Subgroups — #24918347

Order 10624.

390 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128279	C₂	1	1	yes	maximal, characteristic
2	#19128272	C₂	1	1	yes	maximal, characteristic, complemented
3	#19128043	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128037	C₂	1	1	yes	maximal, characteristic, complemented
5	#19128032	C₂	1	1	yes	maximal, characteristic, complemented
6	#19128026	C₂	1	1	yes	maximal, characteristic, complemented
7	#19127959	C₂	1	1	yes	maximal, characteristic, complemented
8	#19127947	C₂	1	1	yes	maximal, characteristic, complemented
9	#19127942	C₂	1	1	yes	maximal, characteristic, complemented
10	#19127930	C₂	1	1	yes	maximal, characteristic, complemented
11	#19127865	C₂	1	1	yes	maximal, characteristic, complemented
12	#19127865	C₂	1	1	yes	maximal, characteristic, complemented
13	#19127859	C₂	1	1	yes	maximal, characteristic, complemented
14	#19127842	C₂	1	1	yes	maximal, characteristic, complemented
15	#408×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13050590	C₂²	1	1	yes	characteristic
17	C₂²×Dic₁₆₆	C₂²	1	1	yes	characteristic
18	C₂×Dic₃₃₂	C₂²	1	1	yes	characteristic
19	C₂×Dic₃₃₂	C₂²	1	1	yes	characteristic
20	C₂×#13050579	C₂²	1	1	yes	characteristic
21	C₂×#13050579	C₂²	1	1	yes	characteristic
22	#14581790	—	2	1	no	—
23	#14581779	—	2	1	no	—
24	#14581773	—	2	1	no	—
25	#14581773	—	2	1	no	—
26	#14581771	C₂²	1	1	yes	characteristic
27	#14581771	C₂²	1	1	yes	characteristic
28	#14581771	C₂²	1	1	yes	characteristic
29	#14581771	C₂²	1	1	yes	characteristic
30	#14581771	C₂²	1	1	yes	characteristic
31	#14581771	C₂²	1	1	yes	characteristic
32	#14581768	—	2	1	no	—
33	#14581766	C₂²	1	1	yes	characteristic
34	#14581766	C₂²	1	1	yes	characteristic
35	#14581764	—	2	1	no	—
36	#14581764	—	2	1	no	—
37	#14581763	—	2	1	no	—
38	#14581751	C₂²	1	1	yes	characteristic
39	C₂×C₄⋊Dic₈₃	C₂²	1	1	yes	characteristic
40	#14581726	C₂²	1	1	yes	characteristic
41	#14581726	C₂²	1	1	yes	characteristic
42	#14581724	C₂²	1	1	yes	characteristic
43	#14581724	C₂²	1	1	yes	characteristic
44	#14581717	C₂²	1	1	yes	characteristic
45	#14581717	C₂²	1	1	yes	characteristic
46	#14581717	C₂²	1	1	yes	characteristic
47	#14581717	C₂²	1	1	yes	characteristic
48	#14581717	C₂²	1	1	yes	characteristic
49	#14581717	C₂²	1	1	yes	characteristic
50	#14581714	C₂²	1	1	yes	characteristic
51	#14581714	C₂²	1	1	yes	characteristic
52	C₂×#41×C₈₃	C₂²	1	1	yes	characteristic
53	C₂×M₁₆×C₈₃	C₂²	1	1	yes	characteristic
54	C₂²×C₈×C₈₃	C₂²	1	1	yes	characteristic
55	C₂²⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
56	C₂²⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
57	C₂²⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
58	C₂²⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
59	#13050590	—	4	1	no	—
60	#13050590	—	4	1	no	—
61	C₂×Dic₁₆₆	—	2	1	no	—
62	C₂×Dic₁₆₆	—	4	1	no	—
63	C₂×Dic₁₆₆	—	4	1	no	—
64	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
65	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
66	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
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	C₂²×Dic₈₃	—	2	1	no	—
