Subgroups — #25115553

Order 11008.

734 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Dic₈₆	C₂	1	1	yes	maximal, characteristic
2	#19137550	C₂	1	2	yes	maximal
3	#19135360	C₂	1	4	yes	maximal
4	#19135358	C₂	1	2	yes	maximal
5	#19134724	C₂	1	2	yes	maximal
6	#19134723	C₂	1	1	yes	maximal, characteristic
7	#19134483	C₂	1	2	yes	maximal
8	C₄×#107×C₄₃	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₈₆	C₂²	1	1	yes	characteristic
10	C₈×Dic₈₆	C₄	1	8	yes	complemented
11	C₂×#13063256	C₂²	1	4	yes	—
12	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
13	C₂×Dic₄₃⋊C₈	C₂²	1	2	yes	—
14	C₂×C₈×Dic₄₃	C₂²	1	2	yes	—
15	#14582805	—	2	2	no	—
16	#14582786	—	2	4	no	—
17	#14582769	C₂²	1	2	yes	—
18	#14582764	C₂²	1	4	yes	—
19	C₄⋊C₄×Dic₄₃	C₂²	1	4	yes	—
20	C₄×C₄⋊Dic₄₃	—	2	2	no	—
21	#14582662	C₂²	1	2	yes	—
22	#14582662	C₂²	1	2	yes	—
23	#14582661	C₂²	1	4	yes	—
24	C₂×#107×C₄₃	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
26	C₄×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
27	C₄₃×#340	C₂²	1	1	yes	characteristic
28	C₄₃×#340	C₂²	1	1	yes	characteristic
29	C₂²×Dic₈₆	D₄	1	1	yes	characteristic
30	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
31	C₄×Dic₈₆	C₂×C₄	1	4	yes	—
32	C₄×Dic₈₆	Q₈	1	4	yes	—
33	C₂×C₄⋊Dic₄₃	—	2	2	no	—
34	C₂×C₄⋊Dic₄₃	—	2	2	no	—
35	C₂×C₄⋊Dic₄₃	D₄	1	1	yes	characteristic
36	C₂×C₄⋊Dic₄₃	C₂³	1	4	yes	—
37	C₂×Dic₄₃⋊C₄	D₄	1	2	yes	—
38	C₂×C₄×Dic₄₃	—	2	4	no	—
39	C₂×C₄×Dic₄₃	C₂³	1	2	yes	—
40	#13063256	—	2	8	no	—
41	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	4	yes	—
42	Dic₄₃⋊C₈	C₂×C₄	1	8	yes	—
43	C₈×Dic₄₃	C₂×C₄	1	8	yes	—
44	C₄₃⋊#95	—	2	4	no	—
45	C₄₃⋊#95	—	2	4	no	—
46	C₄₃⋊#95	—	2	4	no	—
47	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
49	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
50	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
51	C₂×C₄²×C₄₃	—	2	2	no	—
52	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
53	#107×C₄₃	—	2	4	no	—
54	C₄×C₈×C₄₃	C₂×C₄	1	4	yes	—
55	#95×C₄₃	—	2	2	no	—
56	C₂²×C₄×C₄₃	—	2	2	no	—
57	C₂²×C₄×C₄₃	—	2	2	no	—
58	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
61	C₂²×Dic₄₃	—	2	4	no	—
62	C₂²×Dic₄₃	C₂×D₄	1	2	yes	—
63	C₂²×Dic₄₃	C₂×D₄	1	2	yes	—
64	C₂×Dic₈₆	—	2	2	no	—
65	C₂×Dic₈₆	—	2	2	no	—
66	C₂×Dic₈₆	C₄⋊C₄	1	4	yes	—
67	C₂×C₈×C₄₃	—	2	2	no	—
68	C₂×C₈×C₄₃	—	2	2	no	—
69	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
70	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
71	C₄⋊C₄×C₄₃	—	2	2	no	—
72	C₄⋊C₄×C₄₃	—	2	2	no	—
73	C₄⋊C₄×C₄₃	—	2	2	no	—
74	C₄⋊C₄×C₄₃	#41	1	4	yes	—
75	C₄²×C₄₃	—	4	4	no	—
76	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
77	C₄²×C₄₃	C₂×Q₈	1	2	yes	—
78	C₄⋊Dic₄₃	—	2	4	no	—
79	C₄⋊Dic₄₃	—	2	4	no	—
80	C₄⋊Dic₄₃	—	4	4	no	—
81	C₄⋊Dic₄₃	—	8	2	no	—
82	C₄⋊Dic₄₃	C₄⋊C₄	1	4	yes	—
83	Dic₄₃⋊C₄	C₄⋊C₄	1	8	yes	—
84	C₄×Dic₄₃	—	8	4	no	—
85	C₄×Dic₄₃	C₂²×C₄	1	4	yes	—
86	C₄×Dic₄₃	C₂×Q₈	1	4	yes	—
87	C₂×C₄₃⋊C₈	—	2	4	no	—
88	C₂×C₄₃⋊C₈	C₂²×C₄	1	4	yes	—
