Subgroups — #24919646

Order 10624.

392 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128417	C₂	1	1	yes	maximal, characteristic, complemented
2	#19128363	C₂	1	1	yes	maximal, characteristic, complemented
3	#19128063	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128045	C₂	1	1	yes	maximal, characteristic, complemented
5	#19128027	C₂	1	1	yes	maximal, characteristic, complemented
6	#19127955	C₂	1	1	yes	maximal, characteristic, complemented
7	#19127931	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	#19127708	C₂	1	1	yes	maximal, characteristic, complemented
14	#19127696	C₂	1	1	yes	maximal, characteristic, complemented
15	#475×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13050591	C₂²	1	1	yes	characteristic
17	C₂×Q₈×D₈₃	C₂²	1	1	yes	characteristic
18	C₂×#13050579	C₂²	1	1	yes	characteristic
19	C₂×#13050577	C₂²	1	1	yes	characteristic
20	C₂×D₆₆₄	C₂²	1	1	yes	characteristic
21	C₂×#13050573	C₂²	1	1	yes	characteristic
22	C₂×C₈×D₈₃	C₂²	1	1	yes	characteristic
23	#14581781	—	2	1	no	—
24	#14581781	—	2	1	no	—
25	#14581780	—	2	1	no	—
26	#14581780	C₂²	1	1	yes	characteristic
27	#14581772	—	2	1	no	—
28	#14581772	C₂²	1	1	yes	characteristic
29	#14581767	—	2	1	no	—
30	#14581767	—	2	1	no	—
31	#14581767	C₂²	1	1	yes	characteristic
32	#14581767	C₂²	1	1	yes	characteristic
33	#14581767	C₂²	1	1	yes	characteristic
34	C₄×D₃₃₂	—	2	1	no	—
35	C₄×D₃₃₂	C₂²	1	1	yes	characteristic
36	C₄×D₃₃₂	C₂²	1	1	yes	characteristic
37	#14581750	—	2	1	no	—
38	#14581750	C₂²	1	1	yes	characteristic
39	#14581750	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	#106×C₈₃	C₂²	1	1	yes	characteristic
55	#106×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	C₂×Dic₁₆₆	—	2	1	no	—
62	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
63	Q₈×D₈₃	—	2	2	no	—
64	Q₈×D₈₃	—	4	1	no	—
65	C₂×D₃₃₂	—	2	1	no	—
66	C₂×D₃₃₂	—	2	1	no	—
67	C₂×D₃₃₂	—	2	1	no	—
68	C₂×D₃₃₂	—	2	1	no	—
69	C₂×D₃₃₂	D₄	1	1	yes	characteristic
70	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
71	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
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	#13050579	—	4	1	no	—
78	#13050577	—	2	2	no	—
79	D₆₆₄	—	4	1	no	—
80	#13050573	—	4	1	no	—
81	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
82	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
83	C₈×D₈₃	—	4	1	no	—
84	C₄⋊Dic₈₃	—	2	1	no	—
85	C₄⋊Dic₈₃	D₄	1	1	yes	characteristic
86	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
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	(C₂×D₈₃)⋊C₄	—	2	1	no	—
94	(C₂×D₈₃)⋊C₄	—	2	1	no	—
95	(C₂×D₈₃)⋊C₄	—	2	1	no	—
96	(C₂×D₈₃)⋊C₄	D₄	1	1	yes	characteristic
97	(C₂×D₈₃)⋊C₄	D₄	1	1	yes	characteristic
