Subgroups — C₄₃×#17057 #24937074

Order 11008.

1450 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	C₂×C₄₃×#380	C₂	1	2	yes	maximal
2	C₄₃×#1445	C₂	1	4	yes	maximal, complemented
3	C₄₃×#1444	C₂	1	4	yes	maximal, complemented
4	C₄₃×#1439	C₂	1	4	yes	maximal
5	C₄₃×C₂²⋊#95	C₂	1	1	yes	maximal, characteristic
6	C₂⁴×C₄×C₄₃	C₂²	1	1	yes	characteristic
7	C₂²×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
8	C₂²×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
9	C₂²×C₄⋊C₄×C₄₃	C₂²	1	4	yes	—
10	C₂²×C₂²⋊C₄×C₄₃	C₂²	1	4	yes	—
11	C₂²×C₂²⋊C₄×C₄₃	C₂²	1	2	yes	—
12	C₄₃×#380	—	2	8	no	—
13	C₂×#95×C₄₃	C₂²	1	4	yes	—
14	C₂×#95×C₄₃	C₂²	1	4	yes	—
15	C₂×#95×C₄₃	C₂²	1	8	yes	—
16	C₂×#95×C₄₃	C₂²	1	4	yes	—
17	C₂⁵×C₄₃	C₂³	1	1	yes	characteristic
18	C₂³×C₄×C₄₃	—	2	8	no	—
19	C₂³×C₄×C₄₃	—	4	2	no	—
20	C₂³×C₄×C₄₃	D₄	1	4	yes	—
21	C₂³×C₄×C₄₃	C₂³	1	4	yes	—
22	C₂³×C₄×C₄₃	C₂³	1	4	yes	—
23	C₂³×C₄×C₄₃	C₂³	1	4	yes	—
24	C₂³×C₄×C₄₃	C₂³	1	2	yes	—
25	C₂×C₄⋊C₄×C₄₃	—	2	4	no	—
26	C₂×C₄⋊C₄×C₄₃	—	2	4	no	—
27	C₂×C₄⋊C₄×C₄₃	—	2	8	no	—
28	C₂×C₄⋊C₄×C₄₃	—	4	4	no	—
29	C₂×C₄⋊C₄×C₄₃	—	4	8	no	—
30	C₂×C₄⋊C₄×C₄₃	—	4	4	no	—
31	C₂×C₂²⋊C₄×C₄₃	—	2	8	no	—
32	C₂×C₂²⋊C₄×C₄₃	—	4	2	no	—
33	C₂×C₂²⋊C₄×C₄₃	—	4	2	no	—
34	C₂×C₂²⋊C₄×C₄₃	—	4	8	no	—
35	C₂×C₂²⋊C₄×C₄₃	Q₈	1	8	yes	—
36	#95×C₄₃	—	2	8	no	—
37	#95×C₄₃	—	2	8	no	—
38	#95×C₄₃	—	4	4	no	—
39	#95×C₄₃	—	4	8	no	—
40	C₂⁴×C₄₃	—	2	4	no	—
41	C₂⁴×C₄₃	—	2	4	no	—
42	C₂⁴×C₄₃	—	2	1	no	—
43	C₂⁴×C₄₃	—	2	2	no	—
44	C₂⁴×C₄₃	—	2	1	no	—
45	C₂⁴×C₄₃	C₂×D₄	1	2	yes	—
46	C₂⁴×C₄₃	C₂×Q₈	1	4	yes	—
47	C₂⁴×C₄₃	C₂⁴	1	1	yes	characteristic
48	C₂²×C₄×C₄₃	—	2	16	no	—
49	C₂²×C₄×C₄₃	—	2	4	no	—
50	C₂²×C₄×C₄₃	—	2	4	no	—
51	C₂²×C₄×C₄₃	—	2	4	no	—
52	C₂²×C₄×C₄₃	—	2	8	no	—
53	C₂²×C₄×C₄₃	—	2	8	no	—
54	C₂²×C₄×C₄₃	—	2	4	no	—
55	C₂²×C₄×C₄₃	—	2	4	no	—
56	C₂²×C₄×C₄₃	—	2	8	no	—
57	C₂²×C₄×C₄₃	—	2	4	no	—
58	C₂²×C₄×C₄₃	—	2	4	no	—
59	C₂²×C₄×C₄₃	—	2	4	no	—
60	C₂²×C₄×C₄₃	—	2	8	no	—
61	C₂²×C₄×C₄₃	—	2	8	no	—
62	C₂²×C₄×C₄₃	—	2	8	no	—
63	C₂²×C₄×C₄₃	—	2	4	no	—
64	C₂²×C₄×C₄₃	—	2	4	no	—
65	C₂²×C₄×C₄₃	—	4	4	no	—
66	C₂²×C₄×C₄₃	—	4	16	no	—
67	C₂²×C₄×C₄₃	C₂×D₄	1	4	yes	—
68	C₂²×C₄×C₄₃	C₂×Q₈	1	8	yes	—
69	C₂²×C₄×C₄₃	#41	1	8	yes	—
70	C₄⋊C₄×C₄₃	—	4	8	no	—
