Subgroups — C₁₉×#62359 #18068318

Order 4864.

2190 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#3497	C₂	1	2	yes	maximal, complemented
2	C₁₉×#3391	C₂	1	4	yes	maximal, complemented
3	Q₈×D₈×C₁₉	C₂	1	4	yes	maximal, complemented
4	C₁₉×#3358	C₂	1	8	yes	maximal, complemented
5	C₁₉×#3245	C₂	1	6	yes	maximal, complemented
6	C₁₉×#3169	C₂	1	6	yes	maximal, complemented
7	C₂×C₈×Q₈×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₉×2₋¹⁺⁴	—	2	2	no	—
9	C₁₉×#550	C₂²	1	8	yes	—
10	D₄×Q₈×C₁₉	C₂²	1	8	yes	—
11	C₁₉×#548	—	2	8	no	—
12	C₁₉×#537	—	2	6	no	—
13	C₁₉×#532	C₂²	1	12	yes	—
14	C₂×C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
15	C₁₉×#500	C₂²	1	6	yes	—
16	C₁₉×#493	C₂²	1	6	yes	—
17	C₁₉×#486	C₂²	1	12	yes	—
18	C₁₉×#474	C₂²	1	24	yes	—
19	C₁₉×#466	C₂²	1	4	yes	—
20	C₁₉×#466	C₂²	1	12	yes	—
21	C₁₉×#459	C₂²	1	24	yes	—
22	C₈×Q₈×C₁₉	C₂²	1	4	yes	—
23	C₈×Q₈×C₁₉	C₂²	1	4	yes	—
24	C₄×D₈×C₁₉	C₂²	1	12	yes	—
25	C₄×D₈×C₁₉	C₂²	1	12	yes	—
26	C₂×C₁₉×C₄⋊C₈	C₂²	1	3	yes	—
27	C₂×C₄×C₈×C₁₉	C₂²	1	3	yes	—
28	C₁₉×2₋¹⁺⁴	—	4	16	no	—
29	C₂×#41×C₁₉	—	2	8	no	—
30	C₂×#41×C₁₉	—	2	12	no	—
31	C₂²×Q₈×C₁₉	—	2	8	no	—
32	C₂²×Q₈×C₁₉	D₄	1	1	yes	characteristic
33	C₂×D₈×C₁₉	C₂³	1	4	yes	—
34	C₂×D₈×C₁₉	C₂³	1	12	yes	—
35	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₁₉	C₂³	1	3	yes	—
37	C₁₉×C₄⋊Q₈	C₂³	1	12	yes	—
38	C₁₉×#127	C₂³	1	12	yes	—
39	C₁₉×#124	—	2	12	no	—
40	C₁₉×C₂²⋊Q₈	—	2	24	no	—
41	C₁₉×#121	—	2	12	no	—
42	C₁₉×#121	—	2	24	no	—
43	C₁₉×#121	C₂³	1	8	yes	—
44	C₄×Q₈×C₁₉	D₄	1	4	yes	—
45	C₄×Q₈×C₁₉	C₂³	1	4	yes	—
46	C₄×D₄×C₁₉	—	2	12	no	—
47	C₄×D₄×C₁₉	C₂³	1	24	yes	—
48	C₂×C₄⋊C₄×C₁₉	D₄	1	3	yes	—
49	C₂×C₄²×C₁₉	C₂³	1	3	yes	—
50	C₁₉×#107	C₂³	1	4	yes	—
51	C₁₉×#107	C₂³	1	12	yes	—
52	C₁₉×C₄⋊C₈	C₂³	1	12	yes	—
53	C₁₉×#102	C₂³	1	8	yes	—
54	C₁₉×#102	C₂³	1	24	yes	—
55	C₄×C₈×C₁₉	C₂³	1	6	yes	—
56	C₄×C₈×C₁₉	C₂³	1	6	yes	—
57	#41×C₁₉	—	4	32	no	—
58	#41×C₁₉	—	4	48	no	—
59	C₂×Q₈×C₁₉	—	2	12	no	—
60	C₂×Q₈×C₁₉	—	4	32	no	—
61	C₂×Q₈×C₁₉	D₈	1	8	yes	complemented
62	C₂×Q₈×C₁₉	C₂×D₄	1	4	yes	—
63	C₂×D₄×C₁₉	—	2	8	no	—
64	C₂×D₄×C₁₉	—	2	12	no	—
65	C₂×D₄×C₁₉	—	2	12	no	—
66	C₂×D₄×C₁₉	C₂⁴	1	8	yes	—
67	C₂²×C₄×C₁₉	—	2	24	no	—
