Subgroups — #24990372

Order 11008.

422 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19134754	C₂	1	2	yes	maximal
2	#19134723	C₂	1	1	yes	maximal, characteristic
3	#19134722	C₂	1	2	yes	maximal
4	C₂×C₈×C₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₄₃×#1834	C₂	1	1	yes	maximal, characteristic
6	C₂×#13063258	—	2	4	no	—
7	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
9	#14582769	C₂²	1	2	yes	—
10	#14582662	—	2	2	no	—
11	#14582661	—	2	4	no	—
12	C₈×C₄₃⋊C₈	C₄	1	4	yes	complemented
13	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
14	C₄₃×#384	C₂²	1	2	yes	—
15	C₄₃×#340	—	2	2	no	—
16	C₂²×C₄₃⋊C₈	D₄	1	2	yes	—
17	C₂²×C₄₃⋊C₈	D₄	1	2	yes	—
18	C₂×C₄⋊Dic₄₃	—	2	2	no	—
19	C₂×C₄⋊Dic₄₃	—	2	2	no	—
20	C₂×C₄⋊Dic₄₃	—	2	2	no	—
21	#13063258	—	4	8	no	—
22	C₄×C₄₃⋊C₈	C₂×C₄	1	2	yes	—
23	C₄×C₄₃⋊C₈	C₂×C₄	1	2	yes	—
24	C₄×C₄₃⋊C₈	Q₈	1	2	yes	—
25	C₄×C₄₃⋊C₈	Q₈	1	2	yes	—
26	C₂²×C₈×C₄₃	D₄	1	2	yes	—
27	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
28	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
29	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
30	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
31	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₄₃	Q₈	1	2	yes	—
33	C₂²×C₄×C₄₃	—	4	2	no	—
34	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
36	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
37	C₂²×Dic₄₃	—	4	2	no	—
38	C₂×C₈×C₄₃	—	2	4	no	—
39	C₂×C₈×C₄₃	C₄⋊C₄	1	4	yes	—
40	C₄⋊C₄×C₄₃	—	4	4	no	—
41	C₄⋊C₄×C₄₃	—	4	4	no	—
42	C₄⋊C₄×C₄₃	—	4	4	no	—
43	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
44	C₄²×C₄₃	C₂×Q₈	1	1	yes	characteristic
45	C₄²×C₄₃	C₂×Q₈	1	1	yes	characteristic
46	C₄²×C₄₃	C₂×Q₈	1	1	yes	characteristic
47	C₄⋊Dic₄₃	—	4	4	no	—
48	C₄⋊Dic₄₃	—	4	4	no	—
49	C₄⋊Dic₄₃	—	4	4	no	—
50	C₂×C₄₃⋊C₈	—	2	4	no	—
51	C₂×C₄₃⋊C₈	C₄⋊C₄	1	4	yes	—
52	C₂×C₄₃⋊C₈	C₄⋊C₄	1	4	yes	—
53	C₂×C₄₃⋊C₈	D₈	1	4	yes	—
54	C₂×C₄₃⋊C₈	Q₁₆	1	4	yes	—
55	C₂³×C₄₃	#127	1	1	yes	characteristic
56	C₂×C₄×C₄₃	—	8	2	no	—
57	C₂×C₄×C₄₃	—	8	2	no	—
58	C₂×C₄×C₄₃	—	8	2	no	—
59	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
63	C₂×C₄×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
64	C₂×C₄×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
66	C₂×C₄×C₄₃	C₂×Q₁₆	1	2	yes	—
67	C₂×C₄×C₄₃	#135	1	4	yes	—
68	C₂×C₄×C₄₃	#135	1	4	yes	—
69	C₂×Dic₄₃	—	8	2	no	—
70	C₂×Dic₄₃	—	8	2	no	—
71	C₂×Dic₄₃	—	8	2	no	—
72	C₈×C₄₃	—	2	4	no	—
73	C₄₃⋊C₈	—	2	4	no	—
74	C₄₃⋊C₈	#107	1	8	yes	—
75	#8816	—	43	1	no	maximal, Sylow
76	C₂²×C₄₃	#384	1	1	yes	characteristic
77	C₂²×C₄₃	#493	1	1	yes	characteristic
78	C₂²×C₄₃	#494	1	1	yes	characteristic
79	C₂²×C₄₃	#495	1	2	yes	—
80	C₂²×C₄₃	#495	1	2	yes	—
81	C₄×C₄₃	—	16	2	no	—
82	C₄×C₄₃	C₂×#107	1	2	yes	—
83	C₄×C₄₃	#427	1	2	yes	—
84	C₄×C₄₃	#427	1	2	yes	—
85	C₄×C₄₃	#499	1	2	yes	—
86	C₄×C₄₃	#499	1	2	yes	—
87	C₄×C₄₃	#500	1	2	yes	—
88	Dic₄₃	—	16	2	no	—
89	#1845	—	43	2	no	—
90	#1834	—	43	1	no	—
91	#1834	—	43	2	no	—
92	#1834	D₄₃	1	1	yes	characteristic
93	C₂×C₈²	—	43	1	no	—
94	C₂×C₄₃	#1709	1	2	yes	—
95	C₂×C₄₃	#1710	1	2	yes	—
96	C₂×C₄₃	#1845	1	1	yes	characteristic
97	C₂×C₄₃	#1846	1	1	yes	characteristic
98	C₂×C₄₃	#1846	1	1	yes	characteristic
99	C₂×#107	—	86	4	no	—
100	C₂×C₄×C₈	—	43	1	no	—
