Subgroups — C₁₉×#33388 #18039347

Order 4864.

1128 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#3584	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#3407	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₉×#2912	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₉×#2145	C₂	1	6	yes	maximal, complemented
5	C₂×C₁₉×#360	C₂	1	6	yes	maximal, complemented
6	C₁₉×#585	—	2	1	no	—
7	C₁₉×#578	C₄	1	4	yes	complemented
8	C₁₉×#577	—	2	4	no	—
9	C₁₉×#576	C₄	1	4	yes	complemented
10	C₁₉×#575	—	2	3	no	—
11	C₂×C₁₉×#135	C₂²	1	6	yes	—
12	C₂×C₁₉×#131	C₂²	1	1	yes	characteristic
13	C₁₉×#520	—	2	1	no	—
14	C₁₉×#504	—	2	4	no	—
15	C₂×C₁₉×M₃₂	C₂²	1	6	yes	complemented
16	C₁₉×#428	—	2	3	no	—
17	C₁₉×#427	C₂²	1	3	yes	complemented
18	C₂×C₁₉×#106	C₂²	1	3	yes	complemented
19	C₁₉×#360	C₂²	1	12	yes	complemented
20	C₁₉×#360	C₂²	1	4	yes	complemented
21	C₁₉×2₋¹⁺⁴	—	2	4	no	—
22	C₁₉×2₋¹⁺⁴	—	4	1	no	—
23	C₁₉×H(5,𝔽₂)	—	2	4	no	—
24	C₁₉×H(5,𝔽₂)	—	4	3	no	—
25	C₂×#41×C₁₉	—	2	6	no	—
26	C₂×#41×C₁₉	—	4	4	no	—
27	C₂×#41×C₁₉	D₄	1	1	yes	characteristic
28	C₁₉×#137	—	2	12	no	—
29	C₁₉×AGL(1,ℤ/8)	—	2	12	no	—
30	C₁₉×#135	—	2	6	no	—
31	C₁₉×#135	—	2	6	no	—
32	C₁₉×#135	C₂×C₄	1	12	yes	complemented
33	C₁₉×#135	C₂³	1	4	yes	—
34	C₂×Q₁₆×C₁₉	C₂×C₄	1	6	yes	complemented
35	C₂×SD₁₆×C₁₉	—	2	6	no	—
36	C₂×D₈×C₁₉	C₂×C₄	1	6	yes	complemented
37	C₁₉×#131	C₂×C₄	1	4	yes	—
38	C₁₉×#131	D₄	1	4	yes	—
39	C₂×M₁₆×C₁₉	D₄	1	3	yes	—
40	C₂²×C₈×C₁₉	C₂³	1	3	yes	—
41	C₄×Q₈×C₁₉	—	4	1	no	—
42	C₄×D₄×C₁₉	—	4	3	no	—
43	C₁₉×#117	—	2	3	no	—
44	C₂×C₄⋊C₄×C₁₉	—	2	3	no	—
45	C₁₉×M₃₂	—	2	6	no	—
46	C₁₉×M₃₂	C₂³	1	4	yes	—
47	C₂×C₁₆×C₁₉	—	2	6	no	—
48	C₁₉×#107	—	2	3	no	—
49	C₁₉×#106	—	2	3	no	—
50	C₁₉×#106	C₂³	1	3	yes	—
51	C₁₉×#106	C₂³	1	1	yes	characteristic
52	#41×C₁₉	—	2	12	no	—
53	#41×C₁₉	—	2	4	no	—
54	#41×C₁₉	—	4	4	no	—
55	#41×C₁₉	—	4	12	no	—
56	#41×C₁₉	—	4	6	no	—
57	#41×C₁₉	—	4	6	no	—
58	#41×C₁₉	C₂²⋊C₄	1	4	yes	—
59	#41×C₁₉	SD₁₆	1	4	yes	—
60	C₂×Q₈×C₁₉	—	2	6	no	—
61	C₂×Q₈×C₁₉	—	4	4	no	—
62	C₂×Q₈×C₁₉	D₈	1	1	yes	characteristic
63	C₂×D₄×C₁₉	—	2	6	no	—
64	C₂×D₄×C₁₉	—	4	6	no	—
65	C₂×D₄×C₁₉	—	4	12	no	—
66	C₂×D₄×C₁₉	D₈	1	3	yes	—
67	C₂²×C₄×C₁₉	—	4	6	no	—
68	C₂²×C₄×C₁₉	—	4	3	no	—
69	C₂²×C₄×C₁₉	C₂×D₄	1	3	yes	—
