Subgroups — C₁₉×#33464 #18039423

Order 4864.

642 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#3414	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2957	C₂	1	2	yes	maximal, complemented
3	C₁₉×#2179	C₂	1	4	yes	maximal, complemented
4	C₁₉×#2178	C₂	1	4	yes	maximal, complemented
5	C₁₉×#2161	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₁₉×#359	C₂	1	2	yes	maximal, complemented
7	C₁₉×#2108	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₉×#135	C₂²	1	2	yes	—
9	C₁₉×#568	—	2	2	no	—
10	C₁₉×#510	C₄	1	4	yes	complemented
11	C₁₉×#509	C₄	1	4	yes	complemented
12	C₁₉×#508	C₂²	1	4	yes	—
13	C₂×C₁₉×M₃₂	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆×C₁₉	C₂²	1	1	yes	characteristic
15	C₁₉×#444	—	2	4	no	—
16	C₁₉×#443	C₂²	1	8	yes	complemented
17	C₂×C₁₉×#108	C₂²	1	2	yes	—
18	C₁₉×#421	—	2	2	no	—
19	C₁₉×#405	C₂²	1	1	yes	characteristic
20	C₁₉×#368	C₂²	1	4	yes	complemented
21	C₁₉×#359	C₂²	1	4	yes	complemented
22	C₁₉×#359	C₂²	1	4	yes	complemented
23	C₁₉×#346	C₂²	1	2	yes	complemented
24	C₄×C₁₆×C₁₉	C₂²	1	2	yes	complemented
25	C₂×#41×C₁₉	—	2	2	no	—
26	C₁₉×#135	—	2	2	no	—
27	C₁₉×#135	—	2	2	no	—
28	C₁₉×#135	C₂×C₄	1	4	yes	—
29	C₁₉×#135	C₂³	1	4	yes	—
30	C₂×Q₁₆×C₁₉	C₂×C₄	1	2	yes	—
31	C₂×SD₁₆×C₁₉	—	2	2	no	—
32	C₂×D₈×C₁₉	C₂×C₄	1	2	yes	—
33	C₁₉×#131	—	2	8	no	—
34	C₁₉×#131	—	4	4	no	—
35	C₂×M₁₆×C₁₉	—	2	2	no	—
36	C₂×M₁₆×C₁₉	—	4	4	no	—
37	C₂×M₁₆×C₁₉	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
39	C₁₉×#117	D₄	1	1	yes	characteristic
40	C₁₉×Q₃₂	C₂×C₄	1	4	yes	—
41	C₁₉×SD₃₂	C₂×C₄	1	8	yes	—
42	C₁₉×D₁₆	C₂×C₄	1	4	yes	—
43	C₁₉×M₃₂	C₂×C₄	1	4	yes	—
44	C₂×C₁₆×C₁₉	—	2	2	no	—
45	C₂×C₁₆×C₁₉	C₂³	1	1	yes	characteristic
46	C₂×C₁₆×C₁₉	C₂³	1	1	yes	characteristic
47	C₂×C₁₆×C₁₉	C₂³	1	2	yes	—
48	C₁₉×#108	—	2	2	no	—
49	C₁₉×#108	C₂³	1	4	yes	—
50	C₁₉×#104	—	2	8	no	—
51	C₁₉×#97	D₄	1	2	yes	—
52	C₄×C₈×C₁₉	C₂³	1	2	yes	—
53	#41×C₁₉	—	2	4	no	—
54	#41×C₁₉	—	2	4	no	—
55	#41×C₁₉	—	4	2	no	—
56	#41×C₁₉	—	4	2	no	—
57	C₂×Q₈×C₁₉	—	2	2	no	—
58	C₂×D₄×C₁₉	—	2	2	no	—
59	C₂×D₄×C₁₉	—	4	2	no	—
60	C₂²×C₄×C₁₉	—	4	2	no	—
61	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
62	Q₁₆×C₁₉	—	2	2	no	—
63	Q₁₆×C₁₉	C₂²×C₄	1	4	yes	—
64	SD₁₆×C₁₉	—	2	4	no	—
65	SD₁₆×C₁₉	—	2	4	no	—
