Subgroups — #18071893

Order 4864.

606 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#10597260	C₂	1	1	yes	maximal, characteristic
2	C₄²×C₁₉⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×#11926266	C₂	1	2	yes	maximal
4	C₂×C₈×C₁₉⋊C₈	C₂	1	2	yes	maximal
5	C₁₉×#1744	C₂	1	1	yes	maximal, characteristic
6	C₂×#10597260	C₄	1	4	yes	—
7	C₂×C₄×C₁₉⋊C₈	C₄	1	4	yes	—
8	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₁₉⋊C₈	C₂²	1	2	yes	—
10	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
11	#11926266	C₄	1	8	yes	complemented
12	C₈×C₁₉⋊C₈	C₄	1	8	yes	complemented
13	C₂×C₄×C₈×C₁₉	C₂²	1	2	yes	—
14	C₄³×C₁₉	C₂²	1	1	yes	characteristic
15	C₁₉×#336	C₄	1	4	yes	—
16	C₂²×C₈×C₁₉	C₂×C₄	1	4	yes	—
17	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
19	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
20	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
21	C₂²×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
22	C₂²×C₁₉⋊C₈	C₂×C₄	1	4	yes	—
23	C₂²×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
24	C₄×C₈×C₁₉	C₂×C₄	1	4	yes	—
25	C₄×C₈×C₁₉	C₂×C₄	1	4	yes	—
26	#10597260	C₈	1	16	yes	complemented
27	C₄×C₁₉⋊C₈	—	2	8	no	—
28	C₄×C₁₉⋊C₈	C₂×C₄	1	4	yes	—
29	C₄×C₁₉⋊C₈	C₂×C₄	1	4	yes	—
30	C₄×C₁₉⋊C₈	C₂×C₄	1	4	yes	—
31	C₄×C₁₉⋊C₈	C₂×C₄	1	4	yes	—
32	C₂²×C₄×C₁₉	C₄²	1	4	yes	—
33	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₈×C₁₉	—	2	8	no	—
37	C₂×C₈×C₁₉	C₄²	1	8	yes	—
38	C₄²×C₁₉	—	2	4	no	—
39	C₄²×C₁₉	—	2	4	no	—
40	C₄²×C₁₉	C₂×C₈	1	8	yes	—
41	C₄²×C₁₉	C₂²×C₄	1	2	yes	—
42	C₄²×C₁₉	C₂²×C₄	1	2	yes	—
43	C₂×C₁₉⋊C₈	—	2	4	no	—
44	C₂×C₁₉⋊C₈	C₄²	1	4	yes	—
45	C₂×C₁₉⋊C₈	C₄²	1	8	yes	—
46	C₂×C₁₉⋊C₈	C₄²	1	4	yes	—
47	C₂×C₁₉⋊C₈	C₂×C₈	1	16	yes	—
48	C₂×C₁₉⋊C₈	M₁₆	1	8	yes	—
49	C₈×#97	—	19	1	no	maximal, Sylow
50	C₂³×C₁₉	C₂×C₄²	1	1	yes	characteristic
51	C₂×C₄×C₁₉	—	2	4	no	—
52	C₂×C₄×C₁₉	—	2	4	no	—
53	C₂×C₄×C₁₉	C₄×C₈	1	8	yes	—
54	C₂×C₄×C₁₉	C₂×C₄²	1	2	yes	—
55	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
56	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
57	C₂×C₄×C₁₉	C₂×C₄²	1	2	yes	—
58	C₂×C₄×C₁₉	C₂²×C₈	1	4	yes	—
59	C₂×C₄×C₁₉	C₂×M₁₆	1	4	yes	—
60	C₂×C₄×C₁₉	#131	1	4	yes	—
61	C₈×C₁₉	—	2	8	no	—
62	C₁₉⋊C₈	—	2	4	no	—
63	C₁₉⋊C₈	C₄×C₈	1	16	yes	complemented
64	C₁₉⋊C₈	#97	1	8	yes	complemented
65	#1744	D₁₉	1	1	yes	characteristic
66	C₄×#97	—	19	1	no	—
67	C₄²×C₈	—	19	1	no	—
68	C₂×#322	—	19	2	no	—
69	C₂×C₈²	—	19	2	no	—
70	C₂²×C₁₉	C₄³	1	1	yes	characteristic
71	C₂²×C₁₉	C₂×C₄×C₈	1	2	yes	—
72	C₂²×C₁₉	C₄×M₁₆	1	2	yes	—
73	C₂²×C₁₉	#405	1	2	yes	—
74	C₄×C₁₉	—	2	8	no	—
75	C₄×C₁₉	C₂×C₄×C₈	1	4	yes	—
76	C₄×C₁₉	C₂×#97	1	2	yes	—
77	C₄×C₁₉	C₄×M₁₆	1	2	yes	—
78	C₄×C₁₉	#405	1	4	yes	—
79	C₂×#97	—	19	4	no	—
80	C₂×C₄×C₈	—	19	1	no	—
81	C₂×C₄×C₈	—	19	2	no	—
82	C₂×C₄×C₈	—	19	1	no	—
83	C₂×C₄×C₈	—	19	4	no	—
84	C₂×C₄×C₈	C₂×D₁₉	1	2	yes	—
