Subgroups — #25088201

Order 11008.

638 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19147570	C₂	1	1	yes	maximal, characteristic
2	#19147464	C₂	1	1	yes	maximal, characteristic, complemented
3	#19146930	C₂	1	2	yes	maximal, complemented
4	#19134188	C₂	1	1	yes	maximal, characteristic, complemented
5	#19134131	C₂	1	2	yes	maximal, complemented
6	#19133956	C₂	1	1	yes	maximal, characteristic, complemented
7	#19133720	C₂	1	2	yes	maximal, complemented
8	#19133668	C₂	1	2	yes	maximal, complemented
9	#19133479	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×#10621885	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₃×#1676	C₂	1	1	yes	maximal, characteristic, complemented
12	#14583610	C₂²	1	2	yes	—
13	#14583571	C₂²	1	2	yes	—
14	#14583300	—	2	1	no	—
15	#14583276	—	2	1	no	—
16	#14583276	C₂²	1	1	yes	characteristic
17	#14583273	C₂²	1	2	yes	—
18	#14583247	C₂²	1	2	yes	—
19	#14583214	—	2	2	no	—
20	#14583202	—	2	2	no	—
21	#14583186	C₂²	1	2	yes	—
22	#14583146	—	2	2	no	—
23	#14583100	C₂²	1	2	yes	—
24	#14583095	—	2	2	no	—
25	#14583095	C₂²	1	2	yes	—
26	#14583064	—	2	2	no	—
27	C₈×Dic₈₆	C₂²	1	1	yes	characteristic
28	C₄×#10621885	C₂²	1	1	yes	characteristic
29	C₄×D₃₄₄	—	2	1	no	—
30	C₄×D₃₄₄	C₂²	1	1	yes	characteristic
31	#14582981	—	2	1	no	—
32	#14582976	C₂²	1	2	yes	—
33	#14582899	—	2	2	no	—
34	#14582651	C₂²	1	1	yes	characteristic
35	#14582642	C₂²	1	2	yes	—
36	Q₈⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
37	#14582629	C₂²	1	1	yes	characteristic
38	#14582626	C₂²	1	2	yes	—
39	C₈×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
40	C₄₃×#500	C₂²	1	1	yes	characteristic
41	C₄₃×#477	C₂²	1	2	yes	—
42	C₈×Q₈×C₄₃	C₂²	1	1	yes	characteristic
43	C₄₃×#335	C₂²	1	1	yes	characteristic
44	C₄₃×#333	C₂²	1	2	yes	—
45	C₂×#10621902	—	2	2	no	—
46	C₂×#10621885	D₄	1	1	yes	characteristic
47	C₂×#10621873	—	2	2	no	—
48	C₂×#10621884	—	4	2	no	—
49	C₂×D₃₄₄	—	2	1	no	—
50	C₂×D₃₄₄	—	2	2	no	—
51	C₂×D₃₄₄	—	2	1	no	—
52	C₂×D₃₄₄	D₄	1	1	yes	characteristic
53	C₂×#10621872	—	2	2	no	—
54	#13063322	—	2	2	no	—
55	#13063322	C₂³	1	2	yes	—
56	#13063316	—	2	2	no	—
57	#13063305	—	2	2	no	—
58	#13063301	—	4	2	no	—
59	#13063300	—	2	2	no	—
60	#13063300	—	4	2	no	—
61	C₄×Dic₈₆	C₂³	1	1	yes	characteristic
62	C₄×D₁₇₂	—	2	2	no	—
63	C₄×D₁₇₂	C₂³	1	2	yes	—
64	#13063287	—	2	2	no	—
65	C₄⋊C₄×D₄₃	—	2	2	no	—
66	#13063259	D₄	1	1	yes	characteristic
