Subgroups — #24919675

Order 10624.

402 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	#19128390	C₂	1	1	yes	maximal, characteristic, complemented
2	#19128345	C₂	1	1	yes	maximal, characteristic
3	#19128054	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128042	C₂	1	1	yes	maximal, characteristic, complemented
5	#19128018	C₂	1	1	yes	maximal, characteristic, complemented
6	#19128003	C₂	1	1	yes	maximal, characteristic, complemented
7	#19127959	C₂	1	1	yes	maximal, characteristic, complemented
8	#19127946	C₂	1	1	yes	maximal, characteristic, complemented
9	#19127922	C₂	1	1	yes	maximal, characteristic, complemented
10	#19127864	C₂	1	1	yes	maximal, characteristic, complemented
11	#19127852	C₂	1	1	yes	maximal, characteristic, complemented
12	#19127792	C₂	1	1	yes	maximal, characteristic, complemented
13	#19127777	C₂	1	1	yes	maximal, characteristic, complemented
14	#19127699	C₂	1	1	yes	maximal, characteristic, complemented
15	#418×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13050589	C₂²	1	1	yes	characteristic
17	C₂×Dic₃₃₂	C₂²	1	1	yes	characteristic, complemented
18	C₂×#13050578	C₂²	1	1	yes	characteristic
19	C₂×#13050578	C₂²	1	1	yes	characteristic
20	C₂×#13050573	C₂²	1	1	yes	characteristic, complemented
21	#14581785	—	2	1	no	—
22	#14581784	—	2	1	no	—
23	#14581783	C₂²	1	1	yes	characteristic, complemented
24	#14581779	C₂²	1	1	yes	characteristic
25	#14581774	—	2	1	no	—
26	#14581773	—	2	1	no	—
27	#14581772	C₂²	1	1	yes	characteristic, complemented
28	#14581771	C₂²	1	1	yes	characteristic
29	#14581771	C₂²	1	1	yes	characteristic
30	#14581764	—	2	1	no	—
31	#14581763	C₂²	1	1	yes	characteristic
32	#14581761	—	2	1	no	—
33	#14581757	C₂²	1	1	yes	characteristic, complemented
34	D₄×Dic₈₃	C₂²	1	1	yes	characteristic
35	C₄×C₂²⋊D₈₃	C₂²	1	1	yes	characteristic
36	#14581749	—	2	1	no	—
37	C₄⋊C₄×D₈₃	C₂²	1	1	yes	characteristic, complemented
38	C₂×C₂²⋊Dic₈₃	—	2	1	no	—
39	#14581725	C₂²	1	1	yes	characteristic, complemented
40	#14581724	C₂²	1	1	yes	characteristic, complemented
41	#14581723	C₂²	1	1	yes	characteristic, complemented
42	Dic₈₃⋊C₈	C₂²	1	1	yes	characteristic, complemented
43	#14581717	C₂²	1	1	yes	characteristic, complemented
44	#14581717	C₂²	1	1	yes	characteristic, complemented
45	#14581715	C₂²	1	1	yes	characteristic, complemented
46	#14581715	C₂²	1	1	yes	characteristic, complemented
47	#14581713	C₂²	1	1	yes	characteristic
48	#14581713	C₂²	1	1	yes	characteristic
49	C₂²⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
50	(C₂×D₈₃)⋊C₈	C₂²	1	1	yes	characteristic, complemented
51	C₈×Dic₈₃	C₂²	1	1	yes	characteristic, complemented
52	C₂²×D₄×C₈₃	C₂²	1	1	yes	characteristic
53	C₂×M₁₆×C₈₃	C₂²	1	1	yes	characteristic
54	#117×C₈₃	C₂²	1	1	yes	characteristic
55	#102×C₈₃	C₂²	1	1	yes	characteristic
56	#102×C₈₃	C₂²	1	1	yes	characteristic
57	#102×C₈₃	C₂²	1	1	yes	characteristic
58	#102×C₈₃	C₂²	1	1	yes	characteristic
59	#13050589	—	2	2	no	—
60	#13050589	—	4	1	no	—
