Subgroups — #25253222

Order 11008.

664 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583338	C₂	1	2	yes	maximal, complemented
2	Q₁₆×D₁₇₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#19146316	C₂	1	1	yes	maximal, characteristic
4	#19136848	C₂	1	1	yes	maximal, characteristic, complemented
5	#19136704	C₂	1	2	yes	maximal, complemented
6	#19136645	C₂	1	1	yes	maximal, characteristic, complemented
7	#19136640	C₂	1	1	yes	maximal, characteristic, complemented
8	#19136536	C₂	1	2	yes	maximal, complemented
9	#19136530	C₂	1	2	yes	maximal, complemented
10	#19136428	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×SD₃₂×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13063344	C₂²	1	2	yes	—
13	C₂×Q₁₆×D₄₃	C₂²	1	2	yes	—
14	Q₈×D₁₇₂	—	2	1	no	—
15	#14583553	—	2	1	no	—
16	#14583338	—	2	8	no	—
17	C₂×Dic₃₄₄	C₂²	1	2	yes	—
18	C₂×Dic₃₄₄	C₂²	1	2	yes	—
19	C₂×#13063271	C₂²	1	2	yes	—
20	C₂×#13063269	C₂²	1	2	yes	—
21	C₂×#13063264	C₂²	1	2	yes	—
22	#14583279	C₂²	1	1	yes	characteristic
23	#14583279	C₂²	1	1	yes	characteristic
24	#14583141	C₂²	1	2	yes	—
25	#14583141	C₂²	1	2	yes	—
26	#14583109	—	2	1	no	—
27	#14583104	—	2	1	no	—
28	#14583062	—	2	2	no	—
29	#14583056	—	2	2	no	—
30	C₈×D₁₇₂	C₂²	1	1	yes	characteristic
31	C₄⋊(C₄₃⋊C₁₆)	C₂²	1	1	yes	characteristic
32	#14582702	C₂²	1	2	yes	—
33	#14582702	C₂²	1	2	yes	—
34	(C₂×D₄₃)⋊C₁₆	C₂²	1	2	yes	—
35	C₂×SD₃₂×C₄₃	C₂²	1	1	yes	characteristic
36	C₄×Q₁₆×C₄₃	C₂²	1	1	yes	characteristic
37	C₄×D₈×C₄₃	C₂²	1	1	yes	characteristic
38	C₄₃×#367	C₂²	1	1	yes	characteristic
39	C₄₃×#358	C₂²	1	1	yes	characteristic
40	C₄₃×#357	C₂²	1	1	yes	characteristic
41	C₄×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
42	C₂×#10621900	—	2	2	no	—
43	C₂×Q₈×D₄₃	—	2	2	no	—
44	#13063344	—	2	4	no	—
45	#13063344	—	4	2	no	—
46	C₂×Dic₁₇₂	—	2	1	no	—
47	C₂×Dic₁₇₂	—	2	1	no	—
48	C₂×Dic₁₇₂	C₂³	1	2	yes	—
49	C₂×Dic₁₇₂	C₂³	1	2	yes	—
50	C₂×#10621885	—	2	2	no	—
51	Q₁₆×D₄₃	—	2	4	no	—
52	Q₁₆×D₄₃	—	4	2	no	—
53	C₂×#10621883	—	2	2	no	—
54	C₂×C₈×D₄₃	C₂³	1	2	yes	—
55	#13063325	—	2	1	no	—
56	#13063325	—	2	1	no	—
57	#13063325	—	4	1	no	—
58	#13063309	—	4	2	no	—
59	#13063305	—	2	2	no	—
60	#13063305	—	2	2	no	—
61	#13063305	—	4	2	no	—
62	#13063299	—	4	1	no	—
63	C₄×D₁₇₂	—	4	1	no	—
64	C₄×D₁₇₂	D₄	1	1	yes	characteristic
65	C₂×C₄⋊Dic₄₃	—	4	1	no	—
66	Dic₃₄₄	—	2	4	no	—
67	Dic₃₄₄	—	4	2	no	—
68	#13063271	—	2	4	no	—
69	#13063269	—	2	4	no	—