71	C₄⋊C₄×C₄₃	—	4	16	no	—
72	C₄⋊C₄×C₄₃	—	4	8	no	—
73	C₂²⋊C₄×C₄₃	—	4	8	no	—
74	C₂²⋊C₄×C₄₃	—	4	16	no	—
75	C₂³×C₄₃	—	2	4	no	—
76	C₂³×C₄₃	—	2	2	no	—
77	C₂³×C₄₃	—	2	4	no	—
78	C₂³×C₄₃	—	2	2	no	—
79	C₂³×C₄₃	—	2	4	no	—
80	C₂³×C₄₃	—	2	4	no	—
81	C₂³×C₄₃	—	2	16	no	—
82	C₂³×C₄₃	—	4	4	no	—
83	C₂³×C₄₃	—	4	4	no	—
84	C₂³×C₄₃	—	4	4	no	—
85	C₂³×C₄₃	—	4	4	no	—
86	C₂³×C₄₃	C₄⋊Q₈	1	4	yes	—
87	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
88	C₂³×C₄₃	C₂²×Q₈	1	2	yes	—
89	C₂³×C₄₃	C₂×#41	1	4	yes	—
90	C₂³×C₄₃	H(5,𝔽₂)	1	2	yes	—
91	C₂³×C₄₃	H(5,𝔽₂)	1	4	yes	—
92	C₂³×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
93	C₂³×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
94	C₂×C₄×C₄₃	—	2	4	no	—
95	C₂×C₄×C₄₃	—	2	4	no	—
96	C₂×C₄×C₄₃	—	2	16	no	—
97	C₂×C₄×C₄₃	—	2	8	no	—
98	C₂×C₄×C₄₃	—	2	16	no	—
99	C₂×C₄×C₄₃	—	4	16	no	—
100	C₂×C₄×C₄₃	—	4	8	no	—
101	C₂×C₄×C₄₃	—	4	4	no	—
102	C₂×C₄×C₄₃	—	4	4	no	—
103	C₂×C₄×C₄₃	—	4	4	no	—
104	C₂×C₄×C₄₃	—	4	4	no	—
105	C₂×C₄×C₄₃	—	4	4	no	—
106	C₂×C₄×C₄₃	—	4	4	no	—
107	C₂×C₄×C₄₃	—	4	4	no	—
108	C₂×C₄×C₄₃	—	4	4	no	—
109	C₂×C₄×C₄₃	—	4	8	no	—
110	C₂×C₄×C₄₃	—	4	8	no	—
111	C₂×C₄×C₄₃	—	4	8	no	—
112	C₂×C₄×C₄₃	—	4	8	no	—
113	C₂×C₄×C₄₃	#123	1	8	yes	—
114	#17057	C₄₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
115	C₂²×C₄₃	—	2	4	no	—
116	C₂²×C₄₃	—	2	8	no	—
117	C₂²×C₄₃	—	4	4	no	—
118	C₂²×C₄₃	—	4	2	no	—
119	C₂²×C₄₃	—	4	8	no	—
120	C₂²×C₄₃	—	4	8	no	—
121	C₂²×C₄₃	—	4	2	no	—
122	C₂²×C₄₃	C₂×#123	1	2	yes	—
123	C₂²×C₄₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
124	C₂²×C₄₃	#534	1	1	yes	characteristic
125	C₂²×C₄₃	#535	1	2	yes	—
126	C₂²×C₄₃	#537	1	1	yes	characteristic
127	C₂²×C₄₃	#540	1	4	yes	—
128	C₂²×C₄₃	#542	1	4	yes	—
129	C₂²×C₄₃	#543	1	2	yes	—
130	C₂²×C₄₃	#544	1	2	yes	—
131	C₂²×C₄₃	#551	1	4	yes	—
132	C₂²×C₄₃	#554	1	8	yes	—
133	C₂²×C₄₃	#560	1	4	yes	—
134	C₄×C₄₃	—	2	8	no	—
135	C₄×C₄₃	—	4	8	no	—
136	C₄×C₄₃	—	8	4	no	—
137	C₄×C₄₃	—	8	4	no	—
138	C₂×#380	C₂×C₄₃	1	2	yes	—
139	#1445	C₂×C₄₃	1	4	yes	complemented
140	#1444	C₂×C₄₃	1	4	yes	complemented
141	#1439	C₂×C₄₃	1	4	yes	—
142	C₂²⋊#95	C₂×C₄₃	1	1	yes	characteristic
143	C₂×C₄₃	—	4	4	no	—
144	C₂×C₄₃	#2483	1	1	yes	characteristic
145	C₂×C₄₃	#2502	1	2	yes	—
