Subgroups — #25006506

Order 11008.

430 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×D₄×D₄₃	C₂	1	1	yes	maximal, characteristic
2	#19136393	C₂	1	1	yes	maximal, characteristic, complemented
3	#19136392	C₂	1	1	yes	maximal, characteristic, complemented
4	#19132551	C₂	1	1	yes	maximal, characteristic, complemented
5	D₄⋊(C₄₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
6	#19132549	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×D₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₄₃	—	2	1	no	—
9	C₄×D₄×D₄₃	C₄	1	1	yes	characteristic
10	C₈×D₁₇₂	C₂²	1	1	yes	characteristic
11	D₄×C₄₃⋊C₈	C₄	1	1	yes	characteristic
12	#14582964	C₄	1	1	yes	characteristic
13	C₈×C₂²⋊D₄₃	—	2	1	no	—
14	(C₂²⋊D₄₃)⋊C₈	—	2	1	no	—
15	C₄⋊C₈×D₄₃	C₄	1	1	yes	characteristic
16	C₂²⋊C₈×D₄₃	—	2	1	no	—
17	C₄×C₈×D₄₃	C₂²	1	1	yes	characteristic
18	C₄⋊(C₄₃⋊C₁₆)	C₂²	1	1	yes	characteristic, complemented
19	Dic₄₃⋊C₁₆	—	2	1	no	—
20	(C₂×D₄₃)⋊C₁₆	—	2	1	no	—
21	(C₂×D₄₃)⋊C₁₆	—	2	1	no	—
22	C₄×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic, complemented
23	C₁₆×Dic₄₃	—	2	1	no	—
24	C₈×D₄×C₄₃	C₂²	1	1	yes	characteristic
25	C₄₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic, complemented
26	C₄×C₁₆×C₄₃	C₂²	1	1	yes	characteristic, complemented
27	C₂×D₄×D₄₃	C₈	1	1	yes	characteristic
28	C₂²×C₄×D₄₃	—	2	1	no	—
29	C₂²×C₄₃⋊C₈	—	2	1	no	—
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	—	4	2	no	—
34	C₂×C₈×D₄₃	—	4	2	no	—
35	C₂×C₈×D₄₃	D₄	1	1	yes	characteristic
36	C₂×C₈×D₄₃	D₄	1	1	yes	characteristic
37	C₄×D₁₇₂	C₂×C₄	1	1	yes	characteristic
38	D₄×Dic₄₃	C₈	1	1	yes	characteristic
39	#13063287	C₈	1	1	yes	characteristic
40	C₄×C₂²⋊D₄₃	—	2	1	no	—
41	(C₂²⋊D₄₃)⋊C₄	—	2	1	no	—
42	C₄⋊C₄×D₄₃	C₈	1	1	yes	characteristic
43	C₂²⋊C₄×D₄₃	—	2	1	no	—
44	C₄²×D₄₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄₃⋊C₁₆	—	2	1	no	—
46	C₂×C₄₃⋊C₁₆	—	2	1	no	—
47	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
48	Dic₄₃⋊C₈	—	2	1	no	—
49	C₂²⋊(C₄₃⋊C₈)	—	2	1	no	—
50	(C₂×D₄₃)⋊C₈	—	2	1	no	—
51	(C₂×D₄₃)⋊C₈	—	2	1	no	—
52	C₄×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
53	C₈×Dic₄₃	D₄	1	1	yes	characteristic
54	C₈×Dic₄₃	D₄	1	1	yes	characteristic
55	C₂²×C₈×C₄₃	—	2	1	no	—
56	C₄×D₄×C₄₃	C₂×C₄	1	1	yes	characteristic
