Subgroups — C₂²×C₁₉×#424 #18066055

Order 4864.

3006 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	C₂³×M₁₆×C₁₉	C₂	1	1	yes	maximal, characteristic
2	C₂⁴×C₈×C₁₉	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₁₉×#117	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₉×#424	C₂	1	24	yes	maximal, direct product factor
5	C₂²×C₁₉×C₄⋊C₈	C₂	1	4	yes	maximal, complemented
6	C₂⁴×C₄×C₁₉	C₂²	1	1	yes	characteristic
7	C₂²×M₁₆×C₁₉	—	2	8	no	—
8	C₂²×M₁₆×C₁₉	C₂²	1	12	yes	—
9	C₂³×C₈×C₁₉	—	2	8	no	—
10	C₂³×C₈×C₁₉	C₂²	1	12	yes	—
11	C₂³×C₈×C₁₉	C₂²	1	2	yes	—
12	C₂³×C₈×C₁₉	C₂²	1	2	yes	—
13	C₂×C₁₉×#117	C₄	1	12	yes	—
14	C₂×C₁₉×#117	C₂²	1	12	yes	—
15	C₂²×C₄⋊C₄×C₁₉	C₄	1	2	yes	—
16	C₂²×C₂²⋊C₄×C₁₉	C₄	1	2	yes	—
17	C₂²×C₄²×C₁₉	C₂²	1	2	yes	—
18	C₁₉×#424	C₂²	1	64	yes	direct product factor
19	C₂×C₁₉×C₄⋊C₈	C₂²	1	48	yes	complemented
20	C₂⁵×C₁₉	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₁₉	C₂×C₄	1	4	yes	—
22	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₄×C₁₉	C₂×C₄	1	6	yes	—
24	C₂³×C₄×C₁₉	D₄	1	8	yes	—
25	C₂³×C₄×C₁₉	Q₈	1	8	yes	—
26	C₂³×C₄×C₁₉	C₂³	1	6	yes	—
27	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
28	C₂×M₁₆×C₁₉	—	2	48	no	—
29	C₂×M₁₆×C₁₉	C₂³	1	16	yes	—
30	C₂²×C₈×C₁₉	—	2	8	no	—
31	C₂²×C₈×C₁₉	—	2	48	no	—
32	C₂²×C₈×C₁₉	—	2	8	no	—
33	C₂²×C₈×C₁₉	C₂³	1	12	yes	—
34	C₂²×C₈×C₁₉	C₂³	1	12	yes	—
35	C₂²×C₈×C₁₉	C₂³	1	16	yes	—
36	C₁₉×#117	C₂×C₄	1	48	yes	—
37	C₁₉×#117	C₂³	1	16	yes	—
38	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	24	yes	—
39	C₂×C₂²⋊C₄×C₁₉	C₂×C₄	1	24	yes	—
40	C₂×C₄²×C₁₉	C₂×C₄	1	12	yes	—
41	C₂×C₄²×C₁₉	C₂³	1	12	yes	—
42	C₁₉×C₄⋊C₈	C₂³	1	64	yes	complemented
43	C₂⁴×C₁₉	—	2	8	no	—
44	C₂⁴×C₁₉	C₄⋊C₄	1	8	yes	—
45	C₂⁴×C₁₉	C₂²×C₄	1	6	yes	—
46	C₂⁴×C₁₉	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₉	—	2	16	no	—
48	C₂²×C₄×C₁₉	C₄⋊C₄	1	48	yes	—
49	C₂²×C₄×C₁₉	C₄⋊C₄	1	8	yes	—
50	C₂²×C₄×C₁₉	C₂²×C₄	1	24	yes	—
51	C₂²×C₄×C₁₉	C₂²×C₄	1	3	yes	—
52	C₂²×C₄×C₁₉	C₂²×C₄	1	6	yes	—
53	C₂²×C₄×C₁₉	C₂²×C₄	1	12	yes	—
54	C₂²×C₄×C₁₉	C₂×D₄	1	4	yes	—
55	C₂²×C₄×C₁₉	C₂×D₄	1	24	yes	—
56	C₂²×C₄×C₁₉	C₂×Q₈	1	24	yes	—
