Subgroups — #25065767

Order 11008.

642 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#432×D₄₃	C₂	1	2	yes	maximal
2	C₂×#14582950	C₂	1	1	yes	maximal, characteristic
3	#19133379	C₂	1	4	yes	maximal, complemented
4	#19133377	C₂	1	4	yes	maximal, complemented
5	#19133358	C₂	1	1	yes	maximal, characteristic, complemented
6	#19133356	C₂	1	2	yes	maximal, complemented
7	C₄₃×#1568	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄²×D₄₃	C₂²	1	1	yes	characteristic
9	#14582956	C₂²	1	2	yes	—
10	#14582955	C₂²	1	4	yes	—
11	#14582954	—	2	2	no	—
12	#14582953	—	2	2	no	—
13	#14582950	C₄	1	4	yes	—
14	#14582950	C₂²	1	2	yes	—
15	#14582950	C₂²	1	2	yes	—
16	C₂×Dic₄₃⋊C₈	C₄	1	2	yes	—
17	C₄⋊C₈×D₄₃	C₂²	1	4	yes	—
18	C₂×(C₂×D₄₃)⋊C₈	C₄	1	2	yes	—
19	C₂²⋊C₈×D₄₃	—	2	2	no	—
20	#97×D₄₃	—	2	2	no	—
21	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
22	#14582652	C₂²	1	2	yes	complemented
23	#14582651	C₂²	1	4	yes	complemented
24	#14582650	C₂²	1	2	yes	complemented
25	#14582649	C₂²	1	4	yes	complemented
26	C₄₃×#432	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
28	C₄₃×#335	C₂²	1	2	yes	complemented
29	C₄₃×#334	C₂²	1	2	yes	complemented
30	C₂²×C₄×D₄₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×D₄₃	D₄	1	1	yes	characteristic
32	C₂²×C₄×D₄₃	Q₈	1	1	yes	characteristic
33	C₂²×C₄₃⋊C₈	C₂×C₄	1	2	yes	—
34	C₂×C₈×D₄₃	—	2	4	no	—
35	C₂×C₄×Dic₄₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×Dic₄₃	D₄	1	1	yes	characteristic
37	C₂×C₄×Dic₄₃	Q₈	1	1	yes	characteristic
38	C₄²×D₄₃	C₂×C₄	1	2	yes	—
39	C₄²×D₄₃	D₄	1	2	yes	—
40	C₄²×D₄₃	Q₈	1	2	yes	—
41	C₄²×D₄₃	C₂³	1	2	yes	—
42	C₄⋊(C₄₃⋊C₈)	C₂³	1	4	yes	—
43	Dic₄₃⋊C₈	—	2	4	no	—
44	Dic₄₃⋊C₈	C₂×C₄	1	4	yes	—
45	Dic₄₃⋊C₈	C₂×C₄	1	4	yes	—
46	C₂²⋊(C₄₃⋊C₈)	—	2	2	no	—
47	(C₂×D₄₃)⋊C₈	—	2	4	no	—
48	(C₂×D₄₃)⋊C₈	C₂×C₄	1	4	yes	—
49	(C₂×D₄₃)⋊C₈	C₂×C₄	1	4	yes	—
50	#13063236	—	2	2	no	—
51	#13063235	—	2	4	no	—
52	C₄×C₄₃⋊C₈	C₂×C₄	1	2	yes	—
53	C₄×C₄₃⋊C₈	C₂³	1	2	yes	—
54	C₂²×C₈×C₄₃	C₂×C₄	1	2	yes	—
55	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
56	C₄⋊C₈×C₄₃	C₂³	1	4	yes	—
57	C₂²⋊C₈×C₄₃	—	2	2	no	—
58	#97×C₄₃	—	2	2	no	—
59	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
60	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
61	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
62	C₂³×D₄₃	C₄⋊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×Q₈	1	1	yes	characteristic
66	C₂²×Dic₄₃	C₄⋊C₄	1	1	yes	characteristic
67	C₂²×Dic₄₃	C₄⋊C₄	1	1	yes	characteristic
