Subgroups — #18069379

Order 4864.

254 conjugacy classes of subgroups.

The subgroup lattice has 152 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₁₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	#14561918	C₂	1	1	yes	maximal, characteristic
3	C₄×C₁₉×M₃₂	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₁₉⋊C₁₆	C₄	1	2	yes	—
5	#11926597	C₄	1	2	yes	—
6	C₄×C₁₉⋊C₁₆	—	2	1	no	—
7	C₄×C₁₉⋊C₁₆	—	2	1	no	—
8	#11926265	—	2	1	no	—
9	C₈×C₁₉⋊C₈	—	2	1	no	—
10	C₂×C₁₉×M₃₂	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
12	C₁₉×#346	—	2	1	no	—
13	C₄×C₁₆×C₁₉	—	2	1	no	—
14	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
17	C₁₉×M₃₂	—	2	2	no	—
18	C₁₉×M₃₂	C₈	1	4	yes	complemented
19	C₂×C₁₆×C₁₉	—	2	2	no	—
20	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
21	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
22	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
23	C₄×C₈×C₁₉	Q₈	1	1	yes	characteristic
24	C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
25	C₂×C₁₉⋊C₁₆	—	2	2	no	—
26	C₂×C₁₉⋊C₁₆	—	2	2	no	—
27	C₂×C₁₉⋊C₁₆	C₈	1	4	yes	complemented
28	C₄⋊(C₁₉⋊C₈)	C₈	1	4	yes	—
29	C₄×C₁₉⋊C₈	—	2	2	no	—
30	C₂²×C₄×C₁₉	C₄²	1	1	yes	characteristic
31	C₂²×C₄×C₁₉	M₁₆	1	2	yes	—
32	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₈×C₁₉	C₂×C₈	1	2	yes	—
37	C₂×C₈×C₁₉	C₂×C₈	1	2	yes	—
38	C₂×C₈×C₁₉	M₁₆	1	2	yes	—
39	C₂×C₈×C₁₉	M₁₆	1	2	yes	—
40	C₄²×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
41	C₄²×C₁₉	C₄⋊C₄	1	1	yes	characteristic
42	C₄²×C₁₉	C₂×C₈	1	2	yes	—
43	C₂×C₁₉⋊C₈	—	2	4	no	—
44	C₁₆×C₁₉	—	2	4	no	—
45	C₁₉⋊C₁₆	—	2	4	no	—
46	#7141	—	19	1	no	maximal, Sylow
47	C₂³×C₁₉	#97	1	1	yes	characteristic
48	C₂×C₄×C₁₉	—	2	4	no	—
49	C₂×C₄×C₁₉	#95	1	1	yes	characteristic
50	C₂×C₄×C₁₉	C₄×C₈	1	1	yes	characteristic
51	C₂×C₄×C₁₉	C₄×C₈	1	1	yes	characteristic
52	C₂×C₄×C₁₉	C₄×C₈	1	1	yes	characteristic
53	C₂×C₄×C₁₉	#97	1	1	yes	characteristic
54	C₂×C₄×C₁₉	#97	1	1	yes	characteristic
55	C₂×C₄×C₁₉	C₂²⋊C₈	1	2	yes	—
56	C₂×C₄×C₁₉	C₄⋊C₈	1	2	yes	—
57	C₈×C₁₉	C₂²⋊C₈	1	2	yes	—
58	C₈×C₁₉	C₂²⋊C₈	1	2	yes	—
59	C₈×C₁₉	C₄⋊C₈	1	2	yes	—
60	C₈×C₁₉	C₄⋊C₈	1	2	yes	—
61	C₁₉⋊C₈	—	4	4	no	—
62	C₄×M₃₂	D₁₉	1	1	yes	characteristic
63	C₂×C₄×C₁₆	—	19	1	no	—
64	#1450	—	19	1	no	—
65	C₂²×C₁₉	—	2	2	no	—
66	C₂²×C₁₉	#322	1	1	yes	characteristic
67	C₂²×C₁₉	#322	1	1	yes	characteristic
68	C₂²×C₁₉	#336	1	1	yes	characteristic
69	C₄×C₁₉	—	2	2	no	—
70	C₄×C₁₉	C₈²	1	1	yes	characteristic
71	C₄×C₁₉	#322	1	1	yes	characteristic
72	C₄×C₁₉	#336	1	1	yes	characteristic
73	C₄×C₁₉	#336	1	1	yes	characteristic
74	C₄×C₁₉	#350	1	4	yes	—
75	C₂×M₃₂	Dic₁₉	1	2	yes	—
76	C₂²×C₁₆	—	19	2	no	—
77	#431	—	19	2	no	—
78	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
79	#346	—	2	1	no	—
80	C₄×C₁₆	—	2	1	no	—
81	C₄×C₁₆	—	38	1	no	—
82	C₄×C₁₆	—	38	1	no	—
83	#322	—	38	1	no	—
