Subgroups — #25011429

Order 11008.

350 conjugacy classes of subgroups.

The subgroup lattice has 166 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19136200	C₂	1	1	yes	maximal, characteristic
2	#19136199	C₂	1	2	yes	maximal
3	#19136191	C₂	1	2	yes	maximal
4	#19134350	C₂	1	1	yes	maximal, characteristic
5	C₄₃×#2108	C₂	1	1	yes	maximal, characteristic
6	C₂×#13063265	C₂²	1	2	yes	—
7	#14582854	C₂²	1	2	yes	—
8	#14582677	C₄	1	4	yes	complemented
9	#14582676	C₄	1	8	yes	complemented
10	#14582674	C₄	1	4	yes	complemented
11	C₁₆×Dic₄₃	C₄	1	4	yes	complemented
12	C₄₃⋊#338	C₄	1	1	yes	characteristic
13	#14582656	C₄	1	1	yes	characteristic
14	#14582655	C₄	1	2	yes	—
15	C₂×M₃₂×C₄₃	C₂²	1	1	yes	characteristic
16	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
17	C₄₃×#405	C₂²	1	1	yes	characteristic
18	C₄₃×#346	C₄	1	2	yes	complemented
19	C₄×C₁₆×C₄₃	C₄	1	2	yes	complemented
20	C₂×#10621877	C₂×C₄	1	2	yes	—
21	#13063284	C₂×C₄	1	2	yes	—
22	#13063265	C₂×C₄	1	4	yes	—
23	#13063265	C₂×C₄	1	4	yes	—
24	C₂×C₄₃⋊C₁₆	C₂×C₄	1	4	yes	—
25	#13063235	C₂×C₄	1	4	yes	—
26	C₈×Dic₄₃	C₂×C₄	1	4	yes	—
27	C₂×M₁₆×C₄₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
29	#117×C₄₃	C₂×C₄	1	1	yes	characteristic
30	M₃₂×C₄₃	C₂×C₄	1	4	yes	—
31	C₂×C₁₆×C₄₃	C₂×C₄	1	2	yes	—
32	C₂×C₁₆×C₄₃	C₂×C₄	1	2	yes	—
33	C₂×C₁₆×C₄₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₁₆×C₄₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₁₆×C₄₃	C₂×C₄	1	2	yes	—
36	#97×C₄₃	C₂×C₄	1	2	yes	—
37	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
38	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
39	M₁₆×C₄₃	—	2	2	no	—
40	C₂×C₈×C₄₃	C₄²	1	2	yes	—
41	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
43	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
44	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
45	C₄⋊C₄×C₄₃	—	2	1	no	—
46	C₂²⋊C₄×C₄₃	—	2	1	no	—
47	C₄²×C₄₃	C₄²	1	2	yes	—
48	C₂²⋊Dic₄₃	—	2	2	no	—
49	C₄⋊Dic₄₃	—	2	2	no	—
50	C₄×Dic₄₃	C₄²	1	4	yes	—
51	#10621877	—	2	2	no	—
52	#10621877	—	2	2	no	—
53	C₂×C₄₃⋊C₈	C₄²	1	4	yes	—
54	C₁₆×C₄₃	C₄²	1	2	yes	—
55	C₁₆×C₄₃	C₄²	1	1	yes	characteristic
56	C₁₆×C₄₃	C₄²	1	4	yes	—
57	C₁₆×C₄₃	C₄²	1	1	yes	characteristic
58	C₄₃⋊C₁₆	C₄²	1	8	yes	—
59	C₂³×C₄₃	#131	1	1	yes	characteristic
60	C₂×C₄×C₄₃	—	2	2	no	—
61	C₂×C₄×C₄₃	C₂×C₄²	1	2	yes	—
62	C₂×C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
63	C₂×C₄×C₄₃	#131	1	1	yes	characteristic
64	C₂×C₄×C₄₃	#131	1	2	yes	—
65	C₂×Dic₄₃	—	2	4	no	—
66	C₈×C₄₃	—	2	2	no	—
67	C₈×C₄₃	C₂×C₄²	1	2	yes	—
68	C₈×C₄₃	C₂×C₄²	1	1	yes	characteristic
69	C₈×C₄₃	C₂×C₄²	1	1	yes	characteristic
70	C₄₃⋊C₈	—	2	4	no	—
71	#12542	—	43	1	no	maximal, Sylow
72	C₂²×C₄₃	—	4	1	no	—
73	C₂²×C₄₃	#405	1	2	yes	—
74	C₂²×C₄₃	#405	1	1	yes	characteristic
75	C₄×C₄₃	—	4	2	no	—
76	C₄×C₄₃	C₄³	1	1	yes	characteristic
77	C₄×C₄₃	#405	1	2	yes	—
78	C₄×C₄₃	#405	1	1	yes	characteristic
79	Dic₄₃	—	4	4	no	—
80	#2110	—	43	2	no	—
81	#2110	—	43	1	no	—
82	#2108	—	43	2	no	—
83	#2108	D₄₃	1	1	yes	characteristic
84	#1747	—	43	1	no	—
85	C₂×C₄₃	—	2	2	no	—
86	C₂×C₄₃	—	2	1	no	—
87	C₂×C₄₃	#1726	1	1	yes	characteristic
88	C₂×M₃₂	—	43	2	no	—
89	C₂×M₃₂	C₂×D₄₃	1	1	yes	characteristic
90	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
