Subgroups — #18075871

Order 4864.

642 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14563703	C₂	1	2	yes	maximal
2	#14561478	C₂	1	4	yes	maximal
3	C₂×C₄×C₄⋊C₄×C₁₉	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂²⋊(C₁₉⋊C₈)	C₂²	1	4	yes	—
5	#11926299	—	2	2	no	—
6	#11926299	—	2	2	no	—
7	C₂²×C₄⋊C₄×C₁₉	C₄	1	1	yes	characteristic
8	C₂²×C₄⋊C₄×C₁₉	C₄	1	1	yes	characteristic
9	C₂²×C₄²×C₁₉	C₄	1	2	yes	—
10	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
11	C₄×C₄⋊C₄×C₁₉	—	2	2	no	—
12	C₄×C₄⋊C₄×C₁₉	—	2	2	no	—
13	C₂×C₁₉×#95	C₂²	1	2	yes	—
14	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
15	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
16	C₂³×C₄×C₁₉	C₂×C₄	1	2	yes	—
17	C₂³×C₄×C₁₉	C₂×C₄	1	2	yes	—
18	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
20	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
21	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
22	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
23	C₂×C₄⋊C₄×C₁₉	C₈	1	4	yes	complemented
24	C₂×C₄²×C₁₉	—	2	4	no	—
25	C₂×C₄²×C₁₉	—	2	2	no	—
26	C₂×C₄²×C₁₉	—	2	2	no	—
27	C₂×C₄²×C₁₉	—	4	4	no	—
28	C₂×C₄²×C₁₉	D₄	1	2	yes	—
29	C₂×C₄²×C₁₉	D₄	1	2	yes	—
30	C₂²×C₁₉⋊C₈	—	2	4	no	—
31	C₁₉×#95	—	2	4	no	—
32	C₂²⋊(C₁₉⋊C₈)	—	2	8	no	—
33	C₂⁴×C₁₉	C₂²×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₉	—	2	2	no	—
35	C₂²×C₄×C₁₉	—	2	2	no	—
36	C₂²×C₄×C₁₉	—	2	2	no	—
37	C₂²×C₄×C₁₉	—	2	2	no	—
38	C₂²×C₄×C₁₉	—	2	2	no	—
39	C₂²×C₄×C₁₉	—	2	2	no	—
40	C₂²×C₄×C₁₉	—	2	2	no	—
41	C₂²×C₄×C₁₉	—	2	2	no	—
42	C₂²×C₄×C₁₉	—	2	2	no	—
43	C₂²×C₄×C₁₉	—	4	1	no	—
44	C₂²×C₄×C₁₉	—	4	1	no	—
45	C₂²×C₄×C₁₉	—	4	2	no	—
46	C₂²×C₄×C₁₉	—	4	2	no	—
47	C₂²×C₄×C₁₉	—	4	2	no	—
48	C₂²×C₄×C₁₉	—	4	2	no	—
49	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
50	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₉	C₂×C₈	1	2	yes	—
53	C₂²×C₄×C₁₉	C₂×C₈	1	4	yes	—
54	C₂²×C₄×C₁₉	M₁₆	1	2	yes	—
55	C₂²×C₄×C₁₉	M₁₆	1	4	yes	—
56	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₁₉	#41	1	4	yes	—
58	C₄⋊C₄×C₁₉	—	2	8	no	—
59	C₄⋊C₄×C₁₉	—	2	8	no	—
60	C₄²×C₁₉	—	2	4	no	—
61	C₄²×C₁₉	—	2	4	no	—
62	C₄²×C₁₉	—	4	4	no	—
63	C₄²×C₁₉	—	4	4	no	—
64	C₂×C₁₉⋊C₈	—	4	4	no	—
65	C₂×C₁₉⋊C₈	—	4	4	no	—
66	C₂×C₁₉⋊C₈	—	4	4	no	—
67	#8701	—	19	1	no	maximal, Sylow
68	C₂³×C₁₉	—	2	1	no	—
69	C₂³×C₁₉	—	2	1	no	—
70	C₂³×C₁₉	—	2	1	no	—
71	C₂³×C₁₉	—	2	1	no	—
72	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₂³×C₁₉	#117	1	2	yes	—
74	C₂³×C₁₉	C₂²×C₈	1	1	yes	characteristic
75	C₂³×C₁₉	C₂×M₁₆	1	1	yes	characteristic
76	C₂³×C₁₉	C₂×M₁₆	1	2	yes	—
77	C₂×C₄×C₁₉	—	2	8	no	—
78	C₂×C₄×C₁₉	—	2	2	no	—
79	C₂×C₄×C₁₉	—	2	1	no	—
80	C₂×C₄×C₁₉	—	2	1	no	—
81	C₂×C₄×C₁₉	—	2	2	no	—
82	C₂×C₄×C₁₉	—	2	2	no	—
83	C₂×C₄×C₁₉	—	2	2	no	—
84	C₂×C₄×C₁₉	—	2	1	no	—
85	C₂×C₄×C₁₉	—	2	1	no	—
86	C₂×C₄×C₁₉	—	2	4	no	—
87	C₂×C₄×C₁₉	—	2	4	no	—
88	C₂×C₄×C₁₉	—	2	4	no	—
