Subgroups — #18081259

Order 4864.

322 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14563807	C₂	1	1	yes	maximal, characteristic
2	C₈×C₄⋊Dic₁₉	C₂	1	1	yes	maximal, characteristic
3	#14562410	C₂	1	2	yes	maximal
4	#14561537	C₂	1	2	yes	maximal
5	C₁₉×#1564	C₂	1	1	yes	maximal, characteristic
6	C₂×#10597276	—	2	1	no	—
7	C₂×#10597276	—	2	1	no	—
8	#11926568	C₂²	1	2	yes	—
9	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
10	C₂×C₈×Dic₁₉	—	2	1	no	—
11	C₄×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
12	#11926298	—	2	1	no	—
13	#11926297	—	2	4	no	—
14	C₁₉×#431	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
16	C₁₉×#335	C₄	1	2	yes	complemented
17	C₁₉×#334	C₄	1	2	yes	complemented
18	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
20	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
21	C₂×C₄⋊Dic₁₉	—	2	1	no	—
22	C₂×C₄×Dic₁₉	—	2	1	no	—
23	C₂×#64520	—	2	2	no	—
24	C₂²×C₁₉⋊C₈	—	2	1	no	—
25	C₁₉×C₄⋊C₈	C₂×C₄	1	4	yes	—
26	C₁₉×C₂²⋊C₈	—	2	2	no	—
27	C₄×C₈×C₁₉	—	2	2	no	—
28	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₁₉	Q₈	1	1	yes	characteristic
31	C₄×C₈×C₁₉	Q₈	1	1	yes	characteristic
32	C₁₉⋊#95	Q₈	1	2	yes	—
33	#10597276	—	4	2	no	—
34	#10597276	—	4	2	no	—
35	C₈×Dic₁₉	—	4	2	no	—
36	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
37	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
38	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
39	C₄⋊(C₁₉⋊C₈)	D₄	1	2	yes	—
40	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
42	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
43	C₂²×Dic₁₉	—	2	2	no	—
44	C₂×C₈×C₁₉	—	2	1	no	—
45	C₂×C₈×C₁₉	—	2	1	no	—
46	C₂×C₈×C₁₉	—	2	4	no	—
47	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₁₉	D₈	1	2	yes	—
52	C₂×C₈×C₁₉	Q₁₆	1	2	yes	—
53	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
54	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
55	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
56	C₄²×C₁₉	#41	1	1	yes	characteristic
57	C₄⋊Dic₁₉	—	2	4	no	—
58	C₄×Dic₁₉	—	4	1	no	—
59	C₄×Dic₁₉	—	4	1	no	—
60	#64520	—	4	2	no	—
61	#64520	—	4	2	no	—
62	C₂×C₁₉⋊C₈	—	2	2	no	—
63	C₂×C₁₉⋊C₈	—	4	1	no	—
64	C₂×C₁₉⋊C₈	—	4	1	no	—
65	C₂×C₁₉⋊C₈	—	4	2	no	—
66	#9627	—	19	1	no	maximal, Sylow
67	C₂³×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
68	C₂×C₄×C₁₉	—	2	2	no	—
69	C₂×C₄×C₁₉	—	4	1	no	—
70	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
72	C₂×C₄×C₁₉	C₄×Q₈	1	1	yes	characteristic
73	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
74	C₂×C₄×C₁₉	#125	1	1	yes	characteristic
