Subgroups — #25115471

Order 11008.

730 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583029	C₂	1	1	yes	maximal, characteristic
2	#19137553	C₂	1	2	yes	maximal
3	#19135363	C₂	1	2	yes	maximal
4	#19135362	C₂	1	2	yes	maximal
5	#19135357	C₂	1	2	yes	maximal
6	#19134760	C₂	1	2	yes	maximal
7	#19134759	C₂	1	1	yes	maximal, characteristic
8	#19134495	C₂	1	2	yes	maximal
9	C₄×#106×C₄₃	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄×Dic₈₆	C₂²	1	1	yes	characteristic
11	#14583029	C₄	1	8	yes	complemented
12	C₂×#13063259	C₂²	1	2	yes	—
13	C₂×#13063258	C₂²	1	2	yes	—
14	C₂×#13063257	C₂²	1	1	yes	characteristic
15	C₂×#13063257	C₂²	1	1	yes	characteristic
16	C₂×#13063256	C₂²	1	2	yes	—
17	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
18	C₂×Dic₄₃⋊C₈	C₂²	1	2	yes	—
19	C₂×#13063235	C₂²	1	2	yes	—
20	#14582803	—	2	2	no	—
21	#14582783	—	2	2	no	—
22	#14582782	—	2	2	no	—
23	#14582768	C₂²	1	2	yes	—
24	#14582765	C₂²	1	2	yes	—
25	#14582764	C₂²	1	2	yes	—
26	C₄⋊C₄×Dic₄₃	C₂²	1	2	yes	—
27	C₄⋊C₄×Dic₄₃	C₂²	1	2	yes	—
28	C₄×Dic₄₃⋊C₄	—	2	2	no	—
29	#14582661	C₂²	1	2	yes	—
30	#14582661	C₂²	1	2	yes	—
31	C₂×#106×C₄₃	C₂²	1	1	yes	characteristic
32	C₂×#106×C₄₃	C₂²	1	1	yes	characteristic
33	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
34	C₄×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
35	C₄₃×#340	C₂²	1	2	yes	—
36	C₂²×Dic₈₆	D₄	1	1	yes	characteristic
37	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
38	C₄×Dic₈₆	C₂×C₄	1	4	yes	—
39	C₄×Dic₈₆	Q₈	1	4	yes	—
40	C₂×C₄⋊Dic₄₃	D₄	1	1	yes	characteristic
41	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
42	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
43	C₂×Dic₄₃⋊C₄	—	2	2	no	—
44	C₂×Dic₄₃⋊C₄	—	2	2	no	—
45	C₂×Dic₄₃⋊C₄	D₄	1	2	yes	—
46	C₂×C₄×Dic₄₃	—	2	2	no	—
47	C₂×C₄×Dic₄₃	—	2	2	no	—
48	C₂×C₄×Dic₄₃	C₂³	1	2	yes	—
49	#13063259	—	2	4	no	—
50	#13063258	—	4	2	no	—
51	#13063257	—	4	1	no	—
52	#13063257	—	4	1	no	—
53	#13063256	—	4	2	no	—
54	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	4	yes	—
55	Dic₄₃⋊C₈	C₂×C₄	1	8	yes	—
56	#13063235	C₂×C₄	1	8	yes	—
57	C₄₃⋊#95	—	2	2	no	—
58	C₄₃⋊#95	—	2	2	no	—
59	C₄₃⋊#95	—	2	2	no	—
60	C₄₃⋊#95	—	2	2	no	—
61	C₄₃⋊#95	—	2	2	no	—
62	C₄₃⋊#95	—	2	2	no	—
63	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
64	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
65	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
66	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
67	C₂×C₄²×C₄₃	—	2	2	no	—
68	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
69	#106×C₄₃	—	2	2	no	—
70	#106×C₄₃	Q₈	1	4	yes	complemented
71	C₄×C₈×C₄₃	C₂×C₄	1	4	yes	—
72	#95×C₄₃	—	2	2	no	—
73	C₂²×C₄×C₄₃	—	2	2	no	—
74	C₂²×C₄×C₄₃	—	2	2	no	—
