Subgroups — #25250280

Order 11008.

536 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#10621900	C₂	1	1	yes	maximal, characteristic
2	#19136503	C₂	1	1	yes	maximal, characteristic, complemented
3	#19136500	C₂	1	1	yes	maximal, characteristic, complemented
4	#19136420	C₂	1	2	yes	maximal, complemented
5	#19136417	C₂	1	2	yes	maximal, complemented
6	D₄×C₄₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	#19136395	C₂	1	2	yes	maximal, complemented
8	#19136392	C₂	1	1	yes	maximal, characteristic, complemented
9	#19136349	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Dic₄₃⋊C₁₆	C₂	1	2	yes	maximal, complemented
11	D₄×C₁₆×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#10621900	C₄	1	1	yes	characteristic
13	C₂×#13063264	C₂²	1	2	yes	—
14	C₂²×C₄₃⋊C₁₆	C₂²	1	2	yes	—
15	C₈×Dic₈₆	C₂²	1	1	yes	characteristic
16	#14583023	C₄	1	1	yes	characteristic
17	D₄×C₄₃⋊C₈	C₄	1	1	yes	characteristic
18	C₈×C₂²⋊D₄₃	C₂²	1	2	yes	—
19	(C₂²⋊D₄₃)⋊C₈	C₄	1	2	yes	—
20	#14582951	C₄	1	1	yes	characteristic
21	#14582949	C₄	1	2	yes	—
22	C₂×C₈×Dic₄₃	C₂²	1	2	yes	—
23	C₄×C₈×D₄₃	C₂²	1	1	yes	characteristic
24	C₄⋊(C₄₃⋊C₁₆)	C₂²	1	1	yes	characteristic
25	Dic₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
26	Dic₄₃⋊C₁₆	C₂²	1	2	yes	—
27	Dic₄₃⋊C₁₆	C₂²	1	2	yes	—
28	Dic₄₃⋊C₁₆	C₂²	1	2	yes	—
29	Dic₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
30	C₂²⋊(C₄₃⋊C₁₆)	C₂²	1	2	yes	—
31	(C₂×D₄₃)⋊C₁₆	C₂²	1	1	yes	characteristic
32	(C₂×D₄₃)⋊C₁₆	C₂²	1	1	yes	characteristic
33	#14582674	C₂²	1	2	yes	—
34	#14582674	C₂²	1	2	yes	—
35	C₄×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
36	C₂²×C₁₆×C₄₃	C₂²	1	2	yes	—
37	C₈×D₄×C₄₃	C₂²	1	1	yes	characteristic
38	C₄₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
39	C₄₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
40	C₄×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
41	C₂×#10621900	C₈	1	1	yes	characteristic
42	C₂²×C₄₃⋊C₈	C₂×C₄	1	2	yes	—
43	C₂×C₈×D₄₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₈×D₄₃	C₂³	1	1	yes	characteristic
45	C₄×Dic₈₆	C₂×C₄	1	1	yes	characteristic
46	#13063290	C₈	1	1	yes	characteristic
47	D₄×Dic₄₃	C₈	1	1	yes	characteristic
48	C₄×C₂²⋊D₄₃	C₂×C₄	1	2	yes	—
49	(C₂²⋊D₄₃)⋊C₄	C₈	1	2	yes	—
50	#13063283	C₈	1	1	yes	characteristic
51	#13063281	C₈	1	2	yes	—
52	C₂×C₄×Dic₄₃	C₂×C₄	1	2	yes	—
53	C₄²×D₄₃	C₂×C₄	1	1	yes	characteristic
54	#13063264	—	4	2	no	—
55	C₂×C₄₃⋊C₁₆	—	4	2	no	—
