Subgroups — #25065750

Order 11008.

638 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19143227	C₂	1	2	yes	maximal
2	C₂×#14582950	C₂	1	1	yes	maximal, characteristic
3	#19133374	C₂	1	8	yes	maximal, complemented
4	#19133354	C₂	1	1	yes	maximal, characteristic, complemented
5	#19133350	C₂	1	2	yes	maximal, complemented
6	C₄₃×#1567	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄²×D₄₃	C₂²	1	1	yes	characteristic
8	#14582955	—	2	2	no	—
9	#14582955	C₂²	1	4	yes	—
10	#14582951	—	2	2	no	—
11	#14582951	C₂²	1	4	yes	—
12	#14582950	C₄	1	4	yes	—
13	#14582950	C₂²	1	4	yes	—
14	#14582923	C₂²	1	2	yes	—
15	C₂×Dic₄₃⋊C₈	C₄	1	2	yes	—
16	#14582862	—	2	2	no	—
17	C₂×(C₂×D₄₃)⋊C₈	C₄	1	2	yes	—
18	M₁₆×Dic₄₃	—	2	2	no	—
19	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
20	#14582652	C₂²	1	4	yes	complemented
21	#14582649	C₂²	1	8	yes	complemented
22	C₄₃×#423	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
24	C₄₃×#335	C₂²	1	4	yes	complemented
25	C₂²×C₄×D₄₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×D₄₃	D₄	1	1	yes	characteristic
27	C₂²×C₄×D₄₃	Q₈	1	1	yes	characteristic
28	C₂×#10621877	—	2	2	no	—
29	C₂²×C₄₃⋊C₈	C₂×C₄	1	2	yes	—
30	C₂×C₄×Dic₄₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×Dic₄₃	D₄	1	1	yes	characteristic
32	C₂×C₄×Dic₄₃	Q₈	1	1	yes	characteristic
33	C₄²×D₄₃	C₂×C₄	1	2	yes	—
34	C₄²×D₄₃	D₄	1	2	yes	—
35	C₄²×D₄₃	Q₈	1	2	yes	—
36	C₄²×D₄₃	C₂³	1	2	yes	—
37	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
38	C₄⋊(C₄₃⋊C₈)	C₂³	1	4	yes	—
39	Dic₄₃⋊C₈	C₂×C₄	1	8	yes	—
40	(C₂×D₄₃)⋊C₈	—	2	4	no	—
41	(C₂×D₄₃)⋊C₈	—	2	4	no	—
42	(C₂×D₄₃)⋊C₈	C₂×C₄	1	8	yes	—
43	#13063235	—	2	4	no	—
44	C₄×C₄₃⋊C₈	C₂×C₄	1	2	yes	—
45	C₄×C₄₃⋊C₈	C₂³	1	2	yes	—
46	C₈×Dic₄₃	—	2	4	no	—
47	C₂×M₁₆×C₄₃	—	2	2	no	—
48	C₂²×C₈×C₄₃	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
50	C₄⋊C₈×C₄₃	—	2	2	no	—
51	C₄⋊C₈×C₄₃	C₂³	1	4	yes	—
52	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
53	C₄×C₈×C₄₃	C₂³	1	2	yes	—
54	C₂³×D₄₃	C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
58	C₂²×Dic₄₃	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×Dic₄₃	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×Dic₄₃	C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×D₄₃	—	2	2	no	—
62	C₂×C₄×D₄₃	—	2	2	no	—
63	C₂×C₄×D₄₃	—	4	2	no	—
64	C₂×C₄×D₄₃	—	4	2	no	—
65	C₂×C₄×D₄₃	C₄⋊C₄	1	2	yes	—
66	C₂×C₄×D₄₃	C₄⋊C₄	1	2	yes	—
