Subgroups — #25249209

Order 11008.

806 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583309	C₂	1	2	yes	maximal
2	C₂×#14583297	C₂	1	1	yes	maximal, characteristic
3	#19140716	C₂	1	2	yes	maximal
4	#19134762	C₂	1	4	yes	maximal
5	#19134756	C₂	1	2	yes	maximal
6	#19134750	C₂	1	2	yes	maximal
7	#19134281	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄₃×#499	C₂	1	1	yes	maximal, characteristic
9	C₂×#13063325	C₂²	1	2	yes	—
10	#14583309	—	2	8	no	—
11	#14583297	—	2	4	no	—
12	C₂×#13063258	C₂²	1	4	yes	—
13	C₂×#13063257	C₂²	1	2	yes	—
14	C₂×#13063257	C₂²	1	2	yes	—
15	C₂×#13063256	C₂²	1	4	yes	—
16	C₂×#13063236	C₂²	1	1	yes	characteristic
17	C₂×#13063235	C₂²	1	2	yes	—
18	C₂×C₈×Dic₄₃	C₂²	1	2	yes	—
19	#14582829	—	2	4	no	—
20	#14582819	—	2	4	no	—
21	#14582768	C₂²	1	4	yes	—
22	#14582764	C₂²	1	4	yes	—
23	#14582745	C₂²	1	1	yes	characteristic
24	C₂×#125×C₄₃	C₂²	1	2	yes	—
25	C₄₃×#499	—	2	4	no	—
26	C₂×#107×C₄₃	C₂²	1	2	yes	—
27	C₂×#106×C₄₃	C₂²	1	1	yes	characteristic
28	C₂×#106×C₄₃	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
30	C₂²×Dic₈₆	—	2	2	no	—
31	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
32	#13063325	—	2	4	no	—
33	#13063325	—	2	4	no	—
34	C₂×C₄⋊Dic₄₃	—	2	2	no	—
35	C₂×C₄⋊Dic₄₃	C₂³	1	4	yes	—
36	C₂×Dic₄₃⋊C₄	—	2	8	no	—
37	C₂×C₄×Dic₄₃	C₂³	1	2	yes	—
38	#13063258	—	2	8	no	—
39	#13063257	—	2	4	no	—
40	#13063257	—	2	4	no	—
41	#13063256	—	2	8	no	—
42	#13063236	D₄	1	4	yes	—
43	#13063235	Q₈	1	8	yes	complemented
44	C₈×Dic₄₃	Q₈	1	8	yes	complemented
45	C₄₃⋊#95	—	2	8	no	—
46	C₄₃⋊#95	D₄	1	4	yes	—
47	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
49	#125×C₄₃	—	2	4	no	—
50	#125×C₄₃	—	2	4	no	—
51	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
52	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
53	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
54	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
55	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
56	#107×C₄₃	—	2	4	no	—
57	#106×C₄₃	—	2	2	no	—
58	#106×C₄₃	—	2	2	no	—
59	C₄×C₈×C₄₃	D₄	1	4	yes	—
60	C₂²×C₄×C₄₃	—	2	2	no	—
61	C₂²×C₄×C₄₃	—	2	2	no	—
62	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
64	C₂²×Dic₄₃	—	2	4	no	—
65	C₂²×Dic₄₃	C₂×D₄	1	4	yes	—
66	C₂×Dic₈₆	—	4	4	no	—
67	C₂×Dic₈₆	—	8	4	no	—
68	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
69	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
70	C₂×C₈×C₄₃	C₂×Q₈	1	4	yes	—
71	C₂×C₈×C₄₃	C₂×Q₈	1	4	yes	—
72	C₄⋊C₄×C₄₃	—	2	2	no	—
73	C₄⋊C₄×C₄₃	—	2	2	no	—
74	C₄⋊C₄×C₄₃	—	2	2	no	—
75	C₄⋊C₄×C₄₃	—	2	2	no	—
76	C₄⋊C₄×C₄₃	—	4	4	no	—
77	C₄⋊C₄×C₄₃	—	4	4	no	—
78	C₄²×C₄₃	C₂×D₄	1	2	yes	—
79	C₄²×C₄₃	C₂×D₄	1	2	yes	—
80	C₄⋊Dic₄₃	—	2	4	no	—
81	C₄⋊Dic₄₃	—	2	4	no	—
