Subgroups — #25105206

Order 11008.

1046 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19145023	C₂	1	1	yes	maximal, characteristic
2	C₂×#14583015	C₂	1	4	yes	maximal, complemented
3	#19143306	C₂	1	2	yes	maximal, complemented
4	#19135481	C₂	1	4	yes	maximal, complemented
5	#19135422	C₂	1	2	yes	maximal, complemented
6	C₄×C₈×Dic₄₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#1848	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13063347	C₂²	1	4	yes	—
9	C₂×#13063333	C₂²	1	4	yes	—
10	#14583449	C₂²	1	2	yes	—
11	#14583283	C₄	1	4	yes	complemented
12	#14583279	C₄	1	1	yes	characteristic, complemented
13	#14583276	C₄	1	1	yes	characteristic, complemented
14	#14583272	C₄	1	2	yes	complemented
15	#14583015	—	2	16	no	—
16	#14582976	—	2	4	no	—
17	#14582964	—	2	4	no	—
18	C₂×#13063251	C₂²	1	8	yes	—
19	#14582867	C₂²	1	4	yes	—
20	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₈×Dic₄₃	C₂²	1	2	yes	—
22	C₂×C₈×Dic₄₃	C₂²	1	2	yes	—
23	C₄²×Dic₄₃	C₂²	1	1	yes	characteristic
24	C₂×#108×C₄₃	C₂²	1	4	yes	—
25	C₄₃×#423	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
27	C₂×#10621898	C₂³	1	4	yes	—
28	#13063347	—	2	8	no	—
29	#13063347	—	2	8	no	—
30	C₂×Dic₁₇₂	C₂×C₄	1	4	yes	—
31	C₂×#10621873	C₂×C₄	1	8	yes	—
32	C₂×D₃₄₄	C₂×C₄	1	4	yes	—
33	#13063333	—	2	16	no	—
34	C₂×#10621872	—	2	4	no	—
35	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
36	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
37	C₂×C₈×D₄₃	—	2	4	no	—
38	#13063325	C₂×C₄	1	2	yes	—
39	#13063322	C₂×C₄	1	2	yes	—
40	#13063313	C₂×C₄	1	4	yes	—
41	#13063299	—	2	4	no	—
42	C₄×D₁₇₂	—	2	4	no	—
43	C₂×C₄×Dic₄₃	—	2	2	no	—
44	C₂×C₄×Dic₄₃	C₂³	1	2	yes	—
45	#13063251	—	2	16	no	—
46	(C₂×D₄₃)⋊C₈	—	2	8	no	—
47	C₄×C₄₃⋊C₈	D₄	1	2	yes	—
48	C₄×C₄₃⋊C₈	D₄	1	2	yes	—
49	C₈×Dic₄₃	—	2	4	no	—
50	C₈×Dic₄₃	D₄	1	8	yes	complemented
51	C₂×M₁₆×C₄₃	C₂³	1	4	yes	—
52	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
53	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
54	#108×C₄₃	—	2	8	no	—
55	C₄⋊C₈×C₄₃	—	2	4	no	—
56	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
57	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
58	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
59	C₂²×C₄×C₄₃	#41	1	2	yes	—
60	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
61	C₂×C₂²⋊D₄₃	—	2	4	no	—
62	C₂²×Dic₄₃	—	2	2	no	—
63	#10621898	—	2	8	no	—
64	#10621898	—	2	8	no	—
65	C₂×D₁₇₂	—	2	4	no	—
66	C₂×D₁₇₂	C₂²×C₄	1	4	yes	—
67	C₂×C₄×D₄₃	—	2	4	no	—
68	C₂×Dic₈₆	C₂²×C₄	1	4	yes	—
69	M₁₆×C₄₃	—	2	8	no	—
70	C₂×C₈×C₄₃	—	2	4	no	—
71	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
72	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
73	C₂×C₈×C₄₃	C₂×D₄	1	4	yes	—
74	C₂×C₈×C₄₃	#41	1	4	yes	—
75	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
76	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
77	C₄²×C₄₃	C₂×D₄	1	2	yes	—
78	(C₂×D₄₃)⋊C₄	—	2	8	no	—
79	C₄⋊Dic₄₃	—	2	4	no	—
80	C₄×Dic₄₃	—	2	8	no	—
81	C₄×Dic₄₃	—	2	2	no	—
82	C₄×Dic₄₃	C₂×D₄	1	4	yes	—
83	C₂×C₄₃⋊C₈	—	2	2	no	—
84	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
85	C₂×C₄₃⋊C₈	C₂×D₄	1	4	yes	—
