Subgroups — #24918418

Order 10624.

394 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128255	C₂	1	1	yes	maximal, characteristic, complemented
2	#19128250	C₂	1	1	yes	maximal, characteristic
3	#19128079	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128072	C₂	1	1	yes	maximal, characteristic, complemented
5	#19128050	C₂	1	1	yes	maximal, characteristic, complemented
6	#19128046	C₂	1	1	yes	maximal, characteristic, complemented
7	#19127924	C₂	1	1	yes	maximal, characteristic, complemented
8	#19127923	C₂	1	1	yes	maximal, characteristic, complemented
9	#19127920	C₂	1	1	yes	maximal, characteristic, complemented
10	#19127866	C₂	1	1	yes	maximal, characteristic, complemented
11	#19127864	C₂	1	1	yes	maximal, characteristic, complemented
12	#19127862	C₂	1	1	yes	maximal, characteristic, complemented
13	#19127860	C₂	1	1	yes	maximal, characteristic, complemented
14	#19127753	C₂	1	1	yes	maximal, characteristic, complemented
15	#464×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13050591	C₂²	1	1	yes	characteristic
17	C₂×#13050589	C₂²	1	1	yes	characteristic
18	C₂×Dic₃₃₂	C₂²	1	1	yes	characteristic
19	C₂×#13050580	C₂²	1	1	yes	characteristic
20	C₂×#13050580	C₂²	1	1	yes	characteristic
21	C₂×#13050579	C₂²	1	1	yes	characteristic
22	C₂×#13050578	C₂²	1	1	yes	characteristic
23	C₂×#13050577	C₂²	1	1	yes	characteristic
24	#14581791	C₂²	1	1	yes	characteristic
25	#14581790	C₂²	1	1	yes	characteristic
26	#14581790	C₂²	1	1	yes	characteristic
27	#14581788	C₂²	1	1	yes	characteristic
28	#14581782	—	2	1	no	—
29	#14581780	C₂²	1	1	yes	characteristic
30	#14581772	C₂²	1	1	yes	characteristic
31	#14581770	—	2	1	no	—
32	#14581768	C₂²	1	1	yes	characteristic
33	#14581765	—	2	1	no	—
34	#14581765	—	2	1	no	—
35	C₄×Dic₁₆₆	—	2	1	no	—
36	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
37	(C₂²⋊D₈₃)⋊C₄	—	2	1	no	—
38	(C₂²⋊D₈₃)⋊C₄	—	2	1	no	—
39	#14581749	—	2	1	no	—
40	C₄²×D₈₃	C₂²	1	1	yes	characteristic
41	C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
42	#14581717	C₂²	1	1	yes	characteristic
43	#14581716	C₂²	1	1	yes	characteristic
44	#14581715	C₂²	1	1	yes	characteristic
45	#14581714	C₂²	1	1	yes	characteristic
46	#14581713	C₂²	1	1	yes	characteristic
47	#14581712	C₂²	1	1	yes	characteristic
48	(C₂×D₈₃)⋊C₈	C₂²	1	1	yes	characteristic
49	(C₂×D₈₃)⋊C₈	C₂²	1	1	yes	characteristic
50	C₈×Dic₈₃	C₂²	1	1	yes	characteristic
51	C₈×Dic₈₃	C₂²	1	1	yes	characteristic
52	C₂×Q₁₆×C₈₃	C₂²	1	1	yes	characteristic
53	C₂×SD₁₆×C₈₃	C₂²	1	1	yes	characteristic
54	#124×C₈₃	C₂²	1	1	yes	characteristic
55	C₄×Q₈×C₈₃	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
57	#103×C₈₃	C₂²	1	1	yes	characteristic
58	#102×C₈₃	C₂²	1	1	yes	characteristic
59	#13050591	—	4	1	no	—
60	#13050591	—	4	1	no	—
61	#13050589	—	4	1	no	—
62	#13050589	—	4	1	no	—
63	C₂×Dic₁₆₆	—	2	1	no	—
64	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
