Subgroups — #25105209

Order 11008.

1046 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19145021	C₂	1	1	yes	maximal, characteristic
2	C₂×#14583014	C₂	1	4	yes	maximal, complemented
3	#19143308	C₂	1	1	yes	maximal, characteristic, complemented
4	#19135482	C₂	1	4	yes	maximal, complemented
5	#19135424	C₂	1	1	yes	maximal, characteristic, complemented
6	#19135422	C₂	1	1	yes	maximal, characteristic, complemented
7	#19134766	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×C₈×Dic₄₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₃×#2931	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13063345	C₂²	1	4	yes	—
11	C₂×#13063331	C₂²	1	2	yes	—
12	#14583449	C₂²	1	1	yes	characteristic
13	#14583281	C₄	1	4	yes	complemented
14	#14583278	C₄	1	1	yes	characteristic, complemented
15	#14583275	C₄	1	1	yes	characteristic, complemented
16	#14583273	C₄	1	1	yes	characteristic, complemented
17	#14583271	C₄	1	1	yes	characteristic, complemented
18	#14583014	—	2	16	no	—
19	#14582978	—	2	2	no	—
20	C₈×D₁₇₂	—	2	2	no	—
21	C₂×#13063260	C₂²	1	4	yes	—
22	#14582923	C₂²	1	1	yes	characteristic
23	C₂×#13063252	C₂²	1	4	yes	—
24	C₂×#13063251	C₂²	1	4	yes	—
25	#14582865	C₂²	1	2	yes	—
26	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
27	C₂×C₈×Dic₄₃	C₂²	1	2	yes	—
28	C₂×C₈×Dic₄₃	C₂²	1	2	yes	—
29	C₄²×Dic₄₃	C₂²	1	1	yes	characteristic
30	C₂×#131×C₄₃	C₂²	1	2	yes	—
31	C₄₃×#532	C₂²	1	1	yes	characteristic
32	C₄₃×#436	—	2	2	no	—
33	C₈×D₄×C₄₃	—	2	2	no	—
34	C₄₃×#423	C₂²	1	1	yes	characteristic
35	C₄₃×#408	C₂²	1	2	yes	—
36	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
37	C₂×#10621898	C₂³	1	2	yes	—
38	#13063345	—	2	8	no	—
39	#13063345	—	2	8	no	—
40	C₂×#10621885	C₂×C₄	1	4	yes	—
41	C₂×#10621883	C₂×C₄	1	4	yes	—
42	C₂×#10621884	C₂×C₄	1	4	yes	—
43	C₂×#10621882	C₂×C₄	1	4	yes	—
44	#13063331	—	2	8	no	—
45	C₂×#10621877	C₂³	1	2	yes	—
46	C₂×#10621872	—	2	2	no	—
47	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
48	C₂×C₈×D₄₃	—	2	2	no	—
49	#13063325	C₂×C₄	1	1	yes	characteristic
50	#13063322	C₂×C₄	1	1	yes	characteristic
51	#13063313	C₂×C₄	1	2	yes	—
52	#13063299	—	2	2	no	—
53	C₄×D₁₇₂	—	2	2	no	—
54	C₂×C₄×Dic₄₃	—	2	2	no	—
55	C₂×C₄×Dic₄₃	C₂³	1	2	yes	—
56	#13063260	—	2	8	no	—
57	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
58	#13063252	—	2	8	no	—
59	#13063251	—	2	8	no	—
60	(C₂×D₄₃)⋊C₈	—	2	4	no	—
61	C₄×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
