Subgroups — #18073729

Order 4864.

438 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14563488	C₂	1	1	yes	maximal, characteristic
2	C₄²×C₁₉⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×#11926294	C₂	1	2	yes	maximal
4	#14561469	C₂	1	2	yes	maximal
5	C₁₉×#1767	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
7	C₂×#10597260	—	2	1	no	—
8	C₂×C₄×C₁₉⋊C₈	C₄	1	2	yes	—
9	C₂×C₄×C₁₉⋊C₈	C₄	1	2	yes	—
10	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
12	#11926299	—	2	1	no	—
13	#11926294	C₄	1	8	yes	complemented
14	C₂×C₁₉×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₁₉×#97	—	2	1	no	—
16	C₄³×C₁₉	C₂²	1	1	yes	characteristic
17	C₁₉×#336	—	2	1	no	—
18	C₂²×C₈×C₁₉	—	2	2	no	—
19	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
20	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
21	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
22	C₂×C₄²×C₁₉	Q₈	1	2	yes	—
23	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
24	C₂²×C₁₉⋊C₈	—	2	2	no	—
25	C₂²×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
27	C₂²×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₁₉×C₄⋊C₈	—	2	2	no	—
29	C₁₉×C₄⋊C₈	C₂×C₄	1	4	yes	—
30	C₁₉×#97	—	4	2	no	—
31	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
32	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	4	yes	—
33	#10597260	—	4	2	no	—
34	C₄×C₁₉⋊C₈	—	2	2	no	—
35	C₄×C₁₉⋊C₈	—	2	4	no	—
36	C₄×C₁₉⋊C₈	—	2	4	no	—
37	C₄×C₁₉⋊C₈	C₂×C₄	1	4	yes	—
38	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
39	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
40	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₈×C₁₉	—	2	4	no	—
44	C₂×C₈×C₁₉	—	4	2	no	—
45	C₂×C₈×C₁₉	—	4	2	no	—
46	C₄²×C₁₉	—	2	4	no	—
47	C₄²×C₁₉	—	2	2	no	—
48	C₄²×C₁₉	—	2	2	no	—
49	C₄²×C₁₉	—	2	2	no	—
50	C₄²×C₁₉	SD₁₆	1	2	yes	—
51	C₄²×C₁₉	SD₁₆	1	2	yes	—
52	C₄²×C₁₉	C₂²×C₄	1	2	yes	—
53	C₄²×C₁₉	#41	1	2	yes	—
54	C₂×C₁₉⋊C₈	—	2	2	no	—
55	C₂×C₁₉⋊C₈	—	2	2	no	—
56	C₂×C₁₉⋊C₈	—	2	2	no	—
57	C₂×C₁₉⋊C₈	—	2	4	no	—
58	C₂×C₁₉⋊C₈	—	4	2	no	—
59	C₂×C₁₉⋊C₈	—	4	2	no	—
60	C₂×C₁₉⋊C₈	C₄²	1	2	yes	—
61	C₂×C₁₉⋊C₈	C₄²	1	2	yes	—
62	C₂×C₁₉⋊C₈	M₁₆	1	4	yes	—
63	C₂×C₁₉⋊C₈	M₁₆	1	4	yes	—
64	#8273	—	19	1	no	maximal, Sylow
65	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₁₉	—	2	2	no	—
67	C₂×C₄×C₁₉	—	2	2	no	—
68	C₂×C₄×C₁₉	—	2	2	no	—
69	C₂×C₄×C₁₉	—	2	2	no	—
70	C₂×C₄×C₁₉	—	2	2	no	—
71	C₂×C₄×C₁₉	#106	1	4	yes	—
72	C₂×C₄×C₁₉	#108	1	4	yes	—
73	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
74	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
75	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
76	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
77	C₂×C₄×C₁₉	C₂×M₁₆	1	2	yes	—
78	C₂×C₄×C₁₉	C₂×M₁₆	1	2	yes	—
79	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
80	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
81	C₂×C₄×C₁₉	#135	1	2	yes	—
82	C₈×C₁₉	—	4	4	no	—
83	C₁₉⋊C₈	—	2	2	no	—
84	C₁₉⋊C₈	—	2	4	no	—
85	C₁₉⋊C₈	—	4	4	no	—
86	C₁₉⋊C₈	#97	1	4	yes	complemented
87	#1767	—	19	1	no	—
88	#1767	D₁₉	1	1	yes	characteristic
89	C₄²×C₈	—	19	1	no	—
90	C₂×#334	—	19	2	no	—
91	#1277	—	19	2	no	—
92	C₂²×C₁₉	C₄×C₄⋊C₄	1	1	yes	characteristic
93	C₂²×C₁₉	#423	1	2	yes	—
94	C₂²×C₁₉	C₂×#106	1	1	yes	characteristic
95	C₂²×C₁₉	#427	1	1	yes	characteristic
96	C₂²×C₁₉	C₂×#108	1	2	yes	—
97	C₄×C₁₉	—	2	4	no	—
98	C₄×C₁₉	—	2	4	no	—
99	C₄×C₁₉	—	2	2	no	—
100	C₄×C₁₉	C₂×#97	1	1	yes	characteristic
101	C₄×C₁₉	C₄×M₁₆	1	1	yes	characteristic
102	C₄×C₁₉	#432	1	2	yes	—
103	C₄×C₁₉	C₄×SD₁₆	1	2	yes	—
104	C₄×C₁₉	C₄×SD₁₆	1	2	yes	—
105	C₂×C₄⋊C₈	—	19	2	no	—