68	C₂²×C₄×C₁₉	C₂×D₄	1	3	yes	—
69	C₂²×C₄×C₁₉	C₂×D₄	1	3	yes	—
70	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
71	D₈×C₁₉	—	2	8	no	—
72	D₈×C₁₉	—	4	12	no	—
73	C₂×C₈×C₁₉	—	2	2	no	—
74	C₂×C₈×C₁₉	—	4	3	no	—
75	C₂×C₈×C₁₉	C₂⁴	1	6	yes	—
76	C₂×C₈×C₁₉	C₂⁴	1	1	yes	characteristic
77	C₂×C₈×C₁₉	C₂⁴	1	1	yes	characteristic
78	C₄⋊C₄×C₁₉	—	2	12	no	—
79	C₄⋊C₄×C₁₉	C₂×D₄	1	12	yes	—
80	C₄⋊C₄×C₁₉	C₂⁴	1	8	yes	—
81	C₂²⋊C₄×C₁₉	—	2	8	no	—
82	C₂²⋊C₄×C₁₉	—	2	24	no	—
83	C₄²×C₁₉	C₂×D₄	1	6	yes	—
84	C₄²×C₁₉	C₂⁴	1	6	yes	—
85	#62359	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
86	C₂³×C₁₉	—	2	8	no	—
87	C₂³×C₁₉	C₂²×D₄	1	1	yes	characteristic
88	Q₈×C₁₉	—	4	24	no	—
89	Q₈×C₁₉	C₂×D₈	1	16	yes	—
90	D₄×C₁₉	—	2	8	no	—
91	D₄×C₁₉	—	2	8	no	—
92	D₄×C₁₉	—	4	16	no	—
93	D₄×C₁₉	—	4	24	no	—
94	D₄×C₁₉	—	8	12	no	—
95	C₂×C₄×C₁₉	—	2	1	no	—
96	C₂×C₄×C₁₉	—	2	1	no	—
97	C₂×C₄×C₁₉	—	2	8	no	—
98	C₂×C₄×C₁₉	—	4	3	no	—
99	C₂×C₄×C₁₉	—	4	48	no	—
100	C₂×C₄×C₁₉	C₂×D₈	1	12	yes	—
101	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
102	C₂×C₄×C₁₉	C₂²×D₄	1	6	yes	—
103	C₂×C₄×C₁₉	C₂²×D₄	1	6	yes	—
104	C₂×C₄×C₁₉	C₂⁵	1	1	yes	characteristic
105	C₈×C₁₉	—	2	1	no	—
106	C₈×C₁₉	—	2	6	no	—
107	C₈×C₁₉	2₋¹⁺⁴	1	2	yes	complemented
108	#3497	C₂×C₁₉	1	2	yes	complemented
109	#3391	C₂×C₁₉	1	4	yes	complemented
110	Q₈×D₈	C₂×C₁₉	1	4	yes	complemented
111	#3358	C₂×C₁₉	1	8	yes	complemented
112	#3245	C₂×C₁₉	1	6	yes	complemented
113	#3169	C₂×C₁₉	1	6	yes	complemented
114	C₂×C₈×Q₈	C₂×C₁₉	1	1	yes	characteristic, complemented
115	C₂²×C₁₉	—	2	1	no	—
116	C₂²×C₁₉	—	2	1	no	—
117	C₂²×C₁₉	—	4	8	no	—
118	C₂²×C₁₉	—	4	8	no	—
119	C₂²×C₁₉	—	4	8	no	—
120	C₂²×C₁₉	C₂²×D₈	1	2	yes	—
121	C₂²×C₁₉	C₂³×D₄	1	1	yes	characteristic
122	C₄×C₁₉	—	2	6	no	—
123	C₄×C₁₉	—	2	1	no	—
124	C₄×C₁₉	—	4	8	no	—
125	C₄×C₁₉	C₂²×D₈	1	12	yes	—
126	C₄×C₁₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
127	C₄×C₁₉	#585	1	1	yes	characteristic
128	C₂×2₋¹⁺⁴	—	2	2	no	—
129	#550	C₂²×C₁₉	1	8	yes	—
130	D₄×Q₈	C₂²×C₁₉	1	8	yes	—
131	#548	—	2	8	no	—
132	#537	—	2	6	no	—
133	#532	C₂²×C₁₉	1	12	yes	—
134	C₂×C₄×Q₈	C₂²×C₁₉	1	1	yes	characteristic
135	#500	C₂²×C₁₉	1	6	yes	—
