Subgroups — C₁₉×#36674 #18042633

Order 4864.

1778 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#2842	C₂	1	3	yes	maximal
2	C₁₉×#2841	C₂	1	12	yes	maximal, complemented
3	C₁₉×#2840	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#2396	C₂	1	12	yes	maximal, complemented
5	C₁₉×#2280	C₂	1	3	yes	maximal
6	C₂⁴×C₄×C₁₉	C₂²	1	1	yes	characteristic
7	C₂×C₄×Q₈×C₁₉	C₂²	1	4	yes	complemented
8	C₂×C₁₉×#117	C₂²	1	6	yes	—
9	C₁₉×#400	C₂²	1	12	yes	complemented
10	C₁₉×#399	C₂²	1	12	yes	—
11	C₁₉×#398	C₂²	1	24	yes	complemented
12	C₁₉×#396	C₂²	1	24	yes	—
13	C₁₉×#395	C₂²	1	12	yes	complemented
14	C₁₉×#393	C₂²	1	12	yes	—
15	C₁₉×#381	C₂²	1	6	yes	—
16	C₁₉×#380	C₂²	1	12	yes	—
17	C₁₉×#379	C₂²	1	3	yes	—
18	C₁₉×#379	C₂²	1	3	yes	—
19	C₄×C₄⋊C₄×C₁₉	C₂²	1	12	yes	complemented
20	C₄×C₂²⋊C₄×C₁₉	C₂²	1	12	yes	—
21	C₂⁵×C₁₉	C₂³	1	1	yes	characteristic
22	C₂²×Q₈×C₁₉	C₂³	1	4	yes	—
23	C₂³×C₄×C₁₉	—	4	4	no	—
24	C₂³×C₄×C₁₉	Q₈	1	8	yes	complemented
25	C₂³×C₄×C₁₉	C₂³	1	3	yes	—
26	C₂³×C₄×C₁₉	C₂³	1	3	yes	—
27	C₄×Q₈×C₁₉	—	4	8	no	—
28	C₁₉×#117	—	4	12	no	—
29	C₂×C₄⋊C₄×C₁₉	C₂³	1	24	yes	—
30	C₂×C₄⋊C₄×C₁₉	C₂³	1	12	yes	—
31	C₂×C₄⋊C₄×C₁₉	C₂³	1	12	yes	—
32	C₂×C₂²⋊C₄×C₁₉	C₂³	1	12	yes	—
33	C₂×C₂²⋊C₄×C₁₉	C₂³	1	12	yes	—
34	C₂×C₂²⋊C₄×C₁₉	C₂³	1	6	yes	—
35	C₂×C₂²⋊C₄×C₁₉	C₂³	1	6	yes	—
36	C₂×C₄²×C₁₉	C₂³	1	12	yes	—
37	C₁₉×#95	C₂³	1	24	yes	—
38	C₁₉×#95	C₂³	1	12	yes	—
39	C₁₉×#95	C₂³	1	12	yes	—
40	C₂⁴×C₁₉	—	4	2	no	—
41	C₂⁴×C₁₉	—	4	4	no	—
42	C₂⁴×C₁₉	C₂×Q₈	1	4	yes	—
43	C₂⁴×C₁₉	C₂⁴	1	3	yes	—
44	C₂×Q₈×C₁₉	—	4	4	no	—
45	C₂×Q₈×C₁₉	—	4	8	no	—
46	C₂²×C₄×C₁₉	—	2	12	no	—
47	C₂²×C₄×C₁₉	—	4	16	no	—
48	C₂²×C₄×C₁₉	—	4	6	no	—
49	C₂²×C₄×C₁₉	C₂×Q₈	1	12	yes	—
50	C₂²×C₄×C₁₉	C₂×Q₈	1	12	yes	—
51	C₂²×C₄×C₁₉	C₂⁴	1	12	yes	—
52	C₂²×C₄×C₁₉	C₂⁴	1	12	yes	—
53	C₂²×C₄×C₁₉	C₂⁴	1	3	yes	—
54	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
55	C₄⋊C₄×C₁₉	—	4	24	no	—
56	C₄⋊C₄×C₁₉	—	4	12	no	—
57	C₄⋊C₄×C₁₉	—	4	12	no	—
58	C₂²⋊C₄×C₁₉	—	4	12	no	—
59	C₂²⋊C₄×C₁₉	—	4	12	no	—
60	C₂²⋊C₄×C₁₉	—	4	6	no	—
61	C₂²⋊C₄×C₁₉	—	4	6	no	—
62	C₄²×C₁₉	—	2	24	no	—
63	#36674	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
64	C₂³×C₁₉	—	2	6	no	—
65	C₂³×C₁₉	—	2	12	no	—
66	C₂³×C₁₉	—	4	4	no	—
67	C₂³×C₁₉	—	4	8	no	—
68	C₂³×C₁₉	—	4	16	no	—
69	C₂³×C₁₉	C₂²×Q₈	1	6	yes	—
