Subgroups — C₁₉×#3092 #14560949

Order 2432.

458 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×#525	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊C₄×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₉×#484	C₂	1	2	yes	maximal, complemented
4	C₁₉×#483	C₂	1	2	yes	maximal, complemented
5	C₁₉×#481	C₂	1	2	yes	maximal, complemented
6	C₁₉×#480	C₂	1	2	yes	maximal, complemented
7	C₁₉×#428	C₂	1	2	yes	maximal, complemented
8	C₁₉×#417	C₂	1	2	yes	maximal, complemented
9	C₁₉×#407	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#41×C₁₉	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₁₉	—	2	1	no	—
12	C₂³×C₄×C₁₉	C₂²	1	1	yes	characteristic
13	C₂×M₁₆×C₁₉	C₂²	1	2	yes	—
14	C₁₉×#123	—	2	1	no	—
15	C₁₉×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
16	C₁₉×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
17	C₁₉×#121	C₂²	1	1	yes	characteristic
18	C₁₉×#121	C₂²	1	1	yes	characteristic
19	C₁₉×#120	—	2	1	no	—
20	C₄×D₄×C₁₉	—	2	1	no	—
21	C₄×D₄×C₁₉	—	2	1	no	—
22	C₁₉×#117	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
24	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
25	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
26	C₂×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
27	C₂×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
28	C₂×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
29	C₁₉×#107	C₂²	1	4	yes	complemented
30	C₁₉×#106	C₂²	1	4	yes	complemented
31	C₁₉×#103	C₂²	1	4	yes	complemented
32	C₁₉×#102	C₂²	1	4	yes	complemented
33	C₁₉×C₂²⋊C₈	C₂²	1	2	yes	—
34	C₁₉×C₂²⋊C₈	C₂²	1	2	yes	—
35	C₂⁴×C₁₉	D₄	1	1	yes	characteristic
36	#41×C₁₉	—	2	4	no	—
37	C₂×Q₈×C₁₉	C₂³	1	1	yes	characteristic
38	C₂×D₄×C₁₉	—	2	1	no	—
39	C₂×D₄×C₁₉	—	2	1	no	—
40	C₂×D₄×C₁₉	—	2	1	no	—
41	C₂×D₄×C₁₉	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₁₉	—	2	4	no	—
43	C₂²×C₄×C₁₉	—	2	2	no	—
44	C₂²×C₄×C₁₉	—	2	2	no	—
45	C₂²×C₄×C₁₉	—	2	2	no	—
46	C₂²×C₄×C₁₉	—	2	1	no	—
47	C₂²×C₄×C₁₉	—	4	4	no	—
48	C₂²×C₄×C₁₉	D₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₉	D₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₉	D₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
54	M₁₆×C₁₉	—	2	4	no	—
55	C₂×C₈×C₁₉	C₂³	1	4	yes	—
56	C₄⋊C₄×C₁₉	—	2	2	no	—
57	C₄⋊C₄×C₁₉	—	2	2	no	—
58	C₄⋊C₄×C₁₉	—	2	2	no	—
59	C₄⋊C₄×C₁₉	—	2	1	no	—
