Subgroups — #25094787

Order 11008.

1002 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×#14582956	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582745	C₂	1	1	yes	maximal, characteristic
3	#19134454	C₂	1	4	yes	maximal, complemented
4	#19134359	C₂	1	2	yes	maximal
5	C₂²⋊C₈×Dic₄₃	C₂	1	2	yes	maximal
6	#19134346	C₂	1	2	yes	maximal, complemented
7	#19134281	C₂	1	2	yes	maximal, complemented
8	C₂×C₄₃×#431	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₄×Dic₄₃	C₂²	1	2	yes	—
10	#14582956	C₄	1	8	yes	complemented
11	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
12	C₂×C₂²⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
13	C₂×#13063236	C₂²	1	2	yes	—
14	C₂×#13063235	C₂²	1	4	yes	—
15	C₂×C₈×Dic₄₃	C₂²	1	4	yes	—
16	#14582745	C₄	1	2	yes	—
17	#14582745	C₄	1	2	yes	—
18	#14582745	C₂²	1	1	yes	characteristic
19	#14582745	C₂²	1	1	yes	characteristic
20	#14582745	C₂²	1	2	yes	—
21	C₂×C₄₃⋊#95	C₄	1	2	yes	—
22	C₂×C₄₃⋊#95	C₄	1	2	yes	—
23	#14582653	C₂²	1	4	yes	—
24	#14582653	C₂²	1	4	yes	—
25	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
26	C₄₃×#431	C₄	1	8	yes	complemented
27	C₂×C₄⋊C₈×C₄₃	C₂²	1	2	yes	—
28	C₂×C₂²⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
29	C₂×C₂²⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₄₃	C₂²	1	2	yes	—
31	C₂³×Dic₄₃	C₂×C₄	1	4	yes	—
32	C₂²×C₄₃⋊C₈	C₂³	1	4	yes	—
33	C₂×C₄×Dic₄₃	—	2	8	no	—
34	C₂×C₄×Dic₄₃	C₂×C₄	1	4	yes	—
35	C₂×C₄×Dic₄₃	C₂³	1	2	yes	—
36	C₂×C₄×Dic₄₃	C₂³	1	2	yes	—
37	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	8	yes	complemented
38	C₂²⋊(C₄₃⋊C₈)	C₂×C₄	1	8	yes	—
39	#13063236	C₂×C₄	1	8	yes	complemented
40	#13063235	—	2	8	no	—
41	C₈×Dic₄₃	—	2	8	no	—
42	C₄₃⋊#95	C₂×C₄	1	2	yes	—
43	C₄₃⋊#95	C₂×C₄	1	2	yes	—
44	C₄₃⋊#95	C₂×C₄	1	4	yes	—
45	C₄₃⋊#95	C₂×C₄	1	4	yes	—
46	C₄₃⋊#95	C₂×C₄	1	2	yes	—
47	C₄₃⋊#95	C₂×C₄	1	2	yes	—
48	C₂³×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
49	C₂³×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
50	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
52	C₂²×C₈×C₄₃	C₂³	1	2	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	C₂×C₄	1	4	yes	—
56	C₂×C₄²×C₄₃	C₂×C₄	1	4	yes	—
57	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
59	C₄⋊C₈×C₄₃	C₂×C₄	1	8	yes	complemented
60	C₂²⋊C₈×C₄₃	C₂×C₄	1	4	yes	—
61	C₂²⋊C₈×C₄₃	C₂×C₄	1	4	yes	—
62	C₄×C₈×C₄₃	C₂×C₄	1	8	yes	complemented
63	C₂⁴×C₄₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₃	—	2	4	no	—
65	C₂²×C₄×C₄₃	—	2	4	no	—
