Subgroups — #24985011

Order 11008.

438 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19134738	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×C₄⋊(C₄₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
3	#19132418	C₂	1	2	yes	maximal, complemented
4	#19132416	C₂	1	2	yes	maximal
5	C₄₃×#2860	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
7	C₂×C₄×C₄₃⋊C₈	—	2	1	no	—
8	#14582654	—	2	1	no	—
9	#14582654	—	2	1	no	—
10	#14582654	C₂²	1	2	yes	—
11	#14582654	C₂²	1	2	yes	—
12	C₂×#126×C₄₃	—	2	1	no	—
13	C₄₃×#390	C₄	1	1	yes	characteristic
14	C₄₃×#390	C₄	1	1	yes	characteristic
15	C₄₃×#390	C₂²	1	1	yes	characteristic
16	C₄₃×#389	C₄	1	1	yes	characteristic
17	C₄₃×#389	C₄	1	1	yes	characteristic
18	C₄₃×#389	C₂²	1	1	yes	characteristic
19	C₄₃×#388	—	2	1	no	—
20	C₄₃×#388	—	2	1	no	—
21	C₄₃×#388	—	2	1	no	—
22	C₄³×C₄₃	C₂²	1	1	yes	characteristic
23	C₂²×C₄₃⋊C₈	—	2	2	no	—
24	C₂²×C₄₃⋊C₈	—	2	2	no	—
25	C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
26	C₄×C₄₃⋊C₈	—	4	2	no	—
27	C₂²×D₄×C₄₃	C₂×C₄	1	1	yes	characteristic
28	#126×C₄₃	—	4	2	no	—
29	#126×C₄₃	—	4	2	no	—
30	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
31	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
32	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
33	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₂²⋊C₄×C₄₃	—	2	1	no	—
37	C₂×C₂²⋊C₄×C₄₃	—	2	1	no	—
38	C₂×C₂²⋊C₄×C₄₃	—	2	1	no	—
39	C₂×C₂²⋊C₄×C₄₃	—	2	1	no	—
40	C₂×C₂²⋊C₄×C₄₃	—	2	1	no	—
41	C₂×C₂²⋊C₄×C₄₃	—	2	1	no	—
42	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
49	#95×C₄₃	—	2	1	no	—
50	#95×C₄₃	—	2	1	no	—
51	#95×C₄₃	—	2	1	no	—
52	C₂⁴×C₄₃	—	2	1	no	—
53	C₂×D₄×C₄₃	—	2	2	no	—
54	C₂×D₄×C₄₃	—	4	1	no	—
55	C₂×D₄×C₄₃	—	4	1	no	—
56	C₂²×C₄×C₄₃	—	2	1	no	—
57	C₂²×C₄×C₄₃	—	2	1	no	—
58	C₂²×C₄×C₄₃	—	2	1	no	—
59	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
66	C₄⋊C₄×C₄₃	—	2	2	no	—
67	C₄⋊C₄×C₄₃	—	4	1	no	—
68	C₄⋊C₄×C₄₃	—	4	1	no	—
69	C₄⋊C₄×C₄₃	—	4	2	no	—
70	C₄⋊C₄×C₄₃	—	4	2	no	—
71	C₄⋊C₄×C₄₃	—	4	1	no	—
72	C₄⋊C₄×C₄₃	—	4	1	no	—
73	C₂²⋊C₄×C₄₃	—	4	2	no	—
74	C₂²⋊C₄×C₄₃	—	4	2	no	—
75	C₂²⋊C₄×C₄₃	—	4	2	no	—
76	C₂²⋊C₄×C₄₃	—	4	2	no	—
77	C₂²⋊C₄×C₄₃	—	4	1	no	—
78	C₂²⋊C₄×C₄₃	—	4	1	no	—
79	C₂²⋊C₄×C₄₃	—	4	2	no	—
80	C₄²×C₄₃	—	2	2	no	—
81	C₄²×C₄₃	—	2	2	no	—
82	C₄²×C₄₃	—	2	2	no	—
83	C₄²×C₄₃	—	4	1	no	—
84	C₄²×C₄₃	—	4	1	no	—
85	C₄²×C₄₃	—	4	1	no	—
86	C₄²×C₄₃	D₈	1	2	yes	—
87	C₄²×C₄₃	SD₁₆	1	2	yes	—
88	C₂×C₄₃⋊C₈	—	2	4	no	—
89	C₂×C₄₃⋊C₈	—	4	2	no	—
90	C₂×C₄₃⋊C₈	—	4	2	no	—
