Subgroups — C₄₃×#14272 #24934289

Order 11008.

2782 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#127×C₄₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₃×#390	C₂	1	4	yes	maximal, complemented
3	C₂×C₄₃×#382	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄₃×#376	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₃×#1438	C₂	1	8	yes	maximal, complemented
6	C₂³×D₄×C₄₃	C₂²	1	2	yes	—
7	C₂³×D₄×C₄₃	C₂²	1	4	yes	—
8	C₂×#127×C₄₃	—	2	4	no	—
9	C₂×#127×C₄₃	—	2	4	no	—
10	C₂×#127×C₄₃	C₄	1	8	yes	complemented
11	C₂²×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
12	C₂²×C₂²⋊C₄×C₄₃	C₂²	1	16	yes	—
13	C₂²×C₄²×C₄₃	C₂²	1	2	yes	—
14	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
15	C₄₃×#390	—	2	16	no	—
16	C₄₃×#382	—	2	4	no	—
17	C₄₃×#376	—	2	4	no	—
18	C₂×#95×C₄₃	C₂²	1	4	yes	—
19	C₂×#95×C₄₃	C₂²	1	4	yes	—
20	C₂⁵×C₄₃	C₂³	1	4	yes	—
21	C₂²×D₄×C₄₃	—	2	8	no	—
22	C₂²×D₄×C₄₃	—	2	8	no	—
23	C₂²×D₄×C₄₃	—	2	4	no	—
24	C₂²×D₄×C₄₃	—	2	4	no	—
25	C₂²×D₄×C₄₃	—	2	8	no	—
26	C₂²×D₄×C₄₃	—	2	8	no	—
27	C₂²×D₄×C₄₃	—	4	16	no	—
28	C₂²×D₄×C₄₃	C₂×C₄	1	8	yes	—
29	C₂²×D₄×C₄₃	C₂×C₄	1	16	yes	—
30	C₂³×C₄×C₄₃	C₂³	1	4	yes	—
31	C₂³×C₄×C₄₃	C₂³	1	4	yes	—
32	C₂³×C₄×C₄₃	C₂³	1	2	yes	—
33	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
34	#127×C₄₃	—	2	32	no	—
35	C₂×C₄⋊C₄×C₄₃	—	2	4	no	—
36	C₂×C₄⋊C₄×C₄₃	—	2	8	no	—
37	C₂×C₂²⋊C₄×C₄₃	—	2	32	no	—
38	C₂×C₂²⋊C₄×C₄₃	—	4	32	no	—
39	C₂×C₄²×C₄₃	—	2	4	no	—
40	C₂×C₄²×C₄₃	—	2	4	no	—
41	C₂×C₄²×C₄₃	C₂×C₄	1	4	yes	—
42	C₂×C₄²×C₄₃	D₄	1	4	yes	—
43	C₂×C₄²×C₄₃	D₄	1	4	yes	—
44	C₂×C₄²×C₄₃	D₄	1	8	yes	complemented
45	#95×C₄₃	—	2	8	no	—
46	#95×C₄₃	—	2	8	no	—
47	C₂⁴×C₄₃	—	2	8	no	—
48	C₂⁴×C₄₃	—	2	4	no	—
49	C₂⁴×C₄₃	—	2	4	no	—
50	C₂⁴×C₄₃	—	2	8	no	—
51	C₂⁴×C₄₃	—	2	32	no	—
52	C₂⁴×C₄₃	C₂²×C₄	1	8	yes	—
53	C₂⁴×C₄₃	C₂⁴	1	1	yes	characteristic
54	C₂×D₄×C₄₃	—	2	32	no	—
55	C₂×D₄×C₄₃	—	2	16	no	—
56	C₂×D₄×C₄₃	—	2	32	no	—
57	C₂×D₄×C₄₃	—	4	32	no	—