62	C₄×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
63	C₄×C₄₃⋊C₈	C₂×C₄	1	2	yes	—
64	C₈×Dic₄₃	—	2	4	no	—
65	C₈×Dic₄₃	D₄	1	8	yes	complemented
66	C₂×#41×C₄₃	C₂³	1	2	yes	—
67	#131×C₄₃	—	2	8	no	—
68	C₂×M₁₆×C₄₃	—	2	2	no	—
69	C₂×M₁₆×C₄₃	C₂³	1	2	yes	—
70	C₂²×C₈×C₄₃	—	2	2	no	—
71	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
72	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
73	C₄⋊Q₈×C₄₃	C₂×C₄	1	1	yes	characteristic
74	#127×C₄₃	C₂×C₄	1	1	yes	characteristic
75	#124×C₄₃	C₂×C₄	1	2	yes	—
76	#121×C₄₃	—	2	2	no	—
77	C₄×D₄×C₄₃	—	2	2	no	—
78	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
79	C₄⋊C₈×C₄₃	—	2	2	no	—
80	C₂²⋊C₈×C₄₃	—	2	4	no	—
81	C₄×C₈×C₄₃	D₄	1	2	yes	—
82	C₄×C₈×C₄₃	D₄	1	2	yes	—
83	#41×C₄₃	—	2	4	no	—
84	#41×C₄₃	—	2	4	no	—
85	C₂×Q₈×C₄₃	C₂²×C₄	1	2	yes	—
86	C₂×D₄×C₄₃	—	2	2	no	—
87	C₂×D₄×C₄₃	—	2	2	no	—
88	C₂×D₄×C₄₃	C₂²×C₄	1	2	yes	—
89	C₂²×C₄×C₄₃	—	2	2	no	—
90	C₂²×C₄×C₄₃	#41	1	2	yes	—
91	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
92	C₂×C₂²⋊D₄₃	—	2	2	no	—
93	C₂²×Dic₄₃	—	2	2	no	—
94	#10621898	—	2	4	no	—
95	#10621898	—	2	4	no	—
96	C₂×D₁₇₂	—	2	2	no	—
97	C₂×D₁₇₂	C₂²×C₄	1	2	yes	—
98	C₂×C₄×D₄₃	—	2	2	no	—
99	C₂×Dic₈₆	C₂²×C₄	1	2	yes	—
100	M₁₆×C₄₃	—	2	4	no	—
101	M₁₆×C₄₃	—	4	4	no	—
102	C₂×C₈×C₄₃	—	2	2	no	—
103	C₂×C₈×C₄₃	—	2	2	no	—
104	C₂×C₈×C₄₃	—	4	4	no	—
105	C₂×C₈×C₄₃	C₂×D₄	1	4	yes	—
106	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
107	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
109	C₄⋊C₄×C₄₃	—	2	2	no	—
110	C₂²⋊C₄×C₄₃	—	2	4	no	—
111	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
112	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
113	C₄²×C₄₃	C₂×D₄	1	2	yes	—
114	#10621885	—	2	8	no	—
115	#10621884	—	2	8	no	—
116	#10621883	—	2	8	no	—
117	#10621882	—	2	8	no	—
118	(C₂×D₄₃)⋊C₄	—	2	4	no	—
119	C₄⋊Dic₄₃	—	2	2	no	—
120	C₄×Dic₄₃	—	2	8	no	—
121	C₄×Dic₄₃	—	2	2	no	—
122	C₄×Dic₄₃	C₂×D₄	1	4	yes	—
123	#10621877	—	2	4	no	—
124	C₂×C₄₃⋊C₈	—	2	2	no	—
125	C₂×C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
126	C₂×C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
127	C₂×C₄₃⋊C₈	C₂×D₄	1	4	yes	—
128	C₂×C₄₃⋊C₈	#41	1	4	yes	—
129	#10621872	—	4	4	no	—
130	C₈×D₄₃	—	4	4	no	—
131	C₂³×C₄₃	—	2	2	no	—
132	C₂³×C₄₃	C₂×#41	1	1	yes	characteristic
133	C₂²×D₄₃	—	2	2	no	—
134	Q₈×C₄₃	—	2	4	no	—
135	D₄×C₄₃	—	2	4	no	—
136	D₄×C₄₃	—	4	4	no	—
137	D₄×C₄₃	—	4	4	no	—
138	C₂×C₄×C₄₃	—	2	4	no	—
139	C₂×C₄×C₄₃	—	2	2	no	—
140	C₂×C₄×C₄₃	—	4	2	no	—
