Subgroups — #14572327

Order 2432.

520 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₈×D₁₉	C₂	1	1	yes	maximal, characteristic, complemented
2	#11927051	C₂	1	1	yes	maximal, characteristic
3	#11926613	C₂	1	4	yes	maximal, complemented
4	C₈×C₂²⋊D₁₉	C₂	1	4	yes	maximal, complemented
5	#11926572	C₂	1	1	yes	maximal, characteristic, complemented
6	#11926511	C₂	1	1	yes	maximal, characteristic, complemented
7	#11926507	C₂	1	1	yes	maximal, characteristic, complemented
8	#11926499	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₁₉×#131	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#41×C₁₉	C₂²	1	1	yes	characteristic
11	C₂²×C₄×D₁₉	C₂²	1	1	yes	characteristic
12	C₁₉×#131	—	2	4	no	—
13	C₂×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
14	C₂×M₁₆×C₁₉	C₂²	1	2	yes	complemented
15	C₂²×C₈×C₁₉	C₂²	1	1	yes	characteristic
16	C₂²×C₈×C₁₉	C₂²	1	2	yes	complemented
17	#10597392	C₄	1	2	yes	—
18	#10597384	C₄	1	2	yes	—
19	#10597383	C₄	1	2	yes	—
20	#10597381	C₄	1	2	yes	—
21	C₄×C₂²⋊D₁₉	C₂²	1	4	yes	—
22	#10597372	C₂²	1	1	yes	characteristic
23	C₂×#64520	C₂²	1	1	yes	characteristic
24	C₂²×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
25	C₂×C₈×D₁₉	—	2	4	no	—
26	C₂×C₈×D₁₉	C₂²	1	2	yes	—
27	C₂×C₈×D₁₉	C₂²	1	2	yes	—
28	C₂²⋊(C₁₉⋊C₈)	C₂²	1	4	yes	complemented
29	(C₂×D₁₉)⋊C₈	C₂²	1	4	yes	—
30	#10597272	C₂²	1	2	yes	complemented
31	Dic₁₉⋊C₈	C₂²	1	4	yes	complemented
32	C₈×Dic₁₉	C₂²	1	2	yes	complemented
33	C₂³×D₁₉	C₂×C₄	1	1	yes	characteristic
34	#41×C₁₉	—	2	2	no	—
35	#41×C₁₉	—	2	2	no	—
36	C₂×Q₈×C₁₉	C₂×C₄	1	1	yes	characteristic
37	C₂×D₄×C₁₉	C₂×C₄	1	2	yes	—
38	C₂×D₄×C₁₉	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₉	C₂³	1	2	yes	—
40	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
41	C₂×C₂²⋊D₁₉	C₂×C₄	1	4	yes	—
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	2	no	—
48	M₁₆×C₁₉	—	2	4	no	—
49	C₂×C₈×C₁₉	—	2	4	no	—
50	C₂×C₈×C₁₉	D₄	1	4	yes	complemented
51	C₂×C₈×C₁₉	C₂³	1	2	yes	—
52	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
53	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
54	C₂²⋊Dic₁₉	C₂×C₄	1	2	yes	—
55	C₂²⋊Dic₁₉	C₂×C₄	1	4	yes	—
56	(C₂×D₁₉)⋊C₄	C₂×C₄	1	4	yes	—
57	C₄⋊Dic₁₉	C₂×C₄	1	2	yes	—
58	Dic₁₉⋊C₄	C₂×C₄	1	4	yes	—
59	C₄×Dic₁₉	C₂³	1	2	yes	—
60	#64520	—	2	2	no	—
61	C₂×C₁₉⋊C₈	—	2	2	no	—
62	C₂×C₁₉⋊C₈	C₂³	1	2	yes	—
63	C₂×C₁₉⋊C₈	C₂³	1	2	yes	—
64	C₈×D₁₉	—	2	4	no	—
65	C₈×D₁₉	—	2	4	no	—
66	C₂³×C₁₉	C₂²×C₄	1	2	yes	—
67	C₂³×C₁₉	C₂²×C₄	1	1	yes	characteristic
68	C₂²×D₁₉	—	2	2	no	—
69	C₂²×D₁₉	—	2	4	no	—
70	C₂²×D₁₉	C₂²×C₄	1	2	yes	—
71	Q₈×C₁₉	—	2	2	no	—
72	D₄×C₁₉	—	2	4	no	—
73	D₄×C₁₉	—	2	2	no	—
74	C₂×C₄×C₁₉	—	2	2	no	—