98	C₄×Dic₈₃	—	2	1	no	—
99	C₄×Dic₈₃	—	2	1	no	—
100	C₂×Q₈×C₈₃	—	2	1	no	—
101	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
102	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
103	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
104	C₄⋊C₄×C₈₃	—	2	1	no	—
105	C₄⋊C₄×C₈₃	—	2	1	no	—
106	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
107	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
108	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
109	C₄²×C₈₃	—	2	1	no	—
110	C₄²×C₈₃	C₂³	1	1	yes	characteristic
111	C₂²×D₈₃	—	2	1	no	—
112	C₂²×D₈₃	—	2	1	no	—
113	C₂²×D₈₃	C₂×D₄	1	1	yes	characteristic
114	C₂²×D₈₃	C₂×D₄	1	1	yes	characteristic
115	Q₈×C₈₃	—	2	1	no	—
116	Q₈×C₈₃	—	8	1	no	—
117	Q₈×C₈₃	#41	1	2	yes	—
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	—	2	1	no	—
122	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
125	C₂×Dic₈₃	—	2	1	no	—
126	C₂×Dic₈₃	—	2	1	no	—
127	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
128	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
129	D₃₃₂	—	2	1	no	—
130	D₃₃₂	—	2	1	no	—
131	D₃₃₂	—	2	1	no	—
132	D₃₃₂	—	2	1	no	—
133	D₃₃₂	—	4	1	no	—
134	D₃₃₂	—	4	2	no	—
135	D₃₃₂	—	4	2	no	—
136	D₃₃₂	—	8	1	no	—
137	D₃₃₂	—	8	1	no	—
138	C₄×D₈₃	—	2	1	no	—
139	C₄×D₈₃	—	2	1	no	—
140	C₄×D₈₃	—	2	1	no	—
141	C₄×D₈₃	—	2	1	no	—
142	C₄×D₈₃	—	4	2	no	—
143	C₄×D₈₃	—	4	2	no	—
144	C₄×D₈₃	—	4	2	no	—
145	Dic₁₆₆	—	2	1	no	—
146	Dic₁₆₆	—	2	1	no	—
147	Dic₁₆₆	—	4	2	no	—
148	C₈×C₈₃	—	2	1	no	—
149	C₈×C₈₃	—	2	1	no	—
150	C₈₃⋊C₈	—	2	1	no	—
151	C₈₃⋊C₈	—	2	1	no	—
152	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
153	C₂×D₈₃	—	2	1	no	—
154	C₂×D₈₃	—	2	1	no	—
155	C₂×D₈₃	—	2	1	no	—
156	C₂×D₈₃	—	2	1	no	—
157	C₂×D₈₃	—	2	1	no	—
158	C₂×D₈₃	—	2	1	no	—
159	C₂×D₈₃	—	4	1	no	—
160	C₂×D₈₃	—	4	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	C₄×C₈₃	—	2	2	no	—
166	C₄×C₈₃	—	2	1	no	—
167	C₄×C₈₃	—	2	1	no	—
168	C₄×C₈₃	—	4	1	no	—
169	C₄×C₈₃	—	4	1	no	—
170	C₄×C₈₃	—	4	1	no	—
171	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
172	C₄×C₈₃	H(5,𝔽₂)	1	1	yes	characteristic
173	Dic₈₃	—	2	1	no	—
174	Dic₈₃	—	2	1	no	—
175	Dic₈₃	—	4	1	no	—
176	Dic₈₃	—	4	1	no	—
177	C₂×C₈₃	#546	1	1	yes	characteristic
178	C₂×C₈₃	#575	1	1	yes	characteristic
179	C₂×C₈₃	#577	1	1	yes	characteristic
180	D₈₃	—	4	1	no	—
181	D₈₃	—	4	1	no	—
182	D₈₃	—	8	1	no	—
183	D₈₃	—	8	1	no	—
184	#3321	—	83	1	no	maximal, Sylow
185	C₈₃	#3321	1	1	yes	characteristic, complemented, Sylow
186	#555	—	83	1	no	—
187	#548	—	83	1	no	—
188	#489	—	83	1	no	—
189	#485	—	83	1	no	—
190	#482	—	83	1	no	—
191	#475	D₈₃	1	1	yes	characteristic, complemented
192	#469	—	83	1	no	—
193	#465	—	83	1	no	—
194	#463	—	83	1	no	—
