Subgroups — #103×C₄₃⋊C₈ #24987369

Order 11008.

518 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×C₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₄⋊C₄×C₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₈⋊(C₄₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈×C₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic
5	#19132436	C₂	1	1	yes	maximal, characteristic
6	#19132430	C₂	1	1	yes	maximal, characteristic
7	C₄×#103×C₄₃	C₂	1	1	yes	maximal, characteristic
8	Q₈×C₄₃⋊C₈	—	2	2	no	—
9	Q₈×C₄₃⋊C₈	C₄	1	4	yes	complemented
10	C₂×C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
11	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₄₃⋊C₈	—	2	1	no	—
14	C₂×C₄×C₄₃⋊C₈	—	2	1	no	—
15	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
17	#14582654	—	2	1	no	—
18	Q₈⋊(C₄₃⋊C₈)	C₄	1	4	yes	complemented
19	C₈×C₄₃⋊C₈	—	2	2	no	—
20	C₂×C₄×Q₈×C₄₃	C₂²	1	1	yes	characteristic
21	C₂×#103×C₄₃	C₄	1	2	yes	—
22	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
23	C₄×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
24	C₄₃×#340	C₄	1	2	yes	—
25	C₂²×C₄₃⋊C₈	—	2	1	no	—
26	C₂²×C₄₃⋊C₈	—	2	1	no	—
27	C₂²×C₄₃⋊C₈	—	2	1	no	—
28	C₂²×C₄₃⋊C₈	D₄	1	1	yes	characteristic
29	C₂²×C₄₃⋊C₈	D₄	1	1	yes	characteristic
30	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
31	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
32	C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
33	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	2	yes	—
34	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	2	yes	—
35	C₄×C₄₃⋊C₈	—	2	1	no	—
36	C₄×C₄₃⋊C₈	—	2	4	no	—
37	C₄×C₄₃⋊C₈	—	4	2	no	—
38	C₄×C₄₃⋊C₈	C₂×C₄	1	2	yes	—
39	C₄×C₄₃⋊C₈	C₂×C₄	1	2	yes	—
40	C₄×C₄₃⋊C₈	D₄	1	2	yes	—
41	C₂²×Q₈×C₄₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₄₃	C₂×C₄	1	2	yes	—
43	C₄×Q₈×C₄₃	—	2	1	no	—
44	C₄×Q₈×C₄₃	—	2	1	no	—
45	C₄×Q₈×C₄₃	C₂×C₄	1	2	yes	—
46	C₄×Q₈×C₄₃	C₂×C₄	1	2	yes	—
47	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
48	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₄₃	—	2	1	no	—
51	C₂×C₄²×C₄₃	—	2	1	no	—
52	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
53	#103×C₄₃	C₈	1	8	yes	complemented
54	C₄×C₈×C₄₃	—	2	1	no	—
55	C₄×C₈×C₄₃	—	2	1	no	—
56	#95×C₄₃	—	2	1	no	—
57	C₂×Q₈×C₄₃	—	2	1	no	—
58	C₂×Q₈×C₄₃	—	2	2	no	—
59	C₂×Q₈×C₄₃	C₄²	1	2	yes	—
60	C₂×Q₈×C₄₃	C₂×C₈	1	4	yes	—
61	C₂²×C₄×C₄₃	—	2	1	no	—
62	C₂²×C₄×C₄₃	—	2	2	no	—
63	C₂²×C₄×C₄₃	—	2	1	no	—
64	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
67	C₂×C₈×C₄₃	—	2	2	no	—
68	C₂×C₈×C₄₃	—	2	2	no	—
69	C₂×C₈×C₄₃	C₂×C₈	1	4	yes	—
70	C₄⋊C₄×C₄₃	—	2	2	no	—
71	C₄⋊C₄×C₄₃	—	2	2	no	—