86	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
87	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
88	Dic₁₇₂	—	2	8	no	—
89	D₃₄₄	—	2	8	no	—
90	#10621873	—	2	16	no	—
91	#10621872	—	4	8	no	—
92	C₈×D₄₃	—	4	8	no	—
93	C₂³×C₄₃	C₂×#41	1	1	yes	characteristic
94	C₂²×D₄₃	—	2	4	no	—
95	C₂×C₄×C₄₃	—	2	4	no	—
96	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
97	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
98	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
99	C₂×C₄×C₄₃	C₂²×D₄	1	2	yes	—
100	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
101	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
102	C₂²⋊D₄₃	—	4	8	no	—
103	C₂×Dic₄₃	—	2	8	no	—
104	C₂×Dic₄₃	—	2	4	no	—
105	C₂×Dic₄₃	—	2	4	no	—
106	D₁₇₂	—	2	8	no	—
107	D₁₇₂	—	4	8	no	—
108	C₄×D₄₃	—	4	4	no	—
109	C₄×D₄₃	—	4	4	no	—
110	Dic₈₆	—	2	8	no	—
111	C₈×C₄₃	—	4	4	no	—
112	C₈×C₄₃	C₄×D₄	1	4	yes	—
113	C₈×C₄₃	C₄×D₄	1	4	yes	—
114	C₄₃⋊C₈	—	2	2	no	—
115	C₄₃⋊C₈	#127	1	2	yes	—
116	C₄₃⋊C₈	#127	1	2	yes	—
117	#21705	—	43	1	no	maximal, Sylow
118	C₂²×C₄₃	—	2	2	no	—
119	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
120	C₂²×C₄₃	#532	1	2	yes	—
121	C₂×D₄₃	—	4	4	no	—
122	C₂×D₄₃	—	4	8	no	—
123	C₄×C₄₃	—	2	2	no	—
124	C₄×C₄₃	#443	1	4	yes	complemented
125	C₄×C₄₃	C₂×C₄×D₄	1	2	yes	—
126	C₄×C₄₃	C₂×#127	1	1	yes	characteristic
127	C₄×C₄₃	#532	1	1	yes	characteristic
128	Dic₄₃	—	2	8	no	—
129	Dic₄₃	—	4	4	no	—
130	#3147	—	43	1	no	—
131	C₂×#443	—	43	4	no	—
132	#2931	—	43	2	no	—
133	#1974	—	43	4	no	—
134	#1962	—	43	2	no	—
135	#1848	D₄₃	1	1	yes	characteristic, complemented
136	C₄²×C₈	—	43	1	no	—
137	C₂×C₄₃	—	2	2	no	—
138	C₂×C₄₃	C₄×#127	1	1	yes	characteristic
139	C₂×C₄₃	C₂×#443	1	2	yes	—
140	D₄₃	—	8	4	no	—
141	C₂×#135	—	43	4	no	—
142	C₂×#131	—	43	4	no	—
143	#532	—	43	2	no	—
144	#495	—	43	4	no	—
145	#494	—	43	1	no	—
146	#493	—	43	1	no	—
147	#492	—	43	2	no	—
148	#443	—	86	16	no	—
149	#436	—	86	4	no	—
150	C₈×D₄	—	86	4	no	—
151	C₂×#108	C₂×D₄₃	1	4	yes	—
152	#423	C₂×D₄₃	1	2	yes	—
153	C₂×#104	—	43	8	no	—
154	#408	—	43	4	no	—
155	C₂×C₄×C₈	—	43	2	no	—
156	C₂×C₄×C₈	—	43	1	no	—
157	C₂×C₄×C₈	—	43	2	no	—
158	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
159	C₄³	—	43	1	no	—
160	C₄₃	#21705	1	1	yes	characteristic, complemented, Sylow
161	C₂×#41	—	43	4	no	—
162	#135	—	86	8	no	—
163	#135	—	86	8	no	—
164	C₂×Q₁₆	—	43	4	no	—
165	C₂×SD₁₆	—	43	8	no	—
166	C₂×D₈	—	43	4	no	—
167	#131	—	86	16	no	—
168	C₂×M₁₆	—	86	4	no	—
169	C₂×M₁₆	C₂²×D₄₃	1	4	yes	—
170	C₂²×C₈	—	43	1	no	—
171	C₂²×C₈	—	43	1	no	—
172	C₂²×C₈	—	86	4	no	—
173	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
174	C₄⋊Q₈	—	43	2	no	—
175	#127	—	43	2	no	—
176	#124	—	43	4	no	—
177	#121	—	86	4	no	—
178	C₄×D₄	—	86	4	no	—
179	C₂×C₄²	—	43	2	no	—
180	C₂×C₄²	—	86	2	no	—
181	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
182	#108	—	2	8	no	—
183	C₄⋊C₈	—	2	4	no	—
184	#104	—	86	16	no	—
185	C₂²⋊C₈	—	86	8	no	—
186	C₄×C₈	—	43	8	no	—
187	C₄×C₈	—	43	2	no	—
188	C₄×C₈	—	43	2	no	—
189	C₄×C₈	—	86	4	no	—
190	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
191	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
