Subgroups — #24987040

Order 11008.

354 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19134281	C₂	1	1	yes	maximal, characteristic
2	#19132878	C₂	1	1	yes	maximal, characteristic
3	#19132425	C₂	1	2	yes	maximal
4	#19132423	C₂	1	2	yes	maximal
5	C₄₃×#1450	C₂	1	1	yes	maximal, characteristic
6	#14582952	C₂²	1	2	yes	—
7	C₂×#13063236	C₂²	1	1	yes	characteristic
8	C₂×#13063235	C₄	1	2	yes	—
9	C₂×C₈×Dic₄₃	C₄	1	2	yes	—
10	#14582745	C₂²	1	1	yes	characteristic
11	#14582620	C₄	1	2	yes	complemented
12	C₈×C₄₃⋊C₈	C₄	1	2	yes	complemented
13	C₄₃×#431	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
15	C₄₃×#322	C₄	1	2	yes	complemented
16	C₈²×C₄₃	C₄	1	2	yes	complemented
17	C₂²×C₄₃⋊C₈	C₂×C₄	1	2	yes	—
18	C₂×C₄×Dic₄₃	C₂×C₄	1	2	yes	—
19	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
20	C₂²⋊(C₄₃⋊C₈)	—	2	2	no	—
21	#13063236	—	2	2	no	—
22	#13063235	C₈	1	8	yes	complemented
23	C₄×C₄₃⋊C₈	C₂×C₄	1	4	yes	—
24	C₈×Dic₄₃	—	2	4	no	—
25	C₄₃⋊#95	—	2	2	no	—
26	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
29	C₄⋊C₈×C₄₃	—	2	2	no	—
30	C₂²⋊C₈×C₄₃	—	2	2	no	—
31	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₄₃	C₂×C₄	1	4	yes	—
34	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
35	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₄₃	#41	1	2	yes	—
38	C₂²×Dic₄₃	—	2	2	no	—
39	C₂×C₈×C₄₃	—	2	4	no	—
40	C₂×C₈×C₄₃	C₄²	1	1	yes	characteristic
41	C₂×C₈×C₄₃	C₄²	1	1	yes	characteristic
42	C₂×C₈×C₄₃	C₄²	1	1	yes	characteristic
43	C₂×C₈×C₄₃	C₄²	1	1	yes	characteristic
44	C₂×C₈×C₄₃	C₂×C₈	1	4	yes	—
45	C₂×C₈×C₄₃	M₁₆	1	4	yes	—
46	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
47	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
48	C₄²×C₄₃	#41	1	2	yes	—
49	C₄×Dic₄₃	—	2	2	no	—
50	C₄×Dic₄₃	C₂×C₈	1	4	yes	—
51	C₂×C₄₃⋊C₈	—	2	2	no	—
52	C₂×C₄₃⋊C₈	—	2	2	no	—
53	C₂×C₄₃⋊C₈	—	2	4	no	—
54	C₂×C₄₃⋊C₈	C₂×C₈	1	4	yes	—
55	C₂³×C₄₃	#117	1	1	yes	characteristic
56	C₂×C₄×C₄₃	—	4	2	no	—
57	C₂×C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
58	C₂×C₄×C₄₃	#117	1	2	yes	—
59	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
60	C₂×C₄×C₄₃	#117	1	2	yes	—
61	C₂×C₄×C₄₃	C₂²×C₈	1	2	yes	—
62	C₂×C₄×C₄₃	C₂×M₁₆	1	2	yes	—
63	C₂×Dic₄₃	—	2	4	no	—
64	C₂×Dic₄₃	—	4	4	no	—
65	C₈×C₄₃	—	4	4	no	—
66	C₈×C₄₃	C₄×C₈	1	2	yes	—
67	C₈×C₄₃	C₄×C₈	1	2	yes	—
68	C₈×C₄₃	#97	1	2	yes	—
69	C₈×C₄₃	#97	1	2	yes	—
70	C₄₃⋊C₈	—	2	4	no	—
71	C₄₃⋊C₈	—	4	4	no	—
72	#8138	—	43	1	no	maximal, Sylow
73	C₂²×C₄₃	—	2	2	no	—
74	C₂²×C₄₃	#376	1	1	yes	characteristic
75	C₂²×C₄₃	#431	1	2	yes	—
76	C₄×C₄₃	—	2	2	no	—
77	C₄×C₄₃	—	2	2	no	—
78	C₄×C₄₃	C₂×C₄×C₈	1	1	yes	characteristic
79	C₄×C₄₃	C₂×#97	1	1	yes	characteristic
80	C₄×C₄₃	#431	1	2	yes	—
81	Dic₄₃	—	4	4	no	—
82	#1726	—	43	1	no	—
83	#1450	—	43	1	no	—
84	#1450	D₄₃	1	1	yes	characteristic
85	#1275	—	43	2	no	—
86	#1274	—	43	2	no	—
87	C₂×C₄₃	—	2	2	no	—
88	C₂×C₄₃	#1744	1	1	yes	characteristic
89	C₂×C₄₃	#1747	1	2	yes	—
90	#431	—	43	2	no	—
91	#431	C₂×D₄₃	1	2	yes	—
92	C₂×#97	—	43	2	no	—
93	C₂×#97	—	43	1	no	—
94	C₂×C₄×C₈	—	43	2	no	—
95	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
96	#376	—	43	1	no	—
97	#322	—	43	2	no	—