136	#493	C₂²×C₁₉	1	6	yes	—
137	#486	C₂²×C₁₉	1	12	yes	—
138	#474	C₂²×C₁₉	1	24	yes	—
139	#466	C₂²×C₁₉	1	4	yes	—
140	#466	C₂²×C₁₉	1	12	yes	—
141	#459	C₂²×C₁₉	1	24	yes	—
142	C₈×Q₈	C₂²×C₁₉	1	4	yes	—
143	C₈×Q₈	C₂²×C₁₉	1	4	yes	—
144	C₄×D₈	C₂²×C₁₉	1	12	yes	—
145	C₄×D₈	C₂²×C₁₉	1	12	yes	—
146	C₂×C₄⋊C₈	C₂²×C₁₉	1	3	yes	—
147	C₂×C₄×C₈	C₂²×C₁₉	1	3	yes	—
148	C₂×C₁₉	—	2	2	no	—
149	C₂×C₁₉	—	8	8	no	—
150	C₂×C₁₉	#3497	1	1	yes	characteristic
151	C₂×C₁₉	C₂³×D₈	1	1	yes	characteristic
152	C₂×C₁₉	#3584	1	1	yes	characteristic
153	2₋¹⁺⁴	—	4	16	no	—
154	C₂×#41	—	2	8	no	—
155	C₂×#41	—	2	12	no	—
156	C₂²×Q₈	—	2	8	no	—
157	C₂²×Q₈	D₄×C₁₉	1	1	yes	characteristic
158	C₂×D₈	C₂³×C₁₉	1	4	yes	—
159	C₂×D₈	C₂³×C₁₉	1	12	yes	—
160	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
161	C₂²×C₈	C₂³×C₁₉	1	3	yes	—
162	C₄⋊Q₈	C₂³×C₁₉	1	12	yes	—
163	#127	C₂³×C₁₉	1	12	yes	—
164	#124	—	2	12	no	—
165	C₂²⋊Q₈	—	2	24	no	—
166	#121	—	2	12	no	—
167	#121	—	2	24	no	—
168	#121	C₂³×C₁₉	1	8	yes	—
169	C₄×Q₈	D₄×C₁₉	1	4	yes	—
170	C₄×Q₈	C₂³×C₁₉	1	4	yes	—
171	C₄×D₄	—	2	12	no	—
172	C₄×D₄	C₂³×C₁₉	1	24	yes	—
173	C₂×C₄⋊C₄	D₄×C₁₉	1	3	yes	—
174	C₂×C₄²	C₂³×C₁₉	1	3	yes	—
175	#107	C₂³×C₁₉	1	4	yes	—
176	#107	C₂³×C₁₉	1	12	yes	—
177	C₄⋊C₈	C₂³×C₁₉	1	12	yes	—
178	#102	C₂³×C₁₉	1	8	yes	—
179	#102	C₂³×C₁₉	1	24	yes	—
180	C₄×C₈	C₂³×C₁₉	1	6	yes	—
181	C₄×C₈	C₂³×C₁₉	1	6	yes	—
182	C₁₉	#62359	1	1	yes	characteristic, direct product factor, Sylow
183	#41	—	4	32	no	—
184	#41	—	4	48	no	—
185	C₂×Q₈	—	2	12	no	—
186	C₂×Q₈	—	4	32	no	—
187	C₂×Q₈	D₈×C₁₉	1	8	yes	complemented
188	C₂×Q₈	C₂×D₄×C₁₉	1	4	yes	—
189	C₂×D₄	—	2	8	no	—
190	C₂×D₄	—	2	12	no	—
191	C₂×D₄	—	2	12	no	—
192	C₂×D₄	C₂⁴×C₁₉	1	8	yes	—
193	C₂²×C₄	—	2	24	no	—
194	C₂²×C₄	C₂×D₄×C₁₉	1	3	yes	—
195	C₂²×C₄	C₂×D₄×C₁₉	1	3	yes	—
196	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
197	D₈	—	2	8	no	—
198	D₈	—	4	12	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	4	3	no	—
201	C₂×C₈	C₂⁴×C₁₉	1	1	yes	characteristic
202	C₂×C₈	C₂⁴×C₁₉	1	6	yes	—
203	C₂×C₈	C₂⁴×C₁₉	1	1	yes	characteristic
204	C₄⋊C₄	—	2	12	no	—
205	C₄⋊C₄	C₂×D₄×C₁₉	1	12	yes	—
206	C₄⋊C₄	C₂⁴×C₁₉	1	8	yes	—
207	C₂²⋊C₄	—	2	8	no	—
