Subgroups — #25249637

Order 11008.

658 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583315	C₂	1	1	yes	maximal, characteristic
2	#19143191	C₂	1	2	yes	maximal
3	#19136381	C₂	1	2	yes	maximal, complemented
4	#19136380	C₂	1	4	yes	maximal, complemented
5	#19136379	C₂	1	1	yes	maximal, characteristic, complemented
6	#19136376	C₂	1	2	yes	maximal, complemented
7	#366×D₄₃	C₂	1	2	yes	maximal, complemented
8	C₂×C₄₃×#366	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13063331	C₂²	1	1	yes	characteristic
10	#14583380	—	2	2	no	—
11	#14583315	C₄	1	8	yes	complemented
12	C₂×#13063265	C₂²	1	1	yes	characteristic
13	C₂×#13063264	C₂²	1	2	yes	—
14	C₂×C₁₆×D₄₃	C₂²	1	2	yes	—
15	#14582940	—	2	4	no	—
16	#14582939	—	2	2	no	—
17	#14582937	C₂²	1	2	yes	—
18	#14582936	C₂²	1	4	yes	—
19	#107×D₄₃	C₂²	1	4	yes	—
20	#14582727	C₂²	1	2	yes	—
21	#14582726	C₂²	1	4	yes	—
22	#14582725	C₂²	1	1	yes	characteristic
23	#14582725	C₂²	1	1	yes	characteristic
24	#14582724	C₂²	1	4	yes	—
25	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
26	C₂×#107×C₄₃	C₂²	1	2	yes	—
27	C₄₃×#366	C₂²	1	2	yes	—
28	C₄₃×#366	C₂²	1	2	yes	—
29	C₂×#10621898	D₄	1	1	yes	characteristic
30	#13063331	C₂×C₄	1	4	yes	—
31	#13063331	Q₈	1	4	yes	—
32	C₂×#10621877	D₄	1	1	yes	characteristic
33	C₂×#10621872	D₄	1	2	yes	—
34	C₂×C₈×D₄₃	C₂³	1	2	yes	—
35	#13063290	—	4	2	no	—
36	#13063287	—	4	2	no	—
37	C₄×C₂²⋊D₄₃	—	4	4	no	—
38	#13063284	—	2	2	no	—
39	#13063283	—	2	4	no	—
40	C₄⋊C₄×D₄₃	—	2	4	no	—
41	#13063265	C₂×C₄	1	4	yes	complemented
42	#13063264	C₂×C₄	1	8	yes	complemented
43	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
44	C₁₆×D₄₃	C₂×C₄	1	8	yes	complemented
45	#13063259	C₂³	1	2	yes	—
46	#13063259	C₂³	1	2	yes	—
47	#13063258	—	2	4	no	—
48	#13063257	—	2	2	no	—
49	#13063256	—	2	4	no	—
50	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
51	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
52	C₂×C₁₆×C₄₃	C₂×C₄	1	4	yes	complemented
53	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
54	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
55	#107×C₄₃	—	2	2	no	—
56	#107×C₄₃	C₂³	1	4	yes	—
57	C₂²×C₄×C₄₃	—	4	2	no	—
58	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₂²⋊D₄₃	D₈	1	2	yes	—
60	#10621898	C₄⋊C₄	1	4	yes	—
61	#10621898	Q₁₆	1	4	yes	—
62	C₂×D₁₇₂	D₈	1	1	yes	characteristic
63	C₂×C₄×D₄₃	—	4	4	no	—
64	C₂×C₄×D₄₃	C₂×D₄	1	2	yes	—
65	C₂×Dic₈₆	D₈	1	1	yes	characteristic
66	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
67	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₄₃	C₂×Q₈	1	2	yes	—
69	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
70	C₄⋊C₄×C₄₃	—	2	4	no	—
71	C₄⋊C₄×C₄₃	—	4	2	no	—
72	C₂²⋊Dic₄₃	—	4	2	no	—
73	(C₂×D₄₃)⋊C₄	—	4	4	no	—
74	C₄⋊Dic₄₃	—	2	2	no	—
75	C₄⋊Dic₄₃	—	2	2	no	—
76	Dic₄₃⋊C₄	—	4	4	no	—
77	C₄×Dic₄₃	—	4	2	no	—
78	C₄×Dic₄₃	—	4	4	no	—
79	#10621877	C₄⋊C₄	1	4	yes	—
80	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
81	#10621872	C₄⋊C₄	1	8	yes	—
82	C₈×D₄₃	C₂²×C₄	1	4	yes	—
83	C₈×D₄₃	C₂×Q₈	1	4	yes	—