89	C₂×C₄×C₁₉	—	2	4	no	—
90	C₂×C₄×C₁₉	—	4	4	no	—
91	C₂×C₄×C₁₉	—	4	4	no	—
92	C₂×C₄×C₁₉	—	4	4	no	—
93	C₂×C₄×C₁₉	—	4	2	no	—
94	C₂×C₄×C₁₉	—	4	2	no	—
95	C₂×C₄×C₁₉	—	4	2	no	—
96	C₂×C₄×C₁₉	—	4	2	no	—
97	C₂×C₄×C₁₉	—	4	4	no	—
98	C₂×C₄×C₁₉	—	4	4	no	—
99	C₂×C₄×C₁₉	C₂²⋊C₈	1	4	yes	—
100	C₂×C₄×C₁₉	#100	1	2	yes	—
101	C₂×C₄×C₁₉	#101	1	2	yes	—
102	C₂×C₄×C₁₉	#123	1	4	yes	—
103	C₁₉⋊C₈	—	8	4	no	—
104	C₂×C₄×C₄⋊C₄	D₁₉	1	1	yes	characteristic
105	#1817	—	19	2	no	—
106	#1280	—	19	4	no	—
107	C₂²×C₁₉	—	2	2	no	—
108	C₂²×C₁₉	—	2	2	no	—
109	C₂²×C₁₉	—	2	1	no	—
110	C₂²×C₁₉	—	2	1	no	—
111	C₂²×C₁₉	—	2	1	no	—
112	C₂²×C₁₉	—	2	1	no	—
113	C₂²×C₁₉	—	2	2	no	—
114	C₂²×C₁₉	—	2	2	no	—
115	C₂²×C₁₉	#381	1	1	yes	characteristic
116	C₂²×C₁₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
117	C₂²×C₁₉	#407	1	1	yes	characteristic
118	C₂²×C₁₉	#410	1	1	yes	characteristic
119	C₂²×C₁₉	#410	1	1	yes	characteristic
120	C₂²×C₁₉	C₂×#100	1	1	yes	characteristic
121	C₂²×C₁₉	C₂×#101	1	1	yes	characteristic
122	C₂²×C₁₉	#431	1	2	yes	—
123	C₂²×C₁₉	#432	1	2	yes	—
124	C₄×C₁₉	—	2	4	no	—
125	C₄×C₁₉	—	2	4	no	—
126	C₄×C₁₉	—	2	2	no	—
127	C₄×C₁₉	—	2	2	no	—
128	C₄×C₁₉	—	4	4	no	—
129	C₄×C₁₉	—	4	4	no	—
130	C₂²×C₄⋊C₄	Dic₁₉	1	1	yes	characteristic
131	C₂²×C₄⋊C₄	Dic₁₉	1	1	yes	characteristic
132	C₂²×C₄²	Dic₁₉	1	2	yes	—
133	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
134	C₂×C₂²⋊C₈	—	19	4	no	—
135	C₄×C₄⋊C₄	—	2	2	no	—
136	C₄×C₄⋊C₄	—	2	2	no	—
137	C₂×#95	C₂×D₁₉	1	2	yes	—
138	#336	—	38	2	no	—
139	#336	—	38	2	no	—
140	C₂×C₁₉	—	2	2	no	—
141	C₂×C₁₉	—	2	2	no	—
142	C₂×C₁₉	#1460	1	1	yes	characteristic
143	C₂×C₁₉	#1461	1	1	yes	characteristic
144	C₂×C₁₉	#1466	1	1	yes	characteristic
145	C₂×C₁₉	#1467	1	1	yes	characteristic
146	C₂×C₁₉	#1817	1	1	yes	characteristic
147	C₂×C₁₉	#1820	1	1	yes	characteristic
148	C₂×C₁₉	#1821	1	1	yes	characteristic
149	C₂³×C₄	C₂×Dic₁₉	1	1	yes	characteristic
150	C₂³×C₄	C₂×Dic₁₉	1	1	yes	characteristic
151	C₂³×C₄	C₂×Dic₁₉	1	2	yes	—
152	C₂³×C₄	C₂×Dic₁₉	1	2	yes	—
153	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
154	C₂²×C₈	—	38	4	no	—
155	C₂×C₄⋊C₄	—	2	4	no	—
156	C₂×C₄⋊C₄	—	2	2	no	—
157	C₂×C₄⋊C₄	—	2	2	no	—
158	C₂×C₄⋊C₄	—	2	2	no	—
159	C₂×C₄⋊C₄	C₁₉⋊C₈	1	4	yes	complemented
160	C₂×C₄²	—	2	4	no	—
161	C₂×C₄²	—	2	2	no	—
162	C₂×C₄²	—	2	2	no	—
163	C₂×C₄²	—	4	4	no	—
164	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
165	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
166	C₂²⋊C₈	—	38	8	no	—
167	#95	—	2	4	no	—
168	C₁₉	#8701	1	1	yes	characteristic, complemented, Sylow
169	C₂⁴	C₂²×Dic₁₉	1	1	yes	characteristic
170	C₂²×C₄	—	2	2	no	—
171	C₂²×C₄	—	2	2	no	—
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	—	2	2	no	—
175	C₂²×C₄	—	2	2	no	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	—	2	2	no	—
179	C₂²×C₄	—	4	1	no	—
180	C₂²×C₄	—	4	1	no	—
181	C₂²×C₄	—	4	2	no	—
