Subgroups — C₂×C₁₉×#2274 #18039954

Order 4864.

4110 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₉×#117	C₂	1	2	yes	maximal
2	C₂³×C₄⋊C₄×C₁₉	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#2274	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₄×C₄⋊C₄×C₁₉	C₂	1	8	yes	maximal, complemented
5	C₂×C₄×C₂²⋊C₄×C₁₉	C₂	1	4	yes	maximal
6	C₂⁴×C₄×C₁₉	C₂²	1	2	yes	—
7	C₂⁴×C₄×C₁₉	C₂²	1	1	yes	characteristic
8	C₂×C₁₉×#117	C₄	1	32	yes	complemented
9	C₂×C₁₉×#117	C₂²	1	16	yes	—
10	C₂²×C₄⋊C₄×C₁₉	C₄	1	16	yes	complemented
11	C₂²×C₄⋊C₄×C₁₉	C₂²	1	4	yes	—
12	C₂²×C₄⋊C₄×C₁₉	C₂²	1	4	yes	—
13	C₂²×C₄⋊C₄×C₁₉	C₂²	1	8	yes	—
14	C₂²×C₂²⋊C₄×C₁₉	C₂²	1	4	yes	—
15	C₂²×C₄²×C₁₉	C₂²	1	8	yes	—
16	C₂²×C₄²×C₁₉	C₂²	1	4	yes	—
17	C₄×C₄⋊C₄×C₁₉	C₂²	1	64	yes	complemented
18	C₄×C₂²⋊C₄×C₁₉	C₂²	1	32	yes	—
19	C₂×C₁₉×#95	C₂²	1	8	yes	—
20	C₂⁵×C₁₉	C₂³	1	1	yes	characteristic
21	C₂³×C₄×C₁₉	—	4	8	no	—
22	C₂³×C₄×C₁₉	C₂×C₄	1	16	yes	—
23	C₂³×C₄×C₁₉	C₂×C₄	1	8	yes	—
24	C₂³×C₄×C₁₉	C₂×C₄	1	16	yes	—
25	C₂³×C₄×C₁₉	C₂³	1	8	yes	—
26	C₂³×C₄×C₁₉	C₂³	1	4	yes	—
27	C₂³×C₄×C₁₉	C₂³	1	8	yes	—
28	C₂³×C₄×C₁₉	C₂³	1	4	yes	—
29	C₂³×C₄×C₁₉	C₂³	1	2	yes	—
30	C₁₉×#117	C₂×C₄	1	128	yes	complemented
31	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	32	yes	—
32	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	64	yes	complemented
33	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	32	yes	complemented
34	C₂×C₄⋊C₄×C₁₉	C₂³	1	16	yes	—
35	C₂×C₄⋊C₄×C₁₉	C₂³	1	16	yes	—
36	C₂×C₂²⋊C₄×C₁₉	C₂×C₄	1	32	yes	—
37	C₂×C₂²⋊C₄×C₁₉	C₂³	1	16	yes	—
38	C₂×C₄²×C₁₉	—	4	16	no	—
39	C₂×C₄²×C₁₉	C₂×C₄	1	32	yes	complemented
40	C₂×C₄²×C₁₉	C₂³	1	32	yes	—
41	C₂×C₄²×C₁₉	C₂³	1	16	yes	—
42	C₁₉×#95	C₂³	1	32	yes	—
43	C₂⁴×C₁₉	—	2	4	no	—
44	C₂⁴×C₁₉	—	2	4	no	—
45	C₂⁴×C₁₉	C₂²×C₄	1	8	yes	—
46	C₂⁴×C₁₉	C₂²×C₄	1	4	yes	—
47	C₂⁴×C₁₉	C₂⁴	1	2	yes	—
48	C₂⁴×C₁₉	C₂⁴	1	1	yes	characteristic
49	C₂²×C₄×C₁₉	—	2	16	no	—
