Subgroups — C₁₉×#21903 #18027862

Order 4864.

894 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#2297	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#1923	C₂	1	8	yes	maximal, complemented
3	C₁₉×#1798	C₂	1	2	yes	maximal, complemented
4	C₁₉×#1780	C₂	1	2	yes	maximal
5	C₄×C₁₉×C₂²⋊C₈	C₂	1	2	yes	maximal, complemented
6	C₂²×#41×C₁₉	C₂²	1	1	yes	characteristic
7	C₂×C₁₉×#117	C₂²	1	1	yes	characteristic
8	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
9	C₂×C₁₉×C₄⋊C₈	C₂²	1	4	yes	—
10	C₂×C₁₉×C₂²⋊C₈	C₂²	1	4	yes	—
11	C₂×C₁₉×C₂²⋊C₈	C₂²	1	4	yes	—
12	C₂×C₁₉×C₂²⋊C₈	C₂²	1	4	yes	—
13	C₂×C₄×C₈×C₁₉	C₂²	1	4	yes	—
14	C₁₉×#391	C₄	1	2	yes	—
15	C₁₉×#390	C₄	1	2	yes	—
16	C₁₉×#386	C₄	1	4	yes	—
17	C₄×C₂²⋊C₄×C₁₉	C₂²	1	4	yes	—
18	C₁₉×#336	C₂²	1	4	yes	—
19	C₁₉×#336	C₂²	1	4	yes	—
20	C₂×#41×C₁₉	—	2	4	no	—
21	C₂×#41×C₁₉	—	4	2	no	—
22	C₂×#41×C₁₉	—	4	2	no	—
23	C₂²×Q₈×C₁₉	C₂×C₄	1	1	yes	characteristic
24	C₂²×D₄×C₁₉	C₂×C₄	1	2	yes	—
25	C₂²×D₄×C₁₉	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₁₉	C₂×C₄	1	2	yes	—
27	C₂³×C₄×C₁₉	C₂³	1	2	yes	—
28	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
29	C₂²×C₈×C₁₉	C₂³	1	8	yes	—
30	C₁₉×#117	—	4	2	no	—
31	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
32	C₂×C₂²⋊C₄×C₁₉	C₂×C₄	1	8	yes	—
33	C₂×C₂²⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₁₉	—	4	2	no	—
35	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₁₉	C₂³	1	2	yes	—
37	C₂×C₄²×C₁₉	C₂³	1	2	yes	—
38	C₁₉×C₄⋊C₈	—	4	4	no	—
39	C₁₉×C₂²⋊C₈	—	2	8	no	—
40	C₁₉×C₂²⋊C₈	—	4	4	no	—
41	C₁₉×C₂²⋊C₈	—	4	4	no	—
42	C₄×C₈×C₁₉	—	4	4	no	—
43	C₁₉×#95	C₂×C₄	1	8	yes	—
44	C₂⁴×C₁₉	C₂²×C₄	1	2	yes	—
45	C₂⁴×C₁₉	C₂²×C₄	1	1	yes	characteristic
46	#41×C₁₉	—	4	16	no	—
47	C₂×Q₈×C₁₉	—	2	2	no	—
48	C₂×Q₈×C₁₉	—	4	1	no	—
49	C₂×Q₈×C₁₉	—	4	1	no	—
50	C₂×D₄×C₁₉	—	2	4	no	—
51	C₂×D₄×C₁₉	—	2	2	no	—
52	C₂×D₄×C₁₉	—	4	1	no	—
53	C₂×D₄×C₁₉	—	4	1	no	—
54	C₂×D₄×C₁₉	—	4	2	no	—
55	C₂×D₄×C₁₉	—	4	2	no	—
56	C₂²×C₄×C₁₉	—	2	2	no	—
57	C₂²×C₄×C₁₉	—	2	2	no	—
58	C₂²×C₄×C₁₉	—	2	8	no	—
