Subgroups — C₂×C₁₉×#562 #14561384

Order 2432.

1152 conjugacy classes of subgroups.

The subgroup lattice has 152 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#562	C₂	1	16	yes	maximal, direct product factor
2	C₂×C₁₉×#125	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₉×#124	C₂	1	2	yes	maximal, complemented
4	C₂×C₁₉×#123	C₂	1	4	yes	maximal, complemented
5	C₂×C₁₉×C₂²⋊Q₈	C₂	1	4	yes	maximal, complemented
6	C₂×C₁₉×#121	C₂	1	4	yes	maximal, complemented
7	C₂²×Q₈×C₁₉	C₂²	1	2	yes	—
8	C₂²×D₄×C₁₉	C₂²	1	4	yes	—
9	C₂²×D₄×C₁₉	C₂²	1	2	yes	—
10	C₂³×C₄×C₁₉	C₂²	1	4	yes	—
11	C₁₉×#125	C₂²	1	8	yes	complemented
12	C₁₉×#124	C₂²	1	16	yes	complemented
13	C₁₉×#123	C₂²	1	32	yes	complemented
14	C₁₉×C₂²⋊Q₈	C₂²	1	32	yes	complemented
15	C₁₉×#121	C₂²	1	32	yes	complemented
16	C₂×C₄⋊C₄×C₁₉	C₂²	1	4	yes	—
17	C₂×C₄⋊C₄×C₁₉	C₂²	1	4	yes	—
18	C₂×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
19	C₂×C₂²⋊C₄×C₁₉	C₂²	1	4	yes	—
20	C₂×C₂²⋊C₄×C₁₉	C₂²	1	4	yes	—
21	C₂×C₂²⋊C₄×C₁₉	C₂²	1	4	yes	—
22	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
23	C₂⁴×C₁₉	C₂³	1	4	yes	—
24	C₂×Q₈×C₁₉	—	4	4	no	—
25	C₂×Q₈×C₁₉	C₂³	1	8	yes	—
26	C₂×D₄×C₁₉	—	4	8	no	—
27	C₂×D₄×C₁₉	—	4	4	no	—
28	C₂×D₄×C₁₉	C₂³	1	16	yes	—
29	C₂×D₄×C₁₉	C₂³	1	8	yes	—
30	C₂²×C₄×C₁₉	—	4	8	no	—
31	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₁₉	C₂³	1	2	yes	—
33	C₂²×C₄×C₁₉	C₂³	1	4	yes	—
34	C₂²×C₄×C₁₉	C₂³	1	4	yes	—
35	C₂²×C₄×C₁₉	C₂³	1	16	yes	—
36	C₄⋊C₄×C₁₉	C₂³	1	16	yes	—
37	C₄⋊C₄×C₁₉	C₂³	1	16	yes	—
38	C₄⋊C₄×C₁₉	C₂³	1	8	yes	—
39	C₂²⋊C₄×C₁₉	C₂³	1	16	yes	—
40	C₂²⋊C₄×C₁₉	C₂³	1	16	yes	—
41	C₂²⋊C₄×C₁₉	C₂³	1	16	yes	—
42	C₄²×C₁₉	C₂³	1	4	yes	—
43	C₂³×C₁₉	—	2	8	no	—
44	C₂³×C₁₉	—	2	8	no	—
45	C₂³×C₁₉	—	2	8	no	—
46	C₂³×C₁₉	C₂⁴	1	8	yes	—
47	C₂³×C₁₉	C₂⁴	1	1	yes	characteristic
48	Q₈×C₁₉	—	4	8	no	—
49	D₄×C₁₉	—	4	16	no	—
50	D₄×C₁₉	—	4	8	no	—
51	C₂×C₄×C₁₉	—	2	8	no	—
52	C₂×C₄×C₁₉	—	2	8	no	—
53	C₂×C₄×C₁₉	—	2	4	no	—
54	C₂×C₄×C₁₉	—	2	2	no	—
55	C₂×C₄×C₁₉	—	4	16	no	—
56	C₂×C₄×C₁₉	C₂⁴	1	2	yes	—
