Subgroups — C₄₃×#3330 #19132147

Order 5504.

420 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₃×#547	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄₃×#547	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₃×#487	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₃×#484	C₂	1	2	yes	maximal, complemented
5	C₄₃×#474	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₃×#468	C₂	1	2	yes	maximal, complemented
7	C₄₃×#461	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₃×#449	C₂	1	2	yes	maximal, complemented
9	C₄×D₈×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄₃×#436	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₃×#417	C₂	1	2	yes	maximal, complemented
12	C₂×#41×C₄₃	—	2	1	no	—
13	C₂×#41×C₄₃	C₂²	1	2	yes	—
14	C₂×SD₁₆×C₄₃	C₂²	1	2	yes	—
15	C₂×D₈×C₄₃	C₂²	1	1	yes	characteristic
16	C₂×M₁₆×C₄₃	C₂²	1	2	yes	—
17	C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
18	C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
19	#123×C₄₃	—	2	2	no	—
20	#123×C₄₃	—	2	2	no	—
21	C₂²⋊Q₈×C₄₃	—	2	1	no	—
22	C₂²⋊Q₈×C₄₃	C₂²	1	2	yes	—
23	#121×C₄₃	—	2	1	no	—
24	#121×C₄₃	C₂²	1	2	yes	—
25	C₄×D₄×C₄₃	C₂²	1	1	yes	characteristic
26	C₄×D₄×C₄₃	C₂²	1	1	yes	characteristic
27	C₄×D₄×C₄₃	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
29	C₂×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
30	#107×C₄₃	C₂²	1	1	yes	characteristic
31	#107×C₄₃	C₂²	1	2	yes	—
32	C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
33	#103×C₄₃	C₂²	1	2	yes	—
34	#103×C₄₃	C₂²	1	2	yes	—
35	#103×C₄₃	C₂²	1	2	yes	—
36	#102×C₄₃	C₂²	1	1	yes	characteristic
37	#102×C₄₃	C₂²	1	2	yes	—
38	#102×C₄₃	C₂²	1	1	yes	characteristic
39	C₂²⋊C₈×C₄₃	C₂²	1	2	yes	—
40	C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
41	#41×C₄₃	—	2	4	no	—
42	#41×C₄₃	—	4	2	no	—
43	#41×C₄₃	—	4	4	no	—
44	C₂×Q₈×C₄₃	—	2	1	no	—
45	C₂×Q₈×C₄₃	C₂³	1	2	yes	—
46	C₂×D₄×C₄₃	—	2	2	no	—
47	C₂×D₄×C₄₃	—	2	2	no	—
48	C₂×D₄×C₄₃	D₄	1	1	yes	characteristic
49	C₂×D₄×C₄₃	C₂³	1	1	yes	characteristic
50	C₂×D₄×C₄₃	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₄₃	—	2	1	no	—
52	C₂²×C₄×C₄₃	—	2	2	no	—
53	C₂²×C₄×C₄₃	—	2	1	no	—
54	C₂²×C₄×C₄₃	—	2	2	no	—
55	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
56	SD₁₆×C₄₃	—	4	2	no	—
57	D₈×C₄₃	—	2	2	no	—
58	M₁₆×C₄₃	—	4	2	no	—
59	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
60	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
