Subgroups — #25001323

Order 11008.

418 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19135388	C₂	1	1	yes	maximal, characteristic
2	#19134751	C₂	1	1	yes	maximal, characteristic, complemented
3	#19134596	C₂	1	1	yes	maximal, characteristic
4	#19134592	C₂	1	1	yes	maximal, characteristic
5	#19132479	C₂	1	1	yes	maximal, characteristic, complemented
6	#19132429	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#1296	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₁₇₂	C₂²	1	1	yes	characteristic
9	C₂×#13063256	—	2	1	no	—
10	C₂×#13063256	—	2	1	no	—
11	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
12	C₂×#13063247	—	2	1	no	—
13	C₂×#13063245	—	2	1	no	—
14	C₂×(C₂×D₄₃)⋊C₈	—	2	1	no	—
15	C₂×(C₂×D₄₃)⋊C₈	—	2	1	no	—
16	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
17	#14582769	C₂²	1	1	yes	characteristic
18	#14582662	—	2	1	no	—
19	#14582661	—	2	1	no	—
20	C₂×C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
22	C₄₃×#384	C₂²	1	1	yes	characteristic
23	C₂²×D₁₇₂	D₄	1	1	yes	characteristic
24	C₂²×C₄×D₄₃	—	2	1	no	—
25	C₂²×C₄₃⋊C₈	—	2	1	no	—
26	C₂²×C₄₃⋊C₈	D₄	1	1	yes	characteristic
27	C₂²×C₄₃⋊C₈	D₄	1	1	yes	characteristic
28	C₄×D₁₇₂	—	2	1	no	—
29	C₄×D₁₇₂	—	2	1	no	—
30	C₄×D₁₇₂	—	2	1	no	—
31	C₄×D₁₇₂	—	2	1	no	—
32	C₂×C₄⋊Dic₄₃	—	2	1	no	—
33	C₂×C₄⋊Dic₄₃	—	2	1	no	—
34	C₂×C₄⋊Dic₄₃	—	2	1	no	—
35	C₂×C₄⋊Dic₄₃	D₄	1	1	yes	characteristic
36	C₂×(C₂×D₄₃)⋊C₄	—	2	1	no	—
37	#13063256	—	4	2	no	—
38	#13063256	—	4	2	no	—
39	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
40	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
41	#13063247	—	4	2	no	—
42	#13063245	—	4	2	no	—
43	(C₂×D₄₃)⋊C₈	—	4	2	no	—
44	(C₂×D₄₃)⋊C₈	—	4	2	no	—
45	C₄×C₄₃⋊C₈	—	2	1	no	—
46	C₄×C₄₃⋊C₈	—	2	1	no	—
47	C₂²×C₈×C₄₃	—	2	1	no	—
48	C₂²×C₈×C₄₃	—	2	1	no	—
49	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
50	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
51	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
52	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
53	C₄⋊C₈×C₄₃	—	2	1	no	—
54	C₄⋊C₈×C₄₃	—	2	1	no	—
55	C₄⋊C₈×C₄₃	—	2	1	no	—
56	C₄⋊C₈×C₄₃	—	2	1	no	—
57	C₂³×D₄₃	—	2	1	no	—
58	C₂²×C₄×C₄₃	—	4	1	no	—
59	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
62	C₂²×Dic₄₃	—	2	1	no	—
63	C₂²×Dic₄₃	—	4	1	no	—
64	C₂×D₁₇₂	—	2	1	no	—
65	C₂×D₁₇₂	—	2	1	no	—
66	C₂×D₁₇₂	—	2	1	no	—
67	C₂×D₁₇₂	—	2	1	no	—
68	C₂×D₁₇₂	—	2	1	no	—
69	C₂×D₁₇₂	SD₁₆	1	2	yes	—
70	C₂×C₄×D₄₃	—	4	1	no	—
