Subgroups — #24995517

Order 11008.

478 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#10621884	C₂	1	1	yes	maximal, characteristic
2	#19133341	C₂	1	1	yes	maximal, characteristic, complemented
3	#19132990	C₂	1	1	yes	maximal, characteristic, complemented
4	#19132987	C₂	1	1	yes	maximal, characteristic
5	#19132915	C₂	1	1	yes	maximal, characteristic
6	#19132879	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#1461	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#10621884	C₄	1	1	yes	characteristic
9	C₂×C₄×D₁₇₂	C₂²	1	1	yes	characteristic
10	C₄×#10621884	—	2	1	no	—
11	C₄×#10621884	—	2	1	no	—
12	C₄×#10621884	—	2	1	no	—
13	C₄×#10621884	—	2	1	no	—
14	#14582952	C₂²	1	1	yes	characteristic
15	#14582952	C₂²	1	1	yes	characteristic
16	C₂×#13063256	C₄	1	1	yes	characteristic
17	C₂×#13063249	C₄	1	1	yes	characteristic
18	C₂×#13063245	C₄	1	1	yes	characteristic
19	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
20	#14582651	—	2	1	no	—
21	#14582649	—	2	1	no	—
22	Q₈⋊(C₄₃⋊C₈)	—	2	2	no	—
23	#14582628	—	2	2	no	—
24	#14582622	—	2	2	no	—
25	#14582621	—	2	1	no	—
26	C₈×C₄₃⋊C₈	—	2	1	no	—
27	C₂×C₄×Q₈×C₄₃	C₂²	1	1	yes	characteristic
28	C₄₃×#431	C₂²	1	1	yes	characteristic
29	C₄₃×#431	C₂²	1	1	yes	characteristic
30	C₄₃×#324	—	2	2	no	—
31	C₂²×D₁₇₂	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×D₄₃	—	2	1	no	—
33	C₂×#10621884	—	2	1	no	—
34	C₂×#10621884	—	2	1	no	—
35	C₂×#10621884	—	2	2	no	—
36	C₂×#10621884	C₈	1	4	yes	complemented
37	C₂²×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
39	C₄×D₁₇₂	—	2	1	no	—
40	C₄×D₁₇₂	—	2	1	no	—
41	C₄×D₁₇₂	—	2	1	no	—
42	C₄×D₁₇₂	—	2	1	no	—
43	C₂×C₄⋊Dic₄₃	C₂×C₄	1	1	yes	characteristic
44	C₂×(C₂×D₄₃)⋊C₄	—	2	1	no	—
45	#13063256	—	2	1	no	—
46	#13063256	—	2	1	no	—
47	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
48	C₄⋊(C₄₃⋊C₈)	D₄	1	2	yes	—
49	#13063249	—	2	1	no	—
50	#13063249	—	2	1	no	—
51	#13063245	—	2	1	no	—
52	#13063245	—	2	1	no	—
53	C₂²⋊(C₄₃⋊C₈)	—	2	1	no	—
54	C₂²⋊(C₄₃⋊C₈)	—	2	1	no	—
55	C₄×C₄₃⋊C₈	—	2	1	no	—
56	C₄×C₄₃⋊C₈	—	2	1	no	—
57	C₄×C₄₃⋊C₈	D₄	1	1	yes	characteristic
58	C₄×C₄₃⋊C₈	D₄	1	1	yes	characteristic
59	C₄×C₄₃⋊C₈	D₄	1	1	yes	characteristic
60	C₄×C₄₃⋊C₈	D₄	1	1	yes	characteristic
61	C₂²×Q₈×C₄₃	C₂×C₄	1	1	yes	characteristic
62	C₄×Q₈×C₄₃	—	2	1	no	—
63	C₄×Q₈×C₄₃	—	2	1	no	—
64	C₄×Q₈×C₄₃	—	2	1	no	—
65	C₄×Q₈×C₄₃	—	2	1	no	—
66	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
