Subgroups — #25065744

Order 11008.

730 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14582950	C₂	1	1	yes	maximal, characteristic
2	#423×D₄₃	C₂	1	2	yes	maximal
3	#19133368	C₂	1	8	yes	maximal, complemented
4	#19133354	C₂	1	1	yes	maximal, characteristic, complemented
5	#19133353	C₂	1	2	yes	maximal, complemented
6	C₄₃×#1567	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×M₁₆×D₄₃	—	2	2	no	—
8	C₂×C₄²×D₄₃	C₂²	1	1	yes	characteristic
9	#14582950	C₄	1	4	yes	—
10	#14582950	C₂²	1	4	yes	—
11	#14582923	C₂²	1	2	yes	—
12	#14582922	—	2	2	no	—
13	#14582921	—	2	2	no	—
14	#14582921	C₂²	1	4	yes	—
15	C₂×Dic₄₃⋊C₈	C₄	1	2	yes	—
16	C₄⋊C₈×D₄₃	—	2	2	no	—
17	C₄⋊C₈×D₄₃	C₂²	1	4	yes	—
18	C₂×(C₂×D₄₃)⋊C₈	C₄	1	2	yes	—
19	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
20	#14582652	C₂²	1	4	yes	complemented
21	#14582651	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	C₂×#10621872	—	2	4	no	—
30	M₁₆×D₄₃	—	4	8	no	—
31	C₂²×C₄₃⋊C₈	C₂×C₄	1	2	yes	—
32	C₂×C₈×D₄₃	—	2	4	no	—
33	C₂×C₄×Dic₄₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×Dic₄₃	D₄	1	1	yes	characteristic
35	C₂×C₄×Dic₄₃	Q₈	1	1	yes	characteristic
36	C₄²×D₄₃	C₂×C₄	1	2	yes	—
37	C₄²×D₄₃	D₄	1	2	yes	—
38	C₄²×D₄₃	Q₈	1	2	yes	—
39	C₄²×D₄₃	C₂³	1	2	yes	—
40	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
41	C₄⋊(C₄₃⋊C₈)	C₂³	1	4	yes	—
42	Dic₄₃⋊C₈	—	2	4	no	—
43	Dic₄₃⋊C₈	—	2	4	no	—
44	Dic₄₃⋊C₈	C₂×C₄	1	8	yes	—
45	(C₂×D₄₃)⋊C₈	C₂×C₄	1	8	yes	—
46	C₄×C₄₃⋊C₈	C₂×C₄	1	2	yes	—
47	C₄×C₄₃⋊C₈	C₂³	1	2	yes	—
48	C₂×M₁₆×C₄₃	—	2	2	no	—
49	C₂²×C₈×C₄₃	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
51	C₄⋊C₈×C₄₃	—	2	2	no	—
52	C₄⋊C₈×C₄₃	C₂³	1	4	yes	—
53	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
54	C₄×C₈×C₄₃	C₂³	1	2	yes	—
55	C₂³×D₄₃	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
59	C₂²×Dic₄₃	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×Dic₄₃	C₄⋊C₄	1	1	yes	characteristic
61	C₂²×Dic₄₃	C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×D₄₃	—	4	2	no	—
63	C₂×C₄×D₄₃	—	4	2	no	—
64	C₂×C₄×D₄₃	C₄⋊C₄	1	2	yes	—
65	C₂×C₄×D₄₃	C₄⋊C₄	1	2	yes	—
66	C₂×C₄×D₄₃	C₄⋊C₄	1	2	yes	—
67	C₂×C₄×D₄₃	D₈	1	4	yes	—
68	C₂×C₄×D₄₃	Q₁₆	1	4	yes	—
69	C₂×C₄×D₄₃	C₂²×C₄	1	2	yes	—
70	C₂×C₄×D₄₃	C₂×D₄	1	2	yes	—
71	C₂×C₄×D₄₃	C₂×Q₈	1	2	yes	—
