Subgroups — C₁₉×#13196 #18019155

Order 4864.

430 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₉×#443	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₉×#364	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₉×#350	C₂	1	2	yes	maximal, complemented
4	C₁₉×#2113	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₉×#346	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₁₉×#135	C₄	1	1	yes	characteristic
7	C₂×C₁₉×#131	C₂²	1	2	yes	—
8	C₂×C₁₉×M₃₂	C₂²	1	2	yes	—
9	C₂²×C₁₆×C₁₉	C₂²	1	1	yes	characteristic
10	C₂²×C₁₆×C₁₉	C₂²	1	1	yes	characteristic
11	C₁₉×#443	—	2	2	no	—
12	C₁₉×#443	—	2	2	no	—
13	C₂×C₁₉×#108	C₄	1	1	yes	characteristic
14	C₂×C₁₉×#104	C₄	1	2	yes	—
15	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
16	C₁₉×#364	—	2	2	no	—
17	C₁₉×#350	—	2	4	no	—
18	C₁₉×C₂²⋊C₁₆	—	2	4	no	—
19	C₁₉×#346	—	2	2	no	—
20	C₂×#41×C₁₉	C₂×C₄	1	2	yes	—
21	C₁₉×#135	—	2	2	no	—
22	C₁₉×#135	—	2	2	no	—
23	C₂×Q₁₆×C₁₉	C₈	1	1	yes	characteristic
24	C₂×SD₁₆×C₁₉	C₈	1	2	yes	—
25	C₂×D₈×C₁₉	C₈	1	1	yes	characteristic
26	C₁₉×#131	—	2	2	no	—
27	C₁₉×#131	—	2	2	no	—
28	C₁₉×#131	—	2	2	no	—
29	C₁₉×#131	—	2	2	no	—
30	C₂×M₁₆×C₁₉	—	2	2	no	—
31	C₂×M₁₆×C₁₉	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₁₉	—	2	2	no	—
33	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
36	C₁₉×M₃₂	—	2	2	no	—
37	C₁₉×M₃₂	—	2	2	no	—
38	C₂×C₁₆×C₁₉	—	2	1	no	—
39	C₂×C₁₆×C₁₉	—	2	1	no	—
40	C₂×C₁₆×C₁₉	—	2	1	no	—
41	C₂×C₁₆×C₁₉	—	2	1	no	—
42	C₂×C₁₆×C₁₉	—	2	2	no	—
43	C₂×C₁₆×C₁₉	D₄	1	2	yes	—
44	C₂×C₁₆×C₁₉	D₄	1	2	yes	—
45	C₁₉×#108	—	2	2	no	—
46	C₁₉×#104	—	2	4	no	—
47	C₄×C₈×C₁₉	D₄	1	2	yes	—
48	C₄×C₈×C₁₉	D₄	1	2	yes	—
49	#41×C₁₉	—	2	2	no	—
50	#41×C₁₉	—	2	2	no	—
51	#41×C₁₉	—	2	2	no	—
52	#41×C₁₉	—	2	2	no	—
53	C₂×Q₈×C₁₉	C₂×C₈	1	2	yes	—
54	C₂×D₄×C₁₉	—	2	2	no	—
55	C₂×D₄×C₁₉	C₂×C₈	1	2	yes	—
56	C₂²×C₄×C₁₉	—	2	1	no	—
57	C₂²×C₄×C₁₉	—	2	2	no	—
58	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
59	Q₁₆×C₁₉	—	2	2	no	—
60	SD₁₆×C₁₉	—	2	4	no	—
61	D₈×C₁₉	—	2	2	no	—
62	M₁₆×C₁₉	—	2	2	no	—
63	M₁₆×C₁₉	—	2	2	no	—
64	M₁₆×C₁₉	—	4	2	no	—
65	M₁₆×C₁₉	—	4	2	no	—
66	C₂×C₈×C₁₉	—	2	2	no	—
67	C₂×C₈×C₁₉	—	4	2	no	—
68	C₂×C₈×C₁₉	—	4	2	no	—
69	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
70	C₂×C₈×C₁₉	C₂×C₈	1	1	yes	characteristic
71	C₂×C₈×C₁₉	C₂×C₈	1	1	yes	characteristic
72	C₂×C₈×C₁₉	M₁₆	1	2	yes	—
73	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
78	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
79	C₄²×C₁₉	—	2	1	no	—
80	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
81	C₁₆×C₁₉	—	2	2	no	—
82	C₁₆×C₁₉	—	2	2	no	—
83	C₁₆×C₁₉	—	4	2	no	—
84	#13196	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
85	C₂³×C₁₉	—	2	2	no	—
86	C₂³×C₁₉	#131	1	1	yes	characteristic
87	Q₈×C₁₉	—	2	2	no	—
88	Q₈×C₁₉	—	2	2	no	—
89	D₄×C₁₉	—	2	2	no	—
90	D₄×C₁₉	—	2	2	no	—
91	D₄×C₁₉	—	4	2	no	—
92	D₄×C₁₉	—	4	2	no	—
93	C₂×C₄×C₁₉	—	2	2	no	—
94	C₂×C₄×C₁₉	—	2	2	no	—
