Subgroups — #25088124

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	#19146944	C₂	1	1	yes	maximal, characteristic
2	#19146905	C₂	1	2	yes	maximal, complemented
3	#19134204	C₂	1	1	yes	maximal, characteristic, complemented
4	#19134172	C₂	1	2	yes	maximal, complemented
5	#19133991	C₂	1	1	yes	maximal, characteristic, complemented
6	#19133989	C₂	1	1	yes	maximal, characteristic, complemented
7	#19133809	C₂	1	2	yes	maximal, complemented
8	#19133745	C₂	1	2	yes	maximal, complemented
9	#19133480	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×#10621884	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₃×#3380	C₂	1	1	yes	maximal, characteristic, complemented
12	#14583554	C₂²	1	2	yes	—
13	#14583304	C₂²	1	1	yes	characteristic
14	#14583251	C₂²	1	2	yes	—
15	#14583249	C₂²	1	2	yes	—
16	#14583229	—	2	2	no	—
17	#14583211	—	2	2	no	—
18	#14583186	C₂²	1	2	yes	—
19	#14583179	C₂²	1	2	yes	—
20	#14583155	—	2	2	no	—
21	#14583132	—	2	1	no	—
22	#14583109	C₂²	1	2	yes	—
23	#14583059	—	2	2	no	—
24	C₄×#10621884	C₂²	1	1	yes	characteristic
25	#14582977	C₂²	1	2	yes	—
26	C₈×D₁₇₂	C₂²	1	1	yes	characteristic
27	#107×D₄₃	—	2	1	no	—
28	#14582898	—	2	2	no	—
29	#14582652	C₂²	1	1	yes	characteristic
30	#14582649	C₂²	1	1	yes	characteristic
31	#14582648	C₂²	1	2	yes	—
32	#14582644	C₂²	1	2	yes	—
33	Q₈⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
34	D₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
35	#14582626	C₂²	1	1	yes	characteristic
36	C₈×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
37	C₄₃×#554	C₂²	1	2	yes	—
38	C₄₃×#500	C₂²	1	1	yes	characteristic
39	C₄₃×#499	—	2	1	no	—
40	C₄₃×#478	—	2	2	no	—
41	C₄₃×#476	C₂²	1	2	yes	—
42	C₈×Q₈×C₄₃	C₂²	1	1	yes	characteristic
43	C₄×D₈×C₄₃	—	2	1	no	—
44	C₄×D₈×C₄₃	C₂²	1	1	yes	characteristic
45	C₂×#10621898	—	2	2	no	—
46	C₂×#10621885	—	2	2	no	—
47	C₂×#10621884	D₄	1	1	yes	characteristic
48	C₂×#10621877	—	2	2	no	—
49	C₂×C₈×D₄₃	—	2	1	no	—
50	#13063325	C₂³	1	2	yes	—
51	#13063307	—	4	2	no	—
52	#13063304	—	2	2	no	—
53	#13063296	—	2	2	no	—
54	C₄×D₁₇₂	C₂³	1	1	yes	characteristic
55	C₄⋊C₄×D₄₃	—	2	2	no	—
56	#13063259	—	2	1	no	—
57	#13063259	—	2	1	no	—
58	#13063258	—	2	2	no	—
59	#13063256	—	2	2	no	—
60	#13063256	D₄	1	1	yes	characteristic
61	C₄⋊(C₄₃⋊C₈)	C₂³	1	2	yes	—