67	C₂×C₄×D₄₃	C₄⋊C₄	1	2	yes	—
68	C₂×C₄×D₄₃	C₂²×C₄	1	2	yes	—
69	C₂×C₄×D₄₃	C₂×D₄	1	2	yes	—
70	C₂×C₄×D₄₃	C₂×Q₈	1	2	yes	—
71	M₁₆×C₄₃	—	4	4	no	—
72	C₂×C₈×C₄₃	—	2	4	no	—
73	C₂×C₈×C₄₃	—	4	2	no	—
74	C₂×C₈×C₄₃	C₂²×C₄	1	4	yes	—
75	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
76	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₄₃	C₂×Q₈	1	1	yes	characteristic
78	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
79	C₄×Dic₄₃	C₄⋊C₄	1	2	yes	—
80	C₄×Dic₄₃	C₄⋊C₄	1	2	yes	—
81	C₄×Dic₄₃	C₄⋊C₄	1	2	yes	—
82	C₄×Dic₄₃	C₂²×C₄	1	2	yes	—
83	C₄×Dic₄₃	C₂×D₄	1	2	yes	—
84	C₄×Dic₄₃	C₂×Q₈	1	2	yes	—
85	#10621877	—	4	4	no	—
86	C₂×C₄₃⋊C₈	—	2	4	no	—
87	C₂×C₄₃⋊C₈	—	4	2	no	—
88	C₂×C₄₃⋊C₈	C₂²×C₄	1	4	yes	—
89	C₂³×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₂²×D₄₃	—	2	2	no	—
91	C₂²×D₄₃	—	4	1	no	—
92	C₂²×D₄₃	—	4	1	no	—
93	C₂²×D₄₃	C₂×C₄⋊C₄	1	2	yes	—
94	C₂×C₄×C₄₃	—	2	2	no	—
95	C₂×C₄×C₄₃	—	2	2	no	—
96	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₄₃	#135	1	2	yes	—
100	C₂×C₄×C₄₃	#135	1	2	yes	—
101	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
102	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
103	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
104	C₂×Dic₄₃	—	2	2	no	—
105	C₂×Dic₄₃	—	4	1	no	—
106	C₂×Dic₄₃	—	4	1	no	—
107	C₂×Dic₄₃	C₂×C₄⋊C₄	1	2	yes	—
108	C₂×Dic₄₃	C₂×C₄⋊C₄	1	2	yes	—
109	C₂×Dic₄₃	C₂×C₄⋊C₄	1	2	yes	—
110	C₄×D₄₃	—	2	4	no	—
111	C₄×D₄₃	—	2	2	no	—
112	C₄×D₄₃	—	2	2	no	—
113	C₄×D₄₃	—	2	4	no	—
114	C₄×D₄₃	—	2	4	no	—
115	C₄×D₄₃	—	4	2	no	—
116	C₄×D₄₃	—	4	2	no	—
117	C₈×C₄₃	—	2	4	no	—
118	C₈×C₄₃	—	4	4	no	—
119	C₄₃⋊C₈	—	2	4	no	—
120	C₄₃⋊C₈	—	4	4	no	—
121	#17815	—	43	1	no	maximal, Sylow
122	C₂²×C₄₃	—	2	1	no	—
123	C₂²×C₄₃	—	2	1	no	—
124	C₂²×C₄₃	#427	1	2	yes	—
125	C₂²×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
126	C₂×D₄₃	—	2	2	no	—
127	C₂×D₄₃	—	2	2	no	—
128	C₂×D₄₃	—	2	2	no	—
129	C₂×D₄₃	—	4	4	no	—
130	C₄×C₄₃	—	2	1	no	—
131	C₄×C₄₃	—	2	1	no	—
132	C₄×C₄₃	#427	1	2	yes	—
133	C₄×C₄₃	#430	1	2	yes	—
134	C₄×C₄₃	#568	1	2	yes	—
135	C₄×C₄₃	C₂×#135	1	1	yes	characteristic
136	C₄×C₄₃	C₂×#135	1	1	yes	characteristic
137	Dic₄₃	—	2	2	no	—
138	Dic₄₃	—	2	2	no	—
139	Dic₄₃	—	2	2	no	—
140	#2920	—	43	2	no	—
141	C₂×#431	—	43	1	no	—
142	#1572	—	43	8	no	—
143	#1567	—	43	1	no	—
144	#1567	D₄₃	1	1	yes	characteristic, complemented
145	#1566	—	43	2	no	—
146	C₂×C₄₃	—	2	2	no	—
147	C₂×C₄₃	#2905	1	1	yes	characteristic
