Subgroups — #18069408

Order 4864.

242 conjugacy classes of subgroups.

The subgroup lattice has 184 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	C₂×#11926297	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₉×#335	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₁₉⋊C₁₆	C₄	1	2	yes	—
5	C₂×C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
6	C₄×C₁₉⋊C₁₆	—	2	1	no	—
7	C₄×C₁₉⋊C₁₆	—	2	1	no	—
8	#11926297	—	2	1	no	—
9	#11926297	C₄	1	2	yes	—
10	C₂×C₁₉×C₄⋊C₈	—	2	1	no	—
11	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
12	C₁₉×#335	—	2	1	no	—
13	C₁₉×#335	C₄	1	2	yes	—
14	C₂²×C₈×C₁₉	—	4	1	no	—
15	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₁₉	Q₈	1	1	yes	characteristic
17	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
18	C₂²×C₁₉⋊C₈	—	4	1	no	—
19	C₁₉×C₄⋊C₈	—	2	2	no	—
20	C₁₉×C₄⋊C₈	—	4	1	no	—
21	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
22	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
23	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
24	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
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)	—	2	2	no	—
29	C₄⋊(C₁₉⋊C₈)	—	4	1	no	—
30	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₉	D₈	1	1	yes	characteristic
32	C₂²×C₄×C₁₉	Q₁₆	1	1	yes	characteristic
33	C₂×C₈×C₁₉	—	2	2	no	—
34	C₂×C₈×C₁₉	—	4	2	no	—
35	C₂×C₈×C₁₉	—	8	1	no	—
36	C₂×C₈×C₁₉	—	8	1	no	—
37	C₂×C₈×C₁₉	C₄²	1	1	yes	characteristic
38	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₁₉	C₂×C₈	1	2	yes	—
42	C₂×C₈×C₁₉	M₁₆	1	2	yes	—
43	C₄²×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
44	C₄²×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
45	C₄²×C₁₉	SD₁₆	1	1	yes	characteristic
46	C₄²×C₁₉	SD₁₆	1	1	yes	characteristic
47	C₂×C₁₉⋊C₈	—	4	2	no	—
48	C₂×C₁₉⋊C₈	—	8	1	no	—
49	C₂×C₁₉⋊C₈	—	8	1	no	—
50	C₁₉⋊C₁₆	—	2	4	no	—
51	#7154	—	19	1	no	maximal, Sylow
52	C₂³×C₁₉	#107	1	1	yes	characteristic
53	C₂×C₄×C₁₉	—	2	2	no	—
54	C₂×C₄×C₁₉	#95	1	1	yes	characteristic
55	C₂×C₄×C₁₉	#102	1	1	yes	characteristic
56	C₂×C₄×C₁₉	#102	1	1	yes	characteristic
57	C₂×C₄×C₁₉	#103	1	1	yes	characteristic
58	C₂×C₄×C₁₉	#103	1	1	yes	characteristic
59	C₂×C₄×C₁₉	C₄⋊C₈	1	2	yes	—
60	C₂×C₄×C₁₉	#106	1	1	yes	characteristic
61	C₂×C₄×C₁₉	#108	1	2	yes	—
62	C₂×C₄×C₁₉	D₁₆	1	1	yes	characteristic
63	C₂×C₄×C₁₉	SD₃₂	1	1	yes	characteristic
64	C₂×C₄×C₁₉	SD₃₂	1	1	yes	characteristic
65	C₂×C₄×C₁₉	Q₃₂	1	1	yes	characteristic
66	C₈×C₁₉	—	2	1	no	—
67	C₈×C₁₉	—	2	1	no	—
68	C₈×C₁₉	—	8	2	no	—
69	C₈×C₁₉	C₄×C₈	1	1	yes	characteristic
70	C₈×C₁₉	#97	1	1	yes	characteristic
71	C₈×C₁₉	C₂²⋊C₈	1	2	yes	—
72	C₁₉⋊C₈	—	8	2	no	—
73	C₂×C₄×C₁₆	—	19	1	no	—
74	C₂×#335	—	19	1	no	—
75	C₂×#335	D₁₉	1	1	yes	characteristic
76	C₂²×C₁₉	#335	1	2	yes	—
77	C₂²×C₁₉	#340	1	1	yes	characteristic
78	C₂²×C₁₉	#366	1	1	yes	characteristic
79	C₂²×C₁₉	#367	1	1	yes	characteristic
80	C₂²×C₁₉	#368	1	2	yes	—
81	C₄×C₁₉	—	2	1	no	—
82	C₄×C₁₉	—	2	1	no	—
83	C₄×C₁₉	#334	1	2	yes	—
84	C₄×C₁₉	#336	1	1	yes	characteristic
85	C₄×C₁₉	#341	1	1	yes	characteristic
86	C₄×C₁₉	#357	1	1	yes	characteristic
87	C₄×C₁₉	#357	1	1	yes	characteristic
88	C₄×C₁₉	#358	1	1	yes	characteristic
89	C₄×C₁₉	#358	1	1	yes	characteristic
90	C₂²×C₁₆	—	19	2	no	—
91	C₂×C₄⋊C₈	—	2	1	no	—
92	C₂×C₄⋊C₈	—	38	1	no	—
93	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
94	C₄×C₁₆	—	38	1	no	—
95	C₄×C₁₆	—	38	1	no	—
96	#335	—	2	1	no	—
97	#335	—	19	2	no	—
98	#335	—	38	1	no	—
99	#335	Dic₁₉	1	2	yes	—