71	C₂×C₄×D₄₃	—	4	1	no	—
72	C₂×C₄×D₄₃	—	4	1	no	—
73	C₂×C₄×D₄₃	—	4	1	no	—
74	C₂×C₄×D₄₃	—	4	1	no	—
75	C₂×C₄×D₄₃	—	4	1	no	—
76	C₂×C₈×C₄₃	—	4	1	no	—
77	C₂×C₈×C₄₃	—	4	1	no	—
78	C₂×C₈×C₄₃	—	4	1	no	—
79	C₂×C₈×C₄₃	—	4	1	no	—
80	C₂×C₈×C₄₃	—	4	1	no	—
81	C₂×C₈×C₄₃	—	4	1	no	—
82	C₄⋊C₄×C₄₃	—	4	1	no	—
83	C₄⋊C₄×C₄₃	—	4	1	no	—
84	C₄⋊C₄×C₄₃	—	4	1	no	—
85	C₄⋊C₄×C₄₃	—	4	1	no	—
86	C₄⋊C₄×C₄₃	—	8	1	no	—
87	C₄²×C₄₃	#41	1	1	yes	characteristic
88	C₄²×C₄₃	#41	1	1	yes	characteristic
89	C₄²×C₄₃	#41	1	1	yes	characteristic
90	C₄²×C₄₃	#41	1	1	yes	characteristic
91	(C₂×D₄₃)⋊C₄	—	4	1	no	—
92	(C₂×D₄₃)⋊C₄	—	4	1	no	—
93	C₄⋊Dic₄₃	—	2	1	no	—
94	C₄⋊Dic₄₃	—	2	1	no	—
95	C₄⋊Dic₄₃	—	4	1	no	—
96	C₄⋊Dic₄₃	—	4	1	no	—
97	C₄⋊Dic₄₃	—	4	1	no	—
98	C₄⋊Dic₄₃	—	4	1	no	—
99	C₄⋊Dic₄₃	—	8	1	no	—
100	C₂×C₄₃⋊C₈	—	2	1	no	—
101	C₂×C₄₃⋊C₈	—	2	1	no	—
102	C₂×C₄₃⋊C₈	—	2	1	no	—
103	C₂×C₄₃⋊C₈	—	2	1	no	—
104	C₂×C₄₃⋊C₈	—	4	1	no	—
105	C₂×C₄₃⋊C₈	—	4	1	no	—
106	C₂×C₄₃⋊C₈	—	4	1	no	—
107	C₂×C₄₃⋊C₈	D₈	1	2	yes	—
108	C₂×C₄₃⋊C₈	SD₁₆	1	2	yes	—
109	C₂³×C₄₃	#121	1	1	yes	characteristic
110	C₂²×D₄₃	—	2	2	no	—
111	C₂²×D₄₃	—	4	1	no	—
112	C₂²×D₄₃	—	4	1	no	—
113	C₂²×D₄₃	—	4	1	no	—
114	C₂²×D₄₃	—	4	1	no	—
115	C₂²×D₄₃	—	4	1	no	—
116	C₂²×D₄₃	—	4	1	no	—
117	C₂×C₄×C₄₃	—	2	1	no	—
118	C₂×C₄×C₄₃	—	2	1	no	—
119	C₂×C₄×C₄₃	—	8	1	no	—
120	C₂×C₄×C₄₃	—	8	1	no	—
121	C₂×C₄×C₄₃	—	8	1	no	—
122	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
123	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
124	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
125	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
126	C₂×C₄×C₄₃	#126	1	1	yes	characteristic
127	C₂×C₄×C₄₃	#126	1	1	yes	characteristic
128	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
130	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
131	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
132	C₂×C₄×C₄₃	AGL(1,ℤ/8)	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×C₄×C₄₃	#137	1	1	yes	characteristic
136	C₂×Dic₄₃	—	4	1	no	—
137	C₂×Dic₄₃	—	4	1	no	—
138	C₂×Dic₄₃	—	4	1	no	—
139	C₂×Dic₄₃	—	8	1	no	—
140	C₂×Dic₄₃	—	8	1	no	—
141	C₂×Dic₄₃	—	8	1	no	—
142	D₁₇₂	—	2	2	no	—
143	D₁₇₂	—	2	2	no	—
144	D₁₇₂	—	4	1	no	—
145	D₁₇₂	—	4	1	no	—
146	C₄×D₄₃	—	8	1	no	—
147	C₄×D₄₃	—	8	1	no	—
148	C₄×D₄₃	—	8	1	no	—
149	C₄×D₄₃	—	8	1	no	—
150	C₈×C₄₃	—	8	1	no	—
151	C₈×C₄₃	—	8	1	no	—
152	C₄₃⋊C₈	—	2	2	no	—
153	C₄₃⋊C₈	—	2	2	no	—
154	C₄₃⋊C₈	—	8	1	no	—
155	#10619	—	43	1	no	maximal, Sylow
