Subgroups — #25088123

Order 11008.

606 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19147464	C₂	1	1	yes	maximal, characteristic, complemented
2	#19146942	C₂	1	1	yes	maximal, characteristic
3	#19146910	C₂	1	2	yes	maximal, complemented
4	#19134167	C₂	1	2	yes	maximal, complemented
5	#19133991	C₂	1	1	yes	maximal, characteristic, complemented
6	#19133986	C₂	1	1	yes	maximal, characteristic, complemented
7	#19133806	C₂	1	2	yes	maximal, complemented
8	#19133743	C₂	1	2	yes	maximal, complemented
9	#19133481	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×#10621873	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₃×#1710	C₂	1	1	yes	maximal, characteristic, complemented
12	#14583610	C₂²	1	2	yes	—
13	#14583552	C₂²	1	2	yes	—
14	#14583300	—	2	1	no	—
15	#14583228	—	2	2	no	—
16	#14583214	—	2	2	no	—
17	#14583202	—	2	2	no	—
18	#14583186	C₂²	1	2	yes	—
19	#14583186	C₂²	1	2	yes	—
20	#14583176	C₂²	1	2	yes	—
21	#14583156	—	2	2	no	—
22	#14583136	—	2	1	no	—
23	#14583106	C₂²	1	2	yes	—
24	#14583064	—	2	2	no	—
25	C₈×Dic₈₆	C₂²	1	1	yes	characteristic
26	C₄×#10621873	C₂²	1	1	yes	characteristic
27	C₄×D₃₄₄	C₂²	1	1	yes	characteristic
28	#14582981	—	2	1	no	—
29	#14582978	C₂²	1	2	yes	—
30	C₈×D₁₇₂	C₂²	1	1	yes	characteristic
31	#107×D₄₃	—	2	1	no	—
32	#14582899	—	2	2	no	—
33	#14582652	C₂²	1	1	yes	characteristic
34	#14582650	C₂²	1	1	yes	characteristic
35	#14582648	C₂²	1	2	yes	—
36	#14582645	C₂²	1	2	yes	—
37	#14582631	C₂²	1	1	yes	characteristic
38	#14582628	C₂²	1	2	yes	—
39	#14582628	C₂²	1	1	yes	characteristic
40	C₄₃×#500	C₂²	1	1	yes	characteristic
41	C₄₃×#500	C₂²	1	1	yes	characteristic
42	C₄₃×#487	C₂²	1	2	yes	—
43	C₄₃×#487	C₂²	1	2	yes	—
44	C₈²×C₄₃	C₂²	1	1	yes	characteristic
45	C₂×#10621902	—	2	2	no	—
46	C₂×#10621885	—	2	2	no	—
47	C₂×#10621873	D₄	1	1	yes	characteristic
48	C₂×D₃₄₄	D₄	1	1	yes	characteristic
49	C₂×#10621872	—	2	2	no	—
50	C₂×C₈×D₄₃	—	2	1	no	—
51	#13063316	—	2	2	no	—
52	#13063310	—	4	2	no	—
53	#13063305	—	2	2	no	—
54	#13063301	—	2	2	no	—
55	#13063301	—	4	2	no	—
56	#13063300	—	2	2	no	—
57	C₄×Dic₈₆	C₂³	1	1	yes	characteristic
58	C₄×D₁₇₂	C₂³	1	1	yes	characteristic
59	C₄×D₁₇₂	C₂³	1	2	yes	—
60	#13063287	—	2	2	no	—
61	C₄⋊C₄×D₄₃	—	2	2	no	—
62	C₄⋊C₄×D₄₃	—	2	2	no	—
63	#13063259	D₄	1	1	yes	characteristic
64	#13063258	—	2	2	no	—
65	#13063257	D₄	1	1	yes	characteristic