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	C₂²×Dic₈₃	—	2	1	no	—
72	C₂×C₄×D₈₃	—	2	1	no	—
73	C₂×C₄×D₈₃	D₄	1	1	yes	characteristic
74	C₂×C₄×D₈₃	D₄	1	1	yes	characteristic
75	C₂×C₄×D₈₃	C₂³	1	1	yes	characteristic
76	Dic₃₃₂	—	4	1	no	—
77	#13050580	—	4	1	no	—
78	#13050580	—	4	1	no	—
79	#13050579	—	4	1	no	—
80	#13050578	—	4	1	no	—
81	#13050577	—	4	1	no	—
82	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
83	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
84	C₄⋊Dic₈₃	—	2	1	no	—
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	Dic₈₃⋊C₄	—	2	1	no	—
91	Dic₈₃⋊C₄	—	2	1	no	—
92	Dic₈₃⋊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	C₄×Dic₈₃	—	2	1	no	—
98	C₄×Dic₈₃	D₄	1	1	yes	characteristic
99	C₄×Dic₈₃	D₄	1	1	yes	characteristic
100	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
101	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
102	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
103	C₂×D₄×C₈₃	C₂³	1	1	yes	characteristic
104	Q₁₆×C₈₃	—	4	1	no	—
105	SD₁₆×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	—	2	1	no	—
109	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
110	C₂²⋊C₄×C₈₃	—	2	1	no	—
111	C₂²⋊C₄×C₈₃	—	2	1	no	—
112	C₄²×C₈₃	—	2	1	no	—
113	C₄²×C₈₃	C₂³	1	1	yes	characteristic
114	C₂³×C₈₃	—	2	1	no	—
115	C₂²×D₈₃	—	2	1	no	—
116	C₂²×D₈₃	C₂×D₄	1	1	yes	characteristic
117	Q₈×C₈₃	—	2	1	no	—
118	Q₈×C₈₃	—	2	1	no	—
119	Q₈×C₈₃	—	2	1	no	—
120	Q₈×C₈₃	—	2	1	no	—
121	D₄×C₈₃	—	2	1	no	—
122	D₄×C₈₃	—	2	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	C₂×D₄	1	1	yes	characteristic
127	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
128	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
129	C₂²⋊D₈₃	—	4	2	no	—
130	C₂²⋊D₈₃	—	4	2	no	—
131	C₂²⋊D₈₃	—	4	2	no	—
132	C₂×Dic₈₃	—	2	1	no	—
133	C₂×Dic₈₃	—	2	1	no	—
134	C₂×Dic₈₃	—	2	1	no	—
135	C₂×Dic₈₃	—	4	2	no	—
136	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
137	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
138	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
139	D₃₃₂	—	2	1	no	—
140	D₃₃₂	—	2	1	no	—
141	D₃₃₂	—	4	2	no	—
142	C₄×D₈₃	—	2	1	no	—
143	C₄×D₈₃	—	2	1	no	—
144	C₄×D₈₃	—	2	2	no	—
145	C₄×D₈₃	—	2	2	no	—
146	C₄×D₈₃	—	4	2	no	—
147	Dic₁₆₆	—	2	1	no	—
148	Dic₁₆₆	—	2	1	no	—
149	Dic₁₆₆	—	2	1	no	—
150	Dic₁₆₆	—	2	1	no	—
151	Dic₁₆₆	—	4	2	no	—
152	C₈×C₈₃	—	2	1	no	—
153	C₈×C₈₃	—	2	1	no	—
154	C₈₃⋊C₈	—	2	1	no	—
155	C₈₃⋊C₈	—	2	1	no	—
156	C₂²×C₈₃	—	4	1	no	—
157	C₂²×C₈₃	—	4	1	no	—
158	C₂²×C₈₃	—	4	1	no	—
159	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
160	C₂×D₈₃	—	2	1	no	—
161	C₂×D₈₃	—	2	1	no	—
162	C₂×D₈₃	—	2	1	no	—
163	C₂×D₈₃	—	4	1	no	—
164	C₂×D₈₃	—	4	1	no	—
165	C₂×D₈₃	—	4	1	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	H(5,𝔽₂)	1	1	yes	characteristic
172	C₄×C₈₃	H(5,𝔽₂)	1	1	yes	characteristic
173	Dic₈₃	—	2	1	no	—
174	Dic₈₃	—	4	1	no	—
