Subgroups — #24983181

Order 11008.

476 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19134663	C₂	1	1	yes	maximal, characteristic
2	#19132822	C₂	1	1	yes	maximal, characteristic
3	C₄₃×#1728	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂²⋊(C₄₃⋊C₈)	—	2	1	no	—
5	C₂×C₄₃⋊#95	—	2	1	no	—
6	C₂×C₄₃⋊#95	—	2	1	no	—
7	C₂×C₄₃⋊#95	C₄	1	2	yes	—
8	#14582654	—	2	1	no	—
9	#14582654	C₄	1	2	yes	complemented
10	C₂²×M₁₆×C₄₃	—	2	1	no	—
11	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
12	C₄₃×#336	—	2	1	no	—
13	C₄₃×#336	C₄	1	2	yes	complemented
14	C₂³×Dic₄₃	—	2	2	no	—
15	C₂²×C₄₃⋊C₈	—	2	2	no	—
16	C₂²⋊(C₄₃⋊C₈)	—	4	2	no	—
17	C₄₃⋊#95	—	2	4	no	—
18	C₄₃⋊#95	—	4	1	no	—
19	C₄₃⋊#95	—	4	1	no	—
20	C₄₃⋊#95	—	4	2	no	—
21	C₂³×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₄₃	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₄₃	D₄	1	1	yes	characteristic
24	C₂×M₁₆×C₄₃	—	2	4	no	—
25	C₂×M₁₆×C₄₃	—	4	2	no	—
26	C₂×M₁₆×C₄₃	—	4	1	no	—
27	C₂×M₁₆×C₄₃	—	4	1	no	—
28	C₂²×C₈×C₄₃	—	2	2	no	—
29	C₂×C₄²×C₄₃	—	4	2	no	—
30	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₄₃	Q₈	1	1	yes	characteristic
34	C₂⁴×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
35	C₂²×C₄×C₄₃	—	2	2	no	—
36	C₂²×C₄×C₄₃	—	2	2	no	—
37	C₂²×C₄×C₄₃	—	2	2	no	—
38	C₂²×C₄×C₄₃	—	2	2	no	—
39	C₂²×C₄×C₄₃	—	2	2	no	—
40	C₂²×C₄×C₄₃	—	4	1	no	—
41	C₂²×C₄×C₄₃	—	4	1	no	—
42	C₂²×C₄×C₄₃	—	4	1	no	—
43	C₂²×C₄×C₄₃	C₄²	1	1	yes	characteristic
44	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₄₃	C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₄₃	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₄₃	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
50	C₂²×C₄×C₄₃	SD₁₆	1	1	yes	characteristic
51	C₂²×C₄×C₄₃	SD₁₆	1	1	yes	characteristic
52	C₂²×C₄×C₄₃	Q₁₆	1	1	yes	characteristic
53	C₂²×Dic₄₃	—	2	4	no	—
54	C₂²×Dic₄₃	—	4	2	no	—
55	C₂²×Dic₄₃	—	4	2	no	—
56	C₂²×Dic₄₃	—	4	2	no	—
57	C₂²×Dic₄₃	—	4	2	no	—
58	C₂²×Dic₄₃	—	4	2	no	—
59	C₂²×Dic₄₃	—	4	2	no	—
60	M₁₆×C₄₃	—	4	4	no	—
61	M₁₆×C₄₃	—	4	4	no	—
62	C₂×C₈×C₄₃	—	2	4	no	—
63	C₂×C₈×C₄₃	—	4	2	no	—
64	C₂×C₈×C₄₃	—	4	2	no	—
65	C₄²×C₄₃	—	4	1	no	—
66	C₄²×C₄₃	—	4	1	no	—
67	C₄²×C₄₃	—	4	1	no	—
68	C₄²×C₄₃	—	4	1	no	—
69	C₂×C₄₃⋊C₈	—	4	2	no	—
70	C₂×C₄₃⋊C₈	—	4	2	no	—
