Subgroups — #25110298

Order 11008.

578 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19145105	C₂	1	1	yes	maximal, characteristic
2	#19143536	C₂	1	1	yes	maximal, characteristic
3	#19143532	C₂	1	1	yes	maximal, characteristic
4	#19135576	C₂	1	1	yes	maximal, characteristic
5	#19135197	C₂	1	4	yes	maximal
6	#19135192	C₂	1	4	yes	maximal
7	#19134732	C₂	1	1	yes	maximal, characteristic
8	#19134347	C₂	1	1	yes	maximal, characteristic
9	C₄₃×#1995	C₂	1	1	yes	maximal, characteristic
10	#14583456	C₂²	1	1	yes	characteristic
11	#14583302	C₄	1	2	yes	complemented
12	#14583298	C₄	1	4	yes	complemented
13	#14583295	C₄	1	2	yes	complemented
14	#14583030	—	2	4	no	—
15	#14583028	—	2	4	no	—
16	#14582935	C₂²	1	4	yes	—
17	#14582927	C₂²	1	2	yes	—
18	#14582926	C₂²	1	2	yes	—
19	#14582923	C₂²	1	1	yes	characteristic
20	#14582857	C₂²	1	2	yes	—
21	#14582855	C₂²	1	2	yes	—
22	C₂×#13063236	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
24	#14582663	C₂²	1	4	yes	—
25	#14582659	C₂²	1	4	yes	—
26	#14582654	C₂²	1	2	yes	—
27	#14582654	C₂²	1	2	yes	—
28	C₄₃×#533	C₂²	1	1	yes	characteristic
29	C₄₃×#423	C₂²	1	1	yes	characteristic
30	C₄₃×#376	C₂²	1	1	yes	characteristic
31	C₄₃×#339	C₂²	1	4	yes	—
32	C₂×#10621877	C₂³	1	2	yes	—
33	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
34	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
35	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
36	#13063325	C₂×C₄	1	2	yes	—
37	#13063317	C₂×C₄	1	2	yes	—
38	#13063304	—	2	2	no	—
39	C₄×Dic₈₆	—	2	2	no	—
40	#13063284	C₂³	1	2	yes	—
41	#13063258	C₂×C₄	1	8	yes	—
42	#13063256	C₂×C₄	1	8	yes	—
43	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
44	C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
45	Dic₄₃⋊C₈	—	2	4	no	—
46	#13063236	—	2	2	no	—
47	#13063236	D₄	1	4	yes	—
48	#13063235	—	2	2	no	—
49	C₄×C₄₃⋊C₈	—	2	2	no	—
50	C₄×C₄₃⋊C₈	C₂×C₄	1	2	yes	—
51	C₄×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
52	C₄×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
53	C₈×Dic₄₃	—	2	2	no	—
54	C₂×M₁₆×C₄₃	C₂³	1	2	yes	—
55	C₄⋊Q₈×C₄₃	C₂×C₄	1	2	yes	—
56	#125×C₄₃	C₂×C₄	1	2	yes	—
57	C₂²⋊Q₈×C₄₃	—	2	2	no	—
58	C₄×Q₈×C₄₃	—	2	2	no	—
59	#117×C₄₃	C₂³	1	2	yes	—
60	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
61	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
62	C₄⋊C₈×C₄₃	—	2	2	no	—
63	#95×C₄₃	—	2	2	no	—
64	C₂×Q₈×C₄₃	—	2	2	no	—
65	C₂²×C₄×C₄₃	—	2	2	no	—
66	C₂²×C₄×C₄₃	#41	1	2	yes	—
67	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
68	C₂×Dic₈₆	—	2	2	no	—
69	M₁₆×C₄₃	—	2	4	no	—
70	C₂×C₈×C₄₃	—	2	2	no	—
71	C₄⋊C₄×C₄₃	—	2	4	no	—
72	C₄⋊C₄×C₄₃	—	2	2	no	—
73	C₄⋊C₄×C₄₃	C₂²×C₄	1	4	yes	—
74	C₂²⋊C₄×C₄₃	—	2	2	no	—
75	C₄²×C₄₃	—	2	2	no	—
76	C₄²×C₄₃	—	2	2	no	—
77	C₄²×C₄₃	—	2	2	no	—
78	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
79	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
