Subgroups — #25155498

Order 11008.

690 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19145123	C₂	1	1	yes	maximal, characteristic, complemented
2	#19144462	C₂	1	2	yes	maximal
3	#19143186	C₂	1	1	yes	maximal, characteristic
4	#19136106	C₂	1	2	yes	maximal, complemented
5	#19135940	C₂	1	2	yes	maximal, complemented
6	#19135373	C₂	1	1	yes	maximal, characteristic, complemented
7	#19135343	C₂	1	1	yes	maximal, characteristic, complemented
8	#19134820	C₂	1	2	yes	maximal
9	#19134655	C₂	1	1	yes	maximal, characteristic
10	#19134432	C₂	1	1	yes	maximal, characteristic
11	C₄₃×#1844	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13063331	C₂²	1	1	yes	characteristic
13	#14583530	C₂²	1	2	yes	—
14	#14583380	C₂²	1	2	yes	—
15	#14583306	—	2	2	no	—
16	#14583299	—	2	1	no	—
17	#14583296	—	2	1	no	—
18	#14583209	—	2	2	no	—
19	#14583198	—	2	2	no	—
20	#14583188	—	2	2	no	—
21	#14583183	—	2	2	no	—
22	C₂×#13063261	C₂²	1	1	yes	characteristic
23	#14582939	C₂²	1	2	yes	—
24	#14582939	C₂²	1	2	yes	—
25	#14582939	C₂²	1	2	yes	—
26	#14582938	—	2	2	no	—
27	#14582937	C₂²	1	1	yes	characteristic
28	#14582934	—	2	1	no	—
29	#106×D₄₃	—	2	1	no	—
30	#14582925	C₂²	1	2	yes	—
31	C₂×#13063253	C₂²	1	1	yes	characteristic
32	#14582898	C₂²	1	2	yes	—
33	#14582896	C₂²	1	1	yes	characteristic
34	#14582854	C₂²	1	1	yes	characteristic
35	#14582668	C₂²	1	2	yes	—
36	#14582668	C₂²	1	2	yes	—
37	#14582664	C₂²	1	1	yes	characteristic
38	#14582659	C₂²	1	2	yes	—
39	#14582659	C₂²	1	1	yes	characteristic
40	C₂×#106×C₄₃	C₂²	1	2	yes	—
41	C₂×#106×C₄₃	C₂²	1	1	yes	characteristic
42	C₂×#106×C₄₃	C₂²	1	1	yes	characteristic
43	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
44	C₄₃×#384	C₂²	1	2	yes	—
45	C₂×#10621898	C₂³	1	1	yes	characteristic
46	#13063331	—	2	2	no	—
47	#13063331	Q₈	1	4	yes	complemented
48	C₂×#10621877	C₂³	1	2	yes	—
49	C₂×#10621877	C₂³	1	1	yes	characteristic
50	C₂×#10621877	C₂³	1	1	yes	characteristic
51	C₂×#10621872	—	2	1	no	—
52	C₂×C₈×D₄₃	—	2	1	no	—
53	#13063324	—	2	2	no	—
54	#13063324	—	2	2	no	—
55	#13063316	—	2	2	no	—
56	#13063316	—	2	2	no	—
57	#13063304	—	4	2	no	—
58	#13063290	—	2	2	no	—
59	#13063287	—	2	2	no	—
60	C₄×C₂²⋊D₄₃	—	4	2	no	—
61	#13063284	C₂³	1	1	yes	characteristic
62	#13063284	C₂³	1	2	yes	—
63	#13063283	—	2	2	no	—
64	C₄⋊C₄×D₄₃	—	2	2	no	—
65	#13063261	—	2	2	no	—
66	#13063259	D₄	1	2	yes	—
67	#13063258	—	2	2	no	—
68	#13063258	—	2	2	no	—
69	#13063258	—	2	2	no	—
70	#13063257	D₄	1	1	yes	characteristic
71	#13063257	D₄	1	1	yes	characteristic
72	#13063256	—	2	2	no	—
73	#13063256	—	2	2	no	—
74	#13063256	—	2	2	no	—