56	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
57	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
58	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
59	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
60	Dic₄₃⋊C₈	C₂×C₄	1	2	yes	—
61	Dic₄₃⋊C₈	C₂×C₄	1	2	yes	—
62	C₂²⋊(C₄₃⋊C₈)	C₂×C₄	1	2	yes	—
63	(C₂×D₄₃)⋊C₈	C₂×C₄	1	2	yes	—
64	C₄×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
65	C₈×Dic₄₃	C₂×C₄	1	1	yes	characteristic
66	C₈×Dic₄₃	C₂×C₄	1	2	yes	—
67	C₈×Dic₄₃	C₂×C₄	1	2	yes	—
68	C₈×Dic₄₃	C₂³	1	1	yes	characteristic
69	C₈×Dic₄₃	C₂³	1	2	yes	—
70	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
71	C₄×D₄×C₄₃	C₂×C₄	1	1	yes	characteristic
72	C₂×C₁₆×C₄₃	—	4	2	no	—
73	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
74	C₂×C₁₆×C₄₃	C₂³	1	2	yes	—
75	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
76	C₄⋊C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
77	C₂²⋊C₈×C₄₃	C₂×C₄	1	2	yes	—
78	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
79	C₂×D₄×C₄₃	C₂×C₈	1	1	yes	characteristic
80	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
81	C₂×C₂²⋊D₄₃	C₂×C₈	1	2	yes	—
82	C₂²×Dic₄₃	C₂×C₈	1	2	yes	—
83	C₂²×Dic₄₃	C₂×C₈	1	2	yes	—
84	#10621900	—	2	4	no	—
85	C₂×C₄×D₄₃	C₂×C₈	1	1	yes	characteristic
86	C₂×C₄×D₄₃	C₂×C₈	1	1	yes	characteristic
87	C₂×C₄×D₄₃	C₂²×C₄	1	1	yes	characteristic
88	C₂×Dic₈₆	C₂×C₈	1	1	yes	characteristic
89	C₂×C₈×C₄₃	—	2	2	no	—
90	C₂×C₈×C₄₃	—	2	2	no	—
91	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
93	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
94	C₄⋊C₄×C₄₃	C₂×C₈	1	1	yes	characteristic
95	C₂²⋊C₄×C₄₃	C₂×C₈	1	2	yes	—
96	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
97	C₂²⋊Dic₄₃	C₂×C₈	1	2	yes	—
98	(C₂×D₄₃)⋊C₄	C₂×C₈	1	2	yes	—
99	C₄⋊Dic₄₃	C₂×C₈	1	1	yes	characteristic
100	Dic₄₃⋊C₄	C₂×C₈	1	2	yes	—
101	Dic₄₃⋊C₄	C₂×C₈	1	2	yes	—
102	C₄×Dic₄₃	C₂×C₈	1	1	yes	characteristic
103	C₄×Dic₄₃	C₂×C₈	1	1	yes	characteristic
104	C₄×Dic₄₃	C₂×C₈	1	2	yes	—
105	C₄×Dic₄₃	C₂×C₈	1	2	yes	—
106	C₄×Dic₄₃	C₂²×C₄	1	1	yes	characteristic
107	C₄×Dic₄₃	C₂²×C₄	1	2	yes	—
108	C₂×C₄₃⋊C₈	—	4	2	no	—
109	C₂×C₄₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
111	C₂×C₄₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
112	C₈×D₄₃	—	2	1	no	—
113	C₈×D₄₃	—	2	1	no	—
114	C₈×D₄₃	—	4	1	no	—
115	C₁₆×C₄₃	—	2	1	no	—
116	C₁₆×C₄₃	—	2	2	no	—
117	C₁₆×C₄₃	—	2	1	no	—
118	C₄₃⋊C₁₆	—	2	1	no	—
119	C₄₃⋊C₁₆	—	2	2	no	—
