Subgroups — #18082352

Order 4864.

694 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14564442	C₂	1	2	yes	maximal, complemented
2	#14564442	C₂	1	2	yes	maximal, complemented
3	#14563763	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#11926266	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₁₉×#493	C₂	1	1	yes	maximal, characteristic
6	C₂×#10597289	—	2	4	no	—
7	C₂×#10597289	—	2	4	no	—
8	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
10	#11926414	C₂²	1	2	yes	—
11	#11926307	—	2	2	no	—
12	#11926266	—	2	2	no	—
13	C₂²×D₈×C₁₉	—	2	2	no	—
14	C₂×C₁₉×#127	C₂²	1	2	yes	—
15	C₁₉×#493	—	2	2	no	—
16	C₁₉×#493	C₄	1	4	yes	complemented
17	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
18	C₂²×D₄×C₁₉	—	2	2	no	—
19	C₂²×D₄×C₁₉	—	2	2	no	—
20	C₂²×D₄×C₁₉	—	2	2	no	—
21	C₂×D₈×C₁₉	—	2	8	no	—
22	C₂×D₈×C₁₉	—	4	8	no	—
23	C₂²×C₈×C₁₉	D₄	1	2	yes	—
24	C₁₉×#127	—	2	2	no	—
25	C₁₉×#127	—	2	2	no	—
26	C₁₉×#127	—	2	2	no	—
27	C₁₉×#127	C₂×C₄	1	4	yes	—
28	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊Dic₁₉	—	2	2	no	—
30	C₂×C₄⋊Dic₁₉	—	2	2	no	—
31	C₂×C₄⋊Dic₁₉	—	2	2	no	—
32	C₂²×C₁₉⋊C₈	D₄	1	2	yes	—
33	C₂²×C₁₉⋊C₈	D₄	1	2	yes	—
34	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₁₉	D₄	1	2	yes	—
36	#10597289	—	4	8	no	—
37	#10597289	—	4	8	no	—
38	C₄×C₁₉⋊C₈	—	2	1	no	—
39	C₄×C₁₉⋊C₈	—	2	1	no	—
40	C₄×C₁₉⋊C₈	—	2	1	no	—
41	C₄×C₁₉⋊C₈	—	2	1	no	—
42	C₂⁴×C₁₉	—	4	2	no	—
43	C₂×D₄×C₁₉	—	2	4	no	—
44	C₂×D₄×C₁₉	—	2	4	no	—
45	C₂×D₄×C₁₉	—	2	4	no	—
46	C₂×D₄×C₁₉	—	4	2	no	—
47	C₂×D₄×C₁₉	—	4	2	no	—
48	C₂×D₄×C₁₉	—	4	2	no	—
49	C₂×D₄×C₁₉	—	4	4	no	—
50	C₂×D₄×C₁₉	—	4	4	no	—
51	C₂×D₄×C₁₉	—	4	4	no	—
52	C₂×D₄×C₁₉	—	4	4	no	—
53	C₂×D₄×C₁₉	—	4	4	no	—
54	C₂×D₄×C₁₉	—	4	4	no	—
55	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
58	C₂²×Dic₁₉	—	4	2	no	—
59	D₈×C₁₉	—	4	16	no	—
60	C₂×C₈×C₁₉	—	2	4	no	—
61	C₂×C₈×C₁₉	C₂²⋊C₄	1	4	yes	—
62	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
63	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
64	C₄²×C₁₉	#41	1	1	yes	characteristic
65	C₄²×C₁₉	#41	1	1	yes	characteristic
66	C₄⋊Dic₁₉	—	4	4	no	—
67	C₄⋊Dic₁₉	—	4	4	no	—
68	C₄⋊Dic₁₉	—	4	4	no	—
69	C₂×C₁₉⋊C₈	—	2	2	no	—
70	C₂×C₁₉⋊C₈	—	2	2	no	—
71	C₂×C₁₉⋊C₈	—	2	2	no	—
72	C₂×C₁₉⋊C₈	—	2	4	no	—
73	C₂×C₁₉⋊C₈	D₈	1	4	yes	complemented
74	#9832	—	19	1	no	maximal, Sylow
75	C₂³×C₁₉	—	4	4	no	—
76	C₂³×C₁₉	—	8	2	no	—
77	C₂³×C₁₉	—	8	2	no	—
78	C₂³×C₁₉	—	8	2	no	—
79	C₂³×C₁₉	—	8	2	no	—
80	C₂³×C₁₉	—	8	2	no	—
81	C₂³×C₁₉	—	8	2	no	—
82	C₂³×C₁₉	#127	1	1	yes	characteristic
83	D₄×C₁₉	—	4	8	no	—
84	D₄×C₁₉	—	4	8	no	—
85	D₄×C₁₉	—	4	8	no	—
86	D₄×C₁₉	—	8	4	no	—
87	D₄×C₁₉	—	8	4	no	—
88	D₄×C₁₉	—	8	4	no	—
89	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
91	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
92	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
