Subgroups — C₂²×C₁₉×#137 #14561436

Order 2432.

1464 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#41×C₁₉	C₂	1	1	yes	maximal, characteristic
2	C₂³×Q₈×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₉×#137	C₂	1	24	yes	maximal, direct product factor
4	C₂²×Q₁₆×C₁₉	C₂	1	2	yes	maximal, complemented
5	C₂²×SD₁₆×C₁₉	C₂	1	2	yes	maximal, complemented
6	C₂²×M₁₆×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#41×C₁₉	—	2	6	no	—
8	C₂×#41×C₁₉	C₂²	1	12	yes	—
9	C₂²×Q₈×C₁₉	—	2	6	no	—
10	C₂²×Q₈×C₁₉	—	2	1	no	—
11	C₂²×Q₈×C₁₉	C₂²	1	2	yes	—
12	C₂²×Q₈×C₁₉	C₂²	1	1	yes	characteristic
13	C₂²×Q₈×C₁₉	C₂²	1	12	yes	complemented
14	C₂²×D₄×C₁₉	—	2	1	no	—
15	C₂²×D₄×C₁₉	C₂²	1	1	yes	characteristic
16	C₂³×C₄×C₁₉	—	2	1	no	—
17	C₂³×C₄×C₁₉	—	2	1	no	—
18	C₂³×C₄×C₁₉	C₂²	1	1	yes	characteristic
19	C₁₉×#137	C₂²	1	64	yes	direct product factor
20	C₂×Q₁₆×C₁₉	C₂²	1	24	yes	complemented
21	C₂×SD₁₆×C₁₉	C₂²	1	24	yes	complemented
22	C₂×M₁₆×C₁₉	C₂²	1	12	yes	complemented
23	C₂²×C₈×C₁₉	C₂²	1	2	yes	—
24	C₂⁴×C₁₉	—	2	1	no	—
25	C₂⁴×C₁₉	D₄	1	1	yes	characteristic
26	#41×C₁₉	—	2	24	no	—
27	#41×C₁₉	C₂³	1	16	yes	—
28	C₂×Q₈×C₁₉	—	2	6	no	—
29	C₂×Q₈×C₁₉	—	2	12	no	—
30	C₂×Q₈×C₁₉	—	2	24	no	—
31	C₂×Q₈×C₁₉	—	2	3	no	—
32	C₂×Q₈×C₁₉	C₂³	1	12	yes	—
33	C₂×Q₈×C₁₉	C₂³	1	6	yes	—
34	C₂×Q₈×C₁₉	C₂³	1	16	yes	complemented
35	C₂×D₄×C₁₉	—	2	12	no	—
36	C₂×D₄×C₁₉	—	2	3	no	—
37	C₂×D₄×C₁₉	C₂³	1	6	yes	—
38	C₂²×C₄×C₁₉	—	2	12	no	—
39	C₂²×C₄×C₁₉	—	2	1	no	—
40	C₂²×C₄×C₁₉	—	2	2	no	—
41	C₂²×C₄×C₁₉	—	2	12	no	—
42	C₂²×C₄×C₁₉	D₄	1	6	yes	—
43	C₂²×C₄×C₁₉	D₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₉	C₂³	1	6	yes	—
45	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
46	Q₁₆×C₁₉	C₂³	1	32	yes	complemented
47	SD₁₆×C₁₉	C₂³	1	32	yes	complemented
48	M₁₆×C₁₉	C₂³	1	16	yes	complemented
49	C₂×C₈×C₁₉	C₂³	1	12	yes	—
50	C₂³×C₁₉	—	2	4	no	—
51	C₂³×C₁₉	—	2	6	no	—
52	C₂³×C₁₉	—	4	4	no	—
53	C₂³×C₁₉	C₂×D₄	1	6	yes	—
54	C₂³×C₁₉	C₂×D₄	1	1	yes	characteristic
55	Q₈×C₁₉	—	2	12	no	—
56	Q₈×C₁₉	—	2	16	no	—
57	Q₈×C₁₉	—	2	6	no	—
58	Q₈×C₁₉	C₂⁴	1	8	yes	—
59	Q₈×C₁₉	C₂⁴	1	4	yes	—
60	D₄×C₁₉	—	2	16	no	—
61	D₄×C₁₉	—	2	6	no	—
62	D₄×C₁₉	C₂⁴	1	4	yes	—
63	C₂×C₄×C₁₉	—	2	6	no	—
64	C₂×C₄×C₁₉	—	2	12	no	—
65	C₂×C₄×C₁₉	—	2	16	no	—
66	C₂×C₄×C₁₉	—	2	16	no	—
67	C₂×C₄×C₁₉	C₂×D₄	1	6	yes	—
68	C₂×C₄×C₁₉	C₂×D₄	1	3	yes	—
69	C₂×C₄×C₁₉	C₂×D₄	1	12	yes	—
70	C₂×C₄×C₁₉	C₂⁴	1	3	yes	—
71	C₂×C₄×C₁₉	C₂⁴	1	4	yes	—