72	M₁₆×C₄₃	—	4	4	no	—
73	C₂×C₈×C₄₃	—	2	4	no	—
74	C₂×C₈×C₄₃	—	4	2	no	—
75	C₂×C₈×C₄₃	C₂²×C₄	1	4	yes	—
76	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
77	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₄₃	C₂×Q₈	1	1	yes	characteristic
79	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
80	C₄×Dic₄₃	C₄⋊C₄	1	2	yes	—
81	C₄×Dic₄₃	C₄⋊C₄	1	2	yes	—
82	C₄×Dic₄₃	C₄⋊C₄	1	2	yes	—
83	C₄×Dic₄₃	C₂²×C₄	1	2	yes	—
84	C₄×Dic₄₃	C₂×D₄	1	2	yes	—
85	C₄×Dic₄₃	C₂×Q₈	1	2	yes	—
86	#10621877	—	4	4	no	—
87	C₂×C₄₃⋊C₈	—	2	4	no	—
88	C₂×C₄₃⋊C₈	—	4	2	no	—
89	C₂×C₄₃⋊C₈	C₂²×C₄	1	4	yes	—
90	#10621872	—	4	8	no	—
91	C₈×D₄₃	—	4	8	no	—
92	C₂³×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂²×D₄₃	—	4	1	no	—
94	C₂²×D₄₃	—	4	1	no	—
95	C₂²×D₄₃	#107	1	4	yes	—
96	C₂²×D₄₃	C₂×C₄⋊C₄	1	2	yes	—
97	C₂×C₄×C₄₃	—	2	2	no	—
98	C₂×C₄×C₄₃	—	2	2	no	—
99	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
103	C₂×C₄×C₄₃	C₂×Q₁₆	1	2	yes	—
104	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
105	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
106	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
107	C₂×Dic₄₃	—	4	1	no	—
108	C₂×Dic₄₃	—	4	1	no	—
109	C₂×Dic₄₃	#107	1	4	yes	—
110	C₂×Dic₄₃	C₂×C₄⋊C₄	1	2	yes	—
111	C₂×Dic₄₃	C₂×C₄⋊C₄	1	2	yes	—
112	C₂×Dic₄₃	C₂×C₄⋊C₄	1	2	yes	—
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	C₄×D₄₃	#107	1	8	yes	—
118	C₄×D₄₃	C₂×D₈	1	4	yes	—
119	C₄×D₄₃	C₂×Q₁₆	1	4	yes	—
120	C₈×C₄₃	—	2	4	no	—
121	C₈×C₄₃	—	4	4	no	—
122	C₄₃⋊C₈	—	2	4	no	—
123	C₄₃⋊C₈	—	4	4	no	—
124	#17814	—	43	1	no	maximal, Sylow
125	C₂²×C₄₃	—	2	1	no	—
126	C₂²×C₄₃	—	2	1	no	—
127	C₂²×C₄₃	C₂×#107	1	2	yes	—
128	C₂²×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
129	C₂×D₄₃	—	2	2	no	—
130	C₂×D₄₃	—	2	2	no	—
131	C₂×D₄₃	—	4	4	no	—
132	C₂×D₄₃	C₂×#107	1	4	yes	—
133	C₄×C₄₃	—	2	1	no	—
134	C₄×C₄₃	—	2	1	no	—
135	C₄×C₄₃	C₂×#107	1	2	yes	—
136	C₄×C₄₃	#430	1	2	yes	—
137	C₄×C₄₃	#568	1	2	yes	—
138	C₄×C₄₃	C₂²×D₈	1	1	yes	characteristic
139	C₄×C₄₃	C₂²×Q₁₆	1	1	yes	characteristic
140	Dic₄₃	—	2	2	no	—
141	Dic₄₃	—	2	2	no	—
142	Dic₄₃	C₂×#107	1	4	yes	—
143	C₂×#431	—	43	1	no	—
144	C₂×#423	—	43	2	no	—
145	#1571	—	43	8	no	—
146	#1567	—	43	2	no	—
147	#1567	—	43	1	no	—
148	#1567	D₄₃	1	1	yes	characteristic, complemented
