Subgroups — C₂²×C₁₉×#550 #18068659

Order 4864.

12918 conjugacy classes of subgroups.

The subgroup lattice has 234 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	2	yes	maximal, complemented
2	C₂×C₁₉×#550	C₂	1	48	yes	maximal, direct product factor
3	C₂²×C₁₉×#127	C₂	1	3	yes	maximal, complemented
4	C₂²×C₁₉×#121	C₂	1	6	yes	maximal, complemented
5	C₂²×C₄×Q₈×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×D₄×C₁₉	C₂	1	3	yes	maximal, complemented
7	C₂²×#41×C₁₉	—	2	32	no	—
8	C₂²×#41×C₁₉	C₂²	1	48	yes	complemented
9	C₂³×Q₈×C₁₉	C₂²	1	1	yes	characteristic
10	C₂³×D₄×C₁₉	C₂²	1	6	yes	—
11	C₂³×D₄×C₁₉	C₂²	1	3	yes	—
12	C₂³×D₄×C₁₉	C₂²	1	6	yes	—
13	C₂⁴×C₄×C₁₉	C₂²	1	6	yes	—
14	C₁₉×#550	C₂²	1	256	yes	direct product factor
15	C₂×C₁₉×#127	C₂²	1	72	yes	complemented
16	C₂×C₁₉×#121	C₂²	1	144	yes	complemented
17	C₂×C₄×Q₈×C₁₉	C₂²	1	24	yes	complemented
18	C₂×C₄×D₄×C₁₉	C₂²	1	72	yes	complemented
19	C₂²×C₄⋊C₄×C₁₉	C₂²	1	3	yes	—
20	C₂²×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
21	C₂²×C₂²⋊C₄×C₁₉	C₂²	1	6	yes	—
22	C₂²×C₄²×C₁₉	C₂²	1	3	yes	—
23	C₂⁵×C₁₉	C₂³	1	6	yes	—
24	C₂×#41×C₁₉	—	2	384	no	—
25	C₂×#41×C₁₉	C₂³	1	128	yes	complemented
26	C₂²×Q₈×C₁₉	D₄	1	16	yes	complemented
27	C₂²×Q₈×C₁₉	C₂³	1	12	yes	—
28	C₂²×D₄×C₁₉	—	2	48	no	—
29	C₂²×D₄×C₁₉	—	2	24	no	—
30	C₂²×D₄×C₁₉	—	4	24	no	—
31	C₂²×D₄×C₁₉	C₂³	1	36	yes	—
32	C₂²×D₄×C₁₉	C₂³	1	72	yes	—
33	C₂²×D₄×C₁₉	C₂³	1	72	yes	—
34	C₂³×C₄×C₁₉	—	2	48	no	—
35	C₂³×C₄×C₁₉	C₂³	1	72	yes	—
36	C₂³×C₄×C₁₉	C₂³	1	3	yes	—
37	C₂³×C₄×C₁₉	C₂³	1	2	yes	—
38	C₂³×C₄×C₁₉	C₂³	1	3	yes	—
39	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
40	C₁₉×#127	C₂³	1	192	yes	complemented
41	C₁₉×#121	C₂³	1	384	yes	complemented
42	C₄×Q₈×C₁₉	C₂³	1	64	yes	complemented
43	C₄×D₄×C₁₉	C₂³	1	192	yes	complemented
44	C₂×C₄⋊C₄×C₁₉	C₂³	1	12	yes	—
45	C₂×C₄⋊C₄×C₁₉	C₂³	1	36	yes	—
46	C₂×C₂²⋊C₄×C₁₉	C₂³	1	72	yes	—
47	C₂×C₄²×C₁₉	C₂³	1	36	yes	—
48	C₂⁴×C₁₉	—	2	24	no	—
49	C₂⁴×C₁₉	—	2	24	no	—
50	C₂⁴×C₁₉	—	2	24	no	—