72	C₈×C₁₉	C₂⁴	1	8	yes	—
73	C₂²×#137	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
74	C₂²×C₁₉	—	2	12	no	—
75	C₂²×C₁₉	—	2	4	no	—
76	C₂²×C₁₉	—	4	12	no	—
77	C₂²×C₁₉	#137	1	4	yes	direct product factor
78	C₂²×C₁₉	C₂²×D₄	1	4	yes	—
79	C₂²×C₁₉	C₂²×D₄	1	3	yes	—
80	C₄×C₁₉	—	2	4	no	—
81	C₄×C₁₉	—	2	8	no	—
82	C₄×C₁₉	C₂²×D₄	1	4	yes	—
83	C₄×C₁₉	C₂²×D₄	1	3	yes	—
84	C₄×C₁₉	C₂⁵	1	1	yes	characteristic
85	C₂²×#41	C₂×C₁₉	1	1	yes	characteristic
86	C₂³×Q₈	C₂×C₁₉	1	1	yes	characteristic, complemented
87	C₂×#137	C₂×C₁₉	1	24	yes	direct product factor
88	C₂²×Q₁₆	C₂×C₁₉	1	2	yes	complemented
89	C₂²×SD₁₆	C₂×C₁₉	1	2	yes	complemented
90	C₂²×M₁₆	C₂×C₁₉	1	1	yes	characteristic, complemented
91	C₂×C₁₉	—	2	4	no	—
92	C₂×C₁₉	—	4	4	no	—
93	C₂×C₁₉	C₂×#137	1	6	yes	direct product factor
94	C₂×C₁₉	C₂³×D₄	1	1	yes	characteristic
95	C₂×#41	—	2	6	no	—
96	C₂×#41	C₂²×C₁₉	1	12	yes	—
97	C₂²×Q₈	—	2	1	no	—
98	C₂²×Q₈	—	2	6	no	—
99	C₂²×Q₈	C₂²×C₁₉	1	2	yes	—
100	C₂²×Q₈	C₂²×C₁₉	1	1	yes	characteristic
101	C₂²×Q₈	C₂²×C₁₉	1	12	yes	complemented
102	C₂²×D₄	—	2	1	no	—
103	C₂²×D₄	C₂²×C₁₉	1	1	yes	characteristic
104	C₂³×C₄	—	2	1	no	—
105	C₂³×C₄	—	2	1	no	—
106	C₂³×C₄	C₂²×C₁₉	1	1	yes	characteristic
107	#137	C₂²×C₁₉	1	64	yes	direct product factor
108	C₂×Q₁₆	C₂²×C₁₉	1	24	yes	complemented
109	C₂×SD₁₆	C₂²×C₁₉	1	24	yes	complemented
110	C₂×M₁₆	C₂²×C₁₉	1	12	yes	complemented
111	C₂²×C₈	C₂²×C₁₉	1	2	yes	—
112	C₁₉	C₂²×#137	1	1	yes	characteristic, direct product factor, Sylow
113	C₂⁴	—	2	1	no	—
114	C₂⁴	D₄×C₁₉	1	1	yes	characteristic
115	#41	—	2	24	no	—
116	#41	C₂³×C₁₉	1	16	yes	—
117	C₂×Q₈	—	2	6	no	—
118	C₂×Q₈	—	2	12	no	—
119	C₂×Q₈	—	2	24	no	—
120	C₂×Q₈	—	2	3	no	—
121	C₂×Q₈	C₂³×C₁₉	1	12	yes	—
122	C₂×Q₈	C₂³×C₁₉	1	6	yes	—
123	C₂×Q₈	C₂³×C₁₉	1	16	yes	complemented
124	C₂×D₄	—	2	12	no	—
125	C₂×D₄	—	2	3	no	—
126	C₂×D₄	C₂³×C₁₉	1	6	yes	—
127	C₂²×C₄	—	2	1	no	—
128	C₂²×C₄	—	2	2	no	—
129	C₂²×C₄	—	2	12	no	—
130	C₂²×C₄	—	2	12	no	—
131	C₂²×C₄	D₄×C₁₉	1	1	yes	characteristic
132	C₂²×C₄	D₄×C₁₉	1	6	yes	—
133	C₂²×C₄	C₂³×C₁₉	1	1	yes	characteristic
134	C₂²×C₄	C₂³×C₁₉	1	6	yes	—
135	Q₁₆	C₂³×C₁₉	1	32	yes	complemented
136	SD₁₆	C₂³×C₁₉	1	32	yes	complemented
137	M₁₆	C₂³×C₁₉	1	16	yes	complemented
138	C₂×C₈	C₂³×C₁₉	1	12	yes	—
139	C₂³	—	2	4	no	—
140	C₂³	—	2	6	no	—
141	C₂³	—	4	4	no	—
142	C₂³	C₂×D₄×C₁₉	1	6	yes	—
143	C₂³	C₂×D₄×C₁₉	1	1	yes	characteristic
144	Q₈	—	2	12	no	—