75	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
78	C₂²×Dic₄₃	—	2	2	no	—
79	C₂²×Dic₄₃	—	2	2	no	—
80	C₂²×Dic₄₃	C₂×D₄	1	2	yes	—
81	C₂²×Dic₄₃	C₂×D₄	1	2	yes	—
82	C₂×Dic₈₆	—	4	2	no	—
83	C₂×Dic₈₆	C₄⋊C₄	1	4	yes	—
84	C₂×C₈×C₄₃	—	2	1	no	—
85	C₂×C₈×C₄₃	—	2	1	no	—
86	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
87	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
88	C₂×C₈×C₄₃	C₂×Q₈	1	2	yes	—
89	C₂×C₈×C₄₃	#41	1	2	yes	—
90	C₄⋊C₄×C₄₃	—	2	2	no	—
91	C₄⋊C₄×C₄₃	—	2	2	no	—
92	C₄⋊C₄×C₄₃	—	2	2	no	—
93	C₄⋊C₄×C₄₃	C₂×Q₈	1	4	yes	—
94	C₄²×C₄₃	—	8	2	no	—
95	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
96	C₄²×C₄₃	C₂×Q₈	1	2	yes	—
97	C₄⋊Dic₄₃	—	2	2	no	—
98	C₄⋊Dic₄₃	—	2	2	no	—
99	C₄⋊Dic₄₃	—	2	2	no	—
100	C₄⋊Dic₄₃	—	2	2	no	—
101	C₄⋊Dic₄₃	C₄⋊C₄	1	4	yes	—
102	Dic₄₃⋊C₄	—	4	4	no	—
103	Dic₄₃⋊C₄	—	8	2	no	—
104	Dic₄₃⋊C₄	C₄⋊C₄	1	8	yes	—
105	C₄×Dic₄₃	—	4	4	no	—
106	C₄×Dic₄₃	—	8	2	no	—
107	C₄×Dic₄₃	C₂²×C₄	1	4	yes	—
108	C₄×Dic₄₃	C₂×Q₈	1	4	yes	—
109	C₂×C₄₃⋊C₈	—	2	2	no	—
110	C₂×C₄₃⋊C₈	—	2	2	no	—
111	C₂×C₄₃⋊C₈	C₂²×C₄	1	4	yes	—
112	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₄₃	—	2	2	no	—
114	C₂×C₄×C₄₃	—	2	2	no	—
115	C₂×C₄×C₄₃	—	2	4	no	—
116	C₂×C₄×C₄₃	—	4	2	no	—
117	C₂×C₄×C₄₃	—	4	2	no	—
118	C₂×C₄×C₄₃	—	4	2	no	—
119	C₂×C₄×C₄₃	—	4	2	no	—
120	C₂×C₄×C₄₃	—	4	2	no	—
121	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
122	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
123	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
124	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
125	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
126	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
127	C₂×C₄×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
128	C₂×C₄×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
129	C₂×Dic₄₃	—	2	2	no	—
130	C₂×Dic₄₃	—	2	2	no	—
131	C₂×Dic₄₃	—	2	2	no	—
132	C₂×Dic₄₃	—	2	2	no	—
133	C₂×Dic₄₃	—	4	2	no	—
134	C₂×Dic₄₃	—	4	2	no	—
135	C₂×Dic₄₃	—	4	2	no	—
136	C₂×Dic₄₃	—	4	2	no	—
137	C₂×Dic₄₃	—	4	2	no	—
138	C₂×Dic₄₃	—	4	2	no	—
139	C₂×Dic₄₃	C₂×C₄⋊C₄	1	4	yes	—
140	C₂×Dic₄₃	C₂×C₄⋊C₄	1	4	yes	—
141	Dic₈₆	—	4	4	no	—
142	C₈×C₄₃	—	2	2	no	—
143	C₈×C₄₃	C₄×Q₈	1	4	yes	—
144	C₄₃⋊C₈	—	2	4	no	—
145	#22730	—	43	1	no	maximal, Sylow
146	C₂²×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
147	C₂²×C₄₃	D₄×Q₈	1	1	yes	characteristic
148	C₂²×C₄₃	#550	1	1	yes	characteristic
149	C₂²×C₄₃	#575	1	2	yes	—
150	C₂²×C₄₃	#577	1	2	yes	—
151	C₄×C₄₃	—	2	2	no	—
152	C₄×C₄₃	—	2	2	no	—
153	C₄×C₄₃	—	4	4	no	—
154	C₄×C₄₃	—	8	2	no	—
155	C₄×C₄₃	C₂×C₄×Q₈	1	1	yes	characteristic
156	C₄×C₄₃	#520	1	1	yes	characteristic
157	C₄×C₄₃	#557	1	1	yes	characteristic
158	C₄×C₄₃	Q₈²	1	1	yes	characteristic
159	Dic₄₃	—	2	4	no	—
160	Dic₄₃	—	2	4	no	—