208	C₂²⋊C₄	—	2	24	no	—
209	C₄²	C₂×D₄×C₁₉	1	6	yes	—
210	C₄²	C₂⁴×C₁₉	1	6	yes	—
211	C₂³	—	2	8	no	—
212	C₂³	C₂²×D₄×C₁₉	1	1	yes	characteristic
213	Q₈	—	4	24	no	—
214	Q₈	C₂×D₈×C₁₉	1	16	yes	—
215	D₄	—	2	8	no	—
216	D₄	—	2	8	no	—
217	D₄	—	4	16	no	—
218	D₄	—	4	24	no	—
219	D₄	—	8	12	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	8	no	—
223	C₂×C₄	—	4	3	no	—
224	C₂×C₄	—	4	48	no	—
225	C₂×C₄	C₂×D₈×C₁₉	1	12	yes	—
226	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
227	C₂×C₄	C₂²×D₄×C₁₉	1	6	yes	—
228	C₂×C₄	C₂²×D₄×C₁₉	1	6	yes	—
229	C₂×C₄	C₂⁵×C₁₉	1	1	yes	characteristic
230	C₈	—	2	1	no	—
231	C₈	—	2	6	no	—
232	C₈	C₁₉×2₋¹⁺⁴	1	2	yes	complemented
233	C₂²	—	2	1	no	—
234	C₂²	—	2	1	no	—
235	C₂²	—	4	8	no	—
236	C₂²	—	4	8	no	—
237	C₂²	—	4	8	no	—
238	C₂²	C₂²×D₈×C₁₉	1	2	yes	—
239	C₂²	C₂³×D₄×C₁₉	1	1	yes	characteristic
240	C₄	—	2	1	no	—
241	C₄	—	2	6	no	—
242	C₄	—	4	8	no	—
243	C₄	C₂²×D₈×C₁₉	1	12	yes	—
244	C₄	C₂×C₁₉×2₋¹⁺⁴	1	1	yes	characteristic
245	C₄	C₁₉×#585	1	1	yes	characteristic
246	C₂	—	2	2	no	—
247	C₂	—	8	8	no	—
248	C₂	C₁₉×#3497	1	1	yes	characteristic
249	C₂	C₂³×D₈×C₁₉	1	1	yes	characteristic
250	C₂	C₁₉×#3584	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₁₉
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
1. C₁₉×#3497	C₂	semidirect
2. C₁₉×#3391	C₂	semidirect
3. Q₈×D₈×C₁₉	C₂	semidirect
4. C₁₉×#3358	C₂	semidirect
5. C₁₉×#3245	C₂	semidirect
6. C₁₉×#3169	C₂	semidirect
7. C₂×C₈×Q₈×C₁₉	C₂	semidirect
61. C₂×Q₈×C₁₉	D₈	semidirect
85. #62359	C₁₉	direct (N × H)
107. C₈×C₁₉	2₋¹⁺⁴	semidirect
108. #3497	C₂×C₁₉	semidirect
109. #3391	C₂×C₁₉	semidirect
110. Q₈×D₈	C₂×C₁₉	semidirect
111. #3358	C₂×C₁₉	semidirect
112. #3245	C₂×C₁₉	semidirect
113. #3169	C₂×C₁₉	semidirect
114. C₂×C₈×Q₈	C₂×C₁₉	semidirect
182. C₁₉	#62359	direct (N × H)
187. C₂×Q₈	D₈×C₁₉	semidirect
232. C₈	C₁₉×2₋¹⁺⁴	semidirect

Maximal subgroups

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

subgroup	index
C₁₉×#3497	2
C₁₉×#3391	2
Q₈×D₈×C₁₉	2
C₁₉×#3358	2
C₁₉×#3245	2
C₁₉×#3169	2
C₂×C₈×Q₈×C₁₉	2
#62359	19

Subgroups — C19×#62359 #18068318

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#62359 #18068318