67	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄²×C₄₃	—	2	1	no	—
69	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
70	C₄⋊C₈×C₄₃	—	2	1	no	—
71	C₄⋊C₈×C₄₃	—	2	1	no	—
72	C₂²⋊C₈×C₄₃	—	2	1	no	—
73	C₂²⋊C₈×C₄₃	D₄	1	2	yes	—
74	C₄×C₈×C₄₃	—	2	1	no	—
75	C₄×C₈×C₄₃	—	2	1	no	—
76	C₂³×D₄₃	—	2	1	no	—
77	C₂×Q₈×C₄₃	—	2	1	no	—
78	C₂×Q₈×C₄₃	—	2	1	no	—
79	C₂×Q₈×C₄₃	—	2	1	no	—
80	C₂×Q₈×C₄₃	—	2	1	no	—
81	C₂×Q₈×C₄₃	—	2	1	no	—
82	C₂×Q₈×C₄₃	C₂×C₈	1	2	yes	—
83	C₂²×C₄×C₄₃	—	2	1	no	—
84	C₂²×C₄×C₄₃	—	2	1	no	—
85	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
88	C₂²×Dic₄₃	—	2	1	no	—
89	C₂×D₁₇₂	—	2	1	no	—
90	C₂×D₁₇₂	—	2	1	no	—
91	C₂×D₁₇₂	—	2	1	no	—
92	C₂×D₁₇₂	—	2	1	no	—
93	C₂×D₁₇₂	—	2	1	no	—
94	C₂×D₁₇₂	C₂×C₈	1	2	yes	—
95	C₂×C₄×D₄₃	—	4	1	no	—
96	C₂×C₄×D₄₃	—	4	1	no	—
97	C₂×C₄×D₄₃	—	8	2	no	—
98	C₂×C₈×C₄₃	—	2	2	no	—
99	C₂×C₈×C₄₃	—	4	1	no	—
100	C₄⋊C₄×C₄₃	—	2	1	no	—
101	C₄⋊C₄×C₄₃	—	2	1	no	—
102	C₄⋊C₄×C₄₃	—	4	1	no	—
103	C₄⋊C₄×C₄₃	—	4	1	no	—
104	C₄²×C₄₃	—	4	1	no	—
105	C₄²×C₄₃	—	4	1	no	—
106	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₄₃	#41	1	1	yes	characteristic
110	#10621884	—	2	4	no	—
111	#10621884	—	2	4	no	—
112	(C₂×D₄₃)⋊C₄	—	4	1	no	—
113	(C₂×D₄₃)⋊C₄	—	4	1	no	—
114	C₄⋊Dic₄₃	—	2	1	no	—
115	C₄⋊Dic₄₃	—	2	1	no	—
116	C₂×C₄₃⋊C₈	—	2	1	no	—
117	C₂×C₄₃⋊C₈	—	2	1	no	—
118	C₂×C₄₃⋊C₈	—	2	2	no	—
119	C₂×C₄₃⋊C₈	—	4	1	no	—
120	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄₃⋊C₈	C₂×C₈	1	2	yes	—
125	C₂×C₄₃⋊C₈	M₁₆	1	2	yes	—
126	C₂³×C₄₃	C₄×D₄	1	1	yes	characteristic
127	C₂²×D₄₃	—	2	2	no	—
128	C₂²×D₄₃	—	4	1	no	—
129	C₂²×D₄₃	—	4	1	no	—
130	C₂²×D₄₃	—	8	1	no	—
131	C₂²×D₄₃	—	8	1	no	—
132	Q₈×C₄₃	—	2	2	no	—
133	Q₈×C₄₃	—	2	2	no	—
134	Q₈×C₄₃	—	4	1	no	—
135	Q₈×C₄₃	—	4	1	no	—
136	C₂×C₄×C₄₃	—	2	2	no	—
137	C₂×C₄×C₄₃	—	2	2	no	—
138	C₂×C₄×C₄₃	—	4	1	no	—
139	C₂×C₄×C₄₃	—	4	1	no	—
140	C₂×C₄×C₄₃	—	4	1	no	—
141	C₂×C₄×C₄₃	—	4	1	no	—
142	C₂×C₄×C₄₃	—	8	1	no	—
143	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
144	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
145	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
146	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
147	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
148	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
149	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
150	C₂×C₄×C₄₃	C₂×M₁₆	1	1	yes	characteristic
151	C₂×C₄×C₄₃	#131	1	1	yes	characteristic
