Subgroups — #24987061

Order 11008.

726 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₈×D₄₃	C₂	1	1	yes	maximal, characteristic
2	C₄×(C₂×D₄₃)⋊C₈	C₂	1	2	yes	maximal
3	C₂×C₈×C₄₃⋊C₈	C₂	1	2	yes	maximal, complemented
4	C₄₃⋊#1273	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₃×#1273	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₈×D₄₃	C₄	1	2	yes	—
7	C₂×C₄²×D₄₃	C₂²	1	1	yes	characteristic
8	C₂×(C₂×D₄₃)⋊C₈	C₄	1	4	yes	—
9	C₂×C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
10	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₄₃	C₄	1	2	yes	—
12	C₄×C₈×D₄₃	—	2	4	no	—
13	#14582653	C₄	1	4	yes	—
14	C₈×C₄₃⋊C₈	—	2	4	no	—
15	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₄₃	C₂²	1	2	yes	—
17	C₂²×C₄×D₄₃	C₂×C₄	1	2	yes	—
18	C₂²×C₄×D₄₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄₃⋊C₈	C₂×C₄	1	4	yes	—
20	C₂²×C₄₃⋊C₈	C₂×C₄	1	2	yes	—
21	C₂×C₈×D₄₃	—	2	4	no	—
22	C₂×C₈×D₄₃	—	2	4	no	—
23	C₂×C₈×D₄₃	C₈	1	8	yes	complemented
24	C₂×C₄×Dic₄₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×Dic₄₃	C₂×C₄	1	2	yes	—
26	C₄²×D₄₃	—	2	2	no	—
27	C₄²×D₄₃	—	2	1	no	—
28	C₄²×D₄₃	—	2	1	no	—
29	(C₂×D₄₃)⋊C₈	C₈	1	16	yes	complemented
30	C₄×C₄₃⋊C₈	—	2	2	no	—
31	C₄×C₄₃⋊C₈	—	2	2	no	—
32	C₄×C₄₃⋊C₈	D₄	1	4	yes	—
33	C₈×Dic₄₃	—	2	4	no	—
34	C₂²×C₈×C₄₃	C₂×C₄	1	2	yes	—
35	C₂²×C₈×C₄₃	C₂×C₄	1	4	yes	—
36	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
37	C₄×C₈×C₄₃	—	2	2	no	—
38	C₄×C₈×C₄₃	—	2	2	no	—
39	C₄×C₈×C₄₃	—	2	2	no	—
40	C₂³×D₄₃	C₄²	1	1	yes	characteristic
41	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
43	C₂²×Dic₄₃	C₄²	1	1	yes	characteristic
44	C₂²×Dic₄₃	C₄²	1	2	yes	—
45	C₂×C₄×D₄₃	—	2	1	no	—
46	C₂×C₄×D₄₃	—	2	1	no	—
47	C₂×C₄×D₄₃	—	2	2	no	—
48	C₂×C₄×D₄₃	—	2	2	no	—
49	C₂×C₄×D₄₃	—	2	2	no	—
50	C₂×C₄×D₄₃	—	2	4	no	—
51	C₂×C₄×D₄₃	C₂×C₈	1	8	yes	—
52	C₂×C₄×D₄₃	C₂×C₈	1	4	yes	—
53	C₂×C₈×C₄₃	—	2	2	no	—
54	C₂×C₈×C₄₃	—	2	4	no	—
55	C₂×C₈×C₄₃	—	2	2	no	—
56	C₂×C₈×C₄₃	—	2	4	no	—
57	C₂×C₈×C₄₃	C₂×C₈	1	4	yes	—
58	C₂×C₈×C₄₃	C₂×C₈	1	8	yes	complemented
59	C₄²×C₄₃	C₂×D₄	1	2	yes	—
60	C₄²×C₄₃	#41	1	1	yes	characteristic