120	C₄₃⋊C₁₆	—	2	1	no	—
121	C₂³×C₄₃	C₂²×C₈	1	2	yes	—
122	C₂²×D₄₃	C₂²×C₈	1	1	yes	characteristic
123	D₄×C₄₃	M₃₂	1	4	yes	complemented
124	C₂×C₄×C₄₃	—	2	2	no	—
125	C₂×C₄×C₄₃	—	2	2	no	—
126	C₂×C₄×C₄₃	C₂²×C₈	1	2	yes	—
127	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
128	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
129	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
130	C₂²⋊D₄₃	—	2	4	no	—
131	C₂×Dic₄₃	—	2	4	no	—
132	C₂×Dic₄₃	—	2	4	no	—
133	C₂×Dic₄₃	C₂²×C₈	1	1	yes	characteristic
134	C₂×Dic₄₃	C₂²×C₈	1	1	yes	characteristic
135	C₂×Dic₄₃	C₂²×C₈	1	2	yes	—
136	C₂×Dic₄₃	C₂²×C₈	1	2	yes	—
137	C₂×Dic₄₃	C₂²×C₈	1	1	yes	characteristic
138	C₄×D₄₃	—	2	1	no	—
139	C₄×D₄₃	—	2	1	no	—
140	C₄×D₄₃	—	2	2	no	—
141	C₄×D₄₃	—	2	2	no	—
142	Dic₈₆	—	2	2	no	—
143	C₈×C₄₃	—	2	2	no	—
144	C₈×C₄₃	—	2	1	no	—
145	C₈×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
146	C₈×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
147	C₄₃⋊C₈	—	2	1	no	—
148	C₄₃⋊C₈	—	2	1	no	—
149	C₄₃⋊C₈	—	2	2	no	—
150	#33504	—	43	1	no	maximal, Sylow
151	C₂²×C₄₃	—	2	2	no	—
152	C₂²×C₄₃	—	2	2	no	—
153	C₂²×C₄₃	C₂×M₃₂	1	4	yes	—
154	C₂²×C₄₃	C₂³×C₈	1	1	yes	characteristic
155	C₂×D₄₃	—	2	1	no	—
156	C₂×D₄₃	—	2	1	no	—
157	C₂×D₄₃	—	2	1	no	—
158	C₄×C₄₃	—	2	1	no	—
159	C₄×C₄₃	—	2	2	no	—
160	C₄×C₄₃	C₂×M₃₂	1	2	yes	—
161	C₄×C₄₃	#520	1	1	yes	characteristic
162	C₄×C₄₃	#568	1	1	yes	characteristic
163	Dic₄₃	—	2	2	no	—
164	Dic₄₃	—	2	1	no	—
165	Dic₄₃	—	2	2	no	—
166	Dic₄₃	—	2	1	no	—
167	Dic₄₃	#504	1	2	yes	—
168	C₈×#41	—	43	1	no	—
169	#2183	—	43	1	no	—
170	#2183	—	43	1	no	—
171	#2168	—	43	2	no	—
172	#2168	—	43	2	no	—
173	D₄×C₁₆	—	43	1	no	—
174	D₄×C₁₆	D₄₃	1	1	yes	characteristic, complemented
175	#2163	—	43	2	no	—
176	#2162	—	43	1	no	—
177	#2150	—	43	1	no	—
178	C₂×C₄⋊C₁₆	—	43	2	no	—
179	C₂×C₄₃	—	2	4	no	—
180	C₂×C₄₃	#2988	1	1	yes	characteristic
181	C₂×C₄₃	C₂²×M₃₂	1	1	yes	characteristic
182	C₂×C₄₃	C₂×#504	1	1	yes	characteristic
183	D₄₃	—	4	1	no	—
184	C₄×#41	—	43	1	no	—
185	C₂×M₃₂	—	43	2	no	—
186	C₂²×C₁₆	—	43	2	no	—
187	C₂²×C₁₆	C₂×D₄₃	1	2	yes	—
188	C₈×Q₈	—	43	1	no	—
189	C₈×Q₈	—	43	1	no	—
190	C₈×D₄	—	43	2	no	—
191	C₈×D₄	—	43	1	no	—
192	C₈×D₄	—	43	2	no	—
193	C₈×D₄	C₂×D₄₃	1	1	yes	characteristic
194	#431	—	43	1	no	—
195	#431	—	43	2	no	—
196	C₂×C₄×C₈	—	43	2	no	—
197	C₂×C₄×C₈	—	43	1	no	—
198	C₄⋊C₁₆	—	43	1	no	—
199	C₄⋊C₁₆	—	43	2	no	—
200	C₄⋊C₁₆	—	43	2	no	—
201	C₄⋊C₁₆	—	43	2	no	—
202	C₄⋊C₁₆	—	43	1	no	—
