Subgroups — #18076221

Order 4864.

294 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14563799	C₂	1	1	yes	maximal, characteristic
2	#14561918	C₂	1	1	yes	maximal, characteristic
3	#14561495	C₂	1	4	yes	maximal
4	C₁₉×#1848	C₂	1	1	yes	maximal, characteristic
5	#11926597	C₂²	1	2	yes	—
6	C₂×#10597275	—	2	1	no	—
7	C₂×#10597274	—	2	1	no	—
8	#11926410	C₂²	1	2	yes	—
9	#11926265	C₄	1	2	yes	complemented
10	C₈×C₁₉⋊C₈	C₄	1	2	yes	complemented
11	C₂×C₁₉×#108	—	2	1	no	—
12	C₂×C₁₉×#108	—	2	1	no	—
13	C₁₉×#423	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
15	C₂×M₁₆×C₁₉	—	2	2	no	—
16	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
18	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
19	C₂×C₄⋊Dic₁₉	—	2	2	no	—
20	C₁₉×#108	—	4	2	no	—
21	C₁₉×#108	—	4	2	no	—
22	C₁₉×C₄⋊C₈	—	2	4	no	—
23	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₁₉	Q₈	1	1	yes	characteristic
26	C₄×C₈×C₁₉	Q₈	1	1	yes	characteristic
27	C₁₉⋊#95	—	2	4	no	—
28	C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
29	#10597275	—	4	2	no	—
30	#10597274	—	4	2	no	—
31	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
32	C₄×C₁₉⋊C₈	C₂×C₄	1	4	yes	—
33	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₉	#41	1	2	yes	—
35	C₂²×Dic₁₉	—	4	2	no	—
36	M₁₆×C₁₉	—	4	2	no	—
37	M₁₆×C₁₉	—	4	2	no	—
38	C₂×C₈×C₁₉	—	4	2	no	—
39	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₁₉	D₈	1	2	yes	—
44	C₂×C₈×C₁₉	SD₁₆	1	2	yes	—
45	C₂×C₈×C₁₉	SD₁₆	1	2	yes	—
46	C₂×C₈×C₁₉	Q₁₆	1	2	yes	—
47	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
48	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
49	C₄²×C₁₉	#41	1	2	yes	—
50	C₄⋊Dic₁₉	—	4	2	no	—
51	C₄⋊Dic₁₉	—	4	2	no	—
52	C₂×C₁₉⋊C₈	—	2	4	no	—
53	#8790	—	19	1	no	maximal, Sylow
54	C₂³×C₁₉	#124	1	1	yes	characteristic
55	C₂×C₄×C₁₉	—	4	2	no	—
56	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₁₉	#117	1	2	yes	—
58	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
59	C₂×C₄×C₁₉	#125	1	2	yes	—
60	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
61	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
62	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
63	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
64	C₂×Dic₁₉	—	8	4	no	—
65	C₂×Dic₁₉	—	8	2	no	—
66	C₈×C₁₉	—	8	2	no	—
67	C₈×C₁₉	#106	1	2	yes	—
68	C₈×C₁₉	#106	1	2	yes	—
69	C₈×C₁₉	#107	1	2	yes	—
70	C₈×C₁₉	#107	1	2	yes	—
71	C₁₉⋊C₈	—	4	4	no	—
72	#1848	D₁₉	1	1	yes	characteristic
73	#1846	—	19	1	no	—
74	#1450	—	19	1	no	—
75	#1286	—	19	4	no	—
76	C₂²×C₁₉	—	2	2	no	—
77	C₂²×C₁₉	#382	1	1	yes	characteristic
78	C₂²×C₁₉	#486	1	1	yes	characteristic
79	C₂²×C₁₉	#487	1	1	yes	characteristic
80	C₄×C₁₉	—	2	2	no	—
81	C₄×C₁₉	—	2	2	no	—
82	C₄×C₁₉	C₂×#106	1	1	yes	characteristic
83	C₄×C₁₉	C₂×#107	1	1	yes	characteristic
84	C₄×C₁₉	#486	1	1	yes	characteristic
85	C₄×C₁₉	#487	1	1	yes	characteristic
86	Dic₁₉	—	16	2	no	—
87	#431	—	19	2	no	—
88	C₂×#108	—	2	1	no	—
89	C₂×#108	—	2	1	no	—
90	C₂×#107	—	38	1	no	—
91	C₂×#106	—	38	1	no	—
92	#423	C₂×D₁₉	1	2	yes	—
93	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
94	#382	—	19	2	no	—
95	#322	—	19	2	no	—
96	C₈²	—	19	2	no	—
