Subgroups — #25096828

Order 11008.

778 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19143157	C₂	1	2	yes	maximal
2	C₂×#14582853	C₂	1	1	yes	maximal, characteristic
3	#19134735	C₂	1	2	yes	maximal, complemented
4	#19134734	C₂	1	2	yes	maximal, complemented
5	#19134733	C₂	1	1	yes	maximal, characteristic, complemented
6	#19134731	C₂	1	2	yes	maximal, complemented
7	#19134320	C₂	1	2	yes	maximal
8	#341×D₄₃	C₂	1	2	yes	maximal
9	C₄₃×#1837	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#10621872	C₂²	1	1	yes	characteristic
11	C₂×M₁₆×D₄₃	C₂²	1	4	yes	—
12	C₂²×C₈×D₄₃	C₂²	1	1	yes	characteristic
13	C₂×#13063282	C₂²	1	1	yes	characteristic
14	#14582924	—	2	8	no	—
15	#14582923	C₂²	1	2	yes	—
16	#14582922	C₂²	1	4	yes	—
17	#14582853	C₄	1	8	yes	—
18	C₂×#13063236	C₂²	1	1	yes	characteristic
19	C₂×#13063235	C₂²	1	1	yes	characteristic
20	C₂×C₈×Dic₄₃	C₂²	1	1	yes	characteristic
21	#14582664	C₂²	1	2	yes	—
22	#14582664	C₂²	1	2	yes	—
23	#14582663	C₂²	1	4	yes	—
24	#14582663	C₂²	1	4	yes	—
25	C₄₃×#423	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
27	C₄₃×#341	C₂²	1	2	yes	—
28	C₄₃×#341	C₂²	1	2	yes	—
29	C₂²×C₄×D₄₃	C₂³	1	1	yes	characteristic
30	C₂×#10621877	C₂³	1	4	yes	—
31	C₂×#10621872	—	2	4	no	—
32	C₂×#10621872	—	2	4	no	—
33	C₂×#10621872	C₂×C₄	1	4	yes	—
34	M₁₆×D₄₃	—	2	8	no	—
35	M₁₆×D₄₃	—	2	8	no	—
36	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
37	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
38	C₂×C₈×D₄₃	—	2	4	no	—
39	C₂×C₈×D₄₃	—	2	4	no	—
40	C₂×C₈×D₄₃	C₂×C₄	1	4	yes	—
41	#13063282	C₂×C₄	1	4	yes	—
42	#13063282	D₄	1	4	yes	—
43	C₂×Dic₄₃⋊C₄	Q₈	1	2	yes	—
44	C₂×(C₂×D₄₃)⋊C₄	Q₈	1	2	yes	—
45	C₂×C₄×Dic₄₃	C₂³	1	1	yes	characteristic
46	C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
47	Dic₄₃⋊C₈	—	2	8	no	—
48	#13063236	C₂×C₄	1	4	yes	—
49	#13063235	C₂×C₄	1	4	yes	—
50	C₈×Dic₄₃	C₂×C₄	1	4	yes	—
51	C₂×M₁₆×C₄₃	C₂³	1	4	yes	—
52	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
55	C₄⋊C₈×C₄₃	—	2	4	no	—
56	C₄×C₈×C₄₃	C₂×C₄	1	4	yes	—
57	C₂³×D₄₃	C₂×Q₈	1	1	yes	characteristic
58	C₂²×C₄×C₄₃	C₂×Q₈	1	2	yes	—
59	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
60	C₂²×Dic₄₃	C₂×Q₈	1	2	yes	—
61	C₂²×Dic₄₃	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₄×D₄₃	C₂²×C₄	1	2	yes	—
63	C₂×C₄×D₄₃	C₂×D₄	1	2	yes	—
64	C₂×C₄×D₄₃	#41	1	4	yes	—
65	C₂×C₄×D₄₃	#41	1	4	yes	—
66	M₁₆×C₄₃	—	2	4	no	—
67	M₁₆×C₄₃	—	2	4	no	—
68	C₂×C₈×C₄₃	—	2	2	no	—
69	C₂×C₈×C₄₃	—	2	2	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	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
74	C₄²×C₄₃	C₂×D₄	1	2	yes	—
75	(C₂×D₄₃)⋊C₄	C₄⋊C₄	1	8	yes	—
76	Dic₄₃⋊C₄	C₄⋊C₄	1	8	yes	—
77	C₄×Dic₄₃	C₂²×C₄	1	2	yes	—
78	C₄×Dic₄₃	C₂×D₄	1	2	yes	—
79	#10621877	—	2	4	no	—
80	#10621877	—	2	4	no	—
81	C₂×C₄₃⋊C₈	—	2	2	no	—
82	C₂×C₄₃⋊C₈	—	2	2	no	—
83	C₂×C₄₃⋊C₈	—	2	4	no	—
84	C₂×C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