93	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
94	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
95	C₂×C₄×C₁₉	C₂×D₈	1	2	yes	—
96	C₂×C₄×C₁₉	#135	1	2	yes	—
97	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	4	yes	—
98	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	4	yes	—
99	C₂×Dic₁₉	—	8	2	no	—
100	C₂×Dic₁₉	—	8	2	no	—
101	C₂×Dic₁₉	—	8	2	no	—
102	C₈×C₁₉	—	2	4	no	—
103	C₁₉⋊C₈	—	2	4	no	—
104	C₁₉⋊C₈	—	4	2	no	—
105	C₂×#493	D₁₉	1	1	yes	characteristic
106	#1957	—	19	2	no	—
107	#1957	—	19	2	no	—
108	#1834	—	19	1	no	—
109	C₂×#322	—	19	1	no	—
110	C₂²×C₁₉	—	8	4	no	—
111	C₂²×C₁₉	—	8	4	no	—
112	C₂²×C₁₉	—	8	4	no	—
113	C₂²×C₁₉	—	16	2	no	—
114	C₂²×C₁₉	—	16	2	no	—
115	C₂²×C₁₉	—	16	2	no	—
116	C₂²×C₁₉	—	16	2	no	—
117	C₂²×C₁₉	#390	1	1	yes	characteristic
118	C₂²×C₁₉	#493	1	1	yes	characteristic
119	C₂²×C₁₉	#495	1	1	yes	characteristic
120	C₂²×C₁₉	#496	1	2	yes	—
121	C₂²×C₁₉	#496	1	2	yes	—
122	C₄×C₁₉	#418	1	2	yes	—
123	C₄×C₁₉	C₄×D₈	1	2	yes	—
124	C₄×C₁₉	#442	1	2	yes	—
125	C₄×C₁₉	#466	1	2	yes	—
126	C₄×C₁₉	#469	1	2	yes	—
127	C₄×C₁₉	#490	1	2	yes	—
128	Dic₁₉	—	16	2	no	—
129	C₂²×D₈	—	2	2	no	—
130	C₂×#127	C₂×D₁₉	1	2	yes	—
131	#493	—	2	2	no	—
132	#493	Dic₁₉	1	4	yes	complemented
133	C₂×#102	—	38	4	no	—
134	C₂×#102	—	38	4	no	—
135	C₂×C₄×C₈	—	19	1	no	—
136	C₂×C₄×C₈	—	19	1	no	—
137	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
138	#384	—	19	2	no	—
139	#340	—	38	2	no	—
140	#322	—	38	2	no	—
141	C₂×C₁₉	—	16	4	no	—
142	C₂×C₁₉	#1717	1	2	yes	—
143	C₂×C₁₉	#1721	1	2	yes	—
144	C₂×C₁₉	#1961	1	1	yes	characteristic
145	C₂×C₁₉	#1970	1	1	yes	characteristic
146	C₂×C₁₉	#1972	1	1	yes	characteristic
147	C₂²×D₄	—	2	2	no	—
148	C₂²×D₄	—	2	2	no	—
149	C₂²×D₄	—	2	2	no	—
150	C₂×D₈	—	2	8	no	—
151	C₂×D₈	—	4	8	no	—
152	C₂²×C₈	—	19	2	no	—
153	C₂²×C₈	—	19	2	no	—
154	C₂²×C₈	C₂²⋊D₁₉	1	2	yes	—
155	#127	—	2	2	no	—
156	#127	—	2	2	no	—
157	#127	—	2	2	no	—
158	#127	C₂×Dic₁₉	1	4	yes	—
159	C₂×C₄⋊C₄	—	38	2	no	—
160	C₂×C₄⋊C₄	—	38	2	no	—
161	C₂×C₄⋊C₄	—	38	2	no	—
162	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
163	#102	—	76	8	no	—
164	#102	—	76	8	no	—
165	C₄×C₈	—	38	1	no	—
166	C₄×C₈	—	38	1	no	—
167	C₄×C₈	—	38	1	no	—
168	C₄×C₈	—	38	1	no	—
169	C₄×C₈	C₂×Dic₁₉	1	2	yes	—
170	C₄×C₈	C₂²⋊D₁₉	1	2	yes	—
171	C₁₉	#9832	1	1	yes	characteristic, complemented, Sylow
172	C₂⁴	—	4	2	no	—
173	C₂×D₄	—	2	4	no	—
174	C₂×D₄	—	2	4	no	—
175	C₂×D₄	—	2	4	no	—
176	C₂×D₄	—	4	4	no	—
177	C₂×D₄	—	4	4	no	—
178	C₂×D₄	—	4	2	no	—
179	C₂×D₄	—	4	2	no	—
180	C₂×D₄	—	4	2	no	—
181	C₂×D₄	—	4	4	no	—
182	C₂×D₄	—	4	4	no	—
183	C₂×D₄	—	4	4	no	—
184	C₂×D₄	—	4	4	no	—
185	C₂²×C₄	—	76	2	no	—
186	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
187	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
