Subgroups — C₂²×C₁₉×#430 #18066070

Order 4864.

2718 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×M₁₆×C₁₉	C₂	1	2	yes	maximal
2	C₂³×M₁₆×C₁₉	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₉×#430	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₁₉×#108	C₂	1	4	yes	maximal, complemented
5	C₂⁴×C₄×C₁₉	C₂²	1	1	yes	characteristic
6	C₂²×M₁₆×C₁₉	—	2	12	no	—
7	C₂²×M₁₆×C₁₉	—	2	2	no	—
8	C₂²×M₁₆×C₁₉	C₄	1	4	yes	—
9	C₂²×M₁₆×C₁₉	C₄	1	12	yes	—
10	C₂²×M₁₆×C₁₉	C₂²	1	4	yes	—
11	C₂²×M₁₆×C₁₉	C₂²	1	24	yes	—
12	C₂²×M₁₆×C₁₉	C₂²	1	12	yes	—
13	C₂³×C₈×C₁₉	—	2	2	no	—
14	C₂³×C₈×C₁₉	C₂²	1	2	yes	—
15	C₁₉×#430	C₂²	1	64	yes	direct product factor
16	C₂×C₁₉×#108	C₂²	1	48	yes	complemented
17	C₂⁵×C₁₉	Q₈	1	1	yes	characteristic
18	C₂³×C₄×C₁₉	C₂×C₄	1	2	yes	—
19	C₂³×C₄×C₁₉	C₂×C₄	1	6	yes	—
20	C₂³×C₄×C₁₉	D₄	1	6	yes	—
21	C₂³×C₄×C₁₉	D₄	1	2	yes	—
22	C₂³×C₄×C₁₉	Q₈	1	1	yes	characteristic
23	C₂³×C₄×C₁₉	Q₈	1	6	yes	—
24	C₂³×C₄×C₁₉	C₂³	1	6	yes	—
25	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
26	C₂×M₁₆×C₁₉	—	2	24	no	—
27	C₂×M₁₆×C₁₉	—	2	12	no	—
28	C₂×M₁₆×C₁₉	—	2	48	no	—
29	C₂×M₁₆×C₁₉	C₂×C₄	1	48	yes	—
30	C₂×M₁₆×C₁₉	C₂×C₄	1	48	yes	—
31	C₂×M₁₆×C₁₉	C₂³	1	24	yes	—
32	C₂×M₁₆×C₁₉	C₂³	1	32	yes	—
33	C₂×M₁₆×C₁₉	C₂³	1	16	yes	—
34	C₂²×C₈×C₁₉	—	2	24	no	—
35	C₂²×C₈×C₁₉	—	2	4	no	—
36	C₂²×C₈×C₁₉	C₂×C₄	1	4	yes	—
37	C₂²×C₈×C₁₉	C₂×C₄	1	12	yes	—
38	C₂²×C₈×C₁₉	C₂³	1	12	yes	—
39	C₁₉×#108	C₂³	1	64	yes	complemented
40	C₂⁴×C₁₉	—	4	4	no	—
41	C₂⁴×C₁₉	C₄⋊C₄	1	2	yes	—
42	C₂⁴×C₁₉	C₄⋊C₄	1	6	yes	—
43	C₂⁴×C₁₉	C₂×Q₈	1	1	yes	characteristic
44	C₂⁴×C₁₉	C₂×Q₈	1	6	yes	—
45	C₂²×C₄×C₁₉	C₄⋊C₄	1	12	yes	—
46	C₂²×C₄×C₁₉	C₄⋊C₄	1	6	yes	—
47	C₂²×C₄×C₁₉	C₄⋊C₄	1	24	yes	—
48	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₁₉	C₄⋊C₄	1	12	yes	—
50	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₉	C₂²×C₄	1	3	yes	—
52	C₂²×C₄×C₁₉	C₂²×C₄	1	12	yes	—
53	C₂²×C₄×C₁₉	C₂²×C₄	1	12	yes	—
54	C₂²×C₄×C₁₉	C₂×D₄	1	12	yes	—
55	C₂²×C₄×C₁₉	C₂×D₄	1	3	yes	—
56	C₂²×C₄×C₁₉	C₂×D₄	1	12	yes	—
57	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₉	C₂×Q₈	1	6	yes	—
59	C₂²×C₄×C₁₉	C₂×Q₈	1	12	yes	—
60	C₂²×C₄×C₁₉	C₂×Q₈	1	3	yes	—
61	C₂²×C₄×C₁₉	C₂⁴	1	3	yes	—
62	C₂²×C₄×C₁₉	C₂⁴	1	4	yes	—
63	M₁₆×C₁₉	—	2	32	no	—
64	M₁₆×C₁₉	—	2	24	no	—
65	M₁₆×C₁₉	C₂²×C₄	1	64	yes	—
66	M₁₆×C₁₉	C₂⁴	1	16	yes	—
67	C₂×C₈×C₁₉	—	2	24	no	—
68	C₂×C₈×C₁₉	—	2	32	no	—
