Subgroups — #24987020

Order 11008.

574 conjugacy classes of subgroups.

The subgroup lattice has 144 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×C₄₃⋊C₈	C₂	1	2	yes	maximal
2	C₄₃⋊#1273	C₂	1	4	yes	maximal
3	C₄²×C₈×C₄₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₄₃⋊C₈	C₄	1	4	yes	—
5	C₂×C₄×C₄₃⋊C₈	C₄	1	4	yes	—
6	C₂×C₄×C₄₃⋊C₈	C₂²	1	4	yes	—
7	C₂×C₄×C₈×C₄₃	C₄	1	2	yes	—
8	C₂×C₄×C₈×C₄₃	C₄	1	2	yes	—
9	C₂×C₄×C₈×C₄₃	C₂²	1	2	yes	—
10	C₄³×C₄₃	C₂²	1	1	yes	characteristic
11	C₂²×C₄₃⋊C₈	C₂×C₄	1	8	yes	—
12	C₄×C₄₃⋊C₈	—	2	8	no	—
13	C₄×C₄₃⋊C₈	—	2	8	no	—
14	C₄×C₄₃⋊C₈	C₈	1	16	yes	complemented
15	C₂²×C₈×C₄₃	C₂×C₄	1	4	yes	—
16	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
17	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
21	C₄×C₈×C₄₃	—	2	4	no	—
22	C₄×C₈×C₄₃	—	2	4	no	—
23	C₄×C₈×C₄₃	C₈	1	8	yes	complemented
24	C₂²×C₄×C₄₃	C₄²	1	1	yes	characteristic
25	C₂²×C₄×C₄₃	C₄²	1	2	yes	—
26	C₂²×C₄×C₄₃	C₄²	1	1	yes	characteristic
27	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
28	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₈×C₄₃	—	2	4	no	—
30	C₂×C₈×C₄₃	—	2	4	no	—
31	C₂×C₈×C₄₃	C₂×C₈	1	8	yes	—
32	C₄²×C₄₃	C₂×C₈	1	8	yes	—
33	C₄²×C₄₃	C₂×C₈	1	4	yes	—
34	C₄²×C₄₃	M₁₆	1	8	yes	—
35	C₄²×C₄₃	M₁₆	1	4	yes	—
36	C₄²×C₄₃	#41	1	4	yes	—
37	C₂×C₄₃⋊C₈	—	2	8	no	—
38	C₂×C₄₃⋊C₈	—	2	8	no	—
39	C₂×C₄₃⋊C₈	C₂×C₈	1	16	yes	—
40	C₂³×C₄₃	C₂×C₄²	1	1	yes	characteristic
41	C₂×C₄×C₄₃	C₄×C₈	1	4	yes	—
42	C₂×C₄×C₄₃	C₄×C₈	1	4	yes	—
43	C₂×C₄×C₄₃	C₄×C₈	1	2	yes	—
44	C₂×C₄×C₄₃	C₄×C₈	1	2	yes	—
45	C₂×C₄×C₄₃	#97	1	4	yes	—
46	C₂×C₄×C₄₃	#97	1	4	yes	—
47	C₂×C₄×C₄₃	#97	1	4	yes	—
48	C₂×C₄×C₄₃	#117	1	4	yes	—
49	C₂×C₄×C₄₃	#117	1	2	yes	—
50	C₂×C₄×C₄₃	C₂²×C₈	1	4	yes	—
51	C₂×C₄×C₄₃	C₂²×C₈	1	2	yes	—
52	C₂×C₄×C₄₃	C₂×M₁₆	1	4	yes	—
53	C₂×C₄×C₄₃	C₂×M₁₆	1	2	yes	—
54	C₈×C₄₃	—	2	8	no	—
55	C₄₃⋊C₈	—	2	16	no	—
56	#8131	—	43	1	no	maximal, Sylow
57	C₂²×C₄₃	#376	1	1	yes	characteristic
58	C₂²×C₄₃	C₂×C₄×C₈	1	2	yes	—
59	C₂²×C₄₃	C₂×C₄×C₈	1	1	yes	characteristic
60	C₂²×C₄₃	C₂×#97	1	2	yes	—
61	C₂²×C₄₃	C₂×#97	1	1	yes	characteristic
62	C₄×C₄₃	C₈²	1	4	yes	—
63	C₄×C₄₃	#322	1	4	yes	—
64	C₄×C₄₃	#322	1	8	yes	—
65	C₄×C₄₃	#431	1	8	yes	—
66	C₄×C₄₃	#431	1	4	yes	—
67	C₄²×C₈	—	43	2	no	—
68	C₄²×C₈	D₄₃	1	1	yes	characteristic
69	#1273	—	43	4	no	—
70	C₂×C₄₃	C₂×C₈²	1	1	yes	characteristic
71	C₂×C₄₃	C₂×#322	1	1	yes	characteristic
72	C₂×C₄₃	C₂×#322	1	2	yes	—
73	C₂×C₄₃	#1744	1	2	yes	—
74	C₂×C₄₃	#1744	1	1	yes	characteristic
75	C₂×C₄×C₈	—	43	4	no	—
76	C₂×C₄×C₈	—	43	4	no	—
77	C₂×C₄×C₈	—	43	4	no	—
78	C₂×C₄×C₈	Dic₄₃	1	2	yes	—
79	C₂×C₄×C₈	Dic₄₃	1	2	yes	—