106	C₂×C₄⋊C₈	C₂×D₁₉	1	2	yes	—
107	C₂×#97	—	2	1	no	—
108	C₂×#97	—	38	1	no	—
109	C₂×C₄×C₈	—	19	1	no	—
110	C₂×C₄×C₈	—	19	2	no	—
111	C₂×C₄×C₈	—	19	2	no	—
112	C₂×C₄×C₈	—	19	1	no	—
113	C₄³	C₂×D₁₉	1	1	yes	characteristic
114	#336	—	2	1	no	—
115	#336	—	38	1	no	—
116	#334	—	19	8	no	—
117	C₂×C₁₉	#1566	1	2	yes	—
118	C₂×C₁₉	#1568	1	2	yes	—
119	C₂×C₁₉	#1767	1	1	yes	characteristic
120	C₂×C₁₉	C₄×#106	1	1	yes	characteristic
121	C₂×C₁₉	C₄×#108	1	1	yes	characteristic
122	C₂²×C₈	—	2	2	no	—
123	C₂²×C₈	—	19	1	no	—
124	C₂²×C₈	—	19	1	no	—
125	C₂²×C₈	—	19	2	no	—
126	C₂²×C₈	—	38	2	no	—
127	C₂×C₄²	Dic₃₈	1	2	yes	—
128	C₂×C₄²	C₄×D₁₉	1	2	yes	—
129	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
130	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
131	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
132	C₄⋊C₈	—	2	2	no	—
133	C₄⋊C₈	—	19	4	no	—
134	C₄⋊C₈	—	38	2	no	—
135	C₄⋊C₈	C₄×D₁₉	1	4	yes	—
136	#97	—	4	2	no	—
137	#97	—	76	2	no	—
138	C₄×C₈	—	19	4	no	—
139	C₄×C₈	—	38	4	no	—
140	C₄×C₈	—	38	4	no	—
141	C₄×C₈	—	38	2	no	—
142	C₁₉	#8273	1	1	yes	characteristic, complemented, Sylow
143	C₂²×C₄	Dic₁₉⋊C₄	1	2	yes	—
144	C₂²×C₄	Dic₁₉⋊C₄	1	2	yes	—
145	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
146	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
147	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
148	C₂×C₈	—	2	4	no	—
149	C₂×C₈	—	4	2	no	—
150	C₂×C₈	—	4	2	no	—
151	C₂×C₈	—	19	2	no	—
152	C₂×C₈	—	19	4	no	—
153	C₂×C₈	—	19	2	no	—
154	C₂×C₈	—	19	4	no	—
155	C₂×C₈	—	38	2	no	—
156	C₂×C₈	—	38	2	no	—
157	C₂×C₈	—	38	2	no	—
158	C₂×C₈	—	38	4	no	—
159	C₂×C₈	—	76	2	no	—
160	C₂×C₈	—	76	2	no	—
161	C₄²	—	2	2	no	—
162	C₄²	—	2	4	no	—
163	C₄²	—	2	2	no	—
164	C₄²	—	2	2	no	—
165	C₄²	#64526	1	2	yes	—
166	C₄²	#64527	1	2	yes	—
167	C₄²	C₂×C₄×D₁₉	1	2	yes	—
168	C₄²	#64541	1	2	yes	—
169	C₂³	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	#10597265	1	4	yes	—
176	C₂×C₄	#10597279	1	4	yes	—
177	C₂×C₄	C₄×Dic₃₈	1	2	yes	—
178	C₂×C₄	C₄²×D₁₉	1	1	yes	characteristic
179	C₂×C₄	#10597318	1	1	yes	characteristic
180	C₂×C₄	M₁₆×D₁₉	1	2	yes	—
181	C₂×C₄	M₁₆×D₁₉	1	2	yes	—
182	C₂×C₄	C₄×C₂²⋊D₁₉	1	2	yes	—
183	C₂×C₄	C₂×#64526	1	1	yes	characteristic
184	C₂×C₄	C₂×#64527	1	1	yes	characteristic
185	C₂×C₄	#10597396	1	2	yes	—
186	C₈	—	4	4	no	—
187	C₈	—	19	4	no	—
188	C₈	—	38	4	no	—
189	C₈	—	38	2	no	—
190	C₈	—	76	4	no	—
191	C₂²	C₄×Dic₁₉⋊C₄	1	1	yes	characteristic
192	C₂²	#11926567	1	2	yes	—
193	C₂²	C₂×#10597265	1	1	yes	characteristic
194	C₂²	#11926580	1	1	yes	characteristic
195	C₂²	C₂×#10597279	1	2	yes	—
196	C₄	—	2	4	no	—
197	C₄	—	2	4	no	—
198	C₄	—	2	2	no	—
199	C₄	#97×D₁₉	1	1	yes	characteristic
200	C₄	#11926496	1	1	yes	characteristic
201	C₄	#11926599	1	2	yes	—
202	C₄	C₄×#64526	1	2	yes	—
203	C₄	C₄×#64527	1	2	yes	—
204	C₂	#14562390	1	2	yes	—
205	C₂	#14562396	1	2	yes	—
206	C₂	#14563489	1	1	yes	characteristic
207	C₂	C₄×#10597265	1	1	yes	characteristic
208	C₂	C₄×#10597279	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₁₉
commutator	C₂×C₄×C₁₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1767	C₂
socle, prime radical	C₂³×C₁₉	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1767	D₁₉
hypocenter	C₁₉	#8273
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1767

Complements

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

N	H ≅ G/N	type
13. #11926294	C₄	semidirect
86. C₁₉⋊C₈	#97	semidirect
142. C₁₉	#8273	semidirect

Maximal subgroups

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

subgroup	index
#14563488	2
C₄²×C₁₉⋊C₈	2
C₂×#11926294	2
#14561469	2
C₁₉×#1767	2
#8273	19