98	#322	Dic₄₃	1	2	yes	complemented
99	C₈²	—	43	2	no	—
100	C₈²	Dic₄₃	1	2	yes	complemented
101	C₄₃	#8138	1	1	yes	characteristic, complemented, Sylow
102	C₂²×C₈	—	43	2	no	—
103	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
104	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
105	C₂×C₄²	—	43	2	no	—
106	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
107	C₄⋊C₈	—	2	2	no	—
108	C₄⋊C₈	—	86	2	no	—
109	C₂²⋊C₈	—	2	2	no	—
110	C₂²⋊C₈	—	86	2	no	—
111	#97	—	43	8	no	—
112	#97	—	86	2	no	—
113	C₄×C₈	—	43	4	no	—
114	C₄×C₈	—	86	4	no	—
115	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
116	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
117	C₄×C₈	C₂×Dic₄₃	1	4	yes	—
118	C₄×C₈	C₂×Dic₄₃	1	1	yes	characteristic
119	C₄×C₈	C₂×Dic₄₃	1	1	yes	characteristic
120	#95	—	86	2	no	—
121	C₂²×C₄	—	86	2	no	—
122	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
123	C₂²×C₄	#10621898	1	2	yes	—
124	C₂×C₈	—	2	4	no	—
125	C₂×C₈	—	43	4	no	—
126	C₂×C₈	—	86	2	no	—
127	C₂×C₈	—	86	2	no	—
128	C₂×C₈	—	86	4	no	—
129	C₂×C₈	C₈×D₄₃	1	4	yes	—
130	C₂×C₈	#10621872	1	4	yes	—
131	C₂×C₈	C₄×Dic₄₃	1	1	yes	characteristic
132	C₂×C₈	C₄×Dic₄₃	1	1	yes	characteristic
133	C₂×C₈	C₄×Dic₄₃	1	1	yes	characteristic
134	C₂×C₈	C₄×Dic₄₃	1	1	yes	characteristic
135	C₄²	—	43	4	no	—
136	C₄²	—	86	2	no	—
137	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
138	C₄²	#10621898	1	2	yes	—
139	C₄²	C₂²×Dic₄₃	1	1	yes	characteristic
140	C₂³	#13063282	1	1	yes	characteristic
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	86	4	no	—
143	C₂×C₄	—	172	4	no	—
144	C₂×C₄	C₂×C₄×Dic₄₃	1	1	yes	characteristic
145	C₂×C₄	#13063282	1	2	yes	—
146	C₂×C₄	#13063282	1	1	yes	characteristic
147	C₂×C₄	#13063284	1	2	yes	—
148	C₂×C₄	C₂×C₈×D₄₃	1	2	yes	—
149	C₂×C₄	C₂×#10621872	1	2	yes	—
150	C₈	—	4	4	no	—
151	C₈	—	86	4	no	—
152	C₈	—	172	4	no	—
153	C₈	C₈×Dic₄₃	1	2	yes	—
154	C₈	C₈×Dic₄₃	1	2	yes	—
155	C₈	#13063235	1	2	yes	—
156	C₈	#13063235	1	2	yes	—
157	C₂²	—	2	2	no	—
158	C₂²	#14582745	1	1	yes	characteristic
159	C₂²	#14582950	1	2	yes	—
160	C₄	—	2	2	no	—
161	C₄	—	2	2	no	—
162	C₄	—	172	4	no	—
163	C₄	C₂×C₈×Dic₄₃	1	1	yes	characteristic
164	C₄	C₂×#13063235	1	1	yes	characteristic
165	C₄	#14582950	1	2	yes	—
166	C₂	—	2	2	no	—
167	C₂	#19134340	1	1	yes	characteristic
168	C₂	#19134350	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₄×Dic₄₃
commutator	C₄×C₄₃	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1450	C₂
socle	C₂²×C₄₃	#376
prime radical	C₂³×C₄₃	#117
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#1450	D₄₃
hypocenter	C₄₃	#8138
LCS	G ▷ C₄×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₈ ◁ #1450

Complements

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

N	H ≅ G/N	type
11. #14582620	C₄	semidirect
12. C₈×C₄₃⋊C₈	C₄	semidirect
15. C₄₃×#322	C₄	semidirect
16. C₈²×C₄₃	C₄	semidirect
22. #13063235	C₈	semidirect
98. #322	Dic₄₃	semidirect
100. C₈²	Dic₄₃	semidirect
101. C₄₃	#8138	semidirect

Maximal subgroups

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

subgroup	index
#19134281	2
#19132878	2
#19132425	2
#19132423	2
C₄₃×#1450	2
#8138	43