62	C₄⋊(C₄₃⋊C₈)	C₂³	1	1	yes	characteristic
63	#13063249	—	4	2	no	—
64	#13063249	D₄	1	1	yes	characteristic
65	#13063248	—	2	2	no	—
66	#13063248	—	2	2	no	—
67	#13063248	—	2	2	no	—
68	#13063246	—	2	2	no	—
69	#13063245	—	2	2	no	—
70	#13063245	D₄	1	1	yes	characteristic
71	C₂²⋊(C₄₃⋊C₈)	—	2	2	no	—
72	(C₂×D₄₃)⋊C₈	—	2	1	no	—
73	C₄×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
74	C₄×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
75	C₂×D₈×C₄₃	D₄	1	1	yes	characteristic
76	C₄⋊Q₈×C₄₃	C₂³	1	2	yes	—
77	#125×C₄₃	—	2	2	no	—
78	C₂²⋊Q₈×C₄₃	—	2	2	no	—
79	C₂²⋊Q₈×C₄₃	—	4	2	no	—
80	C₄×Q₈×C₄₃	C₂³	1	1	yes	characteristic
81	C₄×D₄×C₄₃	—	2	2	no	—
82	C₄×D₄×C₄₃	C₂³	1	2	yes	—
83	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
84	#107×C₄₃	—	2	1	no	—
85	#107×C₄₃	—	2	1	no	—
86	#107×C₄₃	D₄	1	1	yes	characteristic
87	#106×C₄₃	—	2	2	no	—
88	C₄⋊C₈×C₄₃	—	2	1	no	—
89	C₄⋊C₈×C₄₃	C₂³	1	1	yes	characteristic
90	#102×C₄₃	—	2	2	no	—
91	#102×C₄₃	D₄	1	2	yes	—
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	C₂×D₄×C₄₃	C₂×D₄	1	2	yes	—
97	C₂²×C₄×C₄₃	—	2	2	no	—
98	C₂²×C₄×C₄₃	—	4	2	no	—
99	C₂×C₂²⋊D₄₃	—	4	2	no	—
100	#10621898	—	4	4	no	—
101	#10621898	—	8	2	no	—
102	C₂×D₁₇₂	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×D₄₃	—	2	1	no	—
104	C₂×C₄×D₄₃	—	4	2	no	—
105	C₂×Dic₈₆	—	2	2	no	—
106	D₈×C₄₃	—	2	2	no	—
107	C₂×C₈×C₄₃	—	2	1	no	—
108	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
110	C₄⋊C₄×C₄₃	—	2	1	no	—
111	C₄⋊C₄×C₄₃	—	2	2	no	—
112	C₄⋊C₄×C₄₃	—	2	2	no	—
113	C₄⋊C₄×C₄₃	—	4	2	no	—
114	C₄⋊C₄×C₄₃	—	4	2	no	—
115	C₄⋊C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
116	C₄⋊C₄×C₄₃	C₂×D₄	1	2	yes	—
117	C₂²⋊C₄×C₄₃	—	2	2	no	—
118	C₂²⋊C₄×C₄₃	—	4	2	no	—
119	C₄²×C₄₃	—	2	1	no	—
120	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
121	C₂²⋊Dic₄₃	—	4	2	no	—
122	#10621885	—	8	2	no	—
123	#10621884	—	2	2	no	—
124	(C₂×D₄₃)⋊C₄	—	2	1	no	—
125	C₄⋊Dic₄₃	—	2	2	no	—
126	C₄⋊Dic₄₃	—	2	2	no	—
127	C₄⋊Dic₄₃	C₂×D₄	1	1	yes	characteristic
128	Dic₄₃⋊C₄	—	4	2	no	—
129	#10621877	—	8	2	no	—
130	C₂×C₄₃⋊C₈	—	2	1	no	—
131	C₂×C₄₃⋊C₈	—	2	2	no	—
132	C₂×C₄₃⋊C₈	—	2	1	no	—
133	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
134	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
135	C₈×D₄₃	—	4	2	no	—
136	C₂³×C₄₃	—	2	2	no	—
