Subgroups — C₁₉×#33495 #18039454

Order 4864.

564 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#2931	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2169	C₂	1	2	yes	maximal, complemented
3	C₁₉×#2168	C₂	1	4	yes	maximal, complemented
4	D₄×C₁₆×C₁₉	C₂	1	2	yes	maximal, complemented
5	C₁₉×#2163	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₉×#2162	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#2113	C₂	1	4	yes	maximal, complemented
8	C₂×C₁₉×#131	C₂²	1	2	yes	—
9	C₁₉×#532	C₄	1	1	yes	characteristic
10	C₂×C₁₉×M₃₂	C₂²	1	4	yes	—
11	C₂²×C₁₆×C₁₉	C₂²	1	4	yes	—
12	C₁₉×#436	C₄	1	2	yes	—
13	C₁₉×#436	C₄	1	2	yes	—
14	C₈×D₄×C₁₉	C₂²	1	2	yes	—
15	C₈×D₄×C₁₉	C₂²	1	2	yes	—
16	C₁₉×#423	C₄	1	1	yes	characteristic
17	C₁₉×#408	C₄	1	2	yes	—
18	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
19	C₁₉×C₄⋊C₁₆	C₂²	1	2	yes	—
20	C₁₉×C₄⋊C₁₆	C₂²	1	2	yes	—
21	C₁₉×C₂²⋊C₁₆	C₂²	1	2	yes	—
22	C₁₉×C₂²⋊C₁₆	C₂²	1	2	yes	—
23	C₁₉×C₂²⋊C₁₆	C₂²	1	4	yes	—
24	C₁₉×C₂²⋊C₁₆	C₂²	1	4	yes	—
25	C₁₉×#346	C₂²	1	2	yes	—
26	C₄×C₁₆×C₁₉	C₂²	1	1	yes	characteristic
27	C₄×C₁₆×C₁₉	C₂²	1	1	yes	characteristic
28	C₂×#41×C₁₉	C₂×C₄	1	2	yes	—
29	C₁₉×#131	—	4	4	no	—
30	C₂×M₁₆×C₁₉	C₂×C₄	1	2	yes	—
31	C₂×M₁₆×C₁₉	C₂×C₄	1	4	yes	—
32	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₁₉	C₂³	1	4	yes	—
35	C₁₉×C₄⋊Q₈	C₈	1	1	yes	characteristic
36	C₁₉×#127	C₈	1	1	yes	characteristic
37	C₁₉×#124	C₈	1	2	yes	—
38	C₁₉×#121	C₈	1	2	yes	—
39	C₁₉×#121	C₈	1	2	yes	—
40	C₄×D₄×C₁₉	C₂×C₄	1	2	yes	—
41	C₄×D₄×C₁₉	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
43	C₁₉×M₃₂	—	4	4	no	—
44	C₂×C₁₆×C₁₉	—	4	4	no	—
45	C₂×C₁₆×C₁₉	C₂³	1	2	yes	—
46	C₂×C₁₆×C₁₉	C₂³	1	2	yes	—
47	C₂×C₁₆×C₁₉	C₂³	1	4	yes	—
48	C₁₉×C₄⋊C₈	C₂×C₄	1	2	yes	—
49	C₁₉×C₄⋊C₈	C₂×C₄	1	2	yes	—
50	C₁₉×C₂²⋊C₈	C₂×C₄	1	4	yes	—
51	C₁₉×C₂²⋊C₈	C₂×C₄	1	4	yes	—
52	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₁₉	C₂³	1	1	yes	characteristic
55	C₄×C₈×C₁₉	C₂³	1	1	yes	characteristic
56	#41×C₁₉	—	4	2	no	—
57	#41×C₁₉	—	4	2	no	—
58	C₂×Q₈×C₁₉	C₂×C₈	1	2	yes	—
59	C₂×D₄×C₁₉	C₂×C₈	1	2	yes	—
60	C₂×D₄×C₁₉	C₂×C₈	1	2	yes	—
61	C₂×D₄×C₁₉	C₂×C₈	1	4	yes	—
62	C₂×D₄×C₁₉	C₂×C₈	1	2	yes	—
63	C₂²×C₄×C₁₉	C₂×C₈	1	1	yes	characteristic
64	C₂²×C₄×C₁₉	C₂×C₈	1	1	yes	characteristic
65	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₉	C₂²×C₄	1	4	yes	—
67	M₁₆×C₁₉	—	4	2	no	—
68	M₁₆×C₁₉	—	4	4	no	—
69	C₂×C₈×C₁₉	—	2	2	no	—
70	C₂×C₈×C₁₉	—	2	4	no	—
