Subgroups — #18076314

Order 4864.

422 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14563794	C₂	1	2	yes	maximal
2	#14563763	C₂	1	1	yes	maximal, characteristic
3	#14563762	C₂	1	2	yes	maximal
4	C₂×C₈×C₁₉⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#1834	C₂	1	1	yes	maximal, characteristic
6	C₂×#10597264	—	2	4	no	—
7	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
9	#11926414	C₂²	1	2	yes	—
10	#11926307	—	2	2	no	—
11	#11926306	—	2	4	no	—
12	C₈×C₁₉⋊C₈	C₄	1	4	yes	complemented
13	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
14	C₁₉×#384	C₂²	1	2	yes	—
15	C₁₉×#340	—	2	2	no	—
16	C₂²×C₈×C₁₉	D₄	1	2	yes	—
17	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
18	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
19	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
20	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
21	C₂×C₄⋊Dic₁₉	—	2	2	no	—
22	C₂×C₄⋊Dic₁₉	—	2	2	no	—
23	C₂×C₄⋊Dic₁₉	—	2	2	no	—
24	C₂²×C₁₉⋊C₈	D₄	1	2	yes	—
25	C₂²×C₁₉⋊C₈	D₄	1	2	yes	—
26	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
27	C₄×C₈×C₁₉	Q₈	1	2	yes	—
28	#10597264	—	4	8	no	—
29	C₄×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
30	C₄×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
31	C₄×C₁₉⋊C₈	Q₈	1	2	yes	—
32	C₄×C₁₉⋊C₈	Q₈	1	2	yes	—
33	C₂²×C₄×C₁₉	—	4	2	no	—
34	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
36	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
37	C₂²×Dic₁₉	—	4	2	no	—
38	C₂×C₈×C₁₉	—	2	4	no	—
39	C₂×C₈×C₁₉	C₄⋊C₄	1	4	yes	—
40	C₄⋊C₄×C₁₉	—	4	4	no	—
41	C₄⋊C₄×C₁₉	—	4	4	no	—
42	C₄⋊C₄×C₁₉	—	4	4	no	—
43	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
44	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
45	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
46	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
47	C₄⋊Dic₁₉	—	4	4	no	—
48	C₄⋊Dic₁₉	—	4	4	no	—
49	C₄⋊Dic₁₉	—	4	4	no	—
50	C₂×C₁₉⋊C₈	—	2	4	no	—
51	C₂×C₁₉⋊C₈	C₄⋊C₄	1	4	yes	—
52	C₂×C₁₉⋊C₈	C₄⋊C₄	1	4	yes	—
53	C₂×C₁₉⋊C₈	D₈	1	4	yes	—
54	C₂×C₁₉⋊C₈	Q₁₆	1	4	yes	—
55	#8816	—	19	1	no	maximal, Sylow
56	C₂³×C₁₉	#127	1	1	yes	characteristic
57	C₂×C₄×C₁₉	—	8	2	no	—
58	C₂×C₄×C₁₉	—	8	2	no	—
59	C₂×C₄×C₁₉	—	8	2	no	—
60	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
64	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
66	C₂×C₄×C₁₉	C₂×D₈	1	2	yes	—
67	C₂×C₄×C₁₉	C₂×Q₁₆	1	2	yes	—
68	C₂×C₄×C₁₉	#135	1	4	yes	—
69	C₂×C₄×C₁₉	#135	1	4	yes	—
70	C₂×Dic₁₉	—	8	2	no	—
71	C₂×Dic₁₉	—	8	2	no	—
72	C₂×Dic₁₉	—	8	2	no	—
73	C₈×C₁₉	—	2	4	no	—
74	C₁₉⋊C₈	—	2	4	no	—
75	C₁₉⋊C₈	#107	1	8	yes	—
76	#1845	—	19	2	no	—
77	#1834	—	19	1	no	—
78	#1834	—	19	2	no	—
79	#1834	D₁₉	1	1	yes	characteristic
80	C₂×C₈²	—	19	1	no	—
81	C₂²×C₁₉	#384	1	1	yes	characteristic
82	C₂²×C₁₉	#493	1	1	yes	characteristic
83	C₂²×C₁₉	#494	1	1	yes	characteristic
84	C₂²×C₁₉	#495	1	2	yes	—
85	C₂²×C₁₉	#495	1	2	yes	—
86	C₄×C₁₉	—	16	2	no	—
87	C₄×C₁₉	C₂×#107	1	2	yes	—
88	C₄×C₁₉	#427	1	2	yes	—
89	C₄×C₁₉	#427	1	2	yes	—
90	C₄×C₁₉	#499	1	2	yes	—
91	C₄×C₁₉	#499	1	2	yes	—
92	C₄×C₁₉	#500	1	2	yes	—
93	Dic₁₉	—	16	2	no	—
94	C₂×#107	—	38	4	no	—
95	C₂×C₄×C₈	—	19	1	no	—
96	C₂×C₄×C₈	—	19	1	no	—
97	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
98	#384	—	19	2	no	—
99	#384	C₂×D₁₉	1	2	yes	—
100	#340	—	2	2	no	—