137	C₂²×D₄₃	—	2	1	no	—
138	Q₈×C₄₃	—	2	1	no	—
139	Q₈×C₄₃	—	2	1	no	—
140	Q₈×C₄₃	—	8	2	no	—
141	D₄×C₄₃	—	2	2	no	—
142	D₄×C₄₃	—	2	2	no	—
143	C₂×C₄×C₄₃	—	2	2	no	—
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	—	4	2	no	—
148	C₂×C₄×C₄₃	—	4	2	no	—
149	C₂×C₄×C₄₃	—	16	2	no	—
150	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
152	C₂×C₄×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
153	C₂²⋊D₄₃	—	8	4	no	—
154	C₂×Dic₄₃	—	2	1	no	—
155	C₂×Dic₄₃	—	4	2	no	—
156	D₁₇₂	—	2	1	no	—
157	D₁₇₂	#135	1	2	yes	—
158	C₄×D₄₃	—	4	1	no	—
159	C₄×D₄₃	—	4	1	no	—
160	C₄×D₄₃	—	8	4	no	—
161	Dic₈₆	—	4	2	no	—
162	Dic₈₆	—	4	2	no	—
163	C₈×C₄₃	—	2	1	no	—
164	C₈×C₄₃	—	4	1	no	—
165	C₈×C₄₃	#137	1	2	yes	—
166	C₄₃⋊C₈	—	2	1	no	—
167	C₄₃⋊C₈	—	2	1	no	—
168	C₄₃⋊C₈	—	4	1	no	—
169	C₄₃⋊C₈	—	4	1	no	—
170	C₄₃⋊C₈	—	4	2	no	—
171	#19903	—	43	1	no	maximal, Sylow
172	C₂²×C₄₃	—	4	2	no	—
173	C₂²×C₄₃	—	4	2	no	—
174	C₂²×C₄₃	—	4	2	no	—
175	C₂²×C₄₃	D₄²	1	1	yes	characteristic
176	C₂×D₄₃	—	2	2	no	—
177	C₂×D₄₃	—	4	1	no	—
178	C₂×D₄₃	—	4	1	no	—
179	C₄×C₄₃	—	2	1	no	—
180	C₄×C₄₃	—	4	2	no	—
181	C₄×C₄₃	—	4	1	no	—
182	C₄×C₄₃	—	4	1	no	—
183	C₄×C₄₃	—	8	2	no	—
184	C₄×C₄₃	C₂×#135	1	1	yes	characteristic
185	C₄×C₄₃	C₂×#137	1	1	yes	characteristic
186	C₄×C₄₃	#576	1	1	yes	characteristic
187	C₄×C₄₃	#578	1	1	yes	characteristic
188	Dic₄₃	—	4	1	no	—
189	Dic₄₃	—	8	2	no	—
190	#3380	D₄₃	1	1	yes	characteristic, complemented
191	#3334	—	43	1	no	—
192	#3331	—	43	2	no	—
193	#1710	—	43	1	no	—
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	—	8	2	no	—
202	C₂×C₄₃	#3276	1	1	yes	characteristic
203	C₂×C₄₃	#3280	1	1	yes	characteristic
204	C₂×C₄₃	#3290	1	1	yes	characteristic
205	D₄₃	—	4	2	no	—
206	#554	C₂×D₄₃	1	2	yes	—
207	#547	—	43	2	no	—
208	#500	—	43	1	no	—
209	#500	C₂×D₄₃	1	1	yes	characteristic
210	#499	—	2	1	no	—
211	#487	—	43	2	no	—
212	#487	—	43	2	no	—
213	#483	—	86	2	no	—
214	#480	—	86	2	no	—
215	#478	—	2	2	no	—
216	#476	—	43	2	no	—
217	#476	C₂×D₄₃	1	2	yes	—
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	C₂×D₄₃	1	1	yes	characteristic
224	C₄×SD₁₆	—	43	1	no	—
225	C₄×D₈	—	2	1	no	—
226	C₄×D₈	C₂×D₄₃	1	1	yes	characteristic
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	2	no	—
237	D₄⋊C₈	—	43	1	no	—
238	C₈²	—	43	1	no	—
