Subgroups — #25013549

Order 11008.

530 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583141	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582936	C₂	1	1	yes	maximal, characteristic
3	C₂×#14582704	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14582702	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×(C₂×D₄₃)⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	#19132686	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#1380	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Dic₁₇₂	C₂²	1	1	yes	characteristic
9	C₂²×C₈×D₄₃	C₂²	1	1	yes	characteristic
10	C₂×#13063305	—	2	1	no	—
11	C₂×#13063283	—	2	1	no	—
12	C₂²×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
13	#14583141	—	2	2	no	—
14	#14583141	C₄	1	4	yes	complemented
15	#14582936	—	2	2	no	—
16	#14582936	—	2	2	no	—
17	C₂×#13063259	C₂²	1	1	yes	characteristic
18	C₂×#13063256	—	2	1	no	—
19	C₂×#13063248	—	2	1	no	—
20	#14582704	—	2	2	no	—
21	#14582702	—	2	2	no	—
22	(C₂×D₄₃)⋊C₁₆	—	2	2	no	—
23	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
24	C₂×#107×C₄₃	C₂²	1	1	yes	characteristic
25	C₂×#107×C₄₃	C₂²	1	1	yes	characteristic
26	C₂²×Dic₈₆	—	2	1	no	—
27	C₂²×C₄×D₄₃	D₄	1	1	yes	characteristic
28	C₂×Dic₁₇₂	—	2	1	no	—
29	C₂×Dic₁₇₂	—	2	1	no	—
30	C₂×Dic₁₇₂	—	2	1	no	—
31	C₂×Dic₁₇₂	—	2	1	no	—
32	C₂×Dic₁₇₂	—	2	1	no	—
33	C₂×Dic₁₇₂	C₂×C₄	1	2	yes	—
34	C₂²×C₄₃⋊C₈	D₄	1	1	yes	characteristic
35	C₂×C₈×D₄₃	—	2	1	no	—
36	C₂×C₈×D₄₃	—	2	1	no	—
37	C₂×C₈×D₄₃	—	2	1	no	—
38	C₂×C₈×D₄₃	—	2	1	no	—
39	C₂×C₈×D₄₃	C₂×C₄	1	2	yes	—
40	C₂×C₈×D₄₃	D₄	1	2	yes	—
41	#13063305	—	2	4	no	—
42	#13063305	—	4	2	no	—
43	#13063283	—	2	4	no	—
44	#13063283	—	4	2	no	—
45	C₂×C₄⋊Dic₄₃	—	2	1	no	—
46	C₂×C₄⋊Dic₄₃	—	4	1	no	—
47	C₂×(C₂×D₄₃)⋊C₄	—	4	1	no	—
48	C₂×(C₂×D₄₃)⋊C₄	—	4	1	no	—
49	C₂×C₄×Dic₄₃	—	4	1	no	—
50	C₂×C₄₃⋊C₁₆	—	2	1	no	—
51	C₂×C₄₃⋊C₁₆	—	2	1	no	—
52	C₂×C₄₃⋊C₁₆	—	2	1	no	—
53	#13063259	—	2	1	no	—
54	#13063259	D₄	1	2	yes	—
55	#13063256	—	2	4	no	—
56	#13063248	—	2	4	no	—
57	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
58	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
59	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
60	C₂×C₁₆×C₄₃	—	2	1	no	—
61	C₂×C₁₆×C₄₃	—	2	1	no	—
62	C₂×C₁₆×C₄₃	—	2	1	no	—
63	#107×C₄₃	—	2	1	no	—
64	#107×C₄₃	—	2	1	no	—
65	#107×C₄₃	C₂×C₄	1	2	yes	—
66	#107×C₄₃	D₄	1	2	yes	—
67	C₂³×D₄₃	D₈	1	1	yes	characteristic
