Subgroups — #24987717

Order 11008.

550 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Dic₄₃⋊C₈	C₂	1	2	yes	maximal
2	C₄×C₈×Dic₄₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×C₄₃⋊C₈	C₂	1	2	yes	maximal
4	C₄₃⋊#1273	C₂	1	1	yes	maximal, characteristic
5	C₄₃×#1273	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₄₃⋊C₈	C₄	1	4	yes	—
7	C₂×C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
8	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₈×Dic₄₃	C₄	1	2	yes	—
10	C₂×C₈×Dic₄₃	C₄	1	2	yes	—
11	C₄²×Dic₄₃	C₂²	1	1	yes	characteristic
12	#14582653	C₄	1	4	yes	—
13	C₈×C₄₃⋊C₈	—	2	4	no	—
14	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₄₃	C₂²	1	2	yes	—
16	C₂²×C₄₃⋊C₈	C₂×C₄	1	4	yes	—
17	C₂²×C₄₃⋊C₈	C₂×C₄	1	2	yes	—
18	C₂×C₄×Dic₄₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×Dic₄₃	C₂×C₄	1	2	yes	—
20	C₂×C₄×Dic₄₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₄₃	C₂×C₄	1	2	yes	—
22	Dic₄₃⋊C₈	C₈	1	16	yes	complemented
23	C₄×C₄₃⋊C₈	—	2	2	no	—
24	C₄×C₄₃⋊C₈	—	2	2	no	—
25	C₄×C₄₃⋊C₈	D₄	1	2	yes	—
26	C₄×C₄₃⋊C₈	Q₈	1	2	yes	—
27	C₈×Dic₄₃	—	2	4	no	—
28	C₈×Dic₄₃	C₈	1	8	yes	complemented
29	C₂²×C₈×C₄₃	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₄₃	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
32	C₄×C₈×C₄₃	—	2	2	no	—
33	C₄×C₈×C₄₃	—	2	2	no	—
34	C₄×C₈×C₄₃	—	2	2	no	—
35	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
37	C₂²×Dic₄₃	C₄²	1	2	yes	—
38	C₂²×Dic₄₃	C₄²	1	1	yes	characteristic
39	C₂²×Dic₄₃	C₄²	1	1	yes	characteristic
40	C₂×C₈×C₄₃	—	2	2	no	—
41	C₂×C₈×C₄₃	—	2	4	no	—
42	C₂×C₈×C₄₃	—	2	2	no	—
43	C₂×C₈×C₄₃	—	2	4	no	—
44	C₂×C₈×C₄₃	C₂×C₈	1	4	yes	—
45	C₂×C₈×C₄₃	C₂×C₈	1	8	yes	—
46	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
47	C₄²×C₄₃	C₂×Q₈	1	1	yes	characteristic
48	C₄²×C₄₃	#41	1	2	yes	—
49	C₄×Dic₄₃	—	2	2	no	—
50	C₄×Dic₄₃	—	2	4	no	—
51	C₄×Dic₄₃	C₂×C₈	1	4	yes	—
52	C₄×Dic₄₃	C₂×C₈	1	8	yes	—
53	C₂×C₄₃⋊C₈	—	2	4	no	—
54	C₂×C₄₃⋊C₈	—	2	4	no	—
55	C₂×C₄₃⋊C₈	C₄⋊C₄	1	4	yes	—
56	C₂×C₄₃⋊C₈	C₂×C₈	1	8	yes	—
57	C₂×C₄₃⋊C₈	C₂×C₈	1	4	yes	—
58	C₂×C₄₃⋊C₈	M₁₆	1	4	yes	—
59	C₂³×C₄₃	C₂×C₄²	1	1	yes	characteristic
60	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
62	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
