Subgroups — C₂×#11926620 #14572320

Order 2432.

552 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#64515	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×C₂²⋊D₁₉	C₂	1	1	yes	maximal, characteristic
3	#11926620	C₂	1	8	yes	maximal, direct product factor
4	C₂×Dic₁₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₂²⋊(C₁₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×(C₂×D₁₉)⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₈×Dic₁₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×M₁₆×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×C₄×C₁₉	C₂²	1	1	yes	characteristic
10	C₂²×C₂²⋊D₁₉	C₄	1	1	yes	characteristic
11	C₂²×C₄×D₁₉	C₂²	1	1	yes	characteristic
12	C₂×M₁₆×C₁₉	—	2	4	no	—
13	C₂×M₁₆×C₁₉	C₂²	1	4	yes	complemented
14	C₂²×C₈×C₁₉	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₁₉	C₂²	1	1	yes	characteristic
16	C₂×C₂²⋊Dic₁₉	C₄	1	1	yes	characteristic
17	C₄×C₂²⋊D₁₉	C₄	1	4	yes	—
18	C₄×C₂²⋊D₁₉	C₂²	1	4	yes	—
19	C₂×(C₂×D₁₉)⋊C₄	C₄	1	1	yes	characteristic
20	C₂×Dic₁₉⋊C₄	C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₁₉	C₂²	1	1	yes	characteristic
22	C₂²×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
23	C₂²×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
24	C₂×#64515	—	2	4	no	—
25	C₂×#64515	C₂²	1	4	yes	complemented
26	C₂²⋊(C₁₉⋊C₈)	C₂²	1	4	yes	complemented
27	(C₂×D₁₉)⋊C₈	C₂²	1	4	yes	complemented
28	Dic₁₉⋊C₈	C₂²	1	4	yes	complemented
29	C₈×Dic₁₉	C₂²	1	4	yes	complemented
30	C₂⁴×C₁₉	C₂×C₄	1	1	yes	characteristic
31	C₂³×D₁₉	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₉	—	2	2	no	—
33	C₂²×C₄×C₁₉	—	2	2	no	—
34	C₂²×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₉	C₂×C₄	1	2	yes	—
36	C₂²×C₄×C₁₉	C₂³	1	2	yes	—
37	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
38	C₂×C₂²⋊D₁₉	—	2	4	no	—
39	C₂×C₂²⋊D₁₉	C₂×C₄	1	4	yes	—
40	C₂²×Dic₁₉	C₂×C₄	1	1	yes	characteristic
41	C₂²×Dic₁₉	C₂×C₄	1	1	yes	characteristic
42	C₂²×Dic₁₉	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×D₁₉	—	2	2	no	—
44	C₂×C₄×D₁₉	—	2	2	no	—
45	C₂×C₄×D₁₉	C₂×C₄	1	2	yes	—
46	C₂×C₄×D₁₉	C₂³	1	2	yes	—
47	M₁₆×C₁₉	—	2	8	no	—
48	C₂×C₈×C₁₉	—	2	2	no	—
49	C₂×C₈×C₁₉	D₄	1	4	yes	complemented