75	Dic₄₃⋊C₈	—	2	2	no	—
76	Dic₄₃⋊C₈	—	2	2	no	—
77	#13063253	Q₈	1	4	yes	complemented
78	#13063250	D₄	1	2	yes	—
79	#13063248	—	2	2	no	—
80	#13063247	D₄	1	2	yes	—
81	#13063245	—	2	2	no	—
82	#13063235	—	2	1	no	—
83	C₈×Dic₄₃	—	2	1	no	—
84	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
85	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
86	C₂×C₄⋊C₄×C₄₃	—	4	2	no	—
87	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
88	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
89	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
90	#106×C₄₃	—	2	1	no	—
91	#106×C₄₃	—	2	1	no	—
92	#106×C₄₃	—	2	2	no	—
93	#106×C₄₃	—	2	1	no	—
94	#106×C₄₃	—	2	2	no	—
95	#106×C₄₃	—	2	1	no	—
96	C₄×C₈×C₄₃	—	2	2	no	—
97	C₂²×C₄×C₄₃	—	2	1	no	—
98	C₂²×C₄×C₄₃	—	2	2	no	—
99	C₂²×C₄×C₄₃	—	2	2	no	—
100	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
101	C₂×C₂²⋊D₄₃	—	2	1	no	—
102	#10621898	—	2	1	no	—
103	#10621898	—	2	1	no	—
104	#10621898	C₂×Q₈	1	2	yes	—
105	#10621898	C₂×Q₈	1	2	yes	—
106	C₂×D₁₇₂	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×D₄₃	—	2	1	no	—
108	C₂×C₄×D₄₃	—	4	2	no	—
109	C₂×Dic₈₆	—	4	2	no	—
110	C₂×Dic₈₆	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₄₃	—	2	1	no	—
112	C₂×C₈×C₄₃	—	2	1	no	—
113	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
114	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
115	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₄₃	C₂×Q₈	1	2	yes	—
117	C₂×C₈×C₄₃	#41	1	2	yes	—
118	C₄⋊C₄×C₄₃	—	2	2	no	—
119	C₄⋊C₄×C₄₃	—	2	2	no	—
120	C₄⋊C₄×C₄₃	—	2	2	no	—
121	C₄⋊C₄×C₄₃	—	2	2	no	—
122	C₄⋊C₄×C₄₃	—	8	4	no	—
123	C₄²×C₄₃	—	2	1	no	—
124	C₄²×C₄₃	C₂×Q₈	1	2	yes	—
125	C₂²⋊Dic₄₃	—	2	2	no	—
126	C₂²⋊Dic₄₃	—	2	1	no	—
127	(C₂×D₄₃)⋊C₄	—	4	2	no	—
128	C₄⋊Dic₄₃	—	4	2	no	—
129	C₄⋊Dic₄₃	C₂×D₄	1	2	yes	—
130	C₄⋊Dic₄₃	C₂×D₄	1	2	yes	—
131	C₄⋊Dic₄₃	C₂×D₄	1	2	yes	—
132	Dic₄₃⋊C₄	—	4	2	no	—
133	Dic₄₃⋊C₄	—	4	2	no	—
134	Dic₄₃⋊C₄	—	4	2	no	—
135	C₄×Dic₄₃	—	2	1	no	—
136	C₄×Dic₄₃	—	2	2	no	—
137	C₄×Dic₄₃	—	4	2	no	—
138	#10621877	—	2	1	no	—
139	#10621877	—	2	1	no	—
140	#10621877	—	2	2	no	—
141	#10621877	—	2	2	no	—
142	#10621877	C₂×Q₈	1	2	yes	—
143	#10621877	C₂×Q₈	1	2	yes	—
144	C₂×C₄₃⋊C₈	—	2	1	no	—
145	C₂×C₄₃⋊C₈	—	2	1	no	—
146	C₂×C₄₃⋊C₈	—	2	2	no	—
147	#10621872	—	2	4	no	—
148	C₈×D₄₃	—	2	4	no	—
149	C₂³×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
150	C₂²×D₄₃	—	2	1	no	—
151	C₂×C₄×C₄₃	—	2	4	no	—
152	C₂×C₄×C₄₃	—	2	2	no	—
153	C₂×C₄×C₄₃	—	4	2	no	—
154	C₂×C₄×C₄₃	—	4	2	no	—
155	C₂×C₄×C₄₃	—	8	2	no	—
156	C₂×C₄×C₄₃	—	8	2	no	—
157	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