95	C₂×C₄×C₁₉	—	4	2	no	—
96	C₂×C₄×C₁₉	—	4	2	no	—
97	C₂×C₄×C₁₉	—	4	2	no	—
98	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
100	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
101	C₂×C₄×C₁₉	C₂²×C₈	1	1	yes	characteristic
102	C₂×C₄×C₁₉	C₂×M₁₆	1	1	yes	characteristic
103	C₂×C₄×C₁₉	#131	1	1	yes	characteristic
104	C₈×C₁₉	—	4	2	no	—
105	C₈×C₁₉	C₂²⋊C₈	1	2	yes	—
106	C₈×C₁₉	C₂²⋊C₈	1	2	yes	—
107	C₈×C₁₉	#121	1	1	yes	characteristic
108	C₈×C₁₉	#121	1	1	yes	characteristic
109	C₈×C₁₉	#124	1	1	yes	characteristic
110	C₈×C₁₉	#127	1	1	yes	characteristic
111	C₂×#443	C₂×C₁₉	1	1	yes	characteristic
112	C₂×#364	C₂×C₁₉	1	1	yes	characteristic, complemented
113	C₂×#350	C₂×C₁₉	1	2	yes	complemented
114	#2113	C₂×C₁₉	1	2	yes	complemented
115	C₂×#346	C₂×C₁₉	1	1	yes	characteristic, complemented
116	C₂²×C₁₉	—	2	2	no	—
117	C₂²×C₁₉	—	4	4	no	—
118	C₂²×C₁₉	—	4	2	no	—
119	C₂²×C₁₉	#408	1	1	yes	characteristic
120	C₂²×C₁₉	C₈×D₄	1	1	yes	characteristic
121	C₂²×C₁₉	#436	1	1	yes	characteristic
122	C₄×C₁₉	—	4	2	no	—
123	C₄×C₁₉	—	4	2	no	—
124	C₄×C₁₉	#390	1	1	yes	characteristic
125	C₄×C₁₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
126	C₄×C₁₉	C₈×D₄	1	1	yes	characteristic
127	C₄×C₁₉	#436	1	1	yes	characteristic
128	C₂×#135	C₄×C₁₉	1	1	yes	characteristic
129	C₂×#131	C₂²×C₁₉	1	2	yes	—
130	C₂×M₃₂	C₂²×C₁₉	1	2	yes	—
131	C₂²×C₁₆	C₂²×C₁₉	1	1	yes	characteristic
132	C₂²×C₁₆	C₂²×C₁₉	1	1	yes	characteristic
133	#443	—	2	2	no	—
134	#443	—	2	2	no	—
135	C₂×#108	C₄×C₁₉	1	1	yes	characteristic
136	C₂×#104	C₄×C₁₉	1	2	yes	—
137	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
138	#364	—	2	2	no	—
139	#350	—	2	4	no	—
140	C₂²⋊C₁₆	—	2	4	no	—
141	#346	—	2	2	no	—
142	C₂×C₁₉	—	2	1	no	—
143	C₂×C₁₉	—	2	1	no	—
144	C₂×C₁₉	—	8	2	no	—
145	C₂×C₁₉	#1954	1	1	yes	characteristic
146	C₂×C₁₉	#2171	1	2	yes	—
147	C₂×#41	C₂×C₄×C₁₉	1	2	yes	—
148	#135	—	2	2	no	—
149	#135	—	2	2	no	—
150	C₂×Q₁₆	C₈×C₁₉	1	1	yes	characteristic
151	C₂×SD₁₆	C₈×C₁₉	1	2	yes	—
152	C₂×D₈	C₈×C₁₉	1	1	yes	characteristic
153	#131	—	2	2	no	—
154	#131	—	2	2	no	—
155	#131	—	2	2	no	—
156	#131	—	2	2	no	—
157	C₂×M₁₆	—	2	2	no	—
158	C₂×M₁₆	C₂×C₄×C₁₉	1	2	yes	—
159	C₂²×C₈	—	2	2	no	—
160	C₂²×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
161	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
162	C₂×C₄²	C₂×C₄×C₁₉	1	1	yes	characteristic
163	M₃₂	—	2	2	no	—
164	M₃₂	—	2	2	no	—
165	C₂×C₁₆	—	2	1	no	—
166	C₂×C₁₆	—	2	1	no	—
167	C₂×C₁₆	—	2	1	no	—
168	C₂×C₁₆	—	2	1	no	—
169	C₂×C₁₆	—	2	2	no	—
170	C₂×C₁₆	D₄×C₁₉	1	2	yes	—
171	C₂×C₁₆	D₄×C₁₉	1	2	yes	—
172	#108	—	2	2	no	—
173	#104	—	2	4	no	—
174	C₄×C₈	D₄×C₁₉	1	2	yes	—
175	C₄×C₈	D₄×C₁₉	1	2	yes	—
176	C₁₉	#13196	1	1	yes	characteristic, direct product factor, Sylow
177	#41	—	2	2	no	—
178	#41	—	2	2	no	—
179	#41	—	2	2	no	—
180	#41	—	2	2	no	—
181	C₂×Q₈	C₂×C₈×C₁₉	1	2	yes	—
182	C₂×D₄	—	2	2	no	—
183	C₂×D₄	C₂×C₈×C₁₉	1	2	yes	—
184	C₂²×C₄	—	2	1	no	—
