Subgroups — #25251437

Order 11008.

588 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19144239	C₂	1	1	yes	maximal, characteristic
2	#19143191	C₂	1	1	yes	maximal, characteristic, complemented
3	#19136785	C₂	1	2	yes	maximal, complemented
4	#19136784	C₂	1	2	yes	maximal, complemented
5	#19136779	C₂	1	2	yes	maximal, complemented
6	#19136774	C₂	1	2	yes	maximal, complemented
7	#19136381	C₂	1	1	yes	maximal, characteristic, complemented
8	#19136379	C₂	1	1	yes	maximal, characteristic, complemented
9	#19136311	C₂	1	1	yes	maximal, characteristic, complemented
10	#19136208	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄₃×#357	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13063331	C₂²	1	1	yes	characteristic
13	#14583466	—	2	1	no	—
14	#14583380	—	2	1	no	—
15	C₂×#13063265	C₂²	1	1	yes	characteristic
16	#14583149	—	2	2	no	—
17	#14583132	C₂²	1	2	yes	—
18	#14583132	C₂²	1	2	yes	—
19	#14582940	—	2	2	no	—
20	#14582939	—	2	1	no	—
21	#14582937	C₂²	1	1	yes	characteristic
22	#14582937	C₂²	1	1	yes	characteristic
23	#14582936	C₂²	1	2	yes	complemented
24	#107×D₄₃	C₂²	1	2	yes	complemented
25	#14582904	—	2	1	no	—
26	#14582727	C₂²	1	1	yes	characteristic, complemented
27	#14582727	C₂²	1	1	yes	characteristic, complemented
28	#14582726	C₂²	1	2	yes	complemented
29	#14582725	C₂²	1	1	yes	characteristic, complemented
30	#14582725	C₂²	1	1	yes	characteristic, complemented
31	#14582724	C₂²	1	2	yes	complemented
32	#14582700	C₂²	1	2	yes	—
33	#14582700	C₂²	1	2	yes	—
34	(C₂×D₄₃)⋊C₁₆	C₂²	1	2	yes	complemented
35	(C₂×D₄₃)⋊C₁₆	C₂²	1	2	yes	complemented
36	C₂²×D₈×C₄₃	C₂²	1	1	yes	characteristic
37	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
38	C₂×#107×C₄₃	C₂²	1	1	yes	characteristic
39	C₄₃×#357	C₂²	1	2	yes	—
40	C₄₃×#357	C₂²	1	2	yes	—
41	C₂×#10621898	D₄	1	1	yes	characteristic
42	#13063331	—	2	2	no	—
43	#13063331	—	2	2	no	—
44	C₂×#10621877	D₄	1	1	yes	characteristic
45	C₂×#10621872	D₄	1	2	yes	—
46	C₂×C₈×D₄₃	C₂³	1	2	yes	—
47	#13063321	—	4	1	no	—
48	#13063312	—	4	1	no	—
49	#13063296	—	2	2	no	—
50	#13063296	—	2	2	no	—
51	#13063294	—	4	2	no	—
52	#13063290	—	4	1	no	—
53	#13063287	—	4	1	no	—
54	C₄×C₂²⋊D₄₃	—	4	2	no	—
55	#13063284	—	2	1	no	—
56	#13063284	—	2	1	no	—
57	#13063283	—	2	2	no	—
58	C₄⋊C₄×D₄₃	—	2	2	no	—
59	#13063265	—	2	2	no	—
60	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
61	#13063259	C₂³	1	1	yes	characteristic
62	#13063259	C₂³	1	1	yes	characteristic
