Subgroups — #24989533

Order 11008.

454 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19134587	C₂	1	2	yes	maximal
2	#19133354	C₂	1	1	yes	maximal, characteristic, complemented
3	#19133351	C₂	1	1	yes	maximal, characteristic, complemented
4	#19132470	C₂	1	2	yes	maximal, complemented
5	C₂×C₄₃×#436	C₂	1	1	yes	maximal, characteristic
6	#14582923	C₂²	1	2	yes	—
7	#14582923	C₂²	1	2	yes	—
8	C₂×#13063242	—	2	2	no	—
9	#14582668	—	2	2	no	—
10	#14582652	—	2	1	no	—
11	#14582652	—	2	1	no	—
12	#14582650	—	2	1	no	—
13	#14582650	—	2	1	no	—
14	C₂²×M₁₆×C₄₃	C₄	1	2	yes	—
15	C₂×C₄×D₄×C₄₃	C₂²	1	1	yes	characteristic
16	C₄₃×#436	—	2	1	no	—
17	C₄₃×#436	—	2	1	no	—
18	C₄₃×#436	—	2	1	no	—
19	C₄₃×#436	—	2	1	no	—
20	C₂×C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
21	C₂×C₂²⋊C₈×C₄₃	C₄	1	2	yes	—
22	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
23	C₂×#10621877	—	2	2	no	—
24	C₂×#10621877	—	2	2	no	—
25	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
26	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
27	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
28	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
29	#13063242	—	4	4	no	—
30	C₂²×D₄×C₄₃	D₄	1	1	yes	characteristic
31	C₂³×C₄×C₄₃	C₂×C₄	1	2	yes	—
32	C₂×M₁₆×C₄₃	—	2	2	no	—
33	C₂×M₁₆×C₄₃	—	2	2	no	—
34	C₂×M₁₆×C₄₃	—	4	4	no	—
35	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₄₃	C₂×C₄	1	2	yes	—
37	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
38	C₄×D₄×C₄₃	—	2	1	no	—
39	C₄×D₄×C₄₃	—	2	1	no	—
40	C₄×D₄×C₄₃	—	2	1	no	—
41	C₄×D₄×C₄₃	—	2	1	no	—
42	C₂×C₄⋊C₄×C₄₃	D₄	1	1	yes	characteristic
43	C₂×C₂²⋊C₄×C₄₃	Q₈	1	2	yes	—
44	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
45	C₄⋊C₈×C₄₃	—	2	1	no	—
46	C₄⋊C₈×C₄₃	—	2	1	no	—
47	C₂²⋊C₈×C₄₃	—	2	2	no	—
48	C₂²⋊C₈×C₄₃	—	2	2	no	—
49	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
50	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
51	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
52	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
53	C₂⁴×C₄₃	C₄⋊C₄	1	2	yes	—
54	C₂×D₄×C₄₃	—	2	1	no	—
55	C₂×D₄×C₄₃	—	2	1	no	—
56	C₂×D₄×C₄₃	—	2	2	no	—
57	C₂×D₄×C₄₃	—	2	2	no	—
58	C₂²×C₄×C₄₃	—	2	2	no	—
59	C₂²×C₄×C₄₃	—	2	2	no	—
60	C₂²×C₄×C₄₃	—	4	2	no	—
61	C₂²×C₄×C₄₃	—	4	2	no	—
62	C₂²×C₄×C₄₃	C₄⋊C₄	1	2	yes	—
63	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
66	M₁₆×C₄₃	—	4	4	no	—
67	M₁₆×C₄₃	—	4	4	no	—
68	C₂×C₈×C₄₃	—	2	2	no	—
69	C₂×C₈×C₄₃	—	2	2	no	—
70	C₂×C₈×C₄₃	—	4	1	no	—
71	C₂×C₈×C₄₃	—	4	2	no	—
72	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
76	C₄⋊C₄×C₄₃	—	2	1	no	—
77	C₄⋊C₄×C₄₃	—	2	1	no	—
78	C₂²⋊C₄×C₄₃	—	2	2	no	—
79	C₂²⋊C₄×C₄₃	—	2	2	no	—
80	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₄₃	#41	1	1	yes	characteristic
83	C₄²×C₄₃	#41	1	1	yes	characteristic
84	#10621877	—	4	2	no	—
85	#10621877	—	4	2	no	—
86	#10621877	—	4	2	no	—
87	#10621877	—	4	2	no	—
88	C₂×C₄₃⋊C₈	—	4	2	no	—
89	C₂×C₄₃⋊C₈	—	4	2	no	—
90	C₂³×C₄₃	—	2	2	no	—
91	C₂³×C₄₃	—	2	4	no	—
92	C₂³×C₄₃	—	4	2	no	—
93	C₂³×C₄₃	—	4	2	no	—
94	C₂³×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
95	D₄×C₄₃	—	4	2	no	—
96	D₄×C₄₃	—	4	1	no	—
97	D₄×C₄₃	—	4	1	no	—
98	C₂×C₄×C₄₃	—	2	1	no	—