66	#13063256	—	2	2	no	—
67	#13063256	—	2	2	no	—
68	C₄⋊(C₄₃⋊C₈)	C₂³	1	2	yes	—
69	C₄⋊(C₄₃⋊C₈)	C₂³	1	1	yes	characteristic
70	C₄⋊(C₄₃⋊C₈)	C₂³	1	1	yes	characteristic
71	Dic₄₃⋊C₈	—	2	1	no	—
72	#13063250	D₄	1	1	yes	characteristic
73	#13063249	—	2	2	no	—
74	#13063247	D₄	1	1	yes	characteristic
75	#13063247	D₄	1	2	yes	—
76	#13063245	—	2	2	no	—
77	#13063245	—	2	2	no	—
78	#13063245	—	2	2	no	—
79	(C₂×D₄₃)⋊C₈	—	2	2	no	—
80	(C₂×D₄₃)⋊C₈	—	2	1	no	—
81	C₈×Dic₄₃	—	2	1	no	—
82	C₄⋊Q₈×C₄₃	C₂³	1	2	yes	—
83	C₄⋊Q₈×C₄₃	C₂³	1	2	yes	—
84	#107×C₄₃	—	2	1	no	—
85	#107×C₄₃	—	2	1	no	—
86	#107×C₄₃	—	2	1	no	—
87	#107×C₄₃	—	2	1	no	—
88	#103×C₄₃	—	2	2	no	—
89	#103×C₄₃	—	2	2	no	—
90	#103×C₄₃	—	4	2	no	—
91	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
92	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
94	C₂×Q₈×C₄₃	—	2	2	no	—
95	C₂×Q₈×C₄₃	—	2	2	no	—
96	#10621902	—	4	4	no	—
97	#10621902	—	8	2	no	—
98	C₂×D₁₇₂	—	4	2	no	—
99	C₂×D₁₇₂	C₂×D₄	1	2	yes	—
100	C₂×D₁₇₂	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×D₄₃	—	2	1	no	—
102	C₂×C₄×D₄₃	—	2	2	no	—
103	C₂×C₄×D₄₃	—	4	2	no	—
104	C₂×C₄×D₄₃	—	4	2	no	—
105	C₂×Dic₈₆	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₄₃	—	2	1	no	—
107	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
111	C₄⋊C₄×C₄₃	—	2	2	no	—
112	C₄⋊C₄×C₄₃	—	2	2	no	—
113	C₄⋊C₄×C₄₃	—	2	2	no	—
114	C₄⋊C₄×C₄₃	—	2	2	no	—
115	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
116	#10621885	—	8	2	no	—
117	(C₂×D₄₃)⋊C₄	—	2	2	no	—
118	(C₂×D₄₃)⋊C₄	—	2	1	no	—
119	(C₂×D₄₃)⋊C₄	—	4	2	no	—
120	(C₂×D₄₃)⋊C₄	—	4	2	no	—
121	C₄⋊Dic₄₃	—	4	2	no	—
122	C₄⋊Dic₄₃	C₂×D₄	1	1	yes	characteristic
123	C₄⋊Dic₄₃	C₂×D₄	1	2	yes	—
124	C₄⋊Dic₄₃	C₂×D₄	1	1	yes	characteristic
125	Dic₄₃⋊C₄	—	2	1	no	—
126	Dic₄₃⋊C₄	—	4	2	no	—
127	Dic₄₃⋊C₄	—	4	2	no	—
128	C₄×Dic₄₃	—	2	1	no	—
129	C₂×C₄₃⋊C₈	—	2	1	no	—
130	C₂×C₄₃⋊C₈	—	2	1	no	—
131	C₂×C₄₃⋊C₈	—	2	2	no	—
132	D₃₄₄	—	2	2	no	—
133	#10621873	—	2	2	no	—
134	#10621872	—	8	2	no	—
135	C₈×D₄₃	—	4	2	no	—
136	C₂²×D₄₃	—	2	1	no	—
137	C₂²×D₄₃	—	2	2	no	—
138	Q₈×C₄₃	—	4	2	no	—
139	Q₈×C₄₃	—	4	2	no	—
140	Q₈×C₄₃	—	8	2	no	—
141	C₂×C₄×C₄₃	—	4	2	no	—
142	C₂×C₄×C₄₃	—	4	2	no	—
143	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
146	C₂×Dic₄₃	—	2	1	no	—
147	C₂×Dic₄₃	—	2	1	no	—
148	C₂×Dic₄₃	—	2	2	no	—
149	C₂×Dic₄₃	—	2	1	no	—
150	D₁₇₂	—	2	2	no	—