70	C₂³×C₁₉	C₂⁵	1	1	yes	characteristic
71	Q₈×C₁₉	—	4	16	no	—
72	C₂×C₄×C₁₉	—	2	6	no	—
73	C₂×C₄×C₁₉	—	2	24	no	—
74	C₂×C₄×C₁₉	—	2	12	no	—
75	C₂×C₄×C₁₉	—	2	12	no	—
76	C₂×C₄×C₁₉	—	2	24	no	—
77	C₂×C₄×C₁₉	—	2	24	no	—
78	C₂×C₄×C₁₉	—	4	12	no	—
79	C₂×C₄×C₁₉	C₂²×Q₈	1	6	yes	—
80	C₂×C₄×C₁₉	C₂²×Q₈	1	2	yes	—
81	C₂×C₄×C₁₉	H(5,𝔽₂)	1	4	yes	—
82	#2842	C₂×C₁₉	1	3	yes	—
83	#2841	C₂×C₁₉	1	12	yes	complemented
84	#2840	C₂×C₁₉	1	1	yes	characteristic
85	#2396	C₂×C₁₉	1	12	yes	complemented
86	#2280	C₂×C₁₉	1	3	yes	—
87	C₂²×C₁₉	—	2	6	no	—
88	C₂²×C₁₉	—	2	12	no	—
89	C₂²×C₁₉	—	4	16	no	—
90	C₂²×C₁₉	—	4	12	no	—
91	C₂²×C₁₉	C₂³×Q₈	1	1	yes	characteristic
92	C₂²×C₁₉	C₂×H(5,𝔽₂)	1	2	yes	—
93	C₂²×C₁₉	#585	1	4	yes	—
94	C₄×C₁₉	—	2	6	no	—
95	C₄×C₁₉	—	4	12	no	—
96	C₄×C₁₉	—	4	12	no	—
97	C₄×C₁₉	#543	1	4	yes	—
98	C₂⁴×C₄	C₂²×C₁₉	1	1	yes	characteristic
99	C₂×C₄×Q₈	C₂²×C₁₉	1	4	yes	complemented
100	C₂×#117	C₂²×C₁₉	1	6	yes	—
101	#400	C₂²×C₁₉	1	12	yes	complemented
102	#399	C₂²×C₁₉	1	12	yes	—
103	#398	C₂²×C₁₉	1	24	yes	complemented
104	#396	C₂²×C₁₉	1	24	yes	—
105	#395	C₂²×C₁₉	1	12	yes	complemented
106	#393	C₂²×C₁₉	1	12	yes	—
107	#381	C₂²×C₁₉	1	6	yes	—
108	#380	C₂²×C₁₉	1	12	yes	—
109	#379	C₂²×C₁₉	1	3	yes	—
110	#379	C₂²×C₁₉	1	3	yes	—
111	C₄×C₄⋊C₄	C₂²×C₁₉	1	12	yes	complemented
112	C₄×C₂²⋊C₄	C₂²×C₁₉	1	12	yes	—
113	C₂×C₁₉	—	4	6	no	—
114	C₂×C₁₉	C₂×#543	1	1	yes	characteristic
115	C₂×C₁₉	#3458	1	2	yes	—
116	C₂×C₁₉	#3529	1	4	yes	—
117	C₂⁵	C₂³×C₁₉	1	1	yes	characteristic
118	C₂²×Q₈	C₂³×C₁₉	1	4	yes	—
119	C₂³×C₄	—	4	4	no	—
120	C₂³×C₄	Q₈×C₁₉	1	8	yes	complemented
121	C₂³×C₄	C₂³×C₁₉	1	3	yes	—
122	C₂³×C₄	C₂³×C₁₉	1	3	yes	—
123	C₄×Q₈	—	4	8	no	—
124	#117	—	4	12	no	—
125	C₂×C₄⋊C₄	C₂³×C₁₉	1	12	yes	—
126	C₂×C₄⋊C₄	C₂³×C₁₉	1	12	yes	—
127	C₂×C₄⋊C₄	C₂³×C₁₉	1	24	yes	—
128	C₂×C₂²⋊C₄	C₂³×C₁₉	1	6	yes	—
129	C₂×C₂²⋊C₄	C₂³×C₁₉	1	6	yes	—
130	C₂×C₂²⋊C₄	C₂³×C₁₉	1	12	yes	—
131	C₂×C₂²⋊C₄	C₂³×C₁₉	1	12	yes	—
132	C₂×C₄²	C₂³×C₁₉	1	12	yes	—
133	#95	C₂³×C₁₉	1	24	yes	—
134	#95	C₂³×C₁₉	1	12	yes	—
135	#95	C₂³×C₁₉	1	12	yes	—
136	C₁₉	#36674	1	1	yes	characteristic, direct product factor, Sylow
137	C₂⁴	—	4	2	no	—
138	C₂⁴	—	4	4	no	—
139	C₂⁴	C₂×Q₈×C₁₉	1	4	yes	—
140	C₂⁴	C₂⁴×C₁₉	1	3	yes	—
141	C₂×Q₈	—	4	4	no	—
142	C₂×Q₈	—	4	8	no	—
143	C₂²×C₄	—	2	12	no	—
