Subgroups — C₁₉×#62843 #18068802

Order 4864.

7130 conjugacy classes of subgroups.

The subgroup lattice has 164 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#3501	C₂	1	48	yes	maximal, complemented
2	C₁₉×#3458	C₂	1	4	yes	maximal, complemented
3	C₂×C₁₉×#543	C₂	1	8	yes	maximal, complemented
4	C₂×C₁₉×#518	C₂	1	3	yes	maximal
5	C₂³×D₄×C₁₉	C₂²	1	1	yes	characteristic
6	C₁₉×#554	C₂²	1	192	yes	complemented
7	D₄×Q₈×C₁₉	C₂²	1	64	yes	complemented
8	C₁₉×#544	C₂²	1	48	yes	complemented
9	C₁₉×#543	C₂²	1	16	yes	—
10	C₁₉×#543	C₂²	1	96	yes	complemented
11	C₁₉×#533	C₂²	1	48	yes	complemented
12	C₂×C₁₉×C₂²⋊Q₈	C₂²	1	96	yes	complemented
13	C₁₉×#518	C₂²	1	48	yes	—
14	C₂×C₄×D₄×C₁₉	C₂²	1	24	yes	—
15	C₂×C₁₉×#117	C₂²	1	6	yes	—
16	C₂²×C₂²⋊C₄×C₁₉	C₂²	1	12	yes	—
17	C₂⁵×C₁₉	C₂³	1	2	yes	—
18	C₂²×Q₈×C₁₉	C₂³	1	32	yes	complemented
19	C₂²×D₄×C₁₉	Q₈	1	16	yes	complemented
20	C₂²×D₄×C₁₉	C₂³	1	12	yes	—
21	C₂³×C₄×C₁₉	C₂³	1	24	yes	—
22	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
23	C₁₉×C₄⋊Q₈	C₂³	1	96	yes	complemented
24	C₁₉×#125	C₂³	1	96	yes	complemented
25	C₁₉×C₂²⋊Q₈	C₂³	1	192	yes	—
26	C₁₉×C₂²⋊Q₈	C₂³	1	384	yes	complemented
27	C₄×Q₈×C₁₉	C₂³	1	64	yes	complemented
28	C₄×D₄×C₁₉	C₂³	1	192	yes	—
29	C₁₉×#117	C₂³	1	48	yes	—
30	C₂×C₄⋊C₄×C₁₉	C₂³	1	12	yes	—
31	C₂×C₄⋊C₄×C₁₉	C₂³	1	96	yes	complemented
32	C₂×C₂²⋊C₄×C₁₉	C₂³	1	12	yes	—
33	C₂×C₂²⋊C₄×C₁₉	C₂³	1	24	yes	—
34	C₂×C₂²⋊C₄×C₁₉	C₂³	1	96	yes	—
35	C₂×C₄²×C₁₉	C₂³	1	12	yes	—
36	C₂⁴×C₁₉	—	8	4	no	—
37	C₂⁴×C₁₉	C₂×Q₈	1	16	yes	complemented
38	C₂⁴×C₁₉	C₂⁴	1	1	yes	characteristic
39	C₂⁴×C₁₉	C₂⁴	1	12	yes	—
40	C₂×Q₈×C₁₉	—	8	32	no	—
41	C₂×Q₈×C₁₉	C₂⁴	1	64	yes	—
42	C₂×D₄×C₁₉	C₂×Q₈	1	96	yes	complemented
43	C₂×D₄×C₁₉	C₂⁴	1	16	yes	—
44	C₂²×C₄×C₁₉	—	8	24	no	—
45	C₂²×C₄×C₁₉	C₂×Q₈	1	8	yes	complemented
46	C₂²×C₄×C₁₉	C₂⁴	1	24	yes	—
47	C₂²×C₄×C₁₉	C₂⁴	1	96	yes	—
48	C₂²×C₄×C₁₉	C₂⁴	1	6	yes	—
49	C₄⋊C₄×C₁₉	C₂⁴	1	48	yes	—
50	C₄⋊C₄×C₁₉	C₂⁴	1	192	yes	—
51	C₂²⋊C₄×C₁₉	C₂⁴	1	48	yes	—
52	C₂²⋊C₄×C₁₉	C₂⁴	1	96	yes	—
53	C₄²×C₁₉	C₂⁴	1	48	yes	—
54	#62843	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
55	C₂³×C₁₉	—	4	24	no	—
56	C₂³×C₁₉	—	4	2	no	—
57	C₂³×C₁₉	—	8	16	no	—
58	C₂³×C₁₉	C₂²×Q₈	1	4	yes	—
59	C₂³×C₁₉	C₂²×Q₈	1	48	yes	—
60	C₂³×C₁₉	C₂⁵	1	3	yes	—
61	C₂³×C₁₉	C₂⁵	1	8	yes	—
62	Q₈×C₁₉	—	4	64	no	—
63	D₄×C₁₉	C₂²×Q₈	1	64	yes	—
64	C₂×C₄×C₁₉	—	4	24	no	—
65	C₂×C₄×C₁₉	—	4	96	no	—