151	D₁₇₂	—	2	1	no	—
152	D₁₇₂	—	2	2	no	—
153	D₁₇₂	—	8	4	no	—
154	D₁₇₂	#135	1	2	yes	—
155	C₄×D₄₃	—	4	2	no	—
156	C₄×D₄₃	—	4	1	no	—
157	C₄×D₄₃	—	4	2	no	—
158	C₄×D₄₃	—	4	1	no	—
159	C₄×D₄₃	—	8	4	no	—
160	C₄×D₄₃	—	16	2	no	—
161	Dic₈₆	—	2	1	no	—
162	Dic₈₆	—	2	1	no	—
163	C₈×C₄₃	—	2	1	no	—
164	C₈×C₄₃	—	2	1	no	—
165	C₈×C₄₃	—	2	1	no	—
166	C₈×C₄₃	—	4	1	no	—
167	C₈×C₄₃	#137	1	2	yes	—
168	C₄₃⋊C₈	—	4	1	no	—
169	C₄₃⋊C₈	—	4	2	no	—
170	C₄₃⋊C₈	—	4	1	no	—
171	#19903	—	43	1	no	maximal, Sylow
172	C₂²×C₄₃	D₄²	1	1	yes	characteristic
173	C₂×D₄₃	—	2	2	no	—
174	C₂×D₄₃	—	4	1	no	—
175	C₂×D₄₃	—	4	1	no	—
176	C₂×D₄₃	—	4	2	no	—
177	C₂×D₄₃	—	4	2	no	—
178	C₂×D₄₃	—	4	2	no	—
179	C₄×C₄₃	—	2	1	no	—
180	C₄×C₄₃	—	8	2	no	—
181	C₄×C₄₃	—	8	2	no	—
182	C₄×C₄₃	C₂×#135	1	1	yes	characteristic
183	C₄×C₄₃	C₂×#137	1	1	yes	characteristic
184	C₄×C₄₃	#576	1	1	yes	characteristic
185	C₄×C₄₃	#578	1	1	yes	characteristic
186	Dic₄₃	—	4	1	no	—
187	Dic₄₃	—	4	1	no	—
188	Dic₄₃	—	4	2	no	—
189	Dic₄₃	—	4	1	no	—
190	#3380	—	43	1	no	—
191	#3334	—	43	1	no	—
192	#3331	—	43	2	no	—
193	#1710	D₄₃	1	1	yes	characteristic, complemented
194	#1703	—	43	2	no	—
195	#1675	—	43	1	no	—
196	#1674	—	43	1	no	—
197	#1644	—	43	2	no	—
198	#1635	—	43	2	no	—
199	#1589	—	43	1	no	—
200	C₈×SD₁₆	—	43	1	no	—
201	C₂×C₄₃	#3276	1	1	yes	characteristic
202	C₂×C₄₃	#3280	1	1	yes	characteristic
203	C₂×C₄₃	#3290	1	1	yes	characteristic
204	D₄₃	—	4	2	no	—
205	D₄₃	—	8	2	no	—
206	#554	—	43	2	no	—
207	#547	—	43	2	no	—
208	#500	C₂×D₄₃	1	1	yes	characteristic
209	#500	C₂×D₄₃	1	1	yes	characteristic
210	#499	—	86	1	no	—
211	#487	C₂×D₄₃	1	2	yes	—
212	#487	C₂×D₄₃	1	2	yes	—
213	#483	—	86	2	no	—
214	#480	—	86	2	no	—
215	#478	—	86	2	no	—
216	#476	—	43	2	no	—
217	#476	—	43	2	no	—
218	#474	—	43	2	no	—
219	#470	—	86	2	no	—
220	#466	—	86	1	no	—
221	#462	—	43	2	no	—
222	#452	—	86	2	no	—
223	C₈×Q₈	—	43	1	no	—
224	C₄×SD₁₆	—	43	1	no	—
225	C₄×D₈	—	43	1	no	—
226	C₄×D₈	—	86	1	no	—
227	#436	—	43	2	no	—
228	C₈×D₄	—	43	1	no	—
229	C₂×#107	—	86	1	no	—
230	#417	—	86	2	no	—
231	#335	—	43	1	no	—
232	#334	—	43	1	no	—
233	#333	—	43	2	no	—
234	#332	—	43	2	no	—
235	Q₈⋊C₈	—	43	1	no	—
236	D₄⋊C₈	—	43	1	no	—
237	D₄⋊C₈	—	43	2	no	—
238	C₈²	C₂×D₄₃	1	1	yes	characteristic
239	C₄₃	#19903	1	1	yes	characteristic, complemented, Sylow
240	C₂×#41	—	86	2	no	—
241	C₂×Q₁₆	—	86	2	no	—
242	C₂×SD₁₆	—	43	1	no	—