70	#13063264	—	2	4	no	—
71	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
72	C₄⋊(C₄₃⋊C₈)	D₄	1	1	yes	characteristic
73	#13063248	—	2	2	no	—
74	#13063248	—	2	2	no	—
75	(C₂×D₄₃)⋊C₈	D₄	1	2	yes	—
76	C₂×Q₁₆×C₄₃	C₂³	1	1	yes	characteristic
77	C₂×D₈×C₄₃	C₂³	1	1	yes	characteristic
78	C₄×Q₈×C₄₃	—	2	1	no	—
79	C₄×D₄×C₄₃	—	2	1	no	—
80	SD₃₂×C₄₃	D₄	1	4	yes	complemented
81	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
82	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
83	#107×C₄₃	C₂³	1	1	yes	characteristic
84	#107×C₄₃	C₂³	1	1	yes	characteristic
85	#103×C₄₃	—	2	1	no	—
86	#102×C₄₃	—	2	1	no	—
87	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
88	C₂×Q₈×C₄₃	—	2	1	no	—
89	C₂×D₄×C₄₃	—	2	1	no	—
90	C₂²×C₄×C₄₃	—	4	1	no	—
91	C₂×C₂²⋊D₄₃	—	4	2	no	—
92	C₂²×Dic₄₃	—	4	2	no	—
93	Q₈×D₄₃	—	4	4	no	—
94	Q₈×D₄₃	—	8	2	no	—
95	#10621900	—	4	4	no	—
96	#10621900	—	8	2	no	—
97	C₂×D₁₇₂	D₈	1	1	yes	characteristic
98	C₂×C₄×D₄₃	—	4	2	no	—
99	C₂×C₄×D₄₃	C₂×D₄	1	2	yes	—
100	C₂×Dic₈₆	—	2	2	no	—
101	C₂×Dic₈₆	—	2	2	no	—
102	C₂×Dic₈₆	—	4	2	no	—
103	Q₁₆×C₄₃	—	2	1	no	—
104	Q₁₆×C₄₃	C₂×D₄	1	2	yes	—
105	D₈×C₄₃	—	2	1	no	—
106	D₈×C₄₃	C₂×D₄	1	2	yes	—
107	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
109	C₄⋊C₄×C₄₃	—	2	1	no	—
110	C₄⋊C₄×C₄₃	—	2	1	no	—
111	C₄⋊C₄×C₄₃	—	4	1	no	—
112	C₂²⋊C₄×C₄₃	—	4	1	no	—
113	C₄²×C₄₃	—	4	1	no	—
114	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
115	#10621885	—	4	4	no	—
116	#10621883	—	4	4	no	—
117	(C₂×D₄₃)⋊C₄	—	4	2	no	—
118	(C₂×D₄₃)⋊C₄	—	4	2	no	—
119	(C₂×D₄₃)⋊C₄	D₈	1	2	yes	—
120	C₄⋊Dic₄₃	—	4	1	no	—
121	C₄⋊Dic₄₃	—	4	1	no	—
122	C₄⋊Dic₄₃	—	4	1	no	—
123	C₄⋊Dic₄₃	—	4	1	no	—
124	C₄⋊Dic₄₃	—	4	2	no	—
125	C₄⋊Dic₄₃	—	4	2	no	—
126	C₄⋊Dic₄₃	D₈	1	1	yes	characteristic
127	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
128	Dic₁₇₂	—	2	2	no	—
129	Dic₁₇₂	—	2	2	no	—
130	Dic₁₇₂	—	2	2	no	—
131	Dic₁₇₂	—	2	2	no	—
132	Dic₁₇₂	—	8	1	no	—
133	Dic₁₇₂	—	8	1	no	—
134	C₈×D₄₃	—	2	2	no	—
135	C₈×D₄₃	—	2	2	no	—
136	C₁₆×C₄₃	—	2	1	no	—
137	C₁₆×C₄₃	C₂×D₄	1	2	yes	—
138	C₄₃⋊C₁₆	—	2	2	no	—
139	C₂³×C₄₃	—	4	1	no	—
140	C₂²×D₄₃	C₂×D₈	1	2	yes	—
141	Q₈×C₄₃	—	2	2	no	—
142	Q₈×C₄₃	—	4	1	no	—
143	D₄×C₄₃	—	2	2	no	—
144	D₄×C₄₃	—	4	1	no	—
145	C₂×C₄×C₄₃	—	4	1	no	—
146	C₂×C₄×C₄₃	—	4	1	no	—
147	C₂×C₄×C₄₃	—	4	1	no	—
148	C₂×C₄×C₄₃	—	8	2	no	—
149	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
150	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