158	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
159	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
160	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
161	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
162	C₂×C₄×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
163	C₂²⋊D₄₃	—	2	2	no	—
164	C₂²⋊D₄₃	—	2	2	no	—
165	C₂×Dic₄₃	—	2	1	no	—
166	C₂×Dic₄₃	—	2	2	no	—
167	C₂×Dic₄₃	—	2	2	no	—
168	C₂×Dic₄₃	—	2	2	no	—
169	D₁₇₂	—	2	1	no	—
170	D₁₇₂	C₄⋊Q₈	1	2	yes	—
171	C₄×D₄₃	—	2	2	no	—
172	C₄×D₄₃	—	2	2	no	—
173	C₄×D₄₃	—	16	2	no	—
174	Dic₈₆	—	2	1	no	—
175	Dic₈₆	—	16	2	no	—
176	Dic₈₆	C₄⋊Q₈	1	2	yes	—
177	C₈×C₄₃	—	2	2	no	—
178	C₈×C₄₃	C₂²⋊Q₈	1	2	yes	—
179	C₈×C₄₃	C₂²⋊Q₈	1	2	yes	—
180	C₄₃⋊C₈	—	2	2	no	—
181	C₄₃⋊C₈	—	2	2	no	—
182	C₄₃⋊C₈	—	4	2	no	—
183	#26081	—	43	1	no	maximal, Sylow
184	C₂²×C₄₃	—	2	2	no	—
185	C₂²×C₄₃	#544	1	1	yes	characteristic
186	C₂²×C₄₃	D₄²	1	1	yes	characteristic
187	C₂²×C₄₃	#554	1	1	yes	characteristic
188	C₂×D₄₃	—	2	2	no	—
189	C₂×D₄₃	—	4	2	no	—
190	C₄×C₄₃	—	2	4	no	—
191	C₄×C₄₃	—	8	2	no	—
192	C₄×C₄₃	—	8	2	no	—
193	C₄×C₄₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
194	C₄×C₄₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
195	C₄×C₄₃	#547	1	1	yes	characteristic
196	C₄×C₄₃	D₄×Q₈	1	1	yes	characteristic
197	Dic₄₃	—	2	2	no	—
198	Dic₄₃	—	4	2	no	—
199	Dic₄₃	—	4	2	no	—
200	Dic₄₃	—	4	2	no	—
201	#3163	—	43	1	no	—
202	#3084	—	43	2	no	—
203	#2912	—	43	1	no	—
204	#2090	—	43	2	no	—
205	#2060	—	43	2	no	—
206	#1951	—	43	1	no	—
207	#1945	—	43	1	no	—
208	#1859	—	43	2	no	—
209	#1844	D₄₃	1	1	yes	characteristic, complemented
210	#1815	—	43	1	no	—
211	#1764	—	43	1	no	—
212	C₂×C₄₃	—	2	1	no	—
213	C₂×C₄₃	—	2	1	no	—
214	C₂×C₄₃	#2440	1	1	yes	characteristic
215	C₂×C₄₃	#3291	1	2	yes	—
216	D₄₃	—	4	2	no	—
217	C₂×#131	—	43	1	no	—
218	#544	—	43	2	no	—
219	#520	—	43	2	no	—
220	#501	—	86	2	no	—
221	#499	—	86	1	no	—
222	#498	—	86	1	no	—
223	#479	—	86	2	no	—
224	#478	—	86	2	no	—
225	#476	—	86	2	no	—
226	#475	—	86	2	no	—
227	C₂×#108	—	43	1	no	—
228	#428	—	43	2	no	—
229	#428	—	43	2	no	—
230	#428	—	43	2	no	—
231	#428	—	86	2	no	—
232	#427	—	43	1	no	—
233	#427	—	86	1	no	—
234	C₂×#106	—	86	1	no	—
235	C₂×#106	C₂×D₄₃	1	2	yes	—
236	C₂×#106	C₂×D₄₃	1	1	yes	characteristic
237	C₂×#106	C₂×D₄₃	1	1	yes	characteristic
238	#424	—	43	2	no	—
239	C₂×#104	—	43	1	no	—
240	#417	—	43	2	no	—
241	#416	—	43	1	no	—
242	#405	—	43	1	no	—
243	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
244	#384	C₂×D₄₃	1	2	yes	—
245	#343	—	43	2	no	—
246	#343	—	43	2	no	—
247	#341	—	43	1	no	—
248	#339	—	43	1	no	—
