Subgroups — #25105235

Order 11008.

1006 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583016	C₂	1	4	yes	maximal, complemented
2	#19143307	C₂	1	2	yes	maximal, complemented
3	#19135489	C₂	1	4	yes	maximal, complemented
4	#19135428	C₂	1	2	yes	maximal, complemented
5	#19134770	C₂	1	1	yes	maximal, characteristic, complemented
6	#19134283	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#3150	C₂	1	1	yes	maximal, characteristic
8	C₂×#13063332	C₂²	1	4	yes	—
9	#14583016	—	2	16	no	—
10	#14582977	—	2	4	no	—
11	D₄×C₄₃⋊C₈	—	2	4	no	—
12	C₂×#13063261	C₂²	1	4	yes	—
13	#14582923	C₂²	1	2	yes	—
14	C₂×#13063252	C₂²	1	8	yes	—
15	#14582866	C₂²	1	4	yes	—
16	C₂×#13063235	C₂²	1	2	yes	—
17	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₈×Dic₄₃	C₂²	1	2	yes	—
19	#14582745	C₂²	1	1	yes	characteristic
20	C₂×#135×C₄₃	C₂²	1	4	yes	—
21	C₄₃×#532	C₂²	1	2	yes	—
22	C₄₃×#497	C₄	1	4	yes	complemented
23	C₄₃×#496	C₄	1	4	yes	complemented
24	C₂×#97×C₄₃	C₂²	1	1	yes	characteristic
25	#13063332	—	2	16	no	—
26	C₂×#10621877	—	2	4	no	—
27	C₂×#10621877	C₂³	1	4	yes	—
28	C₂²×C₄₃⋊C₈	—	2	4	no	—
29	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
30	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
31	C₂×C₄×Dic₄₃	C₂³	1	2	yes	—
32	#13063261	—	2	8	no	—
33	C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
34	#13063252	—	2	16	no	—
35	C₂²⋊(C₄₃⋊C₈)	—	2	8	no	—
36	#13063235	—	2	4	no	—
37	C₄×C₄₃⋊C₈	D₄	1	2	yes	—
38	C₄×C₄₃⋊C₈	D₄	1	2	yes	—
39	C₈×Dic₄₃	D₄	1	8	yes	complemented
40	C₄₃⋊#95	—	2	2	no	—
41	C₂×#41×C₄₃	C₂³	1	4	yes	—
42	#135×C₄₃	—	2	8	no	—
43	#135×C₄₃	—	2	8	no	—
44	C₂×Q₁₆×C₄₃	C₂×C₄	1	4	yes	—
45	C₂×SD₁₆×C₄₃	C₂×C₄	1	8	yes	—
46	C₂×D₈×C₄₃	C₂×C₄	1	4	yes	—
47	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
48	C₄⋊Q₈×C₄₃	C₂×C₄	1	2	yes	—
49	#127×C₄₃	C₂×C₄	1	2	yes	—
50	#124×C₄₃	C₂×C₄	1	4	yes	—
51	#121×C₄₃	—	2	4	no	—
52	C₄×D₄×C₄₃	—	2	4	no	—
53	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
54	#97×C₄₃	C₂×C₄	1	4	yes	—
55	#41×C₄₃	—	2	8	no	—
56	#41×C₄₃	—	2	8	no	—
57	C₂×Q₈×C₄₃	C₂²×C₄	1	4	yes	—
58	C₂×D₄×C₄₃	—	2	4	no	—
59	C₂×D₄×C₄₃	—	2	4	no	—
60	C₂×D₄×C₄₃	C₂²×C₄	1	4	yes	—
61	C₂²×C₄×C₄₃	—	2	4	no	—
62	C₂²×C₄×C₄₃	#41	1	2	yes	—
63	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
64	C₂²×Dic₄₃	—	2	2	no	—
65	Q₁₆×C₄₃	—	2	8	no	—
66	SD₁₆×C₄₃	—	2	16	no	—
67	D₈×C₄₃	—	2	8	no	—
68	C₂×C₈×C₄₃	C₂²×C₄	1	4	yes	—
69	C₂×C₈×C₄₃	C₂×D₄	1	4	yes	—
70	C₂×C₈×C₄₃	#41	1	4	yes	—
71	C₄⋊C₄×C₄₃	—	2	4	no	—
72	C₂²⋊C₄×C₄₃	—	2	8	no	—
73	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
74	C₄²×C₄₃	C₂×D₄	1	2	yes	—
75	C₄×Dic₄₃	—	2	2	no	—
76	C₄×Dic₄₃	C₂×D₄	1	4	yes	—
77	#10621877	—	2	8	no	—
78	#10621877	—	4	8	no	—
79	C₂×C₄₃⋊C₈	—	2	2	no	—
80	C₂×C₄₃⋊C₈	—	2	4	no	—
