Subgroups — #25252087

Order 11008.

490 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	#19145107	C₂	1	2	yes	maximal
2	#19136876	C₂	1	2	yes	maximal, complemented
3	#19136873	C₂	1	2	yes	maximal, complemented
4	#19136872	C₂	1	2	yes	maximal, complemented
5	#19136869	C₂	1	2	yes	maximal, complemented
6	#19136379	C₂	1	1	yes	maximal, characteristic
7	#19136377	C₂	1	2	yes	maximal
8	C₂×C₁₆×Dic₄₃	C₂	1	1	yes	maximal, characteristic
9	C₄₃×#2158	C₂	1	1	yes	maximal, characteristic
10	#14583452	—	2	2	no	—
11	C₂²×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
12	#14583303	C₂²	1	4	yes	—
13	#14583300	C₂²	1	4	yes	—
14	#14583299	—	2	2	no	—
15	#14583296	—	2	2	no	—
16	#14582937	C₂²	1	2	yes	—
17	#14582935	—	2	2	no	—
18	C₂×C₈×Dic₄₃	C₂²	1	1	yes	characteristic
19	#14582727	C₂²	1	2	yes	—
20	#14582726	C₂²	1	4	yes	—
21	#14582725	C₂²	1	2	yes	—
22	#14582724	C₂²	1	4	yes	—
23	C₁₆×Dic₄₃	C₂²	1	1	yes	characteristic
24	C₁₆×Dic₄₃	C₂²	1	1	yes	characteristic
25	C₁₆×Dic₄₃	C₂²	1	1	yes	characteristic
26	C₁₆×Dic₄₃	C₂²	1	1	yes	characteristic
27	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
28	C₄₃×#427	C₂²	1	2	yes	—
29	C₄₃×#367	C₂²	1	2	yes	—
30	C₄₃×#366	C₂²	1	2	yes	—
31	C₂²×C₄₃⋊C₈	D₄	1	1	yes	characteristic
32	#13063324	—	2	4	no	—
33	#13063316	—	2	4	no	—
34	#13063302	—	4	4	no	—
35	C₄×Dic₈₆	—	4	4	no	—
36	#13063284	—	2	2	no	—
37	C₂×C₄×Dic₄₃	D₄	1	1	yes	characteristic
38	C₂×C₄₃⋊C₁₆	Q₈	1	4	yes	complemented
39	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
40	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
41	#13063259	C₂³	1	4	yes	—
42	#13063258	—	2	4	no	—
43	#13063257	—	2	2	no	—
44	#13063256	—	2	4	no	—
45	C₈×Dic₄₃	D₄	1	1	yes	characteristic
46	C₈×Dic₄₃	D₄	1	1	yes	characteristic
47	C₈×Dic₄₃	C₂³	1	1	yes	characteristic
48	C₈×Dic₄₃	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
50	#117×C₄₃	—	2	2	no	—
51	C₂×C₁₆×C₄₃	Q₈	1	4	yes	complemented
52	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
53	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
54	#107×C₄₃	C₂³	1	4	yes	—
55	#106×C₄₃	—	2	2	no	—
56	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×Dic₄₃	D₈	1	2	yes	—
58	C₂×Dic₈₆	—	4	4	no	—
59	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₄₃	C₂×Q₈	1	2	yes	—
61	C₂×C₈×C₄₃	C₂×Q₈	1	2	yes	—
62	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
63	C₄⋊C₄×C₄₃	—	2	4	no	—
64	C₂²⋊C₄×C₄₃	—	4	2	no	—
65	C₄²×C₄₃	—	4	2	no	—
66	C₂²⋊Dic₄₃	—	4	2	no	—
67	C₄⋊Dic₄₃	—	2	4	no	—
68	C₄⋊Dic₄₃	—	4	4	no	—
69	Dic₄₃⋊C₄	—	4	4	no	—
70	Dic₄₃⋊C₄	—	4	4	no	—
71	C₄×Dic₄₃	—	4	2	no	—
72	C₄×Dic₄₃	D₈	1	2	yes	—
