Subgroups — #334×Dic₁₉ #18073722

Order 4864.

470 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	C₄⋊C₈×Dic₁₉	C₂	1	2	yes	maximal
2	C₄×C₈×Dic₁₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#11926295	C₂	1	1	yes	maximal, characteristic
4	#14561469	C₂	1	2	yes	maximal
5	C₂×C₁₉×#334	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₈×Dic₁₉	—	2	2	no	—
9	C₂×C₈×Dic₁₉	C₄	1	2	yes	—
10	C₂×C₈×Dic₁₉	C₄	1	2	yes	—
11	C₄²×Dic₁₉	C₂²	1	1	yes	characteristic
12	#11926298	—	2	2	no	—
13	#11926295	C₄	1	4	yes	complemented
14	C₂×C₁₉×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
16	C₁₉×#334	C₄	1	4	yes	complemented
17	C₂²×C₈×C₁₉	—	2	2	no	—
18	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
21	C₂×C₄×Dic₁₉	C₂×C₄	1	2	yes	—
22	C₂×C₄×Dic₁₉	D₄	1	1	yes	characteristic
23	C₂×C₄×Dic₁₉	D₄	1	1	yes	characteristic
24	C₂×C₄×Dic₁₉	Q₈	1	2	yes	—
25	C₂²×C₁₉⋊C₈	—	2	2	no	—
26	C₂²×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
27	C₁₉×C₄⋊C₈	—	2	2	no	—
28	C₁₉×C₄⋊C₈	C₂×C₄	1	4	yes	—
29	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
30	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
31	C₈×Dic₁₉	—	2	4	no	—
32	C₈×Dic₁₉	—	4	4	no	—
33	C₈×Dic₁₉	C₈	1	8	yes	complemented
34	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
35	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	4	yes	—
36	C₄×C₁₉⋊C₈	—	2	1	no	—
37	C₄×C₁₉⋊C₈	—	2	1	no	—
38	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
41	C₂²×Dic₁₉	C₄⋊C₄	1	2	yes	—
42	C₂²×Dic₁₉	C₄⋊C₄	1	2	yes	—
43	C₂×C₈×C₁₉	—	2	2	no	—
44	C₂×C₈×C₁₉	—	2	4	no	—
45	C₂×C₈×C₁₉	—	4	2	no	—
46	C₂×C₈×C₁₉	—	4	2	no	—
47	C₂×C₈×C₁₉	C₄²	1	2	yes	—
48	C₂×C₈×C₁₉	C₄²	1	2	yes	—
49	C₂×C₈×C₁₉	C₂×C₈	1	4	yes	—
50	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
51	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
52	C₄²×C₁₉	#41	1	1	yes	characteristic
53	C₄²×C₁₉	#41	1	1	yes	characteristic
54	C₄×Dic₁₉	—	2	2	no	—
55	C₄×Dic₁₉	—	2	4	no	—
56	C₄×Dic₁₉	C₂×C₈	1	4	yes	—
57	C₄×Dic₁₉	M₁₆	1	4	yes	—
58	C₄×Dic₁₉	SD₁₆	1	4	yes	—
59	C₂×C₁₉⋊C₈	—	2	2	no	—
60	C₂×C₁₉⋊C₈	—	2	2	no	—
61	C₂×C₁₉⋊C₈	—	2	4	no	—
62	C₂×C₁₉⋊C₈	—	4	2	no	—
63	C₂×C₁₉⋊C₈	—	4	2	no	—
64	C₂×C₁₉⋊C₈	C₂×C₈	1	4	yes	—
65	C₄×#334	—	19	1	no	maximal, Sylow
66	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₁₉	—	2	2	no	—
68	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
69	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
70	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
71	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
72	C₂×C₄×C₁₉	C₄×Q₈	1	1	yes	characteristic
