Subgroups — #25014628

Order 11008.

326 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14582725	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆×Dic₄₃	C₂	1	1	yes	maximal, characteristic
3	#19135329	C₂	1	2	yes	maximal
4	#19132686	C₂	1	2	yes	maximal
5	C₂×C₄₃×#366	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
7	C₂×#13063259	C₂²	1	2	yes	—
8	C₂×C₈×Dic₄₃	C₂²	1	1	yes	characteristic
9	#14582798	—	2	2	no	—
10	#14582725	—	2	2	no	—
11	C₁₆×Dic₄₃	—	2	2	no	—
12	#14582661	—	2	2	no	—
13	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
14	C₂×#107×C₄₃	C₂²	1	2	yes	—
15	C₄₃×#366	C₄	1	4	yes	complemented
16	C₂²×C₄₃⋊C₈	D₄	1	1	yes	characteristic
17	C₂×C₄⋊Dic₄₃	—	2	2	no	—
18	C₂×C₄⋊Dic₄₃	—	4	2	no	—
19	C₂×C₄×Dic₄₃	D₄	1	1	yes	characteristic
20	C₂×C₄₃⋊C₁₆	—	2	1	no	—
21	C₂×C₄₃⋊C₁₆	—	2	1	no	—
22	C₂×C₄₃⋊C₁₆	Q₈	1	2	yes	—
23	#13063259	—	2	2	no	—
24	#13063259	—	2	2	no	—
25	C₈×Dic₄₃	—	2	1	no	—
26	C₈×Dic₄₃	—	2	1	no	—
27	C₄₃⋊#95	—	4	2	no	—
28	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
30	C₂×C₁₆×C₄₃	C₂×C₄	1	2	yes	—
31	C₂×C₁₆×C₄₃	D₄	1	2	yes	—
32	C₂×C₁₆×C₄₃	Q₈	1	2	yes	—
33	#107×C₄₃	—	2	2	no	—
34	#107×C₄₃	C₂×C₄	1	4	yes	—
35	C₂²×C₄×C₄₃	—	4	2	no	—
36	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
37	C₂²×Dic₄₃	—	4	2	no	—
38	C₂²×Dic₄₃	D₈	1	1	yes	characteristic
39	C₂²×Dic₄₃	D₈	1	1	yes	characteristic
40	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₈×C₄₃	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₈×C₄₃	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
45	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
46	C₄⋊C₄×C₄₃	—	2	4	no	—
47	C₄⋊C₄×C₄₃	—	4	2	no	—
48	C₄⋊Dic₄₃	—	4	2	no	—
49	C₄⋊Dic₄₃	—	4	2	no	—
50	C₄⋊Dic₄₃	—	8	4	no	—
51	C₄×Dic₄₃	—	2	1	no	—
52	C₄×Dic₄₃	—	2	1	no	—
53	C₂×C₄₃⋊C₈	—	2	1	no	—
54	C₂×C₄₃⋊C₈	—	2	1	no	—
55	C₂×C₄₃⋊C₈	Q₁₆	1	2	yes	—
56	C₁₆×C₄₃	C₄⋊C₄	1	4	yes	complemented
57	C₄₃⋊C₁₆	—	2	2	no	—
58	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
59	C₂×C₄×C₄₃	—	4	4	no	—
60	C₂×C₄×C₄₃	—	8	2	no	—
61	C₂×C₄×C₄₃	—	8	2	no	—
62	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
63	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
64	C₂×C₄×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₄×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
66	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
67	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
68	C₂×Dic₄₃	—	2	1	no	—
69	C₂×Dic₄₃	—	2	1	no	—
70	C₂×Dic₄₃	—	2	1	no	—
71	C₂×Dic₄₃	—	2	1	no	—
72	C₂×Dic₄₃	—	8	2	no	—
73	C₂×Dic₄₃	—	8	2	no	—
