Subgroups — C₂×C₂²⋊C₈×D₁₉ #14572065

Order 2432.

1072 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄×D₁₉	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₈×D₁₉	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊(C₁₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×(C₂×D₁₉)⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₂²⋊C₈×D₁₉	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₁₉×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁴×D₁₉	C₄	1	1	yes	characteristic
8	C₂³×C₄×C₁₉	C₂²	1	1	yes	characteristic
9	C₂³×Dic₁₉	C₄	1	1	yes	characteristic
10	C₂²×C₄×D₁₉	—	2	8	no	—
11	C₂²×C₄×D₁₉	C₄	1	1	yes	characteristic
12	C₂²×C₄×D₁₉	C₄	1	1	yes	characteristic
13	C₂²×C₄×D₁₉	C₄	1	4	yes	—
14	C₂²×C₄×D₁₉	C₂²	1	4	yes	—
15	C₂²×C₄×D₁₉	C₂²	1	1	yes	characteristic
16	C₂²×C₄×D₁₉	C₂²	1	1	yes	characteristic
17	C₂²×C₈×C₁₉	C₂²	1	2	yes	complemented
18	C₂²×C₁₉⋊C₈	C₂²	1	2	yes	complemented
19	C₂×C₈×D₁₉	—	2	8	no	—
20	C₂×C₈×D₁₉	C₂²	1	8	yes	complemented
21	C₁₉×C₂²⋊C₈	C₂²	1	4	yes	complemented
22	C₂²⋊(C₁₉⋊C₈)	C₂²	1	4	yes	complemented
23	(C₂×D₁₉)⋊C₈	C₂²	1	8	yes	complemented
24	C₂⁴×C₁₉	C₂×C₄	1	1	yes	characteristic
25	C₂³×D₁₉	—	2	4	no	—
26	C₂³×D₁₉	—	2	4	no	—
27	C₂³×D₁₉	C₈	1	8	yes	complemented
28	C₂³×D₁₉	C₂×C₄	1	1	yes	characteristic
29	C₂³×D₁₉	C₂×C₄	1	4	yes	—
30	C₂³×D₁₉	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₉	—	2	4	no	—
32	C₂²×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₉	C₂×C₄	1	2	yes	—
34	C₂²×C₄×C₁₉	C₂³	1	2	yes	—
35	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
36	C₂²×Dic₁₉	—	2	4	no	—
37	C₂²×Dic₁₉	C₂×C₄	1	1	yes	characteristic
38	C₂²×Dic₁₉	C₂×C₄	1	4	yes	—
39	C₂²×Dic₁₉	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄×D₁₉	—	2	4	no	—
41	C₂×C₄×D₁₉	—	2	4	no	—
42	C₂×C₄×D₁₉	—	2	4	no	—
43	C₂×C₄×D₁₉	—	2	16	no	—
44	C₂×C₄×D₁₉	—	2	16	no	—
45	C₂×C₄×D₁₉	C₂×C₄	1	4	yes	—
46	C₂×C₄×D₁₉	C₂×C₄	1	2	yes	—
47	C₂×C₄×D₁₉	C₂×C₄	1	4	yes	—
48	C₂×C₄×D₁₉	C₂×C₄	1	2	yes	—
49	C₂×C₄×D₁₉	D₄	1	8	yes	—
50	C₂×C₄×D₁₉	C₂³	1	2	yes	—
51	C₂×C₄×D₁₉	C₂³	1	2	yes	—
52	C₂×C₈×C₁₉	—	2	4	no	—
53	C₂×C₈×C₁₉	C₂³	1	4	yes	complemented
54	C₂×C₁₉⋊C₈	—	2	4	no	—
55	C₂×C₁₉⋊C₈	C₂³	1	4	yes	complemented
56	C₈×D₁₉	—	2	16	no	—