156	C₂²×C₄₃	#397	1	1	yes	characteristic
157	C₂²×C₄₃	#459	1	1	yes	characteristic
158	C₂²×C₄₃	#461	1	1	yes	characteristic
159	C₂²×C₄₃	#465	1	1	yes	characteristic
160	C₂²×C₄₃	#468	1	1	yes	characteristic
161	C₂²×C₄₃	#471	1	1	yes	characteristic
162	C₂²×C₄₃	#471	1	1	yes	characteristic
163	C₂×D₄₃	—	4	2	no	—
164	C₂×D₄₃	—	4	2	no	—
165	C₂×D₄₃	—	4	2	no	—
166	C₂×D₄₃	—	8	1	no	—
167	C₂×D₄₃	—	8	1	no	—
168	C₂×D₄₃	—	8	1	no	—
169	C₂×D₄₃	—	8	1	no	—
170	C₄×C₄₃	—	2	1	no	—
171	C₄×C₄₃	—	2	1	no	—
172	C₄×C₄₃	—	16	1	no	—
173	C₄×C₄₃	#480	1	1	yes	characteristic
174	C₄×C₄₃	#481	1	1	yes	characteristic
175	C₄×C₄₃	#481	1	1	yes	characteristic
176	C₄×C₄₃	#482	1	1	yes	characteristic
177	C₄×C₄₃	#483	1	1	yes	characteristic
178	C₄×C₄₃	#485	1	1	yes	characteristic
179	C₄×C₄₃	#486	1	1	yes	characteristic
180	C₄×C₄₃	#489	1	1	yes	characteristic
181	Dic₄₃	—	8	1	no	—
182	Dic₄₃	—	16	1	no	—
183	#1954	—	43	1	no	—
184	#1844	—	43	1	no	—
185	#1802	—	43	1	no	—
186	#1802	—	43	1	no	—
187	#1296	—	43	1	no	—
188	#1296	D₄₃	1	1	yes	characteristic, complemented
189	#1277	—	43	1	no	—
190	C₂×C₄₃	#1665	1	1	yes	characteristic
191	C₂×C₄₃	#1671	1	1	yes	characteristic
192	C₂×C₄₃	#1685	1	1	yes	characteristic
193	C₂×C₄₃	#1691	1	1	yes	characteristic
194	C₂×C₄₃	#2071	1	1	yes	characteristic
195	C₂×C₄₃	#2077	1	1	yes	characteristic
196	C₂×C₄₃	#2079	1	1	yes	characteristic
197	D₄₃	—	8	2	no	—
198	C₂×C₄×D₄	—	43	1	no	—
199	C₂×#106	—	86	1	no	—
200	C₂×#106	—	86	1	no	—
201	C₂×C₄⋊C₈	—	43	1	no	—
202	C₂×C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
203	C₂×C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
204	C₂×#102	—	86	1	no	—
205	C₂×#102	—	86	1	no	—
206	C₂×C₂²⋊C₈	—	86	1	no	—
207	C₂×C₂²⋊C₈	—	86	1	no	—
208	C₂×C₄×C₈	—	43	1	no	—
209	#384	—	43	1	no	—
210	#384	C₂×D₄₃	1	1	yes	characteristic
211	#340	—	86	1	no	—
212	#340	—	86	1	no	—
213	C₄₃	#10619	1	1	yes	characteristic, complemented, Sylow
214	C₂²×D₄	—	43	1	no	—
215	C₂³×C₄	—	86	1	no	—
216	C₂²×C₈	—	2	1	no	—
217	C₂²×C₈	—	2	1	no	—
218	C₂²×C₈	—	43	1	no	—
219	C₂²×C₈	—	43	1	no	—
220	C₂²×C₈	—	86	1	no	—
221	C₄×D₄	—	86	1	no	—
222	C₄×D₄	—	86	1	no	—
223	C₄×D₄	—	86	1	no	—
224	C₄×D₄	—	86	1	no	—
225	C₂×C₄⋊C₄	—	2	1	no	—
226	C₂×C₄⋊C₄	—	2	1	no	—
227	C₂×C₄⋊C₄	—	2	1	no	—
228	C₂×C₄⋊C₄	—	43	1	no	—
229	C₂×C₄⋊C₄	—	86	1	no	—
230	C₂×C₄⋊C₄	—	86	1	no	—
231	C₂×C₄⋊C₄	—	86	1	no	—
232	C₂×C₂²⋊C₄	—	86	1	no	—
233	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
234	#106	—	172	2	no	—
235	#106	—	172	2	no	—
