Subgroups — #25250261

Order 11008.

444 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19143527	C₂	1	1	yes	maximal, characteristic
2	#19136503	C₂	1	2	yes	maximal, complemented
3	#19136502	C₂	1	2	yes	maximal, complemented
4	#19136501	C₂	1	2	yes	maximal, complemented
5	#19136500	C₂	1	2	yes	maximal, complemented
6	#19136402	C₂	1	1	yes	maximal, characteristic
7	#19136399	C₂	1	2	yes	maximal
8	#19136395	C₂	1	2	yes	maximal
9	C₄₃×#2164	C₂	1	1	yes	maximal, characteristic
10	#14583452	C₄	1	1	yes	characteristic
11	#14583027	C₄	1	2	yes	—
12	#14583027	C₄	1	2	yes	—
13	C₈×Dic₈₆	C₂²	1	2	yes	—
14	C₈×Dic₈₆	C₂²	1	2	yes	—
15	#14582927	C₄	1	1	yes	characteristic
16	#14582926	C₄	1	1	yes	characteristic
17	#14582922	C₄	1	1	yes	characteristic
18	#14582854	C₂²	1	1	yes	characteristic
19	C₂×C₈×Dic₄₃	C₂²	1	1	yes	characteristic
20	C₄⋊(C₄₃⋊C₁₆)	C₂²	1	2	yes	—
21	Dic₄₃⋊C₁₆	C₂²	1	2	yes	—
22	Dic₄₃⋊C₁₆	C₂²	1	2	yes	—
23	Dic₄₃⋊C₁₆	C₂²	1	2	yes	—
24	Dic₄₃⋊C₁₆	C₂²	1	2	yes	—
25	C₂²⋊(C₄₃⋊C₁₆)	C₂²	1	2	yes	—
26	#14582675	C₂²	1	1	yes	characteristic
27	#14582674	C₂²	1	2	yes	—
28	#14582674	C₂²	1	2	yes	—
29	#14582674	C₂²	1	2	yes	—
30	C₄×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
31	C₁₆×Dic₄₃	C₂²	1	2	yes	—
32	C₄₃×#405	C₂²	1	1	yes	characteristic
33	C₄₃×C₄⋊C₁₆	C₂²	1	2	yes	—
34	C₄₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
35	C₄₃×#346	C₂²	1	1	yes	characteristic
36	C₄×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
37	C₂×#10621877	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
39	#13063324	C₈	1	2	yes	—
40	#13063316	C₈	1	2	yes	—
41	#13063302	C₈	1	2	yes	—
42	#13063302	C₈	1	2	yes	—
43	C₄×Dic₈₆	C₂×C₄	1	2	yes	—
44	C₄×Dic₈₆	C₂×C₄	1	2	yes	—
45	#13063284	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×Dic₄₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
48	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
49	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	2	yes	—
50	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	2	yes	—
51	Dic₄₃⋊C₈	C₂×C₄	1	2	yes	—
52	Dic₄₃⋊C₈	C₂×C₄	1	2	yes	—
53	Dic₄₃⋊C₈	C₂×C₄	1	2	yes	—
54	Dic₄₃⋊C₈	C₂×C₄	1	2	yes	—
55	#13063235	C₂×C₄	1	2	yes	—
56	C₈×Dic₄₃	C₂×C₄	1	2	yes	—
57	C₈×Dic₄₃	C₂³	1	2	yes	—
58	C₈×Dic₄₃	C₂³	1	2	yes	—
59	C₂×M₁₆×C₄₃	C₂×C₄	1	1	yes	characteristic
60	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
