Subgroups — #18073558

Order 4864.

368 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14572569	C₂	1	1	yes	maximal, characteristic
2	#14562299	C₂	1	1	yes	maximal, characteristic
3	#14562229	C₂	1	1	yes	maximal, characteristic
4	#14561924	C₂	1	1	yes	maximal, characteristic
5	#14561526	C₂	1	1	yes	maximal, characteristic
6	#14561525	C₂	1	1	yes	maximal, characteristic
7	C₁₉×#1451	C₂	1	1	yes	maximal, characteristic
8	#11927101	C₂²	1	1	yes	characteristic
9	#11926674	—	2	1	no	—
10	#11926674	—	2	1	no	—
11	C₈×Dic₃₈	—	2	1	no	—
12	C₈×Dic₃₈	—	2	1	no	—
13	#11926597	C₂²	1	1	yes	characteristic
14	#11926570	C₄	1	2	yes	—
15	#11926568	C₂²	1	1	yes	characteristic
16	#11926499	C₄	1	2	yes	—
17	C₄×#64520	C₂²	1	1	yes	characteristic
18	#11926291	C₄	1	2	yes	—
19	#11926289	C₄	1	2	yes	—
20	#11926285	C₄	1	4	yes	—
21	#11926266	—	2	1	no	—
22	#11926265	—	2	1	no	—
23	C₁₉×#431	C₂²	1	1	yes	characteristic
24	C₄×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
26	C₁₉×#322	—	2	1	no	—
27	C₁₉×#322	—	2	1	no	—
28	C₂×M₁₆×C₁₉	—	2	1	no	—
29	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
32	#10597372	C₂×C₄	1	2	yes	—
33	#10597370	—	2	1	no	—
34	#10597370	—	2	1	no	—
35	C₂×#64520	—	2	1	no	—
36	C₂×#64520	C₂×C₄	1	2	yes	—
37	#10597316	C₂×C₄	1	2	yes	—
38	#10597315	C₂×C₄	1	1	yes	characteristic
39	#10597315	C₂×C₄	1	1	yes	characteristic
40	C₄×Dic₃₈	—	2	1	no	—
41	C₄×Dic₃₈	—	2	1	no	—
42	C₁₉×C₄⋊C₈	C₂×C₄	1	2	yes	—
43	C₁₉×C₂²⋊C₈	—	2	1	no	—
44	C₁₉×#97	C₂×C₄	1	2	yes	—
45	C₄×C₈×C₁₉	—	2	1	no	—
46	C₄×C₈×C₁₉	—	2	1	no	—
47	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
48	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
49	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
50	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
51	C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
52	#10597272	—	2	2	no	—
53	Dic₁₉⋊C₈	—	2	2	no	—
54	Dic₁₉⋊C₈	—	2	2	no	—
55	C₈×Dic₁₉	—	2	2	no	—
56	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
57	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
58	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	2	yes	—
59	#10597260	C₂×C₄	1	2	yes	—
60	C₄×C₁₉⋊C₈	—	2	1	no	—
61	C₄×C₁₉⋊C₈	—	2	1	no	—
62	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
64	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
65	C₂×Dic₃₈	—	2	1	no	—
66	C₂×Dic₃₈	—	2	1	no	—
67	M₁₆×C₁₉	—	4	2	no	—
68	C₂×C₈×C₁₉	—	2	2	no	—
69	C₂×C₈×C₁₉	—	2	2	no	—
70	C₂×C₈×C₁₉	—	2	2	no	—
71	C₂×C₈×C₁₉	—	4	1	no	—
72	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
76	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
77	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
78	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
80	C₂²⋊Dic₁₉	—	2	2	no	—
81	C₄⋊Dic₁₉	—	2	1	no	—
82	C₄⋊Dic₁₉	—	2	1	no	—
83	C₄⋊Dic₁₉	C₄²	1	2	yes	—
84	C₄⋊Dic₁₉	C₄²	1	2	yes	—
85	Dic₁₉⋊C₄	—	2	2	no	—
86	Dic₁₉⋊C₄	—	2	2	no	—
87	C₄×Dic₁₉	—	2	2	no	—