59	C₂²×C₄×C₁₉	—	2	4	no	—
60	C₂²×C₄×C₁₉	—	2	4	no	—
61	C₂²×C₄×C₁₉	—	2	4	no	—
62	C₂²×C₄×C₁₉	—	4	2	no	—
63	C₂²×C₄×C₁₉	C₂²×C₄	1	4	yes	—
64	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₉	C₂²×C₄	1	4	yes	—
67	C₂²×C₄×C₁₉	#41	1	4	yes	—
68	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
69	C₂×C₈×C₁₉	—	2	8	no	—
70	C₂×C₈×C₁₉	—	2	8	no	—
71	C₂×C₈×C₁₉	—	2	8	no	—
72	C₄⋊C₄×C₁₉	—	4	2	no	—
73	C₂²⋊C₄×C₁₉	—	2	16	no	—
74	C₂²⋊C₄×C₁₉	—	4	2	no	—
75	C₄²×C₁₉	—	2	4	no	—
76	C₄²×C₁₉	—	2	4	no	—
77	C₄²×C₁₉	—	4	2	no	—
78	#21903	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
79	C₂³×C₁₉	—	2	1	no	—
80	C₂³×C₁₉	—	2	1	no	—
81	C₂³×C₁₉	—	2	4	no	—
82	C₂³×C₁₉	—	2	2	no	—
83	C₂³×C₁₉	—	2	2	no	—
84	C₂³×C₁₉	—	2	2	no	—
85	C₂³×C₁₉	—	2	8	no	—
86	C₂³×C₁₉	#117	1	2	yes	—
87	C₂³×C₁₉	C₂³×C₄	1	1	yes	characteristic
88	Q₈×C₁₉	—	4	4	no	—
89	D₄×C₁₉	—	4	4	no	—
90	D₄×C₁₉	—	4	8	no	—
91	C₂×C₄×C₁₉	—	2	4	no	—
92	C₂×C₄×C₁₉	—	2	1	no	—
93	C₂×C₄×C₁₉	—	2	1	no	—
94	C₂×C₄×C₁₉	—	2	4	no	—
95	C₂×C₄×C₁₉	—	2	8	no	—
96	C₂×C₄×C₁₉	—	2	2	no	—
97	C₂×C₄×C₁₉	—	2	4	no	—
98	C₂×C₄×C₁₉	—	2	2	no	—
99	C₂×C₄×C₁₉	—	2	2	no	—
100	C₂×C₄×C₁₉	—	2	8	no	—
101	C₂×C₄×C₁₉	—	4	8	no	—
102	C₂×C₄×C₁₉	—	4	8	no	—
103	C₂×C₄×C₁₉	#117	1	2	yes	—
104	C₂×C₄×C₁₉	#131	1	8	yes	—
105	C₂×C₄×C₁₉	C₂×#41	1	2	yes	—
106	C₂×C₄×C₁₉	H(5,𝔽₂)	1	2	yes	—
107	C₂×C₄×C₁₉	H(5,𝔽₂)	1	2	yes	—
108	C₈×C₁₉	—	4	8	no	—
109	#2297	C₂×C₁₉	1	1	yes	characteristic
110	#1923	C₂×C₁₉	1	8	yes	complemented
111	#1798	C₂×C₁₉	1	2	yes	complemented
112	#1780	C₂×C₁₉	1	2	yes	—
113	C₄×C₂²⋊C₈	C₂×C₁₉	1	2	yes	complemented
114	C₂²×C₁₉	—	2	2	no	—
115	C₂²×C₁₉	—	2	4	no	—
116	C₂²×C₁₉	—	4	2	no	—
117	C₂²×C₁₉	—	4	2	no	—
118	C₂²×C₁₉	—	4	2	no	—
119	C₂²×C₁₉	—	4	4	no	—
120	C₂²×C₁₉	—	4	4	no	—
121	C₂²×C₁₉	—	4	4	no	—
122	C₂²×C₁₉	C₂×#117	1	1	yes	characteristic
123	C₂²×C₁₉	#518	1	1	yes	characteristic
124	C₂²×C₁₉	#518	1	1	yes	characteristic
125	C₂²×C₁₉	C₂×#131	1	4	yes	—
126	C₄×C₁₉	—	2	2	no	—
127	C₄×C₁₉	—	2	8	no	—
128	C₄×C₁₉	—	4	2	no	—
129	C₄×C₁₉	—	4	4	no	—
130	C₄×C₁₉	#538	1	4	yes	—