67	#13063258	D₄	1	1	yes	characteristic
68	#13063256	—	2	2	no	—
69	C₄⋊(C₄₃⋊C₈)	C₂³	1	1	yes	characteristic
70	Dic₄₃⋊C₈	—	2	1	no	—
71	#13063249	D₄	1	1	yes	characteristic
72	#13063248	D₄	1	1	yes	characteristic
73	#13063247	—	2	2	no	—
74	#13063247	—	2	2	no	—
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	C₄×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
81	C₄×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
82	C₈×Dic₄₃	—	2	1	no	—
83	C₄⋊Q₈×C₄₃	C₂³	1	2	yes	—
84	C₄×Q₈×C₄₃	C₂³	1	1	yes	characteristic
85	#107×C₄₃	—	2	1	no	—
86	#107×C₄₃	—	2	1	no	—
87	#107×C₄₃	—	2	2	no	—
88	C₄⋊C₈×C₄₃	—	2	1	no	—
89	C₄⋊C₈×C₄₃	C₂³	1	1	yes	characteristic
90	C₄⋊C₈×C₄₃	C₂³	1	2	yes	—
91	#103×C₄₃	—	2	2	no	—
92	C₄×C₈×C₄₃	—	2	1	no	—
93	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
94	C₂×Q₈×C₄₃	—	2	2	no	—
95	C₂×Q₈×C₄₃	C₂×D₄	1	1	yes	characteristic
96	#10621902	—	4	4	no	—
97	#10621902	—	8	2	no	—
98	C₂×D₁₇₂	—	2	2	no	—
99	C₂×D₁₇₂	—	2	2	no	—
100	C₂×D₁₇₂	—	2	2	no	—
101	C₂×D₁₇₂	—	4	2	no	—
102	C₂×D₁₇₂	—	4	2	no	—
103	C₂×D₁₇₂	C₂×D₄	1	2	yes	—
104	C₂×C₄×D₄₃	—	2	2	no	—
105	C₂×C₄×D₄₃	—	4	2	no	—
106	C₂×C₄×D₄₃	—	4	2	no	—
107	C₂×Dic₈₆	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₄₃	—	2	1	no	—
109	C₂×C₈×C₄₃	—	2	2	no	—
110	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
112	C₄⋊C₄×C₄₃	—	2	1	no	—
113	C₄⋊C₄×C₄₃	—	2	2	no	—
114	C₄⋊C₄×C₄₃	—	2	2	no	—
115	C₄⋊C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₄₃	—	2	1	no	—
117	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
118	#10621885	—	2	2	no	—
119	#10621884	—	8	4	no	—
120	(C₂×D₄₃)⋊C₄	—	2	2	no	—
121	(C₂×D₄₃)⋊C₄	—	4	2	no	—
122	(C₂×D₄₃)⋊C₄	—	4	2	no	—
123	C₄⋊Dic₄₃	—	4	2	no	—
124	C₄⋊Dic₄₃	C₂×D₄	1	1	yes	characteristic
125	C₄⋊Dic₄₃	C₂×D₄	1	2	yes	—
126	Dic₄₃⋊C₄	—	2	1	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	C₂×D₄	1	1	yes	characteristic
132	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
133	D₃₄₄	—	2	2	no	—
134	D₃₄₄	—	4	2	no	—
135	D₃₄₄	—	8	1	no	—
136	D₃₄₄	—	8	2	no	—
137	#10621873	—	8	2	no	—
138	#10621872	—	8	2	no	—
139	C₂²×D₄₃	—	2	2	no	—
140	C₂²×D₄₃	—	4	2	no	—
141	Q₈×C₄₃	—	2	1	no	—
142	Q₈×C₄₃	—	8	2	no	—
143	Q₈×C₄₃	#135	1	2	yes	—
144	C₂×C₄×C₄₃	—	2	1	no	—
145	C₂×C₄×C₄₃	—	2	1	no	—
146	C₂×C₄×C₄₃	—	4	2	no	—
147	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
149	C₂×C₄×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
150	C₂×Dic₄₃	—	2	1	no	—
