Subgroups — C₂²×C₁₉×#520 #18068606

Order 4864.

12270 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#41×C₁₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₉×#520	C₂	1	48	yes	maximal, direct product factor
3	C₂²×C₄×Q₈×C₁₉	C₂	1	2	yes	maximal, complemented
4	C₂²×C₄×D₄×C₁₉	C₂	1	6	yes	maximal, complemented
5	C₂²×C₁₉×#117	C₂	1	3	yes	maximal, complemented
6	C₂³×C₄⋊C₄×C₁₉	C₂	1	3	yes	maximal, complemented
7	C₂²×#41×C₁₉	C₄	1	32	yes	complemented
8	C₂²×#41×C₁₉	C₂²	1	24	yes	—
9	C₂³×Q₈×C₁₉	C₂²	1	1	yes	characteristic
10	C₂³×D₄×C₁₉	C₂²	1	3	yes	—
11	C₂⁴×C₄×C₁₉	C₂²	1	6	yes	—
12	C₂⁴×C₄×C₁₉	C₂²	1	3	yes	—
13	C₁₉×#520	C₂²	1	256	yes	direct product factor
14	C₂×C₄×Q₈×C₁₉	C₂²	1	48	yes	complemented
15	C₂×C₄×D₄×C₁₉	C₂²	1	144	yes	complemented
16	C₂×C₁₉×#117	C₂²	1	72	yes	complemented
17	C₂²×C₄⋊C₄×C₁₉	C₂²	1	6	yes	—
18	C₂²×C₄⋊C₄×C₁₉	C₂²	1	3	yes	—
19	C₂²×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
20	C₂²×C₄⋊C₄×C₁₉	C₂²	1	72	yes	complemented
21	C₂²×C₂²⋊C₄×C₁₉	C₂²	1	6	yes	—
22	C₂²×C₄²×C₁₉	C₂²	1	6	yes	—
23	C₂⁵×C₁₉	C₂³	1	3	yes	—
24	C₂×#41×C₁₉	C₂×C₄	1	384	yes	complemented
25	C₂×#41×C₁₉	C₂³	1	64	yes	—
26	C₂²×Q₈×C₁₉	C₂×C₄	1	16	yes	complemented
27	C₂²×Q₈×C₁₉	C₂³	1	12	yes	—
28	C₂²×D₄×C₁₉	C₂×C₄	1	48	yes	complemented
29	C₂²×D₄×C₁₉	C₂³	1	36	yes	—
30	C₂³×C₄×C₁₉	—	4	24	no	—
31	C₂³×C₄×C₁₉	C₂×C₄	1	48	yes	complemented
32	C₂³×C₄×C₁₉	C₂³	1	2	yes	—
33	C₂³×C₄×C₁₉	C₂³	1	6	yes	—
34	C₂³×C₄×C₁₉	C₂³	1	72	yes	—
35	C₂³×C₄×C₁₉	C₂³	1	3	yes	—
36	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
37	C₂³×C₄×C₁₉	C₂³	1	36	yes	—
38	C₄×Q₈×C₁₉	C₂³	1	128	yes	complemented
39	C₄×D₄×C₁₉	C₂³	1	384	yes	complemented
40	C₁₉×#117	C₂³	1	192	yes	complemented
41	C₂×C₄⋊C₄×C₁₉	C₂³	1	36	yes	—
42	C₂×C₄⋊C₄×C₁₉	C₂³	1	72	yes	—
43	C₂×C₄⋊C₄×C₁₉	C₂³	1	12	yes	—
44	C₂×C₄⋊C₄×C₁₉	C₂³	1	192	yes	complemented
45	C₂×C₂²⋊C₄×C₁₉	C₂³	1	72	yes	—
46	C₂×C₄²×C₁₉	C₂³	1	72	yes	—
47	C₂⁴×C₁₉	—	2	12	no	—
48	C₂⁴×C₁₉	—	2	12	no	—
49	C₂⁴×C₁₉	C₂²×C₄	1	24	yes	—
50	C₂⁴×C₁₉	C₂⁴	1	18	yes	—
51	C₂⁴×C₁₉	C₂⁴	1	1	yes	characteristic
