₄	—	166	1	no	—
274	C₄⋊C₄	—	166	1	no	—
275	C₄⋊C₄	—	166	1	no	—
276	C₄⋊C₄	—	166	1	no	—
277	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
278	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
279	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
280	C₂²⋊C₄	—	2	1	no	—
281	C₂²⋊C₄	—	2	1	no	—
282	C₂²⋊C₄	—	166	1	no	—
283	C₂²⋊C₄	—	166	1	no	—
284	C₂²⋊C₄	—	166	1	no	—
285	C₂²⋊C₄	—	166	1	no	—
286	C₂²⋊C₄	—	166	1	no	—
287	C₄²	—	2	1	no	—
288	C₄²	—	166	1	no	—
289	C₄²	—	166	1	no	—
290	C₂³	—	2	1	no	—
291	C₂³	—	83	1	no	—
292	C₂³	—	83	1	no	—
293	C₂³	D₄×D₈₃	1	1	yes	characteristic
294	Q₈	—	166	2	no	—
295	Q₈	—	166	1	no	—
296	Q₈	—	166	1	no	—
297	D₄	—	2	1	no	—
298	D₄	—	2	1	no	—
299	D₄	—	8	1	no	—
300	D₄	—	332	2	no	—
301	D₄	—	664	1	no	—
302	D₄	—	166	2	no	—
303	D₄	—	166	2	no	—
304	D₄	—	166	2	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	332	2	no	—
311	C₂×C₄	—	83	1	no	—
312	C₂×C₄	—	83	2	no	—
313	C₂×C₄	—	83	2	no	—
314	C₂×C₄	—	83	1	no	—
315	C₂×C₄	—	664	1	no	—
316	C₂×C₄	—	664	1	no	—
317	C₂×C₄	—	664	1	no	—
318	C₂×C₄	—	664	1	no	—
319	C₂×C₄	—	166	1	no	—
320	C₂×C₄	—	166	1	no	—
321	C₂×C₄	—	166	1	no	—
322	C₂×C₄	—	166	1	no	—
323	C₂×C₄	—	166	1	no	—
324	C₂×C₄	—	166	1	no	—
325	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
326	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
327	C₈	—	2	1	no	—
328	C₈	—	2	1	no	—
329	C₈	—	166	1	no	—
330	C₈	—	166	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	2	1	no	—
334	C₂²	—	4	1	no	—
335	C₂²	—	4	1	no	—
336	C₂²	—	4	1	no	—
337	C₂²	—	83	2	no	—
338	C₂²	—	166	1	no	—
339	C₂²	—	166	1	no	—
340	C₂²	—	166	1	no	—
341	C₂²	—	166	1	no	—
342	C₂²	—	166	1	no	—
343	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
344	C₄	—	2	1	no	—
345	C₄	—	4	1	no	—
346	C₄	—	4	1	no	—
347	C₄	—	4	1	no	—
348	C₄	—	332	1	no	—
349	C₄	—	332	1	no	—
350	C₄	—	332	1	no	—
351	C₄	—	332	1	no	—
352	C₄	—	83	2	no	—
353	C₄	—	166	1	no	—
354	C₄	#41×D₈₃	1	1	yes	characteristic
355	C₄	#41×D₈₃	1	1	yes	characteristic
356	C₂	—	4	1	no	—
357	C₂	—	8	1	no	—
358	C₂	—	332	1	no	—
359	C₂	—	166	2	no	—
360	C₂	#123×D₈₃	1	1	yes	characteristic
361	C₂	#19128505	1	1	yes	characteristic
362	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	#482×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	#19128280	C₂
hypercenter	#482	D₈₃
hypocenter	C₈₃	#3098
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #482

Complements

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

N	H ≅ G/N	type
2. #19128181	C₂	semidirect
3. #19128045	C₂	semidirect
4. #19128042	C₂	semidirect
5. #19128038	C₂	semidirect
6. #19128037	C₂	semidirect
7. #19128023	C₂	semidirect
8. #19128013	C₂	semidirect
9. #19128012	C₂	semidirect
10. #19127740	C₂	semidirect
11. #107×D₈₃	C₂	semidirect
12. #19127699	C₂	semidirect
13. #19127694	C₂	semidirect
14. #19127666	C₂	semidirect
15. #482×C₈₃	C₂	semidirect
182. C₈₃	#3098	semidirect
190. #482	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128280	2
#19128181	2
#19128045	2
#19128042	2
#19128038	2
#19128037	2
#19128023	2
#19128013	2
#19128012	2
#19127740	2
#107×D₈₃	2
#19127699	2
#19127694	2
#19127666	2
#482×C₈₃	2
#3098	83