68	C₂²×Dic₄₃	C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×D₄₃	—	2	4	no	—
70	C₂×C₄×D₄₃	—	4	2	no	—
71	C₂×C₄×D₄₃	—	4	2	no	—
72	C₂×C₄×D₄₃	C₄⋊C₄	1	2	yes	—
73	C₂×C₄×D₄₃	C₄⋊C₄	1	2	yes	—
74	C₂×C₄×D₄₃	C₄⋊C₄	1	2	yes	—
75	C₂×C₄×D₄₃	C₂²×C₄	1	2	yes	—
76	C₂×C₄×D₄₃	C₂×D₄	1	2	yes	—
77	C₂×C₄×D₄₃	C₂×Q₈	1	2	yes	—
78	C₂×C₈×C₄₃	—	2	4	no	—
79	C₂×C₈×C₄₃	—	4	2	no	—
80	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
82	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₄₃	C₂×Q₈	1	1	yes	characteristic
85	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
86	C₄×Dic₄₃	C₄⋊C₄	1	2	yes	—
87	C₄×Dic₄₃	C₄⋊C₄	1	2	yes	—
88	C₄×Dic₄₃	C₄⋊C₄	1	2	yes	—
89	C₄×Dic₄₃	C₂²×C₄	1	2	yes	—
90	C₄×Dic₄₃	C₂×D₄	1	2	yes	—
91	C₄×Dic₄₃	C₂×Q₈	1	2	yes	—
92	C₂×C₄₃⋊C₈	—	2	4	no	—
93	C₂×C₄₃⋊C₈	—	4	2	no	—
94	C₂×C₄₃⋊C₈	C₂²×C₄	1	4	yes	—
95	C₈×D₄₃	—	4	8	no	—
96	C₂³×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂²×D₄₃	—	4	1	no	—
98	C₂²×D₄₃	—	4	1	no	—
99	C₂²×D₄₃	#108	1	4	yes	—
100	C₂²×D₄₃	C₂×C₄⋊C₄	1	2	yes	—
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	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
108	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
110	C₂×Dic₄₃	—	2	2	no	—
111	C₂×Dic₄₃	—	4	1	no	—
112	C₂×Dic₄₃	—	4	1	no	—
113	C₂×Dic₄₃	C₂×C₄⋊C₄	1	2	yes	—
114	C₂×Dic₄₃	C₂×C₄⋊C₄	1	2	yes	—
115	C₂×Dic₄₃	C₂×C₄⋊C₄	1	2	yes	—
116	C₄×D₄₃	—	2	4	no	—
117	C₄×D₄₃	—	2	4	no	—
118	C₄×D₄₃	—	2	4	no	—
119	C₄×D₄₃	—	2	4	no	—
120	C₄×D₄₃	—	4	2	no	—
121	C₄×D₄₃	—	4	2	no	—
122	C₈×C₄₃	—	2	2	no	—
123	C₈×C₄₃	—	2	2	no	—
124	C₈×C₄₃	—	4	4	no	—
125	C₄₃⋊C₈	—	2	4	no	—
126	C₄₃⋊C₈	—	4	4	no	—
127	#17818	—	43	1	no	maximal, Sylow
128	C₂²×C₄₃	—	2	1	no	—
129	C₂²×C₄₃	—	2	1	no	—
130	C₂²×C₄₃	C₂×#108	1	2	yes	—
131	C₂²×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
132	C₂×D₄₃	—	2	2	no	—
133	C₂×D₄₃	—	2	2	no	—
134	C₂×D₄₃	—	4	4	no	—
135	C₂×D₄₃	C₂×#108	1	4	yes	—
136	C₄×C₄₃	—	2	1	no	—
137	C₄×C₄₃	—	2	1	no	—
138	C₄×C₄₃	—	2	1	no	—
139	C₄×C₄₃	—	2	1	no	—
140	C₄×C₄₃	#568	1	2	yes	—
141	C₄×C₄₃	#575	1	2	yes	—
142	Dic₄₃	—	2	2	no	—
143	Dic₄₃	—	2	2	no	—
144	Dic₄₃	—	2	2	no	—
145	C₂×#432	—	43	2	no	—
146	C₂×#431	—	43	1	no	—
147	#1573	—	43	4	no	—
148	#1573	—	43	4	no	—
149	#1568	—	43	2	no	—
150	#1568	—	43	1	no	—
151	#1568	D₄₃	1	1	yes	characteristic, complemented
152	C₂×C₄₃	—	2	2	no	—
153	C₂×C₄₃	#2905	1	1	yes	characteristic
154	C₂×C₄₃	#2911	1	1	yes	characteristic
155	C₂×C₄₃	C₂²×#108	1	1	yes	characteristic
156	D₄₃	—	2	4	no	—
