Subgroups — C₂×C₁₉×#2582 #18040908

Order 4864.

2702 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×Q₈×C₁₉	C₂	1	2	yes	maximal
2	C₁₉×#2582	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₁₉×#398	C₂	1	4	yes	maximal
4	C₂×C₁₉×#391	C₂	1	4	yes	maximal
5	C₂×C₁₉×#389	C₂	1	4	yes	maximal
6	C₂×C₁₉×#384	C₂	1	1	yes	maximal, characteristic
7	C₂³×Q₈×C₁₉	C₂²	1	2	yes	—
8	C₂×C₄×Q₈×C₁₉	—	2	16	no	—
9	C₂×C₄×Q₈×C₁₉	C₂²	1	16	yes	—
10	C₂²×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
11	C₂²×C₄⋊C₄×C₁₉	C₂²	1	4	yes	—
12	C₂²×C₄⋊C₄×C₁₉	C₂²	1	4	yes	—
13	C₂²×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
14	C₂²×C₄⋊C₄×C₁₉	C₂²	1	4	yes	—
15	C₂²×C₄²×C₁₉	C₂²	1	4	yes	—
16	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
17	C₁₉×#398	C₂²	1	32	yes	—
18	C₁₉×#391	C₂²	1	32	yes	—
19	C₁₉×#389	C₂²	1	32	yes	—
20	C₁₉×#384	C₂²	1	8	yes	—
21	C₂×C₁₉×#95	C₂²	1	8	yes	—
22	C₂×C₁₉×#95	C₂²	1	4	yes	—
23	C₂²×Q₈×C₁₉	—	2	8	no	—
24	C₂²×Q₈×C₁₉	—	2	8	no	—
25	C₂²×Q₈×C₁₉	Q₈	1	16	yes	complemented
26	C₂²×Q₈×C₁₉	C₂³	1	8	yes	—
27	C₂³×C₄×C₁₉	C₂³	1	4	yes	—
28	C₂³×C₄×C₁₉	C₂³	1	4	yes	—
29	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
30	C₂³×C₄×C₁₉	C₂³	1	4	yes	—
31	C₂³×C₄×C₁₉	C₂³	1	2	yes	—
32	C₄×Q₈×C₁₉	—	2	64	no	—
33	C₂×C₄⋊C₄×C₁₉	—	2	16	no	—
34	C₂×C₄⋊C₄×C₁₉	—	2	8	no	—
35	C₂×C₄⋊C₄×C₁₉	—	2	16	no	—
36	C₂×C₄⋊C₄×C₁₉	—	4	4	no	—
37	C₂×C₄⋊C₄×C₁₉	—	4	8	no	—
38	C₂×C₄⋊C₄×C₁₉	C₂³	1	16	yes	—
39	C₂×C₄⋊C₄×C₁₉	C₂³	1	8	yes	—
40	C₂×C₄⋊C₄×C₁₉	C₂³	1	16	yes	—
41	C₂×C₄⋊C₄×C₁₉	C₂³	1	8	yes	—
42	C₂×C₄⋊C₄×C₁₉	C₂³	1	16	yes	—
43	C₂×C₄²×C₁₉	—	2	16	no	—
44	C₂×C₄²×C₁₉	D₄	1	8	yes	—
45	C₂×C₄²×C₁₉	C₂³	1	16	yes	—
46	C₂×C₄²×C₁₉	C₂³	1	4	yes	—
47	C₁₉×#95	C₂³	1	32	yes	—
48	C₁₉×#95	C₂³	1	16	yes	—
49	C₂⁴×C₁₉	C₂⁴	1	1	yes	characteristic
50	C₂×Q₈×C₁₉	—	2	64	no	—
51	C₂×Q₈×C₁₉	—	2	16	no	—
52	C₂×Q₈×C₁₉	—	2	16	no	—
53	C₂×Q₈×C₁₉	C₂×Q₈	1	32	yes	complemented
54	C₂²×C₄×C₁₉	—	2	8	no	—
55	C₂²×C₄×C₁₉	—	2	8	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	16	no	—
60	C₂²×C₄×C₁₉	—	2	8	no	—
61	C₂²×C₄×C₁₉	—	2	8	no	—
62	C₂²×C₄×C₁₉	—	2	8	no	—
63	C₂²×C₄×C₁₉	C₂×D₄	1	4	yes	—
64	C₂²×C₄×C₁₉	C₂×D₄	1	8	yes	—
65	C₂²×C₄×C₁₉	C₂×Q₈	1	16	yes	—
