Subgroups — #25005601

Order 11008.

282 conjugacy classes of subgroups.

The subgroup lattice has 136 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13063265	C₂	1	1	yes	maximal, characteristic
2	#19132878	C₂	1	1	yes	maximal, characteristic
3	#19132515	C₂	1	4	yes	maximal, complemented
4	C₂×C₄₃×#346	C₂	1	1	yes	maximal, characteristic
5	C₂×#13063265	C₄	1	2	yes	—
6	#14582952	C₄	1	2	yes	—
7	#14582675	C₂²	1	2	yes	—
8	C₄×C₄₃⋊C₁₆	C₄	1	2	yes	—
9	#14582620	C₂²	1	2	yes	—
10	C₈×C₄₃⋊C₈	C₄	1	2	yes	—
11	C₂²×C₁₆×C₄₃	C₄	1	2	yes	—
12	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
13	C₄₃×#346	C₄	1	2	yes	—
14	C₄₃×#346	C₂²	1	2	yes	—
15	#13063265	—	2	4	no	—
16	C₂×C₄₃⋊C₁₆	C₂×C₄	1	4	yes	—
17	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
18	C₂²⋊(C₄₃⋊C₈)	—	2	2	no	—
19	C₄×C₄₃⋊C₈	C₂×C₄	1	4	yes	—
20	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₁₆×C₄₃	C₈	1	8	yes	complemented
24	C₂×C₁₆×C₄₃	C₂×C₄	1	4	yes	—
25	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₄₃	C₄²	1	1	yes	characteristic
30	C₂²×C₄×C₄₃	M₁₆	1	2	yes	—
31	C₂×C₈×C₄₃	C₄²	1	1	yes	characteristic
32	C₂×C₈×C₄₃	C₄²	1	1	yes	characteristic
33	C₂×C₈×C₄₃	C₂×C₈	1	4	yes	—
34	C₂×C₈×C₄₃	M₁₆	1	4	yes	—
35	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
37	C₄²×C₄₃	C₄²	1	1	yes	characteristic
38	C₄²×C₄₃	M₁₆	1	2	yes	—
39	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄₃⋊C₈	—	2	4	no	—
41	C₁₆×C₄₃	C₂×C₈	1	8	yes	—
42	C₄₃⋊C₁₆	—	2	4	no	—
43	C₂³×C₄₃	#97	1	1	yes	characteristic
44	C₂×C₄×C₄₃	—	4	2	no	—
45	C₂×C₄×C₄₃	C₄×C₈	1	2	yes	—
46	C₂×C₄×C₄₃	#97	1	2	yes	—
47	C₂×C₄×C₄₃	#97	1	1	yes	characteristic
48	C₂×C₄×C₄₃	#97	1	2	yes	—
49	C₂×C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
50	C₂×C₄×C₄₃	C₂×M₁₆	1	2	yes	—
51	C₈×C₄₃	C₄×C₈	1	2	yes	—
52	C₈×C₄₃	#97	1	2	yes	—
53	C₈×C₄₃	C₂²×C₈	1	2	yes	—
54	C₈×C₄₃	C₂×M₁₆	1	2	yes	—
55	C₄₃⋊C₈	—	4	4	no	—
56	#11405	—	43	1	no	maximal, Sylow
57	C₂²×C₄₃	—	2	2	no	—
58	C₂²×C₄₃	#322	1	2	yes	—
59	C₂²×C₄₃	C₂×#97	1	1	yes	characteristic
60	C₄×C₄₃	—	2	2	no	—
61	C₄×C₄₃	—	2	2	no	—
62	C₄×C₄₃	#322	1	2	yes	—
63	C₄×C₄₃	C₂×C₄×C₈	1	1	yes	characteristic
64	C₄×C₄₃	C₂×#97	1	1	yes	characteristic
65	C₄×M₃₂	—	43	1	no	—
66	C₂×#346	D₄₃	1	1	yes	characteristic
67	#1450	—	43	1	no	—
68	#1313	—	43	4	no	—
69	C₂×C₄₃	—	2	2	no	—
70	C₂×C₄₃	C₂×#322	1	1	yes	characteristic
71	C₂×C₄₃	#2110	1	2	yes	—
72	C₂×M₃₂	—	43	2	no	—
73	C₂²×C₁₆	Dic₄₃	1	2	yes	—
74	#431	—	43	2	no	—
75	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
76	#346	—	43	2	no	—
77	#346	Dic₄₃	1	2	yes	—