75	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
76	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
77	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
78	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
79	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
80	C₂×Dic₁₉	—	2	4	no	—
81	C₂×Dic₁₉	—	4	2	no	—
82	C₂×Dic₁₉	—	4	2	no	—
83	C₈×C₁₉	—	2	1	no	—
84	C₈×C₁₉	—	2	1	no	—
85	C₈×C₁₉	—	4	4	no	—
86	C₈×C₁₉	#107	1	2	yes	—
87	C₈×C₁₉	#107	1	2	yes	—
88	C₁₉⋊C₈	—	4	2	no	—
89	C₁₉⋊C₈	—	8	2	no	—
90	#1848	—	19	1	no	—
91	C₈×C₄⋊C₄	—	19	1	no	—
92	#1571	—	19	2	no	—
93	#1564	D₁₉	1	1	yes	characteristic
94	#1303	—	19	2	no	—
95	C₂²×C₁₉	—	2	1	no	—
96	C₂²×C₁₉	—	2	1	no	—
97	C₂²×C₁₉	#389	1	1	yes	characteristic
98	C₂²×C₁₉	#474	1	1	yes	characteristic
99	C₂²×C₁₉	#477	1	1	yes	characteristic
100	C₄×C₁₉	—	2	1	no	—
101	C₄×C₁₉	—	2	1	no	—
102	C₄×C₁₉	—	2	1	no	—
103	C₄×C₁₉	C₂×#107	1	1	yes	characteristic
104	C₄×C₁₉	#428	1	1	yes	characteristic
105	C₄×C₁₉	#443	1	2	yes	—
106	C₄×C₁₉	#459	1	1	yes	characteristic
107	C₄×C₁₉	#462	1	1	yes	characteristic
108	Dic₁₉	—	4	4	no	—
109	#431	C₂×D₁₉	1	2	yes	—
110	C₂×#108	—	38	1	no	—
111	C₂×#108	—	38	1	no	—
112	#423	—	19	2	no	—
113	C₂×C₄⋊C₈	—	19	1	no	—
114	C₂×C₄×C₈	—	38	1	no	—
115	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
116	C₄×C₄⋊C₄	—	19	1	no	—
117	#336	—	38	1	no	—
118	#335	—	38	4	no	—
119	#335	Dic₁₉	1	2	yes	complemented
120	#334	Dic₁₉	1	2	yes	complemented
121	C₂×C₁₉	—	2	1	no	—
122	C₂×C₁₉	—	2	1	no	—
123	C₂×C₁₉	#1945	1	1	yes	characteristic
124	C₂×C₁₉	#1947	1	1	yes	characteristic
125	C₂×C₁₉	#1953	1	1	yes	characteristic
126	C₂×M₁₆	—	38	2	no	—
127	C₂²×C₈	—	38	1	no	—
128	C₂²×C₈	D₇₆	1	1	yes	characteristic
129	C₂²×C₈	D₇₆	1	1	yes	characteristic
130	C₂×C₄⋊C₄	—	38	1	no	—
131	C₂×C₄²	—	38	1	no	—
132	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
133	#108	—	76	2	no	—
134	#108	—	76	2	no	—
135	C₄⋊C₈	—	19	2	no	—
136	C₄⋊C₈	—	38	2	no	—
137	C₄⋊C₈	—	38	1	no	—
138	C₄⋊C₈	—	38	2	no	—
139	C₄⋊C₈	C₂×Dic₁₉	1	4	yes	—
140	C₂²⋊C₈	—	2	2	no	—
141	C₄×C₈	—	2	2	no	—
142	C₄×C₈	—	76	2	no	—
143	C₄×C₈	Dic₃₈	1	1	yes	characteristic
144	C₄×C₈	Dic₃₈	1	1	yes	characteristic
145	C₄×C₈	C₂×Dic₁₉	1	1	yes	characteristic
146	C₄×C₈	C₂×Dic₁₉	1	1	yes	characteristic
147	#95	—	19	2	no	—
148	C₁₉	#9627	1	1	yes	characteristic, complemented, Sylow
149	C₂²×C₄	—	38	2	no	—
150	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
151	C₂²×C₄	#64543	1	1	yes	characteristic
152	C₂²×C₄	Q₈×D₁₉	1	1	yes	characteristic
153	M₁₆	—	76	2	no	—
154	M₁₆	—	76	2	no	—
155	C₂×C₈	—	2	1	no	—
