Subgroups — C₁₉×#17773 #18023732

Order 4864.

1108 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#3499	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2991	C₂	1	2	yes	maximal, complemented
3	C₁₉×#1560	C₂	1	3	yes	maximal, complemented
4	C₁₉×#1548	C₂	1	6	yes	maximal, complemented
5	C₁₉×#1543	C₂	1	3	yes	maximal, complemented
6	C₂×C₁₉×2₋¹⁺⁴	C₄	1	2	yes	—
7	D₄×Q₈×C₁₉	—	2	2	no	—
8	C₁₉×#547	—	2	6	no	—
9	C₁₉×#536	C₄	1	6	yes	—
10	C₁₉×#532	C₂²	1	3	yes	—
11	C₂×C₁₉×C₄⋊Q₈	C₂²	1	3	yes	—
12	C₄×#41×C₁₉	C₂²	1	1	yes	characteristic
13	C₈×Q₈×C₁₉	—	2	2	no	—
14	C₁₉×#436	—	2	6	no	—
15	C₁₉×#431	C₂²	1	6	yes	—
16	C₁₉×#423	C₂²	1	6	yes	—
17	C₁₉×#333	C₂²	1	6	yes	—
18	C₁₉×#333	C₂²	1	6	yes	—
19	C₁₉×#331	C₂²	1	1	yes	characteristic
20	C₁₉×#331	C₂²	1	3	yes	—
21	C₁₉×2₋¹⁺⁴	—	4	8	no	—
22	C₂×#41×C₁₉	—	2	6	no	—
23	C₂×#41×C₁₉	C₂×C₄	1	6	yes	—
24	C₂×#41×C₁₉	C₂×C₄	1	1	yes	characteristic
25	C₂²×Q₈×C₁₉	—	2	2	no	—
26	C₂²×Q₈×C₁₉	C₂×C₄	1	6	yes	—
27	C₂×M₁₆×C₁₉	—	2	6	no	—
28	C₁₉×C₄⋊Q₈	—	2	6	no	—
29	C₁₉×C₄⋊Q₈	C₂×C₄	1	6	yes	—
30	C₁₉×C₄⋊Q₈	C₂³	1	3	yes	—
31	C₁₉×#127	C₂³	1	1	yes	characteristic
32	C₁₉×#124	C₂×C₄	1	6	yes	—
33	C₁₉×#123	—	2	12	no	—
34	C₁₉×C₂²⋊Q₈	—	2	12	no	—
35	C₁₉×#121	—	2	6	no	—
36	C₄×Q₈×C₁₉	D₄	1	2	yes	—
37	C₄×D₄×C₁₉	—	2	6	no	—
38	C₄×D₄×C₁₉	D₄	1	6	yes	—
39	C₁₉×#117	C₂×C₄	1	3	yes	—
40	C₂×C₄⋊C₄×C₁₉	—	2	6	no	—
41	C₂×C₄²×C₁₉	C₂³	1	3	yes	—
42	C₁₉×C₄⋊C₈	—	2	6	no	—
43	C₁₉×C₄⋊C₈	C₂³	1	6	yes	—
44	C₁₉×C₄⋊C₈	C₂³	1	2	yes	—
45	C₁₉×C₂²⋊C₈	—	2	6	no	—
46	C₄×C₈×C₁₉	—	2	6	no	—
47	#41×C₁₉	—	2	4	no	—
48	#41×C₁₉	—	4	12	no	—
49	#41×C₁₉	—	4	24	no	—
50	C₂×Q₈×C₁₉	—	2	6	no	—
51	C₂×Q₈×C₁₉	—	4	12	no	—
52	C₂×Q₈×C₁₉	—	4	8	no	—
53	C₂×Q₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×Q₈×C₁₉	C₂²×C₄	1	6	yes	—
55	C₂×D₄×C₁₉	—	2	6	no	—
56	C₂×D₄×C₁₉	—	2	2	no	—
57	C₂×D₄×C₁₉	C₂²⋊C₄	1	3	yes	—
58	C₂×D₄×C₁₉	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₁₉	—	2	6	no	—
60	C₂²×C₄×C₁₉	—	2	6	no	—
61	C₂²×C₄×C₁₉	C₂²×C₄	1	3	yes	—
62	C₂²×C₄×C₁₉	C₂×D₄	1	6	yes	—
63	M₁₆×C₁₉	—	8	6	no	—
64	C₂×C₈×C₁₉	—	2	6	no	—
65	C₂×C₈×C₁₉	—	2	2	no	—
66	C₄⋊C₄×C₁₉	—	2	12	no	—
67	C₄⋊C₄×C₁₉	—	2	6	no	—
68	C₄⋊C₄×C₁₉	C₂²⋊C₄	1	6	yes	—
69	C₂²⋊C₄×C₁₉	—	2	12	no	—
70	C₂²⋊C₄×C₁₉	C₂²⋊C₄	1	6	yes	—
71	C₄²×C₁₉	C₂²×C₄	1	3	yes	—
72	C₄²×C₁₉	C₂×D₄	1	6	yes	—
73	C₄²×C₁₉	C₂⁴	1	1	yes	characteristic
74	#17773	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₂³×C₁₉	—	2	2	no	—
76	C₂³×C₁₉	C₂×C₂²⋊C₄	1	3	yes	—
77	Q₈×C₁₉	—	4	6	no	—