203	C₄⋊C₁₆	—	43	1	no	—
204	C₄⋊C₁₆	C₂×D₄₃	1	1	yes	characteristic
205	C₂²⋊C₁₆	—	43	2	no	—
206	C₂²⋊C₁₆	—	43	1	no	—
207	C₂²⋊C₁₆	—	43	1	no	—
208	C₂²⋊C₁₆	C₂×D₄₃	1	2	yes	—
209	#346	—	43	2	no	—
210	#346	—	43	2	no	—
211	C₄×C₁₆	—	43	1	no	—
212	C₄×C₁₆	C₂×D₄₃	1	1	yes	characteristic
213	C₄₃	#33504	1	1	yes	characteristic, complemented, Sylow
214	C₂×#41	—	43	1	no	—
215	C₂²×C₈	—	43	2	no	—
216	C₂²×C₈	—	43	1	no	—
217	C₂²×C₈	—	43	1	no	—
218	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
219	C₄×Q₈	—	43	1	no	—
220	C₄×Q₈	—	43	1	no	—
221	C₄×D₄	—	43	1	no	—
222	C₄×D₄	—	43	2	no	—
223	C₄×D₄	—	43	2	no	—
224	C₄×D₄	C₄×D₄₃	1	1	yes	characteristic
225	#117	—	43	1	no	—
226	#117	—	43	2	no	—
227	C₂×C₄²	—	43	2	no	—
228	C₂×C₄²	—	43	1	no	—
229	M₃₂	—	172	2	no	—
230	C₂×C₁₆	—	4	2	no	—
231	C₂×C₁₆	—	43	1	no	—
232	C₂×C₁₆	—	43	1	no	—
233	C₂×C₁₆	—	43	2	no	—
234	C₂×C₁₆	—	172	2	no	—
235	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
236	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
237	C₂×C₁₆	C₂²×D₄₃	1	2	yes	—
238	C₄⋊C₈	—	43	2	no	—
239	C₄⋊C₈	—	43	2	no	—
240	C₄⋊C₈	—	43	1	no	—
241	C₄⋊C₈	C₄×D₄₃	1	1	yes	characteristic
242	C₂²⋊C₈	—	43	2	no	—
243	C₂²⋊C₈	—	43	2	no	—
244	C₂²⋊C₈	C₄×D₄₃	1	2	yes	—
245	C₄×C₈	—	43	2	no	—
246	C₄×C₈	—	43	1	no	—
247	C₄×C₈	—	43	2	no	—
248	C₄×C₈	—	43	2	no	—
249	C₄×C₈	—	43	1	no	—
250	C₄×C₈	—	43	1	no	—
251	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
252	#41	—	86	4	no	—
253	C₂×Q₈	—	43	1	no	—
254	C₂×D₄	—	43	2	no	—
255	C₂×D₄	C₈×D₄₃	1	1	yes	characteristic
256	C₂²×C₄	—	43	2	no	—
257	C₂²×C₄	—	43	2	no	—
258	C₂²×C₄	—	43	1	no	—
259	C₂²×C₄	—	43	1	no	—
260	C₂²×C₄	—	43	1	no	—
261	C₂²×C₄	C₂×C₄×D₄₃	1	2	yes	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	43	1	no	—
265	C₂×C₈	—	43	1	no	—
266	C₂×C₈	—	43	2	no	—
267	C₂×C₈	—	86	1	no	—
268	C₂×C₈	—	86	1	no	—
269	C₂×C₈	—	172	1	no	—
270	C₂×C₈	—	172	2	no	—
271	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
272	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
273	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
274	C₄⋊C₄	—	43	2	no	—
275	C₄⋊C₄	—	43	2	no	—
276	C₄⋊C₄	—	43	1	no	—
277	C₄⋊C₄	C₈×D₄₃	1	1	yes	characteristic
278	C₂²⋊C₄	—	43	2	no	—
279	C₂²⋊C₄	—	43	2	no	—
280	C₂²⋊C₄	C₈×D₄₃	1	2	yes	—
281	C₄²	—	43	2	no	—
282	C₄²	—	43	1	no	—
283	C₄²	—	43	1	no	—
284	C₄²	—	43	2	no	—
285	C₄²	—	43	1	no	—
286	C₄²	—	43	2	no	—
287	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