72	C₄⋊C₄×C₄₃	—	2	1	no	—
73	C₄⋊C₄×C₄₃	—	2	2	no	—
74	C₄⋊C₄×C₄₃	C₄²	1	2	yes	—
75	C₄⋊C₄×C₄₃	C₂×C₈	1	4	yes	—
76	C₄²×C₄₃	—	2	2	no	—
77	C₄²×C₄₃	—	2	2	no	—
78	C₄²×C₄₃	—	4	1	no	—
79	C₄²×C₄₃	—	4	1	no	—
80	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
81	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₄₃	#41	1	1	yes	characteristic
84	C₂×C₄₃⋊C₈	—	2	4	no	—
85	C₂×C₄₃⋊C₈	—	2	2	no	—
86	C₂×C₄₃⋊C₈	—	2	1	no	—
87	C₂×C₄₃⋊C₈	—	2	1	no	—
88	C₂×C₄₃⋊C₈	—	2	2	no	—
89	C₂×C₄₃⋊C₈	—	2	2	no	—
90	C₂×C₄₃⋊C₈	—	4	1	no	—
91	C₂×C₄₃⋊C₈	—	4	1	no	—
92	C₂×C₄₃⋊C₈	—	4	1	no	—
93	C₂×C₄₃⋊C₈	—	4	1	no	—
94	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	2	yes	—
95	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	2	yes	—
96	C₂×C₄₃⋊C₈	SD₁₆	1	2	yes	—
97	C₂×C₄₃⋊C₈	Q₁₆	1	2	yes	—
98	C₂³×C₄₃	C₄×D₄	1	1	yes	characteristic
99	Q₈×C₄₃	—	2	2	no	—
100	Q₈×C₄₃	—	2	4	no	—
101	Q₈×C₄₃	C₄×C₈	1	4	yes	complemented
102	C₂×C₄×C₄₃	—	2	2	no	—
103	C₂×C₄×C₄₃	—	2	4	no	—
104	C₂×C₄×C₄₃	—	2	2	no	—
105	C₂×C₄×C₄₃	—	2	2	no	—
106	C₂×C₄×C₄₃	—	2	4	no	—
107	C₂×C₄×C₄₃	—	4	2	no	—
108	C₂×C₄×C₄₃	—	4	2	no	—
109	C₂×C₄×C₄₃	—	4	2	no	—
110	C₂×C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
111	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
113	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₄₃	C₂²×C₈	1	2	yes	—
117	C₂×C₄×C₄₃	#131	1	2	yes	—
118	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
119	C₂×C₄×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
120	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
121	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
122	C₈×C₄₃	—	2	4	no	—
123	C₄₃⋊C₈	—	2	4	no	—
124	C₄₃⋊C₈	—	2	2	no	—
125	C₄₃⋊C₈	—	4	2	no	—
126	C₄₃⋊C₈	—	4	2	no	—
127	C₄₃⋊C₈	#103	1	4	yes	direct product factor
128	C₈×#103	—	43	1	no	maximal, Sylow
129	C₂²×C₄₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
130	C₂²×C₄₃	C₈×D₄	1	2	yes	—
131	C₂²×C₄₃	C₄×SD₁₆	1	1	yes	characteristic
132	C₂²×C₄₃	C₄×Q₁₆	1	1	yes	characteristic
133	C₂²×C₄₃	#443	1	2	yes	—
134	C₄×C₄₃	—	2	4	no	—
135	C₄×C₄₃	—	2	1	no	—
136	C₄×C₄₃	—	2	1	no	—
137	C₄×C₄₃	—	4	4	no	—
138	C₄×C₄₃	—	4	2	no	—
139	C₄×C₄₃	C₂×C₄×C₈	1	1	yes	characteristic
140	C₄×C₄₃	#405	1	1	yes	characteristic
141	C₄×C₄₃	C₂×#103	1	1	yes	characteristic
142	C₄×C₄₃	#416	1	1	yes	characteristic
143	C₄×C₄₃	C₈×D₄	1	2	yes	—
144	C₄×C₄₃	C₄×SD₁₆	1	1	yes	characteristic
145	C₄×C₄₃	C₄×Q₁₆	1	1	yes	characteristic
146	C₂×C₈×Q₈	—	43	1	no	—
147	C₈×C₄⋊C₄	—	43	1	no	—
148	C₄×#103	D₄₃	1	1	yes	characteristic
149	C₂×Q₈⋊C₈	—	43	1	no	—
150	C₂×C₈²	—	43	1	no	—
151	#1279	—	43	1	no	—
152	#1277	—	43	1	no	—
153	C₂×C₄₃	C₈×SD₁₆	1	2	yes	—