151	C₂×Dic₄₃	—	2	1	no	—
152	C₂×Dic₄₃	—	2	2	no	—
153	D₁₇₂	—	2	2	no	—
154	D₁₇₂	—	2	2	no	—
155	D₁₇₂	—	4	2	no	—
156	D₁₇₂	—	4	2	no	—
157	D₁₇₂	—	4	2	no	—
158	D₁₇₂	—	4	2	no	—
159	D₁₇₂	—	8	4	no	—
160	D₁₇₂	—	8	2	no	—
161	D₁₇₂	—	16	2	no	—
162	C₄×D₄₃	—	4	2	no	—
163	C₄×D₄₃	—	4	2	no	—
164	C₄×D₄₃	—	8	4	no	—
165	C₄×D₄₃	—	16	2	no	—
166	Dic₈₆	—	2	1	no	—
167	Dic₈₆	—	2	1	no	—
168	C₈×C₄₃	—	2	1	no	—
169	C₈×C₄₃	—	4	2	no	—
170	C₈×C₄₃	—	4	1	no	—
171	C₈×C₄₃	#137	1	2	yes	—
172	C₄₃⋊C₈	—	2	2	no	—
173	C₄₃⋊C₈	—	2	1	no	—
174	C₄₃⋊C₈	—	2	1	no	—
175	C₄₃⋊C₈	—	4	1	no	—
176	#19909	—	43	1	no	maximal, Sylow
177	C₂²×C₄₃	D₄²	1	1	yes	characteristic
178	C₂×D₄₃	—	4	2	no	—
179	C₂×D₄₃	—	4	2	no	—
180	C₂×D₄₃	—	4	2	no	—
181	C₂×D₄₃	—	8	2	no	—
182	C₂×D₄₃	—	8	2	no	—
183	C₂×D₄₃	—	8	2	no	—
184	C₄×C₄₃	—	2	2	no	—
185	C₄×C₄₃	—	2	1	no	—
186	C₄×C₄₃	—	4	1	no	—
187	C₄×C₄₃	—	8	2	no	—
188	C₄×C₄₃	C₂×#135	1	1	yes	characteristic
189	C₄×C₄₃	C₂×#137	1	1	yes	characteristic
190	C₄×C₄₃	#576	1	1	yes	characteristic
191	C₄×C₄₃	#577	1	1	yes	characteristic
192	Dic₄₃	—	4	1	no	—
193	Dic₄₃	—	4	2	no	—
194	Dic₄₃	—	4	1	no	—
195	#3391	—	43	1	no	—
196	#3380	—	43	1	no	—
197	#3333	—	43	2	no	—
198	#1706	—	43	1	no	—
199	#1697	—	43	2	no	—
200	#1676	D₄₃	1	1	yes	characteristic, complemented
201	#1668	—	43	1	no	—
202	#1631	—	43	2	no	—
203	#1622	—	43	2	no	—
204	#1589	—	43	1	no	—
205	C₈×Q₁₆	—	43	1	no	—
206	C₂×C₄₃	#3278	1	1	yes	characteristic
207	C₂×C₄₃	#3280	1	1	yes	characteristic
208	C₂×C₄₃	#3292	1	1	yes	characteristic
209	D₄₃	—	8	2	no	—
210	D₄₃	—	16	2	no	—
211	#554	—	43	2	no	—
212	#550	—	43	2	no	—
213	#500	C₂×D₄₃	1	1	yes	characteristic
214	#499	—	86	1	no	—
215	#493	—	43	1	no	—
216	#493	—	86	1	no	—
217	#492	—	43	2	no	—
218	#486	—	43	2	no	—
219	#480	—	86	2	no	—
220	#478	—	86	2	no	—
221	#477	C₂×D₄₃	1	2	yes	—
222	#476	—	43	2	no	—
223	#468	—	86	2	no	—
224	#460	—	43	2	no	—
225	#459	—	43	2	no	—
226	#459	—	86	2	no	—
227	#452	—	86	2	no	—
228	C₈×Q₈	—	43	1	no	—
229	C₈×Q₈	C₂×D₄₃	1	1	yes	characteristic
230	C₄×Q₁₆	—	43	1	no	—
231	C₄×D₈	—	43	1	no	—
232	C₄×D₈	—	86	1	no	—
233	C₄×D₈	—	86	1	no	—
234	#436	—	43	2	no	—
235	#417	—	86	2	no	—
236	#335	—	43	1	no	—
237	#335	C₂×D₄₃	1	1	yes	characteristic
238	#333	C₂×D₄₃	1	2	yes	—
239	#331	—	43	2	no	—
240	Q₈⋊C₈	—	43	1	no	—
241	Q₈⋊C₈	—	43	1	no	—
242	D₄⋊C₈	—	43	2	no	—
243	C₈²	—	43	1	no	—
244	C₄₃	#19909	1	1	yes	characteristic, complemented, Sylow
