Subgroups — #25014277

Order 11008.

262 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19135344	C₂	1	1	yes	maximal, characteristic
2	#19134476	C₂	1	1	yes	maximal, characteristic
3	#19132690	C₂	1	1	yes	maximal, characteristic
4	#19132690	C₂	1	1	yes	maximal, characteristic
5	#19132689	C₂	1	1	yes	maximal, characteristic
6	#19132672	C₂	1	1	yes	maximal, characteristic
7	C₄₃×#2158	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
9	C₂×#13063261	C₂²	1	1	yes	characteristic
10	C₂×#13063261	C₂²	1	1	yes	characteristic
11	#14582926	—	2	1	no	—
12	#14582855	—	2	1	no	—
13	C₂×C₈×Dic₄₃	C₂²	1	1	yes	characteristic
14	#14582658	—	2	1	no	—
15	#14582658	—	2	1	no	—
16	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
17	C₄₃×#427	C₂²	1	1	yes	characteristic
18	C₄₃×#427	C₂²	1	1	yes	characteristic
19	C₄₃×#367	C₄	1	1	yes	characteristic, complemented
20	C₄₃×#367	C₄	1	1	yes	characteristic, complemented
21	C₄₃×#366	C₄	1	1	yes	characteristic, complemented
22	C₄₃×#366	C₄	1	1	yes	characteristic, complemented
23	C₂×#10621877	—	2	1	no	—
24	C₂×#10621877	—	2	1	no	—
25	C₂²×C₄₃⋊C₈	D₄	1	1	yes	characteristic
26	C₂×C₄×Dic₄₃	D₄	1	1	yes	characteristic
27	C₂×C₄₃⋊C₁₆	—	2	1	no	—
28	C₂×C₄₃⋊C₁₆	—	2	1	no	—
29	C₂×C₄₃⋊C₁₆	—	2	1	no	—
30	#13063261	—	2	1	no	—
31	#13063261	—	2	1	no	—
32	#13063261	—	2	1	no	—
33	#13063261	Q₈	1	2	yes	—
34	C₄⋊(C₄₃⋊C₈)	—	4	1	no	—
35	C₄⋊(C₄₃⋊C₈)	—	4	1	no	—
36	#13063236	—	4	1	no	—
37	C₄×C₄₃⋊C₈	—	4	1	no	—
38	C₈×Dic₄₃	—	2	1	no	—
39	C₈×Dic₄₃	—	2	1	no	—
40	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
41	#117×C₄₃	—	2	1	no	—
42	#117×C₄₃	—	2	1	no	—
43	C₂×C₁₆×C₄₃	—	2	1	no	—
44	C₂×C₁₆×C₄₃	—	2	1	no	—
45	C₂×C₁₆×C₄₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₁₆×C₄₃	C₂×C₄	1	1	yes	characteristic
47	#107×C₄₃	C₂×C₄	1	1	yes	characteristic
48	#107×C₄₃	C₂×C₄	1	1	yes	characteristic
49	#107×C₄₃	C₂×C₄	1	1	yes	characteristic
50	#107×C₄₃	C₂×C₄	1	1	yes	characteristic
51	#106×C₄₃	—	2	1	no	—
52	#106×C₄₃	—	2	1	no	—
53	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
54	C₂²×Dic₄₃	—	2	1	no	—
55	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₈×C₄₃	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
58	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
59	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
60	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
61	C₄⋊C₄×C₄₃	—	2	1	no	—
62	C₄⋊C₄×C₄₃	—	2	1	no	—
63	C₄⋊C₄×C₄₃	—	2	1	no	—
64	C₄⋊C₄×C₄₃	—	2	1	no	—
65	C₂²⋊C₄×C₄₃	—	4	1	no	—
66	C₂²⋊C₄×C₄₃	—	4	1	no	—
67	C₄²×C₄₃	—	4	1	no	—
68	C₄²×C₄₃	—	4	1	no	—
69	C₄×Dic₄₃	—	2	1	no	—
70	C₄×Dic₄₃	—	2	1	no	—
71	#10621877	—	2	2	no	—
72	#10621877	—	4	1	no	—
73	#10621877	—	4	1	no	—
74	#10621877	—	4	1	no	—
75	C₂×C₄₃⋊C₈	—	2	1	no	—
76	C₂×C₄₃⋊C₈	—	2	1	no	—
77	C₂×C₄₃⋊C₈	—	4	1	no	—
78	C₂×C₄₃⋊C₈	—	4	1	no	—
79	C₂×C₄₃⋊C₈	D₈	1	1	yes	characteristic
80	C₂×C₄₃⋊C₈	D₈	1	1	yes	characteristic
81	C₁₆×C₄₃	—	2	1	no	—
82	C₁₆×C₄₃	—	2	1	no	—
83	C₄₃⋊C₁₆	—	4	1	no	—
84	C₂³×C₄₃	#135	1	1	yes	characteristic