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	C₂²⋊D₄₃	1	1	yes	characteristic
244	C₂×M₁₆	—	86	2	no	—
245	C₂²×C₈	—	86	1	no	—
246	C₄⋊Q₈	—	43	2	no	—
247	C₄⋊Q₈	C₂²×D₄₃	1	2	yes	—
248	#125	—	2	2	no	—
249	#123	—	172	2	no	—
250	C₂²⋊Q₈	—	2	2	no	—
251	C₂²⋊Q₈	—	4	2	no	—
252	C₂²⋊Q₈	—	86	2	no	—
253	#121	—	86	2	no	—
254	C₄×Q₈	C₂²×D₄₃	1	1	yes	characteristic
255	C₄×D₄	—	2	2	no	—
256	C₄×D₄	—	43	1	no	—
257	C₄×D₄	C₂²×D₄₃	1	2	yes	—
258	C₂×C₄⋊C₄	—	2	2	no	—
259	C₂×C₄⋊C₄	—	86	2	no	—
260	#107	—	2	1	no	—
261	#107	—	2	1	no	—
262	#107	—	86	1	no	—
263	#107	—	86	1	no	—
264	#107	—	86	2	no	—
265	#107	C₂²⋊D₄₃	1	1	yes	characteristic
266	#106	—	2	2	no	—
267	#106	—	43	1	no	—
268	#106	—	86	2	no	—
269	C₄⋊C₈	—	2	1	no	—
270	C₄⋊C₈	—	43	2	no	—
271	C₄⋊C₈	—	43	1	no	—
272	C₄⋊C₈	C₂²×D₄₃	1	1	yes	characteristic
273	#103	—	43	1	no	—
274	#103	—	86	2	no	—
275	#103	—	86	2	no	—
276	#103	—	86	2	no	—
277	#103	—	172	2	no	—
278	#102	—	2	2	no	—
279	#102	—	43	1	no	—
280	#102	—	86	2	no	—
281	#102	—	86	2	no	—
282	#102	C₂²⋊D₄₃	1	2	yes	—
283	C₂²⋊C₈	—	86	2	no	—
284	C₂²⋊C₈	—	86	1	no	—
285	C₄×C₈	—	2	1	no	—
286	C₄×C₈	—	43	1	no	—
287	C₄×C₈	—	43	1	no	—
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	—	86	2	no	—
293	C₂×Q₈	D₄×D₄₃	1	1	yes	characteristic
294	C₂×D₄	—	43	1	no	—
295	C₂×D₄	—	172	2	no	—
296	C₂×D₄	C₂×C₂²⋊D₄₃	1	2	yes	—
297	C₂²×C₄	—	2	2	no	—
298	C₂²×C₄	—	4	2	no	—
299	C₂²×C₄	—	86	1	no	—
300	C₂²×C₄	—	172	2	no	—
301	Q₁₆	—	344	2	no	—
302	SD₁₆	—	86	2	no	—
303	D₈	—	2	2	no	—
304	M₁₆	—	344	2	no	—
305	C₂×C₈	—	2	1	no	—
306	C₂×C₈	—	43	1	no	—
307	C₂×C₈	—	43	1	no	—
308	C₂×C₈	—	86	2	no	—
309	C₂×C₈	—	86	1	no	—
310	C₂×C₈	—	86	1	no	—
311	C₂×C₈	—	172	2	no	—
312	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
313	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
314	C₄⋊C₄	—	2	1	no	—
315	C₄⋊C₄	—	2	2	no	—
316	C₄⋊C₄	—	2	2	no	—
317	C₄⋊C₄	—	4	2	no	—
318	C₄⋊C₄	—	4	2	no	—
319	C₄⋊C₄	—	43	1	no	—
320	C₄⋊C₄	—	86	2	no	—
321	C₄⋊C₄	—	86	2	no	—
322	C₄⋊C₄	—	172	2	no	—
323	C₄⋊C₄	D₄×D₄₃	1	1	yes	characteristic
324	C₄⋊C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
325	C₂²⋊C₄	—	2	2	no	—
326	C₂²⋊C₄	—	4	2	no	—
327	C₂²⋊C₄	—	86	1	no	—
328	C₂²⋊C₄	—	172	2	no	—
329	C₄²	—	2	1	no	—
330	C₄²	C₂³×D₄₃	1	1	yes	characteristic
331	C₂³	—	2	2	no	—
332	C₂³	—	86	1	no	—
