Subgroups — C₂³×C₃×C₈²×C₁₉ #72206113

Order 29184.

16648 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₄×C₈×C₁₉	C₂	1	3	yes	maximal
2	C₂²×C₃×C₈²×C₁₉	C₂	1	28	yes	maximal, direct product factor
3	C₂³×C₈²×C₁₉	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₂³×C₃×C₄²×C₁₉	C₂²	1	1	yes	characteristic
5	C₂⁴×C₃×C₈×C₁₉	C₄	1	6	yes	—
6	C₂²×C₃×C₄×C₈×C₁₉	C₄	1	42	yes	—
7	C₂²×C₃×C₄×C₈×C₁₉	C₂²	1	42	yes	—
8	C₂×C₃×C₈²×C₁₉	C₂²	1	112	yes	direct product factor
9	C₂⁴×C₃×C₄×C₁₉	C₂×C₄	1	3	yes	—
10	C₂²×C₃×C₄²×C₁₉	C₂×C₄	1	21	yes	—
11	C₂²×C₃×C₄²×C₁₉	C₂³	1	7	yes	—
12	C₂³×C₃×C₈×C₁₉	C₈	1	96	yes	direct product factor
13	C₂³×C₃×C₈×C₁₉	C₂×C₄	1	84	yes	—
14	C₂×C₃×C₄×C₈×C₁₉	C₂×C₄	1	252	yes	—
15	C₂×C₃×C₄×C₈×C₁₉	C₂³	1	84	yes	—
16	C₃×C₈²×C₁₉	C₂³	1	64	yes	direct product factor
17	C₂³×C₃×C₈²	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₂³×C₄×C₈×C₁₉	C₂×C₃	1	3	yes	—
19	C₂²×C₈²×C₁₉	C₂×C₃	1	28	yes	direct product factor
20	C₂⁴×C₈×C₁₉	C₃×C₄	1	6	yes	—
21	C₂³×C₄²×C₁₉	C₂²×C₃	1	1	yes	characteristic
22	C₂²×C₄×C₈×C₁₉	C₃×C₄	1	42	yes	—
23	C₂²×C₄×C₈×C₁₉	C₂²×C₃	1	42	yes	—
24	C₂×C₈²×C₁₉	C₂²×C₃	1	112	yes	direct product factor
25	C₂⁵×C₃×C₁₉	C₄²	1	1	yes	characteristic
26	C₂³×C₃×C₄×C₁₉	C₄²	1	21	yes	—
27	C₂³×C₃×C₄×C₁₉	C₂×C₈	1	48	yes	—
28	C₂³×C₃×C₄×C₁₉	C₂²×C₄	1	21	yes	—
29	C₂²×C₃×C₈×C₁₉	C₂×C₈	1	672	yes	direct product factor
30	C₂²×C₃×C₈×C₁₉	C₂²×C₄	1	168	yes	—
31	C₂×C₃×C₄²×C₁₉	C₄²	1	42	yes	—
32	C₂×C₃×C₄²×C₁₉	C₂²×C₄	1	63	yes	—
33	C₂×C₃×C₄²×C₁₉	C₂⁴	1	7	yes	—
34	C₃×C₄×C₈×C₁₉	C₂²×C₄	1	168	yes	—
35	C₃×C₄×C₈×C₁₉	C₂⁴	1	24	yes	—
36	C₂⁴×C₄×C₁₉	C₂×C₃×C₄	1	3	yes	—
37	C₂³×C₈×C₁₉	C₃×C₈	1	96	yes	direct product factor
38	C₂³×C₈×C₁₉	C₂×C₃×C₄	1	84	yes	—
39	C₂²×C₄²×C₁₉	C₂×C₃×C₄	1	21	yes	—
40	C₂²×C₄²×C₁₉	C₂³×C₃	1	7	yes	—
41	C₂×C₄×C₈×C₁₉	C₂×C₃×C₄	1	252	yes	—
42	C₂×C₄×C₈×C₁₉	C₂³×C₃	1	84	yes	—
