Subgroups — C₁₉×#62360 #18068319

Order 4864.

1662 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#3502	C₂	1	2	yes	maximal
2	C₁₉×#3395	C₂	1	4	yes	maximal, complemented
3	Q₈×Q₁₆×C₁₉	C₂	1	4	yes	maximal, complemented
4	C₁₉×#3363	C₂	1	8	yes	maximal, complemented
5	C₁₉×#3249	C₂	1	6	yes	maximal
6	C₁₉×#3170	C₂	1	6	yes	maximal
7	C₂×C₈×Q₈×C₁₉	C₂	1	1	yes	maximal, characteristic
8	Q₈²×C₁₉	C₂²	1	8	yes	—
9	C₁₉×#557	—	2	8	no	—
10	C₁₉×#557	C₂²	1	8	yes	—
11	C₁₉×#544	—	2	6	no	—
12	C₁₉×#533	C₂²	1	12	yes	—
13	C₁₉×#519	—	2	2	no	—
14	C₂×C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
15	C₁₉×#500	C₂²	1	6	yes	—
16	C₁₉×#494	C₂²	1	6	yes	—
17	C₁₉×#487	C₂²	1	12	yes	—
18	C₁₉×#477	C₂²	1	24	yes	—
19	C₁₉×#467	C₂²	1	4	yes	—
20	C₁₉×#467	C₂²	1	12	yes	—
21	C₁₉×#462	C₂²	1	24	yes	—
22	C₈×Q₈×C₁₉	C₂²	1	4	yes	—
23	C₈×Q₈×C₁₉	C₂²	1	4	yes	—
24	C₄×Q₁₆×C₁₉	C₂²	1	12	yes	—
25	C₄×Q₁₆×C₁₉	C₂²	1	12	yes	—
26	C₂×C₁₉×C₄⋊C₈	C₂²	1	3	yes	—
27	C₂×C₄×C₈×C₁₉	C₂²	1	3	yes	—
28	C₂²×Q₈×C₁₉	D₄	1	1	yes	characteristic
29	C₂×Q₁₆×C₁₉	C₂³	1	4	yes	—
30	C₂×Q₁₆×C₁₉	C₂³	1	12	yes	—
31	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₁₉	C₂³	1	3	yes	—
33	C₁₉×C₄⋊Q₈	—	2	24	no	—
34	C₁₉×C₄⋊Q₈	C₂³	1	12	yes	—
35	C₁₉×C₄⋊Q₈	C₂³	1	12	yes	—
36	C₁₉×#125	—	2	24	no	—
37	C₁₉×#125	—	2	12	no	—
38	C₁₉×C₂²⋊Q₈	—	2	12	no	—
39	C₁₉×C₂²⋊Q₈	C₂³	1	8	yes	—
40	C₄×Q₈×C₁₉	—	2	8	no	—
41	C₄×Q₈×C₁₉	—	2	8	no	—
42	C₄×Q₈×C₁₉	—	2	12	no	—
43	C₄×Q₈×C₁₉	D₄	1	4	yes	—
44	C₄×Q₈×C₁₉	C₂³	1	24	yes	—
45	C₄×Q₈×C₁₉	C₂³	1	4	yes	—
46	C₁₉×#117	—	2	12	no	—
47	C₂×C₄⋊C₄×C₁₉	D₄	1	3	yes	—
48	C₂×C₄²×C₁₉	C₂³	1	3	yes	—
49	C₁₉×#107	C₂³	1	4	yes	—
50	C₁₉×#107	C₂³	1	12	yes	—
51	C₁₉×C₄⋊C₈	C₂³	1	12	yes	—
52	C₁₉×#103	C₂³	1	8	yes	—
53	C₁₉×#103	C₂³	1	24	yes	—
54	C₄×C₈×C₁₉	C₂³	1	6	yes	—
55	C₄×C₈×C₁₉	C₂³	1	6	yes	—
56	C₂×Q₈×C₁₉	—	2	12	no	—
57	C₂×Q₈×C₁₉	Q₁₆	1	8	yes	complemented
58	C₂×Q₈×C₁₉	C₂×D₄	1	4	yes	—
59	C₂×Q₈×C₁₉	C₂⁴	1	8	yes	—
60	C₂²×C₄×C₁₉	C₂×D₄	1	3	yes	—
