Subgroups — #24919451

Order 10624.

420 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128363	C₂	1	1	yes	maximal, characteristic, complemented
2	#19128354	C₂	1	1	yes	maximal, characteristic, complemented
3	#19128063	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128038	C₂	1	1	yes	maximal, characteristic, complemented
5	#19128016	C₂	1	1	yes	maximal, characteristic, complemented
6	#19127948	C₂	1	1	yes	maximal, characteristic, complemented
7	#19127938	C₂	1	1	yes	maximal, characteristic, complemented
8	#19127917	C₂	1	1	yes	maximal, characteristic, complemented
9	#19127849	C₂	1	1	yes	maximal, characteristic, complemented
10	#19127841	C₂	1	1	yes	maximal, characteristic, complemented
11	#19127799	C₂	1	1	yes	maximal, characteristic, complemented
12	#19127771	C₂	1	1	yes	maximal, characteristic, complemented
13	#19127701	C₂	1	1	yes	maximal, characteristic, complemented
14	#103×D₈₃	C₂	1	1	yes	maximal, characteristic, complemented
15	#477×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13050591	—	2	1	no	—
17	C₂×#13050591	C₂²	1	1	yes	characteristic
18	C₂×Q₈×D₈₃	C₂²	1	1	yes	characteristic
19	C₂×#13050579	C₂²	1	1	yes	characteristic
20	C₂×#13050577	C₂²	1	1	yes	characteristic
21	C₂×D₆₆₄	C₂²	1	1	yes	characteristic
22	C₂×#13050573	C₂²	1	1	yes	characteristic
23	C₂×C₈×D₈₃	C₂²	1	1	yes	characteristic
24	#14581780	—	2	1	no	—
25	#14581780	C₂²	1	1	yes	characteristic
26	#14581777	—	2	1	no	—
27	#14581777	—	2	1	no	—
28	#14581772	—	2	1	no	—
29	#14581772	C₂²	1	1	yes	characteristic
30	#14581768	—	2	1	no	—
31	#14581767	—	2	1	no	—
32	#14581767	C₂²	1	1	yes	characteristic
33	#14581767	C₂²	1	1	yes	characteristic
34	#14581767	C₂²	1	1	yes	characteristic
35	C₄×D₃₃₂	—	2	1	no	—
36	C₄×D₃₃₂	C₂²	1	1	yes	characteristic
37	C₄×D₃₃₂	C₂²	1	1	yes	characteristic
38	C₄⋊C₄×D₈₃	C₂²	1	1	yes	characteristic
39	C₄⋊C₄×D₈₃	C₂²	1	1	yes	characteristic
40	#14581725	C₂²	1	1	yes	characteristic
41	#14581717	C₂²	1	1	yes	characteristic
42	#14581716	C₂²	1	1	yes	characteristic
43	#14581715	C₂²	1	1	yes	characteristic
44	#14581715	C₂²	1	1	yes	characteristic
45	#14581714	C₂²	1	1	yes	characteristic
46	#14581712	C₂²	1	1	yes	characteristic
47	#14581712	C₂²	1	1	yes	characteristic
48	#14581712	C₂²	1	1	yes	characteristic
49	(C₂×D₈₃)⋊C₈	C₂²	1	1	yes	characteristic
50	(C₂×D₈₃)⋊C₈	C₂²	1	1	yes	characteristic
51	#14581703	C₂²	1	1	yes	characteristic
52	C₄⋊Q₈×C₈₃	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₈₃	C₂²	1	1	yes	characteristic
54	#107×C₈₃	C₂²	1	1	yes	characteristic
55	#107×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	#13050591	—	2	2	no	—
60	#13050591	—	4	1	no	—
61	#13050591	—	4	4	no	—
62	C₂×Dic₁₆₆	—	2	1	no	—
63	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
64	Q₈×D₈₃	—	2	2	no	—
65	Q₈×D₈₃	—	4	1	no	—
66	C₂×D₃₃₂	—	2	1	no	—
67	C₂×D₃₃₂	—	2	1	no	—
68	C₂×D₃₃₂	—	2	1	no	—
69	C₂×D₃₃₂	—	2	1	no	—
70	C₂×D₃₃₂	D₄	1	1	yes	characteristic
71	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