243	C₂×D₈	—	43	1	no	—
244	C₂×M₁₆	—	86	2	no	—
245	C₂²×C₈	—	86	1	no	—
246	C₄⋊Q₈	C₂²×D₄₃	1	2	yes	—
247	C₄⋊Q₈	C₂²×D₄₃	1	2	yes	—
248	#125	—	86	2	no	—
249	#123	—	172	2	no	—
250	C₂²⋊Q₈	—	86	2	no	—
251	C₂²⋊Q₈	—	86	2	no	—
252	C₂²⋊Q₈	—	172	2	no	—
253	#121	—	86	2	no	—
254	C₄×Q₈	—	43	1	no	—
255	C₄×D₄	—	43	1	no	—
256	C₄×D₄	—	43	2	no	—
257	C₄×D₄	—	86	2	no	—
258	C₂×C₄⋊C₄	—	86	2	no	—
259	C₂×C₄⋊C₄	—	86	2	no	—
260	#107	—	2	1	no	—
261	#107	—	2	1	no	—
262	#107	—	2	1	no	—
263	#107	—	2	1	no	—
264	#107	—	43	1	no	—
265	#107	—	86	2	no	—
266	#106	—	43	1	no	—
267	#106	—	86	2	no	—
268	#106	—	86	2	no	—
269	C₄⋊C₈	—	43	1	no	—
270	C₄⋊C₈	—	43	2	no	—
271	C₄⋊C₈	—	43	1	no	—
272	C₄⋊C₈	—	86	1	no	—
273	#103	—	2	2	no	—
274	#103	—	2	2	no	—
275	#103	—	4	2	no	—
276	#103	—	43	1	no	—
277	#103	—	86	2	no	—
278	#102	—	43	1	no	—
279	#102	—	43	2	no	—
280	#102	—	86	2	no	—
281	#102	—	86	2	no	—
282	#102	—	86	2	no	—
283	C₂²⋊C₈	—	86	1	no	—
284	C₂²⋊C₈	—	86	2	no	—
285	C₄×C₈	—	86	1	no	—
286	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
287	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
288	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
289	#41	—	344	2	no	—
290	#41	—	172	4	no	—
291	C₂×Q₈	—	2	2	no	—
292	C₂×Q₈	—	2	2	no	—
293	C₂×Q₈	—	43	1	no	—
294	C₂×D₄	—	43	1	no	—
295	C₂×D₄	—	43	2	no	—
296	C₂×D₄	—	172	2	no	—
297	C₂²×C₄	—	86	1	no	—
298	C₂²×C₄	—	86	2	no	—
299	C₂²×C₄	—	172	2	no	—
300	C₂²×C₄	—	172	2	no	—
301	Q₁₆	—	344	2	no	—
302	SD₁₆	—	86	2	no	—
303	D₈	—	86	2	no	—
304	M₁₆	—	344	2	no	—
305	C₂×C₈	—	2	1	no	—
306	C₂×C₈	—	86	1	no	—
307	C₂×C₈	—	86	1	no	—
308	C₂×C₈	—	86	2	no	—
309	C₂×C₈	—	172	2	no	—
310	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
311	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
312	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
313	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
314	C₄⋊C₄	—	2	2	no	—
315	C₄⋊C₄	—	2	2	no	—
316	C₄⋊C₄	—	2	2	no	—
317	C₄⋊C₄	—	2	2	no	—
318	C₄⋊C₄	—	43	1	no	—
319	C₄⋊C₄	—	43	2	no	—
320	C₄⋊C₄	—	43	1	no	—
321	C₄⋊C₄	—	86	1	no	—
322	C₄⋊C₄	—	172	2	no	—
323	C₄⋊C₄	—	172	2	no	—
324	C₄⋊C₄	—	172	2	no	—
325	C₂²⋊C₄	—	86	2	no	—
326	C₂²⋊C₄	—	86	1	no	—
327	C₂²⋊C₄	—	172	2	no	—
328	C₂²⋊C₄	—	172	2	no	—
329	C₄²	—	86	1	no	—
330	C₄²	C₂³×D₄₃	1	1	yes	characteristic
331	C₂³	—	86	1	no	—
332	C₂³	—	86	2	no	—
333	Q₈	—	4	2	no	—
334	Q₈	—	4	2	no	—
