Subgroups — #24983130

Order 11008.

294 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19134712	C₂	1	1	yes	maximal, characteristic
2	#19134341	C₂	1	1	yes	maximal, characteristic
3	C₄₃×#1831	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
5	C₂×C₄×C₄₃⋊C₈	C₄	1	2	yes	—
6	#14582654	—	2	1	no	—
7	#14582654	—	2	1	no	—
8	#14582654	—	2	1	no	—
9	#14582654	C₄	1	2	yes	—
10	C₂×C₄⋊C₈×C₄₃	—	2	1	no	—
11	C₄³×C₄₃	C₂²	1	1	yes	characteristic
12	C₄₃×#336	—	2	1	no	—
13	C₄₃×#336	C₄	1	2	yes	—
14	C₂²×C₄₃⋊C₈	—	2	2	no	—
15	C₂²×C₄₃⋊C₈	—	2	2	no	—
16	C₄⋊(C₄₃⋊C₈)	—	4	1	no	—
17	C₄⋊(C₄₃⋊C₈)	—	4	1	no	—
18	C₄×C₄₃⋊C₈	—	2	2	no	—
19	C₄×C₄₃⋊C₈	C₈	1	4	yes	complemented
20	C₂²×C₈×C₄₃	—	2	2	no	—
21	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
26	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₄₃	Q₈	1	1	yes	characteristic
28	C₄⋊C₈×C₄₃	—	4	1	no	—
29	C₄⋊C₈×C₄₃	—	4	1	no	—
30	C₂²×C₄×C₄₃	C₄²	1	1	yes	characteristic
31	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
32	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
33	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₄₃	C₄⋊C₄	1	1	yes	characteristic
35	C₂²×C₄×C₄₃	C₄⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₄₃	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₄₃	—	4	2	no	—
38	C₂×C₈×C₄₃	—	4	2	no	—
39	C₂×C₈×C₄₃	—	4	2	no	—
40	C₄²×C₄₃	—	2	2	no	—
41	C₄²×C₄₃	—	4	1	no	—
42	C₄²×C₄₃	—	4	1	no	—
43	C₄²×C₄₃	—	4	1	no	—
44	C₄²×C₄₃	—	4	1	no	—
45	C₄²×C₄₃	C₂×C₈	1	2	yes	—
46	C₄²×C₄₃	M₁₆	1	2	yes	—
47	C₄²×C₄₃	D₈	1	1	yes	characteristic
48	C₄²×C₄₃	SD₁₆	1	1	yes	characteristic
49	C₄²×C₄₃	SD₁₆	1	1	yes	characteristic
50	C₄²×C₄₃	Q₁₆	1	1	yes	characteristic
51	C₂×C₄₃⋊C₈	—	2	4	no	—
52	C₂×C₄₃⋊C₈	—	4	2	no	—
53	C₂×C₄₃⋊C₈	—	4	2	no	—
54	C₂×C₄₃⋊C₈	—	4	2	no	—
55	C₂×C₄₃⋊C₈	—	4	2	no	—
56	C₂×C₄₃⋊C₈	—	4	2	no	—
57	C₂³×C₄₃	#95	1	1	yes	characteristic
58	C₂×C₄×C₄₃	—	2	1	no	—
59	C₂×C₄×C₄₃	—	2	1	no	—
60	C₂×C₄×C₄₃	—	2	1	no	—
61	C₂×C₄×C₄₃	—	2	1	no	—
62	C₂×C₄×C₄₃	—	2	1	no	—
63	C₂×C₄×C₄₃	—	2	1	no	—
64	C₂×C₄×C₄₃	—	2	1	no	—
65	C₂×C₄×C₄₃	—	2	1	no	—
66	C₂×C₄×C₄₃	—	2	1	no	—
67	C₂×C₄×C₄₃	—	2	1	no	—
68	C₂×C₄×C₄₃	—	2	1	no	—
69	C₂×C₄×C₄₃	—	2	1	no	—
70	C₂×C₄×C₄₃	—	2	1	no	—
71	C₂×C₄×C₄₃	—	2	1	no	—
72	C₂×C₄×C₄₃	C₄×C₈	1	1	yes	characteristic
73	C₂×C₄×C₄₃	#97	1	1	yes	characteristic
74	C₂×C₄×C₄₃	C₂²⋊C₈	1	2	yes	—
75	C₂×C₄×C₄₃	#102	1	1	yes	characteristic
76	C₂×C₄×C₄₃	#102	1	1	yes	characteristic
77	C₂×C₄×C₄₃	#103	1	1	yes	characteristic
78	C₂×C₄×C₄₃	#103	1	1	yes	characteristic