81	C₂×C₄₃⋊C₈	—	4	8	no	—
82	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
83	C₂×C₄₃⋊C₈	C₂×D₄	1	4	yes	—
84	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
85	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
86	C₂³×C₄₃	—	2	4	no	—
87	C₂³×C₄₃	C₂×#41	1	1	yes	characteristic
88	Q₈×C₄₃	—	2	8	no	—
89	D₄×C₄₃	—	2	8	no	—
90	D₄×C₄₃	—	4	8	no	—
91	D₄×C₄₃	—	4	8	no	—
92	C₂×C₄×C₄₃	—	2	4	no	—
93	C₂×C₄×C₄₃	—	4	2	no	—
94	C₂×C₄×C₄₃	—	4	4	no	—
95	C₂×C₄×C₄₃	—	4	4	no	—
96	C₂×C₄×C₄₃	C₄×D₄	1	4	yes	—
97	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
98	C₂×C₄×C₄₃	C₂²×D₄	1	2	yes	—
99	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
100	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
101	C₂×Dic₄₃	—	2	4	no	—
102	C₂×Dic₄₃	—	4	4	no	—
103	C₈×C₄₃	C₄×D₄	1	8	yes	—
104	C₄₃⋊C₈	—	2	2	no	—
105	C₄₃⋊C₈	—	4	4	no	—
106	C₄₃⋊C₈	#127	1	2	yes	—
107	C₄₃⋊C₈	#127	1	2	yes	—
108	#21708	—	43	1	no	maximal, Sylow
109	C₂²×C₄₃	—	2	2	no	—
110	C₂²×C₄₃	—	4	4	no	—
111	C₂²×C₄₃	—	4	8	no	—
112	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
113	C₂²×C₄₃	#532	1	2	yes	—
114	C₄×C₄₃	—	2	4	no	—
115	C₄×C₄₃	—	4	4	no	—
116	C₄×C₄₃	C₂×C₄×D₄	1	2	yes	—
117	C₄×C₄₃	C₂×#127	1	1	yes	characteristic
118	C₄×C₄₃	#532	1	1	yes	characteristic
119	Dic₄₃	—	4	4	no	—
120	#3150	D₄₃	1	1	yes	characteristic
121	C₂×#443	—	43	4	no	—
122	#2931	—	43	2	no	—
123	#1975	—	43	4	no	—
124	#1963	—	43	2	no	—
125	#1849	—	43	1	no	—
126	#1726	—	43	1	no	—
127	C₂×C₄₃	—	2	2	no	—
128	C₂×C₄₃	—	8	4	no	—
129	C₂×C₄₃	C₄×#127	1	1	yes	characteristic
130	C₂×C₄₃	#2955	1	2	yes	—
131	C₂×#135	C₂×D₄₃	1	4	yes	—
132	C₂×#131	—	43	4	no	—
133	#532	C₂×D₄₃	1	2	yes	—
134	#497	Dic₄₃	1	4	yes	complemented
135	#496	Dic₄₃	1	4	yes	complemented
136	#443	—	86	16	no	—
137	#436	—	86	4	no	—
138	C₈×D₄	—	86	4	no	—
139	C₂×#108	—	43	4	no	—
140	#423	—	43	2	no	—
141	C₂×#104	—	43	8	no	—
142	#408	—	43	4	no	—
143	C₂×#97	—	43	2	no	—
144	C₂×#97	C₂×D₄₃	1	1	yes	characteristic
145	C₂×C₄×C₈	—	43	2	no	—
146	C₂×C₄×C₈	—	43	1	no	—
147	#376	—	43	1	no	—
148	C₄₃	#21708	1	1	yes	characteristic, complemented, Sylow
149	C₂×#41	C₂²×D₄₃	1	4	yes	—
150	#135	—	2	8	no	—
151	#135	—	2	8	no	—
152	C₂×Q₁₆	C₂×Dic₄₃	1	4	yes	—
153	C₂×SD₁₆	C₂×Dic₄₃	1	8	yes	—
154	C₂×D₈	C₂×Dic₄₃	1	4	yes	—
155	#131	—	86	16	no	—
156	C₂×M₁₆	—	43	4	no	—
157	C₂×M₁₆	—	86	4	no	—
158	C₂²×C₈	—	43	1	no	—
159	C₂²×C₈	—	43	1	no	—
160	C₂²×C₈	—	86	4	no	—
161	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
162	C₄⋊Q₈	C₂×Dic₄₃	1	2	yes	—
163	#127	C₂×Dic₄₃	1	2	yes	—
164	#124	C₂×Dic₄₃	1	4	yes	—
165	#121	—	2	4	no	—
166	C₄×D₄	—	2	4	no	—
167	C₂×C₄²	—	43	2	no	—
168	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
169	#108	—	86	8	no	—
170	C₄⋊C₈	—	86	4	no	—
171	#104	—	86	16	no	—