66	C₂²×C₄×C₄₃	—	4	2	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	1	yes	characteristic
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	C₂⁴	1	1	yes	characteristic
77	C₂²×Dic₄₃	—	2	8	no	—
78	C₂²×Dic₄₃	—	2	8	no	—
79	C₂²×Dic₄₃	—	4	4	no	—
80	C₂²×Dic₄₃	C₂²×C₄	1	4	yes	—
81	C₂²×Dic₄₃	C₂²×C₄	1	4	yes	—
82	C₂×C₈×C₄₃	—	2	4	no	—
83	C₂×C₈×C₄₃	—	2	4	no	—
84	C₂×C₈×C₄₃	C₂²×C₄	1	4	yes	—
85	C₂×C₈×C₄₃	C₂²×C₄	1	4	yes	—
86	C₄²×C₄₃	C₄²	1	4	yes	—
87	C₄²×C₄₃	C₄²	1	4	yes	—
88	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
89	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
90	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
91	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
92	C₄×Dic₄₃	—	2	4	no	—
93	C₄×Dic₄₃	—	2	8	no	—
94	C₄×Dic₄₃	#41	1	8	yes	—
95	C₂×C₄₃⋊C₈	—	2	4	no	—
96	C₂×C₄₃⋊C₈	—	2	4	no	—
97	C₂×C₄₃⋊C₈	C₂²×C₄	1	8	yes	—
98	C₂³×C₄₃	—	2	2	no	—
99	C₂³×C₄₃	—	2	2	no	—
100	C₂³×C₄₃	C₂×C₄²	1	2	yes	—
101	C₂³×C₄₃	#131	1	4	yes	—
102	C₂³×C₄₃	C₂³×C₄	1	1	yes	characteristic
103	C₂×C₄×C₄₃	—	2	2	no	—
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	4	no	—
110	C₂×C₄×C₄₃	—	2	4	no	—
111	C₂×C₄×C₄₃	—	4	4	no	—
112	C₂×C₄×C₄₃	C₂×C₄²	1	2	yes	—
113	C₂×C₄×C₄₃	C₂×C₄²	1	2	yes	—
114	C₂×C₄×C₄₃	C₂×C₄²	1	2	yes	—
115	C₂×C₄×C₄₃	C₂×C₄²	1	2	yes	—
116	C₂×C₄×C₄₃	C₂×C₄²	1	2	yes	—
117	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
118	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
119	C₂×C₄×C₄₃	C₂×#41	1	4	yes	—
120	C₂×Dic₄₃	—	2	4	no	—
121	C₂×Dic₄₃	—	2	4	no	—
122	C₂×Dic₄₃	—	2	4	no	—
123	C₂×Dic₄₃	—	2	4	no	—
124	C₂×Dic₄₃	—	2	4	no	—
125	C₂×Dic₄₃	—	4	16	no	—
126	C₂×Dic₄₃	#117	1	8	yes	—
127	C₈×C₄₃	—	2	4	no	—
128	C₈×C₄₃	—	2	4	no	—
129	C₄₃⋊C₈	—	2	8	no	—
130	#20469	—	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	C₂×#117	1	2	yes	—
137	C₂²×C₄₃	C₂×#131	1	2	yes	—
138	C₂²×C₄₃	#568	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	—	2	2	no	—
144	C₄×C₄₃	C₄×#41	1	2	yes	—
145	C₄×C₄₃	C₄×#41	1	2	yes	—
146	Dic₄₃	—	2	8	no	—
147	Dic₄₃	—	4	4	no	—
148	C₂×#432	—	43	1	no	—
149	C₂×#431	D₄₃	1	1	yes	characteristic
150	C₂×#376	—	43	1	no	—
151	#1768	—	43	4	no	—
152	#1749	—	43	2	no	—
153	C₄×C₂²⋊C₈	—	43	2	no	—
154	#1746	—	43	2	no	—
155	#1726	—	43	2	no	—
156	C₂×C₄₃	—	2	4	no	—
157	C₂×C₄₃	C₄×#117	1	1	yes	characteristic
158	C₂×C₄₃	C₂×#405	1	1	yes	characteristic
159	C₂×C₄₃	#2873	1	1	yes	characteristic
160	C₂×C₄₃	#2924	1	4	yes	—
161	C₂²×C₄²	—	43	2	no	—
162	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
