Subgroups — C₂×C₄×C₁₉×C₁₂₈ #72197116

Order 19456.

378 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₁₉×C₆₄	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₁₉×C₁₂₈	C₂	1	2	yes	maximal
3	C₄×C₁₉×C₁₂₈	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄×C₁₂₈	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
5	C₂²×C₁₉×C₆₄	C₄	1	1	yes	characteristic
6	C₂²×C₁₉×C₆₄	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₁₉×C₃₂	C₄	1	1	yes	characteristic
8	C₂×C₁₉×C₁₂₈	C₄	1	8	yes	direct product factor
9	C₂×C₁₉×C₁₂₈	C₂²	1	4	yes	—
10	C₄×C₁₉×C₆₄	C₄	1	2	yes	—
11	C₄×C₁₉×C₆₄	C₂²	1	2	yes	—
12	C₂²×C₁₉×C₃₂	C₈	1	1	yes	characteristic
13	C₂²×C₁₉×C₃₂	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄×C₁₆×C₁₉	C₈	1	1	yes	characteristic
15	C₂×C₁₉×C₆₄	C₈	1	4	yes	—
16	C₂×C₁₉×C₆₄	C₂×C₄	1	1	yes	characteristic
17	C₂×C₁₉×C₆₄	C₂×C₄	1	2	yes	—
18	C₂×C₁₉×C₆₄	C₂×C₄	1	4	yes	—
19	C₂×C₁₉×C₆₄	C₂³	1	1	yes	characteristic
20	C₄×C₁₉×C₃₂	C₈	1	2	yes	—
21	C₄×C₁₉×C₃₂	C₂×C₄	1	2	yes	—
22	C₁₉×C₁₂₈	C₂×C₄	1	8	yes	direct product factor
23	C₂²×C₁₆×C₁₉	C₁₆	1	1	yes	characteristic
24	C₂²×C₁₆×C₁₉	C₂×C₈	1	1	yes	characteristic
25	C₂×C₄×C₈×C₁₉	C₁₆	1	1	yes	characteristic
26	C₂×C₁₉×C₃₂	C₁₆	1	4	yes	—
27	C₂×C₁₉×C₃₂	C₄²	1	2	yes	—
28	C₂×C₁₉×C₃₂	C₂×C₈	1	1	yes	characteristic
29	C₂×C₁₉×C₃₂	C₂×C₈	1	2	yes	—
30	C₂×C₁₉×C₃₂	C₂×C₈	1	2	yes	—
31	C₂×C₁₉×C₃₂	C₂²×C₄	1	1	yes	characteristic
32	C₄×C₁₆×C₁₉	C₁₆	1	2	yes	—
33	C₄×C₁₆×C₁₉	C₂×C₈	1	2	yes	—
34	C₁₉×C₆₄	C₄²	1	2	yes	—
35	C₁₉×C₆₄	C₂×C₈	1	4	yes	—
36	C₁₉×C₆₄	C₂²×C₄	1	2	yes	—
37	C₂²×C₈×C₁₉	C₃₂	1	1	yes	characteristic
38	C₂²×C₈×C₁₉	C₂×C₁₆	1	1	yes	characteristic
39	C₂×C₄²×C₁₉	C₃₂	1	1	yes	characteristic
40	C₂×C₁₆×C₁₉	C₃₂	1	4	yes	—
41	C₂×C₁₆×C₁₉	C₄×C₈	1	2	yes	—
42	C₂×C₁₆×C₁₉	C₂×C₁₆	1	1	yes	characteristic
43	C₂×C₁₆×C₁₉	C₂×C₁₆	1	2	yes	—
44	C₂×C₁₆×C₁₉	C₂×C₁₆	1	2	yes	—
45	C₂×C₁₆×C₁₉	C₂²×C₈	1	1	yes	characteristic