51	C₂⁴×C₁₉	C₂⁴	1	36	yes	—
52	C₂⁴×C₁₉	C₂⁴	1	1	yes	characteristic
53	#41×C₁₉	—	2	512	no	—
54	C₂×Q₈×C₁₉	C₂×D₄	1	96	yes	complemented
55	C₂×Q₈×C₁₉	C₂⁴	1	16	yes	—
56	C₂×D₄×C₁₉	—	2	288	no	—
57	C₂×D₄×C₁₉	—	2	144	no	—
58	C₂×D₄×C₁₉	—	4	144	no	—
59	C₂×D₄×C₁₉	C₂⁴	1	48	yes	—
60	C₂×D₄×C₁₉	C₂⁴	1	96	yes	—
61	C₂×D₄×C₁₉	C₂⁴	1	96	yes	—
62	C₂²×C₄×C₁₉	—	2	8	no	—
63	C₂²×C₄×C₁₉	—	2	12	no	—
64	C₂²×C₄×C₁₉	—	2	288	no	—
65	C₂²×C₄×C₁₉	C₂×D₄	1	24	yes	—
66	C₂²×C₄×C₁₉	#41	1	8	yes	complemented
67	C₂²×C₄×C₁₉	C₂⁴	1	18	yes	—
68	C₂²×C₄×C₁₉	C₂⁴	1	12	yes	—
69	C₂²×C₄×C₁₉	C₂⁴	1	96	yes	—
70	C₂²×C₄×C₁₉	C₂⁴	1	18	yes	—
71	C₂²×C₄×C₁₉	C₂⁴	1	6	yes	—
72	C₄⋊C₄×C₁₉	C₂⁴	1	16	yes	—
73	C₄⋊C₄×C₁₉	C₂⁴	1	48	yes	—
74	C₂²⋊C₄×C₁₉	C₂⁴	1	96	yes	—
75	C₄²×C₁₉	C₂⁴	1	48	yes	—
76	C₂²×#550	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₂³×C₁₉	—	2	72	no	—
78	C₂³×C₁₉	—	2	72	no	—
79	C₂³×C₁₉	—	2	72	no	—
80	C₂³×C₁₉	—	4	96	no	—
81	C₂³×C₁₉	C₂²×D₄	1	4	yes	—
82	C₂³×C₁₉	C₂×#41	1	4	yes	—
83	C₂³×C₁₉	H(5,𝔽₂)	1	4	yes	—
84	C₂³×C₁₉	C₂⁵	1	24	yes	—
85	C₂³×C₁₉	C₂⁵	1	3	yes	—
86	Q₈×C₁₉	C₂²×D₄	1	64	yes	complemented
87	D₄×C₁₉	—	2	192	no	—
88	D₄×C₁₉	—	2	96	no	—
89	D₄×C₁₉	—	4	96	no	—
90	C₂×C₄×C₁₉	—	2	24	no	—
91	C₂×C₄×C₁₉	—	2	36	no	—
92	C₂×C₄×C₁₉	—	2	192	no	—
93	C₂×C₄×C₁₉	C₂²×D₄	1	72	yes	—
94	C₂×C₄×C₁₉	C₂×#41	1	24	yes	complemented
95	C₂×C₄×C₁₉	C₂⁵	1	12	yes	—
96	C₂×C₄×C₁₉	C₂⁵	1	8	yes	—
97	C₂×C₄×C₁₉	C₂⁵	1	12	yes	—
98	C₂×C₄×C₁₉	C₂⁵	1	4	yes	—
99	C₂³×#41	C₂×C₁₉	1	2	yes	complemented
100	C₂×#550	C₂×C₁₉	1	48	yes	direct product factor
101	C₂²×#127	C₂×C₁₉	1	3	yes	complemented
102	C₂²×#121	C₂×C₁₉	1	6	yes	complemented
103	C₂²×C₄×Q₈	C₂×C₁₉	1	1	yes	characteristic, complemented
104	C₂²×C₄×D₄	C₂×C₁₉	1	3	yes	complemented
105	C₂²×C₁₉	—	2	24	no	—
106	C₂²×C₁₉	—	2	24	no	—
107	C₂²×C₁₉	—	2	24	no	—
108	C₂²×C₁₉	—	4	144	no	—
109	C₂²×C₁₉	#550	1	16	yes	direct product factor
110	C₂²×C₁₉	C₂³×D₄	1	6	yes	—
111	C₂²×C₁₉	C₂²×#41	1	6	yes	—
112	C₂²×C₁₉	C₂×H(5,𝔽₂)	1	6	yes	—
