Subgroups — #25013600

Order 11008.

508 conjugacy classes of subgroups.

The subgroup lattice has 406 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×#108×D₄₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#14582716	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14582714	C₂	1	1	yes	maximal, characteristic, complemented
5	#19136205	C₂	1	1	yes	maximal, characteristic
6	#19132703	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#1387	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Dic₁₇₂	C₂²	1	1	yes	characteristic
9	C₂×M₁₆×D₄₃	—	2	1	no	—
10	C₂²×C₈×D₄₃	C₂²	1	1	yes	characteristic
11	C₂×#13063305	—	2	1	no	—
12	C₂×#13063265	C₂²	1	1	yes	characteristic
13	#14583141	—	2	1	no	—
14	#14583141	—	2	1	no	—
15	#14583141	C₄	1	2	yes	—
16	#14583141	C₄	1	2	yes	—
17	C₂×#13063261	C₂²	1	1	yes	characteristic
18	C₂×#13063260	—	2	1	no	—
19	#108×D₄₃	—	2	2	no	—
20	#108×D₄₃	—	2	2	no	—
21	C₂×#13063248	—	2	1	no	—
22	#14582716	—	2	2	no	—
23	#14582714	—	2	2	no	—
24	(C₂×D₄₃)⋊C₁₆	—	2	1	no	—
25	(C₂×D₄₃)⋊C₁₆	—	2	1	no	—
26	C₂×M₃₂×C₄₃	C₂²	1	1	yes	characteristic
27	C₂×#108×C₄₃	C₂²	1	1	yes	characteristic
28	C₂×#107×C₄₃	C₂²	1	1	yes	characteristic
29	C₂²×Dic₈₆	—	2	1	no	—
30	C₂²×C₄×D₄₃	D₄	1	1	yes	characteristic
31	C₂×Dic₁₇₂	—	2	1	no	—
32	C₂×Dic₁₇₂	—	2	1	no	—
33	C₂×Dic₁₇₂	—	2	1	no	—
34	C₂×Dic₁₇₂	—	2	1	no	—
35	C₂×Dic₁₇₂	—	2	1	no	—
36	C₂×Dic₁₇₂	C₂×C₄	1	2	yes	—
37	C₂×#10621877	—	2	1	no	—
38	C₂×#10621872	—	4	1	no	—
39	M₁₆×D₄₃	—	2	4	no	—
40	M₁₆×D₄₃	—	4	2	no	—
41	C₂²×C₄₃⋊C₈	D₄	1	1	yes	characteristic
42	C₂×C₈×D₄₃	—	2	1	no	—
43	C₂×C₈×D₄₃	—	2	1	no	—
44	C₂×C₈×D₄₃	—	2	1	no	—
45	C₂×C₈×D₄₃	—	2	1	no	—
46	C₂×C₈×D₄₃	—	4	1	no	—
47	C₂×C₈×D₄₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₈×D₄₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₈×D₄₃	D₄	1	1	yes	characteristic
50	C₂×C₈×D₄₃	D₄	1	1	yes	characteristic
51	#13063305	—	2	2	no	—
52	#13063305	—	2	2	no	—
53	#13063305	—	4	1	no	—
54	#13063305	—	4	1	no	—
55	C₂×C₄⋊Dic₄₃	—	4	1	no	—
56	C₂×(C₂×D₄₃)⋊C₄	—	4	1	no	—
57	#13063265	—	2	1	no	—
58	#13063265	—	2	1	no	—
59	C₂×C₄₃⋊C₁₆	—	2	1	no	—
60	#13063261	—	2	1	no	—
61	#13063261	D₄	1	2	yes	—
62	#13063260	—	2	4	no	—
63	#13063248	—	2	2	no	—
64	#13063248	—	2	2	no	—
65	C₂×M₁₆×C₄₃	—	2	1	no	—
66	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
67	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
68	M₃₂×C₄₃	—	2	1	no	—
69	M₃₂×C₄₃	—	2	1	no	—
70	C₂×C₁₆×C₄₃	—	2	1	no	—
71	#108×C₄₃	—	2	1	no	—
72	#108×C₄₃	D₄	1	2	yes	—
