Subgroups — #25005591

Order 11008.

322 conjugacy classes of subgroups.

The subgroup lattice has 146 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	C₂×C₈×C₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#19132514	C₂	1	4	yes	maximal, complemented
4	C₄×M₃₂×C₄₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#13063265	C₄	1	2	yes	—
6	C₂×C₄×C₄₃⋊C₈	C₄	1	2	yes	—
7	#14582675	C₄	1	2	yes	—
8	C₄×C₄₃⋊C₁₆	C₂²	1	2	yes	—
9	C₈×C₄₃⋊C₈	C₄	1	1	yes	characteristic
10	C₈×C₄₃⋊C₈	C₄	1	1	yes	characteristic
11	C₈×C₄₃⋊C₈	C₂²	1	2	yes	—
12	C₂×M₃₂×C₄₃	C₄	1	2	yes	—
13	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
14	C₄₃×#346	C₄	1	2	yes	—
15	C₄×C₁₆×C₄₃	C₂²	1	2	yes	—
16	C₂²×C₄₃⋊C₈	C₈	1	4	yes	—
17	#13063265	—	2	4	no	—
18	C₂×C₄₃⋊C₁₆	C₂×C₄	1	4	yes	—
19	C₄×C₄₃⋊C₈	C₈	1	4	yes	—
20	C₄×C₄₃⋊C₈	C₂×C₄	1	2	yes	—
21	C₄×C₄₃⋊C₈	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
25	M₃₂×C₄₃	—	2	4	no	—
26	C₂×C₁₆×C₄₃	C₂×C₄	1	4	yes	—
27	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
31	C₂²×C₄×C₄₃	C₄²	1	1	yes	characteristic
32	C₂²×C₄×C₄₃	C₂×C₈	1	2	yes	—
33	C₂×C₈×C₄₃	C₄²	1	1	yes	characteristic
34	C₂×C₈×C₄₃	C₄²	1	1	yes	characteristic
35	C₂×C₈×C₄₃	M₁₆	1	4	yes	—
36	C₂×C₈×C₄₃	M₁₆	1	4	yes	—
37	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
39	C₄²×C₄₃	C₄²	1	1	yes	characteristic
40	C₄²×C₄₃	C₂×C₈	1	2	yes	—
41	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄₃⋊C₈	—	2	8	no	—
43	C₂×C₄₃⋊C₈	C₂×C₈	1	4	yes	—
44	C₂×C₄₃⋊C₈	C₂×C₈	1	4	yes	—
45	C₁₆×C₄₃	—	2	4	no	—
46	C₄₃⋊C₁₆	—	2	4	no	—
47	C₂³×C₄₃	C₄×C₈	1	1	yes	characteristic
48	C₂×C₄×C₄₃	—	2	4	no	—
49	C₂×C₄×C₄₃	C₄×C₈	1	2	yes	—
50	C₂×C₄×C₄₃	C₄×C₈	1	1	yes	characteristic
51	C₂×C₄×C₄₃	#97	1	2	yes	—
52	C₂×C₄×C₄₃	#97	1	2	yes	—
53	C₂×C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
54	C₂×C₄×C₄₃	C₂²×C₈	1	2	yes	—
55	C₈×C₄₃	#97	1	2	yes	—
56	C₈×C₄₃	#97	1	2	yes	—
57	C₈×C₄₃	C₂×M₁₆	1	2	yes	—
58	C₈×C₄₃	C₂×M₁₆	1	2	yes	—
59	C₄₃⋊C₈	—	2	4	no	—
60	C₄₃⋊C₈	M₃₂	1	8	yes	complemented
61	#11402	—	43	1	no	maximal, Sylow
62	C₂²×C₄₃	—	2	2	no	—
63	C₂²×C₄₃	#322	1	2	yes	—
64	C₂²×C₄₃	C₂×C₄×C₈	1	1	yes	characteristic
65	C₄×C₄₃	—	2	2	no	—
66	C₄×C₄₃	#322	1	2	yes	—
67	C₄×C₄₃	C₂×#97	1	1	yes	characteristic
68	C₄×C₄₃	C₂×#97	1	1	yes	characteristic
69	C₄×C₄₃	C₂×M₃₂	1	4	yes	—
70	C₄×M₃₂	—	43	1	no	—
71	C₄×M₃₂	D₄₃	1	1	yes	characteristic
72	C₂×C₈²	—	43	1	no	—
73	#1312	—	43	4	no	—
74	C₂×C₄₃	—	2	2	no	—
75	C₂×C₄₃	C₂×#322	1	1	yes	characteristic
76	C₂×C₄₃	C₄×M₃₂	1	2	yes	—
77	C₂×M₃₂	—	43	2	no	—
78	C₂×M₃₂	Dic₄₃	1	2	yes	—
79	C₂×C₄×C₈	—	43	2	no	—
