Subgroups — C₂×C₁₉×#2551 #18040883

Order 4864.

3182 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₉×C₄⋊Q₈	C₂	1	4	yes	maximal
2	C₁₉×#2551	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₁₉×#395	C₂	1	4	yes	maximal
4	C₂×C₁₉×#389	C₂	1	4	yes	maximal
5	C₂×C₁₉×#384	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×C₄⋊C₄×C₁₉	C₂	1	2	yes	maximal
7	C₂³×Q₈×C₁₉	C₂²	1	4	yes	—
8	C₂×C₁₉×C₄⋊Q₈	—	2	32	no	—
9	C₂×C₁₉×C₄⋊Q₈	C₂²	1	32	yes	—
10	C₂²×C₄⋊C₄×C₁₉	C₂²	1	4	yes	—
11	C₂²×C₄⋊C₄×C₁₉	C₂²	1	8	yes	—
12	C₂²×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
13	C₂²×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
14	C₂²×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
15	C₂²×C₄²×C₁₉	C₂²	1	4	yes	—
16	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
17	C₁₉×#395	C₂²	1	32	yes	—
18	C₁₉×#389	C₂²	1	32	yes	—
19	C₁₉×#384	C₂²	1	8	yes	—
20	C₄×C₄⋊C₄×C₁₉	C₂²	1	16	yes	—
21	C₂×C₁₉×#95	C₂²	1	4	yes	—
22	C₂×C₁₉×#95	C₂²	1	4	yes	—
23	C₂²×Q₈×C₁₉	—	2	32	no	—
24	C₂²×Q₈×C₁₉	—	4	8	no	—
25	C₂²×Q₈×C₁₉	C₂³	1	16	yes	—
26	C₂³×C₄×C₁₉	C₂³	1	4	yes	—
27	C₂³×C₄×C₁₉	C₂³	1	4	yes	—
28	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
29	C₂³×C₄×C₁₉	C₂³	1	4	yes	—
30	C₂³×C₄×C₁₉	C₂³	1	2	yes	—
31	C₁₉×C₄⋊Q₈	—	2	128	no	—
32	C₂×C₄⋊C₄×C₁₉	—	2	32	no	—
33	C₂×C₄⋊C₄×C₁₉	—	2	16	no	—
34	C₂×C₄⋊C₄×C₁₉	—	4	4	no	—
35	C₂×C₄⋊C₄×C₁₉	D₄	1	16	yes	—
36	C₂×C₄⋊C₄×C₁₉	Q₈	1	16	yes	complemented
37	C₂×C₄⋊C₄×C₁₉	C₂³	1	16	yes	—
38	C₂×C₄⋊C₄×C₁₉	C₂³	1	32	yes	—
39	C₂×C₄⋊C₄×C₁₉	C₂³	1	8	yes	—
40	C₂×C₄⋊C₄×C₁₉	C₂³	1	8	yes	—
41	C₂×C₄⋊C₄×C₁₉	C₂³	1	8	yes	—
42	C₂×C₄²×C₁₉	—	2	16	no	—
43	C₂×C₄²×C₁₉	Q₈	1	8	yes	complemented
44	C₂×C₄²×C₁₉	C₂³	1	16	yes	—
45	C₂×C₄²×C₁₉	C₂³	1	4	yes	—
46	C₁₉×#95	C₂³	1	16	yes	—
47	C₁₉×#95	C₂³	1	16	yes	—
48	C₂⁴×C₁₉	C₂⁴	1	1	yes	characteristic
49	C₂×Q₈×C₁₉	—	2	64	no	—
50	C₂×Q₈×C₁₉	—	4	16	no	—