69	C₂×C₈×C₁₉	C₂²×C₄	1	24	yes	—
70	C₂×C₈×C₁₉	C₂²×C₄	1	24	yes	—
71	C₂×C₈×C₁₉	C₂⁴	1	8	yes	—
72	C₂²×#430	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
73	C₂³×C₁₉	—	2	8	no	—
74	C₂³×C₁₉	—	2	4	no	—
75	C₂³×C₁₉	—	4	12	no	—
76	C₂³×C₁₉	—	4	12	no	—
77	C₂³×C₁₉	C₂×C₄⋊C₄	1	3	yes	—
78	C₂³×C₁₉	C₂×C₄⋊C₄	1	12	yes	—
79	C₂³×C₁₉	C₂×C₄⋊C₄	1	12	yes	—
80	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂³×C₁₉	C₂²×Q₈	1	3	yes	—
82	C₂³×C₁₉	C₂²×Q₈	1	4	yes	—
83	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	6	yes	—
84	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	24	yes	—
85	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	24	yes	—
86	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	12	yes	—
87	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	3	yes	—
88	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	12	yes	—
89	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	3	yes	—
90	C₂×C₄×C₁₉	C₂³×C₄	1	3	yes	—
91	C₂×C₄×C₁₉	C₂³×C₄	1	3	yes	—
92	C₂×C₄×C₁₉	C₂³×C₄	1	8	yes	—
93	C₂×C₄×C₁₉	C₂²×D₄	1	3	yes	—
94	C₂×C₄×C₁₉	C₂²×D₄	1	8	yes	—
95	C₂×C₄×C₁₉	C₂²×D₄	1	3	yes	—
96	C₂×C₄×C₁₉	C₂²×Q₈	1	3	yes	—
97	C₂×C₄×C₁₉	C₂²×Q₈	1	4	yes	—
98	C₂×C₄×C₁₉	C₂⁵	1	1	yes	characteristic
99	C₈×C₁₉	—	2	16	no	—
100	C₈×C₁₉	C₂³×C₄	1	16	yes	—
101	C₂³×M₁₆	C₂×C₁₉	1	2	yes	—
102	C₂³×M₁₆	C₂×C₁₉	1	1	yes	characteristic
103	C₂×#430	C₂×C₁₉	1	24	yes	direct product factor
104	C₂²×#108	C₂×C₁₉	1	4	yes	complemented
105	C₂²×C₁₉	—	2	24	no	—
106	C₂²×C₁₉	—	2	6	no	—
107	C₂²×C₁₉	—	2	6	no	—
108	C₂²×C₁₉	—	4	12	no	—
109	C₂²×C₁₉	—	4	4	no	—
110	C₂²×C₁₉	#430	1	4	yes	direct product factor
111	C₂²×C₁₉	C₂²×C₄⋊C₄	1	3	yes	—
112	C₂²×C₁₉	C₂²×C₄⋊C₄	1	8	yes	—
113	C₂²×C₁₉	C₂²×C₄⋊C₄	1	3	yes	—
114	C₂²×C₁₉	C₂³×Q₈	1	1	yes	characteristic
115	C₄×C₁₉	C₂²×C₄⋊C₄	1	3	yes	—
116	C₄×C₁₉	C₂²×C₄⋊C₄	1	8	yes	—
117	C₄×C₁₉	C₂²×C₄⋊C₄	1	3	yes	—
118	C₄×C₁₉	C₂⁴×C₄	1	1	yes	characteristic
119	C₄×C₁₉	C₂³×D₄	1	1	yes	characteristic
120	C₂⁴×C₄	C₂²×C₁₉	1	1	yes	characteristic
121	C₂²×M₁₆	—	2	2	no	—
122	C₂²×M₁₆	—	2	12	no	—
123	C₂²×M₁₆	C₄×C₁₉	1	4	yes	—
124	C₂²×M₁₆	C₄×C₁₉	1	12	yes	—
125	C₂²×M₁₆	C₂²×C₁₉	1	4	yes	—
126	C₂²×M₁₆	C₂²×C₁₉	1	24	yes	—
127	C₂²×M₁₆	C₂²×C₁₉	1	12	yes	—
128	C₂³×C₈	—	2	2	no	—
129	C₂³×C₈	C₂²×C₁₉	1	2	yes	—
130	#430	C₂²×C₁₉	1	64	yes	direct product factor
131	C₂×#108	C₂²×C₁₉	1	48	yes	complemented
132	C₂×C₁₉	—	2	8	no	—