85	C₄³	C₂×D₁₉	1	1	yes	characteristic
86	#336	Dic₁₉	1	4	yes	—
87	#322	—	19	8	no	—
88	C₈²	—	19	8	no	—
89	C₂×C₁₉	C₈×M₁₆	1	4	yes	—
90	C₂×C₁₉	C₄²×C₈	1	1	yes	characteristic
91	C₂×C₁₉	C₄×#97	1	1	yes	characteristic
92	C₂×C₁₉	#1726	1	1	yes	characteristic
93	C₂²×C₈	—	19	2	no	—
94	C₂²×C₈	—	19	4	no	—
95	C₂²×C₈	—	19	2	no	—
96	C₂²×C₈	C₂×Dic₁₉	1	4	yes	—
97	C₂×C₄²	C₄×D₁₉	1	2	yes	—
98	C₂×C₄²	C₄×D₁₉	1	2	yes	—
99	C₂×C₄²	C₂×Dic₁₉	1	2	yes	—
100	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
101	#97	—	19	16	no	—
102	C₄×C₈	—	19	4	no	—
103	C₄×C₈	—	19	4	no	—
104	C₄×C₈	—	19	4	no	—
105	C₄×C₈	—	19	4	no	—
106	C₄×C₈	—	38	8	no	—
107	C₄×C₈	C₄×D₁₉	1	4	yes	—
108	C₄×C₈	C₄×D₁₉	1	4	yes	—
109	C₁₉	C₈×#97	1	1	yes	characteristic, complemented, Sylow
110	C₂²×C₄	C₄×Dic₁₉	1	4	yes	—
111	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
112	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
113	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
114	C₂×C₈	—	2	8	no	—
115	C₂×C₈	—	19	8	no	—
116	C₂×C₈	—	19	4	no	—
117	C₂×C₈	—	19	16	no	—
118	C₂×C₈	—	19	8	no	—
119	C₂×C₈	—	19	4	no	—
120	C₂×C₈	—	38	4	no	—
121	C₂×C₈	C₄×Dic₁₉	1	8	yes	—
122	C₄²	—	2	4	no	—
123	C₄²	—	2	4	no	—
124	C₄²	C₈×D₁₉	1	8	yes	—
125	C₄²	C₂×C₄×D₁₉	1	2	yes	—
126	C₄²	C₂×C₄×D₁₉	1	2	yes	—
127	C₂³	C₂×C₄×Dic₁₉	1	1	yes	characteristic
128	C₂×C₄	—	2	4	no	—
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	C₈×Dic₁₉	1	8	yes	—
131	C₂×C₄	C₄²×D₁₉	1	2	yes	—
132	C₂×C₄	C₄²×D₁₉	1	1	yes	characteristic
133	C₂×C₄	C₄²×D₁₉	1	1	yes	characteristic
134	C₂×C₄	C₂×C₈×D₁₉	1	4	yes	—
135	C₂×C₄	M₁₆×D₁₉	1	4	yes	—
136	C₂×C₄	C₂×C₄×Dic₁₉	1	2	yes	—
137	C₂×C₄	#10597394	1	4	yes	—
138	C₈	—	2	8	no	—
139	C₈	—	19	16	no	—
140	C₈	—	19	8	no	—
141	C₈	—	38	4	no	—
142	C₂²	C₄²×Dic₁₉	1	1	yes	characteristic
143	C₂²	C₂×C₈×Dic₁₉	1	2	yes	—
144	C₂²	M₁₆×Dic₁₉	1	2	yes	—
145	C₂²	#11926500	1	2	yes	—
146	C₄	—	2	8	no	—
147	C₄	C₄×C₈×D₁₉	1	4	yes	—
148	C₄	#97×D₁₉	1	2	yes	—
149	C₄	#11926496	1	2	yes	—
150	C₄	#11926500	1	4	yes	—
151	C₂	#14561916	1	4	yes	—
152	C₂	C₄×C₈×Dic₁₉	1	1	yes	characteristic
153	C₂	#97×Dic₁₉	1	1	yes	characteristic
154	C₂	#14563322	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₁₉
commutator	C₂×C₁₉	C₄²×C₈
Fitting	C₁₉×#1744	C₂
socle, prime radical	C₂³×C₁₉	C₂×C₄²
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1744	D₁₉
hypocenter	C₁₉	C₈×#97
LCS	G ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄² ◁ #1744

Complements

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

N	H ≅ G/N	type
11. #11926266	C₄	semidirect
12. C₈×C₁₉⋊C₈	C₄	semidirect
26. #10597260	C₈	semidirect
63. C₁₉⋊C₈	C₄×C₈	semidirect
64. C₁₉⋊C₈	#97	semidirect
109. C₁₉	C₈×#97	semidirect

Maximal subgroups

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

subgroup	index
C₄×#10597260	2
C₄²×C₁₉⋊C₈	2
C₂×#11926266	2
C₂×C₈×C₁₉⋊C₈	2
C₁₉×#1744	2
C₈×#97	19