60	C₄⋊C₄×C₁₉	—	2	1	no	—
61	C₄⋊C₄×C₁₉	C₂³	1	4	yes	—
62	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
63	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
64	C₂²⋊C₄×C₁₉	—	2	1	no	—
65	C₂²⋊C₄×C₁₉	—	2	1	no	—
66	C₂²⋊C₄×C₁₉	—	2	1	no	—
67	C₂²⋊C₄×C₁₉	—	2	1	no	—
68	C₂²⋊C₄×C₁₉	—	2	1	no	—
69	C₄²×C₁₉	—	2	1	no	—
70	C₂³×C₁₉	—	2	2	no	—
71	C₂³×C₁₉	—	2	2	no	—
72	C₂³×C₁₉	—	2	2	no	—
73	C₂³×C₁₉	—	2	1	no	—
74	C₂³×C₁₉	C₂×D₄	1	1	yes	characteristic
75	C₂³×C₁₉	C₂×D₄	1	1	yes	characteristic
76	C₂³×C₁₉	C₂×D₄	1	1	yes	characteristic
77	Q₈×C₁₉	—	2	2	no	—
78	D₄×C₁₉	—	2	2	no	—
79	D₄×C₁₉	—	4	2	no	—
80	D₄×C₁₉	—	4	2	no	—
81	D₄×C₁₉	—	4	2	no	—
82	C₂×C₄×C₁₉	—	2	2	no	—
83	C₂×C₄×C₁₉	—	2	2	no	—
84	C₂×C₄×C₁₉	—	2	2	no	—
85	C₂×C₄×C₁₉	—	2	2	no	—
86	C₂×C₄×C₁₉	—	2	2	no	—
87	C₂×C₄×C₁₉	—	2	1	no	—
88	C₂×C₄×C₁₉	—	2	1	no	—
89	C₂×C₄×C₁₉	—	2	1	no	—
90	C₂×C₄×C₁₉	—	2	4	no	—
91	C₂×C₄×C₁₉	—	4	4	no	—
92	C₂×C₄×C₁₉	—	4	4	no	—
93	C₂×C₄×C₁₉	—	4	4	no	—
94	C₂×C₄×C₁₉	—	4	2	no	—
95	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×C₁₉	#41	1	4	yes	—
99	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
100	C₈×C₁₉	—	2	4	no	—
101	#3092	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
102	C₂²×C₁₉	—	2	8	no	—
103	C₂²×C₁₉	—	2	1	no	—
104	C₂²×C₁₉	—	2	1	no	—
105	C₂²×C₁₉	—	2	1	no	—
106	C₂²×C₁₉	—	2	1	no	—
107	C₂²×C₁₉	—	2	1	no	—
108	C₂²×C₁₉	—	2	1	no	—
109	C₂²×C₁₉	—	4	1	no	—
110	C₂²×C₁₉	—	4	1	no	—
111	C₂²×C₁₉	—	4	1	no	—
112	C₂²×C₁₉	#123	1	2	yes	—
113	C₂²×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
114	C₂²×C₁₉	#137	1	2	yes	—
115	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
116	C₄×C₁₉	—	2	1	no	—
117	C₄×C₁₉	—	2	1	no	—
118	C₄×C₁₉	—	4	1	no	—
119	C₄×C₁₉	—	4	4	no	—
120	C₄×C₁₉	—	4	1	no	—
121	C₄×C₁₉	—	4	1	no	—
122	C₄×C₁₉	#123	1	2	yes	—
123	C₄×C₁₉	C₂×#41	1	2	yes	—
124	#525	C₂×C₁₉	1	1	yes	characteristic
125	C₂²×C₄⋊C₄	C₂×C₁₉	1	1	yes	characteristic, complemented
126	#484	C₂×C₁₉	1	2	yes	complemented
127	#483	C₂×C₁₉	1	2	yes	complemented
128	#481	C₂×C₁₉	1	2	yes	complemented
129	#480	C₂×C₁₉	1	2	yes	complemented
130	#428	C₂×C₁₉	1	2	yes	complemented
131	#417	C₂×C₁₉	1	2	yes	complemented
132	#407	C₂×C₁₉	1	1	yes	characteristic, complemented
133	C₂×C₁₉	—	2	2	no	—
134	C₂×C₁₉	—	2	2	no	—
135	C₂×C₁₉	—	2	2	no	—
136	C₂×C₁₉	—	8	1	no	—
137	C₂×C₁₉	C₂×#123	1	1	yes	characteristic
