Subgroups — #25006538

Order 11008.

342 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×Q₈×D₄₃	C₂	1	1	yes	maximal, characteristic
2	#19136393	C₂	1	1	yes	maximal, characteristic
3	#19136392	C₂	1	1	yes	maximal, characteristic
4	#19132575	C₂	1	1	yes	maximal, characteristic, complemented
5	Q₈⋊(C₄₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
6	#19132573	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×Q₈⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×Q₈×D₄₃	C₄	1	1	yes	characteristic
9	Q₈×C₄₃⋊C₈	C₄	1	1	yes	characteristic
10	C₈×Dic₈₆	—	2	1	no	—
11	C₈×Dic₈₆	C₂²	1	1	yes	characteristic
12	#14583023	—	2	1	no	—
13	#14583023	C₄	1	1	yes	characteristic
14	C₄⋊C₈×D₄₃	—	2	1	no	—
15	C₄⋊C₈×D₄₃	C₄	1	1	yes	characteristic
16	C₄×C₈×D₄₃	—	2	1	no	—
17	C₄×C₈×D₄₃	C₂²	1	1	yes	characteristic
18	C₄⋊(C₄₃⋊C₁₆)	C₂²	1	1	yes	characteristic
19	Dic₄₃⋊C₁₆	—	2	1	no	—
20	(C₂×D₄₃)⋊C₁₆	—	2	1	no	—
21	(C₂×D₄₃)⋊C₁₆	—	2	1	no	—
22	C₄×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
23	C₁₆×Dic₄₃	—	2	1	no	—
24	C₈×Q₈×C₄₃	C₂²	1	1	yes	characteristic
25	C₄₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
26	C₄×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
27	C₂×Q₈×D₄₃	C₈	1	1	yes	characteristic
28	C₂×C₈×D₄₃	—	2	1	no	—
29	C₂×C₈×D₄₃	D₄	1	1	yes	characteristic
30	C₂×C₈×D₄₃	D₄	1	1	yes	characteristic
31	Q₈×Dic₄₃	C₈	1	1	yes	characteristic
32	C₄×Dic₈₆	—	2	1	no	—
33	C₄×Dic₈₆	C₂×C₄	1	1	yes	characteristic
34	#13063290	—	2	1	no	—
35	#13063290	C₈	1	1	yes	characteristic
36	C₄⋊C₄×D₄₃	—	2	1	no	—
37	C₄⋊C₄×D₄₃	C₈	1	1	yes	characteristic
38	C₄²×D₄₃	—	2	1	no	—
39	C₄²×D₄₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄₃⋊C₁₆	—	2	1	no	—
41	C₂×C₄₃⋊C₁₆	—	2	1	no	—
42	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
43	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
44	Dic₄₃⋊C₈	—	2	1	no	—
45	Dic₄₃⋊C₈	—	2	1	no	—
46	Dic₄₃⋊C₈	—	2	1	no	—
47	C₄×C₄₃⋊C₈	—	2	1	no	—
48	C₄×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
49	C₈×Dic₄₃	—	2	1	no	—
50	C₈×Dic₄₃	—	2	1	no	—
51	C₈×Dic₄₃	D₄	1	1	yes	characteristic
52	C₈×Dic₄₃	D₄	1	1	yes	characteristic
53	C₄×Q₈×C₄₃	C₂×C₄	1	1	yes	characteristic
54	C₂×C₁₆×C₄₃	—	2	1	no	—
55	C₂×C₁₆×C₄₃	—	2	1	no	—
56	C₄⋊C₈×C₄₃	—	2	1	no	—
57	C₄⋊C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
58	C₄×C₈×C₄₃	—	2	1	no	—
59	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
60	C₂×Q₈×C₄₃	C₂×C₈	1	1	yes	characteristic
61	Q₈×D₄₃	—	2	2	no	—
62	Q₈×D₄₃	C₁₆	1	4	yes	complemented