154	C₂×C₄₃	C₈×Q₁₆	1	2	yes	—
155	C₂×C₄₃	C₈×C₂²⋊C₄	1	1	yes	characteristic
156	C₂×C₄₃	C₄×#103	1	1	yes	characteristic
157	C₂×C₄₃	#1764	1	1	yes	characteristic
158	C₂×C₄×Q₈	C₂×D₄₃	1	1	yes	characteristic
159	C₈×Q₈	—	43	4	no	—
160	C₈×Q₈	—	86	2	no	—
161	C₂×C₄⋊C₈	—	43	1	no	—
162	C₂×C₄⋊C₈	—	43	1	no	—
163	C₂×C₄⋊C₈	—	86	1	no	—
164	C₂×#103	Dic₄₃	1	2	yes	—
165	C₂×C₄×C₈	—	43	1	no	—
166	C₂×C₄×C₈	—	43	1	no	—
167	C₂×C₄×C₈	—	86	1	no	—
168	C₂×C₄×C₈	—	86	1	no	—
169	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
170	C₄×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
171	#340	Dic₄₃	1	2	yes	—
172	#336	—	86	1	no	—
173	Q₈⋊C₈	—	43	4	no	—
174	C₈²	—	86	2	no	—
175	C₄₃	C₈×#103	1	1	yes	characteristic, complemented, Sylow
176	C₂²×Q₈	C₂×Dic₄₃	1	1	yes	characteristic
177	C₂²×C₈	—	43	1	no	—
178	C₂²×C₈	—	43	1	no	—
179	C₂²×C₈	—	86	1	no	—
180	C₂²×C₈	—	86	1	no	—
181	C₂²×C₈	—	86	1	no	—
182	C₂²×C₈	C₂×Dic₄₃	1	2	yes	—
183	C₄×Q₈	—	2	1	no	—
184	C₄×Q₈	—	2	1	no	—
185	C₄×Q₈	C₄×D₄₃	1	2	yes	—
186	C₄×Q₈	C₄×D₄₃	1	2	yes	—
187	C₂×C₄⋊C₄	—	2	1	no	—
188	C₂×C₄⋊C₄	C₂×Dic₄₃	1	2	yes	—
189	C₂×C₄⋊C₄	C₂×Dic₄₃	1	1	yes	characteristic
190	C₂×C₄²	—	2	1	no	—
191	C₂×C₄²	—	2	1	no	—
192	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
193	C₄⋊C₈	—	43	2	no	—
194	C₄⋊C₈	—	43	2	no	—
195	C₄⋊C₈	—	86	4	no	—
196	C₄⋊C₈	—	86	1	no	—
197	C₄⋊C₈	—	86	1	no	—
198	#103	C₄₃⋊C₈	1	8	yes	direct product factor
199	C₄×C₈	—	2	1	no	—
200	C₄×C₈	—	2	1	no	—
201	C₄×C₈	—	43	2	no	—
202	C₄×C₈	—	43	2	no	—
203	C₄×C₈	—	43	2	no	—
204	C₄×C₈	—	86	4	no	—
205	C₄×C₈	—	86	1	no	—
206	C₄×C₈	—	172	2	no	—
207	#95	—	2	1	no	—
208	C₂×Q₈	—	2	1	no	—
209	C₂×Q₈	—	2	2	no	—
210	C₂×Q₈	C₂×C₄₃⋊C₈	1	4	yes	—
211	C₂×Q₈	C₄×Dic₄₃	1	2	yes	—
212	C₂²×C₄	—	2	1	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	2	1	no	—
215	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
216	C₂²×C₄	#10621900	1	1	yes	characteristic
217	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	43	2	no	—
221	C₂×C₈	—	43	2	no	—
222	C₂×C₈	—	43	2	no	—
223	C₂×C₈	—	43	2	no	—
224	C₂×C₈	—	86	4	no	—
225	C₂×C₈	—	86	2	no	—
226	C₂×C₈	—	86	1	no	—
227	C₂×C₈	—	86	1	no	—
228	C₂×C₈	—	86	2	no	—
229	C₂×C₈	—	86	2	no	—
230	C₂×C₈	—	172	1	no	—
231	C₂×C₈	—	172	1	no	—
232	C₂×C₈	—	172	1	no	—
233	C₂×C₈	—	172	1	no	—
234	C₂×C₈	C₂×C₄₃⋊C₈	1	4	yes	—
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	2	1	no	—
238	C₄⋊C₄	—	2	2	no	—
239	C₄⋊C₄	C₂×C₄₃⋊C₈	1	4	yes	—
240	C₄⋊C₄	C₄×Dic₄₃	1	2	yes	—
241	C₄²	—	2	2	no	—
242	C₄²	—	2	2	no	—
243	C₄²	—	4	1	no	—
244	C₄²	—	4	1	no	—
245	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