57	C₂²×C₄×C₁₉	C₂×Q₈	1	4	yes	—
58	C₂²×C₄×C₁₉	C₂⁴	1	3	yes	—
59	C₂²×C₄×C₁₉	C₂⁴	1	4	yes	—
60	M₁₆×C₁₉	—	2	32	no	—
61	C₂×C₈×C₁₉	—	2	24	no	—
62	C₂×C₈×C₁₉	—	2	32	no	—
63	C₂×C₈×C₁₉	—	2	24	no	—
64	C₂×C₈×C₁₉	C₂⁴	1	8	yes	—
65	C₂×C₈×C₁₉	C₂⁴	1	8	yes	—
66	C₄⋊C₄×C₁₉	C₂²×C₄	1	32	yes	—
67	C₂²⋊C₄×C₁₉	C₂²×C₄	1	32	yes	—
68	C₄²×C₁₉	C₂²×C₄	1	24	yes	—
69	C₄²×C₁₉	C₂⁴	1	8	yes	—
70	C₂²×#424	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
71	C₂³×C₁₉	—	2	24	no	—
72	C₂³×C₁₉	—	2	32	no	—
73	C₂³×C₁₉	C₂×C₄⋊C₄	1	24	yes	—
74	C₂³×C₁₉	C₂×C₄⋊C₄	1	4	yes	—
75	C₂³×C₁₉	#131	1	8	yes	—
76	C₂³×C₁₉	C₂³×C₄	1	4	yes	—
77	C₂³×C₁₉	C₂³×C₄	1	3	yes	—
78	C₂×C₄×C₁₉	—	2	48	no	—
79	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	48	yes	—
80	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	12	yes	—
81	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	24	yes	—
82	C₂×C₄×C₁₉	C₂³×C₄	1	16	yes	—
83	C₂×C₄×C₁₉	C₂³×C₄	1	3	yes	—
84	C₂×C₄×C₁₉	C₂³×C₄	1	4	yes	—
85	C₂×C₄×C₁₉	C₂²×D₄	1	6	yes	—
86	C₂×C₄×C₁₉	C₂²×D₄	1	8	yes	—
87	C₂×C₄×C₁₉	C₂²×Q₈	1	8	yes	—
88	C₂×C₄×C₁₉	C₂²×Q₈	1	6	yes	—
89	C₂×C₄×C₁₉	C₂⁵	1	1	yes	characteristic
90	C₈×C₁₉	—	2	8	no	—
91	C₈×C₁₉	—	2	8	no	—
92	C₂³×M₁₆	C₂×C₁₉	1	1	yes	characteristic
93	C₂⁴×C₈	C₂×C₁₉	1	1	yes	characteristic
94	C₂²×#117	C₂×C₁₉	1	1	yes	characteristic
95	C₂×#424	C₂×C₁₉	1	24	yes	direct product factor
96	C₂²×C₄⋊C₈	C₂×C₁₉	1	4	yes	complemented
97	C₂²×C₁₉	—	2	48	no	—
98	C₂²×C₁₉	—	2	8	no	—
99	C₂²×C₁₉	#424	1	16	yes	direct product factor
100	C₂²×C₁₉	C₂²×C₄⋊C₄	1	8	yes	—
101	C₂²×C₁₉	C₂²×C₄⋊C₄	1	6	yes	—
102	C₂²×C₁₉	C₂×#131	1	12	yes	—
103	C₂²×C₁₉	C₂⁴×C₄	1	1	yes	characteristic
104	C₄×C₁₉	—	2	16	no	—
105	C₄×C₁₉	C₂²×C₄⋊C₄	1	6	yes	—
106	C₄×C₁₉	C₂²×C₄⋊C₄	1	8	yes	—
107	C₄×C₁₉	C₂³×D₄	1	1	yes	characteristic
108	C₄×C₁₉	C₂³×Q₈	1	1	yes	characteristic
109	C₂⁴×C₄	C₂²×C₁₉	1	1	yes	characteristic
110	C₂²×M₁₆	—	2	8	no	—
111	C₂²×M₁₆	C₂²×C₁₉	1	12	yes	—
112	C₂³×C₈	—	2	8	no	—
113	C₂³×C₈	C₂²×C₁₉	1	2	yes	—
114	C₂³×C₈	C₂²×C₁₉	1	2	yes	—
115	C₂³×C₈	C₂²×C₁₉	1	12	yes	—
116	C₂×#117	C₄×C₁₉	1	12	yes	—
117	C₂×#117	C₂²×C₁₉	1	12	yes	—
118	C₂²×C₄⋊C₄	C₄×C₁₉	1	2	yes	—
119	C₂²×C₂²⋊C₄	C₄×C₁₉	1	2	yes	—