74	C₂×Dic₄₃	—	8	2	no	—
75	C₂×Dic₄₃	D₁₆	1	2	yes	—
76	C₂×Dic₄₃	Q₃₂	1	2	yes	—
77	C₈×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₈×C₄₃	C₄×Q₈	1	1	yes	characteristic
79	C₈×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
80	C₈×C₄₃	#125	1	1	yes	characteristic
81	C₄₃⋊C₈	—	2	2	no	—
82	#13165	—	43	1	no	maximal, Sylow
83	C₂²×C₄₃	C₄×D₈	1	1	yes	characteristic
84	C₂²×C₄₃	#466	1	1	yes	characteristic
85	C₂²×C₄₃	#500	1	1	yes	characteristic
86	C₂²×C₄₃	C₂×D₁₆	1	1	yes	characteristic
87	C₂²×C₄₃	C₂×Q₃₂	1	1	yes	characteristic
88	C₂²×C₄₃	#508	1	1	yes	characteristic
89	C₂²×C₄₃	#508	1	1	yes	characteristic
90	C₄×C₄₃	—	8	4	no	—
91	C₄×C₄₃	#389	1	1	yes	characteristic
92	C₄×C₄₃	C₄×Q₁₆	1	1	yes	characteristic
93	C₄×C₄₃	#467	1	1	yes	characteristic
94	C₄×C₄₃	#499	1	1	yes	characteristic
95	Dic₄₃	—	2	2	no	—
96	Dic₄₃	—	2	2	no	—
97	Dic₄₃	—	16	2	no	—
98	C₂×#366	—	43	1	no	—
99	C₂×#366	D₄₃	1	1	yes	characteristic
100	C₂×C₄×C₁₆	—	43	1	no	—
101	#1942	—	43	2	no	—
102	#1380	—	43	2	no	—
103	C₂×C₄₃	#1942	1	1	yes	characteristic
104	C₂×C₄₃	C₄×D₁₆	1	1	yes	characteristic
105	C₂×C₄₃	C₄×Q₃₂	1	1	yes	characteristic
106	C₂×C₄₃	#2215	1	1	yes	characteristic
107	C₂×C₄₃	#2216	1	1	yes	characteristic
108	C₂×C₄₃	#2252	1	1	yes	characteristic
109	C₂×C₄₃	#2253	1	1	yes	characteristic
110	C₂²×C₁₆	—	43	1	no	—
111	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
112	C₂×#107	—	43	2	no	—
113	C₂×#107	C₂×D₄₃	1	2	yes	—
114	C₂×C₄×C₈	—	43	1	no	—
115	#389	—	86	2	no	—
116	#366	—	86	2	no	—
117	#366	Dic₄₃	1	4	yes	complemented
118	C₄×C₁₆	—	86	2	no	—
119	#340	—	86	2	no	—
120	C₄₃	#13165	1	1	yes	characteristic, complemented, Sylow
121	C₂²×C₈	—	43	1	no	—
122	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
123	C₂×C₄⋊C₄	—	2	2	no	—
124	C₂×C₄⋊C₄	—	86	2	no	—
125	C₂×C₄⋊C₄	—	172	2	no	—
126	C₂×C₄²	—	43	1	no	—
127	C₂×C₁₆	—	43	2	no	—
128	C₂×C₁₆	—	86	1	no	—
129	C₂×C₁₆	—	86	1	no	—
130	C₂×C₁₆	Dic₈₆	1	2	yes	—
131	C₂×C₁₆	D₁₇₂	1	2	yes	—
132	C₂×C₁₆	C₂×Dic₄₃	1	2	yes	—
133	#107	—	2	2	no	—
134	#107	—	86	2	no	—
135	#107	—	86	2	no	—
136	#107	C₂×Dic₄₃	1	4	yes	—
137	C₄×C₈	—	86	1	no	—
138	C₄×C₈	—	86	1	no	—
139	#95	—	172	2	no	—
140	C₂²×C₄	—	4	2	no	—
141	C₂²×C₄	—	43	1	no	—
142	C₂²×C₄	—	43	1	no	—
143	C₂²×C₄	—	172	2	no	—
144	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
145	C₂×C₈	—	43	2	no	—
146	C₂×C₈	—	86	1	no	—
147	C₂×C₈	—	86	1	no	—
148	C₂×C₈	C₂×Dic₈₆	1	1	yes	characteristic
149	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
150	C₂×C₈	#10621900	1	1	yes	characteristic
151	C₂×C₈	Q₈×D₄₃	1	1	yes	characteristic