236	C₄⋊C₈	—	2	1	no	—
237	C₄⋊C₈	—	2	1	no	—
238	C₄⋊C₈	—	2	1	no	—
239	C₄⋊C₈	—	2	1	no	—
240	C₄⋊C₈	—	86	1	no	—
241	C₄⋊C₈	—	86	1	no	—
242	#102	—	172	2	no	—
243	#102	—	172	2	no	—
244	C₂²⋊C₈	—	172	2	no	—
245	C₂²⋊C₈	—	172	2	no	—
246	C₄×C₈	—	86	1	no	—
247	C₄×C₈	—	86	1	no	—
248	C₂⁴	—	86	1	no	—
249	C₂×D₄	—	43	2	no	—
250	C₂×D₄	—	86	1	no	—
251	C₂×D₄	—	86	1	no	—
252	C₂×D₄	—	86	1	no	—
253	C₂×D₄	—	86	1	no	—
254	C₂×D₄	—	86	1	no	—
255	C₂²×C₄	—	4	1	no	—
256	C₂²×C₄	—	86	1	no	—
257	C₂²×C₄	—	172	1	no	—
258	C₂²×C₄	—	172	1	no	—
259	C₂²×C₄	—	172	1	no	—
260	C₂²×C₄	—	172	1	no	—
261	C₂²×C₄	—	172	1	no	—
262	C₂²×C₄	—	172	1	no	—
263	C₂²×C₄	—	172	1	no	—
264	C₂²×C₄	#10621898	1	1	yes	characteristic
265	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
266	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
267	C₂×C₈	—	4	1	no	—
268	C₂×C₈	—	4	1	no	—
269	C₂×C₈	—	4	1	no	—
270	C₂×C₈	—	4	1	no	—
271	C₂×C₈	—	4	1	no	—
272	C₂×C₈	—	4	1	no	—
273	C₂×C₈	—	43	2	no	—
274	C₂×C₈	—	43	2	no	—
275	C₂×C₈	—	86	1	no	—
276	C₂×C₈	—	86	1	no	—
277	C₂×C₈	—	86	1	no	—
278	C₂×C₈	—	86	1	no	—
279	C₂×C₈	—	172	1	no	—
280	C₂×C₈	—	172	1	no	—
281	C₂×C₈	—	172	1	no	—
282	C₄⋊C₄	—	4	1	no	—
283	C₄⋊C₄	—	4	1	no	—
284	C₄⋊C₄	—	4	1	no	—
285	C₄⋊C₄	—	4	1	no	—
286	C₄⋊C₄	—	8	1	no	—
287	C₄⋊C₄	—	86	1	no	—
288	C₄⋊C₄	—	86	1	no	—
289	C₄⋊C₄	—	344	1	no	—
290	C₄⋊C₄	—	172	1	no	—
291	C₄⋊C₄	—	172	1	no	—
292	C₄⋊C₄	—	172	1	no	—
293	C₄⋊C₄	—	172	1	no	—
294	C₂²⋊C₄	—	172	1	no	—
295	C₂²⋊C₄	—	172	1	no	—
296	C₄²	#10621898	1	1	yes	characteristic
297	C₄²	#10621898	1	1	yes	characteristic
298	C₄²	#10621900	1	1	yes	characteristic
299	C₄²	#10621902	1	1	yes	characteristic
300	C₂³	—	86	2	no	—
301	C₂³	—	172	1	no	—
302	C₂³	—	172	1	no	—
303	C₂³	—	172	1	no	—
304	C₂³	—	172	1	no	—
305	C₂³	—	172	1	no	—
306	C₂³	—	172	1	no	—
307	C₂³	#13063298	1	1	yes	characteristic
308	D₄	—	86	2	no	—
309	D₄	—	86	2	no	—
310	D₄	—	172	1	no	—
311	D₄	—	172	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	8	1	no	—
315	C₂×C₄	—	8	1	no	—
316	C₂×C₄	—	8	1	no	—
317	C₂×C₄	—	344	1	no	—
318	C₂×C₄	—	344	1	no	—
319	C₂×C₄	—	344	1	no	—
320	C₂×C₄	—	344	1	no	—
321	C₂×C₄	—	344	1	no	—
322	C₂×C₄	—	344	1	no	—
323	C₂×C₄	—	344	1	no	—
324	C₂×C₄	—	172	1	no	—
325	C₂×C₄	—	172	1	no	—
326	C₂×C₄	—	172	1	no	—
327	C₂×C₄	#13063307	1	1	yes	characteristic
328	C₂×C₄	#13063310	1	1	yes	characteristic
329	C₂×C₄	#13063310	1	1	yes	characteristic
330	C₂×C₄	#13063311	1	1	yes	characteristic
331	C₂×C₄	#13063319	1	1	yes	characteristic