43	C₈²×C₁₉	C₂³×C₃	1	64	yes	direct product factor
44	C₂⁴×C₃×C₁₉	C₄×C₈	1	24	yes	—
45	C₂⁴×C₃×C₁₉	C₂×C₄²	1	7	yes	—
46	C₂²×C₃×C₄×C₁₉	C₄×C₈	1	168	yes	—
47	C₂²×C₃×C₄×C₁₉	C₂×C₄²	1	63	yes	—
48	C₂²×C₃×C₄×C₁₉	C₂²×C₈	1	168	yes	—
49	C₂²×C₃×C₄×C₁₉	C₂³×C₄	1	21	yes	—
50	C₂×C₃×C₈×C₁₉	C₂²×C₈	1	672	yes	direct product factor
51	C₂×C₃×C₈×C₁₉	C₂³×C₄	1	48	yes	—
52	C₃×C₄²×C₁₉	C₂×C₄²	1	42	yes	—
53	C₃×C₄²×C₁₉	C₂³×C₄	1	21	yes	—
54	C₃×C₄²×C₁₉	C₂⁵	1	1	yes	characteristic
55	C₂³×C₃×C₄×C₈	C₂×C₁₉	1	3	yes	—
56	C₂²×C₃×C₈²	C₂×C₁₉	1	28	yes	direct product factor
57	C₂⁵×C₁₉	C₃×C₄²	1	1	yes	characteristic
58	C₂³×C₄×C₁₉	C₃×C₄²	1	21	yes	—
59	C₂³×C₄×C₁₉	C₂×C₃×C₈	1	48	yes	—
60	C₂³×C₄×C₁₉	C₂²×C₃×C₄	1	21	yes	—
61	C₂²×C₈×C₁₉	C₂×C₃×C₈	1	672	yes	direct product factor
62	C₂²×C₈×C₁₉	C₂²×C₃×C₄	1	168	yes	—
63	C₂×C₄²×C₁₉	C₃×C₄²	1	42	yes	—
64	C₂×C₄²×C₁₉	C₂²×C₃×C₄	1	63	yes	—
65	C₂×C₄²×C₁₉	C₂⁴×C₃	1	7	yes	—
66	C₄×C₈×C₁₉	C₂²×C₃×C₄	1	168	yes	—
67	C₄×C₈×C₁₉	C₂⁴×C₃	1	24	yes	—
68	C₂³×C₈²	C₃×C₁₉	1	1	yes	characteristic, direct product factor, Sylow
69	C₂³×C₃×C₁₉	C₈²	1	64	yes	direct product factor
70	C₂³×C₃×C₁₉	C₂×C₄×C₈	1	84	yes	—
71	C₂³×C₃×C₁₉	C₂²×C₄²	1	7	yes	—
72	C₂×C₃×C₄×C₁₉	C₂×C₄×C₈	1	252	yes	—
73	C₂×C₃×C₄×C₁₉	C₂²×C₄²	1	21	yes	—
74	C₂×C₃×C₄×C₁₉	C₂³×C₈	1	84	yes	—
75	C₂×C₃×C₄×C₁₉	C₂⁴×C₄	1	3	yes	—
76	C₃×C₈×C₁₉	C₂³×C₈	1	96	yes	direct product factor
77	C₂⁴×C₃×C₈	C₄×C₁₉	1	6	yes	—
78	C₂³×C₃×C₄²	C₂²×C₁₉	1	1	yes	characteristic
79	C₂²×C₃×C₄×C₈	C₄×C₁₉	1	42	yes	—
80	C₂²×C₃×C₄×C₈	C₂²×C₁₉	1	42	yes	—
81	C₂×C₃×C₈²	C₂²×C₁₉	1	112	yes	direct product factor
82	C₂⁴×C₁₉	C₃×C₄×C₈	1	24	yes	—
83	C₂⁴×C₁₉	C₂×C₃×C₄²	1	7	yes	—
84	C₂²×C₄×C₁₉	C₃×C₄×C₈	1	168	yes	—
85	C₂²×C₄×C₁₉	C₂×C₃×C₄²	1	63	yes	—
86	C₂²×C₄×C₁₉	C₂²×C₃×C₈	1	168	yes	—
87	C₂²×C₄×C₁₉	C₂³×C₃×C₄	1	21	yes	—
88	C₂×C₈×C₁₉	C₂²×C₃×C₈	1	672	yes	direct product factor