57	C₂×C₁₆×C₄₃	—	2	1	no	—
58	C₂×C₁₆×C₄₃	—	2	1	no	—
59	C₄⋊C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
60	C₂²⋊C₈×C₄₃	—	2	1	no	—
61	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
62	C₂³×D₄₃	—	2	1	no	—
63	C₂×D₄×C₄₃	C₂×C₈	1	1	yes	characteristic
64	C₂²×C₄×C₄₃	—	2	1	no	—
65	C₂×C₂²⋊D₄₃	—	2	1	no	—
66	C₂²×Dic₄₃	—	2	1	no	—
67	D₄×D₄₃	—	2	2	no	—
68	D₄×D₄₃	C₁₆	1	4	yes	complemented
69	C₂×D₁₇₂	C₂×C₈	1	1	yes	characteristic
70	C₂×C₄×D₄₃	—	2	1	no	—
71	C₂×C₄×D₄₃	—	2	1	no	—
72	C₂×C₄×D₄₃	—	2	1	no	—
73	C₂×C₄×D₄₃	—	4	2	no	—
74	C₂×C₄×D₄₃	—	4	2	no	—
75	C₂×C₄×D₄₃	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₄×D₄₃	C₂×C₈	1	1	yes	characteristic
77	C₂×C₄×D₄₃	M₁₆	1	1	yes	characteristic
78	C₂×C₈×C₄₃	—	2	1	no	—
79	C₂×C₈×C₄₃	—	4	1	no	—
80	C₂×C₈×C₄₃	—	4	1	no	—
81	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
83	C₄⋊C₄×C₄₃	C₂×C₈	1	1	yes	characteristic
84	C₂²⋊C₄×C₄₃	—	2	1	no	—
85	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
86	C₂²⋊Dic₄₃	—	2	1	no	—
87	(C₂×D₄₃)⋊C₄	—	2	1	no	—
88	(C₂×D₄₃)⋊C₄	—	2	1	no	—
89	C₄⋊Dic₄₃	C₂×C₈	1	1	yes	characteristic
90	Dic₄₃⋊C₄	—	2	1	no	—
91	C₄×Dic₄₃	C₂²⋊C₄	1	1	yes	characteristic
92	C₄×Dic₄₃	C₂×C₈	1	1	yes	characteristic
93	C₄×Dic₄₃	M₁₆	1	1	yes	characteristic
94	C₂×C₄₃⋊C₈	—	2	1	no	—
95	C₂×C₄₃⋊C₈	—	4	2	no	—
96	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
98	C₈×D₄₃	—	2	1	no	—
99	C₈×D₄₃	—	2	1	no	—
100	C₈×D₄₃	—	4	1	no	—
101	C₈×D₄₃	—	4	2	no	—
102	C₈×D₄₃	—	4	2	no	—
103	C₈×D₄₃	—	4	1	no	—
104	C₈×D₄₃	—	4	1	no	—
105	C₁₆×C₄₃	—	4	1	no	—
106	C₁₆×C₄₃	—	4	1	no	—
107	C₄₃⋊C₁₆	—	4	1	no	—
108	C₄₃⋊C₁₆	—	4	1	no	—
109	C₂³×C₄₃	—	2	1	no	—
110	C₂²×D₄₃	—	2	1	no	—
111	C₂²×D₄₃	—	2	4	no	—
112	C₂²×D₄₃	—	4	2	no	—
113	C₂²×D₄₃	—	4	2	no	—
114	C₂²×D₄₃	C₂²⋊C₈	1	1	yes	characteristic
115	D₄×C₄₃	—	2	1	no	—
116	D₄×C₄₃	C₂×C₁₆	1	2	yes	—
117	C₂×C₄×C₄₃	—	2	1	no	—
118	C₂×C₄×C₄₃	—	4	1	no	—
119	C₂×C₄×C₄₃	—	4	1	no	—
120	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
122	C₂×C₄×C₄₃	C₂×M₁₆	1	1	yes	characteristic
123	C₂²⋊D₄₃	—	2	4	no	—
124	C₂×Dic₄₃	—	2	1	no	—
125	C₂×Dic₄₃	—	4	2	no	—
126	C₂×Dic₄₃	C₂²⋊C₈	1	1	yes	characteristic
127	C₂×Dic₄₃	C₂²⋊C₈	1	1	yes	characteristic
128	C₂×Dic₄₃	C₂²⋊C₈	1	1	yes	characteristic
129	D₁₇₂	—	2	1	no	—
130	D₁₇₂	C₂×C₁₆	1	2	yes	—
131	C₄×D₄₃	—	2	1	no	—