100	C₂×C₁₉	#1277	1	1	yes	characteristic
101	C₂×C₁₉	#1371	1	2	yes	—
102	C₂×C₁₉	#1372	1	2	yes	—
103	C₂×C₁₉	#1380	1	1	yes	characteristic
104	C₂×C₁₉	#1381	1	1	yes	characteristic
105	C₂²×C₈	—	4	1	no	—
106	C₂²×C₈	—	76	1	no	—
107	C₂²×C₈	Dic₃₈	1	1	yes	characteristic
108	C₂²×C₈	C₄×D₁₉	1	1	yes	characteristic
109	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
110	C₂×C₁₆	—	19	4	no	—
111	C₂×C₁₆	—	38	2	no	—
112	C₂×C₁₆	—	38	2	no	—
113	C₄⋊C₈	—	2	2	no	—
114	C₄⋊C₈	—	4	1	no	—
115	C₄⋊C₈	—	38	2	no	—
116	C₄⋊C₈	—	76	1	no	—
117	C₄×C₈	C₄×D₁₉	1	1	yes	characteristic
118	C₄×C₈	D₇₆	1	1	yes	characteristic
119	C₄×C₈	C₂×Dic₁₉	1	1	yes	characteristic
120	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
121	C₁₉	#7154	1	1	yes	characteristic, complemented, Sylow
122	C₂²×C₄	Dic₁₉⋊C₄	1	1	yes	characteristic
123	C₂²×C₄	#64525	1	1	yes	characteristic
124	C₂²×C₄	#64528	1	1	yes	characteristic
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	4	2	no	—
127	C₂×C₈	—	8	1	no	—
128	C₂×C₈	—	8	1	no	—
129	C₂×C₈	—	76	2	no	—
130	C₂×C₈	—	152	1	no	—
131	C₂×C₈	—	152	1	no	—
132	C₂×C₈	C₈×D₁₉	1	2	yes	—
133	C₂×C₈	#64515	1	2	yes	—
134	C₂×C₈	C₄×Dic₁₉	1	1	yes	characteristic
135	C₂×C₈	Dic₁₉⋊C₄	1	1	yes	characteristic
136	C₂×C₈	C₄⋊Dic₁₉	1	1	yes	characteristic
137	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
138	C₄²	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
139	C₄²	#64526	1	1	yes	characteristic
140	C₄²	#64527	1	1	yes	characteristic
141	C₄²	C₂²⋊Dic₁₉	1	1	yes	characteristic
142	C₁₆	—	38	4	no	—
143	C₂³	#10597264	1	1	yes	characteristic
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	#10597265	1	1	yes	characteristic
146	C₂×C₄	#10597266	1	1	yes	characteristic
147	C₂×C₄	#10597267	1	1	yes	characteristic
148	C₂×C₄	Dic₁₉⋊C₈	1	2	yes	—
149	C₂×C₄	#10597279	1	2	yes	—
150	C₂×C₄	#10597283	1	1	yes	characteristic
151	C₂×C₄	#10597284	1	1	yes	characteristic
152	C₂×C₄	#10597285	1	1	yes	characteristic
153	C₂×C₄	#10597286	1	1	yes	characteristic
154	C₂×C₄	C₁₉⋊#95	1	1	yes	characteristic
155	C₂×C₄	#10597289	1	1	yes	characteristic
156	C₂×C₄	#10597292	1	1	yes	characteristic
157	C₈	—	2	1	no	—
158	C₈	—	2	1	no	—
159	C₈	—	8	2	no	—
160	C₈	—	152	2	no	—
161	C₈	C₈×Dic₁₉	1	1	yes	characteristic
162	C₈	#10597272	1	1	yes	characteristic
163	C₈	(C₂×D₁₉)⋊C₈	1	2	yes	—
164	C₂²	#11926296	1	2	yes	—
165	C₂²	#11926306	1	1	yes	characteristic
166	C₂²	#11926369	1	1	yes	characteristic
167	C₂²	#11926371	1	1	yes	characteristic
168	C₂²	#11926373	1	2	yes	—
169	C₄	—	2	1	no	—
170	C₄	—	2	1	no	—
171	C₄	#11926294	1	2	yes	—
172	C₄	#11926298	1	1	yes	characteristic
173	C₄	#11926308	1	1	yes	characteristic
174	C₄	#11926344	1	1	yes	characteristic
175	C₄	#11926345	1	1	yes	characteristic
176	C₄	#11926347	1	1	yes	characteristic
177	C₄	#11926348	1	1	yes	characteristic
178	C₂	#14561469	1	1	yes	characteristic
179	C₂	#14561706	1	2	yes	—
180	C₂	#14561708	1	2	yes	—
181	C₂	#14561726	1	1	yes	characteristic
182	C₂	#14561728	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#64525
commutator	C₈×C₁₉	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₁₉
Fitting	C₂×C₁₉×#335	C₂
socle, prime radical	C₂³×C₁₉	#107
cosocle	C₂×C₄×C₈×C₁₉	C₂²
hypercenter	C₂×#335	D₁₉
hypocenter	C₁₉	#7154
LCS	G ▷ C₈×C₁₉ ▷ C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
27. C₂×C₁₉⋊C₁₆	C₈	semidirect
121. C₁₉	#7154	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
C₂×#11926297	2
C₂×C₁₉×#335	2
#7154	19