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
187	Q₁₆	—	2	2	no	—
188	SD₁₆	—	2	4	no	—
189	D₈	—	2	2	no	—
190	M₁₆	—	2	2	no	—
191	M₁₆	—	2	2	no	—
192	M₁₆	—	4	2	no	—
193	M₁₆	—	4	2	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	4	2	no	—
196	C₂×C₈	—	4	2	no	—
197	C₂×C₈	C₂²⋊C₄×C₁₉	1	2	yes	—
198	C₂×C₈	C₂×C₈×C₁₉	1	1	yes	characteristic
199	C₂×C₈	C₂×C₈×C₁₉	1	1	yes	characteristic
200	C₂×C₈	M₁₆×C₁₉	1	2	yes	—
201	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
202	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
203	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
204	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
205	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
206	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
207	C₄²	—	2	1	no	—
208	C₄²	C₂²⋊C₄×C₁₉	1	2	yes	—
209	C₁₆	—	2	2	no	—
210	C₁₆	—	2	2	no	—
211	C₁₆	—	4	2	no	—
212	C₂³	—	2	2	no	—
213	C₂³	C₁₉×#131	1	1	yes	characteristic
214	Q₈	—	2	2	no	—
215	Q₈	—	2	2	no	—
216	D₄	—	2	2	no	—
217	D₄	—	2	2	no	—
218	D₄	—	4	2	no	—
219	D₄	—	4	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
226	C₂×C₄	C₄×D₄×C₁₉	1	1	yes	characteristic
227	C₂×C₄	C₄×D₄×C₁₉	1	1	yes	characteristic
228	C₂×C₄	C₂²×C₈×C₁₉	1	1	yes	characteristic
229	C₂×C₄	C₂×M₁₆×C₁₉	1	1	yes	characteristic
230	C₂×C₄	C₁₉×#131	1	1	yes	characteristic
231	C₈	—	4	2	no	—
232	C₈	C₁₉×C₂²⋊C₈	1	2	yes	—
233	C₈	C₁₉×C₂²⋊C₈	1	2	yes	—
234	C₈	C₁₉×#121	1	1	yes	characteristic
235	C₈	C₁₉×#121	1	1	yes	characteristic
236	C₈	C₁₉×#124	1	1	yes	characteristic
237	C₈	C₁₉×#127	1	1	yes	characteristic
238	C₂²	—	2	2	no	—
239	C₂²	—	4	4	no	—
240	C₂²	—	4	2	no	—
241	C₂²	C₁₉×#408	1	1	yes	characteristic
242	C₂²	C₈×D₄×C₁₉	1	1	yes	characteristic
243	C₂²	C₁₉×#436	1	1	yes	characteristic
244	C₄	—	4	2	no	—
245	C₄	—	4	2	no	—
246	C₄	C₁₉×#390	1	1	yes	characteristic
247	C₄	C₂×C₁₉×C₂²⋊C₈	1	1	yes	characteristic
248	C₄	C₈×D₄×C₁₉	1	1	yes	characteristic
249	C₄	C₁₉×#436	1	1	yes	characteristic
250	C₂	—	2	1	no	—
251	C₂	—	2	1	no	—
252	C₂	—	8	2	no	—
253	C₂	C₁₉×#1954	1	1	yes	characteristic
254	C₂	C₁₉×#2171	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₁₉	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₁₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₉
socle	C₂²×C₁₉	C₈×D₄
prime radical	C₂×C₁₉×#135	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₉ ◁ C₂²×C₁₆×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₉×#364	C₂	semidirect
3. C₂×C₁₉×#350	C₂	semidirect
4. C₁₉×#2113	C₂	semidirect
5. C₂×C₁₉×#346	C₂	semidirect
84. #13196	C₁₉	direct (N × H)
112. C₂×#364	C₂×C₁₉	semidirect
113. C₂×#350	C₂×C₁₉	semidirect
114. #2113	C₂×C₁₉	semidirect
115. C₂×#346	C₂×C₁₉	semidirect
176. C₁₉	#13196	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₁₉×#443	2
C₂×C₁₉×#364	2
C₂×C₁₉×#350	2
C₁₉×#2113	2
C₂×C₁₉×#346	2
#13196	19

Subgroups — C19×#13196 #18019155

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#13196 #18019155