73	#107×C₄₃	—	2	1	no	—
74	#107×C₄₃	C₂×C₄	1	2	yes	—
75	C₂³×D₄₃	SD₁₆	1	1	yes	characteristic
76	C₂²×C₄×C₄₃	—	4	1	no	—
77	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×Dic₄₃	—	4	1	no	—
79	C₂²×Dic₄₃	SD₁₆	1	1	yes	characteristic
80	C₂×C₄×D₄₃	—	2	1	no	—
81	C₂×C₄×D₄₃	—	2	1	no	—
82	C₂×C₄×D₄₃	—	2	1	no	—
83	C₂×C₄×D₄₃	—	2	1	no	—
84	C₂×C₄×D₄₃	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₄×D₄₃	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₄×D₄₃	D₈	1	1	yes	characteristic
87	C₂×C₄×D₄₃	D₈	1	1	yes	characteristic
88	C₂×Dic₈₆	—	2	2	no	—
89	C₂×Dic₈₆	—	4	1	no	—
90	C₂×Dic₈₆	—	4	1	no	—
91	C₂×Dic₈₆	—	4	1	no	—
92	C₂×Dic₈₆	—	4	1	no	—
93	C₂×Dic₈₆	—	4	1	no	—
94	M₁₆×C₄₃	—	2	2	no	—
95	M₁₆×C₄₃	—	4	1	no	—
96	C₂×C₈×C₄₃	—	4	1	no	—
97	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
102	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
103	C₄⋊C₄×C₄₃	—	2	2	no	—
104	C₄⋊C₄×C₄₃	—	4	1	no	—
105	(C₂×D₄₃)⋊C₄	—	4	2	no	—
106	(C₂×D₄₃)⋊C₄	—	4	2	no	—
107	C₄⋊Dic₄₃	—	4	2	no	—
108	C₄⋊Dic₄₃	—	4	2	no	—
109	#10621877	—	2	2	no	—
110	#10621877	—	4	1	no	—
111	C₂×C₄₃⋊C₈	—	2	1	no	—
112	C₂×C₄₃⋊C₈	—	2	1	no	—
113	C₂×C₄₃⋊C₈	—	4	1	no	—
114	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
116	Dic₁₇₂	—	2	2	no	—
117	Dic₁₇₂	—	2	2	no	—
118	Dic₁₇₂	—	4	1	no	—
119	Dic₁₇₂	—	4	1	no	—
120	#10621872	—	4	4	no	—
121	C₈×D₄₃	—	2	1	no	—
122	C₈×D₄₃	—	2	1	no	—
123	C₈×D₄₃	—	2	1	no	—
124	C₈×D₄₃	—	2	1	no	—
125	C₈×D₄₃	—	2	1	no	—
126	C₈×D₄₃	—	2	1	no	—
127	C₈×D₄₃	—	2	1	no	—
128	C₈×D₄₃	—	2	1	no	—
129	C₈×D₄₃	—	4	4	no	—
130	C₁₆×C₄₃	—	4	1	no	—
131	C₄₃⋊C₁₆	—	4	1	no	—
132	C₂³×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
133	C₂²×D₄₃	—	2	1	no	—
134	C₂²×D₄₃	—	2	1	no	—
135	C₂²×D₄₃	—	2	2	no	—
136	C₂²×D₄₃	—	4	1	no	—
137	C₂²×D₄₃	#102	1	1	yes	characteristic
138	C₂²×D₄₃	#102	1	1	yes	characteristic
139	C₂×C₄×C₄₃	—	4	2	no	—
140	C₂×C₄×C₄₃	—	8	2	no	—
141	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
142	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
143	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
144	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
145	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
146	C₂×C₄×C₄₃	#137	1	1	yes	characteristic
147	C₂×Dic₄₃	—	2	1	no	—
148	C₂×Dic₄₃	—	2	1	no	—
149	C₂×Dic₄₃	—	4	2	no	—
150	C₂×Dic₄₃	—	8	1	no	—
151	C₂×Dic₄₃	—	8	1	no	—
152	C₂×Dic₄₃	#102	1	1	yes	characteristic
153	C₂×Dic₄₃	#102	1	1	yes	characteristic
154	C₄×D₄₃	—	2	1	no	—
155	C₄×D₄₃	—	2	1	no	—
156	C₄×D₄₃	—	2	1	no	—