63	#13063259	C₂³	1	1	yes	characteristic
64	#13063259	C₂³	1	1	yes	characteristic
65	#13063258	—	2	2	no	—
66	#13063257	—	2	1	no	—
67	#13063257	—	2	1	no	—
68	#13063256	—	2	2	no	—
69	#13063246	—	2	2	no	—
70	#13063246	—	2	2	no	—
71	C₂²×D₄×C₄₃	—	2	1	no	—
72	C₂×D₈×C₄₃	—	2	1	no	—
73	C₂×D₈×C₄₃	—	2	2	no	—
74	C₂×D₈×C₄₃	—	2	2	no	—
75	C₂×D₈×C₄₃	C₂³	1	2	yes	—
76	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
77	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
78	C₂×C₁₆×C₄₃	—	2	2	no	—
79	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
80	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
81	#107×C₄₃	—	2	1	no	—
82	#107×C₄₃	C₂³	1	2	yes	—
83	C₂⁴×C₄₃	—	4	1	no	—
84	C₂×D₄×C₄₃	—	2	2	no	—
85	C₂×D₄×C₄₃	—	4	1	no	—
86	C₂×D₄×C₄₃	—	4	2	no	—
87	C₂×D₄×C₄₃	—	4	2	no	—
88	C₂²×C₄×C₄₃	—	4	1	no	—
89	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₂²⋊D₄₃	—	4	2	no	—
91	C₂×C₂²⋊D₄₃	—	4	2	no	—
92	C₂×C₂²⋊D₄₃	D₈	1	2	yes	—
93	#10621898	—	2	2	no	—
94	#10621898	—	2	2	no	—
95	C₂×D₁₇₂	D₈	1	1	yes	characteristic
96	C₂×C₄×D₄₃	—	4	2	no	—
97	C₂×C₄×D₄₃	C₂×D₄	1	2	yes	—
98	C₂×Dic₈₆	D₈	1	1	yes	characteristic
99	D₈×C₄₃	—	2	2	no	—
100	D₈×C₄₃	—	2	2	no	—
101	D₈×C₄₃	—	4	2	no	—
102	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₄₃	#41	1	2	yes	—
104	C₂×C₈×C₄₃	#41	1	2	yes	—
105	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
106	C₄⋊C₄×C₄₃	—	2	2	no	—
107	C₄⋊C₄×C₄₃	—	4	1	no	—
108	C₂²⋊Dic₄₃	—	4	2	no	—
109	C₂²⋊Dic₄₃	—	4	2	no	—
110	C₂²⋊Dic₄₃	—	4	1	no	—
111	C₂²⋊Dic₄₃	—	4	1	no	—
112	(C₂×D₄₃)⋊C₄	—	4	2	no	—
113	C₄⋊Dic₄₃	—	2	1	no	—
114	C₄⋊Dic₄₃	—	2	1	no	—
115	C₄⋊Dic₄₃	—	2	1	no	—
116	C₄⋊Dic₄₃	—	2	1	no	—
117	Dic₄₃⋊C₄	—	4	2	no	—
118	C₄×Dic₄₃	—	4	1	no	—
119	C₄×Dic₄₃	—	4	1	no	—
120	C₄×Dic₄₃	—	4	2	no	—
121	#10621877	—	2	2	no	—
122	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
123	#10621872	—	2	4	no	—
124	C₈×D₄₃	—	2	2	no	—
125	C₈×D₄₃	—	2	2	no	—
126	C₁₆×C₄₃	—	2	1	no	—
127	C₁₆×C₄₃	—	2	1	no	—
128	C₄₃⋊C₁₆	—	2	2	no	—
129	C₂³×C₄₃	—	4	2	no	—
130	C₂³×C₄₃	—	8	2	no	—
131	C₂³×C₄₃	—	16	1	no	—
132	C₂³×C₄₃	—	16	1	no	—
133	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
134	C₂²×D₄₃	C₂×D₈	1	2	yes	—
135	D₄×C₄₃	—	4	2	no	—
136	D₄×C₄₃	—	4	2	no	—
137	D₄×C₄₃	—	8	2	no	—
138	C₂×C₄×C₄₃	—	4	2	no	—
139	C₂×C₄×C₄₃	—	16	1	no	—
140	C₂×C₄×C₄₃	#123	1	2	yes	—
141	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
142	C₂×C₄×C₄₃	#137	1	2	yes	—
143	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