46	C₄×C₈×C₁₉	C₃₂	1	2	yes	—
47	C₄×C₈×C₁₉	C₂×C₁₆	1	2	yes	—
48	C₁₉×C₃₂	C₄×C₈	1	2	yes	—
49	C₁₉×C₃₂	C₂×C₁₆	1	4	yes	—
50	C₁₉×C₃₂	C₂×C₄²	1	1	yes	characteristic
51	C₁₉×C₃₂	C₂²×C₈	1	1	yes	characteristic
52	C₂²×C₁₂₈	C₂×C₁₉	1	2	yes	—
53	C₂×C₄×C₆₄	C₂×C₁₉	1	1	yes	characteristic
54	C₄×C₁₂₈	C₂×C₁₉	1	4	yes	direct product factor
55	C₂²×C₄×C₁₉	C₆₄	1	2	yes	—
56	C₂²×C₄×C₁₉	C₂×C₃₂	1	1	yes	characteristic
57	C₂×C₈×C₁₉	C₆₄	1	4	yes	—
58	C₂×C₈×C₁₉	C₄×C₁₆	1	2	yes	—
59	C₂×C₈×C₁₉	C₂×C₃₂	1	1	yes	characteristic
60	C₂×C₈×C₁₉	C₂×C₃₂	1	2	yes	—
61	C₂×C₈×C₁₉	C₂×C₃₂	1	2	yes	—
62	C₂×C₈×C₁₉	C₂²×C₁₆	1	1	yes	characteristic
63	C₄²×C₁₉	C₆₄	1	2	yes	—
64	C₄²×C₁₉	C₂×C₃₂	1	2	yes	—
65	C₁₆×C₁₉	C₄×C₁₆	1	2	yes	—
66	C₁₆×C₁₉	C₂×C₃₂	1	4	yes	—
67	C₁₆×C₁₉	C₂×C₄×C₈	1	1	yes	characteristic
68	C₁₆×C₁₉	C₂²×C₁₆	1	1	yes	characteristic
69	C₂²×C₆₄	C₄×C₁₉	1	1	yes	characteristic
70	C₂²×C₆₄	C₂²×C₁₉	1	1	yes	characteristic
71	C₂×C₄×C₃₂	C₄×C₁₉	1	1	yes	characteristic
72	C₂×C₁₂₈	C₄×C₁₉	1	8	yes	direct product factor
73	C₂×C₁₂₈	C₂²×C₁₉	1	4	yes	—
74	C₄×C₆₄	C₄×C₁₉	1	2	yes	—
75	C₄×C₆₄	C₂²×C₁₉	1	2	yes	—
76	C₂³×C₁₉	C₂×C₆₄	1	1	yes	characteristic
77	C₂×C₄×C₁₉	C₁₂₈	1	8	yes	direct product factor
78	C₂×C₄×C₁₉	C₄×C₃₂	1	2	yes	—
79	C₂×C₄×C₁₉	C₂×C₆₄	1	1	yes	characteristic
80	C₂×C₄×C₁₉	C₂×C₆₄	1	2	yes	—
81	C₂×C₄×C₁₉	C₂×C₆₄	1	4	yes	—
82	C₂×C₄×C₁₉	C₂²×C₃₂	1	1	yes	characteristic
83	C₈×C₁₉	C₄×C₃₂	1	2	yes	—
84	C₈×C₁₉	C₂×C₆₄	1	4	yes	—
85	C₈×C₁₉	C₂×C₄×C₁₆	1	1	yes	characteristic
86	C₈×C₁₉	C₂²×C₃₂	1	1	yes	characteristic
87	C₂²×C₃₂	C₈×C₁₉	1	1	yes	characteristic
88	C₂²×C₃₂	C₂×C₄×C₁₉	1	1	yes	characteristic
89	C₂×C₄×C₁₆	C₈×C₁₉	1	1	yes	characteristic
90	C₂×C₆₄	C₈×C₁₉	1	4	yes	—
91	C₂×C₆₄	C₂×C₄×C₁₉	1	1	yes	characteristic
92	C₂×C₆₄	C₂×C₄×C₁₉	1	2	yes	—
93	C₂×C₆₄	C₂×C₄×C₁₉	1	4	yes	—
94	C₂×C₆₄	C₂³×C₁₉	1	1	yes	characteristic
95	C₄×C₃₂	C₈×C₁₉	1	2	yes	—
96	C₄×C₃₂	C₂×C₄×C₁₉	1	2	yes	—
