Subgroups — #126×C₄₃⋊C₈ #25104811

Order 11008.

786 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19135203	C₂	1	3	yes	maximal
2	#19135154	C₂	1	3	yes	maximal, complemented
3	#19134738	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄⋊C₄×C₄₃⋊C₈	C₂	1	3	yes	maximal, complemented
5	C₂²⋊C₄×C₄₃⋊C₈	C₂	1	3	yes	maximal
6	C₄²×C₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×#126×C₄₃	C₂	1	1	yes	maximal, characteristic
8	C₂³×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	3	yes	—
10	C₂×C₂²⋊(C₄₃⋊C₈)	C₂²	1	3	yes	—
11	C₂×C₄×C₄₃⋊C₈	C₂²	1	3	yes	—
12	C₂×C₄×C₄₃⋊C₈	C₂²	1	3	yes	—
13	C₂×C₄×C₄₃⋊C₈	C₂²	1	3	yes	—
14	#14582654	C₂²	1	3	yes	—
15	#14582654	C₂²	1	3	yes	—
16	#14582654	C₂²	1	3	yes	—
17	#14582654	C₂²	1	3	yes	—
18	C₂×#126×C₄₃	C₄	1	1	yes	characteristic
19	C₄₃×#388	C₄	1	3	yes	—
20	C₄₃×#387	C₄	1	3	yes	—
21	C₄₃×#383	C₄	1	1	yes	characteristic
22	C₄×C₄⋊C₄×C₄₃	C₂²	1	3	yes	—
23	C₄×C₂²⋊C₄×C₄₃	C₂²	1	3	yes	—
24	C₄³×C₄₃	C₂²	1	1	yes	characteristic
25	C₂²×C₄₃⋊C₈	—	2	6	no	—
26	C₂²×C₄₃⋊C₈	C₂³	1	3	yes	—
27	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
28	C₂²×C₄₃⋊C₈	C₂³	1	3	yes	—
29	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
30	C₄⋊(C₄₃⋊C₈)	—	2	6	no	—
31	C₂²⋊(C₄₃⋊C₈)	—	2	6	no	—
32	C₄×C₄₃⋊C₈	—	2	6	no	—
33	C₄×C₄₃⋊C₈	—	2	6	no	—
34	C₄×C₄₃⋊C₈	—	2	6	no	—
35	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
36	#126×C₄₃	C₈	1	8	yes	complemented
37	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	3	yes	—
38	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	3	yes	—
39	C₂×C₂²⋊C₄×C₄₃	C₂×C₄	1	3	yes	—
40	C₂×C₂²⋊C₄×C₄₃	C₂×C₄	1	3	yes	—
41	C₂×C₄²×C₄₃	C₂×C₄	1	3	yes	—
42	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₄₃	C₂³	1	3	yes	—
44	C₂×C₄²×C₄₃	C₂³	1	3	yes	—
45	#95×C₄₃	C₂×C₄	1	3	yes	—
46	#95×C₄₃	C₂×C₄	1	3	yes	—
47	#95×C₄₃	C₂×C₄	1	3	yes	—
48	#95×C₄₃	C₂×C₄	1	3	yes	—
49	C₂⁴×C₄₃	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₄₃	—	2	3	no	—
51	C₂²×C₄×C₄₃	—	2	3	no	—
52	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₃	C₂²×C₄	1	3	yes	—