68	C₂²×C₄×C₄₃	—	4	1	no	—
69	C₂²×C₄×C₄₃	—	4	1	no	—
70	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×Dic₄₃	—	4	1	no	—
72	C₂²×Dic₄₃	—	4	1	no	—
73	C₂²×Dic₄₃	D₈	1	1	yes	characteristic
74	C₂×C₄×D₄₃	—	2	1	no	—
75	C₂×C₄×D₄₃	—	2	1	no	—
76	C₂×C₄×D₄₃	—	2	1	no	—
77	C₂×C₄×D₄₃	—	2	1	no	—
78	C₂×C₄×D₄₃	C₂²⋊C₄	1	2	yes	—
79	C₂×C₄×D₄₃	SD₁₆	1	2	yes	—
80	C₂×Dic₈₆	—	2	2	no	—
81	C₂×Dic₈₆	—	4	1	no	—
82	C₂×Dic₈₆	—	4	1	no	—
83	C₂×Dic₈₆	—	4	1	no	—
84	C₂×Dic₈₆	—	4	1	no	—
85	C₂×Dic₈₆	—	4	1	no	—
86	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
91	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
92	C₄⋊C₄×C₄₃	—	2	2	no	—
93	C₄⋊C₄×C₄₃	—	2	2	no	—
94	C₄⋊C₄×C₄₃	—	4	1	no	—
95	C₄⋊C₄×C₄₃	—	4	1	no	—
96	(C₂×D₄₃)⋊C₄	—	4	4	no	—
97	(C₂×D₄₃)⋊C₄	—	4	4	no	—
98	C₄⋊Dic₄₃	—	2	2	no	—
99	C₄⋊Dic₄₃	—	4	1	no	—
100	C₄⋊Dic₄₃	—	4	4	no	—
101	C₄×Dic₄₃	—	4	4	no	—
102	C₂×C₄₃⋊C₈	—	2	1	no	—
103	C₂×C₄₃⋊C₈	—	2	1	no	—
104	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	2	yes	—
105	Dic₁₇₂	—	2	2	no	—
106	Dic₁₇₂	—	2	2	no	—
107	Dic₁₇₂	—	4	1	no	—
108	Dic₁₇₂	—	4	1	no	—
109	C₈×D₄₃	—	2	2	no	—
110	C₈×D₄₃	—	2	2	no	—
111	C₈×D₄₃	—	2	2	no	—
112	C₈×D₄₃	—	2	2	no	—
113	C₁₆×C₄₃	—	4	1	no	—
114	C₄₃⋊C₁₆	—	4	1	no	—
115	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
116	C₂²×D₄₃	—	2	1	no	—
117	C₂²×D₄₃	—	2	1	no	—
118	C₂²×D₄₃	—	2	1	no	—
119	C₂²×D₄₃	—	2	1	no	—
120	C₂²×D₄₃	—	2	1	no	—
121	C₂²×D₄₃	—	2	1	no	—
122	C₂²×D₄₃	#102	1	2	yes	—
123	C₂×C₄×C₄₃	—	4	2	no	—
124	C₂×C₄×C₄₃	—	4	2	no	—
125	C₂×C₄×C₄₃	—	8	1	no	—
126	C₂×C₄×C₄₃	—	8	1	no	—
127	C₂×C₄×C₄₃	—	8	1	no	—
128	C₂×C₄×C₄₃	—	8	1	no	—
129	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
131	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
132	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
133	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
134	C₂×C₄×C₄₃	#137	1	1	yes	characteristic
135	C₂×Dic₄₃	—	2	1	no	—
136	C₂×Dic₄₃	—	2	1	no	—
137	C₂×Dic₄₃	—	4	2	no	—
138	C₂×Dic₄₃	—	4	2	no	—
139	C₂×Dic₄₃	—	8	1	no	—
140	C₂×Dic₄₃	—	8	1	no	—
141	C₂×Dic₄₃	—	8	1	no	—
142	C₂×Dic₄₃	—	8	1	no	—
143	C₂×Dic₄₃	#102	1	2	yes	—
144	C₄×D₄₃	—	2	2	no	—
145	C₄×D₄₃	—	2	2	no	—
146	C₄×D₄₃	—	2	2	no	—
147	C₄×D₄₃	—	2	2	no	—
148	Dic₈₆	—	4	2	no	—
149	Dic₈₆	—	4	2	no	—
150	Dic₈₆	—	8	1	no	—
151	Dic₈₆	—	8	1	no	—
152	C₈×C₄₃	C₄×D₄	1	1	yes	characteristic
153	C₈×C₄₃	C₄×D₄	1	1	yes	characteristic
154	C₈×C₄₃	#121	1	1	yes	characteristic