50	C₂²×C₄×C₁₉	—	2	16	no	—
51	C₂²×C₄×C₁₉	—	2	16	no	—
52	C₂²×C₄×C₁₉	—	4	32	no	—
53	C₂²×C₄×C₁₉	—	4	16	no	—
54	C₂²×C₄×C₁₉	C₄²	1	64	yes	complemented
55	C₂²×C₄×C₁₉	C₂²×C₄	1	32	yes	—
56	C₂²×C₄×C₁₉	C₂²×C₄	1	16	yes	—
57	C₂²×C₄×C₁₉	C₂²×C₄	1	32	yes	—
58	C₂²×C₄×C₁₉	C₂²×C₄	1	8	yes	—
59	C₂²×C₄×C₁₉	C₂²×C₄	1	8	yes	—
60	C₂²×C₄×C₁₉	C₂²×C₄	1	8	yes	—
61	C₂²×C₄×C₁₉	C₂²×C₄	1	16	yes	—
62	C₂²×C₄×C₁₉	C₂²×C₄	1	32	yes	—
63	C₂²×C₄×C₁₉	C₂⁴	1	16	yes	—
64	C₂²×C₄×C₁₉	C₂⁴	1	8	yes	—
65	C₂²×C₄×C₁₉	C₂⁴	1	4	yes	—
66	C₄⋊C₄×C₁₉	C₂²×C₄	1	64	yes	—
67	C₄⋊C₄×C₁₉	C₂²×C₄	1	64	yes	complemented
68	C₂²⋊C₄×C₁₉	C₂²×C₄	1	64	yes	—
69	C₄²×C₁₉	—	4	32	no	—
70	C₄²×C₁₉	C₂²×C₄	1	64	yes	complemented
71	C₂×#2274	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₂³×C₁₉	—	2	8	no	—
73	C₂³×C₁₉	—	2	8	no	—
74	C₂³×C₁₉	—	2	32	no	—
75	C₂³×C₁₉	—	2	4	no	—
76	C₂³×C₁₉	—	2	4	no	—
77	C₂³×C₁₉	C₂×C₄²	1	16	yes	—
78	C₂³×C₁₉	C₂³×C₄	1	8	yes	—
79	C₂³×C₁₉	C₂³×C₄	1	4	yes	—
80	C₂³×C₁₉	C₂³×C₄	1	2	yes	—
81	C₂³×C₁₉	C₂³×C₄	1	4	yes	—
82	C₂³×C₁₉	H(5,𝔽₂)	1	4	yes	—
83	C₂³×C₁₉	2₋¹⁺⁴	1	4	yes	—
84	C₂³×C₁₉	C₂⁵	1	1	yes	characteristic
85	C₂×C₄×C₁₉	—	2	32	no	—
86	C₂×C₄×C₁₉	—	2	64	no	—
87	C₂×C₄×C₁₉	—	2	16	no	—
88	C₂×C₄×C₁₉	—	2	16	no	—
89	C₂×C₄×C₁₉	—	2	16	no	—
90	C₂×C₄×C₁₉	—	4	32	no	—
91	C₂×C₄×C₁₉	C₂×C₄²	1	32	yes	—
92	C₂×C₄×C₁₉	C₂×C₄²	1	64	yes	complemented
93	C₂×C₄×C₁₉	C₂³×C₄	1	32	yes	—
94	C₂×C₄×C₁₉	C₂³×C₄	1	16	yes	—
95	C₂×C₄×C₁₉	C₂³×C₄	1	8	yes	—
96	C₂×C₄×C₁₉	C₂³×C₄	1	8	yes	—
97	C₂×C₄×C₁₉	C₂³×C₄	1	8	yes	—
98	C₂²×#117	C₂×C₁₉	1	2	yes	—
99	C₂³×C₄⋊C₄	C₂×C₁₉	1	1	yes	characteristic
100	#2274	C₂×C₁₉	1	16	yes	direct product factor
101	C₂×C₄×C₄⋊C₄	C₂×C₁₉	1	8	yes	complemented
102	C₂×C₄×C₂²⋊C₄	C₂×C₁₉	1	4	yes	—
103	C₂²×C₁₉	—	2	4	no	—
104	C₂²×C₁₉	—	2	4	no	—
105	C₂²×C₁₉	—	2	16	no	—
106	C₂²×C₁₉	—	2	32	no	—
107	C₂²×C₁₉	C₂²×C₄²	1	8	yes	—
108	C₂²×C₁₉	C₂²×C₄²	1	4	yes	—
109	C₂²×C₁₉	#518	1	4	yes	—
