Subgroups — #18087723

Order 4864.

1142 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₄⋊Dic₁₉	C₂	1	2	yes	maximal
2	C₂×#11926307	C₂	1	4	yes	maximal
3	C₂²×C₁₉×#97	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄⋊Dic₁₉	C₂²	1	4	yes	—
5	C₂²×C₄×Dic₁₉	—	2	2	no	—
6	C₄×C₄⋊Dic₁₉	—	2	4	no	—
7	C₄×C₄⋊Dic₁₉	C₄	1	8	yes	complemented
8	C₂×C₁₉⋊#95	—	2	2	no	—
9	#11926307	C₄	1	16	yes	complemented
10	C₂³×C₈×C₁₉	C₂²	1	2	yes	—
11	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
12	C₂×C₁₉×#97	—	2	4	no	—
13	C₂×C₁₉×#97	C₄	1	4	yes	complemented
14	C₂³×C₄×C₁₉	D₄	1	2	yes	—
15	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
16	C₂³×Dic₁₉	—	2	4	no	—
17	C₂²×C₈×C₁₉	—	2	4	no	—
18	C₂²×C₈×C₁₉	—	2	4	no	—
19	C₂²×C₈×C₁₉	C₂×C₄	1	8	yes	—
20	C₂²×C₈×C₁₉	C₂×C₄	1	4	yes	—
21	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₁₉	C₂×C₄	1	4	yes	—
23	C₂×C₄²×C₁₉	D₄	1	4	yes	—
24	C₂×C₄²×C₁₉	Q₈	1	2	yes	—
25	C₂×C₄⋊Dic₁₉	—	2	4	no	—
26	C₂×C₄⋊Dic₁₉	—	2	8	no	—
27	C₂×C₄⋊Dic₁₉	C₂×C₄	1	16	yes	—
28	C₂×C₄⋊Dic₁₉	C₂×C₄	1	8	yes	—
29	C₂×C₄×Dic₁₉	—	2	8	no	—
30	C₂×C₄×Dic₁₉	—	4	8	no	—
31	C₁₉×#97	—	2	8	no	—
32	C₁₉⋊#95	—	2	8	no	—
33	C₂⁴×C₁₉	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₉	—	2	4	no	—
35	C₂²×C₄×C₁₉	—	2	4	no	—
36	C₂²×C₄×C₁₉	C₂²⋊C₄	1	4	yes	—
37	C₂²×C₄×C₁₉	C₂²⋊C₄	1	4	yes	—
38	C₂²×C₄×C₁₉	C₄⋊C₄	1	4	yes	—
39	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
40	C₂²×C₄×C₁₉	C₂²×C₄	1	4	yes	—
41	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
43	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
44	C₂²×C₄×C₁₉	#41	1	4	yes	—
45	C₂²×Dic₁₉	—	2	8	no	—
46	C₂²×Dic₁₉	—	2	16	no	—
47	C₂²×Dic₁₉	—	4	4	no	—
48	C₂²×Dic₁₉	—	4	8	no	—
49	C₂²×Dic₁₉	—	4	4	no	—
50	C₂×C₈×C₁₉	—	2	4	no	—
51	C₂×C₈×C₁₉	—	2	4	no	—
52	C₂×C₈×C₁₉	—	2	16	no	—
53	C₂×C₈×C₁₉	C₄²	1	8	yes	complemented
54	C₄²×C₁₉	C₄²	1	4	yes	—
55	C₄²×C₁₉	C₂²⋊C₄	1	4	yes	—
56	C₄²×C₁₉	C₄⋊C₄	1	8	yes	—
57	C₄⋊Dic₁₉	—	2	8	no	—
58	C₄⋊Dic₁₉	—	2	16	no	—
59	C₄⋊Dic₁₉	C₄²	1	16	yes	complemented
60	C₄×Dic₁₉	—	4	16	no	—
61	#10703	—	19	1	no	maximal, Sylow
62	C₂³×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
63	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂³×C₁₉	C₄×D₄	1	4	yes	—
65	C₂³×C₁₉	AGL(1,ℤ/8)	1	4	yes	—
66	C₂³×C₁₉	#137	1	4	yes	—
67	C₂×C₄×C₁₉	—	2	8	no	—
68	C₂×C₄×C₁₉	—	2	4	no	—
69	C₂×C₄×C₁₉	—	2	4	no	—
70	C₂×C₄×C₁₉	—	2	8	no	—
71	C₂×C₄×C₁₉	#95	1	8	yes	—
72	C₂×C₄×C₁₉	C₂×C₄²	1	4	yes	—
73	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