58	C₂×D₄×C₄₃	—	4	32	no	—
59	C₂×D₄×C₄₃	—	4	32	no	—
60	C₂×D₄×C₄₃	—	4	16	no	—
61	C₂×D₄×C₄₃	—	4	8	no	—
62	C₂×D₄×C₄₃	—	4	8	no	—
63	C₂²×C₄×C₄₃	—	2	4	no	—
64	C₂²×C₄×C₄₃	—	2	4	no	—
65	C₂²×C₄×C₄₃	—	2	8	no	—
66	C₂²×C₄×C₄₃	—	2	4	no	—
67	C₂²×C₄×C₄₃	—	2	8	no	—
68	C₂²×C₄×C₄₃	—	2	16	no	—
69	C₂²×C₄×C₄₃	—	2	16	no	—
70	C₂²×C₄×C₄₃	C₂²×C₄	1	4	yes	—
71	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
72	C₂²×C₄×C₄₃	C₂×D₄	1	8	yes	—
73	C₂²×C₄×C₄₃	C₂×D₄	1	4	yes	—
74	C₂²×C₄×C₄₃	C₂×D₄	1	4	yes	—
75	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₄₃	C₂×D₄	1	4	yes	—
78	C₂²×C₄×C₄₃	#41	1	4	yes	—
79	C₄⋊C₄×C₄₃	—	4	8	no	—
80	C₂²⋊C₄×C₄₃	—	4	64	no	—
81	C₄²×C₄₃	—	2	16	no	—
82	C₄²×C₄₃	C₂²⋊C₄	1	16	yes	complemented
83	C₂³×C₄₃	—	2	8	no	—
84	C₂³×C₄₃	—	2	16	no	—
85	C₂³×C₄₃	—	2	32	no	—
86	C₂³×C₄₃	—	4	8	no	—
87	C₂³×C₄₃	—	4	4	no	—
88	C₂³×C₄₃	—	4	4	no	—
89	C₂³×C₄₃	—	4	16	no	—
90	C₂³×C₄₃	—	4	16	no	—
91	C₂³×C₄₃	—	4	16	no	—
92	C₂³×C₄₃	—	4	16	no	—
93	C₂³×C₄₃	—	4	32	no	—
94	C₂³×C₄₃	C₂³×C₄	1	1	yes	characteristic
95	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
96	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
97	C₂³×C₄₃	C₂²×D₄	1	2	yes	—
98	C₂³×C₄₃	C₂×#41	1	2	yes	—
99	C₂³×C₄₃	H(5,𝔽₂)	1	8	yes	—
100	D₄×C₄₃	—	4	64	no	—
101	D₄×C₄₃	—	4	32	no	—
102	C₂×C₄×C₄₃	—	2	8	no	—
103	C₂×C₄×C₄₃	—	2	8	no	—
104	C₂×C₄×C₄₃	—	2	8	no	—
105	C₂×C₄×C₄₃	—	2	8	no	—
106	C₂×C₄×C₄₃	—	2	16	no	—
107	C₂×C₄×C₄₃	—	4	8	no	—
108	C₂×C₄×C₄₃	—	4	8	no	—
109	C₂×C₄×C₄₃	—	4	4	no	—
110	C₂×C₄×C₄₃	—	4	16	no	—
111	C₂×C₄×C₄₃	—	4	8	no	—
112	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	8	yes	—
113	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	4	yes	—
114	C₂×C₄×C₄₃	C₄×D₄	1	8	yes	—
115	C₂×C₄×C₄₃	#121	1	8	yes	—
116	C₂×C₄×C₄₃	#124	1	4	yes	—
117	C₂×C₄×C₄₃	#127	1	4	yes	—
118	#14272	C₄₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
119	C₂²×C₄₃	—	4	16	no	—
120	C₂²×C₄₃	—	4	32	no	—