84	C₈²	—	38	1	no	—
85	C₂×C₁₉	—	2	1	no	—
86	C₂×C₁₉	—	2	1	no	—
87	C₂×C₁₉	#1273	1	1	yes	characteristic
88	C₂×C₁₉	#1379	1	2	yes	—
89	C₂²×C₈	C₄×D₁₉	1	1	yes	characteristic
90	C₂²×C₈	C₂×Dic₁₉	1	1	yes	characteristic
91	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
92	M₃₂	—	2	2	no	—
93	M₃₂	C₁₉⋊C₈	1	4	yes	complemented
94	C₂×C₁₆	—	2	2	no	—
95	C₂×C₁₆	—	19	4	no	—
96	C₂×C₁₆	—	38	2	no	—
97	C₂×C₁₆	—	38	2	no	—
98	C₄⋊C₈	—	19	4	no	—
99	C₂²⋊C₈	—	38	2	no	—
100	C₄×C₈	—	38	2	no	—
101	C₄×C₈	Dic₃₈	1	1	yes	characteristic
102	C₄×C₈	D₇₆	1	1	yes	characteristic
103	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
104	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
105	C₁₉	#7141	1	1	yes	characteristic, complemented, Sylow
106	C₂²×C₄	#64515	1	2	yes	—
107	C₂²×C₄	C₄×Dic₁₉	1	1	yes	characteristic
108	C₂×C₈	—	38	4	no	—
109	C₂×C₈	C₈×D₁₉	1	2	yes	—
110	C₂×C₈	#64515	1	2	yes	—
111	C₂×C₈	C₂×C₁₉⋊C₈	1	2	yes	—
112	C₂×C₈	#64520	1	2	yes	—
113	C₂×C₈	Dic₁₉⋊C₄	1	1	yes	characteristic
114	C₂×C₈	C₄⋊Dic₁₉	1	1	yes	characteristic
115	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
116	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
117	C₄²	C₈×D₁₉	1	2	yes	—
118	C₄²	Dic₁₉⋊C₄	1	1	yes	characteristic
119	C₄²	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
120	C₁₆	—	2	4	no	—
121	C₁₆	—	38	4	no	—
122	C₂³	#10597272	1	1	yes	characteristic
123	C₂×C₄	—	2	4	no	—
124	C₂×C₄	C₄×C₁₉⋊C₈	1	1	yes	characteristic
125	C₂×C₄	#10597260	1	1	yes	characteristic
126	C₂×C₄	C₈×Dic₁₉	1	1	yes	characteristic
127	C₂×C₄	C₈×Dic₁₉	1	1	yes	characteristic
128	C₂×C₄	Dic₁₉⋊C₈	1	2	yes	—
129	C₂×C₄	#10597272	1	1	yes	characteristic
130	C₂×C₄	(C₂×D₁₉)⋊C₈	1	2	yes	—
131	C₂×C₄	C₁₉⋊#95	1	1	yes	characteristic
132	C₈	—	76	4	no	—
133	C₈	C₄⋊(C₁₉⋊C₈)	1	2	yes	—
134	C₈	Dic₁₉⋊C₈	1	2	yes	—
135	C₈	(C₂×D₁₉)⋊C₈	1	2	yes	—
136	C₈	C₂²⋊(C₁₉⋊C₈)	1	2	yes	—
137	C₂²	—	2	2	no	—
138	C₂²	#11926265	1	1	yes	characteristic
139	C₂²	#11926266	1	1	yes	characteristic
140	C₂²	#11926298	1	1	yes	characteristic
141	C₄	—	2	2	no	—
142	C₄	C₈×C₁₉⋊C₈	1	1	yes	characteristic
143	C₄	#11926266	1	1	yes	characteristic
144	C₄	#11926298	1	1	yes	characteristic
145	C₄	#11926299	1	1	yes	characteristic
146	C₄	#11926329	1	4	yes	—
147	C₂	—	2	1	no	—
148	C₂	—	2	1	no	—
149	C₂	C₁₉⋊#1273	1	1	yes	characteristic
150	C₂	#14561725	1	2	yes	—

	subgroup	G/H
center	C₄×C₈	C₂²⋊D₁₉
commutator	C₄×C₁₉	C₈²
Frattini	C₂×C₄×C₈	C₂×D₁₉
Fitting	C₄×C₁₉×M₃₂	C₂
socle	C₂²×C₁₉	#336
prime radical	C₂³×C₁₉	#97
cosocle	C₂×C₄×C₈×C₁₉	C₂²
hypercenter	C₄×M₃₂	D₁₉
hypocenter	C₁₉	#7141
LCS	G ▷ C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
18. C₁₉×M₃₂	C₈	semidirect
27. C₂×C₁₉⋊C₁₆	C₈	semidirect
93. M₃₂	C₁₉⋊C₈	semidirect
105. C₁₉	#7141	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₁₉⋊C₁₆	2
#14561918	2
C₄×C₁₉×M₃₂	2
#7141	19