246	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
247	C₄²	D₄×D₄₃	1	1	yes	characteristic
248	C₄²	#10621900	1	1	yes	characteristic
249	C₂³	D₄×Dic₄₃	1	1	yes	characteristic
250	Q₈	—	2	2	no	—
251	Q₈	—	2	4	no	—
252	Q₈	C₄×C₄₃⋊C₈	1	4	yes	complemented
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	C₂×C₄×Dic₄₃	1	1	yes	characteristic
262	C₂×C₄	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
263	C₂×C₄	#13063281	1	1	yes	characteristic
264	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
265	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
266	C₂×C₄	D₄×Dic₄₃	1	1	yes	characteristic
267	C₂×C₄	C₂²×C₄₃⋊C₈	1	2	yes	—
268	C₂×C₄	#13063332	1	2	yes	—
269	C₂×C₄	SD₁₆×D₄₃	1	1	yes	characteristic
270	C₂×C₄	Q₁₆×D₄₃	1	1	yes	characteristic
271	C₂×C₄	#13063346	1	1	yes	characteristic
272	C₂×C₄	#13063348	1	1	yes	characteristic
273	C₈	—	2	4	no	—
274	C₈	—	43	4	no	—
275	C₈	—	86	4	no	—
276	C₈	—	86	2	no	—
277	C₈	—	172	2	no	—
278	C₈	—	172	2	no	—
279	C₂²	C₂²⋊C₄×Dic₄₃	1	1	yes	characteristic
280	C₂²	D₄×C₄₃⋊C₈	1	2	yes	—
281	C₂²	SD₁₆×Dic₄₃	1	1	yes	characteristic
282	C₂²	Q₁₆×Dic₄₃	1	1	yes	characteristic
283	C₂²	#14583016	1	2	yes	—
284	C₄	—	2	4	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	4	4	no	—
288	C₄	—	4	2	no	—
289	C₄	C₂×C₄×C₄₃⋊C₈	1	1	yes	characteristic
290	C₄	#14582855	1	1	yes	characteristic
291	C₄	#103×D₄₃	1	1	yes	characteristic
292	C₄	#14582894	1	1	yes	characteristic
293	C₄	D₄×C₄₃⋊C₈	1	2	yes	—
294	C₄	#14582984	1	1	yes	characteristic
295	C₄	#14582991	1	1	yes	characteristic
296	C₂	SD₁₆×C₄₃⋊C₈	1	2	yes	—
297	C₂	Q₁₆×C₄₃⋊C₈	1	2	yes	—
298	C₂	C₂²⋊C₄×C₄₃⋊C₈	1	1	yes	characteristic
299	C₂	#103×Dic₄₃	1	1	yes	characteristic
300	C₂	#19134435	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₄₃
commutator	C₄×C₄₃	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄×#103×C₄₃	C₂
socle, prime radical	C₂³×C₄₃	C₄×D₄
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	C₄×#103	D₄₃
hypocenter	C₄₃	C₈×#103
LCS	G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×#103

Complements

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

N	H ≅ G/N	type
9. Q₈×C₄₃⋊C₈	C₄	semidirect
18. Q₈⋊(C₄₃⋊C₈)	C₄	semidirect
53. #103×C₄₃	C₈	semidirect
101. Q₈×C₄₃	C₄×C₈	semidirect
127. C₄₃⋊C₈	#103	direct (N × H)
175. C₄₃	C₈×#103	semidirect
198. #103	C₄₃⋊C₈	direct (N × H)
252. Q₈	C₄×C₄₃⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×C₄₃⋊C₈	2
C₄⋊C₄×C₄₃⋊C₈	2
C₂×Q₈⋊(C₄₃⋊C₈)	2
C₂×C₈×C₄₃⋊C₈	2
#19132436	2
#19132430	2
C₄×#103×C₄₃	2
C₈×#103	43

Subgroups — #103×C43⋊C8 #24987369

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #103×C₄₃⋊C₈ #24987369