249	#339	—	43	2	no	—
250	C₄₃	#26081	1	1	yes	characteristic, complemented, Sylow
251	C₂×#41	—	43	1	no	—
252	#131	—	43	4	no	—
253	#131	—	86	2	no	—
254	C₂×M₁₆	—	43	1	no	—
255	C₂×M₁₆	—	43	1	no	—
256	C₂×M₁₆	—	43	2	no	—
257	C₂×M₁₆	—	86	1	no	—
258	C₂²×C₈	—	86	1	no	—
259	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
260	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
261	C₄⋊Q₈	—	86	2	no	—
262	C₄⋊Q₈	—	86	2	no	—
263	#125	—	86	2	no	—
264	#125	—	86	2	no	—
265	C₂²⋊Q₈	—	172	2	no	—
266	C₄×Q₈	—	86	2	no	—
267	C₄×D₄	—	86	2	no	—
268	C₄×D₄	—	172	2	no	—
269	#117	—	43	2	no	—
270	#117	—	43	1	no	—
271	#117	—	86	2	no	—
272	C₂×C₄⋊C₄	—	4	2	no	—
273	C₂×C₄⋊C₄	—	86	2	no	—
274	C₂×C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
275	C₂×C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
276	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
277	#108	—	86	2	no	—
278	#107	—	43	2	no	—
279	#107	—	86	2	no	—
280	#107	—	86	2	no	—
281	#107	—	86	2	no	—
282	#106	—	2	1	no	—
283	#106	—	2	1	no	—
284	#106	—	2	2	no	—
285	#106	—	2	2	no	—
286	#106	—	2	1	no	—
287	#106	—	2	1	no	—
288	#106	—	43	1	no	—
289	#106	—	43	1	no	—
290	#106	—	86	2	no	—
291	#106	—	86	2	no	—
292	#106	—	86	2	no	—
293	C₄⋊C₈	—	86	2	no	—
294	C₄⋊C₈	—	86	2	no	—
295	#104	—	43	4	no	—
296	#103	—	43	2	no	—
297	#103	—	86	2	no	—
298	#102	—	43	2	no	—
299	#102	—	86	2	no	—
300	#97	—	86	1	no	—
301	C₄×C₈	—	2	2	no	—
302	C₄×C₈	—	86	1	no	—
303	#41	—	43	2	no	—
304	#41	—	43	2	no	—
305	#41	—	86	1	no	—
306	#41	—	86	1	no	—
307	C₂×Q₈	—	43	1	no	—
308	C₂×Q₈	—	172	2	no	—
309	C₂×D₄	—	43	1	no	—
310	C₂×D₄	—	86	1	no	—
311	C₂²×C₄	—	2	1	no	—
312	C₂²×C₄	—	2	2	no	—
313	C₂²×C₄	—	2	2	no	—
314	C₂²×C₄	—	86	1	no	—
315	C₂²×C₄	—	172	2	no	—
316	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
317	M₁₆	—	43	2	no	—
318	M₁₆	—	43	2	no	—
319	M₁₆	—	86	1	no	—
320	M₁₆	—	86	1	no	—
321	M₁₆	—	86	4	no	—
322	M₁₆	—	86	2	no	—
323	M₁₆	—	86	2	no	—
324	C₂×C₈	—	2	1	no	—
325	C₂×C₈	—	2	1	no	—
326	C₂×C₈	—	86	1	no	—
327	C₂×C₈	—	86	1	no	—
328	C₂×C₈	—	86	4	no	—
329	C₂×C₈	—	86	2	no	—
330	C₂×C₈	#10621898	1	2	yes	—
331	C₂×C₈	D₄×D₄₃	1	2	yes	—
332	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
333	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
334	C₂×C₈	Q₈×D₄₃	1	2	yes	—
335	C₄⋊C₄	—	2	2	no	—
336	C₄⋊C₄	—	2	2	no	—
337	C₄⋊C₄	—	2	2	no	—
338	C₄⋊C₄	—	2	2	no	—
339	C₄⋊C₄	—	8	4	no	—
340	C₄⋊C₄	—	43	2	no	—
341	C₄⋊C₄	—	43	2	no	—
342	C₄⋊C₄	—	43	2	no	—
343	C₄⋊C₄	—	172	2	no	—
344	C₄⋊C₄	—	172	2	no	—
345	C₄⋊C₄	—	172	2	no	—
346	C₄⋊C₄	—	172	2	no	—
347	C₂²⋊C₄	—	86	2	no	—
348	C₂²⋊C₄	—	86	1	no	—
349	C₂²⋊C₄	—	172	2	no	—