52	#41×C₁₉	C₂²×C₄	1	512	yes	complemented
53	C₂×Q₈×C₁₉	C₂²×C₄	1	96	yes	complemented
54	C₂×Q₈×C₁₉	C₂⁴	1	16	yes	—
55	C₂×D₄×C₁₉	C₂²×C₄	1	288	yes	complemented
56	C₂×D₄×C₁₉	C₂⁴	1	48	yes	—
57	C₂²×C₄×C₁₉	—	2	24	no	—
58	C₂²×C₄×C₁₉	—	2	8	no	—
59	C₂²×C₄×C₁₉	—	4	144	no	—
60	C₂²×C₄×C₁₉	C₂²×C₄	1	8	yes	—
61	C₂²×C₄×C₁₉	C₂²×C₄	1	24	yes	—
62	C₂²×C₄×C₁₉	C₂²×C₄	1	288	yes	complemented
63	C₂²×C₄×C₁₉	C₂⁴	1	12	yes	—
64	C₂²×C₄×C₁₉	C₂⁴	1	36	yes	—
65	C₂²×C₄×C₁₉	C₂⁴	1	96	yes	—
66	C₂²×C₄×C₁₉	C₂⁴	1	18	yes	—
67	C₂²×C₄×C₁₉	C₂⁴	1	6	yes	—
68	C₂²×C₄×C₁₉	C₂⁴	1	48	yes	—
69	C₄⋊C₄×C₁₉	C₂⁴	1	48	yes	—
70	C₄⋊C₄×C₁₉	C₂⁴	1	96	yes	—
71	C₄⋊C₄×C₁₉	C₂⁴	1	16	yes	—
72	C₂²⋊C₄×C₁₉	C₂⁴	1	96	yes	—
73	C₄²×C₁₉	C₂⁴	1	96	yes	—
74	C₂²×#520	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₂³×C₁₉	—	2	96	no	—
76	C₂³×C₁₉	—	2	36	no	—
77	C₂³×C₁₉	—	2	36	no	—
78	C₂³×C₁₉	C₂³×C₄	1	72	yes	—
79	C₂³×C₁₉	C₂³×C₄	1	4	yes	—
80	C₂³×C₁₉	H(5,𝔽₂)	1	4	yes	—
81	C₂³×C₁₉	2₋¹⁺⁴	1	4	yes	—
82	C₂³×C₁₉	C₂⁵	1	12	yes	—
83	C₂³×C₁₉	C₂⁵	1	3	yes	—
84	Q₈×C₁₉	C₂³×C₄	1	64	yes	complemented
85	D₄×C₁₉	C₂³×C₄	1	192	yes	complemented
86	C₂×C₄×C₁₉	—	2	72	no	—
87	C₂×C₄×C₁₉	—	2	24	no	—
88	C₂×C₄×C₁₉	—	4	96	no	—
89	C₂×C₄×C₁₉	C₂³×C₄	1	24	yes	—
90	C₂×C₄×C₁₉	C₂³×C₄	1	72	yes	—
91	C₂×C₄×C₁₉	C₂³×C₄	1	192	yes	complemented
92	C₂×C₄×C₁₉	C₂⁵	1	8	yes	—
93	C₂×C₄×C₁₉	C₂⁵	1	24	yes	—
94	C₂×C₄×C₁₉	C₂⁵	1	12	yes	—
95	C₂×C₄×C₁₉	C₂⁵	1	4	yes	—
96	C₂³×#41	C₂×C₁₉	1	1	yes	characteristic
97	C₂×#520	C₂×C₁₉	1	48	yes	direct product factor
98	C₂²×C₄×Q₈	C₂×C₁₉	1	2	yes	complemented
99	C₂²×C₄×D₄	C₂×C₁₉	1	6	yes	complemented
100	C₂²×#117	C₂×C₁₉	1	3	yes	complemented
101	C₂³×C₄⋊C₄	C₂×C₁₉	1	3	yes	complemented
102	C₂²×C₁₉	—	2	12	no	—
103	C₂²×C₁₉	—	2	12	no	—
104	C₂²×C₁₉	—	2	144	no	—
105	C₂²×C₁₉	#520	1	16	yes	direct product factor
106	C₂²×C₁₉	C₂⁴×C₄	1	24	yes	—
107	C₂²×C₁₉	C₂⁴×C₄	1	6	yes	—
108	C₂²×C₁₉	C₂×H(5,𝔽₂)	1	6	yes	—
109	C₂²×C₁₉	C₂×2₋¹⁺⁴	1	6	yes	—
110	C₂²×C₁₉	C₂⁶	1	1	yes	characteristic
111	C₄×C₁₉	—	2	24	no	—
112	C₄×C₁₉	—	2	8	no	—
113	C₄×C₁₉	C₂⁴×C₄	1	8	yes	—
114	C₄×C₁₉	C₂⁴×C₄	1	24	yes	—
