Subgroups — #24919774

Order 10624.

388 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128409	C₂	1	1	yes	maximal, characteristic, complemented
2	#19128353	C₂	1	1	yes	maximal, characteristic
3	#19128050	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128033	C₂	1	1	yes	maximal, characteristic, complemented
5	#19128029	C₂	1	1	yes	maximal, characteristic, complemented
6	#19127990	C₂	1	1	yes	maximal, characteristic, complemented
7	#19127958	C₂	1	1	yes	maximal, characteristic, complemented
8	#19127909	C₂	1	1	yes	maximal, characteristic, complemented
9	#19127862	C₂	1	1	yes	maximal, characteristic, complemented
10	#19127861	C₂	1	1	yes	maximal, characteristic, complemented
11	#19127786	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	#19127701	C₂	1	1	yes	maximal, characteristic, complemented
15	#470×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13050591	C₂²	1	1	yes	characteristic
17	C₂×#13050590	C₂²	1	1	yes	characteristic
18	C₂×#13050580	C₂²	1	1	yes	characteristic
19	C₂×#13050577	C₂²	1	1	yes	characteristic
20	C₂×#13050573	C₂²	1	1	yes	characteristic
21	#14581791	C₂²	1	1	yes	characteristic
22	#14581790	C₂²	1	1	yes	characteristic
23	#14581782	—	2	1	no	—
24	#14581777	—	2	1	no	—
25	#14581774	—	2	1	no	—
26	#14581772	—	2	1	no	—
27	#14581772	C₂²	1	1	yes	characteristic
28	#14581769	—	2	1	no	—
29	#14581768	C₂²	1	1	yes	characteristic
30	#14581765	—	2	1	no	—
31	Q₈×Dic₈₃	C₂²	1	1	yes	characteristic
32	#14581754	C₂²	1	1	yes	characteristic
33	C₄×C₂²⋊D₈₃	C₂²	1	1	yes	characteristic
34	(C₂²⋊D₈₃)⋊C₄	—	2	1	no	—
35	C₂×C₄⋊Dic₈₃	C₂²	1	1	yes	characteristic
36	C₄⋊C₄×D₈₃	—	2	1	no	—
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	#14581716	C₂²	1	1	yes	characteristic
44	#14581716	C₂²	1	1	yes	characteristic
45	#14581715	C₂²	1	1	yes	characteristic
46	#14581714	C₂²	1	1	yes	characteristic
47	#14581714	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×Q₁₆×C₈₃	C₂²	1	1	yes	characteristic
53	C₂×M₁₆×C₈₃	C₂²	1	1	yes	characteristic
54	C₂²⋊Q₈×C₈₃	C₂²	1	1	yes	characteristic
55	C₂²⋊Q₈×C₈₃	C₂²	1	1	yes	characteristic
56	#106×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	#13050590	—	2	2	no	—
62	#13050590	—	4	1	no	—
63	C₂×Dic₁₆₆	—	2	1	no	—
64	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
65	C₂×D₃₃₂	—	2	1	no	—
66	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
67	C₂×C₂²⋊D₈₃	—	2	1	no	—
68	C₂×C₂²⋊D₈₃	D₄	1	1	yes	characteristic
69	C₂²×Dic₈₃	—	2	1	no	—
70	C₂×C₄×D₈₃	—	2	1	no	—
71	C₂×C₄×D₈₃	—	2	1	no	—
72	C₂×C₄×D₈₃	—	2	1	no	—
73	C₂×C₄×D₈₃	—	2	1	no	—
74	C₂×C₄×D₈₃	C₂³	1	1	yes	characteristic
75	#13050580	—	4	1	no	—
76	#13050577	—	2	2	no	—