57	C₂³×C₁₉	—	2	2	no	—
58	C₂³×C₁₉	—	2	2	no	—
59	C₂³×C₁₉	C₂×C₈	1	4	yes	complemented
60	C₂³×C₁₉	C₂²×C₄	1	2	yes	—
61	C₂³×C₁₉	C₂²×C₄	1	1	yes	characteristic
62	C₂²×D₁₉	—	2	2	no	—
63	C₂²×D₁₉	—	2	2	no	—
64	C₂²×D₁₉	—	2	32	no	—
65	C₂²×D₁₉	—	2	8	no	—
66	C₂²×D₁₉	—	2	8	no	—
67	C₂²×D₁₉	C₂²⋊C₄	1	4	yes	—
68	C₂²×D₁₉	C₂×C₈	1	4	yes	—
69	C₂²×D₁₉	C₂×C₈	1	16	yes	complemented
70	C₂²×D₁₉	C₂×C₈	1	4	yes	complemented
71	C₂²×D₁₉	M₁₆	1	4	yes	—
72	C₂²×D₁₉	C₂²×C₄	1	2	yes	—
73	C₂²×D₁₉	C₂²×C₄	1	2	yes	—
74	C₂×C₄×C₁₉	—	2	2	no	—
75	C₂×C₄×C₁₉	—	2	4	no	—
76	C₂×C₄×C₁₉	—	2	4	no	—
77	C₂×C₄×C₁₉	C₂²×C₄	1	2	yes	—
78	C₂×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₁₉	C₂×D₄	1	4	yes	—
80	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
81	C₂×Dic₁₉	—	2	2	no	—
82	C₂×Dic₁₉	—	2	8	no	—
83	C₂×Dic₁₉	C₂²⋊C₄	1	4	yes	—
84	C₂×Dic₁₉	C₂²×C₄	1	2	yes	—
85	C₂×Dic₁₉	C₂²×C₄	1	2	yes	—
86	C₄×D₁₉	—	2	8	no	—
87	C₄×D₁₉	—	2	4	no	—
88	C₄×D₁₉	—	2	4	no	—
89	C₄×D₁₉	—	2	8	no	—
90	C₄×D₁₉	C₂²⋊C₄	1	8	yes	—
91	C₄×D₁₉	C₂×D₄	1	8	yes	—
92	C₈×C₁₉	—	2	4	no	—
93	C₁₉⋊C₈	—	2	4	no	—
94	C₂²×C₂²⋊C₈	—	19	1	no	maximal, Sylow
95	C₂²×C₁₉	—	2	2	no	—
96	C₂²×C₁₉	—	2	2	no	—
97	C₂²×C₁₉	—	2	8	no	—
98	C₂²×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
99	C₂²×C₁₉	C₂²×C₈	1	2	yes	—
100	C₂²×C₁₉	C₂²×C₈	1	4	yes	complemented
101	C₂²×C₁₉	C₂×M₁₆	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	8	no	—
106	C₂×D₁₉	—	2	32	no	—
107	C₂×D₁₉	C₂²⋊C₈	1	16	yes	direct product factor
108	C₂×D₁₉	C₂×C₂²⋊C₄	1	4	yes	—
109	C₂×D₁₉	C₂²×C₈	1	4	yes	—
110	C₂×D₁₉	C₂²×C₈	1	4	yes	complemented
111	C₂×D₁₉	C₂×M₁₆	1	4	yes	—
112	C₄×C₁₉	—	2	2	no	—
113	C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
114	C₄×C₁₉	C₂²×D₄	1	2	yes	—
115	Dic₁₉	—	2	2	no	—
116	Dic₁₉	C₂×C₂²⋊C₄	1	4	yes	—
117	C₂⁴×C₄	—	19	1	no	—
118	C₂³×C₈	—	19	2	no	—
119	C₂×C₂²⋊C₈	—	19	1	no	—
120	C₂×C₂²⋊C₈	—	19	2	no	—
121	C₂×C₂²⋊C₈	—	19	8	no	—
122	C₂×C₂²⋊C₈	D₁₉	1	1	yes	characteristic, direct product factor
123	C₂×C₁₉	—	2	4	no	—
124	C₂×C₁₉	C₂×C₂²⋊C₈	1	4	yes	complemented
125	C₂×C₁₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
126	C₂×C₁₉	C₂³×C₈	1	1	yes	characteristic
127	C₂×C₁₉	C₂²×M₁₆	1	1	yes	characteristic