91	C₂×C₄₃⋊C₈	—	4	2	no	—
92	C₂×C₄₃⋊C₈	—	4	2	no	—
93	C₂×C₄₃⋊C₈	—	4	2	no	—
94	C₂³×C₄₃	—	4	1	no	—
95	C₂³×C₄₃	—	4	1	no	—
96	C₂³×C₄₃	—	4	1	no	—
97	C₂³×C₄₃	—	4	1	no	—
98	C₂³×C₄₃	—	4	1	no	—
99	C₂³×C₄₃	—	4	1	no	—
100	C₂³×C₄₃	—	4	1	no	—
101	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	D₄×C₄₃	—	8	2	no	—
103	C₂×C₄×C₄₃	—	2	1	no	—
104	C₂×C₄×C₄₃	—	2	1	no	—
105	C₂×C₄×C₄₃	—	2	1	no	—
106	C₂×C₄×C₄₃	—	2	1	no	—
107	C₂×C₄×C₄₃	—	2	1	no	—
108	C₂×C₄×C₄₃	—	2	1	no	—
109	C₂×C₄×C₄₃	—	2	1	no	—
110	C₂×C₄×C₄₃	—	2	1	no	—
111	C₂×C₄×C₄₃	—	2	1	no	—
112	C₂×C₄×C₄₃	—	2	1	no	—
113	C₂×C₄×C₄₃	—	2	1	no	—
114	C₂×C₄×C₄₃	—	2	1	no	—
115	C₂×C₄×C₄₃	—	2	1	no	—
116	C₂×C₄×C₄₃	—	2	1	no	—
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	—	4	1	no	—
121	C₂×C₄×C₄₃	—	4	1	no	—
122	C₂×C₄×C₄₃	—	4	1	no	—
123	C₂×C₄×C₄₃	—	4	1	no	—
124	C₂×C₄×C₄₃	—	4	1	no	—
125	C₂×C₄×C₄₃	—	4	1	no	—
126	C₂×C₄×C₄₃	—	4	1	no	—
127	C₂×C₄×C₄₃	#102	1	2	yes	—
128	C₂×C₄×C₄₃	#102	1	2	yes	—
129	C₂×C₄×C₄₃	#104	1	2	yes	—
130	C₂×C₄×C₄₃	#131	1	2	yes	—
131	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
132	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
133	C₂×C₄×C₄₃	#135	1	2	yes	—
134	C₄₃⋊C₈	—	4	4	no	—
135	C₄₃⋊C₈	—	8	2	no	—
136	#7672	—	43	1	no	maximal, Sylow
137	C₂²×C₄₃	—	8	1	no	—
138	C₂²×C₄₃	—	8	1	no	—
139	C₂²×C₄₃	—	8	1	no	—
140	C₂²×C₄₃	—	8	1	no	—
141	C₂²×C₄₃	—	8	1	no	—
142	C₂²×C₄₃	—	8	1	no	—
143	C₂²×C₄₃	—	8	1	no	—
144	C₂²×C₄₃	#408	1	1	yes	characteristic
145	C₂²×C₄₃	#410	1	1	yes	characteristic
146	C₂²×C₄₃	#413	1	1	yes	characteristic
147	C₂²×C₄₃	C₂×#102	1	1	yes	characteristic
148	C₂²×C₄₃	#416	1	1	yes	characteristic
149	C₂²×C₄₃	C₂×#104	1	1	yes	characteristic
150	C₂²×C₄₃	#421	1	1	yes	characteristic
151	C₄×C₄₃	—	2	2	no	—
152	C₄×C₄₃	—	2	2	no	—
153	C₄×C₄₃	—	2	2	no	—
154	C₄×C₄₃	—	2	2	no	—
155	C₄×C₄₃	—	2	1	no	—
156	C₄×C₄₃	—	2	1	no	—
157	C₄×C₄₃	—	2	2	no	—
158	C₄×C₄₃	—	4	1	no	—
159	C₄×C₄₃	—	8	1	no	—
160	C₄×C₄₃	—	8	1	no	—
161	C₄×C₄₃	—	8	1	no	—
162	#2860	D₄₃	1	1	yes	characteristic
163	#1839	—	43	1	no	—
164	C₄×C₄⋊C₈	—	43	1	no	—
165	#1271	—	43	2	no	—
166	#1270	—	43	2	no	—
167	C₂×C₄₃	—	16	1	no	—
168	C₂×C₄₃	#1473	1	1	yes	characteristic
169	C₂×C₄₃	#1480	1	1	yes	characteristic
170	C₂×C₄₃	#1494	1	1	yes	characteristic
171	C₂×C₄₃	#1502	1	1	yes	characteristic
172	C₂×C₄₃	#1521	1	1	yes	characteristic
173	C₂×C₄₃	#1529	1	1	yes	characteristic
174	C₂×C₄₃	#1540	1	1	yes	characteristic
175	C₂×#126	—	2	1	no	—
176	C₂×C₄⋊C₈	—	86	1	no	—
177	C₂×C₄×C₈	—	86	1	no	—
178	#390	Dic₄₃	1	1	yes	characteristic
