Subgroups — C₁₉×#20399 #18026358

Order 4864.

1442 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₉×#519	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₁₉×#101	C₂	1	2	yes	maximal
3	C₁₉×#2873	C₂	1	2	yes	maximal
4	C₄×C₁₉×#101	C₂	1	8	yes	maximal, complemented
5	C₁₉×#1742	C₂	1	2	yes	maximal
6	C₂³×Q₈×C₁₉	C₂²	1	1	yes	characteristic
7	C₂²×M₁₆×C₁₉	C₂²	1	4	yes	—
8	C₁₉×#519	—	2	8	no	—
9	C₂×C₄×Q₈×C₁₉	C₄	1	4	yes	—
10	C₂×C₄×Q₈×C₁₉	C₂²	1	4	yes	—
11	C₂×C₁₉×#117	C₄	1	4	yes	—
12	C₂×C₁₉×#117	C₂²	1	1	yes	characteristic
13	C₂×C₁₉×#117	C₂²	1	1	yes	characteristic
14	C₂×C₁₉×#101	C₄	1	16	yes	complemented
15	C₂×C₁₉×#101	C₂²	1	8	yes	—
16	C₁₉×#405	—	2	8	no	—
17	C₄×M₁₆×C₁₉	C₂²	1	8	yes	—
18	C₁₉×#343	C₂²	1	8	yes	—
19	C₂²×Q₈×C₁₉	—	2	4	no	—
20	C₂²×Q₈×C₁₉	—	2	4	no	—
21	C₂²×Q₈×C₁₉	C₂×C₄	1	8	yes	—
22	C₂²×Q₈×C₁₉	C₂×C₄	1	2	yes	—
23	C₂²×Q₈×C₁₉	C₂³	1	2	yes	—
24	C₂³×C₄×C₁₉	C₂×C₄	1	2	yes	—
25	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
26	C₂×M₁₆×C₁₉	—	2	4	no	—
27	C₂×M₁₆×C₁₉	—	2	8	no	—
28	C₂×M₁₆×C₁₉	C₂×C₄	1	16	yes	—
29	C₂×M₁₆×C₁₉	C₂³	1	8	yes	—
30	C₂²×C₈×C₁₉	—	2	4	no	—
31	C₄×Q₈×C₁₉	—	2	16	no	—
32	C₄×Q₈×C₁₉	—	2	16	no	—
33	C₁₉×#117	—	2	4	no	—
34	C₁₉×#117	—	2	16	no	—
35	C₁₉×#117	D₄	1	8	yes	—
36	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	8	yes	—
37	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
38	C₂×C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
39	C₂×C₂²⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
40	C₂×C₂²⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₁₉	C₂×C₄	1	8	yes	—
42	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₁₉	C₂³	1	2	yes	—
44	C₁₉×#101	C₂×C₄	1	32	yes	—
45	C₁₉×#97	—	2	8	no	—
46	C₄×C₈×C₁₉	—	2	8	no	—
47	C₂⁴×C₁₉	C₂²×C₄	1	1	yes	characteristic
48	C₂×Q₈×C₁₉	—	2	4	no	—
49	C₂×Q₈×C₁₉	—	2	4	no	—
50	C₂×Q₈×C₁₉	—	2	16	no	—
51	C₂×Q₈×C₁₉	—	2	16	no	—
52	C₂×Q₈×C₁₉	—	2	4	no	—
53	C₂×Q₈×C₁₉	—	2	4	no	—
54	C₂×Q₈×C₁₉	C₄²	1	8	yes	—
55	C₂×Q₈×C₁₉	C₂²×C₄	1	8	yes	—
56	C₂²×C₄×C₁₉	—	2	4	no	—
57	C₂²×C₄×C₁₉	—	2	4	no	—
58	C₂²×C₄×C₁₉	C₄²	1	8	yes	—
59	C₂²×C₄×C₁₉	C₂²×C₄	1	4	yes	—
60	C₂²×C₄×C₁₉	C₂²×C₄	1	4	yes	—
61	C₂²×C₄×C₁₉	C₂²×C₄	1	4	yes	—
