Subgroups — #24919785

Order 10624.

430 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128369	C₂	1	1	yes	maximal, characteristic, complemented
2	#19128351	C₂	1	1	yes	maximal, characteristic, complemented
3	#19128046	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128021	C₂	1	1	yes	maximal, characteristic, complemented
5	#19128017	C₂	1	1	yes	maximal, characteristic, complemented
6	#19127985	C₂	1	1	yes	maximal, characteristic, complemented
7	#19127953	C₂	1	1	yes	maximal, characteristic, complemented
8	#19127894	C₂	1	1	yes	maximal, characteristic, complemented
9	#19127844	C₂	1	1	yes	maximal, characteristic, complemented
10	#19127843	C₂	1	1	yes	maximal, characteristic, complemented
11	#19127792	C₂	1	1	yes	maximal, characteristic, complemented
12	#19127777	C₂	1	1	yes	maximal, characteristic, complemented
13	#19127703	C₂	1	1	yes	maximal, characteristic, complemented
14	#19127699	C₂	1	1	yes	maximal, characteristic, complemented
15	#469×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13050590	C₂²	1	1	yes	characteristic
17	C₂×#13050589	—	2	1	no	—
18	C₂×#13050589	C₂²	1	1	yes	characteristic
19	C₂×#13050578	C₂²	1	1	yes	characteristic
20	C₂×#13050575	C₂²	1	1	yes	characteristic
21	C₂×#13050573	C₂²	1	1	yes	characteristic
22	#14581791	C₂²	1	1	yes	characteristic
23	#14581788	—	2	1	no	—
24	#14581788	C₂²	1	1	yes	characteristic
25	#14581774	—	2	1	no	—
26	#14581773	—	2	1	no	—
27	#14581769	—	2	1	no	—
28	#14581767	C₂²	1	1	yes	characteristic
29	#14581765	—	2	1	no	—
30	#14581764	—	2	1	no	—
31	#14581763	C₂²	1	1	yes	characteristic
32	#14581757	C₂²	1	1	yes	characteristic
33	D₄×Dic₈₃	C₂²	1	1	yes	characteristic
34	C₄×C₂²⋊D₈₃	C₂²	1	1	yes	characteristic
35	(C₂²⋊D₈₃)⋊C₄	—	2	1	no	—
36	C₂×C₄⋊Dic₈₃	C₂²	1	1	yes	characteristic
37	C₄⋊C₄×D₈₃	C₂²	1	1	yes	characteristic
38	#14581723	C₂²	1	1	yes	characteristic
39	#14581723	C₂²	1	1	yes	characteristic
40	Dic₈₃⋊C₈	C₂²	1	1	yes	characteristic
41	#14581717	C₂²	1	1	yes	characteristic
42	#14581717	C₂²	1	1	yes	characteristic
43	#14581715	C₂²	1	1	yes	characteristic
44	#14581714	C₂²	1	1	yes	characteristic
45	#14581714	C₂²	1	1	yes	characteristic
46	#14581713	C₂²	1	1	yes	characteristic
47	#14581713	C₂²	1	1	yes	characteristic
48	#14581712	C₂²	1	1	yes	characteristic
49	C₂²⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
50	(C₂×D₈₃)⋊C₈	C₂²	1	1	yes	characteristic
51	C₈×Dic₈₃	C₂²	1	1	yes	characteristic
52	C₂×D₈×C₈₃	C₂²	1	1	yes	characteristic
53	C₂×M₁₆×C₈₃	C₂²	1	1	yes	characteristic