61	C₂×C₈×C₄₃	C₂³	1	2	yes	—
62	C₄⋊C₄×C₄₃	—	2	1	no	—
63	C₄⋊C₄×C₄₃	—	2	2	no	—
64	C₄⋊C₄×C₄₃	—	2	2	no	—
65	C₄⋊C₄×C₄₃	—	2	1	no	—
66	C₄⋊C₄×C₄₃	—	2	1	no	—
67	C₄⋊C₄×C₄₃	D₄	1	1	yes	characteristic
68	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
69	C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
70	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
71	C₂²⋊C₄×C₄₃	—	2	1	no	—
72	C₂²⋊C₄×C₄₃	—	2	1	no	—
73	C₂²⋊C₄×C₄₃	—	2	2	no	—
74	C₂²⋊C₄×C₄₃	—	2	2	no	—
75	C₂²⋊C₄×C₄₃	D₄	1	2	yes	—
76	C₄²×C₄₃	C₂³	1	1	yes	characteristic
77	C₂³×C₄₃	—	2	1	no	—
78	C₂³×C₄₃	—	2	1	no	—
79	C₂³×C₄₃	C₂×D₄	1	2	yes	—
80	Q₈×C₄₃	—	2	2	no	—
81	Q₈×C₄₃	—	2	2	no	—
82	Q₈×C₄₃	—	4	2	no	—
83	D₄×C₄₃	—	2	2	no	—
84	D₄×C₄₃	—	2	1	no	—
85	D₄×C₄₃	—	2	1	no	—
86	D₄×C₄₃	—	2	1	no	—
87	D₄×C₄₃	—	4	4	no	—
88	D₄×C₄₃	—	4	4	no	—
89	D₄×C₄₃	#41	1	2	yes	complemented
90	C₂×C₄×C₄₃	—	2	2	no	—
91	C₂×C₄×C₄₃	—	2	2	no	—
92	C₂×C₄×C₄₃	—	2	1	no	—
93	C₂×C₄×C₄₃	—	2	2	no	—
94	C₂×C₄×C₄₃	—	2	1	no	—
95	C₂×C₄×C₄₃	—	2	2	no	—
96	C₂×C₄×C₄₃	—	4	4	no	—
97	C₂×C₄×C₄₃	—	4	4	no	—
98	C₂×C₄×C₄₃	—	4	2	no	—
99	C₂×C₄×C₄₃	—	4	2	no	—
100	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
102	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
104	C₈×C₄₃	—	2	1	no	—
105	C₈×C₄₃	—	2	1	no	—
106	C₈×C₄₃	—	2	2	no	—
107	C₂²×C₄₃	—	2	2	no	—
108	C₂²×C₄₃	—	2	2	no	—
109	C₂²×C₄₃	—	2	2	no	—
110	C₂²×C₄₃	—	2	2	no	—
111	C₂²×C₄₃	—	4	1	no	—
112	C₂²×C₄₃	—	4	1	no	—
113	C₂²×C₄₃	—	4	1	no	—
114	C₂²×C₄₃	—	4	1	no	—
115	C₂²×C₄₃	—	4	1	no	—
116	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
117	C₄×C₄₃	—	2	2	no	—
118	C₄×C₄₃	—	2	1	no	—
119	C₄×C₄₃	—	4	2	no	—
120	C₄×C₄₃	—	4	1	no	—
121	C₄×C₄₃	—	4	1	no	—
122	C₄×C₄₃	—	4	2	no	—
123	C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
124	C₄×C₄₃	C₂×#41	1	1	yes	characteristic
125	C₄×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
126	#3330	C₄₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
127	C₂×C₄₃	—	4	2	no	—
128	C₂×C₄₃	—	4	2	no	—
129	C₂×C₄₃	—	8	1	no	—
130	C₂×C₄₃	#546	1	1	yes	characteristic
131	C₂×C₄₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
132	C₂×C₄₃	#578	1	1	yes	characteristic
133	#547	C₂×C₄₃	1	1	yes	characteristic, complemented
134	#547	C₂×C₄₃	1	1	yes	characteristic, complemented
135	#487	C₂×C₄₃	1	1	yes	characteristic, complemented
136	#484	C₂×C₄₃	1	2	yes	complemented
137	#474	C₂×C₄₃	1	1	yes	characteristic, complemented