175	Dic₈₃	—	4	1	no	—
176	Dic₈₃	—	4	1	no	—
177	Dic₈₃	#135	1	2	yes	—
178	Dic₈₃	#137	1	2	yes	—
179	C₂×C₈₃	—	8	1	no	—
180	C₂×C₈₃	#535	1	1	yes	characteristic
181	C₂×C₈₃	C₂×#135	1	1	yes	characteristic
182	C₂×C₈₃	C₂×#137	1	1	yes	characteristic
183	D₈₃	—	4	1	no	—
184	D₈₃	—	8	1	no	—
185	#3210	—	83	1	no	maximal, Sylow
186	C₈₃	#3210	1	1	yes	characteristic, complemented, Sylow
187	#533	—	83	1	no	—
188	#532	—	83	1	no	—
189	#494	—	83	1	no	—
190	#492	—	83	1	no	—
191	#487	—	83	1	no	—
192	#486	—	83	1	no	—
193	#464	—	83	1	no	—
194	#464	—	83	1	no	—
195	#464	—	83	1	no	—
196	#464	D₈₃	1	1	yes	characteristic, complemented
197	#452	—	83	1	no	—
198	#452	—	83	1	no	—
199	#452	—	83	1	no	—
200	#452	—	83	1	no	—
201	#431	—	83	1	no	—
202	C₂×#41	—	83	1	no	—
203	C₂×#41	—	83	1	no	—
204	C₂×Q₁₆	—	83	1	no	—
205	C₂×Q₁₆	—	83	1	no	—
206	C₂×Q₁₆	—	83	1	no	—
207	C₂×Q₁₆	C₂×D₈₃	1	1	yes	characteristic
208	C₂×SD₁₆	—	83	1	no	—
209	C₂×SD₁₆	—	83	1	no	—
210	C₂×SD₁₆	—	83	1	no	—
211	C₂×SD₁₆	C₂×D₈₃	1	1	yes	characteristic
212	C₄⋊Q₈	—	83	1	no	—
213	C₄⋊Q₈	—	83	1	no	—
214	C₄⋊Q₈	—	83	1	no	—
215	#127	—	83	1	no	—
216	#125	—	166	1	no	—
217	#124	—	83	1	no	—
218	#124	C₂×D₈₃	1	1	yes	characteristic
219	C₂²⋊Q₈	—	83	1	no	—
220	C₂²⋊Q₈	—	83	1	no	—
221	C₂²⋊Q₈	—	166	1	no	—
222	#121	—	166	1	no	—
223	#121	—	166	1	no	—
224	C₄×Q₈	—	83	1	no	—
225	C₄×Q₈	—	166	1	no	—
226	C₄×Q₈	C₂×D₈₃	1	1	yes	characteristic
227	C₄×D₄	—	166	1	no	—
228	C₄×D₄	—	166	1	no	—
229	#117	—	166	1	no	—
230	C₂×C₄²	—	83	1	no	—
231	C₄⋊C₈	—	83	1	no	—
232	C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
233	#103	—	83	1	no	—
234	#103	—	83	1	no	—
235	#103	—	83	1	no	—
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	C₂×D₈₃	1	1	yes	characteristic
241	C₂²⋊C₈	—	83	1	no	—
242	C₂²⋊C₈	—	83	1	no	—
243	C₄×C₈	—	83	1	no	—
244	C₄×C₈	—	83	1	no	—
245	#41	—	332	1	no	—
246	#41	—	332	1	no	—
247	#41	—	332	1	no	—
248	#41	—	332	1	no	—
249	C₂×Q₈	—	83	1	no	—
250	C₂×Q₈	—	83	1	no	—
251	C₂×Q₈	—	166	1	no	—
252	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
253	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
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	C₂×D₄	C₂²×D₈₃	1	1	yes	characteristic
260	C₂²×C₄	—	83	1	no	—
261	C₂²×C₄	—	83	1	no	—
262	C₂²×C₄	—	83	1	no	—
263	C₂²×C₄	—	166	1	no	—
264	C₂²×C₄	—	166	1	no	—
265	Q₁₆	—	4	1	no	—
266	Q₁₆	—	332	1	no	—
267	Q₁₆	—	332	1	no	—
268	Q₁₆	—	332	1	no	—
269	SD₁₆	—	4	1	no	—
270	SD₁₆	—	332	1	no	—
271	SD₁₆	—	332	1	no	—
272	SD₁₆	—	332	1	no	—
273	C₂×C₈	—	83	1	no	—
274	C₂×C₈	—	83	1	no	—
275	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
276	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
277	C₄⋊C₄	—	2	1	no	—
278	C₄⋊C₄	—	83	1	no	—
279	C₄⋊C₄	—	166	1	no	—
280	C₄⋊C₄	—	166	1	no	—
281	C₄⋊C₄	—	166	1	no	—
282	C₄⋊C₄	—	166	1	no	—
283	C₄⋊C₄	—	166	1	no	—
284	C₄⋊C₄	—	166	1	no	—
285	C₄⋊C₄	—	166	1	no	—
286	C₄⋊C₄	—	166	1	no	—