73	C₂×C₄×D₈₃	—	2	1	no	—
74	Dic₃₃₂	—	4	1	no	—
75	Dic₃₃₂	—	4	1	no	—
76	#13050579	—	4	1	no	—
77	#13050579	—	4	1	no	—
78	C₄⋊Dic₈₃	—	2	1	no	—
79	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
80	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
81	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
82	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
83	Dic₈₃⋊C₄	—	2	1	no	—
84	Dic₈₃⋊C₄	—	2	1	no	—
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	C₂²⋊Dic₈₃	—	2	1	no	—
90	C₂²⋊Dic₈₃	—	2	1	no	—
91	C₂²⋊Dic₈₃	—	2	1	no	—
92	C₂²⋊Dic₈₃	—	2	1	no	—
93	C₂²⋊Dic₈₃	—	2	1	no	—
94	C₂²⋊Dic₈₃	—	2	1	no	—
95	C₂²⋊Dic₈₃	—	2	1	no	—
96	(C₂×D₈₃)⋊C₄	—	2	1	no	—
97	(C₂×D₈₃)⋊C₄	—	2	1	no	—
98	C₄×Dic₈₃	—	2	1	no	—
99	#41×C₈₃	—	4	1	no	—
100	#41×C₈₃	—	4	1	no	—
101	C₂×Q₈×C₈₃	D₄	1	1	yes	characteristic
102	C₂×D₄×C₈₃	D₄	1	1	yes	characteristic
103	C₂×D₄×C₈₃	D₄	1	1	yes	characteristic
104	C₂×D₄×C₈₃	D₄	1	1	yes	characteristic
105	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
106	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
107	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
108	M₁₆×C₈₃	—	4	1	no	—
109	C₂×C₈×C₈₃	—	4	1	no	—
110	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
111	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
112	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
113	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
114	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
115	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
116	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
117	C₂²×D₈₃	—	2	1	no	—
118	Q₈×C₈₃	—	4	1	no	—
119	D₄×C₈₃	—	4	1	no	—
120	D₄×C₈₃	—	4	1	no	—
121	D₄×C₈₃	—	4	1	no	—
122	C₂×C₄×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	—	2	1	no	—
127	C₂×C₄×C₈₃	—	2	1	no	—
128	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
129	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
130	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
131	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
132	C₂²⋊D₈₃	—	4	2	no	—
133	C₂²⋊D₈₃	—	8	1	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	—	2	1	no	—
139	C₂×Dic₈₃	—	2	1	no	—
140	C₂×Dic₈₃	—	2	1	no	—
141	C₂×Dic₈₃	—	2	1	no	—
142	C₂×Dic₈₃	—	4	2	no	—
143	C₂×Dic₈₃	—	8	1	no	—
144	D₃₃₂	—	2	1	no	—
145	D₃₃₂	—	2	1	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	—	2	1	no	—
152	Dic₁₆₆	—	2	1	no	—
153	Dic₁₆₆	—	4	2	no	—
154	C₈×C₈₃	—	2	1	no	—
155	C₈×C₈₃	—	2	1	no	—
156	C₈×C₈₃	—	2	1	no	—
157	C₈×C₈₃	—	2	1	no	—
158	C₂²×C₈₃	—	2	1	no	—
159	C₂²×C₈₃	—	2	1	no	—
160	C₂²×C₈₃	—	2	1	no	—
161	C₂²×C₈₃	—	2	1	no	—
162	C₂²×C₈₃	—	2	1	no	—
163	C₂²×C₈₃	—	2	1	no	—
164	C₂²×C₈₃	—	2	1	no	—
165	C₂²×C₈₃	—	2	1	no	—
166	C₂²×C₈₃	—	2	1	no	—
167	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
168	C₂×D₈₃	—	4	1	no	—
169	C₂×D₈₃	—	4	1	no	—
170	C₂×D₈₃	—	4	1	no	—
171	C₄×C₈₃	—	2	1	no	—
172	C₄×C₈₃	—	2	1	no	—
173	C₄×C₈₃	—	2	1	no	—
174	C₄×C₈₃	H(5,𝔽₂)	1	1	yes	characteristic
175	C₄×C₈₃	H(5,𝔽₂)	1	1	yes	characteristic
176	Dic₈₃	—	4	1	no	—
177	Dic₈₃	—	4	1	no	—
178	Dic₈₃	—	4	1	no	—
179	Dic₈₃	—	4	1	no	—
180	Dic₈₃	—	4	1	no	—
181	Dic₈₃	—	4	1	no	—