77	#13050573	—	4	1	no	—
78	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
79	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
80	C₄⋊Dic₈₃	—	2	1	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	Dic₈₃⋊C₄	—	2	1	no	—
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	Dic₈₃⋊C₄	D₄	1	1	yes	characteristic
91	C₂²⋊Dic₈₃	D₄	1	1	yes	characteristic
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	D₄	1	1	yes	characteristic
96	C₄×Dic₈₃	—	2	1	no	—
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	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
101	Q₁₆×C₈₃	—	4	1	no	—
102	M₁₆×C₈₃	—	4	1	no	—
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	C₂²⋊C₄×C₈₃	—	2	1	no	—
110	C₂²⋊C₄×C₈₃	—	2	1	no	—
111	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
112	C₂²×D₈₃	—	2	1	no	—
113	C₂²×D₈₃	C₂×D₄	1	1	yes	characteristic
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	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	—	2	1	no	—
123	C₂×C₄×C₈₃	—	2	1	no	—
124	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
126	C₂²⋊D₈₃	—	2	2	no	—
127	C₂²⋊D₈₃	—	4	2	no	—
128	C₂×Dic₈₃	—	2	1	no	—
129	C₂×Dic₈₃	—	2	1	no	—
130	C₂×Dic₈₃	—	2	1	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	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
135	D₃₃₂	—	2	1	no	—
136	D₃₃₂	—	4	2	no	—
137	D₃₃₂	#41	1	2	yes	—
138	C₄×D₈₃	—	2	1	no	—
139	C₄×D₈₃	—	2	1	no	—
140	C₄×D₈₃	—	4	2	no	—
141	C₄×D₈₃	—	4	2	no	—
142	C₄×D₈₃	—	4	2	no	—
143	C₄×D₈₃	—	8	1	no	—
144	Dic₁₆₆	—	2	1	no	—
145	Dic₁₆₆	—	2	1	no	—
146	Dic₁₆₆	—	8	1	no	—
147	C₈×C₈₃	—	2	1	no	—
148	C₈×C₈₃	—	2	1	no	—
149	C₈₃⋊C₈	—	2	1	no	—
150	C₈₃⋊C₈	—	2	1	no	—
151	C₂²×C₈₃	—	2	1	no	—
152	C₂²×C₈₃	—	2	1	no	—
153	C₂²×C₈₃	—	2	1	no	—
154	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
155	C₂×D₈₃	—	2	1	no	—
156	C₂×D₈₃	—	2	2	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	C₄×C₈₃	—	2	1	no	—
162	C₄×C₈₃	—	4	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	H(5,𝔽₂)	1	1	yes	characteristic
168	Dic₈₃	—	2	1	no	—
169	Dic₈₃	—	2	1	no	—
170	Dic₈₃	—	4	1	no	—
171	Dic₈₃	—	4	1	no	—
172	Dic₈₃	—	4	1	no	—
173	Dic₈₃	#137	1	2	yes	—
174	C₂×C₈₃	—	4	1	no	—
175	C₂×C₈₃	#546	1	1	yes	characteristic
176	C₂×C₈₃	C₂×#137	1	1	yes	characteristic
177	C₂×C₈₃	#577	1	1	yes	characteristic
178	D₈₃	—	4	1	no	—
179	D₈₃	—	4	2	no	—
180	#3331	—	83	1	no	maximal, Sylow
181	C₈₃	#3331	1	1	yes	characteristic, complemented, Sylow
182	#554	—	83	1	no	—
183	#547	—	83	1	no	—
184	#487	—	83	1	no	—
185	#483	—	83	1	no	—
186	#483	—	83	1	no	—
187	#476	—	83	1	no	—
188	#470	—	83	1	no	—
189	#470	D₈₃	1	1	yes	characteristic, complemented
