Subgroups — #25084303

Order 11008.

706 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19145182	C₂	1	2	yes	maximal
2	#19145179	C₂	1	2	yes	maximal
3	#19145106	C₂	1	2	yes	maximal
4	#19134213	C₂	1	1	yes	maximal, characteristic, complemented
5	#19134204	C₂	1	2	yes	maximal, complemented
6	#19134193	C₂	1	2	yes	maximal, complemented
7	#19134189	C₂	1	2	yes	maximal, complemented
8	C₈²×D₄₃	C₂	1	1	yes	maximal, characteristic
9	C₄₃×#1713	C₂	1	1	yes	maximal, characteristic, complemented
10	#14583454	C₂²	1	4	yes	—
11	#14583304	C₂²	1	2	yes	—
12	#14583304	C₂²	1	2	yes	—
13	#14583300	—	2	4	no	—
14	#14583295	C₂²	1	4	yes	—
15	#14583279	C₂²	1	2	yes	—
16	#14583251	C₂²	1	4	yes	—
17	#14583249	C₂²	1	4	yes	—
18	#14583141	—	2	4	no	—
19	#14583137	—	2	4	no	—
20	Q₁₆×Dic₄₃	—	2	4	no	—
21	#14582993	—	2	4	no	—
22	#14582936	—	2	4	no	—
23	C₄×C₈×D₄₃	C₂²	1	1	yes	characteristic
24	C₄×C₈×D₄₃	C₂²	1	1	yes	characteristic
25	C₄×C₈×D₄₃	C₂²	1	1	yes	characteristic
26	C₈×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
27	C₈×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
28	C₈×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
29	C₄₃×#500	C₂²	1	2	yes	—
30	C₄₃×#500	C₂²	1	2	yes	—
31	C₄₃×#494	C₂²	1	2	yes	—
32	C₈²×C₄₃	C₂²	1	1	yes	characteristic
33	C₂×Dic₁₇₂	—	2	4	no	—
34	C₂×C₈×D₄₃	D₄	1	2	yes	—
35	C₂×C₈×D₄₃	D₄	1	2	yes	—
36	C₂×C₈×D₄₃	D₄	1	2	yes	—
37	#13063325	C₂³	1	4	yes	—
38	#13063316	—	2	4	no	—
39	#13063305	—	2	4	no	—
40	#13063301	—	2	4	no	—
41	Q₈×Dic₄₃	—	2	4	no	—
42	#13063290	—	2	4	no	—
43	#13063283	—	2	4	no	—
44	C₄²×D₄₃	C₂³	1	1	yes	characteristic
45	#13063259	—	2	4	no	—
46	#13063259	—	2	4	no	—
47	#13063256	—	2	8	no	—
48	#13063249	—	2	8	no	—
49	#13063248	—	2	8	no	—
50	C₄×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
51	C₄×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
52	C₄×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
53	C₈×Dic₄₃	D₄	1	2	yes	—
54	C₈×Dic₄₃	D₄	1	2	yes	—
55	C₈×Dic₄₃	D₄	1	2	yes	—
56	C₂×Q₁₆×C₄₃	—	2	4	no	—
57	C₄⋊Q₈×C₄₃	C₂³	1	4	yes	—
58	#107×C₄₃	—	2	4	no	—
59	#107×C₄₃	—	2	4	no	—
60	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
61	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
62	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
63	C₂×Q₈×C₄₃	—	2	4	no	—
64	C₂×C₄×D₄₃	C₂×D₄	1	1	yes	characteristic
65	C₂×C₄×D₄₃	C₂×D₄	1	1	yes	characteristic
66	C₂×C₄×D₄₃	C₂×D₄	1	1	yes	characteristic
67	C₂×Dic₈₆	—	2	4	no	—
68	Q₁₆×C₄₃	—	4	8	no	—
69	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
70	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
71	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
72	C₄⋊C₄×C₄₃	—	2	4	no	—
73	C₄⋊C₄×C₄₃	—	2	4	no	—
74	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
75	(C₂×D₄₃)⋊C₄	—	4	4	no	—
76	C₄⋊Dic₄₃	—	2	4	no	—
77	C₄⋊Dic₄₃	—	2	4	no	—
78	C₄⋊Dic₄₃	—	4	4	no	—
