Subgroups — #25250223

Order 11008.

572 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19143308	C₂	1	1	yes	maximal, characteristic
2	#19136428	C₂	1	2	yes	maximal, complemented
3	#19136426	C₂	1	2	yes	maximal, complemented
4	C₁₆×D₁₇₂	C₂	1	2	yes	maximal, complemented
5	#19136414	C₂	1	2	yes	maximal, complemented
6	#19136394	C₂	1	1	yes	maximal, characteristic, complemented
7	#19136207	C₂	1	4	yes	maximal, complemented
8	C₄₃×#2162	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13063331	C₂²	1	2	yes	—
10	#14583449	C₄	1	1	yes	characteristic
11	C₂×#13063264	C₂²	1	4	yes	—
12	C₂×C₁₆×D₄₃	C₂²	1	4	yes	—
13	#14582978	C₄	1	2	yes	—
14	#14582978	C₄	1	2	yes	—
15	C₈×D₁₇₂	C₂²	1	2	yes	—
16	C₈×D₁₇₂	C₂²	1	2	yes	—
17	#14582923	C₄	1	1	yes	characteristic
18	#14582865	C₄	1	2	yes	—
19	C₄⋊(C₄₃⋊C₁₆)	C₂²	1	1	yes	characteristic
20	C₄⋊(C₄₃⋊C₁₆)	C₂²	1	1	yes	characteristic
21	C₂²⋊(C₄₃⋊C₁₆)	C₂²	1	2	yes	—
22	(C₂×D₄₃)⋊C₁₆	C₂²	1	4	yes	—
23	(C₂×D₄₃)⋊C₁₆	C₂²	1	4	yes	—
24	C₄×C₄₃⋊C₁₆	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
26	C₄₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
27	C₄₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
28	C₄₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
29	C₄×C₁₆×C₄₃	C₂²	1	2	yes	—
30	C₂×#10621898	C₂×C₄	1	2	yes	—
31	#13063331	—	4	2	no	—
32	#13063331	—	4	2	no	—
33	C₂×#10621877	C₂×C₄	1	2	yes	—
34	C₂×#10621872	C₂×C₄	1	4	yes	—
35	C₂×C₈×D₄₃	C₂³	1	4	yes	—
36	#13063325	C₈	1	1	yes	characteristic
37	#13063322	C₈	1	1	yes	characteristic
38	#13063313	C₈	1	2	yes	—
39	#13063299	C₈	1	2	yes	—
40	#13063299	C₈	1	2	yes	—
41	C₄×D₁₇₂	C₂×C₄	1	2	yes	—
42	C₄×D₁₇₂	C₂×C₄	1	2	yes	—
43	#13063264	—	4	4	no	—
44	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
45	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
46	C₁₆×D₄₃	—	4	4	no	—
47	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	2	yes	—
48	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	2	yes	—
49	(C₂×D₄₃)⋊C₈	C₂×C₄	1	4	yes	—
50	(C₂×D₄₃)⋊C₈	C₂×C₄	1	4	yes	—
51	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
54	C₂×C₁₆×C₄₃	C₂³	1	2	yes	—
55	C₂×C₁₆×C₄₃	C₂³	1	2	yes	—
56	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
57	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
58	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
59	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₄₃	C₂×C₈	1	1	yes	characteristic
61	C₂²×C₄×C₄₃	C₂×C₈	1	1	yes	characteristic
62	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₂²⋊D₄₃	C₂×C₈	1	4	yes	—
64	#10621898	—	4	2	no	—
65	#10621898	—	4	2	no	—
66	C₂×D₁₇₂	C₂×C₈	1	2	yes	—
67	C₂×D₁₇₂	C₂×C₈	1	2	yes	—
68	C₂×D₁₇₂	C₂×C₈	1	2	yes	—
69	C₂×C₄×D₄₃	C₂²×C₄	1	4	yes	—
70	C₂×Dic₈₆	C₂×C₈	1	2	yes	—
71	C₂×C₈×C₄₃	—	2	1	no	—
72	C₂×C₈×C₄₃	—	2	1	no	—
73	C₂×C₈×C₄₃	—	4	1	no	—
74	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
78	C₄²×C₄₃	C₂×C₈	1	1	yes	characteristic
79	C₄²×C₄₃	C₂×C₈	1	1	yes	characteristic
80	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
81	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
82	(C₂×D₄₃)⋊C₄	C₂×C₈	1	4	yes	—
83	(C₂×D₄₃)⋊C₄	C₂×C₈	1	4	yes	—
84	C₄⋊Dic₄₃	C₂×C₈	1	2	yes	—
85	C₄⋊Dic₄₃	C₂×C₈	1	2	yes	—
86	#10621877	—	4	2	no	—
87	C₂×C₄₃⋊C₈	C₂²×C₄	1	4	yes	—
88	#10621872	—	4	4	no	—
89	C₈×D₄₃	—	2	4	no	—
90	C₈×D₄₃	—	2	4	no	—
