Subgroups — #24987444

Order 11008.

550 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14582976	C₂	1	1	yes	maximal, characteristic
2	C₄⋊C₄×C₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#13063245	C₂	1	1	yes	maximal, characteristic
4	C₂×#14582628	C₂	1	1	yes	maximal, characteristic
5	C₂×#14582621	C₂	1	1	yes	maximal, characteristic, complemented
6	#19132429	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#1279	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#10621872	—	2	1	no	—
9	C₂×C₄×D₁₇₂	C₂²	1	1	yes	characteristic
10	#14582976	—	2	2	no	—
11	#14582976	C₄	1	4	yes	complemented
12	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
13	C₂×#13063245	C₄	1	2	yes	—
14	C₂×(C₂×D₄₃)⋊C₈	—	2	1	no	—
15	C₂×C₄×C₄₃⋊C₈	—	2	1	no	—
16	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
18	#14582661	C₄	1	2	yes	—
19	#14582654	—	2	1	no	—
20	#14582628	C₄	1	4	yes	complemented
21	#14582621	—	2	2	no	—
22	C₂×C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
24	C₄×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
25	C₂²×D₁₇₂	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×D₄₃	—	2	1	no	—
27	C₂×#10621872	—	2	4	no	—
28	C₂×#10621872	—	4	2	no	—
29	C₂×#10621872	—	4	2	no	—
30	C₂²×C₄₃⋊C₈	—	2	1	no	—
31	C₂²×C₄₃⋊C₈	C₂×C₄	1	2	yes	—
32	C₂²×C₄₃⋊C₈	D₄	1	1	yes	characteristic
33	C₂²×C₄₃⋊C₈	D₄	1	1	yes	characteristic
34	C₄×D₁₇₂	—	2	1	no	—
35	C₄×D₁₇₂	—	2	1	no	—
36	C₄×D₁₇₂	C₂×C₄	1	2	yes	—
37	C₄×D₁₇₂	C₂×C₄	1	2	yes	—
38	C₂×C₄⋊Dic₄₃	C₂×C₄	1	1	yes	characteristic
39	C₂×(C₂×D₄₃)⋊C₄	—	2	1	no	—
40	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
41	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	2	yes	—
42	#13063245	—	2	4	no	—
43	(C₂×D₄₃)⋊C₈	—	2	4	no	—
44	C₄×C₄₃⋊C₈	—	2	1	no	—
45	C₄×C₄₃⋊C₈	—	2	1	no	—
46	C₄×C₄₃⋊C₈	—	4	2	no	—
47	C₄×C₄₃⋊C₈	C₂×C₄	1	2	yes	—
48	C₄×C₄₃⋊C₈	D₄	1	2	yes	—
49	C₂²×C₈×C₄₃	—	2	1	no	—
50	C₂²×C₈×C₄₃	—	2	1	no	—
51	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₄₃	—	2	1	no	—
53	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
54	C₄⋊C₈×C₄₃	—	2	1	no	—
55	C₄⋊C₈×C₄₃	C₂×C₄	1	2	yes	—
56	C₄×C₈×C₄₃	—	2	1	no	—
57	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
58	#95×C₄₃	—	2	1	no	—
59	C₂³×D₄₃	—	2	1	no	—
60	C₂²×C₄×C₄₃	—	2	2	no	—
61	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
64	C₂²×Dic₄₃	—	2	1	no	—
65	C₂×D₁₇₂	—	2	1	no	—
66	C₂×D₁₇₂	—	2	2	no	—
67	C₂×D₁₇₂	—	2	2	no	—
68	C₂×D₁₇₂	C₄²	1	2	yes	—
69	C₂×C₄×D₄₃	—	2	2	no	—
70	C₂×C₄×D₄₃	—	2	2	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×C₈×C₄₃	—	2	2	no	—
76	C₂×C₈×C₄₃	—	2	2	no	—
77	C₂×C₈×C₄₃	—	4	2	no	—
78	C₂×C₈×C₄₃	—	4	1	no	—
79	C₂×C₈×C₄₃	—	4	1	no	—
80	C₄⋊C₄×C₄₃	—	2	2	no	—
81	C₄⋊C₄×C₄₃	M₁₆	1	4	yes	complemented
82	C₄²×C₄₃	—	4	1	no	—
83	C₄²×C₄₃	—	4	1	no	—
