Subgroups — #24983790

Order 11008.

222 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₄₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	#19134767	C₂	1	1	yes	maximal, characteristic
3	C₄₃×#1564	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄₃⋊C₁₆	—	2	1	no	—
5	C₂×#13063261	—	2	1	no	—
6	C₂×#13063261	C₄	1	2	yes	—
7	#14582923	—	2	1	no	—
8	C₄⋊(C₄₃⋊C₁₆)	—	2	1	no	—
9	C₄⋊(C₄₃⋊C₁₆)	C₄	1	2	yes	—
10	C₄₃×#431	—	2	1	no	—
11	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
12	C₄₃×#335	—	2	1	no	—
13	C₄₃×#334	C₄	1	2	yes	—
14	C₂×#10621877	—	2	2	no	—
15	C₂×C₄₃⋊C₁₆	—	2	2	no	—
16	C₂×C₄₃⋊C₁₆	—	4	1	no	—
17	C₂×C₄₃⋊C₁₆	—	4	1	no	—
18	#13063261	—	4	1	no	—
19	#13063261	—	4	1	no	—
20	#13063261	—	4	2	no	—
21	C₄⋊(C₄₃⋊C₈)	—	4	1	no	—
22	C₄⋊(C₄₃⋊C₈)	—	4	1	no	—
23	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
26	C₄⋊C₈×C₄₃	—	2	2	no	—
27	C₂²⋊C₈×C₄₃	—	4	1	no	—
28	C₄×C₈×C₄₃	—	4	1	no	—
29	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
32	C₄×C₈×C₄₃	Q₈	1	1	yes	characteristic
33	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₄₃	SD₁₆	1	1	yes	characteristic
35	C₂²×C₄×C₄₃	Q₁₆	1	1	yes	characteristic
36	C₂×C₈×C₄₃	—	2	1	no	—
37	C₂×C₈×C₄₃	—	2	1	no	—
38	C₂×C₈×C₄₃	—	4	2	no	—
39	C₂×C₈×C₄₃	C₄²	1	1	yes	characteristic
40	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₄₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₄₃	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₄₃	D₈	1	1	yes	characteristic
44	C₂×C₈×C₄₃	SD₁₆	1	1	yes	characteristic
45	C₂×C₈×C₄₃	SD₁₆	1	1	yes	characteristic
46	C₂×C₈×C₄₃	Q₁₆	1	1	yes	characteristic
47	C₄²×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₄²×C₄₃	C₄⋊C₄	1	1	yes	characteristic
49	C₄²×C₄₃	D₈	1	1	yes	characteristic
50	C₄²×C₄₃	SD₁₆	1	1	yes	characteristic
51	#10621877	—	4	2	no	—
52	#10621877	—	4	2	no	—
53	C₂×C₄₃⋊C₈	—	4	2	no	—
54	C₄₃⋊C₁₆	—	4	2	no	—
55	C₂³×C₄₃	#103	1	1	yes	characteristic
56	C₂×C₄×C₄₃	—	2	2	no	—
57	C₂×C₄×C₄₃	—	4	1	no	—
58	C₂×C₄×C₄₃	#95	1	1	yes	characteristic
59	C₂×C₄×C₄₃	#100	1	1	yes	characteristic
60	C₂×C₄×C₄₃	#101	1	1	yes	characteristic
61	C₂×C₄×C₄₃	#102	1	1	yes	characteristic
62	C₂×C₄×C₄₃	#102	1	1	yes	characteristic
63	C₂×C₄×C₄₃	#102	1	1	yes	characteristic
64	C₂×C₄×C₄₃	#103	1	1	yes	characteristic
65	C₂×C₄×C₄₃	#103	1	1	yes	characteristic
66	C₂×C₄×C₄₃	#106	1	1	yes	characteristic
67	C₂×C₄×C₄₃	#107	1	1	yes	characteristic
68	C₈×C₄₃	—	2	1	no	—
69	C₈×C₄₃	—	2	1	no	—
70	C₈×C₄₃	—	8	2	no	—
71	C₈×C₄₃	#102	1	1	yes	characteristic
72	C₈×C₄₃	#103	1	1	yes	characteristic
73	C₈×C₄₃	#106	1	1	yes	characteristic
74	C₈×C₄₃	#107	1	1	yes	characteristic
75	C₄₃⋊C₈	—	8	2	no	—
76	#7276	—	43	1	no	maximal, Sylow
77	C₂²×C₄₃	—	2	1	no	—
78	C₂²×C₄₃	—	2	1	no	—
79	C₂²×C₄₃	#332	1	1	yes	characteristic
80	C₂²×C₄₃	#333	1	1	yes	characteristic
81	C₂²×C₄₃	#340	1	1	yes	characteristic
82	C₄×C₄₃	—	2	1	no	—
83	C₄×C₄₃	—	2	1	no	—
84	C₄×C₄₃	—	2	1	no	—
85	C₄×C₄₃	#331	1	1	yes	characteristic
86	C₄×C₄₃	#332	1	1	yes	characteristic
87	C₄×C₄₃	#340	1	1	yes	characteristic
88	C₄×C₄₃	#343	1	1	yes	characteristic
89	C₄×C₄₃	#368	1	2	yes	—
90	C₂×C₄⋊C₁₆	—	43	1	no	—
91	#1848	—	43	1	no	—
92	#1564	D₄₃	1	1	yes	characteristic
93	C₂×C₄₃	—	2	1	no	—
94	C₂×C₄₃	—	2	1	no	—
95	C₂×C₄₃	#1296	1	1	yes	characteristic
96	C₂×C₄₃	#1381	1	1	yes	characteristic
97	C₂×C₄₃	#1389	1	1	yes	characteristic
98	C₂²×C₁₆	—	86	1	no	—
99	#431	—	2	1	no	—
100	C₂×#108	—	43	2	no	—
101	C₂×#108	—	86	1	no	—
102	#423	—	86	1	no	—
103	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