138	#468	C₂×C₄₃	1	2	yes	complemented
139	#461	C₂×C₄₃	1	1	yes	characteristic, complemented
140	#449	C₂×C₄₃	1	2	yes	complemented
141	C₄×D₈	C₂×C₄₃	1	1	yes	characteristic, complemented
142	#436	C₂×C₄₃	1	1	yes	characteristic, complemented
143	#417	C₂×C₄₃	1	2	yes	complemented
144	C₄₃	#3330	1	1	yes	characteristic, direct product factor, Sylow
145	C₂×#41	—	2	1	no	—
146	C₂×#41	C₂²×C₄₃	1	2	yes	—
147	C₂×SD₁₆	C₂²×C₄₃	1	2	yes	—
148	C₂×D₈	C₂²×C₄₃	1	1	yes	characteristic
149	C₂×M₁₆	C₂²×C₄₃	1	2	yes	—
150	C₄⋊Q₈	C₂²×C₄₃	1	1	yes	characteristic
151	C₄⋊Q₈	C₂²×C₄₃	1	1	yes	characteristic
152	#123	—	2	2	no	—
153	#123	—	2	2	no	—
154	C₂²⋊Q₈	—	2	1	no	—
155	C₂²⋊Q₈	C₂²×C₄₃	1	2	yes	—
156	#121	—	2	1	no	—
157	#121	C₂²×C₄₃	1	2	yes	—
158	C₄×D₄	C₂²×C₄₃	1	1	yes	characteristic
159	C₄×D₄	C₂²×C₄₃	1	1	yes	characteristic
160	C₄×D₄	C₂²×C₄₃	1	1	yes	characteristic
161	C₂×C₄⋊C₄	—	2	1	no	—
162	C₂×C₄⋊C₄	C₂²×C₄₃	1	2	yes	—
163	#107	C₂²×C₄₃	1	1	yes	characteristic
164	#107	C₂²×C₄₃	1	2	yes	—
165	C₄⋊C₈	C₂²×C₄₃	1	1	yes	characteristic
166	#103	C₂²×C₄₃	1	2	yes	—
167	#103	C₂²×C₄₃	1	2	yes	—
168	#103	C₂²×C₄₃	1	2	yes	—
169	#102	C₂²×C₄₃	1	1	yes	characteristic
170	#102	C₂²×C₄₃	1	2	yes	—
171	#102	C₂²×C₄₃	1	1	yes	characteristic
172	C₂²⋊C₈	C₂²×C₄₃	1	2	yes	—
173	C₄×C₈	C₂²×C₄₃	1	1	yes	characteristic
174	#41	—	2	4	no	—
175	#41	—	4	2	no	—
176	#41	—	4	4	no	—
177	C₂×Q₈	—	2	1	no	—
178	C₂×Q₈	C₂³×C₄₃	1	2	yes	—
179	C₂×D₄	—	2	2	no	—
180	C₂×D₄	—	2	2	no	—
181	C₂×D₄	D₄×C₄₃	1	1	yes	characteristic
182	C₂×D₄	C₂³×C₄₃	1	1	yes	characteristic
183	C₂×D₄	C₂³×C₄₃	1	1	yes	characteristic
184	C₂²×C₄	—	2	2	no	—
185	C₂²×C₄	—	2	1	no	—
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	C₂³×C₄₃	1	2	yes	—
189	SD₁₆	—	4	2	no	—
190	D₈	—	2	2	no	—
191	M₁₆	—	4	2	no	—
192	C₂×C₈	C₂³×C₄₃	1	1	yes	characteristic
193	C₂×C₈	C₂³×C₄₃	1	1	yes	characteristic
194	C₂×C₈	C₂³×C₄₃	1	2	yes	—
195	C₄⋊C₄	—	2	1	no	—
196	C₄⋊C₄	—	2	2	no	—
197	C₄⋊C₄	—	2	2	no	—
198	C₄⋊C₄	—	2	1	no	—
199	C₄⋊C₄	—	2	1	no	—
200	C₄⋊C₄	D₄×C₄₃	1	1	yes	characteristic
201	C₄⋊C₄	C₂³×C₄₃	1	1	yes	characteristic
202	C₄⋊C₄	C₂³×C₄₃	1	2	yes	—
203	C₄⋊C₄	C₂³×C₄₃	1	1	yes	characteristic
204	C₂²⋊C₄	—	2	2	no	—
205	C₂²⋊C₄	—	2	1	no	—
206	C₂²⋊C₄	—	2	1	no	—
207	C₂²⋊C₄	—	2	2	no	—
208	C₂²⋊C₄	D₄×C₄₃	1	2	yes	—
209	C₄²	C₂³×C₄₃	1	1	yes	characteristic
210	C₂³	—	2	1	no	—
211	C₂³	—	2	1	no	—
212	C₂³	C₂×D₄×C₄₃	1	2	yes	—
213	Q₈	—	2	2	no	—