332	C₂×C₄	#13063320	1	1	yes	characteristic
333	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
334	C₂×C₄	SD₁₆×D₄₃	1	1	yes	characteristic
335	C₂×C₄	SD₁₆×D₄₃	1	1	yes	characteristic
336	C₂×C₄	#13063348	1	1	yes	characteristic
337	C₂×C₄	#13063349	1	1	yes	characteristic
338	C₂×C₄	#13063352	1	1	yes	characteristic
339	C₂×C₄	#13063354	1	1	yes	characteristic
340	C₂×C₄	#13063357	1	1	yes	characteristic
341	C₈	—	8	1	no	—
342	C₈	—	8	1	no	—
343	C₈	—	86	2	no	—
344	C₈	—	86	2	no	—
345	C₈	—	344	1	no	—
346	C₂²	—	344	1	no	—
347	C₂²	—	344	1	no	—
348	C₂²	—	344	1	no	—
349	C₂²	—	344	1	no	—
350	C₂²	—	172	2	no	—
351	C₂²	—	172	2	no	—
352	C₂²	—	172	2	no	—
353	C₂²	#14582828	1	1	yes	characteristic
354	C₂²	#14583093	1	1	yes	characteristic
355	C₂²	#14583103	1	1	yes	characteristic
356	C₂²	#14583131	1	1	yes	characteristic
357	C₂²	#14583144	1	1	yes	characteristic
358	C₂²	#14583161	1	1	yes	characteristic
359	C₂²	#14583165	1	1	yes	characteristic
360	C₄	—	2	1	no	—
361	C₄	—	2	1	no	—
362	C₄	—	16	1	no	—
363	C₄	—	344	1	no	—
364	C₄	—	688	1	no	—
365	C₄	#14583214	1	1	yes	characteristic
366	C₄	#14583216	1	1	yes	characteristic
367	C₄	#14583219	1	1	yes	characteristic
368	C₄	#14583226	1	1	yes	characteristic
369	C₄	#14583231	1	1	yes	characteristic
370	C₄	#14583239	1	1	yes	characteristic
371	C₄	#14583244	1	1	yes	characteristic
372	C₄	#14583260	1	1	yes	characteristic
373	C₂	—	344	2	no	—
374	C₂	#19133939	1	1	yes	characteristic
375	C₂	#19133973	1	1	yes	characteristic
376	C₂	#19134059	1	1	yes	characteristic
377	C₂	#19134093	1	1	yes	characteristic
378	C₂	#19136002	1	1	yes	characteristic
379	C₂	#19136034	1	1	yes	characteristic
380	C₂	#19136044	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13063298
commutator, cosocle	C₂×C₄²×C₄₃	C₂³
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1296	C₂
socle	C₂³×C₄₃	#121
prime radical	C₂²×D₁₇₂	D₄
hypercenter	#1296	D₄₃
hypocenter	C₄₃	#10619
LCS	G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1296

Complements

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

N	H ≅ G/N	type
2. #19134751	C₂	semidirect
5. #19132479	C₂	semidirect
6. #19132429	C₂	semidirect
7. C₄₃×#1296	C₂	semidirect
188. #1296	D₄₃	semidirect
213. C₄₃	#10619	semidirect

Maximal subgroups

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

subgroup	index
#19135388	2
#19134751	2
#19134596	2
#19134592	2
#19132479	2
#19132429	2
C₄₃×#1296	2
#10619	43