Subgroups — C₂×C₁₉×#1561 #18023695

Order 4864.

1042 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₉×#533	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₉×#423	C₂	1	2	yes	maximal
3	C₁₉×#1561	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₁₉×#332	C₂	1	4	yes	maximal, complemented
5	C₂²×M₁₆×C₁₉	—	2	2	no	—
6	C₁₉×#533	—	2	4	no	—
7	C₁₉×#533	C₄	1	4	yes	—
8	C₁₉×#533	C₂²	1	4	yes	—
9	C₂×C₁₉×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₂×C₁₉×#125	C₄	1	2	yes	—
11	C₂×C₁₉×C₂²⋊Q₈	—	2	2	no	—
12	C₂×C₄×Q₈×C₁₉	—	2	2	no	—
13	C₂×C₁₉×#117	C₄	1	2	yes	—
14	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
15	C₁₉×#423	—	2	4	no	—
16	C₁₉×#423	C₂²	1	8	yes	—
17	C₂×C₁₉×C₄⋊C₈	—	2	2	no	—
18	C₂×C₁₉×C₄⋊C₈	C₂²	1	4	yes	—
19	C₁₉×#332	C₂²	1	16	yes	complemented
20	C₂²×Q₈×C₁₉	—	2	2	no	—
21	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₁₉	D₄	1	2	yes	—
23	C₂×M₁₆×C₁₉	—	2	8	no	—
24	C₂×M₁₆×C₁₉	—	4	8	no	—
25	C₂²×C₈×C₁₉	—	2	4	no	—
26	C₁₉×C₄⋊Q₈	—	2	2	no	—
27	C₁₉×C₄⋊Q₈	—	2	2	no	—
28	C₁₉×C₄⋊Q₈	C₂×C₄	1	4	yes	—
29	C₁₉×C₄⋊Q₈	C₂³	1	4	yes	—
30	C₁₉×#125	—	2	4	no	—
31	C₁₉×#125	C₂×C₄	1	8	yes	—
32	C₁₉×C₂²⋊Q₈	—	2	8	no	—
33	C₁₉×C₂²⋊Q₈	—	2	8	no	—
34	C₄×Q₈×C₁₉	—	2	8	no	—
35	C₄×Q₈×C₁₉	—	2	8	no	—
36	C₁₉×#117	—	2	4	no	—
37	C₁₉×#117	C₂×C₄	1	8	yes	—
38	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
39	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
40	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	4	yes	—
41	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
42	C₂×C₄²×C₁₉	—	2	2	no	—
43	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₁₉	D₄	1	2	yes	—
46	C₂×C₄²×C₁₉	D₄	1	4	yes	—
47	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₁₉	C₂³	1	2	yes	—
49	C₁₉×C₄⋊C₈	—	2	8	no	—
50	C₁₉×C₄⋊C₈	—	2	4	no	—
51	C₁₉×C₄⋊C₈	C₂³	1	8	yes	—
52	C₂⁴×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×Q₈×C₁₉	—	2	4	no	—
54	C₂×Q₈×C₁₉	—	2	4	no	—
55	C₂×Q₈×C₁₉	—	4	4	no	—
56	C₂×Q₈×C₁₉	—	4	4	no	—
57	C₂²×C₄×C₁₉	—	2	4	no	—
58	C₂²×C₄×C₁₉	—	2	4	no	—
59	C₂²×C₄×C₁₉	—	2	4	no	—
60	C₂²×C₄×C₁₉	—	2	4	no	—
61	C₂²×C₄×C₁₉	C₂²⋊C₄	1	4	yes	—
62	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₉	C₂×D₄	1	4	yes	—
68	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
69	M₁₆×C₁₉	—	4	16	no	—