288	C₁₆	—	2	1	no	—
289	C₁₆	—	2	2	no	—
290	C₁₆	—	2	1	no	—
291	C₁₆	—	86	1	no	—
292	C₁₆	—	86	2	no	—
293	C₁₆	—	86	1	no	—
294	C₂³	—	43	1	no	—
295	C₂³	C₂×C₈×D₄₃	1	2	yes	—
296	Q₈	—	86	2	no	—
297	D₄	—	86	4	no	—
298	D₄	#13063264	1	4	yes	complemented
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	43	1	no	—
302	C₂×C₄	—	43	1	no	—
303	C₂×C₄	—	43	1	no	—
304	C₂×C₄	—	43	2	no	—
305	C₂×C₄	—	43	2	no	—
306	C₂×C₄	—	86	2	no	—
307	C₂×C₄	—	86	4	no	—
308	C₂×C₄	—	86	4	no	—
309	C₂×C₄	—	86	2	no	—
310	C₂×C₄	—	86	1	no	—
311	C₂×C₄	—	86	1	no	—
312	C₂×C₄	C₂×C₈×D₄₃	1	2	yes	—
313	C₂×C₄	C₂×C₈×D₄₃	1	1	yes	characteristic
314	C₂×C₄	C₂×C₈×D₄₃	1	1	yes	characteristic
315	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
316	C₈	—	2	2	no	—
317	C₈	—	2	1	no	—
318	C₈	—	86	1	no	—
319	C₈	—	86	1	no	—
320	C₈	—	86	2	no	—
321	C₈	#13063370	1	1	yes	characteristic
322	C₈	#13063372	1	1	yes	characteristic
323	C₂²	—	2	2	no	—
324	C₂²	—	2	2	no	—
325	C₂²	—	86	1	no	—
326	C₂²	—	86	1	no	—
327	C₂²	—	86	1	no	—
328	C₂²	C₂×#13063264	1	4	yes	—
329	C₂²	C₂²×C₈×D₄₃	1	1	yes	characteristic
330	C₄	—	2	2	no	—
331	C₄	—	2	1	no	—
332	C₄	—	43	2	no	—
333	C₄	—	86	1	no	—
334	C₄	—	86	1	no	—
335	C₄	—	86	2	no	—
336	C₄	—	86	2	no	—
337	C₄	C₂×#13063264	1	2	yes	—
338	C₄	#14583378	1	1	yes	characteristic
339	C₄	#14583664	1	1	yes	characteristic
340	C₂	—	2	4	no	—
341	C₂	—	172	1	no	—
342	C₂	#19143702	1	1	yes	characteristic
343	C₂	C₂²×#13063264	1	1	yes	characteristic
344	C₂	#504×D₄₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₄₃
commutator, socle	C₂²×C₄₃	C₂³×C₈
Fitting	D₄×C₁₆×C₄₃	C₂
prime radical	C₂×#10621900	C₈
cosocle	C₂×C₈×C₄₃	C₂⁴
hypercenter	D₄×C₁₆	D₄₃
hypocenter	C₄₃	#33504
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₈ ◁ D₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. #19136503	C₂	semidirect
3. #19136500	C₂	semidirect
4. #19136420	C₂	semidirect
5. #19136417	C₂	semidirect
6. D₄×C₄₃⋊C₁₆	C₂	semidirect
7. #19136395	C₂	semidirect
8. #19136392	C₂	semidirect
9. #19136349	C₂	semidirect
10. C₂×Dic₄₃⋊C₁₆	C₂	semidirect
11. D₄×C₁₆×C₄₃	C₂	semidirect
123. D₄×C₄₃	M₃₂	semidirect
174. D₄×C₁₆	D₄₃	semidirect
213. C₄₃	#33504	semidirect
298. D₄	#13063264	semidirect

Maximal subgroups

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

subgroup	index
C₈×#10621900	2
#19136503	2
#19136500	2
#19136420	2
#19136417	2
D₄×C₄₃⋊C₁₆	2
#19136395	2
#19136392	2
#19136349	2
C₂×Dic₄₃⋊C₁₆	2
D₄×C₁₆×C₄₃	2
#33504	43