63	C₂×C₄×D₄₃	—	2	1	no	—
64	C₂×C₄×D₄₃	—	2	1	no	—
65	C₂×C₄×D₄₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₄×D₄₃	C₂×C₈	1	1	yes	characteristic
67	C₂×C₄×D₄₃	M₁₆	1	1	yes	characteristic
68	C₂×Dic₈₆	—	2	1	no	—
69	C₂×Dic₈₆	C₂×C₈	1	1	yes	characteristic
70	C₂×C₈×C₄₃	—	2	1	no	—
71	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
73	C₄⋊C₄×C₄₃	—	2	1	no	—
74	C₄⋊C₄×C₄₃	C₂×C₈	1	1	yes	characteristic
75	C₄²×C₄₃	—	2	1	no	—
76	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
77	C₄⋊Dic₄₃	—	2	1	no	—
78	C₄⋊Dic₄₃	C₂×C₈	1	1	yes	characteristic
79	Dic₄₃⋊C₄	—	2	1	no	—
80	Dic₄₃⋊C₄	—	2	1	no	—
81	Dic₄₃⋊C₄	—	2	1	no	—
82	C₄×Dic₄₃	—	2	1	no	—
83	C₄×Dic₄₃	—	2	1	no	—
84	C₄×Dic₄₃	—	2	1	no	—
85	C₄×Dic₄₃	C₂²⋊C₄	1	1	yes	characteristic
86	C₄×Dic₄₃	C₂×C₈	1	1	yes	characteristic
87	C₄×Dic₄₃	M₁₆	1	1	yes	characteristic
88	C₂×C₄₃⋊C₈	—	2	1	no	—
89	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
91	C₈×D₄₃	—	2	1	no	—
92	C₈×D₄₃	—	2	1	no	—
93	C₈×D₄₃	—	4	1	no	—
94	C₈×D₄₃	—	4	2	no	—
95	C₁₆×C₄₃	—	4	1	no	—
96	C₁₆×C₄₃	—	4	1	no	—
97	C₄₃⋊C₁₆	—	4	1	no	—
98	C₄₃⋊C₁₆	—	4	1	no	—
99	C₂²×D₄₃	C₂²⋊C₈	1	1	yes	characteristic
100	Q₈×C₄₃	—	2	1	no	—
101	Q₈×C₄₃	C₂×C₁₆	1	2	yes	—
102	C₂×C₄×C₄₃	—	2	1	no	—
103	C₂×C₄×C₄₃	—	2	1	no	—
104	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
106	C₂×C₄×C₄₃	C₂×M₁₆	1	1	yes	characteristic
107	C₂×Dic₄₃	—	2	1	no	—
108	C₂×Dic₄₃	—	2	1	no	—
109	C₂×Dic₄₃	C₂²⋊C₈	1	1	yes	characteristic
110	C₂×Dic₄₃	C₂²⋊C₈	1	1	yes	characteristic
111	C₂×Dic₄₃	C₂²⋊C₈	1	1	yes	characteristic
112	C₄×D₄₃	—	2	4	no	—
113	C₄×D₄₃	—	2	1	no	—
114	C₄×D₄₃	—	2	1	no	—
115	C₄×D₄₃	—	4	1	no	—
116	C₄×D₄₃	—	4	2	no	—
117	C₄×D₄₃	C₂×C₁₆	1	2	yes	—
118	C₄×D₄₃	M₃₂	1	2	yes	—
119	Dic₈₆	—	2	1	no	—
120	Dic₈₆	—	2	4	no	—
121	Dic₈₆	C₂×C₁₆	1	2	yes	—
122	C₈×C₄₃	—	2	1	no	—
123	C₈×C₄₃	—	4	1	no	—
124	C₈×C₄₃	#137	1	1	yes	characteristic
125	C₈×C₄₃	#137	1	1	yes	characteristic
126	C₄₃⋊C₈	—	2	1	no	—
127	C₄₃⋊C₈	—	2	1	no	—
128	C₄₃⋊C₈	—	4	1	no	—
129	#11637	—	43	1	no	maximal, Sylow
130	C₂²×C₄₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
131	C₂×D₄₃	—	2	1	no	—
132	C₂×D₄₃	—	2	1	no	—
133	C₂×D₄₃	C₂²⋊C₁₆	1	2	yes	—
134	C₄×C₄₃	—	2	2	no	—
135	C₄×C₄₃	—	2	1	no	—
136	C₄×C₄₃	—	4	1	no	—
137	C₄×C₄₃	#419	1	1	yes	characteristic
138	C₄×C₄₃	#421	1	1	yes	characteristic
139	C₄×C₄₃	C₂²×C₁₆	1	1	yes	characteristic
140	C₄×C₄₃	C₂×M₃₂	1	1	yes	characteristic
141	Dic₄₃	—	2	2	no	—
142	Dic₄₃	—	2	1	no	—
143	Dic₄₃	—	2	1	no	—
144	Dic₄₃	—	4	1	no	—