149	C₂×C₄₃	—	2	2	no	—
150	C₂×C₄₃	#2905	1	1	yes	characteristic
151	C₂×C₄₃	C₂²×#107	1	1	yes	characteristic
152	C₂×C₄₃	C₂×#430	1	1	yes	characteristic
153	D₄₃	—	2	4	no	—
154	C₂²×M₁₆	—	86	2	no	—
155	C₂²×C₄²	—	43	1	no	—
156	#431	—	43	4	no	—
157	#431	—	43	4	no	—
158	#423	—	43	2	no	—
159	#423	—	43	4	no	—
160	#423	—	86	2	no	—
161	#423	—	86	2	no	—
162	#423	C₂×D₄₃	1	2	yes	—
163	C₂×C₄⋊C₈	—	43	2	no	—
164	C₂×C₄⋊C₈	—	43	4	no	—
165	C₂×C₄⋊C₈	—	86	2	no	—
166	C₂×C₂²⋊C₈	—	43	2	no	—
167	C₂×C₄×C₈	—	43	1	no	—
168	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
169	#335	—	43	8	no	—
170	#335	—	43	4	no	—
171	#335	C₂×D₄₃	1	4	yes	complemented
172	C₄₃	#17814	1	1	yes	characteristic, complemented, Sylow
173	C₂³×C₄	—	43	1	no	—
174	C₂³×C₄	—	43	1	no	—
175	C₂³×C₄	—	43	1	no	—
176	C₂×M₁₆	—	2	2	no	—
177	C₂×M₁₆	—	86	4	no	—
178	C₂×M₁₆	—	86	2	no	—
179	C₂×M₁₆	—	172	8	no	—
180	C₂²×C₈	—	43	2	no	—
181	C₂²×C₈	—	86	4	no	—
182	C₂²×C₈	C₄×D₄₃	1	2	yes	—
183	C₂×C₄²	—	43	1	no	—
184	C₂×C₄²	—	43	1	no	—
185	C₂×C₄²	—	43	1	no	—
186	C₂×C₄²	—	43	2	no	—
187	C₂×C₄²	—	43	2	no	—
188	C₂×C₄²	—	43	2	no	—
189	C₂×C₄²	—	43	2	no	—
190	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
191	C₄⋊C₈	—	2	2	no	—
192	C₄⋊C₈	—	43	8	no	—
193	C₄⋊C₈	—	43	4	no	—
194	C₄⋊C₈	—	86	4	no	—
195	C₄⋊C₈	—	86	4	no	—
196	C₄⋊C₈	—	86	2	no	—
197	C₄⋊C₈	C₂²×D₄₃	1	4	yes	—
198	C₂²⋊C₈	—	43	8	no	—
199	C₄×C₈	—	43	2	no	—
200	C₄×C₈	—	43	2	no	—
201	C₄×C₈	C₄×D₄₃	1	2	yes	—
202	C₄×C₈	C₂²×D₄₃	1	2	yes	—
203	C₂⁴	—	43	1	no	—
204	C₂²×C₄	—	43	1	no	—
205	C₂²×C₄	—	43	2	no	—
206	C₂²×C₄	—	43	4	no	—
207	C₂²×C₄	—	43	4	no	—
208	C₂²×C₄	—	43	2	no	—
209	C₂²×C₄	—	43	1	no	—
210	C₂²×C₄	—	43	1	no	—
211	C₂²×C₄	—	43	2	no	—
212	C₂²×C₄	—	43	2	no	—
213	C₂²×C₄	—	43	2	no	—
214	C₂²×C₄	—	43	2	no	—
215	C₂²×C₄	—	172	2	no	—
216	C₂²×C₄	—	172	2	no	—
217	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
218	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
219	C₂²×C₄	Q₈×D₄₃	1	1	yes	characteristic
220	M₁₆	—	4	4	no	—
221	M₁₆	—	172	8	no	—
222	M₁₆	—	172	4	no	—
223	C₂×C₈	—	2	4	no	—
224	C₂×C₈	—	4	2	no	—
225	C₂×C₈	—	43	4	no	—
226	C₂×C₈	—	86	4	no	—
227	C₂×C₈	—	172	8	no	—
228	C₂×C₈	—	172	2	no	—
229	C₂×C₈	C₂×C₄×D₄₃	1	4	yes	—
230	C₄²	—	43	2	no	—
231	C₄²	—	43	2	no	—
232	C₄²	—	43	2	no	—
233	C₄²	—	43	2	no	—
