Subgroups — C₁₉×#12929 #18018888

Order 4864.

458 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₉×#106	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₉×#365	C₂	1	2	yes	maximal, complemented
3	C₁₉×#2112	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1810	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#1383	C₂	1	2	yes	maximal, complemented
6	C₂³×C₈×C₁₉	C₂²	1	1	yes	characteristic
7	C₂²×C₄⋊C₄×C₁₉	—	2	1	no	—
8	C₂×C₁₉×M₃₂	C₂²	1	2	yes	—
9	C₂×C₁₉×#108	C₂²	1	1	yes	characteristic
10	C₂×C₁₉×#108	C₂²	1	1	yes	characteristic
11	C₂×C₁₉×#106	—	2	4	no	—
12	C₂×C₁₉×#106	—	2	1	no	—
13	C₂×C₁₉×#106	C₂²	1	2	yes	—
14	C₁₉×#407	—	2	1	no	—
15	C₁₉×#365	—	2	4	no	—
16	C₁₉×C₂²⋊C₁₆	C₄	1	4	yes	complemented
17	C₁₉×#341	—	2	1	no	—
18	C₂³×C₄×C₁₉	—	4	1	no	—
19	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
20	C₂×M₁₆×C₁₉	—	2	1	no	—
21	C₂×M₁₆×C₁₉	—	2	1	no	—
22	C₂²×C₈×C₁₉	—	2	2	no	—
23	C₂²×C₈×C₁₉	—	2	2	no	—
24	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₉	Q₈	1	2	yes	—
27	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
29	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
30	C₂×C₄⋊C₄×C₁₉	—	4	2	no	—
31	C₂×C₄⋊C₄×C₁₉	—	4	1	no	—
32	C₁₉×M₃₂	—	2	2	no	—
33	C₁₉×M₃₂	—	2	2	no	—
34	C₂×C₁₆×C₁₉	C₂×C₄	1	4	yes	complemented
35	C₁₉×#108	—	2	2	no	—
36	C₁₉×#108	—	2	2	no	—
37	C₁₉×#106	—	2	4	no	—
38	C₁₉×#106	—	2	2	no	—
39	C₁₉×#106	D₄	1	4	yes	—
40	C₁₉×C₂²⋊C₈	—	4	2	no	—
41	C₂⁴×C₁₉	SD₁₆	1	1	yes	characteristic
42	C₂²×C₄×C₁₉	—	2	1	no	—
43	C₂²×C₄×C₁₉	—	2	1	no	—
44	C₂²×C₄×C₁₉	—	2	1	no	—
45	C₂²×C₄×C₁₉	—	2	1	no	—
46	C₂²×C₄×C₁₉	—	4	4	no	—
47	C₂²×C₄×C₁₉	—	4	2	no	—
48	C₂²×C₄×C₁₉	—	8	4	no	—
49	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
50	C₂²×C₄×C₁₉	SD₁₆	1	1	yes	characteristic
51	C₂²×C₄×C₁₉	SD₁₆	1	2	yes	—
52	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
53	M₁₆×C₁₉	—	4	1	no	—
54	M₁₆×C₁₉	—	4	1	no	—
55	M₁₆×C₁₉	—	4	1	no	—
56	M₁₆×C₁₉	—	4	1	no	—
57	C₂×C₈×C₁₉	—	2	1	no	—
58	C₂×C₈×C₁₉	—	2	1	no	—
59	C₂×C₈×C₁₉	—	2	4	no	—
60	C₂×C₈×C₁₉	—	2	4	no	—
61	C₂×C₈×C₁₉	—	4	1	no	—
62	C₂×C₈×C₁₉	—	4	1	no	—
63	C₂×C₈×C₁₉	C₄⋊C₄	1	2	yes	—
64	C₂×C₈×C₁₉	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₈×C₁₉	C₂×D₄	1	2	yes	—
66	C₂×C₈×C₁₉	C₂×Q₈	1	1	yes	characteristic
67	C₂×C₈×C₁₉	#41	1	2	yes	—
68	C₄⋊C₄×C₁₉	—	2	4	no	—
69	C₄⋊C₄×C₁₉	—	4	4	no	—
70	C₄⋊C₄×C₁₉	—	4	2	no	—
71	C₁₆×C₁₉	—	2	4	no	—
72	#12929	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
73	C₂³×C₁₉	—	2	4	no	—
74	C₂³×C₁₉	—	2	1	no	—
75	C₂³×C₁₉	—	2	1	no	—
76	C₂³×C₁₉	#106	1	2	yes	—
77	C₂³×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
78	C₂×C₄×C₁₉	—	2	1	no	—
79	C₂×C₄×C₁₉	—	2	1	no	—
80	C₂×C₄×C₁₉	—	2	2	no	—
81	C₂×C₄×C₁₉	—	2	2	no	—
82	C₂×C₄×C₁₉	—	2	2	no	—
83	C₂×C₄×C₁₉	—	2	2	no	—