61	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
62	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
63	C₂×C₂²⋊D₈₃	—	2	1	no	—
64	C₂×C₂²⋊D₈₃	—	2	1	no	—
65	C₂×C₂²⋊D₈₃	D₄	1	1	yes	characteristic
66	C₂²×Dic₈₃	—	2	1	no	—
67	C₂²×Dic₈₃	—	2	1	no	—
68	C₂×C₄×D₈₃	—	2	1	no	—
69	C₂×C₄×D₈₃	C₂³	1	1	yes	characteristic
70	Dic₃₃₂	—	4	1	no	—
71	#13050578	—	2	2	no	—
72	#13050578	—	4	1	no	—
73	#13050573	—	4	1	no	—
74	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
75	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
76	C₄⋊Dic₈₃	—	2	1	no	—
77	C₄⋊Dic₈₃	—	2	1	no	—
78	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
79	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
80	Dic₈₃⋊C₄	—	2	1	no	—
81	Dic₈₃⋊C₄	—	2	1	no	—
82	Dic₈₃⋊C₄	—	2	1	no	—
83	Dic₈₃⋊C₄	—	2	1	no	—
84	Dic₈₃⋊C₄	—	2	1	no	—
85	Dic₈₃⋊C₄	D₄	1	1	yes	characteristic
86	C₂²⋊Dic₈₃	—	2	1	no	—
87	C₂²⋊Dic₈₃	—	2	1	no	—
88	C₂²⋊Dic₈₃	—	2	1	no	—
89	C₂²⋊Dic₈₃	—	2	1	no	—
90	C₂²⋊Dic₈₃	—	2	1	no	—
91	C₂²⋊Dic₈₃	—	2	1	no	—
92	C₂²⋊Dic₈₃	D₄	1	1	yes	characteristic
93	(C₂×D₈₃)⋊C₄	—	2	1	no	—
94	(C₂×D₈₃)⋊C₄	D₄	1	1	yes	characteristic
95	C₄×Dic₈₃	—	2	1	no	—
96	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
97	C₂⁴×C₈₃	—	2	1	no	—
98	C₂×D₄×C₈₃	—	2	2	no	—
99	C₂×D₄×C₈₃	—	2	1	no	—
100	C₂×D₄×C₈₃	—	4	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	M₁₆×C₈₃	—	4	1	no	—
105	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
106	C₂×C₈×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	C₂²⋊C₄×C₈₃	—	2	1	no	—
110	C₄²×C₈₃	—	2	1	no	—
111	C₂³×C₈₃	—	2	1	no	—
112	C₂³×C₈₃	—	2	1	no	—
113	C₂³×C₈₃	—	4	2	no	—
114	C₂³×C₈₃	—	4	2	no	—
115	C₂³×C₈₃	—	8	1	no	—
116	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
117	C₂²×D₈₃	C₂×D₄	1	1	yes	characteristic
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	#41	1	2	yes	—
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	—	4	1	no	—
130	C₂²⋊D₈₃	—	4	2	no	—
131	C₂²⋊D₈₃	—	8	1	no	—
132	C₂×Dic₈₃	—	2	1	no	—
133	C₂×Dic₈₃	—	2	1	no	—
134	C₂×Dic₈₃	—	2	1	no	—
135	C₂×Dic₈₃	—	2	1	no	—
136	C₂×Dic₈₃	—	4	2	no	—
137	C₂×Dic₈₃	—	8	1	no	—
138	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
139	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
140	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
141	C₄×D₈₃	—	2	1	no	—
142	C₄×D₈₃	—	2	1	no	—
143	C₄×D₈₃	—	4	2	no	—
144	Dic₁₆₆	—	2	1	no	—
145	Dic₁₆₆	—	2	1	no	—
146	Dic₁₆₆	—	2	1	no	—
147	Dic₁₆₆	—	2	1	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	—	2	2	no	—
153	C₂²×C₈₃	—	2	1	no	—
154	C₂²×C₈₃	—	2	1	no	—
155	C₂²×C₈₃	—	2	1	no	—
156	C₂²×C₈₃	—	4	1	no	—
157	C₂²×C₈₃	—	4	1	no	—
158	C₂²×C₈₃	—	4	1	no	—
159	C₂²×C₈₃	—	4	1	no	—
160	C₂²×C₈₃	—	4	1	no	—
161	C₂²×C₈₃	—	8	4	no	—
162	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
163	C₂×D₈₃	—	2	1	no	—
164	C₂×D₈₃	—	2	1	no	—
165	C₂×D₈₃	—	2	1	no	—
166	C₄×C₈₃	—	2	1	no	—
167	C₄×C₈₃	—	4	1	no	—
168	C₄×C₈₃	—	4	1	no	—
169	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
170	C₄×C₈₃	H(5,𝔽₂)	1	1	yes	characteristic
171	Dic₈₃	—	2	1	no	—