54	C₂²×C₄×C₄₃	C₂²×C₄	1	3	yes	—
55	C₂²×C₄×C₄₃	C₂²×C₄	1	3	yes	—
56	C₂²×C₄×C₄₃	C₂²×C₄	1	3	yes	—
57	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
58	C₄⋊C₄×C₄₃	—	2	6	no	—
59	C₄⋊C₄×C₄₃	C₂×C₈	1	12	yes	complemented
60	C₂²⋊C₄×C₄₃	—	2	6	no	—
61	C₂²⋊C₄×C₄₃	C₂×C₈	1	12	yes	—
62	C₄²×C₄₃	—	2	6	no	—
63	C₄²×C₄₃	—	2	3	no	—
64	C₄²×C₄₃	—	2	3	no	—
65	C₄²×C₄₃	—	2	3	no	—
66	C₄²×C₄₃	—	2	3	no	—
67	C₄²×C₄₃	C₂×C₈	1	4	yes	complemented
68	C₂×C₄₃⋊C₈	—	2	3	no	—
69	C₂×C₄₃⋊C₈	—	2	3	no	—
70	C₂×C₄₃⋊C₈	—	2	3	no	—
71	C₂×C₄₃⋊C₈	—	2	3	no	—
72	C₂×C₄₃⋊C₈	—	2	3	no	—
73	C₂×C₄₃⋊C₈	—	2	3	no	—
74	C₂×C₄₃⋊C₈	—	4	4	no	—
75	C₂×C₄₃⋊C₈	#41	1	6	yes	—
76	C₂×C₄₃⋊C₈	#41	1	6	yes	—
77	C₂³×C₄₃	—	2	3	no	—
78	C₂³×C₄₃	—	2	3	no	—
79	C₂³×C₄₃	C₂²×C₈	1	2	yes	—
80	C₂³×C₄₃	C₂³×C₄	1	1	yes	characteristic
81	C₂×C₄×C₄₃	—	2	3	no	—
82	C₂×C₄×C₄₃	—	2	3	no	—
83	C₂×C₄×C₄₃	—	2	3	no	—
84	C₂×C₄×C₄₃	—	2	3	no	—
85	C₂×C₄×C₄₃	—	2	3	no	—
86	C₂×C₄×C₄₃	—	2	3	no	—
87	C₂×C₄×C₄₃	—	2	3	no	—
88	C₂×C₄×C₄₃	—	2	3	no	—
89	C₂×C₄×C₄₃	—	2	3	no	—
90	C₂×C₄×C₄₃	—	2	3	no	—
91	C₂×C₄×C₄₃	—	4	1	no	—
92	C₂×C₄×C₄₃	—	4	3	no	—
93	C₂×C₄×C₄₃	C₂²×C₈	1	6	yes	—
94	C₂×C₄×C₄₃	C₂²×C₈	1	6	yes	—
95	C₂×C₄×C₄₃	#131	1	6	yes	—
96	C₂×C₄×C₄₃	C₂×#41	1	3	yes	—
97	C₂×C₄×C₄₃	C₂×#41	1	3	yes	—
98	C₄₃⋊C₈	—	2	6	no	—
99	C₄₃⋊C₈	—	4	1	no	—
100	C₄₃⋊C₈	—	4	3	no	—
101	C₄₃⋊C₈	#126	1	4	yes	direct product factor
102	C₈×#126	—	43	1	no	maximal, Sylow
103	C₂²×C₄₃	—	2	6	no	—
104	C₂²×C₄₃	—	4	1	no	—
105	C₂²×C₄₃	—	4	3	no	—
106	C₂²×C₄₃	C₄×#41	1	3	yes	—
107	C₂²×C₄₃	C₂³×C₈	1	1	yes	characteristic
108	C₂²×C₄₃	C₂×#131	1	3	yes	—
109	C₄×C₄₃	—	2	6	no	—
110	C₄×C₄₃	—	2	6	no	—
111	C₄×C₄₃	—	2	6	no	—
112	C₄×C₄₃	—	4	1	no	—
113	C₄×C₄₃	—	4	3	no	—
114	C₄×C₄₃	C₂×#126	1	1	yes	characteristic
115	C₄×C₄₃	#529	1	3	yes	—
116	C₄×#126	D₄₃	1	1	yes	characteristic
117	#1923	—	43	3	no	—
118	#1916	—	43	3	no	—
119	#1839	—	43	1	no	—
120	C₈×C₄⋊C₄	—	43	3	no	—
121	C₈×C₂²⋊C₄	—	43	3	no	—
122	C₄²×C₈	—	43	1	no	—
123	C₂×C₄₃	—	4	2	no	—
124	C₂×C₄₃	C₄×#126	1	1	yes	characteristic
125	C₂×C₄₃	C₈×#41	1	3	yes	—
126	C₂×C₄₃	#2966	1	3	yes	—