333	Q₈	—	2	1	no	—
334	Q₈	—	2	1	no	—
335	Q₈	—	8	2	no	—
336	Q₈	—	172	2	no	—
337	Q₈	—	172	2	no	—
338	D₄	—	2	2	no	—
339	D₄	—	2	2	no	—
340	D₄	—	43	2	no	—
341	D₄	—	86	1	no	—
342	D₄	—	344	4	no	—
343	C₂×C₄	—	2	1	no	—
344	C₂×C₄	—	2	1	no	—
345	C₂×C₄	—	2	2	no	—
346	C₂×C₄	—	4	2	no	—
347	C₂×C₄	—	4	2	no	—
348	C₂×C₄	—	4	2	no	—
349	C₂×C₄	—	16	2	no	—
350	C₂×C₄	—	86	1	no	—
351	C₂×C₄	—	344	4	no	—
352	C₂×C₄	—	172	2	no	—
353	C₂×C₄	—	172	1	no	—
354	C₂×C₄	—	172	1	no	—
355	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
356	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
357	C₂×C₄	#13063370	1	1	yes	characteristic
358	C₈	—	2	1	no	—
359	C₈	—	4	1	no	—
360	C₈	—	86	1	no	—
361	C₈	—	86	1	no	—
362	C₈	—	172	1	no	—
363	C₈	—	172	1	no	—
364	C₈	—	172	2	no	—
365	C₈	#13063356	1	2	yes	—
366	C₂²	—	4	2	no	—
367	C₂²	—	4	2	no	—
368	C₂²	—	4	2	no	—
369	C₂²	—	86	2	no	—
370	C₂²	—	172	1	no	—
371	C₂²	—	172	1	no	—
372	C₂²	D₄×C₂²⋊D₄₃	1	1	yes	characteristic
373	C₄	—	2	1	no	—
374	C₄	—	4	2	no	—
375	C₄	—	4	1	no	—
376	C₄	—	4	1	no	—
377	C₄	—	8	2	no	—
378	C₄	—	344	2	no	—
379	C₄	—	172	1	no	—
380	C₄	#135×D₄₃	1	1	yes	characteristic
381	C₄	C₂×#13063356	1	1	yes	characteristic
382	C₄	#14583701	1	1	yes	characteristic
383	C₄	#14583714	1	1	yes	characteristic
384	C₂	—	8	2	no	—
385	C₂	—	172	2	no	—
386	C₂	#19146241	1	1	yes	characteristic
387	C₂	#19146274	1	1	yes	characteristic
388	C₂	#19146377	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×C₂²⋊D₄₃
commutator, cosocle	C₄²×C₄₃	C₂⁴
Frattini	C₄²	C₂³×D₄₃
Fitting	C₄₃×#3380	C₂
socle	C₂²×C₄₃	D₄²
prime radical	#19146944	C₂
hypercenter	#3380	D₄₃
hypocenter	C₄₃	#19903
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #3380

Complements

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

N	H ≅ G/N	type
2. #19146905	C₂	semidirect
3. #19134204	C₂	semidirect
4. #19134172	C₂	semidirect
5. #19133991	C₂	semidirect
6. #19133989	C₂	semidirect
7. #19133809	C₂	semidirect
8. #19133745	C₂	semidirect
9. #19133480	C₂	semidirect
10. C₈×#10621884	C₂	semidirect
11. C₄₃×#3380	C₂	semidirect
190. #3380	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
#19146944	2
#19146905	2
#19134204	2
#19134172	2
#19133991	2
#19133989	2
#19133809	2
#19133745	2
#19133480	2
C₈×#10621884	2
C₄₃×#3380	2
#19903	43