151	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
152	C₂²⋊D₄₃	—	8	4	no	—
153	C₂×Dic₄₃	—	4	2	no	—
154	C₂×Dic₄₃	—	4	2	no	—
155	C₂×Dic₄₃	—	8	4	no	—
156	C₂×Dic₄₃	C₂×D₈	1	2	yes	—
157	D₁₇₂	—	2	2	no	—
158	C₄×D₄₃	—	2	2	no	—
159	C₄×D₄₃	—	2	2	no	—
160	C₄×D₄₃	—	8	4	no	—
161	Dic₈₆	—	4	2	no	—
162	Dic₈₆	—	4	2	no	—
163	Dic₈₆	—	4	2	no	—
164	Dic₈₆	—	4	2	no	—
165	Dic₈₆	—	8	4	no	—
166	C₈×C₄₃	—	2	1	no	—
167	C₈×C₄₃	C₂²×D₄	1	1	yes	characteristic
168	C₈×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
169	C₄₃⋊C₈	—	2	2	no	—
170	#33842	—	43	1	no	maximal, Sylow
171	C₂²×C₄₃	—	4	2	no	—
172	C₂²×C₄₃	—	8	1	no	—
173	C₂²×C₄₃	—	8	1	no	—
174	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
175	C₂×D₄₃	—	2	2	no	—
176	C₂×D₄₃	—	2	2	no	—
177	C₂×D₄₃	—	2	2	no	—
178	C₄×C₄₃	—	2	1	no	—
179	C₄×C₄₃	—	4	2	no	—
180	C₄×C₄₃	—	8	1	no	—
181	C₄×C₄₃	—	8	1	no	—
182	C₄×C₄₃	#510	1	2	yes	complemented
183	C₄×C₄₃	D₄²	1	1	yes	characteristic
184	C₄×C₄₃	#576	1	1	yes	characteristic
185	Dic₄₃	—	2	2	no	—
186	Dic₄₃	—	8	2	no	—
187	Dic₄₃	—	8	2	no	—
188	C₂×#510	—	43	2	no	—
189	D₄×Q₁₆	—	43	1	no	—
190	#3283	—	43	1	no	—
191	#2247	—	43	1	no	—
192	#2219	—	43	2	no	—
193	#2209	—	43	1	no	—
194	#2208	—	43	1	no	—
195	#2189	—	43	2	no	—
196	#2188	—	43	2	no	—
197	C₄×SD₃₂	D₄₃	1	1	yes	characteristic, complemented
198	#2169	—	43	1	no	—
199	C₂×C₄₃	—	8	2	no	—
200	C₂×C₄₃	D₄×D₈	1	1	yes	characteristic
201	C₂×C₄₃	C₂×#510	1	1	yes	characteristic
202	C₂×C₄₃	#3417	1	1	yes	characteristic
203	D₄₃	—	4	2	no	—
204	C₂×#135	—	43	2	no	—
205	C₂²×Q₁₆	—	43	2	no	—
206	D₄×Q₈	—	86	1	no	—
207	#547	—	86	1	no	—
208	#510	—	86	8	no	—
209	C₂×Q₃₂	—	43	2	no	—
210	C₂×Q₃₂	—	43	2	no	—
211	C₂×Q₃₂	—	43	2	no	—
212	C₂×SD₃₂	—	43	2	no	—
213	C₂×SD₃₂	C₂×D₄₃	1	1	yes	characteristic
214	C₂×M₃₂	—	43	2	no	—
215	#494	—	43	1	no	—
216	#494	—	43	1	no	—
217	#467	—	43	2	no	—
218	#467	—	43	2	no	—
219	#462	—	86	1	no	—
220	#461	—	86	1	no	—
221	#452	—	86	2	no	—
222	#451	—	86	2	no	—
223	C₄×Q₁₆	C₂×D₄₃	1	1	yes	characteristic
224	C₄×D₈	C₂×D₄₃	1	1	yes	characteristic
225	C₈×D₄	—	43	1	no	—
226	#367	C₂×D₄₃	1	1	yes	characteristic
227	C₄⋊C₁₆	—	43	1	no	—
228	#358	—	43	2	no	—
229	#358	—	43	2	no	—
230	#358	C₂×D₄₃	1	1	yes	characteristic
231	#357	C₂×D₄₃	1	1	yes	characteristic
232	C₂²⋊C₁₆	—	43	2	no	—
233	C₄×C₁₆	C₂×D₄₃	1	1	yes	characteristic
234	C₄₃	#33842	1	1	yes	characteristic, complemented, Sylow
235	C₂×#41	—	86	2	no	—
236	C₂²×Q₈	—	86	2	no	—