120	C₂²×C₄²	C₂²×C₁₉	1	2	yes	—
121	#424	C₂²×C₁₉	1	64	yes	direct product factor
122	C₂×C₄⋊C₈	C₂²×C₁₉	1	48	yes	complemented
123	C₂×C₁₉	—	2	8	no	—
124	C₂×C₁₉	C₂×#424	1	12	yes	direct product factor
125	C₂×C₁₉	C₂³×C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₁₉	C₂²×#131	1	2	yes	—
127	C₂⁵	C₂×C₄×C₁₉	1	1	yes	characteristic
128	C₂³×C₄	C₂×C₄×C₁₉	1	4	yes	—
129	C₂³×C₄	C₂×C₄×C₁₉	1	1	yes	characteristic
130	C₂³×C₄	C₂×C₄×C₁₉	1	6	yes	—
131	C₂³×C₄	D₄×C₁₉	1	8	yes	—
132	C₂³×C₄	Q₈×C₁₉	1	8	yes	—
133	C₂³×C₄	C₂³×C₁₉	1	6	yes	—
134	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
135	C₂×M₁₆	—	2	48	no	—
136	C₂×M₁₆	C₂³×C₁₉	1	16	yes	—
137	C₂²×C₈	—	2	8	no	—
138	C₂²×C₈	—	2	48	no	—
139	C₂²×C₈	—	2	8	no	—
140	C₂²×C₈	C₂³×C₁₉	1	12	yes	—
141	C₂²×C₈	C₂³×C₁₉	1	16	yes	—
142	C₂²×C₈	C₂³×C₁₉	1	12	yes	—
143	#117	C₂×C₄×C₁₉	1	48	yes	—
144	#117	C₂³×C₁₉	1	16	yes	—
145	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	24	yes	—
146	C₂×C₂²⋊C₄	C₂×C₄×C₁₉	1	24	yes	—
147	C₂×C₄²	C₂×C₄×C₁₉	1	12	yes	—
148	C₂×C₄²	C₂³×C₁₉	1	12	yes	—
149	C₄⋊C₈	C₂³×C₁₉	1	64	yes	complemented
150	C₁₉	C₂²×#424	1	1	yes	characteristic, direct product factor, Sylow
151	C₂⁴	—	2	8	no	—
152	C₂⁴	C₄⋊C₄×C₁₉	1	8	yes	—
153	C₂⁴	C₂²×C₄×C₁₉	1	6	yes	—
154	C₂⁴	C₂²×C₄×C₁₉	1	1	yes	characteristic
155	C₂²×C₄	—	2	16	no	—
156	C₂²×C₄	C₄⋊C₄×C₁₉	1	48	yes	—
157	C₂²×C₄	C₄⋊C₄×C₁₉	1	8	yes	—
158	C₂²×C₄	C₂²×C₄×C₁₉	1	24	yes	—
159	C₂²×C₄	C₂²×C₄×C₁₉	1	3	yes	—
160	C₂²×C₄	C₂²×C₄×C₁₉	1	6	yes	—
161	C₂²×C₄	C₂²×C₄×C₁₉	1	12	yes	—
162	C₂²×C₄	C₂×D₄×C₁₉	1	4	yes	—
163	C₂²×C₄	C₂×D₄×C₁₉	1	24	yes	—
164	C₂²×C₄	C₂×Q₈×C₁₉	1	24	yes	—
165	C₂²×C₄	C₂×Q₈×C₁₉	1	4	yes	—
166	C₂²×C₄	C₂⁴×C₁₉	1	3	yes	—
167	C₂²×C₄	C₂⁴×C₁₉	1	4	yes	—
168	M₁₆	—	2	32	no	—
169	C₂×C₈	—	2	24	no	—
170	C₂×C₈	—	2	32	no	—
171	C₂×C₈	—	2	24	no	—
172	C₂×C₈	C₂⁴×C₁₉	1	8	yes	—
173	C₂×C₈	C₂⁴×C₁₉	1	8	yes	—
174	C₄⋊C₄	C₂²×C₄×C₁₉	1	32	yes	—
175	C₂²⋊C₄	C₂²×C₄×C₁₉	1	32	yes	—
176	C₄²	C₂²×C₄×C₁₉	1	24	yes	—
177	C₄²	C₂⁴×C₁₉	1	8	yes	—
178	C₂³	—	2	32	no	—
179	C₂³	—	2	24	no	—
180	C₂³	C₂×C₄⋊C₄×C₁₉	1	24	yes	—
181	C₂³	C₂×C₄⋊C₄×C₁₉	1	4	yes	—
182	C₂³	C₁₉×#131	1	8	yes	—