141	C₂×C₄×C₄₃	—	4	2	no	—
142	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
143	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
144	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
145	C₂×C₄×C₄₃	C₂²×D₄	1	2	yes	—
146	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
148	C₂²⋊D₄₃	—	4	4	no	—
149	C₂×Dic₄₃	—	2	8	no	—
150	C₂×Dic₄₃	—	2	4	no	—
151	C₂×Dic₄₃	—	2	2	no	—
152	D₁₇₂	—	2	4	no	—
153	D₁₇₂	—	4	4	no	—
154	C₄×D₄₃	—	4	2	no	—
155	C₄×D₄₃	—	4	2	no	—
156	Dic₈₆	—	2	4	no	—
157	C₈×C₄₃	—	2	2	no	—
158	C₈×C₄₃	—	4	2	no	—
159	C₈×C₄₃	#127	1	2	yes	—
160	C₈×C₄₃	#127	1	2	yes	—
161	C₄₃⋊C₈	—	4	2	no	—
162	C₄₃⋊C₈	C₄×D₄	1	4	yes	—
163	C₄₃⋊C₈	C₄×D₄	1	4	yes	—
164	#21705	—	43	1	no	maximal, Sylow
165	C₂²×C₄₃	—	2	2	no	—
166	C₂²×C₄₃	—	4	2	no	—
167	C₂²×C₄₃	—	4	4	no	—
168	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
169	C₂²×C₄₃	#532	1	2	yes	—
170	C₂×D₄₃	—	4	2	no	—
171	C₂×D₄₃	—	4	4	no	—
172	C₄×C₄₃	—	2	2	no	—
173	C₄×C₄₃	—	4	2	no	—
174	C₄×C₄₃	#443	1	4	yes	complemented
175	C₄×C₄₃	C₂×C₄×D₄	1	2	yes	—
176	C₄×C₄₃	C₂×#127	1	1	yes	characteristic
177	C₄×C₄₃	#532	1	1	yes	characteristic
178	Dic₄₃	—	2	8	no	—
179	Dic₄₃	—	4	2	no	—
180	#3147	—	43	1	no	—
181	C₂×#443	—	43	4	no	—
182	#2931	—	43	1	no	—
183	#2931	D₄₃	1	1	yes	characteristic, complemented
184	#1974	—	43	4	no	—
185	#1962	—	43	1	no	—
186	#1962	—	43	1	no	—
187	#1848	—	43	1	no	—
188	C₄²×C₈	—	43	1	no	—
189	C₂×C₄₃	—	2	2	no	—
190	C₂×C₄₃	—	8	2	no	—
191	C₂×C₄₃	C₄×#127	1	1	yes	characteristic
192	C₂×C₄₃	C₂×#443	1	2	yes	—
193	D₄₃	—	8	2	no	—
194	C₂×#135	—	43	4	no	—
195	C₂×#131	—	43	2	no	—
196	C₂×#131	C₂×D₄₃	1	2	yes	—
197	#532	—	43	1	no	—
198	#532	C₂×D₄₃	1	1	yes	characteristic
199	#495	—	43	4	no	—
200	#494	—	43	1	no	—
201	#493	—	43	1	no	—
202	#492	—	43	1	no	—
203	#492	—	43	1	no	—
204	#443	—	86	16	no	—
205	#436	—	2	2	no	—
206	#436	—	86	2	no	—
207	C₈×D₄	—	2	2	no	—
208	C₈×D₄	—	86	2	no	—
209	C₂×#108	—	43	4	no	—
210	#423	—	43	1	no	—
211	#423	C₂×D₄₃	1	1	yes	characteristic
212	C₂×#104	—	43	4	no	—
213	C₂×#104	—	43	4	no	—
214	#408	—	43	2	no	—
215	#408	C₂×D₄₃	1	2	yes	—
216	C₂×C₄×C₈	—	43	2	no	—
217	C₂×C₄×C₈	—	43	1	no	—
218	C₂×C₄×C₈	—	43	2	no	—
219	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
220	C₄³	—	43	1	no	—
221	C₄₃	#21705	1	1	yes	characteristic, complemented, Sylow
222	C₂×#41	—	43	2	no	—
223	C₂×#41	C₂²×D₄₃	1	2	yes	—
224	#135	—	86	8	no	—
225	#135	—	86	8	no	—
226	C₂×Q₁₆	—	43	4	no	—
227	C₂×SD₁₆	—	43	4	no	—
