Subgroups — #25110207

Order 11008.

790 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊Q₈×Dic₄₃	C₂	1	2	yes	maximal
2	#19135557	C₂	1	2	yes	maximal
3	#19135168	C₂	1	8	yes	maximal
4	#19134723	C₂	1	1	yes	maximal, characteristic
5	#19134340	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄₃×#500	C₂	1	1	yes	maximal, characteristic
7	C₂×Q₈×Dic₄₃	—	2	2	no	—
8	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
9	#14582806	—	2	2	no	—
10	#14582798	—	2	2	no	—
11	#14582798	C₂²	1	4	yes	—
12	#14582769	C₂²	1	2	yes	—
13	C₄⋊C₄×Dic₄₃	—	2	2	no	—
14	C₄⋊C₄×Dic₄₃	C₂²	1	4	yes	—
15	C₄²×Dic₄₃	C₂²	1	1	yes	characteristic
16	#14582662	C₂²	1	4	yes	—
17	#14582661	C₂²	1	8	yes	—
18	#14582653	C₂²	1	4	yes	—
19	C₂×C₄⋊Q₈×C₄₃	C₂²	1	2	yes	—
20	C₄₃×#500	C₄	1	8	yes	complemented
21	C₂×#107×C₄₃	C₂²	1	4	yes	—
22	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
23	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
24	Q₈×Dic₄₃	—	4	8	no	—
25	C₂×C₄⋊Dic₄₃	—	2	2	no	—
26	C₂×C₄⋊Dic₄₃	—	2	2	no	—
27	C₂×C₄⋊Dic₄₃	—	2	4	no	—
28	C₂×C₄⋊Dic₄₃	C₂³	1	4	yes	—
29	C₂×C₄×Dic₄₃	—	2	4	no	—
30	C₂×C₄×Dic₄₃	D₄	1	2	yes	—
31	C₂×C₄×Dic₄₃	D₄	1	1	yes	characteristic
32	C₂×C₄×Dic₄₃	D₄	1	1	yes	characteristic
33	C₂×C₄×Dic₄₃	C₂³	1	2	yes	—
34	C₄×C₄₃⋊C₈	—	2	2	no	—
35	C₄₃⋊#95	—	2	4	no	—
36	C₄₃⋊#95	—	2	4	no	—
37	C₂²×Q₈×C₄₃	—	2	2	no	—
38	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
39	C₄⋊Q₈×C₄₃	—	2	4	no	—
40	C₄⋊Q₈×C₄₃	C₂×C₄	1	8	yes	—
41	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
42	C₂×C₄⋊C₄×C₄₃	C₂³	1	4	yes	—
43	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
44	#107×C₄₃	C₂×C₄	1	16	yes	—
45	C₄×C₈×C₄₃	C₂×C₄	1	4	yes	—
46	C₂×Q₈×C₄₃	—	2	8	no	—
47	C₂×Q₈×C₄₃	—	4	4	no	—
48	C₂×Q₈×C₄₃	—	4	4	no	—
49	C₂²×C₄×C₄₃	—	2	4	no	—
50	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
53	C₂²×Dic₄₃	—	2	4	no	—
54	C₂²×Dic₄₃	C₂×D₄	1	2	yes	—
55	C₂²×Dic₄₃	C₂×D₄	1	2	yes	—
56	C₂×C₈×C₄₃	—	2	4	no	—
57	C₂×C₈×C₄₃	C₂²×C₄	1	4	yes	—
58	C₄⋊C₄×C₄₃	—	2	8	no	—
59	C₄⋊C₄×C₄₃	—	2	4	no	—
60	C₄⋊C₄×C₄₃	C₂²×C₄	1	8	yes	—
61	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
62	C₄²×C₄₃	#41	1	2	yes	—
63	C₄⋊Dic₄₃	—	2	8	no	—
64	C₄⋊Dic₄₃	—	4	4	no	—
65	C₄⋊Dic₄₃	—	4	4	no	—
66	C₄⋊Dic₄₃	—	4	8	no	—
67	C₄×Dic₄₃	—	2	2	no	—
68	C₄×Dic₄₃	—	2	4	no	—
69	C₄×Dic₄₃	—	2	4	no	—
70	C₄×Dic₄₃	—	4	8	no	—
71	C₄×Dic₄₃	D₈	1	4	yes	—
72	C₂×C₄₃⋊C₈	—	2	2	no	—
73	C₂×C₄₃⋊C₈	—	2	4	no	—
74	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
75	Q₈×C₄₃	—	4	16	no	—
76	C₂×C₄×C₄₃	—	2	8	no	—
77	C₂×C₄×C₄₃	—	4	4	no	—