84	C₂×C₄×C₁₉	—	4	4	no	—
85	C₂×C₄×C₁₉	—	8	8	no	—
86	C₂×C₄×C₁₉	—	8	4	no	—
87	C₂×C₄×C₁₉	#106	1	2	yes	—
88	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
90	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
91	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
92	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
93	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
94	C₈×C₁₉	—	2	2	no	—
95	C₈×C₁₉	—	8	1	no	—
96	C₈×C₁₉	—	8	1	no	—
97	C₈×C₁₉	C₄×D₄	1	2	yes	—
98	C₈×C₁₉	C₂²⋊Q₈	1	2	yes	—
99	C₂²×#106	C₂×C₁₉	1	1	yes	characteristic
100	C₂×#365	C₂×C₁₉	1	2	yes	complemented
101	#2112	C₂×C₁₉	1	1	yes	characteristic
102	#1810	C₂×C₁₉	1	1	yes	characteristic
103	#1383	C₂×C₁₉	1	2	yes	complemented
104	C₂²×C₁₉	—	2	4	no	—
105	C₂²×C₁₉	—	2	4	no	—
106	C₂²×C₁₉	—	2	2	no	—
107	C₂²×C₁₉	—	2	2	no	—
108	C₂²×C₁₉	—	2	2	no	—
109	C₂²×C₁₉	#365	1	4	yes	complemented
110	C₂²×C₁₉	C₂×#106	1	1	yes	characteristic
111	C₂²×C₁₉	#450	1	1	yes	characteristic
112	C₂²×C₁₉	#483	1	1	yes	characteristic
113	C₄×C₁₉	—	2	2	no	—
114	C₄×C₁₉	—	8	4	no	—
115	C₄×C₁₉	#385	1	1	yes	characteristic
116	C₄×C₁₉	#428	1	1	yes	characteristic
117	C₄×C₁₉	#448	1	1	yes	characteristic
118	C₄×C₁₉	#481	1	1	yes	characteristic
119	C₂³×C₈	C₂²×C₁₉	1	1	yes	characteristic
120	C₂²×C₄⋊C₄	—	2	1	no	—
121	C₂×M₃₂	C₂²×C₁₉	1	2	yes	—
122	C₂×#108	C₂²×C₁₉	1	1	yes	characteristic
123	C₂×#108	C₂²×C₁₉	1	1	yes	characteristic
124	C₂×#106	—	2	1	no	—
125	C₂×#106	—	2	4	no	—
126	C₂×#106	C₂²×C₁₉	1	2	yes	—
127	#407	—	2	1	no	—
128	#365	—	2	4	no	—
129	C₂²⋊C₁₆	C₄×C₁₉	1	4	yes	complemented
130	#341	—	2	1	no	—
131	C₂×C₁₉	—	2	1	no	—
132	C₂×C₁₉	—	2	4	no	—
133	C₂×C₁₉	—	2	1	no	—
134	C₂×C₁₉	#1853	1	1	yes	characteristic
135	C₂×C₁₉	C₂×#365	1	2	yes	—
136	C₂³×C₄	—	4	1	no	—
137	C₂³×C₄	D₄×C₁₉	1	1	yes	characteristic
138	C₂×M₁₆	—	2	1	no	—
139	C₂×M₁₆	—	2	1	no	—
140	C₂²×C₈	—	2	2	no	—
141	C₂²×C₈	—	2	2	no	—
142	C₂²×C₈	C₂×C₄×C₁₉	1	2	yes	—
143	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
144	C₂²×C₈	Q₈×C₁₉	1	2	yes	—
145	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
146	C₂×C₄⋊C₄	—	2	4	no	—
147	C₂×C₄⋊C₄	—	2	2	no	—
148	C₂×C₄⋊C₄	—	4	2	no	—
149	C₂×C₄⋊C₄	—	4	1	no	—
150	M₃₂	—	2	2	no	—
151	M₃₂	—	2	2	no	—
152	C₂×C₁₆	C₂×C₄×C₁₉	1	4	yes	complemented
153	#108	—	2	2	no	—
154	#108	—	2	2	no	—
155	#106	—	2	4	no	—
156	#106	—	2	2	no	—
157	#106	D₄×C₁₉	1	4	yes	—
158	C₂²⋊C₈	—	4	2	no	—
159	C₁₉	#12929	1	1	yes	characteristic, direct product factor, Sylow
160	C₂⁴	SD₁₆×C₁₉	1	1	yes	characteristic
161	C₂²×C₄	—	2	1	no	—
162	C₂²×C₄	—	2	1	no	—
163	C₂²×C₄	—	2	1	no	—
164	C₂²×C₄	—	2	1	no	—
165	C₂²×C₄	—	4	4	no	—
166	C₂²×C₄	—	4	2	no	—
167	C₂²×C₄	—	8	4	no	—
168	C₂²×C₄	C₄⋊C₄×C₁₉	1	2	yes	—
169	C₂²×C₄	SD₁₆×C₁₉	1	2	yes	—
170	C₂²×C₄	SD₁₆×C₁₉	1	1	yes	characteristic