71	C₂×C₄₃⋊C₈	—	4	2	no	—
72	C₂³×C₄₃	—	2	1	no	—
73	C₂³×C₄₃	—	2	1	no	—
74	C₂³×C₄₃	—	2	1	no	—
75	C₂³×C₄₃	—	2	1	no	—
76	C₂³×C₄₃	#95	1	1	yes	characteristic
77	C₂³×C₄₃	#99	1	2	yes	—
78	C₂³×C₄₃	#102	1	1	yes	characteristic
79	C₂³×C₄₃	#103	1	1	yes	characteristic
80	C₂³×C₄₃	#104	1	2	yes	—
81	C₂×C₄×C₄₃	—	2	2	no	—
82	C₂×C₄×C₄₃	—	2	2	no	—
83	C₂×C₄×C₄₃	—	2	2	no	—
84	C₂×C₄×C₄₃	—	2	2	no	—
85	C₂×C₄×C₄₃	—	2	1	no	—
86	C₂×C₄×C₄₃	—	2	1	no	—
87	C₂×C₄×C₄₃	—	2	1	no	—
88	C₂×C₄×C₄₃	—	2	1	no	—
89	C₂×C₄×C₄₃	—	2	1	no	—
90	C₂×C₄×C₄₃	—	2	1	no	—
91	C₂×C₄×C₄₃	—	2	1	no	—
92	C₂×C₄×C₄₃	—	2	1	no	—
93	C₂×C₄×C₄₃	—	2	1	no	—
94	C₂×C₄×C₄₃	—	2	1	no	—
95	C₂×C₄×C₄₃	—	2	1	no	—
96	C₂×C₄×C₄₃	—	2	1	no	—
97	C₂×C₄×C₄₃	—	2	1	no	—
98	C₂×C₄×C₄₃	—	2	1	no	—
99	C₂×C₄×C₄₃	—	2	1	no	—
100	C₂×C₄×C₄₃	—	4	4	no	—
101	C₂×C₄×C₄₃	—	4	4	no	—
102	C₂×C₄×C₄₃	#102	1	1	yes	characteristic
103	C₂×C₄×C₄₃	#103	1	1	yes	characteristic
104	C₂×C₄×C₄₃	#104	1	2	yes	—
105	C₂×C₄×C₄₃	#106	1	1	yes	characteristic
106	C₂×C₄×C₄₃	#107	1	1	yes	characteristic
107	C₂×Dic₄₃	—	4	4	no	—
108	C₂×Dic₄₃	—	4	4	no	—
109	C₂×Dic₄₃	—	4	4	no	—
110	C₂×Dic₄₃	—	8	2	no	—
111	C₂×Dic₄₃	—	8	2	no	—
112	C₂×Dic₄₃	—	8	2	no	—
113	C₂×Dic₄₃	—	8	2	no	—
114	C₈×C₄₃	—	4	4	no	—
115	C₄₃⋊C₈	—	8	2	no	—
116	#7037	—	43	1	no	maximal, Sylow
117	C₂²×C₄₃	—	2	2	no	—
118	C₂²×C₄₃	—	2	1	no	—
119	C₂²×C₄₃	—	2	1	no	—
120	C₂²×C₄₃	—	2	2	no	—
121	C₂²×C₄₃	—	2	2	no	—
122	C₂²×C₄₃	—	2	1	no	—
123	C₂²×C₄₃	—	2	1	no	—
124	C₂²×C₄₃	—	2	2	no	—
125	C₂²×C₄₃	#327	1	2	yes	—
126	C₂²×C₄₃	#328	1	2	yes	—
127	C₂²×C₄₃	H(3,ℤ/4)	1	1	yes	characteristic
128	C₂²×C₄₃	#339	1	1	yes	characteristic
129	C₂²×C₄₃	#339	1	1	yes	characteristic
130	C₂²×C₄₃	#340	1	1	yes	characteristic
131	C₂²×C₄₃	#342	1	1	yes	characteristic
132	C₂²×C₄₃	#344	1	1	yes	characteristic
133	C₂²×C₄₃	#344	1	1	yes	characteristic
134	C₄×C₄₃	—	2	2	no	—
135	C₄×C₄₃	—	2	1	no	—
136	C₄×C₄₃	—	2	1	no	—
137	C₄×C₄₃	—	2	1	no	—
138	C₄×C₄₃	—	2	1	no	—
139	C₄×C₄₃	—	4	1	no	—
140	C₄×C₄₃	—	4	1	no	—
141	C₄×C₄₃	—	4	1	no	—
142	Dic₄₃	—	8	4	no	—
143	#1817	—	43	1	no	—
144	#1728	D₄₃	1	1	yes	characteristic
145	#1434	—	43	1	no	—
146	C₂×C₄₃	—	2	2	no	—
147	C₂×C₄₃	—	2	2	no	—
148	C₂×C₄₃	#1284	1	1	yes	characteristic
149	C₂×C₄₃	#1286	1	1	yes	characteristic
150	C₂×C₄₃	#1288	1	1	yes	characteristic
