Subgroups — C₂²×#14582950 #25960302

Order 11008.

3086 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄²×D₄₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582950	C₂	1	24	yes	maximal, direct product factor
3	C₂²×Dic₄₃⋊C₈	C₂	1	2	yes	maximal, complemented
4	C₂²×(C₂×D₄₃)⋊C₈	C₂	1	2	yes	maximal
5	C₂²×C₄×C₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×C₈×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×C₄×D₄₃	C₄	1	1	yes	characteristic
8	C₂³×C₄×D₄₃	C₄	1	1	yes	characteristic
9	C₂³×C₄×D₄₃	C₂²	1	1	yes	characteristic
10	C₂³×C₄₃⋊C₈	C₂²	1	2	yes	—
11	C₂²×C₄×Dic₄₃	C₄	1	1	yes	characteristic
12	C₂²×C₄×Dic₄₃	C₄	1	1	yes	characteristic
13	C₂²×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
14	C₂×C₄²×D₄₃	C₄	1	12	yes	—
15	C₂×C₄²×D₄₃	C₂²	1	12	yes	—
16	#14582950	C₂²	1	64	yes	direct product factor
17	C₂×Dic₄₃⋊C₈	C₂²	1	24	yes	complemented
18	C₂×(C₂×D₄₃)⋊C₈	C₂²	1	24	yes	—
19	C₂×C₄×C₄₃⋊C₈	C₂²	1	12	yes	complemented
20	C₂³×C₈×C₄₃	C₂²	1	2	yes	—
21	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₄₃	C₂²	1	12	yes	complemented
23	C₂⁴×D₄₃	C₂×C₄	1	1	yes	characteristic
24	C₂³×Dic₄₃	C₂×C₄	1	1	yes	characteristic
25	C₂³×Dic₄₃	C₂×C₄	1	1	yes	characteristic
26	C₂³×Dic₄₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×D₄₃	—	2	8	no	—
28	C₂²×C₄×D₄₃	—	2	8	no	—
29	C₂²×C₄×D₄₃	C₈	1	16	yes	complemented
30	C₂²×C₄×D₄₃	C₂×C₄	1	6	yes	—
31	C₂²×C₄×D₄₃	C₂×C₄	1	12	yes	—
32	C₂²×C₄×D₄₃	C₂×C₄	1	12	yes	—
33	C₂²×C₄×D₄₃	C₂³	1	6	yes	—
34	C₂²×C₄₃⋊C₈	—	2	8	no	—
35	C₂²×C₄₃⋊C₈	C₂³	1	12	yes	—
36	C₂×C₄×Dic₄₃	C₂×C₄	1	6	yes	—
37	C₂×C₄×Dic₄₃	C₂×C₄	1	12	yes	—
38	C₂×C₄×Dic₄₃	C₂×C₄	1	12	yes	—
39	C₂×C₄×Dic₄₃	C₂³	1	6	yes	—
40	C₄²×D₄₃	C₂×C₄	1	48	yes	—
41	C₄²×D₄₃	C₂³	1	16	yes	—
42	Dic₄₃⋊C₈	C₂³	1	32	yes	complemented
43	(C₂×D₄₃)⋊C₈	C₂³	1	32	yes	—
44	C₄×C₄₃⋊C₈	C₂³	1	16	yes	complemented
45	C₂³×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
46	C₂³×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