145	Q₈	—	2	16	no	—
146	Q₈	—	2	6	no	—
147	Q₈	C₂⁴×C₁₉	1	8	yes	—
148	Q₈	C₂⁴×C₁₉	1	4	yes	—
149	D₄	—	2	16	no	—
150	D₄	—	2	6	no	—
151	D₄	C₂⁴×C₁₉	1	4	yes	—
152	C₂×C₄	—	2	6	no	—
153	C₂×C₄	—	2	12	no	—
154	C₂×C₄	—	2	16	no	—
155	C₂×C₄	—	2	16	no	—
156	C₂×C₄	C₂×D₄×C₁₉	1	6	yes	—
157	C₂×C₄	C₂×D₄×C₁₉	1	3	yes	—
158	C₂×C₄	C₂×D₄×C₁₉	1	12	yes	—
159	C₂×C₄	C₂⁴×C₁₉	1	3	yes	—
160	C₂×C₄	C₂⁴×C₁₉	1	4	yes	—
161	C₈	C₂⁴×C₁₉	1	8	yes	—
162	C₂²	—	2	12	no	—
163	C₂²	—	2	4	no	—
164	C₂²	—	4	12	no	—
165	C₂²	C₁₉×#137	1	4	yes	direct product factor
166	C₂²	C₂²×D₄×C₁₉	1	4	yes	—
167	C₂²	C₂²×D₄×C₁₉	1	3	yes	—
168	C₄	—	2	4	no	—
169	C₄	—	2	8	no	—
170	C₄	C₂²×D₄×C₁₉	1	4	yes	—
171	C₄	C₂²×D₄×C₁₉	1	3	yes	—
172	C₄	C₂⁵×C₁₉	1	1	yes	characteristic
173	C₂	—	2	4	no	—
174	C₂	—	4	4	no	—
175	C₂	C₂×C₁₉×#137	1	6	yes	direct product factor
176	C₂	C₂³×D₄×C₁₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₉	C₂×D₄
commutator, Frattini, cosocle	C₄	C₂⁵×C₁₉
prime radical	C₂²×#41×C₁₉	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₉ ◁ C₂³×C₄×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂³×Q₈×C₁₉	C₂	semidirect
3. C₂×C₁₉×#137	C₂	direct (N × H)
4. C₂²×Q₁₆×C₁₉	C₂	semidirect
5. C₂²×SD₁₆×C₁₉	C₂	semidirect
6. C₂²×M₁₆×C₁₉	C₂	semidirect
13. C₂²×Q₈×C₁₉	C₂²	semidirect
19. C₁₉×#137	C₂²	direct (N × H)
20. C₂×Q₁₆×C₁₉	C₂²	semidirect
21. C₂×SD₁₆×C₁₉	C₂²	semidirect
22. C₂×M₁₆×C₁₉	C₂²	semidirect
34. C₂×Q₈×C₁₉	C₂³	semidirect
46. Q₁₆×C₁₉	C₂³	semidirect
47. SD₁₆×C₁₉	C₂³	semidirect
48. M₁₆×C₁₉	C₂³	semidirect
73. C₂²×#137	C₁₉	direct (N × H)
77. C₂²×C₁₉	#137	direct (N × H)
86. C₂³×Q₈	C₂×C₁₉	semidirect
87. C₂×#137	C₂×C₁₉	direct (N × H)
88. C₂²×Q₁₆	C₂×C₁₉	semidirect
89. C₂²×SD₁₆	C₂×C₁₉	semidirect
90. C₂²×M₁₆	C₂×C₁₉	semidirect
93. C₂×C₁₉	C₂×#137	direct (N × H)
101. C₂²×Q₈	C₂²×C₁₉	semidirect
107. #137	C₂²×C₁₉	direct (N × H)
108. C₂×Q₁₆	C₂²×C₁₉	semidirect
109. C₂×SD₁₆	C₂²×C₁₉	semidirect
110. C₂×M₁₆	C₂²×C₁₉	semidirect
112. C₁₉	C₂²×#137	direct (N × H)
123. C₂×Q₈	C₂³×C₁₉	semidirect
135. Q₁₆	C₂³×C₁₉	semidirect
136. SD₁₆	C₂³×C₁₉	semidirect
137. M₁₆	C₂³×C₁₉	semidirect
165. C₂²	C₁₉×#137	direct (N × H)
175. C₂	C₂×C₁₉×#137	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#41×C₁₉	2
C₂³×Q₈×C₁₉	2
C₂×C₁₉×#137	2
C₂²×Q₁₆×C₁₉	2
C₂²×SD₁₆×C₁₉	2
C₂²×M₁₆×C₁₉	2
C₂²×#137	19

Subgroups — C22×C19×#137 #14561436

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₁₉×#137 #14561436