144	C₂²⋊D₄₃	—	2	4	no	—
145	C₂²⋊D₄₃	—	16	2	no	—
146	C₂²⋊D₄₃	—	16	2	no	—
147	C₂×Dic₄₃	—	4	1	no	—
148	C₂×Dic₄₃	—	4	1	no	—
149	C₂×Dic₄₃	—	4	1	no	—
150	C₂×Dic₄₃	—	4	1	no	—
151	C₂×Dic₄₃	C₂×D₈	1	2	yes	—
152	D₁₇₂	—	2	2	no	—
153	C₄×D₄₃	—	2	2	no	—
154	C₄×D₄₃	—	2	2	no	—
155	C₄×D₄₃	—	16	2	no	—
156	Dic₈₆	—	2	2	no	—
157	C₈×C₄₃	#123	1	2	yes	—
158	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
159	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
160	C₄₃⋊C₈	—	2	2	no	—
161	#33650	—	43	1	no	maximal, Sylow
162	C₂²×C₄₃	—	2	1	no	—
163	C₂²×C₄₃	—	2	1	no	—
164	C₂²×C₄₃	—	8	2	no	—
165	C₂²×C₄₃	—	8	2	no	—
166	C₂²×C₄₃	—	8	2	no	—
167	C₂²×C₄₃	—	16	4	no	—
168	C₂²×C₄₃	#480	1	2	yes	—
169	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
170	C₂×D₄₃	—	2	2	no	—
171	C₂×D₄₃	—	2	2	no	—
172	C₂×D₄₃	—	2	2	no	—
173	C₄×C₄₃	—	8	2	no	—
174	C₄×C₄₃	#480	1	2	yes	—
175	C₄×C₄₃	C₂×#123	1	1	yes	characteristic
176	C₄×C₄₃	C₂×#137	1	1	yes	characteristic
177	Dic₄₃	—	2	2	no	—
178	Dic₄₃	—	8	1	no	—
179	Dic₄₃	—	8	1	no	—
180	Dic₄₃	—	8	1	no	—
181	Dic₄₃	—	8	1	no	—
182	#3058	—	43	1	no	—
183	#2913	—	43	1	no	—
184	#2234	—	43	2	no	—
185	#2234	—	43	2	no	—
186	#2233	—	43	2	no	—
187	#2232	—	43	2	no	—
188	#2159	—	43	1	no	—
189	#2158	—	43	1	no	—
190	#2140	—	43	1	no	—
191	C₂×#357	D₄₃	1	1	yes	characteristic, complemented
192	#2113	—	43	1	no	—
193	C₂×C₄₃	—	2	2	no	—
194	C₂×C₄₃	—	16	2	no	—
195	C₂×C₄₃	C₂×#480	1	1	yes	characteristic
196	C₂×C₄₃	#3415	1	1	yes	characteristic
197	C₂×C₄₃	#3416	1	1	yes	characteristic
198	D₄₃	—	4	2	no	—
199	C₂²×D₈	C₂×D₄₃	1	1	yes	characteristic
200	C₂×#131	—	43	1	no	—
201	#535	—	86	1	no	—
202	#520	—	86	1	no	—
203	C₂×M₃₂	—	43	1	no	—
204	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
205	#469	—	86	2	no	—
206	#466	—	43	2	no	—
207	#466	—	43	2	no	—
208	#428	—	86	2	no	—
209	#428	—	86	1	no	—
210	#427	—	43	1	no	—
211	#427	—	43	1	no	—
212	#427	—	43	2	no	—
213	C₂×#107	—	43	2	no	—
214	C₂×#107	C₂×D₄₃	1	1	yes	characteristic
215	#418	—	86	1	no	—
216	#367	—	43	1	no	—
217	#367	—	43	1	no	—
218	#367	—	43	2	no	—
219	#366	—	43	1	no	—
220	#366	—	43	1	no	—
221	#366	—	43	2	no	—
222	#357	—	43	2	no	—
223	#357	—	43	2	no	—
224	#357	C₂×D₄₃	1	2	yes	—
225	#357	C₂×D₄₃	1	2	yes	—
226	C₂²⋊C₁₆	—	43	2	no	—
227	C₂²⋊C₁₆	—	43	2	no	—
228	C₄₃	#33650	1	1	yes	characteristic, complemented, Sylow
229	C₂×#41	—	43	1	no	—
230	C₂²×D₄	—	2	1	no	—
231	C₂×D₈	—	2	2	no	—
232	C₂×D₈	—	2	1	no	—