70	C₂×C₈×C₁₉	—	2	8	no	—
71	C₂×C₈×C₁₉	—	4	8	no	—
72	C₄⋊C₄×C₁₉	—	2	16	no	—
73	C₄⋊C₄×C₁₉	—	2	4	no	—
74	C₄⋊C₄×C₁₉	—	2	4	no	—
75	C₄⋊C₄×C₁₉	—	2	4	no	—
76	C₄⋊C₄×C₁₉	C₂²×C₄	1	8	yes	—
77	C₂²⋊C₄×C₁₉	—	2	8	no	—
78	C₄²×C₁₉	—	2	8	no	—
79	C₄²×C₁₉	C₂²⋊C₄	1	4	yes	—
80	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
81	C₄²×C₁₉	C₂²×C₄	1	2	yes	—
82	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₁₉	C₂×D₄	1	4	yes	—
84	C₄²×C₁₉	C₂×D₄	1	2	yes	—
85	C₄²×C₁₉	C₂⁴	1	1	yes	characteristic
86	C₂×#1561	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
87	C₂³×C₁₉	—	2	2	no	—
88	C₂³×C₁₉	—	2	4	no	—
89	C₂³×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
90	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	Q₈×C₁₉	—	4	4	no	—
92	Q₈×C₁₉	—	4	4	no	—
93	Q₈×C₁₉	—	4	8	no	—
94	C₂×C₄×C₁₉	—	2	2	no	—
95	C₂×C₄×C₁₉	—	2	4	no	—
96	C₂×C₄×C₁₉	—	2	4	no	—
97	C₂×C₄×C₁₉	—	2	4	no	—
98	C₂×C₄×C₁₉	—	2	4	no	—
99	C₂×C₄×C₁₉	—	2	4	no	—
100	C₂×C₄×C₁₉	—	2	8	no	—
101	C₂×C₄×C₁₉	—	2	8	no	—
102	C₂×C₄×C₁₉	—	4	8	no	—
103	C₂×C₄×C₁₉	#101	1	4	yes	—
104	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	4	yes	—
105	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
107	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
108	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	4	yes	—
109	C₂×C₄×C₁₉	#137	1	4	yes	—
110	C₂×C₄×C₁₉	C₂³×C₄	1	1	yes	characteristic
111	C₂×C₄×C₁₉	C₂²×D₄	1	2	yes	—
112	C₈×C₁₉	—	4	8	no	—
113	C₂×#533	C₂×C₁₉	1	1	yes	characteristic
114	C₂×#423	C₂×C₁₉	1	2	yes	—
115	#1561	C₂×C₁₉	1	8	yes	direct product factor
116	C₂×#332	C₂×C₁₉	1	4	yes	complemented
117	C₂²×C₁₉	—	2	2	no	—
118	C₂²×C₁₉	—	2	4	no	—
119	C₂²×C₁₉	—	4	4	no	—
120	C₂²×C₁₉	C₂×#101	1	2	yes	—
121	C₂²×C₁₉	#417	1	4	yes	—
122	C₂²×C₁₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
123	C₄×C₁₉	—	2	1	no	—
124	C₄×C₁₉	—	2	1	no	—
125	C₄×C₁₉	—	2	4	no	—
126	C₄×C₁₉	—	4	8	no	—
127	C₄×C₁₉	C₂×#101	1	4	yes	—
128	C₄×C₁₉	#417	1	4	yes	—
129	C₄×C₁₉	C₂×AGL(1,ℤ/8)	1	2	yes	—
130	C₄×C₁₉	C₂×#137	1	2	yes	—
131	C₂²×M₁₆	—	2	2	no	—
132	#533	—	2	4	no	—
133	#533	C₄×C₁₉	1	4	yes	—
134	#533	C₂²×C₁₉	1	4	yes	—
135	C₂×C₄⋊Q₈	C₂²×C₁₉	1	2	yes	—
136	C₂×#125	C₄×C₁₉	1	2	yes	—
137	C₂×C₂²⋊Q₈	—	2	2	no	—
138	C₂×C₄×Q₈	—	2	2	no	—
139	C₂×#117	C₄×C₁₉	1	2	yes	—
140	C₂²×C₄²	C₂²×C₁₉	1	1	yes	characteristic