91	C₁₆×C₄₃	—	2	2	no	—
92	C₁₆×C₄₃	—	2	2	no	—
93	C₄₃⋊C₁₆	—	2	2	no	—
94	C₄₃⋊C₁₆	—	2	2	no	—
95	C₂³×C₄₃	C₂²×C₈	1	1	yes	characteristic
96	C₂²×D₄₃	C₂²×C₈	1	4	yes	—
97	C₂×C₄×C₄₃	—	2	1	no	—
98	C₂×C₄×C₄₃	—	2	1	no	—
99	C₂×C₄×C₄₃	—	2	2	no	—
100	C₂×C₄×C₄₃	—	2	2	no	—
101	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
102	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
103	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
104	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
105	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
106	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
107	C₂²⋊D₄₃	—	4	4	no	—
108	C₂×Dic₄₃	C₂²×C₈	1	4	yes	—
109	D₁₇₂	—	2	4	no	—
110	D₁₇₂	—	2	4	no	—
111	D₁₇₂	—	4	2	no	—
112	C₄×D₄₃	—	2	4	no	—
113	C₄×D₄₃	—	2	4	no	—
114	Dic₈₆	—	4	2	no	—
115	C₈×C₄₃	—	2	1	no	—
116	C₈×C₄₃	—	2	1	no	—
117	C₈×C₄₃	—	2	1	no	—
118	C₈×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
119	C₈×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
120	C₄₃⋊C₈	—	2	4	no	—
121	#33493	—	43	1	no	maximal, Sylow
122	C₂²×C₄₃	—	2	1	no	—
123	C₂²×C₄₃	—	2	1	no	—
124	C₂²×C₄₃	—	2	1	no	—
125	C₂²×C₄₃	C₂³×C₈	1	1	yes	characteristic
126	C₂×D₄₃	—	2	4	no	—
127	C₂×D₄₃	—	2	4	no	—
128	C₂×D₄₃	—	2	4	no	—
129	C₄×C₄₃	—	2	1	no	—
130	C₄×C₄₃	—	2	1	no	—
131	C₄×C₄₃	—	2	1	no	—
132	C₄×C₄₃	#504	1	2	yes	—
133	C₄×C₄₃	#504	1	2	yes	—
134	C₄×C₄₃	#518	1	1	yes	characteristic
135	C₄×C₄₃	#568	1	1	yes	characteristic
136	Dic₄₃	—	2	4	no	—
137	#2931	—	43	1	no	—
138	#2169	—	43	2	no	—
139	#2169	—	43	2	no	—
140	D₄×C₁₆	—	43	2	no	—
141	D₄×C₁₆	—	43	2	no	—
142	#2162	—	43	1	no	—
143	#2162	D₄₃	1	1	yes	characteristic, complemented
144	#2113	—	43	4	no	—
145	C₂×C₄₃	—	4	1	no	—
146	C₂×C₄₃	#2972	1	1	yes	characteristic
147	C₂×C₄₃	C₂×#504	1	1	yes	characteristic
148	C₂×C₄₃	C₂×#504	1	1	yes	characteristic
149	D₄₃	—	4	4	no	—
150	C₂×#131	—	43	2	no	—
151	#532	—	43	1	no	—
152	C₂×M₃₂	—	43	4	no	—
153	C₂²×C₁₆	—	43	4	no	—
154	#436	—	43	2	no	—
155	#436	—	43	2	no	—
156	C₈×D₄	—	43	2	no	—
157	C₈×D₄	—	43	2	no	—
158	#423	—	43	1	no	—
159	#408	—	43	2	no	—
160	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
161	C₄⋊C₁₆	—	43	1	no	—
162	C₄⋊C₁₆	—	43	1	no	—
163	C₄⋊C₁₆	C₂×D₄₃	1	1	yes	characteristic
164	C₄⋊C₁₆	C₂×D₄₃	1	1	yes	characteristic
165	C₂²⋊C₁₆	—	43	2	no	—
166	C₂²⋊C₁₆	—	43	4	no	—
167	C₂²⋊C₁₆	—	43	4	no	—
168	C₂²⋊C₁₆	C₂×D₄₃	1	2	yes	—
169	C₄×C₁₆	—	43	2	no	—
170	C₄×C₁₆	C₂×D₄₃	1	2	yes	—
171	C₄₃	#33493	1	1	yes	characteristic, complemented, Sylow
172	C₂×#41	—	43	2	no	—
173	#131	—	172	2	no	—
174	#131	—	172	2	no	—
175	C₂×M₁₆	—	43	2	no	—
176	C₂×M₁₆	—	43	4	no	—
177	C₂²×C₈	—	43	4	no	—
178	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
179	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
180	C₄⋊Q₈	—	43	1	no	—
181	#127	—	43	1	no	—
182	#124	—	43	2	no	—
183	#121	—	43	2	no	—
184	#121	—	43	2	no	—
185	C₄×D₄	—	43	2	no	—
186	C₄×D₄	—	43	2	no	—
187	C₂×C₄²	C₄×D₄₃	1	1	yes	characteristic
188	M₃₂	—	172	4	no	—
189	C₂×C₁₆	—	43	2	no	—
190	C₂×C₁₆	—	43	2	no	—
191	C₂×C₁₆	—	172	4	no	—
192	C₂×C₁₆	C₂²×D₄₃	1	2	yes	—
193	C₂×C₁₆	C₂²×D₄₃	1	2	yes	—
194	C₄⋊C₈	—	43	2	no	—
195	C₄⋊C₈	—	43	2	no	—
196	C₂²⋊C₈	—	43	4	no	—
197	C₂²⋊C₈	—	43	4	no	—
198	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
199	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