47	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₄₃	—	2	8	no	—
49	C₂²×C₈×C₄₃	C₂³	1	12	yes	—
50	C₂×C₄²×C₄₃	C₂×C₄	1	6	yes	—
51	C₂×C₄²×C₄₃	C₂³	1	6	yes	—
52	C₄×C₈×C₄₃	C₂³	1	16	yes	complemented
53	C₂⁴×C₄₃	C₂²×C₄	1	1	yes	characteristic
54	C₂³×D₄₃	—	2	4	no	—
55	C₂³×D₄₃	—	2	4	no	—
56	C₂³×D₄₃	C₂×C₈	1	8	yes	—
57	C₂³×D₄₃	C₂²×C₄	1	6	yes	—
58	C₂²×C₄×C₄₃	C₂×C₈	1	8	yes	—
59	C₂²×C₄×C₄₃	M₁₆	1	8	yes	—
60	C₂²×C₄×C₄₃	C₂²×C₄	1	3	yes	—
61	C₂²×C₄×C₄₃	C₂²×C₄	1	6	yes	—
62	C₂²×C₄×C₄₃	C₂²×C₄	1	6	yes	—
63	C₂²×C₄×C₄₃	#41	1	8	yes	—
64	C₂²×C₄×C₄₃	C₂⁴	1	3	yes	—
65	C₂²×Dic₄₃	—	2	4	no	—
66	C₂²×Dic₄₃	—	2	4	no	—
67	C₂²×Dic₄₃	C₂×C₈	1	8	yes	—
68	C₂²×Dic₄₃	C₂²×C₄	1	6	yes	—
69	C₂²×Dic₄₃	C₂²×C₄	1	6	yes	—
70	C₂²×Dic₄₃	C₂²×C₄	1	6	yes	—
71	C₂×C₄×D₄₃	—	2	24	no	—
72	C₂×C₄×D₄₃	—	2	48	no	—
73	C₂×C₄×D₄₃	—	2	24	no	—
74	C₂×C₄×D₄₃	C₂×C₈	1	96	yes	complemented
75	C₂×C₄×D₄₃	C₂²×C₄	1	12	yes	—
76	C₂×C₄×D₄₃	C₂²×C₄	1	16	yes	—
77	C₂×C₄×D₄₃	C₂²×C₄	1	16	yes	—
78	C₂×C₄×D₄₃	C₂⁴	1	4	yes	—
79	C₂×C₈×C₄₃	—	2	24	no	—
80	C₂×C₈×C₄₃	C₂⁴	1	8	yes	—
81	C₄²×C₄₃	C₂²×C₄	1	12	yes	—
82	C₄²×C₄₃	C₂⁴	1	4	yes	—
83	C₄×Dic₄₃	C₂²×C₄	1	12	yes	—
84	C₄×Dic₄₃	C₂²×C₄	1	16	yes	—
85	C₄×Dic₄₃	C₂²×C₄	1	16	yes	—
86	C₄×Dic₄₃	C₂⁴	1	4	yes	—
87	C₂×C₄₃⋊C₈	—	2	24	no	—
88	C₂×C₄₃⋊C₈	C₂⁴	1	8	yes	—
89	C₂³×C₄₃	#117	1	4	yes	—
90	C₂³×C₄₃	C₂²×C₈	1	4	yes	—
91	C₂³×C₄₃	C₂×M₁₆	1	4	yes	—
92	C₂³×C₄₃	C₂³×C₄	1	3	yes	—
93	C₂²×D₄₃	—	2	12	no	—
94	C₂²×D₄₃	—	2	12	no	—
95	C₂²×D₄₃	—	2	32	no	—
96	C₂²×D₄₃	C₂²×C₈	1	24	yes	—
97	C₂²×D₄₃	C₂³×C₄	1	4	yes	—
98	C₂×C₄×C₄₃	#117	1	12	yes	—
99	C₂×C₄×C₄₃	C₂²×C₈	1	24	yes	—
100	C₂×C₄×C₄₃	C₂×M₁₆	1	24	yes	—
101	C₂×C₄×C₄₃	C₂³×C₄	1	3	yes	—
102	C₂×C₄×C₄₃	C₂³×C₄	1	4	yes	—
103	C₂×C₄×C₄₃	C₂³×C₄	1	4	yes	—
104	C₂×C₄×C₄₃	C₂×#41	1	12	yes	—
105	C₂×C₄×C₄₃	C₂⁵	1	1	yes	characteristic
106	C₂×Dic₄₃	—	2	12	no	—
107	C₂×Dic₄₃	—	2	12	no	—
108	C₂×Dic₄₃	C₂²×C₈	1	24	yes	—
109	C₂×Dic₄₃	C₂³×C₄	1	4	yes	—
110	C₂×Dic₄₃	C₂³×C₄	1	4	yes	—
111	C₂×Dic₄₃	C₂³×C₄	1	4	yes	—
112	C₄×D₄₃	—	2	24	no	—
113	C₄×D₄₃	—	2	32	no	—
114	C₄×D₄₃	—	2	8	no	—
