Subgroups — #25005384

Order 11008.

886 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14582661	C₂	1	4	yes	maximal
2	C₂×#14582654	C₂	1	1	yes	maximal, characteristic
3	#19132825	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄₃×#384	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₄₃⋊C₈	C₂²	1	2	yes	—
6	C₂²×C₄⋊Dic₄₃	C₂²	1	2	yes	—
7	C₂×C₄₃⋊#95	—	2	1	no	—
8	C₂×C₄₃⋊#95	—	2	1	no	—
9	#14582661	—	2	8	no	—
10	#14582654	—	2	2	no	—
11	C₂²×C₄⋊C₄×C₄₃	—	2	1	no	—
12	C₂²×C₄⋊C₄×C₄₃	—	2	1	no	—
13	C₂²×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
14	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
15	C₄₃×#384	—	2	2	no	—
16	C₄₃×#384	C₄	1	4	yes	complemented
17	C₂³×Dic₄₃	—	2	2	no	—
18	C₂²×C₄₃⋊C₈	—	2	2	no	—
19	C₂²×C₄₃⋊C₈	—	2	4	no	—
20	C₂²×C₄₃⋊C₈	—	2	4	no	—
21	C₂²×C₄₃⋊C₈	—	2	2	no	—
22	C₂²×C₄₃⋊C₈	Q₈	1	4	yes	—
23	C₂×C₄⋊Dic₄₃	—	2	4	no	—
24	C₂×C₄⋊Dic₄₃	—	2	4	no	—
25	C₂×C₄⋊Dic₄₃	—	2	2	no	—
26	C₂×C₄⋊Dic₄₃	D₄	1	4	yes	—
27	C₄₃⋊#95	—	4	2	no	—
28	C₄₃⋊#95	—	4	2	no	—
29	C₂³×C₄×C₄₃	—	2	2	no	—
30	C₂³×C₄×C₄₃	D₄	1	1	yes	characteristic
31	C₂³×C₄×C₄₃	D₄	1	1	yes	characteristic
32	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
33	C₂×C₄⋊C₄×C₄₃	—	2	4	no	—
34	C₂×C₄⋊C₄×C₄₃	—	2	4	no	—
35	C₂×C₄⋊C₄×C₄₃	—	2	4	no	—
36	C₂×C₄⋊C₄×C₄₃	—	2	4	no	—
37	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
38	C₂×C₄⋊C₄×C₄₃	—	4	4	no	—
39	C₂×C₄⋊C₄×C₄₃	—	4	2	no	—
40	C₂×C₄⋊C₄×C₄₃	—	4	2	no	—
41	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	4	yes	—
42	C₂×C₄²×C₄₃	—	2	4	no	—
43	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₄₃	D₄	1	2	yes	—
45	C₂⁴×C₄₃	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₄₃	—	2	4	no	—
47	C₂²×C₄×C₄₃	—	2	2	no	—
48	C₂²×C₄×C₄₃	—	2	2	no	—
49	C₂²×C₄×C₄₃	—	2	2	no	—
50	C₂²×C₄×C₄₃	—	2	2	no	—
51	C₂²×C₄×C₄₃	—	4	4	no	—
52	C₂²×C₄×C₄₃	—	4	2	no	—
53	C₂²×C₄×C₄₃	—	4	2	no	—
54	C₂²×C₄×C₄₃	—	4	4	no	—
55	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₄×C₄₃	D₈	1	2	yes	—
58	C₂²×C₄×C₄₃	SD₁₆	1	2	yes	—
59	C₂²×C₄×C₄₃	SD₁₆	1	2	yes	—
60	C₂²×C₄×C₄₃	Q₁₆	1	2	yes	—
61	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₄₃	C₂×Q₈	1	2	yes	—
65	C₂²×C₄×C₄₃	#41	1	2	yes	—
66	C₂²×C₄×C₄₃	#41	1	4	yes	—
67	C₂²×C₄×C₄₃	#41	1	2	yes	—
68	C₂²×Dic₄₃	—	2	4	no	—
69	C₂²×Dic₄₃	—	4	2	no	—
70	C₂²×Dic₄₃	—	4	2	no	—