73	C₂×C₄×C₁₉	C₄×Q₈	1	1	yes	characteristic
74	C₂×C₄×C₁₉	C₂²×C₈	1	2	yes	—
75	C₂×C₄×C₁₉	C₂×M₁₆	1	2	yes	—
76	C₂×C₄×C₁₉	C₂×SD₁₆	1	2	yes	—
77	C₂×C₄×C₁₉	#135	1	2	yes	—
78	C₂×Dic₁₉	—	2	2	no	—
79	C₂×Dic₁₉	—	2	2	no	—
80	C₂×Dic₁₉	C₄⋊C₈	1	4	yes	—
81	C₂×Dic₁₉	C₄⋊C₈	1	4	yes	—
82	C₂×Dic₁₉	#106	1	4	yes	—
83	C₂×Dic₁₉	#108	1	4	yes	—
84	C₈×C₁₉	—	2	2	no	—
85	C₈×C₁₉	—	4	4	no	—
86	C₈×C₁₉	C₄×C₈	1	4	yes	complemented
87	C₁₉⋊C₈	—	2	4	no	—
88	C₁₉⋊C₈	—	4	4	no	—
89	C₄×C₄⋊C₈	—	19	2	no	—
90	C₄²×C₈	—	19	1	no	—
91	C₂×#334	—	19	1	no	—
92	C₂×#334	D₁₉	1	1	yes	characteristic
93	#1277	—	19	2	no	—
94	C₂²×C₁₉	C₄×C₄⋊C₄	1	1	yes	characteristic
95	C₂²×C₁₉	C₂×C₄⋊C₈	1	2	yes	—
96	C₂²×C₁₉	C₂×#106	1	1	yes	characteristic
97	C₂²×C₁₉	#427	1	1	yes	characteristic
98	C₂²×C₁₉	C₂×#108	1	2	yes	—
99	C₄×C₁₉	—	2	1	no	—
100	C₄×C₁₉	—	2	1	no	—
101	C₄×C₁₉	C₂×C₄×C₈	1	1	yes	characteristic
102	C₄×C₁₉	C₄×M₁₆	1	1	yes	characteristic
103	C₄×C₁₉	#431	1	2	yes	—
104	C₄×C₁₉	C₄×SD₁₆	1	2	yes	—
105	C₄×C₁₉	C₄×SD₁₆	1	2	yes	—
106	Dic₁₉	—	2	4	no	—
107	Dic₁₉	#334	1	8	yes	direct product factor
108	C₂×C₄⋊C₈	—	19	2	no	—
109	C₂×C₄⋊C₈	C₂×D₁₉	1	2	yes	—
110	C₂×C₄×C₈	—	19	2	no	—
111	C₂×C₄×C₈	—	19	2	no	—
112	C₂×C₄×C₈	—	19	1	no	—
113	C₂×C₄×C₈	—	38	2	no	—
114	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
115	C₄³	—	19	1	no	—
116	#336	—	38	2	no	—
117	#334	—	19	4	no	—
118	#334	Dic₁₉	1	4	yes	direct product factor
119	C₂×C₁₉	C₂×#334	1	2	yes	—
120	C₂×C₁₉	#1564	1	2	yes	—
121	C₂×C₁₉	C₄×C₄⋊C₈	1	1	yes	characteristic
122	C₂×C₁₉	C₄×#106	1	1	yes	characteristic
123	C₂×C₁₉	C₄×#108	1	1	yes	characteristic
124	C₂²×C₈	—	2	2	no	—
125	C₂²×C₈	—	19	2	no	—
126	C₂²×C₈	—	38	2	no	—
127	C₂²×C₈	C₄×D₁₉	1	1	yes	characteristic
128	C₂²×C₈	C₄×D₁₉	1	1	yes	characteristic
129	C₂×C₄²	—	19	2	no	—
130	C₂×C₄²	—	19	1	no	—
131	C₂×C₄²	—	19	1	no	—
132	C₂×C₄²	—	19	2	no	—
133	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
134	C₄⋊C₈	—	2	2	no	—
135	C₄⋊C₈	—	19	4	no	—
136	C₄⋊C₈	—	38	2	no	—
137	C₄⋊C₈	C₂×Dic₁₉	1	4	yes	—
138	C₄×C₈	—	19	8	no	—
139	C₄×C₈	—	38	4	no	—
140	C₄×C₈	—	38	1	no	—
141	C₄×C₈	—	38	1	no	—
142	C₄×C₈	—	76	4	no	—
143	C₄×C₈	C₄×D₁₉	1	2	yes	—
144	C₄×C₈	C₂×Dic₁₉	1	2	yes	—
145	C₁₉	C₄×#334	1	1	yes	characteristic, complemented, Sylow
146	C₂²×C₄	—	19	2	no	—
147	C₂²×C₄	—	19	2	no	—
148	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
149	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
150	C₂²×C₄	Q₈×D₁₉	1	1	yes	characteristic
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	2	4	no	—
153	C₂×C₈	—	4	2	no	—
154	C₂×C₈	—	4	2	no	—