88	#64520	—	4	2	no	—
89	#64520	—	4	2	no	—
90	C₂×C₁₉⋊C₈	—	2	2	no	—
91	C₂×C₁₉⋊C₈	—	2	2	no	—
92	C₂×C₁₉⋊C₈	—	2	2	no	—
93	#8244	—	19	1	no	maximal, Sylow
94	C₂³×C₁₉	#117	1	1	yes	characteristic
95	C₂×C₄×C₁₉	—	2	1	no	—
96	C₂×C₄×C₁₉	—	2	1	no	—
97	C₂×C₄×C₁₉	—	2	2	no	—
98	C₂×C₄×C₁₉	—	2	2	no	—
99	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
100	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
102	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
105	C₂×Dic₁₉	—	2	2	no	—
106	C₂×Dic₁₉	—	2	2	no	—
107	Dic₃₈	—	4	2	no	—
108	Dic₃₈	—	4	2	no	—
109	C₈×C₁₉	—	2	1	no	—
110	C₈×C₁₉	—	2	1	no	—
111	C₈×C₁₉	—	2	1	no	—
112	C₈×C₁₉	—	4	2	no	—
113	C₈×C₁₉	—	4	2	no	—
114	C₈×C₁₉	#101	1	2	yes	—
115	C₁₉⋊C₈	—	4	2	no	—
116	C₁₉⋊C₈	—	4	2	no	—
117	C₁₉⋊C₈	—	4	2	no	—
118	#2960	—	19	1	no	—
119	#1544	—	19	1	no	—
120	#1526	—	19	1	no	—
121	#1451	—	19	1	no	—
122	#1451	D₁₉	1	1	yes	characteristic
123	#1299	—	19	1	no	—
124	#1299	—	19	1	no	—
125	C₂²×C₁₉	—	2	1	no	—
126	C₂²×C₁₉	—	2	1	no	—
127	C₂²×C₁₉	—	2	1	no	—
128	C₂²×C₁₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
129	C₄×C₁₉	—	2	1	no	—
130	C₄×C₁₉	—	2	1	no	—
131	C₄×C₁₉	—	2	1	no	—
132	C₄×C₁₉	C₂×#101	1	1	yes	characteristic
133	C₄×C₁₉	#413	1	1	yes	characteristic
134	C₄×C₁₉	#444	1	2	yes	—
135	C₄×C₁₉	#444	1	2	yes	—
136	Dic₁₉	—	4	2	no	—
137	Dic₁₉	—	4	2	no	—
138	#533	—	19	1	no	—
139	#446	—	38	1	no	—
140	#446	—	38	1	no	—
141	C₈×Q₈	—	38	1	no	—
142	C₈×Q₈	—	38	1	no	—
143	#431	—	19	1	no	—
144	#431	C₂×D₁₉	1	1	yes	characteristic
145	#424	—	19	2	no	—
146	#423	—	19	1	no	—
147	#405	—	19	2	no	—
148	C₄×M₁₆	—	19	1	no	—
149	C₄×M₁₆	C₂×D₁₉	1	1	yes	characteristic
150	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
151	#332	—	19	2	no	—
152	#332	—	19	2	no	—
153	#330	—	19	4	no	—
154	#322	—	2	1	no	—
155	#322	—	2	1	no	—
156	#322	—	38	1	no	—
157	#322	—	38	1	no	—
158	C₂×C₁₉	—	4	1	no	—
159	C₂×C₁₉	C₄×#101	1	1	yes	characteristic
160	C₂×C₁₉	#1765	1	1	yes	characteristic
161	C₂×C₁₉	#1765	1	1	yes	characteristic
162	C₂×M₁₆	—	2	1	no	—
163	C₂×M₁₆	—	19	2	no	—
164	C₂×M₁₆	—	38	1	no	—
165	C₂²×C₈	C₄×D₁₉	1	1	yes	characteristic
166	C₂²×C₈	C₄×D₁₉	1	1	yes	characteristic
167	C₄⋊Q₈	—	19	1	no	—
168	C₄⋊Q₈	—	19	1	no	—
169	#125	—	19	2	no	—
170	C₂²⋊Q₈	—	38	1	no	—
171	C₂²⋊Q₈	—	38	1	no	—
172	C₄×Q₈	—	38	1	no	—
173	C₄×Q₈	—	38	1	no	—
174	#117	—	19	2	no	—
175	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
176	C₄⋊C₈	—	19	2	no	—
177	C₄⋊C₈	—	38	2	no	—
178	C₄⋊C₈	—	38	2	no	—
179	C₄⋊C₈	—	38	1	no	—
180	C₄⋊C₈	—	38	1	no	—
181	C₄⋊C₈	C₄×D₁₉	1	2	yes	—
182	C₂²⋊C₈	—	2	1	no	—
183	C₂²⋊C₈	—	38	1	no	—
184	#97	—	19	2	no	—
185	#97	—	38	2	no	—
186	#97	C₄×D₁₉	1	2	yes	—
187	C₄×C₈	—	2	1	no	—
188	C₄×C₈	—	2	1	no	—
189	C₄×C₈	—	38	2	no	—
190	C₄×C₈	—	38	1	no	—
191	C₄×C₈	—	38	1	no	—
192	C₄×C₈	D₇₆	1	1	yes	characteristic
193	C₄×C₈	D₇₆	1	1	yes	characteristic
194	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