157	C₄×D₄₃	—	2	1	no	—
158	C₄×D₄₃	—	2	1	no	—
159	C₄×D₄₃	—	2	1	no	—
160	C₄×D₄₃	—	2	1	no	—
161	C₄×D₄₃	—	2	1	no	—
162	Dic₈₆	—	4	2	no	—
163	Dic₈₆	—	4	2	no	—
164	Dic₈₆	—	8	1	no	—
165	Dic₈₆	—	8	1	no	—
166	C₈×C₄₃	—	4	2	no	—
167	C₈×C₄₃	C₄×D₄	1	1	yes	characteristic
168	C₈×C₄₃	C₄×D₄	1	1	yes	characteristic
169	C₈×C₄₃	#121	1	1	yes	characteristic
170	C₈×C₄₃	#121	1	1	yes	characteristic
171	C₄₃⋊C₈	—	2	1	no	—
172	C₄₃⋊C₈	—	2	1	no	—
173	C₄₃⋊C₈	—	4	2	no	—
174	#13003	—	43	1	no	maximal, Sylow
175	C₂²×C₄₃	—	2	1	no	—
176	C₂²×C₄₃	—	2	1	no	—
177	C₂²×C₄₃	C₂×#102	1	1	yes	characteristic
178	C₂²×C₄₃	#448	1	1	yes	characteristic
179	C₂²×C₄₃	#481	1	1	yes	characteristic
180	C₂×D₄₃	—	2	1	no	—
181	C₂×D₄₃	—	2	1	no	—
182	C₂×D₄₃	—	2	1	no	—
183	C₂×D₄₃	—	2	1	no	—
184	C₂×D₄₃	—	2	1	no	—
185	C₂×D₄₃	—	4	2	no	—
186	C₂×D₄₃	—	4	2	no	—
187	C₂×D₄₃	#362	1	2	yes	complemented
188	C₄×C₄₃	—	8	2	no	—
189	C₄×C₄₃	#386	1	1	yes	characteristic
190	C₄×C₄₃	#417	1	1	yes	characteristic
191	C₄×C₄₃	#447	1	1	yes	characteristic
192	C₄×C₄₃	#480	1	1	yes	characteristic
193	Dic₄₃	—	2	1	no	—
194	Dic₄₃	—	2	1	no	—
195	Dic₄₃	—	8	2	no	—
196	C₂×#467	—	43	1	no	—
197	C₂²×#108	—	43	1	no	—
198	C₂×#362	—	43	1	no	—
199	C₂×#362	—	43	1	no	—
200	#2112	—	43	1	no	—
201	#1387	—	43	1	no	—
202	#1387	D₄₃	1	1	yes	characteristic, complemented
203	C₂×C₄₃	—	2	1	no	—
204	C₂×C₄₃	—	2	1	no	—
205	C₂×C₄₃	#1874	1	1	yes	characteristic
206	C₂×C₄₃	C₂×#362	1	1	yes	characteristic
207	C₂×C₄₃	#2148	1	1	yes	characteristic
208	D₄₃	—	2	2	no	—
209	D₄₃	—	4	1	no	—
210	C₂²×Q₁₆	—	43	1	no	—
211	C₂²×M₁₆	—	86	1	no	—
212	C₂³×C₈	—	43	1	no	—
213	C₂×C₂²⋊Q₈	—	86	1	no	—
214	C₂×M₃₂	—	43	1	no	—
215	C₂×M₃₂	C₂×D₄₃	1	1	yes	characteristic
216	#467	—	43	2	no	—
217	#467	—	43	2	no	—
218	#467	—	86	1	no	—
219	#467	—	86	1	no	—
220	C₂×#108	—	43	1	no	—
221	C₂×#108	—	86	2	no	—
222	C₂×#108	—	86	1	no	—
223	C₂×#108	—	86	2	no	—
224	C₂×#108	C₂×D₄₃	1	1	yes	characteristic
225	C₂×#107	C₂×D₄₃	1	1	yes	characteristic
226	C₂×#103	—	86	1	no	—
227	#362	—	86	2	no	—
228	#362	—	86	2	no	—
229	C₂²⋊C₁₆	—	86	1	no	—
230	C₂²⋊C₁₆	—	86	1	no	—
231	C₄₃	#13003	1	1	yes	characteristic, complemented, Sylow
232	C₂²×Q₈	—	86	1	no	—
233	C₂³×C₄	—	43	1	no	—
234	C₂×Q₁₆	—	43	2	no	—
235	C₂×Q₁₆	—	86	1	no	—
236	C₂×Q₁₆	—	86	1	no	—
237	C₂×Q₁₆	—	86	1	no	—
238	C₂×Q₁₆	—	86	1	no	—
239	C₂×Q₁₆	—	86	1	no	—
240	C₂×M₁₆	—	2	1	no	—
241	C₂×M₁₆	—	86	4	no	—
242	C₂×M₁₆	—	86	1	no	—
243	C₂×M₁₆	—	172	1	no	—
244	C₂×M₁₆	—	172	2	no	—
245	C₂²×C₈	—	43	1	no	—
246	C₂²×C₈	—	43	1	no	—
