Subgroups — #25013493

Order 11008.

488 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13063264	C₂	1	1	yes	maximal, characteristic, complemented
2	#19143287	C₂	1	1	yes	maximal, characteristic
3	C₂×#14582683	C₂	1	1	yes	maximal, characteristic, complemented
4	#19136205	C₂	1	1	yes	maximal, characteristic, complemented
5	#19132685	C₂	1	1	yes	maximal, characteristic, complemented
6	#19132683	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄₃×#350	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₄₃	C₂²	1	1	yes	characteristic
9	#14583406	C₄	1	1	yes	characteristic
10	C₂×#13063265	C₂²	1	1	yes	characteristic
11	C₂×#13063264	—	2	2	no	—
12	C₂×#13063264	—	2	2	no	—
13	C₂×#13063264	—	2	2	no	—
14	C₂²×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
15	#14582968	—	2	2	no	—
16	C₈×C₂²⋊D₄₃	—	2	2	no	—
17	#14582927	C₄	1	1	yes	characteristic
18	#14582866	C₄	1	1	yes	characteristic
19	#14582862	C₄	1	1	yes	characteristic
20	#14582854	C₂²	1	1	yes	characteristic
21	#14582683	—	2	2	no	—
22	(C₂×D₄₃)⋊C₁₆	—	2	2	no	—
23	C₂×#131×C₄₃	C₂²	1	1	yes	characteristic
24	C₂×M₃₂×C₄₃	C₂²	1	1	yes	characteristic
25	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
26	C₄₃×#350	—	2	2	no	—
27	C₂²×C₄×D₄₃	C₂×C₄	1	1	yes	characteristic
28	C₂×#10621877	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×D₄₃	—	2	1	no	—
31	C₂×C₈×D₄₃	—	2	1	no	—
32	C₂×C₈×D₄₃	—	2	1	no	—
33	C₂×C₈×D₄₃	—	2	1	no	—
34	C₂×C₈×D₄₃	D₄	1	2	yes	—
35	C₂×C₈×D₄₃	D₄	1	2	yes	—
36	#13063306	—	2	1	no	—
37	#13063301	C₈	1	2	yes	—
38	#13063296	C₈	1	2	yes	—
39	#13063293	—	2	1	no	—
40	C₄×C₂²⋊D₄₃	—	2	2	no	—
41	#13063284	C₂×C₄	1	1	yes	characteristic
42	#13063265	—	2	1	no	—
43	#13063265	—	2	1	no	—
44	#13063264	—	2	8	no	—
45	C₂×C₄₃⋊C₁₆	—	2	1	no	—
46	C₂×C₄₃⋊C₁₆	—	2	1	no	—
47	C₂×C₄₃⋊C₁₆	—	2	1	no	—
48	C₂×C₄₃⋊C₁₆	D₄	1	2	yes	complemented
49	Dic₄₃⋊C₈	—	2	2	no	—
50	C₂²⋊(C₄₃⋊C₈)	—	2	2	no	—
51	(C₂×D₄₃)⋊C₈	—	2	2	no	—
52	#13063235	—	2	1	no	—
53	C₈×Dic₄₃	—	2	1	no	—
54	C₂×#41×C₄₃	C₂×C₄	1	1	yes	characteristic
55	#131×C₄₃	—	2	1	no	—
56	#131×C₄₃	—	2	1	no	—
57	#131×C₄₃	—	2	1	no	—
58	#131×C₄₃	—	2	1	no	—
59	C₂×M₁₆×C₄₃	—	2	1	no	—
60	C₂×M₁₆×C₄₃	C₂×C₄	1	1	yes	characteristic
61	C₂²×C₈×C₄₃	—	2	1	no	—
62	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
63	M₃₂×C₄₃	—	2	1	no	—