70	Q₁₆×C₁₉	—	2	6	no	—
71	Q₁₆×C₁₉	C₂²×C₄	1	4	yes	—
72	SD₁₆×C₁₉	—	2	12	no	—
73	SD₁₆×C₁₉	—	2	4	no	—
74	D₈×C₁₉	—	2	6	no	—
75	D₈×C₁₉	C₂²×C₄	1	4	yes	—
76	M₁₆×C₁₉	C₂²⋊C₄	1	12	yes	—
77	C₂×C₈×C₁₉	C₂²×C₄	1	6	yes	—
78	C₂×C₈×C₁₉	C₂×D₄	1	3	yes	—
79	C₂×C₈×C₁₉	C₂×D₄	1	6	yes	—
80	C₂×C₈×C₁₉	C₂⁴	1	1	yes	characteristic
81	C₄⋊C₄×C₁₉	—	2	3	no	—
82	C₄⋊C₄×C₁₉	—	2	1	no	—
83	C₄⋊C₄×C₁₉	—	4	3	no	—
84	C₂²⋊C₄×C₁₉	—	4	3	no	—
85	C₄²×C₁₉	—	4	3	no	—
86	C₁₆×C₁₉	—	2	4	no	—
87	#33388	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₂³×C₁₉	—	4	6	no	—
89	C₂³×C₁₉	C₂×D₈	1	3	yes	—
90	Q₈×C₁₉	—	2	4	no	—
91	Q₈×C₁₉	—	4	6	no	—
92	Q₈×C₁₉	#102	1	4	yes	—
93	D₄×C₁₉	—	2	4	no	—
94	D₄×C₁₉	—	4	4	no	—
95	D₄×C₁₉	—	4	12	no	—
96	D₄×C₁₉	—	4	6	no	—
97	D₄×C₁₉	#102	1	12	yes	—
98	C₂×C₄×C₁₉	—	4	6	no	—
99	C₂×C₄×C₁₉	—	4	4	no	—
100	C₂×C₄×C₁₉	—	4	12	no	—
101	C₂×C₄×C₁₉	—	4	1	no	—
102	C₂×C₄×C₁₉	—	4	3	no	—
103	C₂×C₄×C₁₉	—	16	3	no	—
104	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	6	yes	—
105	C₂×C₄×C₁₉	C₂×D₈	1	3	yes	—
106	C₂×C₄×C₁₉	C₂×SD₁₆	1	6	yes	—
107	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
108	C₈×C₁₉	C₂×C₂²⋊C₄	1	6	yes	—
109	C₈×C₁₉	C₂³×C₄	1	1	yes	characteristic
110	C₈×C₁₉	C₂²×D₄	1	1	yes	characteristic
111	#3584	C₂×C₁₉	1	1	yes	characteristic
112	#3407	C₂×C₁₉	1	1	yes	characteristic, complemented
113	#2912	C₂×C₁₉	1	1	yes	characteristic, complemented
114	#2145	C₂×C₁₉	1	6	yes	complemented
115	C₂×#360	C₂×C₁₉	1	6	yes	complemented
116	C₂²×C₁₉	—	4	3	no	—
117	C₂²×C₁₉	—	4	4	no	—
118	C₂²×C₁₉	—	16	6	no	—
119	C₂²×C₁₉	C₂×#102	1	6	yes	—
120	C₂²×C₁₉	C₂²×D₈	1	1	yes	characteristic
121	C₄×C₁₉	—	4	4	no	—
122	C₄×C₁₉	—	8	1	no	—
123	C₄×C₁₉	—	8	3	no	—
124	C₄×C₁₉	C₂×#102	1	6	yes	—
125	C₄×C₁₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
126	C₄×C₁₉	C₂²×SD₁₆	1	1	yes	characteristic
127	#585	—	2	1	no	—
128	#578	C₄×C₁₉	1	4	yes	complemented
129	#577	—	2	4	no	—
130	#576	C₄×C₁₉	1	4	yes	complemented
131	#575	—	2	3	no	—
132	C₂×#135	C₂²×C₁₉	1	6	yes	—
133	C₂×#131	C₂²×C₁₉	1	1	yes	characteristic
134	#520	—	2	1	no	—
135	#504	—	2	4	no	—
136	C₂×M₃₂	C₂²×C₁₉	1	6	yes	complemented
137	#428	—	2	3	no	—
138	#427	C₂²×C₁₉	1	3	yes	complemented
139	C₂×#106	C₂²×C₁₉	1	3	yes	complemented