97	C₁₂₈	C₂×C₄×C₁₉	1	8	yes	direct product factor
98	C₂²×C₁₉	C₄×C₆₄	1	2	yes	—
99	C₂²×C₁₉	C₂×C₁₂₈	1	4	yes	—
100	C₂²×C₁₉	C₂²×C₆₄	1	1	yes	characteristic
101	C₄×C₁₉	C₄×C₆₄	1	2	yes	—
102	C₄×C₁₉	C₂×C₁₂₈	1	8	yes	direct product factor
103	C₄×C₁₉	C₂×C₄×C₃₂	1	1	yes	characteristic
104	C₄×C₁₉	C₂²×C₆₄	1	1	yes	characteristic
105	C₂²×C₁₆	C₁₆×C₁₉	1	1	yes	characteristic
106	C₂²×C₁₆	C₂×C₈×C₁₉	1	1	yes	characteristic
107	C₂×C₄×C₈	C₁₆×C₁₉	1	1	yes	characteristic
108	C₂×C₃₂	C₁₆×C₁₉	1	4	yes	—
109	C₂×C₃₂	C₄²×C₁₉	1	2	yes	—
110	C₂×C₃₂	C₂×C₈×C₁₉	1	1	yes	characteristic
111	C₂×C₃₂	C₂×C₈×C₁₉	1	2	yes	—
112	C₂×C₃₂	C₂×C₈×C₁₉	1	2	yes	—
113	C₂×C₃₂	C₂²×C₄×C₁₉	1	1	yes	characteristic
114	C₄×C₁₆	C₁₆×C₁₉	1	2	yes	—
115	C₄×C₁₆	C₂×C₈×C₁₉	1	2	yes	—
116	C₆₄	C₄²×C₁₉	1	2	yes	—
117	C₆₄	C₂×C₈×C₁₉	1	4	yes	—
118	C₆₄	C₂²×C₄×C₁₉	1	2	yes	—
119	C₂×C₁₉	C₄×C₁₂₈	1	4	yes	direct product factor
120	C₂×C₁₉	C₂×C₄×C₆₄	1	1	yes	characteristic
121	C₂×C₁₉	C₂²×C₁₂₈	1	2	yes	—
122	C₂²×C₈	C₁₉×C₃₂	1	1	yes	characteristic
123	C₂²×C₈	C₂×C₁₆×C₁₉	1	1	yes	characteristic
124	C₂×C₄²	C₁₉×C₃₂	1	1	yes	characteristic
125	C₂×C₁₆	C₁₉×C₃₂	1	4	yes	—
126	C₂×C₁₆	C₄×C₈×C₁₉	1	2	yes	—
127	C₂×C₁₆	C₂×C₁₆×C₁₉	1	1	yes	characteristic
128	C₂×C₁₆	C₂×C₁₆×C₁₉	1	2	yes	—
129	C₂×C₁₆	C₂×C₁₆×C₁₉	1	2	yes	—
130	C₂×C₁₆	C₂²×C₈×C₁₉	1	1	yes	characteristic
131	C₄×C₈	C₁₉×C₃₂	1	2	yes	—
132	C₄×C₈	C₂×C₁₆×C₁₉	1	2	yes	—
133	C₃₂	C₄×C₈×C₁₉	1	2	yes	—
134	C₃₂	C₂×C₁₆×C₁₉	1	4	yes	—
135	C₃₂	C₂×C₄²×C₁₉	1	1	yes	characteristic
136	C₃₂	C₂²×C₈×C₁₉	1	1	yes	characteristic
137	C₁₉	C₂×C₄×C₁₂₈	1	1	yes	characteristic, direct product factor, Sylow
138	C₂²×C₄	C₁₉×C₆₄	1	2	yes	—
139	C₂²×C₄	C₂×C₁₉×C₃₂	1	1	yes	characteristic
140	C₂×C₈	C₁₉×C₆₄	1	4	yes	—
141	C₂×C₈	C₄×C₁₆×C₁₉	1	2	yes	—
142	C₂×C₈	C₂×C₁₉×C₃₂	1	1	yes	characteristic
143	C₂×C₈	C₂×C₁₉×C₃₂	1	2	yes	—
144	C₂×C₈	C₂×C₁₉×C₃₂	1	2	yes	—
145	C₂×C₈	C₂²×C₁₆×C₁₉	1	1	yes	characteristic