287	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
288	C₂²⋊C₄	—	2	1	no	—
289	C₂²⋊C₄	—	2	1	no	—
290	C₂²⋊C₄	—	166	1	no	—
291	C₂²⋊C₄	—	166	1	no	—
292	C₂²⋊C₄	—	166	1	no	—
293	C₂²⋊C₄	—	166	1	no	—
294	C₄²	—	2	1	no	—
295	C₄²	—	83	1	no	—
296	C₄²	—	83	1	no	—
297	C₄²	—	83	1	no	—
298	C₄²	—	166	1	no	—
299	C₄²	C₂²×D₈₃	1	1	yes	characteristic
300	C₂³	—	2	1	no	—
301	C₂³	—	83	1	no	—
302	C₂³	—	166	1	no	—
303	Q₈	—	2	1	no	—
304	Q₈	—	2	1	no	—
305	Q₈	—	2	1	no	—
306	Q₈	—	2	1	no	—
307	Q₈	—	332	2	no	—
308	Q₈	—	166	1	no	—
309	Q₈	—	166	1	no	—
310	Q₈	—	166	1	no	—
311	Q₈	—	166	1	no	—
312	D₄	—	2	1	no	—
313	D₄	—	2	1	no	—
314	D₄	—	332	2	no	—
315	D₄	—	332	2	no	—
316	D₄	—	332	2	no	—
317	D₄	—	332	2	no	—
318	D₄	—	166	1	no	—
319	D₄	—	166	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	—	83	1	no	—
326	C₂×C₄	—	83	1	no	—
327	C₂×C₄	—	83	1	no	—
328	C₂×C₄	—	166	2	no	—
329	C₂×C₄	—	166	2	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	D₄×D₈₃	1	1	yes	characteristic
336	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
337	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
338	C₈	—	2	1	no	—
339	C₈	—	2	1	no	—
340	C₈	—	166	1	no	—
341	C₈	—	166	1	no	—
342	C₂²	—	4	1	no	—
343	C₂²	—	4	1	no	—
344	C₂²	—	4	1	no	—
345	C₂²	—	332	1	no	—
346	C₂²	—	332	1	no	—
347	C₂²	—	332	1	no	—
348	C₂²	—	166	1	no	—
349	C₂²	—	166	1	no	—
350	C₂²	—	166	1	no	—
351	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	4	1	no	—
355	C₄	—	4	1	no	—
356	C₄	—	4	1	no	—
357	C₄	—	332	1	no	—
358	C₄	—	332	1	no	—
359	C₄	—	332	1	no	—
360	C₄	—	83	2	no	—
361	C₄	—	83	2	no	—
362	C₄	—	166	1	no	—
363	C₄	#14581837	1	1	yes	characteristic
364	C₄	#14581838	1	1	yes	characteristic
365	C₂	—	8	1	no	—
366	C₂	—	332	1	no	—
367	C₂	—	664	1	no	—
368	C₂	#19128267	1	1	yes	characteristic
369	C₂	#135×D₈₃	1	1	yes	characteristic
370	C₂	#137×D₈₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₃
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#464×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	#19128250	C₂
hypercenter	#464	D₈₃
hypocenter	C₈₃	#3210
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #464

Complements

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

N	H ≅ G/N	type
1. #19128255	C₂	semidirect
3. #19128079	C₂	semidirect
4. #19128072	C₂	semidirect
5. #19128050	C₂	semidirect
6. #19128046	C₂	semidirect
7. #19127924	C₂	semidirect
8. #19127923	C₂	semidirect
9. #19127920	C₂	semidirect
10. #19127866	C₂	semidirect
11. #19127864	C₂	semidirect
12. #19127862	C₂	semidirect
13. #19127860	C₂	semidirect
14. #19127753	C₂	semidirect
15. #464×C₈₃	C₂	semidirect
186. C₈₃	#3210	semidirect
196. #464	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128255	2
#19128250	2
#19128079	2
#19128072	2
#19128050	2
#19128046	2
#19127924	2
#19127923	2
#19127920	2
#19127866	2
#19127864	2
#19127862	2
#19127860	2
#19127753	2
#464×C₈₃	2
#3210	83