146	C₂×C₄₃	#2624	1	2	yes	—
147	C₂×C₄₃	#2659	1	1	yes	characteristic
148	C₂×C₄₃	#2742	1	4	yes	—
149	C₂×C₄₃	#2748	1	4	yes	—
150	C₂×C₄₃	#2815	1	1	yes	characteristic
151	C₂⁴×C₄	C₂²×C₄₃	1	1	yes	characteristic
152	C₂²×C₄⋊C₄	C₂²×C₄₃	1	2	yes	—
153	C₂²×C₄⋊C₄	C₂²×C₄₃	1	2	yes	—
154	C₂²×C₄⋊C₄	C₂²×C₄₃	1	4	yes	—
155	C₂²×C₂²⋊C₄	C₂²×C₄₃	1	4	yes	—
156	C₂²×C₂²⋊C₄	C₂²×C₄₃	1	2	yes	—
157	#380	—	2	8	no	—
158	C₂×#95	C₂²×C₄₃	1	4	yes	—
159	C₂×#95	C₂²×C₄₃	1	4	yes	—
160	C₂×#95	C₂²×C₄₃	1	8	yes	—
161	C₂×#95	C₂²×C₄₃	1	4	yes	—
162	C₄₃	#17057	1	1	yes	characteristic, direct product factor, Sylow
163	C₂⁵	C₂³×C₄₃	1	1	yes	characteristic
164	C₂³×C₄	—	2	8	no	—
165	C₂³×C₄	—	4	2	no	—
166	C₂³×C₄	D₄×C₄₃	1	4	yes	—
167	C₂³×C₄	C₂³×C₄₃	1	4	yes	—
168	C₂³×C₄	C₂³×C₄₃	1	4	yes	—
169	C₂³×C₄	C₂³×C₄₃	1	4	yes	—
170	C₂³×C₄	C₂³×C₄₃	1	2	yes	—
171	C₂×C₄⋊C₄	—	2	4	no	—
172	C₂×C₄⋊C₄	—	2	4	no	—
173	C₂×C₄⋊C₄	—	2	8	no	—
174	C₂×C₄⋊C₄	—	4	4	no	—
175	C₂×C₄⋊C₄	—	4	8	no	—
176	C₂×C₄⋊C₄	—	4	4	no	—
177	C₂×C₂²⋊C₄	—	2	8	no	—
178	C₂×C₂²⋊C₄	—	4	2	no	—
179	C₂×C₂²⋊C₄	—	4	2	no	—
180	C₂×C₂²⋊C₄	—	4	8	no	—
181	C₂×C₂²⋊C₄	Q₈×C₄₃	1	8	yes	—
182	#95	—	2	8	no	—
183	#95	—	2	8	no	—
184	#95	—	4	4	no	—
185	#95	—	4	8	no	—
186	C₂⁴	—	2	4	no	—
187	C₂⁴	—	2	4	no	—
188	C₂⁴	—	2	1	no	—
189	C₂⁴	—	2	2	no	—
190	C₂⁴	—	2	1	no	—
191	C₂⁴	C₂×D₄×C₄₃	1	2	yes	—
192	C₂⁴	C₂×Q₈×C₄₃	1	4	yes	—
193	C₂⁴	C₂⁴×C₄₃	1	1	yes	characteristic
194	C₂²×C₄	—	2	4	no	—
195	C₂²×C₄	—	2	4	no	—
196	C₂²×C₄	—	2	8	no	—
197	C₂²×C₄	—	2	4	no	—
198	C₂²×C₄	—	2	8	no	—
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	—	2	16	no	—
201	C₂²×C₄	—	2	4	no	—
202	C₂²×C₄	—	2	4	no	—
203	C₂²×C₄	—	2	8	no	—
204	C₂²×C₄	—	2	4	no	—
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	2	8	no	—
207	C₂²×C₄	—	2	8	no	—
208	C₂²×C₄	—	2	8	no	—
209	C₂²×C₄	—	2	4	no	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	4	4	no	—
212	C₂²×C₄	—	4	16	no	—
213	C₂²×C₄	C₂×D₄×C₄₃	1	4	yes	—
214	C₂²×C₄	C₂×Q₈×C₄₃	1	8	yes	—
215	C₂²×C₄	#41×C₄₃	1	8	yes	—
216	C₄⋊C₄	—	4	8	no	—
217	C₄⋊C₄	—	4	16	no	—
218	C₄⋊C₄	—	4	8	no	—
219	C₂²⋊C₄	—	4	8	no	—
220	C₂²⋊C₄	—	4	16	no	—
221	C₂³	—	2	4	no	—
222	C₂³	—	2	2	no	—
223	C₂³	—	2	4	no	—
224	C₂³	—	2	2	no	—
225	C₂³	—	2	4	no	—