66	C₂²×C₄×C₁₉	C₂×Q₈	1	8	yes	—
67	C₂²×C₄×C₁₉	#41	1	8	yes	—
68	C₂²×C₄×C₁₉	C₂⁴	1	8	yes	—
69	C₂²×C₄×C₁₉	C₂⁴	1	8	yes	—
70	C₂²×C₄×C₁₉	C₂⁴	1	2	yes	—
71	C₂²×C₄×C₁₉	C₂⁴	1	8	yes	—
72	C₂²×C₄×C₁₉	C₂⁴	1	4	yes	—
73	C₄⋊C₄×C₁₉	—	2	32	no	—
74	C₄⋊C₄×C₁₉	—	2	16	no	—
75	C₄⋊C₄×C₁₉	—	2	32	no	—
76	C₄⋊C₄×C₁₉	—	4	8	no	—
77	C₄⋊C₄×C₁₉	—	4	16	no	—
78	C₄²×C₁₉	—	2	32	no	—
79	C₄²×C₁₉	C₂×D₄	1	16	yes	—
80	C₂×#2582	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
81	C₂³×C₁₉	C₂²×D₄	1	2	yes	—
82	C₂³×C₁₉	C₂²×Q₈	1	4	yes	—
83	C₂³×C₁₉	C₂×#41	1	4	yes	—
84	C₂³×C₁₉	H(5,𝔽₂)	1	2	yes	—
85	C₂³×C₁₉	2₋¹⁺⁴	1	2	yes	—
86	C₂³×C₁₉	C₂⁵	1	1	yes	characteristic
87	Q₈×C₁₉	—	2	64	no	—
88	C₂×C₄×C₁₉	—	2	32	no	—
89	C₂×C₄×C₁₉	—	2	8	no	—
90	C₂×C₄×C₁₉	—	2	8	no	—
91	C₂×C₄×C₁₉	—	2	8	no	—
92	C₂×C₄×C₁₉	—	2	2	no	—
93	C₂×C₄×C₁₉	—	2	2	no	—
94	C₂×C₄×C₁₉	—	2	8	no	—
95	C₂×C₄×C₁₉	—	2	16	no	—
96	C₂×C₄×C₁₉	—	2	8	no	—
97	C₂×C₄×C₁₉	—	2	8	no	—
98	C₂×C₄×C₁₉	—	2	8	no	—
99	C₂×C₄×C₁₉	—	4	16	no	—
100	C₂×C₄×C₁₉	—	4	16	no	—
101	C₂×C₄×C₁₉	C₂²⋊Q₈	1	32	yes	—
102	C₂×C₄×C₁₉	C₂²×D₄	1	4	yes	—
103	C₂×C₄×C₁₉	C₂²×D₄	1	8	yes	—
104	C₂×C₄×C₁₉	C₂²×Q₈	1	16	yes	—
105	C₂×C₄×C₁₉	C₂²×Q₈	1	8	yes	—
106	C₂×C₄×C₁₉	C₂×#41	1	8	yes	—
107	C₂²×C₄×Q₈	C₂×C₁₉	1	2	yes	—
108	#2582	C₂×C₁₉	1	16	yes	direct product factor
109	C₂×#398	C₂×C₁₉	1	4	yes	—
110	C₂×#391	C₂×C₁₉	1	4	yes	—
111	C₂×#389	C₂×C₁₉	1	4	yes	—
112	C₂×#384	C₂×C₁₉	1	1	yes	characteristic
113	C₂²×C₁₉	C₂×C₂²⋊Q₈	1	8	yes	—
114	C₂²×C₁₉	#537	1	4	yes	—
115	C₂²×C₁₉	#557	1	8	yes	—
116	C₂²×C₁₉	Q₈²	1	4	yes	—
117	C₂²×C₁₉	#559	1	4	yes	—
118	C₂²×C₁₉	C₂³×D₄	1	1	yes	characteristic
119	C₂²×C₁₉	C₂³×Q₈	1	2	yes	—
120	C₂²×C₁₉	C₂²×#41	1	2	yes	—
121	C₂²×C₁₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
122	C₂²×C₁₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
123	C₄×C₁₉	—	2	16	no	—
124	C₄×C₁₉	—	2	4	no	—
125	C₄×C₁₉	—	4	8	no	—
126	C₄×C₁₉	—	4	8	no	—
127	C₄×C₁₉	C₂×C₂²⋊Q₈	1	16	yes	—
128	C₂³×Q₈	C₂²×C₁₉	1	2	yes	—
129	C₂×C₄×Q₈	—	2	16	no	—
130	C₂×C₄×Q₈	C₂²×C₁₉	1	16	yes	—
131	C₂²×C₄⋊C₄	C₂²×C₁₉	1	4	yes	—
132	C₂²×C₄⋊C₄	C₂²×C₁₉	1	2	yes	—
133	C₂²×C₄⋊C₄	C₂²×C₁₉	1	4	yes	—
134	C₂²×C₄⋊C₄	C₂²×C₁₉	1	2	yes	—
135	C₂²×C₄⋊C₄	C₂²×C₁₉	1	4	yes	—