84	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₄₃	#41	1	1	yes	characteristic
88	(C₂×D₄₃)⋊C₄	—	2	2	no	—
89	(C₂×D₄₃)⋊C₄	—	2	2	no	—
90	C₄⋊Dic₄₃	—	2	1	no	—
91	C₄⋊Dic₄₃	C₄²	1	2	yes	—
92	C₂×C₄₃⋊C₈	—	2	1	no	—
93	C₂×C₄₃⋊C₈	—	2	1	no	—
94	C₂×C₄₃⋊C₈	—	2	2	no	—
95	C₂×C₄₃⋊C₈	—	2	2	no	—
96	C₂×C₄₃⋊C₈	—	2	2	no	—
97	C₂×C₄₃⋊C₈	—	2	2	no	—
98	C₂×C₄₃⋊C₈	—	4	1	no	—
99	C₂×C₄₃⋊C₈	—	4	1	no	—
100	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	2	yes	—
101	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	2	yes	—
102	C₂×C₄₃⋊C₈	D₈	1	2	yes	—
103	C₂×C₄₃⋊C₈	SD₁₆	1	2	yes	—
104	#10621872	—	4	8	no	—
105	C₂³×C₄₃	C₄×D₄	1	1	yes	characteristic
106	C₂²×D₄₃	—	2	2	no	—
107	C₂²×D₄₃	—	4	2	no	—
108	C₂²×D₄₃	—	4	1	no	—
109	C₂²×D₄₃	—	4	1	no	—
110	C₂²×D₄₃	—	4	2	no	—
111	C₂×C₄×C₄₃	—	2	2	no	—
112	C₂×C₄×C₄₃	—	2	4	no	—
113	C₂×C₄×C₄₃	—	4	2	no	—
114	C₂×C₄×C₄₃	—	4	2	no	—
115	C₂×C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
116	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
118	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₄₃	C₂×M₁₆	1	2	yes	—
122	C₂×C₄×C₄₃	#131	1	2	yes	—
123	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
124	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
125	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
126	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
127	C₂×Dic₄₃	—	2	2	no	—
128	C₂×Dic₄₃	—	4	1	no	—
129	C₂×Dic₄₃	—	4	1	no	—
130	D₁₇₂	—	2	2	no	—
131	D₁₇₂	—	2	2	no	—
132	D₁₇₂	—	4	2	no	—
133	C₄×D₄₃	—	4	2	no	—
134	C₄×D₄₃	—	4	2	no	—
135	C₄×D₄₃	—	4	2	no	—
136	C₄×D₄₃	—	4	2	no	—
137	C₈×C₄₃	—	4	2	no	—
138	C₈×C₄₃	—	4	2	no	—
139	C₄₃⋊C₈	—	2	2	no	—
140	C₄₃⋊C₈	—	4	2	no	—
141	C₄₃⋊C₈	—	4	2	no	—
142	C₄₃⋊C₈	#102	1	4	yes	complemented
143	#8210	—	43	1	no	maximal, Sylow
144	C₂²×C₄₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
145	C₂²×C₄₃	#435	1	2	yes	—
146	C₂²×C₄₃	C₄×D₈	1	1	yes	characteristic
147	C₂²×C₄₃	C₄×SD₁₆	1	1	yes	characteristic
148	C₂²×C₄₃	#444	1	2	yes	—
149	C₂×D₄₃	—	4	2	no	—
150	C₂×D₄₃	—	4	2	no	—
151	C₂×D₄₃	—	4	2	no	—
152	C₂×D₄₃	—	8	2	no	—
153	C₂×D₄₃	—	8	2	no	—
154	C₄×C₄₃	—	2	1	no	—
155	C₄×C₄₃	—	2	1	no	—
156	C₄×C₄₃	—	4	4	no	—
157	C₄×C₄₃	C₄×M₁₆	1	1	yes	characteristic
158	C₄×C₄₃	#405	1	1	yes	characteristic
159	C₄×C₄₃	C₂×#102	1	1	yes	characteristic
160	C₄×C₄₃	#416	1	1	yes	characteristic
161	C₄×C₄₃	#436	1	2	yes	—
162	C₄×C₄₃	C₄×D₈	1	1	yes	characteristic
163	C₄×C₄₃	C₄×SD₁₆	1	1	yes	characteristic
164	Dic₄₃	—	4	2	no	—
165	C₂×#436	—	43	1	no	—
166	C₈×C₄⋊C₄	—	43	1	no	—
167	C₄×#102	—	43	1	no	—
168	C₂×D₄⋊C₈	—	43	1	no	—
169	C₂×#322	—	43	1	no	—
170	#1279	D₄₃	1	1	yes	characteristic, complemented
171	#1277	—	43	1	no	—
172	C₂×C₄₃	#1581	1	2	yes	—
173	C₂×C₄₃	#1583	1	2	yes	—
174	C₂×C₄₃	#1753	1	1	yes	characteristic
175	C₂×C₄₃	C₄×#102	1	1	yes	characteristic
