Subgroups — #18073009

Order 4864.

678 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×C₄×C₈×D₁₉	C₂	1	1	yes	maximal, characteristic
2	C₄×(C₂×D₁₉)⋊C₈	C₂	1	2	yes	maximal
3	C₂×#11926265	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₉⋊#1273	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₉×#322	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₈×D₁₉	C₄	1	2	yes	—
7	C₂×C₄²×D₁₉	C₂²	1	1	yes	characteristic
8	C₂×(C₂×D₁₉)⋊C₈	C₄	1	4	yes	—
9	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₁₉⋊C₈	C₂²	1	2	yes	—
11	C₂×C₈×Dic₁₉	C₄	1	2	yes	—
12	C₄×C₈×D₁₉	—	2	2	no	—
13	C₄×C₈×D₁₉	—	2	2	no	—
14	#11926298	C₄	1	4	yes	—
15	#11926265	—	2	2	no	—
16	C₂×C₄×C₈×C₁₉	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
18	C₁₉×#322	—	2	2	no	—
19	C₂²×C₄×D₁₉	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×D₁₉	C₂×C₄	1	2	yes	—
21	C₂²×C₈×C₁₉	C₂×C₄	1	4	yes	—
22	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
24	C₂×C₄×Dic₁₉	C₂×C₄	1	2	yes	—
25	C₂×C₄×Dic₁₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₁₉⋊C₈	C₂×C₄	1	4	yes	—
27	C₂²×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
28	C₂×C₈×D₁₉	—	2	4	no	—
29	C₂×C₈×D₁₉	—	2	4	no	—
30	C₂×C₈×D₁₉	C₈	1	8	yes	complemented
31	C₄²×D₁₉	—	2	1	no	—
32	C₄²×D₁₉	—	2	1	no	—
33	C₄²×D₁₉	—	2	2	no	—
34	C₄×C₈×C₁₉	—	2	2	no	—
35	C₄×C₈×C₁₉	—	2	2	no	—
36	C₄×C₈×C₁₉	D₄	1	2	yes	—
37	C₄×C₈×C₁₉	D₄	1	2	yes	—
38	(C₂×D₁₉)⋊C₈	—	2	8	no	—
39	C₈×Dic₁₉	—	2	4	no	—
40	C₄×C₁₉⋊C₈	—	2	1	no	—
41	C₄×C₁₉⋊C₈	—	2	1	no	—
42	C₄×C₁₉⋊C₈	—	2	2	no	—
43	C₄×C₁₉⋊C₈	—	2	2	no	—
44	C₂³×D₁₉	C₄²	1	1	yes	characteristic
45	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
47	C₂²×Dic₁₉	C₄²	1	1	yes	characteristic
48	C₂²×Dic₁₉	C₄²	1	2	yes	—
49	C₂×C₄×D₁₉	—	2	1	no	—
50	C₂×C₄×D₁₉	—	2	1	no	—
51	C₂×C₄×D₁₉	—	2	2	no	—
52	C₂×C₄×D₁₉	—	2	2	no	—
53	C₂×C₄×D₁₉	—	2	2	no	—
54	C₂×C₄×D₁₉	—	2	4	no	—
55	C₂×C₄×D₁₉	C₂×C₈	1	4	yes	—
56	C₂×C₄×D₁₉	M₁₆	1	8	yes	—
57	C₂×C₈×C₁₉	—	2	4	no	—
58	C₂×C₈×C₁₉	—	2	4	no	—
59	C₂×C₈×C₁₉	—	2	4	no	—
60	C₂×C₈×C₁₉	C₂²⋊C₄	1	4	yes	—
61	C₂×C₈×C₁₉	C₂×C₈	1	4	yes	—
62	C₂×C₈×C₁₉	M₁₆	1	4	yes	—
63	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
64	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
65	C₄²×C₁₉	#41	1	2	yes	—
66	C₄×Dic₁₉	—	2	1	no	—
67	C₄×Dic₁₉	—	2	1	no	—
68	C₄×Dic₁₉	—	2	2	no	—
69	C₄×Dic₁₉	—	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	—	2	4	no	—
74	C₂×C₁₉⋊C₈	—	2	4	no	—
75	C₂×C₁₉⋊C₈	C₂×C₈	1	4	yes	—
76	C₈×D₁₉	—	2	16	no	—
77	#8144	—	19	1	no	maximal, Sylow
78	C₂³×C₁₉	C₂×C₄²	1	1	yes	characteristic
79	C₂²×D₁₉	—	2	1	no	—
80	C₂²×D₁₉	—	2	1	no	—
81	C₂²×D₁₉	—	2	2	no	—
82	C₂²×D₁₉	C₄×C₈	1	2	yes	—
83	C₂²×D₁₉	#97	1	4	yes	—
84	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
86	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