63	C₂×C₄×C₄₃	C₄×Q₈	1	2	yes	—
64	C₂×C₄×C₄₃	C₂²×C₈	1	2	yes	—
65	C₂×C₄×C₄₃	C₂²×C₈	1	4	yes	—
66	C₂×C₄×C₄₃	C₂×M₁₆	1	2	yes	—
67	C₂×C₄×C₄₃	#131	1	4	yes	—
68	C₂×Dic₄₃	—	2	2	no	—
69	C₂×Dic₄₃	—	2	2	no	—
70	C₂×Dic₄₃	—	2	2	no	—
71	C₂×Dic₄₃	C₄×C₈	1	4	yes	—
72	C₂×Dic₄₃	C₄×C₈	1	2	yes	—
73	C₂×Dic₄₃	C₄×C₈	1	4	yes	—
74	C₂×Dic₄₃	C₄×C₈	1	2	yes	—
75	C₈×C₄₃	—	2	4	no	—
76	C₈×C₄₃	—	2	8	no	—
77	C₄₃⋊C₈	—	2	8	no	—
78	C₄₃⋊C₈	C₄⋊C₈	1	8	yes	complemented
79	C₈×C₄⋊C₈	—	43	1	no	maximal, Sylow
80	C₂²×C₄₃	C₄×C₄⋊C₄	1	1	yes	characteristic
81	C₂²×C₄₃	C₂×C₄×C₈	1	2	yes	—
82	C₂²×C₄₃	C₂×C₄×C₈	1	1	yes	characteristic
83	C₂²×C₄₃	C₄×M₁₆	1	1	yes	characteristic
84	C₂²×C₄₃	#405	1	2	yes	—
85	C₄×C₄₃	C₂×C₄⋊C₈	1	2	yes	—
86	C₄×C₄₃	#431	1	2	yes	—
87	C₄×C₄₃	C₈×D₄	1	4	yes	—
88	C₄×C₄₃	C₈×Q₈	1	4	yes	—
89	Dic₄₃	—	2	4	no	—
90	Dic₄₃	—	2	2	no	—
91	Dic₄₃	C₈²	1	4	yes	complemented
92	C₄×C₄⋊C₈	—	43	2	no	—
93	C₄²×C₈	—	43	1	no	—
94	C₂×C₈²	—	43	2	no	—
95	#1273	—	43	1	no	—
96	#1273	D₄₃	1	1	yes	characteristic
97	C₂×C₄₃	C₂×C₈²	1	1	yes	characteristic
98	C₂×C₄₃	C₈×M₁₆	1	2	yes	—
99	C₂×C₄₃	#1450	1	1	yes	characteristic
100	C₂×C₄₃	C₄×C₄⋊C₈	1	1	yes	characteristic
101	C₂×C₄₃	C₈×C₄⋊C₄	1	2	yes	—
102	C₂×C₄⋊C₈	—	43	4	no	—
103	C₂×C₄×C₈	—	43	2	no	—
104	C₂×C₄×C₈	—	43	1	no	—
105	C₂×C₄×C₈	—	43	2	no	—
106	C₂×C₄×C₈	—	43	2	no	—
107	C₂×C₄×C₈	C₂×D₄₃	1	2	yes	—
108	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
109	C₄³	—	43	1	no	—
110	#336	—	43	4	no	—
111	C₈²	—	86	4	no	—
112	C₄₃	C₈×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
113	C₂²×C₈	—	43	2	no	—
114	C₂²×C₈	—	43	4	no	—
115	C₂²×C₈	C₄×D₄₃	1	4	yes	—
116	C₂²×C₈	C₄×D₄₃	1	2	yes	—
117	C₂×C₄²	—	43	1	no	—
118	C₂×C₄²	—	43	2	no	—
119	C₂×C₄²	—	43	2	no	—
120	C₂×C₄²	—	43	1	no	—
121	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
122	C₄⋊C₈	—	43	16	no	—
123	C₄×C₈	—	2	2	no	—
124	C₄×C₈	—	2	2	no	—
125	C₄×C₈	—	2	2	no	—
126	C₄×C₈	—	43	8	no	—
127	C₄×C₈	—	43	2	no	—
128	C₄×C₈	—	43	2	no	—
129	C₄×C₈	—	86	4	no	—
130	C₄×C₈	—	86	2	no	—
131	C₄×C₈	—	86	2	no	—
132	C₂²×C₄	—	43	1	no	—
133	C₂²×C₄	—	43	2	no	—
134	C₂²×C₄	—	43	1	no	—
135	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
136	C₂²×C₄	C₂×C₄×D₄₃	1	2	yes	—