188	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
189	D₈	—	4	16	no	—
190	C₂×C₈	—	2	4	no	—
191	C₂×C₈	—	19	4	no	—
192	C₂×C₈	—	38	2	no	—
193	C₂×C₈	—	38	2	no	—
194	C₂×C₈	—	38	2	no	—
195	C₂×C₈	—	38	4	no	—
196	C₂×C₈	C₂²⋊Dic₁₉	1	4	yes	—
197	C₄⋊C₄	—	76	4	no	—
198	C₄⋊C₄	—	76	4	no	—
199	C₄⋊C₄	—	76	4	no	—
200	C₄²	#64543	1	1	yes	characteristic
201	C₄²	#64543	1	1	yes	characteristic
202	C₄²	C₂²×Dic₁₉	1	1	yes	characteristic
203	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
204	C₂³	—	4	4	no	—
205	C₂³	—	8	2	no	—
206	C₂³	—	8	2	no	—
207	C₂³	—	8	2	no	—
208	C₂³	—	8	2	no	—
209	C₂³	—	8	2	no	—
210	C₂³	—	8	2	no	—
211	C₂³	#10597386	1	1	yes	characteristic
212	D₄	—	4	8	no	—
213	D₄	—	4	8	no	—
214	D₄	—	4	8	no	—
215	D₄	—	8	4	no	—
216	D₄	—	8	4	no	—
217	D₄	—	8	4	no	—
218	C₂×C₄	—	152	2	no	—
219	C₂×C₄	—	152	2	no	—
220	C₂×C₄	—	152	2	no	—
221	C₂×C₄	D₈×D₁₉	1	2	yes	—
222	C₂×C₄	#10597357	1	4	yes	—
223	C₂×C₄	#10597358	1	2	yes	—
224	C₂×C₄	#10597378	1	4	yes	—
225	C₂×C₄	D₄×Dic₁₉	1	1	yes	characteristic
226	C₂×C₄	D₄×Dic₁₉	1	1	yes	characteristic
227	C₂×C₄	#10597382	1	1	yes	characteristic
228	C₂×C₄	#10597384	1	1	yes	characteristic
229	C₂×C₄	#10597384	1	1	yes	characteristic
230	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
231	C₈	—	2	4	no	—
232	C₈	—	38	4	no	—
233	C₈	—	76	2	no	—
234	C₂²	—	8	4	no	—
235	C₂²	—	8	4	no	—
236	C₂²	—	8	4	no	—
237	C₂²	—	16	2	no	—
238	C₂²	—	16	2	no	—
239	C₂²	—	16	2	no	—
240	C₂²	—	16	2	no	—
241	C₂²	#11926447	1	1	yes	characteristic
242	C₂²	#11926919	1	1	yes	characteristic
243	C₂²	#11926927	1	1	yes	characteristic
244	C₂²	#11926931	1	2	yes	—
245	C₂²	#11926932	1	2	yes	—
246	C₄	—	304	2	no	—
247	C₄	#11926549	1	2	yes	—
248	C₄	D₈×Dic₁₉	1	2	yes	—
249	C₄	#11926656	1	2	yes	—
250	C₄	#11926777	1	2	yes	—
251	C₄	#11926794	1	2	yes	—
252	C₄	#11926908	1	2	yes	—
253	C₂	—	16	4	no	—
254	C₂	#14563273	1	2	yes	—
255	C₂	#14563300	1	2	yes	—
256	C₂	#14564460	1	1	yes	characteristic
257	C₂	#14564502	1	1	yes	characteristic
258	C₂	#14564512	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#10597386
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting, prime radical	C₂×C₁₉×#493	C₂
socle	C₂³×C₁₉	#127
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂×#493	D₁₉
hypocenter	C₁₉	#9832
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#493

Complements

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

N	H ≅ G/N	type
1. #14564442	C₂	semidirect
2. #14564442	C₂	semidirect
3. #14563763	C₂	semidirect
4. C₂×#11926266	C₂	semidirect
16. C₁₉×#493	C₄	semidirect
73. C₂×C₁₉⋊C₈	D₈	semidirect
132. #493	Dic₁₉	semidirect
171. C₁₉	#9832	semidirect

Maximal subgroups

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

subgroup	index
#14564442	2
#14563763	2
C₂×#11926266	2
C₂×C₁₉×#493	2
#9832	19