152	C₂×C₄×C₄₃	#131	1	1	yes	characteristic
153	C₂×Dic₄₃	—	4	1	no	—
154	C₂×Dic₄₃	—	8	1	no	—
155	D₁₇₂	—	2	2	no	—
156	D₁₇₂	—	2	2	no	—
157	D₁₇₂	—	4	1	no	—
158	D₁₇₂	—	4	1	no	—
159	C₄×D₄₃	—	8	2	no	—
160	C₄×D₄₃	—	8	2	no	—
161	C₈×C₄₃	—	4	2	no	—
162	C₈×C₄₃	—	8	1	no	—
163	C₄₃⋊C₈	—	2	1	no	—
164	C₄₃⋊C₈	—	2	1	no	—
165	C₄₃⋊C₈	—	4	2	no	—
166	C₄₃⋊C₈	—	8	1	no	—
167	C₄₃⋊C₈	C₂²⋊C₈	1	2	yes	—
168	C₄₃⋊C₈	C₂²⋊C₈	1	2	yes	—
169	#9667	—	43	1	no	maximal, Sylow
170	C₂²×C₄₃	—	2	1	no	—
171	C₂²×C₄₃	—	2	1	no	—
172	C₂²×C₄₃	#390	1	1	yes	characteristic
173	C₂²×C₄₃	C₈×D₄	1	1	yes	characteristic
174	C₂²×C₄₃	#436	1	1	yes	characteristic
175	C₂×D₄₃	—	4	2	no	—
176	C₂×D₄₃	—	8	2	no	—
177	C₂×D₄₃	—	8	1	no	—
178	C₂×D₄₃	—	8	1	no	—
179	C₂×D₄₃	—	8	2	no	—
180	C₄×C₄₃	—	2	1	no	—
181	C₄×C₄₃	—	2	1	no	—
182	C₄×C₄₃	—	2	1	no	—
183	C₄×C₄₃	—	4	2	no	—
184	C₄×C₄₃	—	8	1	no	—
185	C₄×C₄₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
186	C₄×C₄₃	#408	1	1	yes	characteristic
187	C₄×C₄₃	C₈×D₄	1	1	yes	characteristic
188	C₄×C₄₃	#436	1	1	yes	characteristic
189	C₄×C₄₃	#471	1	2	yes	—
190	Dic₄₃	—	8	1	no	—
191	C₂×C₄×SD₁₆	—	43	1	no	—
192	#1564	—	43	1	no	—
193	#1479	—	43	1	no	—
194	#1478	—	43	1	no	—
195	#1461	D₄₃	1	1	yes	characteristic, complemented
196	#1460	—	43	1	no	—
197	#1450	—	43	1	no	—
198	C₂×C₄₃	—	2	1	no	—
199	C₂×C₄₃	—	2	1	no	—
200	C₂×C₄₃	#1954	1	1	yes	characteristic
201	C₂×C₄₃	#1976	1	1	yes	characteristic
202	C₂×C₄₃	#1977	1	1	yes	characteristic
203	D₄₃	—	8	2	no	—
204	C₂²×SD₁₆	—	43	1	no	—
205	C₂×C₄×Q₈	C₂×D₄₃	1	1	yes	characteristic
206	C₂×C₄×D₄	—	43	1	no	—
207	C₄×SD₁₆	—	86	1	no	—
208	C₄×SD₁₆	—	86	1	no	—
209	C₄×SD₁₆	—	86	1	no	—
210	C₄×SD₁₆	—	86	1	no	—
211	#431	—	43	1	no	—
212	#431	—	43	1	no	—
213	#431	C₂×D₄₃	1	1	yes	characteristic
214	#431	C₂×D₄₃	1	1	yes	characteristic
215	C₂×#106	—	43	1	no	—
216	C₂×#103	—	43	1	no	—
217	C₂×#102	—	43	1	no	—
218	C₂×C₄×C₈	—	43	1	no	—
219	#335	—	86	1	no	—
220	#334	—	86	1	no	—
221	Q₈⋊C₈	—	86	2	no	—
222	D₄⋊C₈	—	86	2	no	—
223	#324	—	2	2	no	—
224	#323	—	86	2	no	—
225	#322	—	86	1	no	—
226	C₈²	—	86	1	no	—
227	C₄₃	#9667	1	1	yes	characteristic, complemented, Sylow
228	C₂²×Q₈	C₄×D₄₃	1	1	yes	characteristic
229	C₂²×D₄	—	43	1	no	—
230	C₂³×C₄	—	86	1	no	—
231	C₂×SD₁₆	—	43	4	no	—
232	C₂×SD₁₆	—	86	2	no	—
233	C₂×SD₁₆	—	86	1	no	—
234	C₂×SD₁₆	—	86	1	no	—
235	C₂²×C₈	—	43	1	no	—
236	C₂²×C₈	—	43	1	no	—
237	C₄×Q₈	—	2	1	no	—
238	C₄×Q₈	—	2	1	no	—