161	Dic₄₃	—	8	2	no	—
162	Dic₄₃	—	8	2	no	—
163	C₂×#446	—	43	1	no	—
164	#2355	—	43	2	no	—
165	#1949	—	43	2	no	—
166	#1949	—	43	2	no	—
167	#1948	—	43	2	no	—
168	#1847	—	43	2	no	—
169	#1846	—	43	1	no	—
170	#1776	—	43	2	no	—
171	C₄×#106	D₄₃	1	1	yes	characteristic
172	C₂×C₄₃	Q₈×C₄⋊C₄	1	1	yes	characteristic
173	C₂×C₄₃	#2911	1	1	yes	characteristic
174	C₂×C₄₃	#2912	1	1	yes	characteristic
175	C₂×C₄₃	#3388	1	2	yes	—
176	C₂×C₄₃	#3401	1	2	yes	—
177	C₂×C₄×Q₈	—	43	1	no	—
178	#446	—	43	8	no	—
179	C₂×#107	—	43	2	no	—
180	C₂×#107	—	43	2	no	—
181	C₂×#106	—	43	1	no	—
182	C₂×#106	—	43	1	no	—
183	C₂×#106	—	43	2	no	—
184	C₂×#106	C₂×D₄₃	1	1	yes	characteristic
185	C₂×#106	C₂×D₄₃	1	1	yes	characteristic
186	C₂×C₄⋊C₈	—	43	2	no	—
187	C₂×C₄⋊C₈	—	43	1	no	—
188	C₂×#97	—	43	2	no	—
189	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
190	#391	—	86	2	no	—
191	#387	—	86	2	no	—
192	#387	—	86	2	no	—
193	#384	—	43	2	no	—
194	#382	—	43	2	no	—
195	#382	—	43	2	no	—
196	C₄×C₄⋊C₄	—	43	2	no	—
197	C₄×C₄⋊C₄	—	43	2	no	—
198	C₄×C₄⋊C₄	—	86	2	no	—
199	C₄×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
200	#340	—	43	2	no	—
201	#340	—	43	2	no	—
202	#340	C₂×D₄₃	1	2	yes	—
203	C₄₃	#22730	1	1	yes	characteristic, complemented, Sylow
204	C₂²×Q₈	—	43	1	no	—
205	C₂²×C₈	—	43	2	no	—
206	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
207	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
208	C₄×Q₈	—	43	4	no	—
209	C₄×Q₈	—	43	4	no	—
210	C₂×C₄⋊C₄	—	43	2	no	—
211	C₂×C₄⋊C₄	—	43	1	no	—
212	C₂×C₄⋊C₄	—	43	2	no	—
213	C₂×C₄⋊C₄	—	43	2	no	—
214	C₂×C₄⋊C₄	—	86	2	no	—
215	C₂×C₄⋊C₄	—	86	2	no	—
216	C₂×C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
217	C₂×C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
218	C₂×C₄²	—	2	2	no	—
219	C₂×C₄²	—	43	2	no	—
220	C₂×C₄²	—	86	2	no	—
221	C₂×C₄²	—	86	2	no	—
222	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
223	#107	—	86	4	no	—
224	#107	—	172	2	no	—
225	#106	—	2	2	no	—
226	#106	—	172	2	no	—
227	#106	—	172	1	no	—
228	#106	—	172	1	no	—
229	#106	Dic₈₆	1	4	yes	complemented
230	C₄⋊C₈	—	43	8	no	—
231	C₄⋊C₈	—	43	4	no	—
232	#97	—	43	8	no	—
233	C₄×C₈	C₄×D₄₃	1	4	yes	—
234	#95	—	2	2	no	—
235	#95	—	86	2	no	—
236	#95	—	86	2	no	—
237	#95	—	86	2	no	—
238	#95	—	86	2	no	—
239	#95	—	86	2	no	—
240	#95	—	86	2	no	—
241	C₂×Q₈	—	43	4	no	—
242	C₂×Q₈	—	172	2	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	—	43	2	no	—
246	C₂²×C₄	—	43	2	no	—
247	C₂²×C₄	—	86	2	no	—
248	C₂²×C₄	—	86	2	no	—
249	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
250	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
251	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
252	C₂×C₈	—	2	1	no	—
253	C₂×C₈	—	2	1	no	—
254	C₂×C₈	—	43	4	no	—
255	C₂×C₈	—	86	2	no	—