115	C₄×D₄₃	C₂²×C₈	1	64	yes	complemented
116	C₈×C₄₃	—	2	8	no	—
117	C₄₃⋊C₈	—	2	8	no	—
118	C₂²×#431	—	43	1	no	maximal, Sylow
119	C₂²×C₄₃	#431	1	16	yes	complemented
120	C₂²×C₄₃	C₂×#117	1	6	yes	—
121	C₂²×C₄₃	C₂³×C₈	1	6	yes	—
122	C₂²×C₄₃	C₂²×M₁₆	1	6	yes	—
123	C₂²×C₄₃	C₂⁴×C₄	1	1	yes	characteristic
124	C₂×D₄₃	—	2	4	no	—
125	C₂×D₄₃	—	2	48	no	—
126	C₂×D₄₃	—	2	4	no	—
127	C₂×D₄₃	C₂³×C₈	1	8	yes	—
128	C₄×C₄₃	C₂×#117	1	6	yes	—
129	C₄×C₄₃	C₂³×C₈	1	8	yes	—
130	C₄×C₄₃	C₂²×M₁₆	1	8	yes	—
131	C₄×C₄₃	C₂²×#41	1	2	yes	—
132	Dic₄₃	—	2	4	no	—
133	Dic₄₃	—	2	4	no	—
134	Dic₄₃	C₂³×C₈	1	8	yes	—
135	C₂³×C₄²	—	43	1	no	—
136	C₂×#431	—	43	24	no	—
137	C₂²×C₄⋊C₈	—	43	2	no	—
138	C₂²×C₂²⋊C₈	—	43	2	no	—
139	C₂²×C₄×C₈	—	43	1	no	—
140	C₂²×C₄×C₈	D₄₃	1	1	yes	characteristic, complemented
141	C₂×C₄₃	C₂×#431	1	12	yes	complemented
142	C₂×C₄₃	C₂²×#117	1	1	yes	characteristic
143	C₂×C₄₃	C₂⁴×C₈	1	1	yes	characteristic
144	C₂×C₄₃	C₂³×M₁₆	1	1	yes	characteristic
145	D₄₃	—	2	8	no	—
146	C₂⁴×C₄	—	43	1	no	—
147	C₂⁴×C₄	—	43	1	no	—
148	C₂⁴×C₄	—	43	1	no	—
149	C₂³×C₈	—	43	2	no	—
150	C₂³×C₈	C₂×D₄₃	1	2	yes	—
151	C₂²×C₄²	—	43	1	no	—
152	C₂²×C₄²	—	43	1	no	—
153	C₂²×C₄²	—	43	1	no	—
154	C₂²×C₄²	—	43	12	no	—
155	C₂²×C₄²	—	43	12	no	—
156	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
157	#431	—	43	64	no	—
158	C₂×C₄⋊C₈	—	43	24	no	—
159	C₂×C₂²⋊C₈	—	43	24	no	—
160	C₂×C₄×C₈	—	43	12	no	—
161	C₂×C₄×C₈	C₂×D₄₃	1	12	yes	complemented
162	C₄₃	C₂²×#431	1	1	yes	characteristic, complemented, Sylow
163	C₂⁵	—	43	1	no	—
164	C₂³×C₄	—	43	1	no	—
165	C₂³×C₄	—	43	16	no	—
166	C₂³×C₄	—	43	12	no	—
167	C₂³×C₄	—	43	1	no	—
168	C₂³×C₄	—	43	1	no	—
169	C₂³×C₄	—	43	6	no	—
170	C₂³×C₄	—	43	12	no	—
171	C₂³×C₄	—	43	6	no	—
172	C₂³×C₄	—	86	8	no	—
173	C₂³×C₄	—	86	8	no	—
174	C₂³×C₄	C₄×D₄₃	1	1	yes	characteristic
175	C₂³×C₄	C₄×D₄₃	1	1	yes	characteristic
176	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
177	C₂²×C₈	—	2	8	no	—
178	C₂²×C₈	—	43	12	no	—
179	C₂²×C₈	—	86	8	no	—