85	C₂×C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
86	#10621872	—	2	8	no	—
87	#10621872	—	4	8	no	—
88	C₈×D₄₃	—	2	8	no	—
89	C₈×D₄₃	—	4	8	no	—
90	C₂³×C₄₃	C₂²×Q₈	1	1	yes	characteristic
91	C₂²×D₄₃	—	4	2	no	—
92	C₂²×D₄₃	C₂×C₄⋊C₄	1	2	yes	—
93	C₂²×D₄₃	#125	1	4	yes	—
94	C₂×C₄×C₄₃	—	4	2	no	—
95	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	4	yes	—
96	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
97	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
98	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
99	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
100	C₂×Dic₄₃	—	4	2	no	—
101	C₂×Dic₄₃	C₂×C₄⋊C₄	1	4	yes	—
102	C₂×Dic₄₃	C₂×C₄⋊C₄	1	2	yes	—
103	C₂×Dic₄₃	#125	1	4	yes	—
104	C₄×D₄₃	#117	1	4	yes	—
105	C₄×D₄₃	#117	1	4	yes	—
106	C₄×D₄₃	#124	1	8	yes	—
107	C₈×C₄₃	—	2	2	no	—
108	C₈×C₄₃	—	2	2	no	—
109	C₈×C₄₃	—	4	4	no	—
110	C₄₃⋊C₈	—	2	2	no	—
111	C₄₃⋊C₈	—	2	2	no	—
112	C₄₃⋊C₈	—	4	4	no	—
113	#20768	—	43	1	no	maximal, Sylow
114	C₂²×C₄₃	—	2	2	no	—
115	C₂²×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
116	C₂²×C₄₃	C₂×#125	1	2	yes	—
117	C₂×D₄₃	—	2	4	no	—
118	C₂×D₄₃	—	4	4	no	—
119	C₂×D₄₃	#382	1	4	yes	—
120	C₄×C₄₃	—	2	4	no	—
121	C₄×C₄₃	C₂×#117	1	1	yes	characteristic
122	C₄×C₄₃	C₂×#117	1	1	yes	characteristic
123	C₄×C₄₃	C₂×#124	1	2	yes	—
124	Dic₄₃	—	2	4	no	—
125	Dic₄₃	#382	1	4	yes	—
126	#2905	—	43	2	no	—
127	C₂×#405	—	43	1	no	—
128	#1838	—	43	2	no	—
129	#1838	—	43	2	no	—
130	#1837	—	43	1	no	—
131	#1837	—	43	2	no	—
132	#1837	D₄₃	1	1	yes	characteristic, complemented
133	#1738	—	43	2	no	—
134	C₂×#341	—	43	2	no	—
135	C₂×C₄₃	—	2	2	no	—
136	C₂×C₄₃	C₂×#382	1	1	yes	characteristic
137	C₂×C₄₃	#2916	1	2	yes	—
138	D₄₃	—	2	4	no	—
139	C₂²×M₁₆	—	43	1	no	—
140	C₂²×M₁₆	—	43	4	no	—
141	C₂³×C₈	—	43	1	no	—
142	C₂×#117	—	43	1	no	—
143	#424	—	86	8	no	—
144	#423	—	43	4	no	—
145	#423	—	43	2	no	—
146	#423	C₂×D₄₃	1	2	yes	—
147	#405	—	43	8	no	—
148	C₂×#97	—	43	1	no	—
149	C₂×#97	—	43	1	no	—
150	C₂×C₄×C₈	—	43	1	no	—
151	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
152	#341	—	43	4	no	—
153	#341	—	43	4	no	—
154	#341	—	43	2	no	—
155	#341	—	43	2	no	—
156	#341	C₂×D₄₃	1	2	yes	—
157	#341	C₂×D₄₃	1	2	yes	—
158	C₄₃	#20768	1	1	yes	characteristic, complemented, Sylow
159	C₂³×C₄	—	43	1	no	—
160	C₂×M₁₆	—	43	4	no	—
161	C₂×M₁₆	—	43	4	no	—
162	C₂×M₁₆	—	86	4	no	—
163	C₂×M₁₆	—	86	4	no	—
164	C₂×M₁₆	—	86	8	no	—
165	C₂×M₁₆	—	86	8	no	—
166	C₂×M₁₆	C₂²×D₄₃	1	4	yes	—
167	C₂²×C₈	—	43	1	no	—
168	C₂²×C₈	—	43	1	no	—
169	C₂²×C₈	—	43	4	no	—
170	C₂²×C₈	—	86	4	no	—
171	C₂²×C₈	—	86	4	no	—
172	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
173	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
174	#117	—	43	4	no	—
175	#117	—	43	4	no	—
176	C₂×C₄⋊C₄	—	43	2	no	—
177	C₂×C₂²⋊C₄	—	43	2	no	—
178	C₂×C₄²	—	43	1	no	—
179	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