61	#117×C₄₃	C₂×C₄	1	1	yes	characteristic
62	C₂×C₁₆×C₄₃	C₂³	1	2	yes	—
63	C₂×C₁₆×C₄₃	C₂³	1	2	yes	—
64	#97×C₄₃	C₂×C₄	1	1	yes	characteristic
65	#97×C₄₃	C₂×C₄	1	1	yes	characteristic
66	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
68	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×Dic₄₃	C₂×C₈	1	2	yes	—
70	C₂×Dic₈₆	C₂×C₈	1	2	yes	—
71	C₂×Dic₈₆	C₂×C₈	1	2	yes	—
72	M₁₆×C₄₃	—	4	1	no	—
73	C₂×C₈×C₄₃	—	2	2	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	C₄⋊C₄×C₄₃	C₂×C₈	1	2	yes	—
79	C₂²⋊C₄×C₄₃	C₂×C₈	1	2	yes	—
80	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
81	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
82	C₂²⋊Dic₄₃	C₂×C₈	1	1	yes	characteristic
83	C₂²⋊Dic₄₃	C₂×C₈	1	1	yes	characteristic
84	C₄⋊Dic₄₃	C₂×C₈	1	2	yes	—
85	C₄⋊Dic₄₃	C₂×C₈	1	2	yes	—
86	C₄⋊Dic₄₃	C₂×C₈	1	2	yes	—
87	Dic₄₃⋊C₄	C₂×C₈	1	2	yes	—
88	Dic₄₃⋊C₄	C₂×C₈	1	2	yes	—
89	Dic₄₃⋊C₄	C₂×C₈	1	2	yes	—
90	Dic₄₃⋊C₄	C₂×C₈	1	2	yes	—
91	C₄×Dic₄₃	C₂×C₈	1	1	yes	characteristic
92	C₄×Dic₄₃	C₂×C₈	1	1	yes	characteristic
93	C₄×Dic₄₃	C₂²×C₄	1	2	yes	—
94	C₄×Dic₄₃	C₂²×C₄	1	2	yes	—
95	#10621877	—	4	1	no	—
96	C₂×C₄₃⋊C₈	—	4	1	no	—
97	C₂×C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
98	C₂×C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
99	C₁₆×C₄₃	—	2	2	no	—
100	C₁₆×C₄₃	—	2	2	no	—
101	C₄₃⋊C₁₆	—	2	2	no	—
102	C₄₃⋊C₁₆	—	2	2	no	—
103	C₂³×C₄₃	C₂²×C₈	1	1	yes	characteristic
104	C₂×C₄×C₄₃	—	2	1	no	—
105	C₂×C₄×C₄₃	—	2	1	no	—
106	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
107	C₂×C₄×C₄₃	C₂²×C₈	1	2	yes	—
108	C₂×C₄×C₄₃	C₂²×C₈	1	2	yes	—
109	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
110	C₂×Dic₄₃	—	4	2	no	—
111	C₂×Dic₄₃	C₂²×C₈	1	2	yes	—
112	C₂×Dic₄₃	C₂²×C₈	1	2	yes	—
113	C₂×Dic₄₃	C₂²×C₈	1	2	yes	—
114	C₂×Dic₄₃	C₂²×C₈	1	2	yes	—
115	Dic₈₆	—	4	2	no	—
116	Dic₈₆	—	4	2	no	—
117	C₈×C₄₃	—	2	1	no	—
118	C₈×C₄₃	—	2	1	no	—
119	C₈×C₄₃	—	2	1	no	—
120	C₈×C₄₃	2₋¹⁺⁴	1	2	yes	—
121	C₄₃⋊C₈	—	2	2	no	—
122	C₄₃⋊C₈	—	2	2	no	—
123	#33501	—	43	1	no	maximal, Sylow
124	C₂²×C₄₃	—	2	2	no	—
125	C₂²×C₄₃	—	2	1	no	—
126	C₂²×C₄₃	C₂³×C₈	1	1	yes	characteristic
127	C₄×C₄₃	—	2	2	no	—
128	C₄×C₄₃	—	2	2	no	—
129	C₄×C₄₃	—	2	1	no	—
130	C₄×C₄₃	#519	1	1	yes	characteristic
131	C₄×C₄₃	#568	1	1	yes	characteristic
132	Dic₄₃	—	2	2	no	—
133	Dic₄₃	—	2	2	no	—
134	Dic₄₃	—	2	2	no	—