155	C₈×C₄₃	#121	1	1	yes	characteristic
156	C₄₃⋊C₈	—	2	2	no	—
157	#12994	—	43	1	no	maximal, Sylow
158	C₂²×C₄₃	C₂×#102	1	1	yes	characteristic
159	C₂²×C₄₃	#447	1	1	yes	characteristic
160	C₂²×C₄₃	#480	1	1	yes	characteristic
161	C₂²×C₄₃	#508	1	1	yes	characteristic
162	C₂²×C₄₃	#508	1	1	yes	characteristic
163	C₂²×C₄₃	#510	1	1	yes	characteristic
164	C₂²×C₄₃	#510	1	1	yes	characteristic
165	C₂×D₄₃	—	2	2	no	—
166	C₂×D₄₃	—	2	2	no	—
167	C₂×D₄₃	—	2	2	no	—
168	C₂×D₄₃	—	4	1	no	—
169	C₂×D₄₃	—	4	1	no	—
170	C₂×D₄₃	—	4	1	no	—
171	C₂×D₄₃	—	4	1	no	—
172	C₄×C₄₃	—	8	2	no	—
173	C₄×C₄₃	—	8	2	no	—
174	C₄×C₄₃	#386	1	1	yes	characteristic
175	C₄×C₄₃	#417	1	1	yes	characteristic
176	C₄×C₄₃	#448	1	1	yes	characteristic
177	C₄×C₄₃	#481	1	1	yes	characteristic
178	Dic₄₃	—	2	2	no	—
179	Dic₄₃	—	8	2	no	—
180	Dic₄₃	—	8	2	no	—
181	C₂×#467	—	43	1	no	—
182	C₂×#427	—	43	1	no	—
183	C₂×#358	—	43	1	no	—
184	C₂×#358	—	43	1	no	—
185	C₂×C₂²⋊C₁₆	—	43	1	no	—
186	#1380	—	43	1	no	—
187	#1380	D₄₃	1	1	yes	characteristic, complemented
188	C₂×C₄₃	#1874	1	1	yes	characteristic
189	C₂×C₄₃	#2138	1	1	yes	characteristic
190	C₂×C₄₃	#2141	1	1	yes	characteristic
191	C₂×C₄₃	#2189	1	1	yes	characteristic
192	C₂×C₄₃	#2189	1	1	yes	characteristic
193	C₂×C₄₃	#2237	1	1	yes	characteristic
194	C₂×C₄₃	#2237	1	1	yes	characteristic
195	D₄₃	—	4	2	no	—
196	C₂²×Q₁₆	—	43	1	no	—
197	C₂³×C₈	—	43	1	no	—
198	C₂×C₂²⋊Q₈	—	86	1	no	—
199	C₂×#117	—	86	1	no	—
200	C₂²×C₁₆	—	43	1	no	—
201	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
202	#467	—	43	4	no	—
203	#467	—	86	2	no	—
204	#427	—	86	2	no	—
205	#427	—	86	2	no	—
206	C₂×#107	—	43	1	no	—
207	C₂×#107	C₂×D₄₃	1	1	yes	characteristic
208	C₂×#107	C₂×D₄₃	1	1	yes	characteristic
209	C₂×#106	—	86	1	no	—
210	C₂×#103	—	86	1	no	—
211	#358	—	86	2	no	—
212	#358	—	86	2	no	—
213	C₂²⋊C₁₆	—	86	2	no	—
214	C₄₃	#12994	1	1	yes	characteristic, complemented, Sylow
215	C₂²×Q₈	—	86	1	no	—
216	C₂³×C₄	—	43	1	no	—
217	C₂×Q₁₆	—	43	2	no	—
218	C₂×Q₁₆	—	86	1	no	—
219	C₂×Q₁₆	—	86	1	no	—
220	C₂×Q₁₆	—	86	1	no	—
221	C₂×Q₁₆	—	86	1	no	—
222	C₂×Q₁₆	—	86	1	no	—
223	C₂²×C₈	—	43	1	no	—
224	C₂²×C₈	—	43	2	no	—
225	C₂²×C₈	—	43	2	no	—
226	C₂²×C₈	—	86	1	no	—
227	C₂²×C₈	—	86	1	no	—
228	C₂²×C₈	—	86	1	no	—
229	C₂²×C₈	—	86	1	no	—
230	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
231	C₂²⋊Q₈	—	86	4	no	—
232	C₂²⋊Q₈	—	172	2	no	—
233	#117	—	86	4	no	—
234	#117	—	172	2	no	—