141	#423	—	2	4	no	—
142	#423	C₂²×C₁₉	1	8	yes	—
143	C₂×C₄⋊C₈	—	2	2	no	—
144	C₂×C₄⋊C₈	C₂²×C₁₉	1	4	yes	—
145	#332	C₂²×C₁₉	1	16	yes	complemented
146	C₂×C₁₉	—	4	2	no	—
147	C₂×C₁₉	#1561	1	4	yes	direct product factor
148	C₂×C₁₉	C₂²×#101	1	1	yes	characteristic
149	C₂×C₁₉	C₂×#417	1	2	yes	—
150	C₂²×Q₈	—	2	2	no	—
151	C₂³×C₄	C₂×C₄×C₁₉	1	1	yes	characteristic
152	C₂³×C₄	D₄×C₁₉	1	2	yes	—
153	C₂×M₁₆	—	2	8	no	—
154	C₂×M₁₆	—	4	8	no	—
155	C₂²×C₈	—	2	4	no	—
156	C₄⋊Q₈	—	2	2	no	—
157	C₄⋊Q₈	—	2	2	no	—
158	C₄⋊Q₈	C₂×C₄×C₁₉	1	4	yes	—
159	C₄⋊Q₈	C₂³×C₁₉	1	4	yes	—
160	#125	—	2	4	no	—
161	#125	C₂×C₄×C₁₉	1	8	yes	—
162	C₂²⋊Q₈	—	2	8	no	—
163	C₂²⋊Q₈	—	2	8	no	—
164	C₄×Q₈	—	2	8	no	—
165	C₄×Q₈	—	2	8	no	—
166	#117	—	2	4	no	—
167	#117	C₂×C₄×C₁₉	1	8	yes	—
168	C₂×C₄⋊C₄	—	2	4	no	—
169	C₂×C₄⋊C₄	—	2	2	no	—
170	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	4	yes	—
171	C₂×C₂²⋊C₄	—	2	2	no	—
172	C₂×C₄²	—	2	2	no	—
173	C₂×C₄²	C₂×C₄×C₁₉	1	1	yes	characteristic
174	C₂×C₄²	C₂×C₄×C₁₉	1	2	yes	—
175	C₂×C₄²	D₄×C₁₉	1	2	yes	—
176	C₂×C₄²	D₄×C₁₉	1	4	yes	—
177	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
178	C₂×C₄²	C₂³×C₁₉	1	2	yes	—
179	C₄⋊C₈	—	2	8	no	—
180	C₄⋊C₈	—	2	4	no	—
181	C₄⋊C₈	C₂³×C₁₉	1	8	yes	—
182	C₁₉	C₂×#1561	1	1	yes	characteristic, direct product factor, Sylow
183	C₂⁴	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
184	C₂×Q₈	—	2	4	no	—
185	C₂×Q₈	—	2	4	no	—
186	C₂×Q₈	—	4	4	no	—
187	C₂×Q₈	—	4	4	no	—
188	C₂²×C₄	—	2	4	no	—
189	C₂²×C₄	—	2	4	no	—
190	C₂²×C₄	—	2	4	no	—
191	C₂²×C₄	—	2	4	no	—
192	C₂²×C₄	C₂²⋊C₄×C₁₉	1	2	yes	—
193	C₂²×C₄	C₂²⋊C₄×C₁₉	1	4	yes	—
194	C₂²×C₄	C₂²⋊C₄×C₁₉	1	2	yes	—
195	C₂²×C₄	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
196	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
197	C₂²×C₄	C₂²×C₄×C₁₉	1	2	yes	—
198	C₂²×C₄	C₂×D₄×C₁₉	1	4	yes	—
199	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
200	M₁₆	—	4	16	no	—
201	C₂×C₈	—	2	8	no	—
202	C₂×C₈	—	4	8	no	—
203	C₄⋊C₄	—	2	16	no	—
204	C₄⋊C₄	—	2	4	no	—
205	C₄⋊C₄	—	2	4	no	—
206	C₄⋊C₄	—	2	4	no	—
207	C₄⋊C₄	C₂²×C₄×C₁₉	1	8	yes	—
208	C₂²⋊C₄	—	2	8	no	—
209	C₄²	—	2	8	no	—
210	C₄²	C₂²⋊C₄×C₁₉	1	4	yes	—
211	C₄²	C₂²⋊C₄×C₁₉	1	2	yes	—
212	C₄²	C₂²×C₄×C₁₉	1	2	yes	—
213	C₄²	C₂²×C₄×C₁₉	1	1	yes	characteristic