135	Dic₄₃	—	2	2	no	—
136	#2960	—	43	1	no	—
137	#2183	—	43	2	no	—
138	#2183	—	43	2	no	—
139	#2183	—	43	2	no	—
140	#2183	—	43	2	no	—
141	#2164	—	43	1	no	—
142	#2164	—	43	2	no	—
143	#2164	D₄₃	1	1	yes	characteristic
144	#2163	—	43	2	no	—
145	C₂×C₄₃	—	4	1	no	—
146	C₂×C₄₃	#2982	1	1	yes	characteristic
147	C₂×C₄₃	#3407	1	2	yes	—
148	#533	—	43	1	no	—
149	#446	—	43	2	no	—
150	#446	—	43	2	no	—
151	C₈×Q₈	—	43	2	no	—
152	C₈×Q₈	—	43	2	no	—
153	#424	—	43	1	no	—
154	#424	—	43	1	no	—
155	#423	—	43	1	no	—
156	#405	—	43	1	no	—
157	#405	C₂×D₄₃	1	1	yes	characteristic
158	C₂×C₄×C₈	—	43	1	no	—
159	C₄⋊C₁₆	—	43	2	no	—
160	C₄⋊C₁₆	—	43	2	no	—
161	C₄⋊C₁₆	—	43	2	no	—
162	C₄⋊C₁₆	—	43	2	no	—
163	C₄⋊C₁₆	—	43	2	no	—
164	C₄⋊C₁₆	C₂×D₄₃	1	2	yes	—
165	C₂²⋊C₁₆	—	43	2	no	—
166	C₂²⋊C₁₆	C₂×D₄₃	1	2	yes	—
167	#346	—	43	2	no	—
168	#346	—	43	2	no	—
169	#346	—	43	2	no	—
170	#346	—	43	1	no	—
171	#346	C₂×D₄₃	1	1	yes	characteristic
172	C₄×C₁₆	—	43	2	no	—
173	C₄×C₁₆	—	43	1	no	—
174	C₄×C₁₆	C₂×D₄₃	1	1	yes	characteristic
175	C₄₃	#33501	1	1	yes	characteristic, complemented, Sylow
176	C₂×M₁₆	—	43	1	no	—
177	C₂×M₁₆	C₄×D₄₃	1	1	yes	characteristic
178	C₂²×C₈	—	43	1	no	—
179	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
180	C₄⋊Q₈	—	43	2	no	—
181	#125	—	43	2	no	—
182	C₂²⋊Q₈	—	43	2	no	—
183	C₂²⋊Q₈	—	43	2	no	—
184	C₄×Q₈	—	43	2	no	—
185	C₄×Q₈	—	43	2	no	—
186	#117	—	43	1	no	—
187	#117	C₄×D₄₃	1	1	yes	characteristic
188	C₂×C₄²	—	43	1	no	—
189	C₂×C₁₆	—	43	2	no	—
190	C₂×C₁₆	—	43	2	no	—
191	C₂×C₁₆	C₂²×D₄₃	1	2	yes	—
192	C₂×C₁₆	C₂²×D₄₃	1	2	yes	—
193	C₄⋊C₈	—	43	2	no	—
194	C₄⋊C₈	—	43	2	no	—
195	C₄⋊C₈	—	43	2	no	—
196	C₄⋊C₈	—	43	2	no	—
197	C₄⋊C₈	—	43	2	no	—
198	C₄⋊C₈	—	43	2	no	—
199	#97	—	43	2	no	—
200	#97	C₄×D₄₃	1	1	yes	characteristic
201	#97	C₄×D₄₃	1	1	yes	characteristic
202	C₄×C₈	—	43	2	no	—
203	C₄×C₈	—	43	2	no	—
204	C₄×C₈	—	43	2	no	—
205	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
206	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
207	C₂×Q₈	—	43	2	no	—
208	C₂×Q₈	—	43	2	no	—
209	C₂²×C₄	—	43	2	no	—
210	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
211	M₁₆	—	4	1	no	—
212	M₁₆	—	172	1	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	43	2	no	—
215	C₂×C₈	—	43	2	no	—
216	C₂×C₈	—	172	1	no	—
217	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
