Subgroups — #18071998

Order 4864.

718 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	C₂×#11926597	C₂	1	2	yes	maximal
2	#14561470	C₂	1	4	yes	maximal, complemented
3	C₂²×C₄×C₈×C₁₉	C₂	1	1	yes	maximal, characteristic
4	#11926597	—	2	4	no	—
5	#11926597	C₄	1	8	yes	—
6	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	4	yes	—
7	C₂×C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
8	C₂×C₄×C₁₉⋊C₈	—	2	2	no	—
9	C₂³×C₈×C₁₉	C₄	1	2	yes	—
10	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₁₉	—	2	4	no	—
12	C₂×C₄×C₈×C₁₉	C₄	1	2	yes	—
13	C₂×C₄×C₈×C₁₉	C₂²	1	2	yes	—
14	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
15	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
16	C₂³×C₄×C₁₉	Q₈	1	1	yes	characteristic
17	C₂²×C₈×C₁₉	—	2	4	no	—
18	C₂²×C₈×C₁₉	—	2	4	no	—
19	C₂²×C₈×C₁₉	C₈	1	8	yes	complemented
20	C₂²×C₈×C₁₉	C₂×C₄	1	4	yes	—
21	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₁₉	C₂×C₄	1	4	yes	—
23	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
24	C₂×C₄²×C₁₉	D₄	1	4	yes	—
25	C₂×C₄²×C₁₉	Q₈	1	1	yes	characteristic
26	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
27	C₂²×C₁₉⋊C₈	—	2	4	no	—
28	C₄×C₈×C₁₉	—	2	4	no	—
29	C₄×C₈×C₁₉	—	2	4	no	—
30	C₂²⋊(C₁₉⋊C₈)	—	2	8	no	—
31	C₄⋊(C₁₉⋊C₈)	—	2	4	no	—
32	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	8	yes	—
33	C₄×C₁₉⋊C₈	—	2	8	no	—
34	C₂⁴×C₁₉	C₄⋊C₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₉	—	2	4	no	—
36	C₂²×C₄×C₁₉	—	2	4	no	—
37	C₂²×C₄×C₁₉	C₄²	1	2	yes	—
38	C₂²×C₄×C₁₉	C₂²⋊C₄	1	4	yes	—
39	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
40	C₂²×C₄×C₁₉	C₄⋊C₄	1	4	yes	—
41	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₉	C₂×C₈	1	4	yes	—
45	C₂²×C₄×C₁₉	M₁₆	1	4	yes	—
46	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₈×C₁₉	—	2	16	no	—
50	C₂×C₈×C₁₉	—	2	4	no	—
51	C₂×C₈×C₁₉	—	2	4	no	—
52	C₂×C₈×C₁₉	C₂×C₈	1	8	yes	—
53	C₄²×C₁₉	C₄²	1	2	yes	—
54	C₄²×C₁₉	C₂²⋊C₄	1	4	yes	—
55	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
56	C₄²×C₁₉	C₄⋊C₄	1	4	yes	—
57	C₄²×C₁₉	C₂²×C₄	1	2	yes	—
58	C₄²×C₁₉	C₂×D₄	1	2	yes	—
59	C₂×C₁₉⋊C₈	—	2	8	no	—
60	C₂×C₁₉⋊C₈	—	4	8	no	—
61	#7872	—	19	1	no	maximal, Sylow
62	C₂³×C₁₉	—	2	2	no	—
63	C₂³×C₁₉	—	2	2	no	—
64	C₂³×C₁₉	#95	1	2	yes	—
65	C₂³×C₁₉	C₄⋊C₈	1	4	yes	—
66	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₁₉	—	2	2	no	—
68	C₂×C₄×C₁₉	—	2	2	no	—
69	C₂×C₄×C₁₉	—	2	8	no	—
70	C₂×C₄×C₁₉	—	2	8	no	—
71	C₂×C₄×C₁₉	#95	1	2	yes	—
72	C₂×C₄×C₁₉	#95	1	2	yes	—
73	C₂×C₄×C₁₉	#95	1	2	yes	—
74	C₂×C₄×C₁₉	C₄×C₈	1	4	yes	—
75	C₂×C₄×C₁₉	#97	1	4	yes	—
