Subgroups — #25095403

Order 11008.

934 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14582952	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊(C₄₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#19134464	C₂	1	2	yes	maximal, complemented
4	C₄⋊C₄×C₄₃⋊C₈	C₂	1	2	yes	maximal, complemented
5	#19134452	C₂	1	4	yes	maximal, complemented
6	#19134373	C₂	1	2	yes	maximal
7	C₂²⋊C₄×C₄₃⋊C₈	C₂	1	2	yes	maximal
8	C₄₃×#2283	C₂	1	1	yes	maximal, characteristic
9	C₂³×C₄₃⋊C₈	C₂²	1	2	yes	—
10	#14582952	C₄	1	8	yes	complemented
11	C₂×C₄⋊(C₄₃⋊C₈)	C₄	1	8	yes	complemented
12	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
14	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
15	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
16	C₂×C₂²⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
17	C₂×C₂²⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
18	C₂×#13063236	C₂²	1	4	yes	—
19	C₂×C₄×C₄₃⋊C₈	C₂²	1	4	yes	—
20	C₂×C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
21	#14582654	C₂²	1	4	yes	—
22	#14582654	C₂²	1	4	yes	—
23	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
24	C₄₃×#383	C₄	1	2	yes	—
25	C₄₃×#382	C₄	1	2	yes	—
26	C₄₃×#381	C₄	1	2	yes	—
27	C₄₃×#380	C₄	1	2	yes	—
28	C₄×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
29	C₄×C₂²⋊C₄×C₄₃	C₂²	1	2	yes	—
30	C₄₃×#376	C₂²	1	2	yes	—
31	C₂²×C₄₃⋊C₈	—	4	4	no	—
32	C₂²×C₄₃⋊C₈	C₂×C₄	1	8	yes	—
33	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
34	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
35	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
36	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
37	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	4	yes	—
38	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	8	yes	—
39	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	4	yes	—
40	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	8	yes	—
41	C₂²⋊(C₄₃⋊C₈)	C₂×C₄	1	4	yes	—
42	C₂²⋊(C₄₃⋊C₈)	C₂×C₄	1	4	yes	—
43	#13063236	—	4	4	no	—
44	C₄×C₄₃⋊C₈	—	4	4	no	—
45	C₄×C₄₃⋊C₈	C₂×C₄	1	8	yes	—
46	C₂³×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
47	C₂³×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
48	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
49	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	4	yes	—
50	C₂×C₂²⋊C₄×C₄₃	C₂×C₄	1	4	yes	—
51	C₂×C₄²×C₄₃	C₂×C₄	1	4	yes	—
52	C₂×C₄²×C₄₃	C₂×C₄	1	4	yes	—
53	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄²×C₄₃	C₂³	1	4	yes	—
56	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
58	#95×C₄₃	C₂×C₄	1	4	yes	—
59	#95×C₄₃	C₂×C₄	1	4	yes	—
60	#95×C₄₃	C₂×C₄	1	4	yes	—