218	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
219	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
220	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
221	C₄⋊C₄	—	43	2	no	—
222	C₄⋊C₄	—	43	2	no	—
223	C₄⋊C₄	—	43	2	no	—
224	C₄⋊C₄	—	43	2	no	—
225	C₄⋊C₄	—	43	2	no	—
226	C₄⋊C₄	—	43	2	no	—
227	C₄⋊C₄	—	43	2	no	—
228	C₄⋊C₄	C₈×D₄₃	1	2	yes	—
229	C₂²⋊C₄	—	43	1	no	—
230	C₂²⋊C₄	—	43	1	no	—
231	C₂²⋊C₄	C₈×D₄₃	1	2	yes	—
232	C₄²	—	43	1	no	—
233	C₄²	—	43	1	no	—
234	C₄²	—	43	2	no	—
235	C₄²	—	43	2	no	—
236	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
237	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
238	C₁₆	—	2	2	no	—
239	C₁₆	—	2	2	no	—
240	C₁₆	—	86	2	no	—
241	C₁₆	—	86	2	no	—
242	C₂³	C₂×C₈×D₄₃	1	1	yes	characteristic
243	Q₈	—	172	2	no	—
244	Q₈	—	172	2	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	43	2	no	—
248	C₂×C₄	—	43	2	no	—
249	C₂×C₄	—	43	2	no	—
250	C₂×C₄	—	43	2	no	—
251	C₂×C₄	—	172	2	no	—
252	C₂×C₄	C₂×C₈×D₄₃	1	1	yes	characteristic
253	C₂×C₄	C₂×C₈×D₄₃	1	2	yes	—
254	C₂×C₄	C₂×C₈×D₄₃	1	2	yes	—
255	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
256	C₈	—	2	1	no	—
257	C₈	—	2	1	no	—
258	C₈	—	2	1	no	—
259	C₈	—	86	2	no	—
260	C₈	—	86	2	no	—
261	C₈	#13063372	1	2	yes	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	1	no	—
264	C₂²	C₂²×C₈×D₄₃	1	1	yes	characteristic
265	C₄	—	2	2	no	—
266	C₄	—	2	2	no	—
267	C₄	—	2	1	no	—
268	C₄	—	86	2	no	—
269	C₄	—	86	2	no	—
270	C₄	—	86	2	no	—
271	C₄	—	86	2	no	—
272	C₄	#14583376	1	1	yes	characteristic
273	C₄	#14583665	1	1	yes	characteristic
274	C₂	—	4	1	no	—
275	C₂	#19143675	1	1	yes	characteristic
276	C₂	#19147696	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₄₃
commutator, socle	C₂²×C₄₃	C₂³×C₈
Fitting	C₄₃×#2164	C₂
prime radical	C₂³×C₄₃	C₂²×C₈
cosocle	C₂×C₈×C₄₃	C₂⁴
hypercenter	#2164	D₄₃
hypocenter	C₄₃	#33501
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₈ ◁ #2164

Complements

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

N	H ≅ G/N	type
2. #19136503	C₂	semidirect
3. #19136502	C₂	semidirect
4. #19136501	C₂	semidirect
5. #19136500	C₂	semidirect
175. C₄₃	#33501	semidirect

Maximal subgroups

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

subgroup	index
#19143527	2
#19136503	2
#19136502	2
#19136501	2
#19136500	2
#19136402	2
#19136399	2
#19136395	2
C₄₃×#2164	2
#33501	43