76	C₂×C₄×C₁₉	C₂²⋊C₈	1	8	yes	—
77	C₂×C₄×C₁₉	C₄⋊C₈	1	4	yes	—
78	C₂×C₄×C₁₉	#108	1	4	yes	—
79	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
80	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
82	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
83	C₂×C₄×C₁₉	C₂²×C₈	1	2	yes	—
84	C₂×C₄×C₁₉	C₂×M₁₆	1	2	yes	—
85	C₂×C₄×C₁₉	#135	1	4	yes	—
86	C₈×C₁₉	—	2	8	no	—
87	C₁₉⋊C₈	—	4	8	no	—
88	C₂×#431	—	19	2	no	—
89	C₂²×C₄×C₈	D₁₉	1	1	yes	characteristic
90	#1277	—	19	4	no	—
91	C₂²×C₁₉	—	2	8	no	—
92	C₂²×C₁₉	—	2	2	no	—
93	C₂²×C₁₉	—	2	2	no	—
94	C₂²×C₁₉	#336	1	4	yes	—
95	C₂²×C₁₉	C₂×#95	1	1	yes	characteristic
96	C₂²×C₁₉	C₂×C₄⋊C₈	1	2	yes	—
97	C₂²×C₁₉	#427	1	2	yes	—
98	C₂²×C₁₉	C₂×#108	1	2	yes	—
99	C₄×C₁₉	—	2	2	no	—
100	C₄×C₁₉	—	2	2	no	—
101	C₄×C₁₉	#336	1	4	yes	—
102	C₄×C₁₉	#341	1	4	yes	—
103	C₄×C₁₉	C₂×C₄×C₈	1	1	yes	characteristic
104	C₄×C₁₉	C₂×#97	1	1	yes	characteristic
105	C₄×C₁₉	C₂×C₂²⋊C₈	1	2	yes	—
106	C₄×C₁₉	#416	1	4	yes	—
107	C₂³×C₈	Dic₁₉	1	2	yes	—
108	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
109	#431	—	19	8	no	—
110	#431	—	38	4	no	—
111	C₂×C₄⋊C₈	—	19	4	no	—
112	C₂×C₂²⋊C₈	—	38	2	no	—
113	C₂×C₄×C₈	—	2	4	no	—
114	C₂×C₄×C₈	—	38	2	no	—
115	C₂×C₄×C₈	Dic₁₉	1	2	yes	—
116	C₂×C₄×C₈	C₂×D₁₉	1	2	yes	—
117	C₂×C₁₉	—	2	4	no	—
118	C₂×C₁₉	#1564	1	4	yes	—
119	C₂×C₁₉	C₂×#336	1	1	yes	characteristic
120	C₂×C₁₉	#1735	1	1	yes	characteristic
121	C₂×C₁₉	C₂×#341	1	1	yes	characteristic
122	C₂³×C₄	Dic₃₈	1	1	yes	characteristic
123	C₂³×C₄	D₇₆	1	1	yes	characteristic
124	C₂³×C₄	C₂×Dic₁₉	1	1	yes	characteristic
125	C₂²×C₈	—	2	4	no	—
126	C₂²×C₈	—	2	4	no	—
127	C₂²×C₈	—	38	4	no	—
128	C₂²×C₈	C₁₉⋊C₈	1	8	yes	complemented
129	C₂²×C₈	C₂×Dic₁₉	1	4	yes	—
130	C₂×C₄²	Dic₃₈	1	1	yes	characteristic
131	C₂×C₄²	C₄×D₁₉	1	4	yes	—
132	C₂×C₄²	D₇₆	1	1	yes	characteristic
133	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
134	C₂×C₄²	C₂²⋊D₁₉	1	4	yes	—
135	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
136	C₄⋊C₈	—	19	8	no	—
137	C₄⋊C₈	—	38	4	no	—
138	C₂²⋊C₈	—	38	8	no	—
139	C₄×C₈	—	2	4	no	—
140	C₄×C₈	—	2	4	no	—
141	C₄×C₈	—	38	8	no	—
142	C₁₉	#7872	1	1	yes	characteristic, complemented, Sylow
143	C₂⁴	C₄⋊Dic₁₉	1	1	yes	characteristic
144	C₂²×C₄	—	2	4	no	—
145	C₂²×C₄	—	2	4	no	—
146	C₂²×C₄	C₂×C₁₉⋊C₈	1	4	yes	—
147	C₂²×C₄	#64520	1	4	yes	—
148	C₂²×C₄	C₄×Dic₁₉	1	2	yes	—
149	C₂²×C₄	Dic₁₉⋊C₄	1	4	yes	—
150	C₂²×C₄	C₄⋊Dic₁₉	1	1	yes	characteristic
151	C₂²×C₄	C₄⋊Dic₁₉	1	1	yes	characteristic
152	C₂²×C₄	C₄⋊Dic₁₉	1	1	yes	characteristic
153	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	4	yes	—