80	C₂×C₄×C₈	C₂×D₄₃	1	2	yes	—
81	C₄³	C₂×D₄₃	1	1	yes	characteristic
82	C₄₃	#8131	1	1	yes	characteristic, complemented, Sylow
83	C₂²×C₈	—	43	8	no	—
84	C₂²×C₈	C₂×Dic₄₃	1	4	yes	—
85	C₂×C₄²	C₄×D₄₃	1	2	yes	—
86	C₂×C₄²	C₄×D₄₃	1	2	yes	—
87	C₂×C₄²	C₂×Dic₄₃	1	1	yes	characteristic
88	C₂×C₄²	C₂×Dic₄₃	1	1	yes	characteristic
89	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
90	C₄×C₈	—	2	4	no	—
91	C₄×C₈	—	2	4	no	—
92	C₄×C₈	—	43	16	no	—
93	C₄×C₈	—	86	8	no	—
94	C₄×C₈	—	86	8	no	—
95	C₄×C₈	C₄₃⋊C₈	1	8	yes	complemented
96	C₂²×C₄	C₄×Dic₄₃	1	1	yes	characteristic
97	C₂²×C₄	C₄×Dic₄₃	1	2	yes	—
98	C₂²×C₄	C₄×Dic₄₃	1	1	yes	characteristic
99	C₂²×C₄	C₂×C₄×D₄₃	1	2	yes	—
100	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
101	C₂×C₈	—	2	4	no	—
102	C₂×C₈	—	2	4	no	—
103	C₂×C₈	—	43	16	no	—
104	C₂×C₈	—	86	8	no	—
105	C₂×C₈	—	86	8	no	—
106	C₂×C₈	C₂×C₄₃⋊C₈	1	8	yes	—
107	C₄²	C₈×D₄₃	1	8	yes	—
108	C₄²	#10621872	1	8	yes	—
109	C₄²	C₂×C₄₃⋊C₈	1	4	yes	—
110	C₄²	#10621877	1	4	yes	—
111	C₄²	#10621898	1	4	yes	—
112	C₂³	C₂×C₄×Dic₄₃	1	1	yes	characteristic
113	C₂×C₄	C₈×Dic₄₃	1	4	yes	—
114	C₂×C₄	C₈×Dic₄₃	1	4	yes	—
115	C₂×C₄	C₄×C₄₃⋊C₈	1	2	yes	—
116	C₂×C₄	C₄×C₄₃⋊C₈	1	2	yes	—
117	C₂×C₄	#13063235	1	4	yes	—
118	C₂×C₄	#13063235	1	4	yes	—
119	C₂×C₄	#13063236	1	4	yes	—
120	C₂×C₄	#13063282	1	4	yes	—
121	C₂×C₄	#13063284	1	2	yes	—
122	C₂×C₄	C₂×C₈×D₄₃	1	4	yes	—
123	C₂×C₄	C₂²×C₄₃⋊C₈	1	2	yes	—
124	C₂×C₄	C₂×#10621872	1	4	yes	—
125	C₂×C₄	C₂×#10621877	1	2	yes	—
126	C₈	—	2	8	no	—
127	C₈	—	86	16	no	—
128	C₂²	#14582745	1	1	yes	characteristic
129	C₂²	C₂×C₈×Dic₄₃	1	2	yes	—
130	C₂²	C₂×C₄×C₄₃⋊C₈	1	1	yes	characteristic
131	C₂²	C₂×#13063235	1	2	yes	—
132	C₂²	C₂×#13063236	1	1	yes	characteristic
133	C₄	C₈×C₄₃⋊C₈	1	4	yes	—
134	C₄	#14582620	1	4	yes	—
135	C₄	#14582621	1	8	yes	—
136	C₄	#14582950	1	8	yes	—
137	C₄	#14582952	1	4	yes	—
138	C₂	C₂×C₈×C₄₃⋊C₈	1	1	yes	characteristic
139	C₂	C₂×#14582620	1	1	yes	characteristic
140	C₂	C₂×#14582621	1	2	yes	—
141	C₂	#19134340	1	2	yes	—
142	C₂	#19134341	1	1	yes	characteristic

	subgroup	G/H
center	C₄³	C₂×D₄₃
commutator	C₂×C₄₃	C₂×C₈²
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄²×C₈×C₄₃	C₂
socle, prime radical	C₂³×C₄₃	C₂×C₄²
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	C₄²×C₈	D₄₃
hypocenter	C₄₃	#8131
LCS	G ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₄³ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
14. C₄×C₄₃⋊C₈	C₈	semidirect
23. C₄×C₈×C₄₃	C₈	semidirect
82. C₄₃	#8131	semidirect
95. C₄×C₈	C₄₃⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄²×C₄₃⋊C₈	2
C₄₃⋊#1273	2
C₄²×C₈×C₄₃	2
#8131	43