163	#432	—	43	8	no	—
164	#431	Dic₄₃	1	8	yes	complemented
165	C₂×C₄⋊C₈	—	43	2	no	—
166	C₂×C₄⋊C₈	C₂×D₄₃	1	2	yes	—
167	C₂×C₂²⋊C₈	—	43	2	no	—
168	C₂×C₂²⋊C₈	C₂×D₄₃	1	1	yes	characteristic
169	C₂×C₂²⋊C₈	C₂×D₄₃	1	1	yes	characteristic
170	C₂×#97	—	43	2	no	—
171	C₂×#97	—	43	4	no	—
172	C₂×C₄×C₈	—	43	4	no	—
173	C₂×C₄×C₈	C₂×D₄₃	1	2	yes	—
174	#376	—	43	1	no	—
175	#376	—	43	1	no	—
176	#376	—	43	2	no	—
177	#376	—	43	2	no	—
178	#376	—	43	2	no	—
179	C₂×#95	—	43	2	no	—
180	C₂×#95	—	43	2	no	—
181	#336	—	43	4	no	—
182	#336	—	43	4	no	—
183	C₄₃	#20469	1	1	yes	characteristic, complemented, Sylow
184	C₂³×C₄	—	43	4	no	—
185	C₂³×C₄	C₄×D₄₃	1	1	yes	characteristic
186	C₂³×C₄	C₄×D₄₃	1	1	yes	characteristic
187	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
188	C₂²×C₈	—	43	4	no	—
189	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
190	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
191	C₂×C₄²	—	43	2	no	—
192	C₂×C₄²	—	43	4	no	—
193	C₂×C₄²	—	43	2	no	—
194	C₂×C₄²	—	86	8	no	—
195	C₂×C₄²	C₄×D₄₃	1	1	yes	characteristic
196	C₂×C₄²	C₄×D₄₃	1	1	yes	characteristic
197	C₂×C₄²	C₄×D₄₃	1	4	yes	—
198	C₂×C₄²	C₂×Dic₄₃	1	4	yes	—
199	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
200	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
201	C₄⋊C₈	—	43	8	no	—
202	C₄⋊C₈	C₂×Dic₄₃	1	8	yes	complemented
203	C₂²⋊C₈	—	43	8	no	—
204	C₂²⋊C₈	C₂×Dic₄₃	1	4	yes	—
205	C₂²⋊C₈	C₂×Dic₄₃	1	4	yes	—
206	#97	—	43	8	no	—
207	#97	—	86	8	no	—
208	C₄×C₈	—	86	8	no	—
209	C₄×C₈	C₂×Dic₄₃	1	8	yes	complemented
210	#95	—	43	2	no	—
211	#95	—	43	2	no	—
212	#95	—	43	4	no	—
213	#95	—	43	4	no	—
214	#95	—	43	2	no	—
215	#95	—	43	2	no	—
216	C₂⁴	C₂×C₄×D₄₃	1	1	yes	characteristic
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	4	2	no	—
220	C₂²×C₄	—	43	4	no	—
221	C₂²×C₄	—	43	4	no	—
222	C₂²×C₄	—	86	8	no	—
223	C₂²×C₄	—	86	8	no	—
224	C₂²×C₄	—	172	4	no	—
225	C₂²×C₄	C₄×Dic₄₃	1	4	yes	—
226	C₂²×C₄	C₄×Dic₄₃	1	4	yes	—
227	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
228	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
229	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
230	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
231	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
232	C₂²×C₄	C₂×C₄×D₄₃	1	2	yes	—
233	C₂²×C₄	C₂²×Dic₄₃	1	2	yes	—
234	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
235	C₂×C₈	—	2	4	no	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	43	8	no	—
238	C₂×C₈	—	86	4	no	—
239	C₂×C₈	—	86	4	no	—