151	C₂×C₄₃	#1294	1	1	yes	characteristic
152	C₂×C₄₃	#1300	1	1	yes	characteristic
153	C₂×C₄₃	#1303	1	1	yes	characteristic
154	C₂×C₄₃	#1306	1	1	yes	characteristic
155	C₂²×M₁₆	—	2	1	no	—
156	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
157	C₂×C₂²⋊C₈	—	86	1	no	—
158	C₂×#95	—	43	2	no	—
159	C₂×#95	—	86	1	no	—
160	C₂×#95	—	86	1	no	—
161	#336	—	2	1	no	—
162	#336	—	43	2	no	—
163	#336	—	86	1	no	—
164	#336	Dic₄₃	1	2	yes	complemented
165	C₄₃	#7037	1	1	yes	characteristic, complemented, Sylow
166	C₂³×C₄	—	86	2	no	—
167	C₂³×C₄	C₄×D₄₃	1	1	yes	characteristic
168	C₂³×C₄	D₁₇₂	1	1	yes	characteristic
169	C₂³×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
170	C₂×M₁₆	—	2	4	no	—
171	C₂×M₁₆	—	4	2	no	—
172	C₂×M₁₆	—	4	1	no	—
173	C₂×M₁₆	—	4	1	no	—
174	C₂²×C₈	—	2	2	no	—
175	C₂²×C₈	—	86	2	no	—
176	C₂×C₄²	—	4	2	no	—
177	C₂×C₄²	Dic₈₆	1	1	yes	characteristic
178	C₂×C₄²	C₄×D₄₃	1	1	yes	characteristic
179	C₂×C₄²	C₂×Dic₄₃	1	1	yes	characteristic
180	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
181	C₂²⋊C₈	—	172	2	no	—
182	#95	—	86	4	no	—
183	#95	—	172	2	no	—
184	#95	—	172	1	no	—
185	#95	—	172	1	no	—
186	C₂⁴	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	4	1	no	—
193	C₂²×C₄	—	4	1	no	—
194	C₂²×C₄	—	4	1	no	—
195	C₂²×C₄	—	86	4	no	—
196	C₂²×C₄	—	172	2	no	—
197	C₂²×C₄	—	172	2	no	—
198	C₂²×C₄	—	172	2	no	—
199	C₂²×C₄	—	172	2	no	—
200	C₂²×C₄	—	172	2	no	—
201	C₂²×C₄	—	172	2	no	—
202	C₂²×C₄	C₄×Dic₄₃	1	1	yes	characteristic
203	C₂²×C₄	Dic₄₃⋊C₄	1	1	yes	characteristic
204	C₂²×C₄	Dic₄₃⋊C₄	1	1	yes	characteristic
205	C₂²×C₄	C₄⋊Dic₄₃	1	1	yes	characteristic
206	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
207	C₂²×C₄	#10621882	1	1	yes	characteristic
208	C₂²×C₄	#10621883	1	1	yes	characteristic
209	C₂²×C₄	#10621884	1	1	yes	characteristic
210	C₂²×C₄	#10621885	1	1	yes	characteristic
211	C₂²×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
212	M₁₆	—	4	4	no	—
213	M₁₆	—	4	4	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	4	2	no	—
216	C₂×C₈	—	4	2	no	—
217	C₂×C₈	—	172	2	no	—
218	C₂×C₈	—	172	2	no	—
219	C₂×C₈	—	172	2	no	—
220	C₄²	—	4	1	no	—
221	C₄²	—	4	1	no	—
222	C₄²	—	4	1	no	—
223	C₄²	—	4	1	no	—
224	C₂³	—	2	1	no	—
225	C₂³	—	2	1	no	—
