Subgroups — C₁₉×#21706 #18027665

Order 4864.

1022 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#3147	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₉×#444	C₂	1	4	yes	maximal, complemented
3	C₁₉×#2931	C₂	1	2	yes	maximal, complemented
4	C₁₉×#1975	C₂	1	4	yes	maximal, complemented
5	C₁₉×#1962	C₂	1	2	yes	maximal, complemented
6	C₁₉×#1848	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₁₉×#97	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₉×#135	C₂²	1	4	yes	—
9	C₂×C₁₉×#131	C₂²	1	4	yes	—
10	C₁₉×#532	C₂²	1	2	yes	—
11	C₁₉×#495	C₄	1	4	yes	complemented
12	C₁₉×#494	C₄	1	1	yes	characteristic, complemented
13	C₁₉×#493	C₄	1	1	yes	characteristic, complemented
14	C₁₉×#492	C₄	1	2	yes	complemented
15	C₁₉×#444	—	2	16	no	—
16	C₁₉×#436	—	2	4	no	—
17	C₈×D₄×C₁₉	—	2	4	no	—
18	C₂×C₁₉×#108	C₂²	1	4	yes	—
19	C₁₉×#423	C₂²	1	2	yes	—
20	C₂×C₁₉×#104	C₂²	1	8	yes	—
21	C₁₉×#408	C₂²	1	4	yes	—
22	C₂×C₁₉×#97	C₂²	1	2	yes	—
23	C₂×C₁₉×#97	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
26	C₄³×C₁₉	C₂²	1	1	yes	characteristic
27	C₂×#41×C₁₉	C₂³	1	4	yes	—
28	C₁₉×#135	—	2	8	no	—
29	C₁₉×#135	—	2	8	no	—
30	C₂×Q₁₆×C₁₉	C₂×C₄	1	4	yes	—
31	C₂×SD₁₆×C₁₉	C₂×C₄	1	8	yes	—
32	C₂×D₈×C₁₉	C₂×C₄	1	4	yes	—
33	C₁₉×#131	—	2	16	no	—
34	C₂×M₁₆×C₁₉	—	2	4	no	—
35	C₂×M₁₆×C₁₉	C₂³	1	4	yes	—
36	C₂²×C₈×C₁₉	—	2	4	no	—
37	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₁₉	C₂³	1	2	yes	—
39	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
40	C₁₉×C₄⋊Q₈	C₂×C₄	1	2	yes	—
41	C₁₉×#127	C₂×C₄	1	2	yes	—
42	C₁₉×#124	C₂×C₄	1	4	yes	—
43	C₁₉×#121	—	2	4	no	—
44	C₄×D₄×C₁₉	—	2	4	no	—
45	C₂×C₄²×C₁₉	—	2	2	no	—
46	C₂×C₄²×C₁₉	C₂³	1	2	yes	—
47	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
48	C₁₉×#108	—	2	8	no	—
49	C₁₉×C₄⋊C₈	—	2	4	no	—
50	C₁₉×#104	—	2	16	no	—
51	C₁₉×C₂²⋊C₈	—	2	8	no	—
52	C₁₉×#97	—	2	4	no	—
53	C₁₉×#97	D₄	1	8	yes	complemented
54	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
55	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
56	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
57	C₄×C₈×C₁₉	D₄	1	4	yes	—
58	#41×C₁₉	—	2	8	no	—
59	#41×C₁₉	—	2	8	no	—
60	C₂×Q₈×C₁₉	C₂²×C₄	1	4	yes	—
61	C₂×D₄×C₁₉	—	2	4	no	—
62	C₂×D₄×C₁₉	—	2	4	no	—
63	C₂×D₄×C₁₉	C₂²×C₄	1	4	yes	—
64	C₂²×C₄×C₁₉	—	2	2	no	—
65	C₂²×C₄×C₁₉	—	2	4	no	—
66	C₂²×C₄×C₁₉	#41	1	2	yes	—
67	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
68	Q₁₆×C₁₉	—	2	8	no	—