247	C₂²×C₈	—	43	1	no	—
248	C₂²×C₈	—	43	1	no	—
249	C₂²×C₈	—	43	1	no	—
250	C₂²×C₈	—	86	1	no	—
251	C₂²×C₈	—	86	1	no	—
252	C₂²×C₈	—	86	1	no	—
253	C₂²×C₈	—	86	1	no	—
254	C₂²×C₈	—	172	1	no	—
255	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
256	C₂²⋊Q₈	—	86	2	no	—
257	C₂²⋊Q₈	—	86	2	no	—
258	C₂²⋊Q₈	—	172	1	no	—
259	C₂²⋊Q₈	—	172	1	no	—
260	C₂×C₄⋊C₄	—	2	1	no	—
261	C₂×C₄⋊C₄	—	172	1	no	—
262	C₂×C₂²⋊C₄	—	172	1	no	—
263	M₃₂	—	2	1	no	—
264	M₃₂	—	2	1	no	—
265	M₃₂	—	86	1	no	—
266	M₃₂	—	86	1	no	—
267	C₂×C₁₆	—	2	1	no	—
268	C₂×C₁₆	—	86	1	no	—
269	#108	—	2	1	no	—
270	#108	—	43	2	no	—
271	#108	—	86	4	no	—
272	#108	—	86	1	no	—
273	#108	D₁₇₂	1	2	yes	—
274	#107	—	2	1	no	—
275	#107	C₄×D₄₃	1	2	yes	—
276	#103	—	86	2	no	—
277	#103	—	86	2	no	—
278	C₂⁴	—	43	1	no	—
279	C₂×Q₈	—	86	2	no	—
280	C₂×Q₈	—	172	1	no	—
281	C₂×Q₈	—	172	1	no	—
282	C₂×Q₈	—	172	1	no	—
283	C₂×Q₈	—	172	1	no	—
284	C₂×Q₈	—	172	1	no	—
285	C₂²×C₄	—	4	1	no	—
286	C₂²×C₄	—	43	1	no	—
287	C₂²×C₄	—	43	1	no	—
288	C₂²×C₄	—	43	1	no	—
289	C₂²×C₄	—	43	1	no	—
290	C₂²×C₄	—	43	1	no	—
291	C₂²×C₄	—	86	1	no	—
292	C₂²×C₄	—	86	1	no	—
293	C₂²×C₄	—	86	1	no	—
294	C₂²×C₄	—	86	1	no	—
295	C₂²×C₄	—	172	1	no	—
296	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
297	Q₁₆	—	86	2	no	—
298	Q₁₆	—	86	2	no	—
299	Q₁₆	—	172	1	no	—
300	Q₁₆	—	172	1	no	—
301	M₁₆	—	2	2	no	—
302	M₁₆	—	4	1	no	—
303	M₁₆	—	86	2	no	—
304	M₁₆	—	172	4	no	—
305	M₁₆	—	172	1	no	—
306	C₂×C₈	—	4	1	no	—
307	C₂×C₈	—	43	1	no	—
308	C₂×C₈	—	43	1	no	—
309	C₂×C₈	—	86	1	no	—
310	C₂×C₈	—	86	1	no	—
311	C₂×C₈	—	86	1	no	—
312	C₂×C₈	—	86	1	no	—
313	C₂×C₈	—	86	1	no	—
314	C₂×C₈	—	86	1	no	—
315	C₂×C₈	—	86	1	no	—
316	C₂×C₈	—	86	1	no	—
317	C₂×C₈	—	86	1	no	—
318	C₂×C₈	—	86	1	no	—
319	C₂×C₈	—	172	4	no	—
320	C₂×C₈	—	172	1	no	—
321	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
322	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
323	C₂×C₈	#10621898	1	1	yes	characteristic
324	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
325	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
326	C₂×C₈	#10621902	1	1	yes	characteristic
327	C₄⋊C₄	—	2	2	no	—
328	C₄⋊C₄	—	4	1	no	—
329	C₄⋊C₄	—	172	2	no	—
330	C₄⋊C₄	—	172	2	no	—
331	C₂²⋊C₄	—	172	2	no	—
332	C₂²⋊C₄	—	172	2	no	—
333	C₁₆	—	4	1	no	—
334	C₁₆	—	172	1	no	—
335	C₂³	—	43	1	no	—
336	C₂³	—	43	1	no	—
337	C₂³	—	86	2	no	—
338	C₂³	—	86	1	no	—
339	C₂³	—	86	1	no	—
340	C₂³	—	172	1	no	—
341	C₂³	SD₁₆×D₄₃	1	1	yes	characteristic
342	Q₈	—	344	1	no	—
343	Q₈	—	344	1	no	—