237	#135	—	86	4	no	—
238	#135	—	172	2	no	—
239	C₂×Q₁₆	—	43	2	no	—
240	C₂×Q₁₆	—	43	2	no	—
241	C₂×Q₁₆	—	86	2	no	—
242	C₂×Q₁₆	—	86	4	no	—
243	C₂×Q₁₆	—	86	1	no	—
244	C₂×Q₁₆	—	86	1	no	—
245	C₂×Q₁₆	—	172	2	no	—
246	C₂×Q₁₆	C₂²×D₄₃	1	1	yes	characteristic
247	C₂×SD₁₆	—	86	2	no	—
248	C₂×D₈	C₂²×D₄₃	1	1	yes	characteristic
249	C₂²×C₈	—	43	2	no	—
250	C₄⋊Q₈	—	86	1	no	—
251	C₄⋊Q₈	—	86	1	no	—
252	C₄⋊Q₈	—	172	1	no	—
253	#123	—	172	2	no	—
254	C₂²⋊Q₈	—	86	2	no	—
255	C₂²⋊Q₈	—	86	2	no	—
256	C₂²⋊Q₈	—	172	2	no	—
257	#121	—	172	1	no	—
258	C₄×Q₈	—	2	1	no	—
259	C₄×D₄	—	2	1	no	—
260	C₄×D₄	—	43	1	no	—
261	C₄×D₄	—	172	1	no	—
262	C₂×C₄⋊C₄	—	172	1	no	—
263	Q₃₂	—	86	4	no	—
264	Q₃₂	—	86	4	no	—
265	Q₃₂	—	172	2	no	—
266	SD₃₂	—	86	4	no	—
267	SD₃₂	D₁₇₂	1	4	yes	complemented
268	M₃₂	—	86	4	no	—
269	C₂×C₁₆	—	43	2	no	—
270	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
271	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
272	#107	C₂²×D₄₃	1	1	yes	characteristic
273	#107	C₂²×D₄₃	1	1	yes	characteristic
274	C₄⋊C₈	—	43	1	no	—
275	#103	—	2	1	no	—
276	#103	—	86	2	no	—
277	#103	—	86	2	no	—
278	#102	—	2	1	no	—
279	C₂²⋊C₈	—	43	2	no	—
280	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
281	#41	—	344	2	no	—
282	#41	—	172	4	no	—
283	C₂×Q₈	—	2	1	no	—
284	C₂×Q₈	—	86	2	no	—
285	C₂×Q₈	—	86	2	no	—
286	C₂×Q₈	—	344	2	no	—
287	C₂×Q₈	—	172	2	no	—
288	C₂×Q₈	—	172	4	no	—
289	C₂×D₄	—	2	1	no	—
290	C₂×D₄	—	43	1	no	—
291	C₂×D₄	—	172	2	no	—
292	C₂²×C₄	—	4	1	no	—
293	C₂²×C₄	—	43	2	no	—
294	C₂²×C₄	—	172	2	no	—
295	C₂²×C₄	—	172	2	no	—
296	Q₁₆	—	2	1	no	—
297	Q₁₆	—	86	2	no	—
298	Q₁₆	—	86	2	no	—
299	Q₁₆	—	86	2	no	—
300	Q₁₆	—	86	2	no	—
301	Q₁₆	—	344	1	no	—
302	Q₁₆	—	344	1	no	—
303	Q₁₆	—	172	4	no	—
304	Q₁₆	C₂×D₁₇₂	1	2	yes	—
305	SD₁₆	—	172	4	no	—
306	D₈	—	2	1	no	—
307	D₈	C₂×D₁₇₂	1	2	yes	—
308	C₂×C₈	—	43	2	no	—
309	C₂×C₈	—	86	2	no	—
310	C₂×C₈	—	86	2	no	—
311	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
312	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
313	C₄⋊C₄	—	2	1	no	—
314	C₄⋊C₄	—	2	1	no	—
315	C₄⋊C₄	—	4	1	no	—
316	C₄⋊C₄	—	43	1	no	—
317	C₄⋊C₄	—	172	2	no	—
318	C₄⋊C₄	—	172	2	no	—
319	C₄⋊C₄	—	172	1	no	—
320	C₄⋊C₄	—	172	1	no	—
321	C₄⋊C₄	—	172	1	no	—
322	C₄⋊C₄	—	172	1	no	—
323	C₂²⋊C₄	—	4	1	no	—
324	C₂²⋊C₄	—	43	2	no	—
325	C₂²⋊C₄	—	172	2	no	—
326	C₂²⋊C₄	—	172	2	no	—
327	C₄²	—	4	1	no	—
328	C₄²	D₄×D₄₃	1	1	yes	characteristic
329	C₁₆	—	2	1	no	—