131	C₂²×#41	C₂²×C₁₉	1	1	yes	characteristic
132	C₂×#117	C₂²×C₁₉	1	1	yes	characteristic
133	C₂²×C₄²	C₂²×C₁₉	1	1	yes	characteristic
134	C₂×C₄⋊C₈	C₂²×C₁₉	1	4	yes	—
135	C₂×C₂²⋊C₈	C₂²×C₁₉	1	4	yes	—
136	C₂×C₂²⋊C₈	C₂²×C₁₉	1	4	yes	—
137	C₂×C₂²⋊C₈	C₂²×C₁₉	1	4	yes	—
138	C₂×C₄×C₈	C₂²×C₁₉	1	4	yes	—
139	#391	C₄×C₁₉	1	2	yes	—
140	#390	C₄×C₁₉	1	2	yes	—
141	#386	C₄×C₁₉	1	4	yes	—
142	C₄×C₂²⋊C₄	C₂²×C₁₉	1	4	yes	—
143	#336	C₂²×C₁₉	1	4	yes	—
144	#336	C₂²×C₁₉	1	4	yes	—
145	C₂×C₁₉	—	4	2	no	—
146	C₂×C₁₉	—	8	2	no	—
147	C₂×C₁₉	#2312	1	1	yes	characteristic
148	C₂×C₁₉	#2922	1	2	yes	—
149	C₂×C₁₉	#2977	1	2	yes	—
150	C₂×C₁₉	#2979	1	2	yes	—
151	C₂×#41	—	2	4	no	—
152	C₂×#41	—	4	2	no	—
153	C₂×#41	—	4	2	no	—
154	C₂²×Q₈	C₂×C₄×C₁₉	1	1	yes	characteristic
155	C₂²×D₄	C₂×C₄×C₁₉	1	2	yes	—
156	C₂²×D₄	C₂×C₄×C₁₉	1	1	yes	characteristic
157	C₂³×C₄	C₂×C₄×C₁₉	1	2	yes	—
158	C₂³×C₄	C₂³×C₁₉	1	2	yes	—
159	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
160	C₂²×C₈	C₂³×C₁₉	1	8	yes	—
161	#117	—	4	2	no	—
162	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	2	yes	—
163	C₂×C₂²⋊C₄	C₂×C₄×C₁₉	1	8	yes	—
164	C₂×C₂²⋊C₄	C₂×C₄×C₁₉	1	2	yes	—
165	C₂×C₄²	—	4	2	no	—
166	C₂×C₄²	C₂×C₄×C₁₉	1	2	yes	—
167	C₂×C₄²	C₂³×C₁₉	1	2	yes	—
168	C₂×C₄²	C₂³×C₁₉	1	2	yes	—
169	C₄⋊C₈	—	4	4	no	—
170	C₂²⋊C₈	—	2	8	no	—
171	C₂²⋊C₈	—	4	4	no	—
172	C₂²⋊C₈	—	4	4	no	—
173	C₄×C₈	—	4	4	no	—
174	#95	C₂×C₄×C₁₉	1	8	yes	—
175	C₁₉	#21903	1	1	yes	characteristic, direct product factor, Sylow
176	C₂⁴	C₂²×C₄×C₁₉	1	2	yes	—
177	C₂⁴	C₂²×C₄×C₁₉	1	1	yes	characteristic
178	#41	—	4	16	no	—
179	C₂×Q₈	—	2	2	no	—
180	C₂×Q₈	—	4	1	no	—
181	C₂×Q₈	—	4	1	no	—
182	C₂×D₄	—	2	4	no	—
183	C₂×D₄	—	2	2	no	—
184	C₂×D₄	—	4	1	no	—
185	C₂×D₄	—	4	1	no	—
186	C₂×D₄	—	4	2	no	—
187	C₂×D₄	—	4	2	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	2	8	no	—
191	C₂²×C₄	—	2	4	no	—
192	C₂²×C₄	—	2	4	no	—
193	C₂²×C₄	—	2	4	no	—
194	C₂²×C₄	—	4	2	no	—
195	C₂²×C₄	C₂²×C₄×C₁₉	1	2	yes	—
196	C₂²×C₄	C₂²×C₄×C₁₉	1	4	yes	—
197	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