146	C₄²	C₁₉×C₆₄	1	2	yes	—
147	C₄²	C₂×C₁₉×C₃₂	1	2	yes	—
148	C₁₆	C₄×C₁₆×C₁₉	1	2	yes	—
149	C₁₆	C₂×C₁₉×C₃₂	1	4	yes	—
150	C₁₆	C₂×C₄×C₈×C₁₉	1	1	yes	characteristic
151	C₁₆	C₂²×C₁₆×C₁₉	1	1	yes	characteristic
152	C₂³	C₂×C₁₉×C₆₄	1	1	yes	characteristic
153	C₂×C₄	C₁₉×C₁₂₈	1	8	yes	direct product factor
154	C₂×C₄	C₄×C₁₉×C₃₂	1	2	yes	—
155	C₂×C₄	C₂×C₁₉×C₆₄	1	1	yes	characteristic
156	C₂×C₄	C₂×C₁₉×C₆₄	1	2	yes	—
157	C₂×C₄	C₂×C₁₉×C₆₄	1	4	yes	—
158	C₂×C₄	C₂²×C₁₉×C₃₂	1	1	yes	characteristic
159	C₈	C₄×C₁₉×C₃₂	1	2	yes	—
160	C₈	C₂×C₁₉×C₆₄	1	4	yes	—
161	C₈	C₂×C₄×C₁₆×C₁₉	1	1	yes	characteristic
162	C₈	C₂²×C₁₉×C₃₂	1	1	yes	characteristic
163	C₂²	C₄×C₁₉×C₆₄	1	2	yes	—
164	C₂²	C₂×C₁₉×C₁₂₈	1	4	yes	—
165	C₂²	C₂²×C₁₉×C₆₄	1	1	yes	characteristic
166	C₄	C₄×C₁₉×C₆₄	1	2	yes	—
167	C₄	C₂×C₁₉×C₁₂₈	1	8	yes	direct product factor
168	C₄	C₂×C₄×C₁₉×C₃₂	1	1	yes	characteristic
169	C₄	C₂²×C₁₉×C₆₄	1	1	yes	characteristic
170	C₂	C₄×C₁₉×C₁₂₈	1	4	yes	direct product factor
171	C₂	C₂²×C₁₉×C₁₂₈	1	2	yes	—
172	C₂	C₂×C₄×C₁₉×C₆₄	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₆₄	C₂³×C₁₉
socle, prime radical	C₂³×C₁₉	C₂×C₆₄
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄×C₁₉×C₁₂₈	C₂	direct (N × H)
4. C₂×C₄×C₁₂₈	C₁₉	direct (N × H)
8. C₂×C₁₉×C₁₂₈	C₄	direct (N × H)
22. C₁₉×C₁₂₈	C₂×C₄	direct (N × H)
54. C₄×C₁₂₈	C₂×C₁₉	direct (N × H)
72. C₂×C₁₂₈	C₄×C₁₉	direct (N × H)
77. C₂×C₄×C₁₉	C₁₂₈	direct (N × H)
97. C₁₂₈	C₂×C₄×C₁₉	direct (N × H)
102. C₄×C₁₉	C₂×C₁₂₈	direct (N × H)
119. C₂×C₁₉	C₄×C₁₂₈	direct (N × H)
137. C₁₉	C₂×C₄×C₁₂₈	direct (N × H)
153. C₂×C₄	C₁₉×C₁₂₈	direct (N × H)
167. C₄	C₂×C₁₉×C₁₂₈	direct (N × H)
170. C₂	C₄×C₁₉×C₁₂₈	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×C₁₉×C₆₄	2
C₂²×C₁₉×C₁₂₈	2
C₄×C₁₉×C₁₂₈	2
C₂×C₄×C₁₂₈	19

Subgroups — C2×C4×C19×C128 #72197116

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×C₁₉×C₁₂₈ #72197116