132	C₄×D₄₃	—	2	1	no	—
133	C₄×D₄₃	—	4	1	no	—
134	C₄×D₄₃	—	4	2	no	—
135	C₄×D₄₃	—	4	2	no	—
136	C₄×D₄₃	—	4	1	no	—
137	C₄×D₄₃	—	4	1	no	—
138	C₄×D₄₃	C₂×C₁₆	1	2	yes	—
139	C₄×D₄₃	M₃₂	1	2	yes	—
140	C₈×C₄₃	—	2	1	no	—
141	C₈×C₄₃	—	4	1	no	—
142	C₈×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
143	C₈×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
144	C₄₃⋊C₈	—	2	1	no	—
145	C₄₃⋊C₈	—	2	1	no	—
146	C₄₃⋊C₈	—	4	1	no	—
147	#11629	—	43	1	no	maximal, Sylow
148	C₂²×C₄₃	—	2	2	no	—
149	C₂²×C₄₃	—	4	1	no	—
150	C₂²×C₄₃	—	4	1	no	—
151	C₂²×C₄₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
152	C₂×D₄₃	—	2	1	no	—
153	C₂×D₄₃	—	2	1	no	—
154	C₂×D₄₃	—	2	2	no	—
155	C₂×D₄₃	—	4	8	no	—
156	C₂×D₄₃	—	4	1	no	—
157	C₂×D₄₃	—	4	1	no	—
158	C₂×D₄₃	C₂²⋊C₁₆	1	2	yes	—
159	C₄×C₄₃	—	2	1	no	—
160	C₄×C₄₃	—	4	1	no	—
161	C₄×C₄₃	#418	1	1	yes	characteristic
162	C₄×C₄₃	#421	1	1	yes	characteristic
163	C₄×C₄₃	C₂²×C₁₆	1	1	yes	characteristic
164	C₄×C₄₃	C₂×M₃₂	1	1	yes	characteristic
165	Dic₄₃	—	2	1	no	—
166	Dic₄₃	—	2	1	no	—
167	Dic₄₃	—	4	1	no	—
168	Dic₄₃	C₂²⋊C₁₆	1	2	yes	—
169	C₂×C₈×D₄	—	43	1	no	—
170	#2162	—	43	1	no	—
171	#2162	—	43	1	no	—
172	D₄⋊C₁₆	—	43	1	no	—
173	D₄⋊C₁₆	—	43	1	no	—
174	D₄⋊C₁₆	—	43	1	no	—
175	D₄⋊C₁₆	D₄₃	1	1	yes	characteristic, complemented
176	C₂×C₄₃	—	4	2	no	—
177	C₂×C₄₃	#1478	1	1	yes	characteristic
178	C₂×C₄₃	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
179	C₂×C₄₃	#2117	1	1	yes	characteristic
180	D₄₃	—	2	2	no	—
181	D₄₃	—	4	2	no	—
182	C₂³×C₈	—	86	1	no	—
183	C₂×C₄×D₄	—	43	1	no	—
184	C₈×D₄	—	43	1	no	—
185	C₈×D₄	—	43	1	no	—
186	C₈×D₄	—	43	1	no	—
187	C₈×D₄	—	86	1	no	—
188	C₈×D₄	—	86	1	no	—
189	C₈×D₄	C₂×D₄₃	1	1	yes	characteristic
190	C₂×C₄⋊C₈	—	43	1	no	—
191	C₂×C₂²⋊C₈	—	86	1	no	—
192	C₂×C₄×C₈	—	43	1	no	—
193	C₄⋊C₁₆	—	43	1	no	—
194	C₄⋊C₁₆	—	86	1	no	—
195	C₄⋊C₁₆	C₂×D₄₃	1	1	yes	characteristic, complemented
196	C₂²⋊C₁₆	—	86	1	no	—
197	C₂²⋊C₁₆	—	86	1	no	—
198	C₄×C₁₆	—	43	1	no	—
199	C₄×C₁₆	—	86	1	no	—
200	C₄×C₁₆	C₂×D₄₃	1	1	yes	characteristic, complemented
201	C₄₃	#11629	1	1	yes	characteristic, complemented, Sylow
202	C₂²×D₄	—	43	1	no	—
203	C₂³×C₄	—	86	1	no	—
204	C₂²×C₈	—	2	1	no	—
205	C₂²×C₈	—	43	1	no	—
206	C₂²×C₈	—	43	1	no	—
207	C₂²×C₈	—	86	1	no	—
208	C₂²×C₈	—	86	1	no	—
209	C₂²×C₈	—	86	1	no	—