62	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₉	C₂²×C₄	1	4	yes	—
64	C₂²×C₄×C₁₉	C₂×D₄	1	4	yes	—
65	C₂²×C₄×C₁₉	#41	1	4	yes	—
66	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
67	M₁₆×C₁₉	—	2	16	no	—
68	M₁₆×C₁₉	—	2	8	no	—
69	M₁₆×C₁₉	C₂²×C₄	1	16	yes	—
70	C₂×C₈×C₁₉	—	2	16	no	—
71	C₂×C₈×C₁₉	—	2	8	no	—
72	C₄⋊C₄×C₁₉	—	2	16	no	—
73	C₄⋊C₄×C₁₉	—	2	4	no	—
74	C₄⋊C₄×C₁₉	C₂²⋊C₄	1	8	yes	—
75	C₂²⋊C₄×C₁₉	—	2	4	no	—
76	C₂²⋊C₄×C₁₉	C₂²⋊C₄	1	8	yes	—
77	C₄²×C₁₉	—	2	16	no	—
78	C₄²×C₁₉	—	2	4	no	—
79	C₄²×C₁₉	C₂×D₄	1	8	yes	—
80	#20399	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
81	C₂³×C₁₉	—	4	2	no	—
82	C₂³×C₁₉	C₂×C₄²	1	2	yes	—
83	C₂³×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
84	C₂³×C₁₉	#117	1	2	yes	—
85	C₂³×C₁₉	C₂³×C₄	1	1	yes	characteristic
86	Q₈×C₁₉	—	2	16	no	—
87	Q₈×C₁₉	—	2	16	no	—
88	C₂×C₄×C₁₉	—	2	4	no	—
89	C₂×C₄×C₁₉	—	2	4	no	—
90	C₂×C₄×C₁₉	—	2	16	no	—
91	C₂×C₄×C₁₉	—	2	4	no	—
92	C₂×C₄×C₁₉	—	4	4	no	—
93	C₂×C₄×C₁₉	—	4	4	no	—
94	C₂×C₄×C₁₉	C₂×C₄²	1	8	yes	—
95	C₂×C₄×C₁₉	C₂×C₄²	1	2	yes	—
96	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	8	yes	—
97	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
98	C₂×C₄×C₁₉	#117	1	2	yes	—
99	C₂×C₄×C₁₉	C₄×D₄	1	8	yes	—
100	C₂×C₄×C₁₉	C₄×D₄	1	8	yes	—
101	C₂×C₄×C₁₉	C₂³×C₄	1	2	yes	—
102	C₂×C₄×C₁₉	C₂²×D₄	1	2	yes	—
103	C₂×C₄×C₁₉	C₂×#41	1	2	yes	—
104	C₈×C₁₉	—	2	16	no	—
105	C₂×#519	C₂×C₁₉	1	1	yes	characteristic
106	C₂²×#101	C₂×C₁₉	1	2	yes	—
107	#2873	C₂×C₁₉	1	2	yes	—
108	C₄×#101	C₂×C₁₉	1	8	yes	complemented
109	#1742	C₂×C₁₉	1	2	yes	—
110	C₂²×C₁₉	—	2	2	no	—
111	C₂²×C₁₉	—	2	4	no	—
112	C₂²×C₁₉	—	4	2	no	—
113	C₂²×C₁₉	—	4	2	no	—
114	C₂²×C₁₉	C₄×C₂²⋊C₄	1	4	yes	—
115	C₂²×C₁₉	C₂²×C₄²	1	1	yes	characteristic
116	C₂²×C₁₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
117	C₂²×C₁₉	C₂×#117	1	1	yes	characteristic
118	C₄×C₁₉	—	2	8	no	—
119	C₄×C₁₉	—	2	8	no	—
120	C₄×C₁₉	—	4	4	no	—
121	C₄×C₁₉	C₄×C₂²⋊C₄	1	4	yes	—
122	C₄×C₁₉	C₂×C₄×D₄	1	4	yes	—
123	C₂³×Q₈	C₂²×C₁₉	1	1	yes	characteristic
124	C₂²×M₁₆	C₂²×C₁₉	1	4	yes	—
125	#519	—	2	8	no	—
126	C₂×C₄×Q₈	C₄×C₁₉	1	4	yes	—
127	C₂×C₄×Q₈	C₂²×C₁₉	1	4	yes	—
128	C₂×#117	C₄×C₁₉	1	4	yes	—