198	C₂²×C₄	C₂²×C₄×C₁₉	1	4	yes	—
199	C₂²×C₄	#41×C₁₉	1	4	yes	—
200	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
201	C₂×C₈	—	2	8	no	—
202	C₂×C₈	—	2	8	no	—
203	C₂×C₈	—	2	8	no	—
204	C₄⋊C₄	—	4	2	no	—
205	C₂²⋊C₄	—	2	16	no	—
206	C₂²⋊C₄	—	4	2	no	—
207	C₄²	—	2	4	no	—
208	C₄²	—	2	4	no	—
209	C₄²	—	4	2	no	—
210	C₂³	—	2	1	no	—
211	C₂³	—	2	1	no	—
212	C₂³	—	2	4	no	—
213	C₂³	—	2	2	no	—
214	C₂³	—	2	2	no	—
215	C₂³	—	2	2	no	—
216	C₂³	—	2	8	no	—
217	C₂³	C₁₉×#117	1	2	yes	—
218	C₂³	C₂³×C₄×C₁₉	1	1	yes	characteristic
219	Q₈	—	4	4	no	—
220	D₄	—	4	4	no	—
221	D₄	—	4	8	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	2	8	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	8	no	—
232	C₂×C₄	—	4	8	no	—
233	C₂×C₄	—	4	8	no	—
234	C₂×C₄	C₁₉×#117	1	2	yes	—
235	C₂×C₄	C₁₉×#131	1	8	yes	—
236	C₂×C₄	C₂×#41×C₁₉	1	2	yes	—
237	C₂×C₄	C₁₉×H(5,𝔽₂)	1	2	yes	—
238	C₂×C₄	C₁₉×H(5,𝔽₂)	1	2	yes	—
239	C₈	—	4	8	no	—
240	C₂²	—	2	4	no	—
241	C₂²	—	2	2	no	—
242	C₂²	—	4	2	no	—
243	C₂²	—	4	2	no	—
244	C₂²	—	4	4	no	—
245	C₂²	—	4	4	no	—
246	C₂²	—	4	4	no	—
247	C₂²	—	4	2	no	—
248	C₂²	C₂×C₁₉×#117	1	1	yes	characteristic
249	C₂²	C₁₉×#518	1	1	yes	characteristic
250	C₂²	C₁₉×#518	1	1	yes	characteristic
251	C₂²	C₂×C₁₉×#131	1	4	yes	—
252	C₄	—	2	8	no	—
253	C₄	—	2	2	no	—
254	C₄	—	4	4	no	—
255	C₄	—	4	2	no	—
256	C₄	C₁₉×#538	1	4	yes	—
257	C₂	—	4	2	no	—
258	C₂	—	8	2	no	—
259	C₂	C₁₉×#2312	1	1	yes	characteristic
260	C₂	C₁₉×#2922	1	2	yes	—
261	C₂	C₁₉×#2977	1	2	yes	—
262	C₂	C₁₉×#2979	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₁₉×#1923	C₂	semidirect
3. C₁₉×#1798	C₂	semidirect
5. C₄×C₁₉×C₂²⋊C₈	C₂	semidirect
78. #21903	C₁₉	direct (N × H)
110. #1923	C₂×C₁₉	semidirect
111. #1798	C₂×C₁₉	semidirect
113. C₄×C₂²⋊C₈	C₂×C₁₉	semidirect
175. C₁₉	#21903	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×#2297	2
C₁₉×#1923	2
C₁₉×#1798	2
C₁₉×#1780	2
C₄×C₁₉×C₂²⋊C₈	2
#21903	19

Subgroups — C19×#21903 #18027862

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#21903 #18027862