69	SD₁₆×C₁₉	—	2	16	no	—
70	D₈×C₁₉	—	2	8	no	—
71	M₁₆×C₁₉	—	2	8	no	—
72	M₁₆×C₁₉	—	4	8	no	—
73	C₂×C₈×C₁₉	—	2	2	no	—
74	C₂×C₈×C₁₉	—	2	4	no	—
75	C₂×C₈×C₁₉	—	4	8	no	—
76	C₂×C₈×C₁₉	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₁₉	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₁₉	C₂×D₄	1	2	yes	—
79	C₂×C₈×C₁₉	C₂×D₄	1	4	yes	—
80	C₂×C₈×C₁₉	C₂×D₄	1	4	yes	—
81	C₂×C₈×C₁₉	C₂×D₄	1	2	yes	—
82	C₂×C₈×C₁₉	#41	1	4	yes	—
83	C₄⋊C₄×C₁₉	—	2	4	no	—
84	C₂²⋊C₄×C₁₉	—	2	8	no	—
85	C₄²×C₁₉	—	2	2	no	—
86	C₄²×C₁₉	—	4	4	no	—
87	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₁₉	C₂×D₄	1	4	yes	—
90	C₄²×C₁₉	C₂×D₄	1	2	yes	—
91	#21706	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
92	C₂³×C₁₉	—	2	4	no	—
93	C₂³×C₁₉	C₂×#41	1	1	yes	characteristic
94	Q₈×C₁₉	—	2	8	no	—
95	D₄×C₁₉	—	2	8	no	—
96	D₄×C₁₉	—	4	8	no	—
97	D₄×C₁₉	—	4	8	no	—
98	C₂×C₄×C₁₉	—	2	4	no	—
99	C₂×C₄×C₁₉	—	2	4	no	—
100	C₂×C₄×C₁₉	—	2	4	no	—
101	C₂×C₄×C₁₉	—	4	4	no	—
102	C₂×C₄×C₁₉	—	4	4	no	—
103	C₂×C₄×C₁₉	—	4	4	no	—
104	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
105	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
106	C₂×C₄×C₁₉	C₂³×C₄	1	1	yes	characteristic
107	C₂×C₄×C₁₉	C₂²×D₄	1	2	yes	—
108	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₁₉	C₂×#41	1	2	yes	—
110	C₈×C₁₉	—	2	2	no	—
111	C₈×C₁₉	—	4	4	no	—
112	C₈×C₁₉	C₄×D₄	1	4	yes	—
113	C₈×C₁₉	C₄×D₄	1	4	yes	—
114	C₈×C₁₉	#127	1	4	yes	—
115	#3147	C₂×C₁₉	1	1	yes	characteristic
116	C₂×#444	C₂×C₁₉	1	4	yes	complemented
117	#2931	C₂×C₁₉	1	2	yes	complemented
118	#1975	C₂×C₁₉	1	4	yes	complemented
119	#1962	C₂×C₁₉	1	2	yes	complemented
120	#1848	C₂×C₁₉	1	1	yes	characteristic, complemented
121	C₄×#97	C₂×C₁₉	1	1	yes	characteristic, complemented
122	C₂²×C₁₉	—	2	2	no	—
123	C₂²×C₁₉	—	4	4	no	—
124	C₂²×C₁₉	—	4	8	no	—
125	C₂²×C₁₉	C₂×C₄×D₄	1	1	yes	characteristic
126	C₂²×C₁₉	#532	1	2	yes	—
127	C₄×C₁₉	—	2	2	no	—
128	C₄×C₁₉	—	4	4	no	—
129	C₄×C₁₉	—	4	4	no	—
130	C₄×C₁₉	#444	1	4	yes	complemented
131	C₄×C₁₉	C₂×C₄×D₄	1	2	yes	—
132	C₄×C₁₉	C₂×#127	1	1	yes	characteristic
133	C₄×C₁₉	#532	1	1	yes	characteristic
134	C₂×#135	C₂²×C₁₉	1	4	yes	—
135	C₂×#131	C₂²×C₁₉	1	4	yes	—
136	#532	C₂²×C₁₉	1	2	yes	—
137	#495	C₄×C₁₉	1	4	yes	complemented
138	#494	C₄×C₁₉	1	1	yes	characteristic, complemented
139	#493	C₄×C₁₉	1	1	yes	characteristic, complemented
140	#492	C₄×C₁₉	1	2	yes	complemented
141	#444	—	2	16	no	—
142	#436	—	2	4	no	—