85	C₂×C₄×C₄₃	—	4	1	no	—
86	C₂×C₄×C₄₃	—	4	1	no	—
87	C₂×C₄×C₄₃	—	4	1	no	—
88	C₂×C₄×C₄₃	—	4	1	no	—
89	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
90	C₂×C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
91	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
92	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
93	C₂×C₄×C₄₃	#137	1	1	yes	characteristic
94	C₂×C₄×C₄₃	#137	1	1	yes	characteristic
95	C₂×Dic₄₃	—	2	2	no	—
96	C₂×Dic₄₃	—	4	1	no	—
97	C₂×Dic₄₃	—	4	1	no	—
98	C₈×C₄₃	#117	1	1	yes	characteristic
99	C₈×C₄₃	C₄×Q₈	1	1	yes	characteristic
100	C₈×C₄₃	#125	1	1	yes	characteristic
101	C₈×C₄₃	#126	1	1	yes	characteristic
102	C₄₃⋊C₈	—	2	1	no	—
103	C₄₃⋊C₈	—	2	1	no	—
104	C₄₃⋊C₈	—	4	2	no	—
105	C₄₃⋊C₈	—	8	1	no	—
106	#13113	—	43	1	no	maximal, Sylow
107	C₂²×C₄₃	—	2	1	no	—
108	C₂²×C₄₃	—	2	1	no	—
109	C₂²×C₄₃	C₄×D₈	1	1	yes	characteristic
110	C₂²×C₄₃	#479	1	1	yes	characteristic
111	C₂²×C₄₃	#485	1	1	yes	characteristic
112	C₄×C₄₃	—	8	1	no	—
113	C₄×C₄₃	—	8	1	no	—
114	C₄×C₄₃	—	8	1	no	—
115	C₄×C₄₃	—	8	1	no	—
116	C₄×C₄₃	#387	1	1	yes	characteristic
117	C₄×C₄₃	#441	1	1	yes	characteristic
118	C₄×C₄₃	#474	1	1	yes	characteristic
119	C₄×C₄₃	#480	1	1	yes	characteristic
120	Dic₄₃	—	4	2	no	—
121	#2158	D₄₃	1	1	yes	characteristic
122	#1945	—	43	1	no	—
123	#1772	—	43	1	no	—
124	#1382	—	43	1	no	—
125	#1382	—	43	1	no	—
126	#1381	—	43	1	no	—
127	#1374	—	43	1	no	—
128	C₂×C₄₃	—	2	1	no	—
129	C₂×C₄₃	—	2	1	no	—
130	C₂×C₄₃	#1918	1	1	yes	characteristic
131	C₂×C₄₃	#2178	1	1	yes	characteristic
132	C₂×C₄₃	#2179	1	1	yes	characteristic
133	C₂²×C₁₆	—	43	1	no	—
134	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
135	C₂×#108	—	43	1	no	—
136	C₂×#108	—	43	1	no	—
137	#427	C₂×D₄₃	1	1	yes	characteristic
138	#427	C₂×D₄₃	1	1	yes	characteristic
139	#424	—	86	1	no	—
140	#405	—	86	1	no	—
141	C₂×C₄×C₈	—	43	1	no	—
142	#367	Dic₄₃	1	1	yes	characteristic, complemented
143	#367	Dic₄₃	1	1	yes	characteristic, complemented
144	#366	Dic₄₃	1	1	yes	characteristic, complemented
145	#366	Dic₄₃	1	1	yes	characteristic, complemented
146	#339	—	86	1	no	—
147	#339	—	86	1	no	—
148	C₄₃	#13113	1	1	yes	characteristic, complemented, Sylow
149	C₂×M₁₆	—	86	1	no	—
150	C₂×M₁₆	—	86	1	no	—
151	C₂²×C₈	—	43	1	no	—
152	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
153	#117	—	2	1	no	—
154	#117	—	2	1	no	—
155	C₂×C₄²	—	43	1	no	—
156	C₂×C₁₆	—	2	1	no	—
157	C₂×C₁₆	—	2	1	no	—
158	C₂×C₁₆	—	86	1	no	—
159	C₂×C₁₆	—	86	1	no	—
160	C₂×C₁₆	—	86	1	no	—
161	C₂×C₁₆	C₂×Dic₄₃	1	1	yes	characteristic
162	C₂×C₁₆	C₂×Dic₄₃	1	1	yes	characteristic
163	#108	—	43	2	no	—
164	#108	—	86	1	no	—
165	#108	—	86	1	no	—
166	#108	—	86	1	no	—
167	#107	C₂×Dic₄₃	1	1	yes	characteristic
168	#107	C₂×Dic₄₃	1	1	yes	characteristic
169	#107	C₂×Dic₄₃	1	1	yes	characteristic
170	#107	C₂×Dic₄₃	1	1	yes	characteristic
171	#106	—	2	1	no	—
172	#106	—	2	1	no	—
173	C₄⋊C₈	—	172	1	no	—
174	C₄⋊C₈	—	172	1	no	—
175	#97	—	172	1	no	—
176	C₄×C₈	—	86	1	no	—
177	C₄×C₈	—	86	1	no	—
178	C₄×C₈	—	172	1	no	—