148	C₂×C₄₃	C₂×#427	1	1	yes	characteristic
149	C₂×C₄₃	C₂×#430	1	1	yes	characteristic
150	D₄₃	—	4	2	no	—
151	C₂²×C₄²	—	43	1	no	—
152	#432	—	43	4	no	—
153	#432	—	86	2	no	—
154	#431	—	43	4	no	—
155	#431	—	43	4	no	—
156	#431	—	43	4	no	—
157	#431	—	86	2	no	—
158	#423	—	43	2	no	—
159	#423	C₂×D₄₃	1	2	yes	—
160	C₂×C₄⋊C₈	—	43	2	no	—
161	#407	—	86	2	no	—
162	C₂×C₂²⋊C₈	—	43	2	no	—
163	C₄×M₁₆	—	86	2	no	—
164	C₂×C₄×C₈	—	43	1	no	—
165	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
166	#335	—	43	4	no	—
167	#335	C₂×D₄₃	1	4	yes	complemented
168	#334	—	43	8	no	—
169	C₄₃	#17815	1	1	yes	characteristic, complemented, Sylow
170	C₂³×C₄	—	43	1	no	—
171	C₂³×C₄	—	43	1	no	—
172	C₂³×C₄	—	43	1	no	—
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	—	86	2	no	—
175	C₂²×C₈	—	43	2	no	—
176	C₂²×C₈	C₄×D₄₃	1	2	yes	—
177	C₂×C₄²	—	43	1	no	—
178	C₂×C₄²	—	43	1	no	—
179	C₂×C₄²	—	43	1	no	—
180	C₂×C₄²	—	43	2	no	—
181	C₂×C₄²	—	43	2	no	—
182	C₂×C₄²	—	43	2	no	—
183	C₂×C₄²	—	43	2	no	—
184	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
185	C₄⋊C₈	—	2	2	no	—
186	C₄⋊C₈	—	43	8	no	—
187	C₄⋊C₈	—	43	4	no	—
188	C₄⋊C₈	—	86	2	no	—
189	C₄⋊C₈	C₂²×D₄₃	1	4	yes	—
190	C₂²⋊C₈	—	43	8	no	—
191	C₂²⋊C₈	—	86	4	no	—
192	C₂²⋊C₈	—	86	4	no	—
193	#97	—	86	4	no	—
194	C₄×C₈	—	43	2	no	—
195	C₄×C₈	—	43	2	no	—
196	C₄×C₈	—	86	4	no	—
197	C₄×C₈	C₄×D₄₃	1	2	yes	—
198	C₄×C₈	C₂²×D₄₃	1	2	yes	—
199	C₂⁴	—	43	1	no	—
200	C₂²×C₄	—	43	1	no	—
201	C₂²×C₄	—	43	1	no	—
202	C₂²×C₄	—	43	1	no	—
203	C₂²×C₄	—	43	2	no	—
204	C₂²×C₄	—	43	2	no	—
205	C₂²×C₄	—	43	2	no	—
206	C₂²×C₄	—	43	2	no	—
207	C₂²×C₄	—	43	2	no	—
208	C₂²×C₄	—	43	2	no	—
209	C₂²×C₄	—	86	2	no	—
210	C₂²×C₄	—	86	2	no	—
211	C₂²×C₄	—	172	2	no	—
212	C₂²×C₄	—	172	2	no	—
213	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
214	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
215	C₂²×C₄	Q₈×D₄₃	1	1	yes	characteristic
216	M₁₆	—	4	4	no	—
217	M₁₆	—	172	4	no	—
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	—	4	2	no	—
220	C₂×C₈	—	43	4	no	—
221	C₂×C₈	—	86	4	no	—
222	C₂×C₈	—	172	2	no	—
223	C₂×C₈	C₂×C₄×D₄₃	1	4	yes	—
224	C₄²	—	43	2	no	—
225	C₄²	—	43	2	no	—
226	C₄²	—	43	2	no	—
227	C₄²	—	43	2	no	—
228	C₄²	—	43	2	no	—
229	C₄²	—	43	2	no	—
230	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
231	C₄²	D₄×D₄₃	1	1	yes	characteristic
232	C₄²	Q₈×D₄₃	1	1	yes	characteristic
233	C₄²	C₂³×D₄₃	1	1	yes	characteristic