87	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
88	C₂×C₄×C₁₉	C₂²×C₈	1	2	yes	—
89	C₂×C₄×C₁₉	C₂×M₁₆	1	2	yes	—
90	C₂×C₄×C₁₉	C₂×M₁₆	1	4	yes	—
91	C₂×C₄×C₁₉	#131	1	4	yes	—
92	C₂×Dic₁₉	—	2	1	no	—
93	C₂×Dic₁₉	—	2	2	no	—
94	C₂×Dic₁₉	—	2	1	no	—
95	C₂×Dic₁₉	—	2	2	no	—
96	C₂×Dic₁₉	C₄×C₈	1	2	yes	—
97	C₂×Dic₁₉	#97	1	4	yes	—
98	C₄×D₁₉	—	2	4	no	—
99	C₄×D₁₉	—	2	4	no	—
100	C₄×D₁₉	—	2	8	no	—
101	C₄×D₁₉	—	2	8	no	—
102	C₈×C₁₉	—	4	4	no	—
103	C₈×C₁₉	C₂²⋊C₈	1	8	yes	complemented
104	C₁₉⋊C₈	—	2	4	no	—
105	C₁₉⋊C₈	—	4	4	no	—
106	C₂²×C₄×C₈	—	19	1	no	—
107	C₄×C₂²⋊C₈	—	19	2	no	—
108	C₂×#322	—	19	1	no	—
109	C₂×#322	D₁₉	1	1	yes	characteristic, complemented
110	#1273	—	19	2	no	—
111	C₂²×C₁₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
112	C₂²×C₁₉	C₂×C₄×C₈	1	1	yes	characteristic
113	C₂²×C₁₉	C₂×#97	1	2	yes	—
114	C₂²×C₁₉	C₄×M₁₆	1	1	yes	characteristic
115	C₂²×C₁₉	#405	1	2	yes	—
116	C₂×D₁₉	—	2	2	no	—
117	C₂×D₁₉	—	2	4	no	—
118	C₂×D₁₉	—	2	2	no	—
119	C₂×D₁₉	—	2	4	no	—
120	C₂×D₁₉	#322	1	4	yes	complemented
121	C₄×C₁₉	C₂×C₂²⋊C₈	1	2	yes	—
122	C₄×C₁₉	#431	1	2	yes	—
123	C₄×C₁₉	#435	1	4	yes	—
124	C₄×C₁₉	#435	1	4	yes	—
125	Dic₁₉	—	2	2	no	—
126	Dic₁₉	—	2	4	no	—
127	C₂³×C₈	—	19	2	no	—
128	C₂²×C₄²	—	19	1	no	—
129	C₂×C₂²⋊C₈	—	19	4	no	—
130	C₂×C₄×C₈	—	19	2	no	—
131	C₂×C₄×C₈	—	19	1	no	—
132	C₂×C₄×C₈	—	19	2	no	—
133	C₂×C₄×C₈	—	38	2	no	—
134	C₂×C₄×C₈	—	38	2	no	—
135	C₂×C₄×C₈	C₂×D₁₉	1	2	yes	—
136	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
137	#336	—	19	4	no	—
138	#322	—	2	2	no	—
139	#322	—	38	2	no	—
140	C₂×C₁₉	C₂×#322	1	1	yes	characteristic
141	C₂×C₁₉	#1450	1	1	yes	characteristic
142	C₂×C₁₉	#1451	1	2	yes	—
143	C₂×C₁₉	C₄×C₂²⋊C₈	1	1	yes	characteristic
144	C₂×C₁₉	#1752	1	2	yes	—
145	D₁₉	—	2	4	no	—
146	C₂³×C₄	—	19	1	no	—
147	C₂³×C₄	—	19	2	no	—
148	C₂²×C₈	—	19	2	no	—
149	C₂²×C₈	—	19	8	no	—
150	C₂²×C₈	—	19	4	no	—
151	C₂²×C₈	—	38	4	no	—
152	C₂²×C₈	—	38	4	no	—
153	C₂²×C₈	C₄×D₁₉	1	4	yes	—
154	C₂²×C₈	C₄×D₁₉	1	2	yes	—
155	C₂×C₄²	—	19	2	no	—
156	C₂×C₄²	—	19	1	no	—
157	C₂×C₄²	—	38	1	no	—
158	C₂×C₄²	—	38	2	no	—
159	C₂×C₄²	—	38	1	no	—
160	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
161	C₂²⋊C₈	—	38	8	no	—
162	C₄×C₈	—	2	2	no	—
163	C₄×C₈	—	2	2	no	—
164	C₄×C₈	—	38	4	no	—
165	C₄×C₈	—	38	2	no	—
166	C₄×C₈	—	38	1	no	—
167	C₄×C₈	—	38	2	no	—
168	C₄×C₈	—	38	1	no	—
169	C₄×C₈	D₇₆	1	2	yes	—
170	C₄×C₈	C₂²⋊D₁₉	1	2	yes	—
171	C₁₉	#8144	1	1	yes	characteristic, complemented, Sylow
172	C₂⁴	—	19	1	no	—
173	C₂²×C₄	—	19	1	no	—
174	C₂²×C₄	—	19	2	no	—
175	C₂²×C₄	—	19	4	no	—
176	C₂²×C₄	—	19	8	no	—
177	C₂²×C₄	—	38	1	no	—
178	C₂²×C₄	—	38	2	no	—
179	C₂²×C₄	—	38	2	no	—
180	C₂²×C₄	—	38	4	no	—
181	C₂²×C₄	—	38	1	no	—
182	C₂²×C₄	—	38	2	no	—