226	C₂³	—	2	1	no	—
227	C₂³	—	2	1	no	—
228	C₂³	C₄₃⋊#95	1	1	yes	characteristic
229	C₂³	#13063239	1	2	yes	—
230	C₂³	#13063245	1	1	yes	characteristic
231	C₂³	#13063248	1	1	yes	characteristic
232	C₂³	#13063253	1	2	yes	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	344	2	no	—
255	C₂×C₄	—	344	2	no	—
256	C₂×C₄	—	344	2	no	—
257	C₂×C₄	—	344	2	no	—
258	C₂×C₄	—	172	4	no	—
259	C₂×C₄	—	172	4	no	—
260	C₂×C₄	—	172	4	no	—
261	C₂×C₄	#13063246	1	1	yes	characteristic
262	C₂×C₄	#13063249	1	1	yes	characteristic
263	C₂×C₄	#13063251	1	2	yes	—
264	C₂×C₄	#13063256	1	1	yes	characteristic
265	C₂×C₄	#13063258	1	1	yes	characteristic
266	C₈	—	4	4	no	—
267	C₈	—	344	2	no	—
268	C₂²	—	2	2	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	2	2	no	—
276	C₂²	#14582632	1	2	yes	—
277	C₂²	#14582635	1	2	yes	—
278	C₂²	#14582656	1	1	yes	characteristic
279	C₂²	#14582659	1	1	yes	characteristic
280	C₂²	#14582660	1	1	yes	characteristic
281	C₂²	#14582661	1	1	yes	characteristic
282	C₂²	#14582665	1	1	yes	characteristic
283	C₂²	#14582670	1	1	yes	characteristic
284	C₂²	#14582671	1	1	yes	characteristic
285	C₄	—	2	2	no	—
286	C₄	—	2	1	no	—
287	C₄	—	2	1	no	—
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	4	1	no	—
291	C₄	—	4	1	no	—
292	C₄	—	4	1	no	—
293	C₄	—	344	4	no	—
294	C₂	—	2	2	no	—
295	C₂	—	2	2	no	—
296	C₂	#19132449	1	1	yes	characteristic
297	C₂	#19132455	1	1	yes	characteristic
298	C₂	#19132460	1	1	yes	characteristic
299	C₂	#19132474	1	1	yes	characteristic
300	C₂	#19132489	1	1	yes	characteristic
301	C₂	#19132496	1	1	yes	characteristic
302	C₂	#19132504	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₄₃⋊#95
commutator	C₂²×C₄×C₄₃	C₄²
Frattini	C₂²×C₄²	C₂×D₄₃
Fitting	C₄₃×#1728	C₂
socle	C₂³×C₄₃	#95
prime radical	C₂⁴×C₄₃	C₂²⋊C₄
cosocle	C₂²×C₄²×C₄₃	C₂²
hypercenter	#1728	D₄₃
hypocenter	C₄₃	#7037
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #1728

Complements

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

N	H ≅ G/N	type
9. #14582654	C₄	semidirect
13. C₄₃×#336	C₄	semidirect
164. #336	Dic₄₃	semidirect
165. C₄₃	#7037	semidirect

Maximal subgroups

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

subgroup	index
#19134663	2
#19132822	2
C₄₃×#1728	2
#7037	43