71	C₂²×Dic₄₃	—	4	2	no	—
72	C₂²×Dic₄₃	—	4	2	no	—
73	C₂²×Dic₄₃	—	4	2	no	—
74	C₂²×Dic₄₃	—	4	2	no	—
75	C₄⋊C₄×C₄₃	—	2	8	no	—
76	C₄⋊C₄×C₄₃	—	4	8	no	—
77	C₄⋊C₄×C₄₃	—	4	8	no	—
78	C₄⋊C₄×C₄₃	—	4	2	no	—
79	C₄⋊C₄×C₄₃	—	4	2	no	—
80	C₄²×C₄₃	—	2	4	no	—
81	C₄²×C₄₃	—	2	4	no	—
82	C₄⋊Dic₄₃	—	2	8	no	—
83	C₄⋊Dic₄₃	—	4	4	no	—
84	C₂×C₄₃⋊C₈	—	2	4	no	—
85	C₂×C₄₃⋊C₈	—	2	8	no	—
86	C₂×C₄₃⋊C₈	—	4	2	no	—
87	C₂×C₄₃⋊C₈	—	4	4	no	—
88	C₂×C₄₃⋊C₈	—	4	2	no	—
89	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂³×C₄₃	#121	1	2	yes	—
91	C₂³×C₄₃	#123	1	2	yes	—
92	C₂³×C₄₃	#124	1	1	yes	characteristic
93	C₂³×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
94	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
95	C₂³×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
96	C₂³×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
97	C₂³×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
98	C₂³×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
99	C₂³×C₄₃	#137	1	2	yes	—
100	C₂×C₄×C₄₃	—	2	2	no	—
101	C₂×C₄×C₄₃	—	2	2	no	—
102	C₂×C₄×C₄₃	—	2	2	no	—
103	C₂×C₄×C₄₃	—	2	2	no	—
104	C₂×C₄×C₄₃	—	2	4	no	—
105	C₂×C₄×C₄₃	—	2	4	no	—
106	C₂×C₄×C₄₃	—	2	4	no	—
107	C₂×C₄×C₄₃	—	2	4	no	—
108	C₂×C₄×C₄₃	—	4	4	no	—
109	C₂×C₄×C₄₃	—	4	4	no	—
110	C₂×C₄×C₄₃	—	4	4	no	—
111	C₂×C₄×C₄₃	—	8	4	no	—
112	C₂×C₄×C₄₃	—	8	4	no	—
113	C₂×C₄×C₄₃	#102	1	4	yes	—
114	C₂×C₄×C₄₃	#103	1	4	yes	—
115	C₂×C₄×C₄₃	#117	1	2	yes	—
116	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
117	C₂×C₄×C₄₃	C₄×Q₈	1	2	yes	—
118	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
119	C₂×C₄×C₄₃	C₂²⋊Q₈	1	4	yes	—
120	C₂×C₄×C₄₃	#123	1	2	yes	—
121	C₂×C₄×C₄₃	#123	1	4	yes	—
122	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
123	C₂×C₄×C₄₃	#125	1	4	yes	—
124	C₂×C₄×C₄₃	#126	1	4	yes	—
125	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
126	C₂×Dic₄₃	—	4	4	no	—
127	C₂×Dic₄₃	—	4	4	no	—
128	C₂×Dic₄₃	—	4	4	no	—
129	C₂×Dic₄₃	—	8	2	no	—
130	C₂×Dic₄₃	—	8	2	no	—
131	C₂×Dic₄₃	—	8	2	no	—
132	C₂×Dic₄₃	—	8	2	no	—
133	C₄₃⋊C₈	—	4	4	no	—
134	#11357	—	43	1	no	maximal, Sylow
135	C₂²×C₄₃	#381	1	1	yes	characteristic
136	C₂²×C₄₃	#390	1	1	yes	characteristic
137	C₂²×C₄₃	#391	1	1	yes	characteristic
138	C₂²×C₄₃	#397	1	2	yes	—
139	C₂²×C₄₃	#398	1	2	yes	—
