Subgroups — #24995972

Order 11008.

394 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19134759	C₂	1	1	yes	maximal, characteristic
2	#19133159	C₂	1	4	yes	maximal
3	#19132878	C₂	1	1	yes	maximal, characteristic
4	C₄₃×#3160	C₂	1	1	yes	maximal, characteristic
5	#14582952	C₂²	1	2	yes	—
6	C₂×#13063259	—	2	1	no	—
7	C₂×#13063257	—	2	1	no	—
8	#14582765	C₂²	1	2	yes	—
9	#14582636	—	2	8	no	—
10	#14582620	—	2	1	no	—
11	C₈×C₄₃⋊C₈	—	2	1	no	—
12	C₄₃×#533	C₂²	1	2	yes	—
13	C₄₃×#500	C₄	1	2	yes	complemented
14	C₄₃×#499	—	2	1	no	—
15	C₄₃×#499	—	2	1	no	—
16	C₄₃×#498	C₄	1	2	yes	complemented
17	C₄₃×#427	—	2	1	no	—
18	C₄₃×#427	—	2	1	no	—
19	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊Dic₄₃	—	2	2	no	—
21	#13063259	—	4	2	no	—
22	#13063257	—	4	2	no	—
23	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
24	C₂²⋊(C₄₃⋊C₈)	D₄	1	4	yes	—
25	C₄×C₄₃⋊C₈	—	2	2	no	—
26	C₄₃⋊#95	—	2	4	no	—
27	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
29	C₄⋊Q₈×C₄₃	C₂×C₄	1	4	yes	—
30	#125×C₄₃	—	2	2	no	—
31	C₂²⋊Q₈×C₄₃	—	2	4	no	—
32	C₄×Q₈×C₄₃	—	2	4	no	—
33	#117×C₄₃	—	2	2	no	—
34	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
35	#107×C₄₃	—	2	2	no	—
36	#107×C₄₃	—	2	2	no	—
37	#106×C₄₃	—	2	2	no	—
38	#106×C₄₃	—	2	2	no	—
39	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
41	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
42	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
43	C₂×Q₈×C₄₃	—	2	4	no	—
44	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
45	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
46	C₂²×Dic₄₃	—	4	2	no	—
47	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₄₃	SD₁₆	1	2	yes	—
52	C₂×C₈×C₄₃	SD₁₆	1	2	yes	—
53	C₂×C₈×C₄₃	Q₁₆	1	2	yes	—
54	C₂×C₈×C₄₃	Q₁₆	1	2	yes	—
55	C₄⋊C₄×C₄₃	—	2	4	no	—
56	C₄⋊C₄×C₄₃	—	2	4	no	—
57	C₄⋊C₄×C₄₃	—	4	4	no	—
58	C₂²⋊C₄×C₄₃	—	4	2	no	—
59	C₄²×C₄₃	—	4	2	no	—
60	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
61	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
62	C₄²×C₄₃	#41	1	2	yes	—
63	C₄⋊Dic₄₃	—	4	2	no	—
64	C₄⋊Dic₄₃	—	4	2	no	—
65	C₂×C₄₃⋊C₈	—	2	4	no	—
66	C₂³×C₄₃	#127	1	1	yes	characteristic
67	Q₈×C₄₃	—	8	4	no	—
68	C₂×C₄×C₄₃	—	4	2	no	—
69	C₂×C₄×C₄₃	—	4	4	no	—
70	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
72	C₂×C₄×C₄₃	#121	1	2	yes	—
73	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
74	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
75	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
76	C₂×C₄×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
77	C₂×C₄×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
78	C₂×Dic₄₃	—	8	4	no	—
79	C₂×Dic₄₃	—	8	2	no	—
80	C₈×C₄₃	#103	1	2	yes	—
81	C₈×C₄₃	#103	1	2	yes	—
82	C₈×C₄₃	#103	1	2	yes	—
83	C₈×C₄₃	#103	1	2	yes	—
84	C₄₃⋊C₈	—	4	4	no	—
85	#9755	—	43	1	no	maximal, Sylow
86	C₂²×C₄₃	—	2	2	no	—
87	C₂²×C₄₃	#390	1	1	yes	characteristic
88	C₂²×C₄₃	#492	1	1	yes	characteristic
89	C₂²×C₄₃	#494	1	1	yes	characteristic
90	C₄×C₄₃	—	2	2	no	—
91	C₄×C₄₃	—	2	2	no	—
92	C₄×C₄₃	—	8	4	no	—
93	C₄×C₄₃	C₂×#103	1	1	yes	characteristic
94	C₄×C₄₃	C₂×#103	1	1	yes	characteristic
95	C₄×C₄₃	#487	1	1	yes	characteristic
96	C₄×C₄₃	#487	1	1	yes	characteristic
97	Dic₄₃	—	16	2	no	—
98	#3160	D₄₃	1	1	yes	characteristic
99	#1846	—	43	1	no	—
100	#1518	—	43	4	no	—
101	#1450	—	43	1	no	—
102	C₂×C₄₃	—	2	1	no	—
103	C₂×C₄₃	—	2	1	no	—
104	C₂×C₄₃	#1956	1	1	yes	characteristic
105	C₂×C₄₃	#1976	1	2	yes	—
106	#533	C₂×D₄₃	1	2	yes	—
107	#500	Dic₄₃	1	2	yes	complemented
108	#499	—	2	1	no	—
109	#499	—	2	1	no	—
110	#498	Dic₄₃	1	2	yes	complemented
111	#431	—	43	2	no	—
112	#427	—	2	1	no	—