140	#360	C₂²×C₁₉	1	12	yes	complemented
141	#360	C₂²×C₁₉	1	4	yes	complemented
142	C₂×C₁₉	—	2	1	no	—
143	C₂×C₁₉	—	2	6	no	—
144	C₂×C₁₉	—	8	4	no	—
145	C₂×C₁₉	C₂²×#102	1	1	yes	characteristic
146	2₋¹⁺⁴	—	2	4	no	—
147	2₋¹⁺⁴	—	4	1	no	—
148	H(5,𝔽₂)	—	2	4	no	—
149	H(5,𝔽₂)	—	4	3	no	—
150	C₂×#41	—	2	6	no	—
151	C₂×#41	—	4	4	no	—
152	C₂×#41	D₄×C₁₉	1	1	yes	characteristic
153	#137	—	2	12	no	—
154	AGL(1,ℤ/8)	—	2	12	no	—
155	#135	—	2	6	no	—
156	#135	—	2	6	no	—
157	#135	C₂×C₄×C₁₉	1	12	yes	complemented
158	#135	C₂³×C₁₉	1	4	yes	—
159	C₂×Q₁₆	C₂×C₄×C₁₉	1	6	yes	complemented
160	C₂×SD₁₆	—	2	6	no	—
161	C₂×D₈	C₂×C₄×C₁₉	1	6	yes	complemented
162	#131	C₂×C₄×C₁₉	1	4	yes	—
163	#131	D₄×C₁₉	1	4	yes	—
164	C₂×M₁₆	D₄×C₁₉	1	3	yes	—
165	C₂²×C₈	C₂³×C₁₉	1	3	yes	—
166	C₄×Q₈	—	4	1	no	—
167	C₄×D₄	—	4	3	no	—
168	#117	—	2	3	no	—
169	C₂×C₄⋊C₄	—	2	3	no	—
170	M₃₂	—	2	6	no	—
171	M₃₂	C₂³×C₁₉	1	4	yes	—
172	C₂×C₁₆	—	2	6	no	—
173	#107	—	2	3	no	—
174	#106	—	2	3	no	—
175	#106	C₂³×C₁₉	1	3	yes	—
176	#106	C₂³×C₁₉	1	1	yes	characteristic
177	C₁₉	#33388	1	1	yes	characteristic, direct product factor, Sylow
178	#41	—	2	12	no	—
179	#41	—	2	4	no	—
180	#41	—	4	4	no	—
181	#41	—	4	12	no	—
182	#41	—	4	6	no	—
183	#41	—	4	6	no	—
184	#41	C₂²⋊C₄×C₁₉	1	4	yes	—
185	#41	SD₁₆×C₁₉	1	4	yes	—
186	C₂×Q₈	—	2	6	no	—
187	C₂×Q₈	—	4	4	no	—
188	C₂×Q₈	D₈×C₁₉	1	1	yes	characteristic
189	C₂×D₄	—	2	6	no	—
190	C₂×D₄	—	4	6	no	—
191	C₂×D₄	—	4	12	no	—
192	C₂×D₄	D₈×C₁₉	1	3	yes	—
193	C₂²×C₄	—	4	6	no	—
194	C₂²×C₄	—	4	3	no	—
195	C₂²×C₄	C₂×D₄×C₁₉	1	3	yes	—
196	Q₁₆	—	2	6	no	—
197	Q₁₆	C₂²×C₄×C₁₉	1	4	yes	—
198	SD₁₆	—	2	12	no	—
199	SD₁₆	—	2	4	no	—
200	D₈	—	2	6	no	—
201	D₈	C₂²×C₄×C₁₉	1	4	yes	—
202	M₁₆	C₂²⋊C₄×C₁₉	1	12	yes	—
203	C₂×C₈	C₂²×C₄×C₁₉	1	6	yes	—
204	C₂×C₈	C₂×D₄×C₁₉	1	3	yes	—
205	C₂×C₈	C₂×D₄×C₁₉	1	6	yes	—
206	C₂×C₈	C₂⁴×C₁₉	1	1	yes	characteristic
207	C₄⋊C₄	—	2	3	no	—
208	C₄⋊C₄	—	2	1	no	—
209	C₄⋊C₄	—	4	3	no	—
210	C₂²⋊C₄	—	4	3	no	—
211	C₄²	—	4	3	no	—
212	C₁₆	—	2	4	no	—
213	C₂³	—	4	6	no	—
214	C₂³	C₂×D₈×C₁₉	1	3	yes	—
215	Q₈	—	2	4	no	—
216	Q₈	—	4	6	no	—
217	Q₈	C₁₉×#102	1	4	yes	—
218	D₄	—	2	4	no	—
219	D₄	—	4	12	no	—
220	D₄	—	4	4	no	—
221	D₄	—	4	6	no	—
222	D₄	C₁₉×#102	1	12	yes	—
223	C₂×C₄	—	4	6	no	—
224	C₂×C₄	—	4	12	no	—