226	C₂³	—	2	4	no	—
227	C₂³	—	2	16	no	—
228	C₂³	—	4	4	no	—
229	C₂³	—	4	4	no	—
230	C₂³	—	4	4	no	—
231	C₂³	—	4	4	no	—
232	C₂³	C₄⋊Q₈×C₄₃	1	4	yes	—
233	C₂³	C₂²×D₄×C₄₃	1	1	yes	characteristic
234	C₂³	C₂²×Q₈×C₄₃	1	2	yes	—
235	C₂³	C₂×#41×C₄₃	1	4	yes	—
236	C₂³	H(5,𝔽₂)×C₄₃	1	2	yes	—
237	C₂³	H(5,𝔽₂)×C₄₃	1	4	yes	—
238	C₂³	H(5,𝔽₂)×C₄₃	1	1	yes	characteristic
239	C₂³	2₋¹⁺⁴×C₄₃	1	1	yes	characteristic
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	16	no	—
243	C₂×C₄	—	2	8	no	—
244	C₂×C₄	—	2	16	no	—
245	C₂×C₄	—	4	16	no	—
246	C₂×C₄	—	4	8	no	—
247	C₂×C₄	—	4	4	no	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	4	8	no	—
256	C₂×C₄	—	4	8	no	—
257	C₂×C₄	—	4	8	no	—
258	C₂×C₄	—	4	8	no	—
259	C₂×C₄	#123×C₄₃	1	8	yes	—
260	C₂²	—	2	4	no	—
261	C₂²	—	2	8	no	—
262	C₂²	—	4	4	no	—
263	C₂²	—	4	2	no	—
264	C₂²	—	4	8	no	—
265	C₂²	—	4	8	no	—
266	C₂²	—	4	2	no	—
267	C₂²	C₂×#123×C₄₃	1	2	yes	—
268	C₂²	C₂×C₄⋊Q₈×C₄₃	1	1	yes	characteristic
269	C₂²	C₄₃×#534	1	1	yes	characteristic
270	C₂²	C₄₃×#535	1	2	yes	—
271	C₂²	C₄₃×#537	1	1	yes	characteristic
272	C₂²	C₄₃×#540	1	4	yes	—
273	C₂²	C₄₃×#542	1	4	yes	—
274	C₂²	C₄₃×#543	1	2	yes	—
275	C₂²	C₄₃×#544	1	2	yes	—
276	C₂²	C₄₃×#551	1	4	yes	—
277	C₂²	C₄₃×#554	1	8	yes	—
278	C₂²	C₄₃×#560	1	4	yes	—
279	C₄	—	2	8	no	—
280	C₄	—	4	8	no	—
281	C₄	—	8	4	no	—
282	C₄	—	8	4	no	—
283	C₂	—	4	4	no	—
284	C₂	C₄₃×#2483	1	1	yes	characteristic
285	C₂	C₄₃×#2502	1	2	yes	—
286	C₂	C₄₃×#2624	1	2	yes	—
287	C₂	C₄₃×#2659	1	1	yes	characteristic
288	C₂	C₄₃×#2742	1	4	yes	—
289	C₂	C₄₃×#2748	1	4	yes	—
290	C₂	C₄₃×#2815	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₄₃	C₂⁴
commutator, Frattini, cosocle	C₂⁴	C₂⁴×C₄₃
prime radical	C₂⁵×C₄₃	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₄₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄₃×#1445	C₂	semidirect
3. C₄₃×#1444	C₂	semidirect
114. #17057	C₄₃	direct (N × H)
139. #1445	C₂×C₄₃	semidirect
140. #1444	C₂×C₄₃	semidirect
162. C₄₃	#17057	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄₃×#380	2
C₄₃×#1445	2
C₄₃×#1444	2
C₄₃×#1439	2
C₄₃×C₂²⋊#95	2
#17057	43

Subgroups — C43×#17057 #24937074

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₃×#17057 #24937074