57	C₂×C₄×C₁₉	C₂⁴	1	4	yes	—
58	C₂×C₄×C₁₉	C₂⁴	1	8	yes	—
59	C₂×C₄×C₁₉	C₂⁴	1	8	yes	—
60	C₂×#562	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₂²×C₁₉	—	2	8	no	—
62	C₂²×C₁₉	—	2	8	no	—
63	C₂²×C₁₉	—	2	8	no	—
64	C₂²×C₁₉	—	4	16	no	—
65	C₂²×C₁₉	H(5,𝔽₂)	1	4	yes	—
66	C₂²×C₁₉	2₋¹⁺⁴	1	2	yes	—
67	C₂²×C₁₉	C₂⁵	1	1	yes	characteristic
68	C₄×C₁₉	—	2	8	no	—
69	C₄×C₁₉	—	2	8	no	—
70	C₄×C₁₉	—	2	4	no	—
71	C₄×C₁₉	—	2	2	no	—
72	#562	C₂×C₁₉	1	16	yes	direct product factor
73	C₂×#125	C₂×C₁₉	1	1	yes	characteristic, complemented
74	C₂×#124	C₂×C₁₉	1	2	yes	complemented
75	C₂×#123	C₂×C₁₉	1	4	yes	complemented
76	C₂×C₂²⋊Q₈	C₂×C₁₉	1	4	yes	complemented
77	C₂×#121	C₂×C₁₉	1	4	yes	complemented
78	C₂×C₁₉	—	4	8	no	—
79	C₂×C₁₉	#562	1	4	yes	direct product factor
80	C₂×C₁₉	C₂×H(5,𝔽₂)	1	2	yes	—
81	C₂×C₁₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
82	C₂²×Q₈	C₂²×C₁₉	1	2	yes	—
83	C₂²×D₄	C₂²×C₁₉	1	4	yes	—
84	C₂²×D₄	C₂²×C₁₉	1	2	yes	—
85	C₂³×C₄	C₂²×C₁₉	1	4	yes	—
86	#125	C₂²×C₁₉	1	8	yes	complemented
87	#124	C₂²×C₁₉	1	16	yes	complemented
88	#123	C₂²×C₁₉	1	32	yes	complemented
89	C₂²⋊Q₈	C₂²×C₁₉	1	32	yes	complemented
90	#121	C₂²×C₁₉	1	32	yes	complemented
91	C₂×C₄⋊C₄	C₂²×C₁₉	1	4	yes	—
92	C₂×C₄⋊C₄	C₂²×C₁₉	1	4	yes	—
93	C₂×C₄⋊C₄	C₂²×C₁₉	1	2	yes	—
94	C₂×C₂²⋊C₄	C₂²×C₁₉	1	4	yes	—
95	C₂×C₂²⋊C₄	C₂²×C₁₉	1	4	yes	—
96	C₂×C₂²⋊C₄	C₂²×C₁₉	1	4	yes	—
97	C₂×C₄²	C₂²×C₁₉	1	1	yes	characteristic
98	C₁₉	C₂×#562	1	1	yes	characteristic, direct product factor, Sylow
99	C₂⁴	C₂³×C₁₉	1	4	yes	—
100	C₂×Q₈	—	4	4	no	—
101	C₂×Q₈	C₂³×C₁₉	1	8	yes	—
102	C₂×D₄	—	4	8	no	—
103	C₂×D₄	—	4	4	no	—
104	C₂×D₄	C₂³×C₁₉	1	16	yes	—
105	C₂×D₄	C₂³×C₁₉	1	8	yes	—
106	C₂²×C₄	—	4	8	no	—
107	C₂²×C₄	C₂³×C₁₉	1	1	yes	characteristic
108	C₂²×C₄	C₂³×C₁₉	1	2	yes	—
109	C₂²×C₄	C₂³×C₁₉	1	4	yes	—
110	C₂²×C₄	C₂³×C₁₉	1	4	yes	—
111	C₂²×C₄	C₂³×C₁₉	1	16	yes	—
112	C₄⋊C₄	C₂³×C₁₉	1	16	yes	—
113	C₄⋊C₄	C₂³×C₁₉	1	16	yes	—
114	C₄⋊C₄	C₂³×C₁₉	1	8	yes	—
115	C₂²⋊C₄	C₂³×C₁₉	1	16	yes	—
116	C₂²⋊C₄	C₂³×C₁₉	1	16	yes	—
117	C₂²⋊C₄	C₂³×C₁₉	1	16	yes	—