195	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
196	C₁₉	#8244	1	1	yes	characteristic, complemented, Sylow
197	C₂×Q₈	—	38	1	no	—
198	C₂×Q₈	—	38	1	no	—
199	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
200	C₂²×C₄	#64541	1	1	yes	characteristic
201	C₂²×C₄	#64541	1	1	yes	characteristic
202	M₁₆	—	4	2	no	—
203	M₁₆	—	76	2	no	—
204	M₁₆	—	76	2	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	4	1	no	—
209	C₂×C₈	—	38	2	no	—
210	C₂×C₈	—	38	2	no	—
211	C₂×C₈	—	38	2	no	—
212	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
213	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
214	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
215	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
216	C₄⋊C₄	—	19	2	no	—
217	C₄⋊C₄	—	19	2	no	—
218	C₄⋊C₄	—	38	2	no	—
219	C₄⋊C₄	—	38	1	no	—
220	C₄⋊C₄	—	38	2	no	—
221	C₄⋊C₄	—	38	1	no	—
222	C₂²⋊C₄	—	38	2	no	—
223	C₄²	—	38	2	no	—
224	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
225	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
226	C₄²	C₂×D₇₆	1	1	yes	characteristic
227	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
228	C₂³	#10597318	1	1	yes	characteristic
229	Q₈	—	76	2	no	—
230	Q₈	—	76	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	38	2	no	—
236	C₂×C₄	—	38	2	no	—
237	C₂×C₄	C₄²×D₁₉	1	1	yes	characteristic
238	C₂×C₄	C₄×D₇₆	1	1	yes	characteristic
239	C₂×C₄	C₄×D₇₆	1	1	yes	characteristic
240	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
241	C₂×C₄	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
242	C₂×C₄	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
243	C₈	—	2	1	no	—
244	C₈	—	2	1	no	—
245	C₈	—	2	1	no	—
246	C₈	—	4	2	no	—
247	C₈	—	4	2	no	—
248	C₈	—	76	2	no	—
249	C₈	—	76	2	no	—
250	C₈	—	76	2	no	—
251	C₈	#10597281	1	2	yes	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	1	no	—
255	C₂²	C₄×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
256	C₄	—	2	1	no	—
257	C₄	—	2	1	no	—
258	C₄	—	2	1	no	—
259	C₄	—	76	2	no	—
260	C₄	—	76	2	no	—
261	C₄	C₂×#10597281	1	1	yes	characteristic
262	C₄	#11926526	1	1	yes	characteristic
263	C₄	#11926664	1	2	yes	—
264	C₄	#11926667	1	2	yes	—
265	C₂	—	4	1	no	—
266	C₂	C₄×#10597281	1	1	yes	characteristic
267	C₂	#14563479	1	1	yes	characteristic
268	C₂	#14563481	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄
commutator	C₂×C₄×C₁₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1451	C₂
socle	C₂²×C₁₉	C₄×C₂²⋊C₄
prime radical	C₂³×C₁₉	#117
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1451	D₁₉
hypocenter	C₁₉	#8244
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #1451

Complements

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

N	H ≅ G/N	type
196. C₁₉	#8244	semidirect

Maximal subgroups

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

subgroup	index
#14572569	2
#14562299	2
#14562229	2
#14561924	2
#14561526	2
#14561525	2
C₁₉×#1451	2
#8244	19