210	C₂²×C₈	—	86	1	no	—
211	C₂²×C₈	—	172	2	no	—
212	C₂²×C₈	—	172	2	no	—
213	C₄×D₄	—	43	1	no	—
214	C₄×D₄	—	43	1	no	—
215	C₄×D₄	—	43	1	no	—
216	C₄×D₄	—	86	1	no	—
217	C₄×D₄	—	86	1	no	—
218	C₄×D₄	C₄×D₄₃	1	1	yes	characteristic
219	C₂×C₄⋊C₄	—	43	1	no	—
220	C₂×C₂²⋊C₄	—	86	1	no	—
221	C₂×C₄²	—	43	1	no	—
222	C₂×C₁₆	—	2	1	no	—
223	C₂×C₁₆	—	2	1	no	—
224	C₂×C₁₆	—	86	1	no	—
225	C₂×C₁₆	—	86	1	no	—
226	C₄⋊C₈	—	43	1	no	—
227	C₄⋊C₈	—	86	1	no	—
228	C₄⋊C₈	C₄×D₄₃	1	1	yes	characteristic
229	C₂²⋊C₈	—	2	1	no	—
230	C₂²⋊C₈	—	86	1	no	—
231	C₂²⋊C₈	—	86	1	no	—
232	C₂²⋊C₈	—	86	1	no	—
233	C₄×C₈	—	43	1	no	—
234	C₄×C₈	—	43	1	no	—
235	C₄×C₈	—	43	1	no	—
236	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
237	C₂⁴	—	86	1	no	—
238	C₂×D₄	—	43	4	no	—
239	C₂×D₄	—	43	1	no	—
240	C₂×D₄	—	86	2	no	—
241	C₂×D₄	—	86	1	no	—
242	C₂×D₄	C₈×D₄₃	1	1	yes	characteristic
243	C₂²×C₄	—	2	1	no	—
244	C₂²×C₄	—	43	1	no	—
245	C₂²×C₄	—	43	1	no	—
246	C₂²×C₄	—	43	1	no	—
247	C₂²×C₄	—	86	1	no	—
248	C₂²×C₄	—	86	1	no	—
249	C₂²×C₄	—	86	1	no	—
250	C₂²×C₄	—	86	1	no	—
251	C₂²×C₄	—	172	2	no	—
252	C₂²×C₄	—	172	2	no	—
253	C₂×C₈	—	2	1	no	—
254	C₂×C₈	—	4	1	no	—
255	C₂×C₈	—	4	1	no	—
256	C₂×C₈	—	43	1	no	—
257	C₂×C₈	—	43	1	no	—
258	C₂×C₈	—	86	1	no	—
259	C₂×C₈	—	86	1	no	—
260	C₂×C₈	—	86	1	no	—
261	C₂×C₈	—	172	1	no	—
262	C₂×C₈	—	172	2	no	—
263	C₂×C₈	—	172	2	no	—
264	C₂×C₈	—	172	2	no	—
265	C₂×C₈	—	172	1	no	—
266	C₂×C₈	—	172	1	no	—
267	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
268	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
269	C₄⋊C₄	—	43	1	no	—
270	C₄⋊C₄	—	86	1	no	—
271	C₄⋊C₄	C₈×D₄₃	1	1	yes	characteristic
272	C₂²⋊C₄	—	2	1	no	—
273	C₂²⋊C₄	—	86	1	no	—
274	C₂²⋊C₄	—	86	1	no	—
275	C₂²⋊C₄	—	86	1	no	—
276	C₄²	—	43	1	no	—
277	C₄²	—	43	1	no	—
278	C₄²	—	43	1	no	—
279	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
280	C₁₆	—	4	1	no	—
281	C₁₆	—	4	1	no	—
282	C₁₆	—	172	1	no	—
283	C₁₆	—	172	1	no	—
284	C₂³	—	2	1	no	—
285	C₂³	—	43	1	no	—
286	C₂³	—	86	4	no	—
287	C₂³	—	86	1	no	—
288	C₂³	—	172	2	no	—
289	C₂³	—	172	2	no	—
290	D₄	—	2	1	no	—
291	D₄	—	43	2	no	—
292	D₄	—	86	4	no	—
293	D₄	—	86	1	no	—
294	D₄	C₁₆×D₄₃	1	2	yes	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	43	1	no	—
299	C₂×C₄	—	43	2	no	—
300	C₂×C₄	—	43	2	no	—
301	C₂×C₄	—	43	1	no	—