61	#95×C₄₃	C₂×C₄	1	4	yes	—
62	C₂⁴×C₄₃	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₃	—	2	4	no	—
64	C₂²×C₄×C₄₃	—	2	2	no	—
65	C₂²×C₄×C₄₃	—	2	2	no	—
66	C₂²×C₄×C₄₃	—	2	4	no	—
67	C₂²×C₄×C₄₃	C₄²	1	4	yes	—
68	C₂²×C₄×C₄₃	C₄²	1	4	yes	—
69	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
70	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
71	C₂²×C₄×C₄₃	C₂²×C₄	1	8	yes	—
72	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
78	C₄⋊C₄×C₄₃	—	2	8	no	—
79	C₂²⋊C₄×C₄₃	—	2	8	no	—
80	C₄²×C₄₃	—	2	4	no	—
81	C₄²×C₄₃	—	2	4	no	—
82	C₄²×C₄₃	C₄²	1	4	yes	—
83	C₄²×C₄₃	C₄²	1	4	yes	—
84	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
85	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
86	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
87	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
88	C₂×C₄₃⋊C₈	—	2	2	no	—
89	C₂×C₄₃⋊C₈	—	2	4	no	—
90	C₂×C₄₃⋊C₈	—	2	2	no	—
91	C₂×C₄₃⋊C₈	—	2	4	no	—
92	C₂×C₄₃⋊C₈	—	2	2	no	—
93	C₂×C₄₃⋊C₈	—	2	2	no	—
94	C₂×C₄₃⋊C₈	—	4	8	no	—
95	C₂×C₄₃⋊C₈	C₂²×C₄	1	4	yes	—
96	C₂×C₄₃⋊C₈	C₂²×C₄	1	4	yes	—
97	C₂×C₄₃⋊C₈	C₂²×C₄	1	4	yes	—
98	C₂×C₄₃⋊C₈	C₂²×C₄	1	4	yes	—
99	C₂³×C₄₃	—	2	2	no	—
100	C₂³×C₄₃	—	2	2	no	—
101	C₂³×C₄₃	C₂×C₄²	1	2	yes	—
102	C₂³×C₄₃	#131	1	4	yes	—
103	C₂³×C₄₃	C₂³×C₄	1	1	yes	characteristic
104	C₂×C₄×C₄₃	—	2	2	no	—
105	C₂×C₄×C₄₃	—	2	2	no	—
106	C₂×C₄×C₄₃	—	2	2	no	—
107	C₂×C₄×C₄₃	—	2	2	no	—
108	C₂×C₄×C₄₃	—	2	8	no	—
109	C₂×C₄×C₄₃	—	2	2	no	—
110	C₂×C₄×C₄₃	—	2	2	no	—
111	C₂×C₄×C₄₃	—	2	4	no	—
112	C₂×C₄×C₄₃	—	2	8	no	—
113	C₂×C₄×C₄₃	—	2	8	no	—
114	C₂×C₄×C₄₃	—	2	8	no	—
115	C₂×C₄×C₄₃	—	2	8	no	—
116	C₂×C₄×C₄₃	C₂×C₄²	1	2	yes	—
117	C₂×C₄×C₄₃	C₂×C₄²	1	2	yes	—
118	C₂×C₄×C₄₃	C₂×C₄²	1	2	yes	—
119	C₂×C₄×C₄₃	C₂×C₄²	1	2	yes	—
120	C₂×C₄×C₄₃	C₂×C₄²	1	2	yes	—
121	C₂×C₄×C₄₃	#131	1	4	yes	—
122	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
124	C₂×C₄×C₄₃	H(5,𝔽₂)	1	2	yes	—
125	C₂×C₄×C₄₃	2₋¹⁺⁴	1	2	yes	—
126	C₄₃⋊C₈	—	2	4	no	—
127	C₄₃⋊C₈	—	2	4	no	—
128	C₄₃⋊C₈	—	2	4	no	—
129	C₄₃⋊C₈	—	2	4	no	—
130	#20545	—	43	1	no	maximal, Sylow
131	C₂²×C₄₃	—	2	2	no	—
132	C₂²×C₄₃	—	2	8	no	—
133	C₂²×C₄₃	—	2	2	no	—
134	C₂²×C₄₃	#405	1	4	yes	—
135	C₂²×C₄₃	C₂²×C₄²	1	1	yes	characteristic
136	C₂²×C₄₃	#520	1	2	yes	—
137	C₂²×C₄₃	C₂×#131	1	2	yes	—
138	C₂²×C₄₃	C₂×#131	1	2	yes	—
139	C₄×C₄₃	—	2	2	no	—
140	C₄×C₄₃	—	2	2	no	—
141	C₄×C₄₃	—	2	2	no	—
142	C₄×C₄₃	—	2	2	no	—
143	C₄×C₄₃	—	4	8	no	—
144	C₄×C₄₃	#405	1	4	yes	—
145	C₄×C₄₃	#518	1	1	yes	characteristic
146	C₄×C₄₃	#519	1	1	yes	characteristic
147	C₄×C₄₃	#520	1	2	yes	—
148	C₂×#431	—	43	1	no	—
149	C₂²×C₄⋊C₈	—	43	1	no	—
150	#2283	D₄₃	1	1	yes	characteristic