54	#121×C₈₃	C₂²	1	1	yes	characteristic
55	#121×C₈₃	C₂²	1	1	yes	characteristic
56	#106×C₈₃	C₂²	1	1	yes	characteristic
57	#102×C₈₃	C₂²	1	1	yes	characteristic
58	#102×C₈₃	C₂²	1	1	yes	characteristic
59	#13050590	—	2	2	no	—
60	#13050590	—	4	1	no	—
61	#13050589	—	2	2	no	—
62	#13050589	—	4	1	no	—
63	#13050589	—	4	4	no	—
64	C₂×Dic₁₆₆	—	2	1	no	—
65	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
66	C₂×D₃₃₂	—	2	1	no	—
67	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
68	C₂×C₂²⋊D₈₃	—	2	1	no	—
69	C₂×C₂²⋊D₈₃	—	2	1	no	—
70	C₂×C₂²⋊D₈₃	—	2	1	no	—
71	C₂×C₂²⋊D₈₃	—	2	1	no	—
72	C₂×C₂²⋊D₈₃	—	2	1	no	—
73	C₂×C₂²⋊D₈₃	D₄	1	1	yes	characteristic
74	C₂²×Dic₈₃	—	2	1	no	—
75	C₂²×Dic₈₃	—	2	1	no	—
76	C₂²×Dic₈₃	—	2	1	no	—
77	C₂×C₄×D₈₃	—	2	1	no	—
78	C₂×C₄×D₈₃	—	2	1	no	—
79	C₂×C₄×D₈₃	C₂³	1	1	yes	characteristic
80	#13050578	—	2	2	no	—
81	#13050575	—	4	1	no	—
82	#13050573	—	4	1	no	—
83	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
84	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
85	C₄⋊Dic₈₃	—	2	1	no	—
86	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
87	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
88	Dic₈₃⋊C₄	—	2	1	no	—
89	Dic₈₃⋊C₄	—	2	1	no	—
90	Dic₈₃⋊C₄	—	2	1	no	—
91	Dic₈₃⋊C₄	D₄	1	1	yes	characteristic
92	C₂²⋊Dic₈₃	—	2	1	no	—
93	C₂²⋊Dic₈₃	—	2	1	no	—
94	C₂²⋊Dic₈₃	D₄	1	1	yes	characteristic
95	(C₂×D₈₃)⋊C₄	—	2	1	no	—
96	(C₂×D₈₃)⋊C₄	D₄	1	1	yes	characteristic
97	C₄×Dic₈₃	—	2	1	no	—
98	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
99	C₂×D₄×C₈₃	—	2	1	no	—
100	C₂×D₄×C₈₃	—	2	1	no	—
101	C₂×D₄×C₈₃	C₂³	1	1	yes	characteristic
102	C₂×D₄×C₈₃	C₂³	1	1	yes	characteristic
103	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
104	D₈×C₈₃	—	4	1	no	—
105	M₁₆×C₈₃	—	4	1	no	—
106	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
107	C₂×C₈×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	C₂²⋊C₄×C₈₃	—	2	1	no	—
111	C₂²⋊C₄×C₈₃	—	2	1	no	—
112	C₂³×C₈₃	—	2	1	no	—
113	C₂³×C₈₃	—	2	1	no	—
114	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
115	C₂²×D₈₃	—	2	1	no	—
116	C₂²×D₈₃	C₂×D₄	1	1	yes	characteristic
117	D₄×C₈₃	—	2	1	no	—
118	D₄×C₈₃	—	2	1	no	—
119	D₄×C₈₃	—	2	1	no	—
120	D₄×C₈₃	—	2	1	no	—
121	D₄×C₈₃	—	4	2	no	—
122	D₄×C₈₃	—	8	1	no	—
123	C₂×C₄×C₈₃	—	2	1	no	—
124	C₂×C₄×C₈₃	—	2	1	no	—
125	C₂×C₄×C₈₃	—	2	1	no	—
126	C₂×C₄×C₈₃	—	2	1	no	—
127	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
128	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
129	C₂²⋊D₈₃	—	2	2	no	—
130	C₂²⋊D₈₃	—	4	2	no	—
131	C₂²⋊D₈₃	—	4	2	no	—
132	C₂²⋊D₈₃	—	4	2	no	—