172	C₂²⋊C₈	—	86	8	no	—
173	#97	—	86	4	no	—
174	#97	C₂×Dic₄₃	1	4	yes	—
175	C₄×C₈	—	43	8	no	—
176	C₄×C₈	—	43	2	no	—
177	C₄×C₈	—	43	2	no	—
178	#95	—	86	2	no	—
179	#41	—	2	8	no	—
180	#41	—	2	8	no	—
181	C₂×Q₈	C₂²×Dic₄₃	1	4	yes	—
182	C₂×D₄	—	2	4	no	—
183	C₂×D₄	—	2	4	no	—
184	C₂×D₄	C₂²×Dic₄₃	1	4	yes	—
185	C₂²×C₄	—	2	4	no	—
186	C₂²×C₄	—	86	2	no	—
187	C₂²×C₄	#10621900	1	2	yes	—
188	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
189	Q₁₆	—	2	8	no	—
190	SD₁₆	—	2	16	no	—
191	D₈	—	2	8	no	—
192	M₁₆	—	86	8	no	—
193	M₁₆	—	172	8	no	—
194	C₂×C₈	—	43	4	no	—
195	C₂×C₈	—	43	1	no	—
196	C₂×C₈	—	43	1	no	—
197	C₂×C₈	—	43	2	no	—
198	C₂×C₈	—	86	2	no	—
199	C₂×C₈	—	86	4	no	—
200	C₂×C₈	—	172	8	no	—
201	C₂×C₈	D₄×D₄₃	1	4	yes	—
202	C₂×C₈	#10621900	1	4	yes	—
203	C₂×C₈	C₂²×Dic₄₃	1	4	yes	—
204	C₄⋊C₄	—	2	4	no	—
205	C₂²⋊C₄	—	2	8	no	—
206	C₄²	—	43	4	no	—
207	C₄²	—	86	2	no	—
208	C₄²	D₄×D₄₃	1	2	yes	—
209	C₄²	C₂²×Dic₄₃	1	2	yes	—
210	C₂³	—	2	4	no	—
211	C₂³	C₂×#10621900	1	1	yes	characteristic
212	Q₈	—	2	8	no	—
213	D₄	—	2	8	no	—
214	D₄	—	4	8	no	—
215	D₄	—	4	8	no	—
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	—	4	4	no	—
219	C₂×C₄	—	4	4	no	—
220	C₂×C₄	—	86	4	no	—
221	C₂×C₄	—	172	4	no	—
222	C₂×C₄	D₄×Dic₄₃	1	4	yes	—
223	C₂×C₄	C₂³×Dic₄₃	1	1	yes	characteristic
224	C₂×C₄	C₂×D₄×D₄₃	1	2	yes	—
225	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
226	C₂×C₄	C₂×#10621900	1	2	yes	—
227	C₈	—	43	2	no	—
228	C₈	—	43	2	no	—
229	C₈	—	86	2	no	—
230	C₈	—	172	4	no	—
231	C₈	D₄×Dic₄₃	1	8	yes	—
232	C₂²	—	2	2	no	—
233	C₂²	—	4	8	no	—
234	C₂²	—	4	4	no	—
235	C₂²	C₂×D₄×Dic₄₃	1	1	yes	characteristic
236	C₂²	#14583447	1	2	yes	—
237	C₄	—	2	4	no	—
238	C₄	—	4	4	no	—
239	C₄	—	172	4	no	—
240	C₄	C₂×D₄×Dic₄₃	1	2	yes	—
241	C₄	#127×D₄₃	1	1	yes	characteristic
242	C₄	#14583447	1	1	yes	characteristic
243	C₂	—	2	2	no	—
244	C₂	—	8	4	no	—
245	C₂	#127×Dic₄₃	1	1	yes	characteristic
246	C₂	#19143498	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting, prime radical	C₄₃×#3150	C₂
socle	C₂²×C₄₃	C₂×C₄×D₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#3150	D₄₃
hypocenter	C₄₃	#21708
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3150

Complements

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

N	H ≅ G/N	type
1. C₂×#14583016	C₂	semidirect
2. #19143307	C₂	semidirect
3. #19135489	C₂	semidirect
4. #19135428	C₂	semidirect
5. #19134770	C₂	semidirect
6. #19134283	C₂	semidirect
22. C₄₃×#497	C₄	semidirect
23. C₄₃×#496	C₄	semidirect
39. C₈×Dic₄₃	D₄	semidirect
134. #497	Dic₄₃	semidirect
135. #496	Dic₄₃	semidirect
148. C₄₃	#21708	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583016	2
#19143307	2
#19135489	2
#19135428	2
#19134770	2
#19134283	2
C₄₃×#3150	2
#21708	43