152	C₂×C₈	#10621902	1	1	yes	characteristic
153	C₂×C₈	C₂²×Dic₄₃	1	1	yes	characteristic
154	C₄⋊C₄	—	2	4	no	—
155	C₄⋊C₄	—	4	2	no	—
156	C₄⋊C₄	—	344	4	no	—
157	C₄⋊C₄	—	172	2	no	—
158	C₄⋊C₄	—	172	2	no	—
159	C₄²	—	86	1	no	—
160	C₄²	—	86	1	no	—
161	C₁₆	—	86	2	no	—
162	C₁₆	C₄⋊Dic₄₃	1	4	yes	complemented
163	C₂³	D₈×D₄₃	1	1	yes	characteristic
164	C₂×C₄	—	4	4	no	—
165	C₂×C₄	—	8	2	no	—
166	C₂×C₄	—	8	2	no	—
167	C₂×C₄	—	43	2	no	—
168	C₂×C₄	—	43	2	no	—
169	C₂×C₄	—	86	1	no	—
170	C₂×C₄	—	86	1	no	—
171	C₂×C₄	—	86	1	no	—
172	C₂×C₄	—	86	1	no	—
173	C₂×C₄	—	344	2	no	—
174	C₂×C₄	—	344	2	no	—
175	C₂×C₄	—	344	2	no	—
176	C₂×C₄	D₄×Dic₄₃	1	1	yes	characteristic
177	C₂×C₄	#13063300	1	1	yes	characteristic
178	C₂×C₄	#13063324	1	1	yes	characteristic
179	C₂×C₄	Q₁₆×D₄₃	1	1	yes	characteristic
180	C₂×C₄	#13063344	1	1	yes	characteristic
181	C₂×C₄	#13063348	1	1	yes	characteristic
182	C₈	—	86	2	no	—
183	C₈	C₂×C₄⋊Dic₄₃	1	1	yes	characteristic
184	C₈	Q₈×Dic₄₃	1	1	yes	characteristic
185	C₈	#13063305	1	1	yes	characteristic
186	C₈	#13063316	1	1	yes	characteristic
187	C₂²	D₈×Dic₄₃	1	1	yes	characteristic
188	C₂²	#14583136	1	1	yes	characteristic
189	C₂²	#14583303	1	1	yes	characteristic
190	C₂²	D₁₆×D₄₃	1	1	yes	characteristic
191	C₂²	Q₃₂×D₄₃	1	1	yes	characteristic
192	C₂²	#14583328	1	1	yes	characteristic
193	C₂²	#14583332	1	1	yes	characteristic
194	C₄	—	8	4	no	—
195	C₄	—	86	2	no	—
196	C₄	—	86	2	no	—
197	C₄	—	688	2	no	—
198	C₄	#14582798	1	1	yes	characteristic
199	C₄	Q₁₆×Dic₄₃	1	1	yes	characteristic
200	C₄	#14583141	1	1	yes	characteristic
201	C₄	#14583300	1	1	yes	characteristic
202	C₂	#19135329	1	1	yes	characteristic
203	C₂	D₁₆×Dic₄₃	1	1	yes	characteristic
204	C₂	Q₃₂×Dic₄₃	1	1	yes	characteristic
205	C₂	#19136682	1	1	yes	characteristic
206	C₂	#19136687	1	1	yes	characteristic
207	C₂	#19136869	1	1	yes	characteristic
208	C₂	#19136872	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₈×D₄₃
commutator	C₂×C₈×C₄₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₂×C₄₃×#366	C₂
socle, prime radical	C₂³×C₄₃	C₂×D₈
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	C₂×#366	D₄₃
hypocenter	C₄₃	#13165
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

Complements

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

N	H ≅ G/N	type
15. C₄₃×#366	C₄	semidirect
56. C₁₆×C₄₃	C₄⋊C₄	semidirect
117. #366	Dic₄₃	semidirect
120. C₄₃	#13165	semidirect
162. C₁₆	C₄⋊Dic₄₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14582725	2
C₂×C₁₆×Dic₄₃	2
#19135329	2
#19132686	2
C₂×C₄₃×#366	2
#13165	43