72	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
73	C₂×C₄×D₈₃	—	2	1	no	—
74	C₂×C₄×D₈₃	—	2	1	no	—
75	C₂×C₄×D₈₃	—	2	1	no	—
76	C₂×C₄×D₈₃	—	2	1	no	—
77	C₂×C₄×D₈₃	—	2	1	no	—
78	C₂×C₄×D₈₃	C₂³	1	1	yes	characteristic
79	C₂×C₄×D₈₃	C₂³	1	1	yes	characteristic
80	#13050579	—	4	1	no	—
81	#13050577	—	2	2	no	—
82	D₆₆₄	—	4	1	no	—
83	#13050573	—	4	1	no	—
84	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
85	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
86	C₈×D₈₃	—	4	1	no	—
87	C₄⋊Dic₈₃	—	2	1	no	—
88	C₄⋊Dic₈₃	D₄	1	1	yes	characteristic
89	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
90	Dic₈₃⋊C₄	—	2	1	no	—
91	Dic₈₃⋊C₄	—	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	(C₂×D₈₃)⋊C₄	—	2	1	no	—
96	(C₂×D₈₃)⋊C₄	—	2	1	no	—
97	(C₂×D₈₃)⋊C₄	—	2	1	no	—
98	(C₂×D₈₃)⋊C₄	—	2	1	no	—
99	(C₂×D₈₃)⋊C₄	D₄	1	1	yes	characteristic
100	(C₂×D₈₃)⋊C₄	D₄	1	1	yes	characteristic
101	C₂×Q₈×C₈₃	—	2	1	no	—
102	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
107	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
108	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
109	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
110	C₄²×C₈₃	—	2	1	no	—
111	C₄²×C₈₃	C₂³	1	1	yes	characteristic
112	C₂²×D₈₃	—	2	1	no	—
113	C₂²×D₈₃	—	2	1	no	—
114	C₂²×D₈₃	C₂×D₄	1	1	yes	characteristic
115	C₂²×D₈₃	C₂×D₄	1	1	yes	characteristic
116	Q₈×C₈₃	—	2	1	no	—
117	Q₈×C₈₃	—	4	2	no	—
118	Q₈×C₈₃	#41	1	2	yes	complemented
119	C₂×C₄×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	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
126	C₂×Dic₈₃	—	2	1	no	—
127	C₂×Dic₈₃	—	2	1	no	—
128	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
129	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
130	D₃₃₂	—	2	2	no	—
131	D₃₃₂	—	2	1	no	—
132	D₃₃₂	—	2	1	no	—
133	D₃₃₂	—	2	1	no	—
134	D₃₃₂	—	2	1	no	—
135	D₃₃₂	—	4	2	no	—
136	D₃₃₂	—	4	2	no	—
137	D₃₃₂	—	4	2	no	—
138	D₃₃₂	—	4	2	no	—
139	C₄×D₈₃	—	2	1	no	—
140	C₄×D₈₃	—	2	1	no	—
141	C₄×D₈₃	—	2	1	no	—
142	C₄×D₈₃	—	2	1	no	—
143	C₄×D₈₃	—	4	2	no	—
144	C₄×D₈₃	—	4	2	no	—
145	C₄×D₈₃	—	4	2	no	—
146	C₄×D₈₃	—	4	2	no	—
147	C₄×D₈₃	—	4	2	no	—
148	Dic₁₆₆	—	2	1	no	—
149	Dic₁₆₆	—	2	1	no	—
150	Dic₁₆₆	—	4	2	no	—
151	C₈×C₈₃	—	2	1	no	—
152	C₈×C₈₃	—	2	1	no	—
153	C₈₃⋊C₈	—	2	1	no	—
154	C₈₃⋊C₈	—	2	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	—	4	1	no	—
162	C₂×D₈₃	—	4	1	no	—
163	C₂×D₈₃	—	4	1	no	—
164	C₂×D₈₃	—	4	1	no	—
165	C₂×D₈₃	—	4	1	no	—
166	C₂×D₈₃	—	4	1	no	—
167	C₂×D₈₃	AGL(1,ℤ/8)	1	2	yes	—
168	C₄×C₈₃	—	2	2	no	—
169	C₄×C₈₃	—	2	1	no	—
170	C₄×C₈₃	—	2	1	no	—
171	C₄×C₈₃	—	4	1	no	—
172	C₄×C₈₃	—	4	1	no	—
173	C₄×C₈₃	—	4	1	no	—
174	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
175	C₄×C₈₃	H(5,𝔽₂)	1	1	yes	characteristic
176	Dic₈₃	—	2	1	no	—
177	Dic₈₃	—	2	1	no	—
178	Dic₈₃	—	4	1	no	—
179	Dic₈₃	—	4	1	no	—
180	C₂×C₈₃	#546	1	1	yes	characteristic
181	C₂×C₈₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
182	C₂×C₈₃	#578	1	1	yes	characteristic
183	D₈₃	—	2	2	no	—
184	D₈₃	—	4	1	no	—
185	D₈₃	—	8	1	no	—
186	D₈₃	—	8	1	no	—