79	C₂×C₄×C₄₃	C₄⋊C₈	1	2	yes	—
80	C₂×C₄×C₄₃	#106	1	1	yes	characteristic
81	C₂×C₄×C₄₃	#107	1	1	yes	characteristic
82	C₂×C₄×C₄₃	#108	1	2	yes	—
83	C₈×C₄₃	—	8	2	no	—
84	C₄₃⋊C₈	—	4	4	no	—
85	C₄₃⋊C₈	—	8	2	no	—
86	#7018	—	43	1	no	maximal, Sylow
87	C₂²×C₄₃	#336	1	1	yes	characteristic
88	C₂²×C₄₃	H(3,ℤ/4)	1	1	yes	characteristic
89	C₂²×C₄₃	H(3,ℤ/4)	1	1	yes	characteristic
90	C₂²×C₄₃	#340	1	1	yes	characteristic
91	C₂²×C₄₃	#341	1	1	yes	characteristic
92	C₂²×C₄₃	#344	1	1	yes	characteristic
93	C₂²×C₄₃	#344	1	1	yes	characteristic
94	C₄×C₄₃	—	2	1	no	—
95	C₄×C₄₃	—	2	1	no	—
96	C₄×C₄₃	—	2	1	no	—
97	C₄×C₄₃	—	2	1	no	—
98	C₄×C₄₃	—	2	1	no	—
99	C₄×C₄₃	—	2	1	no	—
100	C₄×C₄₃	—	2	2	no	—
101	C₄×C₄₃	—	4	1	no	—
102	C₄×C₄₃	—	4	1	no	—
103	C₄×C₄₃	#334	1	2	yes	—
104	C₄×C₄₃	#335	1	2	yes	—
105	#1831	—	43	1	no	—
106	#1831	D₄₃	1	1	yes	characteristic
107	#1744	—	43	1	no	—
108	C₂×C₄₃	#1274	1	1	yes	characteristic
109	C₂×C₄₃	#1277	1	1	yes	characteristic
110	C₂×C₄₃	#1281	1	1	yes	characteristic
111	C₂×C₄₃	#1286	1	1	yes	characteristic
112	C₂×C₄₃	#1286	1	1	yes	characteristic
113	C₂×C₄₃	#1287	1	1	yes	characteristic
114	C₂×C₄₃	#1287	1	1	yes	characteristic
115	C₂×C₄⋊C₈	—	2	1	no	—
116	C₂×C₄⋊C₈	—	86	1	no	—
117	C₂×C₄×C₈	—	43	2	no	—
118	C₄³	C₂×D₄₃	1	1	yes	characteristic
119	#336	—	2	1	no	—
120	#336	—	43	2	no	—
121	#336	—	86	1	no	—
122	#336	—	86	1	no	—
123	#336	—	86	1	no	—
124	#336	Dic₄₃	1	2	yes	—
125	C₄₃	#7018	1	1	yes	characteristic, complemented, Sylow
126	C₂²×C₈	—	2	2	no	—
127	C₂²×C₈	—	86	2	no	—
128	C₂²×C₈	—	86	2	no	—
129	C₂×C₄²	Dic₈₆	1	1	yes	characteristic
130	C₂×C₄²	C₄×D₄₃	1	1	yes	characteristic
131	C₂×C₄²	C₄×D₄₃	1	1	yes	characteristic
132	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
133	C₂×C₄²	C₂×Dic₄₃	1	1	yes	characteristic
134	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
135	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
136	C₄⋊C₈	—	4	1	no	—
137	C₄⋊C₈	—	4	1	no	—
138	C₄⋊C₈	—	172	1	no	—
139	C₄⋊C₈	—	172	1	no	—
140	C₄×C₈	—	43	4	no	—
141	C₄×C₈	—	86	2	no	—
142	C₂²×C₄	C₄×Dic₄₃	1	1	yes	characteristic
143	C₂²×C₄	Dic₄₃⋊C₄	1	1	yes	characteristic
144	C₂²×C₄	Dic₄₃⋊C₄	1	1	yes	characteristic
145	C₂²×C₄	C₄⋊Dic₄₃	1	1	yes	characteristic
146	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
147	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
148	C₂²×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
149	C₂×C₈	—	4	2	no	—
150	C₂×C₈	—	4	2	no	—
151	C₂×C₈	—	4	2	no	—