113	C₂²×C₁₉	C₂⁶	1	1	yes	characteristic
114	C₄×C₁₉	—	2	8	no	—
115	C₄×C₁₉	—	2	12	no	—
116	C₄×C₁₉	C₂³×D₄	1	24	yes	—
117	C₄×C₁₉	C₂²×#41	1	8	yes	complemented
118	C₂²×#41	—	2	32	no	—
119	C₂²×#41	C₂²×C₁₉	1	48	yes	complemented
120	C₂³×Q₈	C₂²×C₁₉	1	1	yes	characteristic
121	C₂³×D₄	C₂²×C₁₉	1	6	yes	—
122	C₂³×D₄	C₂²×C₁₉	1	3	yes	—
123	C₂³×D₄	C₂²×C₁₉	1	6	yes	—
124	C₂⁴×C₄	C₂²×C₁₉	1	6	yes	—
125	#550	C₂²×C₁₉	1	256	yes	direct product factor
126	C₂×#127	C₂²×C₁₉	1	72	yes	complemented
127	C₂×#121	C₂²×C₁₉	1	144	yes	complemented
128	C₂×C₄×Q₈	C₂²×C₁₉	1	24	yes	complemented
129	C₂×C₄×D₄	C₂²×C₁₉	1	72	yes	complemented
130	C₂²×C₄⋊C₄	C₂²×C₁₉	1	1	yes	characteristic
131	C₂²×C₄⋊C₄	C₂²×C₁₉	1	3	yes	—
132	C₂²×C₂²⋊C₄	C₂²×C₁₉	1	6	yes	—
133	C₂²×C₄²	C₂²×C₁₉	1	3	yes	—
134	C₂×C₁₉	—	4	24	no	—
135	C₂×C₁₉	C₂×#550	1	12	yes	direct product factor
136	C₂×C₁₉	C₂⁴×D₄	1	1	yes	characteristic
137	C₂×C₁₉	C₂³×#41	1	1	yes	characteristic
138	C₂×C₁₉	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
139	C₂⁵	C₂³×C₁₉	1	6	yes	—
140	C₂×#41	—	2	384	no	—
141	C₂×#41	C₂³×C₁₉	1	128	yes	complemented
142	C₂²×Q₈	D₄×C₁₉	1	16	yes	complemented
143	C₂²×Q₈	C₂³×C₁₉	1	12	yes	—
144	C₂²×D₄	—	2	48	no	—
145	C₂²×D₄	—	2	24	no	—
146	C₂²×D₄	—	4	24	no	—
147	C₂²×D₄	C₂³×C₁₉	1	36	yes	—
148	C₂²×D₄	C₂³×C₁₉	1	72	yes	—
149	C₂²×D₄	C₂³×C₁₉	1	72	yes	—
150	C₂³×C₄	—	2	48	no	—
151	C₂³×C₄	C₂³×C₁₉	1	2	yes	—
152	C₂³×C₄	C₂³×C₁₉	1	3	yes	—
153	C₂³×C₄	C₂³×C₁₉	1	72	yes	—
154	C₂³×C₄	C₂³×C₁₉	1	3	yes	—
155	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
156	#127	C₂³×C₁₉	1	192	yes	complemented
157	#121	C₂³×C₁₉	1	384	yes	complemented
158	C₄×Q₈	C₂³×C₁₉	1	64	yes	complemented
159	C₄×D₄	C₂³×C₁₉	1	192	yes	complemented
160	C₂×C₄⋊C₄	C₂³×C₁₉	1	12	yes	—
161	C₂×C₄⋊C₄	C₂³×C₁₉	1	36	yes	—
162	C₂×C₂²⋊C₄	C₂³×C₁₉	1	72	yes	—
163	C₂×C₄²	C₂³×C₁₉	1	36	yes	—
164	C₁₉	C₂²×#550	1	1	yes	characteristic, direct product factor, Sylow
165	C₂⁴	—	2	24	no	—
166	C₂⁴	—	2	24	no	—
167	C₂⁴	—	2	24	no	—
168	C₂⁴	C₂⁴×C₁₉	1	36	yes	—