245	C₂×#41	—	86	2	no	—
246	C₂×Q₁₆	—	43	1	no	—
247	C₂×SD₁₆	—	86	2	no	—
248	C₂×SD₁₆	—	172	2	no	—
249	C₂×D₈	—	43	1	no	—
250	C₂×D₈	—	86	2	no	—
251	C₂×D₈	—	86	1	no	—
252	C₂×D₈	—	86	1	no	—
253	C₂×M₁₆	—	86	2	no	—
254	C₄⋊Q₈	C₂²×D₄₃	1	2	yes	—
255	#127	—	43	2	no	—
256	#127	—	86	2	no	—
257	#125	—	86	2	no	—
258	C₂²⋊Q₈	—	86	2	no	—
259	C₂²⋊Q₈	—	172	2	no	—
260	#121	—	86	2	no	—
261	#121	—	172	2	no	—
262	C₄×Q₈	—	43	1	no	—
263	C₄×Q₈	C₂²×D₄₃	1	1	yes	characteristic
264	C₄×D₄	—	43	2	no	—
265	C₄×D₄	—	86	2	no	—
266	C₄×D₄	—	86	2	no	—
267	C₂×C₄⋊C₄	—	86	2	no	—
268	#107	—	2	1	no	—
269	#107	—	2	1	no	—
270	#107	—	2	2	no	—
271	#107	—	43	1	no	—
272	#107	—	43	1	no	—
273	#106	—	86	2	no	—
274	C₄⋊C₈	—	2	1	no	—
275	C₄⋊C₈	—	43	1	no	—
276	C₄⋊C₈	—	86	1	no	—
277	C₄⋊C₈	C₂²×D₄₃	1	2	yes	—
278	C₄⋊C₈	C₂²×D₄₃	1	1	yes	characteristic
279	#103	—	2	2	no	—
280	#103	—	43	1	no	—
281	#103	—	43	1	no	—
282	#102	—	43	2	no	—
283	#102	—	86	2	no	—
284	#102	—	86	2	no	—
285	#102	—	86	2	no	—
286	#102	—	86	2	no	—
287	#102	—	86	2	no	—
288	C₂²⋊C₈	—	86	2	no	—
289	C₄×C₈	—	2	1	no	—
290	C₄×C₈	—	43	1	no	—
291	C₄×C₈	—	43	1	no	—
292	C₄×C₈	—	86	1	no	—
293	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
294	#41	—	344	2	no	—
295	#41	—	172	4	no	—
296	C₂×Q₈	—	2	2	no	—
297	C₂×Q₈	—	43	1	no	—
298	C₂×Q₈	D₄×D₄₃	1	1	yes	characteristic
299	C₂×D₄	—	43	2	no	—
300	C₂×D₄	—	86	2	no	—
301	C₂×D₄	—	86	2	no	—
302	C₂×D₄	—	86	2	no	—
303	C₂×D₄	—	172	2	no	—
304	C₂×D₄	—	172	2	no	—
305	C₂²×C₄	—	86	2	no	—
306	C₂²×C₄	—	172	2	no	—
307	C₂²×C₄	—	172	2	no	—
308	Q₁₆	—	86	2	no	—
309	SD₁₆	—	344	2	no	—
310	SD₁₆	—	344	4	no	—
311	D₈	—	86	2	no	—
312	D₈	—	344	1	no	—
313	D₈	—	344	2	no	—
314	D₈	—	172	2	no	—
315	M₁₆	—	344	2	no	—
316	C₂×C₈	—	2	1	no	—
317	C₂×C₈	—	2	2	no	—
318	C₂×C₈	—	43	1	no	—
319	C₂×C₈	—	43	1	no	—
320	C₂×C₈	—	86	1	no	—
321	C₂×C₈	—	86	1	no	—
322	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
323	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
324	C₄⋊C₄	—	2	1	no	—
325	C₄⋊C₄	—	2	2	no	—
326	C₄⋊C₄	—	2	2	no	—
327	C₄⋊C₄	—	43	2	no	—
328	C₄⋊C₄	—	43	1	no	—
329	C₄⋊C₄	—	86	1	no	—
330	C₄⋊C₄	—	172	2	no	—
331	C₄⋊C₄	—	172	2	no	—
332	C₄⋊C₄	D₄×D₄₃	1	1	yes	characteristic
333	C₂²⋊C₄	—	86	2	no	—
334	C₂²⋊C₄	—	172	2	no	—
335	C₂²⋊C₄	—	172	2	no	—
336	C₄²	—	2	1	no	—
337	C₄²	—	86	1	no	—
338	C₄²	C₂³×D₄₃	1	1	yes	characteristic
339	C₂³	—	86	2	no	—
340	C₂³	—	172	2	no	—
341	Q₈	—	2	1	no	—