80	C₄²×C₄₃	C₂×D₄	1	2	yes	—
81	C₂²⋊Dic₄₃	—	2	2	no	—
82	C₄⋊Dic₄₃	—	2	2	no	—
83	C₄⋊Dic₄₃	C₂²×C₄	1	4	yes	—
84	Dic₄₃⋊C₄	—	2	4	no	—
85	C₄×Dic₄₃	—	2	2	no	—
86	#10621877	—	2	4	no	—
87	C₂×C₄₃⋊C₈	—	2	2	no	—
88	C₂×C₄₃⋊C₈	—	2	2	no	—
89	C₂×C₄₃⋊C₈	—	2	2	no	—
90	C₂×C₄₃⋊C₈	—	2	2	no	—
91	C₂×C₄₃⋊C₈	—	2	2	no	—
92	C₂×C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
93	C₂×C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
94	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
95	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
96	C₂³×C₄₃	C₂×#41	1	1	yes	characteristic
97	Q₈×C₄₃	—	8	2	no	—
98	C₂×C₄×C₄₃	—	2	4	no	—
99	C₂×C₄×C₄₃	—	4	2	no	—
100	C₂×C₄×C₄₃	—	4	4	no	—
101	C₂×C₄×C₄₃	—	4	2	no	—
102	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
103	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
104	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
105	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
106	C₂×C₄×C₄₃	2₋¹⁺⁴	1	2	yes	—
107	C₂×C₄×C₄₃	2₋¹⁺⁴	1	2	yes	—
108	C₂×Dic₄₃	—	2	4	no	—
109	Dic₈₆	—	8	2	no	—
110	C₈×C₄₃	—	4	2	no	—
111	C₄₃⋊C₈	—	2	2	no	—
112	C₄₃⋊C₈	—	2	2	no	—
113	C₄₃⋊C₈	—	2	2	no	—
114	C₄₃⋊C₈	—	2	2	no	—
115	C₄₃⋊C₈	—	4	2	no	—
116	#22258	—	43	1	no	maximal, Sylow
117	C₂²×C₄₃	—	2	2	no	—
118	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
119	C₂²×C₄₃	#541	1	2	yes	—
120	C₄×C₄₃	—	2	2	no	—
121	C₄×C₄₃	—	2	2	no	—
122	C₄×C₄₃	—	4	4	no	—
123	C₄×C₄₃	—	8	2	no	—
124	C₄×C₄₃	#519	1	2	yes	—
125	C₄×C₄₃	#536	1	1	yes	characteristic
126	C₄×C₄₃	#536	1	1	yes	characteristic
127	Dic₄₃	—	4	4	no	—
128	#3160	—	43	1	no	—
129	#2961	—	43	1	no	—
130	#2961	—	43	1	no	—
131	#1995	—	43	1	no	—
132	#1995	D₄₃	1	1	yes	characteristic
133	#1922	—	43	4	no	—
134	#1922	—	43	4	no	—
135	#1837	—	43	1	no	—
136	#1746	—	43	1	no	—
137	C₂×C₄₃	—	2	2	no	—
138	C₂×C₄₃	#2327	1	1	yes	characteristic
139	C₂×C₄₃	#2957	1	2	yes	—
140	#533	—	43	1	no	—
141	#533	C₂×D₄₃	1	1	yes	characteristic
142	#500	—	43	2	no	—
143	#499	—	43	4	no	—
144	#498	—	43	2	no	—
145	#446	—	86	4	no	—
146	#446	—	86	4	no	—
147	#427	—	43	4	no	—
148	#424	—	43	2	no	—
149	#424	—	43	2	no	—
150	#423	—	43	1	no	—
151	#423	C₂×D₄₃	1	1	yes	characteristic
152	#405	—	43	2	no	—
153	#405	—	43	2	no	—
154	C₂×#97	—	43	1	no	—
155	C₂×C₄×C₈	—	43	1	no	—
156	#376	C₂×D₄₃	1	1	yes	characteristic
157	#341	—	43	4	no	—
158	#339	—	43	4	no	—
159	#339	C₂×D₄₃	1	4	yes	—
160	#336	—	43	2	no	—
161	#336	—	43	2	no	—
162	C₄₃	#22258	1	1	yes	characteristic, complemented, Sylow
163	C₂×M₁₆	—	43	2	no	—
164	C₂×M₁₆	C₂²×D₄₃	1	2	yes	—
165	C₂²×C₈	—	43	2	no	—
166	C₂²×C₈	—	43	1	no	—
167	C₂²×C₈	—	43	1	no	—
168	C₄⋊Q₈	—	43	2	no	—
169	C₄⋊Q₈	C₄×D₄₃	1	2	yes	—
170	#125	—	43	2	no	—
171	#125	C₄×D₄₃	1	2	yes	—
172	C₂²⋊Q₈	—	2	2	no	—
173	C₂²⋊Q₈	—	86	2	no	—
174	C₄×Q₈	—	2	2	no	—
175	C₄×Q₈	—	86	2	no	—
176	#117	—	43	2	no	—
177	#117	C₂²×D₄₃	1	2	yes	—
178	C₂×C₄²	C₂²×D₄₃	1	2	yes	—