180	C₂²×C₈	C₂²×D₄₃	1	12	yes	—
181	C₂×C₄²	—	43	6	no	—
182	C₂×C₄²	—	43	12	no	—
183	C₂×C₄²	—	43	12	no	—
184	C₂×C₄²	—	43	6	no	—
185	C₂×C₄²	—	43	48	no	—
186	C₂×C₄²	—	43	16	no	—
187	C₂×C₄²	C₄×D₄₃	1	6	yes	—
188	C₂×C₄²	C₂²×D₄₃	1	6	yes	—
189	C₄⋊C₈	—	43	32	no	—
190	C₂²⋊C₈	—	43	32	no	—
191	C₄×C₈	—	43	16	no	—
192	C₄×C₈	C₂²×D₄₃	1	16	yes	complemented
193	C₂⁴	—	43	8	no	—
194	C₂⁴	—	43	6	no	—
195	C₂⁴	—	86	4	no	—
196	C₂⁴	—	86	4	no	—
197	C₂⁴	C₂×C₄×D₄₃	1	1	yes	characteristic
198	C₂²×C₄	—	43	8	no	—
199	C₂²×C₄	—	43	6	no	—
200	C₂²×C₄	—	43	96	no	—
201	C₂²×C₄	—	43	16	no	—
202	C₂²×C₄	—	43	6	no	—
203	C₂²×C₄	—	43	6	no	—
204	C₂²×C₄	—	43	12	no	—
205	C₂²×C₄	—	43	16	no	—
206	C₂²×C₄	—	43	4	no	—
207	C₂²×C₄	—	86	4	no	—
208	C₂²×C₄	—	86	4	no	—
209	C₂²×C₄	—	86	24	no	—
210	C₂²×C₄	—	86	48	no	—
211	C₂²×C₄	—	86	24	no	—
212	C₂²×C₄	C₈×D₄₃	1	8	yes	—
213	C₂²×C₄	#10621872	1	8	yes	—
214	C₂²×C₄	C₂×C₄×D₄₃	1	3	yes	—
215	C₂²×C₄	C₂×C₄×D₄₃	1	6	yes	—
216	C₂²×C₄	C₂×C₄×D₄₃	1	6	yes	—
217	C₂²×C₄	#10621898	1	8	yes	—
218	C₂²×C₄	C₂³×D₄₃	1	3	yes	—
219	C₂×C₈	—	2	24	no	—
220	C₂×C₈	—	43	8	no	—
221	C₂×C₈	—	86	24	no	—
222	C₂×C₈	C₂³×D₄₃	1	8	yes	—
223	C₄²	—	43	12	no	—
224	C₄²	—	43	16	no	—
225	C₄²	—	43	16	no	—
226	C₄²	—	43	4	no	—
227	C₄²	C₂×C₄×D₄₃	1	12	yes	—
228	C₄²	C₂³×D₄₃	1	4	yes	—
229	C₂³	—	43	24	no	—
230	C₂³	—	43	4	no	—
231	C₂³	—	86	12	no	—
232	C₂³	—	86	12	no	—
233	C₂³	—	86	32	no	—
234	C₂³	#13063282	1	4	yes	—
235	C₂³	C₂×C₈×D₄₃	1	4	yes	—
236	C₂³	C₂×#10621872	1	4	yes	—
237	C₂³	C₂²×C₄×D₄₃	1	3	yes	—
238	C₂×C₄	—	43	24	no	—
239	C₂×C₄	—	43	4	no	—
240	C₂×C₄	—	43	64	no	—
241	C₂×C₄	—	43	4	no	—
242	C₂×C₄	—	43	4	no	—
243	C₂×C₄	—	86	12	no	—
244	C₂×C₄	—	86	12	no	—
245	C₂×C₄	—	86	24	no	—
246	C₂×C₄	—	86	32	no	—
247	C₂×C₄	—	86	8	no	—
248	C₂×C₄	#13063282	1	12	yes	—
249	C₂×C₄	C₂×C₈×D₄₃	1	24	yes	—
250	C₂×C₄	C₂×#10621872	1	24	yes	—
251	C₂×C₄	C₂²×C₄×D₄₃	1	3	yes	—
252	C₂×C₄	C₂²×C₄×D₄₃	1	4	yes	—
253	C₂×C₄	C₂²×C₄×D₄₃	1	4	yes	—
254	C₂×C₄	C₂×#10621898	1	12	yes	—