110	C₂²×C₁₉	#519	1	4	yes	—
111	C₂²×C₁₉	#520	1	16	yes	—
112	C₂²×C₁₉	C₂⁴×C₄	1	1	yes	characteristic
113	C₂²×C₁₉	C₂⁴×C₄	1	2	yes	—
114	C₂²×C₁₉	C₂×H(5,𝔽₂)	1	2	yes	—
115	C₂²×C₁₉	C₂×2₋¹⁺⁴	1	2	yes	—
116	C₄×C₁₉	—	2	16	no	—
117	C₄×C₁₉	—	2	32	no	—
118	C₄×C₁₉	C₂²×C₄²	1	16	yes	—
119	C₂⁴×C₄	C₂²×C₁₉	1	2	yes	—
120	C₂⁴×C₄	C₂²×C₁₉	1	1	yes	characteristic
121	C₂×#117	C₄×C₁₉	1	32	yes	complemented
122	C₂×#117	C₂²×C₁₉	1	16	yes	—
123	C₂²×C₄⋊C₄	C₄×C₁₉	1	16	yes	complemented
124	C₂²×C₄⋊C₄	C₂²×C₁₉	1	4	yes	—
125	C₂²×C₄⋊C₄	C₂²×C₁₉	1	8	yes	—
126	C₂²×C₄⋊C₄	C₂²×C₁₉	1	4	yes	—
127	C₂²×C₂²⋊C₄	C₂²×C₁₉	1	4	yes	—
128	C₂²×C₄²	C₂²×C₁₉	1	8	yes	—
129	C₂²×C₄²	C₂²×C₁₉	1	4	yes	—
130	C₄×C₄⋊C₄	C₂²×C₁₉	1	64	yes	complemented
131	C₄×C₂²⋊C₄	C₂²×C₁₉	1	32	yes	—
132	C₂×#95	C₂²×C₁₉	1	8	yes	—
133	C₂×C₁₉	—	2	8	no	—
134	C₂×C₁₉	#2274	1	8	yes	direct product factor
135	C₂×C₁₉	C₂³×C₄²	1	1	yes	characteristic
136	C₂×C₁₉	C₂×#518	1	1	yes	characteristic
137	C₂×C₁₉	C₂×#519	1	1	yes	characteristic
138	C₂×C₁₉	C₂×#520	1	4	yes	—
139	C₂⁵	C₂³×C₁₉	1	1	yes	characteristic
140	C₂³×C₄	—	4	8	no	—
141	C₂³×C₄	C₂×C₄×C₁₉	1	16	yes	—
142	C₂³×C₄	C₂×C₄×C₁₉	1	8	yes	—
143	C₂³×C₄	C₂×C₄×C₁₉	1	16	yes	—
144	C₂³×C₄	C₂³×C₁₉	1	8	yes	—
145	C₂³×C₄	C₂³×C₁₉	1	4	yes	—
146	C₂³×C₄	C₂³×C₁₉	1	8	yes	—
147	C₂³×C₄	C₂³×C₁₉	1	2	yes	—
148	C₂³×C₄	C₂³×C₁₉	1	4	yes	—
149	#117	C₂×C₄×C₁₉	1	128	yes	complemented
150	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	32	yes	—
151	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	64	yes	complemented
152	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	32	yes	complemented
153	C₂×C₄⋊C₄	C₂³×C₁₉	1	16	yes	—
154	C₂×C₄⋊C₄	C₂³×C₁₉	1	16	yes	—
155	C₂×C₂²⋊C₄	C₂×C₄×C₁₉	1	32	yes	—
156	C₂×C₂²⋊C₄	C₂³×C₁₉	1	16	yes	—
157	C₂×C₄²	—	4	16	no	—
158	C₂×C₄²	C₂×C₄×C₁₉	1	32	yes	complemented
159	C₂×C₄²	C₂³×C₁₉	1	32	yes	—
160	C₂×C₄²	C₂³×C₁₉	1	16	yes	—
161	#95	C₂³×C₁₉	1	32	yes	—