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	—	2	4	no	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	2	4	no	—
141	C₂×C₈	—	43	4	no	—
142	C₂×C₈	—	43	4	no	—
143	C₂×C₈	—	43	4	no	—
144	C₂×C₈	—	43	8	no	—
145	C₂×C₈	—	86	4	no	—
146	C₂×C₈	—	86	4	no	—
147	C₂×C₈	C₈×D₄₃	1	4	yes	—
148	C₂×C₈	C₈×D₄₃	1	8	yes	—
149	C₄²	—	43	4	no	—
150	C₄²	—	43	8	no	—
151	C₄²	—	86	2	no	—
152	C₄²	—	86	4	no	—
153	C₄²	D₄×D₄₃	1	1	yes	characteristic
154	C₄²	#10621900	1	2	yes	—
155	C₄²	Q₈×D₄₃	1	1	yes	characteristic
156	C₂³	C₄²×D₄₃	1	1	yes	characteristic
157	C₂×C₄	—	43	4	no	—
158	C₂×C₄	—	43	2	no	—
159	C₂×C₄	—	43	4	no	—
160	C₂×C₄	—	43	2	no	—
161	C₂×C₄	—	86	2	no	—
162	C₂×C₄	—	86	2	no	—
163	C₂×C₄	—	86	2	no	—
164	C₂×C₄	C₄⋊C₄×D₄₃	1	1	yes	characteristic
165	C₂×C₄	#13063281	1	1	yes	characteristic
166	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	2	yes	—
167	C₂×C₄	#13063290	1	2	yes	—
168	C₂×C₄	C₂×C₈×D₄₃	1	2	yes	—
169	C₂×C₄	C₂×C₈×D₄₃	1	4	yes	—
170	C₂×C₄	M₁₆×D₄₃	1	2	yes	—
171	C₂×C₄	#13063333	1	4	yes	—
172	C₈	—	2	4	no	—
173	C₈	—	2	8	no	—
174	C₈	—	43	8	no	—
175	C₈	—	86	8	no	—
176	C₂²	#14582750	1	1	yes	characteristic
177	C₂²	C₄×C₈×D₄₃	1	2	yes	—
178	C₂²	C₄×C₈×D₄₃	1	1	yes	characteristic
179	C₂²	#14582851	1	1	yes	characteristic
180	C₂²	#14582856	1	2	yes	—
181	C₄	—	43	4	no	—
182	C₄	—	86	4	no	—
183	C₄	—	86	2	no	—
184	C₄	C₄⋊C₈×D₄₃	1	2	yes	—
185	C₄	#14582949	1	2	yes	—
186	C₄	(C₂²⋊D₄₃)⋊C₈	1	4	yes	—
187	C₄	#14583023	1	4	yes	—
188	C₂	C₈²×D₄₃	1	1	yes	characteristic
189	C₂	#19132874	1	2	yes	—
190	C₂	#19132880	1	1	yes	characteristic
191	C₂	#19134443	1	1	yes	characteristic
192	C₂	#19134456	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₄₃
commutator	C₂×C₄₃	C₂×C₈²
Fitting	C₄₃×#1273	C₂
socle, prime radical	C₂³×C₄₃	C₂×C₄²
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#1273	D₄₃
hypocenter	C₄₃	C₈×C₄⋊C₈
LCS	G ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄² ◁ #1273

Complements

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

N	H ≅ G/N	type
22. Dic₄₃⋊C₈	C₈	semidirect
28. C₈×Dic₄₃	C₈	semidirect
78. C₄₃⋊C₈	C₄⋊C₈	semidirect
91. Dic₄₃	C₈²	semidirect
112. C₄₃	C₈×C₄⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×Dic₄₃⋊C₈	2
C₄×C₈×Dic₄₃	2
C₂×C₈×C₄₃⋊C₈	2
C₄₃⋊#1273	2
C₄₃×#1273	2
C₈×C₄⋊C₈	43