64	M₃₂×C₄₃	—	2	1	no	—
65	C₂×C₁₆×C₄₃	—	2	1	no	—
66	C₂×C₁₆×C₄₃	—	2	1	no	—
67	C₂×C₁₆×C₄₃	—	2	1	no	—
68	C₂×C₁₆×C₄₃	D₄	1	2	yes	complemented
69	C₂³×D₄₃	M₁₆	1	1	yes	characteristic
70	#41×C₄₃	—	2	1	no	—
71	#41×C₄₃	—	2	1	no	—
72	#41×C₄₃	—	2	1	no	—
73	#41×C₄₃	—	2	1	no	—
74	C₂×Q₈×C₄₃	C₂×C₈	1	1	yes	characteristic
75	C₂×D₄×C₄₃	—	2	1	no	—
76	C₂×D₄×C₄₃	C₂×C₈	1	1	yes	characteristic
77	C₂²×C₄×C₄₃	—	2	1	no	—
78	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₂²⋊D₄₃	—	2	2	no	—
80	C₂²×Dic₄₃	M₁₆	1	1	yes	characteristic
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	—	2	1	no	—
85	C₂×C₄×D₄₃	C₂²⋊C₄	1	2	yes	—
86	C₂×C₄×D₄₃	C₂×C₈	1	2	yes	—
87	M₁₆×C₄₃	—	2	1	no	—
88	M₁₆×C₄₃	—	2	1	no	—
89	M₁₆×C₄₃	—	4	1	no	—
90	M₁₆×C₄₃	—	4	1	no	—
91	C₂×C₈×C₄₃	—	2	1	no	—
92	C₂×C₈×C₄₃	—	4	1	no	—
93	C₂×C₈×C₄₃	—	4	1	no	—
94	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
99	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
100	C₂²⋊Dic₄₃	—	2	2	no	—
101	C₂²⋊Dic₄₃	—	2	1	no	—
102	(C₂×D₄₃)⋊C₄	—	2	2	no	—
103	C₄⋊Dic₄₃	C₂×C₈	1	2	yes	—
104	Dic₄₃⋊C₄	—	2	2	no	—
105	C₄×Dic₄₃	—	2	1	no	—
106	#10621877	—	2	1	no	—
107	#10621877	—	2	1	no	—
108	C₂×C₄₃⋊C₈	—	2	1	no	—
109	C₂×C₄₃⋊C₈	—	2	1	no	—
110	C₂×C₄₃⋊C₈	—	2	1	no	—
111	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	2	yes	—
112	C₈×D₄₃	—	2	2	no	—
113	C₈×D₄₃	—	2	2	no	—
114	C₈×D₄₃	—	2	2	no	—
115	C₈×D₄₃	—	2	2	no	—
116	C₁₆×C₄₃	—	2	2	no	—
117	C₁₆×C₄₃	—	4	1	no	—
118	C₄₃⋊C₁₆	—	2	2	no	—
119	C₄₃⋊C₁₆	—	4	1	no	—
120	C₂³×C₄₃	—	2	1	no	—
121	C₂³×C₄₃	C₂×M₁₆	1	1	yes	characteristic
122	C₂²×D₄₃	—	2	1	no	—
123	C₂²×D₄₃	—	2	1	no	—
124	C₂²×D₄₃	—	4	2	no	—
125	C₂²×D₄₃	C₂²⋊C₈	1	2	yes	—
126	Q₈×C₄₃	—	2	1	no	—
127	Q₈×C₄₃	—	2	1	no	—
128	D₄×C₄₃	—	2	1	no	—
129	D₄×C₄₃	—	2	1	no	—
130	D₄×C₄₃	—	4	1	no	—
131	D₄×C₄₃	—	4	1	no	—
132	C₂×C₄×C₄₃	—	2	1	no	—
133	C₂×C₄×C₄₃	—	4	1	no	—
134	C₂×C₄×C₄₃	—	4	1	no	—
135	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
136	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
137	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
138	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
139	C₂×C₄×C₄₃	#131	1	1	yes	characteristic
140	C₂×C₄×C₄₃	#131	1	1	yes	characteristic
141	C₂²⋊D₄₃	—	4	4	no	—
142	C₂×Dic₄₃	—	2	1	no	—
143	C₂×Dic₄₃	—	2	1	no	—
144	C₂×Dic₄₃	—	2	2	no	—