89	C₂×C₈×C₁₉	C₂³×C₃×C₄	1	48	yes	—
90	C₄²×C₁₉	C₂×C₃×C₄²	1	42	yes	—
91	C₄²×C₁₉	C₂³×C₃×C₄	1	21	yes	—
92	C₄²×C₁₉	C₂⁵×C₃	1	1	yes	characteristic
93	C₂³×C₄×C₈	C₂×C₃×C₁₉	1	3	yes	—
94	C₂²×C₈²	C₂×C₃×C₁₉	1	28	yes	direct product factor
95	C₂²×C₃×C₁₉	C₂×C₈²	1	112	yes	direct product factor
96	C₂²×C₃×C₁₉	C₂²×C₄×C₈	1	42	yes	—
97	C₂²×C₃×C₁₉	C₂³×C₄²	1	1	yes	characteristic
98	C₃×C₄×C₁₉	C₂²×C₄×C₈	1	42	yes	—
99	C₃×C₄×C₁₉	C₂⁴×C₈	1	6	yes	—
100	C₂⁴×C₃×C₄	C₂×C₄×C₁₉	1	3	yes	—
101	C₂³×C₃×C₈	C₈×C₁₉	1	96	yes	direct product factor
102	C₂³×C₃×C₈	C₂×C₄×C₁₉	1	84	yes	—
103	C₂²×C₃×C₄²	C₂×C₄×C₁₉	1	21	yes	—
104	C₂²×C₃×C₄²	C₂³×C₁₉	1	7	yes	—
105	C₂×C₃×C₄×C₈	C₂×C₄×C₁₉	1	252	yes	—
106	C₂×C₃×C₄×C₈	C₂³×C₁₉	1	84	yes	—
107	C₃×C₈²	C₂³×C₁₉	1	64	yes	direct product factor
108	C₂³×C₁₉	C₃×C₈²	1	64	yes	direct product factor
109	C₂³×C₁₉	C₂×C₃×C₄×C₈	1	84	yes	—
110	C₂³×C₁₉	C₂²×C₃×C₄²	1	7	yes	—
111	C₂×C₄×C₁₉	C₂×C₃×C₄×C₈	1	252	yes	—
112	C₂×C₄×C₁₉	C₂²×C₃×C₄²	1	21	yes	—
113	C₂×C₄×C₁₉	C₂³×C₃×C₈	1	84	yes	—
114	C₂×C₄×C₁₉	C₂⁴×C₃×C₄	1	3	yes	—
115	C₈×C₁₉	C₂³×C₃×C₈	1	96	yes	direct product factor
116	C₂⁴×C₈	C₃×C₄×C₁₉	1	6	yes	—
117	C₂³×C₄²	C₂²×C₃×C₁₉	1	1	yes	characteristic
118	C₂²×C₄×C₈	C₃×C₄×C₁₉	1	42	yes	—
119	C₂²×C₄×C₈	C₂²×C₃×C₁₉	1	42	yes	—
120	C₂×C₈²	C₂²×C₃×C₁₉	1	112	yes	direct product factor
121	C₂×C₃×C₁₉	C₂²×C₈²	1	28	yes	direct product factor
122	C₂×C₃×C₁₉	C₂³×C₄×C₈	1	3	yes	—
123	C₂⁵×C₃	C₄²×C₁₉	1	1	yes	characteristic
124	C₂³×C₃×C₄	C₄²×C₁₉	1	21	yes	—
125	C₂³×C₃×C₄	C₂×C₈×C₁₉	1	48	yes	—
126	C₂³×C₃×C₄	C₂²×C₄×C₁₉	1	21	yes	—
127	C₂²×C₃×C₈	C₂×C₈×C₁₉	1	672	yes	direct product factor
128	C₂²×C₃×C₈	C₂²×C₄×C₁₉	1	168	yes	—
129	C₂×C₃×C₄²	C₄²×C₁₉	1	42	yes	—
130	C₂×C₃×C₄²	C₂²×C₄×C₁₉	1	63	yes	—
131	C₂×C₃×C₄²	C₂⁴×C₁₉	1	7	yes	—
132	C₃×C₄×C₈	C₂²×C₄×C₁₉	1	168	yes	—
133	C₃×C₄×C₈	C₂⁴×C₁₉	1	24	yes	—