136	C₂²×C₄²	C₂²×C₁₉	1	4	yes	—
137	C₂²×C₄²	C₂²×C₁₉	1	1	yes	characteristic
138	#398	C₂²×C₁₉	1	32	yes	—
139	#391	C₂²×C₁₉	1	32	yes	—
140	#389	C₂²×C₁₉	1	32	yes	—
141	#384	C₂²×C₁₉	1	8	yes	—
142	C₂×#95	C₂²×C₁₉	1	8	yes	—
143	C₂×#95	C₂²×C₁₉	1	4	yes	—
144	C₂×C₁₉	#2582	1	8	yes	direct product factor
145	C₂×C₁₉	C₂²×C₂²⋊Q₈	1	2	yes	—
146	C₂×C₁₉	C₂×#537	1	1	yes	characteristic
147	C₂×C₁₉	C₂×#557	1	2	yes	—
148	C₂×C₁₉	C₂×Q₈²	1	1	yes	characteristic
149	C₂×C₁₉	C₂×#559	1	1	yes	characteristic
150	C₂²×Q₈	—	2	8	no	—
151	C₂²×Q₈	—	2	8	no	—
152	C₂²×Q₈	Q₈×C₁₉	1	16	yes	complemented
153	C₂²×Q₈	C₂³×C₁₉	1	8	yes	—
154	C₂³×C₄	C₂³×C₁₉	1	4	yes	—
155	C₂³×C₄	C₂³×C₁₉	1	4	yes	—
156	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
157	C₂³×C₄	C₂³×C₁₉	1	4	yes	—
158	C₂³×C₄	C₂³×C₁₉	1	2	yes	—
159	C₄×Q₈	—	2	64	no	—
160	C₂×C₄⋊C₄	—	2	16	no	—
161	C₂×C₄⋊C₄	—	2	8	no	—
162	C₂×C₄⋊C₄	—	2	16	no	—
163	C₂×C₄⋊C₄	—	4	4	no	—
164	C₂×C₄⋊C₄	—	4	8	no	—
165	C₂×C₄⋊C₄	C₂³×C₁₉	1	8	yes	—
166	C₂×C₄⋊C₄	C₂³×C₁₉	1	16	yes	—
167	C₂×C₄⋊C₄	C₂³×C₁₉	1	16	yes	—
168	C₂×C₄⋊C₄	C₂³×C₁₉	1	8	yes	—
169	C₂×C₄⋊C₄	C₂³×C₁₉	1	16	yes	—
170	C₂×C₄²	—	2	16	no	—
171	C₂×C₄²	D₄×C₁₉	1	8	yes	—
172	C₂×C₄²	C₂³×C₁₉	1	16	yes	—
173	C₂×C₄²	C₂³×C₁₉	1	4	yes	—
174	#95	C₂³×C₁₉	1	32	yes	—
175	#95	C₂³×C₁₉	1	16	yes	—
176	C₁₉	C₂×#2582	1	1	yes	characteristic, direct product factor, Sylow
177	C₂⁴	C₂⁴×C₁₉	1	1	yes	characteristic
178	C₂×Q₈	—	2	64	no	—
179	C₂×Q₈	—	2	16	no	—
180	C₂×Q₈	—	2	16	no	—
181	C₂×Q₈	C₂×Q₈×C₁₉	1	32	yes	complemented
182	C₂²×C₄	—	2	8	no	—
183	C₂²×C₄	—	2	8	no	—
184	C₂²×C₄	—	2	2	no	—
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	—	2	8	no	—
187	C₂²×C₄	—	2	16	no	—
188	C₂²×C₄	—	2	8	no	—
189	C₂²×C₄	—	2	8	no	—
190	C₂²×C₄	—	2	8	no	—
191	C₂²×C₄	C₂×D₄×C₁₉	1	4	yes	—
192	C₂²×C₄	C₂×D₄×C₁₉	1	8	yes	—
193	C₂²×C₄	C₂×Q₈×C₁₉	1	16	yes	—
194	C₂²×C₄	C₂×Q₈×C₁₉	1	8	yes	—
195	C₂²×C₄	#41×C₁₉	1	8	yes	—
196	C₂²×C₄	C₂⁴×C₁₉	1	8	yes	—
197	C₂²×C₄	C₂⁴×C₁₉	1	8	yes	—
198	C₂²×C₄	C₂⁴×C₁₉	1	2	yes	—
199	C₂²×C₄	C₂⁴×C₁₉	1	8	yes	—
200	C₂²×C₄	C₂⁴×C₁₉	1	4	yes	—
201	C₄⋊C₄	—	2	32	no	—
202	C₄⋊C₄	—	2	16	no	—
203	C₄⋊C₄	—	2	32	no	—
204	C₄⋊C₄	—	4	8	no	—
205	C₄⋊C₄	—	4	16	no	—