187	#3308	—	83	1	no	maximal, Sylow
188	C₈₃	#3308	1	1	yes	characteristic, complemented, Sylow
189	#548	—	83	1	no	—
190	#547	—	83	1	no	—
191	#489	—	83	1	no	—
192	#484	—	83	1	no	—
193	#480	—	83	1	no	—
194	#477	D₈₃	1	1	yes	characteristic, complemented
195	#468	—	83	1	no	—
196	#466	—	83	1	no	—
197	#463	—	83	1	no	—
198	#449	—	83	1	no	—
199	#448	—	83	1	no	—
200	#440	—	83	1	no	—
201	#435	—	83	1	no	—
202	#417	—	83	1	no	—
203	C₂×#103	—	83	1	no	—
204	C₂×#41	—	83	1	no	—
205	C₂×#41	—	166	1	no	—
206	C₂²×Q₈	—	83	1	no	—
207	C₂×SD₁₆	—	83	1	no	—
208	C₂×SD₁₆	—	83	1	no	—
209	C₂×D₈	—	83	1	no	—
210	C₂×M₁₆	—	83	1	no	—
211	C₂²×C₈	—	83	1	no	—
212	C₄⋊Q₈	C₂×D₈₃	1	1	yes	characteristic
213	#124	—	83	1	no	—
214	#124	—	166	1	no	—
215	#123	—	166	1	no	—
216	#123	—	166	1	no	—
217	C₂²⋊Q₈	—	83	1	no	—
218	C₂²⋊Q₈	—	166	1	no	—
219	C₂²⋊Q₈	—	166	1	no	—
220	#121	—	83	1	no	—
221	#121	—	83	1	no	—
222	#121	—	83	1	no	—
223	#121	—	166	1	no	—
224	C₄×Q₈	C₂×D₈₃	1	1	yes	characteristic
225	C₄×D₄	—	83	1	no	—
226	C₄×D₄	—	83	1	no	—
227	C₄×D₄	—	166	1	no	—
228	C₂×C₄⋊C₄	—	83	1	no	—
229	C₂×C₄⋊C₄	—	83	1	no	—
230	#107	—	83	1	no	—
231	#107	C₂×D₈₃	1	1	yes	characteristic
232	#107	C₂×D₈₃	1	1	yes	characteristic
233	C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
234	#103	—	83	1	no	—
235	#103	—	83	1	no	—
236	#103	—	83	1	no	—
237	#103	—	83	1	no	—
238	#103	C₂×D₈₃	1	1	yes	characteristic
239	#103	C₂×D₈₃	1	1	yes	characteristic
240	#102	—	83	1	no	—
241	#102	—	83	1	no	—
242	#102	—	83	1	no	—
243	#102	—	83	1	no	—
244	C₂²⋊C₈	—	83	1	no	—
245	C₂²⋊C₈	—	83	1	no	—
246	#97	—	83	1	no	—
247	#41	—	332	4	no	—
248	#41	—	332	1	no	—
249	#41	—	166	2	no	—
250	C₂×Q₈	—	2	1	no	—
251	C₂×Q₈	—	332	1	no	—
252	C₂×Q₈	—	83	1	no	—
253	C₂×Q₈	—	166	1	no	—
254	C₂×Q₈	—	166	2	no	—
255	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
256	C₂×D₄	—	83	1	no	—
257	C₂×D₄	—	83	1	no	—
258	C₂×D₄	—	83	1	no	—
259	C₂×D₄	—	166	1	no	—
260	C₂×D₄	—	166	1	no	—
261	C₂×D₄	—	166	1	no	—
262	C₂×D₄	—	166	1	no	—
263	C₂²×C₄	—	83	1	no	—
264	C₂²×C₄	—	83	1	no	—
265	C₂²×C₄	—	166	1	no	—
266	C₂²×C₄	—	166	1	no	—
267	C₂²×C₄	—	166	1	no	—
268	C₂²×C₄	—	166	1	no	—
269	C₂²×C₄	—	166	1	no	—
270	SD₁₆	—	332	1	no	—
271	SD₁₆	—	166	2	no	—
272	D₈	—	332	1	no	—
273	M₁₆	—	332	1	no	—
274	C₂×C₈	—	332	1	no	—
275	C₂×C₈	—	83	1	no	—
276	C₂×C₈	—	83	1	no	—
277	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
278	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
279	C₄⋊C₄	—	2	1	no	—
280	C₄⋊C₄	—	2	1	no	—
281	C₄⋊C₄	—	83	1	no	—
282	C₄⋊C₄	—	83	1	no	—
283	C₄⋊C₄	—	166	1	no	—
284	C₄⋊C₄	—	166	1	no	—
285	C₄⋊C₄	—	166	1	no	—
286	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
287	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
288	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
289	C₂²⋊C₄	—	83	1	no	—
290	C₂²⋊C₄	—	83	1	no	—
291	C₂²⋊C₄	—	166	1	no	—
292	C₂²⋊C₄	—	166	1	no	—
293	C₂²⋊C₄	—	166	1	no	—
294	C₂²⋊C₄	—	166	1	no	—
295	C₂²⋊C₄	—	166	1	no	—
296	C₂²⋊C₄	—	166	1	no	—
297	C₂²⋊C₄	—	166	1	no	—
298	C₄²	—	2	1	no	—