89	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
90	C₂×C₄×C₄₃	—	2	2	no	—
91	C₂×C₄×C₄₃	—	2	4	no	—
92	C₂×C₄×C₄₃	—	4	2	no	—
93	C₂×C₄×C₄₃	—	4	2	no	—
94	C₂×C₄×C₄₃	—	4	2	no	—
95	C₂×C₄×C₄₃	—	4	2	no	—
96	C₂×C₄×C₄₃	—	4	2	no	—
97	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
98	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
99	C₂×C₄×C₄₃	#135	1	2	yes	—
100	C₂×C₄×C₄₃	#135	1	2	yes	—
101	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
102	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
103	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
104	C₂×C₄×C₄₃	2₋¹⁺⁴	1	2	yes	—
105	C₂×Dic₄₃	—	2	2	no	—
106	C₂×Dic₄₃	—	2	2	no	—
107	C₂×Dic₄₃	—	2	4	no	—
108	C₂×Dic₄₃	—	4	4	no	—
109	C₂×Dic₄₃	—	4	4	no	—
110	C₂×Dic₄₃	—	4	4	no	—
111	C₂×Dic₄₃	C₂×C₄⋊C₄	1	4	yes	—
112	C₂×Dic₄₃	C₂×C₄⋊C₄	1	4	yes	—
113	Dic₈₆	—	2	8	no	—
114	C₈×C₄₃	—	2	2	no	—
115	C₈×C₄₃	—	2	2	no	—
116	C₄₃⋊C₈	—	2	4	no	—
117	#22741	—	43	1	no	maximal, Sylow
118	C₂²×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
119	C₂²×C₄₃	#548	1	2	yes	—
120	C₂²×C₄₃	C₂×#135	1	1	yes	characteristic
121	C₂²×C₄₃	C₂×#135	1	1	yes	characteristic
122	C₂²×C₄₃	#577	1	2	yes	—
123	C₄×C₄₃	—	2	2	no	—
124	C₄×C₄₃	—	4	4	no	—
125	C₄×C₄₃	—	8	2	no	—
126	C₄×C₄₃	#427	1	4	yes	—
127	C₄×C₄₃	C₄×#41	1	1	yes	characteristic
128	C₄×C₄₃	#519	1	1	yes	characteristic
129	C₄×C₄₃	#557	1	2	yes	—
130	Dic₄₃	—	2	4	no	—
131	Dic₄₃	—	2	4	no	—
132	Dic₄₃	—	8	4	no	—
133	C₂×C₈×Q₈	—	43	1	no	—
134	#2355	—	43	2	no	—
135	#1949	—	43	4	no	—
136	#1948	—	43	2	no	—
137	#1835	—	43	2	no	—
138	#1834	—	43	1	no	—
139	C₄×#107	D₄₃	1	1	yes	characteristic
140	C₄×#106	—	43	2	no	—
141	C₂×C₄₃	#2351	1	1	yes	characteristic
142	C₂×C₄₃	C₂×#427	1	1	yes	characteristic
143	C₂×C₄₃	#2912	1	1	yes	characteristic
144	C₂×C₄₃	#3360	1	2	yes	—
145	C₂×C₄₃	#3365	1	2	yes	—
146	C₂×C₄×Q₈	—	43	1	no	—
147	C₈×Q₈	—	43	8	no	—
148	C₂×#107	C₂×D₄₃	1	2	yes	—
149	C₂×#106	—	43	4	no	—
150	C₂×C₄⋊C₈	—	43	2	no	—
151	C₂×C₄⋊C₈	—	43	1	no	—
152	C₂×C₄×C₈	—	43	2	no	—
153	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
154	#391	—	86	2	no	—
155	#387	—	86	4	no	—
156	#384	—	43	2	no	—
157	#382	—	43	4	no	—
158	C₄×C₄⋊C₄	—	43	4	no	—
159	C₄×C₄⋊C₄	—	86	2	no	—
160	C₄×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
161	#340	—	43	2	no	—
162	#340	—	43	2	no	—
163	#340	—	43	4	no	—
164	#340	C₂×D₄₃	1	1	yes	characteristic
165	#340	C₂×D₄₃	1	1	yes	characteristic
166	C₄₃	#22741	1	1	yes	characteristic, complemented, Sylow
167	C₂²×Q₈	—	43	1	no	—
168	C₂²×C₈	—	43	2	no	—
169	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
170	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
171	C₄×Q₈	—	43	4	no	—
172	C₄×Q₈	—	43	4	no	—
173	C₂×C₄⋊C₄	—	43	2	no	—
174	C₂×C₄⋊C₄	—	43	1	no	—
175	C₂×C₄⋊C₄	—	43	4	no	—
176	C₂×C₄⋊C₄	—	86	2	no	—
177	C₂×C₄⋊C₄	—	86	2	no	—
178	C₂×C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
179	C₂×C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
180	C₂×C₄²	—	2	2	no	—
181	C₂×C₄²	—	43	2	no	—