143	C₈×D₄	—	2	4	no	—
144	C₂×#108	C₂²×C₁₉	1	4	yes	—
145	#423	C₂²×C₁₉	1	2	yes	—
146	C₂×#104	C₂²×C₁₉	1	8	yes	—
147	#408	C₂²×C₁₉	1	4	yes	—
148	C₂×#97	C₂²×C₁₉	1	2	yes	—
149	C₂×#97	C₂²×C₁₉	1	2	yes	—
150	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
151	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
152	C₄³	C₂²×C₁₉	1	1	yes	characteristic
153	C₂×C₁₉	—	2	2	no	—
154	C₂×C₁₉	—	8	4	no	—
155	C₂×C₁₉	C₄×#127	1	1	yes	characteristic
156	C₂×C₁₉	C₂×#444	1	2	yes	—
157	C₂×#41	C₂³×C₁₉	1	4	yes	—
158	#135	—	2	8	no	—
159	#135	—	2	8	no	—
160	C₂×Q₁₆	C₂×C₄×C₁₉	1	4	yes	—
161	C₂×SD₁₆	C₂×C₄×C₁₉	1	8	yes	—
162	C₂×D₈	C₂×C₄×C₁₉	1	4	yes	—
163	#131	—	2	16	no	—
164	C₂×M₁₆	—	2	4	no	—
165	C₂×M₁₆	C₂³×C₁₉	1	4	yes	—
166	C₂²×C₈	—	2	4	no	—
167	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
168	C₂²×C₈	C₂³×C₁₉	1	2	yes	—
169	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
170	C₄⋊Q₈	C₂×C₄×C₁₉	1	2	yes	—
171	#127	C₂×C₄×C₁₉	1	2	yes	—
172	#124	C₂×C₄×C₁₉	1	4	yes	—
173	#121	—	2	4	no	—
174	C₄×D₄	—	2	4	no	—
175	C₂×C₄²	—	2	2	no	—
176	C₂×C₄²	C₂³×C₁₉	1	2	yes	—
177	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
178	#108	—	2	8	no	—
179	C₄⋊C₈	—	2	4	no	—
180	#104	—	2	16	no	—
181	C₂²⋊C₈	—	2	8	no	—
182	#97	—	2	4	no	—
183	#97	D₄×C₁₉	1	8	yes	complemented
184	C₄×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
185	C₄×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
186	C₄×C₈	C₂×C₄×C₁₉	1	2	yes	—
187	C₄×C₈	D₄×C₁₉	1	4	yes	—
188	C₁₉	#21706	1	1	yes	characteristic, direct product factor, Sylow
189	#41	—	2	8	no	—
190	#41	—	2	8	no	—
191	C₂×Q₈	C₂²×C₄×C₁₉	1	4	yes	—
192	C₂×D₄	—	2	4	no	—
193	C₂×D₄	—	2	4	no	—
194	C₂×D₄	C₂²×C₄×C₁₉	1	4	yes	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	2	4	no	—
197	C₂²×C₄	#41×C₁₉	1	2	yes	—
198	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
199	Q₁₆	—	2	8	no	—
200	SD₁₆	—	2	16	no	—
201	D₈	—	2	8	no	—
202	M₁₆	—	2	8	no	—
203	M₁₆	—	4	8	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	—	4	8	no	—
207	C₂×C₈	C₂²×C₄×C₁₉	1	2	yes	—
208	C₂×C₈	C₂²×C₄×C₁₉	1	2	yes	—
209	C₂×C₈	C₂×D₄×C₁₉	1	2	yes	—
210	C₂×C₈	C₂×D₄×C₁₉	1	4	yes	—
211	C₂×C₈	C₂×D₄×C₁₉	1	2	yes	—
212	C₂×C₈	C₂×D₄×C₁₉	1	4	yes	—
213	C₂×C₈	#41×C₁₉	1	4	yes	—
214	C₄⋊C₄	—	2	4	no	—
215	C₂²⋊C₄	—	2	8	no	—
216	C₄²	—	2	2	no	—
217	C₄²	—	4	4	no	—
218	C₄²	C₂²×C₄×C₁₉	1	1	yes	characteristic
219	C₄²	C₂²×C₄×C₁₉	1	1	yes	characteristic
220	C₄²	C₂×D₄×C₁₉	1	4	yes	—