299	C₄²	C₂²×D₈₃	1	1	yes	characteristic
300	C₂³	—	83	1	no	—
301	C₂³	—	83	1	no	—
302	C₂³	—	166	1	no	—
303	C₂³	—	166	1	no	—
304	Q₈	—	2	1	no	—
305	Q₈	—	4	2	no	—
306	Q₈	—	332	2	no	—
307	Q₈	—	166	1	no	—
308	Q₈	—	166	1	no	—
309	Q₈	#13050591	1	2	yes	complemented
310	D₄	—	332	2	no	—
311	D₄	—	332	2	no	—
312	D₄	—	332	2	no	—
313	D₄	—	332	2	no	—
314	D₄	—	166	2	no	—
315	D₄	—	166	1	no	—
316	D₄	—	166	1	no	—
317	D₄	—	166	1	no	—
318	D₄	—	166	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	332	2	no	—
324	C₂×C₄	—	332	2	no	—
325	C₂×C₄	—	332	2	no	—
326	C₂×C₄	—	332	2	no	—
327	C₂×C₄	—	332	2	no	—
328	C₂×C₄	—	83	1	no	—
329	C₂×C₄	—	83	1	no	—
330	C₂×C₄	—	166	1	no	—
331	C₂×C₄	—	166	1	no	—
332	C₂×C₄	—	166	1	no	—
333	C₂×C₄	—	166	1	no	—
334	C₂×C₄	—	166	1	no	—
335	C₂×C₄	—	166	1	no	—
336	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
337	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
338	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
339	C₈	—	2	1	no	—
340	C₈	—	2	1	no	—
341	C₈	—	166	1	no	—
342	C₈	—	166	1	no	—
343	C₂²	—	332	1	no	—
344	C₂²	—	332	1	no	—
345	C₂²	—	332	1	no	—
346	C₂²	—	332	1	no	—
347	C₂²	—	332	1	no	—
348	C₂²	—	332	1	no	—
349	C₂²	—	83	2	no	—
350	C₂²	—	166	1	no	—
351	C₂²	—	166	1	no	—
352	C₂²	—	166	1	no	—
353	C₂²	—	166	1	no	—
354	C₂²	—	166	1	no	—
355	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
356	C₄	—	2	2	no	—
357	C₄	—	2	1	no	—
358	C₄	—	2	1	no	—
359	C₄	—	4	1	no	—
360	C₄	—	4	1	no	—
361	C₄	—	4	1	no	—
362	C₄	—	332	1	no	—
363	C₄	—	332	1	no	—
364	C₄	—	166	1	no	—
365	C₄	—	166	1	no	—
366	C₄	C₂×#13050591	1	1	yes	characteristic
367	C₄	#14581838	1	1	yes	characteristic
368	C₂	—	332	1	no	—
369	C₂	—	664	1	no	—
370	C₂	—	664	1	no	—
371	C₂	—	166	2	no	—
372	C₂	#19128342	1	1	yes	characteristic
373	C₂	AGL(1,ℤ/8)×D₈₃	1	1	yes	characteristic
374	C₂	#19128514	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₃
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#477×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
hypercenter	#477	D₈₃
hypocenter	C₈₃	#3308
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #477

Complements

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

N	H ≅ G/N	type
1. #19128363	C₂	semidirect
2. #19128354	C₂	semidirect
3. #19128063	C₂	semidirect
4. #19128038	C₂	semidirect
5. #19128016	C₂	semidirect
6. #19127948	C₂	semidirect
7. #19127938	C₂	semidirect
8. #19127917	C₂	semidirect
9. #19127849	C₂	semidirect
10. #19127841	C₂	semidirect
11. #19127799	C₂	semidirect
12. #19127771	C₂	semidirect
13. #19127701	C₂	semidirect
14. #103×D₈₃	C₂	semidirect
15. #477×C₈₃	C₂	semidirect
118. Q₈×C₈₃	#41	semidirect
188. C₈₃	#3308	semidirect
194. #477	D₈₃	semidirect
309. Q₈	#13050591	semidirect

Maximal subgroups

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

subgroup	index
#19128363	2
#19128354	2
#19128063	2
#19128038	2
#19128016	2
#19127948	2
#19127938	2
#19127917	2
#19127849	2
#19127841	2
#19127799	2
#19127771	2
#19127701	2
#103×D₈₃	2
#477×C₈₃	2
#3308	83