228	C₂×SD₁₆	—	43	4	no	—
229	C₂×D₈	—	43	4	no	—
230	#131	—	2	8	no	—
231	#131	—	86	8	no	—
232	C₂×M₁₆	—	2	2	no	—
233	C₂×M₁₆	—	43	2	no	—
234	C₂×M₁₆	—	86	2	no	—
235	C₂×M₁₆	C₂²×D₄₃	1	2	yes	—
236	C₂²×C₈	—	2	2	no	—
237	C₂²×C₈	—	43	2	no	—
238	C₂²×C₈	—	86	2	no	—
239	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
240	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
241	C₄⋊Q₈	—	43	1	no	—
242	C₄⋊Q₈	C₄×D₄₃	1	1	yes	characteristic
243	#127	—	43	1	no	—
244	#127	C₄×D₄₃	1	1	yes	characteristic
245	#124	—	43	2	no	—
246	#124	C₄×D₄₃	1	2	yes	—
247	#121	—	2	2	no	—
248	#121	—	86	2	no	—
249	C₄×D₄	—	2	2	no	—
250	C₄×D₄	—	86	2	no	—
251	C₂×C₄²	—	43	2	no	—
252	C₂×C₄²	—	86	2	no	—
253	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
254	#108	—	86	8	no	—
255	C₄⋊C₈	—	2	2	no	—
256	C₄⋊C₈	—	86	2	no	—
257	#104	—	86	8	no	—
258	#104	—	86	8	no	—
259	C₂²⋊C₈	—	2	4	no	—
260	C₂²⋊C₈	—	86	4	no	—
261	C₄×C₈	—	43	8	no	—
262	C₄×C₈	—	43	1	no	—
263	C₄×C₈	—	43	1	no	—
264	C₄×C₈	—	43	2	no	—
265	C₄×C₈	—	86	4	no	—
266	C₄×C₈	C₂²⋊D₄₃	1	2	yes	—
267	C₄×C₈	C₂²⋊D₄₃	1	2	yes	—
268	#41	—	2	4	no	—
269	#41	—	2	4	no	—
270	#41	—	86	4	no	—
271	#41	—	86	4	no	—
272	C₂×Q₈	—	43	2	no	—
273	C₂×Q₈	C₂×C₄×D₄₃	1	2	yes	—
274	C₂×D₄	—	2	2	no	—
275	C₂×D₄	—	2	2	no	—
276	C₂×D₄	—	43	2	no	—
277	C₂×D₄	—	86	2	no	—
278	C₂×D₄	—	86	2	no	—
279	C₂×D₄	C₂×C₄×D₄₃	1	2	yes	—
280	C₂²×C₄	—	2	2	no	—
281	C₂²×C₄	—	86	2	no	—
282	C₂²×C₄	—	86	2	no	—
283	C₂²×C₄	#10621898	1	2	yes	—
284	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
285	Q₁₆	—	86	8	no	—
286	SD₁₆	—	86	8	no	—
287	SD₁₆	—	86	8	no	—
288	D₈	—	86	8	no	—
289	M₁₆	—	2	4	no	—
290	M₁₆	—	4	4	no	—
291	M₁₆	—	86	4	no	—
292	M₁₆	—	172	4	no	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	—	2	2	no	—
295	C₂×C₈	—	4	4	no	—
296	C₂×C₈	—	43	2	no	—
297	C₂×C₈	—	43	2	no	—
298	C₂×C₈	—	43	4	no	—
299	C₂×C₈	—	43	4	no	—
300	C₂×C₈	—	86	2	no	—
301	C₂×C₈	—	172	4	no	—
302	C₂×C₈	D₄×D₄₃	1	4	yes	—
303	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
304	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
305	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
306	C₄⋊C₄	—	2	2	no	—
307	C₄⋊C₄	—	86	2	no	—
308	C₂²⋊C₄	—	2	4	no	—
309	C₂²⋊C₄	—	86	4	no	—
310	C₄²	—	43	4	no	—
311	C₄²	—	86	8	no	—
312	C₄²	—	86	2	no	—
313	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
314	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
315	C₄²	C₂×C₂²⋊D₄₃	1	2	yes	—
316	C₂³	—	2	2	no	—