128	D₁₉	—	2	4	no	—
129	D₁₉	C₂×C₂²⋊C₈	1	8	yes	direct product factor
130	C₂⁵	—	19	1	no	—
131	C₂³×C₄	—	19	1	no	—
132	C₂³×C₄	—	19	1	no	—
133	C₂³×C₄	—	19	1	no	—
134	C₂³×C₄	—	19	4	no	—
135	C₂³×C₄	—	19	1	no	—
136	C₂³×C₄	—	19	4	no	—
137	C₂³×C₄	—	19	1	no	—
138	C₂³×C₄	—	38	8	no	—
139	C₂³×C₄	C₂×D₁₉	1	1	yes	characteristic
140	C₂²×C₈	—	19	2	no	—
141	C₂²×C₈	—	19	8	no	—
142	C₂²×C₈	—	38	8	no	—
143	C₂²×C₈	C₂×D₁₉	1	2	yes	complemented
144	C₂²⋊C₈	—	19	4	no	—
145	C₂²⋊C₈	—	19	8	no	—
146	C₂²⋊C₈	C₂×D₁₉	1	4	yes	direct product factor
147	C₁₉	C₂²×C₂²⋊C₈	1	1	yes	characteristic, complemented, Sylow
148	C₂⁴	—	19	1	no	—
149	C₂⁴	—	19	8	no	—
150	C₂⁴	—	19	4	no	—
151	C₂⁴	—	19	1	no	—
152	C₂⁴	—	38	4	no	—
153	C₂⁴	—	38	4	no	—
154	C₂⁴	C₄×D₁₉	1	1	yes	characteristic
155	C₂²×C₄	—	2	4	no	—
156	C₂²×C₄	—	19	1	no	—
157	C₂²×C₄	—	19	2	no	—
158	C₂²×C₄	—	19	8	no	—
159	C₂²×C₄	—	19	2	no	—
160	C₂²×C₄	—	19	4	no	—
161	C₂²×C₄	—	19	1	no	—
162	C₂²×C₄	—	19	4	no	—
163	C₂²×C₄	—	19	4	no	—
164	C₂²×C₄	—	19	2	no	—
165	C₂²×C₄	—	19	2	no	—
166	C₂²×C₄	—	38	4	no	—
167	C₂²×C₄	—	38	16	no	—
168	C₂²×C₄	—	38	16	no	—
169	C₂²×C₄	—	38	4	no	—
170	C₂²×C₄	—	38	4	no	—
171	C₂²×C₄	—	38	4	no	—
172	C₂²×C₄	C₄×D₁₉	1	1	yes	characteristic
173	C₂²×C₄	C₄×D₁₉	1	2	yes	—
174	C₂²×C₄	C₂²×D₁₉	1	2	yes	—
175	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
176	C₂×C₈	—	2	4	no	—
177	C₂×C₈	—	19	4	no	—
178	C₂×C₈	—	38	16	no	—
179	C₂×C₈	—	38	4	no	—
180	C₂×C₈	C₂²×D₁₉	1	4	yes	complemented
181	C₂³	—	2	2	no	—
182	C₂³	—	2	2	no	—
183	C₂³	—	19	2	no	—
184	C₂³	—	19	4	no	—
185	C₂³	—	19	4	no	—
186	C₂³	—	19	4	no	—
187	C₂³	—	19	2	no	—
188	C₂³	—	19	4	no	—
189	C₂³	—	19	16	no	—
190	C₂³	—	38	8	no	—
191	C₂³	—	38	8	no	—
192	C₂³	—	38	32	no	—
193	C₂³	—	38	2	no	—
194	C₂³	—	38	2	no	—
195	C₂³	C₈×D₁₉	1	4	yes	complemented
196	C₂³	C₂×C₄×D₁₉	1	2	yes	—
197	C₂³	C₂×C₄×D₁₉	1	1	yes	characteristic
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	19	2	no	—
202	C₂×C₄	—	19	4	no	—
203	C₂×C₄	—	19	8	no	—
204	C₂×C₄	—	19	8	no	—
205	C₂×C₄	—	19	2	no	—
206	C₂×C₄	—	38	8	no	—
207	C₂×C₄	—	38	8	no	—
208	C₂×C₄	—	38	8	no	—
209	C₂×C₄	—	38	2	no	—
210	C₂×C₄	—	38	4	no	—
211	C₂×C₄	—	38	4	no	—
212	C₂×C₄	C₂×C₄×D₁₉	1	2	yes	—