78	#346	C₂×D₄₃	1	2	yes	—
79	C₄×C₁₆	—	43	2	no	—
80	#322	—	43	2	no	—
81	C₈²	—	43	2	no	—
82	C₄₃	#11405	1	1	yes	characteristic, complemented, Sylow
83	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
84	C₂²×C₈	C₂×Dic₄₃	1	1	yes	characteristic
85	C₂×C₄²	C₄×D₄₃	1	1	yes	characteristic
86	M₃₂	—	86	4	no	—
87	C₂×C₁₆	—	43	4	no	—
88	C₂×C₁₆	C₄₃⋊C₈	1	8	yes	complemented
89	C₂×C₁₆	C₂×Dic₄₃	1	4	yes	—
90	C₄⋊C₈	—	86	2	no	—
91	C₂²⋊C₈	—	86	2	no	—
92	C₄×C₈	—	43	4	no	—
93	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
94	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
95	C₄×C₈	C₂×Dic₄₃	1	1	yes	characteristic
96	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
97	C₂²×C₄	#10621872	1	2	yes	—
98	C₂²×C₄	C₄×Dic₄₃	1	1	yes	characteristic
99	C₂×C₈	—	86	4	no	—
100	C₂×C₈	#10621872	1	4	yes	—
101	C₂×C₈	C₂×C₄₃⋊C₈	1	4	yes	—
102	C₂×C₈	C₄×Dic₄₃	1	1	yes	characteristic
103	C₂×C₈	C₄×Dic₄₃	1	1	yes	characteristic
104	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
105	C₂×C₈	C₂²×Dic₄₃	1	1	yes	characteristic
106	C₄²	#10621872	1	2	yes	—
107	C₄²	C₄×Dic₄₃	1	1	yes	characteristic
108	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
109	C₁₆	—	86	4	no	—
110	C₁₆	C₂×C₄₃⋊C₈	1	8	yes	—
111	C₂³	#13063235	1	1	yes	characteristic
112	C₂×C₄	—	4	2	no	—
113	C₂×C₄	C₄×C₄₃⋊C₈	1	2	yes	—
114	C₂×C₄	#13063235	1	2	yes	—
115	C₂×C₄	#13063235	1	2	yes	—
116	C₂×C₄	#13063235	1	1	yes	characteristic
117	C₂×C₄	C₂×C₄×Dic₄₃	1	1	yes	characteristic
118	C₂×C₄	C₂×#10621872	1	2	yes	—
119	C₈	—	172	4	no	—
120	C₈	C₄×C₄₃⋊C₈	1	2	yes	—
121	C₈	#13063235	1	2	yes	—
122	C₈	C₂²×C₄₃⋊C₈	1	2	yes	—
123	C₈	C₂×#10621872	1	2	yes	—
124	C₂²	—	2	2	no	—
125	C₂²	#14582620	1	2	yes	—
126	C₂²	C₂×#13063235	1	1	yes	characteristic
127	C₄	—	2	2	no	—
128	C₄	—	2	2	no	—
129	C₄	#14582620	1	2	yes	—
130	C₄	C₂×C₄×C₄₃⋊C₈	1	1	yes	characteristic
131	C₄	C₂×#13063235	1	1	yes	characteristic
132	C₂	—	2	2	no	—
133	C₂	C₂×#14582620	1	1	yes	characteristic
134	C₂	#19136199	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₄×Dic₄₃
commutator	C₄×C₄₃	C₂×C₄×C₈
Frattini	C₄×C₈	C₂²×D₄₃
Fitting	C₂×C₄₃×#346	C₂
socle	C₂²×C₄₃	C₂×#97
prime radical	C₂³×C₄₃	#97
cosocle	C₄×C₈×C₄₃	C₂³
hypercenter	C₂×#346	D₄₃
hypocenter	C₄₃	#11405
LCS	G ▷ C₄×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
3. #19132515	C₂	semidirect
23. C₂×C₁₆×C₄₃	C₈	semidirect
82. C₄₃	#11405	semidirect
88. C₂×C₁₆	C₄₃⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13063265	2
#19132878	2
#19132515	2
C₂×C₄₃×#346	2
#11405	43