78	C₂×C₄×C₄₃	—	4	4	no	—
79	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
80	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
81	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
82	C₂×C₄×C₄₃	#135	1	2	yes	—
83	C₂×C₄×C₄₃	#137	1	4	yes	—
84	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
85	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
86	C₂×C₄×C₄₃	2₋¹⁺⁴	1	4	yes	—
87	C₂×Dic₄₃	—	2	2	no	—
88	C₂×Dic₄₃	—	2	2	no	—
89	C₂×Dic₄₃	—	2	2	no	—
90	C₂×Dic₄₃	—	2	2	no	—
91	C₂×Dic₄₃	—	2	2	no	—
92	C₂×Dic₄₃	—	4	4	no	—
93	C₂×Dic₄₃	—	4	4	no	—
94	C₂×Dic₄₃	—	4	4	no	—
95	C₂×Dic₄₃	C₂×D₈	1	4	yes	—
96	C₈×C₄₃	—	2	4	no	—
97	C₄₃⋊C₈	—	4	2	no	—
98	#22247	—	43	1	no	maximal, Sylow
99	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
100	C₂²×C₄₃	#536	1	2	yes	—
101	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
102	C₂²×C₄₃	C₂×#135	1	1	yes	characteristic
103	C₂²×C₄₃	C₂×#137	1	2	yes	—
104	C₄×C₄₃	—	4	8	no	—
105	C₄×C₄₃	C₄×D₈	1	4	yes	—
106	C₄×C₄₃	#441	1	4	yes	—
107	C₄×C₄₃	#519	1	2	yes	—
108	C₄×C₄₃	#541	1	2	yes	—
109	Dic₄₃	—	2	4	no	—
110	Dic₄₃	—	4	2	no	—
111	Dic₄₃	—	8	4	no	—
112	C₂×#500	D₄₃	1	1	yes	characteristic
113	C₄×C₄⋊Q₈	—	43	2	no	—
114	#1990	—	43	2	no	—
115	#1918	—	43	8	no	—
116	#1834	—	43	1	no	—
117	#1744	—	43	1	no	—
118	C₂×C₄₃	#2327	1	1	yes	characteristic
119	C₂×C₄₃	C₂×C₄×D₈	1	1	yes	characteristic
120	C₂×C₄₃	C₂×#441	1	1	yes	characteristic
121	C₂×C₄₃	#3226	1	2	yes	—
122	C₂×C₄₃	#3231	1	2	yes	—
123	C₂×C₄⋊Q₈	C₂×D₄₃	1	2	yes	—
124	C₂×C₄×Q₈	—	86	2	no	—
125	#500	Dic₄₃	1	8	yes	complemented
126	C₂×#107	C₂×D₄₃	1	4	yes	—
127	C₂×C₄×C₈	—	43	1	no	—
128	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
129	#391	—	86	2	no	—
130	#389	—	43	4	no	—
131	#389	—	86	2	no	—
132	#384	—	43	2	no	—
133	C₄×C₄⋊C₄	—	43	4	no	—
134	C₄×C₄⋊C₄	—	86	2	no	—
135	C₄³	—	43	1	no	—
136	#340	—	43	4	no	—
137	#340	—	43	8	no	—
138	#336	—	43	4	no	—
139	C₄₃	#22247	1	1	yes	characteristic, complemented, Sylow
140	C₂²×Q₈	—	2	2	no	—
141	C₂²×C₈	—	43	2	no	—
142	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
143	C₄⋊Q₈	—	2	4	no	—
144	C₄⋊Q₈	C₂×Dic₄₃	1	8	yes	—
145	C₄×Q₈	—	172	8	no	—
146	C₂×C₄⋊C₄	—	2	2	no	—
147	C₂×C₄⋊C₄	—	43	4	no	—
148	C₂×C₄⋊C₄	—	86	4	no	—
149	C₂×C₄⋊C₄	—	86	2	no	—
150	C₂×C₄⋊C₄	—	86	2	no	—
151	C₂×C₄⋊C₄	C₂²×D₄₃	1	4	yes	—
152	C₂×C₄²	—	43	1	no	—
153	C₂×C₄²	—	43	2	no	—
154	C₂×C₄²	—	43	1	no	—
155	C₂×C₄²	—	43	2	no	—
156	C₂×C₄²	—	86	4	no	—
157	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
158	#107	C₂×Dic₄₃	1	16	yes	—
159	C₄×C₈	—	86	2	no	—