156	C₂×C₈	—	2	1	no	—
157	C₂×C₈	—	2	4	no	—
158	C₂×C₈	—	38	2	no	—
159	C₂×C₈	—	76	1	no	—
160	C₂×C₈	—	76	1	no	—
161	C₂×C₈	—	76	2	no	—
162	C₂×C₈	D₁₅₂	1	2	yes	—
163	C₂×C₈	Dic₇₆	1	2	yes	—
164	C₂×C₈	C₄⋊Dic₁₉	1	1	yes	characteristic
165	C₂×C₈	C₄⋊Dic₁₉	1	1	yes	characteristic
166	C₂×C₈	C₄⋊Dic₁₉	1	1	yes	characteristic
167	C₂×C₈	C₄⋊Dic₁₉	1	1	yes	characteristic
168	C₄⋊C₄	—	38	4	no	—
169	C₄²	—	76	1	no	—
170	C₄²	—	76	1	no	—
171	C₄²	C₂×Dic₃₈	1	1	yes	characteristic
172	C₄²	D₄×D₁₉	1	1	yes	characteristic
173	C₄²	#64545	1	1	yes	characteristic
174	C₄²	C₂²×Dic₁₉	1	1	yes	characteristic
175	C₂³	#10597343	1	1	yes	characteristic
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	4	1	no	—
178	C₂×C₄	—	38	4	no	—
179	C₂×C₄	—	76	2	no	—
180	C₂×C₄	—	76	2	no	—
181	C₂×C₄	#10597334	1	1	yes	characteristic
182	C₂×C₄	#10597336	1	1	yes	characteristic
183	C₂×C₄	#10597341	1	1	yes	characteristic
184	C₂×C₄	C₂×D₁₅₂	1	1	yes	characteristic
185	C₂×C₄	C₂×Dic₇₆	1	1	yes	characteristic
186	C₂×C₄	#10597354	1	1	yes	characteristic
187	C₂×C₄	#10597355	1	1	yes	characteristic
188	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
189	C₂×C₄	D₄×Dic₁₉	1	1	yes	characteristic
190	C₂×C₄	Q₈×Dic₁₉	1	1	yes	characteristic
191	C₈	—	2	1	no	—
192	C₈	—	2	1	no	—
193	C₈	—	4	4	no	—
194	C₈	—	76	2	no	—
195	C₈	—	152	2	no	—
196	C₈	#10597275	1	2	yes	—
197	C₈	#10597275	1	2	yes	—
198	C₂²	—	2	1	no	—
199	C₂²	—	2	1	no	—
200	C₂²	#11926443	1	1	yes	characteristic
201	C₂²	#11926825	1	1	yes	characteristic
202	C₂²	#11926840	1	1	yes	characteristic
203	C₄	—	2	1	no	—
204	C₄	—	2	1	no	—
205	C₄	—	2	1	no	—
206	C₄	—	76	4	no	—
207	C₄	C₂×#10597275	1	1	yes	characteristic
208	C₄	#11926586	1	1	yes	characteristic
209	C₄	#11926661	1	2	yes	—
210	C₄	#11926740	1	1	yes	characteristic
211	C₄	#11926755	1	1	yes	characteristic
212	C₂	—	2	1	no	—
213	C₂	—	2	1	no	—
214	C₂	#14564384	1	1	yes	characteristic
215	C₂	#14564394	1	1	yes	characteristic
216	C₂	#14564426	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#10597341
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1564	C₂
socle	C₂²×C₁₉	#389
prime radical	C₂³×C₁₉	C₂²⋊Q₈
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1564	D₁₉
hypocenter	C₁₉	#9627
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
16. C₁₉×#335	C₄	semidirect
17. C₁₉×#334	C₄	semidirect
119. #335	Dic₁₉	semidirect
120. #334	Dic₁₉	semidirect
148. C₁₉	#9627	semidirect

Maximal subgroups

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

subgroup	index
#14563807	2
C₈×C₄⋊Dic₁₉	2
#14562410	2
#14561537	2
C₁₉×#1564	2
#9627	19