172	Dic₈₃	—	2	1	no	—
173	Dic₈₃	—	2	1	no	—
174	Dic₈₃	—	4	1	no	—
175	Dic₈₃	—	4	1	no	—
176	Dic₈₃	—	4	1	no	—
177	Dic₈₃	—	4	1	no	—
178	C₂×C₈₃	—	4	1	no	—
179	C₂×C₈₃	—	4	2	no	—
180	C₂×C₈₃	—	8	1	no	—
181	C₂×C₈₃	#546	1	1	yes	characteristic
182	C₂×C₈₃	#575	1	1	yes	characteristic
183	C₂×C₈₃	#577	1	1	yes	characteristic
184	D₈₃	—	4	1	no	—
185	#3323	—	83	1	no	maximal, Sylow
186	C₈₃	#3323	1	1	yes	characteristic, complemented, Sylow
187	#552	—	83	1	no	—
188	#546	—	83	1	no	—
189	#488	—	83	1	no	—
190	#485	—	83	1	no	—
191	#481	—	83	1	no	—
192	#478	—	83	1	no	—
193	#470	—	83	1	no	—
194	#468	—	83	1	no	—
195	#464	—	83	1	no	—
196	#452	—	83	1	no	—
197	#450	—	83	1	no	—
198	C₄×SD₁₆	—	83	1	no	—
199	#436	—	83	1	no	—
200	#418	D₈₃	1	1	yes	characteristic, complemented
201	#417	—	83	1	no	—
202	C₂×#41	—	83	1	no	—
203	C₂²×D₄	C₂×D₈₃	1	1	yes	characteristic
204	C₂×Q₁₆	—	83	1	no	—
205	C₂×SD₁₆	—	83	1	no	—
206	C₂×SD₁₆	—	83	1	no	—
207	C₂×M₁₆	—	83	1	no	—
208	C₂×M₁₆	C₂×D₈₃	1	1	yes	characteristic
209	#126	—	166	1	no	—
210	#125	—	83	1	no	—
211	#125	—	166	1	no	—
212	#124	—	83	1	no	—
213	#123	—	166	1	no	—
214	#123	—	166	1	no	—
215	C₂²⋊Q₈	—	83	1	no	—
216	C₂²⋊Q₈	—	83	1	no	—
217	C₂²⋊Q₈	—	83	1	no	—
218	#121	—	83	1	no	—
219	#121	—	166	1	no	—
220	#120	—	166	1	no	—
221	C₄×Q₈	—	83	1	no	—
222	C₄×D₄	—	83	1	no	—
223	C₄×D₄	—	83	1	no	—
224	#117	—	166	1	no	—
225	#117	C₂×D₈₃	1	1	yes	characteristic
226	C₂×C₄⋊C₄	—	83	1	no	—
227	C₂×C₂²⋊C₄	—	166	1	no	—
228	#107	—	83	1	no	—
229	#106	—	83	1	no	—
230	#106	—	83	1	no	—
231	C₄⋊C₈	—	83	1	no	—
232	#103	—	83	1	no	—
233	#103	—	83	1	no	—
234	#103	—	83	1	no	—
235	#103	—	83	1	no	—
236	#102	—	83	1	no	—
237	#102	—	83	1	no	—
238	#102	C₂×D₈₃	1	1	yes	characteristic
239	#102	C₂×D₈₃	1	1	yes	characteristic
240	#102	C₂×D₈₃	1	1	yes	characteristic
241	#102	C₂×D₈₃	1	1	yes	characteristic
242	C₂²⋊C₈	—	83	1	no	—
243	C₂²⋊C₈	—	83	1	no	—
244	C₄×C₈	—	83	1	no	—
245	C₂⁴	—	2	1	no	—
246	#41	—	332	1	no	—
247	#41	—	166	2	no	—
248	C₂×Q₈	—	83	1	no	—
249	C₂×Q₈	—	83	1	no	—
250	C₂×D₄	—	2	2	no	—
251	C₂×D₄	—	2	1	no	—
252	C₂×D₄	—	4	1	no	—
253	C₂×D₄	—	83	1	no	—
254	C₂×D₄	—	166	1	no	—
255	C₂×D₄	—	166	1	no	—
256	C₂×D₄	C₂²×D₈₃	1	1	yes	characteristic
257	C₂×D₄	C₂²×D₈₃	1	1	yes	characteristic
258	C₂²×C₄	—	83	1	no	—
259	C₂²×C₄	—	166	1	no	—
260	C₂²×C₄	—	166	1	no	—
261	C₂²×C₄	—	166	1	no	—
262	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
263	Q₁₆	—	332	1	no	—
264	SD₁₆	—	332	1	no	—
265	SD₁₆	—	166	2	no	—
266	M₁₆	—	4	1	no	—
267	M₁₆	—	332	1	no	—
268	C₂×C₈	—	83	1	no	—
269	C₂×C₈	—	83	1	no	—
270	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
271	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
272	C₄⋊C₄	—	83	1	no	—
273	C₄⋊C₄	—	83	1	no	—
274	C₄⋊C₄	—	83	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	—	166	1	no	—
282	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
283	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
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	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	—	166	1	no	—