61	C₄²×C₄₃	#41	1	1	yes	characteristic
62	C₄×Dic₄₃	—	2	1	no	—
63	C₄×Dic₄₃	—	2	1	no	—
64	C₄×Dic₄₃	—	2	2	no	—
65	C₄×Dic₄₃	—	2	2	no	—
66	C₂×C₄₃⋊C₈	—	2	4	no	—
67	C₂×C₄₃⋊C₈	—	2	4	no	—
68	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	4	yes	—
69	C₂×C₄₃⋊C₈	C₂×C₈	1	4	yes	—
70	C₂×C₄₃⋊C₈	C₂×C₈	1	8	yes	complemented
71	C₂×C₄₃⋊C₈	M₁₆	1	4	yes	—
72	C₈×D₄₃	—	2	16	no	—
73	C₂³×C₄₃	C₂×C₄²	1	1	yes	characteristic
74	C₂²×D₄₃	—	2	1	no	—
75	C₂²×D₄₃	—	2	1	no	—
76	C₂²×D₄₃	—	2	2	no	—
77	C₂²×D₄₃	C₄×C₈	1	4	yes	—
78	C₂²×D₄₃	C₄×C₈	1	2	yes	—
79	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
81	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
82	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
83	C₂×C₄×C₄₃	C₂²×C₈	1	2	yes	—
84	C₂×C₄×C₄₃	C₂²×C₈	1	4	yes	—
85	C₂×C₄×C₄₃	C₂×M₁₆	1	2	yes	—
86	C₂×C₄×C₄₃	#131	1	4	yes	—
87	C₂×Dic₄₃	—	2	1	no	—
88	C₂×Dic₄₃	—	2	2	no	—
89	C₂×Dic₄₃	—	2	1	no	—
90	C₂×Dic₄₃	—	2	2	no	—
91	C₂×Dic₄₃	C₄×C₈	1	2	yes	—
92	C₂×Dic₄₃	C₄×C₈	1	4	yes	—
93	C₄×D₄₃	—	2	4	no	—
94	C₄×D₄₃	—	2	4	no	—
95	C₄×D₄₃	—	2	8	no	—
96	C₄×D₄₃	—	2	8	no	—
97	C₈×C₄₃	—	2	4	no	—
98	C₈×C₄₃	—	2	8	no	—
99	C₄₃⋊C₈	—	2	8	no	—
100	C₄₃⋊C₈	C₂²⋊C₈	1	8	yes	complemented
101	C₈×C₂²⋊C₈	—	43	1	no	maximal, Sylow
102	C₂²×C₄₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
103	C₂²×C₄₃	C₂×C₄×C₈	1	2	yes	—
104	C₂²×C₄₃	C₂×C₄×C₈	1	1	yes	characteristic
105	C₂²×C₄₃	C₄×M₁₆	1	1	yes	characteristic
106	C₂²×C₄₃	#405	1	2	yes	—
107	C₂×D₄₃	—	2	4	no	—
108	C₂×D₄₃	—	2	2	no	—
109	C₂×D₄₃	—	2	2	no	—
110	C₂×D₄₃	—	2	4	no	—
111	C₂×D₄₃	C₈²	1	4	yes	complemented
112	C₄×C₄₃	C₂×C₂²⋊C₈	1	2	yes	—
113	C₄×C₄₃	#431	1	2	yes	—
114	C₄×C₄₃	C₈×D₄	1	4	yes	—
115	C₄×C₄₃	C₈×D₄	1	4	yes	—
116	Dic₄₃	—	2	2	no	—
117	Dic₄₃	—	2	4	no	—
118	C₂²×C₄×C₈	—	43	1	no	—
119	C₄×C₂²⋊C₈	—	43	2	no	—
120	C₂×C₈²	—	43	2	no	—
121	#1273	—	43	1	no	—
122	#1273	D₄₃	1	1	yes	characteristic, complemented
123	C₂×C₄₃	C₂×C₈²	1	1	yes	characteristic
124	C₂×C₄₃	C₈×M₁₆	1	2	yes	—
125	C₂×C₄₃	#1450	1	1	yes	characteristic
126	C₂×C₄₃	C₄×C₂²⋊C₈	1	1	yes	characteristic
127	C₂×C₄₃	C₈×C₂²⋊C₄	1	2	yes	—