118	C₄²	C₂³×C₁₉	1	4	yes	—
119	C₂³	—	2	8	no	—
120	C₂³	—	2	8	no	—
121	C₂³	—	2	8	no	—
122	C₂³	C₂⁴×C₁₉	1	8	yes	—
123	C₂³	C₂⁴×C₁₉	1	1	yes	characteristic
124	Q₈	—	4	8	no	—
125	D₄	—	4	16	no	—
126	D₄	—	4	8	no	—
127	C₂×C₄	—	2	8	no	—
128	C₂×C₄	—	2	8	no	—
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	4	16	no	—
132	C₂×C₄	C₂⁴×C₁₉	1	2	yes	—
133	C₂×C₄	C₂⁴×C₁₉	1	4	yes	—
134	C₂×C₄	C₂⁴×C₁₉	1	8	yes	—
135	C₂×C₄	C₂⁴×C₁₉	1	8	yes	—
136	C₂²	—	2	8	no	—
137	C₂²	—	2	8	no	—
138	C₂²	—	2	8	no	—
139	C₂²	—	4	16	no	—
140	C₂²	C₁₉×H(5,𝔽₂)	1	4	yes	—
141	C₂²	C₁₉×2₋¹⁺⁴	1	2	yes	—
142	C₂²	C₂⁵×C₁₉	1	1	yes	characteristic
143	C₄	—	2	8	no	—
144	C₄	—	2	8	no	—
145	C₄	—	2	4	no	—
146	C₄	—	2	2	no	—
147	C₂	—	4	8	no	—
148	C₂	C₁₉×#562	1	4	yes	direct product factor
149	C₂	C₂×C₁₉×H(5,𝔽₂)	1	2	yes	—
150	C₂	C₂×C₁₉×2₋¹⁺⁴	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₉	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵×C₁₉
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₉×#562	C₂	direct (N × H)
2. C₂×C₁₉×#125	C₂	semidirect
3. C₂×C₁₉×#124	C₂	semidirect
4. C₂×C₁₉×#123	C₂	semidirect
5. C₂×C₁₉×C₂²⋊Q₈	C₂	semidirect
6. C₂×C₁₉×#121	C₂	semidirect
11. C₁₉×#125	C₂²	semidirect
12. C₁₉×#124	C₂²	semidirect
13. C₁₉×#123	C₂²	semidirect
14. C₁₉×C₂²⋊Q₈	C₂²	semidirect
15. C₁₉×#121	C₂²	semidirect
60. C₂×#562	C₁₉	direct (N × H)
72. #562	C₂×C₁₉	direct (N × H)
73. C₂×#125	C₂×C₁₉	semidirect
74. C₂×#124	C₂×C₁₉	semidirect
75. C₂×#123	C₂×C₁₉	semidirect
76. C₂×C₂²⋊Q₈	C₂×C₁₉	semidirect
77. C₂×#121	C₂×C₁₉	semidirect
79. C₂×C₁₉	#562	direct (N × H)
86. #125	C₂²×C₁₉	semidirect
87. #124	C₂²×C₁₉	semidirect
88. #123	C₂²×C₁₉	semidirect
89. C₂²⋊Q₈	C₂²×C₁₉	semidirect
90. #121	C₂²×C₁₉	semidirect
98. C₁₉	C₂×#562	direct (N × H)
148. C₂	C₁₉×#562	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×#562	2
C₂×C₁₉×#125	2
C₂×C₁₉×#124	2
C₂×C₁₉×#123	2
C₂×C₁₉×C₂²⋊Q₈	2
C₂×C₁₉×#121	2
C₂×#562	19

Subgroups — C2×C19×#562 #14561384

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₉×#562 #14561384