127	C₂³×C₈	—	43	1	no	—
128	C₂×#126	Dic₄₃	1	1	yes	characteristic
129	C₂×C₄⋊C₈	—	43	3	no	—
130	C₂×C₂²⋊C₈	—	43	3	no	—
131	C₂×C₄×C₈	—	43	3	no	—
132	C₂×C₄×C₈	—	43	3	no	—
133	C₂×C₄×C₈	—	43	3	no	—
134	#388	Dic₄₃	1	3	yes	—
135	#387	Dic₄₃	1	3	yes	—
136	#383	Dic₄₃	1	1	yes	characteristic
137	C₄×C₄⋊C₄	C₂×D₄₃	1	3	yes	—
138	C₄×C₂²⋊C₄	C₂×D₄₃	1	3	yes	—
139	C₄³	C₂×D₄₃	1	1	yes	characteristic
140	#336	—	43	3	no	—
141	#336	—	43	3	no	—
142	#336	—	43	3	no	—
143	#336	—	43	3	no	—
144	C₄₃	C₈×#126	1	1	yes	characteristic, complemented, Sylow
145	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
146	C₂²×C₈	—	43	1	no	—
147	C₂²×C₈	—	43	3	no	—
148	C₂²×C₈	—	43	1	no	—
149	C₂²×C₈	—	43	3	no	—
150	C₂²×C₈	—	86	6	no	—
151	#126	C₄₃⋊C₈	1	8	yes	direct product factor
152	C₂×C₄⋊C₄	C₂×Dic₄₃	1	3	yes	—
153	C₂×C₄⋊C₄	C₂×Dic₄₃	1	3	yes	—
154	C₂×C₂²⋊C₄	C₂×Dic₄₃	1	3	yes	—
155	C₂×C₂²⋊C₄	C₂×Dic₄₃	1	3	yes	—
156	C₂×C₄²	C₂×Dic₄₃	1	3	yes	—
157	C₂×C₄²	C₂×Dic₄₃	1	1	yes	characteristic
158	C₂×C₄²	C₂²×D₄₃	1	3	yes	—
159	C₂×C₄²	C₂²×D₄₃	1	3	yes	—
160	C₄⋊C₈	—	86	6	no	—
161	C₂²⋊C₈	—	86	6	no	—
162	C₄×C₈	—	86	6	no	—
163	C₄×C₈	—	86	6	no	—
164	C₄×C₈	—	86	6	no	—
165	#95	C₂×Dic₄₃	1	3	yes	—
166	#95	C₂×Dic₄₃	1	3	yes	—
167	#95	C₂×Dic₄₃	1	3	yes	—
168	#95	C₂×Dic₄₃	1	3	yes	—
169	C₂⁴	C₂²×Dic₄₃	1	1	yes	characteristic
170	C₂²×C₄	—	2	3	no	—
171	C₂²×C₄	—	2	3	no	—
172	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
173	C₂²×C₄	C₂²×Dic₄₃	1	3	yes	—
174	C₂²×C₄	C₂²×Dic₄₃	1	3	yes	—
175	C₂²×C₄	C₂²×Dic₄₃	1	3	yes	—
176	C₂²×C₄	C₂²×Dic₄₃	1	3	yes	—
177	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
178	C₂×C₈	—	43	6	no	—
179	C₂×C₈	—	43	6	no	—
180	C₂×C₈	—	86	3	no	—
181	C₂×C₈	—	86	3	no	—
182	C₂×C₈	—	86	3	no	—
183	C₂×C₈	—	86	3	no	—
184	C₂×C₈	—	86	3	no	—
185	C₂×C₈	—	86	3	no	—
186	C₂×C₈	—	172	4	no	—
187	C₄⋊C₄	—	2	6	no	—
188	C₄⋊C₄	C₂×C₄₃⋊C₈	1	12	yes	complemented
189	C₂²⋊C₄	—	2	6	no	—
190	C₂²⋊C₄	C₂×C₄₃⋊C₈	1	12	yes	—
191	C₄²	—	2	6	no	—
192	C₄²	—	2	3	no	—
193	C₄²	—	2	3	no	—
194	C₄²	—	2	3	no	—
195	C₄²	—	2	3	no	—
196	C₄²	C₂×C₄₃⋊C₈	1	4	yes	complemented