66	C₂×C₄×C₁₉	C₂²×Q₈	1	24	yes	—
67	C₂×C₄×C₁₉	C₂⁵	1	48	yes	—
68	C₂×C₄×C₁₉	C₂⁵	1	4	yes	—
69	#3501	C₂×C₁₉	1	48	yes	complemented
70	#3458	C₂×C₁₉	1	4	yes	complemented
71	C₂×#543	C₂×C₁₉	1	8	yes	complemented
72	C₂×#518	C₂×C₁₉	1	3	yes	—
73	C₂²×C₁₉	—	2	16	no	—
74	C₂²×C₁₉	—	2	6	no	—
75	C₂²×C₁₉	—	4	4	no	—
76	C₂²×C₁₉	—	4	48	no	—
77	C₂²×C₁₉	C₂³×Q₈	1	6	yes	—
78	C₂²×C₁₉	C₂³×Q₈	1	16	yes	—
79	C₂²×C₁₉	C₂⁶	1	1	yes	characteristic
80	C₄×C₁₉	—	2	48	no	—
81	C₄×C₁₉	C₂³×Q₈	1	8	yes	—
82	C₂³×D₄	C₂²×C₁₉	1	1	yes	characteristic
83	#554	C₂²×C₁₉	1	192	yes	complemented
84	D₄×Q₈	C₂²×C₁₉	1	64	yes	complemented
85	#544	C₂²×C₁₉	1	48	yes	complemented
86	#543	C₂²×C₁₉	1	16	yes	—
87	#543	C₂²×C₁₉	1	96	yes	complemented
88	#533	C₂²×C₁₉	1	48	yes	complemented
89	C₂×C₂²⋊Q₈	C₂²×C₁₉	1	96	yes	complemented
90	#518	C₂²×C₁₉	1	48	yes	—
91	C₂×C₄×D₄	C₂²×C₁₉	1	24	yes	—
92	C₂×#117	C₂²×C₁₉	1	6	yes	—
93	C₂²×C₂²⋊C₄	C₂²×C₁₉	1	12	yes	—
94	C₂×C₁₉	—	2	6	no	—
95	C₂×C₁₉	—	2	16	no	—
96	C₂×C₁₉	C₂⁴×Q₈	1	1	yes	characteristic
97	C₂×C₁₉	H(7,𝔽₂)	1	2	yes	—
98	C₂⁵	C₂³×C₁₉	1	2	yes	—
99	C₂²×Q₈	C₂³×C₁₉	1	32	yes	complemented
100	C₂²×D₄	Q₈×C₁₉	1	16	yes	complemented
101	C₂²×D₄	C₂³×C₁₉	1	12	yes	—
102	C₂³×C₄	C₂³×C₁₉	1	24	yes	—
103	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
104	C₄⋊Q₈	C₂³×C₁₉	1	96	yes	complemented
105	#125	C₂³×C₁₉	1	96	yes	complemented
106	C₂²⋊Q₈	C₂³×C₁₉	1	192	yes	—
107	C₂²⋊Q₈	C₂³×C₁₉	1	384	yes	complemented
108	C₄×Q₈	C₂³×C₁₉	1	64	yes	complemented
109	C₄×D₄	C₂³×C₁₉	1	192	yes	—
110	#117	C₂³×C₁₉	1	48	yes	—
111	C₂×C₄⋊C₄	C₂³×C₁₉	1	12	yes	—
112	C₂×C₄⋊C₄	C₂³×C₁₉	1	96	yes	complemented
113	C₂×C₂²⋊C₄	C₂³×C₁₉	1	12	yes	—
114	C₂×C₂²⋊C₄	C₂³×C₁₉	1	24	yes	—
115	C₂×C₂²⋊C₄	C₂³×C₁₉	1	96	yes	—
116	C₂×C₄²	C₂³×C₁₉	1	12	yes	—
117	C₁₉	#62843	1	1	yes	characteristic, direct product factor, Sylow
118	C₂⁴	—	8	4	no	—
119	C₂⁴	C₂×Q₈×C₁₉	1	16	yes	complemented
120	C₂⁴	C₂⁴×C₁₉	1	1	yes	characteristic
121	C₂⁴	C₂⁴×C₁₉	1	12	yes	—
122	C₂×Q₈	—	8	32	no	—
123	C₂×Q₈	C₂⁴×C₁₉	1	64	yes	—
124	C₂×D₄	C₂×Q₈×C₁₉	1	96	yes	complemented
125	C₂×D₄	C₂⁴×C₁₉	1	16	yes	—
126	C₂²×C₄	—	8	24	no	—
127	C₂²×C₄	C₂×Q₈×C₁₉	1	8	yes	complemented
128	C₂²×C₄	C₂⁴×C₁₉	1	24	yes	—
129	C₂²×C₄	C₂⁴×C₁₉	1	96	yes	—
130	C₂²×C₄	C₂⁴×C₁₉	1	6	yes	—
131	C₄⋊C₄	C₂⁴×C₁₉	1	48	yes	—
132	C₄⋊C₄	C₂⁴×C₁₉	1	192	yes	—
133	C₂²⋊C₄	C₂⁴×C₁₉	1	48	yes	—
134	C₂²⋊C₄	C₂⁴×C₁₉	1	96	yes	—
135	C₄²	C₂⁴×C₁₉	1	48	yes	—