157	C₂²×C₄²	—	43	1	no	—
158	#432	—	43	2	no	—
159	#432	—	43	4	no	—
160	#432	—	86	2	no	—
161	#432	—	86	2	no	—
162	#432	C₂×D₄₃	1	2	yes	—
163	#431	—	43	2	no	—
164	#431	—	43	2	no	—
165	#431	—	43	4	no	—
166	C₂×C₄⋊C₈	—	43	2	no	—
167	C₂×C₄⋊C₈	—	43	4	no	—
168	C₂×C₂²⋊C₈	—	43	2	no	—
169	C₂×C₂²⋊C₈	—	86	2	no	—
170	C₂×#97	—	86	2	no	—
171	C₂×C₄×C₈	—	43	1	no	—
172	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
173	#335	—	43	4	no	—
174	#335	—	43	2	no	—
175	#335	C₂×D₄₃	1	2	yes	complemented
176	#334	—	43	4	no	—
177	#334	—	43	2	no	—
178	#334	C₂×D₄₃	1	2	yes	complemented
179	C₄₃	#17818	1	1	yes	characteristic, complemented, Sylow
180	C₂³×C₄	—	43	1	no	—
181	C₂³×C₄	—	43	1	no	—
182	C₂³×C₄	—	43	1	no	—
183	C₂²×C₈	—	43	2	no	—
184	C₂²×C₈	—	86	4	no	—
185	C₂²×C₈	C₄×D₄₃	1	2	yes	—
186	C₂×C₄²	—	43	1	no	—
187	C₂×C₄²	—	43	1	no	—
188	C₂×C₄²	—	43	1	no	—
189	C₂×C₄²	—	43	2	no	—
190	C₂×C₄²	—	43	2	no	—
191	C₂×C₄²	—	43	2	no	—
192	C₂×C₄²	—	43	2	no	—
193	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
194	C₄⋊C₈	—	43	4	no	—
195	C₄⋊C₈	—	43	4	no	—
196	C₄⋊C₈	—	43	4	no	—
197	C₄⋊C₈	—	86	4	no	—
198	C₄⋊C₈	C₂²×D₄₃	1	4	yes	—
199	C₂²⋊C₈	—	2	2	no	—
200	C₂²⋊C₈	—	43	4	no	—
201	C₂²⋊C₈	—	43	4	no	—
202	C₂²⋊C₈	—	86	4	no	—
203	C₂²⋊C₈	—	86	2	no	—
204	#97	—	2	2	no	—
205	#97	—	86	4	no	—
206	#97	—	86	2	no	—
207	C₄×C₈	—	43	2	no	—
208	C₄×C₈	—	43	2	no	—
209	C₄×C₈	C₄×D₄₃	1	2	yes	—
210	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
211	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
212	C₂⁴	—	43	1	no	—
213	C₂²×C₄	—	43	1	no	—
214	C₂²×C₄	—	43	1	no	—
215	C₂²×C₄	—	43	1	no	—
216	C₂²×C₄	—	43	2	no	—
217	C₂²×C₄	—	43	2	no	—
218	C₂²×C₄	—	43	2	no	—
219	C₂²×C₄	—	43	2	no	—
220	C₂²×C₄	—	43	2	no	—
221	C₂²×C₄	—	43	2	no	—
222	C₂²×C₄	—	86	4	no	—
223	C₂²×C₄	—	172	2	no	—
224	C₂²×C₄	—	172	2	no	—
225	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
226	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
227	C₂²×C₄	Q₈×D₄₃	1	1	yes	characteristic
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	4	2	no	—
230	C₂×C₈	—	43	4	no	—
231	C₂×C₈	—	86	4	no	—
232	C₂×C₈	—	172	8	no	—
233	C₂×C₈	—	172	2	no	—
234	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
235	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
236	C₄²	—	43	2	no	—
237	C₄²	—	43	2	no	—
238	C₄²	—	43	2	no	—
239	C₄²	—	43	2	no	—
240	C₄²	—	43	2	no	—
241	C₄²	—	43	2	no	—
242	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
243	C₄²	D₄×D₄₃	1	1	yes	characteristic