145	Dic₄₃	C₂²⋊C₁₆	1	2	yes	—
146	C₂×C₈×Q₈	—	43	1	no	—
147	#2162	—	43	1	no	—
148	#2162	—	43	1	no	—
149	Q₈⋊C₁₆	—	43	1	no	—
150	Q₈⋊C₁₆	—	43	1	no	—
151	Q₈⋊C₁₆	—	43	1	no	—
152	Q₈⋊C₁₆	D₄₃	1	1	yes	characteristic, complemented
153	C₂×C₄₃	#1479	1	1	yes	characteristic
154	C₂×C₄₃	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
155	C₂×C₄₃	#2117	1	1	yes	characteristic
156	D₄₃	—	2	2	no	—
157	C₂×C₄×Q₈	—	43	1	no	—
158	C₈×Q₈	—	43	1	no	—
159	C₈×Q₈	—	43	1	no	—
160	C₈×Q₈	—	43	1	no	—
161	C₈×Q₈	—	86	1	no	—
162	C₈×Q₈	—	86	1	no	—
163	C₈×Q₈	C₂×D₄₃	1	1	yes	characteristic
164	C₂×C₄⋊C₈	—	43	1	no	—
165	C₂×C₄⋊C₈	—	86	1	no	—
166	C₂×C₄×C₈	—	43	1	no	—
167	C₂×C₄×C₈	—	86	1	no	—
168	C₄⋊C₁₆	—	43	1	no	—
169	C₄⋊C₁₆	—	86	1	no	—
170	C₄⋊C₁₆	C₂×D₄₃	1	1	yes	characteristic
171	C₂²⋊C₁₆	—	86	1	no	—
172	C₂²⋊C₁₆	—	86	1	no	—
173	C₄×C₁₆	—	43	1	no	—
174	C₄×C₁₆	—	86	1	no	—
175	C₄×C₁₆	C₂×D₄₃	1	1	yes	characteristic
176	C₄₃	#11637	1	1	yes	characteristic, complemented, Sylow
177	C₂²×Q₈	—	43	1	no	—
178	C₂²×C₈	—	43	1	no	—
179	C₂²×C₈	—	43	1	no	—
180	C₂²×C₈	—	86	1	no	—
181	C₄×Q₈	—	43	1	no	—
182	C₄×Q₈	—	43	1	no	—
183	C₄×Q₈	—	43	1	no	—
184	C₄×Q₈	—	86	1	no	—
185	C₄×Q₈	—	86	1	no	—
186	C₄×Q₈	C₄×D₄₃	1	1	yes	characteristic
187	C₂×C₄⋊C₄	—	43	1	no	—
188	C₂×C₄⋊C₄	—	86	1	no	—
189	C₂×C₄²	—	43	1	no	—
190	C₂×C₄²	—	86	1	no	—
191	C₂×C₁₆	—	2	1	no	—
192	C₂×C₁₆	—	2	1	no	—
193	C₂×C₁₆	—	86	1	no	—
194	C₂×C₁₆	—	86	1	no	—
195	C₄⋊C₈	—	2	1	no	—
196	C₄⋊C₈	—	43	1	no	—
197	C₄⋊C₈	—	86	1	no	—
198	C₄⋊C₈	—	86	1	no	—
199	C₄⋊C₈	—	86	1	no	—
200	C₄⋊C₈	—	86	1	no	—
201	C₄⋊C₈	C₄×D₄₃	1	1	yes	characteristic
202	C₄×C₈	—	2	1	no	—
203	C₄×C₈	—	43	1	no	—
204	C₄×C₈	—	43	1	no	—
205	C₄×C₈	—	43	1	no	—
206	C₄×C₈	—	86	1	no	—
207	C₄×C₈	—	86	1	no	—
208	C₄×C₈	—	86	1	no	—
209	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
210	C₂×Q₈	—	43	4	no	—
211	C₂×Q₈	—	43	1	no	—
212	C₂×Q₈	—	86	1	no	—
213	C₂×Q₈	—	86	2	no	—
214	C₂×Q₈	C₈×D₄₃	1	1	yes	characteristic
215	C₂²×C₄	—	43	1	no	—
216	C₂²×C₄	—	43	1	no	—
217	C₂²×C₄	—	43	1	no	—
218	C₂²×C₄	—	86	1	no	—
219	C₂²×C₄	—	86	1	no	—
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	43	1	no	—
222	C₂×C₈	—	43	1	no	—
223	C₂×C₈	—	86	1	no	—
224	C₂×C₈	—	86	1	no	—
225	C₂×C₈	—	86	1	no	—
226	C₂×C₈	—	172	1	no	—
227	C₂×C₈	—	172	2	no	—
228	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
229	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
230	C₄⋊C₄	—	2	1	no	—
231	C₄⋊C₄	—	43	1	no	—
232	C₄⋊C₄	—	86	1	no	—
233	C₄⋊C₄	—	86	1	no	—
234	C₄⋊C₄	—	86	1	no	—