162	C₁₉	C₂×#2274	1	1	yes	characteristic, direct product factor, Sylow
163	C₂⁴	—	2	4	no	—
164	C₂⁴	—	2	4	no	—
165	C₂⁴	C₂²×C₄×C₁₉	1	8	yes	—
166	C₂⁴	C₂²×C₄×C₁₉	1	4	yes	—
167	C₂⁴	C₂⁴×C₁₉	1	2	yes	—
168	C₂⁴	C₂⁴×C₁₉	1	1	yes	characteristic
169	C₂²×C₄	—	2	16	no	—
170	C₂²×C₄	—	2	16	no	—
171	C₂²×C₄	—	2	16	no	—
172	C₂²×C₄	—	4	32	no	—
173	C₂²×C₄	—	4	16	no	—
174	C₂²×C₄	C₄²×C₁₉	1	64	yes	complemented
175	C₂²×C₄	C₂²×C₄×C₁₉	1	32	yes	—
176	C₂²×C₄	C₂²×C₄×C₁₉	1	16	yes	—
177	C₂²×C₄	C₂²×C₄×C₁₉	1	32	yes	—
178	C₂²×C₄	C₂²×C₄×C₁₉	1	8	yes	—
179	C₂²×C₄	C₂²×C₄×C₁₉	1	8	yes	—
180	C₂²×C₄	C₂²×C₄×C₁₉	1	8	yes	—
181	C₂²×C₄	C₂²×C₄×C₁₉	1	16	yes	—
182	C₂²×C₄	C₂²×C₄×C₁₉	1	32	yes	—
183	C₂²×C₄	C₂⁴×C₁₉	1	16	yes	—
184	C₂²×C₄	C₂⁴×C₁₉	1	8	yes	—
185	C₂²×C₄	C₂⁴×C₁₉	1	4	yes	—
186	C₄⋊C₄	C₂²×C₄×C₁₉	1	64	yes	—
187	C₄⋊C₄	C₂²×C₄×C₁₉	1	64	yes	complemented
188	C₂²⋊C₄	C₂²×C₄×C₁₉	1	64	yes	—
189	C₄²	—	4	32	no	—
190	C₄²	C₂²×C₄×C₁₉	1	64	yes	complemented
191	C₂³	—	2	8	no	—
192	C₂³	—	2	8	no	—
193	C₂³	—	2	32	no	—
194	C₂³	—	2	4	no	—
195	C₂³	—	2	4	no	—
196	C₂³	C₂×C₄²×C₁₉	1	16	yes	—
197	C₂³	C₂³×C₄×C₁₉	1	8	yes	—
198	C₂³	C₂³×C₄×C₁₉	1	4	yes	—
199	C₂³	C₂³×C₄×C₁₉	1	2	yes	—
200	C₂³	C₂³×C₄×C₁₉	1	4	yes	—
201	C₂³	C₁₉×H(5,𝔽₂)	1	4	yes	—
202	C₂³	C₁₉×2₋¹⁺⁴	1	4	yes	—
203	C₂³	C₂⁵×C₁₉	1	1	yes	characteristic
204	C₂×C₄	—	2	32	no	—
205	C₂×C₄	—	2	64	no	—
206	C₂×C₄	—	2	16	no	—
207	C₂×C₄	—	2	16	no	—
208	C₂×C₄	—	2	16	no	—
209	C₂×C₄	—	4	32	no	—
210	C₂×C₄	C₂×C₄²×C₁₉	1	32	yes	—
211	C₂×C₄	C₂×C₄²×C₁₉	1	64	yes	complemented
212	C₂×C₄	C₂³×C₄×C₁₉	1	32	yes	—
213	C₂×C₄	C₂³×C₄×C₁₉	1	16	yes	—
214	C₂×C₄	C₂³×C₄×C₁₉	1	8	yes	—
215	C₂×C₄	C₂³×C₄×C₁₉	1	8	yes	—
216	C₂×C₄	C₂³×C₄×C₁₉	1	8	yes	—
217	C₂²	—	2	4	no	—
218	C₂²	—	2	4	no	—
219	C₂²	—	2	16	no	—
220	C₂²	—	2	32	no	—
221	C₂²	C₂²×C₄²×C₁₉	1	8	yes	—
222	C₂²	C₂²×C₄²×C₁₉	1	4	yes	—
223	C₂²	C₁₉×#518	1	4	yes	—
224	C₂²	C₁₉×#519	1	4	yes	—
225	C₂²	C₁₉×#520	1	16	yes	—