256	C₂×C₈	—	86	2	no	—
257	C₂×C₈	C₂×Dic₈₆	1	2	yes	—
258	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
259	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
260	C₂×C₈	#10621898	1	2	yes	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	43	8	no	—
265	C₄⋊C₄	—	43	4	no	—
266	C₄⋊C₄	—	86	2	no	—
267	C₄⋊C₄	—	86	2	no	—
268	C₄⋊C₄	—	86	2	no	—
269	C₄⋊C₄	—	86	2	no	—
270	C₄⋊C₄	—	344	2	no	—
271	C₄⋊C₄	—	172	4	no	—
272	C₄⋊C₄	C₂×Dic₈₆	1	4	yes	—
273	C₄²	—	8	2	no	—
274	C₄²	—	43	4	no	—
275	C₄²	—	43	4	no	—
276	C₄²	—	344	2	no	—
277	C₄²	—	172	4	no	—
278	C₄²	C₂×C₄×D₄₃	1	2	yes	—
279	C₄²	Q₈×D₄₃	1	2	yes	—
280	C₂³	C₂×D₄×D₄₃	1	1	yes	characteristic
281	Q₈	—	172	4	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	43	4	no	—
291	C₂×C₄	—	43	4	no	—
292	C₂×C₄	—	86	2	no	—
293	C₂×C₄	—	86	2	no	—
294	C₂×C₄	—	86	2	no	—
295	C₂×C₄	—	86	2	no	—
296	C₂×C₄	—	172	2	no	—
297	C₂×C₄	—	172	2	no	—
298	C₂×C₄	—	172	2	no	—
299	C₂×C₄	—	172	2	no	—
300	C₂×C₄	—	172	2	no	—
301	C₂×C₄	—	172	2	no	—
302	C₂×C₄	C₄⋊C₄×D₄₃	1	2	yes	—
303	C₂×C₄	C₄⋊C₄×D₄₃	1	2	yes	—
304	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
305	C₂×C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
306	C₂×C₄	C₂²×Dic₈₆	1	1	yes	characteristic
307	C₂×C₄	C₂×#10621898	1	1	yes	characteristic
308	C₂×C₄	#13063371	1	1	yes	characteristic
309	C₂×C₄	#13063372	1	1	yes	characteristic
310	C₈	—	2	2	no	—
311	C₈	—	86	4	no	—
312	C₈	C₄×Dic₈₆	1	4	yes	—
313	C₂²	C₂×C₄⋊C₄×D₄₃	1	1	yes	characteristic
314	C₂²	D₄×Dic₈₆	1	1	yes	characteristic
315	C₂²	#14583570	1	1	yes	characteristic
316	C₂²	#14583696	1	2	yes	—
317	C₂²	#14583705	1	2	yes	—
318	C₄	—	2	2	no	—
319	C₄	—	2	2	no	—
320	C₄	—	4	4	no	—
321	C₄	—	8	2	no	—
322	C₄	—	86	4	no	—
323	C₄	—	86	4	no	—
324	C₄	—	344	2	no	—
325	C₄	—	344	2	no	—
326	C₄	C₂×C₄×Dic₈₆	1	1	yes	characteristic
327	C₄	#14583379	1	1	yes	characteristic
328	C₄	#14583627	1	1	yes	characteristic
329	C₄	Q₈×Dic₈₆	1	1	yes	characteristic
330	C₂	C₄⋊C₄×Dic₈₆	1	1	yes	characteristic
331	C₂	#19143183	1	1	yes	characteristic
332	C₂	#19143186	1	1	yes	characteristic
333	C₂	#19147536	1	2	yes	—
334	C₂	#19147657	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄×#106×C₄₃	C₂
socle, prime radical	C₂³×C₄₃	C₂²×D₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₄×#106	D₄₃
hypocenter	C₄₃	#22730
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×#106

Complements

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

N	H ≅ G/N	type
11. #14583029	C₄	semidirect
70. #106×C₄₃	Q₈	semidirect
203. C₄₃	#22730	semidirect
229. #106	Dic₈₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583029	2
#19137553	2
#19135363	2
#19135362	2
#19135357	2
#19134760	2
#19134759	2
#19134495	2
C₄×#106×C₄₃	2
#22730	43