239	C₄×Q₈	—	2	1	no	—
240	C₄×Q₈	—	2	1	no	—
241	C₄×D₄	—	86	1	no	—
242	C₄×D₄	—	86	1	no	—
243	C₄×D₄	—	86	1	no	—
244	C₄×D₄	—	86	1	no	—
245	C₂×C₄⋊C₄	—	2	1	no	—
246	C₂×C₄⋊C₄	—	43	1	no	—
247	C₂×C₄⋊C₄	C₄×D₄₃	1	1	yes	characteristic
248	C₂×C₂²⋊C₄	—	86	1	no	—
249	C₂×C₄²	—	2	1	no	—
250	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
251	#106	—	86	1	no	—
252	#106	—	86	1	no	—
253	C₄⋊C₈	—	2	1	no	—
254	C₄⋊C₈	—	2	1	no	—
255	C₄⋊C₈	—	43	2	no	—
256	C₄⋊C₈	—	86	1	no	—
257	#103	—	86	1	no	—
258	#103	—	86	1	no	—
259	#102	—	86	1	no	—
260	#102	—	86	1	no	—
261	C₂²⋊C₈	—	2	1	no	—
262	C₂²⋊C₈	—	86	1	no	—
263	C₂²⋊C₈	—	86	1	no	—
264	C₂²⋊C₈	C₂²⋊D₄₃	1	2	yes	—
265	C₄×C₈	—	2	1	no	—
266	C₄×C₈	—	2	1	no	—
267	C₄×C₈	—	43	1	no	—
268	C₄×C₈	—	43	1	no	—
269	C₄×C₈	—	43	1	no	—
270	C₄×C₈	—	43	1	no	—
271	C₄×C₈	—	86	1	no	—
272	C₄×C₈	—	86	1	no	—
273	C₂⁴	—	86	1	no	—
274	C₂×Q₈	—	2	1	no	—
275	C₂×Q₈	—	2	1	no	—
276	C₂×Q₈	—	2	1	no	—
277	C₂×Q₈	—	2	1	no	—
278	C₂×Q₈	—	2	1	no	—
279	C₂×Q₈	C₈×D₄₃	1	2	yes	—
280	C₂×D₄	—	43	2	no	—
281	C₂×D₄	—	86	1	no	—
282	C₂×D₄	—	86	1	no	—
283	C₂×D₄	—	86	1	no	—
284	C₂×D₄	—	86	1	no	—
285	C₂×D₄	—	86	1	no	—
286	C₂²×C₄	—	2	1	no	—
287	C₂²×C₄	—	2	1	no	—
288	C₂²×C₄	—	86	1	no	—
289	C₂²×C₄	—	344	2	no	—
290	C₂²×C₄	—	172	1	no	—
291	C₂²×C₄	—	172	1	no	—
292	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
293	C₂²×C₄	#10621898	1	1	yes	characteristic
294	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
295	SD₁₆	—	86	4	no	—
296	SD₁₆	—	86	4	no	—
297	C₂×C₈	—	2	2	no	—
298	C₂×C₈	—	4	1	no	—
299	C₂×C₈	—	43	1	no	—
300	C₂×C₈	—	43	1	no	—
301	C₂×C₈	—	43	2	no	—
302	C₂×C₈	—	43	2	no	—
303	C₂×C₈	—	43	1	no	—
304	C₂×C₈	—	43	1	no	—
305	C₂×C₈	—	86	1	no	—
306	C₂×C₈	—	86	1	no	—
307	C₂×C₈	—	86	2	no	—
308	C₂×C₈	—	172	1	no	—
309	C₄⋊C₄	—	2	1	no	—
310	C₄⋊C₄	—	2	1	no	—
311	C₄⋊C₄	—	4	1	no	—
312	C₄⋊C₄	—	4	1	no	—
313	C₄⋊C₄	—	86	1	no	—
314	C₄⋊C₄	—	86	1	no	—
315	C₂²⋊C₄	—	172	1	no	—
316	C₂²⋊C₄	—	172	1	no	—
317	C₄²	—	4	1	no	—
318	C₄²	—	4	1	no	—
319	C₄²	D₄×D₄₃	1	1	yes	characteristic
320	C₄²	D₄×D₄₃	1	1	yes	characteristic
321	C₄²	D₄×D₄₃	1	1	yes	characteristic
322	C₄²	#10621900	1	1	yes	characteristic
323	C₂³	—	86	2	no	—
324	C₂³	—	344	1	no	—
325	C₂³	—	344	1	no	—
326	C₂³	—	172	1	no	—
327	C₂³	—	172	1	no	—
328	C₂³	C₄×C₂²⋊D₄₃	1	1	yes	characteristic
329	Q₈	—	2	2	no	—
330	Q₈	—	2	2	no	—
331	Q₈	—	4	1	no	—
332	Q₈	—	4	1	no	—
333	D₄	—	86	2	no	—
334	D₄	—	86	2	no	—
335	D₄	—	172	1	no	—