115	C₂²×#41	C₄×C₁₉	1	32	yes	complemented
116	C₂²×#41	C₂²×C₁₉	1	24	yes	—
117	C₂³×Q₈	C₂²×C₁₉	1	1	yes	characteristic
118	C₂³×D₄	C₂²×C₁₉	1	3	yes	—
119	C₂⁴×C₄	C₂²×C₁₉	1	6	yes	—
120	C₂⁴×C₄	C₂²×C₁₉	1	3	yes	—
121	#520	C₂²×C₁₉	1	256	yes	direct product factor
122	C₂×C₄×Q₈	C₂²×C₁₉	1	48	yes	complemented
123	C₂×C₄×D₄	C₂²×C₁₉	1	144	yes	complemented
124	C₂×#117	C₂²×C₁₉	1	72	yes	complemented
125	C₂²×C₄⋊C₄	C₂²×C₁₉	1	3	yes	—
126	C₂²×C₄⋊C₄	C₂²×C₁₉	1	6	yes	—
127	C₂²×C₄⋊C₄	C₂²×C₁₉	1	1	yes	characteristic
128	C₂²×C₄⋊C₄	C₂²×C₁₉	1	72	yes	complemented
129	C₂²×C₂²⋊C₄	C₂²×C₁₉	1	6	yes	—
130	C₂²×C₄²	C₂²×C₁₉	1	6	yes	—
131	C₂×C₁₉	—	2	24	no	—
132	C₂×C₁₉	C₂×#520	1	12	yes	direct product factor
133	C₂×C₁₉	C₂⁵×C₄	1	1	yes	characteristic
134	C₂×C₁₉	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
135	C₂×C₁₉	C₂²×2₋¹⁺⁴	1	1	yes	characteristic
136	C₂⁵	C₂³×C₁₉	1	3	yes	—
137	C₂×#41	C₂×C₄×C₁₉	1	384	yes	complemented
138	C₂×#41	C₂³×C₁₉	1	64	yes	—
139	C₂²×Q₈	C₂×C₄×C₁₉	1	16	yes	complemented
140	C₂²×Q₈	C₂³×C₁₉	1	12	yes	—
141	C₂²×D₄	C₂×C₄×C₁₉	1	48	yes	complemented
142	C₂²×D₄	C₂³×C₁₉	1	36	yes	—
143	C₂³×C₄	—	4	24	no	—
144	C₂³×C₄	C₂×C₄×C₁₉	1	48	yes	complemented
145	C₂³×C₄	C₂³×C₁₉	1	2	yes	—
146	C₂³×C₄	C₂³×C₁₉	1	6	yes	—
147	C₂³×C₄	C₂³×C₁₉	1	72	yes	—
148	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
149	C₂³×C₄	C₂³×C₁₉	1	3	yes	—
150	C₂³×C₄	C₂³×C₁₉	1	36	yes	—
151	C₄×Q₈	C₂³×C₁₉	1	128	yes	complemented
152	C₄×D₄	C₂³×C₁₉	1	384	yes	complemented
153	#117	C₂³×C₁₉	1	192	yes	complemented
154	C₂×C₄⋊C₄	C₂³×C₁₉	1	36	yes	—
155	C₂×C₄⋊C₄	C₂³×C₁₉	1	72	yes	—
156	C₂×C₄⋊C₄	C₂³×C₁₉	1	12	yes	—
157	C₂×C₄⋊C₄	C₂³×C₁₉	1	192	yes	complemented
158	C₂×C₂²⋊C₄	C₂³×C₁₉	1	72	yes	—
159	C₂×C₄²	C₂³×C₁₉	1	72	yes	—
160	C₁₉	C₂²×#520	1	1	yes	characteristic, direct product factor, Sylow
161	C₂⁴	—	2	12	no	—
162	C₂⁴	—	2	12	no	—
163	C₂⁴	C₂²×C₄×C₁₉	1	24	yes	—
164	C₂⁴	C₂⁴×C₁₉	1	18	yes	—
165	C₂⁴	C₂⁴×C₁₉	1	1	yes	characteristic
166	#41	C₂²×C₄×C₁₉	1	512	yes	complemented
167	C₂×Q₈	C₂²×C₄×C₁₉	1	96	yes	complemented
168	C₂×Q₈	C₂⁴×C₁₉	1	16	yes	—
169	C₂×D₄	C₂²×C₄×C₁₉	1	288	yes	complemented