140	C₂²×C₄₃	C₂×#102	1	1	yes	characteristic
141	C₂²×C₄₃	C₂×#103	1	1	yes	characteristic
142	C₂²×C₄₃	#418	1	1	yes	characteristic
143	C₂²×C₄₃	#419	1	1	yes	characteristic
144	C₂²×C₄₃	#459	1	2	yes	—
145	C₂²×C₄₃	#460	1	2	yes	—
146	C₂²×C₄₃	#461	1	2	yes	—
147	C₂²×C₄₃	#462	1	2	yes	—
148	C₂²×C₄₃	#480	1	2	yes	—
149	C₂²×C₄₃	#481	1	2	yes	—
150	C₂²×C₄₃	#483	1	2	yes	—
151	C₂²×C₄₃	#484	1	2	yes	—
152	C₂²×C₄₃	#486	1	1	yes	characteristic
153	C₂²×C₄₃	#487	1	1	yes	characteristic
154	C₂²×C₄₃	#489	1	2	yes	—
155	C₂²×C₄₃	#498	1	1	yes	characteristic
156	C₂²×C₄₃	#500	1	1	yes	characteristic
157	C₂²×C₄₃	#501	1	2	yes	—
158	C₄×C₄₃	—	2	4	no	—
159	C₄×C₄₃	—	2	4	no	—
160	C₄×C₄₃	—	8	4	no	—
161	C₄×C₄₃	#387	1	4	yes	—
162	C₄×C₄₃	#399	1	2	yes	—
163	C₄×C₄₃	#400	1	2	yes	—
164	Dic₄₃	—	8	4	no	—
165	C₂×#384	D₄₃	1	1	yes	characteristic
166	C₂×#340	—	43	4	no	—
167	C₂×#336	—	43	1	no	—
168	C₄⋊#95	—	43	1	no	—
169	C₂×C₄₃	#1445	1	1	yes	characteristic
170	C₂×C₄₃	#1823	1	1	yes	characteristic
171	C₂×C₄₃	#1824	1	1	yes	characteristic
172	C₂×C₄₃	#1981	1	1	yes	characteristic
173	C₂×C₄₃	#1982	1	1	yes	characteristic
174	C₂×C₄₃	#1990	1	1	yes	characteristic
175	C₂×C₄₃	#1993	1	1	yes	characteristic
176	C₂×C₄₃	#2071	1	2	yes	—
177	C₂×C₄₃	#2072	1	2	yes	—
178	C₂×C₄₃	#2083	1	2	yes	—
179	C₂×C₄₃	#2084	1	2	yes	—
180	C₂³×C₈	—	43	2	no	—
181	C₂²×C₄⋊C₄	—	2	1	no	—
182	C₂²×C₄⋊C₄	—	2	1	no	—
183	C₂²×C₄⋊C₄	—	43	2	no	—
184	C₂²×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
185	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
186	#384	—	2	2	no	—
187	#384	Dic₄₃	1	4	yes	complemented
188	C₂×#95	—	86	1	no	—
189	C₂×#95	—	86	1	no	—
190	#340	—	86	8	no	—
191	#336	—	86	2	no	—
192	C₄₃	#11357	1	1	yes	characteristic, complemented, Sylow
193	C₂³×C₄	—	2	2	no	—
194	C₂³×C₄	—	86	2	no	—
195	C₂³×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
196	C₂³×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
197	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
198	C₂²×C₈	—	43	4	no	—
199	C₂²×C₈	—	86	4	no	—
200	C₂²×C₈	—	86	4	no	—
201	C₂²×C₈	—	86	2	no	—
202	C₂²×C₈	—	86	2	no	—
203	C₂×C₄⋊C₄	—	2	4	no	—
204	C₂×C₄⋊C₄	—	2	4	no	—
205	C₂×C₄⋊C₄	—	2	4	no	—
206	C₂×C₄⋊C₄	—	2	4	no	—
207	C₂×C₄⋊C₄	—	2	2	no	—
208	C₂×C₄⋊C₄	—	4	4	no	—
209	C₂×C₄⋊C₄	—	4	2	no	—
210	C₂×C₄⋊C₄	—	4	2	no	—