302	C₂×C₄	—	43	1	no	—
303	C₂×C₄	—	86	1	no	—
304	C₂×C₄	—	86	1	no	—
305	C₂×C₄	—	86	1	no	—
306	C₂×C₄	—	172	1	no	—
307	C₂×C₄	—	172	2	no	—
308	C₂×C₄	—	172	2	no	—
309	C₂×C₄	—	172	2	no	—
310	C₂×C₄	—	172	1	no	—
311	C₂×C₄	—	172	1	no	—
312	C₂×C₄	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
313	C₂×C₄	C₂×C₈×D₄₃	1	1	yes	characteristic
314	C₂×C₄	M₁₆×D₄₃	1	1	yes	characteristic
315	C₈	—	2	1	no	—
316	C₈	—	4	1	no	—
317	C₈	—	86	1	no	—
318	C₈	—	86	1	no	—
319	C₈	—	172	1	no	—
320	C₈	#13063349	1	1	yes	characteristic
321	C₈	#13063352	1	1	yes	characteristic
322	C₂²	—	2	2	no	—
323	C₂²	—	4	1	no	—
324	C₂²	—	4	1	no	—
325	C₂²	—	43	2	no	—
326	C₂²	—	86	1	no	—
327	C₂²	—	86	1	no	—
328	C₂²	—	86	2	no	—
329	C₂²	—	172	8	no	—
330	C₂²	—	172	1	no	—
331	C₂²	—	172	1	no	—
332	C₂²	C₂²⋊C₈×D₄₃	1	1	yes	characteristic
333	C₄	—	2	1	no	—
334	C₄	—	4	1	no	—
335	C₄	—	43	2	no	—
336	C₄	—	86	1	no	—
337	C₄	—	86	1	no	—
338	C₄	—	172	1	no	—
339	C₄	#14582902	1	1	yes	characteristic
340	C₄	#14582914	1	1	yes	characteristic
341	C₄	C₂×C₁₆×D₄₃	1	1	yes	characteristic
342	C₄	M₃₂×D₄₃	1	1	yes	characteristic
343	C₂	—	4	2	no	—
344	C₂	—	86	2	no	—
345	C₂	—	172	2	no	—
346	C₂	#19132984	1	1	yes	characteristic
347	C₂	C₂²⋊C₁₆×D₄₃	1	1	yes	characteristic
348	C₂	#19136221	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₄₃
commutator	C₄×C₄₃	C₂²×C₁₆
Frattini	C₄×C₈	C₂²×D₄₃
Fitting	C₄₃×D₄⋊C₁₆	C₂
socle	C₂²×C₄₃	C₂×C₂²⋊C₈
prime radical	C₂×D₄×D₄₃	C₈
cosocle	C₄×C₈×C₄₃	C₂³
hypercenter	D₄⋊C₁₆	D₄₃
hypocenter	C₄₃	#11629
LCS	G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ D₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
2. #19136393	C₂	semidirect
3. #19136392	C₂	semidirect
4. #19132551	C₂	semidirect
5. D₄⋊(C₄₃⋊C₁₆)	C₂	semidirect
6. #19132549	C₂	semidirect
7. C₄₃×D₄⋊C₁₆	C₂	semidirect
18. C₄⋊(C₄₃⋊C₁₆)	C₂²	semidirect
22. C₄×C₄₃⋊C₁₆	C₂²	semidirect
25. C₄₃×C₄⋊C₁₆	C₂²	semidirect
26. C₄×C₁₆×C₄₃	C₂²	semidirect
68. D₄×D₄₃	C₁₆	semidirect
175. D₄⋊C₁₆	D₄₃	semidirect
195. C₄⋊C₁₆	C₂×D₄₃	semidirect
200. C₄×C₁₆	C₂×D₄₃	semidirect
201. C₄₃	#11629	semidirect

Maximal subgroups

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

subgroup	index
C₈×D₄×D₄₃	2
#19136393	2
#19136392	2
#19132551	2
D₄⋊(C₄₃⋊C₁₆)	2
#19132549	2
C₄₃×D₄⋊C₁₆	2
#11629	43