221	C₄²	C₂×D₄×C₁₉	1	2	yes	—
222	C₂³	—	2	4	no	—
223	C₂³	C₂×#41×C₁₉	1	1	yes	characteristic
224	Q₈	—	2	8	no	—
225	D₄	—	2	8	no	—
226	D₄	—	4	8	no	—
227	D₄	—	4	8	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	4	4	no	—
232	C₂×C₄	—	4	4	no	—
233	C₂×C₄	—	4	4	no	—
234	C₂×C₄	C₄×D₄×C₁₉	1	2	yes	—
235	C₂×C₄	C₄×D₄×C₁₉	1	2	yes	—
236	C₂×C₄	C₂³×C₄×C₁₉	1	1	yes	characteristic
237	C₂×C₄	C₂²×D₄×C₁₉	1	2	yes	—
238	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
239	C₂×C₄	C₂×#41×C₁₉	1	2	yes	—
240	C₈	—	2	2	no	—
241	C₈	—	4	4	no	—
242	C₈	C₄×D₄×C₁₉	1	4	yes	—
243	C₈	C₄×D₄×C₁₉	1	4	yes	—
244	C₈	C₁₉×#127	1	4	yes	—
245	C₂²	—	2	2	no	—
246	C₂²	—	4	8	no	—
247	C₂²	—	4	4	no	—
248	C₂²	C₂×C₄×D₄×C₁₉	1	1	yes	characteristic
249	C₂²	C₁₉×#532	1	2	yes	—
250	C₄	—	2	2	no	—
251	C₄	—	4	4	no	—
252	C₄	—	4	4	no	—
253	C₄	C₁₉×#444	1	4	yes	complemented
254	C₄	C₂×C₄×D₄×C₁₉	1	2	yes	—
255	C₄	C₂×C₁₉×#127	1	1	yes	characteristic
256	C₄	C₁₉×#532	1	1	yes	characteristic
257	C₂	—	2	2	no	—
258	C₂	—	8	4	no	—
259	C₂	C₄×C₁₉×#127	1	1	yes	characteristic
260	C₂	C₂×C₁₉×#444	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₉	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₉
socle	C₂²×C₁₉	C₂×C₄×D₄
prime radical	C₁₉×#3147	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₉ ◁ C₂×C₄×C₈×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₉×#444	C₂	semidirect
3. C₁₉×#2931	C₂	semidirect
4. C₁₉×#1975	C₂	semidirect
5. C₁₉×#1962	C₂	semidirect
6. C₁₉×#1848	C₂	semidirect
7. C₄×C₁₉×#97	C₂	semidirect
11. C₁₉×#495	C₄	semidirect
12. C₁₉×#494	C₄	semidirect
13. C₁₉×#493	C₄	semidirect
14. C₁₉×#492	C₄	semidirect
53. C₁₉×#97	D₄	semidirect
91. #21706	C₁₉	direct (N × H)
116. C₂×#444	C₂×C₁₉	semidirect
117. #2931	C₂×C₁₉	semidirect
118. #1975	C₂×C₁₉	semidirect
119. #1962	C₂×C₁₉	semidirect
120. #1848	C₂×C₁₉	semidirect
121. C₄×#97	C₂×C₁₉	semidirect
130. C₄×C₁₉	#444	semidirect
137. #495	C₄×C₁₉	semidirect
138. #494	C₄×C₁₉	semidirect
139. #493	C₄×C₁₉	semidirect
140. #492	C₄×C₁₉	semidirect
183. #97	D₄×C₁₉	semidirect
188. C₁₉	#21706	direct (N × H)
253. C₄	C₁₉×#444	semidirect

Maximal subgroups

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

subgroup	index
C₁₉×#3147	2
C₂×C₁₉×#444	2
C₁₉×#2931	2
C₁₉×#1975	2
C₁₉×#1962	2
C₁₉×#1848	2
C₄×C₁₉×#97	2
#21706	19

Subgroups — C19×#21706 #18027665

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#21706 #18027665