133	C₂²⋊D₈₃	—	4	2	no	—
134	C₂²⋊D₈₃	—	8	1	no	—
135	C₂×Dic₈₃	—	2	1	no	—
136	C₂×Dic₈₃	—	2	1	no	—
137	C₂×Dic₈₃	—	2	1	no	—
138	C₂×Dic₈₃	—	4	2	no	—
139	C₂×Dic₈₃	—	4	2	no	—
140	C₂×Dic₈₃	—	4	2	no	—
141	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
142	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
143	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
144	D₃₃₂	—	2	1	no	—
145	D₃₃₂	—	2	1	no	—
146	D₃₃₂	—	8	1	no	—
147	C₄×D₈₃	—	2	1	no	—
148	C₄×D₈₃	—	2	1	no	—
149	C₄×D₈₃	—	4	2	no	—
150	C₄×D₈₃	—	4	2	no	—
151	Dic₁₆₆	—	2	1	no	—
152	Dic₁₆₆	—	4	2	no	—
153	Dic₁₆₆	#41	1	2	yes	complemented
154	C₈×C₈₃	—	2	1	no	—
155	C₈×C₈₃	—	2	1	no	—
156	C₈₃⋊C₈	—	2	1	no	—
157	C₈₃⋊C₈	—	2	1	no	—
158	C₂²×C₈₃	—	2	1	no	—
159	C₂²×C₈₃	—	2	1	no	—
160	C₂²×C₈₃	—	2	1	no	—
161	C₂²×C₈₃	—	4	1	no	—
162	C₂²×C₈₃	—	4	1	no	—
163	C₂²×C₈₃	—	4	1	no	—
164	C₂²×C₈₃	—	4	1	no	—
165	C₂²×C₈₃	—	4	1	no	—
166	C₂²×C₈₃	—	4	1	no	—
167	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
168	C₂×D₈₃	—	2	1	no	—
169	C₂×D₈₃	—	2	1	no	—
170	C₂×D₈₃	—	2	1	no	—
171	C₂×D₈₃	—	4	1	no	—
172	C₂×D₈₃	—	4	1	no	—
173	C₂×D₈₃	—	4	1	no	—
174	C₄×C₈₃	—	2	1	no	—
175	C₄×C₈₃	—	4	1	no	—
176	C₄×C₈₃	—	4	1	no	—
177	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
178	C₄×C₈₃	H(5,𝔽₂)	1	1	yes	characteristic
179	Dic₈₃	—	2	2	no	—
180	Dic₈₃	—	2	1	no	—
181	Dic₈₃	—	2	1	no	—
182	Dic₈₃	—	4	1	no	—
183	Dic₈₃	—	4	1	no	—
184	Dic₈₃	AGL(1,ℤ/8)	1	2	yes	—
185	C₂×C₈₃	—	4	1	no	—
186	C₂×C₈₃	—	8	1	no	—
187	C₂×C₈₃	—	8	1	no	—
188	C₂×C₈₃	#546	1	1	yes	characteristic
189	C₂×C₈₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
190	C₂×C₈₃	#577	1	1	yes	characteristic
191	D₈₃	—	4	1	no	—
192	D₈₃	—	8	1	no	—
193	#3332	—	83	1	no	maximal, Sylow
194	C₈₃	#3332	1	1	yes	characteristic, complemented, Sylow
195	#550	—	83	1	no	—
196	#547	—	83	1	no	—
197	#486	—	83	1	no	—
198	#481	—	83	1	no	—
199	#481	—	83	1	no	—
200	#475	—	83	1	no	—
201	#469	—	83	1	no	—
202	#469	D₈₃	1	1	yes	characteristic, complemented
203	#459	—	83	1	no	—
204	#449	—	83	1	no	—
205	#449	—	83	1	no	—
206	C₄×SD₁₆	—	83	1	no	—
207	#436	—	83	1	no	—
208	#417	—	83	1	no	—
209	#417	—	83	1	no	—
210	C₂×#41	—	83	1	no	—
211	C₂×#41	—	83	1	no	—
212	C₂×#41	—	166	1	no	—
213	C₂×SD₁₆	—	83	1	no	—
214	C₂×D₈	—	83	1	no	—
215	C₂×D₈	C₂×D₈₃	1	1	yes	characteristic
216	C₂×M₁₆	—	83	1	no	—
217	C₂×M₁₆	C₂×D₈₃	1	1	yes	characteristic
218	C₄⋊Q₈	—	83	1	no	—
219	#127	—	83	1	no	—
220	#127	—	166	1	no	—
221	#123	—	166	1	no	—
222	#123	—	166	1	no	—
223	C₂²⋊Q₈	—	166	1	no	—
224	#121	—	83	1	no	—