190	#462	—	83	1	no	—
191	#452	—	83	1	no	—
192	#452	—	83	1	no	—
193	C₄×SD₁₆	—	83	1	no	—
194	#436	—	83	1	no	—
195	#417	—	83	1	no	—
196	#417	—	83	1	no	—
197	C₂×#41	—	83	1	no	—
198	C₂×#41	—	83	1	no	—
199	C₂×Q₁₆	—	83	1	no	—
200	C₂×Q₁₆	C₂×D₈₃	1	1	yes	characteristic
201	C₂×SD₁₆	—	83	1	no	—
202	C₂×M₁₆	—	83	1	no	—
203	C₂×M₁₆	C₂×D₈₃	1	1	yes	characteristic
204	C₄⋊Q₈	—	83	1	no	—
205	C₄⋊Q₈	—	83	1	no	—
206	#125	—	166	1	no	—
207	#123	—	166	1	no	—
208	#123	—	166	1	no	—
209	C₂²⋊Q₈	—	83	1	no	—
210	C₂²⋊Q₈	—	83	1	no	—
211	C₂²⋊Q₈	—	166	1	no	—
212	C₂²⋊Q₈	—	166	1	no	—
213	C₂²⋊Q₈	C₂×D₈₃	1	1	yes	characteristic
214	C₂²⋊Q₈	C₂×D₈₃	1	1	yes	characteristic
215	#121	—	166	1	no	—
216	C₄×Q₈	—	83	1	no	—
217	C₄×D₄	—	83	1	no	—
218	C₄×D₄	—	83	1	no	—
219	C₄×D₄	—	166	1	no	—
220	C₂×C₄⋊C₄	—	83	1	no	—
221	C₂×C₄⋊C₄	—	83	1	no	—
222	C₂×C₄⋊C₄	—	166	1	no	—
223	#106	—	83	1	no	—
224	#106	—	83	1	no	—
225	#106	C₂×D₈₃	1	1	yes	characteristic
226	C₄⋊C₈	—	83	1	no	—
227	#103	—	83	1	no	—
228	#103	—	83	1	no	—
229	#103	—	83	1	no	—
230	#103	—	83	1	no	—
231	#103	—	83	1	no	—
232	#103	C₂×D₈₃	1	1	yes	characteristic
233	#103	C₂×D₈₃	1	1	yes	characteristic
234	#102	—	83	1	no	—
235	#102	—	83	1	no	—
236	#102	—	83	1	no	—
237	C₂²⋊C₈	—	83	1	no	—
238	C₂²⋊C₈	—	83	1	no	—
239	C₄×C₈	—	83	1	no	—
240	#41	—	332	1	no	—
241	#41	—	332	1	no	—
242	#41	—	166	2	no	—
243	#41	—	166	2	no	—
244	C₂×Q₈	—	83	1	no	—
245	C₂×Q₈	—	166	1	no	—
246	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
247	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
248	C₂×D₄	—	83	1	no	—
249	C₂×D₄	—	83	1	no	—
250	C₂×D₄	—	166	1	no	—
251	C₂×D₄	—	166	1	no	—
252	C₂²×C₄	—	83	1	no	—
253	C₂²×C₄	—	166	1	no	—
254	C₂²×C₄	—	166	1	no	—
255	C₂²×C₄	—	166	1	no	—
256	C₂²×C₄	—	166	1	no	—
257	C₂²×C₄	—	166	1	no	—
258	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
259	Q₁₆	—	4	1	no	—
260	Q₁₆	—	332	1	no	—
261	SD₁₆	—	166	2	no	—
262	M₁₆	—	4	1	no	—
263	M₁₆	—	332	1	no	—
264	C₂×C₈	—	83	1	no	—
265	C₂×C₈	—	83	1	no	—
266	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
267	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
268	C₄⋊C₄	—	2	1	no	—
269	C₄⋊C₄	—	2	1	no	—
270	C₄⋊C₄	—	83	1	no	—
271	C₄⋊C₄	—	83	1	no	—
272	C₄⋊C₄	—	83	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	—	166	1	no	—
278	C₄⋊C₄	—	166	1	no	—
279	C₄⋊C₄	—	166	1	no	—
280	C₄⋊C₄	—	166	1	no	—
281	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
282	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
283	C₂²⋊C₄	—	2	1	no	—
284	C₂²⋊C₄	—	2	1	no	—
285	C₂²⋊C₄	—	83	1	no	—
286	C₂²⋊C₄	—	83	1	no	—
287	C₂²⋊C₄	—	166	1	no	—
288	C₂²⋊C₄	—	166	1	no	—
289	C₂²⋊C₄	—	166	1	no	—
290	C₄²	—	83	1	no	—