176	C₂×C₄₃	#1765	1	1	yes	characteristic
177	D₄₃	—	8	2	no	—
178	C₂²×M₁₆	—	86	1	no	—
179	C₂×C₄×D₄	—	43	1	no	—
180	#436	—	43	4	no	—
181	#436	—	86	2	no	—
182	C₂×C₄⋊C₈	—	43	1	no	—
183	C₂×C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
184	C₂×#102	—	43	2	no	—
185	C₂×C₂²⋊C₈	—	86	1	no	—
186	C₂×C₄×C₈	—	43	1	no	—
187	C₂×C₄×C₈	—	43	1	no	—
188	C₂×C₄×C₈	—	86	1	no	—
189	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
190	C₄×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
191	#340	—	43	2	no	—
192	#336	—	86	1	no	—
193	D₄⋊C₈	—	43	4	no	—
194	#322	—	86	2	no	—
195	C₄₃	#8210	1	1	yes	characteristic, complemented, Sylow
196	C₂²×D₄	—	43	1	no	—
197	C₂³×C₄	—	86	1	no	—
198	C₂×M₁₆	—	86	4	no	—
199	C₂×M₁₆	—	172	2	no	—
200	C₂×M₁₆	—	172	2	no	—
201	C₂²×C₈	—	2	1	no	—
202	C₂²×C₈	—	2	1	no	—
203	C₂²×C₈	—	43	1	no	—
204	C₂²×C₈	—	43	1	no	—
205	C₂²×C₈	—	43	2	no	—
206	C₂²×C₈	—	86	1	no	—
207	C₄×D₄	—	43	2	no	—
208	C₄×D₄	—	43	2	no	—
209	C₄×D₄	—	86	1	no	—
210	C₄×D₄	—	86	1	no	—
211	C₂×C₄⋊C₄	—	43	1	no	—
212	C₂×C₄⋊C₄	C₄×D₄₃	1	2	yes	—
213	C₂×C₂²⋊C₄	—	86	1	no	—
214	C₂×C₄²	—	2	1	no	—
215	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
216	C₄⋊C₈	—	2	1	no	—
217	C₄⋊C₈	—	43	2	no	—
218	C₄⋊C₈	—	86	1	no	—
219	C₄⋊C₈	C₄×D₄₃	1	2	yes	—
220	#102	—	86	4	no	—
221	C₂²⋊C₈	—	86	4	no	—
222	C₄×C₈	—	2	1	no	—
223	C₄×C₈	—	43	2	no	—
224	C₄×C₈	—	43	2	no	—
225	C₄×C₈	—	86	1	no	—
226	C₄×C₈	—	86	1	no	—
227	C₄×C₈	—	172	2	no	—
228	C₄×C₈	C₄×D₄₃	1	2	yes	—
229	#95	—	2	1	no	—
230	C₂⁴	—	86	1	no	—
231	C₂×D₄	—	43	2	no	—
232	C₂×D₄	—	86	1	no	—
233	C₂×D₄	—	86	2	no	—
234	C₂×D₄	—	86	2	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	86	1	no	—
237	C₂²×C₄	—	86	2	no	—
238	C₂²×C₄	—	86	2	no	—
239	C₂²×C₄	—	172	1	no	—
240	C₂²×C₄	—	172	1	no	—
241	C₂²×C₄	—	172	1	no	—
242	C₂²×C₄	—	172	1	no	—
243	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
244	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
245	C₂²×C₄	#10621900	1	1	yes	characteristic
246	M₁₆	—	172	8	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	4	2	no	—
250	C₂×C₈	—	4	1	no	—
251	C₂×C₈	—	4	1	no	—
252	C₂×C₈	—	43	2	no	—
253	C₂×C₈	—	43	2	no	—
254	C₂×C₈	—	43	2	no	—
255	C₂×C₈	—	43	2	no	—
256	C₂×C₈	—	86	1	no	—
257	C₂×C₈	—	86	1	no	—
258	C₂×C₈	—	86	2	no	—
259	C₂×C₈	—	86	2	no	—
260	C₂×C₈	—	86	2	no	—
261	C₂×C₈	—	86	2	no	—
262	C₂×C₈	—	172	1	no	—
263	C₂×C₈	—	172	1	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	43	2	no	—
266	C₄⋊C₄	—	86	1	no	—
267	C₄⋊C₄	#10621872	1	4	yes	complemented
268	C₂²⋊C₄	—	86	2	no	—
269	C₂²⋊C₄	—	86	2	no	—
270	C₄²	—	4	1	no	—
271	C₄²	—	4	1	no	—
272	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
273	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
274	C₄²	D₄×D₄₃	1	1	yes	characteristic
275	C₄²	#10621902	1	1	yes	characteristic