133	C₂×C₁₉	—	2	3	no	—
134	C₂×C₁₉	—	2	1	no	—
135	C₂×C₁₉	C₂×#430	1	6	yes	direct product factor
136	C₂×C₁₉	C₂³×C₄⋊C₄	1	1	yes	characteristic
137	C₂⁵	Q₈×C₁₉	1	1	yes	characteristic
138	C₂³×C₄	C₂×C₄×C₁₉	1	2	yes	—
139	C₂³×C₄	C₂×C₄×C₁₉	1	6	yes	—
140	C₂³×C₄	D₄×C₁₉	1	2	yes	—
141	C₂³×C₄	D₄×C₁₉	1	6	yes	—
142	C₂³×C₄	Q₈×C₁₉	1	1	yes	characteristic
143	C₂³×C₄	Q₈×C₁₉	1	6	yes	—
144	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
145	C₂³×C₄	C₂³×C₁₉	1	6	yes	—
146	C₂×M₁₆	—	2	24	no	—
147	C₂×M₁₆	—	2	12	no	—
148	C₂×M₁₆	—	2	48	no	—
149	C₂×M₁₆	C₂×C₄×C₁₉	1	48	yes	—
150	C₂×M₁₆	C₂×C₄×C₁₉	1	48	yes	—
151	C₂×M₁₆	C₂³×C₁₉	1	24	yes	—
152	C₂×M₁₆	C₂³×C₁₉	1	32	yes	—
153	C₂×M₁₆	C₂³×C₁₉	1	16	yes	—
154	C₂²×C₈	—	2	24	no	—
155	C₂²×C₈	—	2	4	no	—
156	C₂²×C₈	C₂×C₄×C₁₉	1	12	yes	—
157	C₂²×C₈	C₂×C₄×C₁₉	1	4	yes	—
158	C₂²×C₈	C₂³×C₁₉	1	12	yes	—
159	#108	C₂³×C₁₉	1	64	yes	complemented
160	C₁₉	C₂²×#430	1	1	yes	characteristic, direct product factor, Sylow
161	C₂⁴	—	4	4	no	—
162	C₂⁴	C₄⋊C₄×C₁₉	1	2	yes	—
163	C₂⁴	C₄⋊C₄×C₁₉	1	6	yes	—
164	C₂⁴	C₂×Q₈×C₁₉	1	1	yes	characteristic
165	C₂⁴	C₂×Q₈×C₁₉	1	6	yes	—
166	C₂²×C₄	C₄⋊C₄×C₁₉	1	12	yes	—
167	C₂²×C₄	C₄⋊C₄×C₁₉	1	12	yes	—
168	C₂²×C₄	C₄⋊C₄×C₁₉	1	6	yes	—
169	C₂²×C₄	C₄⋊C₄×C₁₉	1	24	yes	—
170	C₂²×C₄	C₄⋊C₄×C₁₉	1	2	yes	—
171	C₂²×C₄	C₂²×C₄×C₁₉	1	3	yes	—
172	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
173	C₂²×C₄	C₂²×C₄×C₁₉	1	12	yes	—
174	C₂²×C₄	C₂²×C₄×C₁₉	1	12	yes	—
175	C₂²×C₄	C₂×D₄×C₁₉	1	12	yes	—
176	C₂²×C₄	C₂×D₄×C₁₉	1	3	yes	—
177	C₂²×C₄	C₂×D₄×C₁₉	1	12	yes	—
178	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
179	C₂²×C₄	C₂×Q₈×C₁₉	1	3	yes	—
180	C₂²×C₄	C₂×Q₈×C₁₉	1	6	yes	—
181	C₂²×C₄	C₂×Q₈×C₁₉	1	12	yes	—
182	C₂²×C₄	C₂⁴×C₁₉	1	3	yes	—
183	C₂²×C₄	C₂⁴×C₁₉	1	4	yes	—
184	M₁₆	—	2	32	no	—
185	M₁₆	—	2	24	no	—
186	M₁₆	C₂²×C₄×C₁₉	1	64	yes	—
187	M₁₆	C₂⁴×C₁₉	1	16	yes	—
188	C₂×C₈	—	2	24	no	—
189	C₂×C₈	—	2	32	no	—
190	C₂×C₈	C₂²×C₄×C₁₉	1	24	yes	—
191	C₂×C₈	C₂²×C₄×C₁₉	1	24	yes	—
192	C₂×C₈	C₂⁴×C₁₉	1	8	yes	—
193	C₂³	—	2	8	no	—
194	C₂³	—	2	4	no	—
195	C₂³	—	4	12	no	—
196	C₂³	—	4	12	no	—
197	C₂³	C₂×C₄⋊C₄×C₁₉	1	3	yes	—
198	C₂³	C₂×C₄⋊C₄×C₁₉	1	12	yes	—
199	C₂³	C₂×C₄⋊C₄×C₁₉	1	12	yes	—
200	C₂³	C₂×C₄⋊C₄×C₁₉	1	1	yes	characteristic
201	C₂³	C₂²×Q₈×C₁₉	1	3	yes	—
202	C₂³	C₂²×Q₈×C₁₉	1	4	yes	—
203	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	6	yes	—