195	#449	—	83	1	no	—
196	#448	—	83	1	no	—
197	#440	—	83	1	no	—
198	#435	—	83	1	no	—
199	#419	—	83	1	no	—
200	#416	—	83	1	no	—
201	C₂×#41	—	83	1	no	—
202	C₂²×Q₈	—	83	1	no	—
203	C₂×SD₁₆	—	83	1	no	—
204	C₂×SD₁₆	—	83	1	no	—
205	C₂×D₈	—	83	1	no	—
206	C₂×M₁₆	—	83	1	no	—
207	C₂²×C₈	—	83	1	no	—
208	C₄⋊Q₈	C₂×D₈₃	1	1	yes	characteristic
209	#124	—	83	1	no	—
210	#124	—	166	1	no	—
211	#124	—	166	1	no	—
212	#124	—	166	1	no	—
213	C₂²⋊Q₈	—	83	1	no	—
214	C₂²⋊Q₈	—	166	1	no	—
215	#121	—	83	1	no	—
216	#121	—	83	1	no	—
217	#121	—	83	1	no	—
218	#121	—	166	1	no	—
219	#121	—	166	1	no	—
220	C₄×Q₈	C₂×D₈₃	1	1	yes	characteristic
221	C₄×D₄	—	83	1	no	—
222	C₄×D₄	—	83	1	no	—
223	C₄×D₄	—	166	1	no	—
224	#117	—	83	1	no	—
225	#117	—	83	1	no	—
226	#117	—	166	1	no	—
227	#107	—	83	1	no	—
228	#106	C₂×D₈₃	1	1	yes	characteristic
229	#106	C₂×D₈₃	1	1	yes	characteristic
230	C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
231	#103	—	83	1	no	—
232	#103	—	83	1	no	—
233	#103	—	83	1	no	—
234	#103	—	83	1	no	—
235	#103	C₂×D₈₃	1	1	yes	characteristic
236	#103	C₂×D₈₃	1	1	yes	characteristic
237	#102	—	83	1	no	—
238	#102	—	83	1	no	—
239	#102	—	83	1	no	—
240	#102	—	83	1	no	—
241	C₂²⋊C₈	—	83	1	no	—
242	C₂²⋊C₈	—	83	1	no	—
243	#97	—	83	1	no	—
244	#41	—	332	1	no	—
245	#41	—	166	2	no	—
246	C₂×Q₈	—	2	1	no	—
247	C₂×Q₈	—	332	1	no	—
248	C₂×Q₈	—	83	1	no	—
249	C₂×Q₈	—	166	1	no	—
250	C₂×Q₈	—	166	2	no	—
251	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
252	C₂×D₄	—	83	1	no	—
253	C₂×D₄	—	83	1	no	—
254	C₂×D₄	—	83	1	no	—
255	C₂×D₄	—	166	1	no	—
256	C₂×D₄	—	166	1	no	—
257	C₂×D₄	—	166	1	no	—
258	C₂×D₄	—	166	1	no	—
259	C₂²×C₄	—	83	1	no	—
260	C₂²×C₄	—	83	1	no	—
261	C₂²×C₄	—	166	1	no	—
262	C₂²×C₄	—	166	1	no	—
263	C₂²×C₄	—	166	1	no	—
264	SD₁₆	—	332	1	no	—
265	SD₁₆	—	166	2	no	—
266	D₈	—	332	1	no	—
267	M₁₆	—	332	1	no	—
268	C₂×C₈	—	332	1	no	—
269	C₂×C₈	—	83	1	no	—
270	C₂×C₈	—	83	1	no	—
271	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
272	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
273	C₄⋊C₄	—	2	1	no	—
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	—	83	1	no	—
276	C₄⋊C₄	—	83	1	no	—
277	C₄⋊C₄	—	166	1	no	—
278	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
279	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
280	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
281	C₂²⋊C₄	—	83	1	no	—
282	C₂²⋊C₄	—	83	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	C₂²⋊C₄	—	166	1	no	—
288	C₂²⋊C₄	—	166	1	no	—
289	C₂²⋊C₄	—	166	1	no	—
290	C₂²⋊C₄	—	166	1	no	—
291	C₂²⋊C₄	—	166	1	no	—
292	C₄²	—	2	1	no	—
293	C₄²	—	166	1	no	—
294	C₄²	—	166	1	no	—
295	C₄²	C₂²×D₈₃	1	1	yes	characteristic
296	C₂³	—	83	1	no	—
297	C₂³	—	83	1	no	—