182	C₂²×C₄	—	4	2	no	—
183	C₂²×C₄	—	4	2	no	—
184	C₂²×C₄	—	4	2	no	—
185	C₂²×C₄	C₂×C₁₉⋊C₈	1	2	yes	—
186	C₂²×C₄	C₂×C₁₉⋊C₈	1	4	yes	—
187	C₂²×C₄	#64520	1	2	yes	—
188	C₂²×C₄	#64520	1	4	yes	—
189	C₂²×C₄	C₂²⋊Dic₁₉	1	2	yes	—
190	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
191	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
192	C₂²×C₄	#64541	1	4	yes	—
193	C₂²×C₄	C₂×C₂²⋊D₁₉	1	2	yes	—
194	C₂×C₈	—	76	4	no	—
195	C₂×C₈	—	76	4	no	—
196	C₂×C₈	—	76	4	no	—
197	C₄⋊C₄	—	2	8	no	—
198	C₄⋊C₄	—	2	8	no	—
199	C₄²	—	2	4	no	—
200	C₄²	—	2	4	no	—
201	C₄²	—	4	4	no	—
202	C₄²	—	4	4	no	—
203	C₂³	—	2	1	no	—
204	C₂³	—	2	1	no	—
205	C₂³	—	2	1	no	—
206	C₂³	—	2	1	no	—
207	C₂³	C₂²×C₁₉⋊C₈	1	1	yes	characteristic
208	C₂³	C₂×#64520	1	2	yes	—
209	C₂³	C₂×#64520	1	1	yes	characteristic
210	C₂³	#10597372	1	2	yes	—
211	C₂³	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
212	C₂×C₄	—	2	8	no	—
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	4	4	no	—
226	C₂×C₄	—	4	4	no	—
227	C₂×C₄	—	4	4	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	4	no	—
233	C₂×C₄	—	4	4	no	—
234	C₂×C₄	C₂²⋊(C₁₉⋊C₈)	1	4	yes	—
235	C₂×C₄	#10597290	1	2	yes	—
236	C₂×C₄	#10597293	1	2	yes	—
237	C₂×C₄	#10597375	1	4	yes	—
238	C₈	—	152	4	no	—
239	C₂²	—	2	2	no	—
240	C₂²	—	2	2	no	—
241	C₂²	—	2	1	no	—
242	C₂²	—	2	1	no	—
243	C₂²	—	2	1	no	—
244	C₂²	—	2	1	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	2	2	no	—
247	C₂²	#11926407	1	1	yes	characteristic
248	C₂²	C₂×C₂²⋊(C₁₉⋊C₈)	1	1	yes	characteristic
249	C₂²	#11926508	1	1	yes	characteristic
250	C₂²	#11926518	1	1	yes	characteristic
251	C₂²	#11926518	1	1	yes	characteristic
252	C₂²	C₂×#10597290	1	1	yes	characteristic
253	C₂²	C₂×#10597293	1	1	yes	characteristic
254	C₂²	#11926597	1	2	yes	—
255	C₂²	#11926601	1	2	yes	—
256	C₄	—	2	4	no	—
257	C₄	—	2	4	no	—
258	C₄	—	2	2	no	—
259	C₄	—	2	2	no	—
260	C₄	—	4	4	no	—
261	C₄	—	4	4	no	—
262	C₂	—	2	2	no	—
263	C₂	—	2	2	no	—
264	C₂	#14561956	1	1	yes	characteristic
265	C₂	#14561959	1	1	yes	characteristic
266	C₂	#14561976	1	1	yes	characteristic
267	C₂	#14561979	1	1	yes	characteristic
268	C₂	#14563703	1	1	yes	characteristic
269	C₂	#14563712	1	1	yes	characteristic
270	C₂	#14563715	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₁₉
commutator	C₂³×C₁₉	C₂²×C₈
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₂×C₄×C₄⋊C₄×C₁₉	C₂
socle	C₂³×C₁₉	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₁₉	C₂²×C₄
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	C₂×C₄×C₄⋊C₄	D₁₉
hypocenter	C₁₉	#8701
LCS	G ▷ C₂³×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
23. C₂×C₄⋊C₄×C₁₉	C₈	semidirect
159. C₂×C₄⋊C₄	C₁₉⋊C₈	semidirect
168. C₁₉	#8701	semidirect

Maximal subgroups

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

subgroup	index
#14563703	2
#14561478	2
C₂×C₄×C₄⋊C₄×C₁₉	2
#8701	19