84	C₁₆×C₄₃	C₂²×C₄	1	2	yes	—
85	C₁₆×C₄₃	C₂²×C₄	1	2	yes	—
86	C₄₃⋊C₁₆	C₂²×C₄	1	4	yes	—
87	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
88	C₂²×D₄₃	C₂×D₈	1	2	yes	—
89	C₂×C₄×C₄₃	—	4	4	no	—
90	C₂×C₄×C₄₃	—	16	2	no	—
91	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
92	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
93	C₂×C₄×C₄₃	C₂×Q₁₆	1	2	yes	—
94	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
95	C₂²⋊D₄₃	#107	1	8	yes	—
96	C₂×Dic₄₃	—	4	2	no	—
97	C₂×Dic₄₃	—	4	2	no	—
98	C₂×Dic₄₃	C₂×D₈	1	2	yes	—
99	D₁₇₂	#107	1	4	yes	—
100	C₄×D₄₃	—	16	4	no	—
101	C₄×D₄₃	C₂×C₄⋊C₄	1	4	yes	—
102	C₄×D₄₃	C₂×Q₁₆	1	4	yes	—
103	Dic₈₆	#107	1	4	yes	—
104	C₈×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
105	C₈×C₄₃	C₂³×C₄	1	1	yes	characteristic
106	C₈×C₄₃	C₂²×Q₈	1	1	yes	characteristic
107	C₄₃⋊C₈	C₂×C₄⋊C₄	1	4	yes	—
108	#33410	—	43	1	no	maximal, Sylow
109	C₂²×C₄₃	—	2	1	no	—
110	C₂²×C₄₃	—	2	1	no	—
111	C₂²×C₄₃	C₂×#107	1	2	yes	—
112	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
113	C₂×D₄₃	—	2	2	no	—
114	C₂×D₄₃	—	2	2	no	—
115	C₂×D₄₃	C₂×#107	1	4	yes	—
116	C₄×C₄₃	—	8	4	no	—
117	C₄×C₄₃	C₂×#107	1	2	yes	—
118	C₄×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
119	C₄×C₄₃	C₂²×Q₁₆	1	1	yes	characteristic
120	Dic₄₃	—	8	2	no	—
121	Dic₄₃	—	8	2	no	—
122	Dic₄₃	C₂×#107	1	4	yes	—
123	C₂×#504	—	43	1	no	—
124	#2913	—	43	2	no	—
125	#2159	—	43	2	no	—
126	#2159	—	43	4	no	—
127	#2158	—	43	1	no	—
128	#2158	—	43	2	no	—
129	C₂×#366	—	43	2	no	—
130	C₂×#366	D₄₃	1	1	yes	characteristic, complemented
131	C₂×C₄₃	—	2	2	no	—
132	C₂×C₄₃	C₂²×#107	1	1	yes	characteristic
133	C₂×C₄₃	#3415	1	1	yes	characteristic
134	C₂×C₄₃	#3417	1	1	yes	characteristic
135	D₄₃	—	2	4	no	—
136	C₂×#131	—	43	1	no	—
137	#520	—	86	2	no	—
138	#504	—	43	8	no	—
139	C₂×M₃₂	—	43	1	no	—
140	C₂×M₃₂	—	43	2	no	—
141	C₂²×C₁₆	—	43	2	no	—
142	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
143	#428	—	86	4	no	—
144	#428	—	86	2	no	—
145	#427	—	43	2	no	—
146	#427	—	43	4	no	—
147	C₂×#107	—	43	4	no	—
148	C₂×#107	C₂×D₄₃	1	2	yes	—
149	#367	—	43	2	no	—
150	#367	—	43	4	no	—
151	#366	—	43	1	no	—
152	#366	—	43	1	no	—
153	#366	—	43	4	no	—
154	#366	C₂×D₄₃	1	2	yes	—
155	#366	C₂×D₄₃	1	2	yes	—
156	C₄₃	#33410	1	1	yes	characteristic, complemented, Sylow
157	C₂×#41	—	43	1	no	—
158	#131	—	43	4	no	—
159	#131	—	43	4	no	—
160	C₂×M₁₆	—	43	1	no	—
161	C₂×M₁₆	—	43	2	no	—
162	C₂²×C₈	—	43	2	no	—
163	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
164	C₄×Q₈	—	172	2	no	—
165	C₄×D₄	—	172	2	no	—
166	C₄×D₄	—	172	4	no	—
167	#117	—	86	4	no	—
168	#117	—	86	2	no	—
169	C₂×C₄⋊C₄	—	2	2	no	—
170	C₂×C₄⋊C₄	—	86	4	no	—
171	M₃₂	—	43	4	no	—
172	M₃₂	—	43	8	no	—
173	C₂×C₁₆	—	43	2	no	—
174	C₂×C₁₆	—	43	8	no	—
175	C₂×C₁₆	C₄×D₄₃	1	4	yes	complemented
176	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
177	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
178	#107	—	2	2	no	—
179	#107	—	43	2	no	—
180	#107	—	43	2	no	—
181	#107	—	86	4	no	—
