Subgroups — #24987829

Order 11008.

446 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13063259	C₂	1	1	yes	maximal, characteristic
2	#19134460	C₂	1	2	yes	maximal
3	C₂×C₈×C₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#19132434	C₂	1	2	yes	maximal
5	C₂×C₄₃×#335	C₂	1	1	yes	maximal, characteristic
6	C₂×#13063259	C₄	1	2	yes	—
7	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₈×Dic₄₃	—	2	2	no	—
10	C₄×C₄⋊Dic₄₃	C₂²	1	2	yes	—
11	#14582662	C₄	1	2	yes	—
12	#14582653	—	2	2	no	—
13	C₈×C₄₃⋊C₈	C₄	1	4	yes	complemented
14	C₂×C₄⋊C₈×C₄₃	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
16	C₄₃×#335	C₄	1	4	yes	complemented
17	C₂²×C₄₃⋊C₈	—	2	1	no	—
18	C₂²×C₄₃⋊C₈	D₄	1	1	yes	characteristic
19	C₂²×C₄₃⋊C₈	D₄	1	1	yes	characteristic
20	C₂×C₄⋊Dic₄₃	C₂×C₄	1	4	yes	—
21	C₂×C₄×Dic₄₃	—	2	2	no	—
22	#13063259	C₈	1	8	yes	complemented
23	C₄×C₄₃⋊C₈	—	2	1	no	—
24	C₄×C₄₃⋊C₈	C₂×C₄	1	2	yes	—
25	C₄×C₄₃⋊C₈	C₂×C₄	1	2	yes	—
26	C₄×C₄₃⋊C₈	Q₈	1	2	yes	—
27	C₈×Dic₄₃	—	4	4	no	—
28	C₄₃⋊#95	—	2	2	no	—
29	C₂²×C₈×C₄₃	—	2	2	no	—
30	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
33	C₄⋊C₈×C₄₃	—	2	2	no	—
34	C₄⋊C₈×C₄₃	C₂×C₄	1	4	yes	—
35	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
36	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
37	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
40	C₂²×Dic₄₃	—	2	4	no	—
41	C₂×C₈×C₄₃	—	2	2	no	—
42	C₂×C₈×C₄₃	—	2	4	no	—
43	C₂×C₈×C₄₃	—	4	2	no	—
44	C₂×C₈×C₄₃	—	4	2	no	—
45	C₂×C₈×C₄₃	C₄²	1	2	yes	—
46	C₂×C₈×C₄₃	C₄²	1	2	yes	—
47	C₂×C₈×C₄₃	C₂×C₈	1	4	yes	—
48	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
49	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
50	C₄²×C₄₃	C₂×Q₈	1	1	yes	characteristic
51	C₄²×C₄₃	#41	1	1	yes	characteristic
52	C₄⋊Dic₄₃	—	2	4	no	—
53	C₄⋊Dic₄₃	C₂×C₈	1	8	yes	—
54	C₄×Dic₄₃	—	4	2	no	—
55	C₄×Dic₄₃	—	4	2	no	—
56	C₂×C₄₃⋊C₈	—	2	4	no	—
57	C₂×C₄₃⋊C₈	—	2	2	no	—
58	C₂×C₄₃⋊C₈	—	2	1	no	—
59	C₂×C₄₃⋊C₈	—	2	1	no	—
60	C₂×C₄₃⋊C₈	C₄⋊C₄	1	2	yes	—
61	C₂×C₄₃⋊C₈	C₄⋊C₄	1	2	yes	—
62	C₂×C₄₃⋊C₈	D₈	1	2	yes	—
63	C₂×C₄₃⋊C₈	Q₁₆	1	2	yes	—
64	C₂³×C₄₃	C₄×D₄	1	1	yes	characteristic
65	C₂×C₄×C₄₃	—	2	2	no	—
66	C₂×C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
67	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
69	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₄₃	C₄×Q₈	1	1	yes	characteristic
71	C₂×C₄×C₄₃	C₄×Q₈	1	1	yes	characteristic
72	C₂×C₄×C₄₃	C₂²×C₈	1	2	yes	—
73	C₂×C₄×C₄₃	#131	1	2	yes	—
74	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
75	C₂×C₄×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
76	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