101	C₂×C₄×C₈	—	43	1	no	—
102	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
103	#384	—	43	2	no	—
104	#384	C₂×D₄₃	1	2	yes	—
105	#340	—	2	2	no	—
106	#340	—	86	4	no	—
107	#340	—	86	2	no	—
108	C₈²	—	43	4	no	—
109	C₄₃	#8816	1	1	yes	characteristic, complemented, Sylow
110	C₂²×C₈	—	43	2	no	—
111	C₂²×C₈	—	43	2	no	—
112	C₂²×C₈	C₂²⋊D₄₃	1	2	yes	—
113	C₂×C₄⋊C₄	—	2	2	no	—
114	C₂×C₄⋊C₄	—	2	2	no	—
115	C₂×C₄⋊C₄	—	2	2	no	—
116	C₂×C₄⋊C₄	—	86	2	no	—
117	C₂×C₄⋊C₄	—	86	2	no	—
118	C₂×C₄⋊C₄	—	86	2	no	—
119	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
120	#107	—	172	8	no	—
121	C₄×C₈	—	43	2	no	—
122	C₄×C₈	—	43	2	no	—
123	C₄×C₈	—	43	2	no	—
124	C₄×C₈	—	43	2	no	—
125	C₄×C₈	Dic₈₆	1	2	yes	—
126	C₄×C₈	C₄×D₄₃	1	2	yes	—
127	C₂²×C₄	—	4	2	no	—
128	C₂²×C₄	—	172	2	no	—
129	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
130	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
131	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
132	C₂×C₈	—	2	4	no	—
133	C₂×C₈	—	43	4	no	—
134	C₂×C₈	—	43	4	no	—
135	C₂×C₈	—	43	4	no	—
136	C₂×C₈	—	43	4	no	—
137	C₂×C₈	—	86	4	no	—
138	C₂×C₈	Dic₄₃⋊C₄	1	4	yes	—
139	C₄⋊C₄	—	4	4	no	—
140	C₄⋊C₄	—	4	4	no	—
141	C₄⋊C₄	—	4	4	no	—
142	C₄⋊C₄	—	172	4	no	—
143	C₄⋊C₄	—	172	4	no	—
144	C₄⋊C₄	—	172	4	no	—
145	C₄²	C₂×Dic₈₆	1	1	yes	characteristic
146	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
147	C₄²	Q₈×D₄₃	1	1	yes	characteristic
148	C₄²	Q₈×D₄₃	1	1	yes	characteristic
149	C₂³	#13063321	1	1	yes	characteristic
150	C₂×C₄	—	8	2	no	—
151	C₂×C₄	—	8	2	no	—
152	C₂×C₄	—	8	2	no	—
153	C₂×C₄	—	344	2	no	—
154	C₂×C₄	—	344	2	no	—
155	C₂×C₄	—	344	2	no	—
156	C₂×C₄	C₄⋊C₄×D₄₃	1	1	yes	characteristic
157	C₂×C₄	C₄⋊C₄×D₄₃	1	1	yes	characteristic
158	C₂×C₄	C₂×Dic₄₃⋊C₄	1	1	yes	characteristic
159	C₂×C₄	#13063323	1	1	yes	characteristic
160	C₂×C₄	#13063324	1	1	yes	characteristic
161	C₂×C₄	#13063324	1	1	yes	characteristic
162	C₂×C₄	D₈×D₄₃	1	2	yes	—
163	C₂×C₄	Q₁₆×D₄₃	1	2	yes	—
164	C₂×C₄	#13063345	1	4	yes	—
165	C₂×C₄	#13063346	1	4	yes	—
166	C₈	—	2	4	no	—
167	C₈	—	43	8	no	—
168	C₈	—	86	4	no	—
169	C₂²	#14582768	1	1	yes	characteristic
170	C₂²	#14583274	1	1	yes	characteristic
171	C₂²	#14583277	1	1	yes	characteristic
172	C₂²	#14583280	1	2	yes	—
173	C₂²	#14583281	1	2	yes	—
174	C₄	—	16	2	no	—
175	C₄	—	688	2	no	—
176	C₄	#107×D₄₃	1	2	yes	—
177	C₄	#14582934	1	2	yes	—
178	C₄	#14582935	1	2	yes	—
179	C₄	#14583298	1	2	yes	—
180	C₄	#14583299	1	2	yes	—
181	C₄	#14583303	1	2	yes	—
182	C₂	#19134199	1	2	yes	—
183	C₂	#19134203	1	2	yes	—
184	C₂	#19134753	1	1	yes	characteristic
185	C₂	#19134756	1	1	yes	characteristic
186	C₂	#19134757	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13063321
commutator	C₄²×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1834	C₂
socle, prime radical	C₂³×C₄₃	#127
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#1834	D₄₃
hypocenter	C₄₃	#8816
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1834

Complements

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

N	H ≅ G/N	type
12. C₈×C₄₃⋊C₈	C₄	semidirect
109. C₄₃	#8816	semidirect

Maximal subgroups

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

subgroup	index
#19134754	2
#19134723	2
#19134722	2
C₂×C₈×C₄₃⋊C₈	2
C₄₃×#1834	2
#8816	43