235	C₂×C₄⋊C₄	—	2	1	no	—
236	C₂×C₄⋊C₄	—	2	1	no	—
237	C₂×C₄⋊C₄	—	86	1	no	—
238	C₂×C₄⋊C₄	—	172	1	no	—
239	C₂×C₂²⋊C₄	—	172	1	no	—
240	C₂×C₂²⋊C₄	—	172	1	no	—
241	C₂×C₄²	—	172	1	no	—
242	C₂×C₁₆	—	2	1	no	—
243	C₂×C₁₆	—	2	1	no	—
244	C₂×C₁₆	—	2	1	no	—
245	C₂×C₁₆	—	86	1	no	—
246	C₂×C₁₆	—	86	1	no	—
247	C₂×C₁₆	—	86	1	no	—
248	#107	—	2	1	no	—
249	#107	—	2	1	no	—
250	#107	—	43	2	no	—
251	#107	—	86	1	no	—
252	#107	C₄×D₄₃	1	2	yes	—
253	#107	D₁₇₂	1	2	yes	—
254	#106	—	86	4	no	—
255	#103	—	86	4	no	—
256	C₂⁴	—	43	1	no	—
257	C₂×Q₈	—	86	2	no	—
258	C₂×Q₈	—	172	1	no	—
259	C₂×Q₈	—	172	1	no	—
260	C₂×Q₈	—	172	1	no	—
261	C₂×Q₈	—	172	1	no	—
262	C₂×Q₈	—	172	1	no	—
263	C₂²×C₄	—	4	1	no	—
264	C₂²×C₄	—	4	1	no	—
265	C₂²×C₄	—	43	1	no	—
266	C₂²×C₄	—	43	2	no	—
267	C₂²×C₄	—	43	2	no	—
268	C₂²×C₄	—	86	1	no	—
269	C₂²×C₄	—	86	1	no	—
270	C₂²×C₄	—	86	1	no	—
271	C₂²×C₄	—	86	1	no	—
272	C₂²×C₄	—	172	1	no	—
273	C₂²×C₄	—	172	1	no	—
274	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
275	Q₁₆	—	86	2	no	—
276	Q₁₆	—	86	2	no	—
277	Q₁₆	—	172	1	no	—
278	Q₁₆	—	172	1	no	—
279	C₂×C₈	—	43	2	no	—
280	C₂×C₈	—	86	1	no	—
281	C₂×C₈	—	86	1	no	—
282	C₂×C₈	—	86	2	no	—
283	C₂×C₈	—	86	2	no	—
284	C₂×C₈	—	86	2	no	—
285	C₂×C₈	—	86	2	no	—
286	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
287	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
288	C₂×C₈	#10621898	1	1	yes	characteristic
289	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
290	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
291	C₂×C₈	#10621902	1	1	yes	characteristic
292	C₄⋊C₄	—	2	2	no	—
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	—	4	1	no	—
295	C₄⋊C₄	—	4	1	no	—
296	C₄⋊C₄	—	86	2	no	—
297	C₄⋊C₄	—	172	4	no	—
298	C₄⋊C₄	—	172	1	no	—
299	C₂²⋊C₄	—	172	4	no	—
300	C₂²⋊C₄	—	172	4	no	—
301	C₄²	—	172	4	no	—
302	C₁₆	—	4	1	no	—
303	C₁₆	—	172	1	no	—
304	C₂³	—	43	2	no	—
305	C₂³	—	86	1	no	—
306	C₂³	—	86	1	no	—
307	C₂³	—	86	1	no	—
308	C₂³	—	86	1	no	—
309	C₂³	—	86	1	no	—
310	C₂³	—	86	1	no	—
311	C₂³	D₈×D₄₃	1	1	yes	characteristic
312	Q₈	—	344	1	no	—
313	Q₈	—	344	1	no	—
314	Q₈	—	172	2	no	—
315	Q₈	—	172	2	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	8	1	no	—
319	C₂×C₄	—	8	1	no	—
320	C₂×C₄	—	8	1	no	—
321	C₂×C₄	—	8	1	no	—
322	C₂×C₄	—	43	2	no	—
323	C₂×C₄	—	86	1	no	—
324	C₂×C₄	—	86	1	no	—
325	C₂×C₄	—	86	2	no	—
326	C₂×C₄	—	86	2	no	—
327	C₂×C₄	—	86	2	no	—
328	C₂×C₄	—	86	2	no	—