298	C₂³	—	166	1	no	—
299	C₂³	—	166	1	no	—
300	Q₈	—	2	1	no	—
301	Q₈	—	8	1	no	—
302	Q₈	—	332	2	no	—
303	Q₈	—	166	1	no	—
304	Q₈	—	166	1	no	—
305	Q₈	#13050591	1	2	yes	—
306	D₄	—	332	2	no	—
307	D₄	—	332	1	no	—
308	D₄	—	332	2	no	—
309	D₄	—	664	1	no	—
310	D₄	—	664	1	no	—
311	D₄	—	166	1	no	—
312	D₄	—	166	1	no	—
313	D₄	—	166	1	no	—
314	D₄	—	166	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	332	2	no	—
320	C₂×C₄	—	332	2	no	—
321	C₂×C₄	—	332	2	no	—
322	C₂×C₄	—	83	1	no	—
323	C₂×C₄	—	83	1	no	—
324	C₂×C₄	—	166	1	no	—
325	C₂×C₄	—	166	1	no	—
326	C₂×C₄	—	166	1	no	—
327	C₂×C₄	—	166	1	no	—
328	C₂×C₄	—	166	1	no	—
329	C₂×C₄	—	166	1	no	—
330	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
331	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
332	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
333	C₈	—	2	1	no	—
334	C₈	—	2	1	no	—
335	C₈	—	166	1	no	—
336	C₈	—	166	1	no	—
337	C₂²	—	332	1	no	—
338	C₂²	—	332	1	no	—
339	C₂²	—	332	1	no	—
340	C₂²	—	332	1	no	—
341	C₂²	—	332	1	no	—
342	C₂²	—	332	1	no	—
343	C₂²	—	166	1	no	—
344	C₂²	—	166	1	no	—
345	C₂²	—	166	1	no	—
346	C₂²	—	166	1	no	—
347	C₂²	—	166	1	no	—
348	C₂²	—	166	1	no	—
349	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
350	C₄	—	2	2	no	—
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	4	1	no	—
354	C₄	—	4	1	no	—
355	C₄	—	4	1	no	—
356	C₄	—	332	1	no	—
357	C₄	—	332	1	no	—
358	C₄	—	166	1	no	—
359	C₄	—	166	1	no	—
360	C₄	C₂×#13050591	1	1	yes	characteristic
361	C₄	#14581838	1	1	yes	characteristic
362	C₂	—	332	1	no	—
363	C₂	—	332	1	no	—
364	C₂	—	664	1	no	—
365	C₂	—	664	1	no	—
366	C₂	#19128342	1	1	yes	characteristic
367	C₂	#19128500	1	1	yes	characteristic
368	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	#475×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
hypercenter	#475	D₈₃
hypocenter	C₈₃	#3321
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #475

Complements

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

N	H ≅ G/N	type
1. #19128417	C₂	semidirect
2. #19128363	C₂	semidirect
3. #19128063	C₂	semidirect
4. #19128045	C₂	semidirect
5. #19128027	C₂	semidirect
6. #19127955	C₂	semidirect
7. #19127931	C₂	semidirect
8. #19127917	C₂	semidirect
9. #19127849	C₂	semidirect
10. #19127841	C₂	semidirect
11. #19127799	C₂	semidirect
12. #19127771	C₂	semidirect
13. #19127708	C₂	semidirect
14. #19127696	C₂	semidirect
15. #475×C₈₃	C₂	semidirect
185. C₈₃	#3321	semidirect
191. #475	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128417	2
#19128363	2
#19128063	2
#19128045	2
#19128027	2
#19127955	2
#19127931	2
#19127917	2
#19127849	2
#19127841	2
#19127799	2
#19127771	2
#19127708	2
#19127696	2
#475×C₈₃	2
#3321	83