Subgroups — #25005368

Order 11008.

774 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14582769	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582667	C₂	1	4	yes	maximal
3	#19134289	C₂	1	1	yes	maximal, characteristic
4	C₄₃×#1728	C₂	1	1	yes	maximal, characteristic
5	C₂²×#10621877	C₂²	1	2	yes	—
6	C₂²×C₄⋊Dic₄₃	C₄	1	4	yes	—
7	C₂²×C₄⋊Dic₄₃	C₂²	1	2	yes	—
8	#14582769	—	2	2	no	—
9	#14582769	—	2	1	no	—
10	#14582769	—	2	1	no	—
11	#14582667	—	2	8	no	—
12	#14582654	—	2	2	no	—
13	C₂²×M₁₆×C₄₃	C₂²	1	2	yes	—
14	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
15	C₄₃×#336	—	2	1	no	—
16	C₄₃×#336	—	2	1	no	—
17	C₂³×Dic₄₃	C₂×C₄	1	4	yes	—
18	C₂×#10621877	—	2	2	no	—
19	C₂×#10621877	—	2	4	no	—
20	C₂×#10621877	—	2	4	no	—
21	C₂×#10621877	Q₈	1	4	yes	—
22	C₂²×C₄₃⋊C₈	—	2	2	no	—
23	C₂×C₄⋊Dic₄₃	—	2	4	no	—
24	C₂×C₄⋊Dic₄₃	—	2	4	no	—
25	C₂×C₄⋊Dic₄₃	—	2	4	no	—
26	C₂×C₄⋊Dic₄₃	—	2	2	no	—
27	C₂×C₄⋊Dic₄₃	—	2	2	no	—
28	C₂×C₄⋊Dic₄₃	—	2	4	no	—
29	C₂×C₄⋊Dic₄₃	—	4	8	no	—
30	C₂³×C₄×C₄₃	C₂×C₄	1	2	yes	—
31	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
32	C₂×M₁₆×C₄₃	—	2	2	no	—
33	C₂×M₁₆×C₄₃	—	2	4	no	—
34	C₂×M₁₆×C₄₃	—	2	4	no	—
35	C₂×M₁₆×C₄₃	D₄	1	4	yes	—
36	C₂²×C₈×C₄₃	—	2	2	no	—
37	C₂×C₄²×C₄₃	—	2	4	no	—
38	C₂×C₄²×C₄₃	D₄	1	2	yes	—
39	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
41	C₂⁴×C₄₃	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₄₃	—	2	4	no	—
43	C₂²×C₄×C₄₃	—	2	4	no	—
44	C₂²×C₄×C₄₃	—	2	4	no	—
45	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
46	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
50	C₂²×C₄×C₄₃	C₂×Q₈	1	2	yes	—
51	C₂²×C₄×C₄₃	#41	1	4	yes	—
52	C₂²×C₄×C₄₃	#41	1	2	yes	—
53	C₂²×C₄×C₄₃	#41	1	2	yes	—
54	C₂²×Dic₄₃	—	2	4	no	—
55	C₂²×Dic₄₃	—	2	8	no	—
56	C₂²×Dic₄₃	—	4	8	no	—
57	M₁₆×C₄₃	—	2	8	no	—
58	M₁₆×C₄₃	—	4	4	no	—
59	C₂×C₈×C₄₃	—	2	4	no	—
60	C₂×C₈×C₄₃	—	4	2	no	—
61	C₂×C₈×C₄₃	—	4	2	no	—
62	C₄²×C₄₃	—	2	2	no	—
63	C₄²×C₄₃	—	2	2	no	—
64	C₄²×C₄₃	—	2	4	no	—
65	C₄⋊Dic₄₃	—	4	8	no	—
66	C₄⋊Dic₄₃	—	4	8	no	—
67	C₄⋊Dic₄₃	—	4	4	no	—
68	C₄⋊Dic₄₃	—	4	2	no	—
69	C₄⋊Dic₄₃	—	4	2	no	—
70	#10621877	—	2	8	no	—
71	#10621877	—	4	4	no	—
72	C₂×C₄₃⋊C₈	—	2	4	no	—
73	C₂×C₄₃⋊C₈	—	4	2	no	—
74	C₂×C₄₃⋊C₈	—	4	2	no	—
75	C₂³×C₄₃	—	2	2	no	—
76	C₂³×C₄₃	—	2	2	no	—
77	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂³×C₄₃	#117	1	2	yes	—
79	C₂³×C₄₃	C₄×D₄	1	2	yes	—
80	C₂³×C₄₃	C₄×Q₈	1	2	yes	—
81	C₂×C₄×C₄₃	—	2	16	no	—
82	C₂×C₄×C₄₃	—	2	2	no	—
83	C₂×C₄×C₄₃	—	2	2	no	—
84	C₂×C₄×C₄₃	—	2	2	no	—
85	C₂×C₄×C₄₃	—	2	2	no	—
86	C₂×C₄×C₄₃	#100	1	4	yes	—
87	C₂×C₄×C₄₃	#101	1	4	yes	—
88	C₂×C₄×C₄₃	#121	1	2	yes	—
89	C₂×C₄×C₄₃	C₂²⋊Q₈	1	4	yes	—
90	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
91	C₂×C₄×C₄₃	#123	1	2	yes	—
92	C₂×C₄×C₄₃	#123	1	4	yes	—