74	C₂×C₄×C₁₉	C₂×C₄²	1	2	yes	—
75	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
77	C₂×C₄×C₁₉	#117	1	4	yes	—
78	C₂×C₄×C₁₉	#117	1	2	yes	—
79	C₂×C₄×C₁₉	C₄×D₄	1	8	yes	—
80	C₂×C₄×C₁₉	C₄×Q₈	1	4	yes	—
81	C₂×Dic₁₉	—	2	16	no	—
82	C₂×Dic₁₉	—	4	8	no	—
83	C₂×Dic₁₉	—	4	8	no	—
84	C₂×Dic₁₉	—	4	16	no	—
85	C₂×Dic₁₉	—	4	16	no	—
86	C₈×C₁₉	—	2	8	no	—
87	C₂²×#97	D₁₉	1	1	yes	characteristic
88	C₂×C₄×C₄⋊C₄	—	19	2	no	—
89	C₂×#340	—	19	4	no	—
90	C₂²×C₁₉	C₂×#95	1	1	yes	characteristic
91	C₂²×C₁₉	C₄×C₂²⋊C₄	1	4	yes	—
92	C₂²×C₁₉	C₄×C₄⋊C₄	1	2	yes	—
93	C₂²×C₁₉	#417	1	4	yes	—
94	C₂²×C₁₉	#418	1	2	yes	—
95	C₂²×C₁₉	#419	1	2	yes	—
96	C₂²×C₁₉	#428	1	4	yes	—
97	C₂²×C₁₉	#440	1	8	yes	—
98	C₂²×C₁₉	#441	1	4	yes	—
99	C₂²×C₁₉	#442	1	4	yes	—
100	C₄×C₁₉	—	2	8	no	—
101	C₄×C₁₉	C₄³	1	1	yes	characteristic
102	C₄×C₁₉	#376	1	1	yes	characteristic
103	C₄×C₁₉	C₄×C₂²⋊C₄	1	2	yes	—
104	C₄×C₁₉	C₄×C₄⋊C₄	1	4	yes	—
105	Dic₁₉	—	4	16	no	—
106	C₂³×C₈	C₂×D₁₉	1	2	yes	—
107	C₂²×C₄⋊C₄	—	19	4	no	—
108	C₂²×C₄²	—	38	2	no	—
109	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
110	C₂×#97	—	2	4	no	—
111	C₂×#97	Dic₁₉	1	4	yes	complemented
112	C₄×C₄⋊C₄	—	19	8	no	—
113	C₄×C₄⋊C₄	—	38	4	no	—
114	C₂×#95	—	38	2	no	—
115	#340	—	19	16	no	—
116	C₂×C₁₉	C₄×#95	1	1	yes	characteristic
117	C₂×C₁₉	#1736	1	2	yes	—
118	C₂×C₁₉	#1762	1	4	yes	—
119	C₂×C₁₉	#1763	1	4	yes	—
120	C₂×C₁₉	#1776	1	4	yes	—
121	C₂³×C₄	—	38	4	no	—
122	C₂³×C₄	D₇₆	1	2	yes	—
123	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
124	C₂²×C₈	—	2	4	no	—
125	C₂²×C₈	—	2	4	no	—
126	C₂²×C₈	C₄×D₁₉	1	8	yes	—
127	C₂²×C₈	C₂×Dic₁₉	1	4	yes	—
128	C₂×C₄⋊C₄	—	19	16	no	—
129	C₂×C₄⋊C₄	—	19	8	no	—
130	C₂×C₄⋊C₄	—	38	8	no	—
131	C₂×C₄⋊C₄	—	38	4	no	—
132	C₂×C₄²	—	38	8	no	—
133	C₂×C₄²	—	76	8	no	—
134	C₂×C₄²	Dic₃₈	1	2	yes	—
135	C₂×C₄²	C₄×D₁₉	1	4	yes	—
136	C₂×C₄²	C₂×Dic₁₉	1	2	yes	—
137	C₂×C₄²	C₂²⋊D₁₉	1	4	yes	—
138	#97	—	2	8	no	—
139	#95	—	38	8	no	—
140	C₁₉	#10703	1	1	yes	characteristic, complemented, Sylow
141	C₂⁴	C₂×D₇₆	1	1	yes	characteristic
142	C₂²×C₄	—	2	4	no	—
143	C₂²×C₄	—	2	4	no	—
144	C₂²×C₄	—	38	16	no	—
145	C₂²×C₄	—	38	8	no	—
146	C₂²×C₄	—	76	8	no	—
147	C₂²×C₄	—	76	4	no	—
148	C₂²×C₄	—	76	4	no	—
149	C₂²×C₄	C₄⋊Dic₁₉	1	4	yes	—
150	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	4	yes	—
151	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	4	yes	—
152	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
153	C₂²×C₄	C₂×C₄×D₁₉	1	4	yes	—
154	C₂²×C₄	C₂×C₄×D₁₉	1	2	yes	—