336	D₄	—	172	1	no	—
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	4	1	no	—
340	C₂×C₄	—	4	1	no	—
341	C₂×C₄	—	4	1	no	—
342	C₂×C₄	—	4	1	no	—
343	C₂×C₄	—	8	1	no	—
344	C₂×C₄	—	344	2	no	—
345	C₂×C₄	—	344	2	no	—
346	C₂×C₄	—	344	1	no	—
347	C₂×C₄	—	172	1	no	—
348	C₂×C₄	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
349	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
350	C₂×C₄	#13063297	1	1	yes	characteristic
351	C₂×C₄	#13063298	1	1	yes	characteristic
352	C₂×C₄	#13063311	1	1	yes	characteristic
353	C₂×C₄	#13063321	1	1	yes	characteristic
354	C₂×C₄	C₂×C₈×D₄₃	1	1	yes	characteristic
355	C₂×C₄	M₁₆×D₄₃	1	1	yes	characteristic
356	C₂×C₄	#13063331	1	1	yes	characteristic
357	C₂×C₄	#13063333	1	1	yes	characteristic
358	C₈	—	4	2	no	—
359	C₈	—	8	1	no	—
360	C₈	—	43	2	no	—
361	C₈	—	43	2	no	—
362	C₈	—	86	1	no	—
363	C₈	—	86	1	no	—
364	C₈	—	344	1	no	—
365	C₈	—	172	2	no	—
366	C₂²	—	2	1	no	—
367	C₂²	—	2	1	no	—
368	C₂²	—	344	2	no	—
369	C₂²	—	344	2	no	—
370	C₂²	—	344	1	no	—
371	C₂²	—	344	1	no	—
372	C₂²	—	172	2	no	—
373	C₂²	#14582799	1	1	yes	characteristic
374	C₂²	C₈×C₂²⋊D₄₃	1	1	yes	characteristic
375	C₂²	#14582975	1	1	yes	characteristic
376	C₄	—	2	1	no	—
377	C₄	—	2	1	no	—
378	C₄	—	2	1	no	—
379	C₄	—	4	2	no	—
380	C₄	—	8	1	no	—
381	C₄	—	344	1	no	—
382	C₄	C₂²⋊C₈×D₄₃	1	1	yes	characteristic
383	C₄	#14582864	1	1	yes	characteristic
384	C₄	(C₂²⋊D₄₃)⋊C₈	1	1	yes	characteristic
385	C₄	#14582974	1	1	yes	characteristic
386	C₄	#14583165	1	2	yes	—
387	C₂	—	2	1	no	—
388	C₂	—	2	1	no	—
389	C₂	—	344	2	no	—
390	C₂	#19135387	1	1	yes	characteristic
391	C₂	#19135490	1	1	yes	characteristic
392	C₂	#19135499	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13063298
commutator	C₂×C₄×C₄₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1461	C₂
socle	C₂²×C₄₃	#390
prime radical	C₂²×D₁₇₂	C₂×C₄
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#1461	D₄₃
hypocenter	C₄₃	#9667
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1461

Complements

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

N	H ≅ G/N	type
2. #19133341	C₂	semidirect
3. #19132990	C₂	semidirect
6. #19132879	C₂	semidirect
7. C₄₃×#1461	C₂	semidirect
36. C₂×#10621884	C₈	semidirect
195. #1461	D₄₃	semidirect
227. C₄₃	#9667	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#10621884	2
#19133341	2
#19132990	2
#19132987	2
#19132915	2
#19132879	2
C₄₃×#1461	2
#9667	43