329	C₂×C₄	—	344	1	no	—
330	C₂×C₄	—	344	1	no	—
331	C₂×C₄	—	344	1	no	—
332	C₂×C₄	—	344	1	no	—
333	C₂×C₄	—	172	2	no	—
334	C₂×C₄	—	172	2	no	—
335	C₂×C₄	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
336	C₂×C₄	#13063295	1	1	yes	characteristic
337	C₂×C₄	#13063310	1	1	yes	characteristic
338	C₂×C₄	SD₁₆×D₄₃	1	1	yes	characteristic
339	C₂×C₄	#13063352	1	1	yes	characteristic
340	C₂×C₄	#13063357	1	1	yes	characteristic
341	C₈	—	86	2	no	—
342	C₈	#13063287	1	1	yes	characteristic
343	C₈	C₄×D₁₇₂	1	1	yes	characteristic
344	C₈	#13063298	1	1	yes	characteristic
345	C₈	#13063300	1	1	yes	characteristic
346	C₂²	—	86	2	no	—
347	C₂²	—	86	2	no	—
348	C₂²	—	86	2	no	—
349	C₂²	—	172	1	no	—
350	C₂²	—	172	1	no	—
351	C₂²	—	172	1	no	—
352	C₂²	—	172	1	no	—
353	C₂²	#102×D₄₃	1	1	yes	characteristic
354	C₂²	#14583034	1	1	yes	characteristic
355	C₂²	#14583214	1	1	yes	characteristic
356	C₂²	#14583328	1	1	yes	characteristic
357	C₂²	#14583330	1	1	yes	characteristic
358	C₂²	#14583338	1	1	yes	characteristic
359	C₂²	#14583341	1	1	yes	characteristic
360	C₄	—	8	2	no	—
361	C₄	—	8	2	no	—
362	C₄	—	86	2	no	—
363	C₄	—	344	2	no	—
364	C₄	—	344	2	no	—
365	C₄	#14582777	1	1	yes	characteristic
366	C₄	#14582899	1	1	yes	characteristic
367	C₄	#14583039	1	1	yes	characteristic
368	C₄	#14583219	1	1	yes	characteristic
369	C₂	—	172	2	no	—
370	C₂	#19134911	1	1	yes	characteristic
371	C₂	#19136299	1	1	yes	characteristic
372	C₂	#19136313	1	1	yes	characteristic
373	C₂	#19136536	1	1	yes	characteristic
374	C₂	#19136538	1	1	yes	characteristic
375	C₂	#19136805	1	1	yes	characteristic
376	C₂	#19136806	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₈×D₄₃
commutator	C₂×C₈×C₄₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₄₃×#1380	C₂
socle	C₂³×C₄₃	C₂×D₈
prime radical	C₂³×D₄₃	D₈
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	#1380	D₄₃
hypocenter	C₄₃	#12994
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1380

Complements

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

N	H ≅ G/N	type
3. C₂×#14582704	C₂	semidirect
4. C₂×#14582702	C₂	semidirect
6. #19132686	C₂	semidirect
7. C₄₃×#1380	C₂	semidirect
14. #14583141	C₄	semidirect
187. #1380	D₄₃	semidirect
214. C₄₃	#12994	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583141	2
C₂×#14582936	2
C₂×#14582704	2
C₂×#14582702	2
C₂×(C₂×D₄₃)⋊C₁₆	2
#19132686	2
C₄₃×#1380	2
#12994	43