234	C₂³	—	43	2	no	—
235	C₂³	—	86	2	no	—
236	C₂³	—	172	1	no	—
237	C₂³	—	172	1	no	—
238	C₂³	C₄⋊C₄×D₄₃	1	1	yes	characteristic
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	43	2	no	—
242	C₂×C₄	—	43	2	no	—
243	C₂×C₄	—	43	2	no	—
244	C₂×C₄	—	86	2	no	—
245	C₂×C₄	—	86	4	no	—
246	C₂×C₄	—	86	2	no	—
247	C₂×C₄	—	86	2	no	—
248	C₂×C₄	—	86	4	no	—
249	C₂×C₄	—	86	4	no	—
250	C₂×C₄	—	172	2	no	—
251	C₂×C₄	—	172	2	no	—
252	C₂×C₄	—	172	1	no	—
253	C₂×C₄	—	172	1	no	—
254	C₂×C₄	C₄⋊C₄×D₄₃	1	1	yes	characteristic
255	C₂×C₄	C₄⋊C₄×D₄₃	1	1	yes	characteristic
256	C₂×C₄	C₄⋊C₄×D₄₃	1	1	yes	characteristic
257	C₂×C₄	#13063344	1	2	yes	—
258	C₂×C₄	#13063348	1	2	yes	—
259	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
260	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
261	C₂×C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
262	C₈	—	2	4	no	—
263	C₈	—	4	4	no	—
264	C₈	—	86	4	no	—
265	C₈	—	172	4	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	2	1	no	—
268	C₂²	—	86	2	no	—
269	C₂²	—	86	2	no	—
270	C₂²	—	86	2	no	—
271	C₂²	—	172	4	no	—
272	C₂²	#14582936	1	2	yes	—
273	C₂²	C₂×C₄⋊C₄×D₄₃	1	1	yes	characteristic
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	86	2	no	—
277	C₄	—	86	2	no	—
278	C₄	—	86	2	no	—
279	C₄	#14582936	1	2	yes	—
280	C₄	#14582946	1	2	yes	—
281	C₄	#14583665	1	2	yes	—
282	C₄	C₂×#13063344	1	1	yes	characteristic
283	C₄	C₂×#13063348	1	1	yes	characteristic
284	C₂	—	2	2	no	—
285	C₂	—	172	2	no	—
286	C₂	#19143157	1	1	yes	characteristic
287	C₂	C₂×#14582936	1	1	yes	characteristic
288	C₂	C₂×#14582946	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₄²	C₂³×D₄₃
Fitting	C₄₃×#1567	C₂
socle	C₂²×C₄₃	C₂²×C₄⋊C₄
prime radical	C₂³×D₄₃	C₄⋊C₄
cosocle	C₄²×C₄₃	C₂⁴
hypercenter	#1567	D₄₃
hypocenter	C₄₃	#17815
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1567

Complements

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

N	H ≅ G/N	type
3. #19133374	C₂	semidirect
4. #19133354	C₂	semidirect
5. #19133350	C₂	semidirect
6. C₄₃×#1567	C₂	semidirect
20. #14582652	C₂²	semidirect
21. #14582649	C₂²	semidirect
24. C₄₃×#335	C₂²	semidirect
144. #1567	D₄₃	semidirect
167. #335	C₂×D₄₃	semidirect
169. C₄₃	#17815	semidirect

Maximal subgroups

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

subgroup	index
#19143227	2
C₂×#14582950	2
#19133374	2
#19133354	2
#19133350	2
C₄₃×#1567	2
#17815	43