97	C₂×C₁₉	—	2	1	no	—
98	C₂×C₁₉	—	2	1	no	—
99	C₂×C₁₉	#1833	1	2	yes	—
100	C₂×C₁₉	#1834	1	1	yes	characteristic
101	C₂×M₁₆	—	2	2	no	—
102	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
103	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
104	C₂×C₄⋊C₄	—	38	2	no	—
105	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
106	#108	—	4	2	no	—
107	#108	—	4	2	no	—
108	#107	—	76	2	no	—
109	#106	—	76	2	no	—
110	C₄⋊C₈	—	2	4	no	—
111	C₄⋊C₈	—	38	2	no	—
112	C₂²⋊C₈	—	38	2	no	—
113	C₄×C₈	—	19	4	no	—
114	C₄×C₈	Dic₃₈	1	1	yes	characteristic
115	C₄×C₈	Dic₃₈	1	1	yes	characteristic
116	C₄×C₈	C₄×D₁₉	1	1	yes	characteristic
117	C₄×C₈	C₄×D₁₉	1	1	yes	characteristic
118	#95	—	38	4	no	—
119	C₁₉	#8790	1	1	yes	characteristic, complemented, Sylow
120	C₂²×C₄	—	76	2	no	—
121	C₂²×C₄	#64545	1	2	yes	—
122	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
123	M₁₆	—	4	2	no	—
124	M₁₆	—	4	2	no	—
125	C₂×C₈	—	4	2	no	—
126	C₂×C₈	—	38	4	no	—
127	C₂×C₈	Dic₁₉⋊C₄	1	1	yes	characteristic
128	C₂×C₈	Dic₁₉⋊C₄	1	1	yes	characteristic
129	C₂×C₈	Dic₁₉⋊C₄	1	1	yes	characteristic
130	C₂×C₈	Dic₁₉⋊C₄	1	1	yes	characteristic
131	C₂×C₈	#64525	1	2	yes	—
132	C₂×C₈	#64526	1	2	yes	—
133	C₂×C₈	#64527	1	2	yes	—
134	C₂×C₈	#64528	1	2	yes	—
135	C₄⋊C₄	—	76	2	no	—
136	C₄⋊C₄	—	76	2	no	—
137	C₄²	C₂×Dic₃₈	1	1	yes	characteristic
138	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
139	C₄²	#64543	1	2	yes	—
140	C₂³	#10597393	1	1	yes	characteristic
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	152	4	no	—
143	C₂×C₄	—	152	2	no	—
144	C₂×C₄	#10597336	1	2	yes	—
145	C₂×C₄	#10597338	1	2	yes	—
146	C₂×C₄	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
147	C₂×C₄	C₂×#64525	1	1	yes	characteristic
148	C₂×C₄	C₂×#64526	1	1	yes	characteristic
149	C₂×C₄	#10597382	1	1	yes	characteristic
150	C₂×C₄	C₂×#64527	1	1	yes	characteristic
151	C₂×C₄	C₂×#64528	1	1	yes	characteristic
152	C₈	—	8	2	no	—
153	C₈	—	76	4	no	—
154	C₈	#10597264	1	2	yes	—
155	C₈	#10597264	1	2	yes	—
156	C₈	#10597265	1	2	yes	—
157	C₈	#10597265	1	2	yes	—
158	C₂²	—	2	2	no	—
159	C₂²	#11926409	1	1	yes	characteristic
160	C₂²	#11926892	1	1	yes	characteristic
161	C₂²	#11926896	1	1	yes	characteristic
162	C₄	—	2	2	no	—
163	C₄	—	2	2	no	—
164	C₄	—	304	2	no	—
165	C₄	C₂×#10597265	1	1	yes	characteristic
166	C₄	C₂×#10597264	1	1	yes	characteristic
167	C₄	#11926890	1	1	yes	characteristic
168	C₄	#11926894	1	1	yes	characteristic
169	C₂	—	2	1	no	—
170	C₂	—	2	1	no	—
171	C₂	#14563760	1	2	yes	—
172	C₂	#14563762	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11926409
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1848	C₂
socle	C₂²×C₁₉	#382
prime radical	C₂³×C₁₉	#124
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1848	D₁₉
hypocenter	C₁₉	#8790
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1848

Complements

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

N	H ≅ G/N	type
9. #11926265	C₄	semidirect
10. C₈×C₁₉⋊C₈	C₄	semidirect
119. C₁₉	#8790	semidirect

Maximal subgroups

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

subgroup	index
#14563799	2
#14561918	2
#14561495	2
C₁₉×#1848	2
#8790	19