317	C₂³	—	86	2	no	—
318	C₂³	C₂×#10621898	1	1	yes	characteristic
319	Q₈	—	2	4	no	—
320	Q₈	—	86	4	no	—
321	D₄	—	2	4	no	—
322	D₄	—	4	4	no	—
323	D₄	—	4	4	no	—
324	D₄	—	86	4	no	—
325	D₄	—	172	4	no	—
326	D₄	—	172	4	no	—
327	C₂×C₄	—	2	4	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	4	2	no	—
330	C₂×C₄	—	4	2	no	—
331	C₂×C₄	—	86	8	no	—
332	C₂×C₄	—	86	4	no	—
333	C₂×C₄	—	86	2	no	—
334	C₂×C₄	—	172	2	no	—
335	C₂×C₄	—	172	2	no	—
336	C₂×C₄	C₄×C₂²⋊D₄₃	1	2	yes	—
337	C₂×C₄	C₄×C₂²⋊D₄₃	1	2	yes	—
338	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
339	C₂×C₄	C₂×D₄×D₄₃	1	2	yes	—
340	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
341	C₂×C₄	#41×D₄₃	1	2	yes	—
342	C₈	—	2	2	no	—
343	C₈	—	4	2	no	—
344	C₈	—	43	4	no	—
345	C₈	—	43	4	no	—
346	C₈	—	172	2	no	—
347	C₈	#13063321	1	2	yes	—
348	C₈	#13063321	1	2	yes	—
349	C₂²	—	2	2	no	—
350	C₂²	—	4	4	no	—
351	C₂²	—	4	2	no	—
352	C₂²	—	172	4	no	—
353	C₂²	—	172	2	no	—
354	C₂²	C₂×C₄×C₂²⋊D₄₃	1	1	yes	characteristic
355	C₂²	#14583446	1	2	yes	—
356	C₄	—	2	2	no	—
357	C₄	—	4	2	no	—
358	C₄	—	86	8	no	—
359	C₄	—	172	2	no	—
360	C₄	#14583014	1	4	yes	complemented
361	C₄	C₄×D₄×D₄₃	1	2	yes	—
362	C₄	C₂×#13063321	1	1	yes	characteristic
363	C₄	#14583445	1	1	yes	characteristic
364	C₂	—	2	2	no	—
365	C₂	—	8	2	no	—
366	C₂	—	344	2	no	—
367	C₂	C₄×#13063321	1	1	yes	characteristic
368	C₂	C₂×#14583014	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄₃×#2931	C₂
socle	C₂²×C₄₃	C₂×C₄×D₄
prime radical	#19145021	C₂
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#2931	D₄₃
hypocenter	C₄₃	#21705
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2931

Complements

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

N	H ≅ G/N	type
2. C₂×#14583014	C₂	semidirect
3. #19143308	C₂	semidirect
4. #19135482	C₂	semidirect
5. #19135424	C₂	semidirect
6. #19135422	C₂	semidirect
7. #19134766	C₂	semidirect
8. C₄×C₈×Dic₄₃	C₂	semidirect
9. C₄₃×#2931	C₂	semidirect
13. #14583281	C₄	semidirect
14. #14583278	C₄	semidirect
15. #14583275	C₄	semidirect
16. #14583273	C₄	semidirect
17. #14583271	C₄	semidirect
65. C₈×Dic₄₃	D₄	semidirect
174. C₄×C₄₃	#443	semidirect
183. #2931	D₄₃	semidirect
221. C₄₃	#21705	semidirect
360. C₄	#14583014	semidirect

Maximal subgroups

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

subgroup	index
#19145021	2
C₂×#14583014	2
#19143308	2
#19135482	2
#19135424	2
#19135422	2
#19134766	2
C₄×C₈×Dic₄₃	2
C₄₃×#2931	2
#21705	43