104	C₄⋊C₁₆	—	43	2	no	—
105	C₄⋊C₁₆	—	86	1	no	—
106	#335	—	2	1	no	—
107	#334	Dic₄₃	1	2	yes	—
108	C₄₃	#7276	1	1	yes	characteristic, complemented, Sylow
109	C₂×M₁₆	—	86	2	no	—
110	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
111	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
112	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
113	C₂×C₁₆	—	86	2	no	—
114	C₂×C₁₆	—	172	1	no	—
115	C₂×C₁₆	—	172	1	no	—
116	#108	—	172	2	no	—
117	#108	—	172	1	no	—
118	#108	—	172	1	no	—
119	C₄⋊C₈	—	2	2	no	—
120	C₄⋊C₈	—	172	1	no	—
121	C₄⋊C₈	—	172	1	no	—
122	C₂²⋊C₈	—	4	1	no	—
123	C₄×C₈	—	4	1	no	—
124	C₄×C₈	Dic₈₆	1	1	yes	characteristic
125	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
126	C₄×C₈	C₂×Dic₄₃	1	1	yes	characteristic
127	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
128	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
129	C₂²×C₄	#10621883	1	1	yes	characteristic
130	C₂²×C₄	#10621885	1	1	yes	characteristic
131	M₁₆	—	172	2	no	—
132	M₁₆	—	172	2	no	—
133	C₂×C₈	—	2	1	no	—
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	4	2	no	—
136	C₂×C₈	—	172	2	no	—
137	C₂×C₈	#10621873	1	1	yes	characteristic
138	C₂×C₈	#10621873	1	1	yes	characteristic
139	C₂×C₈	D₃₄₄	1	1	yes	characteristic
140	C₂×C₈	Dic₁₇₂	1	1	yes	characteristic
141	C₂×C₈	C₄×Dic₄₃	1	1	yes	characteristic
142	C₂×C₈	Dic₄₃⋊C₄	1	1	yes	characteristic
143	C₂×C₈	C₄⋊Dic₄₃	1	1	yes	characteristic
144	C₂×C₈	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
145	C₄²	Dic₄₃⋊C₄	1	1	yes	characteristic
146	C₄²	#10621882	1	1	yes	characteristic
147	C₄²	#10621884	1	1	yes	characteristic
148	C₄²	C₂²⋊Dic₄₃	1	1	yes	characteristic
149	C₁₆	—	172	2	no	—
150	C₂³	#13063248	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	4	1	no	—
153	C₂×C₄	C₄₃⋊#95	1	1	yes	characteristic
154	C₂×C₄	#13063241	1	1	yes	characteristic
155	C₂×C₄	#13063243	1	1	yes	characteristic
156	C₂×C₄	#13063245	1	1	yes	characteristic
157	C₂×C₄	#13063246	1	1	yes	characteristic
158	C₂×C₄	#13063247	1	1	yes	characteristic
159	C₂×C₄	#13063249	1	1	yes	characteristic
160	C₂×C₄	#13063250	1	1	yes	characteristic
161	C₂×C₄	#13063256	1	1	yes	characteristic
162	C₂×C₄	#13063258	1	1	yes	characteristic
163	C₈	—	2	1	no	—
164	C₈	—	2	1	no	—
165	C₈	—	8	2	no	—
166	C₈	—	344	2	no	—
167	C₈	#13063247	1	1	yes	characteristic
168	C₈	#13063250	1	1	yes	characteristic
169	C₈	#13063257	1	1	yes	characteristic
170	C₈	#13063259	1	1	yes	characteristic
171	C₂²	—	2	1	no	—
172	C₂²	—	2	1	no	—
173	C₂²	#14582644	1	1	yes	characteristic
174	C₂²	#14582647	1	1	yes	characteristic
175	C₂²	#14582661	1	1	yes	characteristic
176	C₄	—	2	1	no	—
177	C₄	—	2	1	no	—
178	C₄	—	2	1	no	—
179	C₄	#14582642	1	1	yes	characteristic
180	C₄	#14582646	1	1	yes	characteristic
181	C₄	#14582662	1	1	yes	characteristic
182	C₄	#14582667	1	1	yes	characteristic
183	C₄	#14582728	1	2	yes	—
184	C₂	—	2	1	no	—
185	C₂	—	2	1	no	—
186	C₂	#19132479	1	1	yes	characteristic
187	C₂	#19132688	1	1	yes	characteristic
188	C₂	#19132711	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13063245
commutator	C₂×C₈×C₄₃	C₄²
Frattini	C₂×C₄×C₈	C₂×D₄₃
Fitting	C₄₃×#1564	C₂
socle	C₂²×C₄₃	#340
prime radical	C₂³×C₄₃	#103
cosocle	C₂×C₄×C₈×C₄₃	C₂²
hypercenter	#1564	D₄₃
hypocenter	C₄₃	#7276
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1564

Complements

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

N	H ≅ G/N	type
108. C₄₃	#7276	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₄₃⋊C₁₆)	2
#19134767	2
C₄₃×#1564	2
#7276	43