121	C₂²×C₄₃	—	4	8	no	—
122	C₂²×C₄₃	—	8	16	no	—
123	C₂²×C₄₃	—	8	8	no	—
124	C₂²×C₄₃	—	8	8	no	—
125	C₂²×C₄₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
126	C₂²×C₄₃	C₂×C₄×D₄	1	2	yes	—
127	C₂²×C₄₃	#518	1	4	yes	—
128	C₂²×C₄₃	C₂×#121	1	2	yes	—
129	C₂²×C₄₃	C₂×#124	1	1	yes	characteristic
130	C₂²×C₄₃	C₂×#127	1	1	yes	characteristic
131	C₂²×C₄₃	#535	1	4	yes	—
132	C₂²×C₄₃	#535	1	4	yes	—
133	C₂²×C₄₃	#535	1	8	yes	—
134	C₂²×C₄₃	#540	1	8	yes	—
135	C₄×C₄₃	—	2	8	no	—
136	C₄×C₄₃	—	4	8	no	—
137	C₄×C₄₃	—	8	4	no	—
138	C₄×C₄₃	#390	1	8	yes	—
139	C₂²×#127	C₂×C₄₃	1	1	yes	characteristic
140	C₂×#390	C₂×C₄₃	1	4	yes	complemented
141	C₂×#382	C₂×C₄₃	1	1	yes	characteristic, complemented
142	C₂×#376	C₂×C₄₃	1	1	yes	characteristic, complemented
143	#1438	C₂×C₄₃	1	8	yes	complemented
144	C₂×C₄₃	—	8	8	no	—
145	C₂×C₄₃	C₂×#390	1	1	yes	characteristic
146	C₂×C₄₃	#2309	1	2	yes	—
147	C₂×C₄₃	#2328	1	4	yes	—
148	C₂×C₄₃	#2619	1	4	yes	—
149	C₂×C₄₃	#2641	1	2	yes	—
150	C₂×C₄₃	#2657	1	2	yes	—
151	C₂³×D₄	C₂²×C₄₃	1	2	yes	—
152	C₂³×D₄	C₂²×C₄₃	1	4	yes	—
153	C₂×#127	—	2	4	no	—
154	C₂×#127	—	2	4	no	—
155	C₂×#127	C₄×C₄₃	1	8	yes	complemented
156	C₂²×C₄⋊C₄	C₂²×C₄₃	1	2	yes	—
157	C₂²×C₂²⋊C₄	C₂²×C₄₃	1	16	yes	—
158	C₂²×C₄²	C₂²×C₄₃	1	2	yes	—
159	C₂²×C₄²	C₂²×C₄₃	1	1	yes	characteristic
160	#390	—	2	16	no	—
161	#382	—	2	4	no	—
162	#376	—	2	4	no	—
163	C₂×#95	C₂²×C₄₃	1	4	yes	—
164	C₂×#95	C₂²×C₄₃	1	4	yes	—
165	C₄₃	#14272	1	1	yes	characteristic, direct product factor, Sylow
166	C₂⁵	C₂³×C₄₃	1	4	yes	—
167	C₂²×D₄	—	2	8	no	—
168	C₂²×D₄	—	2	8	no	—
169	C₂²×D₄	—	2	4	no	—
170	C₂²×D₄	—	2	4	no	—
171	C₂²×D₄	—	2	8	no	—
172	C₂²×D₄	—	2	8	no	—
173	C₂²×D₄	—	4	16	no	—
174	C₂²×D₄	C₂×C₄×C₄₃	1	8	yes	—
175	C₂²×D₄	C₂×C₄×C₄₃	1	16	yes	—
176	C₂³×C₄	C₂³×C₄₃	1	4	yes	—
177	C₂³×C₄	C₂³×C₄₃	1	4	yes	—
178	C₂³×C₄	C₂³×C₄₃	1	2	yes	—
179	C₂³×C₄	C₂³×C₄₃	1	1	yes	characteristic
180	#127	—	2	32	no	—
181	C₂×C₄⋊C₄	—	2	4	no	—