50	C₂×C₈×C₁₉	C₂³	1	2	yes	—
51	C₂×C₈×C₁₉	C₂³	1	2	yes	—
52	C₂²⋊Dic₁₉	C₂×C₄	1	4	yes	—
53	(C₂×D₁₉)⋊C₄	C₂×C₄	1	4	yes	—
54	Dic₁₉⋊C₄	C₂×C₄	1	4	yes	—
55	C₄×Dic₁₉	C₂×C₄	1	2	yes	—
56	C₄×Dic₁₉	C₂³	1	2	yes	—
57	C₂×C₁₉⋊C₈	—	2	2	no	—
58	C₂×C₁₉⋊C₈	—	2	2	no	—
59	C₂×C₁₉⋊C₈	C₂³	1	2	yes	—
60	C₂×C₁₉⋊C₈	C₂³	1	2	yes	—
61	#64515	—	2	8	no	—
62	C₂³×C₁₉	—	2	2	no	—
63	C₂³×C₁₉	—	2	2	no	—
64	C₂³×C₁₉	—	2	2	no	—
65	C₂³×C₁₉	C₂²×C₄	1	2	yes	—
66	C₂³×C₁₉	C₂²×C₄	1	1	yes	characteristic
67	C₂²×D₁₉	—	2	2	no	—
68	C₂²×D₁₉	—	2	2	no	—
69	C₂²×D₁₉	—	2	2	no	—
70	C₂²×D₁₉	C₂²×C₄	1	2	yes	—
71	C₂×C₄×C₁₉	—	2	2	no	—
72	C₂×C₄×C₁₉	—	2	4	no	—
73	C₂×C₄×C₁₉	—	2	2	no	—
74	C₂×C₄×C₁₉	—	2	2	no	—
75	C₂×C₄×C₁₉	C₂²×C₄	1	2	yes	—
76	C₂×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
78	C₂×C₄×C₁₉	#41	1	2	yes	—
79	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
80	C₂²⋊D₁₉	—	2	8	no	—
81	C₂×Dic₁₉	—	2	2	no	—
82	C₂×Dic₁₉	—	2	2	no	—
83	C₂×Dic₁₉	M₁₆	1	4	yes	complemented
84	C₂×Dic₁₉	C₂²×C₄	1	2	yes	—
85	C₂×Dic₁₉	C₂²×C₄	1	2	yes	—
86	C₂×Dic₁₉	C₂²×C₄	1	2	yes	—
87	C₄×D₁₉	—	2	4	no	—
88	C₄×D₁₉	—	2	2	no	—
89	C₄×D₁₉	—	2	2	no	—
90	C₈×C₁₉	—	2	2	no	—
91	C₈×C₁₉	C₂×D₄	1	4	yes	complemented
92	C₁₉⋊C₈	—	2	2	no	—
93	C₁₉⋊C₈	—	2	2	no	—
94	C₂×#436	—	19	1	no	maximal, Sylow
95	C₂²×C₁₉	—	2	2	no	—
96	C₂²×C₁₉	—	2	2	no	—
97	C₂²×C₁₉	—	2	2	no	—
98	C₂²×C₁₉	—	4	4	no	—
99	C₂²×C₁₉	C₄×D₄	1	2	yes	—
100	C₂²×C₁₉	C₂×M₁₆	1	2	yes	—
101	C₂²×C₁₉	#131	1	2	yes	—
102	C₂²×C₁₉	C₂³×C₄	1	1	yes	characteristic
103	C₂×D₁₉	—	2	2	no	—
104	C₂×D₁₉	—	2	2	no	—
105	C₂×D₁₉	—	2	2	no	—
106	C₂×D₁₉	—	4	4	no	—
107	C₄×C₁₉	—	2	2	no	—
108	C₄×C₁₉	C₄×D₄	1	2	yes	—
109	C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
110	C₄×C₁₉	C₂×#41	1	1	yes	characteristic
111	Dic₁₉	—	2	2	no	—
112	Dic₁₉	—	2	2	no	—
113	Dic₁₉	C₂×M₁₆	1	4	yes	complemented
114	C₂²×M₁₆	—	19	1	no	—
115	C₂²×M₁₆	D₁₉	1	1	yes	characteristic, complemented
116	C₂×C₄×D₄	—	19	1	no	—
117	#436	—	19	8	no	—
118	C₂×C₄⋊C₈	—	19	1	no	—
119	C₂×C₂²⋊C₈	—	19	1	no	—
120	C₂×C₂²⋊C₈	—	19	1	no	—
121	C₂×C₄×C₈	—	19	1	no	—
122	C₂×C₁₉	—	4	2	no	—
123	C₂×C₁₉	#436	1	4	yes	complemented
124	C₂×C₁₉	C₂×C₄×D₄	1	1	yes	characteristic
125	C₂×C₁₉	C₂²×M₁₆	1	1	yes	characteristic