211	C₂×C₄⋊C₄	—	43	4	no	—
212	C₂×C₄⋊C₄	—	86	4	no	—
213	C₂×C₄⋊C₄	—	86	4	no	—
214	C₂×C₄⋊C₄	—	86	2	no	—
215	C₂×C₄⋊C₄	C₂×Dic₄₃	1	4	yes	—
216	C₂×C₄²	—	2	4	no	—
217	C₂×C₄²	C₂×Dic₄₃	1	2	yes	—
218	C₂×C₄²	C₂²⋊D₄₃	1	2	yes	—
219	#95	—	172	2	no	—
220	#95	—	172	2	no	—
221	C₂⁴	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	2	4	no	—
227	C₂²×C₄	—	4	4	no	—
228	C₂²×C₄	—	4	2	no	—
229	C₂²×C₄	—	4	2	no	—
230	C₂²×C₄	—	4	4	no	—
231	C₂²×C₄	—	86	4	no	—
232	C₂²×C₄	—	172	2	no	—
233	C₂²×C₄	—	172	2	no	—
234	C₂²×C₄	—	172	2	no	—
235	C₂²×C₄	—	172	2	no	—
236	C₂²×C₄	—	172	2	no	—
237	C₂²×C₄	—	172	2	no	—
238	C₂²×C₄	#10621882	1	2	yes	—
239	C₂²×C₄	#10621883	1	2	yes	—
240	C₂²×C₄	#10621884	1	2	yes	—
241	C₂²×C₄	#10621885	1	2	yes	—
242	C₂²×C₄	C₂²⋊Dic₄₃	1	2	yes	—
243	C₂²×C₄	C₂²⋊Dic₄₃	1	2	yes	—
244	C₂²×C₄	#10621898	1	4	yes	—
245	C₂²×C₄	D₄×D₄₃	1	2	yes	—
246	C₂²×C₄	#10621900	1	2	yes	—
247	C₂²×C₄	Q₈×D₄₃	1	2	yes	—
248	C₂²×C₄	#10621902	1	2	yes	—
249	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
250	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
251	C₂×C₈	—	86	4	no	—
252	C₂×C₈	—	86	8	no	—
253	C₂×C₈	—	172	2	no	—
254	C₂×C₈	—	172	4	no	—
255	C₂×C₈	—	172	2	no	—
256	C₄⋊C₄	—	2	8	no	—
257	C₄⋊C₄	—	4	8	no	—
258	C₄⋊C₄	—	4	8	no	—
259	C₄⋊C₄	—	4	2	no	—
260	C₄⋊C₄	—	4	2	no	—
261	C₄⋊C₄	—	86	8	no	—
262	C₄⋊C₄	—	172	4	no	—
263	C₄²	—	2	4	no	—
264	C₄²	—	2	4	no	—
265	C₂³	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
266	C₂³	#13063296	1	2	yes	—
267	C₂³	#13063308	1	2	yes	—
268	C₂³	#13063314	1	1	yes	characteristic
269	C₂³	#13063323	1	1	yes	characteristic
270	C₂³	C₂×#10621882	1	1	yes	characteristic
271	C₂³	C₂×#10621884	1	1	yes	characteristic
272	C₂³	C₂×#10621883	1	1	yes	characteristic
273	C₂³	C₂×#10621885	1	1	yes	characteristic
274	C₂³	#13063351	1	2	yes	—
275	C₂³	#13063356	1	2	yes	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	4	4	no	—
285	C₂×C₄	—	4	4	no	—
286	C₂×C₄	—	4	4	no	—
287	C₂×C₄	—	8	4	no	—
288	C₂×C₄	—	8	4	no	—
289	C₂×C₄	—	344	2	no	—
290	C₂×C₄	—	344	2	no	—
291	C₂×C₄	—	344	2	no	—
292	C₂×C₄	—	344	2	no	—
293	C₂×C₄	—	172	4	no	—
294	C₂×C₄	—	172	4	no	—
295	C₂×C₄	—	172	4	no	—
296	C₂×C₄	#13063246	1	4	yes	—
297	C₂×C₄	#13063249	1	4	yes	—
298	C₂×C₄	#13063284	1	2	yes	—
299	C₂×C₄	D₄×Dic₄₃	1	2	yes	—
300	C₂×C₄	Q₈×Dic₄₃	1	2	yes	—