154	C₂²×C₄	C₂²⋊Dic₁₉	1	2	yes	—
155	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
156	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
157	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
158	C₂×C₈	—	2	4	no	—
159	C₂×C₈	—	2	16	no	—
160	C₂×C₈	—	2	4	no	—
161	C₂×C₈	—	38	8	no	—
162	C₂×C₈	—	76	8	no	—
163	C₂×C₈	C₂×C₁₉⋊C₈	1	8	yes	—
164	C₄²	C₄×Dic₁₉	1	2	yes	—
165	C₄²	Dic₁₉⋊C₄	1	4	yes	—
166	C₄²	(C₂×D₁₉)⋊C₄	1	4	yes	—
167	C₄²	C₂²⋊Dic₁₉	1	2	yes	—
168	C₄²	C₂×C₄×D₁₉	1	2	yes	—
169	C₄²	C₂×C₂²⋊D₁₉	1	2	yes	—
170	C₂³	—	2	2	no	—
171	C₂³	—	2	2	no	—
172	C₂³	C₄⋊(C₁₉⋊C₈)	1	4	yes	—
173	C₂³	C₁₉⋊#95	1	2	yes	—
174	C₂³	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	8	no	—
178	C₂×C₄	—	2	8	no	—
179	C₂×C₄	C₄×C₁₉⋊C₈	1	4	yes	—
180	C₂×C₄	#10597260	1	4	yes	—
181	C₂×C₄	C₄⋊(C₁₉⋊C₈)	1	4	yes	—
182	C₂×C₄	#10597276	1	4	yes	—
183	C₂×C₄	C₂²⋊(C₁₉⋊C₈)	1	8	yes	—
184	C₂×C₄	C₁₉⋊#95	1	2	yes	—
185	C₂×C₄	C₁₉⋊#95	1	2	yes	—
186	C₂×C₄	C₁₉⋊#95	1	2	yes	—
187	C₂×C₄	#10597350	1	4	yes	—
188	C₂×C₄	C₂²×C₁₉⋊C₈	1	2	yes	—
189	C₂×C₄	C₂×#64520	1	2	yes	—
190	C₂×C₄	C₂×C₄×Dic₁₉	1	1	yes	characteristic
191	C₂×C₄	C₂×Dic₁₉⋊C₄	1	2	yes	—
192	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	2	yes	—
193	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
194	C₈	—	2	8	no	—
195	C₈	—	76	8	no	—
196	C₂²	—	2	2	no	—
197	C₂²	—	2	2	no	—
198	C₂²	—	2	8	no	—
199	C₂²	#11926299	1	4	yes	—
200	C₂²	C₂×C₁₉⋊#95	1	1	yes	characteristic
201	C₂²	C₂×C₄⋊(C₁₉⋊C₈)	1	2	yes	—
202	C₂²	#11926582	1	2	yes	—
203	C₂²	C₂×#10597276	1	2	yes	—
204	C₄	—	2	2	no	—
205	C₄	—	2	2	no	—
206	C₄	#11926299	1	4	yes	—
207	C₄	#11926309	1	4	yes	—
208	C₄	C₂×C₄×C₁₉⋊C₈	1	1	yes	characteristic
209	C₄	C₂×#10597260	1	1	yes	characteristic
210	C₄	C₂×C₂²⋊(C₁₉⋊C₈)	1	2	yes	—
211	C₄	#11926541	1	4	yes	—
212	C₂	—	2	4	no	—
213	C₂	#14562383	1	4	yes	—
214	C₂	C₂×#11926299	1	1	yes	characteristic
215	C₂	#14563352	1	1	yes	characteristic
216	C₂	C₂×#11926309	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²	D₇₆
commutator	C₄×C₁₉	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₂²×C₄×C₈×C₁₉	C₂
socle	C₂³×C₁₉	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₉	C₄⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂²×C₄×C₈	D₁₉
hypocenter	C₁₉	#7872
LCS	G ▷ C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #14561470	C₂	semidirect
19. C₂²×C₈×C₁₉	C₈	semidirect
128. C₂²×C₈	C₁₉⋊C₈	semidirect
142. C₁₉	#7872	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926597	2
#14561470	2
C₂²×C₄×C₈×C₁₉	2
#7872	19