128	D₄₃	—	2	4	no	—
129	C₂³×C₈	—	43	2	no	—
130	C₂²×C₄²	—	43	1	no	—
131	C₂×C₂²⋊C₈	—	43	4	no	—
132	C₂×C₄×C₈	—	43	1	no	—
133	C₂×C₄×C₈	—	43	2	no	—
134	C₂×C₄×C₈	—	43	2	no	—
135	C₂×C₄×C₈	—	86	4	no	—
136	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
137	C₂×C₄×C₈	C₂×D₄₃	1	2	yes	—
138	#336	—	43	4	no	—
139	C₈²	—	86	4	no	—
140	C₄₃	C₈×C₂²⋊C₈	1	1	yes	characteristic, complemented, Sylow
141	C₂³×C₄	—	43	1	no	—
142	C₂³×C₄	—	43	2	no	—
143	C₂²×C₈	—	43	2	no	—
144	C₂²×C₈	—	43	4	no	—
145	C₂²×C₈	—	43	8	no	—
146	C₂²×C₈	—	86	4	no	—
147	C₂²×C₈	—	86	4	no	—
148	C₂²×C₈	C₄×D₄₃	1	2	yes	—
149	C₂²×C₈	C₄×D₄₃	1	4	yes	—
150	C₂×C₄²	—	43	2	no	—
151	C₂×C₄²	—	43	1	no	—
152	C₂×C₄²	—	86	2	no	—
153	C₂×C₄²	—	86	1	no	—
154	C₂×C₄²	—	86	1	no	—
155	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
156	C₂²⋊C₈	—	43	16	no	—
157	C₄×C₈	—	2	2	no	—
158	C₄×C₈	—	2	2	no	—
159	C₄×C₈	—	2	2	no	—
160	C₄×C₈	—	43	4	no	—
161	C₄×C₈	—	86	4	no	—
162	C₄×C₈	—	86	2	no	—
163	C₄×C₈	—	86	2	no	—
164	C₂⁴	—	43	1	no	—
165	C₂²×C₄	—	43	1	no	—
166	C₂²×C₄	—	43	2	no	—
167	C₂²×C₄	—	43	8	no	—
168	C₂²×C₄	—	43	4	no	—
169	C₂²×C₄	—	86	1	no	—
170	C₂²×C₄	—	86	1	no	—
171	C₂²×C₄	—	86	2	no	—
172	C₂²×C₄	—	86	2	no	—
173	C₂²×C₄	—	86	2	no	—
174	C₂²×C₄	—	86	4	no	—
175	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
176	C₂²×C₄	C₂×C₄×D₄₃	1	2	yes	—
177	C₂×C₈	—	2	2	no	—
178	C₂×C₈	—	2	4	no	—
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	—	2	4	no	—
181	C₂×C₈	—	43	4	no	—
182	C₂×C₈	—	43	4	no	—
183	C₂×C₈	—	43	4	no	—
184	C₂×C₈	—	43	8	no	—
185	C₂×C₈	—	86	16	no	—
186	C₂×C₈	—	86	4	no	—
187	C₂×C₈	—	86	4	no	—
188	C₂×C₈	C₈×D₄₃	1	4	yes	—
189	C₂×C₈	C₈×D₄₃	1	8	yes	complemented
190	C₄²	—	86	1	no	—
191	C₄²	—	86	1	no	—
192	C₄²	—	86	2	no	—
193	C₄²	—	86	2	no	—
194	C₄²	D₄×D₄₃	1	2	yes	—
195	C₄²	#10621900	1	1	yes	characteristic
196	C₄²	#10621902	1	1	yes	characteristic
197	C₂³	—	43	2	no	—
198	C₂³	—	43	4	no	—
199	C₂³	—	86	1	no	—
200	C₂³	—	86	2	no	—
201	C₂³	—	86	1	no	—
202	C₂³	C₄²×D₄₃	1	1	yes	characteristic
203	C₂×C₄	—	43	2	no	—
204	C₂×C₄	—	43	4	no	—
205	C₂×C₄	—	86	1	no	—
