Subgroups — #18081261

Order 4864.

322 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×C₄⋊Dic₁₉	C₂	1	1	yes	maximal, characteristic
2	#14562407	C₂	1	2	yes	maximal
3	#14562383	C₂	1	1	yes	maximal, characteristic
4	#14561536	C₂	1	2	yes	maximal
5	C₁₉×#1848	C₂	1	1	yes	maximal, characteristic
6	#11926597	C₂²	1	2	yes	—
7	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
8	C₂×C₈×Dic₁₉	—	2	1	no	—
9	C₄×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
10	#11926298	—	2	1	no	—
11	#11926297	C₄	1	2	yes	complemented
12	#11926295	C₄	1	2	yes	complemented
13	#11926294	—	2	4	no	—
14	C₂×C₁₉×#108	—	2	1	no	—
15	C₂×C₁₉×#108	—	2	1	no	—
16	C₁₉×#423	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
18	C₂×M₁₆×C₁₉	—	2	2	no	—
19	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
21	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
22	C₂×C₄⋊Dic₁₉	—	2	1	no	—
23	C₂×C₄×Dic₁₉	—	2	1	no	—
24	C₂²×C₁₉⋊C₈	—	2	1	no	—
25	C₁₉×#108	—	4	2	no	—
26	C₁₉×#108	—	4	2	no	—
27	C₁₉×C₄⋊C₈	—	2	2	no	—
28	C₁₉×C₄⋊C₈	—	2	2	no	—
29	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₁₉	Q₈	1	1	yes	characteristic
32	C₄×C₈×C₁₉	Q₈	1	1	yes	characteristic
33	C₁₉⋊#95	Q₈	1	2	yes	—
34	C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
35	C₈×Dic₁₉	—	4	2	no	—
36	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
37	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	4	yes	—
38	C₄⋊(C₁₉⋊C₈)	D₄	1	2	yes	—
39	C₄×C₁₉⋊C₈	—	2	2	no	—
40	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
42	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
43	C₂²×Dic₁₉	—	2	2	no	—
44	M₁₆×C₁₉	—	4	2	no	—
45	M₁₆×C₁₉	—	4	2	no	—
46	C₂×C₈×C₁₉	—	2	1	no	—
47	C₂×C₈×C₁₉	—	2	1	no	—
48	C₂×C₈×C₁₉	—	4	2	no	—
49	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₁₉	SD₁₆	1	2	yes	—
54	C₂×C₈×C₁₉	SD₁₆	1	2	yes	—
55	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
56	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
57	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
58	C₄²×C₁₉	#41	1	1	yes	characteristic
59	C₄⋊Dic₁₉	—	2	4	no	—
60	C₄×Dic₁₉	—	4	1	no	—
61	C₄×Dic₁₉	—	4	1	no	—
62	C₂×C₁₉⋊C₈	—	2	4	no	—
63	C₂×C₁₉⋊C₈	—	2	2	no	—
64	C₂×C₁₉⋊C₈	—	4	1	no	—
65	C₂×C₁₉⋊C₈	—	4	1	no	—
66	#9628	—	19	1	no	maximal, Sylow
67	C₂³×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
68	C₂×C₄×C₁₉	—	2	2	no	—
69	C₂×C₄×C₁₉	—	4	1	no	—
70	C₂×C₄×C₁₉	C₂×C₄⋊C₄	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	#121	1	1	yes	characteristic
74	C₂×C₄×C₁₉	#125	1	1	yes	characteristic
75	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
76	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
77	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
78	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
79	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
80	C₂×Dic₁₉	—	2	4	no	—
81	C₂×Dic₁₉	—	4	2	no	—
82	C₂×Dic₁₉	—	4	2	no	—
83	C₈×C₁₉	—	2	1	no	—
84	C₈×C₁₉	—	2	1	no	—
85	C₈×C₁₉	—	8	2	no	—
86	C₈×C₁₉	#106	1	2	yes	—
87	C₈×C₁₉	#106	1	2	yes	—
88	C₁₉⋊C₈	—	4	4	no	—
89	C₁₉⋊C₈	—	4	2	no	—
90	#1848	D₁₉	1	1	yes	characteristic
91	C₈×C₄⋊C₄	—	19	1	no	—
92	#1570	—	19	2	no	—
93	#1564	—	19	1	no	—
94	#1303	—	19	2	no	—
95	C₂²×C₁₉	—	2	1	no	—
96	C₂²×C₁₉	—	2	1	no	—
97	C₂²×C₁₉	#389	1	1	yes	characteristic
98	C₂²×C₁₉	#475	1	1	yes	characteristic
99	C₂²×C₁₉	#476	1	1	yes	characteristic
100	C₄×C₁₉	—	2	1	no	—
101	C₄×C₁₉	—	2	1	no	—
102	C₄×C₁₉	—	2	1	no	—
103	C₄×C₁₉	C₂×#106	1	1	yes	characteristic
104	C₄×C₁₉	#428	1	1	yes	characteristic
105	C₄×C₁₉	#443	1	2	yes	—
106	C₄×C₁₉	#460	1	1	yes	characteristic
107	C₄×C₁₉	#461	1	1	yes	characteristic
108	Dic₁₉	—	4	4	no	—
109	#431	—	19	2	no	—
110	C₂×#108	—	2	1	no	—
111	C₂×#108	—	2	1	no	—
112	#423	C₂×D₁₉	1	2	yes	—
113	C₂×C₄⋊C₈	—	19	1	no	—
114	C₂×C₄×C₈	—	38	1	no	—
115	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
116	C₄×C₄⋊C₄	—	19	1	no	—