225	#121	—	83	1	no	—
226	#121	—	166	1	no	—
227	#121	—	166	1	no	—
228	#121	C₂×D₈₃	1	1	yes	characteristic
229	#121	C₂×D₈₃	1	1	yes	characteristic
230	C₄×Q₈	—	83	1	no	—
231	C₄×D₄	—	83	1	no	—
232	C₄×D₄	—	83	1	no	—
233	C₄×D₄	—	166	1	no	—
234	C₂×C₄⋊C₄	—	83	1	no	—
235	C₂×C₄⋊C₄	—	83	1	no	—
236	#106	—	83	1	no	—
237	#106	—	83	1	no	—
238	#106	C₂×D₈₃	1	1	yes	characteristic
239	C₄⋊C₈	—	83	1	no	—
240	#103	—	83	1	no	—
241	#103	—	83	1	no	—
242	#103	—	83	1	no	—
243	#102	—	83	1	no	—
244	#102	—	83	1	no	—
245	#102	—	83	1	no	—
246	#102	—	83	1	no	—
247	#102	—	83	1	no	—
248	#102	C₂×D₈₃	1	1	yes	characteristic
249	#102	C₂×D₈₃	1	1	yes	characteristic
250	C₂²⋊C₈	—	83	1	no	—
251	C₂²⋊C₈	—	83	1	no	—
252	C₄×C₈	—	83	1	no	—
253	#41	—	332	1	no	—
254	#41	—	332	4	no	—
255	#41	—	332	1	no	—
256	#41	—	166	2	no	—
257	#41	—	166	2	no	—
258	C₂×Q₈	—	83	1	no	—
259	C₂×Q₈	—	166	1	no	—
260	C₂×D₄	—	2	1	no	—
261	C₂×D₄	—	2	1	no	—
262	C₂×D₄	—	83	1	no	—
263	C₂×D₄	—	83	1	no	—
264	C₂×D₄	—	166	1	no	—
265	C₂×D₄	—	166	1	no	—
266	C₂×D₄	—	166	1	no	—
267	C₂×D₄	—	166	1	no	—
268	C₂×D₄	—	166	1	no	—
269	C₂×D₄	—	166	1	no	—
270	C₂×D₄	C₂²×D₈₃	1	1	yes	characteristic
271	C₂×D₄	C₂²×D₈₃	1	1	yes	characteristic
272	C₂²×C₄	—	83	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	C₂²×D₈₃	1	1	yes	characteristic
279	SD₁₆	—	166	2	no	—
280	D₈	—	4	1	no	—
281	D₈	—	332	1	no	—
282	M₁₆	—	4	1	no	—
283	M₁₆	—	332	1	no	—
284	C₂×C₈	—	83	1	no	—
285	C₂×C₈	—	83	1	no	—
286	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
287	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
288	C₄⋊C₄	—	83	1	no	—
289	C₄⋊C₄	—	83	1	no	—
290	C₄⋊C₄	—	83	1	no	—
291	C₄⋊C₄	—	166	1	no	—
292	C₄⋊C₄	—	166	1	no	—
293	C₄⋊C₄	—	166	1	no	—
294	C₄⋊C₄	—	166	1	no	—
295	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
296	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
297	C₂²⋊C₄	—	2	1	no	—
298	C₂²⋊C₄	—	2	1	no	—
299	C₂²⋊C₄	—	83	1	no	—
300	C₂²⋊C₄	—	83	1	no	—
301	C₂²⋊C₄	—	166	1	no	—
302	C₂²⋊C₄	—	166	1	no	—
303	C₂²⋊C₄	—	166	1	no	—
304	C₄²	—	83	1	no	—
305	C₄²	—	166	1	no	—
306	C₂³	—	2	1	no	—
307	C₂³	—	2	1	no	—
308	C₂³	—	83	1	no	—
309	C₂³	—	166	1	no	—
310	C₂³	D₄×D₈₃	1	1	yes	characteristic
311	Q₈	—	332	2	no	—
312	Q₈	—	83	2	no	—
313	Q₈	—	166	1	no	—
314	D₄	—	2	1	no	—
315	D₄	—	2	1	no	—
316	D₄	—	2	1	no	—
317	D₄	—	2	1	no	—
318	D₄	—	4	2	no	—
319	D₄	—	8	1	no	—
320	D₄	—	332	2	no	—
321	D₄	—	332	2	no	—
322	D₄	—	332	2	no	—
323	D₄	—	332	2	no	—
324	D₄	—	664	1	no	—
325	D₄	—	664	1	no	—
326	D₄	—	166	2	no	—
327	D₄	—	166	1	no	—