335	Q₈	—	8	2	no	—
336	Q₈	—	86	1	no	—
337	Q₈	—	86	1	no	—
338	D₄	—	43	2	no	—
339	D₄	—	86	2	no	—
340	D₄	—	86	1	no	—
341	D₄	—	86	2	no	—
342	D₄	—	344	4	no	—
343	C₂×C₄	—	4	2	no	—
344	C₂×C₄	—	4	2	no	—
345	C₂×C₄	—	86	1	no	—
346	C₂×C₄	—	86	1	no	—
347	C₂×C₄	—	86	1	no	—
348	C₂×C₄	—	86	2	no	—
349	C₂×C₄	—	344	4	no	—
350	C₂×C₄	—	172	2	no	—
351	C₂×C₄	—	172	1	no	—
352	C₂×C₄	—	172	2	no	—
353	C₂×C₄	—	172	1	no	—
354	C₂×C₄	—	688	2	no	—
355	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
356	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
357	C₂×C₄	#13063370	1	1	yes	characteristic
358	C₈	—	2	1	no	—
359	C₈	—	2	1	no	—
360	C₈	—	2	1	no	—
361	C₈	—	4	1	no	—
362	C₈	—	172	1	no	—
363	C₈	—	172	2	no	—
364	C₈	—	172	1	no	—
365	C₈	#13063357	1	2	yes	—
366	C₂²	—	86	2	no	—
367	C₂²	—	172	2	no	—
368	C₂²	—	172	2	no	—
369	C₂²	—	172	2	no	—
370	C₂²	—	172	1	no	—
371	C₂²	—	172	1	no	—
372	C₂²	#14583534	1	1	yes	characteristic
373	C₄	—	2	1	no	—
374	C₄	—	8	2	no	—
375	C₄	—	8	2	no	—
376	C₄	—	172	1	no	—
377	C₄	—	172	2	no	—
378	C₄	—	172	1	no	—
379	C₄	—	172	1	no	—
380	C₄	#135×D₄₃	1	1	yes	characteristic
381	C₄	C₂×#13063357	1	1	yes	characteristic
382	C₄	#14583701	1	1	yes	characteristic
383	C₄	#14583712	1	1	yes	characteristic
384	C₂	—	344	2	no	—
385	C₂	—	172	2	no	—
386	C₂	#19146240	1	1	yes	characteristic
387	C₂	#19146279	1	1	yes	characteristic
388	C₂	#19146380	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#14583534
commutator, cosocle	C₄²×C₄₃	C₂⁴
Frattini	C₄²	C₂³×D₄₃
Fitting	C₄₃×#1710	C₂
socle	C₂²×C₄₃	D₄²
prime radical	#19146942	C₂
hypercenter	#1710	D₄₃
hypocenter	C₄₃	#19903
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1710

Complements

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

N	H ≅ G/N	type
1. #19147464	C₂	semidirect
3. #19146910	C₂	semidirect
4. #19134167	C₂	semidirect
5. #19133991	C₂	semidirect
6. #19133986	C₂	semidirect
7. #19133806	C₂	semidirect
8. #19133743	C₂	semidirect
9. #19133481	C₂	semidirect
10. C₈×#10621873	C₂	semidirect
11. C₄₃×#1710	C₂	semidirect
193. #1710	D₄₃	semidirect
239. C₄₃	#19903	semidirect

Maximal subgroups

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

subgroup	index
#19147464	2
#19146942	2
#19146910	2
#19134167	2
#19133991	2
#19133986	2
#19133806	2
#19133743	2
#19133481	2
C₈×#10621873	2
C₄₃×#1710	2
#19903	43