235	C₄⋊C₄	—	86	1	no	—
236	C₄⋊C₄	C₈×D₄₃	1	1	yes	characteristic
237	C₄²	—	2	1	no	—
238	C₄²	—	43	1	no	—
239	C₄²	—	43	1	no	—
240	C₄²	—	43	1	no	—
241	C₄²	—	86	1	no	—
242	C₄²	—	86	1	no	—
243	C₄²	—	86	1	no	—
244	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
245	C₁₆	—	4	1	no	—
246	C₁₆	—	4	1	no	—
247	C₁₆	—	172	1	no	—
248	C₁₆	—	172	1	no	—
249	C₂³	—	43	1	no	—
250	Q₈	—	2	1	no	—
251	Q₈	—	43	2	no	—
252	Q₈	—	86	4	no	—
253	Q₈	—	86	1	no	—
254	Q₈	C₁₆×D₄₃	1	2	yes	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	43	1	no	—
258	C₂×C₄	—	43	2	no	—
259	C₂×C₄	—	43	2	no	—
260	C₂×C₄	—	43	1	no	—
261	C₂×C₄	—	43	1	no	—
262	C₂×C₄	—	86	4	no	—
263	C₂×C₄	—	86	1	no	—
264	C₂×C₄	—	86	1	no	—
265	C₂×C₄	—	86	1	no	—
266	C₂×C₄	—	86	1	no	—
267	C₂×C₄	—	172	1	no	—
268	C₂×C₄	—	172	2	no	—
269	C₂×C₄	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
270	C₂×C₄	C₂×C₈×D₄₃	1	1	yes	characteristic
271	C₂×C₄	M₁₆×D₄₃	1	1	yes	characteristic
272	C₈	—	2	1	no	—
273	C₈	—	4	1	no	—
274	C₈	—	86	1	no	—
275	C₈	—	86	1	no	—
276	C₈	—	172	1	no	—
277	C₈	#13063354	1	1	yes	characteristic
278	C₈	#13063357	1	1	yes	characteristic
279	C₂²	—	43	2	no	—
280	C₂²	—	86	1	no	—
281	C₂²	—	86	1	no	—
282	C₂²	C₂²⋊C₈×D₄₃	1	1	yes	characteristic
283	C₄	—	2	2	no	—
284	C₄	—	2	1	no	—
285	C₄	—	4	1	no	—
286	C₄	—	43	2	no	—
287	C₄	—	86	1	no	—
288	C₄	—	86	2	no	—
289	C₄	—	86	1	no	—
290	C₄	—	172	1	no	—
291	C₄	#14582906	1	1	yes	characteristic
292	C₄	#14582914	1	1	yes	characteristic
293	C₄	C₂×C₁₆×D₄₃	1	1	yes	characteristic
294	C₄	M₃₂×D₄₃	1	1	yes	characteristic
295	C₂	—	86	2	no	—
296	C₂	#19132988	1	1	yes	characteristic
297	C₂	C₂²⋊C₁₆×D₄₃	1	1	yes	characteristic
298	C₂	#19136221	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₄₃
commutator	C₄×C₄₃	C₂²×C₁₆
Frattini	C₄×C₈	C₂²×D₄₃
Fitting	C₄₃×Q₈⋊C₁₆	C₂
socle	C₂²×C₄₃	C₂×C₂²⋊C₈
prime radical	C₂²×D₄₃	C₂²⋊C₈
cosocle	C₄×C₈×C₄₃	C₂³
hypercenter	Q₈⋊C₁₆	D₄₃
hypocenter	C₄₃	#11637
LCS	G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ Q₈⋊C₁₆

Complements

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

N	H ≅ G/N	type
4. #19132575	C₂	semidirect
5. Q₈⋊(C₄₃⋊C₁₆)	C₂	semidirect
6. #19132573	C₂	semidirect
7. C₄₃×Q₈⋊C₁₆	C₂	semidirect
62. Q₈×D₄₃	C₁₆	semidirect
152. Q₈⋊C₁₆	D₄₃	semidirect
176. C₄₃	#11637	semidirect

Maximal subgroups

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

subgroup	index
C₈×Q₈×D₄₃	2
#19136393	2
#19136392	2
#19132575	2
Q₈⋊(C₄₃⋊C₁₆)	2
#19132573	2
C₄₃×Q₈⋊C₁₆	2
#11637	43