134	C₂²×C₁₉	C₂×C₃×C₈²	1	112	yes	direct product factor
135	C₂²×C₁₉	C₂²×C₃×C₄×C₈	1	42	yes	—
136	C₂²×C₁₉	C₂³×C₃×C₄²	1	1	yes	characteristic
137	C₄×C₁₉	C₂²×C₃×C₄×C₈	1	42	yes	—
138	C₄×C₁₉	C₂⁴×C₃×C₈	1	6	yes	—
139	C₂⁴×C₄	C₂×C₃×C₄×C₁₉	1	3	yes	—
140	C₂³×C₈	C₃×C₈×C₁₉	1	96	yes	direct product factor
141	C₂³×C₈	C₂×C₃×C₄×C₁₉	1	84	yes	—
142	C₂²×C₄²	C₂×C₃×C₄×C₁₉	1	21	yes	—
143	C₂²×C₄²	C₂³×C₃×C₁₉	1	7	yes	—
144	C₂×C₄×C₈	C₂×C₃×C₄×C₁₉	1	252	yes	—
145	C₂×C₄×C₈	C₂³×C₃×C₁₉	1	84	yes	—
146	C₈²	C₂³×C₃×C₁₉	1	64	yes	direct product factor
147	C₃×C₁₉	C₂³×C₈²	1	1	yes	characteristic, direct product factor, Hall
148	C₂⁴×C₃	C₄×C₈×C₁₉	1	24	yes	—
149	C₂⁴×C₃	C₂×C₄²×C₁₉	1	7	yes	—
150	C₂²×C₃×C₄	C₄×C₈×C₁₉	1	168	yes	—
151	C₂²×C₃×C₄	C₂×C₄²×C₁₉	1	63	yes	—
152	C₂²×C₃×C₄	C₂²×C₈×C₁₉	1	168	yes	—
153	C₂²×C₃×C₄	C₂³×C₄×C₁₉	1	21	yes	—
154	C₂×C₃×C₈	C₂²×C₈×C₁₉	1	672	yes	direct product factor
155	C₂×C₃×C₈	C₂³×C₄×C₁₉	1	48	yes	—
156	C₃×C₄²	C₂×C₄²×C₁₉	1	42	yes	—
157	C₃×C₄²	C₂³×C₄×C₁₉	1	21	yes	—
158	C₃×C₄²	C₂⁵×C₁₉	1	1	yes	characteristic
159	C₂×C₁₉	C₂²×C₃×C₈²	1	28	yes	direct product factor
160	C₂×C₁₉	C₂³×C₃×C₄×C₈	1	3	yes	—
161	C₂⁵	C₃×C₄²×C₁₉	1	1	yes	characteristic
162	C₂³×C₄	C₃×C₄²×C₁₉	1	21	yes	—
163	C₂³×C₄	C₂×C₃×C₈×C₁₉	1	48	yes	—
164	C₂³×C₄	C₂²×C₃×C₄×C₁₉	1	21	yes	—
165	C₂²×C₈	C₂×C₃×C₈×C₁₉	1	672	yes	direct product factor
166	C₂²×C₈	C₂²×C₃×C₄×C₁₉	1	168	yes	—
167	C₂×C₄²	C₃×C₄²×C₁₉	1	42	yes	—
168	C₂×C₄²	C₂²×C₃×C₄×C₁₉	1	63	yes	—
169	C₂×C₄²	C₂⁴×C₃×C₁₉	1	7	yes	—
170	C₄×C₈	C₂²×C₃×C₄×C₁₉	1	168	yes	—
171	C₄×C₈	C₂⁴×C₃×C₁₉	1	24	yes	—
172	C₂³×C₃	C₈²×C₁₉	1	64	yes	direct product factor
173	C₂³×C₃	C₂×C₄×C₈×C₁₉	1	84	yes	—
174	C₂³×C₃	C₂²×C₄²×C₁₉	1	7	yes	—
175	C₂×C₃×C₄	C₂×C₄×C₈×C₁₉	1	252	yes	—
176	C₂×C₃×C₄	C₂²×C₄²×C₁₉	1	21	yes	—
177	C₂×C₃×C₄	C₂³×C₈×C₁₉	1	84	yes	—
178	C₂×C₃×C₄	C₂⁴×C₄×C₁₉	1	3	yes	—