61	C₂²×C₄×C₁₉	C₂×D₄	1	3	yes	—
62	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
63	Q₁₆×C₁₉	—	2	8	no	—
64	Q₁₆×C₁₉	—	4	12	no	—
65	C₂×C₈×C₁₉	—	2	2	no	—
66	C₂×C₈×C₁₉	—	4	3	no	—
67	C₂×C₈×C₁₉	C₂⁴	1	6	yes	—
68	C₂×C₈×C₁₉	C₂⁴	1	1	yes	characteristic
69	C₂×C₈×C₁₉	C₂⁴	1	1	yes	characteristic
70	C₄⋊C₄×C₁₉	—	2	12	no	—
71	C₄⋊C₄×C₁₉	—	2	12	no	—
72	C₄⋊C₄×C₁₉	—	2	8	no	—
73	C₄⋊C₄×C₁₉	—	2	12	no	—
74	C₄⋊C₄×C₁₉	—	2	24	no	—
75	C₄⋊C₄×C₁₉	C₂×D₄	1	12	yes	—
76	C₄⋊C₄×C₁₉	C₂⁴	1	8	yes	—
77	C₂²⋊C₄×C₁₉	—	2	8	no	—
78	C₄²×C₁₉	—	2	24	no	—
79	C₄²×C₁₉	C₂×D₄	1	6	yes	—
80	C₄²×C₁₉	C₂⁴	1	6	yes	—
81	#62360	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
82	C₂³×C₁₉	C₂²×D₄	1	1	yes	characteristic
83	Q₈×C₁₉	—	2	8	no	—
84	Q₈×C₁₉	—	2	8	no	—
85	Q₈×C₁₉	—	8	12	no	—
86	Q₈×C₁₉	C₂×Q₁₆	1	16	yes	—
87	C₂×C₄×C₁₉	—	2	1	no	—
88	C₂×C₄×C₁₉	—	2	1	no	—
89	C₂×C₄×C₁₉	—	2	8	no	—
90	C₂×C₄×C₁₉	—	2	8	no	—
91	C₂×C₄×C₁₉	—	4	3	no	—
92	C₂×C₄×C₁₉	C₂×Q₁₆	1	12	yes	—
93	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
94	C₂×C₄×C₁₉	C₂²×D₄	1	6	yes	—
95	C₂×C₄×C₁₉	C₂²×D₄	1	6	yes	—
96	C₂×C₄×C₁₉	C₂⁵	1	1	yes	characteristic
97	C₈×C₁₉	—	2	1	no	—
98	C₈×C₁₉	—	2	6	no	—
99	C₈×C₁₉	2₋¹⁺⁴	1	2	yes	—
100	#3502	C₂×C₁₉	1	2	yes	—
101	#3395	C₂×C₁₉	1	4	yes	complemented
102	Q₈×Q₁₆	C₂×C₁₉	1	4	yes	complemented
103	#3363	C₂×C₁₉	1	8	yes	complemented
104	#3249	C₂×C₁₉	1	6	yes	—
105	#3170	C₂×C₁₉	1	6	yes	—
106	C₂×C₈×Q₈	C₂×C₁₉	1	1	yes	characteristic
107	C₂²×C₁₉	—	2	1	no	—
108	C₂²×C₁₉	—	2	1	no	—
109	C₂²×C₁₉	C₂²×Q₁₆	1	2	yes	—
110	C₂²×C₁₉	C₂³×D₄	1	1	yes	characteristic
111	C₄×C₁₉	—	2	6	no	—
112	C₄×C₁₉	—	2	1	no	—
113	C₄×C₁₉	—	4	8	no	—
114	C₄×C₁₉	—	4	8	no	—
115	C₄×C₁₉	C₂²×Q₁₆	1	12	yes	—
116	C₄×C₁₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
117	C₄×C₁₉	#585	1	1	yes	characteristic
118	Q₈²	C₂²×C₁₉	1	8	yes	—
119	#557	—	2	8	no	—
120	#557	C₂²×C₁₉	1	8	yes	—
121	#544	—	2	6	no	—
122	#533	C₂²×C₁₉	1	12	yes	—
123	#519	—	2	2	no	—
124	C₂×C₄×Q₈	C₂²×C₁₉	1	1	yes	characteristic