170	C₂×D₄	C₂⁴×C₁₉	1	48	yes	—
171	C₂²×C₄	—	2	24	no	—
172	C₂²×C₄	—	2	8	no	—
173	C₂²×C₄	—	4	144	no	—
174	C₂²×C₄	C₂²×C₄×C₁₉	1	8	yes	—
175	C₂²×C₄	C₂²×C₄×C₁₉	1	24	yes	—
176	C₂²×C₄	C₂²×C₄×C₁₉	1	288	yes	complemented
177	C₂²×C₄	C₂⁴×C₁₉	1	12	yes	—
178	C₂²×C₄	C₂⁴×C₁₉	1	36	yes	—
179	C₂²×C₄	C₂⁴×C₁₉	1	96	yes	—
180	C₂²×C₄	C₂⁴×C₁₉	1	18	yes	—
181	C₂²×C₄	C₂⁴×C₁₉	1	6	yes	—
182	C₂²×C₄	C₂⁴×C₁₉	1	48	yes	—
183	C₄⋊C₄	C₂⁴×C₁₉	1	48	yes	—
184	C₄⋊C₄	C₂⁴×C₁₉	1	96	yes	—
185	C₄⋊C₄	C₂⁴×C₁₉	1	16	yes	—
186	C₂²⋊C₄	C₂⁴×C₁₉	1	96	yes	—
187	C₄²	C₂⁴×C₁₉	1	96	yes	—
188	C₂³	—	2	96	no	—
189	C₂³	—	2	36	no	—
190	C₂³	—	2	36	no	—
191	C₂³	C₂³×C₄×C₁₉	1	72	yes	—
192	C₂³	C₂³×C₄×C₁₉	1	4	yes	—
193	C₂³	C₁₉×H(5,𝔽₂)	1	4	yes	—
194	C₂³	C₁₉×2₋¹⁺⁴	1	4	yes	—
195	C₂³	C₂⁵×C₁₉	1	12	yes	—
196	C₂³	C₂⁵×C₁₉	1	3	yes	—
197	Q₈	C₂³×C₄×C₁₉	1	64	yes	complemented
198	D₄	C₂³×C₄×C₁₉	1	192	yes	complemented
199	C₂×C₄	—	2	72	no	—
200	C₂×C₄	—	2	24	no	—
201	C₂×C₄	—	4	96	no	—
202	C₂×C₄	C₂³×C₄×C₁₉	1	24	yes	—
203	C₂×C₄	C₂³×C₄×C₁₉	1	72	yes	—
204	C₂×C₄	C₂³×C₄×C₁₉	1	192	yes	complemented
205	C₂×C₄	C₂⁵×C₁₉	1	8	yes	—
206	C₂×C₄	C₂⁵×C₁₉	1	24	yes	—
207	C₂×C₄	C₂⁵×C₁₉	1	12	yes	—
208	C₂×C₄	C₂⁵×C₁₉	1	4	yes	—
209	C₂²	—	2	144	no	—
210	C₂²	—	2	12	no	—
211	C₂²	—	2	12	no	—
212	C₂²	C₁₉×#520	1	16	yes	direct product factor
213	C₂²	C₂⁴×C₄×C₁₉	1	24	yes	—
214	C₂²	C₂⁴×C₄×C₁₉	1	6	yes	—
215	C₂²	C₂×C₁₉×H(5,𝔽₂)	1	6	yes	—
216	C₂²	C₂×C₁₉×2₋¹⁺⁴	1	6	yes	—
217	C₂²	C₂⁶×C₁₉	1	1	yes	characteristic
218	C₄	—	2	24	no	—
219	C₄	—	2	8	no	—
220	C₄	C₂⁴×C₄×C₁₉	1	8	yes	—
221	C₄	C₂⁴×C₄×C₁₉	1	24	yes	—
222	C₂	—	2	24	no	—
223	C₂	C₂×C₁₉×#520	1	12	yes	direct product factor
224	C₂	C₂⁵×C₄×C₁₉	1	1	yes	characteristic
225	C₂	C₂²×C₁₉×H(5,𝔽₂)	1	1	yes	characteristic
226	C₂	C₂²×C₁₉×2₋¹⁺⁴	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₁₉	C₂⁴
commutator	C₂	C₂⁵×C₄×C₁₉
Frattini, cosocle	C₂²	C₂⁶×C₁₉
prime radical	C₂³×#41×C₁₉	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₉×#520	C₂	direct (N × H)
3. C₂²×C₄×Q₈×C₁₉	C₂	semidirect
4. C₂²×C₄×D₄×C₁₉	C₂	semidirect