342	Q₈	—	8	2	no	—
343	Q₈	—	86	1	no	—
344	Q₈	—	86	1	no	—
345	Q₈	#13063348	1	2	yes	—
346	D₄	—	86	2	no	—
347	D₄	—	86	2	no	—
348	D₄	—	344	2	no	—
349	D₄	—	344	4	no	—
350	D₄	—	172	2	no	—
351	D₄	—	172	2	no	—
352	D₄	—	172	2	no	—
353	D₄	—	172	2	no	—
354	D₄	—	688	2	no	—
355	C₂×C₄	—	2	1	no	—
356	C₂×C₄	—	2	1	no	—
357	C₂×C₄	—	4	2	no	—
358	C₂×C₄	—	86	1	no	—
359	C₂×C₄	—	86	1	no	—
360	C₂×C₄	—	86	2	no	—
361	C₂×C₄	—	344	4	no	—
362	C₂×C₄	—	172	2	no	—
363	C₂×C₄	—	172	2	no	—
364	C₂×C₄	—	688	2	no	—
365	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
366	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
367	C₂×C₄	#13063371	1	1	yes	characteristic
368	C₈	—	2	1	no	—
369	C₈	—	4	2	no	—
370	C₈	—	4	1	no	—
371	C₈	—	86	2	no	—
372	C₈	—	86	1	no	—
373	C₈	—	86	1	no	—
374	C₈	—	172	1	no	—
375	C₈	#13063357	1	2	yes	—
376	C₂²	—	344	2	no	—
377	C₂²	—	344	2	no	—
378	C₂²	—	344	2	no	—
379	C₂²	—	172	2	no	—
380	C₂²	—	172	2	no	—
381	C₂²	—	172	2	no	—
382	C₂²	#14583533	1	1	yes	characteristic
383	C₄	—	2	2	no	—
384	C₄	—	2	1	no	—
385	C₄	—	4	1	no	—
386	C₄	—	8	2	no	—
387	C₄	—	172	2	no	—
388	C₄	—	172	1	no	—
389	C₄	—	172	1	no	—
390	C₄	C₂×#13063348	1	1	yes	characteristic
391	C₄	C₂×#13063357	1	1	yes	characteristic
392	C₄	#14583701	1	1	yes	characteristic
393	C₄	#14583708	1	1	yes	characteristic
394	C₂	—	344	2	no	—
395	C₂	—	688	2	no	—
396	C₂	#19146256	1	1	yes	characteristic
397	C₂	#19146277	1	1	yes	characteristic
398	C₂	#19146393	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#14583533
commutator, cosocle	C₄²×C₄₃	C₂⁴
Frattini	C₄²	C₂³×D₄₃
Fitting	C₄₃×#1676	C₂
socle	C₂²×C₄₃	D₄²
prime radical	#19147570	C₂
hypercenter	#1676	D₄₃
hypocenter	C₄₃	#19909
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1676

Complements

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

N	H ≅ G/N	type
2. #19147464	C₂	semidirect
3. #19146930	C₂	semidirect
4. #19134188	C₂	semidirect
5. #19134131	C₂	semidirect
6. #19133956	C₂	semidirect
7. #19133720	C₂	semidirect
8. #19133668	C₂	semidirect
9. #19133479	C₂	semidirect
10. C₈×#10621885	C₂	semidirect
11. C₄₃×#1676	C₂	semidirect
200. #1676	D₄₃	semidirect
244. C₄₃	#19909	semidirect

Maximal subgroups

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

subgroup	index
#19147570	2
#19147464	2
#19146930	2
#19134188	2
#19134131	2
#19133956	2
#19133720	2
#19133668	2
#19133479	2
C₈×#10621885	2
C₄₃×#1676	2
#19909	43