151	#1770	—	43	2	no	—
152	C₈×C₄⋊C₄	—	43	2	no	—
153	#1768	—	43	4	no	—
154	#1752	—	43	2	no	—
155	C₈×C₂²⋊C₄	—	43	2	no	—
156	C₂×C₄₃	—	2	4	no	—
157	C₂×C₄₃	#2274	1	1	yes	characteristic
158	C₂×C₄₃	C₂×#405	1	1	yes	characteristic
159	C₂×C₄₃	C₂×#405	1	1	yes	characteristic
160	C₂×C₄₃	#2994	1	4	yes	—
161	C₂³×C₈	—	43	2	no	—
162	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
163	#431	—	43	8	no	—
164	C₂×C₄⋊C₈	—	43	2	no	—
165	C₂×C₄⋊C₈	—	43	1	no	—
166	C₂×C₄⋊C₈	—	43	1	no	—
167	C₂×C₄⋊C₈	—	43	2	no	—
168	C₂×C₄⋊C₈	—	43	8	no	—
169	C₂×C₂²⋊C₈	—	43	1	no	—
170	C₂×C₂²⋊C₈	—	43	1	no	—
171	C₂×#97	—	43	4	no	—
172	C₂×C₄×C₈	—	43	2	no	—
173	C₂×C₄×C₈	—	43	4	no	—
174	#383	Dic₄₃	1	2	yes	—
175	#382	Dic₄₃	1	2	yes	—
176	#381	Dic₄₃	1	2	yes	—
177	#380	Dic₄₃	1	2	yes	—
178	C₄×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
179	C₄×C₂²⋊C₄	C₂×D₄₃	1	2	yes	—
180	#376	C₂×D₄₃	1	2	yes	—
181	#336	—	43	4	no	—
182	#336	—	43	4	no	—
183	C₄₃	#20545	1	1	yes	characteristic, complemented, Sylow
184	C₂³×C₄	C₂×Dic₄₃	1	1	yes	characteristic
185	C₂³×C₄	C₂×Dic₄₃	1	1	yes	characteristic
186	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
187	C₂²×C₈	—	43	2	no	—
188	C₂²×C₈	—	43	2	no	—
189	C₂²×C₈	—	43	2	no	—
190	C₂²×C₈	—	43	2	no	—
191	C₂²×C₈	—	43	8	no	—
192	C₂²×C₈	—	172	4	no	—
193	C₂×C₄⋊C₄	C₂×Dic₄₃	1	4	yes	—
194	C₂×C₂²⋊C₄	C₂×Dic₄₃	1	4	yes	—
195	C₂×C₄²	C₄×D₄₃	1	4	yes	—
196	C₂×C₄²	C₄×D₄₃	1	4	yes	—
197	C₂×C₄²	C₂×Dic₄₃	1	1	yes	characteristic
198	C₂×C₄²	C₂×Dic₄₃	1	1	yes	characteristic
199	C₂×C₄²	C₂²×D₄₃	1	4	yes	—
200	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
201	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
202	C₄⋊C₈	—	43	8	no	—
203	C₄⋊C₈	—	43	4	no	—
204	C₄⋊C₈	—	43	4	no	—
205	C₄⋊C₈	—	43	8	no	—
206	C₂²⋊C₈	—	43	4	no	—
207	C₂²⋊C₈	—	43	4	no	—
208	#97	—	172	4	no	—
209	C₄×C₈	—	43	8	no	—
210	C₄×C₈	—	172	4	no	—
211	#95	C₂×Dic₄₃	1	4	yes	—
212	#95	C₂×Dic₄₃	1	4	yes	—
213	#95	C₂×Dic₄₃	1	4	yes	—
214	#95	C₂×Dic₄₃	1	4	yes	—
215	C₂⁴	C₂²×Dic₄₃	1	1	yes	characteristic
216	C₂²×C₄	—	2	4	no	—
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	C₄×Dic₄₃	1	4	yes	—
221	C₂²×C₄	C₄×Dic₄₃	1	4	yes	—
222	C₂²×C₄	C₂×C₄×D₄₃	1	2	yes	—
223	C₂²×C₄	C₂×C₄×D₄₃	1	2	yes	—
224	C₂²×C₄	C₂²×Dic₄₃	1	8	yes	—
225	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
226	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
227	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
228	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
229	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
230	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
231	C₂×C₈	—	43	4	no	—
232	C₂×C₈	—	43	4	no	—