182	C₂×C₄²	—	86	4	no	—
183	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
184	#107	—	2	4	no	—
185	#106	—	86	8	no	—
186	C₄⋊C₈	—	43	8	no	—
187	C₄⋊C₈	—	43	4	no	—
188	C₄×C₈	—	43	8	no	—
189	C₄×C₈	C₄×D₄₃	1	4	yes	—
190	#95	—	2	2	no	—
191	#95	—	86	4	no	—
192	#95	—	86	4	no	—
193	#95	—	86	4	no	—
194	C₂×Q₈	—	43	4	no	—
195	C₂×Q₈	—	86	2	no	—
196	C₂×Q₈	—	86	2	no	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	43	2	no	—
200	C₂²×C₄	—	43	2	no	—
201	C₂²×C₄	—	86	4	no	—
202	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
203	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
204	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	43	4	no	—
208	C₂×C₈	—	86	4	no	—
209	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
210	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	2	2	no	—
213	C₄⋊C₄	—	2	2	no	—
214	C₄⋊C₄	—	43	8	no	—
215	C₄⋊C₄	—	43	4	no	—
216	C₄⋊C₄	—	86	4	no	—
217	C₄⋊C₄	—	86	4	no	—
218	C₄⋊C₄	—	344	2	no	—
219	C₄⋊C₄	—	172	4	no	—
220	C₄⋊C₄	#10621898	1	4	yes	—
221	C₄²	—	4	4	no	—
222	C₄²	—	43	4	no	—
223	C₄²	—	43	4	no	—
224	C₄²	—	344	4	no	—
225	C₄²	C₂×C₄×D₄₃	1	2	yes	—
226	C₄²	Q₈×D₄₃	1	2	yes	—
227	C₂³	C₂×D₄×D₄₃	1	1	yes	characteristic
228	Q₈	—	86	8	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	43	4	no	—
237	C₂×C₄	—	43	4	no	—
238	C₂×C₄	—	86	2	no	—
239	C₂×C₄	—	86	2	no	—
240	C₂×C₄	—	86	4	no	—
241	C₂×C₄	—	172	4	no	—
242	C₂×C₄	—	172	4	no	—
243	C₂×C₄	—	172	4	no	—
244	C₂×C₄	C₄⋊C₄×D₄₃	1	2	yes	—
245	C₂×C₄	C₄⋊C₄×D₄₃	1	2	yes	—
246	C₂×C₄	#13063344	1	2	yes	—
247	C₂×C₄	#13063348	1	2	yes	—
248	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
249	C₂×C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
250	C₂×C₄	C₂×#10621898	1	2	yes	—
251	C₂×C₄	#13063372	1	2	yes	—
252	C₈	—	2	2	no	—
253	C₈	—	2	2	no	—
254	C₈	—	86	4	no	—
255	C₂²	C₂×C₄⋊C₄×D₄₃	1	1	yes	characteristic
256	C₂²	#14583558	1	2	yes	—
257	C₂²	C₂×#13063344	1	1	yes	characteristic
258	C₂²	C₂×#13063348	1	1	yes	characteristic
259	C₂²	#14583706	1	2	yes	—
260	C₄	—	2	2	no	—
261	C₄	—	4	4	no	—
262	C₄	—	8	2	no	—
263	C₄	—	86	4	no	—
264	C₄	—	86	4	no	—
265	C₄	—	344	4	no	—
266	C₄	#14582936	1	4	yes	—
267	C₄	C₄×#10621898	1	1	yes	characteristic
268	C₄	#14583376	1	1	yes	characteristic
269	C₄	#14583627	1	2	yes	—
270	C₂	#19137515	1	1	yes	characteristic
271	C₂	C₂×#14582936	1	1	yes	characteristic
272	C₂	#19143187	1	1	yes	characteristic
273	C₂	#19147242	1	2	yes	—
274	C₂	#19147289	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄×#107×C₄₃	C₂
socle, prime radical	C₂³×C₄₃	C₂²×D₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₄×#107	D₄₃
hypocenter	C₄₃	#22741
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×#107

Complements

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

N	H ≅ G/N	type
10. C₈×Dic₈₆	C₄	semidirect
166. C₄₃	#22741	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Dic₈₆	2
#19137550	2
#19135360	2
#19135358	2
#19134724	2
#19134723	2
#19134483	2
C₄×#107×C₄₃	2
#22741	43