204	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	24	yes	—
205	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	24	yes	—
206	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	12	yes	—
207	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	3	yes	—
208	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	3	yes	—
209	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	12	yes	—
210	C₂×C₄	C₂³×C₄×C₁₉	1	3	yes	—
211	C₂×C₄	C₂³×C₄×C₁₉	1	3	yes	—
212	C₂×C₄	C₂³×C₄×C₁₉	1	8	yes	—
213	C₂×C₄	C₂²×D₄×C₁₉	1	3	yes	—
214	C₂×C₄	C₂²×D₄×C₁₉	1	8	yes	—
215	C₂×C₄	C₂²×D₄×C₁₉	1	3	yes	—
216	C₂×C₄	C₂²×Q₈×C₁₉	1	3	yes	—
217	C₂×C₄	C₂²×Q₈×C₁₉	1	4	yes	—
218	C₂×C₄	C₂⁵×C₁₉	1	1	yes	characteristic
219	C₈	—	2	16	no	—
220	C₈	C₂³×C₄×C₁₉	1	16	yes	—
221	C₂²	—	2	24	no	—
222	C₂²	—	2	6	no	—
223	C₂²	—	2	6	no	—
224	C₂²	—	4	12	no	—
225	C₂²	—	4	4	no	—
226	C₂²	C₁₉×#430	1	4	yes	direct product factor
227	C₂²	C₂²×C₄⋊C₄×C₁₉	1	3	yes	—
228	C₂²	C₂²×C₄⋊C₄×C₁₉	1	8	yes	—
229	C₂²	C₂²×C₄⋊C₄×C₁₉	1	3	yes	—
230	C₂²	C₂³×Q₈×C₁₉	1	1	yes	characteristic
231	C₄	C₂²×C₄⋊C₄×C₁₉	1	3	yes	—
232	C₄	C₂²×C₄⋊C₄×C₁₉	1	3	yes	—
233	C₄	C₂²×C₄⋊C₄×C₁₉	1	8	yes	—
234	C₄	C₂⁴×C₄×C₁₉	1	1	yes	characteristic
235	C₄	C₂³×D₄×C₁₉	1	1	yes	characteristic
236	C₂	—	2	8	no	—
237	C₂	—	2	3	no	—
238	C₂	—	2	1	no	—
239	C₂	C₂×C₁₉×#430	1	6	yes	direct product factor
240	C₂	C₂³×C₄⋊C₄×C₁₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₉	C₂×D₄
commutator	C₄	C₂⁴×C₄×C₁₉
Frattini, cosocle	C₂×C₄	C₂⁵×C₁₉
socle	C₂³×C₁₉	C₂×C₄⋊C₄
prime radical	C₂⁵×C₁₉	Q₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₉ ◁ C₂⁴×C₄×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₁₉×#430	C₂	direct (N × H)
4. C₂²×C₁₉×#108	C₂	semidirect
15. C₁₉×#430	C₂²	direct (N × H)
16. C₂×C₁₉×#108	C₂²	semidirect
39. C₁₉×#108	C₂³	semidirect
72. C₂²×#430	C₁₉	direct (N × H)
103. C₂×#430	C₂×C₁₉	direct (N × H)
104. C₂²×#108	C₂×C₁₉	semidirect
110. C₂²×C₁₉	#430	direct (N × H)
130. #430	C₂²×C₁₉	direct (N × H)
131. C₂×#108	C₂²×C₁₉	semidirect
135. C₂×C₁₉	C₂×#430	direct (N × H)
159. #108	C₂³×C₁₉	semidirect
160. C₁₉	C₂²×#430	direct (N × H)
226. C₂²	C₁₉×#430	direct (N × H)
239. C₂	C₂×C₁₉×#430	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×M₁₆×C₁₉	2
C₂×C₁₉×#430	2
C₂²×C₁₉×#108	2
C₂²×#430	19

Subgroups — C22×C19×#430 #18066070

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₁₉×#430 #18066070