126	C₂×C₁₉	C₂×#131	1	1	yes	characteristic
127	D₁₉	—	4	2	no	—
128	C₂²×D₄	—	19	1	no	—
129	C₂³×C₄	—	19	1	no	—
130	C₂³×C₄	C₂×D₁₉	1	1	yes	characteristic
131	C₂×M₁₆	—	2	4	no	—
132	C₂×M₁₆	—	19	4	no	—
133	C₂×M₁₆	—	38	4	no	—
134	C₂×M₁₆	C₂×D₁₉	1	4	yes	complemented
135	C₂²×C₈	—	19	1	no	—
136	C₂²×C₈	—	19	1	no	—
137	C₂²×C₈	C₂×D₁₉	1	1	yes	characteristic
138	C₂²×C₈	C₂×D₁₉	1	1	yes	characteristic
139	C₄×D₄	—	19	4	no	—
140	C₄×D₄	—	19	4	no	—
141	C₂×C₄⋊C₄	—	19	1	no	—
142	C₂×C₂²⋊C₄	—	19	1	no	—
143	C₂×C₂²⋊C₄	—	19	1	no	—
144	C₂×C₄²	—	19	1	no	—
145	C₄⋊C₈	—	19	4	no	—
146	C₂²⋊C₈	—	19	4	no	—
147	C₂²⋊C₈	—	19	4	no	—
148	C₄×C₈	—	19	4	no	—
149	C₁₉	C₂×#436	1	1	yes	characteristic, complemented, Sylow
150	C₂⁴	—	19	1	no	—
151	C₂⁴	C₄×D₁₉	1	1	yes	characteristic
152	C₂×D₄	—	19	4	no	—
153	C₂×D₄	—	38	4	no	—
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	—	19	1	no	—
157	C₂²×C₄	—	19	1	no	—
158	C₂²×C₄	—	19	1	no	—
159	C₂²×C₄	—	19	2	no	—
160	C₂²×C₄	—	19	2	no	—
161	C₂²×C₄	—	38	2	no	—
162	C₂²×C₄	—	38	2	no	—
163	C₂²×C₄	C₄×D₁₉	1	1	yes	characteristic
164	C₂²×C₄	C₄×D₁₉	1	2	yes	—
165	C₂²×C₄	C₂²×D₁₉	1	2	yes	—
166	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
167	M₁₆	—	2	8	no	—
168	M₁₆	—	38	8	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	19	2	no	—
171	C₂×C₈	—	19	2	no	—
172	C₂×C₈	—	38	2	no	—
173	C₂×C₈	—	38	2	no	—
174	C₂×C₈	C₂²⋊D₁₉	1	4	yes	complemented
175	C₂×C₈	C₂²×D₁₉	1	2	yes	—
176	C₂×C₈	C₂²×D₁₉	1	2	yes	—
177	C₄⋊C₄	—	19	4	no	—
178	C₂²⋊C₄	—	19	4	no	—
179	C₂²⋊C₄	—	19	4	no	—
180	C₄²	—	19	2	no	—
181	C₄²	—	19	2	no	—
182	C₂³	—	2	2	no	—
183	C₂³	—	2	2	no	—
184	C₂³	—	2	2	no	—
185	C₂³	—	19	2	no	—
186	C₂³	—	38	2	no	—
187	C₂³	—	38	2	no	—
188	C₂³	—	38	2	no	—
189	C₂³	C₂×C₄×D₁₉	1	2	yes	—
190	C₂³	C₂×C₄×D₁₉	1	1	yes	characteristic
191	D₄	—	38	8	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	4	no	—
196	C₂×C₄	—	19	2	no	—
197	C₂×C₄	—	19	4	no	—
198	C₂×C₄	—	19	2	no	—
199	C₂×C₄	—	19	2	no	—
200	C₂×C₄	—	38	2	no	—
201	C₂×C₄	—	38	2	no	—
202	C₂×C₄	—	38	2	no	—
203	C₂×C₄	—	38	2	no	—
204	C₂×C₄	—	38	4	no	—
205	C₂×C₄	C₂×C₄×D₁₉	1	2	yes	—
206	C₂×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
207	C₂×C₄	#64541	1	2	yes	—
208	C₂×C₄	C₂×C₂²⋊D₁₉	1	2	yes	—