234	C₄²	—	43	2	no	—
235	C₄²	—	43	2	no	—
236	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
237	C₄²	D₄×D₄₃	1	1	yes	characteristic
238	C₄²	Q₈×D₄₃	1	1	yes	characteristic
239	C₄²	C₂³×D₄₃	1	1	yes	characteristic
240	C₂³	—	43	2	no	—
241	C₂³	—	43	4	no	—
242	C₂³	—	172	1	no	—
243	C₂³	—	172	1	no	—
244	C₂³	C₄⋊C₄×D₄₃	1	1	yes	characteristic
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	43	2	no	—
248	C₂×C₄	—	43	4	no	—
249	C₂×C₄	—	43	4	no	—
250	C₂×C₄	—	43	4	no	—
251	C₂×C₄	—	43	8	no	—
252	C₂×C₄	—	43	2	no	—
253	C₂×C₄	—	43	2	no	—
254	C₂×C₄	—	86	4	no	—
255	C₂×C₄	—	86	4	no	—
256	C₂×C₄	—	172	2	no	—
257	C₂×C₄	—	172	2	no	—
258	C₂×C₄	—	172	1	no	—
259	C₂×C₄	—	172	1	no	—
260	C₂×C₄	C₄⋊C₄×D₄₃	1	1	yes	characteristic
261	C₂×C₄	C₄⋊C₄×D₄₃	1	1	yes	characteristic
262	C₂×C₄	C₄⋊C₄×D₄₃	1	1	yes	characteristic
263	C₂×C₄	D₈×D₄₃	1	2	yes	—
264	C₂×C₄	Q₁₆×D₄₃	1	2	yes	—
265	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
266	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
267	C₂×C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
268	C₈	—	2	4	no	—
269	C₈	—	4	4	no	—
270	C₈	—	86	4	no	—
271	C₈	—	172	4	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	43	4	no	—
275	C₂²	—	86	2	no	—
276	C₂²	—	86	2	no	—
277	C₂²	—	172	4	no	—
278	C₂²	#107×D₄₃	1	2	yes	—
279	C₂²	C₂×C₄⋊C₄×D₄₃	1	1	yes	characteristic
280	C₄	—	2	1	no	—
281	C₄	—	2	1	no	—
282	C₄	—	43	4	no	—
283	C₄	—	86	2	no	—
284	C₄	—	86	2	no	—
285	C₄	#107×D₄₃	1	2	yes	—
286	C₄	#14582946	1	2	yes	—
287	C₄	#14583665	1	2	yes	—
288	C₄	C₂×D₈×D₄₃	1	1	yes	characteristic
289	C₄	C₂×Q₁₆×D₄₃	1	1	yes	characteristic
290	C₂	—	2	2	no	—
291	C₂	—	86	4	no	—
292	C₂	#19143157	1	1	yes	characteristic
293	C₂	C₂×#107×D₄₃	1	1	yes	characteristic
294	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	#17814
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. #19133368	C₂	semidirect
4. #19133354	C₂	semidirect
5. #19133353	C₂	semidirect
6. C₄₃×#1567	C₂	semidirect
20. #14582652	C₂²	semidirect
21. #14582651	C₂²	semidirect
24. C₄₃×#335	C₂²	semidirect
148. #1567	D₄₃	semidirect
171. #335	C₂×D₄₃	semidirect
172. C₄₃	#17814	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14582950	2
#423×D₄₃	2
#19133368	2
#19133354	2
#19133353	2
C₄₃×#1567	2
#17814	43