206	C₂×C₄	—	86	2	no	—
207	C₂×C₄	—	86	1	no	—
208	C₂×C₄	—	86	2	no	—
209	C₂×C₄	—	86	4	no	—
210	C₂×C₄	—	86	4	no	—
211	C₂×C₄	—	86	8	no	—
212	C₂×C₄	—	86	8	no	—
213	C₂×C₄	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
214	C₂×C₄	#13063283	1	1	yes	characteristic
215	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	2	yes	—
216	C₂×C₄	#13063287	1	2	yes	—
217	C₂×C₄	C₂×C₈×D₄₃	1	2	yes	—
218	C₂×C₄	C₂×C₈×D₄₃	1	4	yes	—
219	C₂×C₄	M₁₆×D₄₃	1	2	yes	—
220	C₂×C₄	#13063333	1	4	yes	—
221	C₈	—	2	4	no	—
222	C₈	—	2	8	no	—
223	C₈	—	43	8	no	—
224	C₈	—	86	8	no	—
225	C₂²	—	43	4	no	—
226	C₂²	—	86	2	no	—
227	C₂²	—	86	4	no	—
228	C₂²	—	86	4	no	—
229	C₂²	—	86	2	no	—
230	C₂²	#14582746	1	1	yes	characteristic
231	C₂²	C₄×C₈×D₄₃	1	1	yes	characteristic
232	C₂²	C₄×C₈×D₄₃	1	2	yes	—
233	C₂²	#14582851	1	1	yes	characteristic
234	C₂²	#14582856	1	2	yes	—
235	C₄	—	86	2	no	—
236	C₄	—	86	4	no	—
237	C₄	C₂²⋊C₈×D₄₃	1	2	yes	—
238	C₄	#14582951	1	2	yes	—
239	C₄	(C₂²⋊D₄₃)⋊C₈	1	4	yes	—
240	C₄	#14582964	1	4	yes	—
241	C₂	—	86	4	no	—
242	C₂	C₈²×D₄₃	1	1	yes	characteristic
243	C₂	#19132874	1	2	yes	—
244	C₂	#19132880	1	1	yes	characteristic
245	C₂	#19134352	1	1	yes	characteristic
246	C₂	#19134365	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₄₃
commutator	C₂×C₄₃	C₂×C₈²
Fitting	C₄₃×#1273	C₂
socle	C₂³×C₄₃	C₂×C₄²
prime radical	C₂³×D₄₃	C₄²
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#1273	D₄₃
hypocenter	C₄₃	C₈×C₂²⋊C₈
LCS	G ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄² ◁ #1273

Complements

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

N	H ≅ G/N	type
3. C₂×C₈×C₄₃⋊C₈	C₂	semidirect
4. C₄₃⋊#1273	C₂	semidirect
5. C₄₃×#1273	C₂	semidirect
23. C₂×C₈×D₄₃	C₈	semidirect
29. (C₂×D₄₃)⋊C₈	C₈	semidirect
58. C₂×C₈×C₄₃	C₂×C₈	semidirect
70. C₂×C₄₃⋊C₈	C₂×C₈	semidirect
100. C₄₃⋊C₈	C₂²⋊C₈	semidirect
111. C₂×D₄₃	C₈²	semidirect
122. #1273	D₄₃	semidirect
140. C₄₃	C₈×C₂²⋊C₈	semidirect
189. C₂×C₈	C₈×D₄₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₈×D₄₃	2
C₄×(C₂×D₄₃)⋊C₈	2
C₂×C₈×C₄₃⋊C₈	2
C₄₃⋊#1273	2
C₄₃×#1273	2
C₈×C₂²⋊C₈	43