138	C₂×C₁₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
139	C₂×C₁₉	C₂×#137	1	1	yes	characteristic
140	C₂×#41	C₂²×C₁₉	1	1	yes	characteristic
141	C₂³×C₄	—	2	1	no	—
142	C₂³×C₄	C₂²×C₁₉	1	1	yes	characteristic
143	C₂×M₁₆	C₂²×C₁₉	1	2	yes	—
144	#123	—	2	1	no	—
145	C₂²⋊Q₈	C₂²×C₁₉	1	1	yes	characteristic
146	C₂²⋊Q₈	C₂²×C₁₉	1	1	yes	characteristic
147	#121	C₂²×C₁₉	1	1	yes	characteristic
148	#121	C₂²×C₁₉	1	1	yes	characteristic
149	#120	—	2	1	no	—
150	C₄×D₄	—	2	1	no	—
151	C₄×D₄	—	2	1	no	—
152	#117	C₂²×C₁₉	1	1	yes	characteristic
153	C₂×C₄⋊C₄	—	2	1	no	—
154	C₂×C₄⋊C₄	—	2	1	no	—
155	C₂×C₄⋊C₄	—	2	1	no	—
156	C₂×C₄⋊C₄	C₂²×C₁₉	1	2	yes	—
157	C₂×C₄⋊C₄	C₂²×C₁₉	1	2	yes	—
158	C₂×C₄⋊C₄	C₂²×C₁₉	1	2	yes	—
159	#107	C₂²×C₁₉	1	4	yes	complemented
160	#106	C₂²×C₁₉	1	4	yes	complemented
161	#103	C₂²×C₁₉	1	4	yes	complemented
162	#102	C₂²×C₁₉	1	4	yes	complemented
163	C₂²⋊C₈	C₂²×C₁₉	1	2	yes	—
164	C₂²⋊C₈	C₂²×C₁₉	1	2	yes	—
165	C₁₉	#3092	1	1	yes	characteristic, direct product factor, Sylow
166	C₂⁴	D₄×C₁₉	1	1	yes	characteristic
167	#41	—	2	4	no	—
168	C₂×Q₈	C₂³×C₁₉	1	1	yes	characteristic
169	C₂×D₄	—	2	1	no	—
170	C₂×D₄	—	2	1	no	—
171	C₂×D₄	—	2	1	no	—
172	C₂×D₄	C₂³×C₁₉	1	1	yes	characteristic
173	C₂²×C₄	—	2	4	no	—
174	C₂²×C₄	—	2	2	no	—
175	C₂²×C₄	—	2	2	no	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	—	2	1	no	—
178	C₂²×C₄	—	4	4	no	—
179	C₂²×C₄	D₄×C₁₉	1	1	yes	characteristic
180	C₂²×C₄	D₄×C₁₉	1	1	yes	characteristic
181	C₂²×C₄	D₄×C₁₉	1	1	yes	characteristic
182	C₂²×C₄	C₂³×C₁₉	1	1	yes	characteristic
183	C₂²×C₄	C₂³×C₁₉	1	1	yes	characteristic
184	C₂²×C₄	C₂³×C₁₉	1	1	yes	characteristic
185	M₁₆	—	2	4	no	—
186	C₂×C₈	C₂³×C₁₉	1	4	yes	—
187	C₄⋊C₄	—	2	2	no	—
188	C₄⋊C₄	—	2	2	no	—
189	C₄⋊C₄	—	2	2	no	—
190	C₄⋊C₄	—	2	1	no	—
191	C₄⋊C₄	—	2	1	no	—
192	C₄⋊C₄	C₂³×C₁₉	1	4	yes	—
193	C₄⋊C₄	C₂³×C₁₉	1	1	yes	characteristic
194	C₄⋊C₄	C₂³×C₁₉	1	1	yes	characteristic
195	C₂²⋊C₄	—	2	1	no	—
196	C₂²⋊C₄	—	2	1	no	—
197	C₂²⋊C₄	—	2	1	no	—
198	C₂²⋊C₄	—	2	1	no	—
199	C₂²⋊C₄	—	2	1	no	—
200	C₄²	—	2	1	no	—
201	C₂³	—	2	2	no	—
202	C₂³	—	2	2	no	—
203	C₂³	—	2	2	no	—
204	C₂³	—	2	1	no	—
205	C₂³	C₂×D₄×C₁₉	1	1	yes	characteristic
206	C₂³	C₂×D₄×C₁₉	1	1	yes	characteristic
207	C₂³	C₂×D₄×C₁₉	1	1	yes	characteristic
208	Q₈	—	2	2	no	—
209	D₄	—	2	2	no	—
210	D₄	—	4	2	no	—
211	D₄	—	4	2	no	—