179	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
180	#107	—	43	8	no	—
181	#106	—	43	8	no	—
182	C₄⋊C₈	—	2	2	no	—
183	C₄⋊C₈	—	86	4	no	—
184	C₄⋊C₈	—	86	4	no	—
185	C₄⋊C₈	—	86	2	no	—
186	#97	—	43	4	no	—
187	#97	—	86	2	no	—
188	#97	—	86	2	no	—
189	C₄×C₈	—	43	1	no	—
190	C₄×C₈	—	43	2	no	—
191	C₄×C₈	—	43	1	no	—
192	C₄×C₈	—	86	2	no	—
193	C₄×C₈	—	86	2	no	—
194	#95	—	2	2	no	—
195	C₂×Q₈	—	2	2	no	—
196	C₂×Q₈	—	86	2	no	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	#10621900	1	2	yes	—
199	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
200	M₁₆	—	2	4	no	—
201	M₁₆	—	86	4	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	43	2	no	—
204	C₂×C₈	—	43	2	no	—
205	C₂×C₈	—	43	2	no	—
206	C₂×C₈	—	43	2	no	—
207	C₂×C₈	—	86	2	no	—
208	C₂×C₈	—	86	2	no	—
209	C₂×C₈	—	86	2	no	—
210	C₂×C₈	—	86	2	no	—
211	C₂×C₈	—	86	2	no	—
212	C₄⋊C₄	—	2	4	no	—
213	C₄⋊C₄	—	2	2	no	—
214	C₄⋊C₄	—	43	4	no	—
215	C₄⋊C₄	—	86	4	no	—
216	C₄⋊C₄	—	86	2	no	—
217	C₄⋊C₄	C₂×C₄×D₄₃	1	4	yes	—
218	C₂²⋊C₄	—	2	2	no	—
219	C₂²⋊C₄	—	86	2	no	—
220	C₄²	—	2	2	no	—
221	C₄²	—	2	2	no	—
222	C₄²	—	2	2	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	C₂×#10621900	1	1	yes	characteristic
228	Q₈	—	8	2	no	—
229	Q₈	—	344	2	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	4	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	86	4	no	—
235	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	2	yes	—
236	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	2	yes	—
237	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
238	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
239	C₂×C₄	#13063372	1	2	yes	—
240	C₂×C₄	#13063373	1	2	yes	—
241	C₈	—	4	2	no	—
242	C₈	—	86	2	no	—
243	C₈	—	86	2	no	—
244	C₈	—	86	2	no	—
245	C₈	—	86	2	no	—
246	C₈	—	172	2	no	—
247	C₂²	—	2	2	no	—
248	C₂²	C₂×(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
249	C₂²	#14583509	1	2	yes	—
250	C₄	—	2	2	no	—
251	C₄	—	2	2	no	—
252	C₄	—	4	4	no	—
253	C₄	—	8	2	no	—
254	C₄	—	172	4	no	—
255	C₄	#14583375	1	2	yes	—
256	C₄	#14583469	1	1	yes	characteristic
257	C₄	#14583474	1	1	yes	characteristic
258	C₂	—	2	2	no	—
259	C₂	#19137313	1	1	yes	characteristic
260	C₂	#19143508	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×#1995	C₂
socle	C₂²×C₄₃	C₂×C₄×D₄
prime radical	C₂³×C₄₃	C₂×#41
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#1995	D₄₃
hypocenter	C₄₃	#22258
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1995

Complements

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

N	H ≅ G/N	type
11. #14583302	C₄	semidirect
12. #14583298	C₄	semidirect
13. #14583295	C₄	semidirect
162. C₄₃	#22258	semidirect

Maximal subgroups

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

subgroup	index
#19145105	2
#19143536	2
#19143532	2
#19135576	2
#19135197	2
#19135192	2
#19134732	2
#19134347	2
C₄₃×#1995	2
#22258	43