182	Dic₈₃	—	4	1	no	—
183	C₂×C₈₃	—	4	1	no	—
184	C₂×C₈₃	—	4	1	no	—
185	C₂×C₈₃	—	4	1	no	—
186	C₂×C₈₃	#534	1	1	yes	characteristic
187	C₂×C₈₃	#577	1	1	yes	characteristic
188	C₂×C₈₃	#578	1	1	yes	characteristic
189	D₈₃	—	8	1	no	—
190	#3201	—	83	1	no	maximal, Sylow
191	C₈₃	#3201	1	1	yes	characteristic, complemented, Sylow
192	#537	—	83	1	no	—
193	#536	—	83	1	no	—
194	#485	—	83	1	no	—
195	#484	—	83	1	no	—
196	#483	—	83	1	no	—
197	#482	—	83	1	no	—
198	#470	—	83	1	no	—
199	#468	—	83	1	no	—
200	#467	—	83	1	no	—
201	#465	—	83	1	no	—
202	#452	—	83	1	no	—
203	#452	—	83	1	no	—
204	#451	—	83	1	no	—
205	#448	—	83	1	no	—
206	#408	D₈₃	1	1	yes	characteristic, complemented
207	C₂×#41	—	83	1	no	—
208	C₂×#41	C₂×D₈₃	1	1	yes	characteristic
209	C₂²×Q₈	—	83	1	no	—
210	C₂×Q₁₆	—	83	1	no	—
211	C₂×Q₁₆	—	83	1	no	—
212	C₂×SD₁₆	—	83	1	no	—
213	C₂×SD₁₆	—	83	1	no	—
214	C₂×M₁₆	C₂×D₈₃	1	1	yes	characteristic
215	C₂²×C₈	C₂×D₈₃	1	1	yes	characteristic
216	C₄⋊Q₈	—	166	1	no	—
217	#124	—	166	1	no	—
218	#123	—	166	1	no	—
219	#123	—	166	1	no	—
220	C₂²⋊Q₈	—	83	1	no	—
221	C₂²⋊Q₈	—	83	1	no	—
222	C₂²⋊Q₈	—	83	1	no	—
223	C₂²⋊Q₈	—	83	1	no	—
224	C₂²⋊Q₈	—	83	1	no	—
225	C₂²⋊Q₈	—	83	1	no	—
226	C₂²⋊Q₈	—	166	1	no	—
227	#121	—	83	1	no	—
228	#121	—	83	1	no	—
229	#121	—	166	1	no	—
230	#121	—	166	1	no	—
231	#121	—	166	1	no	—
232	#117	—	83	1	no	—
233	C₂×C₄⋊C₄	—	83	1	no	—
234	#107	—	83	1	no	—
235	#107	—	83	1	no	—
236	#106	—	83	1	no	—
237	#106	—	83	1	no	—
238	#103	—	83	1	no	—
239	#103	—	83	1	no	—
240	#103	—	83	1	no	—
241	#103	—	83	1	no	—
242	#103	—	83	1	no	—
243	#103	—	83	1	no	—
244	#102	—	83	1	no	—
245	#102	—	83	1	no	—
246	C₂²⋊C₈	C₂×D₈₃	1	1	yes	characteristic
247	C₂²⋊C₈	C₂×D₈₃	1	1	yes	characteristic
248	C₂²⋊C₈	C₂×D₈₃	1	1	yes	characteristic
249	C₂²⋊C₈	C₂×D₈₃	1	1	yes	characteristic
250	#41	—	4	1	no	—
251	#41	—	4	1	no	—
252	#41	—	332	1	no	—
253	#41	—	332	1	no	—
254	C₂×Q₈	—	332	1	no	—
255	C₂×Q₈	—	332	1	no	—
256	C₂×Q₈	—	83	1	no	—
257	C₂×Q₈	—	83	1	no	—
258	C₂×Q₈	—	83	1	no	—
259	C₂×Q₈	—	166	1	no	—
260	C₂×Q₈	D₃₃₂	1	1	yes	characteristic
261	C₂×D₄	—	83	1	no	—
262	C₂×D₄	—	166	1	no	—
263	C₂×D₄	—	166	1	no	—
264	C₂×D₄	—	166	1	no	—
265	C₂×D₄	D₃₃₂	1	1	yes	characteristic
266	C₂×D₄	D₃₃₂	1	1	yes	characteristic
267	C₂×D₄	D₃₃₂	1	1	yes	characteristic
268	C₂²×C₄	—	166	1	no	—
269	C₂²×C₄	—	166	1	no	—
270	C₂²×C₄	—	166	1	no	—
271	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
272	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
273	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
274	Q₁₆	—	332	1	no	—
275	Q₁₆	—	332	1	no	—
276	SD₁₆	—	332	1	no	—
277	SD₁₆	—	332	1	no	—
278	M₁₆	—	4	1	no	—
279	C₂×C₈	—	4	1	no	—
280	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
281	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
282	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
283	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
284	C₄⋊C₄	—	83	1	no	—
285	C₄⋊C₄	—	83	1	no	—
286	C₄⋊C₄	—	83	1	no	—
287	C₄⋊C₄	—	83	1	no	—
288	C₄⋊C₄	—	166	1	no	—
289	C₄⋊C₄	—	166	1	no	—
290	C₄⋊C₄	—	166	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	C₂²⋊C₄	—	166	1	no	—