179	#390	Dic₄₃	1	1	yes	characteristic
180	#390	C₂×D₄₃	1	1	yes	characteristic
181	#389	Dic₄₃	1	1	yes	characteristic
182	#389	Dic₄₃	1	1	yes	characteristic
183	#389	C₂×D₄₃	1	1	yes	characteristic
184	#388	—	2	1	no	—
185	#388	—	2	1	no	—
186	#388	—	2	1	no	—
187	C₄³	C₂×D₄₃	1	1	yes	characteristic
188	#336	—	43	2	no	—
189	#336	—	43	2	no	—
190	#336	—	86	1	no	—
191	#336	—	86	1	no	—
192	C₄₃	#7672	1	1	yes	characteristic, complemented, Sylow
193	C₂²×D₄	C₂×Dic₄₃	1	1	yes	characteristic
194	C₂²×C₈	—	86	2	no	—
195	C₂²×C₈	—	86	2	no	—
196	#126	—	4	2	no	—
197	#126	—	4	2	no	—
198	C₂×C₄⋊C₄	—	2	1	no	—
199	C₂×C₄⋊C₄	—	2	1	no	—
200	C₂×C₄⋊C₄	—	2	1	no	—
201	C₂×C₄⋊C₄	C₂×Dic₄₃	1	1	yes	characteristic
202	C₂×C₄⋊C₄	C₂×Dic₄₃	1	1	yes	characteristic
203	C₂×C₄⋊C₄	C₂×Dic₄₃	1	1	yes	characteristic
204	C₂×C₂²⋊C₄	—	2	1	no	—
205	C₂×C₂²⋊C₄	—	2	1	no	—
206	C₂×C₂²⋊C₄	—	2	1	no	—
207	C₂×C₂²⋊C₄	—	2	1	no	—
208	C₂×C₂²⋊C₄	—	2	1	no	—
209	C₂×C₂²⋊C₄	—	2	1	no	—
210	C₂×C₄²	C₂×Dic₄₃	1	1	yes	characteristic
211	C₂×C₄²	C₂×Dic₄₃	1	1	yes	characteristic
212	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
213	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
214	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
215	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
216	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
217	C₄⋊C₈	—	86	4	no	—
218	C₄×C₈	—	172	2	no	—
219	#95	—	2	1	no	—
220	#95	—	2	1	no	—
221	#95	—	2	1	no	—
222	C₂⁴	—	2	1	no	—
223	C₂×D₄	—	2	2	no	—
224	C₂×D₄	—	4	1	no	—
225	C₂×D₄	—	4	1	no	—
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	—	2	1	no	—
228	C₂²×C₄	—	2	1	no	—
229	C₂²×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
230	C₂²×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
231	C₂²×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
232	C₂²×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
233	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
234	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
235	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
236	C₂×C₈	—	86	4	no	—
237	C₂×C₈	—	172	2	no	—
238	C₂×C₈	—	172	2	no	—
239	C₂×C₈	—	172	2	no	—
240	C₂×C₈	—	172	2	no	—
241	C₂×C₈	—	172	2	no	—
242	C₄⋊C₄	—	2	2	no	—
243	C₄⋊C₄	—	4	1	no	—
244	C₄⋊C₄	—	4	1	no	—
245	C₄⋊C₄	—	4	2	no	—
246	C₄⋊C₄	—	4	2	no	—
247	C₄⋊C₄	—	4	1	no	—
248	C₄⋊C₄	—	4	1	no	—
249	C₂²⋊C₄	—	4	2	no	—
250	C₂²⋊C₄	—	4	2	no	—
251	C₂²⋊C₄	—	4	2	no	—
252	C₂²⋊C₄	—	4	2	no	—
253	C₂²⋊C₄	—	4	1	no	—
254	C₂²⋊C₄	—	4	1	no	—