145	C₂×Dic₄₃	C₂²⋊C₈	1	2	yes	—
146	C₄×D₄₃	—	2	2	no	—
147	C₄×D₄₃	—	2	2	no	—
148	C₄×D₄₃	—	2	2	no	—
149	C₄×D₄₃	—	2	2	no	—
150	C₈×C₄₃	—	4	1	no	—
151	C₈×C₄₃	#120	1	1	yes	characteristic
152	C₈×C₄₃	#121	1	1	yes	characteristic
153	C₈×C₄₃	#121	1	1	yes	characteristic
154	C₈×C₄₃	#123	1	1	yes	characteristic
155	C₄₃⋊C₈	—	2	2	no	—
156	C₄₃⋊C₈	—	4	1	no	—
157	#12984	—	43	1	no	maximal, Sylow
158	C₂²×C₄₃	—	2	2	no	—
159	C₂²×C₄₃	—	4	2	no	—
160	C₂²×C₄₃	—	4	1	no	—
161	C₂²×C₄₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
162	C₂²×C₄₃	#435	1	1	yes	characteristic
163	C₂²×C₄₃	#435	1	1	yes	characteristic
164	C₂×D₄₃	—	2	2	no	—
165	C₂×D₄₃	—	2	2	no	—
166	C₂×D₄₃	—	2	2	no	—
167	C₂×D₄₃	—	4	2	no	—
168	C₂×D₄₃	—	4	2	no	—
169	C₄×C₄₃	—	4	1	no	—
170	C₄×C₄₃	#386	1	1	yes	characteristic
171	C₄×C₄₃	#408	1	1	yes	characteristic
172	C₄×C₄₃	C₈×D₄	1	1	yes	characteristic
173	C₄×C₄₃	C₈×D₄	1	1	yes	characteristic
174	Dic₄₃	—	2	2	no	—
175	Dic₄₃	—	4	2	no	—
176	C₂²×M₃₂	—	43	1	no	—
177	#2929	—	43	1	no	—
178	C₂×#350	—	43	1	no	—
179	C₂×#350	D₄₃	1	1	yes	characteristic, complemented
180	#2112	—	43	1	no	—
181	#1379	—	43	1	no	—
182	#1379	—	43	1	no	—
183	C₂×C₄₃	—	2	1	no	—
184	C₂×C₄₃	—	2	1	no	—
185	C₂×C₄₃	—	8	1	no	—
186	C₂×C₄₃	#1869	1	1	yes	characteristic
187	C₂×C₄₃	#2117	1	2	yes	—
188	D₄₃	—	4	2	no	—
189	C₂×#131	C₂×D₄₃	1	1	yes	characteristic
190	C₂³×C₈	—	43	1	no	—
191	#525	—	43	1	no	—
192	C₂×M₃₂	—	43	1	no	—
193	C₂×M₃₂	—	86	2	no	—
194	C₂×M₃₂	—	86	2	no	—
195	C₂×M₃₂	—	86	2	no	—
196	C₂×M₃₂	C₂×D₄₃	1	1	yes	characteristic
197	C₂²×C₁₆	—	43	1	no	—
198	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
199	#435	—	86	2	no	—
200	C₈×D₄	—	86	2	no	—
201	#424	—	43	1	no	—
202	#408	—	43	1	no	—
203	#407	—	43	1	no	—
204	#405	—	43	1	no	—
205	#350	—	2	2	no	—
206	#350	—	86	2	no	—
207	C₂²⋊C₁₆	—	86	2	no	—
208	C₄₃	#12984	1	1	yes	characteristic, complemented, Sylow
209	C₂×#41	C₄×D₄₃	1	1	yes	characteristic
210	C₂³×C₄	—	43	1	no	—
211	#131	—	2	1	no	—
212	#131	—	2	1	no	—
213	#131	—	2	1	no	—
214	#131	—	2	1	no	—
215	C₂×M₁₆	—	2	1	no	—
216	C₂×M₁₆	—	43	1	no	—
217	C₂×M₁₆	C₄×D₄₃	1	1	yes	characteristic
218	C₂²×C₈	—	2	1	no	—
219	C₂²×C₈	—	43	1	no	—
220	C₂²×C₈	—	43	2	no	—
221	C₂²×C₈	—	43	2	no	—
222	C₂²×C₈	—	86	1	no	—
223	C₂²×C₈	—	86	1	no	—
224	C₂²×C₈	—	86	1	no	—