71	C₂×C₈×C₁₉	—	2	4	no	—
72	C₂×C₈×C₁₉	—	4	1	no	—
73	C₂×C₈×C₁₉	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₈×C₁₉	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₁₉	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₁₉	C₂²×C₄	1	4	yes	—
77	C₂×C₈×C₁₉	C₂⁴	1	1	yes	characteristic
78	C₄⋊C₄×C₁₉	C₂×C₈	1	2	yes	—
79	C₄⋊C₄×C₁₉	C₂×C₈	1	2	yes	—
80	C₂²⋊C₄×C₁₉	C₂×C₈	1	4	yes	—
81	C₂²⋊C₄×C₁₉	C₂×C₈	1	4	yes	—
82	C₄²×C₁₉	C₂×C₈	1	1	yes	characteristic
83	C₄²×C₁₉	C₂×C₈	1	1	yes	characteristic
84	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
86	C₁₆×C₁₉	—	2	2	no	—
87	C₁₆×C₁₉	—	2	4	no	—
88	C₁₆×C₁₉	—	2	2	no	—
89	#33495	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂³×C₁₉	C₂²×C₈	1	4	yes	—
91	C₂³×C₁₉	C₂²×C₈	1	1	yes	characteristic
92	Q₈×C₁₉	—	4	2	no	—
93	D₄×C₁₉	—	2	4	no	—
94	D₄×C₁₉	—	4	2	no	—
95	D₄×C₁₉	—	4	2	no	—
96	D₄×C₁₉	—	4	4	no	—
97	C₂×C₄×C₁₉	—	2	1	no	—
98	C₂×C₄×C₁₉	—	2	4	no	—
99	C₂×C₄×C₁₉	—	2	1	no	—
100	C₂×C₄×C₁₉	—	2	4	no	—
101	C₂×C₄×C₁₉	—	2	2	no	—
102	C₂×C₄×C₁₉	—	4	1	no	—
103	C₂×C₄×C₁₉	C₂²×C₈	1	1	yes	characteristic
104	C₂×C₄×C₁₉	C₂²×C₈	1	1	yes	characteristic
105	C₂×C₄×C₁₉	C₂²×C₈	1	1	yes	characteristic
106	C₂×C₄×C₁₉	C₂²×C₈	1	1	yes	characteristic
107	C₂×C₄×C₁₉	C₂²×C₈	1	4	yes	—
108	C₂×C₄×C₁₉	C₂²×C₈	1	1	yes	characteristic
109	C₂×C₄×C₁₉	C₂³×C₄	1	1	yes	characteristic
110	C₈×C₁₉	—	2	4	no	—
111	C₈×C₁₉	—	2	1	no	—
112	C₈×C₁₉	—	2	1	no	—
113	C₈×C₁₉	—	2	1	no	—
114	C₈×C₁₉	H(5,𝔽₂)	1	2	yes	—
115	#2931	C₂×C₁₉	1	1	yes	characteristic
116	#2169	C₂×C₁₉	1	2	yes	complemented
117	#2168	C₂×C₁₉	1	4	yes	complemented
118	D₄×C₁₆	C₂×C₁₉	1	2	yes	complemented
119	#2163	C₂×C₁₉	1	1	yes	characteristic, complemented
120	#2162	C₂×C₁₉	1	1	yes	characteristic, complemented
121	#2113	C₂×C₁₉	1	4	yes	complemented
122	C₂²×C₁₉	—	2	1	no	—
123	C₂²×C₁₉	—	2	4	no	—
124	C₂²×C₁₉	—	2	1	no	—
125	C₂²×C₁₉	—	2	4	no	—
126	C₂²×C₁₉	—	2	1	no	—
127	C₂²×C₁₉	—	2	4	no	—
128	C₂²×C₁₉	C₂³×C₈	1	1	yes	characteristic
129	C₄×C₁₉	—	2	1	no	—
130	C₄×C₁₉	—	2	1	no	—
131	C₄×C₁₉	—	2	1	no	—
132	C₄×C₁₉	—	2	1	no	—
133	C₄×C₁₉	—	2	4	no	—
134	C₄×C₁₉	#504	1	2	yes	—
135	C₄×C₁₉	#518	1	1	yes	characteristic
136	C₄×C₁₉	#568	1	1	yes	characteristic
137	C₂×#131	C₂²×C₁₉	1	2	yes	—
138	#532	C₄×C₁₉	1	1	yes	characteristic
139	C₂×M₃₂	C₂²×C₁₉	1	4	yes	—
140	C₂²×C₁₆	C₂²×C₁₉	1	4	yes	—
141	#436	C₄×C₁₉	1	2	yes	—
142	#436	C₄×C₁₉	1	2	yes	—
143	C₈×D₄	C₂²×C₁₉	1	2	yes	—
144	C₈×D₄	C₂²×C₁₉	1	2	yes	—