255	C₂²⋊C₄	—	4	2	no	—
256	C₄²	—	2	2	no	—
257	C₄²	—	2	2	no	—
258	C₄²	—	2	2	no	—
259	C₄²	—	4	1	no	—
260	C₄²	—	4	1	no	—
261	C₄²	—	4	1	no	—
262	C₄²	#10621882	1	2	yes	—
263	C₄²	#10621883	1	2	yes	—
264	C₂³	—	4	1	no	—
265	C₂³	—	4	1	no	—
266	C₂³	—	4	1	no	—
267	C₂³	—	4	1	no	—
268	C₂³	—	4	1	no	—
269	C₂³	—	4	1	no	—
270	C₂³	—	4	1	no	—
271	C₂³	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
272	D₄	—	8	2	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	4	1	no	—
291	C₂×C₄	—	4	1	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	#13063246	1	2	yes	—
298	C₂×C₄	#13063246	1	2	yes	—
299	C₂×C₄	#13063252	1	2	yes	—
300	C₂×C₄	#13063332	1	2	yes	—
301	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
302	C₂×C₄	C₂×#10621883	1	1	yes	characteristic
303	C₂×C₄	#13063345	1	2	yes	—
304	C₈	—	344	2	no	—
305	C₈	—	172	4	no	—
306	C₂²	—	8	1	no	—
307	C₂²	—	8	1	no	—
308	C₂²	—	8	1	no	—
309	C₂²	—	8	1	no	—
310	C₂²	—	8	1	no	—
311	C₂²	—	8	1	no	—
312	C₂²	—	8	1	no	—
313	C₂²	#14582866	1	1	yes	characteristic
314	C₂²	#14582873	1	1	yes	characteristic
315	C₂²	#14582883	1	1	yes	characteristic
316	C₂²	C₂×#13063246	1	1	yes	characteristic
317	C₂²	#14582895	1	1	yes	characteristic
318	C₂²	C₂×#13063252	1	1	yes	characteristic
319	C₂²	#14582916	1	1	yes	characteristic
320	C₄	—	2	2	no	—
321	C₄	—	2	2	no	—
322	C₄	—	2	2	no	—
323	C₄	—	2	2	no	—
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	2	2	no	—
327	C₄	—	4	1	no	—
328	C₄	—	8	1	no	—
329	C₄	—	8	1	no	—
330	C₄	—	8	1	no	—
331	C₂	—	16	1	no	—
332	C₂	#19132964	1	1	yes	characteristic
333	C₂	#19132994	1	1	yes	characteristic
334	C₂	#19133055	1	1	yes	characteristic
335	C₂	#19133090	1	1	yes	characteristic
336	C₂	#19133172	1	1	yes	characteristic
337	C₂	#19133202	1	1	yes	characteristic
338	C₂	#19133244	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₄₃
commutator	C₂²×C₄×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#2860	C₂
socle	C₂³×C₄₃	C₂×C₂²⋊C₄
prime radical	C₂²×D₄×C₄₃	C₂×C₄
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#2860	D₄₃
hypocenter	C₄₃	#7672
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2860

Complements

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

N	H ≅ G/N	type
1. #19134738	C₂	semidirect
2. C₄×C₄⋊(C₄₃⋊C₈)	C₂	semidirect
3. #19132418	C₂	semidirect
192. C₄₃	#7672	semidirect

Maximal subgroups

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

subgroup	index
#19134738	2
C₄×C₄⋊(C₄₃⋊C₈)	2
#19132418	2
#19132416	2
C₄₃×#2860	2
#7672	43