182	C₂×C₄⋊C₄	—	2	8	no	—
183	C₂×C₂²⋊C₄	—	2	32	no	—
184	C₂×C₂²⋊C₄	—	4	32	no	—
185	C₂×C₄²	—	2	4	no	—
186	C₂×C₄²	—	2	4	no	—
187	C₂×C₄²	C₂×C₄×C₄₃	1	4	yes	—
188	C₂×C₄²	D₄×C₄₃	1	4	yes	—
189	C₂×C₄²	D₄×C₄₃	1	4	yes	—
190	C₂×C₄²	D₄×C₄₃	1	8	yes	complemented
191	#95	—	2	8	no	—
192	#95	—	2	8	no	—
193	C₂⁴	—	2	8	no	—
194	C₂⁴	—	2	4	no	—
195	C₂⁴	—	2	4	no	—
196	C₂⁴	—	2	8	no	—
197	C₂⁴	—	2	32	no	—
198	C₂⁴	C₂²×C₄×C₄₃	1	8	yes	—
199	C₂⁴	C₂⁴×C₄₃	1	1	yes	characteristic
200	C₂×D₄	—	2	32	no	—
201	C₂×D₄	—	2	16	no	—
202	C₂×D₄	—	2	32	no	—
203	C₂×D₄	—	4	32	no	—
204	C₂×D₄	—	4	32	no	—
205	C₂×D₄	—	4	32	no	—
206	C₂×D₄	—	4	16	no	—
207	C₂×D₄	—	4	8	no	—
208	C₂×D₄	—	4	8	no	—
209	C₂²×C₄	—	2	8	no	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	—	2	16	no	—
213	C₂²×C₄	—	2	4	no	—
214	C₂²×C₄	—	2	8	no	—
215	C₂²×C₄	—	2	16	no	—
216	C₂²×C₄	C₂²×C₄×C₄₃	1	4	yes	—
217	C₂²×C₄	C₂²×C₄×C₄₃	1	2	yes	—
218	C₂²×C₄	C₂×D₄×C₄₃	1	8	yes	—
219	C₂²×C₄	C₂×D₄×C₄₃	1	4	yes	—
220	C₂²×C₄	C₂×D₄×C₄₃	1	4	yes	—
221	C₂²×C₄	C₂×D₄×C₄₃	1	2	yes	—
222	C₂²×C₄	C₂×D₄×C₄₃	1	2	yes	—
223	C₂²×C₄	C₂×D₄×C₄₃	1	4	yes	—
224	C₂²×C₄	#41×C₄₃	1	4	yes	—
225	C₄⋊C₄	—	4	8	no	—
226	C₂²⋊C₄	—	4	64	no	—
227	C₄²	—	2	16	no	—
228	C₄²	C₂²⋊C₄×C₄₃	1	16	yes	complemented
229	C₂³	—	2	8	no	—
230	C₂³	—	2	16	no	—
231	C₂³	—	2	32	no	—
232	C₂³	—	4	8	no	—
233	C₂³	—	4	4	no	—
234	C₂³	—	4	4	no	—
235	C₂³	—	4	16	no	—
236	C₂³	—	4	16	no	—
237	C₂³	—	4	16	no	—
238	C₂³	—	4	16	no	—
239	C₂³	—	4	32	no	—
240	C₂³	C₂³×C₄×C₄₃	1	1	yes	characteristic
241	C₂³	C₂²×D₄×C₄₃	1	1	yes	characteristic
242	C₂³	C₂²×D₄×C₄₃	1	1	yes	characteristic
243	C₂³	C₂²×D₄×C₄₃	1	2	yes	—
244	C₂³	C₂×#41×C₄₃	1	2	yes	—
245	C₂³	H(5,𝔽₂)×C₄₃	1	8	yes	—
246	D₄	—	4	64	no	—
247	D₄	—	4	32	no	—
248	C₂×C₄	—	2	8	no	—
249	C₂×C₄	—	2	8	no	—
250	C₂×C₄	—	2	8	no	—
251	C₂×C₄	—	2	8	no	—
252	C₂×C₄	—	2	16	no	—
253	C₂×C₄	—	4	8	no	—
254	C₂×C₄	—	4	8	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	—	4	16	no	—