101	#340	—	38	4	no	—
102	#340	—	38	2	no	—
103	C₈²	—	19	4	no	—
104	C₂×C₁₉	#1709	1	2	yes	—
105	C₂×C₁₉	#1710	1	2	yes	—
106	C₂×C₁₉	#1845	1	1	yes	characteristic
107	C₂×C₁₉	#1846	1	1	yes	characteristic
108	C₂×C₁₉	#1846	1	1	yes	characteristic
109	C₂²×C₈	—	19	2	no	—
110	C₂²×C₈	—	19	2	no	—
111	C₂²×C₈	C₂²⋊D₁₉	1	2	yes	—
112	C₂×C₄⋊C₄	—	2	2	no	—
113	C₂×C₄⋊C₄	—	2	2	no	—
114	C₂×C₄⋊C₄	—	2	2	no	—
115	C₂×C₄⋊C₄	—	38	2	no	—
116	C₂×C₄⋊C₄	—	38	2	no	—
117	C₂×C₄⋊C₄	—	38	2	no	—
118	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
119	#107	—	76	8	no	—
120	C₄×C₈	—	19	2	no	—
121	C₄×C₈	—	19	2	no	—
122	C₄×C₈	—	19	2	no	—
123	C₄×C₈	—	19	2	no	—
124	C₄×C₈	Dic₃₈	1	2	yes	—
125	C₄×C₈	C₄×D₁₉	1	2	yes	—
126	C₁₉	#8816	1	1	yes	characteristic, complemented, Sylow
127	C₂²×C₄	—	4	2	no	—
128	C₂²×C₄	—	76	2	no	—
129	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
130	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
131	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
132	C₂×C₈	—	2	4	no	—
133	C₂×C₈	—	19	4	no	—
134	C₂×C₈	—	19	4	no	—
135	C₂×C₈	—	19	4	no	—
136	C₂×C₈	—	19	4	no	—
137	C₂×C₈	—	38	4	no	—
138	C₂×C₈	Dic₁₉⋊C₄	1	4	yes	—
139	C₄⋊C₄	—	4	4	no	—
140	C₄⋊C₄	—	4	4	no	—
141	C₄⋊C₄	—	4	4	no	—
142	C₄⋊C₄	—	76	4	no	—
143	C₄⋊C₄	—	76	4	no	—
144	C₄⋊C₄	—	76	4	no	—
145	C₄²	C₂×Dic₃₈	1	1	yes	characteristic
146	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
147	C₄²	Q₈×D₁₉	1	1	yes	characteristic
148	C₄²	Q₈×D₁₉	1	1	yes	characteristic
149	C₂³	#10597386	1	1	yes	characteristic
150	C₂×C₄	—	8	2	no	—
151	C₂×C₄	—	8	2	no	—
152	C₂×C₄	—	8	2	no	—
153	C₂×C₄	—	152	2	no	—
154	C₂×C₄	—	152	2	no	—
155	C₂×C₄	—	152	2	no	—
156	C₂×C₄	#10597334	1	1	yes	characteristic
157	C₂×C₄	#10597334	1	1	yes	characteristic
158	C₂×C₄	C₄⋊C₄×D₁₉	1	1	yes	characteristic
159	C₂×C₄	C₄⋊C₄×D₁₉	1	1	yes	characteristic
160	C₂×C₄	D₈×D₁₉	1	2	yes	—
161	C₂×C₄	#10597362	1	4	yes	—
162	C₂×C₄	Q₁₆×D₁₉	1	2	yes	—
163	C₂×C₄	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
164	C₂×C₄	#10597390	1	1	yes	characteristic
165	C₂×C₄	#10597396	1	4	yes	—
166	C₈	—	2	4	no	—
167	C₈	—	19	8	no	—
168	C₈	—	38	4	no	—
169	C₂²	#11926413	1	1	yes	characteristic
170	C₂²	#11926919	1	1	yes	characteristic
171	C₂²	#11926922	1	1	yes	characteristic
172	C₂²	#11926925	1	2	yes	—
173	C₂²	#11926926	1	2	yes	—
174	C₄	—	16	2	no	—
175	C₄	—	304	2	no	—
176	C₄	#107×D₁₉	1	2	yes	—
177	C₄	#11926579	1	2	yes	—
178	C₄	#11926580	1	2	yes	—
179	C₄	#11926943	1	2	yes	—
180	C₄	#11926944	1	2	yes	—
181	C₄	#11926948	1	2	yes	—
182	C₂	#14563239	1	2	yes	—
183	C₂	#14563243	1	2	yes	—
184	C₂	#14563793	1	1	yes	characteristic
185	C₂	#14563796	1	1	yes	characteristic
186	C₂	#14563797	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#10597386
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1834	C₂
socle, prime radical	C₂³×C₁₉	#127
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1834	D₁₉
hypocenter	C₁₉	#8816
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1834

Complements

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

N	H ≅ G/N	type
12. C₈×C₁₉⋊C₈	C₄	semidirect
126. C₁₉	#8816	semidirect

Maximal subgroups

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

subgroup	index
#14563794	2
#14563763	2
#14563762	2
C₂×C₈×C₁₉⋊C₈	2
C₁₉×#1834	2
#8816	19