301	C₂×C₄	#13063301	1	2	yes	—
302	C₂×C₄	#13063303	1	4	yes	—
303	C₂×C₄	#13063308	1	2	yes	—
304	C₂×C₄	#13063310	1	4	yes	—
305	C₂×C₄	#13063312	1	1	yes	characteristic
306	C₂×C₄	#13063315	1	4	yes	—
307	C₂×C₄	#13063320	1	4	yes	—
308	C₂×C₄	#13063321	1	1	yes	characteristic
309	C₈	—	172	4	no	—
310	C₂²	#14582762	1	1	yes	characteristic
311	C₂²	#14582802	1	1	yes	characteristic
312	C₂²	#14582806	1	1	yes	characteristic
313	C₂²	#14582825	1	2	yes	—
314	C₂²	#14582830	1	2	yes	—
315	C₂²	C₂×#13063246	1	1	yes	characteristic
316	C₂²	C₂×#13063249	1	1	yes	characteristic
317	C₂²	#14582904	1	1	yes	characteristic
318	C₂²	#14582908	1	1	yes	characteristic
319	C₂²	#14583091	1	2	yes	—
320	C₂²	#14583096	1	2	yes	—
321	C₂²	#14583101	1	2	yes	—
322	C₂²	#14583106	1	2	yes	—
323	C₂²	#14583212	1	2	yes	—
324	C₂²	#14583217	1	2	yes	—
325	C₂²	#14583229	1	2	yes	—
326	C₂²	#14583234	1	2	yes	—
327	C₂²	#14583247	1	1	yes	characteristic
328	C₂²	#14583251	1	1	yes	characteristic
329	C₂²	#14583261	1	2	yes	—
330	C₂²	#14583295	1	1	yes	characteristic
331	C₂²	#14583302	1	1	yes	characteristic
332	C₂²	#14583305	1	2	yes	—
333	C₄	—	2	4	no	—
334	C₄	—	2	4	no	—
335	C₄	—	8	4	no	—
336	C₄	—	344	4	no	—
337	C₄	#14582785	1	4	yes	—
338	C₄	#14582834	1	2	yes	—
339	C₄	#14582837	1	2	yes	—
340	C₂	#19132863	1	1	yes	characteristic
341	C₂	#19134681	1	1	yes	characteristic
342	C₂	#19134685	1	1	yes	characteristic
343	C₂	#19135519	1	1	yes	characteristic
344	C₂	#19135523	1	1	yes	characteristic
345	C₂	#19135557	1	1	yes	characteristic
346	C₂	#19135569	1	1	yes	characteristic
347	C₂	#19135999	1	2	yes	—
348	C₂	#19136004	1	2	yes	—
349	C₂	#19136067	1	2	yes	—
350	C₂	#19136072	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂×C₂²⋊D₄₃
commutator	C₂²×C₄×C₄₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₄₃
Fitting	C₂×C₄₃×#384	C₂
socle, prime radical	C₂⁴×C₄₃	C₂×D₄
cosocle	C₂³×C₄×C₄₃	C₂³
hypercenter	C₂×#384	D₄₃
hypocenter	C₄₃	#11357
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂×#384

Complements

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

N	H ≅ G/N	type
16. C₄₃×#384	C₄	semidirect
187. #384	Dic₄₃	semidirect
192. C₄₃	#11357	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14582661	2
C₂×#14582654	2
#19132825	2
C₂×C₄₃×#384	2
#11357	43