75	C₂×C₄×C₁₉	—	2	2	no	—
76	C₂×C₄×C₁₉	C₂²×C₄	1	2	yes	—
77	C₂×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
79	C₂×C₄×C₁₉	#41	1	2	yes	—
80	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
81	C₂²⋊D₁₉	—	2	8	no	—
82	C₂×Dic₁₉	—	2	2	no	—
83	C₂×Dic₁₉	C₂²×C₄	1	4	yes	—
84	C₂×Dic₁₉	C₂²×C₄	1	2	yes	—
85	C₄×D₁₉	—	2	4	no	—
86	C₄×D₁₉	—	2	2	no	—
87	C₄×D₁₉	—	2	2	no	—
88	C₈×C₁₉	—	2	2	no	—
89	C₈×C₁₉	C₂×D₄	1	2	yes	—
90	C₈×C₁₉	C₂×D₄	1	2	yes	—
91	C₁₉⋊C₈	—	2	2	no	—
92	C₁₉⋊C₈	—	2	2	no	—
93	#2929	—	19	1	no	maximal, Sylow
94	C₂²×C₁₉	—	2	2	no	—
95	C₂²×C₁₉	—	2	2	no	—
96	C₂²×C₁₉	—	2	4	no	—
97	C₂²×C₁₉	C₄×D₄	1	2	yes	—
98	C₂²×C₁₉	C₂³×C₄	1	1	yes	characteristic
99	C₂×D₁₉	—	2	8	no	—
100	C₂×D₁₉	—	2	2	no	—
101	C₂×D₁₉	—	2	2	no	—
102	C₂×D₁₉	#131	1	4	yes	complemented
103	C₄×C₁₉	—	2	2	no	—
104	C₄×C₁₉	C₄×D₄	1	2	yes	—
105	C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
106	C₄×C₁₉	C₂×#41	1	1	yes	characteristic
107	Dic₁₉	—	2	2	no	—
108	Dic₁₉	—	2	4	no	—
109	C₂×#131	D₁₉	1	1	yes	characteristic, complemented
110	C₂³×C₈	—	19	1	no	—
111	#525	—	19	1	no	—
112	#435	—	19	4	no	—
113	C₈×D₄	—	19	4	no	—
114	#424	—	19	1	no	—
115	#408	—	19	1	no	—
116	#407	—	19	1	no	—
117	#405	—	19	1	no	—
118	C₂×C₁₉	—	2	2	no	—
119	C₂×C₁₉	—	4	2	no	—
120	C₂×C₁₉	C₂×C₄×D₄	1	1	yes	characteristic
121	C₂×C₁₉	C₂×#131	1	2	yes	—
122	D₁₉	—	2	4	no	—
123	C₂×#41	C₂×D₁₉	1	1	yes	characteristic
124	C₂³×C₄	—	19	1	no	—
125	#131	—	2	4	no	—
126	C₂×M₁₆	—	19	1	no	—
127	C₂×M₁₆	C₂×D₁₉	1	1	yes	characteristic
128	C₂×M₁₆	C₂×D₁₉	1	2	yes	complemented
129	C₂²×C₈	—	19	1	no	—
130	C₂²×C₈	—	19	2	no	—
131	C₂²×C₈	—	19	2	no	—
132	C₂²×C₈	—	38	4	no	—
133	C₂²×C₈	C₂×D₁₉	1	1	yes	characteristic
134	C₂²×C₈	C₂×D₁₉	1	2	yes	complemented
135	#123	—	19	2	no	—
136	C₂²⋊Q₈	—	19	2	no	—
137	#121	—	19	2	no	—
138	#120	—	19	2	no	—
139	C₄×D₄	—	19	4	no	—
140	#117	—	19	1	no	—
141	C₄⋊C₈	—	19	4	no	—
142	C₂²⋊C₈	—	19	4	no	—
143	C₂²⋊C₈	—	19	4	no	—
144	#97	—	19	2	no	—
145	C₄×C₈	—	19	2	no	—
146	C₁₉	#2929	1	1	yes	characteristic, complemented, Sylow
147	C₂⁴	—	19	1	no	—
148	#41	—	2	2	no	—
149	#41	—	2	2	no	—
150	C₂×Q₈	C₄×D₁₉	1	1	yes	characteristic
151	C₂×D₄	—	19	4	no	—
152	C₂×D₄	C₄×D₁₉	1	2	yes	—
153	C₂×D₄	C₄×D₁₉	1	1	yes	characteristic
154	C₂²×C₄	—	19	1	no	—
155	C₂²×C₄	—	19	2	no	—
156	C₂²×C₄	—	19	2	no	—
157	C₂²×C₄	—	38	2	no	—
158	C₂²×C₄	—	38	2	no	—
159	C₂²×C₄	C₂²×D₁₉	1	2	yes	—
160	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
161	M₁₆	—	2	2	no	—
162	M₁₆	—	2	4	no	—
163	M₁₆	—	38	2	no	—
164	C₂×C₈	—	2	4	no	—
165	C₂×C₈	—	19	2	no	—
166	C₂×C₈	—	19	2	no	—
167	C₂×C₈	—	38	2	no	—
168	C₂×C₈	—	38	4	no	—
169	C₂×C₈	—	38	4	no	—