171	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
172	M₁₆	—	4	1	no	—
173	M₁₆	—	4	1	no	—
174	M₁₆	—	4	1	no	—
175	M₁₆	—	4	1	no	—
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	2	1	no	—
178	C₂×C₈	—	2	4	no	—
179	C₂×C₈	—	2	4	no	—
180	C₂×C₈	—	4	1	no	—
181	C₂×C₈	—	4	1	no	—
182	C₂×C₈	C₄⋊C₄×C₁₉	1	2	yes	—
183	C₂×C₈	C₂²×C₄×C₁₉	1	1	yes	characteristic
184	C₂×C₈	C₂×D₄×C₁₉	1	2	yes	—
185	C₂×C₈	C₂×Q₈×C₁₉	1	1	yes	characteristic
186	C₂×C₈	#41×C₁₉	1	2	yes	—
187	C₄⋊C₄	—	2	4	no	—
188	C₄⋊C₄	—	4	4	no	—
189	C₄⋊C₄	—	4	2	no	—
190	C₁₆	—	2	4	no	—
191	C₂³	—	2	4	no	—
192	C₂³	—	2	1	no	—
193	C₂³	—	2	1	no	—
194	C₂³	C₁₉×#106	1	2	yes	—
195	C₂³	C₂×SD₁₆×C₁₉	1	1	yes	characteristic
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	4	4	no	—
203	C₂×C₄	—	8	8	no	—
204	C₂×C₄	—	8	4	no	—
205	C₂×C₄	C₁₉×#106	1	2	yes	—
206	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	1	yes	characteristic
207	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
208	C₂×C₄	C₁₉×#123	1	1	yes	characteristic
209	C₂×C₄	C₂×SD₁₆×C₁₉	1	1	yes	characteristic
210	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
211	C₂×C₄	C₁₉×#137	1	1	yes	characteristic
212	C₈	—	2	2	no	—
213	C₈	—	8	1	no	—
214	C₈	—	8	1	no	—
215	C₈	C₄×D₄×C₁₉	1	2	yes	—
216	C₈	C₁₉×C₂²⋊Q₈	1	2	yes	—
217	C₂²	—	2	4	no	—
218	C₂²	—	2	4	no	—
219	C₂²	—	2	2	no	—
220	C₂²	—	2	2	no	—
221	C₂²	—	2	2	no	—
222	C₂²	C₁₉×#365	1	4	yes	complemented
223	C₂²	C₂×C₁₉×#106	1	1	yes	characteristic
224	C₂²	C₁₉×#450	1	1	yes	characteristic
225	C₂²	C₁₉×#483	1	1	yes	characteristic
226	C₄	—	2	2	no	—
227	C₄	—	8	4	no	—
228	C₄	C₁₉×#385	1	1	yes	characteristic
229	C₄	C₁₉×#428	1	1	yes	characteristic
230	C₄	C₁₉×#448	1	1	yes	characteristic
231	C₄	C₁₉×#481	1	1	yes	characteristic
232	C₂	—	2	1	no	—
233	C₂	—	2	4	no	—
234	C₂	—	2	1	no	—
235	C₂	C₁₉×#1853	1	1	yes	characteristic
236	C₂	C₂×C₁₉×#365	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂×#106
commutator	C₂×C₈	C₂²×C₄×C₁₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₉
prime radical	C₂⁴×C₁₉	SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂³×C₄×C₁₉ ◁ C₂³×C₈×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₉×#365	C₂	semidirect
5. C₁₉×#1383	C₂	semidirect
16. C₁₉×C₂²⋊C₁₆	C₄	semidirect
34. C₂×C₁₆×C₁₉	C₂×C₄	semidirect
72. #12929	C₁₉	direct (N × H)
100. C₂×#365	C₂×C₁₉	semidirect
103. #1383	C₂×C₁₉	semidirect
109. C₂²×C₁₉	#365	semidirect
129. C₂²⋊C₁₆	C₄×C₁₉	semidirect
152. C₂×C₁₆	C₂×C₄×C₁₉	semidirect
159. C₁₉	#12929	direct (N × H)
222. C₂²	C₁₉×#365	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₁₉×#106	2
C₂×C₁₉×#365	2
C₁₉×#2112	2
C₁₉×#1810	2
C₁₉×#1383	2
#12929	19

Subgroups — C19×#12929 #18018888

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#12929 #18018888