257	C₂×C₄	—	4	8	no	—
258	C₂×C₄	C₂×C₂²⋊C₄×C₄₃	1	8	yes	—
259	C₂×C₄	C₂×C₂²⋊C₄×C₄₃	1	4	yes	—
260	C₂×C₄	C₄×D₄×C₄₃	1	8	yes	—
261	C₂×C₄	#121×C₄₃	1	8	yes	—
262	C₂×C₄	#124×C₄₃	1	4	yes	—
263	C₂×C₄	#127×C₄₃	1	4	yes	—
264	C₂²	—	4	32	no	—
265	C₂²	—	4	8	no	—
266	C₂²	—	4	16	no	—
267	C₂²	—	8	16	no	—
268	C₂²	—	8	8	no	—
269	C₂²	—	8	8	no	—
270	C₂²	C₂²×C₂²⋊C₄×C₄₃	1	1	yes	characteristic
271	C₂²	C₂×C₄×D₄×C₄₃	1	2	yes	—
272	C₂²	C₄₃×#518	1	4	yes	—
273	C₂²	C₂×#121×C₄₃	1	2	yes	—
274	C₂²	C₂×#124×C₄₃	1	1	yes	characteristic
275	C₂²	C₂×#127×C₄₃	1	1	yes	characteristic
276	C₂²	C₄₃×#535	1	8	yes	—
277	C₂²	C₄₃×#535	1	4	yes	—
278	C₂²	C₄₃×#535	1	4	yes	—
279	C₂²	C₄₃×#540	1	8	yes	—
280	C₄	—	2	8	no	—
281	C₄	—	4	8	no	—
282	C₄	—	8	4	no	—
283	C₄	C₄₃×#390	1	8	yes	—
284	C₂	—	8	8	no	—
285	C₂	C₂×C₄₃×#390	1	1	yes	characteristic
286	C₂	C₄₃×#2309	1	2	yes	—
287	C₂	C₄₃×#2328	1	4	yes	—
288	C₂	C₄₃×#2619	1	4	yes	—
289	C₂	C₄₃×#2641	1	2	yes	—
290	C₂	C₄₃×#2657	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₄₃	C₂⁴
commutator	C₂³	C₂³×C₄×C₄₃
Frattini, cosocle	C₂⁴	C₂⁴×C₄₃
prime radical	C₂²×#127×C₄₃	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₄₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄₃×#390	C₂	semidirect
3. C₂×C₄₃×#382	C₂	semidirect
4. C₂×C₄₃×#376	C₂	semidirect
5. C₄₃×#1438	C₂	semidirect
10. C₂×#127×C₄₃	C₄	semidirect
44. C₂×C₄²×C₄₃	D₄	semidirect
82. C₄²×C₄₃	C₂²⋊C₄	semidirect
118. #14272	C₄₃	direct (N × H)
140. C₂×#390	C₂×C₄₃	semidirect
141. C₂×#382	C₂×C₄₃	semidirect
142. C₂×#376	C₂×C₄₃	semidirect
143. #1438	C₂×C₄₃	semidirect
155. C₂×#127	C₄×C₄₃	semidirect
165. C₄₃	#14272	direct (N × H)
190. C₂×C₄²	D₄×C₄₃	semidirect
228. C₄²	C₂²⋊C₄×C₄₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#127×C₄₃	2
C₂×C₄₃×#390	2
C₂×C₄₃×#382	2
C₂×C₄₃×#376	2
C₄₃×#1438	2
#14272	43

Subgroups — C43×#14272 #24934289

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₃×#14272 #24934289