79	Dic₄₃⋊C₄	—	4	4	no	—
80	C₄×Dic₄₃	—	4	4	no	—
81	C₄×Dic₄₃	C₂×D₄	1	1	yes	characteristic
82	C₄×Dic₄₃	C₂×D₄	1	1	yes	characteristic
83	C₄×Dic₄₃	C₂×D₄	1	1	yes	characteristic
84	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
85	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
86	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
87	Dic₁₇₂	—	4	8	no	—
88	C₈×D₄₃	—	2	4	no	—
89	C₈×D₄₃	—	2	4	no	—
90	C₈×D₄₃	Q₁₆	1	8	yes	complemented
91	C₂²×D₄₃	#127	1	1	yes	characteristic
92	Q₈×C₄₃	—	4	8	no	—
93	C₂×C₄×C₄₃	—	4	4	no	—
94	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
95	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
96	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
97	C₂×Dic₄₃	—	4	4	no	—
98	C₂×Dic₄₃	#127	1	1	yes	characteristic
99	C₂×Dic₄₃	#127	1	1	yes	characteristic
100	C₂×Dic₄₃	#127	1	1	yes	characteristic
101	C₄×D₄₃	—	2	2	no	—
102	C₄×D₄₃	—	2	2	no	—
103	C₄×D₄₃	C₂×Q₁₆	1	4	yes	—
104	Dic₈₆	—	4	8	no	—
105	C₈×C₄₃	C₂×Q₁₆	1	4	yes	—
106	C₈×C₄₃	#135	1	4	yes	—
107	C₈×C₄₃	#135	1	4	yes	—
108	C₄₃⋊C₈	—	2	2	no	—
109	C₄₃⋊C₈	—	2	2	no	—
110	C₄₃⋊C₈	C₂×Q₁₆	1	4	yes	—
111	#19604	—	43	1	no	maximal, Sylow
112	C₂²×C₄₃	C₂×#127	1	1	yes	characteristic
113	C₂×D₄₃	—	2	1	no	—
114	C₂×D₄₃	—	2	1	no	—
115	C₂×D₄₃	#494	1	2	yes	—
116	C₄×C₄₃	—	8	4	no	—
117	C₄×C₄₃	C₂²×Q₁₆	1	2	yes	—
118	C₄×C₄₃	C₂×#135	1	2	yes	—
119	C₄×C₄₃	C₂×#135	1	2	yes	—
120	Dic₄₃	—	2	1	no	—
121	Dic₄₃	—	2	1	no	—
122	Dic₄₃	—	8	4	no	—
123	Dic₄₃	#494	1	2	yes	—
124	#3170	—	43	2	no	—
125	#3170	—	43	2	no	—
126	#3160	—	43	2	no	—
127	#1713	—	43	1	no	—
128	#1713	D₄₃	1	1	yes	characteristic, complemented
129	#1710	—	43	2	no	—
130	#1707	—	43	2	no	—
131	#1707	—	43	2	no	—
132	C₂×C₈²	—	43	1	no	—
133	C₂×C₄₃	C₂×#494	1	1	yes	characteristic
134	C₂×C₄₃	#3147	1	1	yes	characteristic
135	C₂×C₄₃	#3147	1	1	yes	characteristic
136	D₄₃	—	2	2	no	—
137	#533	—	43	4	no	—
138	#500	—	43	2	no	—
139	#500	—	43	2	no	—
140	#500	C₂×D₄₃	1	2	yes	—
141	#500	C₂×D₄₃	1	2	yes	—
142	#499	—	86	4	no	—
143	#498	—	43	4	no	—
144	#494	—	43	2	no	—
145	#494	C₂×D₄₃	1	2	yes	—
146	#487	—	43	4	no	—
147	#487	—	43	4	no	—
148	#467	—	86	4	no	—
149	#467	—	86	4	no	—
150	C₄×Q₁₆	—	86	4	no	—
151	C₄×Q₁₆	—	86	4	no	—
152	#427	—	86	4	no	—
153	C₂×C₄×C₈	—	43	1	no	—
154	C₂×C₄×C₈	—	43	1	no	—
155	C₂×C₄×C₈	—	43	1	no	—
156	C₈²	—	43	1	no	—
157	C₈²	—	43	1	no	—
158	C₈²	—	43	1	no	—
159	C₈²	C₂×D₄₃	1	1	yes	characteristic
160	C₄₃	#19604	1	1	yes	characteristic, complemented, Sylow
161	C₂×Q₁₆	—	2	4	no	—
162	C₂×Q₁₆	—	86	4	no	—
163	C₂²×C₈	—	43	2	no	—
164	C₂²×C₈	—	43	2	no	—
165	C₂²×C₈	—	43	2	no	—
166	C₄⋊Q₈	—	43	4	no	—
167	C₄⋊Q₈	C₂²×D₄₃	1	4	yes	—
168	#125	—	86	4	no	—
169	C₂²⋊Q₈	—	86	4	no	—
170	C₂²⋊Q₈	—	86	4	no	—
171	C₄×Q₈	—	86	4	no	—
172	C₄×Q₈	—	86	4	no	—
173	#117	—	86	4	no	—
174	C₂×C₄²	—	43	1	no	—
175	#107	—	2	4	no	—
176	#107	—	2	4	no	—
177	#107	—	86	4	no	—