255	C₂×C₄	C₂⁴×D₄₃	1	1	yes	characteristic
256	C₈	—	2	8	no	—
257	C₈	—	86	8	no	—
258	C₂²	—	43	8	no	—
259	C₂²	—	86	48	no	—
260	C₂²	—	86	4	no	—
261	C₂²	—	86	4	no	—
262	C₂²	#14582950	1	16	yes	direct product factor
263	C₂²	C₂×#13063282	1	6	yes	—
264	C₂²	C₂²×C₈×D₄₃	1	6	yes	—
265	C₂²	C₂²×#10621872	1	6	yes	—
266	C₂²	C₂³×C₄×D₄₃	1	1	yes	characteristic
267	C₄	—	43	8	no	—
268	C₄	—	86	4	no	—
269	C₄	—	86	4	no	—
270	C₄	C₂×#13063282	1	6	yes	—
271	C₄	C₂²×C₈×D₄₃	1	8	yes	—
272	C₄	C₂²×#10621872	1	8	yes	—
273	C₄	C₂²×#10621898	1	2	yes	—
274	C₂	—	86	8	no	—
275	C₂	C₂×#14582950	1	12	yes	direct product factor
276	C₂	C₂²×#13063282	1	1	yes	characteristic
277	C₂	C₂³×C₈×D₄₃	1	1	yes	characteristic
278	C₂	C₂³×#10621872	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄²	C₂×D₄₃
commutator	C₂×C₄₃	C₂⁴×C₈
Frattini	C₂×C₄	C₂⁴×D₄₃
Fitting	C₂²×C₄×C₈×C₄₃	C₂
socle	C₂⁴×C₄₃	C₂²×C₄
prime radical	C₂⁴×D₄₃	C₂×C₄
cosocle	C₂×C₄×C₄₃	C₂⁵
hypercenter	C₂²×C₄×C₈	D₄₃
hypocenter	C₄₃	C₂²×#431
LCS	G ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#14582950	C₂	direct (N × H)
3. C₂²×Dic₄₃⋊C₈	C₂	semidirect
5. C₂²×C₄×C₄₃⋊C₈	C₂	semidirect
6. C₂²×C₄×C₈×C₄₃	C₂	semidirect
16. #14582950	C₂²	direct (N × H)
17. C₂×Dic₄₃⋊C₈	C₂²	semidirect
19. C₂×C₄×C₄₃⋊C₈	C₂²	semidirect
22. C₂×C₄×C₈×C₄₃	C₂²	semidirect
29. C₂²×C₄×D₄₃	C₈	semidirect
42. Dic₄₃⋊C₈	C₂³	semidirect
44. C₄×C₄₃⋊C₈	C₂³	semidirect
52. C₄×C₈×C₄₃	C₂³	semidirect
74. C₂×C₄×D₄₃	C₂×C₈	semidirect
115. C₄×D₄₃	C₂²×C₈	semidirect
119. C₂²×C₄₃	#431	semidirect
140. C₂²×C₄×C₈	D₄₃	semidirect
141. C₂×C₄₃	C₂×#431	semidirect
161. C₂×C₄×C₈	C₂×D₄₃	semidirect
162. C₄₃	C₂²×#431	semidirect
192. C₄×C₈	C₂²×D₄₃	semidirect
262. C₂²	#14582950	direct (N × H)
275. C₂	C₂×#14582950	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄²×D₄₃	2
C₂×#14582950	2
C₂²×Dic₄₃⋊C₈	2
C₂²×(C₂×D₄₃)⋊C₈	2
C₂²×C₄×C₄₃⋊C₈	2
C₂²×C₄×C₈×C₄₃	2
C₂²×#431	43

Subgroups — C22×#14582950 #25960302

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#14582950 #25960302