117	#336	—	38	1	no	—
118	#335	—	19	2	no	—
119	#334	—	19	2	no	—
120	#334	—	38	4	no	—
121	C₂×C₁₉	—	2	1	no	—
122	C₂×C₁₉	—	2	1	no	—
123	C₂×C₁₉	#1945	1	1	yes	characteristic
124	C₂×C₁₉	#1948	1	1	yes	characteristic
125	C₂×C₁₉	#1953	1	1	yes	characteristic
126	C₂×M₁₆	—	2	2	no	—
127	C₂²×C₈	—	38	1	no	—
128	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
129	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
130	C₂×C₄⋊C₄	—	38	1	no	—
131	C₂×C₄²	—	38	1	no	—
132	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
133	#108	—	4	2	no	—
134	#108	—	4	2	no	—
135	C₄⋊C₈	—	2	2	no	—
136	C₄⋊C₈	—	2	2	no	—
137	C₄⋊C₈	—	19	4	no	—
138	C₄⋊C₈	—	19	2	no	—
139	C₄⋊C₈	—	38	1	no	—
140	C₂²⋊C₈	—	38	2	no	—
141	C₄×C₈	—	38	2	no	—
142	C₄×C₈	—	76	2	no	—
143	C₄×C₈	Dic₃₈	1	1	yes	characteristic
144	C₄×C₈	Dic₃₈	1	1	yes	characteristic
145	C₄×C₈	C₄×D₁₉	1	1	yes	characteristic
146	C₄×C₈	C₄×D₁₉	1	1	yes	characteristic
147	#95	—	19	2	no	—
148	C₁₉	#9628	1	1	yes	characteristic, complemented, Sylow
149	C₂²×C₄	—	38	2	no	—
150	C₂²×C₄	Q₈×D₁₉	1	1	yes	characteristic
151	C₂²×C₄	#64545	1	1	yes	characteristic
152	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
153	M₁₆	—	4	2	no	—
154	M₁₆	—	4	2	no	—
155	C₂×C₈	—	2	1	no	—
156	C₂×C₈	—	2	1	no	—
157	C₂×C₈	—	4	2	no	—
158	C₂×C₈	—	38	2	no	—
159	C₂×C₈	—	38	4	no	—
160	C₂×C₈	—	76	1	no	—
161	C₂×C₈	—	76	1	no	—
162	C₂×C₈	Dic₁₉⋊C₄	1	1	yes	characteristic
163	C₂×C₈	Dic₁₉⋊C₄	1	1	yes	characteristic
164	C₂×C₈	Dic₁₉⋊C₄	1	1	yes	characteristic
165	C₂×C₈	Dic₁₉⋊C₄	1	1	yes	characteristic
166	C₂×C₈	#64526	1	2	yes	—
167	C₂×C₈	#64527	1	2	yes	—
168	C₄⋊C₄	—	38	4	no	—
169	C₄²	—	76	1	no	—
170	C₄²	—	76	1	no	—
171	C₄²	C₂×Dic₃₈	1	1	yes	characteristic
172	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
173	C₄²	D₄×D₁₉	1	1	yes	characteristic
174	C₄²	#64543	1	1	yes	characteristic
175	C₂³	#10597392	1	1	yes	characteristic
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	4	1	no	—
178	C₂×C₄	—	38	4	no	—
179	C₂×C₄	—	76	2	no	—
180	C₂×C₄	—	76	2	no	—
181	C₂×C₄	#10597333	1	1	yes	characteristic
182	C₂×C₄	#10597334	1	1	yes	characteristic
183	C₂×C₄	#10597336	1	1	yes	characteristic
184	C₂×C₄	#10597339	1	1	yes	characteristic
185	C₂×C₄	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
186	C₂×C₄	#10597378	1	1	yes	characteristic
187	C₂×C₄	C₂×#64526	1	1	yes	characteristic
188	C₂×C₄	#10597384	1	1	yes	characteristic
189	C₂×C₄	C₂×#64527	1	1	yes	characteristic
190	C₂×C₄	#10597388	1	1	yes	characteristic
191	C₈	—	2	1	no	—
192	C₈	—	2	1	no	—
193	C₈	—	8	2	no	—
194	C₈	—	76	4	no	—
195	C₈	—	76	2	no	—
196	C₈	#10597265	1	2	yes	—
197	C₈	#10597265	1	2	yes	—
198	C₂²	—	2	1	no	—
199	C₂²	—	2	1	no	—
200	C₂²	#11926440	1	1	yes	characteristic
201	C₂²	#11926826	1	1	yes	characteristic
202	C₂²	#11926831	1	1	yes	characteristic
203	C₄	—	2	1	no	—
204	C₄	—	2	1	no	—
205	C₄	—	2	1	no	—
206	C₄	—	76	4	no	—
207	C₄	C₂×#10597265	1	1	yes	characteristic
208	C₄	#11926584	1	1	yes	characteristic
209	C₄	#11926660	1	2	yes	—
210	C₄	#11926741	1	1	yes	characteristic
211	C₄	#11926746	1	1	yes	characteristic
212	C₂	—	2	1	no	—
213	C₂	—	2	1	no	—
214	C₂	#14564381	1	1	yes	characteristic
215	C₂	#14564396	1	1	yes	characteristic
216	C₂	#14564423	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#10597384
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1848	C₂
socle	C₂²×C₁₉	#389
prime radical	C₂³×C₁₉	C₂²⋊Q₈
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1848	D₁₉
hypocenter	C₁₉	#9628
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

Complements

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

N	H ≅ G/N	type
11. #11926297	C₄	semidirect
12. #11926295	C₄	semidirect
148. C₁₉	#9628	semidirect

Maximal subgroups

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

subgroup	index
C₈×C₄⋊Dic₁₉	2
#14562407	2
#14562383	2
#14561536	2
C₁₉×#1848	2
#9628	19