192	C₄×C₈	C₄×D₄₃	1	2	yes	—
193	#41	—	86	8	no	—
194	#41	—	86	8	no	—
195	C₂×Q₈	—	43	4	no	—
196	C₂×D₄	—	43	4	no	—
197	C₂×D₄	—	86	4	no	—
198	C₂×D₄	—	86	4	no	—
199	C₂²×C₄	—	86	2	no	—
200	C₂²×C₄	—	86	4	no	—
201	C₂²×C₄	#10621902	1	2	yes	—
202	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
203	Q₁₆	—	86	8	no	—
204	SD₁₆	—	86	16	no	—
205	D₈	—	86	8	no	—
206	M₁₆	—	2	8	no	—
207	M₁₆	—	172	8	no	—
208	C₂×C₈	—	2	4	no	—
209	C₂×C₈	—	43	4	no	—
210	C₂×C₈	—	43	1	no	—
211	C₂×C₈	—	43	1	no	—
212	C₂×C₈	—	43	2	no	—
213	C₂×C₈	—	86	2	no	—
214	C₂×C₈	—	172	8	no	—
215	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
216	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
217	C₂×C₈	D₄×D₄₃	1	4	yes	—
218	C₂×C₈	#10621902	1	4	yes	—
219	C₄⋊C₄	—	86	4	no	—
220	C₂²⋊C₄	—	86	8	no	—
221	C₄²	—	43	4	no	—
222	C₄²	—	86	8	no	—
223	C₄²	—	86	2	no	—
224	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
225	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
226	C₄²	D₄×D₄₃	1	2	yes	—
227	C₂³	—	86	4	no	—
228	C₂³	C₂×#10621902	1	1	yes	characteristic
229	Q₈	—	86	8	no	—
230	D₄	—	86	8	no	—
231	D₄	—	172	8	no	—
232	D₄	—	172	8	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	86	8	no	—
235	C₂×C₄	—	86	4	no	—
236	C₂×C₄	—	86	4	no	—
237	C₂×C₄	—	172	4	no	—
238	C₂×C₄	—	172	4	no	—
239	C₂×C₄	#13063287	1	2	yes	—
240	C₂×C₄	#13063287	1	2	yes	—
241	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
242	C₂×C₄	C₂×D₄×D₄₃	1	2	yes	—
243	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
244	C₂×C₄	C₂×#10621902	1	2	yes	—
245	C₈	—	4	4	no	—
246	C₈	—	43	2	no	—
247	C₈	—	43	2	no	—
248	C₈	—	86	2	no	—
249	C₈	#13063287	1	4	yes	—
250	C₈	#13063287	1	4	yes	—
251	C₂²	—	2	2	no	—
252	C₂²	—	172	8	no	—
253	C₂²	—	172	4	no	—
254	C₂²	C₂×#13063287	1	1	yes	characteristic
255	C₂²	#14583450	1	2	yes	—
256	C₄	—	2	2	no	—
257	C₄	—	86	8	no	—
258	C₄	—	172	4	no	—
259	C₄	#14583015	1	4	yes	complemented
260	C₄	C₂×#13063287	1	2	yes	—
261	C₄	#127×D₄₃	1	1	yes	characteristic
262	C₄	#14583450	1	1	yes	characteristic
263	C₂	—	2	2	no	—
264	C₂	—	344	4	no	—
265	C₂	#19137157	1	1	yes	characteristic
266	C₂	C₂×#14583015	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄₃×#1848	C₂
socle	C₂²×C₄₃	C₂×C₄×D₄
prime radical	#19145023	C₂
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#1848	D₄₃
hypocenter	C₄₃	#21705
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

Complements

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

N	H ≅ G/N	type
2. C₂×#14583015	C₂	semidirect
3. #19143306	C₂	semidirect
4. #19135481	C₂	semidirect
5. #19135422	C₂	semidirect
6. C₄×C₈×Dic₄₃	C₂	semidirect
7. C₄₃×#1848	C₂	semidirect
11. #14583283	C₄	semidirect
12. #14583279	C₄	semidirect
13. #14583276	C₄	semidirect
14. #14583272	C₄	semidirect
50. C₈×Dic₄₃	D₄	semidirect
124. C₄×C₄₃	#443	semidirect
135. #1848	D₄₃	semidirect
160. C₄₃	#21705	semidirect
259. C₄	#14583015	semidirect

Maximal subgroups

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

subgroup	index
#19145023	2
C₂×#14583015	2
#19143306	2
#19135481	2
#19135422	2
C₄×C₈×Dic₄₃	2
C₄₃×#1848	2
#21705	43