155	C₂×C₈	—	19	4	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	C₂×C₈	C₈×D₁₉	1	4	yes	—
162	C₂×C₈	C₄×Dic₁₉	1	2	yes	—
163	C₂×C₈	C₄×Dic₁₉	1	2	yes	—
164	C₄²	—	19	4	no	—
165	C₄²	—	19	4	no	—
166	C₄²	—	19	4	no	—
167	C₄²	—	38	4	no	—
168	C₄²	—	38	2	no	—
169	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
170	C₄²	#64543	1	1	yes	characteristic
171	C₄²	#64545	1	1	yes	characteristic
172	C₄²	C₂²×Dic₁₉	1	1	yes	characteristic
173	C₂³	C₄⋊C₄×D₁₉	1	1	yes	characteristic
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	19	4	no	—
176	C₂×C₄	—	19	4	no	—
177	C₂×C₄	—	19	4	no	—
178	C₂×C₄	—	19	4	no	—
179	C₂×C₄	—	38	2	no	—
180	C₂×C₄	—	38	2	no	—
181	C₂×C₄	#10597333	1	1	yes	characteristic
182	C₂×C₄	#10597338	1	1	yes	characteristic
183	C₂×C₄	#10597339	1	1	yes	characteristic
184	C₂×C₄	C₂×C₈×D₁₉	1	2	yes	—
185	C₂×C₄	M₁₆×D₁₉	1	2	yes	—
186	C₂×C₄	SD₁₆×D₁₉	1	2	yes	—
187	C₂×C₄	#10597362	1	2	yes	—
188	C₂×C₄	C₂×C₄×Dic₁₉	1	1	yes	characteristic
189	C₂×C₄	D₄×Dic₁₉	1	1	yes	characteristic
190	C₂×C₄	Q₈×Dic₁₉	1	1	yes	characteristic
191	C₈	—	2	2	no	—
192	C₈	—	4	4	no	—
193	C₈	—	38	4	no	—
194	C₈	—	76	4	no	—
195	C₈	C₈×Dic₁₉	1	4	yes	complemented
196	C₂²	C₄⋊C₄×Dic₁₉	1	1	yes	characteristic
197	C₂²	C₄⋊C₈×D₁₉	1	2	yes	—
198	C₂²	#106×D₁₉	1	1	yes	characteristic
199	C₂²	#11926579	1	1	yes	characteristic
200	C₂²	#108×D₁₉	1	2	yes	—
201	C₄	—	2	1	no	—
202	C₄	—	2	1	no	—
203	C₄	—	19	8	no	—
204	C₄	—	38	4	no	—
205	C₄	C₂×C₈×Dic₁₉	1	1	yes	characteristic
206	C₄	M₁₆×Dic₁₉	1	1	yes	characteristic
207	C₄	#11926596	1	2	yes	—
208	C₄	#11926631	1	2	yes	—
209	C₄	SD₁₆×Dic₁₉	1	2	yes	—
210	C₂	D₁₉×#334	1	2	yes	—
211	C₂	#14562380	1	2	yes	—
212	C₂	C₄⋊C₈×Dic₁₉	1	1	yes	characteristic
213	C₂	#106×Dic₁₉	1	1	yes	characteristic
214	C₂	#108×Dic₁₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₁₉
commutator	C₄×C₁₉	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₂×C₁₉×#334	C₂
socle, prime radical	C₂³×C₁₉	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂×#334	D₁₉
hypocenter	C₁₉	C₄×#334
LCS	G ▷ C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
13. #11926295	C₄	semidirect
16. C₁₉×#334	C₄	semidirect
33. C₈×Dic₁₉	C₈	semidirect
86. C₈×C₁₉	C₄×C₈	semidirect
107. Dic₁₉	#334	direct (N × H)
118. #334	Dic₁₉	direct (N × H)
145. C₁₉	C₄×#334	semidirect
195. C₈	C₈×Dic₁₉	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊C₈×Dic₁₉	2
C₄×C₈×Dic₁₉	2
C₂×#11926295	2
#14561469	2
C₂×C₁₉×#334	2
C₄×#334	19

Subgroups — #334×Dic19 #18073722

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #334×Dic₁₉ #18073722