240	C₂×C₈	C₂²×Dic₄₃	1	4	yes	—
241	C₂×C₈	C₂²×Dic₄₃	1	4	yes	—
242	C₄²	—	43	8	no	—
243	C₄²	—	86	8	no	—
244	C₄²	—	86	4	no	—
245	C₄²	C₄×Dic₄₃	1	4	yes	—
246	C₄²	C₄×Dic₄₃	1	4	yes	—
247	C₄²	C₂×C₄×D₄₃	1	2	yes	—
248	C₄²	C₂×C₄×D₄₃	1	2	yes	—
249	C₄²	C₂²×Dic₄₃	1	2	yes	—
250	C₄²	C₂²×Dic₄₃	1	2	yes	—
251	C₂³	—	2	2	no	—
252	C₂³	—	2	2	no	—
253	C₂³	C₂×C₄×Dic₄₃	1	2	yes	—
254	C₂³	#13063331	1	4	yes	—
255	C₂³	C₂²×C₄×D₄₃	1	1	yes	characteristic
256	C₂×C₄	—	2	2	no	—
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	4	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	—	43	8	no	—
266	C₂×C₄	—	86	4	no	—
267	C₂×C₄	—	86	4	no	—
268	C₂×C₄	—	86	4	no	—
269	C₂×C₄	—	86	4	no	—
270	C₂×C₄	—	86	4	no	—
271	C₂×C₄	—	172	16	no	—
272	C₂×C₄	C₂×C₄×Dic₄₃	1	2	yes	—
273	C₂×C₄	C₂×C₄×Dic₄₃	1	2	yes	—
274	C₂×C₄	C₂×C₄×Dic₄₃	1	2	yes	—
275	C₂×C₄	C₂×C₄×Dic₄₃	1	2	yes	—
276	C₂×C₄	C₂×C₄×Dic₄₃	1	2	yes	—
277	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
278	C₂×C₄	C₂³×Dic₄₃	1	1	yes	characteristic
279	C₂×C₄	#41×D₄₃	1	4	yes	—
280	C₈	—	2	4	no	—
281	C₈	—	2	4	no	—
282	C₈	—	86	8	no	—
283	C₂²	—	2	8	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	2	no	—
286	C₂²	#14582854	1	4	yes	—
287	C₂²	C₂²×C₄×Dic₄₃	1	1	yes	characteristic
288	C₂²	#117×D₄₃	1	2	yes	—
289	C₂²	C₂×#13063331	1	2	yes	—
290	C₂²	#14583665	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	—	2	2	no	—
296	C₄	—	86	8	no	—
297	C₄	—	172	4	no	—
298	C₄	#14583367	1	2	yes	—
299	C₄	#41×Dic₄₃	1	2	yes	—
300	C₂	—	2	4	no	—
301	C₂	#117×Dic₄₃	1	1	yes	characteristic
302	C₂	C₂×#14582854	1	1	yes	characteristic
303	C₂	#19143013	1	1	yes	characteristic
304	C₂	#19143248	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₃
commutator	C₂²×C₄₃	C₂²×C₄²
Fitting	C₂×C₄₃×#431	C₂
socle	C₂³×C₄₃	C₂³×C₄
prime radical	C₂⁴×C₄₃	C₂²×C₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₂×#431	D₄₃
hypocenter	C₄₃	#20469
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
3. #19134454	C₂	semidirect
6. #19134346	C₂	semidirect
7. #19134281	C₂	semidirect
10. #14582956	C₄	semidirect
26. C₄₃×#431	C₄	semidirect
37. C₄⋊(C₄₃⋊C₈)	C₂×C₄	semidirect
39. #13063236	C₂×C₄	semidirect
59. C₄⋊C₈×C₄₃	C₂×C₄	semidirect
62. C₄×C₈×C₄₃	C₂×C₄	semidirect
164. #431	Dic₄₃	semidirect
183. C₄₃	#20469	semidirect
202. C₄⋊C₈	C₂×Dic₄₃	semidirect
209. C₄×C₈	C₂×Dic₄₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14582956	2
C₂×#14582745	2
#19134454	2
#19134359	2
C₂²⋊C₈×Dic₄₃	2
#19134346	2
#19134281	2
C₂×C₄₃×#431	2
#20469	43