160	C₄×C₈	C₂×Dic₄₃	1	4	yes	—
161	#95	—	86	4	no	—
162	#95	—	86	4	no	—
163	C₂×Q₈	—	2	8	no	—
164	C₂×Q₈	—	4	4	no	—
165	C₂×Q₈	—	4	4	no	—
166	C₂²×C₄	—	2	4	no	—
167	C₂²×C₄	—	43	2	no	—
168	C₂²×C₄	—	43	2	no	—
169	C₂²×C₄	—	86	4	no	—
170	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
171	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
172	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
173	C₂×C₈	—	2	4	no	—
174	C₂×C₈	—	86	2	no	—
175	C₂×C₈	—	86	4	no	—
176	C₂×C₈	C₂²×Dic₄₃	1	4	yes	—
177	C₄⋊C₄	—	2	8	no	—
178	C₄⋊C₄	—	2	4	no	—
179	C₄⋊C₄	—	86	8	no	—
180	C₄⋊C₄	—	172	8	no	—
181	C₄⋊C₄	—	172	4	no	—
182	C₄⋊C₄	—	172	4	no	—
183	C₄⋊C₄	C₂²×Dic₄₃	1	8	yes	—
184	C₄²	—	43	4	no	—
185	C₄²	—	86	2	no	—
186	C₄²	—	86	4	no	—
187	C₄²	—	86	4	no	—
188	C₄²	—	172	8	no	—
189	C₄²	#10621900	1	2	yes	—
190	C₄²	C₂²×Dic₄₃	1	2	yes	—
191	C₂³	C₂×D₄×D₄₃	1	1	yes	characteristic
192	Q₈	—	4	16	no	—
193	C₂×C₄	—	2	8	no	—
194	C₂×C₄	—	4	4	no	—
195	C₂×C₄	—	4	4	no	—
196	C₂×C₄	—	43	4	no	—
197	C₂×C₄	—	86	2	no	—
198	C₂×C₄	—	86	2	no	—
199	C₂×C₄	—	86	2	no	—
200	C₂×C₄	—	86	2	no	—
201	C₂×C₄	—	86	2	no	—
202	C₂×C₄	—	172	4	no	—
203	C₂×C₄	—	172	4	no	—
204	C₂×C₄	—	172	4	no	—
205	C₂×C₄	D₄×Dic₄₃	1	2	yes	—
206	C₂×C₄	D₄×Dic₄₃	1	2	yes	—
207	C₂×C₄	D₈×D₄₃	1	2	yes	—
208	C₂×C₄	#13063344	1	2	yes	—
209	C₂×C₄	#13063357	1	4	yes	—
210	C₂×C₄	C₂³×Dic₄₃	1	1	yes	characteristic
211	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
212	C₂×C₄	#13063373	1	4	yes	—
213	C₈	—	2	4	no	—
214	C₈	—	172	2	no	—
215	C₂²	C₂×D₄×Dic₄₃	1	1	yes	characteristic
216	C₂²	#14583474	1	2	yes	—
217	C₂²	C₂×D₈×D₄₃	1	1	yes	characteristic
218	C₂²	C₂×#13063344	1	1	yes	characteristic
219	C₂²	C₂×#13063357	1	2	yes	—
220	C₄	—	4	8	no	—
221	C₄	—	86	4	no	—
222	C₄	—	344	4	no	—
223	C₄	—	172	2	no	—
224	C₄	D₈×Dic₄₃	1	4	yes	—
225	C₄	#14583006	1	4	yes	—
226	C₄	#14583377	1	2	yes	—
227	C₄	#14583510	1	2	yes	—
228	C₂	#19137315	1	1	yes	characteristic
229	C₂	C₂×D₈×Dic₄₃	1	1	yes	characteristic
230	C₂	C₂×#14583006	1	1	yes	characteristic
231	C₂	#19145717	1	2	yes	—
232	C₂	#19145755	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₂×C₄₃×#500	C₂
socle, prime radical	C₂³×C₄₃	C₂²×D₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₂×#500	D₄₃
hypocenter	C₄₃	#22247
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
20. C₄₃×#500	C₄	semidirect
125. #500	Dic₄₃	semidirect
139. C₄₃	#22247	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊Q₈×Dic₄₃	2
#19135557	2
#19135168	2
#19134723	2
#19134340	2
C₂×C₄₃×#500	2
#22247	43