244	C₄²	Q₈×D₄₃	1	1	yes	characteristic
245	C₄²	C₂³×D₄₃	1	1	yes	characteristic
246	C₂³	—	43	2	no	—
247	C₂³	—	43	4	no	—
248	C₂³	—	172	1	no	—
249	C₂³	—	172	1	no	—
250	C₂³	C₄⋊C₄×D₄₃	1	1	yes	characteristic
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	43	2	no	—
255	C₂×C₄	—	43	2	no	—
256	C₂×C₄	—	43	2	no	—
257	C₂×C₄	—	86	4	no	—
258	C₂×C₄	—	86	4	no	—
259	C₂×C₄	—	86	2	no	—
260	C₂×C₄	—	86	4	no	—
261	C₂×C₄	—	86	4	no	—
262	C₂×C₄	—	172	2	no	—
263	C₂×C₄	—	172	2	no	—
264	C₂×C₄	—	172	1	no	—
265	C₂×C₄	—	172	1	no	—
266	C₂×C₄	C₄⋊C₄×D₄₃	1	1	yes	characteristic
267	C₂×C₄	C₄⋊C₄×D₄₃	1	1	yes	characteristic
268	C₂×C₄	C₄⋊C₄×D₄₃	1	1	yes	characteristic
269	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
270	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
271	C₂×C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
272	C₈	—	2	2	no	—
273	C₈	—	2	2	no	—
274	C₈	—	4	4	no	—
275	C₈	—	86	4	no	—
276	C₈	—	172	4	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	43	4	no	—
280	C₂²	—	86	2	no	—
281	C₂²	—	86	2	no	—
282	C₂²	—	172	4	no	—
283	C₂²	#108×D₄₃	1	2	yes	—
284	C₂²	C₂×C₄⋊C₄×D₄₃	1	1	yes	characteristic
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	2	1	no	—
288	C₄	—	2	1	no	—
289	C₄	—	86	2	no	—
290	C₄	—	86	2	no	—
291	C₄	—	86	2	no	—
292	C₄	#14583665	1	2	yes	—
293	C₄	#14583696	1	2	yes	—
294	C₂	—	2	2	no	—
295	C₂	—	86	4	no	—
296	C₂	#19143157	1	1	yes	characteristic
297	C₂	#19143183	1	1	yes	characteristic
298	C₂	C₂×#108×D₄₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₄²	C₂³×D₄₃
Fitting	C₄₃×#1568	C₂
socle	C₂²×C₄₃	C₂²×C₄⋊C₄
prime radical	C₂³×D₄₃	C₄⋊C₄
cosocle	C₄²×C₄₃	C₂⁴
hypercenter	#1568	D₄₃
hypocenter	C₄₃	#17818
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1568

Complements

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

N	H ≅ G/N	type
3. #19133379	C₂	semidirect
4. #19133377	C₂	semidirect
5. #19133358	C₂	semidirect
6. #19133356	C₂	semidirect
7. C₄₃×#1568	C₂	semidirect
22. #14582652	C₂²	semidirect
23. #14582651	C₂²	semidirect
24. #14582650	C₂²	semidirect
25. #14582649	C₂²	semidirect
28. C₄₃×#335	C₂²	semidirect
29. C₄₃×#334	C₂²	semidirect
151. #1568	D₄₃	semidirect
175. #335	C₂×D₄₃	semidirect
178. #334	C₂×D₄₃	semidirect
179. C₄₃	#17818	semidirect

Maximal subgroups

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

subgroup	index
#432×D₄₃	2
C₂×#14582950	2
#19133379	2
#19133377	2
#19133358	2
#19133356	2
C₄₃×#1568	2
#17818	43