155	C₂²×C₄	#64541	1	4	yes	—
156	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
157	C₂²×C₄	C₂×C₂²⋊D₁₉	1	2	yes	—
158	C₂×C₈	—	2	4	no	—
159	C₂×C₈	—	2	4	no	—
160	C₂×C₈	—	2	16	no	—
161	C₂×C₈	C₄×Dic₁₉	1	8	yes	complemented
162	C₄⋊C₄	—	19	16	no	—
163	C₄⋊C₄	—	38	8	no	—
164	C₄⋊C₄	—	38	16	no	—
165	C₄²	—	76	16	no	—
166	C₄²	C₄×Dic₁₉	1	4	yes	—
167	C₄²	Dic₁₉⋊C₄	1	8	yes	—
168	C₄²	C₂²⋊Dic₁₉	1	4	yes	—
169	C₂³	C₄×D₇₆	1	4	yes	—
170	C₂³	#10597354	1	4	yes	—
171	C₂³	#10597355	1	4	yes	—
172	C₂³	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
173	C₂³	C₂×(C₂×D₁₉)⋊C₄	1	2	yes	—
174	C₂×C₄	—	2	8	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	8	no	—
178	C₂×C₄	—	38	16	no	—
179	C₂×C₄	—	76	8	no	—
180	C₂×C₄	—	76	8	no	—
181	C₂×C₄	—	76	16	no	—
182	C₂×C₄	—	76	16	no	—
183	C₂×C₄	C₁₉⋊#95	1	8	yes	—
184	C₂×C₄	C₄×Dic₃₈	1	4	yes	—
185	C₂×C₄	C₄²×D₁₉	1	4	yes	—
186	C₂×C₄	#10597318	1	4	yes	—
187	C₂×C₄	C₂×C₄×Dic₁₉	1	1	yes	characteristic
188	C₂×C₄	C₂×C₄×Dic₁₉	1	2	yes	—
189	C₂×C₄	C₂×Dic₁₉⋊C₄	1	2	yes	—
190	C₂×C₄	#10597372	1	2	yes	—
191	C₂×C₄	C₄×C₂²⋊D₁₉	1	8	yes	—
192	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
193	C₈	—	2	8	no	—
194	C₂²	C₂×C₁₉⋊#95	1	1	yes	characteristic
195	C₂²	C₄×(C₂×D₁₉)⋊C₄	1	4	yes	—
196	C₂²	C₄×C₄⋊Dic₁₉	1	2	yes	—
197	C₂²	#11926546	1	4	yes	—
198	C₂²	#11926550	1	2	yes	—
199	C₂²	#11926554	1	2	yes	—
200	C₂²	#11926586	1	4	yes	—
201	C₂²	#11926646	1	8	yes	—
202	C₂²	#11926652	1	4	yes	—
203	C₂²	#11926657	1	4	yes	—
204	C₄	—	2	8	no	—
205	C₄	—	76	16	no	—
206	C₄	C₄²×Dic₁₉	1	1	yes	characteristic
207	C₄	#11926390	1	1	yes	characteristic
208	C₄	C₄×C₂²⋊Dic₁₉	1	2	yes	—
209	C₄	C₄×Dic₁₉⋊C₄	1	4	yes	—
210	C₂	C₄×C₁₉⋊#95	1	1	yes	characteristic
211	C₂	#14563355	1	2	yes	—
212	C₂	#14563461	1	4	yes	—
213	C₂	#14563466	1	4	yes	—
214	C₂	#14563532	1	4	yes	—

	subgroup	G/H
center	C₂⁴	C₂×D₇₆
commutator	C₄×C₁₉	C₄³
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₂²×C₁₉×#97	C₂
socle, prime radical	C₂⁴×C₁₉	C₂×D₄
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	C₂²×#97	D₁₉
hypocenter	C₁₉	#10703
LCS	G ▷ C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
7. C₄×C₄⋊Dic₁₉	C₄	semidirect
9. #11926307	C₄	semidirect
13. C₂×C₁₉×#97	C₄	semidirect
53. C₂×C₈×C₁₉	C₄²	semidirect
59. C₄⋊Dic₁₉	C₄²	semidirect
111. C₂×#97	Dic₁₉	semidirect
140. C₁₉	#10703	semidirect
161. C₂×C₈	C₄×Dic₁₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₄⋊Dic₁₉	2
C₂×#11926307	2
C₂²×C₁₉×#97	2
#10703	19