78	Q₈×C₁₉	—	4	12	no	—
79	Q₈×C₁₉	#100	1	4	yes	—
80	D₄×C₁₉	—	2	6	no	—
81	D₄×C₁₉	—	4	2	no	—
82	D₄×C₁₉	—	4	12	no	—
83	D₄×C₁₉	—	4	2	no	—
84	D₄×C₁₉	—	4	2	no	—
85	C₂×C₄×C₁₉	—	2	6	no	—
86	C₂×C₄×C₁₉	—	2	6	no	—
87	C₂×C₄×C₁₉	—	2	6	no	—
88	C₂×C₄×C₁₉	—	2	6	no	—
89	C₂×C₄×C₁₉	—	4	12	no	—
90	C₂×C₄×C₁₉	—	4	12	no	—
91	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	6	yes	—
92	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	3	yes	—
93	C₂×C₄×C₁₉	C₂³×C₄	1	1	yes	characteristic
94	C₂×C₄×C₁₉	C₂²×D₄	1	2	yes	—
95	C₈×C₁₉	—	4	6	no	—
96	C₈×C₁₉	—	4	2	no	—
97	#3499	C₂×C₁₉	1	1	yes	characteristic
98	#2991	C₂×C₁₉	1	2	yes	complemented
99	#1560	C₂×C₁₉	1	3	yes	complemented
100	#1548	C₂×C₁₉	1	6	yes	complemented
101	#1543	C₂×C₁₉	1	3	yes	complemented
102	C₂²×C₁₉	—	2	3	no	—
103	C₂²×C₁₉	—	2	6	no	—
104	C₂²×C₁₉	—	4	2	no	—
105	C₂²×C₁₉	—	4	4	no	—
106	C₂²×C₁₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
107	C₄×C₁₉	—	2	1	no	—
108	C₄×C₁₉	—	2	6	no	—
109	C₄×C₁₉	—	4	6	no	—
110	C₄×C₁₉	C₂×#100	1	6	yes	—
111	C₄×C₁₉	#577	1	2	yes	—
112	C₄×C₁₉	#578	1	2	yes	—
113	C₂×2₋¹⁺⁴	C₄×C₁₉	1	2	yes	—
114	D₄×Q₈	—	2	2	no	—
115	#547	—	2	6	no	—
116	#536	C₄×C₁₉	1	6	yes	—
117	#532	C₂²×C₁₉	1	3	yes	—
118	C₂×C₄⋊Q₈	C₂²×C₁₉	1	3	yes	—
119	C₄×#41	C₂²×C₁₉	1	1	yes	characteristic
120	C₈×Q₈	—	2	2	no	—
121	#436	—	2	6	no	—
122	#431	C₂²×C₁₉	1	6	yes	—
123	#423	C₂²×C₁₉	1	6	yes	—
124	#333	C₂²×C₁₉	1	6	yes	—
125	#333	C₂²×C₁₉	1	6	yes	—
126	#331	C₂²×C₁₉	1	1	yes	characteristic
127	#331	C₂²×C₁₉	1	3	yes	—
128	C₂×C₁₉	—	4	3	no	—
129	C₂×C₁₉	—	8	2	no	—
130	C₂×C₁₉	C₂²×#100	1	1	yes	characteristic
131	C₂×C₁₉	#2898	1	2	yes	—
132	2₋¹⁺⁴	—	4	8	no	—
133	C₂×#41	—	2	6	no	—
134	C₂×#41	C₂×C₄×C₁₉	1	6	yes	—
135	C₂×#41	C₂×C₄×C₁₉	1	1	yes	characteristic
136	C₂²×Q₈	—	2	2	no	—
137	C₂²×Q₈	C₂×C₄×C₁₉	1	6	yes	—
138	C₂×M₁₆	—	2	6	no	—
139	C₄⋊Q₈	—	2	6	no	—
140	C₄⋊Q₈	C₂×C₄×C₁₉	1	6	yes	—
141	C₄⋊Q₈	C₂³×C₁₉	1	3	yes	—
142	#127	C₂³×C₁₉	1	1	yes	characteristic
143	#124	C₂×C₄×C₁₉	1	6	yes	—
144	#123	—	2	12	no	—
145	C₂²⋊Q₈	—	2	12	no	—
146	#121	—	2	6	no	—
147	C₄×Q₈	D₄×C₁₉	1	2	yes	—
148	C₄×D₄	—	2	6	no	—
149	C₄×D₄	D₄×C₁₉	1	6	yes	—
150	#117	C₂×C₄×C₁₉	1	3	yes	—
151	C₂×C₄⋊C₄	—	2	6	no	—
152	C₂×C₄²	C₂³×C₁₉	1	3	yes	—
153	C₄⋊C₈	—	2	6	no	—
154	C₄⋊C₈	C₂³×C₁₉	1	6	yes	—
155	C₄⋊C₈	C₂³×C₁₉	1	2	yes	—
156	C₂²⋊C₈	—	2	6	no	—
157	C₄×C₈	—	2	6	no	—
158	C₁₉	#17773	1	1	yes	characteristic, direct product factor, Sylow
159	#41	—	2	4	no	—