350	C₄²	—	2	1	no	—
351	C₄²	—	86	1	no	—
352	C₄²	—	86	2	no	—
353	C₄²	—	172	2	no	—
354	C₄²	Q₈×D₄₃	1	2	yes	—
355	C₂³	—	86	1	no	—
356	C₂³	#13063370	1	1	yes	characteristic
357	Q₈	—	43	2	no	—
358	Q₈	—	86	1	no	—
359	Q₈	—	688	2	no	—
360	D₄	—	43	2	no	—
361	D₄	—	86	2	no	—
362	D₄	—	86	2	no	—
363	D₄	—	86	1	no	—
364	C₂×C₄	—	2	4	no	—
365	C₂×C₄	—	2	2	no	—
366	C₂×C₄	—	4	2	no	—
367	C₂×C₄	—	4	2	no	—
368	C₂×C₄	—	8	2	no	—
369	C₂×C₄	—	8	2	no	—
370	C₂×C₄	—	86	1	no	—
371	C₂×C₄	—	86	2	no	—
372	C₂×C₄	—	86	2	no	—
373	C₂×C₄	—	86	2	no	—
374	C₂×C₄	—	86	2	no	—
375	C₂×C₄	—	86	2	no	—
376	C₂×C₄	—	688	2	no	—
377	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
378	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
379	C₂×C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
380	C₂×C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
381	C₂×C₄	C₂×#10621898	1	1	yes	characteristic
382	C₂×C₄	#13063373	1	1	yes	characteristic
383	C₈	—	2	2	no	—
384	C₈	—	86	2	no	—
385	C₈	—	86	2	no	—
386	C₈	—	172	2	no	—
387	C₈	#13063303	1	2	yes	—
388	C₈	#13063303	1	2	yes	—
389	C₂²	—	2	2	no	—
390	C₂²	—	86	2	no	—
391	C₂²	—	172	2	no	—
392	C₂²	#14583526	1	1	yes	characteristic
393	C₂²	#14583534	1	1	yes	characteristic
394	C₂²	#14583604	1	1	yes	characteristic
395	C₄	—	2	4	no	—
396	C₄	—	8	2	no	—
397	C₄	—	8	2	no	—
398	C₄	—	86	2	no	—
399	C₄	—	172	2	no	—
400	C₄	—	172	2	no	—
401	C₄	—	172	2	no	—
402	C₄	C₂×#13063303	1	1	yes	characteristic
403	C₄	C₄⋊Q₈×D₄₃	1	1	yes	characteristic
404	C₄	#14583550	1	1	yes	characteristic
405	C₄	#14583564	1	1	yes	characteristic
406	C₂	—	2	1	no	—
407	C₂	—	2	1	no	—
408	C₂	—	172	2	no	—
409	C₂	#19138300	1	1	yes	characteristic
410	C₂	#19146388	1	2	yes	—

	subgroup	G/H
center	C₂²	#14583534
commutator, cosocle	C₂²×C₄×C₄₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄₃×#1844	C₂
socle	C₂²×C₄₃	D₄²
prime radical	C₂×#10621898	C₂³
hypercenter	#1844	D₄₃
hypocenter	C₄₃	#26081
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1844

Complements

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

N	H ≅ G/N	type
1. #19145123	C₂	semidirect
4. #19136106	C₂	semidirect
5. #19135940	C₂	semidirect
6. #19135373	C₂	semidirect
7. #19135343	C₂	semidirect
11. C₄₃×#1844	C₂	semidirect
47. #13063331	Q₈	semidirect
77. #13063253	Q₈	semidirect
209. #1844	D₄₃	semidirect
250. C₄₃	#26081	semidirect

Maximal subgroups

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

subgroup	index
#19145123	2
#19144462	2
#19143186	2
#19136106	2
#19135940	2
#19135373	2
#19135343	2
#19134820	2
#19134655	2
#19134432	2
C₄₃×#1844	2
#26081	43