99	C₂×C₄×C₄₃	—	2	1	no	—
100	C₂×C₄×C₄₃	—	2	2	no	—
101	C₂×C₄×C₄₃	—	4	1	no	—
102	C₂×C₄×C₄₃	—	4	4	no	—
103	C₂×C₄×C₄₃	—	4	2	no	—
104	C₂×C₄×C₄₃	—	4	2	no	—
105	C₂×C₄×C₄₃	—	4	2	no	—
106	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
108	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
109	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
110	C₂×C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
111	C₂×C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
112	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
113	C₂×C₄×C₄₃	#137	1	2	yes	—
114	C₈×C₄₃	—	2	1	no	—
115	C₈×C₄₃	—	2	1	no	—
116	C₈×C₄₃	—	2	1	no	—
117	C₈×C₄₃	—	4	2	no	—
118	C₈×C₄₃	#100	1	2	yes	complemented
119	C₄₃⋊C₈	—	8	2	no	—
120	C₄₃⋊C₈	—	8	2	no	—
121	#8618	—	43	1	no	maximal, Sylow
122	C₂²×C₄₃	—	2	1	no	—
123	C₂²×C₄₃	—	2	1	no	—
124	C₂²×C₄₃	—	4	2	no	—
125	C₂²×C₄₃	—	4	2	no	—
126	C₂²×C₄₃	—	4	2	no	—
127	C₂²×C₄₃	—	8	4	no	—
128	C₂²×C₄₃	#380	1	1	yes	characteristic
129	C₂²×C₄₃	#428	1	2	yes	—
130	C₄×C₄₃	—	2	1	no	—
131	C₄×C₄₃	—	2	1	no	—
132	C₄×C₄₃	—	2	1	no	—
133	C₄×C₄₃	—	4	2	no	—
134	C₄×C₄₃	C₂×#100	1	1	yes	characteristic
135	C₄×C₄₃	#413	1	1	yes	characteristic
136	C₄×C₄₃	#468	1	2	yes	—
137	C₄×C₄₃	#469	1	1	yes	characteristic
138	C₄×C₄₃	#470	1	1	yes	characteristic
139	C₂×#436	D₄₃	1	1	yes	characteristic
140	#1800	—	43	2	no	—
141	#1567	—	43	1	no	—
142	#1566	—	43	1	no	—
143	#1292	—	43	2	no	—
144	C₂×C₄₃	—	2	2	no	—
145	C₂×C₄₃	—	8	2	no	—
146	C₂×C₄₃	#1800	1	1	yes	characteristic
147	C₂×C₄₃	#1809	1	1	yes	characteristic
148	C₂×C₄₃	#1813	1	1	yes	characteristic
149	C₂²×M₁₆	Dic₄₃	1	2	yes	—
150	C₂×C₄×D₄	C₂×D₄₃	1	1	yes	characteristic
151	#436	—	2	1	no	—
152	#436	—	2	1	no	—
153	#436	—	2	1	no	—
154	#436	—	2	1	no	—
155	#423	—	43	2	no	—
156	#423	—	43	2	no	—
157	C₂×C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
158	C₂×#100	—	86	2	no	—
159	C₂×C₂²⋊C₈	Dic₄₃	1	2	yes	—
160	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
161	#343	—	86	2	no	—
162	#335	—	86	1	no	—
163	#335	—	86	1	no	—
164	#334	—	86	1	no	—
165	#334	—	86	1	no	—
166	C₄₃	#8618	1	1	yes	characteristic, complemented, Sylow
167	C₂²×D₄	D₁₇₂	1	1	yes	characteristic
168	C₂³×C₄	C₂×Dic₄₃	1	2	yes	—
169	C₂×M₁₆	—	2	2	no	—
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	—	4	4	no	—
172	C₂×M₁₆	—	86	2	no	—
173	C₂×M₁₆	—	86	2	no	—
174	C₂²×C₈	C₂×Dic₄₃	1	1	yes	characteristic
175	C₂²×C₈	C₂×Dic₄₃	1	2	yes	—
176	C₂²×C₈	C₂×Dic₄₃	1	1	yes	characteristic
177	C₄×D₄	—	2	1	no	—
178	C₄×D₄	—	2	1	no	—
179	C₄×D₄	—	2	1	no	—
180	C₄×D₄	—	2	1	no	—
181	C₂×C₄⋊C₄	D₁₇₂	1	1	yes	characteristic
182	C₂×C₂²⋊C₄	Dic₈₆	1	2	yes	—
183	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
184	C₄⋊C₈	—	2	1	no	—
185	C₄⋊C₈	—	2	1	no	—
186	C₄⋊C₈	—	86	2	no	—
187	C₄⋊C₈	—	86	2	no	—
188	C₄⋊C₈	—	86	2	no	—
189	C₄⋊C₈	—	86	2	no	—
190	#100	—	172	4	no	—
191	C₂²⋊C₈	—	2	2	no	—
192	C₂²⋊C₈	—	2	2	no	—
193	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
194	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
195	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
196	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
197	C₂⁴	C₄⋊Dic₄₃	1	2	yes	—
198	C₂×D₄	—	2	1	no	—