160	#41	—	4	12	no	—
161	#41	—	4	24	no	—
162	C₂×Q₈	—	2	6	no	—
163	C₂×Q₈	—	4	12	no	—
164	C₂×Q₈	—	4	8	no	—
165	C₂×Q₈	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
166	C₂×Q₈	C₂²×C₄×C₁₉	1	6	yes	—
167	C₂×D₄	—	2	6	no	—
168	C₂×D₄	—	2	2	no	—
169	C₂×D₄	C₂²⋊C₄×C₁₉	1	3	yes	—
170	C₂×D₄	C₂²×C₄×C₁₉	1	2	yes	—
171	C₂²×C₄	—	2	6	no	—
172	C₂²×C₄	—	2	6	no	—
173	C₂²×C₄	C₂²×C₄×C₁₉	1	3	yes	—
174	C₂²×C₄	C₂×D₄×C₁₉	1	6	yes	—
175	M₁₆	—	8	6	no	—
176	C₂×C₈	—	2	2	no	—
177	C₂×C₈	—	2	6	no	—
178	C₄⋊C₄	—	2	12	no	—
179	C₄⋊C₄	—	2	6	no	—
180	C₄⋊C₄	C₂²⋊C₄×C₁₉	1	6	yes	—
181	C₂²⋊C₄	—	2	12	no	—
182	C₂²⋊C₄	C₂²⋊C₄×C₁₉	1	6	yes	—
183	C₄²	C₂²×C₄×C₁₉	1	3	yes	—
184	C₄²	C₂×D₄×C₁₉	1	6	yes	—
185	C₄²	C₂⁴×C₁₉	1	1	yes	characteristic
186	C₂³	—	2	2	no	—
187	C₂³	C₂×C₂²⋊C₄×C₁₉	1	3	yes	—
188	Q₈	—	4	6	no	—
189	Q₈	—	4	12	no	—
190	Q₈	C₁₉×#100	1	4	yes	—
191	D₄	—	2	6	no	—
192	D₄	—	4	2	no	—
193	D₄	—	4	12	no	—
194	D₄	—	4	2	no	—
195	D₄	—	4	2	no	—
196	C₂×C₄	—	2	6	no	—
197	C₂×C₄	—	2	6	no	—
198	C₂×C₄	—	2	6	no	—
199	C₂×C₄	—	2	6	no	—
200	C₂×C₄	—	4	12	no	—
201	C₂×C₄	—	4	12	no	—
202	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	6	yes	—
203	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	3	yes	—
204	C₂×C₄	C₂³×C₄×C₁₉	1	1	yes	characteristic
205	C₂×C₄	C₂²×D₄×C₁₉	1	2	yes	—
206	C₈	—	4	6	no	—
207	C₈	—	4	2	no	—
208	C₂²	—	2	3	no	—
209	C₂²	—	2	6	no	—
210	C₂²	—	4	2	no	—
211	C₂²	—	4	4	no	—
212	C₂²	C₂²×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
213	C₄	—	2	1	no	—
214	C₄	—	2	6	no	—
215	C₄	—	4	6	no	—
216	C₄	C₂×C₁₉×#100	1	6	yes	—
217	C₄	C₁₉×#577	1	2	yes	—
218	C₄	C₁₉×#578	1	2	yes	—
219	C₂	—	4	3	no	—
220	C₂	—	8	2	no	—
221	C₂	C₂²×C₁₉×#100	1	1	yes	characteristic
222	C₂	C₁₉×#2898	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₁₉
Frattini, cosocle	C₄²	C₂⁴×C₁₉
prime radical	C₁₉×#3499	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₄×#41×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₉×#2991	C₂	semidirect
3. C₁₉×#1560	C₂	semidirect
4. C₁₉×#1548	C₂	semidirect
5. C₁₉×#1543	C₂	semidirect
74. #17773	C₁₉	direct (N × H)
98. #2991	C₂×C₁₉	semidirect
99. #1560	C₂×C₁₉	semidirect
100. #1548	C₂×C₁₉	semidirect
101. #1543	C₂×C₁₉	semidirect
158. C₁₉	#17773	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×#3499	2
C₁₉×#2991	2
C₁₉×#1560	2
C₁₉×#1548	2
C₁₉×#1543	2
#17773	19

Subgroups — C19×#17773 #18023732

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#17773 #18023732