296	C₂²⋊C₄	—	166	1	no	—
297	C₂²⋊C₄	—	166	1	no	—
298	C₂²⋊C₄	—	166	1	no	—
299	C₂²⋊C₄	—	166	1	no	—
300	C₂²⋊C₄	—	166	1	no	—
301	C₂²⋊C₄	—	166	1	no	—
302	C₂²⋊C₄	—	166	1	no	—
303	C₂²⋊C₄	—	166	1	no	—
304	C₄²	—	166	1	no	—
305	C₂³	—	166	1	no	—
306	C₂³	C₂×D₃₃₂	1	1	yes	characteristic
307	C₂³	C₂×D₃₃₂	1	1	yes	characteristic
308	C₂³	C₂×D₃₃₂	1	1	yes	characteristic
309	Q₈	—	4	1	no	—
310	Q₈	—	332	2	no	—
311	Q₈	—	166	1	no	—
312	Q₈	—	166	1	no	—
313	Q₈	—	166	1	no	—
314	Q₈	—	166	1	no	—
315	Q₈	—	166	1	no	—
316	Q₈	—	166	1	no	—
317	D₄	—	4	1	no	—
318	D₄	—	4	1	no	—
319	D₄	—	4	1	no	—
320	D₄	—	332	2	no	—
321	D₄	—	664	1	no	—
322	D₄	—	664	1	no	—
323	D₄	—	166	1	no	—
324	D₄	—	166	1	no	—
325	C₂×C₄	—	2	1	no	—
326	C₂×C₄	—	2	1	no	—
327	C₂×C₄	—	2	1	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	—	2	1	no	—
331	C₂×C₄	—	332	2	no	—
332	C₂×C₄	—	332	2	no	—
333	C₂×C₄	—	664	1	no	—
334	C₂×C₄	—	166	1	no	—
335	C₂×C₄	—	166	1	no	—
336	C₂×C₄	—	166	1	no	—
337	C₂×C₄	—	166	1	no	—
338	C₂×C₄	—	166	1	no	—
339	C₂×C₄	—	166	1	no	—
340	C₂×C₄	—	166	1	no	—
341	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
342	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
343	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
344	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
345	C₈	—	2	1	no	—
346	C₈	—	2	1	no	—
347	C₈	—	2	1	no	—
348	C₈	—	2	1	no	—
349	C₂²	—	2	1	no	—
350	C₂²	—	2	1	no	—
351	C₂²	—	2	1	no	—
352	C₂²	—	2	1	no	—
353	C₂²	—	2	1	no	—
354	C₂²	—	2	1	no	—
355	C₂²	—	2	1	no	—
356	C₂²	—	2	1	no	—
357	C₂²	—	2	1	no	—
358	C₂²	—	332	1	no	—
359	C₂²	—	332	1	no	—
360	C₂²	—	332	1	no	—
361	C₂²	C₂²×D₃₃₂	1	1	yes	characteristic
362	C₄	—	2	1	no	—
363	C₄	—	2	1	no	—
364	C₄	—	2	1	no	—
365	C₄	—	332	1	no	—
366	C₄	—	332	1	no	—
367	C₄	—	332	1	no	—
368	C₄	—	332	1	no	—
369	C₄	—	332	1	no	—
370	C₄	—	332	1	no	—
371	C₄	—	332	1	no	—
372	C₄	#14581837	1	1	yes	characteristic
373	C₄	#14581837	1	1	yes	characteristic
374	C₂	—	4	1	no	—
375	C₂	—	4	1	no	—
376	C₂	—	4	1	no	—
377	C₂	—	664	1	no	—
378	C₂	#19128263	1	1	yes	characteristic
379	C₂	#19128513	1	1	yes	characteristic
380	C₂	#19128518	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₃₃₂
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#408×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	#19128279	C₂
hypercenter	#408	D₈₃
hypocenter	C₈₃	#3201
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ #408

Complements

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

N	H ≅ G/N	type
2. #19128272	C₂	semidirect
3. #19128043	C₂	semidirect
4. #19128037	C₂	semidirect
5. #19128032	C₂	semidirect
6. #19128026	C₂	semidirect
7. #19127959	C₂	semidirect
8. #19127947	C₂	semidirect
9. #19127942	C₂	semidirect
10. #19127930	C₂	semidirect
11. #19127865	C₂	semidirect
12. #19127865	C₂	semidirect
13. #19127859	C₂	semidirect
14. #19127842	C₂	semidirect
15. #408×C₈₃	C₂	semidirect
191. C₈₃	#3201	semidirect
206. #408	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128279	2
#19128272	2
#19128043	2
#19128037	2
#19128032	2
#19128026	2
#19127959	2
#19127947	2
#19127942	2
#19127930	2
#19127865	2
#19127859	2
#19127842	2
#408×C₈₃	2
#3201	83