213	C₂×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
214	C₂×C₄	D₄×D₁₉	1	4	yes	—
215	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
216	C₈	—	2	4	no	—
217	C₈	—	38	4	no	—
218	C₂²	—	2	2	no	—
219	C₂²	—	2	2	no	—
220	C₂²	—	2	8	no	—
221	C₂²	—	19	4	no	—
222	C₂²	—	19	4	no	—
223	C₂²	—	19	4	no	—
224	C₂²	—	19	16	no	—
225	C₂²	—	19	4	no	—
226	C₂²	—	38	32	no	—
227	C₂²	—	38	2	no	—
228	C₂²	—	38	2	no	—
229	C₂²	—	38	8	no	—
230	C₂²	C₂²⋊C₄×D₁₉	1	2	yes	—
231	C₂²	C₂×C₈×D₁₉	1	2	yes	—
232	C₂²	C₂×C₈×D₁₉	1	4	yes	complemented
233	C₂²	M₁₆×D₁₉	1	2	yes	—
234	C₂²	C₂²×C₄×D₁₉	1	1	yes	characteristic
235	C₄	—	2	2	no	—
236	C₄	—	19	4	no	—
237	C₄	—	38	2	no	—
238	C₄	C₂²⋊C₄×D₁₉	1	2	yes	—
239	C₄	C₂×D₄×D₁₉	1	2	yes	—
240	C₂	—	2	4	no	—
241	C₂	—	19	8	no	—
242	C₂	—	38	4	no	—
243	C₂	C₂²⋊C₈×D₁₉	1	4	yes	direct product factor
244	C₂	C₂×C₂²⋊C₄×D₁₉	1	1	yes	characteristic
245	C₂	C₂²×C₈×D₁₉	1	1	yes	characteristic
246	C₂	C₂×M₁₆×D₁₉	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂²×C₈×D₁₉	C₂	semidirect
3. C₂×C₂²⋊(C₁₉⋊C₈)	C₂	semidirect
4. C₂×(C₂×D₁₉)⋊C₈	C₂	semidirect
5. C₂²⋊C₈×D₁₉	C₂	direct (N × H)
6. C₂×C₁₉×C₂²⋊C₈	C₂	semidirect
17. C₂²×C₈×C₁₉	C₂²	semidirect
18. C₂²×C₁₉⋊C₈	C₂²	semidirect
20. C₂×C₈×D₁₉	C₂²	semidirect
21. C₁₉×C₂²⋊C₈	C₂²	semidirect
22. C₂²⋊(C₁₉⋊C₈)	C₂²	semidirect
23. (C₂×D₁₉)⋊C₈	C₂²	semidirect
27. C₂³×D₁₉	C₈	semidirect
53. C₂×C₈×C₁₉	C₂³	semidirect
55. C₂×C₁₉⋊C₈	C₂³	semidirect
59. C₂³×C₁₉	C₂×C₈	semidirect
69. C₂²×D₁₉	C₂×C₈	semidirect
70. C₂²×D₁₉	C₂×C₈	semidirect
100. C₂²×C₁₉	C₂²×C₈	semidirect
107. C₂×D₁₉	C₂²⋊C₈	direct (N × H)
110. C₂×D₁₉	C₂²×C₈	semidirect
122. C₂×C₂²⋊C₈	D₁₉	direct (N × H)
124. C₂×C₁₉	C₂×C₂²⋊C₈	semidirect
129. D₁₉	C₂×C₂²⋊C₈	direct (N × H)
143. C₂²×C₈	C₂×D₁₉	semidirect
146. C₂²⋊C₈	C₂×D₁₉	direct (N × H)
147. C₁₉	C₂²×C₂²⋊C₈	semidirect
180. C₂×C₈	C₂²×D₁₉	semidirect
195. C₂³	C₈×D₁₉	semidirect
232. C₂²	C₂×C₈×D₁₉	semidirect
243. C₂	C₂²⋊C₈×D₁₉	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×C₄×D₁₉	2
C₂²×C₈×D₁₉	2
C₂×C₂²⋊(C₁₉⋊C₈)	2
C₂×(C₂×D₁₉)⋊C₈	2
C₂²⋊C₈×D₁₉	2
C₂×C₁₉×C₂²⋊C₈	2
C₂²×C₂²⋊C₈	19

Subgroups — C2×C22⋊C8×D19 #14572065

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂²⋊C₈×D₁₉ #14572065