145	#423	C₄×C₁₉	1	1	yes	characteristic
146	#408	C₄×C₁₉	1	2	yes	—
147	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
148	C₄⋊C₁₆	C₂²×C₁₉	1	2	yes	—
149	C₄⋊C₁₆	C₂²×C₁₉	1	2	yes	—
150	C₂²⋊C₁₆	C₂²×C₁₉	1	2	yes	—
151	C₂²⋊C₁₆	C₂²×C₁₉	1	2	yes	—
152	C₂²⋊C₁₆	C₂²×C₁₉	1	4	yes	—
153	C₂²⋊C₁₆	C₂²×C₁₉	1	4	yes	—
154	#346	C₂²×C₁₉	1	2	yes	—
155	C₄×C₁₆	C₂²×C₁₉	1	1	yes	characteristic
156	C₄×C₁₆	C₂²×C₁₉	1	1	yes	characteristic
157	C₂×C₁₉	—	4	1	no	—
158	C₂×C₁₉	—	4	4	no	—
159	C₂×C₁₉	#2972	1	1	yes	characteristic
160	C₂×C₁₉	C₂×#504	1	1	yes	characteristic
161	C₂×C₁₉	#3407	1	1	yes	characteristic
162	C₂×#41	C₂×C₄×C₁₉	1	2	yes	—
163	#131	—	4	4	no	—
164	C₂×M₁₆	C₂×C₄×C₁₉	1	2	yes	—
165	C₂×M₁₆	C₂×C₄×C₁₉	1	4	yes	—
166	C₂²×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
167	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
168	C₂²×C₈	C₂³×C₁₉	1	4	yes	—
169	C₄⋊Q₈	C₈×C₁₉	1	1	yes	characteristic
170	#127	C₈×C₁₉	1	1	yes	characteristic
171	#124	C₈×C₁₉	1	2	yes	—
172	#121	C₈×C₁₉	1	2	yes	—
173	#121	C₈×C₁₉	1	2	yes	—
174	C₄×D₄	C₂×C₄×C₁₉	1	2	yes	—
175	C₄×D₄	C₂×C₄×C₁₉	1	2	yes	—
176	C₂×C₄²	C₂×C₄×C₁₉	1	1	yes	characteristic
177	M₃₂	—	4	4	no	—
178	C₂×C₁₆	—	4	4	no	—
179	C₂×C₁₆	C₂³×C₁₉	1	2	yes	—
180	C₂×C₁₆	C₂³×C₁₉	1	2	yes	—
181	C₂×C₁₆	C₂³×C₁₉	1	4	yes	—
182	C₄⋊C₈	C₂×C₄×C₁₉	1	2	yes	—
183	C₄⋊C₈	C₂×C₄×C₁₉	1	2	yes	—
184	C₂²⋊C₈	C₂×C₄×C₁₉	1	4	yes	—
185	C₂²⋊C₈	C₂×C₄×C₁₉	1	4	yes	—
186	C₄×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
187	C₄×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
188	C₄×C₈	C₂³×C₁₉	1	1	yes	characteristic
189	C₄×C₈	C₂³×C₁₉	1	1	yes	characteristic
190	C₁₉	#33495	1	1	yes	characteristic, direct product factor, Sylow
191	#41	—	4	2	no	—
192	#41	—	4	2	no	—
193	C₂×Q₈	C₂×C₈×C₁₉	1	2	yes	—
194	C₂×D₄	C₂×C₈×C₁₉	1	2	yes	—
195	C₂×D₄	C₂×C₈×C₁₉	1	2	yes	—
196	C₂×D₄	C₂×C₈×C₁₉	1	4	yes	—
197	C₂×D₄	C₂×C₈×C₁₉	1	2	yes	—
198	C₂²×C₄	C₂×C₈×C₁₉	1	1	yes	characteristic
199	C₂²×C₄	C₂×C₈×C₁₉	1	1	yes	characteristic
200	C₂²×C₄	C₂²×C₄×C₁₉	1	4	yes	—
201	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
202	M₁₆	—	4	4	no	—
203	M₁₆	—	4	2	no	—
204	C₂×C₈	—	2	4	no	—
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	4	1	no	—
208	C₂×C₈	C₂²×C₄×C₁₉	1	1	yes	characteristic
209	C₂×C₈	C₂²×C₄×C₁₉	1	1	yes	characteristic
210	C₂×C₈	C₂²×C₄×C₁₉	1	1	yes	characteristic
211	C₂×C₈	C₂²×C₄×C₁₉	1	4	yes	—
212	C₂×C₈	C₂⁴×C₁₉	1	1	yes	characteristic