209	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
210	C₈	—	2	2	no	—
211	C₈	—	38	2	no	—
212	C₈	—	38	2	no	—
213	C₈	C₂×C₂²⋊D₁₉	1	4	yes	complemented
214	C₂²	—	2	2	no	—
215	C₂²	—	2	2	no	—
216	C₂²	—	2	2	no	—
217	C₂²	—	4	4	no	—
218	C₂²	—	38	2	no	—
219	C₂²	—	38	2	no	—
220	C₂²	—	38	2	no	—
221	C₂²	—	76	4	no	—
222	C₂²	M₁₆×D₁₉	1	2	yes	—
223	C₂²	#10597353	1	2	yes	—
224	C₂²	C₄×C₂²⋊D₁₉	1	2	yes	—
225	C₂²	C₂²×C₄×D₁₉	1	1	yes	characteristic
226	C₄	—	2	2	no	—
227	C₄	—	19	4	no	—
228	C₄	—	38	2	no	—
229	C₄	—	38	2	no	—
230	C₄	C₄×C₂²⋊D₁₉	1	2	yes	—
231	C₄	C₂×#64541	1	1	yes	characteristic
232	C₄	C₂²×C₂²⋊D₁₉	1	1	yes	characteristic
233	C₂	—	4	2	no	—
234	C₂	—	76	2	no	—
235	C₂	#11926620	1	4	yes	direct product factor
236	C₂	C₂×C₄×C₂²⋊D₁₉	1	1	yes	characteristic
237	C₂	C₂×M₁₆×D₁₉	1	1	yes	characteristic
238	C₂	C₂×#10597353	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₉
commutator	C₂²×C₁₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₉
Fitting	C₂²×M₁₆×C₁₉	C₂
socle	C₂³×C₁₉	C₂²×C₄
prime radical	C₂²×C₂²⋊D₁₉	C₄
cosocle	C₂×C₄×C₁₉	C₂⁴
hypercenter	C₂²×M₁₆	D₁₉
hypocenter	C₁₉	C₂×#436
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
1. C₂²×#64515	C₂	semidirect
3. #11926620	C₂	direct (N × H)
4. C₂×Dic₁₉⋊C₈	C₂	semidirect
5. C₂×C₂²⋊(C₁₉⋊C₈)	C₂	semidirect
6. C₂×(C₂×D₁₉)⋊C₈	C₂	semidirect
7. C₂×C₈×Dic₁₉	C₂	semidirect
8. C₂²×M₁₆×C₁₉	C₂	semidirect
13. C₂×M₁₆×C₁₉	C₂²	semidirect
25. C₂×#64515	C₂²	semidirect
26. C₂²⋊(C₁₉⋊C₈)	C₂²	semidirect
27. (C₂×D₁₉)⋊C₈	C₂²	semidirect
28. Dic₁₉⋊C₈	C₂²	semidirect
29. C₈×Dic₁₉	C₂²	semidirect
49. C₂×C₈×C₁₉	D₄	semidirect
83. C₂×Dic₁₉	M₁₆	semidirect
91. C₈×C₁₉	C₂×D₄	semidirect
113. Dic₁₉	C₂×M₁₆	semidirect
115. C₂²×M₁₆	D₁₉	semidirect
123. C₂×C₁₉	#436	semidirect
134. C₂×M₁₆	C₂×D₁₉	semidirect
149. C₁₉	C₂×#436	semidirect
174. C₂×C₈	C₂²⋊D₁₉	semidirect
213. C₈	C₂×C₂²⋊D₁₉	semidirect
235. C₂	#11926620	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#64515	2
C₂×C₄×C₂²⋊D₁₉	2
#11926620	2
C₂×Dic₁₉⋊C₈	2
C₂×C₂²⋊(C₁₉⋊C₈)	2
C₂×(C₂×D₁₉)⋊C₈	2
C₂×C₈×Dic₁₉	2
C₂²×M₁₆×C₁₉	2
C₂×#436	19

Subgroups — C2×#11926620 #14572320

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11926620 #14572320