51	C₂×Q₈×C₁₉	—	4	64	no	—
52	C₂²×C₄×C₁₉	—	2	8	no	—
53	C₂²×C₄×C₁₉	—	2	8	no	—
54	C₂²×C₄×C₁₉	—	2	2	no	—
55	C₂²×C₄×C₁₉	—	2	8	no	—
56	C₂²×C₄×C₁₉	—	2	2	no	—
57	C₂²×C₄×C₁₉	—	2	8	no	—
58	C₂²×C₄×C₁₉	—	2	16	no	—
59	C₂²×C₄×C₁₉	—	2	8	no	—
60	C₂²×C₄×C₁₉	C₂×D₄	1	16	yes	—
61	C₂²×C₄×C₁₉	C₂×D₄	1	8	yes	—
62	C₂²×C₄×C₁₉	C₂×Q₈	1	4	yes	—
63	C₂²×C₄×C₁₉	C₂×Q₈	1	16	yes	—
64	C₂²×C₄×C₁₉	C₂×Q₈	1	8	yes	—
65	C₂²×C₄×C₁₉	C₂×Q₈	1	8	yes	—
66	C₂²×C₄×C₁₉	C₂⁴	1	8	yes	—
67	C₂²×C₄×C₁₉	C₂⁴	1	8	yes	—
68	C₂²×C₄×C₁₉	C₂⁴	1	8	yes	—
69	C₂²×C₄×C₁₉	C₂⁴	1	2	yes	—
70	C₂²×C₄×C₁₉	C₂⁴	1	4	yes	—
71	C₄⋊C₄×C₁₉	—	2	64	no	—
72	C₄⋊C₄×C₁₉	—	2	32	no	—
73	C₄⋊C₄×C₁₉	—	4	8	no	—
74	C₄⋊C₄×C₁₉	C₂×D₄	1	32	yes	—
75	C₄⋊C₄×C₁₉	C₂×Q₈	1	32	yes	complemented
76	C₄²×C₁₉	—	2	32	no	—
77	C₄²×C₁₉	C₂×Q₈	1	16	yes	complemented
78	C₂×#2551	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
79	C₂³×C₁₉	C₂²×D₄	1	4	yes	—
80	C₂³×C₁₉	C₂²×Q₈	1	2	yes	—
81	C₂³×C₁₉	C₂²×Q₈	1	4	yes	—
82	C₂³×C₁₉	H(5,𝔽₂)	1	2	yes	—
83	C₂³×C₁₉	2₋¹⁺⁴	1	2	yes	—
84	C₂³×C₁₉	C₂⁵	1	1	yes	characteristic
85	Q₈×C₁₉	—	4	64	no	—
86	C₂×C₄×C₁₉	—	2	8	no	—
87	C₂×C₄×C₁₉	—	2	32	no	—
88	C₂×C₄×C₁₉	—	2	32	no	—
89	C₂×C₄×C₁₉	—	2	8	no	—
90	C₂×C₄×C₁₉	—	2	8	no	—
91	C₂×C₄×C₁₉	—	2	2	no	—
92	C₂×C₄×C₁₉	—	2	8	no	—
93	C₂×C₄×C₁₉	—	2	2	no	—
94	C₂×C₄×C₁₉	—	2	8	no	—
95	C₂×C₄×C₁₉	—	2	16	no	—
96	C₂×C₄×C₁₉	—	2	8	no	—
97	C₂×C₄×C₁₉	—	4	16	no	—
98	C₂×C₄×C₁₉	C₄⋊Q₈	1	32	yes	complemented
99	C₂×C₄×C₁₉	C₂²×D₄	1	16	yes	—
100	C₂×C₄×C₁₉	C₂²×D₄	1	8	yes	—
101	C₂×C₄×C₁₉	C₂²×Q₈	1	4	yes	—
102	C₂×C₄×C₁₉	C₂²×Q₈	1	16	yes	—
103	C₂×C₄×C₁₉	C₂²×Q₈	1	8	yes	—
104	C₂×C₄×C₁₉	C₂²×Q₈	1	8	yes	—
105	C₂²×C₄⋊Q₈	C₂×C₁₉	1	4	yes	—
106	#2551	C₂×C₁₉	1	16	yes	direct product factor
107	C₂×#395	C₂×C₁₉	1	4	yes	—
108	C₂×#389	C₂×C₁₉	1	4	yes	—
109	C₂×#384	C₂×C₁₉	1	1	yes	characteristic
110	C₂×C₄×C₄⋊C₄	C₂×C₁₉	1	2	yes	—
111	C₂²×C₁₉	C₂×C₄⋊Q₈	1	8	yes	—