344	Q₈	—	172	2	no	—
345	Q₈	—	172	2	no	—
346	C₂×C₄	—	4	2	no	—
347	C₂×C₄	—	8	2	no	—
348	C₂×C₄	—	43	1	no	—
349	C₂×C₄	—	43	1	no	—
350	C₂×C₄	—	86	1	no	—
351	C₂×C₄	—	86	1	no	—
352	C₂×C₄	—	86	1	no	—
353	C₂×C₄	—	86	1	no	—
354	C₂×C₄	—	86	1	no	—
355	C₂×C₄	—	86	1	no	—
356	C₂×C₄	—	86	1	no	—
357	C₂×C₄	—	86	1	no	—
358	C₂×C₄	—	86	1	no	—
359	C₂×C₄	—	86	1	no	—
360	C₂×C₄	—	344	1	no	—
361	C₂×C₄	—	344	1	no	—
362	C₂×C₄	—	172	2	no	—
363	C₂×C₄	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
364	C₂×C₄	#13063295	1	1	yes	characteristic
365	C₂×C₄	#13063310	1	1	yes	characteristic
366	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
367	C₂×C₄	#13063352	1	1	yes	characteristic
368	C₂×C₄	#13063357	1	1	yes	characteristic
369	C₈	—	4	2	no	—
370	C₈	—	86	1	no	—
371	C₈	—	86	1	no	—
372	C₈	—	172	2	no	—
373	C₈	#13063287	1	1	yes	characteristic
374	C₈	C₄×D₁₇₂	1	1	yes	characteristic
375	C₈	#13063298	1	1	yes	characteristic
376	C₈	#13063300	1	1	yes	characteristic
377	C₂²	—	2	1	no	—
378	C₂²	—	2	1	no	—
379	C₂²	—	43	2	no	—
380	C₂²	—	86	1	no	—
381	C₂²	—	86	1	no	—
382	C₂²	—	86	1	no	—
383	C₂²	—	86	1	no	—
384	C₂²	—	86	1	no	—
385	C₂²	—	172	2	no	—
386	C₂²	—	172	2	no	—
387	C₂²	#102×D₄₃	1	1	yes	characteristic
388	C₂²	#14583039	1	1	yes	characteristic
389	C₂²	#14583219	1	1	yes	characteristic
390	C₄	—	8	2	no	—
391	C₄	—	86	1	no	—
392	C₄	—	86	1	no	—
393	C₄	—	344	2	no	—
394	C₄	#14582777	1	1	yes	characteristic
395	C₄	#14582899	1	1	yes	characteristic
396	C₄	#14583034	1	1	yes	characteristic
397	C₄	#14583214	1	1	yes	characteristic
398	C₂	—	2	1	no	—
399	C₂	—	2	1	no	—
400	C₂	—	86	2	no	—
401	C₂	—	172	1	no	—
402	C₂	#19134911	1	1	yes	characteristic
403	C₂	#362×D₄₃	1	1	yes	characteristic
404	C₂	#19136341	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#102×D₄₃
commutator	C₂×C₈×C₄₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₄₃×#1387	C₂
socle	C₂²×C₄₃	C₂×#102
prime radical	C₂³×D₄₃	SD₁₆
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	#1387	D₄₃
hypocenter	C₄₃	#13003
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1387

Complements

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

N	H ≅ G/N	type
3. C₂×#14582716	C₂	semidirect
4. C₂×#14582714	C₂	semidirect
6. #19132703	C₂	semidirect
7. C₄₃×#1387	C₂	semidirect
187. C₂×D₄₃	#362	semidirect
202. #1387	D₄₃	semidirect
231. C₄₃	#13003	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×#108×D₄₃	2
C₂×#14582716	2
C₂×#14582714	2
#19136205	2
#19132703	2
C₄₃×#1387	2
#13003	43