129	C₂×#117	C₂²×C₁₉	1	1	yes	characteristic
130	C₂×#117	C₂²×C₁₉	1	1	yes	characteristic
131	C₂×#101	C₄×C₁₉	1	16	yes	complemented
132	C₂×#101	C₂²×C₁₉	1	8	yes	—
133	#405	—	2	8	no	—
134	C₄×M₁₆	C₂²×C₁₉	1	8	yes	—
135	#343	C₂²×C₁₉	1	8	yes	—
136	C₂×C₁₉	—	2	1	no	—
137	C₂×C₁₉	—	2	4	no	—
138	C₂×C₁₉	—	2	1	no	—
139	C₂×C₁₉	C₂×C₄×C₂²⋊C₄	1	1	yes	characteristic
140	C₂×C₁₉	#2889	1	2	yes	—
141	C₂²×Q₈	—	2	4	no	—
142	C₂²×Q₈	—	2	4	no	—
143	C₂²×Q₈	C₂×C₄×C₁₉	1	8	yes	—
144	C₂²×Q₈	C₂×C₄×C₁₉	1	2	yes	—
145	C₂²×Q₈	C₂³×C₁₉	1	2	yes	—
146	C₂³×C₄	C₂×C₄×C₁₉	1	2	yes	—
147	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
148	C₂×M₁₆	—	2	4	no	—
149	C₂×M₁₆	—	2	8	no	—
150	C₂×M₁₆	C₂×C₄×C₁₉	1	16	yes	—
151	C₂×M₁₆	C₂³×C₁₉	1	8	yes	—
152	C₂²×C₈	—	2	4	no	—
153	C₄×Q₈	—	2	16	no	—
154	C₄×Q₈	—	2	16	no	—
155	#117	—	2	4	no	—
156	#117	—	2	16	no	—
157	#117	D₄×C₁₉	1	8	yes	—
158	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	8	yes	—
159	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	2	yes	—
160	C₂×C₄⋊C₄	C₂³×C₁₉	1	2	yes	—
161	C₂×C₂²⋊C₄	C₂×C₄×C₁₉	1	2	yes	—
162	C₂×C₂²⋊C₄	C₂×C₄×C₁₉	1	2	yes	—
163	C₂×C₄²	C₂×C₄×C₁₉	1	8	yes	—
164	C₂×C₄²	C₂×C₄×C₁₉	1	2	yes	—
165	C₂×C₄²	C₂³×C₁₉	1	2	yes	—
166	#101	C₂×C₄×C₁₉	1	32	yes	—
167	#97	—	2	8	no	—
168	C₄×C₈	—	2	8	no	—
169	C₁₉	#20399	1	1	yes	characteristic, direct product factor, Sylow
170	C₂⁴	C₂²×C₄×C₁₉	1	1	yes	characteristic
171	C₂×Q₈	—	2	4	no	—
172	C₂×Q₈	—	2	4	no	—
173	C₂×Q₈	—	2	16	no	—
174	C₂×Q₈	—	2	16	no	—
175	C₂×Q₈	—	2	4	no	—
176	C₂×Q₈	—	2	4	no	—
177	C₂×Q₈	C₄²×C₁₉	1	8	yes	—
178	C₂×Q₈	C₂²×C₄×C₁₉	1	8	yes	—
179	C₂²×C₄	—	2	4	no	—
180	C₂²×C₄	—	2	4	no	—
181	C₂²×C₄	C₄²×C₁₉	1	8	yes	—
182	C₂²×C₄	C₂²×C₄×C₁₉	1	4	yes	—
183	C₂²×C₄	C₂²×C₄×C₁₉	1	4	yes	—
184	C₂²×C₄	C₂²×C₄×C₁₉	1	4	yes	—
185	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
186	C₂²×C₄	C₂²×C₄×C₁₉	1	4	yes	—
187	C₂²×C₄	C₂×D₄×C₁₉	1	4	yes	—
188	C₂²×C₄	#41×C₁₉	1	4	yes	—
189	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
190	M₁₆	—	2	16	no	—
191	M₁₆	—	2	8	no	—
192	M₁₆	C₂²×C₄×C₁₉	1	16	yes	—
193	C₂×C₈	—	2	16	no	—
194	C₂×C₈	—	2	8	no	—
195	C₄⋊C₄	—	2	16	no	—
196	C₄⋊C₄	—	2	4	no	—
197	C₄⋊C₄	C₂²⋊C₄×C₁₉	1	8	yes	—