233	C₂×C₈	—	43	4	no	—
234	C₂×C₈	—	43	4	no	—
235	C₂×C₈	—	86	2	no	—
236	C₂×C₈	—	86	4	no	—
237	C₂×C₈	—	86	2	no	—
238	C₂×C₈	—	86	4	no	—
239	C₂×C₈	—	86	2	no	—
240	C₂×C₈	—	86	2	no	—
241	C₂×C₈	—	172	8	no	—
242	C₄⋊C₄	—	2	8	no	—
243	C₂²⋊C₄	—	2	8	no	—
244	C₄²	—	2	4	no	—
245	C₄²	—	2	4	no	—
246	C₄²	C₄×Dic₄₃	1	4	yes	—
247	C₄²	C₄×Dic₄₃	1	4	yes	—
248	C₄²	C₂×C₄×D₄₃	1	2	yes	—
249	C₄²	C₂×C₄×D₄₃	1	2	yes	—
250	C₄²	C₂×C₄×D₄₃	1	2	yes	—
251	C₄²	C₂×C₄×D₄₃	1	2	yes	—
252	C₂³	—	2	2	no	—
253	C₂³	—	2	2	no	—
254	C₂³	C₂×C₄×Dic₄₃	1	2	yes	—
255	C₂³	#13063332	1	4	yes	—
256	C₂³	C₂³×Dic₄₃	1	1	yes	characteristic
257	C₂×C₄	—	2	8	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	8	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	8	no	—
267	C₂×C₄	—	2	8	no	—
268	C₂×C₄	—	2	8	no	—
269	C₂×C₄	C₂×C₄×Dic₄₃	1	2	yes	—
270	C₂×C₄	C₂×C₄×Dic₄₃	1	2	yes	—
271	C₂×C₄	C₂×C₄×Dic₄₃	1	2	yes	—
272	C₂×C₄	C₂×C₄×Dic₄₃	1	2	yes	—
273	C₂×C₄	C₂×C₄×Dic₄₃	1	2	yes	—
274	C₂×C₄	#13063332	1	4	yes	—
275	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
276	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
277	C₂×C₄	#13063371	1	2	yes	—
278	C₂×C₄	#13063372	1	2	yes	—
279	C₈	—	86	4	no	—
280	C₈	—	86	4	no	—
281	C₈	—	86	4	no	—
282	C₈	—	86	4	no	—
283	C₂²	—	2	8	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	2	no	—
286	C₂²	#14582855	1	4	yes	—
287	C₂²	C₂²×C₄×Dic₄₃	1	1	yes	characteristic
288	C₂²	#14583382	1	2	yes	—
289	C₂²	C₂×#13063332	1	2	yes	—
290	C₂²	C₂×#13063332	1	2	yes	—
291	C₄	—	2	2	no	—
292	C₄	—	2	2	no	—
293	C₄	—	2	2	no	—
294	C₄	—	2	2	no	—
295	C₄	—	4	8	no	—
296	C₄	#14582855	1	4	yes	—
297	C₄	#14583373	1	1	yes	characteristic
298	C₄	#14583376	1	1	yes	characteristic
299	C₄	#14583379	1	2	yes	—
300	C₂	—	2	4	no	—
301	C₂	#19136951	1	1	yes	characteristic
302	C₂	C₂×#14582855	1	1	yes	characteristic
303	C₂	C₂×#14582855	1	1	yes	characteristic
304	C₂	#19143755	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₃
commutator	C₂²×C₄₃	C₂²×C₄²
Fitting	C₄₃×#2283	C₂
socle	C₂³×C₄₃	C₂³×C₄
prime radical	C₂⁴×C₄₃	C₂²×C₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#2283	D₄₃
hypocenter	C₄₃	#20545
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ #2283

Complements

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

N	H ≅ G/N	type
3. #19134464	C₂	semidirect
4. C₄⋊C₄×C₄₃⋊C₈	C₂	semidirect
5. #19134452	C₂	semidirect
10. #14582952	C₄	semidirect
11. C₂×C₄⋊(C₄₃⋊C₈)	C₄	semidirect
183. C₄₃	#20545	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14582952	2
C₂²×C₄⋊(C₄₃⋊C₈)	2
#19134464	2
C₄⋊C₄×C₄₃⋊C₈	2
#19134452	2
#19134373	2
C₂²⋊C₄×C₄₃⋊C₈	2
C₄₃×#2283	2
#20545	43