178	#107	—	86	4	no	—
179	#106	—	86	8	no	—
180	#103	—	86	8	no	—
181	#103	—	86	8	no	—
182	C₄×C₈	—	43	2	no	—
183	C₄×C₈	—	43	2	no	—
184	C₄×C₈	—	43	2	no	—
185	C₄×C₈	—	43	1	no	—
186	C₄×C₈	—	43	1	no	—
187	C₄×C₈	—	43	1	no	—
188	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
189	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
190	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
191	C₂×Q₈	—	2	4	no	—
192	C₂×Q₈	—	86	4	no	—
193	C₂²×C₄	—	43	1	no	—
194	C₂²×C₄	—	43	1	no	—
195	C₂²×C₄	—	43	1	no	—
196	Q₁₆	—	4	8	no	—
197	Q₁₆	—	172	8	no	—
198	C₂×C₈	—	43	2	no	—
199	C₂×C₈	—	43	8	no	—
200	C₂×C₈	—	43	2	no	—
201	C₂×C₈	—	43	2	no	—
202	C₂×C₈	—	86	4	no	—
203	C₂×C₈	—	86	4	no	—
204	C₂×C₈	D₄×D₄₃	1	2	yes	—
205	C₂×C₈	D₄×D₄₃	1	2	yes	—
206	C₂×C₈	D₄×D₄₃	1	2	yes	—
207	C₄⋊C₄	—	2	4	no	—
208	C₄⋊C₄	—	2	4	no	—
209	C₄⋊C₄	—	86	4	no	—
210	C₄⋊C₄	—	86	4	no	—
211	C₄⋊C₄	—	172	4	no	—
212	C₄⋊C₄	—	172	4	no	—
213	C₂²⋊C₄	—	172	4	no	—
214	C₄²	—	43	1	no	—
215	C₄²	—	43	1	no	—
216	C₄²	—	43	1	no	—
217	C₄²	—	172	4	no	—
218	C₄²	C₂³×D₄₃	1	1	yes	characteristic
219	C₂³	—	43	1	no	—
220	Q₈	—	4	8	no	—
221	Q₈	—	172	8	no	—
222	C₂×C₄	—	4	4	no	—
223	C₂×C₄	—	43	1	no	—
224	C₂×C₄	—	43	4	no	—
225	C₂×C₄	—	43	1	no	—
226	C₂×C₄	—	43	1	no	—
227	C₂×C₄	—	86	2	no	—
228	C₂×C₄	—	86	2	no	—
229	C₂×C₄	—	172	4	no	—
230	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
231	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
232	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
233	C₈	—	43	4	no	—
234	C₈	—	86	2	no	—
235	C₈	—	86	2	no	—
236	C₈	Q₁₆×D₄₃	1	4	yes	—
237	C₈	#13063344	1	4	yes	—
238	C₈	#13063348	1	4	yes	—
239	C₂²	—	43	2	no	—
240	C₂²	—	86	1	no	—
241	C₂²	—	86	1	no	—
242	C₂²	#127×D₄₃	1	1	yes	characteristic
243	C₄	—	8	4	no	—
244	C₄	—	43	2	no	—
245	C₄	—	86	1	no	—
246	C₄	—	86	1	no	—
247	C₄	—	344	4	no	—
248	C₄	C₂×Q₁₆×D₄₃	1	2	yes	—
249	C₄	C₂×#13063344	1	2	yes	—
250	C₄	C₂×#13063348	1	2	yes	—
251	C₂	—	86	2	no	—
252	C₂	#494×D₄₃	1	1	yes	characteristic
253	C₂	#19145022	1	1	yes	characteristic
254	C₂	#19145024	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#127×D₄₃
commutator, cosocle	C₄²×C₄₃	C₂⁴
Frattini	C₄²	C₂³×D₄₃
Fitting	C₄₃×#1713	C₂
socle	C₂²×C₄₃	C₂×#127
prime radical	C₂²×D₄₃	#127
hypercenter	#1713	D₄₃
hypocenter	C₄₃	#19604
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1713

Complements

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

N	H ≅ G/N	type
4. #19134213	C₂	semidirect
5. #19134204	C₂	semidirect
6. #19134193	C₂	semidirect
7. #19134189	C₂	semidirect
9. C₄₃×#1713	C₂	semidirect
90. C₈×D₄₃	Q₁₆	semidirect
128. #1713	D₄₃	semidirect
160. C₄₃	#19604	semidirect

Maximal subgroups

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

subgroup	index
#19145182	2
#19145179	2
#19145106	2
#19134213	2
#19134204	2
#19134193	2
#19134189	2
C₈²×D₄₃	2
C₄₃×#1713	2
#19604	43