276	C₂³	—	86	2	no	—
277	C₂³	—	172	2	no	—
278	C₂³	—	172	1	no	—
279	C₂³	—	172	1	no	—
280	C₂³	—	172	2	no	—
281	C₂³	(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
282	D₄	—	86	2	no	—
283	D₄	—	86	2	no	—
284	D₄	—	172	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	86	2	no	—
290	C₂×C₄	—	172	1	no	—
291	C₂×C₄	—	172	1	no	—
292	C₂×C₄	—	172	2	no	—
293	C₂×C₄	—	172	2	no	—
294	C₂×C₄	—	172	2	no	—
295	C₂×C₄	—	172	2	no	—
296	C₂×C₄	C₄²×D₄₃	1	1	yes	characteristic
297	C₂×C₄	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
298	C₂×C₄	#13063283	1	1	yes	characteristic
299	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
300	C₂×C₄	#13063287	1	1	yes	characteristic
301	C₂×C₄	#13063287	1	1	yes	characteristic
302	C₂×C₄	C₂×#10621872	1	2	yes	—
303	C₂×C₄	#13063333	1	2	yes	—
304	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
305	C₂×C₄	SD₁₆×D₄₃	1	1	yes	characteristic
306	C₂×C₄	#13063346	1	1	yes	characteristic
307	C₂×C₄	#13063348	1	1	yes	characteristic
308	C₈	—	4	2	no	—
309	C₈	—	4	2	no	—
310	C₈	—	43	4	no	—
311	C₈	—	86	2	no	—
312	C₈	—	172	2	no	—
313	C₈	—	172	2	no	—
314	C₂²	—	344	2	no	—
315	C₂²	—	344	2	no	—
316	C₂²	—	172	2	no	—
317	C₂²	—	172	2	no	—
318	C₂²	—	172	2	no	—
319	C₂²	#14582746	1	1	yes	characteristic
320	C₂²	#14582967	1	2	yes	—
321	C₂²	#14582979	1	1	yes	characteristic
322	C₂²	#14582984	1	1	yes	characteristic
323	C₂²	#14583018	1	2	yes	—
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	4	4	no	—
327	C₄	—	172	2	no	—
328	C₄	C₄×#10621872	1	1	yes	characteristic
329	C₄	#14582856	1	1	yes	characteristic
330	C₄	#102×D₄₃	1	1	yes	characteristic
331	C₄	#14582894	1	1	yes	characteristic
332	C₄	#14582976	1	2	yes	—
333	C₄	#14582981	1	1	yes	characteristic
334	C₄	#14582986	1	1	yes	characteristic
335	C₂	—	344	2	no	—
336	C₂	#19133421	1	2	yes	—
337	C₂	#19133435	1	2	yes	—
338	C₂	#19134375	1	1	yes	characteristic
339	C₂	#19134409	1	1	yes	characteristic
340	C₂	#19134436	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1279	C₂
socle	C₂³×C₄₃	C₄×D₄
prime radical	C₂²×D₁₇₂	C₂×C₄
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#1279	D₄₃
hypocenter	C₄₃	#8210
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1279

Complements

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

N	H ≅ G/N	type
2. C₄⋊C₄×C₄₃⋊C₈	C₂	semidirect
5. C₂×#14582621	C₂	semidirect
6. #19132429	C₂	semidirect
7. C₄₃×#1279	C₂	semidirect
11. #14582976	C₄	semidirect
20. #14582628	C₄	semidirect
81. C₄⋊C₄×C₄₃	M₁₆	semidirect
142. C₄₃⋊C₈	#102	semidirect
170. #1279	D₄₃	semidirect
195. C₄₃	#8210	semidirect
267. C₄⋊C₄	#10621872	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14582976	2
C₄⋊C₄×C₄₃⋊C₈	2
C₄×#13063245	2
C₂×#14582628	2
C₂×#14582621	2
#19132429	2
C₄₃×#1279	2
#8210	43