Subgroups — #18076276

Order 4864.

462 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14563787	C₂	1	1	yes	maximal, characteristic
2	C₄²×C₁₉⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×#11926294	C₂	1	4	yes	maximal
4	C₁₉×#1842	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
6	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄×C₁₉⋊C₈	C₄	1	2	yes	—
8	C₂×C₄×C₁₉⋊C₈	C₄	1	1	yes	characteristic
9	C₂×C₄×C₁₉⋊C₈	C₄	1	1	yes	characteristic
10	C₂×C₄×C₁₉⋊C₈	C₂²	1	2	yes	—
11	#11926294	—	2	8	no	—
12	C₂×C₁₉×C₄⋊C₈	—	2	2	no	—
13	C₂×C₁₉×C₄⋊C₈	C₂²	1	2	yes	—
14	C₄³×C₁₉	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₁₉	—	2	2	no	—
16	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄²×C₁₉	D₄	1	2	yes	—
19	C₂×C₄²×C₁₉	Q₈	1	2	yes	—
20	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
21	C₂²×C₁₉⋊C₈	—	2	2	no	—
22	C₂²×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
23	C₂²×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
24	C₁₉×C₄⋊C₈	—	2	4	no	—
25	C₁₉×C₄⋊C₈	—	4	4	no	—
26	C₄⋊(C₁₉⋊C₈)	—	2	4	no	—
27	C₄⋊(C₁₉⋊C₈)	—	4	4	no	—
28	C₄×C₁₉⋊C₈	—	2	2	no	—
29	C₄×C₁₉⋊C₈	—	2	4	no	—
30	C₄×C₁₉⋊C₈	C₈	1	4	yes	complemented
31	C₄×C₁₉⋊C₈	D₄	1	4	yes	—
32	C₄×C₁₉⋊C₈	Q₈	1	4	yes	—
33	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
34	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
35	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₈×C₁₉	—	4	2	no	—
39	C₂×C₈×C₁₉	—	4	4	no	—
40	C₄²×C₁₉	—	2	4	no	—
41	C₄²×C₁₉	—	2	2	no	—
42	C₄²×C₁₉	C₂×C₈	1	2	yes	—
43	C₄²×C₁₉	M₁₆	1	2	yes	—
44	C₄²×C₁₉	SD₁₆	1	4	yes	—
45	C₄²×C₁₉	SD₁₆	1	4	yes	—
46	C₄²×C₁₉	C₂×D₄	1	2	yes	—
47	C₄²×C₁₉	C₂×Q₈	1	2	yes	—
48	C₂×C₁₉⋊C₈	—	2	2	no	—
49	C₂×C₁₉⋊C₈	—	2	2	no	—
50	C₂×C₁₉⋊C₈	—	4	2	no	—
51	C₂×C₁₉⋊C₈	—	4	4	no	—
52	C₂×C₁₉⋊C₈	C₄⋊C₄	1	4	yes	—
53	C₂×C₁₉⋊C₈	C₄⋊C₄	1	4	yes	—
54	C₂×C₁₉⋊C₈	C₂×C₈	1	4	yes	—
55	C₂×C₁₉⋊C₈	M₁₆	1	4	yes	—
56	#8805	—	19	1	no	maximal, Sylow
57	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₁₉	—	2	2	no	—
59	C₂×C₄×C₁₉	—	2	2	no	—
60	C₂×C₄×C₁₉	—	2	2	no	—
61	C₂×C₄×C₁₉	C₄⋊C₈	1	4	yes	—
62	C₂×C₄×C₁₉	#106	1	4	yes	—
63	C₂×C₄×C₁₉	#106	1	4	yes	—
64	C₂×C₄×C₁₉	#108	1	4	yes	—
65	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
66	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
67	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
68	C₂×C₄×C₁₉	C₄⋊Q₈	1	2	yes	—
69	C₂×C₄×C₁₉	C₂²×C₈	1	1	yes	characteristic
70	C₂×C₄×C₁₉	C₂×M₁₆	1	1	yes	characteristic
71	C₂×C₄×C₁₉	C₂×M₁₆	1	2	yes	—
72	C₂×C₄×C₁₉	C₂×SD₁₆	1	2	yes	—
73	C₂×C₄×C₁₉	C₂×SD₁₆	1	2	yes	—
74	C₈×C₁₉	—	8	2	no	—
75	C₁₉⋊C₈	—	2	4	no	—
76	C₁₉⋊C₈	—	8	2	no	—
77	C₁₉⋊C₈	C₄⋊C₈	1	8	yes	complemented
78	#1842	—	19	1	no	—
79	#1842	D₁₉	1	1	yes	characteristic
80	C₄²×C₈	—	19	1	no	—
81	C₂×#334	—	19	4	no	—
82	C₂²×C₁₉	#384	1	1	yes	characteristic
83	C₂²×C₁₉	C₂×C₄⋊C₈	1	1	yes	characteristic
84	C₂²×C₁₉	#423	1	1	yes	characteristic
85	C₂²×C₁₉	C₂×#106	1	2	yes	—
86	C₂²×C₁₉	C₂×#108	1	2	yes	—
87	C₄×C₁₉	—	2	4	no	—
88	C₄×C₁₉	—	2	2	no	—
89	C₄×C₁₉	#334	1	8	yes	complemented
90	C₄×C₁₉	C₂×C₄⋊C₈	1	2	yes	—
91	C₄×C₁₉	#423	1	2	yes	—
92	C₄×C₁₉	#492	1	1	yes	characteristic
93	C₄×C₁₉	#492	1	1	yes	characteristic
94	C₄×C₁₉	#498	1	2	yes	—
95	C₂×C₄⋊C₈	—	2	2	no	—
96	C₂×C₄⋊C₈	—	19	2	no	—
97	C₂×C₄⋊C₈	—	38	2	no	—
98	C₂×C₄⋊C₈	C₂×D₁₉	1	2	yes	—
99	C₂×C₄×C₈	—	19	1	no	—
100	C₂×C₄×C₈	—	19	2	no	—
101	C₂×C₄×C₈	—	19	2	no	—
102	C₂×C₄×C₈	—	19	1	no	—