197	C₂³	—	2	3	no	—
198	C₂³	—	2	3	no	—
199	C₂³	C₂²×C₄₃⋊C₈	1	2	yes	—
200	C₂³	C₂³×Dic₄₃	1	1	yes	characteristic
201	C₂×C₄	—	2	3	no	—
202	C₂×C₄	—	2	3	no	—
203	C₂×C₄	—	2	3	no	—
204	C₂×C₄	—	2	3	no	—
205	C₂×C₄	—	2	3	no	—
206	C₂×C₄	—	2	3	no	—
207	C₂×C₄	—	2	3	no	—
208	C₂×C₄	—	2	3	no	—
209	C₂×C₄	—	2	3	no	—
210	C₂×C₄	—	2	3	no	—
211	C₂×C₄	—	4	1	no	—
212	C₂×C₄	—	4	3	no	—
213	C₂×C₄	C₂²×C₄₃⋊C₈	1	6	yes	—
214	C₂×C₄	C₂²×C₄₃⋊C₈	1	6	yes	—
215	C₂×C₄	#13063332	1	6	yes	—
216	C₂×C₄	#41×D₄₃	1	3	yes	—
217	C₂×C₄	#41×D₄₃	1	3	yes	—
218	C₈	—	43	4	no	—
219	C₈	—	86	6	no	—
220	C₈	—	172	1	no	—
221	C₈	—	172	3	no	—
222	C₂²	—	2	6	no	—
223	C₂²	—	4	1	no	—
224	C₂²	—	4	3	no	—
225	C₂²	#41×Dic₄₃	1	3	yes	—
226	C₂²	C₂³×C₄₃⋊C₈	1	1	yes	characteristic
227	C₂²	C₂×#13063332	1	3	yes	—
228	C₄	—	2	6	no	—
229	C₄	—	2	6	no	—
230	C₄	—	2	6	no	—
231	C₄	—	4	1	no	—
232	C₄	—	4	3	no	—
233	C₄	#126×D₄₃	1	1	yes	characteristic
234	C₄	#14583432	1	3	yes	—
235	C₂	—	4	2	no	—
236	C₂	#126×Dic₄₃	1	1	yes	characteristic
237	C₂	#41×C₄₃⋊C₈	1	3	yes	—
238	C₂	#19143571	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₃
commutator	C₂²×C₄₃	C₂³×C₈
Fitting	C₄×#126×C₄₃	C₂
socle	C₂³×C₄₃	C₂³×C₄
prime radical	C₂⁴×C₄₃	C₂²×C₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₄×#126	D₄₃
hypocenter	C₄₃	C₈×#126
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#126

Complements

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

N	H ≅ G/N	type
2. #19135154	C₂	semidirect
3. #19134738	C₂	semidirect
4. C₄⋊C₄×C₄₃⋊C₈	C₂	semidirect
6. C₄²×C₄₃⋊C₈	C₂	semidirect
36. #126×C₄₃	C₈	semidirect
59. C₄⋊C₄×C₄₃	C₂×C₈	semidirect
67. C₄²×C₄₃	C₂×C₈	semidirect
101. C₄₃⋊C₈	#126	direct (N × H)
144. C₄₃	C₈×#126	semidirect
151. #126	C₄₃⋊C₈	direct (N × H)
188. C₄⋊C₄	C₂×C₄₃⋊C₈	semidirect
196. C₄²	C₂×C₄₃⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#19135203	2
#19135154	2
#19134738	2
C₄⋊C₄×C₄₃⋊C₈	2
C₂²⋊C₄×C₄₃⋊C₈	2
C₄²×C₄₃⋊C₈	2
C₄×#126×C₄₃	2
C₈×#126	43

Subgroups — #126×C43⋊C8 #25104811

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #126×C₄₃⋊C₈ #25104811