144	C₂²×C₄	—	4	16	no	—
145	C₂²×C₄	—	4	6	no	—
146	C₂²×C₄	C₂×Q₈×C₁₉	1	12	yes	—
147	C₂²×C₄	C₂×Q₈×C₁₉	1	12	yes	—
148	C₂²×C₄	C₂⁴×C₁₉	1	12	yes	—
149	C₂²×C₄	C₂⁴×C₁₉	1	12	yes	—
150	C₂²×C₄	C₂⁴×C₁₉	1	3	yes	—
151	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
152	C₄⋊C₄	—	4	24	no	—
153	C₄⋊C₄	—	4	12	no	—
154	C₄⋊C₄	—	4	12	no	—
155	C₂²⋊C₄	—	4	12	no	—
156	C₂²⋊C₄	—	4	12	no	—
157	C₂²⋊C₄	—	4	6	no	—
158	C₂²⋊C₄	—	4	6	no	—
159	C₄²	—	2	24	no	—
160	C₂³	—	2	6	no	—
161	C₂³	—	2	12	no	—
162	C₂³	—	4	4	no	—
163	C₂³	—	4	8	no	—
164	C₂³	—	4	16	no	—
165	C₂³	C₂²×Q₈×C₁₉	1	6	yes	—
166	C₂³	C₂⁵×C₁₉	1	1	yes	characteristic
167	Q₈	—	4	16	no	—
168	C₂×C₄	—	2	6	no	—
169	C₂×C₄	—	2	24	no	—
170	C₂×C₄	—	2	12	no	—
171	C₂×C₄	—	2	12	no	—
172	C₂×C₄	—	2	24	no	—
173	C₂×C₄	—	2	24	no	—
174	C₂×C₄	—	4	12	no	—
175	C₂×C₄	C₂²×Q₈×C₁₉	1	6	yes	—
176	C₂×C₄	C₂²×Q₈×C₁₉	1	2	yes	—
177	C₂×C₄	C₁₉×H(5,𝔽₂)	1	4	yes	—
178	C₂²	—	2	6	no	—
179	C₂²	—	2	12	no	—
180	C₂²	—	4	16	no	—
181	C₂²	—	4	12	no	—
182	C₂²	C₂³×Q₈×C₁₉	1	1	yes	characteristic
183	C₂²	C₂×C₁₉×H(5,𝔽₂)	1	2	yes	—
184	C₂²	C₁₉×#585	1	4	yes	—
185	C₄	—	2	6	no	—
186	C₄	—	4	12	no	—
187	C₄	—	4	12	no	—
188	C₄	C₁₉×#543	1	4	yes	—
189	C₂	—	4	6	no	—
190	C₂	C₂×C₁₉×#543	1	1	yes	characteristic
191	C₂	C₁₉×#3458	1	2	yes	—
192	C₂	C₁₉×#3529	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₉	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₁₉
socle	C₂³×C₁₉	C₂⁵
prime radical	C₂⁵×C₁₉	C₂³
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×#2841	C₂	semidirect
4. C₁₉×#2396	C₂	semidirect
7. C₂×C₄×Q₈×C₁₉	C₂²	semidirect
9. C₁₉×#400	C₂²	semidirect
11. C₁₉×#398	C₂²	semidirect
13. C₁₉×#395	C₂²	semidirect
19. C₄×C₄⋊C₄×C₁₉	C₂²	semidirect
24. C₂³×C₄×C₁₉	Q₈	semidirect
63. #36674	C₁₉	direct (N × H)
83. #2841	C₂×C₁₉	semidirect
85. #2396	C₂×C₁₉	semidirect
99. C₂×C₄×Q₈	C₂²×C₁₉	semidirect
101. #400	C₂²×C₁₉	semidirect
103. #398	C₂²×C₁₉	semidirect
105. #395	C₂²×C₁₉	semidirect
111. C₄×C₄⋊C₄	C₂²×C₁₉	semidirect
120. C₂³×C₄	Q₈×C₁₉	semidirect
136. C₁₉	#36674	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×#2842	2
C₁₉×#2841	2
C₁₉×#2840	2
C₁₉×#2396	2
C₁₉×#2280	2
#36674	19

Subgroups — C19×#36674 #18042633

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#36674 #18042633