291	C₄²	—	166	1	no	—
292	C₂³	—	83	1	no	—
293	C₂³	—	166	1	no	—
294	C₂³	D₄×D₈₃	1	1	yes	characteristic
295	Q₈	—	2	1	no	—
296	Q₈	—	2	1	no	—
297	Q₈	—	2	1	no	—
298	Q₈	—	2	1	no	—
299	Q₈	—	664	1	no	—
300	Q₈	—	166	1	no	—
301	Q₈	—	166	1	no	—
302	D₄	—	332	2	no	—
303	D₄	—	332	2	no	—
304	D₄	—	83	2	no	—
305	D₄	—	166	2	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	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	332	2	no	—
314	C₂×C₄	—	332	2	no	—
315	C₂×C₄	—	332	2	no	—
316	C₂×C₄	—	83	1	no	—
317	C₂×C₄	—	83	1	no	—
318	C₂×C₄	—	83	1	no	—
319	C₂×C₄	—	664	1	no	—
320	C₂×C₄	—	664	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	—	166	1	no	—
326	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
327	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
328	C₈	—	2	1	no	—
329	C₈	—	2	1	no	—
330	C₈	—	166	1	no	—
331	C₈	—	166	1	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	2	1	no	—
334	C₂²	—	2	1	no	—
335	C₂²	—	332	1	no	—
336	C₂²	—	332	1	no	—
337	C₂²	—	166	2	no	—
338	C₂²	—	166	1	no	—
339	C₂²	—	166	1	no	—
340	C₂²	—	166	1	no	—
341	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
342	C₄	—	2	1	no	—
343	C₄	—	4	1	no	—
344	C₄	—	4	1	no	—
345	C₄	—	4	1	no	—
346	C₄	—	4	1	no	—
347	C₄	—	332	1	no	—
348	C₄	—	332	1	no	—
349	C₄	—	332	1	no	—
350	C₄	—	83	2	no	—
351	C₄	—	166	1	no	—
352	C₄	—	166	1	no	—
353	C₄	#41×D₈₃	1	1	yes	characteristic
354	C₄	#14581837	1	1	yes	characteristic
355	C₂	—	4	1	no	—
356	C₂	—	332	2	no	—
357	C₂	—	332	1	no	—
358	C₂	#19128340	1	1	yes	characteristic
359	C₂	#137×D₈₃	1	1	yes	characteristic
360	C₂	#19128510	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₃
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#470×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	#19128353	C₂
hypercenter	#470	D₈₃
hypocenter	C₈₃	#3331
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #470

Complements

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

N	H ≅ G/N	type
1. #19128409	C₂	semidirect
3. #19128050	C₂	semidirect
4. #19128033	C₂	semidirect
5. #19128029	C₂	semidirect
6. #19127990	C₂	semidirect
7. #19127958	C₂	semidirect
8. #19127909	C₂	semidirect
9. #19127862	C₂	semidirect
10. #19127861	C₂	semidirect
11. #19127786	C₂	semidirect
12. #19127777	C₂	semidirect
13. #19127703	C₂	semidirect
14. #19127701	C₂	semidirect
15. #470×C₈₃	C₂	semidirect
181. C₈₃	#3331	semidirect
189. #470	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128409	2
#19128353	2
#19128050	2
#19128033	2
#19128029	2
#19127990	2
#19127958	2
#19127909	2
#19127862	2
#19127861	2
#19127786	2
#19127777	2
#19127703	2
#19127701	2
#470×C₈₃	2
#3331	83