93	C₂×C₄×C₄₃	#123	1	2	yes	—
94	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
95	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
96	C₂×C₄×C₄₃	#125	1	4	yes	—
97	C₂×C₄×C₄₃	#126	1	4	yes	—
98	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
99	C₂×C₄×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
100	C₂×Dic₄₃	—	4	4	no	—
101	C₂×Dic₄₃	—	4	8	no	—
102	C₂×Dic₄₃	—	8	8	no	—
103	C₈×C₄₃	—	4	4	no	—
104	C₄₃⋊C₈	—	4	4	no	—
105	#11354	—	43	1	no	maximal, Sylow
106	C₂²×C₄₃	—	2	4	no	—
107	C₂²×C₄₃	—	2	2	no	—
108	C₂²×C₄₃	—	2	2	no	—
109	C₂²×C₄₃	—	2	4	no	—
110	C₂²×C₄₃	#381	1	1	yes	characteristic
111	C₂²×C₄₃	#387	1	4	yes	—
112	C₂²×C₄₃	#390	1	1	yes	characteristic
113	C₂²×C₄₃	#391	1	1	yes	characteristic
114	C₂²×C₄₃	C₂×#100	1	2	yes	—
115	C₂²×C₄₃	C₂×#101	1	2	yes	—
116	C₄×C₄₃	—	2	4	no	—
117	C₄×C₄₃	—	2	4	no	—
118	C₄×C₄₃	#397	1	2	yes	—
119	C₄×C₄₃	#398	1	2	yes	—
120	C₄×C₄₃	#399	1	2	yes	—
121	C₄×C₄₃	#400	1	2	yes	—
122	Dic₄₃	—	8	4	no	—
123	C₂×#384	—	43	1	no	—
124	C₂×#343	—	43	4	no	—
125	#1728	—	43	1	no	—
126	#1728	D₄₃	1	1	yes	characteristic
127	C₂×C₄₃	—	2	4	no	—
128	C₂×C₄₃	#1445	1	1	yes	characteristic
129	C₂×C₄₃	#1821	1	2	yes	—
130	C₂×C₄₃	#1965	1	1	yes	characteristic
131	C₂×C₄₃	#1966	1	1	yes	characteristic
132	C₂×C₄₃	#1997	1	2	yes	—
133	C₂²×M₁₆	—	43	2	no	—
134	C₂²×M₁₆	C₂×D₄₃	1	2	yes	—
135	C₂²×C₄⋊C₄	—	43	2	no	—
136	C₂²×C₄⋊C₄	—	43	4	no	—
137	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
138	#384	—	86	2	no	—
139	#384	—	86	1	no	—
140	#384	—	86	1	no	—
141	#343	—	86	8	no	—
142	#336	—	2	1	no	—
143	#336	—	2	1	no	—
144	#336	—	86	2	no	—
145	C₄₃	#11354	1	1	yes	characteristic, complemented, Sylow
146	C₂³×C₄	—	43	4	no	—
147	C₂³×C₄	C₄×D₄₃	1	2	yes	—
148	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
149	C₂×M₁₆	—	2	2	no	—
150	C₂×M₁₆	—	2	4	no	—
151	C₂×M₁₆	—	2	4	no	—
152	C₂×M₁₆	—	43	4	no	—
153	C₂×M₁₆	—	86	2	no	—
154	C₂×M₁₆	—	86	4	no	—
155	C₂×M₁₆	—	86	4	no	—
156	C₂×M₁₆	D₁₇₂	1	4	yes	—
157	C₂²×C₈	—	2	2	no	—
158	C₂²×C₈	—	86	2	no	—
159	C₂×C₄⋊C₄	—	86	4	no	—
160	C₂×C₄⋊C₄	—	86	4	no	—
161	C₂×C₄⋊C₄	—	86	4	no	—
162	C₂×C₄⋊C₄	—	86	2	no	—
163	C₂×C₄⋊C₄	—	86	4	no	—
164	C₂×C₄⋊C₄	—	86	2	no	—
165	C₂×C₄⋊C₄	—	172	8	no	—
166	C₂×C₄²	—	2	4	no	—
167	C₂×C₄²	D₁₇₂	1	2	yes	—
168	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
169	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
170	C₂⁴	C₂×C₄×D₄₃	1	1	yes	characteristic
171	C₂²×C₄	—	2	4	no	—
172	C₂²×C₄	—	2	4	no	—
173	C₂²×C₄	—	2	4	no	—
174	C₂²×C₄	—	86	8	no	—
175	C₂²×C₄	—	86	4	no	—
176	C₂²×C₄	—	172	8	no	—
177	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	2	yes	—
178	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	2	yes	—
179	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