77	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
78	C₂×Dic₄₃	—	2	8	no	—
79	C₂×Dic₄₃	—	4	4	no	—
80	C₂×Dic₄₃	—	4	4	no	—
81	C₈×C₄₃	—	2	2	no	—
82	C₈×C₄₃	—	4	4	no	—
83	C₈×C₄₃	C₄×C₈	1	4	yes	complemented
84	C₄₃⋊C₈	—	2	4	no	—
85	C₄₃⋊C₈	—	2	2	no	—
86	C₄₃⋊C₈	#107	1	4	yes	complemented
87	C₈×#107	—	43	1	no	maximal, Sylow
88	C₂²×C₄₃	C₄×C₄⋊C₄	1	1	yes	characteristic
89	C₂²×C₄₃	C₈×D₄	1	2	yes	—
90	C₂²×C₄₃	C₄×D₈	1	1	yes	characteristic
91	C₂²×C₄₃	C₄×Q₁₆	1	1	yes	characteristic
92	C₂²×C₄₃	#443	1	2	yes	—
93	C₄×C₄₃	—	2	1	no	—
94	C₄×C₄₃	—	2	1	no	—
95	C₄×C₄₃	C₂×C₄×C₈	1	1	yes	characteristic
96	C₄×C₄₃	#405	1	1	yes	characteristic
97	C₄×C₄₃	C₂×#107	1	1	yes	characteristic
98	C₄×C₄₃	#427	1	1	yes	characteristic
99	C₄×C₄₃	C₄×D₈	1	1	yes	characteristic
100	C₄×C₄₃	C₄×Q₁₆	1	1	yes	characteristic
101	C₄×C₄₃	C₈×Q₈	1	2	yes	—
102	Dic₄₃	—	4	8	no	—
103	C₄×#107	—	43	1	no	—
104	C₈×C₄⋊C₄	—	43	2	no	—
105	C₂×#335	D₄₃	1	1	yes	characteristic
106	C₂×C₈²	—	43	1	no	—
107	#1279	—	43	2	no	—
108	C₂×C₄₃	C₈×D₈	1	2	yes	—
109	C₂×C₄₃	C₈×Q₁₆	1	2	yes	—
110	C₂×C₄₃	C₈×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄₃	#1772	1	1	yes	characteristic
112	C₂×C₄₃	C₄×#107	1	1	yes	characteristic
113	C₂×#107	—	43	2	no	—
114	C₂×C₄⋊C₈	C₂×D₄₃	1	2	yes	—
115	C₂×C₄×C₈	—	43	1	no	—
116	C₂×C₄×C₈	—	43	1	no	—
117	C₂×C₄×C₈	—	86	2	no	—
118	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
119	C₄×C₄⋊C₄	—	43	2	no	—
120	#340	—	43	2	no	—
121	#336	—	86	2	no	—
122	#335	Dic₄₃	1	4	yes	complemented
123	C₈²	—	43	4	no	—
124	C₄₃	C₈×#107	1	1	yes	characteristic, complemented, Sylow
125	C₂²×C₈	—	2	2	no	—
126	C₂²×C₈	—	43	1	no	—
127	C₂²×C₈	—	43	1	no	—
128	C₂²×C₈	—	86	1	no	—
129	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
130	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
131	C₂×C₄⋊C₄	—	43	4	no	—
132	C₂×C₄²	—	86	2	no	—
133	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
134	#107	—	43	8	no	—
135	C₄⋊C₈	—	2	2	no	—
136	C₄⋊C₈	C₂×Dic₄₃	1	4	yes	—
137	C₄×C₈	—	43	2	no	—
138	C₄×C₈	—	43	2	no	—
139	C₄×C₈	—	43	2	no	—
140	C₄×C₈	—	86	1	no	—
141	C₄×C₈	—	172	4	no	—
142	C₄×C₈	C₄×D₄₃	1	2	yes	—
143	C₄×C₈	C₂×Dic₄₃	1	2	yes	—
144	#95	—	86	2	no	—
145	C₂²×C₄	—	86	4	no	—
146	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
147	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
148	C₂²×C₄	#10621902	1	1	yes	characteristic
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	2	4	no	—
151	C₂×C₈	—	4	2	no	—
152	C₂×C₈	—	4	2	no	—
153	C₂×C₈	—	43	2	no	—
154	C₂×C₈	—	43	2	no	—
155	C₂×C₈	—	43	2	no	—
156	C₂×C₈	—	43	2	no	—
157	C₂×C₈	—	86	4	no	—