125	#500	C₂²×C₁₉	1	6	yes	—
126	#494	C₂²×C₁₉	1	6	yes	—
127	#487	C₂²×C₁₉	1	12	yes	—
128	#477	C₂²×C₁₉	1	24	yes	—
129	#467	C₂²×C₁₉	1	4	yes	—
130	#467	C₂²×C₁₉	1	12	yes	—
131	#462	C₂²×C₁₉	1	24	yes	—
132	C₈×Q₈	C₂²×C₁₉	1	4	yes	—
133	C₈×Q₈	C₂²×C₁₉	1	4	yes	—
134	C₄×Q₁₆	C₂²×C₁₉	1	12	yes	—
135	C₄×Q₁₆	C₂²×C₁₉	1	12	yes	—
136	C₂×C₄⋊C₈	C₂²×C₁₉	1	3	yes	—
137	C₂×C₄×C₈	C₂²×C₁₉	1	3	yes	—
138	C₂×C₁₉	—	2	2	no	—
139	C₂×C₁₉	#3497	1	1	yes	characteristic
140	C₂×C₁₉	C₂³×Q₁₆	1	1	yes	characteristic
141	C₂×C₁₉	#3584	1	1	yes	characteristic
142	C₂²×Q₈	D₄×C₁₉	1	1	yes	characteristic
143	C₂×Q₁₆	C₂³×C₁₉	1	4	yes	—
144	C₂×Q₁₆	C₂³×C₁₉	1	12	yes	—
145	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
146	C₂²×C₈	C₂³×C₁₉	1	3	yes	—
147	C₄⋊Q₈	—	2	24	no	—
148	C₄⋊Q₈	C₂³×C₁₉	1	12	yes	—
149	C₄⋊Q₈	C₂³×C₁₉	1	12	yes	—
150	#125	—	2	24	no	—
151	#125	—	2	12	no	—
152	C₂²⋊Q₈	—	2	12	no	—
153	C₂²⋊Q₈	C₂³×C₁₉	1	8	yes	—
154	C₄×Q₈	—	2	8	no	—
155	C₄×Q₈	—	2	8	no	—
156	C₄×Q₈	—	2	12	no	—
157	C₄×Q₈	D₄×C₁₉	1	4	yes	—
158	C₄×Q₈	C₂³×C₁₉	1	24	yes	—
159	C₄×Q₈	C₂³×C₁₉	1	4	yes	—
160	#117	—	2	12	no	—
161	C₂×C₄⋊C₄	D₄×C₁₉	1	3	yes	—
162	C₂×C₄²	C₂³×C₁₉	1	3	yes	—
163	#107	C₂³×C₁₉	1	4	yes	—
164	#107	C₂³×C₁₉	1	12	yes	—
165	C₄⋊C₈	C₂³×C₁₉	1	12	yes	—
166	#103	C₂³×C₁₉	1	8	yes	—
167	#103	C₂³×C₁₉	1	24	yes	—
168	C₄×C₈	C₂³×C₁₉	1	6	yes	—
169	C₄×C₈	C₂³×C₁₉	1	6	yes	—
170	C₁₉	#62360	1	1	yes	characteristic, direct product factor, Sylow
171	C₂×Q₈	—	2	12	no	—
172	C₂×Q₈	Q₁₆×C₁₉	1	8	yes	complemented
173	C₂×Q₈	C₂×D₄×C₁₉	1	4	yes	—
174	C₂×Q₈	C₂⁴×C₁₉	1	8	yes	—
175	C₂²×C₄	C₂×D₄×C₁₉	1	3	yes	—
176	C₂²×C₄	C₂×D₄×C₁₉	1	3	yes	—
177	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
178	Q₁₆	—	2	8	no	—
179	Q₁₆	—	4	12	no	—
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	4	3	no	—
182	C₂×C₈	C₂⁴×C₁₉	1	1	yes	characteristic
183	C₂×C₈	C₂⁴×C₁₉	1	6	yes	—
184	C₂×C₈	C₂⁴×C₁₉	1	1	yes	characteristic
185	C₄⋊C₄	—	2	12	no	—
186	C₄⋊C₄	—	2	12	no	—