66	D₈×C₁₉	—	2	2	no	—
67	D₈×C₁₉	C₂²×C₄	1	4	yes	—
68	M₁₆×C₁₉	—	2	4	no	—
69	M₁₆×C₁₉	—	4	2	no	—
70	M₁₆×C₁₉	—	4	8	no	—
71	M₁₆×C₁₉	D₈	1	4	yes	—
72	C₂×C₈×C₁₉	—	4	2	no	—
73	C₂×C₈×C₁₉	—	4	8	no	—
74	C₂×C₈×C₁₉	C₂²×C₄	1	2	yes	—
75	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₁₉	C₂×D₄	1	2	yes	—
77	C₂×C₈×C₁₉	#41	1	2	yes	—
78	C₂×C₈×C₁₉	C₂⁴	1	1	yes	characteristic
79	C₄⋊C₄×C₁₉	—	2	1	no	—
80	C₂²⋊C₄×C₁₉	—	2	1	no	—
81	C₄²×C₁₉	C₂×D₄	1	2	yes	—
82	C₁₆×C₁₉	—	2	1	no	—
83	C₁₆×C₁₉	—	2	1	no	—
84	C₁₆×C₁₉	C₂²×C₄	1	4	yes	—
85	#33464	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
86	C₂³×C₁₉	—	4	2	no	—
87	C₂³×C₁₉	#135	1	1	yes	characteristic
88	Q₈×C₁₉	—	2	4	no	—
89	Q₈×C₁₉	—	4	2	no	—
90	D₄×C₁₉	—	2	4	no	—
91	D₄×C₁₉	—	4	4	no	—
92	D₄×C₁₉	—	4	4	no	—
93	D₄×C₁₉	—	4	2	no	—
94	C₂×C₄×C₁₉	—	2	2	no	—
95	C₂×C₄×C₁₉	—	4	2	no	—
96	C₂×C₄×C₁₉	—	4	4	no	—
97	C₂×C₄×C₁₉	—	4	4	no	—
98	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
99	C₂×C₄×C₁₉	C₂×D₈	1	2	yes	—
100	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
101	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
102	C₈×C₁₉	—	4	4	no	—
103	C₈×C₁₉	C₄×D₄	1	2	yes	—
104	C₈×C₁₉	C₂×D₈	1	4	yes	—
105	C₈×C₁₉	C₂³×C₄	1	1	yes	characteristic
106	C₈×C₁₉	C₂×#41	1	1	yes	characteristic
107	#3414	C₂×C₁₉	1	1	yes	characteristic
108	#2957	C₂×C₁₉	1	2	yes	complemented
109	#2179	C₂×C₁₉	1	4	yes	complemented
110	#2178	C₂×C₁₉	1	4	yes	complemented
111	#2161	C₂×C₁₉	1	1	yes	characteristic, complemented
112	C₂×#359	C₂×C₁₉	1	2	yes	complemented
113	#2108	C₂×C₁₉	1	1	yes	characteristic, complemented
114	C₂²×C₁₉	—	4	1	no	—
115	C₂²×C₁₉	—	4	4	no	—
116	C₂²×C₁₉	—	16	2	no	—
117	C₂²×C₁₉	C₄×D₈	1	2	yes	—
118	C₂²×C₁₉	C₂×#135	1	1	yes	characteristic
119	C₄×C₁₉	—	4	2	no	—
120	C₄×C₁₉	—	4	4	no	—
121	C₄×C₁₉	C₄×D₈	1	2	yes	—
122	C₄×C₁₉	C₂×C₄×D₄	1	1	yes	characteristic
123	C₄×C₁₉	C₂²×D₈	1	1	yes	characteristic
124	C₂×#135	C₂²×C₁₉	1	2	yes	—
125	#568	—	2	2	no	—
126	#510	C₄×C₁₉	1	4	yes	complemented
127	#509	C₄×C₁₉	1	4	yes	complemented
128	#508	C₂²×C₁₉	1	4	yes	—
129	C₂×M₃₂	C₂²×C₁₉	1	1	yes	characteristic
130	C₂²×C₁₆	C₂²×C₁₉	1	1	yes	characteristic
131	#444	—	2	4	no	—
132	#443	C₂²×C₁₉	1	8	yes	complemented
133	C₂×#108	C₂²×C₁₉	1	2	yes	—
134	#421	—	2	2	no	—