198	C₂²⋊C₄	—	2	4	no	—
199	C₂²⋊C₄	C₂²⋊C₄×C₁₉	1	8	yes	—
200	C₄²	—	2	16	no	—
201	C₄²	—	2	4	no	—
202	C₄²	C₂×D₄×C₁₉	1	8	yes	—
203	C₂³	—	4	2	no	—
204	C₂³	C₂×C₄²×C₁₉	1	2	yes	—
205	C₂³	C₂×C₂²⋊C₄×C₁₉	1	2	yes	—
206	C₂³	C₁₉×#117	1	2	yes	—
207	C₂³	C₂³×C₄×C₁₉	1	1	yes	characteristic
208	Q₈	—	2	16	no	—
209	Q₈	—	2	16	no	—
210	C₂×C₄	—	2	16	no	—
211	C₂×C₄	—	2	4	no	—
212	C₂×C₄	—	2	4	no	—
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	4	4	no	—
215	C₂×C₄	—	4	4	no	—
216	C₂×C₄	C₂×C₄²×C₁₉	1	8	yes	—
217	C₂×C₄	C₂×C₄²×C₁₉	1	2	yes	—
218	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	8	yes	—
219	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	2	yes	—
220	C₂×C₄	C₁₉×#117	1	2	yes	—
221	C₂×C₄	C₄×D₄×C₁₉	1	8	yes	—
222	C₂×C₄	C₄×D₄×C₁₉	1	8	yes	—
223	C₂×C₄	C₂³×C₄×C₁₉	1	2	yes	—
224	C₂×C₄	C₂²×D₄×C₁₉	1	2	yes	—
225	C₂×C₄	C₂×#41×C₁₉	1	2	yes	—
226	C₈	—	2	16	no	—
227	C₂²	—	2	2	no	—
228	C₂²	—	2	4	no	—
229	C₂²	—	4	2	no	—
230	C₂²	—	4	2	no	—
231	C₂²	C₄×C₂²⋊C₄×C₁₉	1	4	yes	—
232	C₂²	C₂²×C₄²×C₁₉	1	1	yes	characteristic
233	C₂²	C₂²×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
234	C₂²	C₂×C₁₉×#117	1	1	yes	characteristic
235	C₄	—	2	8	no	—
236	C₄	—	2	8	no	—
237	C₄	—	4	4	no	—
238	C₄	C₄×C₂²⋊C₄×C₁₉	1	4	yes	—
239	C₄	C₂×C₄×D₄×C₁₉	1	4	yes	—
240	C₂	—	2	4	no	—
241	C₂	—	2	1	no	—
242	C₂	—	2	1	no	—
243	C₂	C₂×C₄×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
244	C₂	C₁₉×#2889	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂²×C₂²⋊C₄
commutator	C₂²	C₂²×C₄²×C₁₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₉
prime radical	C₂⁴×C₁₉	C₂²×C₄
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂×C₁₉×#117 ◁ G

Complements

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

N	H ≅ G/N	type
4. C₄×C₁₉×#101	C₂	semidirect
14. C₂×C₁₉×#101	C₄	semidirect
80. #20399	C₁₉	direct (N × H)
108. C₄×#101	C₂×C₁₉	semidirect
131. C₂×#101	C₄×C₁₉	semidirect
169. C₁₉	#20399	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₁₉×#519	2
C₂²×C₁₉×#101	2
C₁₉×#2873	2
C₄×C₁₉×#101	2
C₁₉×#1742	2
#20399	19

Subgroups — C19×#20399 #18026358

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#20399 #18026358