183	C₂³	C₂³×C₄×C₁₉	1	4	yes	—
184	C₂³	C₂³×C₄×C₁₉	1	3	yes	—
185	C₂×C₄	—	2	48	no	—
186	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	48	yes	—
187	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	12	yes	—
188	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	24	yes	—
189	C₂×C₄	C₂³×C₄×C₁₉	1	16	yes	—
190	C₂×C₄	C₂³×C₄×C₁₉	1	3	yes	—
191	C₂×C₄	C₂³×C₄×C₁₉	1	4	yes	—
192	C₂×C₄	C₂²×D₄×C₁₉	1	6	yes	—
193	C₂×C₄	C₂²×D₄×C₁₉	1	8	yes	—
194	C₂×C₄	C₂²×Q₈×C₁₉	1	8	yes	—
195	C₂×C₄	C₂²×Q₈×C₁₉	1	6	yes	—
196	C₂×C₄	C₂⁵×C₁₉	1	1	yes	characteristic
197	C₈	—	2	8	no	—
198	C₈	—	2	8	no	—
199	C₂²	—	2	8	no	—
200	C₂²	—	2	48	no	—
201	C₂²	C₁₉×#424	1	16	yes	direct product factor
202	C₂²	C₂²×C₄⋊C₄×C₁₉	1	8	yes	—
203	C₂²	C₂²×C₄⋊C₄×C₁₉	1	6	yes	—
204	C₂²	C₂×C₁₉×#131	1	12	yes	—
205	C₂²	C₂⁴×C₄×C₁₉	1	1	yes	characteristic
206	C₄	—	2	16	no	—
207	C₄	C₂²×C₄⋊C₄×C₁₉	1	6	yes	—
208	C₄	C₂²×C₄⋊C₄×C₁₉	1	8	yes	—
209	C₄	C₂³×D₄×C₁₉	1	1	yes	characteristic
210	C₄	C₂³×Q₈×C₁₉	1	1	yes	characteristic
211	C₂	—	2	8	no	—
212	C₂	C₂×C₁₉×#424	1	12	yes	direct product factor
213	C₂	C₂³×C₄⋊C₄×C₁₉	1	1	yes	characteristic
214	C₂	C₂²×C₁₉×#131	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄×C₁₉	C₂³
commutator	C₂²	C₂⁴×C₄×C₁₉
Frattini, cosocle	C₂×C₄	C₂⁵×C₁₉
socle	C₂⁴×C₁₉	C₂²×C₄
prime radical	C₂⁵×C₁₉	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂×C₁₉×#424	C₂	direct (N × H)
5. C₂²×C₁₉×C₄⋊C₈	C₂	semidirect
18. C₁₉×#424	C₂²	direct (N × H)
19. C₂×C₁₉×C₄⋊C₈	C₂²	semidirect
42. C₁₉×C₄⋊C₈	C₂³	semidirect
70. C₂²×#424	C₁₉	direct (N × H)
95. C₂×#424	C₂×C₁₉	direct (N × H)
96. C₂²×C₄⋊C₈	C₂×C₁₉	semidirect
99. C₂²×C₁₉	#424	direct (N × H)
121. #424	C₂²×C₁₉	direct (N × H)
122. C₂×C₄⋊C₈	C₂²×C₁₉	semidirect
124. C₂×C₁₉	C₂×#424	direct (N × H)
149. C₄⋊C₈	C₂³×C₁₉	semidirect
150. C₁₉	C₂²×#424	direct (N × H)
201. C₂²	C₁₉×#424	direct (N × H)
212. C₂	C₂×C₁₉×#424	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×M₁₆×C₁₉	2
C₂⁴×C₈×C₁₉	2
C₂²×C₁₉×#117	2
C₂×C₁₉×#424	2
C₂²×C₁₉×C₄⋊C₈	2
C₂²×#424	19

Subgroups — C22×C19×#424 #18066055

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₁₉×#424 #18066055