152	C₂×C₈	—	86	4	no	—
153	C₂×C₈	—	172	2	no	—
154	C₂×C₈	—	172	2	no	—
155	C₂×C₈	—	172	2	no	—
156	C₂×C₈	—	172	2	no	—
157	C₂×C₈	—	172	2	no	—
158	C₄²	—	2	2	no	—
159	C₄²	—	4	1	no	—
160	C₄²	—	4	1	no	—
161	C₄²	—	4	1	no	—
162	C₄²	—	4	1	no	—
163	C₄²	C₈×D₄₃	1	2	yes	—
164	C₄²	#10621872	1	2	yes	—
165	C₄²	#10621882	1	1	yes	characteristic
166	C₄²	#10621883	1	1	yes	characteristic
167	C₄²	#10621884	1	1	yes	characteristic
168	C₄²	#10621885	1	1	yes	characteristic
169	C₂³	C₄₃⋊#95	1	1	yes	characteristic
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	2	1	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	C₈×Dic₄₃	1	1	yes	characteristic
185	C₂×C₄	#13063235	1	1	yes	characteristic
186	C₂×C₄	(C₂×D₄₃)⋊C₈	1	2	yes	—
187	C₂×C₄	#13063245	1	1	yes	characteristic
188	C₂×C₄	#13063246	1	1	yes	characteristic
189	C₂×C₄	#13063248	1	1	yes	characteristic
190	C₂×C₄	#13063249	1	1	yes	characteristic
191	C₂×C₄	Dic₄₃⋊C₈	1	2	yes	—
192	C₂×C₄	#13063256	1	1	yes	characteristic
193	C₂×C₄	#13063258	1	1	yes	characteristic
194	C₂×C₄	#13063260	1	2	yes	—
195	C₈	—	8	2	no	—
196	C₈	—	344	2	no	—
197	C₈	—	172	4	no	—
198	C₂²	#14582653	1	1	yes	characteristic
199	C₂²	#14582655	1	1	yes	characteristic
200	C₂²	#14582656	1	1	yes	characteristic
201	C₂²	#14582661	1	1	yes	characteristic
202	C₂²	#14582663	1	1	yes	characteristic
203	C₂²	#14582669	1	1	yes	characteristic
204	C₂²	#14582671	1	1	yes	characteristic
205	C₄	—	2	2	no	—
206	C₄	—	2	1	no	—
207	C₄	—	2	1	no	—
208	C₄	—	2	1	no	—
209	C₄	—	2	1	no	—
210	C₄	—	2	1	no	—
211	C₄	—	2	1	no	—
212	C₄	—	4	1	no	—
213	C₄	—	4	1	no	—
214	C₄	#14582649	1	2	yes	—
215	C₄	#14582651	1	2	yes	—
216	C₂	#19132423	1	1	yes	characteristic
217	C₂	#19132429	1	1	yes	characteristic
218	C₂	#19132439	1	1	yes	characteristic
219	C₂	#19132453	1	1	yes	characteristic
220	C₂	#19132455	1	1	yes	characteristic
221	C₂	#19132456	1	1	yes	characteristic
222	C₂	#19132458	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₄₃⋊#95
commutator	C₂×C₄×C₄₃	C₄×C₈
Frattini	C₄³	C₂×D₄₃
Fitting	C₄₃×#1831	C₂
socle, prime radical	C₂³×C₄₃	#95
cosocle	C₄³×C₄₃	C₂²
hypercenter	#1831	D₄₃
hypocenter	C₄₃	#7018
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1831

Complements

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

N	H ≅ G/N	type
19. C₄×C₄₃⋊C₈	C₈	semidirect
125. C₄₃	#7018	semidirect

Maximal subgroups

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

subgroup	index
#19134712	2
#19134341	2
C₄₃×#1831	2
#7018	43