225	C₂×C₄	—	4	4	no	—
226	C₂×C₄	—	4	1	no	—
227	C₂×C₄	—	4	3	no	—
228	C₂×C₄	—	16	3	no	—
229	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	6	yes	—
230	C₂×C₄	C₂×D₈×C₁₉	1	3	yes	—
231	C₂×C₄	C₂×SD₁₆×C₁₉	1	6	yes	—
232	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
233	C₈	C₂×C₂²⋊C₄×C₁₉	1	6	yes	—
234	C₈	C₂³×C₄×C₁₉	1	1	yes	characteristic
235	C₈	C₂²×D₄×C₁₉	1	1	yes	characteristic
236	C₂²	—	4	3	no	—
237	C₂²	—	4	4	no	—
238	C₂²	—	16	6	no	—
239	C₂²	C₂×C₁₉×#102	1	6	yes	—
240	C₂²	C₂²×D₈×C₁₉	1	1	yes	characteristic
241	C₄	—	4	4	no	—
242	C₄	—	8	3	no	—
243	C₄	—	8	1	no	—
244	C₄	C₂×C₁₉×#102	1	6	yes	—
245	C₄	C₂²×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
246	C₄	C₂²×SD₁₆×C₁₉	1	1	yes	characteristic
247	C₂	—	2	1	no	—
248	C₂	—	2	6	no	—
249	C₂	—	8	4	no	—
250	C₂	C₂²×C₁₉×#102	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₁₉	C₂²×#102
commutator	C₈	C₂³×C₄×C₁₉
Frattini, cosocle	C₂×C₈	C₂⁴×C₁₉
prime radical	C₁₉×#3584	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₉ ◁ C₂×#41×C₁₉ ◁ C₂×C₁₉×#131 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₉×#3407	C₂	semidirect
3. C₁₉×#2912	C₂	semidirect
4. C₁₉×#2145	C₂	semidirect
5. C₂×C₁₉×#360	C₂	semidirect
7. C₁₉×#578	C₄	semidirect
9. C₁₉×#576	C₄	semidirect
15. C₂×C₁₉×M₃₂	C₂²	semidirect
17. C₁₉×#427	C₂²	semidirect
18. C₂×C₁₉×#106	C₂²	semidirect
19. C₁₉×#360	C₂²	semidirect
20. C₁₉×#360	C₂²	semidirect
32. C₁₉×#135	C₂×C₄	semidirect
34. C₂×Q₁₆×C₁₉	C₂×C₄	semidirect
36. C₂×D₈×C₁₉	C₂×C₄	semidirect
87. #33388	C₁₉	direct (N × H)
112. #3407	C₂×C₁₉	semidirect
113. #2912	C₂×C₁₉	semidirect
114. #2145	C₂×C₁₉	semidirect
115. C₂×#360	C₂×C₁₉	semidirect
128. #578	C₄×C₁₉	semidirect
130. #576	C₄×C₁₉	semidirect
136. C₂×M₃₂	C₂²×C₁₉	semidirect
138. #427	C₂²×C₁₉	semidirect
139. C₂×#106	C₂²×C₁₉	semidirect
140. #360	C₂²×C₁₉	semidirect
141. #360	C₂²×C₁₉	semidirect
157. #135	C₂×C₄×C₁₉	semidirect
159. C₂×Q₁₆	C₂×C₄×C₁₉	semidirect
161. C₂×D₈	C₂×C₄×C₁₉	semidirect
177. C₁₉	#33388	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×#3584	2
C₁₉×#3407	2
C₁₉×#2912	2
C₁₉×#2145	2
C₂×C₁₉×#360	2
#33388	19

Subgroups — C19×#33388 #18039347

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#33388 #18039347