180	C₄⋊C₈	—	2	4	no	—
181	C₄⋊C₈	—	86	8	no	—
182	C₄⋊C₈	—	86	4	no	—
183	#97	—	43	4	no	—
184	#97	—	43	4	no	—
185	C₄×C₈	—	43	4	no	—
186	C₄×C₈	C₄×D₄₃	1	4	yes	—
187	C₂⁴	—	43	1	no	—
188	C₂²×C₄	—	43	1	no	—
189	C₂²×C₄	—	43	2	no	—
190	C₂²×C₄	—	43	2	no	—
191	C₂²×C₄	—	43	4	no	—
192	C₂²×C₄	—	43	4	no	—
193	C₂²×C₄	—	43	2	no	—
194	C₂²×C₄	Q₈×D₄₃	1	2	yes	—
195	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
196	M₁₆	—	2	4	no	—
197	M₁₆	—	2	4	no	—
198	M₁₆	—	86	8	no	—
199	M₁₆	—	86	4	no	—
200	M₁₆	—	86	4	no	—
201	M₁₆	—	172	8	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	4	no	—
205	C₂×C₈	—	43	2	no	—
206	C₂×C₈	—	43	2	no	—
207	C₂×C₈	—	86	8	no	—
208	C₂×C₈	—	86	2	no	—
209	C₂×C₈	—	86	2	no	—
210	C₂×C₈	—	86	4	no	—
211	C₂×C₈	—	172	8	no	—
212	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
213	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
214	C₄⋊C₄	—	43	8	no	—
215	C₂²⋊C₄	—	43	8	no	—
216	C₄²	—	43	2	no	—
217	C₄²	—	43	2	no	—
218	C₄²	C₂×C₄×D₄₃	1	2	yes	—
219	C₄²	D₄×D₄₃	1	2	yes	—
220	C₂³	—	43	2	no	—
221	C₂³	—	43	4	no	—
222	C₂³	—	172	2	no	—
223	C₂³	C₂×Q₈×D₄₃	1	1	yes	characteristic
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	—	43	2	no	—
226	C₂×C₄	—	43	4	no	—
227	C₂×C₄	—	43	4	no	—
228	C₂×C₄	—	43	4	no	—
229	C₂×C₄	—	43	8	no	—
230	C₂×C₄	—	43	4	no	—
231	C₂×C₄	—	172	2	no	—
232	C₂×C₄	C₄⋊C₄×D₄₃	1	4	yes	—
233	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
234	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
235	C₂×C₄	#41×D₄₃	1	2	yes	—
236	C₂×C₄	#41×D₄₃	1	2	yes	—
237	C₈	—	2	2	no	—
238	C₈	—	2	2	no	—
239	C₈	—	4	4	no	—
240	C₈	—	86	2	no	—
241	C₈	—	86	2	no	—
242	C₈	—	172	4	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	43	4	no	—
245	C₂²	—	86	4	no	—
246	C₂²	—	172	4	no	—
247	C₂²	C₂×C₄⋊C₄×D₄₃	1	1	yes	characteristic
248	C₂²	#125×D₄₃	1	2	yes	—
249	C₄	—	2	4	no	—
250	C₄	—	43	4	no	—
251	C₄	—	86	4	no	—
252	C₄	#117×D₄₃	1	1	yes	characteristic
253	C₄	#117×D₄₃	1	1	yes	characteristic
254	C₄	#124×D₄₃	1	2	yes	—
255	C₂	—	2	2	no	—
256	C₂	—	86	4	no	—
257	C₂	#382×D₄₃	1	1	yes	characteristic
258	C₂	#19143203	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×#1837	C₂
socle	C₂²×C₄₃	C₂²×C₄⋊C₄
prime radical	C₂³×D₄₃	C₂×Q₈
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#1837	D₄₃
hypocenter	C₄₃	#20768
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1837

Complements

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

N	H ≅ G/N	type
3. #19134735	C₂	semidirect
4. #19134734	C₂	semidirect
5. #19134733	C₂	semidirect
6. #19134731	C₂	semidirect
9. C₄₃×#1837	C₂	semidirect
132. #1837	D₄₃	semidirect
158. C₄₃	#20768	semidirect

Maximal subgroups

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

subgroup	index
#19143157	2
C₂×#14582853	2
#19134735	2
#19134734	2
#19134733	2
#19134731	2
#19134320	2
#341×D₄₃	2
C₄₃×#1837	2
#20768	43