180	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
181	C₂²×C₄	#10621898	1	2	yes	—
182	C₂²×C₄	#10621900	1	4	yes	—
183	C₂²×C₄	#10621900	1	2	yes	—
184	C₂²×C₄	Q₈×D₄₃	1	2	yes	—
185	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
186	M₁₆	—	2	8	no	—
187	M₁₆	—	4	4	no	—
188	M₁₆	—	86	8	no	—
189	M₁₆	—	172	4	no	—
190	C₂×C₈	—	2	4	no	—
191	C₂×C₈	—	4	2	no	—
192	C₂×C₈	—	4	2	no	—
193	C₂×C₈	—	86	4	no	—
194	C₂×C₈	—	172	2	no	—
195	C₂×C₈	—	172	2	no	—
196	C₄⋊C₄	—	172	8	no	—
197	C₄⋊C₄	—	172	8	no	—
198	C₄⋊C₄	—	172	4	no	—
199	C₄⋊C₄	—	172	2	no	—
200	C₄⋊C₄	—	172	2	no	—
201	C₄²	—	2	2	no	—
202	C₄²	—	2	2	no	—
203	C₄²	—	2	4	no	—
204	C₂³	—	2	2	no	—
205	C₂³	—	2	2	no	—
206	C₂³	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
207	C₂³	#13063281	1	2	yes	—
208	C₂³	C₄×D₁₇₂	1	2	yes	—
209	C₂³	#13063290	1	2	yes	—
210	C₂×C₄	—	2	16	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	344	8	no	—
216	C₂×C₄	—	172	4	no	—
217	C₂×C₄	—	172	8	no	—
218	C₂×C₄	#13063241	1	4	yes	—
219	C₂×C₄	#13063243	1	4	yes	—
220	C₂×C₄	#13063299	1	2	yes	—
221	C₂×C₄	#13063305	1	4	yes	—
222	C₂×C₄	#13063305	1	2	yes	—
223	C₂×C₄	#13063306	1	2	yes	—
224	C₂×C₄	#13063309	1	4	yes	—
225	C₂×C₄	#13063309	1	2	yes	—
226	C₂×C₄	#13063312	1	1	yes	characteristic
227	C₂×C₄	#13063313	1	1	yes	characteristic
228	C₂×C₄	#13063315	1	4	yes	—
229	C₂×C₄	#13063318	1	4	yes	—
230	C₂×C₄	#13063322	1	1	yes	characteristic
231	C₂×C₄	#13063323	1	1	yes	characteristic
232	C₈	—	4	4	no	—
233	C₈	—	172	4	no	—
234	C₂²	—	2	4	no	—
235	C₂²	—	2	2	no	—
236	C₂²	—	2	2	no	—
237	C₂²	—	2	4	no	—
238	C₂²	#14582760	1	1	yes	characteristic
239	C₂²	#14582784	1	4	yes	—
240	C₂²	#14582801	1	1	yes	characteristic
241	C₂²	#14582804	1	1	yes	characteristic
242	C₂²	C₂×#13063241	1	2	yes	—
243	C₂²	C₂×#13063243	1	2	yes	—
244	C₄	—	2	4	no	—
245	C₄	—	2	4	no	—
246	C₄	—	344	4	no	—
247	C₄	#14582824	1	2	yes	—
248	C₄	#14582829	1	2	yes	—
249	C₄	#14582836	1	2	yes	—
250	C₄	#14582839	1	2	yes	—
251	C₂	—	2	4	no	—
252	C₂	#19132862	1	1	yes	characteristic
253	C₂	#19134673	1	2	yes	—
254	C₂	#19135435	1	1	yes	characteristic
255	C₂	#19135439	1	1	yes	characteristic
256	C₂	#19135583	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×(C₂×D₄₃)⋊C₄
commutator, prime radical	C₂⁴×C₄₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₄₃
Fitting	C₄₃×#1728	C₂
socle	C₂³×C₄₃	C₂×C₂²⋊C₄
cosocle	C₂³×C₄×C₄₃	C₂³
hypercenter	#1728	D₄₃
hypocenter	C₄₃	#11354
LCS	G ▷ C₂⁴×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #1728

Complements

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

N	H ≅ G/N	type
145. C₄₃	#11354	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14582769	2
C₂×#14582667	2
#19134289	2
C₄₃×#1728	2
#11354	43