225	C₂²×C₈	—	86	1	no	—
226	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
227	#123	—	86	1	no	—
228	C₂²⋊Q₈	—	43	2	no	—
229	#121	—	43	2	no	—
230	#120	—	86	1	no	—
231	C₄×D₄	—	86	2	no	—
232	#117	—	43	1	no	—
233	M₃₂	—	2	1	no	—
234	M₃₂	—	2	1	no	—
235	M₃₂	—	86	8	no	—
236	M₃₂	—	86	1	no	—
237	M₃₂	—	86	1	no	—
238	C₂×C₁₆	—	2	1	no	—
239	C₂×C₁₆	—	2	1	no	—
240	C₂×C₁₆	—	2	1	no	—
241	C₂×C₁₆	—	43	2	no	—
242	C₂×C₁₆	—	86	1	no	—
243	C₂×C₁₆	—	86	1	no	—
244	C₂×C₁₆	—	86	1	no	—
245	C₂×C₁₆	C₂²⋊D₄₃	1	2	yes	complemented
246	C₄⋊C₈	—	86	2	no	—
247	C₂²⋊C₈	—	86	2	no	—
248	C₂²⋊C₈	—	86	2	no	—
249	#97	—	86	1	no	—
250	C₄×C₈	—	86	1	no	—
251	C₂⁴	—	43	1	no	—
252	#41	—	2	1	no	—
253	#41	—	2	1	no	—
254	#41	—	2	1	no	—
255	#41	—	2	1	no	—
256	C₂×Q₈	C₈×D₄₃	1	1	yes	characteristic
257	C₂×D₄	—	2	1	no	—
258	C₂×D₄	—	86	2	no	—
259	C₂×D₄	C₈×D₄₃	1	1	yes	characteristic
260	C₂²×C₄	—	2	1	no	—
261	C₂²×C₄	—	43	1	no	—
262	C₂²×C₄	—	43	2	no	—
263	C₂²×C₄	—	43	2	no	—
264	C₂²×C₄	—	86	1	no	—
265	C₂²×C₄	—	86	1	no	—
266	C₂²×C₄	—	86	1	no	—
267	C₂²×C₄	—	86	1	no	—
268	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
269	M₁₆	—	2	1	no	—
270	M₁₆	—	2	1	no	—
271	M₁₆	—	4	1	no	—
272	M₁₆	—	4	1	no	—
273	M₁₆	—	86	1	no	—
274	M₁₆	—	86	1	no	—
275	C₂×C₈	—	2	1	no	—
276	C₂×C₈	—	4	1	no	—
277	C₂×C₈	—	4	1	no	—
278	C₂×C₈	—	43	2	no	—
279	C₂×C₈	—	86	1	no	—
280	C₂×C₈	—	86	1	no	—
281	C₂×C₈	—	86	2	no	—
282	C₂×C₈	—	86	2	no	—
283	C₂×C₈	—	86	2	no	—
284	C₂×C₈	—	86	2	no	—
285	C₂×C₈	—	86	1	no	—
286	C₂×C₈	#10621898	1	1	yes	characteristic
287	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
288	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
289	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
290	C₂×C₈	#10621900	1	1	yes	characteristic
291	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
292	C₄⋊C₄	—	43	2	no	—
293	C₄⋊C₄	—	86	2	no	—
294	C₂²⋊C₄	—	86	2	no	—
295	C₂²⋊C₄	—	86	2	no	—
296	C₂²⋊C₄	—	86	1	no	—
297	C₄²	—	86	1	no	—
298	C₁₆	—	2	2	no	—
299	C₁₆	—	4	1	no	—
300	C₁₆	—	86	2	no	—
301	C₁₆	—	172	1	no	—
302	C₂³	—	2	1	no	—
303	C₂³	—	43	2	no	—
304	C₂³	—	86	1	no	—
305	C₂³	—	86	1	no	—
306	C₂³	—	172	2	no	—
307	C₂³	M₁₆×D₄₃	1	1	yes	characteristic
308	Q₈	—	2	1	no	—
309	Q₈	—	2	1	no	—
310	D₄	—	2	1	no	—
311	D₄	—	2	1	no	—
312	D₄	—	4	1	no	—
313	D₄	—	4	1	no	—