135	#405	C₂²×C₁₉	1	1	yes	characteristic
136	#368	C₂²×C₁₉	1	4	yes	complemented
137	#359	C₂²×C₁₉	1	4	yes	complemented
138	#359	C₂²×C₁₉	1	4	yes	complemented
139	#346	C₂²×C₁₉	1	2	yes	complemented
140	C₄×C₁₆	C₂²×C₁₉	1	2	yes	complemented
141	C₂×C₁₉	—	2	1	no	—
142	C₂×C₁₉	—	2	2	no	—
143	C₂×C₁₉	—	8	4	no	—
144	C₂×C₁₉	C₂×C₄×D₈	1	1	yes	characteristic
145	C₂×#41	—	2	2	no	—
146	#135	—	2	2	no	—
147	#135	—	2	2	no	—
148	#135	C₂×C₄×C₁₉	1	4	yes	—
149	#135	C₂³×C₁₉	1	4	yes	—
150	C₂×Q₁₆	C₂×C₄×C₁₉	1	2	yes	—
151	C₂×SD₁₆	—	2	2	no	—
152	C₂×D₈	C₂×C₄×C₁₉	1	2	yes	—
153	#131	—	2	8	no	—
154	#131	—	4	4	no	—
155	C₂×M₁₆	—	2	2	no	—
156	C₂×M₁₆	—	4	4	no	—
157	C₂×M₁₆	D₄×C₁₉	1	1	yes	characteristic
158	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
159	#117	D₄×C₁₉	1	1	yes	characteristic
160	Q₃₂	C₂×C₄×C₁₉	1	4	yes	—
161	SD₃₂	C₂×C₄×C₁₉	1	8	yes	—
162	D₁₆	C₂×C₄×C₁₉	1	4	yes	—
163	M₃₂	C₂×C₄×C₁₉	1	4	yes	—
164	C₂×C₁₆	—	2	2	no	—
165	C₂×C₁₆	C₂³×C₁₉	1	1	yes	characteristic
166	C₂×C₁₆	C₂³×C₁₉	1	1	yes	characteristic
167	C₂×C₁₆	C₂³×C₁₉	1	2	yes	—
168	#108	—	2	2	no	—
169	#108	C₂³×C₁₉	1	4	yes	—
170	#104	—	2	8	no	—
171	#97	D₄×C₁₉	1	2	yes	—
172	C₄×C₈	C₂³×C₁₉	1	2	yes	—
173	C₁₉	#33464	1	1	yes	characteristic, direct product factor, Sylow
174	#41	—	2	4	no	—
175	#41	—	2	4	no	—
176	#41	—	4	2	no	—
177	#41	—	4	2	no	—
178	C₂×Q₈	—	2	2	no	—
179	C₂×D₄	—	2	2	no	—
180	C₂×D₄	—	4	2	no	—
181	C₂²×C₄	—	4	2	no	—
182	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
183	Q₁₆	—	2	2	no	—
184	Q₁₆	C₂²×C₄×C₁₉	1	4	yes	—
185	SD₁₆	—	2	4	no	—
186	SD₁₆	—	2	4	no	—
187	D₈	—	2	2	no	—
188	D₈	C₂²×C₄×C₁₉	1	4	yes	—
189	M₁₆	—	2	4	no	—
190	M₁₆	—	4	2	no	—
191	M₁₆	—	4	8	no	—
192	M₁₆	D₈×C₁₉	1	4	yes	—
193	C₂×C₈	—	4	2	no	—
194	C₂×C₈	—	4	8	no	—
195	C₂×C₈	C₂²×C₄×C₁₉	1	2	yes	—
196	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
197	C₂×C₈	C₂×D₄×C₁₉	1	2	yes	—
198	C₂×C₈	#41×C₁₉	1	2	yes	—
199	C₂×C₈	C₂⁴×C₁₉	1	1	yes	characteristic
200	C₄⋊C₄	—	2	1	no	—
201	C₂²⋊C₄	—	2	1	no	—
202	C₄²	C₂×D₄×C₁₉	1	2	yes	—
203	C₁₆	—	2	1	no	—
204	C₁₆	—	2	1	no	—
205	C₁₆	C₂²×C₄×C₁₉	1	4	yes	—
206	C₂³	—	4	2	no	—
207	C₂³	C₁₉×#135	1	1	yes	characteristic
208	Q₈	—	2	4	no	—
209	Q₈	—	4	2	no	—
210	D₄	—	2	4	no	—
211	D₄	—	4	4	no	—
212	D₄	—	4	4	no	—