113	#427	—	2	1	no	—
114	C₂×#107	—	86	1	no	—
115	C₂×#106	—	86	1	no	—
116	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
117	#382	—	43	2	no	—
118	#328	—	86	8	no	—
119	#322	—	86	1	no	—
120	C₈²	—	86	1	no	—
121	C₄₃	#9755	1	1	yes	characteristic, complemented, Sylow
122	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
123	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
124	C₄⋊Q₈	C₂×Dic₄₃	1	4	yes	—
125	#125	—	2	2	no	—
126	C₂²⋊Q₈	—	2	4	no	—
127	C₄×Q₈	—	2	4	no	—
128	#117	—	2	2	no	—
129	C₂×C₄⋊C₄	—	86	2	no	—
130	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
131	#107	—	2	2	no	—
132	#107	—	2	2	no	—
133	#107	—	172	2	no	—
134	#106	—	2	2	no	—
135	#106	—	2	2	no	—
136	#106	—	172	2	no	—
137	C₄⋊C₈	—	86	2	no	—
138	C₂²⋊C₈	—	43	4	no	—
139	C₄×C₈	—	86	2	no	—
140	C₄×C₈	C₂×Dic₄₃	1	1	yes	characteristic
141	C₄×C₈	C₂×Dic₄₃	1	1	yes	characteristic
142	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
143	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
144	#95	—	86	4	no	—
145	C₂×Q₈	—	2	4	no	—
146	C₂²×C₄	—	172	2	no	—
147	C₂²×C₄	D₄×D₄₃	1	2	yes	—
148	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
149	C₂×C₈	—	86	4	no	—
150	C₂×C₈	#10621884	1	2	yes	—
151	C₂×C₈	#10621884	1	2	yes	—
152	C₂×C₈	#10621885	1	2	yes	—
153	C₂×C₈	#10621885	1	2	yes	—
154	C₂×C₈	C₂²⋊Dic₄₃	1	1	yes	characteristic
155	C₂×C₈	C₂²⋊Dic₄₃	1	1	yes	characteristic
156	C₂×C₈	C₂²⋊Dic₄₃	1	1	yes	characteristic
157	C₂×C₈	C₂²⋊Dic₄₃	1	1	yes	characteristic
158	C₄⋊C₄	—	2	4	no	—
159	C₄⋊C₄	—	2	4	no	—
160	C₄⋊C₄	—	4	4	no	—
161	C₄⋊C₄	—	172	2	no	—
162	C₄⋊C₄	—	172	2	no	—
163	C₂²⋊C₄	—	4	2	no	—
164	C₄²	—	4	2	no	—
165	C₄²	#10621900	1	2	yes	—
166	C₄²	C₂²×Dic₄₃	1	1	yes	characteristic
167	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
168	C₂³	#13063321	1	1	yes	characteristic
169	Q₈	—	8	4	no	—
170	C₂×C₄	—	4	2	no	—
171	C₂×C₄	—	4	4	no	—
172	C₂×C₄	—	344	4	no	—
173	C₂×C₄	—	344	2	no	—
174	C₂×C₄	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
175	C₂×C₄	D₄×Dic₄₃	1	2	yes	—
176	C₂×C₄	#13063296	1	2	yes	—
177	C₂×C₄	#13063312	1	1	yes	characteristic
178	C₂×C₄	C₂×#10621884	1	1	yes	characteristic
179	C₂×C₄	C₂×#10621884	1	1	yes	characteristic
180	C₂×C₄	C₂×#10621885	1	1	yes	characteristic
181	C₂×C₄	C₂×#10621885	1	1	yes	characteristic
182	C₈	—	172	4	no	—
183	C₈	#13063249	1	2	yes	—
184	C₈	#13063249	1	2	yes	—
185	C₈	#13063249	1	2	yes	—
186	C₈	#13063249	1	2	yes	—
187	C₂²	—	2	2	no	—
188	C₂²	#14582802	1	1	yes	characteristic
189	C₂²	#14583273	1	1	yes	characteristic
190	C₂²	#14583278	1	1	yes	characteristic
191	C₄	—	2	2	no	—
192	C₄	—	2	2	no	—
193	C₄	—	8	4	no	—
194	C₄	—	688	2	no	—
195	C₄	C₂×#13063249	1	1	yes	characteristic
196	C₄	C₂×#13063249	1	1	yes	characteristic
197	C₄	#14583249	1	1	yes	characteristic
198	C₄	#14583249	1	1	yes	characteristic
199	C₂	—	2	1	no	—
200	C₂	—	2	1	no	—
201	C₂	#19135398	1	1	yes	characteristic
202	C₂	#19135494	1	2	yes	—

	subgroup	G/H
center	C₂²	#14582802
commutator	C₄²×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#3160	C₂
socle	C₂²×C₄₃	#390
prime radical	C₂³×C₄₃	#127
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#3160	D₄₃
hypocenter	C₄₃	#9755
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3160

Complements

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

N	H ≅ G/N	type
13. C₄₃×#500	C₄	semidirect
16. C₄₃×#498	C₄	semidirect
107. #500	Dic₄₃	semidirect
110. #498	Dic₄₃	semidirect
121. C₄₃	#9755	semidirect

Maximal subgroups

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

subgroup	index
#19134759	2
#19133159	2
#19132878	2
C₄₃×#3160	2
#9755	43