183	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
184	C₂²×C₄	C₂×C₄×D₁₉	1	2	yes	—
185	C₂×C₈	—	2	4	no	—
186	C₂×C₈	—	2	4	no	—
187	C₂×C₈	—	2	4	no	—
188	C₂×C₈	—	19	4	no	—
189	C₂×C₈	—	38	2	no	—
190	C₂×C₈	—	38	2	no	—
191	C₂×C₈	—	38	16	no	—
192	C₂×C₈	—	38	4	no	—
193	C₂×C₈	—	38	4	no	—
194	C₂×C₈	—	38	4	no	—
195	C₂×C₈	C₈×D₁₉	1	4	yes	—
196	C₂×C₈	#64515	1	4	yes	—
197	C₂×C₈	(C₂×D₁₉)⋊C₄	1	4	yes	—
198	C₄²	—	38	1	no	—
199	C₄²	—	38	2	no	—
200	C₄²	—	38	1	no	—
201	C₄²	—	38	2	no	—
202	C₄²	C₂×D₇₆	1	1	yes	characteristic
203	C₄²	#64541	1	2	yes	—
204	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
205	C₂³	—	19	2	no	—
206	C₂³	—	19	4	no	—
207	C₂³	—	38	1	no	—
208	C₂³	—	38	1	no	—
209	C₂³	—	38	2	no	—
210	C₂³	C₄²×D₁₉	1	1	yes	characteristic
211	C₂×C₄	—	19	2	no	—
212	C₂×C₄	—	19	4	no	—
213	C₂×C₄	—	38	1	no	—
214	C₂×C₄	—	38	2	no	—
215	C₂×C₄	—	38	1	no	—
216	C₂×C₄	—	38	2	no	—
217	C₂×C₄	—	38	4	no	—
218	C₂×C₄	—	38	8	no	—
219	C₂×C₄	—	38	8	no	—
220	C₂×C₄	—	38	4	no	—
221	C₂×C₄	#10597318	1	1	yes	characteristic
222	C₂×C₄	C₄×D₇₆	1	2	yes	—
223	C₂×C₄	C₂×C₈×D₁₉	1	2	yes	—
224	C₂×C₄	C₂×#64515	1	2	yes	—
225	C₂×C₄	M₁₆×D₁₉	1	4	yes	—
226	C₂×C₄	#10597353	1	4	yes	—
227	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
228	C₂×C₄	C₄×C₂²⋊D₁₉	1	2	yes	—
229	C₈	—	4	4	no	—
230	C₈	—	38	4	no	—
231	C₈	—	76	4	no	—
232	C₈	(C₂×D₁₉)⋊C₈	1	8	yes	complemented
233	C₂²	—	19	4	no	—
234	C₂²	—	38	2	no	—
235	C₂²	—	38	4	no	—
236	C₂²	—	38	2	no	—
237	C₂²	—	38	4	no	—
238	C₂²	C₄×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
239	C₂²	C₄×C₈×D₁₉	1	1	yes	characteristic
240	C₂²	#97×D₁₉	1	2	yes	—
241	C₂²	C₄×#64515	1	1	yes	characteristic
242	C₂²	#11926501	1	2	yes	—
243	C₄	—	38	2	no	—
244	C₄	—	38	4	no	—
245	C₄	C₂×(C₂×D₁₉)⋊C₈	1	2	yes	—
246	C₄	#11926595	1	2	yes	—
247	C₄	#11926613	1	4	yes	—
248	C₄	#11926617	1	4	yes	—
249	C₂	—	38	4	no	—
250	C₂	D₁₉×#322	1	1	yes	characteristic
251	C₂	#14561920	1	1	yes	characteristic
252	C₂	#14561921	1	2	yes	—
253	C₂	C₄×(C₂×D₁₉)⋊C₈	1	1	yes	characteristic
254	C₂	#14563412	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₁₉
commutator	C₂²×C₁₉	C₂×C₄×C₈
Fitting	C₂×C₁₉×#322	C₂
socle	C₂³×C₁₉	C₂×C₄²
prime radical	C₂³×D₁₉	C₄²
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂×#322	D₁₉
hypocenter	C₁₉	#8144
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄² ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
3. C₂×#11926265	C₂	semidirect
4. C₁₉⋊#1273	C₂	semidirect
5. C₂×C₁₉×#322	C₂	semidirect
30. C₂×C₈×D₁₉	C₈	semidirect
103. C₈×C₁₉	C₂²⋊C₈	semidirect
109. C₂×#322	D₁₉	semidirect
120. C₂×D₁₉	#322	semidirect
171. C₁₉	#8144	semidirect
232. C₈	(C₂×D₁₉)⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₈×D₁₉	2
C₄×(C₂×D₁₉)⋊C₈	2
C₂×#11926265	2
C₁₉⋊#1273	2
C₂×C₁₉×#322	2
#8144	19