213	C₄⋊C₄	C₂×C₈×C₁₉	1	2	yes	—
214	C₄⋊C₄	C₂×C₈×C₁₉	1	2	yes	—
215	C₂²⋊C₄	C₂×C₈×C₁₉	1	4	yes	—
216	C₂²⋊C₄	C₂×C₈×C₁₉	1	4	yes	—
217	C₄²	C₂×C₈×C₁₉	1	1	yes	characteristic
218	C₄²	C₂×C₈×C₁₉	1	1	yes	characteristic
219	C₄²	C₂²×C₄×C₁₉	1	1	yes	characteristic
220	C₄²	C₂²×C₄×C₁₉	1	1	yes	characteristic
221	C₁₆	—	2	2	no	—
222	C₁₆	—	2	4	no	—
223	C₁₆	—	2	2	no	—
224	C₂³	C₂²×C₈×C₁₉	1	4	yes	—
225	C₂³	C₂²×C₈×C₁₉	1	1	yes	characteristic
226	Q₈	—	4	2	no	—
227	D₄	—	2	4	no	—
228	D₄	—	4	2	no	—
229	D₄	—	4	4	no	—
230	D₄	—	4	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	4	1	no	—
237	C₂×C₄	C₂²×C₈×C₁₉	1	1	yes	characteristic
238	C₂×C₄	C₂²×C₈×C₁₉	1	1	yes	characteristic
239	C₂×C₄	C₂²×C₈×C₁₉	1	1	yes	characteristic
240	C₂×C₄	C₂²×C₈×C₁₉	1	1	yes	characteristic
241	C₂×C₄	C₂²×C₈×C₁₉	1	4	yes	—
242	C₂×C₄	C₂²×C₈×C₁₉	1	1	yes	characteristic
243	C₂×C₄	C₂³×C₄×C₁₉	1	1	yes	characteristic
244	C₈	—	2	1	no	—
245	C₈	—	2	4	no	—
246	C₈	—	2	1	no	—
247	C₈	—	2	1	no	—
248	C₈	C₁₉×H(5,𝔽₂)	1	2	yes	—
249	C₂²	—	2	1	no	—
250	C₂²	—	2	4	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	4	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	4	no	—
255	C₂²	C₂³×C₈×C₁₉	1	1	yes	characteristic
256	C₄	—	2	1	no	—
257	C₄	—	2	4	no	—
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	2	1	no	—
261	C₄	C₁₉×#504	1	2	yes	—
262	C₄	C₁₉×#518	1	1	yes	characteristic
263	C₄	C₁₉×#568	1	1	yes	characteristic
264	C₂	—	4	1	no	—
265	C₂	—	4	4	no	—
266	C₂	C₁₉×#2972	1	1	yes	characteristic
267	C₂	C₂×C₁₉×#504	1	1	yes	characteristic
268	C₂	C₁₉×#3407	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₁₉	C₂⁴
commutator	C₂²	C₂³×C₈×C₁₉
Frattini, cosocle	C₂×C₈	C₂⁴×C₁₉
socle	C₂²×C₁₉	C₂³×C₈
prime radical	C₁₉×#532	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₉×#2169	C₂	semidirect
3. C₁₉×#2168	C₂	semidirect
4. D₄×C₁₆×C₁₉	C₂	semidirect
5. C₁₉×#2163	C₂	semidirect
6. C₁₉×#2162	C₂	semidirect
7. C₁₉×#2113	C₂	semidirect
89. #33495	C₁₉	direct (N × H)
116. #2169	C₂×C₁₉	semidirect
117. #2168	C₂×C₁₉	semidirect
118. D₄×C₁₆	C₂×C₁₉	semidirect
119. #2163	C₂×C₁₉	semidirect
120. #2162	C₂×C₁₉	semidirect
121. #2113	C₂×C₁₉	semidirect
190. C₁₉	#33495	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₉×#2931	2
C₁₉×#2169	2
C₁₉×#2168	2
D₄×C₁₆×C₁₉	2
C₁₉×#2163	2
C₁₉×#2162	2
C₁₉×#2113	2
#33495	19

Subgroups — C19×#33495 #18039454

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#33495 #18039454