182	#107	C₂²×D₄₃	1	4	yes	—
183	#106	—	86	4	no	—
184	#106	—	86	2	no	—
185	#41	—	43	4	no	—
186	#41	—	43	4	no	—
187	C₂×Q₈	—	43	1	no	—
188	C₂×D₄	—	43	2	no	—
189	C₂×D₄	—	43	1	no	—
190	C₂²×C₄	—	4	2	no	—
191	C₂²×C₄	—	43	2	no	—
192	C₂²×C₄	—	172	4	no	—
193	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
194	M₁₆	—	43	4	no	—
195	M₁₆	—	43	8	no	—
196	C₂×C₈	—	43	2	no	—
197	C₂×C₈	—	43	4	no	—
198	C₂×C₈	—	43	4	no	—
199	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
200	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
201	C₂×C₈	Q₈×D₄₃	1	2	yes	—
202	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
203	C₄⋊C₄	—	2	4	no	—
204	C₄⋊C₄	—	4	2	no	—
205	C₄⋊C₄	—	86	2	no	—
206	C₄⋊C₄	—	86	2	no	—
207	C₄⋊C₄	—	172	4	no	—
208	C₂²⋊C₄	—	172	4	no	—
209	C₂²⋊C₄	—	172	2	no	—
210	C₄²	—	172	4	no	—
211	C₄²	—	172	2	no	—
212	C₁₆	—	43	4	no	—
213	C₁₆	C₂×C₄×D₄₃	1	2	yes	—
214	C₁₆	C₂×C₄×D₄₃	1	2	yes	—
215	C₂³	—	43	2	no	—
216	C₂³	D₈×D₄₃	1	1	yes	characteristic
217	Q₈	—	43	4	no	—
218	D₄	—	43	8	no	—
219	D₄	—	43	4	no	—
220	C₂×C₄	—	4	4	no	—
221	C₂×C₄	—	16	2	no	—
222	C₂×C₄	—	43	2	no	—
223	C₂×C₄	—	43	4	no	—
224	C₂×C₄	—	43	4	no	—
225	C₂×C₄	—	172	2	no	—
226	C₂×C₄	—	172	2	no	—
227	C₂×C₄	—	688	4	no	—
228	C₂×C₄	C₄⋊C₄×D₄₃	1	2	yes	—
229	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
230	C₂×C₄	Q₁₆×D₄₃	1	2	yes	—
231	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
232	C₈	—	43	4	no	—
233	C₈	C₄⋊C₄×D₄₃	1	2	yes	—
234	C₈	C₂²×C₄×D₄₃	1	1	yes	characteristic
235	C₈	C₂×Q₈×D₄₃	1	1	yes	characteristic
236	C₂²	—	2	1	no	—
237	C₂²	—	2	1	no	—
238	C₂²	—	43	4	no	—
239	C₂²	—	86	2	no	—
240	C₂²	—	86	2	no	—
241	C₂²	#107×D₄₃	1	2	yes	—
242	C₂²	C₂×D₈×D₄₃	1	1	yes	characteristic
243	C₄	—	8	4	no	—
244	C₄	—	43	4	no	—
245	C₄	—	344	2	no	—
246	C₄	—	344	2	no	—
247	C₄	#107×D₄₃	1	2	yes	—
248	C₄	C₂×C₄⋊C₄×D₄₃	1	1	yes	characteristic
249	C₄	C₂×Q₁₆×D₄₃	1	1	yes	characteristic
250	C₂	—	2	2	no	—
251	C₂	—	86	4	no	—
252	C₂	C₂×#107×D₄₃	1	1	yes	characteristic
253	C₂	#19147732	1	1	yes	characteristic
254	C₂	#19147744	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₄₃
commutator	C₈×C₄₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₂×C₄₃×#366	C₂
socle	C₂²×C₄₃	C₂²×D₈
prime radical	C₂×#10621898	D₄
cosocle	C₂×C₈×C₄₃	C₂⁴
hypercenter	C₂×#366	D₄₃
hypocenter	C₄₃	#33410
LCS	G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

Complements

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

N	H ≅ G/N	type
3. #19136381	C₂	semidirect
4. #19136380	C₂	semidirect
5. #19136379	C₂	semidirect
6. #19136376	C₂	semidirect
7. #366×D₄₃	C₂	semidirect
8. C₂×C₄₃×#366	C₂	semidirect
11. #14583315	C₄	semidirect
41. #13063265	C₂×C₄	semidirect
42. #13063264	C₂×C₄	semidirect
44. C₁₆×D₄₃	C₂×C₄	semidirect
52. C₂×C₁₆×C₄₃	C₂×C₄	semidirect
130. C₂×#366	D₄₃	semidirect
156. C₄₃	#33410	semidirect
175. C₂×C₁₆	C₄×D₄₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583315	2
#19143191	2
#19136381	2
#19136380	2
#19136379	2
#19136376	2
#366×D₄₃	2
C₂×C₄₃×#366	2
#33410	43