169	C₂⁴	C₂⁴×C₁₉	1	1	yes	characteristic
170	#41	—	2	512	no	—
171	C₂×Q₈	C₂×D₄×C₁₉	1	96	yes	complemented
172	C₂×Q₈	C₂⁴×C₁₉	1	16	yes	—
173	C₂×D₄	—	2	288	no	—
174	C₂×D₄	—	2	144	no	—
175	C₂×D₄	—	4	144	no	—
176	C₂×D₄	C₂⁴×C₁₉	1	48	yes	—
177	C₂×D₄	C₂⁴×C₁₉	1	96	yes	—
178	C₂×D₄	C₂⁴×C₁₉	1	96	yes	—
179	C₂²×C₄	—	2	8	no	—
180	C₂²×C₄	—	2	12	no	—
181	C₂²×C₄	—	2	288	no	—
182	C₂²×C₄	C₂×D₄×C₁₉	1	24	yes	—
183	C₂²×C₄	#41×C₁₉	1	8	yes	complemented
184	C₂²×C₄	C₂⁴×C₁₉	1	18	yes	—
185	C₂²×C₄	C₂⁴×C₁₉	1	12	yes	—
186	C₂²×C₄	C₂⁴×C₁₉	1	96	yes	—
187	C₂²×C₄	C₂⁴×C₁₉	1	18	yes	—
188	C₂²×C₄	C₂⁴×C₁₉	1	6	yes	—
189	C₄⋊C₄	C₂⁴×C₁₉	1	16	yes	—
190	C₄⋊C₄	C₂⁴×C₁₉	1	48	yes	—
191	C₂²⋊C₄	C₂⁴×C₁₉	1	96	yes	—
192	C₄²	C₂⁴×C₁₉	1	48	yes	—
193	C₂³	—	2	72	no	—
194	C₂³	—	2	72	no	—
195	C₂³	—	2	72	no	—
196	C₂³	—	4	96	no	—
197	C₂³	C₂²×D₄×C₁₉	1	4	yes	—
198	C₂³	C₂×#41×C₁₉	1	4	yes	—
199	C₂³	C₁₉×H(5,𝔽₂)	1	4	yes	—
200	C₂³	C₂⁵×C₁₉	1	24	yes	—
201	C₂³	C₂⁵×C₁₉	1	3	yes	—
202	Q₈	C₂²×D₄×C₁₉	1	64	yes	complemented
203	D₄	—	2	192	no	—
204	D₄	—	2	96	no	—
205	D₄	—	4	96	no	—
206	C₂×C₄	—	2	24	no	—
207	C₂×C₄	—	2	36	no	—
208	C₂×C₄	—	2	192	no	—
209	C₂×C₄	C₂²×D₄×C₁₉	1	72	yes	—
210	C₂×C₄	C₂×#41×C₁₉	1	24	yes	complemented
211	C₂×C₄	C₂⁵×C₁₉	1	12	yes	—
212	C₂×C₄	C₂⁵×C₁₉	1	8	yes	—
213	C₂×C₄	C₂⁵×C₁₉	1	12	yes	—
214	C₂×C₄	C₂⁵×C₁₉	1	4	yes	—
215	C₂²	—	2	24	no	—
216	C₂²	—	2	24	no	—
217	C₂²	—	2	24	no	—
218	C₂²	—	4	144	no	—
219	C₂²	C₁₉×#550	1	16	yes	direct product factor
220	C₂²	C₂³×D₄×C₁₉	1	6	yes	—
221	C₂²	C₂²×#41×C₁₉	1	6	yes	—
222	C₂²	C₂×C₁₉×H(5,𝔽₂)	1	6	yes	—
223	C₂²	C₂⁶×C₁₉	1	1	yes	characteristic
224	C₄	—	2	8	no	—
225	C₄	—	2	12	no	—
226	C₄	C₂³×D₄×C₁₉	1	24	yes	—
227	C₄	C₂²×#41×C₁₉	1	8	yes	complemented
228	C₂	—	4	24	no	—
229	C₂	C₂×C₁₉×#550	1	12	yes	direct product factor
230	C₂	C₂⁴×D₄×C₁₉	1	1	yes	characteristic
231	C₂	C₂³×#41×C₁₉	1	1	yes	characteristic
232	C₂	C₂²×C₁₉×H(5,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₁₉	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁶×C₁₉