314	D₄	—	172	4	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	4	1	no	—
317	C₂×C₄	—	4	1	no	—
318	C₂×C₄	—	43	2	no	—
319	C₂×C₄	—	86	1	no	—
320	C₂×C₄	—	86	1	no	—
321	C₂×C₄	—	86	2	no	—
322	C₂×C₄	—	86	2	no	—
323	C₂×C₄	—	86	2	no	—
324	C₂×C₄	—	86	2	no	—
325	C₂×C₄	—	86	2	no	—
326	C₂×C₄	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
327	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
328	C₂×C₄	C₄×C₂²⋊D₄₃	1	1	yes	characteristic
329	C₂×C₄	C₂×C₈×D₄₃	1	1	yes	characteristic
330	C₂×C₄	#13063331	1	1	yes	characteristic
331	C₂×C₄	#13063333	1	1	yes	characteristic
332	C₈	—	4	1	no	—
333	C₈	—	86	2	no	—
334	C₈	—	172	1	no	—
335	C₈	#13063293	1	1	yes	characteristic
336	C₈	#13063296	1	1	yes	characteristic
337	C₈	#13063298	1	1	yes	characteristic
338	C₈	#13063307	1	1	yes	characteristic
339	C₂²	—	2	2	no	—
340	C₂²	—	4	1	no	—
341	C₂²	—	4	2	no	—
342	C₂²	—	86	2	no	—
343	C₂²	—	86	2	no	—
344	C₂²	—	86	2	no	—
345	C₂²	—	172	2	no	—
346	C₂²	—	172	2	no	—
347	C₂²	C₂²⋊C₈×D₄₃	1	1	yes	characteristic
348	C₂²	#14582968	1	1	yes	characteristic
349	C₂²	#14582973	1	1	yes	characteristic
350	C₄	—	4	1	no	—
351	C₄	—	86	2	no	—
352	C₄	—	172	2	no	—
353	C₄	#14582775	1	1	yes	characteristic
354	C₄	#14582864	1	1	yes	characteristic
355	C₄	(C₂²⋊D₄₃)⋊C₈	1	1	yes	characteristic
356	C₄	C₈×C₂²⋊D₄₃	1	1	yes	characteristic
357	C₂	—	2	1	no	—
358	C₂	—	2	1	no	—
359	C₂	—	8	1	no	—
360	C₂	—	172	2	no	—
361	C₂	#19134884	1	1	yes	characteristic
362	C₂	#19136221	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₂×C₄₃×#350	C₂
socle	C₂²×C₄₃	C₂×C₂²⋊C₈
prime radical	#14583406	C₄
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	C₂×#350	D₄₃
hypocenter	C₄₃	#12984
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#350

Complements

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

N	H ≅ G/N	type
1. C₂²×#13063264	C₂	semidirect
3. C₂×#14582683	C₂	semidirect
4. #19136205	C₂	semidirect
5. #19132685	C₂	semidirect
6. #19132683	C₂	semidirect
7. C₂×C₄₃×#350	C₂	semidirect
48. C₂×C₄₃⋊C₁₆	D₄	semidirect
68. C₂×C₁₆×C₄₃	D₄	semidirect
179. C₂×#350	D₄₃	semidirect
208. C₄₃	#12984	semidirect
245. C₂×C₁₆	C₂²⋊D₄₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13063264	2
#19143287	2
C₂×#14582683	2
#19136205	2
#19132685	2
#19132683	2
C₂×C₄₃×#350	2
#12984	43