200	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
201	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
202	#41	—	172	2	no	—
203	#41	—	172	2	no	—
204	C₂×Q₈	—	43	2	no	—
205	C₂×D₄	—	43	4	no	—
206	C₂×D₄	—	43	2	no	—
207	C₂×D₄	—	43	2	no	—
208	C₂×D₄	—	43	2	no	—
209	C₂²×C₄	—	43	4	no	—
210	C₂²×C₄	C₈×D₄₃	1	1	yes	characteristic
211	C₂²×C₄	C₈×D₄₃	1	1	yes	characteristic
212	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
213	M₁₆	—	172	4	no	—
214	M₁₆	—	172	2	no	—
215	C₂×C₈	—	2	1	no	—
216	C₂×C₈	—	2	1	no	—
217	C₂×C₈	—	4	1	no	—
218	C₂×C₈	—	43	4	no	—
219	C₂×C₈	—	86	4	no	—
220	C₂×C₈	—	86	4	no	—
221	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
222	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
223	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
224	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
225	C₄⋊C₄	—	43	2	no	—
226	C₄⋊C₄	—	43	2	no	—
227	C₂²⋊C₄	—	43	4	no	—
228	C₂²⋊C₄	—	43	4	no	—
229	C₄²	C₈×D₄₃	1	1	yes	characteristic
230	C₄²	C₈×D₄₃	1	1	yes	characteristic
231	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
232	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
233	C₁₆	—	2	2	no	—
234	C₁₆	—	2	2	no	—
235	C₁₆	—	86	2	no	—
236	C₁₆	—	86	2	no	—
237	C₂³	—	43	4	no	—
238	C₂³	C₂×C₈×D₄₃	1	1	yes	characteristic
239	Q₈	—	172	2	no	—
240	D₄	—	86	4	no	—
241	D₄	—	86	4	no	—
242	D₄	—	172	2	no	—
243	D₄	—	172	4	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	43	4	no	—
249	C₂×C₄	—	86	4	no	—
250	C₂×C₄	—	86	4	no	—
251	C₂×C₄	C₂×C₈×D₄₃	1	1	yes	characteristic
252	C₂×C₄	C₂×C₈×D₄₃	1	1	yes	characteristic
253	C₂×C₄	C₂×C₈×D₄₃	1	1	yes	characteristic
254	C₂×C₄	C₂×C₈×D₄₃	1	1	yes	characteristic
255	C₂×C₄	C₂×C₈×D₄₃	1	1	yes	characteristic
256	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
257	C₈	—	2	1	no	—
258	C₈	—	2	1	no	—
259	C₈	—	2	1	no	—
260	C₈	—	86	4	no	—
261	C₈	#13063371	1	1	yes	characteristic
262	C₈	#13063371	1	1	yes	characteristic
263	C₂²	—	2	1	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	86	4	no	—
267	C₂²	—	86	4	no	—
268	C₂²	—	86	4	no	—
269	C₂²	C₂²×C₈×D₄₃	1	1	yes	characteristic
270	C₄	—	2	1	no	—
271	C₄	—	2	1	no	—
272	C₄	—	2	1	no	—
273	C₄	—	86	4	no	—
274	C₄	#14583315	1	2	yes	—
275	C₄	#14583317	1	2	yes	—
276	C₄	#14583373	1	1	yes	characteristic
277	C₄	#14583665	1	1	yes	characteristic
278	C₂	—	4	1	no	—
279	C₂	—	172	4	no	—
280	C₂	#19143618	1	1	yes	characteristic
281	C₂	C₂×#14583315	1	1	yes	characteristic
282	C₂	C₂×#14583317	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₄₃
commutator, socle	C₂²×C₄₃	C₂³×C₈
Fitting	C₄₃×#2162	C₂
prime radical	#14583449	C₄
cosocle	C₂×C₈×C₄₃	C₂⁴
hypercenter	#2162	D₄₃
hypocenter	C₄₃	#33493
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₈ ◁ #2162

Complements

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

N	H ≅ G/N	type
2. #19136428	C₂	semidirect
3. #19136426	C₂	semidirect
4. C₁₆×D₁₇₂	C₂	semidirect
5. #19136414	C₂	semidirect
6. #19136394	C₂	semidirect
7. #19136207	C₂	semidirect
8. C₄₃×#2162	C₂	semidirect
143. #2162	D₄₃	semidirect
171. C₄₃	#33493	semidirect

Maximal subgroups

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

subgroup	index
#19143308	2
#19136428	2
#19136426	2
C₁₆×D₁₇₂	2
#19136414	2
#19136394	2
#19136207	2
C₄₃×#2162	2
#33493	43