5. C₂²×C₁₉×#117	C₂	semidirect
6. C₂³×C₄⋊C₄×C₁₉	C₂	semidirect
7. C₂²×#41×C₁₉	C₄	semidirect
13. C₁₉×#520	C₂²	direct (N × H)
14. C₂×C₄×Q₈×C₁₉	C₂²	semidirect
15. C₂×C₄×D₄×C₁₉	C₂²	semidirect
16. C₂×C₁₉×#117	C₂²	semidirect
20. C₂²×C₄⋊C₄×C₁₉	C₂²	semidirect
24. C₂×#41×C₁₉	C₂×C₄	semidirect
26. C₂²×Q₈×C₁₉	C₂×C₄	semidirect
28. C₂²×D₄×C₁₉	C₂×C₄	semidirect
31. C₂³×C₄×C₁₉	C₂×C₄	semidirect
38. C₄×Q₈×C₁₉	C₂³	semidirect
39. C₄×D₄×C₁₉	C₂³	semidirect
40. C₁₉×#117	C₂³	semidirect
44. C₂×C₄⋊C₄×C₁₉	C₂³	semidirect
52. #41×C₁₉	C₂²×C₄	semidirect
53. C₂×Q₈×C₁₉	C₂²×C₄	semidirect
55. C₂×D₄×C₁₉	C₂²×C₄	semidirect
62. C₂²×C₄×C₁₉	C₂²×C₄	semidirect
74. C₂²×#520	C₁₉	direct (N × H)
84. Q₈×C₁₉	C₂³×C₄	semidirect
85. D₄×C₁₉	C₂³×C₄	semidirect
91. C₂×C₄×C₁₉	C₂³×C₄	semidirect
97. C₂×#520	C₂×C₁₉	direct (N × H)
98. C₂²×C₄×Q₈	C₂×C₁₉	semidirect
99. C₂²×C₄×D₄	C₂×C₁₉	semidirect
100. C₂²×#117	C₂×C₁₉	semidirect
101. C₂³×C₄⋊C₄	C₂×C₁₉	semidirect
105. C₂²×C₁₉	#520	direct (N × H)
115. C₂²×#41	C₄×C₁₉	semidirect
121. #520	C₂²×C₁₉	direct (N × H)
122. C₂×C₄×Q₈	C₂²×C₁₉	semidirect
123. C₂×C₄×D₄	C₂²×C₁₉	semidirect
124. C₂×#117	C₂²×C₁₉	semidirect
128. C₂²×C₄⋊C₄	C₂²×C₁₉	semidirect
132. C₂×C₁₉	C₂×#520	direct (N × H)
137. C₂×#41	C₂×C₄×C₁₉	semidirect
139. C₂²×Q₈	C₂×C₄×C₁₉	semidirect
141. C₂²×D₄	C₂×C₄×C₁₉	semidirect
144. C₂³×C₄	C₂×C₄×C₁₉	semidirect
151. C₄×Q₈	C₂³×C₁₉	semidirect
152. C₄×D₄	C₂³×C₁₉	semidirect
153. #117	C₂³×C₁₉	semidirect
157. C₂×C₄⋊C₄	C₂³×C₁₉	semidirect
160. C₁₉	C₂²×#520	direct (N × H)
166. #41	C₂²×C₄×C₁₉	semidirect
167. C₂×Q₈	C₂²×C₄×C₁₉	semidirect
169. C₂×D₄	C₂²×C₄×C₁₉	semidirect
176. C₂²×C₄	C₂²×C₄×C₁₉	semidirect
197. Q₈	C₂³×C₄×C₁₉	semidirect
198. D₄	C₂³×C₄×C₁₉	semidirect
204. C₂×C₄	C₂³×C₄×C₁₉	semidirect
212. C₂²	C₁₉×#520	direct (N × H)
223. C₂	C₂×C₁₉×#520	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×#41×C₁₉	2
C₂×C₁₉×#520	2
C₂²×C₄×Q₈×C₁₉	2
C₂²×C₄×D₄×C₁₉	2
C₂²×C₁₉×#117	2
C₂³×C₄⋊C₄×C₁₉	2
C₂²×#520	19

Subgroups — C22×C19×#520 #18068606

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₁₉×#520 #18068606