82	C₄⋊Dic₄₃	—	4	4	no	—
83	Dic₄₃⋊C₄	—	8	8	no	—
84	C₄×Dic₄₃	C₂×Q₈	1	4	yes	—
85	C₄×Dic₄₃	C₂×Q₈	1	4	yes	—
86	C₂×C₄₃⋊C₈	C₂×D₄	1	4	yes	—
87	C₂×C₄₃⋊C₈	C₂×Q₈	1	4	yes	—
88	C₂×C₄₃⋊C₈	C₂×Q₈	1	4	yes	—
89	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
90	C₂×C₄×C₄₃	—	2	4	no	—
91	C₂×C₄×C₄₃	—	4	2	no	—
92	C₂×C₄×C₄₃	—	4	2	no	—
93	C₂×C₄×C₄₃	—	4	4	no	—
94	C₂×C₄×C₄₃	—	4	4	no	—
95	C₂×C₄×C₄₃	#127	1	4	yes	—
96	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
97	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
98	C₂×C₄×C₄₃	C₂²×Q₈	1	2	yes	—
99	C₂×C₄×C₄₃	C₂²×Q₈	1	2	yes	—
100	C₂×Dic₄₃	—	2	8	no	—
101	C₂×Dic₄₃	—	4	4	no	—
102	C₂×Dic₄₃	—	4	8	no	—
103	C₂×Dic₄₃	C₄⋊Q₈	1	8	yes	—
104	Dic₈₆	—	8	8	no	—
105	C₈×C₄₃	C₄⋊Q₈	1	4	yes	—
106	C₈×C₄₃	C₄⋊Q₈	1	4	yes	—
107	C₄₃⋊C₈	C₄⋊Q₈	1	8	yes	—
108	#33318	—	43	1	no	maximal, Sylow
109	C₂²×C₄₃	C₂×#127	1	1	yes	characteristic
110	C₂²×C₄₃	C₂×C₄⋊Q₈	1	2	yes	—
111	C₂²×C₄₃	#577	1	4	yes	—
112	C₄×C₄₃	—	2	4	no	—
113	C₄×C₄₃	—	8	2	no	—
114	C₄×C₄₃	—	8	2	no	—
115	C₄×C₄₃	C₂×C₄⋊Q₈	1	2	yes	—
116	C₄×C₄₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
117	C₄×C₄₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
118	Dic₄₃	—	2	8	no	—
119	Dic₄₃	—	8	4	no	—
120	C₂×#501	—	43	2	no	—
121	C₂×#499	D₄₃	1	1	yes	characteristic
122	C₂×#498	—	43	1	no	—
123	#2660	—	43	2	no	—
124	#1847	—	43	4	no	—
125	#1846	—	43	2	no	—
126	#1844	—	43	2	no	—
127	#1726	—	43	1	no	—
128	C₂×C₄₃	#2868	1	1	yes	characteristic
129	C₂×C₄₃	#3154	1	2	yes	—
130	C₂×C₄₃	#3163	1	4	yes	—
131	C₂×C₄⋊Q₈	—	43	2	no	—
132	C₂×#125	C₂×D₄₃	1	2	yes	—
133	#501	—	86	8	no	—
134	#499	—	2	4	no	—
135	#498	—	86	4	no	—
136	C₂×#107	—	43	4	no	—
137	C₂×#107	C₂×D₄₃	1	2	yes	—
138	C₂×#106	—	43	2	no	—
139	C₂×#106	—	43	2	no	—
140	C₂×#106	—	43	4	no	—
141	C₂×#106	C₂×D₄₃	1	1	yes	characteristic
142	C₂×#106	C₂×D₄₃	1	1	yes	characteristic
143	C₂×#97	—	43	2	no	—
144	C₂×#97	—	43	1	no	—
145	C₂×C₄×C₈	—	43	2	no	—
146	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
147	#398	—	86	4	no	—
148	#395	—	86	4	no	—
149	#384	—	43	4	no	—
150	#382	—	43	4	no	—
151	#376	—	43	1	no	—
152	C₄₃	#33318	1	1	yes	characteristic, complemented, Sylow
153	C₂²×Q₈	—	86	2	no	—
154	C₂²×C₈	—	43	2	no	—
155	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
156	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
157	C₄⋊Q₈	—	86	4	no	—
158	C₄⋊Q₈	—	86	4	no	—
159	#125	—	2	4	no	—
160	#125	—	2	4	no	—
161	C₂×C₄⋊C₄	—	2	2	no	—
162	C₂×C₄⋊C₄	—	2	2	no	—
163	C₂×C₄⋊C₄	—	43	4	no	—
164	C₂×C₄⋊C₄	—	86	8	no	—
165	C₂×C₄⋊C₄	—	86	2	no	—
166	C₂×C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
167	C₂×C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
168	C₂×C₄²	—	43	2	no	—
169	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