73	C₄×Dic₄₃	C₂×D₄	1	1	yes	characteristic
74	C₄×Dic₄₃	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄₃⋊C₈	Q₁₆	1	4	yes	—
76	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
77	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
78	C₁₆×C₄₃	C₂×Q₈	1	2	yes	—
79	C₁₆×C₄₃	C₂×Q₈	1	2	yes	—
80	C₄₃⋊C₁₆	C₂×Q₈	1	2	yes	—
81	C₄₃⋊C₁₆	C₂×Q₈	1	2	yes	—
82	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
83	C₂×C₄×C₄₃	—	4	4	no	—
84	C₂×C₄×C₄₃	C₄⋊Q₈	1	2	yes	—
85	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
86	C₂×C₄×C₄₃	C₂×Q₁₆	1	2	yes	—
87	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
88	C₂×Dic₄₃	—	2	4	no	—
89	C₂×Dic₄₃	—	4	4	no	—
90	C₂×Dic₄₃	C₂×D₈	1	2	yes	—
91	C₂×Dic₄₃	C₂×D₈	1	2	yes	—
92	Dic₈₆	—	8	8	no	—
93	C₈×C₄₃	C₄⋊Q₈	1	2	yes	—
94	C₈×C₄₃	C₂²×Q₈	1	1	yes	characteristic
95	C₈×C₄₃	C₂²×Q₈	1	1	yes	characteristic
96	C₄₃⋊C₈	C₂×Q₁₆	1	2	yes	—
97	C₄₃⋊C₈	C₂×Q₁₆	1	2	yes	—
98	#33726	—	43	1	no	maximal, Sylow
99	C₂²×C₄₃	—	2	2	no	—
100	C₂²×C₄₃	#500	1	2	yes	—
101	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
102	C₄×C₄₃	—	8	4	no	—
103	C₄×C₄₃	#500	1	2	yes	—
104	C₄×C₄₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
105	C₄×C₄₃	C₂²×Q₁₆	1	1	yes	characteristic
106	Dic₄₃	—	2	2	no	—
107	Dic₄₃	—	8	4	no	—
108	Dic₄₃	#508	1	4	yes	—
109	#3160	—	43	2	no	—
110	#2254	—	43	2	no	—
111	#2253	—	43	2	no	—
112	#2253	—	43	2	no	—
113	#2252	—	43	2	no	—
114	#2158	—	43	1	no	—
115	#2158	—	43	2	no	—
116	#2158	D₄₃	1	1	yes	characteristic
117	C₂×C₄×C₁₆	—	43	1	no	—
118	C₂×C₄₃	—	2	2	no	—
119	C₂×C₄₃	C₂×#500	1	1	yes	characteristic
120	C₂×C₄₃	C₂×#508	1	2	yes	—
121	#533	—	86	2	no	—
122	C₂²×C₁₆	—	43	1	no	—
123	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
124	#500	—	43	4	no	—
125	#499	—	43	4	no	—
126	#499	—	86	2	no	—
127	#498	—	86	2	no	—
128	#427	—	43	2	no	—
129	#427	—	86	2	no	—
130	#427	C₂×D₄₃	1	2	yes	—
131	C₂×C₄×C₈	—	43	1	no	—
132	#367	—	43	2	no	—
133	#367	—	43	4	no	—
134	#367	C₂×D₄₃	1	2	yes	—
135	#366	—	43	2	no	—
136	#366	—	43	4	no	—
137	#366	C₂×D₄₃	1	2	yes	—
138	C₄×C₁₆	—	43	1	no	—
139	C₄×C₁₆	—	43	1	no	—
140	C₄×C₁₆	—	43	1	no	—
141	C₄×C₁₆	—	43	1	no	—
142	C₄₃	#33726	1	1	yes	characteristic, complemented, Sylow
143	C₂²×C₈	—	43	1	no	—
144	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
145	C₄⋊Q₈	—	86	4	no	—
146	#125	—	86	4	no	—
147	C₂²⋊Q₈	—	172	4	no	—
148	C₄×Q₈	—	172	4	no	—
149	#117	—	2	2	no	—
150	#117	—	86	2	no	—
151	C₂×C₄²	—	43	1	no	—
152	C₂×C₁₆	—	43	4	no	—
153	C₂×C₁₆	—	43	1	no	—
154	C₂×C₁₆	—	43	1	no	—
155	C₂×C₁₆	Dic₈₆	1	4	yes	complemented
156	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