112	C₂²×C₁₉	#544	1	4	yes	—
113	C₂²×C₁₉	D₄²	1	4	yes	—
114	C₂²×C₁₉	D₄×Q₈	1	8	yes	—
115	C₂²×C₁₉	Q₈²	1	4	yes	—
116	C₂²×C₁₉	C₂³×D₄	1	2	yes	—
117	C₂²×C₁₉	C₂³×Q₈	1	1	yes	characteristic
118	C₂²×C₁₉	C₂³×Q₈	1	2	yes	—
119	C₂²×C₁₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
120	C₂²×C₁₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
121	C₄×C₁₉	—	2	4	no	—
122	C₄×C₁₉	—	2	16	no	—
123	C₄×C₁₉	—	2	16	no	—
124	C₄×C₁₉	—	4	8	no	—
125	C₄×C₁₉	C₂×C₄⋊Q₈	1	16	yes	complemented
126	C₂³×Q₈	C₂²×C₁₉	1	4	yes	—
127	C₂×C₄⋊Q₈	—	2	32	no	—
128	C₂×C₄⋊Q₈	C₂²×C₁₉	1	32	yes	—
129	C₂²×C₄⋊C₄	C₂²×C₁₉	1	8	yes	—
130	C₂²×C₄⋊C₄	C₂²×C₁₉	1	4	yes	—
131	C₂²×C₄⋊C₄	C₂²×C₁₉	1	2	yes	—
132	C₂²×C₄⋊C₄	C₂²×C₁₉	1	2	yes	—
133	C₂²×C₄⋊C₄	C₂²×C₁₉	1	2	yes	—
134	C₂²×C₄²	C₂²×C₁₉	1	4	yes	—
135	C₂²×C₄²	C₂²×C₁₉	1	1	yes	characteristic
136	#395	C₂²×C₁₉	1	32	yes	—
137	#389	C₂²×C₁₉	1	32	yes	—
138	#384	C₂²×C₁₉	1	8	yes	—
139	C₄×C₄⋊C₄	C₂²×C₁₉	1	16	yes	—
140	C₂×#95	C₂²×C₁₉	1	4	yes	—
141	C₂×#95	C₂²×C₁₉	1	4	yes	—
142	C₂×C₁₉	#2551	1	8	yes	direct product factor
143	C₂×C₁₉	C₂²×C₄⋊Q₈	1	2	yes	—
144	C₂×C₁₉	C₂×#544	1	1	yes	characteristic
145	C₂×C₁₉	C₂×D₄²	1	1	yes	characteristic
146	C₂×C₁₉	C₂×D₄×Q₈	1	2	yes	—
147	C₂×C₁₉	C₂×Q₈²	1	1	yes	characteristic
148	C₂²×Q₈	—	2	32	no	—
149	C₂²×Q₈	—	4	8	no	—
150	C₂²×Q₈	C₂³×C₁₉	1	16	yes	—
151	C₂³×C₄	C₂³×C₁₉	1	4	yes	—
152	C₂³×C₄	C₂³×C₁₉	1	4	yes	—
153	C₂³×C₄	C₂³×C₁₉	1	4	yes	—
154	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
155	C₂³×C₄	C₂³×C₁₉	1	2	yes	—
156	C₄⋊Q₈	—	2	128	no	—
157	C₂×C₄⋊C₄	—	2	32	no	—
158	C₂×C₄⋊C₄	—	2	16	no	—
159	C₂×C₄⋊C₄	—	4	4	no	—
160	C₂×C₄⋊C₄	D₄×C₁₉	1	16	yes	—
161	C₂×C₄⋊C₄	Q₈×C₁₉	1	16	yes	complemented
162	C₂×C₄⋊C₄	C₂³×C₁₉	1	16	yes	—
163	C₂×C₄⋊C₄	C₂³×C₁₉	1	32	yes	—
164	C₂×C₄⋊C₄	C₂³×C₁₉	1	8	yes	—
165	C₂×C₄⋊C₄	C₂³×C₁₉	1	8	yes	—
166	C₂×C₄⋊C₄	C₂³×C₁₉	1	8	yes	—