170	C₂×C₈	C₂²⋊D₁₉	1	4	yes	complemented
171	C₂×C₈	C₂²×D₁₉	1	2	yes	—
172	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
173	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
174	C₄⋊C₄	—	19	4	no	—
175	C₄⋊C₄	—	19	2	no	—
176	C₂²⋊C₄	—	19	2	no	—
177	C₂²⋊C₄	—	19	4	no	—
178	C₂²⋊C₄	—	19	4	no	—
179	C₄²	—	19	2	no	—
180	C₂³	—	19	2	no	—
181	C₂³	—	38	4	no	—
182	C₂³	—	38	2	no	—
183	C₂³	C₂×C₄×D₁₉	1	2	yes	—
184	C₂³	C₂×C₄×D₁₉	1	1	yes	characteristic
185	Q₈	—	2	2	no	—
186	D₄	—	2	4	no	—
187	D₄	—	2	2	no	—
188	D₄	—	38	8	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	19	4	no	—
192	C₂×C₄	—	19	2	no	—
193	C₂×C₄	—	38	2	no	—
194	C₂×C₄	—	38	2	no	—
195	C₂×C₄	—	38	2	no	—
196	C₂×C₄	—	38	4	no	—
197	C₂×C₄	C₂×C₄×D₁₉	1	2	yes	—
198	C₂×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
199	C₂×C₄	#64541	1	2	yes	—
200	C₂×C₄	C₂×C₂²⋊D₁₉	1	2	yes	—
201	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
202	C₈	—	2	2	no	—
203	C₈	—	38	2	no	—
204	C₈	—	38	2	no	—
205	C₈	C₂×C₂²⋊D₁₉	1	2	yes	—
206	C₈	C₂×C₂²⋊D₁₉	1	2	yes	—
207	C₂²	—	2	2	no	—
208	C₂²	—	2	4	no	—
209	C₂²	—	2	2	no	—
210	C₂²	—	19	4	no	—
211	C₂²	—	38	8	no	—
212	C₂²	—	38	2	no	—
213	C₂²	—	38	2	no	—
214	C₂²	C₄×C₂²⋊D₁₉	1	2	yes	—
215	C₂²	C₂²×C₄×D₁₉	1	1	yes	characteristic
216	C₄	—	2	2	no	—
217	C₄	—	38	4	no	—
218	C₄	—	38	2	no	—
219	C₄	C₄×C₂²⋊D₁₉	1	2	yes	—
220	C₄	C₂×#64541	1	1	yes	characteristic
221	C₄	C₂²×C₂²⋊D₁₉	1	1	yes	characteristic
222	C₂	—	2	2	no	—
223	C₂	—	4	2	no	—
224	C₂	—	38	4	no	—
225	C₂	C₂×C₄×C₂²⋊D₁₉	1	1	yes	characteristic
226	C₂	#131×D₁₉	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂²×D₁₉
commutator, socle	C₂²×C₁₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₉
Fitting	C₂×C₁₉×#131	C₂
prime radical	#11927051	C₂
cosocle	C₂×C₄×C₁₉	C₂⁴
hypercenter	C₂×#131	D₁₉
hypocenter	C₁₉	#2929
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
1. C₂²×C₈×D₁₉	C₂	semidirect
3. #11926613	C₂	semidirect
4. C₈×C₂²⋊D₁₉	C₂	semidirect
5. #11926572	C₂	semidirect
6. #11926511	C₂	semidirect
7. #11926507	C₂	semidirect
8. #11926499	C₂	semidirect
9. C₂×C₁₉×#131	C₂	semidirect
14. C₂×M₁₆×C₁₉	C₂²	semidirect
16. C₂²×C₈×C₁₉	C₂²	semidirect
28. C₂²⋊(C₁₉⋊C₈)	C₂²	semidirect
30. #10597272	C₂²	semidirect
31. Dic₁₉⋊C₈	C₂²	semidirect
32. C₈×Dic₁₉	C₂²	semidirect
50. C₂×C₈×C₁₉	D₄	semidirect
102. C₂×D₁₉	#131	semidirect
109. C₂×#131	D₁₉	semidirect
128. C₂×M₁₆	C₂×D₁₉	semidirect
134. C₂²×C₈	C₂×D₁₉	semidirect
146. C₁₉	#2929	semidirect
170. C₂×C₈	C₂²⋊D₁₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₈×D₁₉	2
#11927051	2
#11926613	2
C₈×C₂²⋊D₁₉	2
#11926572	2
#11926511	2
#11926507	2
#11926499	2
C₂×C₁₉×#131	2
#2929	19