158	C₂×C₈	—	86	2	no	—
159	C₂×C₈	—	86	1	no	—
160	C₂×C₈	—	86	1	no	—
161	C₂×C₈	C₈×D₄₃	1	4	yes	—
162	C₂×C₈	C₄×Dic₄₃	1	2	yes	—
163	C₂×C₈	C₄×Dic₄₃	1	2	yes	—
164	C₄⋊C₄	—	43	8	no	—
165	C₄⋊C₄	—	86	4	no	—
166	C₄²	—	172	2	no	—
167	C₄²	—	172	2	no	—
168	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
169	C₄²	#10621900	1	1	yes	characteristic
170	C₄²	Q₈×D₄₃	1	1	yes	characteristic
171	C₄²	C₂²×Dic₄₃	1	1	yes	characteristic
172	C₂³	#13063287	1	1	yes	characteristic
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	86	8	no	—
175	C₂×C₄	—	172	4	no	—
176	C₂×C₄	—	172	4	no	—
177	C₂×C₄	C₂×C₄×Dic₄₃	1	1	yes	characteristic
178	C₂×C₄	C₄⋊C₄×D₄₃	1	1	yes	characteristic
179	C₂×C₄	#13063283	1	1	yes	characteristic
180	C₂×C₄	D₄×Dic₄₃	1	1	yes	characteristic
181	C₂×C₄	#13063290	1	1	yes	characteristic
182	C₂×C₄	Q₈×Dic₄₃	1	1	yes	characteristic
183	C₂×C₄	C₂×C₈×D₄₃	1	2	yes	—
184	C₂×C₄	#13063333	1	2	yes	—
185	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
186	C₂×C₄	Q₁₆×D₄₃	1	1	yes	characteristic
187	C₂×C₄	#13063344	1	1	yes	characteristic
188	C₂×C₄	#13063348	1	1	yes	characteristic
189	C₈	—	2	2	no	—
190	C₈	—	4	4	no	—
191	C₈	—	43	4	no	—
192	C₈	—	86	4	no	—
193	C₈	—	86	2	no	—
194	C₈	C₈×Dic₄₃	1	4	yes	complemented
195	C₂²	C₄⋊C₄×Dic₄₃	1	1	yes	characteristic
196	C₂²	#14582964	1	2	yes	—
197	C₂²	#14582981	1	1	yes	characteristic
198	C₂²	#14582993	1	1	yes	characteristic
199	C₂²	#14583015	1	2	yes	—
200	C₄	—	2	1	no	—
201	C₄	—	2	1	no	—
202	C₄	—	172	8	no	—
203	C₄	C₂×C₈×Dic₄₃	1	1	yes	characteristic
204	C₄	#14582857	1	1	yes	characteristic
205	C₄	#107×D₄₃	1	1	yes	characteristic
206	C₄	#14582936	1	1	yes	characteristic
207	C₄	D₈×Dic₄₃	1	1	yes	characteristic
208	C₄	Q₁₆×Dic₄₃	1	1	yes	characteristic
209	C₄	#14583023	1	2	yes	—
210	C₂	#19133389	1	2	yes	—
211	C₂	#19133401	1	2	yes	—
212	C₂	#19134460	1	1	yes	characteristic
213	C₂	#19134477	1	1	yes	characteristic
214	C₂	#107×Dic₄₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₄₃
commutator	C₄×C₄₃	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₂×C₄₃×#335	C₂
socle, prime radical	C₂³×C₄₃	C₄×D₄
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	C₂×#335	D₄₃
hypocenter	C₄₃	C₈×#107
LCS	G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
13. C₈×C₄₃⋊C₈	C₄	semidirect
16. C₄₃×#335	C₄	semidirect
22. #13063259	C₈	semidirect
83. C₈×C₄₃	C₄×C₈	semidirect
86. C₄₃⋊C₈	#107	semidirect
122. #335	Dic₄₃	semidirect
124. C₄₃	C₈×#107	semidirect
194. C₈	C₈×Dic₄₃	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13063259	2
#19134460	2
C₂×C₈×C₄₃⋊C₈	2
#19132434	2
C₂×C₄₃×#335	2
C₈×#107	43