136	C₂³	—	4	24	no	—
137	C₂³	—	4	2	no	—
138	C₂³	—	8	16	no	—
139	C₂³	C₂²×Q₈×C₁₉	1	4	yes	—
140	C₂³	C₂²×Q₈×C₁₉	1	48	yes	—
141	C₂³	C₂⁵×C₁₉	1	3	yes	—
142	C₂³	C₂⁵×C₁₉	1	8	yes	—
143	Q₈	—	4	64	no	—
144	D₄	C₂²×Q₈×C₁₉	1	64	yes	—
145	C₂×C₄	—	4	24	no	—
146	C₂×C₄	—	4	96	no	—
147	C₂×C₄	C₂²×Q₈×C₁₉	1	24	yes	—
148	C₂×C₄	C₂⁵×C₁₉	1	48	yes	—
149	C₂×C₄	C₂⁵×C₁₉	1	4	yes	—
150	C₂²	—	2	16	no	—
151	C₂²	—	2	6	no	—
152	C₂²	—	4	4	no	—
153	C₂²	—	4	48	no	—
154	C₂²	C₂³×Q₈×C₁₉	1	6	yes	—
155	C₂²	C₂³×Q₈×C₁₉	1	16	yes	—
156	C₂²	C₂⁶×C₁₉	1	1	yes	characteristic
157	C₄	—	2	48	no	—
158	C₄	C₂³×Q₈×C₁₉	1	8	yes	—
159	C₂	—	2	6	no	—
160	C₂	—	2	16	no	—
161	C₂	C₂⁴×Q₈×C₁₉	1	1	yes	characteristic
162	C₂	C₁₉×H(7,𝔽₂)	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂⁶
commutator, Frattini, cosocle	C₂²	C₂⁶×C₁₉
prime radical	C₂³×D₄×C₁₉	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₉×#3501	C₂	semidirect
2. C₁₉×#3458	C₂	semidirect
3. C₂×C₁₉×#543	C₂	semidirect
6. C₁₉×#554	C₂²	semidirect
7. D₄×Q₈×C₁₉	C₂²	semidirect
8. C₁₉×#544	C₂²	semidirect
10. C₁₉×#543	C₂²	semidirect
11. C₁₉×#533	C₂²	semidirect
12. C₂×C₁₉×C₂²⋊Q₈	C₂²	semidirect
18. C₂²×Q₈×C₁₉	C₂³	semidirect
19. C₂²×D₄×C₁₉	Q₈	semidirect
23. C₁₉×C₄⋊Q₈	C₂³	semidirect
24. C₁₉×#125	C₂³	semidirect
26. C₁₉×C₂²⋊Q₈	C₂³	semidirect
27. C₄×Q₈×C₁₉	C₂³	semidirect
31. C₂×C₄⋊C₄×C₁₉	C₂³	semidirect
37. C₂⁴×C₁₉	C₂×Q₈	semidirect
42. C₂×D₄×C₁₉	C₂×Q₈	semidirect
45. C₂²×C₄×C₁₉	C₂×Q₈	semidirect
54. #62843	C₁₉	direct (N × H)
69. #3501	C₂×C₁₉	semidirect
70. #3458	C₂×C₁₉	semidirect
71. C₂×#543	C₂×C₁₉	semidirect
83. #554	C₂²×C₁₉	semidirect
84. D₄×Q₈	C₂²×C₁₉	semidirect
85. #544	C₂²×C₁₉	semidirect
87. #543	C₂²×C₁₉	semidirect
88. #533	C₂²×C₁₉	semidirect
89. C₂×C₂²⋊Q₈	C₂²×C₁₉	semidirect
99. C₂²×Q₈	C₂³×C₁₉	semidirect
100. C₂²×D₄	Q₈×C₁₉	semidirect
104. C₄⋊Q₈	C₂³×C₁₉	semidirect
105. #125	C₂³×C₁₉	semidirect
107. C₂²⋊Q₈	C₂³×C₁₉	semidirect
108. C₄×Q₈	C₂³×C₁₉	semidirect
112. C₂×C₄⋊C₄	C₂³×C₁₉	semidirect
117. C₁₉	#62843	direct (N × H)
119. C₂⁴	C₂×Q₈×C₁₉	semidirect
124. C₂×D₄	C₂×Q₈×C₁₉	semidirect
127. C₂²×C₄	C₂×Q₈×C₁₉	semidirect

Maximal subgroups

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

subgroup	index
C₁₉×#3501	2
C₁₉×#3458	2
C₂×C₁₉×#543	2
C₂×C₁₉×#518	2
#62843	19

Subgroups — C19×#62843 #18068802

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#62843 #18068802