226	C₂²	C₂⁴×C₄×C₁₉	1	1	yes	characteristic
227	C₂²	C₂⁴×C₄×C₁₉	1	2	yes	—
228	C₂²	C₂×C₁₉×H(5,𝔽₂)	1	2	yes	—
229	C₂²	C₂×C₁₉×2₋¹⁺⁴	1	2	yes	—
230	C₄	—	2	16	no	—
231	C₄	—	2	32	no	—
232	C₄	C₂²×C₄²×C₁₉	1	16	yes	—
233	C₂	—	2	8	no	—
234	C₂	C₁₉×#2274	1	8	yes	direct product factor
235	C₂	C₂³×C₄²×C₁₉	1	1	yes	characteristic
236	C₂	C₂×C₁₉×#518	1	1	yes	characteristic
237	C₂	C₂×C₁₉×#519	1	1	yes	characteristic
238	C₂	C₂×C₁₉×#520	1	4	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₁₉	C₂⁴
commutator	C₂	C₂³×C₄²×C₁₉
Frattini, cosocle	C₂³	C₂⁵×C₁₉
prime radical	C₂⁵×C₁₉	C₂³
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
3. C₁₉×#2274	C₂	direct (N × H)
4. C₂×C₄×C₄⋊C₄×C₁₉	C₂	semidirect
8. C₂×C₁₉×#117	C₄	semidirect
10. C₂²×C₄⋊C₄×C₁₉	C₄	semidirect
17. C₄×C₄⋊C₄×C₁₉	C₂²	semidirect
30. C₁₉×#117	C₂×C₄	semidirect
32. C₂×C₄⋊C₄×C₁₉	C₂×C₄	semidirect
33. C₂×C₄⋊C₄×C₁₉	C₂×C₄	semidirect
39. C₂×C₄²×C₁₉	C₂×C₄	semidirect
54. C₂²×C₄×C₁₉	C₄²	semidirect
67. C₄⋊C₄×C₁₉	C₂²×C₄	semidirect
70. C₄²×C₁₉	C₂²×C₄	semidirect
71. C₂×#2274	C₁₉	direct (N × H)
92. C₂×C₄×C₁₉	C₂×C₄²	semidirect
100. #2274	C₂×C₁₉	direct (N × H)
101. C₂×C₄×C₄⋊C₄	C₂×C₁₉	semidirect
121. C₂×#117	C₄×C₁₉	semidirect
123. C₂²×C₄⋊C₄	C₄×C₁₉	semidirect
130. C₄×C₄⋊C₄	C₂²×C₁₉	semidirect
134. C₂×C₁₉	#2274	direct (N × H)
149. #117	C₂×C₄×C₁₉	semidirect
151. C₂×C₄⋊C₄	C₂×C₄×C₁₉	semidirect
152. C₂×C₄⋊C₄	C₂×C₄×C₁₉	semidirect
158. C₂×C₄²	C₂×C₄×C₁₉	semidirect
162. C₁₉	C₂×#2274	direct (N × H)
174. C₂²×C₄	C₄²×C₁₉	semidirect
187. C₄⋊C₄	C₂²×C₄×C₁₉	semidirect
190. C₄²	C₂²×C₄×C₁₉	semidirect
211. C₂×C₄	C₂×C₄²×C₁₉	semidirect
234. C₂	C₁₉×#2274	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₁₉×#117	2
C₂³×C₄⋊C₄×C₁₉	2
C₁₉×#2274	2
C₂×C₄×C₄⋊C₄×C₁₉	2
C₂×C₄×C₂²⋊C₄×C₁₉	2
C₂×#2274	19

Subgroups — C2×C19×#2274 #18039954

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₉×#2274 #18039954