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×#41×C₁₉	C₂	semidirect
2. C₂×C₁₉×#550	C₂	direct (N × H)
3. C₂²×C₁₉×#127	C₂	semidirect
4. C₂²×C₁₉×#121	C₂	semidirect
5. C₂²×C₄×Q₈×C₁₉	C₂	semidirect
6. C₂²×C₄×D₄×C₁₉	C₂	semidirect
8. C₂²×#41×C₁₉	C₂²	semidirect
14. C₁₉×#550	C₂²	direct (N × H)
15. C₂×C₁₉×#127	C₂²	semidirect
16. C₂×C₁₉×#121	C₂²	semidirect
17. C₂×C₄×Q₈×C₁₉	C₂²	semidirect
18. C₂×C₄×D₄×C₁₉	C₂²	semidirect
25. C₂×#41×C₁₉	C₂³	semidirect
26. C₂²×Q₈×C₁₉	D₄	semidirect
40. C₁₉×#127	C₂³	semidirect
41. C₁₉×#121	C₂³	semidirect
42. C₄×Q₈×C₁₉	C₂³	semidirect
43. C₄×D₄×C₁₉	C₂³	semidirect
54. C₂×Q₈×C₁₉	C₂×D₄	semidirect
66. C₂²×C₄×C₁₉	#41	semidirect
76. C₂²×#550	C₁₉	direct (N × H)
86. Q₈×C₁₉	C₂²×D₄	semidirect
94. C₂×C₄×C₁₉	C₂×#41	semidirect
99. C₂³×#41	C₂×C₁₉	semidirect
100. C₂×#550	C₂×C₁₉	direct (N × H)
101. C₂²×#127	C₂×C₁₉	semidirect
102. C₂²×#121	C₂×C₁₉	semidirect
103. C₂²×C₄×Q₈	C₂×C₁₉	semidirect
104. C₂²×C₄×D₄	C₂×C₁₉	semidirect
109. C₂²×C₁₉	#550	direct (N × H)
117. C₄×C₁₉	C₂²×#41	semidirect
119. C₂²×#41	C₂²×C₁₉	semidirect
125. #550	C₂²×C₁₉	direct (N × H)
126. C₂×#127	C₂²×C₁₉	semidirect
127. C₂×#121	C₂²×C₁₉	semidirect
128. C₂×C₄×Q₈	C₂²×C₁₉	semidirect
129. C₂×C₄×D₄	C₂²×C₁₉	semidirect
135. C₂×C₁₉	C₂×#550	direct (N × H)
141. C₂×#41	C₂³×C₁₉	semidirect
142. C₂²×Q₈	D₄×C₁₉	semidirect
156. #127	C₂³×C₁₉	semidirect
157. #121	C₂³×C₁₉	semidirect
158. C₄×Q₈	C₂³×C₁₉	semidirect
159. C₄×D₄	C₂³×C₁₉	semidirect
164. C₁₉	C₂²×#550	direct (N × H)
171. C₂×Q₈	C₂×D₄×C₁₉	semidirect
183. C₂²×C₄	#41×C₁₉	semidirect
202. Q₈	C₂²×D₄×C₁₉	semidirect
210. C₂×C₄	C₂×#41×C₁₉	semidirect
219. C₂²	C₁₉×#550	direct (N × H)
227. C₄	C₂²×#41×C₁₉	semidirect
229. C₂	C₂×C₁₉×#550	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×#550	2
C₂²×C₁₉×#127	2
C₂²×C₁₉×#121	2
C₂²×C₄×Q₈×C₁₉	2
C₂²×C₄×D₄×C₁₉	2
C₂²×#550	19

Subgroups — C22×C19×#550 #18068659

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₁₉×#550 #18068659