214	C₄²	C₂×D₄×C₁₉	1	4	yes	—
215	C₄²	C₂×D₄×C₁₉	1	2	yes	—
216	C₄²	C₂⁴×C₁₉	1	1	yes	characteristic
217	C₂³	—	2	2	no	—
218	C₂³	—	2	4	no	—
219	C₂³	C₂×C₂²⋊C₄×C₁₉	1	2	yes	—
220	C₂³	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
221	Q₈	—	4	8	no	—
222	Q₈	—	4	4	no	—
223	Q₈	—	4	4	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	8	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	8	no	—
232	C₂×C₄	—	4	8	no	—
233	C₂×C₄	C₁₉×#101	1	4	yes	—
234	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	4	yes	—
235	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
236	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	2	yes	—
237	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	2	yes	—
238	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	4	yes	—
239	C₂×C₄	C₁₉×#137	1	4	yes	—
240	C₂×C₄	C₂³×C₄×C₁₉	1	1	yes	characteristic
241	C₂×C₄	C₂²×D₄×C₁₉	1	2	yes	—
242	C₈	—	4	8	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	4	no	—
245	C₂²	—	4	4	no	—
246	C₂²	C₂×C₁₉×#101	1	2	yes	—
247	C₂²	C₁₉×#417	1	4	yes	—
248	C₂²	C₂²×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
249	C₄	—	2	1	no	—
250	C₄	—	2	1	no	—
251	C₄	—	2	4	no	—
252	C₄	—	4	8	no	—
253	C₄	C₂×C₁₉×#101	1	4	yes	—
254	C₄	C₁₉×#417	1	4	yes	—
255	C₄	C₂×C₁₉×AGL(1,ℤ/8)	1	2	yes	—
256	C₄	C₂×C₁₉×#137	1	2	yes	—
257	C₂	—	4	2	no	—
258	C₂	C₁₉×#1561	1	4	yes	direct product factor
259	C₂	C₂²×C₁₉×#101	1	1	yes	characteristic
260	C₂	C₂×C₁₉×#417	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₉	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₁₉
Frattini, cosocle	C₄²	C₂⁴×C₁₉
prime radical	C₂⁴×C₁₉	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₉ ◁ C₂²×C₄²×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₉×#1561	C₂	direct (N × H)
4. C₂×C₁₉×#332	C₂	semidirect
19. C₁₉×#332	C₂²	semidirect
86. C₂×#1561	C₁₉	direct (N × H)
115. #1561	C₂×C₁₉	direct (N × H)
116. C₂×#332	C₂×C₁₉	semidirect
145. #332	C₂²×C₁₉	semidirect
147. C₂×C₁₉	#1561	direct (N × H)
182. C₁₉	C₂×#1561	direct (N × H)
258. C₂	C₁₉×#1561	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×#533	2
C₂×C₁₉×#423	2
C₁₉×#1561	2
C₂×C₁₉×#332	2
C₂×#1561	19

Subgroups — C2×C19×#1561 #18023695

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₉×#1561 #18023695