187	C₄⋊C₄	—	2	8	no	—
188	C₄⋊C₄	—	2	12	no	—
189	C₄⋊C₄	—	2	24	no	—
190	C₄⋊C₄	C₂×D₄×C₁₉	1	12	yes	—
191	C₄⋊C₄	C₂⁴×C₁₉	1	8	yes	—
192	C₂²⋊C₄	—	2	8	no	—
193	C₄²	—	2	24	no	—
194	C₄²	C₂×D₄×C₁₉	1	6	yes	—
195	C₄²	C₂⁴×C₁₉	1	6	yes	—
196	C₂³	C₂²×D₄×C₁₉	1	1	yes	characteristic
197	Q₈	—	2	8	no	—
198	Q₈	—	2	8	no	—
199	Q₈	—	8	12	no	—
200	Q₈	C₂×Q₁₆×C₁₉	1	16	yes	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	8	no	—
204	C₂×C₄	—	2	8	no	—
205	C₂×C₄	—	4	3	no	—
206	C₂×C₄	C₂×Q₁₆×C₁₉	1	12	yes	—
207	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
208	C₂×C₄	C₂²×D₄×C₁₉	1	6	yes	—
209	C₂×C₄	C₂²×D₄×C₁₉	1	6	yes	—
210	C₂×C₄	C₂⁵×C₁₉	1	1	yes	characteristic
211	C₈	—	2	1	no	—
212	C₈	—	2	6	no	—
213	C₈	C₁₉×2₋¹⁺⁴	1	2	yes	—
214	C₂²	—	2	1	no	—
215	C₂²	—	2	1	no	—
216	C₂²	C₂²×Q₁₆×C₁₉	1	2	yes	—
217	C₂²	C₂³×D₄×C₁₉	1	1	yes	characteristic
218	C₄	—	2	1	no	—
219	C₄	—	2	6	no	—
220	C₄	—	4	8	no	—
221	C₄	—	4	8	no	—
222	C₄	C₂²×Q₁₆×C₁₉	1	12	yes	—
223	C₄	C₂×C₁₉×2₋¹⁺⁴	1	1	yes	characteristic
224	C₄	C₁₉×#585	1	1	yes	characteristic
225	C₂	—	2	2	no	—
226	C₂	C₁₉×#3497	1	1	yes	characteristic
227	C₂	C₂³×Q₁₆×C₁₉	1	1	yes	characteristic
228	C₂	C₁₉×#3584	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₁₉
prime radical	C₂³×C₁₉	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂×C₄×Q₈×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₉×#3395	C₂	semidirect
3. Q₈×Q₁₆×C₁₉	C₂	semidirect
4. C₁₉×#3363	C₂	semidirect
57. C₂×Q₈×C₁₉	Q₁₆	semidirect
81. #62360	C₁₉	direct (N × H)
101. #3395	C₂×C₁₉	semidirect
102. Q₈×Q₁₆	C₂×C₁₉	semidirect
103. #3363	C₂×C₁₉	semidirect
170. C₁₉	#62360	direct (N × H)
172. C₂×Q₈	Q₁₆×C₁₉	semidirect

Maximal subgroups

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

subgroup	index
C₁₉×#3502	2
C₁₉×#3395	2
Q₈×Q₁₆×C₁₉	2
C₁₉×#3363	2
C₁₉×#3249	2
C₁₉×#3170	2
C₂×C₈×Q₈×C₁₉	2
#62360	19

Subgroups — C19×#62360 #18068319

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#62360 #18068319