91	#405	—	43	2	no	—
92	#405	C₂×D₄₃	1	1	yes	characteristic
93	#347	—	43	8	no	—
94	#347	—	43	4	no	—
95	#346	—	43	4	no	—
96	#346	Dic₄₃	1	2	yes	complemented
97	C₄×C₁₆	—	43	4	no	—
98	C₄×C₁₆	Dic₄₃	1	2	yes	complemented
99	#338	—	43	1	no	—
100	H(3,ℤ/4)	—	43	1	no	—
101	H(3,ℤ/4)	—	43	2	no	—
102	C₄₃	#12542	1	1	yes	characteristic, complemented, Sylow
103	C₂×M₁₆	—	43	2	no	—
104	C₂×M₁₆	C₄×D₄₃	1	1	yes	characteristic
105	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
106	#117	—	43	2	no	—
107	#117	C₄×D₄₃	1	1	yes	characteristic
108	M₃₂	—	43	4	no	—
109	M₃₂	—	43	4	no	—
110	M₃₂	C₄×D₄₃	1	4	yes	—
111	C₂×C₁₆	—	43	4	no	—
112	C₂×C₁₆	C₄×D₄₃	1	2	yes	—
113	C₂×C₁₆	C₄×D₄₃	1	2	yes	—
114	C₂×C₁₆	C₂×Dic₄₃	1	2	yes	—
115	C₂×C₁₆	C₂×Dic₄₃	1	1	yes	characteristic
116	C₂×C₁₆	C₂×Dic₄₃	1	1	yes	characteristic
117	#97	—	43	4	no	—
118	#97	C₄×D₄₃	1	2	yes	—
119	C₄×C₈	—	43	4	no	—
120	C₄×C₈	C₂×Dic₄₃	1	2	yes	—
121	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
122	M₁₆	—	2	2	no	—
123	M₁₆	—	86	2	no	—
124	M₁₆	—	86	2	no	—
125	C₂×C₈	—	43	4	no	—
126	C₂×C₈	C₄×Dic₄₃	1	2	yes	—
127	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
128	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
129	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
130	C₂×C₈	C₂²×Dic₄₃	1	1	yes	characteristic
131	C₄⋊C₄	—	2	1	no	—
132	C₄⋊C₄	—	86	2	no	—
133	C₂²⋊C₄	—	2	1	no	—
134	C₂²⋊C₄	—	86	2	no	—
135	C₄²	—	43	4	no	—
136	C₄²	C₄×Dic₄₃	1	2	yes	—
137	C₁₆	—	43	8	no	—
138	C₁₆	C₄×Dic₄₃	1	1	yes	characteristic
139	C₁₆	C₄×Dic₄₃	1	2	yes	—
140	C₁₆	C₄×Dic₄₃	1	4	yes	—
141	C₁₆	C₄×Dic₄₃	1	1	yes	characteristic
142	C₂³	#13063331	1	1	yes	characteristic
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	86	4	no	—
145	C₂×C₄	C₄²×D₄₃	1	2	yes	—
146	C₂×C₄	C₂×C₄×Dic₄₃	1	1	yes	characteristic
147	C₂×C₄	#13063331	1	1	yes	characteristic
148	C₂×C₄	#13063331	1	2	yes	—
149	C₈	—	2	2	no	—
150	C₈	—	86	4	no	—
151	C₈	C₄²×D₄₃	1	2	yes	—
152	C₈	C₂×C₄×Dic₄₃	1	1	yes	characteristic
153	C₈	C₂×C₄×Dic₄₃	1	1	yes	characteristic
154	C₂²	—	4	1	no	—
155	C₂²	#14582853	1	2	yes	—
156	C₂²	#14582854	1	1	yes	characteristic
157	C₄	—	4	2	no	—
158	C₄	—	172	4	no	—
159	C₄	C₄²×Dic₄₃	1	1	yes	characteristic
160	C₄	#14582853	1	2	yes	—
161	C₄	#14582854	1	1	yes	characteristic
162	C₂	—	2	1	no	—
163	C₂	—	2	2	no	—
164	C₂	#19134281	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	C₄×Dic₄₃
commutator	C₄×C₄₃	C₄³
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₄₃×#2108	C₂
socle	C₂×C₄₃	#1726
prime radical	C₂³×C₄₃	#131
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	#2108	D₄₃
hypocenter	C₄₃	#12542
LCS	G ▷ C₄×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₁₆ ◁ #2108

Complements

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

N	H ≅ G/N	type
8. #14582677	C₄	semidirect
9. #14582676	C₄	semidirect
10. #14582674	C₄	semidirect
11. C₁₆×Dic₄₃	C₄	semidirect
18. C₄₃×#346	C₄	semidirect
19. C₄×C₁₆×C₄₃	C₄	semidirect
96. #346	Dic₄₃	semidirect
98. C₄×C₁₆	Dic₄₃	semidirect
102. C₄₃	#12542	semidirect

Maximal subgroups

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

subgroup	index
#19136200	2
#19136199	2
#19136191	2
#19134350	2
C₄₃×#2108	2
#12542	43