328	D₄	—	166	1	no	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	—	2	1	no	—
331	C₂×C₄	—	2	1	no	—
332	C₂×C₄	—	2	1	no	—
333	C₂×C₄	—	332	2	no	—
334	C₂×C₄	—	332	2	no	—
335	C₂×C₄	—	332	2	no	—
336	C₂×C₄	—	332	2	no	—
337	C₂×C₄	—	332	2	no	—
338	C₂×C₄	—	83	1	no	—
339	C₂×C₄	—	83	1	no	—
340	C₂×C₄	—	83	1	no	—
341	C₂×C₄	—	166	1	no	—
342	C₂×C₄	—	166	1	no	—
343	C₂×C₄	—	166	1	no	—
344	C₂×C₄	—	166	1	no	—
345	C₂×C₄	—	166	1	no	—
346	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
347	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
348	C₈	—	2	1	no	—
349	C₈	—	2	1	no	—
350	C₈	—	166	1	no	—
351	C₈	—	166	1	no	—
352	C₂²	—	2	1	no	—
353	C₂²	—	2	1	no	—
354	C₂²	—	2	1	no	—
355	C₂²	—	4	1	no	—
356	C₂²	—	4	1	no	—
357	C₂²	—	4	1	no	—
358	C₂²	—	4	1	no	—
359	C₂²	—	4	1	no	—
360	C₂²	—	4	1	no	—
361	C₂²	—	332	1	no	—
362	C₂²	—	332	1	no	—
363	C₂²	—	332	1	no	—
364	C₂²	—	166	1	no	—
365	C₂²	—	166	1	no	—
366	C₂²	—	166	1	no	—
367	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
368	C₄	—	2	1	no	—
369	C₄	—	4	1	no	—
370	C₄	—	4	1	no	—
371	C₄	—	332	1	no	—
372	C₄	—	332	1	no	—
373	C₄	—	83	2	no	—
374	C₄	—	166	1	no	—
375	C₄	—	166	2	no	—
376	C₄	—	166	1	no	—
377	C₄	#41×D₈₃	1	1	yes	characteristic
378	C₄	#14581837	1	1	yes	characteristic
379	C₂	—	4	1	no	—
380	C₂	—	8	1	no	—
381	C₂	—	8	1	no	—
382	C₂	—	332	1	no	—
383	C₂	—	664	1	no	—
384	C₂	#19128340	1	1	yes	characteristic
385	C₂	AGL(1,ℤ/8)×D₈₃	1	1	yes	characteristic
386	C₂	#19128511	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₃
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#469×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
hypercenter	#469	D₈₃
hypocenter	C₈₃	#3332
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #469

Complements

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

N	H ≅ G/N	type
1. #19128369	C₂	semidirect
2. #19128351	C₂	semidirect
3. #19128046	C₂	semidirect
4. #19128021	C₂	semidirect
5. #19128017	C₂	semidirect
6. #19127985	C₂	semidirect
7. #19127953	C₂	semidirect
8. #19127894	C₂	semidirect
9. #19127844	C₂	semidirect
10. #19127843	C₂	semidirect
11. #19127792	C₂	semidirect
12. #19127777	C₂	semidirect
13. #19127703	C₂	semidirect
14. #19127699	C₂	semidirect
15. #469×C₈₃	C₂	semidirect
153. Dic₁₆₆	#41	semidirect
194. C₈₃	#3332	semidirect
202. #469	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128369	2
#19128351	2
#19128046	2
#19128021	2
#19128017	2
#19127985	2
#19127953	2
#19127894	2
#19127844	2
#19127843	2
#19127792	2
#19127777	2
#19127703	2
#19127699	2
#469×C₈₃	2
#3332	83