179	C₃×C₈	C₂³×C₈×C₁₉	1	96	yes	direct product factor
180	C₁₉	C₂³×C₃×C₈²	1	1	yes	characteristic, direct product factor, Sylow
181	C₂⁴	C₃×C₄×C₈×C₁₉	1	24	yes	—
182	C₂⁴	C₂×C₃×C₄²×C₁₉	1	7	yes	—
183	C₂²×C₄	C₃×C₄×C₈×C₁₉	1	168	yes	—
184	C₂²×C₄	C₂×C₃×C₄²×C₁₉	1	63	yes	—
185	C₂²×C₄	C₂²×C₃×C₈×C₁₉	1	168	yes	—
186	C₂²×C₄	C₂³×C₃×C₄×C₁₉	1	21	yes	—
187	C₂×C₈	C₂²×C₃×C₈×C₁₉	1	672	yes	direct product factor
188	C₂×C₈	C₂³×C₃×C₄×C₁₉	1	48	yes	—
189	C₄²	C₂×C₃×C₄²×C₁₉	1	42	yes	—
190	C₄²	C₂³×C₃×C₄×C₁₉	1	21	yes	—
191	C₄²	C₂⁵×C₃×C₁₉	1	1	yes	characteristic
192	C₂²×C₃	C₂×C₈²×C₁₉	1	112	yes	direct product factor
193	C₂²×C₃	C₂²×C₄×C₈×C₁₉	1	42	yes	—
194	C₂²×C₃	C₂³×C₄²×C₁₉	1	1	yes	characteristic
195	C₃×C₄	C₂²×C₄×C₈×C₁₉	1	42	yes	—
196	C₃×C₄	C₂⁴×C₈×C₁₉	1	6	yes	—
197	C₂³	C₃×C₈²×C₁₉	1	64	yes	direct product factor
198	C₂³	C₂×C₃×C₄×C₈×C₁₉	1	84	yes	—
199	C₂³	C₂²×C₃×C₄²×C₁₉	1	7	yes	—
200	C₂×C₄	C₂×C₃×C₄×C₈×C₁₉	1	252	yes	—
201	C₂×C₄	C₂³×C₃×C₈×C₁₉	1	84	yes	—
202	C₂×C₄	C₂²×C₃×C₄²×C₁₉	1	21	yes	—
203	C₂×C₄	C₂⁴×C₃×C₄×C₁₉	1	3	yes	—
204	C₈	C₂³×C₃×C₈×C₁₉	1	96	yes	direct product factor
205	C₂×C₃	C₂²×C₈²×C₁₉	1	28	yes	direct product factor
206	C₂×C₃	C₂³×C₄×C₈×C₁₉	1	3	yes	—
207	C₂²	C₂×C₃×C₈²×C₁₉	1	112	yes	direct product factor
208	C₂²	C₂²×C₃×C₄×C₈×C₁₉	1	42	yes	—
209	C₂²	C₂³×C₃×C₄²×C₁₉	1	1	yes	characteristic
210	C₄	C₂²×C₃×C₄×C₈×C₁₉	1	42	yes	—
211	C₄	C₂⁴×C₃×C₈×C₁₉	1	6	yes	—
212	C₃	C₂³×C₈²×C₁₉	1	1	yes	characteristic, direct product factor, Sylow
213	C₂	C₂²×C₃×C₈²×C₁₉	1	28	yes	direct product factor
214	C₂	C₂³×C₃×C₄×C₈×C₁₉	1	3	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄²	C₂⁵×C₃×C₁₉
socle, prime radical	C₂⁵×C₃×C₁₉	C₄²
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃×C₈²×C₁₉	C₂	direct (N × H)
3. C₂³×C₈²×C₁₉	C₃	direct (N × H)
8. C₂×C₃×C₈²×C₁₉	C₂²	direct (N × H)
12. C₂³×C₃×C₈×C₁₉	C₈	direct (N × H)