213	D₄	—	4	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	4	2	no	—
216	C₂×C₄	—	4	4	no	—
217	C₂×C₄	—	4	4	no	—
218	C₂×C₄	C₄×D₄×C₁₉	1	2	yes	—
219	C₂×C₄	C₂×D₈×C₁₉	1	2	yes	—
220	C₂×C₄	C₁₉×#135	1	1	yes	characteristic
221	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
222	C₈	—	4	4	no	—
223	C₈	C₄×D₄×C₁₉	1	2	yes	—
224	C₈	C₂×D₈×C₁₉	1	4	yes	—
225	C₈	C₂³×C₄×C₁₉	1	1	yes	characteristic
226	C₈	C₂×#41×C₁₉	1	1	yes	characteristic
227	C₂²	—	4	1	no	—
228	C₂²	—	4	4	no	—
229	C₂²	—	16	2	no	—
230	C₂²	C₄×D₈×C₁₉	1	2	yes	—
231	C₂²	C₂×C₁₉×#135	1	1	yes	characteristic
232	C₄	—	4	2	no	—
233	C₄	—	4	4	no	—
234	C₄	C₄×D₈×C₁₉	1	2	yes	—
235	C₄	C₂×C₄×D₄×C₁₉	1	1	yes	characteristic
236	C₄	C₂²×D₈×C₁₉	1	1	yes	characteristic
237	C₂	—	2	1	no	—
238	C₂	—	2	2	no	—
239	C₂	—	8	4	no	—
240	C₂	C₂×C₄×D₈×C₁₉	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₉	C₂²×D₈
commutator	C₈	C₂³×C₄×C₁₉
Frattini, cosocle	C₂×C₈	C₂⁴×C₁₉
socle	C₂×C₁₉	C₂×C₄×D₈
prime radical	C₁₉×#3414	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₉ ◁ C₂×M₁₆×C₁₉ ◁ C₁₉×#405 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₉×#2957	C₂	semidirect
3. C₁₉×#2179	C₂	semidirect
4. C₁₉×#2178	C₂	semidirect
5. C₁₉×#2161	C₂	semidirect
6. C₂×C₁₉×#359	C₂	semidirect
7. C₁₉×#2108	C₂	semidirect
10. C₁₉×#510	C₄	semidirect
11. C₁₉×#509	C₄	semidirect
16. C₁₉×#443	C₂²	semidirect
20. C₁₉×#368	C₂²	semidirect
21. C₁₉×#359	C₂²	semidirect
22. C₁₉×#359	C₂²	semidirect
23. C₁₉×#346	C₂²	semidirect
24. C₄×C₁₆×C₁₉	C₂²	semidirect
85. #33464	C₁₉	direct (N × H)
108. #2957	C₂×C₁₉	semidirect
109. #2179	C₂×C₁₉	semidirect
110. #2178	C₂×C₁₉	semidirect
111. #2161	C₂×C₁₉	semidirect
112. C₂×#359	C₂×C₁₉	semidirect
113. #2108	C₂×C₁₉	semidirect
126. #510	C₄×C₁₉	semidirect
127. #509	C₄×C₁₉	semidirect
132. #443	C₂²×C₁₉	semidirect
136. #368	C₂²×C₁₉	semidirect
137. #359	C₂²×C₁₉	semidirect
138. #359	C₂²×C₁₉	semidirect
139. #346	C₂²×C₁₉	semidirect
140. C₄×C₁₆	C₂²×C₁₉	semidirect
173. C₁₉	#33464	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×#3414	2
C₁₉×#2957	2
C₁₉×#2179	2
C₁₉×#2178	2
C₁₉×#2161	2
C₂×C₁₉×#359	2
C₁₉×#2108	2
#33464	19

Subgroups — C19×#33464 #18039423

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#33464 #18039423