330	C₁₆	—	86	2	no	—
331	C₁₆	C₂×D₁₇₂	1	2	yes	—
332	C₂³	—	4	1	no	—
333	C₂³	—	43	2	no	—
334	Q₈	—	2	2	no	—
335	Q₈	—	4	1	no	—
336	Q₈	—	344	4	no	—
337	Q₈	—	172	2	no	—
338	Q₈	—	172	2	no	—
339	Q₈	—	172	2	no	—
340	Q₈	—	172	2	no	—
341	D₄	—	2	2	no	—
342	D₄	—	4	1	no	—
343	D₄	—	86	2	no	—
344	D₄	—	344	4	no	—
345	C₂×C₄	—	4	1	no	—
346	C₂×C₄	—	4	1	no	—
347	C₂×C₄	—	4	1	no	—
348	C₂×C₄	—	8	2	no	—
349	C₂×C₄	—	43	2	no	—
350	C₂×C₄	—	86	2	no	—
351	C₂×C₄	—	86	2	no	—
352	C₂×C₄	—	344	4	no	—
353	C₂×C₄	—	344	4	no	—
354	C₂×C₄	—	172	2	no	—
355	C₂×C₄	—	172	2	no	—
356	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
357	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
358	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
359	C₈	—	2	1	no	—
360	C₈	—	86	2	no	—
361	C₈	C₂²×D₁₇₂	1	1	yes	characteristic
362	C₈	#13063371	1	1	yes	characteristic
363	C₂²	—	4	2	no	—
364	C₂²	—	8	1	no	—
365	C₂²	—	8	1	no	—
366	C₂²	—	86	2	no	—
367	C₂²	—	86	2	no	—
368	C₂²	—	86	2	no	—
369	C₂²	C₂×D₈×D₄₃	1	1	yes	characteristic
370	C₄	—	2	1	no	—
371	C₄	—	4	2	no	—
372	C₄	—	8	1	no	—
373	C₄	—	8	1	no	—
374	C₄	—	86	2	no	—
375	C₄	—	344	2	no	—
376	C₄	—	344	2	no	—
377	C₄	#14583338	1	2	yes	complemented
378	C₄	D₄×D₁₇₂	1	1	yes	characteristic
379	C₄	#14583701	1	1	yes	characteristic
380	C₂	—	8	2	no	—
381	C₂	—	172	2	no	—
382	C₂	D₈×D₁₇₂	1	1	yes	characteristic
383	C₂	C₂×#14583338	1	1	yes	characteristic
384	C₂	#19147745	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₄₃
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₄×SD₃₂×C₄₃	C₂
socle	C₂²×C₄₃	C₂²×D₈
prime radical	#19146316	C₂
hypercenter	C₄×SD₃₂	D₄₃
hypocenter	C₄₃	#33842
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×SD₃₂

Complements

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

N	H ≅ G/N	type
1. C₂×#14583338	C₂	semidirect
2. Q₁₆×D₁₇₂	C₂	semidirect
4. #19136848	C₂	semidirect
5. #19136704	C₂	semidirect
6. #19136645	C₂	semidirect
7. #19136640	C₂	semidirect
8. #19136536	C₂	semidirect
9. #19136530	C₂	semidirect
10. #19136428	C₂	semidirect
11. C₄×SD₃₂×C₄₃	C₂	semidirect
80. SD₃₂×C₄₃	D₄	semidirect
182. C₄×C₄₃	#510	semidirect
197. C₄×SD₃₂	D₄₃	semidirect
234. C₄₃	#33842	semidirect
267. SD₃₂	D₁₇₂	semidirect
377. C₄	#14583338	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583338	2
Q₁₆×D₁₇₂	2
#19146316	2
#19136848	2
#19136704	2
#19136645	2
#19136640	2
#19136536	2
#19136530	2
#19136428	2
C₄×SD₃₂×C₄₃	2
#33842	43