16. C₃×C₈²×C₁₉	C₂³	direct (N × H)
17. C₂³×C₃×C₈²	C₁₉	direct (N × H)
19. C₂²×C₈²×C₁₉	C₂×C₃	direct (N × H)
24. C₂×C₈²×C₁₉	C₂²×C₃	direct (N × H)
29. C₂²×C₃×C₈×C₁₉	C₂×C₈	direct (N × H)
37. C₂³×C₈×C₁₉	C₃×C₈	direct (N × H)
43. C₈²×C₁₉	C₂³×C₃	direct (N × H)
50. C₂×C₃×C₈×C₁₉	C₂²×C₈	direct (N × H)
56. C₂²×C₃×C₈²	C₂×C₁₉	direct (N × H)
61. C₂²×C₈×C₁₉	C₂×C₃×C₈	direct (N × H)
68. C₂³×C₈²	C₃×C₁₉	direct (N × H)
69. C₂³×C₃×C₁₉	C₈²	direct (N × H)
76. C₃×C₈×C₁₉	C₂³×C₈	direct (N × H)
81. C₂×C₃×C₈²	C₂²×C₁₉	direct (N × H)
88. C₂×C₈×C₁₉	C₂²×C₃×C₈	direct (N × H)
94. C₂²×C₈²	C₂×C₃×C₁₉	direct (N × H)
95. C₂²×C₃×C₁₉	C₂×C₈²	direct (N × H)
101. C₂³×C₃×C₈	C₈×C₁₉	direct (N × H)
107. C₃×C₈²	C₂³×C₁₉	direct (N × H)
108. C₂³×C₁₉	C₃×C₈²	direct (N × H)
115. C₈×C₁₉	C₂³×C₃×C₈	direct (N × H)
120. C₂×C₈²	C₂²×C₃×C₁₉	direct (N × H)
121. C₂×C₃×C₁₉	C₂²×C₈²	direct (N × H)
127. C₂²×C₃×C₈	C₂×C₈×C₁₉	direct (N × H)
134. C₂²×C₁₉	C₂×C₃×C₈²	direct (N × H)
140. C₂³×C₈	C₃×C₈×C₁₉	direct (N × H)
146. C₈²	C₂³×C₃×C₁₉	direct (N × H)
147. C₃×C₁₉	C₂³×C₈²	direct (N × H)
154. C₂×C₃×C₈	C₂²×C₈×C₁₉	direct (N × H)
159. C₂×C₁₉	C₂²×C₃×C₈²	direct (N × H)
165. C₂²×C₈	C₂×C₃×C₈×C₁₉	direct (N × H)
172. C₂³×C₃	C₈²×C₁₉	direct (N × H)
179. C₃×C₈	C₂³×C₈×C₁₉	direct (N × H)
180. C₁₉	C₂³×C₃×C₈²	direct (N × H)
187. C₂×C₈	C₂²×C₃×C₈×C₁₉	direct (N × H)
192. C₂²×C₃	C₂×C₈²×C₁₉	direct (N × H)
197. C₂³	C₃×C₈²×C₁₉	direct (N × H)
204. C₈	C₂³×C₃×C₈×C₁₉	direct (N × H)
205. C₂×C₃	C₂²×C₈²×C₁₉	direct (N × H)
207. C₂²	C₂×C₃×C₈²×C₁₉	direct (N × H)
212. C₃	C₂³×C₈²×C₁₉	direct (N × H)
213. C₂	C₂²×C₃×C₈²×C₁₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃×C₄×C₈×C₁₉	2
C₂²×C₃×C₈²×C₁₉	2
C₂³×C₈²×C₁₉	3
C₂³×C₃×C₈²	19

Subgroups — C23×C3×C82×C19 #72206113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₃×C₈²×C₁₉ #72206113