233	C₂×D₈	—	2	2	no	—
234	C₂×D₈	C₂²×D₄₃	1	2	yes	—
235	#131	—	86	2	no	—
236	#131	—	86	2	no	—
237	C₂×M₁₆	—	43	1	no	—
238	C₂×M₁₆	—	43	2	no	—
239	C₂²×C₈	—	43	2	no	—
240	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
241	#127	—	172	1	no	—
242	#124	—	172	1	no	—
243	#121	—	86	2	no	—
244	#121	—	86	2	no	—
245	#120	—	172	2	no	—
246	C₄×Q₈	—	172	1	no	—
247	C₄×D₄	—	172	1	no	—
248	C₄×D₄	—	172	2	no	—
249	#117	—	86	2	no	—
250	#117	—	86	1	no	—
251	#117	—	86	1	no	—
252	C₂×C₄⋊C₄	—	2	1	no	—
253	C₂×C₄⋊C₄	—	86	2	no	—
254	M₃₂	—	86	2	no	—
255	C₂×C₁₆	—	2	2	no	—
256	C₂×C₁₆	—	43	2	no	—
257	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
258	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
259	#107	—	2	1	no	—
260	#107	—	43	1	no	—
261	#107	—	43	1	no	—
262	#107	—	43	1	no	—
263	#107	—	43	1	no	—
264	#107	—	86	2	no	—
265	#107	C₂²×D₄₃	1	2	yes	—
266	#106	—	86	2	no	—
267	#106	—	86	1	no	—
268	#106	—	86	1	no	—
269	#102	—	86	2	no	—
270	#102	—	86	2	no	—
271	C₂⁴	—	4	1	no	—
272	#41	—	86	2	no	—
273	#41	—	86	2	no	—
274	C₂×Q₈	—	43	1	no	—
275	C₂×D₄	—	2	2	no	—
276	C₂×D₄	—	4	2	no	—
277	C₂×D₄	—	4	1	no	—
278	C₂×D₄	—	4	2	no	—
279	C₂×D₄	—	43	2	no	—
280	C₂×D₄	—	43	1	no	—
281	C₂×D₄	—	172	2	no	—
282	C₂×D₄	—	172	2	no	—
283	C₂²×C₄	—	4	1	no	—
284	C₂²×C₄	—	43	2	no	—
285	C₂²×C₄	—	172	2	no	—
286	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
287	D₈	—	2	2	no	—
288	D₈	—	2	2	no	—
289	D₈	—	4	2	no	—
290	M₁₆	—	86	2	no	—
291	M₁₆	—	86	4	no	—
292	C₂×C₈	—	43	2	no	—
293	C₂×C₈	—	86	2	no	—
294	C₂×C₈	—	86	2	no	—
295	C₂×C₈	#10621898	1	2	yes	—
296	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
297	C₂×C₈	#10621900	1	2	yes	—
298	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
299	C₄⋊C₄	—	2	2	no	—
300	C₄⋊C₄	—	4	1	no	—
301	C₄⋊C₄	—	86	1	no	—
302	C₄⋊C₄	—	86	1	no	—
303	C₄⋊C₄	—	86	1	no	—
304	C₄⋊C₄	—	86	1	no	—
305	C₄⋊C₄	—	172	2	no	—
306	C₂²⋊C₄	—	172	2	no	—
307	C₂²⋊C₄	—	172	2	no	—
308	C₂²⋊C₄	—	172	2	no	—
309	C₂²⋊C₄	—	172	1	no	—
310	C₂²⋊C₄	—	172	1	no	—
311	C₄²	—	172	2	no	—
312	C₄²	—	172	1	no	—
313	C₄²	—	172	1	no	—
314	C₁₆	—	2	1	no	—
315	C₁₆	—	2	1	no	—
316	C₁₆	—	86	2	no	—
317	C₂³	—	4	2	no	—
318	C₂³	—	8	2	no	—
319	C₂³	—	16	1	no	—
320	C₂³	—	16	1	no	—
321	C₂³	—	43	2	no	—
322	C₂³	D₈×D₄₃	1	1	yes	characteristic
323	Q₈	—	86	2	no	—
324	D₄	—	4	2	no	—
325	D₄	—	4	2	no	—
326	D₄	—	8	2	no	—
327	D₄	—	86	4	no	—
328	D₄	—	86	2	no	—
329	D₄	—	688	2	no	—
330	D₄	—	688	2	no	—
331	C₂×C₄	—	4	2	no	—