103	C₄³	C₂×D₁₉	1	1	yes	characteristic
104	#334	—	38	8	no	—
105	C₂×C₁₉	C₂×#334	1	2	yes	—
106	C₂×C₁₉	#1566	1	2	yes	—
107	C₂×C₁₉	#1842	1	1	yes	characteristic
108	C₂×C₁₉	#1844	1	1	yes	characteristic
109	C₂×C₁₉	#1848	1	1	yes	characteristic
110	C₂²×C₈	—	2	2	no	—
111	C₂²×C₈	—	19	2	no	—
112	C₂²×C₈	—	19	2	no	—
113	C₂²×C₈	—	38	2	no	—
114	C₂×C₄²	Dic₃₈	1	2	yes	—
115	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
116	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
117	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
118	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
119	C₄⋊C₈	—	2	4	no	—
120	C₄⋊C₈	—	4	4	no	—
121	C₄⋊C₈	—	38	4	no	—
122	C₄⋊C₈	—	76	4	no	—
123	C₄×C₈	—	19	4	no	—
124	C₄×C₈	—	19	4	no	—
125	C₄×C₈	—	19	4	no	—
126	C₄×C₈	—	38	4	no	—
127	C₄×C₈	—	38	2	no	—
128	C₁₉	#8805	1	1	yes	characteristic, complemented, Sylow
129	C₂²×C₄	Dic₁₉⋊C₄	1	2	yes	—
130	C₂²×C₄	Dic₁₉⋊C₄	1	2	yes	—
131	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
132	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
133	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
134	C₂×C₈	—	4	4	no	—
135	C₂×C₈	—	4	2	no	—
136	C₂×C₈	—	19	4	no	—
137	C₂×C₈	—	19	4	no	—
138	C₂×C₈	—	19	4	no	—
139	C₂×C₈	—	19	4	no	—
140	C₂×C₈	—	38	2	no	—
141	C₂×C₈	—	38	2	no	—
142	C₂×C₈	—	76	2	no	—
143	C₂×C₈	—	76	4	no	—
144	C₄²	—	2	4	no	—
145	C₄²	—	2	2	no	—
146	C₄²	C₈×D₁₉	1	2	yes	—
147	C₄²	#64515	1	2	yes	—
148	C₄²	#64526	1	4	yes	—
149	C₄²	#64527	1	4	yes	—
150	C₄²	D₄×D₁₉	1	2	yes	—
151	C₄²	Q₈×D₁₉	1	2	yes	—
152	C₂³	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	#10597265	1	4	yes	—
157	C₂×C₄	#10597265	1	4	yes	—
158	C₂×C₄	Dic₁₉⋊C₈	1	4	yes	—
159	C₂×C₄	#10597279	1	4	yes	—
160	C₂×C₄	#10597334	1	2	yes	—
161	C₂×C₄	C₄⋊C₄×D₁₉	1	2	yes	—
162	C₂×C₄	C₂×C₈×D₁₉	1	1	yes	characteristic
163	C₂×C₄	C₂×#64515	1	1	yes	characteristic
164	C₂×C₄	M₁₆×D₁₉	1	2	yes	—
165	C₂×C₄	C₂×#64526	1	2	yes	—
166	C₂×C₄	#10597386	1	1	yes	characteristic
167	C₂×C₄	C₂×#64527	1	2	yes	—
168	C₂×C₄	#10597390	1	1	yes	characteristic
169	C₈	—	8	2	no	—
170	C₈	—	19	8	no	—
171	C₈	—	38	4	no	—
172	C₈	—	152	2	no	—
173	C₂²	#11926413	1	1	yes	characteristic
174	C₂²	C₂×Dic₁₉⋊C₈	1	1	yes	characteristic
175	C₂²	#11926567	1	1	yes	characteristic
176	C₂²	C₂×#10597265	1	2	yes	—
177	C₂²	C₂×#10597279	1	2	yes	—
178	C₄	—	2	4	no	—
179	C₄	—	2	2	no	—
180	C₄	#11926294	1	8	yes	complemented
181	C₄	C₄⋊C₈×D₁₉	1	2	yes	—
182	C₄	#11926566	1	2	yes	—
183	C₄	#11926916	1	1	yes	characteristic
184	C₄	#11926918	1	1	yes	characteristic
185	C₄	#11926940	1	2	yes	—
186	C₂	C₂×#11926294	1	2	yes	—
187	C₂	#14562390	1	2	yes	—
188	C₂	#14563786	1	1	yes	characteristic
189	C₂	#14563791	1	1	yes	characteristic
190	C₂	#14563806	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₁₉
commutator	C₂×C₄×C₁₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1842	C₂
socle, prime radical	C₂³×C₁₉	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1842	D₁₉
hypocenter	C₁₉	#8805
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1842

Complements

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

N	H ≅ G/N	type
30. C₄×C₁₉⋊C₈	C₈	semidirect
77. C₁₉⋊C₈	C₄⋊C₈	semidirect
89. C₄×C₁₉	#334	semidirect
128. C₁₉	#8805	semidirect
180. C₄	#11926294	semidirect

Maximal subgroups

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

subgroup	index
#14563787	2
C₄²×C₁₉⋊C₈	2
C₂×#11926294	2
C₁₉×#1842	2
#8805	19