199	C₂×D₄	—	2	1	no	—
200	C₂×D₄	—	2	2	no	—
201	C₂×D₄	—	2	2	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	4	2	no	—
205	C₂²×C₄	—	4	2	no	—
206	C₂²×C₄	C₄⋊Dic₄₃	1	2	yes	—
207	C₂²×C₄	C₂×Dic₈₆	1	1	yes	characteristic
208	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
209	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
210	M₁₆	—	4	4	no	—
211	M₁₆	—	4	4	no	—
212	M₁₆	—	172	2	no	—
213	M₁₆	—	172	2	no	—
214	M₁₆	—	172	2	no	—
215	M₁₆	—	172	2	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	4	1	no	—
219	C₂×C₈	—	4	2	no	—
220	C₂×C₈	—	172	2	no	—
221	C₂×C₈	—	172	2	no	—
222	C₂×C₈	C₂²⋊Dic₄₃	1	1	yes	characteristic
223	C₂×C₈	C₂²⋊Dic₄₃	1	1	yes	characteristic
224	C₂×C₈	C₂²⋊Dic₄₃	1	1	yes	characteristic
225	C₂×C₈	C₂²⋊Dic₄₃	1	1	yes	characteristic
226	C₄⋊C₄	—	2	1	no	—
227	C₄⋊C₄	—	2	1	no	—
228	C₂²⋊C₄	—	2	2	no	—
229	C₂²⋊C₄	—	2	2	no	—
230	C₄²	#10621898	1	1	yes	characteristic
231	C₄²	#10621898	1	1	yes	characteristic
232	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
233	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
234	C₂³	—	2	2	no	—
235	C₂³	—	2	4	no	—
236	C₂³	—	4	2	no	—
237	C₂³	—	4	2	no	—
238	C₂³	C₂×C₄⋊Dic₄₃	1	1	yes	characteristic
239	D₄	—	4	2	no	—
240	D₄	—	4	1	no	—
241	D₄	—	4	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	4	1	no	—
246	C₂×C₄	—	4	4	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
251	C₂×C₄	#13063284	1	1	yes	characteristic
252	C₂×C₄	#13063299	1	1	yes	characteristic
253	C₂×C₄	#13063299	1	1	yes	characteristic
254	C₂×C₄	#13063304	1	1	yes	characteristic
255	C₂×C₄	#13063304	1	1	yes	characteristic
256	C₂×C₄	#13063353	1	2	yes	—
257	C₂×C₄	#13063358	1	2	yes	—
258	C₈	—	2	1	no	—
259	C₈	—	2	1	no	—
260	C₈	—	2	1	no	—
261	C₈	—	4	2	no	—
262	C₈	—	344	2	no	—
263	C₈	—	344	2	no	—
264	C₈	#13063242	1	2	yes	complemented
265	C₂²	—	2	1	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	4	2	no	—
268	C₂²	—	4	2	no	—
269	C₂²	—	4	2	no	—
270	C₂²	—	8	4	no	—
271	C₂²	#14582759	1	1	yes	characteristic
272	C₂²	#14582941	1	2	yes	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	4	2	no	—
277	C₄	C₂×#13063242	1	1	yes	characteristic
278	C₄	#14582883	1	1	yes	characteristic
279	C₄	#14583145	1	2	yes	—
280	C₄	#14583152	1	1	yes	characteristic
281	C₄	#14583157	1	1	yes	characteristic
282	C₂	—	2	2	no	—
283	C₂	—	8	2	no	—
284	C₂	#19134587	1	1	yes	characteristic
285	C₂	#19134628	1	1	yes	characteristic
286	C₂	#19134645	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#14582759
commutator	C₂²×C₄×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₂×C₄₃×#436	C₂
socle	C₂²×C₄₃	#380
prime radical	C₂²×D₄×C₄₃	D₄
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	C₂×#436	D₄₃
hypocenter	C₄₃	#8618
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#436

Complements

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

N	H ≅ G/N	type
2. #19133354	C₂	semidirect
3. #19133351	C₂	semidirect
4. #19132470	C₂	semidirect
118. C₈×C₄₃	#100	semidirect
166. C₄₃	#8618	semidirect
264. C₈	#13063242	semidirect

Maximal subgroups

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

subgroup	index
#19134587	2
#19133354	2
#19133351	2
#19132470	2
C₂×C₄₃×#436	2
#8618	43