206	C₄²	—	2	32	no	—
207	C₄²	C₂×D₄×C₁₉	1	16	yes	—
208	C₂³	C₂²×D₄×C₁₉	1	2	yes	—
209	C₂³	C₂²×Q₈×C₁₉	1	4	yes	—
210	C₂³	C₂×#41×C₁₉	1	4	yes	—
211	C₂³	C₁₉×H(5,𝔽₂)	1	2	yes	—
212	C₂³	C₁₉×2₋¹⁺⁴	1	2	yes	—
213	C₂³	C₂⁵×C₁₉	1	1	yes	characteristic
214	Q₈	—	2	64	no	—
215	C₂×C₄	—	2	32	no	—
216	C₂×C₄	—	2	8	no	—
217	C₂×C₄	—	2	8	no	—
218	C₂×C₄	—	2	8	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	8	no	—
222	C₂×C₄	—	2	16	no	—
223	C₂×C₄	—	2	8	no	—
224	C₂×C₄	—	2	8	no	—
225	C₂×C₄	—	2	8	no	—
226	C₂×C₄	—	4	16	no	—
227	C₂×C₄	—	4	16	no	—
228	C₂×C₄	C₁₉×C₂²⋊Q₈	1	32	yes	—
229	C₂×C₄	C₂²×D₄×C₁₉	1	4	yes	—
230	C₂×C₄	C₂²×D₄×C₁₉	1	8	yes	—
231	C₂×C₄	C₂²×Q₈×C₁₉	1	16	yes	—
232	C₂×C₄	C₂²×Q₈×C₁₉	1	8	yes	—
233	C₂×C₄	C₂×#41×C₁₉	1	8	yes	—
234	C₂²	C₂×C₁₉×C₂²⋊Q₈	1	8	yes	—
235	C₂²	C₁₉×#537	1	4	yes	—
236	C₂²	C₁₉×#557	1	8	yes	—
237	C₂²	Q₈²×C₁₉	1	4	yes	—
238	C₂²	C₁₉×#559	1	4	yes	—
239	C₂²	C₂³×D₄×C₁₉	1	1	yes	characteristic
240	C₂²	C₂³×Q₈×C₁₉	1	2	yes	—
241	C₂²	C₂²×#41×C₁₉	1	2	yes	—
242	C₂²	C₂×C₁₉×H(5,𝔽₂)	1	1	yes	characteristic
243	C₂²	C₂×C₁₉×2₋¹⁺⁴	1	1	yes	characteristic
244	C₄	—	2	16	no	—
245	C₄	—	2	4	no	—
246	C₄	—	4	8	no	—
247	C₄	—	4	8	no	—
248	C₄	C₂×C₁₉×C₂²⋊Q₈	1	16	yes	—
249	C₂	C₁₉×#2582	1	8	yes	direct product factor
250	C₂	C₂²×C₁₉×C₂²⋊Q₈	1	2	yes	—
251	C₂	C₂×C₁₉×#537	1	1	yes	characteristic
252	C₂	C₂×C₁₉×#557	1	2	yes	—
253	C₂	C₂×Q₈²×C₁₉	1	1	yes	characteristic
254	C₂	C₂×C₁₉×#559	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₁₉	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₁₉
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₉×#2582	C₂	direct (N × H)
25. C₂²×Q₈×C₁₉	Q₈	semidirect
53. C₂×Q₈×C₁₉	C₂×Q₈	semidirect
80. C₂×#2582	C₁₉	direct (N × H)
108. #2582	C₂×C₁₉	direct (N × H)
144. C₂×C₁₉	#2582	direct (N × H)
152. C₂²×Q₈	Q₈×C₁₉	semidirect
176. C₁₉	C₂×#2582	direct (N × H)
181. C₂×Q₈	C₂×Q₈×C₁₉	semidirect
249. C₂	C₁₉×#2582	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄×Q₈×C₁₉	2
C₁₉×#2582	2
C₂×C₁₉×#398	2
C₂×C₁₉×#391	2
C₂×C₁₉×#389	2
C₂×C₁₉×#384	2
C₂×#2582	19

Subgroups — C2×C19×#2582 #18040908

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₉×#2582 #18040908