157	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
158	#107	—	43	4	no	—
159	#107	—	86	4	no	—
160	#107	C₂²×D₄₃	1	4	yes	—
161	#106	—	2	2	no	—
162	#106	—	86	4	no	—
163	#106	—	86	2	no	—
164	C₄×C₈	—	43	1	no	—
165	C₄×C₈	—	43	1	no	—
166	C₄×C₈	—	43	1	no	—
167	C₄×C₈	—	43	1	no	—
168	C₂×Q₈	—	172	4	no	—
169	C₂²×C₄	—	43	2	no	—
170	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
171	C₂×C₈	—	43	4	no	—
172	C₂×C₈	—	43	1	no	—
173	C₂×C₈	—	43	1	no	—
174	C₂×C₈	C₂×Dic₈₆	1	2	yes	—
175	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
176	C₂×C₈	Q₈×D₄₃	1	2	yes	—
177	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
178	C₄⋊C₄	—	2	4	no	—
179	C₄⋊C₄	—	86	4	no	—
180	C₄⋊C₄	—	172	4	no	—
181	C₄⋊C₄	—	172	4	no	—
182	C₄⋊C₄	—	172	4	no	—
183	C₂²⋊C₄	—	4	2	no	—
184	C₂²⋊C₄	—	172	2	no	—
185	C₄²	—	4	2	no	—
186	C₄²	—	43	2	no	—
187	C₄²	—	43	1	no	—
188	C₄²	—	43	1	no	—
189	C₄²	—	172	2	no	—
190	C₁₆	—	43	2	no	—
191	C₁₆	—	43	2	no	—
192	C₁₆	C₂×Dic₈₆	1	2	yes	—
193	C₁₆	C₂×Dic₈₆	1	2	yes	—
194	C₂³	D₈×D₄₃	1	1	yes	characteristic
195	Q₈	—	344	8	no	—
196	C₂×C₄	—	4	4	no	—
197	C₂×C₄	—	43	2	no	—
198	C₂×C₄	—	43	2	no	—
199	C₂×C₄	—	86	4	no	—
200	C₂×C₄	—	172	4	no	—
201	C₂×C₄	#13063324	1	2	yes	—
202	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
203	C₂×C₄	Q₁₆×D₄₃	1	2	yes	—
204	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
205	C₈	—	43	2	no	—
206	C₈	—	43	2	no	—
207	C₈	#13063324	1	2	yes	—
208	C₈	C₂×Q₈×D₄₃	1	1	yes	characteristic
209	C₈	C₂²×Dic₈₆	1	1	yes	characteristic
210	C₂²	—	2	2	no	—
211	C₂²	#14583303	1	2	yes	—
212	C₂²	C₂×D₈×D₄₃	1	1	yes	characteristic
213	C₄	—	8	4	no	—
214	C₄	—	43	4	no	—
215	C₄	—	86	2	no	—
216	C₄	—	344	4	no	—
217	C₄	#14583303	1	2	yes	—
218	C₄	C₂×#13063324	1	1	yes	characteristic
219	C₄	C₂×Q₁₆×D₄₃	1	1	yes	characteristic
220	C₂	—	2	2	no	—
221	C₂	C₂×#14583303	1	1	yes	characteristic
222	C₂	#508×D₄₃	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₈×D₄₃
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₄₃×#2158	C₂
socle	C₂²×C₄₃	C₂²×D₈
prime radical	C₂³×C₄₃	C₂×D₈
hypercenter	#2158	D₄₃
hypocenter	C₄₃	#33726
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
2. #19136876	C₂	semidirect
3. #19136873	C₂	semidirect
4. #19136872	C₂	semidirect
5. #19136869	C₂	semidirect
38. C₂×C₄₃⋊C₁₆	Q₈	semidirect
51. C₂×C₁₆×C₄₃	Q₈	semidirect
142. C₄₃	#33726	semidirect
155. C₂×C₁₆	Dic₈₆	semidirect

Maximal subgroups

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

subgroup	index
#19145107	2
#19136876	2
#19136873	2
#19136872	2
#19136869	2
#19136379	2
#19136377	2
C₂×C₁₆×Dic₄₃	2
C₄₃×#2158	2
#33726	43