179	C₂²×C₄	—	86	1	no	—
180	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
181	M₁₆	—	86	2	no	—
182	M₁₆	—	172	1	no	—
183	M₁₆	—	172	1	no	—
184	M₁₆	—	172	1	no	—
185	C₂×C₈	—	43	1	no	—
186	C₂×C₈	—	43	1	no	—
187	C₂×C₈	—	86	1	no	—
188	C₂×C₈	—	86	1	no	—
189	C₂×C₈	—	172	1	no	—
190	C₂×C₈	—	172	1	no	—
191	C₂×C₈	#10621898	1	1	yes	characteristic
192	C₂×C₈	#10621898	1	1	yes	characteristic
193	C₂×C₈	#10621900	1	1	yes	characteristic
194	C₂×C₈	Q₈×D₄₃	1	1	yes	characteristic
195	C₂×C₈	#10621902	1	1	yes	characteristic
196	C₂×C₈	C₂²×Dic₄₃	1	1	yes	characteristic
197	C₄⋊C₄	—	2	1	no	—
198	C₄⋊C₄	—	2	1	no	—
199	C₄⋊C₄	—	2	1	no	—
200	C₄⋊C₄	—	2	1	no	—
201	C₂²⋊C₄	—	4	1	no	—
202	C₂²⋊C₄	—	4	1	no	—
203	C₄²	—	4	1	no	—
204	C₄²	—	4	1	no	—
205	C₄²	—	86	1	no	—
206	C₄²	—	86	1	no	—
207	C₁₆	—	2	1	no	—
208	C₁₆	—	2	1	no	—
209	C₁₆	—	172	1	no	—
210	C₂³	#13063344	1	1	yes	characteristic
211	C₂×C₄	—	4	1	no	—
212	C₂×C₄	—	4	1	no	—
213	C₂×C₄	—	4	1	no	—
214	C₂×C₄	—	4	1	no	—
215	C₂×C₄	—	86	2	no	—
216	C₂×C₄	—	172	1	no	—
217	C₂×C₄	—	172	1	no	—
218	C₂×C₄	D₄×Dic₄₃	1	1	yes	characteristic
219	C₂×C₄	#13063303	1	1	yes	characteristic
220	C₂×C₄	#13063310	1	1	yes	characteristic
221	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
222	C₂×C₄	#13063357	1	1	yes	characteristic
223	C₂×C₄	#13063357	1	1	yes	characteristic
224	C₈	—	86	1	no	—
225	C₈	—	86	1	no	—
226	C₈	—	344	1	no	—
227	C₈	—	172	2	no	—
228	C₈	#13063284	1	1	yes	characteristic
229	C₈	Q₈×Dic₄₃	1	1	yes	characteristic
230	C₈	#13063315	1	1	yes	characteristic
231	C₈	#13063320	1	1	yes	characteristic
232	C₂²	—	2	1	no	—
233	C₂²	—	2	1	no	—
234	C₂²	D₈×Dic₄₃	1	1	yes	characteristic
235	C₂²	#14583205	1	1	yes	characteristic
236	C₂²	#14583242	1	1	yes	characteristic
237	C₄	—	8	1	no	—
238	C₄	—	8	1	no	—
239	C₄	—	8	1	no	—
240	C₄	—	8	1	no	—
241	C₄	—	172	2	no	—
242	C₄	#14582785	1	1	yes	characteristic
243	C₄	#14583006	1	1	yes	characteristic
244	C₄	#14583178	1	1	yes	characteristic
245	C₄	#14583214	1	1	yes	characteristic
246	C₂	—	2	1	no	—
247	C₂	—	2	1	no	—
248	C₂	#19135168	1	1	yes	characteristic
249	C₂	#19136476	1	1	yes	characteristic
250	C₂	#19136481	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₈×D₄₃
commutator	C₂×C₈×C₄₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₄₃×#2158	C₂
socle	C₂²×C₄₃	C₄×D₈
prime radical	C₂³×C₄₃	#135
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	#2158	D₄₃
hypocenter	C₄₃	#13113
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

Complements

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

N	H ≅ G/N	type
19. C₄₃×#367	C₄	semidirect
20. C₄₃×#367	C₄	semidirect
21. C₄₃×#366	C₄	semidirect
22. C₄₃×#366	C₄	semidirect
142. #367	Dic₄₃	semidirect
143. #367	Dic₄₃	semidirect
144. #366	Dic₄₃	semidirect
145. #366	Dic₄₃	semidirect
148. C₄₃	#13113	semidirect

Maximal subgroups

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

subgroup	index
#19135344	2
#19134476	2
#19132690	2
#19132689	2
#19132672	2
C₄₃×#2158	2
#13113	43