80	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
81	#346	—	43	2	no	—
82	#346	Dic₄₃	1	2	yes	—
83	C₄×C₁₆	—	43	2	no	—
84	C₄×C₁₆	C₂×D₄₃	1	2	yes	—
85	C₈²	—	43	2	no	—
86	C₈²	—	43	1	no	—
87	C₈²	—	43	1	no	—
88	C₄₃	#11402	1	1	yes	characteristic, complemented, Sylow
89	C₂²×C₈	—	43	4	no	—
90	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
91	C₂²×C₈	C₂×Dic₄₃	1	1	yes	characteristic
92	C₂×C₄²	C₄×D₄₃	1	1	yes	characteristic
93	M₃₂	—	2	4	no	—
94	M₃₂	—	86	4	no	—
95	C₂×C₁₆	—	43	4	no	—
96	C₂×C₁₆	C₂×Dic₄₃	1	4	yes	—
97	C₄×C₈	—	43	4	no	—
98	C₄×C₈	—	43	2	no	—
99	C₄×C₈	—	43	2	no	—
100	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
101	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
102	C₄×C₈	C₂×Dic₄₃	1	1	yes	characteristic
103	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
104	C₂²×C₄	C₈×D₄₃	1	2	yes	—
105	C₂²×C₄	C₄×Dic₄₃	1	1	yes	characteristic
106	C₂×C₈	—	43	4	no	—
107	C₂×C₈	—	43	4	no	—
108	C₂×C₈	—	86	8	no	—
109	C₂×C₈	#10621872	1	4	yes	—
110	C₂×C₈	#10621877	1	4	yes	—
111	C₂×C₈	C₄×Dic₄₃	1	1	yes	characteristic
112	C₂×C₈	C₄×Dic₄₃	1	1	yes	characteristic
113	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
114	C₂×C₈	C₂²×Dic₄₃	1	1	yes	characteristic
115	C₄²	C₈×D₄₃	1	2	yes	—
116	C₄²	C₄×Dic₄₃	1	1	yes	characteristic
117	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
118	C₁₆	—	2	4	no	—
119	C₁₆	—	86	4	no	—
120	C₂³	C₈×Dic₄₃	1	1	yes	characteristic
121	C₂×C₄	—	2	4	no	—
122	C₂×C₄	C₈×Dic₄₃	1	2	yes	—
123	C₂×C₄	C₈×Dic₄₃	1	1	yes	characteristic
124	C₂×C₄	#13063235	1	2	yes	—
125	C₂×C₄	#13063236	1	2	yes	—
126	C₂×C₄	C₂×C₄×Dic₄₃	1	1	yes	characteristic
127	C₂×C₄	C₂×C₈×D₄₃	1	2	yes	—
128	C₈	—	43	8	no	—
129	C₈	—	86	4	no	—
130	C₈	#13063235	1	2	yes	—
131	C₈	#13063236	1	2	yes	—
132	C₈	C₂×#10621872	1	2	yes	—
133	C₈	C₂×#10621877	1	2	yes	—
134	C₂²	—	2	2	no	—
135	C₂²	#14582621	1	2	yes	—
136	C₂²	C₂×C₈×Dic₄₃	1	1	yes	characteristic
137	C₄	—	2	2	no	—
138	C₄	#14582621	1	2	yes	—
139	C₄	C₂×#13063235	1	1	yes	characteristic
140	C₄	C₂×#13063236	1	1	yes	characteristic
141	C₄	M₃₂×D₄₃	1	4	yes	—
142	C₂	—	2	2	no	—
143	C₂	C₂×#14582621	1	1	yes	characteristic
144	C₂	M₃₂×Dic₄₃	1	2	yes	—

	subgroup	G/H
center, Frattini	C₄×C₈	C₂²×D₄₃
commutator, socle	C₂²×C₄₃	C₂×C₄×C₈
Fitting	C₄×M₃₂×C₄₃	C₂
prime radical	C₂³×C₄₃	C₄×C₈
cosocle	C₄×C₈×C₄₃	C₂³
hypercenter	C₄×M₃₂	D₄₃
hypocenter	C₄₃	#11402
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
3. #19132514	C₂	semidirect
60. C₄₃⋊C₈	M₃₂	semidirect
88. C₄₃	#11402	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
C₂×C₈×C₄₃⋊C₈	2
#19132514	2
C₄×M₃₂×C₄₃	2
#11402	43