212	D₄	—	4	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	4	4	no	—
223	C₂×C₄	—	4	4	no	—
224	C₂×C₄	—	4	4	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
227	C₂×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
228	C₂×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
229	C₂×C₄	#41×C₁₉	1	4	yes	—
230	C₂×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
231	C₈	—	2	4	no	—
232	C₂²	—	2	8	no	—
233	C₂²	—	2	1	no	—
234	C₂²	—	2	1	no	—
235	C₂²	—	2	1	no	—
236	C₂²	—	2	1	no	—
237	C₂²	—	2	1	no	—
238	C₂²	—	2	1	no	—
239	C₂²	—	4	1	no	—
240	C₂²	—	4	1	no	—
241	C₂²	—	4	1	no	—
242	C₂²	C₁₉×#123	1	2	yes	—
243	C₂²	C₁₉×AGL(1,ℤ/8)	1	2	yes	—
244	C₂²	C₁₉×#137	1	2	yes	—
245	C₂²	C₂²×D₄×C₁₉	1	1	yes	characteristic
246	C₄	—	2	1	no	—
247	C₄	—	2	1	no	—
248	C₄	—	4	1	no	—
249	C₄	—	4	4	no	—
250	C₄	—	4	1	no	—
251	C₄	—	4	1	no	—
252	C₄	C₁₉×#123	1	2	yes	—
253	C₄	C₂×#41×C₁₉	1	2	yes	—
254	C₂	—	2	2	no	—
255	C₂	—	2	2	no	—
256	C₂	—	2	2	no	—
257	C₂	—	8	1	no	—
258	C₂	C₂×C₁₉×#123	1	1	yes	characteristic
259	C₂	C₂×C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
260	C₂	C₂×C₁₉×#137	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₉
prime radical	C₁₉×#525	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂³×C₄×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄⋊C₄×C₁₉	C₂	semidirect
3. C₁₉×#484	C₂	semidirect
4. C₁₉×#483	C₂	semidirect
5. C₁₉×#481	C₂	semidirect
6. C₁₉×#480	C₂	semidirect
7. C₁₉×#428	C₂	semidirect
8. C₁₉×#417	C₂	semidirect
9. C₁₉×#407	C₂	semidirect
29. C₁₉×#107	C₂²	semidirect
30. C₁₉×#106	C₂²	semidirect
31. C₁₉×#103	C₂²	semidirect
32. C₁₉×#102	C₂²	semidirect
101. #3092	C₁₉	direct (N × H)
125. C₂²×C₄⋊C₄	C₂×C₁₉	semidirect
126. #484	C₂×C₁₉	semidirect
127. #483	C₂×C₁₉	semidirect
128. #481	C₂×C₁₉	semidirect
129. #480	C₂×C₁₉	semidirect
130. #428	C₂×C₁₉	semidirect
131. #417	C₂×C₁₉	semidirect
132. #407	C₂×C₁₉	semidirect
159. #107	C₂²×C₁₉	semidirect
160. #106	C₂²×C₁₉	semidirect
161. #103	C₂²×C₁₉	semidirect
162. #102	C₂²×C₁₉	semidirect
165. C₁₉	#3092	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×#525	2
C₂²×C₄⋊C₄×C₁₉	2
C₁₉×#484	2
C₁₉×#483	2
C₁₉×#481	2
C₁₉×#480	2
C₁₉×#428	2
C₁₉×#417	2
C₁₉×#407	2
#3092	19

Subgroups — C19×#3092 #14560949

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#3092 #14560949