332	C₂×C₄	—	16	1	no	—
333	C₂×C₄	—	43	2	no	—
334	C₂×C₄	—	86	2	no	—
335	C₂×C₄	—	86	2	no	—
336	C₂×C₄	—	172	1	no	—
337	C₂×C₄	—	172	1	no	—
338	C₂×C₄	—	172	1	no	—
339	C₂×C₄	—	172	1	no	—
340	C₂×C₄	—	688	2	no	—
341	C₂×C₄	#13063307	1	2	yes	—
342	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
343	C₂×C₄	#13063354	1	2	yes	—
344	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
345	C₈	—	86	2	no	—
346	C₈	#13063307	1	2	yes	—
347	C₈	C₂×#10621900	1	1	yes	characteristic
348	C₈	C₂×#10621898	1	1	yes	characteristic
349	C₂²	—	2	1	no	—
350	C₂²	—	2	1	no	—
351	C₂²	—	8	2	no	—
352	C₂²	—	8	2	no	—
353	C₂²	—	8	2	no	—
354	C₂²	—	16	4	no	—
355	C₂²	—	86	2	no	—
356	C₂²	—	86	2	no	—
357	C₂²	—	86	2	no	—
358	C₂²	#14583211	1	2	yes	—
359	C₂²	C₂×D₈×D₄₃	1	1	yes	characteristic
360	C₄	—	8	2	no	—
361	C₄	—	86	2	no	—
362	C₄	—	344	1	no	—
363	C₄	—	344	1	no	—
364	C₄	—	344	1	no	—
365	C₄	—	344	1	no	—
366	C₄	#14583211	1	2	yes	—
367	C₄	C₂×#13063307	1	1	yes	characteristic
368	C₄	C₂×#13063354	1	1	yes	characteristic
369	C₂	—	2	2	no	—
370	C₂	—	16	2	no	—
371	C₂	—	172	2	no	—
372	C₂	C₂×#14583211	1	1	yes	characteristic
373	C₂	#19147732	1	1	yes	characteristic
374	C₂	#19147737	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₄₃
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₂×C₄₃×#357	C₂
socle	C₂²×C₄₃	C₂²×D₈
prime radical	#19144239	C₂
hypercenter	C₂×#357	D₄₃
hypocenter	C₄₃	#33650
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#357

Complements

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

N	H ≅ G/N	type
2. #19143191	C₂	semidirect
3. #19136785	C₂	semidirect
4. #19136784	C₂	semidirect
5. #19136779	C₂	semidirect
6. #19136774	C₂	semidirect
7. #19136381	C₂	semidirect
8. #19136379	C₂	semidirect
9. #19136311	C₂	semidirect
10. #19136208	C₂	semidirect
11. C₂×C₄₃×#357	C₂	semidirect
23. #14582936	C₂²	semidirect
24. #107×D₄₃	C₂²	semidirect
26. #14582727	C₂²	semidirect
27. #14582727	C₂²	semidirect
28. #14582726	C₂²	semidirect
29. #14582725	C₂²	semidirect
30. #14582725	C₂²	semidirect
31. #14582724	C₂²	semidirect
34. (C₂×D₄₃)⋊C₁₆	C₂²	semidirect
35. (C₂×D₄₃)⋊C₁₆	C₂²	semidirect
191. C₂×#357	D₄₃	semidirect
228. C₄₃	#33650	semidirect

Maximal subgroups

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

subgroup	index
#19144239	2
#19143191	2
#19136785	2
#19136784	2
#19136779	2
#19136774	2
#19136381	2
#19136379	2
#19136311	2
#19136208	2
C₂×C₄₃×#357	2
#33650	43