297	C₂³	—	2	1	no	—
298	C₂³	—	2	1	no	—
299	C₂³	—	4	2	no	—
300	C₂³	—	4	2	no	—
301	C₂³	—	8	1	no	—
302	C₂³	—	83	1	no	—
303	C₂³	D₄×D₈₃	1	1	yes	characteristic
304	Q₈	—	166	1	no	—
305	Q₈	—	166	1	no	—
306	Q₈	—	166	1	no	—
307	Q₈	—	166	1	no	—
308	D₄	—	2	1	no	—
309	D₄	—	2	1	no	—
310	D₄	—	2	1	no	—
311	D₄	—	4	2	no	—
312	D₄	—	332	2	no	—
313	D₄	—	332	1	no	—
314	D₄	—	664	1	no	—
315	D₄	#13050590	1	2	yes	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	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	—	83	1	no	—
325	C₂×C₄	—	664	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	—	166	1	no	—
331	C₂×C₄	—	166	1	no	—
332	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
333	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
334	C₈	—	2	1	no	—
335	C₈	—	2	1	no	—
336	C₈	—	166	1	no	—
337	C₈	—	166	1	no	—
338	C₂²	—	2	1	no	—
339	C₂²	—	2	2	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	4	1	no	—
343	C₂²	—	4	1	no	—
344	C₂²	—	4	1	no	—
345	C₂²	—	4	1	no	—
346	C₂²	—	4	1	no	—
347	C₂²	—	8	4	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	—	4	1	no	—
354	C₄	—	4	1	no	—
355	C₄	—	332	1	no	—
356	C₄	—	332	1	no	—
357	C₄	—	332	1	no	—
358	C₄	—	332	1	no	—
359	C₄	—	166	1	no	—
360	C₄	—	166	1	no	—
361	C₄	—	166	1	no	—
362	C₄	C₂×#13050590	1	1	yes	characteristic
363	C₄	#14581837	1	1	yes	characteristic
364	C₂	—	4	1	no	—
365	C₂	—	4	2	no	—
366	C₂	—	8	1	no	—
367	C₂	—	332	1	no	—
368	C₂	#19128341	1	1	yes	characteristic
369	C₂	#19128497	1	1	yes	characteristic
370	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	#418×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	#19128345	C₂
hypercenter	#418	D₈₃
hypocenter	C₈₃	#3323
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #418

Complements

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

N	H ≅ G/N	type
1. #19128390	C₂	semidirect
3. #19128054	C₂	semidirect
4. #19128042	C₂	semidirect
5. #19128018	C₂	semidirect
6. #19128003	C₂	semidirect
7. #19127959	C₂	semidirect
8. #19127946	C₂	semidirect
9. #19127922	C₂	semidirect
10. #19127864	C₂	semidirect
11. #19127852	C₂	semidirect
12. #19127792	C₂	semidirect
13. #19127777	C₂	semidirect
14. #19127699	C₂	semidirect
15. #418×C₈₃	C₂	semidirect
17. C₂×Dic₃₃₂	C₂²	semidirect
20. C₂×#13050573	C₂²	semidirect
23. #14581783	C₂²	semidirect
27. #14581772	C₂²	semidirect
33. #14581757	C₂²	semidirect
37. C₄⋊C₄×D₈₃	C₂²	semidirect
39. #14581725	C₂²	semidirect
40. #14581724	C₂²	semidirect
41. #14581723	C₂²	semidirect
42. Dic₈₃⋊C₈	C₂²	semidirect
43. #14581717	C₂²	semidirect
44. #14581717	C₂²	semidirect
45. #14581715	C₂²	semidirect
46. #14581715	C₂²	semidirect
50. (C₂×D₈₃)⋊C₈	C₂²	semidirect
51. C₈×Dic₈₃	C₂²	semidirect
186. C₈₃	#3323	semidirect
200. #418	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128390	2
#19128345	2
#19128054	2
#19128042	2
#19128018	2
#19128003	2
#19127959	2
#19127946	2
#19127922	2
#19127864	2
#19127852	2
#19127792	2
#19127777	2
#19127699	2
#418×C₈₃	2
#3323	83