170	#107	—	2	4	no	—
171	#107	—	86	8	no	—
172	#106	—	2	2	no	—
173	#106	—	2	2	no	—
174	#106	—	86	8	no	—
175	#106	—	86	4	no	—
176	#106	—	86	4	no	—
177	#97	—	43	8	no	—
178	#97	—	43	4	no	—
179	C₄×C₈	—	43	8	no	—
180	C₄×C₈	C₂²⋊D₄₃	1	4	yes	—
181	#95	—	43	4	no	—
182	#95	—	86	8	no	—
183	C₂×Q₈	—	344	4	no	—
184	C₂×Q₈	—	172	4	no	—
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	—	43	4	no	—
188	C₂²×C₄	—	86	4	no	—
189	C₂²×C₄	D₄×D₄₃	1	2	yes	—
190	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
191	C₂×C₈	—	43	4	no	—
192	C₂×C₈	—	43	4	no	—
193	C₂×C₈	—	43	4	no	—
194	C₂×C₈	Q₈×D₄₃	1	4	yes	—
195	C₂×C₈	Q₈×D₄₃	1	4	yes	—
196	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
197	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
198	C₄⋊C₄	—	2	2	no	—
199	C₄⋊C₄	—	2	2	no	—
200	C₄⋊C₄	—	2	2	no	—
201	C₄⋊C₄	—	2	2	no	—
202	C₄⋊C₄	—	4	4	no	—
203	C₄⋊C₄	—	4	4	no	—
204	C₄⋊C₄	—	86	4	no	—
205	C₄⋊C₄	—	86	4	no	—
206	C₄⋊C₄	—	344	8	no	—
207	C₄⋊C₄	—	172	4	no	—
208	C₄²	—	43	4	no	—
209	C₄²	—	43	4	no	—
210	C₄²	D₄×D₄₃	1	2	yes	—
211	C₄²	C₂×C₂²⋊D₄₃	1	2	yes	—
212	C₂³	C₂×D₄×D₄₃	1	1	yes	characteristic
213	Q₈	—	344	8	no	—
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	4	2	no	—
216	C₂×C₄	—	4	2	no	—
217	C₂×C₄	—	4	4	no	—
218	C₂×C₄	—	4	4	no	—
219	C₂×C₄	—	43	8	no	—
220	C₂×C₄	—	86	8	no	—
221	C₂×C₄	—	172	4	no	—
222	C₂×C₄	—	172	8	no	—
223	C₂×C₄	#13063321	1	4	yes	—
224	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
225	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
226	C₂×C₄	C₂×Q₈×D₄₃	1	2	yes	—
227	C₂×C₄	C₂×Q₈×D₄₃	1	2	yes	—
228	C₈	—	43	8	no	—
229	C₈	#13063323	1	4	yes	—
230	C₈	#13063323	1	4	yes	—
231	C₂²	C₂×#13063321	1	1	yes	characteristic
232	C₂²	C₄⋊Q₈×D₄₃	1	2	yes	—
233	C₂²	#14583706	1	4	yes	—
234	C₄	—	2	4	no	—
235	C₄	—	8	2	no	—
236	C₄	—	8	2	no	—
237	C₄	—	86	8	no	—
238	C₄	—	344	4	no	—
239	C₄	C₄⋊Q₈×D₄₃	1	2	yes	—
240	C₄	C₂×#13063323	1	1	yes	characteristic
241	C₄	C₂×#13063323	1	1	yes	characteristic
242	C₂	#19142991	1	1	yes	characteristic
243	C₂	#19145072	1	2	yes	—
244	C₂	#19145124	1	4	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₃
commutator, cosocle	C₂²×C₄×C₄₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₂×C₄₃×#499	C₂
socle, prime radical	C₂³×C₄₃	C₂²×D₄
hypercenter	C₂×#499	D₄₃
hypocenter	C₄₃	#33318
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#499

Complements

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

N	H ≅ G/N	type
43. #13063235	Q₈	semidirect
44. C₈×Dic₄₃	Q₈	semidirect
152. C₄₃	#33318	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583309	2
C₂×#14583297	2
#19140716	2
#19134762	2
#19134756	2
#19134750	2
#19134281	2
C₂×C₄₃×#499	2
#33318	43