214	Q₈	—	2	2	no	—
215	Q₈	—	4	2	no	—
216	D₄	—	2	2	no	—
217	D₄	—	2	1	no	—
218	D₄	—	2	1	no	—
219	D₄	—	2	1	no	—
220	D₄	—	4	4	no	—
221	D₄	—	4	4	no	—
222	D₄	#41×C₄₃	1	2	yes	complemented
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	4	4	no	—
230	C₂×C₄	—	4	4	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	C₂×D₄×C₄₃	1	1	yes	characteristic
234	C₂×C₄	C₂×D₄×C₄₃	1	2	yes	—
235	C₂×C₄	C₂×D₄×C₄₃	1	1	yes	characteristic
236	C₂×C₄	C₂⁴×C₄₃	1	1	yes	characteristic
237	C₈	—	2	1	no	—
238	C₈	—	2	1	no	—
239	C₈	—	2	2	no	—
240	C₂²	—	2	2	no	—
241	C₂²	—	2	2	no	—
242	C₂²	—	2	2	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	4	1	no	—
245	C₂²	—	4	1	no	—
246	C₂²	—	4	1	no	—
247	C₂²	—	4	1	no	—
248	C₂²	—	4	1	no	—
249	C₂²	C₂²×D₄×C₄₃	1	1	yes	characteristic
250	C₄	—	2	2	no	—
251	C₄	—	2	1	no	—
252	C₄	—	4	2	no	—
253	C₄	—	4	1	no	—
254	C₄	—	4	2	no	—
255	C₄	—	4	1	no	—
256	C₄	AGL(1,ℤ/8)×C₄₃	1	2	yes	—
257	C₄	C₂×#41×C₄₃	1	1	yes	characteristic
258	C₄	H(5,𝔽₂)×C₄₃	1	1	yes	characteristic
259	C₂	—	4	2	no	—
260	C₂	—	4	2	no	—
261	C₂	—	8	1	no	—
262	C₂	C₄₃×#546	1	1	yes	characteristic
263	C₂	C₂×AGL(1,ℤ/8)×C₄₃	1	1	yes	characteristic
264	C₂	C₄₃×#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₄₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄₃ ◁ C₄×D₄×C₄₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₄₃×#547	C₂	semidirect
2. C₄₃×#547	C₂	semidirect
3. C₄₃×#487	C₂	semidirect
4. C₄₃×#484	C₂	semidirect
5. C₄₃×#474	C₂	semidirect
6. C₄₃×#468	C₂	semidirect
7. C₄₃×#461	C₂	semidirect
8. C₄₃×#449	C₂	semidirect
9. C₄×D₈×C₄₃	C₂	semidirect
10. C₄₃×#436	C₂	semidirect
11. C₄₃×#417	C₂	semidirect
89. D₄×C₄₃	#41	semidirect
126. #3330	C₄₃	direct (N × H)
133. #547	C₂×C₄₃	semidirect
134. #547	C₂×C₄₃	semidirect
135. #487	C₂×C₄₃	semidirect
136. #484	C₂×C₄₃	semidirect
137. #474	C₂×C₄₃	semidirect
138. #468	C₂×C₄₃	semidirect
139. #461	C₂×C₄₃	semidirect
140. #449	C₂×C₄₃	semidirect
141. C₄×D₈	C₂×C₄₃	semidirect
142. #436	C₂×C₄₃	semidirect
143. #417	C₂×C₄₃	semidirect
144. C₄₃	#3330	direct (N × H)
222. D₄	#41×C₄₃	semidirect

Maximal subgroups

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

subgroup	index
C₄₃×#547	2
C₄₃×#487	2
C₄₃×#484	2
C₄₃×#474	2
C₄₃×#468	2
C₄₃×#461	2
C₄₃×#449	2
C₄×D₈×C₄₃	2
C₄₃×#436	2
C₄₃×#417	2
#3330	43

Subgroups — C43×#3330 #19132147

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₃×#3330 #19132147