167	C₂×C₄²	—	2	16	no	—
168	C₂×C₄²	Q₈×C₁₉	1	8	yes	complemented
169	C₂×C₄²	C₂³×C₁₉	1	16	yes	—
170	C₂×C₄²	C₂³×C₁₉	1	4	yes	—
171	#95	C₂³×C₁₉	1	16	yes	—
172	#95	C₂³×C₁₉	1	16	yes	—
173	C₁₉	C₂×#2551	1	1	yes	characteristic, direct product factor, Sylow
174	C₂⁴	C₂⁴×C₁₉	1	1	yes	characteristic
175	C₂×Q₈	—	2	64	no	—
176	C₂×Q₈	—	4	16	no	—
177	C₂×Q₈	—	4	64	no	—
178	C₂²×C₄	—	2	8	no	—
179	C₂²×C₄	—	2	8	no	—
180	C₂²×C₄	—	2	2	no	—
181	C₂²×C₄	—	2	8	no	—
182	C₂²×C₄	—	2	2	no	—
183	C₂²×C₄	—	2	8	no	—
184	C₂²×C₄	—	2	16	no	—
185	C₂²×C₄	—	2	8	no	—
186	C₂²×C₄	C₂×D₄×C₁₉	1	16	yes	—
187	C₂²×C₄	C₂×D₄×C₁₉	1	8	yes	—
188	C₂²×C₄	C₂×Q₈×C₁₉	1	4	yes	—
189	C₂²×C₄	C₂×Q₈×C₁₉	1	16	yes	—
190	C₂²×C₄	C₂×Q₈×C₁₉	1	8	yes	—
191	C₂²×C₄	C₂×Q₈×C₁₉	1	8	yes	—
192	C₂²×C₄	C₂⁴×C₁₉	1	8	yes	—
193	C₂²×C₄	C₂⁴×C₁₉	1	8	yes	—
194	C₂²×C₄	C₂⁴×C₁₉	1	2	yes	—
195	C₂²×C₄	C₂⁴×C₁₉	1	8	yes	—
196	C₂²×C₄	C₂⁴×C₁₉	1	4	yes	—
197	C₄⋊C₄	—	2	64	no	—
198	C₄⋊C₄	—	2	32	no	—
199	C₄⋊C₄	—	4	8	no	—
200	C₄⋊C₄	C₂×D₄×C₁₉	1	32	yes	—
201	C₄⋊C₄	C₂×Q₈×C₁₉	1	32	yes	complemented
202	C₄²	—	2	32	no	—
203	C₄²	C₂×Q₈×C₁₉	1	16	yes	complemented
204	C₂³	C₂²×D₄×C₁₉	1	4	yes	—
205	C₂³	C₂²×Q₈×C₁₉	1	2	yes	—
206	C₂³	C₂²×Q₈×C₁₉	1	4	yes	—
207	C₂³	C₁₉×H(5,𝔽₂)	1	2	yes	—
208	C₂³	C₁₉×2₋¹⁺⁴	1	2	yes	—
209	C₂³	C₂⁵×C₁₉	1	1	yes	characteristic
210	Q₈	—	4	64	no	—
211	C₂×C₄	—	2	8	no	—
212	C₂×C₄	—	2	32	no	—
213	C₂×C₄	—	2	32	no	—
214	C₂×C₄	—	2	8	no	—
215	C₂×C₄	—	2	8	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	8	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	8	no	—
220	C₂×C₄	—	2	16	no	—
221	C₂×C₄	—	2	8	no	—
222	C₂×C₄	—	4	16	no	—
223	C₂×C₄	C₁₉×C₄⋊Q₈	1	32	yes	complemented
224	C₂×C₄	C₂²×D₄×C₁₉	1	16	yes	—
225	C₂×C₄	C₂²×D₄×C₁₉	1	8	yes	—
226	C₂×C₄	C₂²×Q₈×C₁₉	1	4	yes	—
227	C₂×C₄	C₂²×Q₈×C₁₉	1	16	yes	—
228	C₂×C₄	C₂²×Q₈×C₁₉	1	8	yes	—
229	C₂×C₄	C₂²×Q₈×C₁₉	1	8	yes	—
230	C₂²	C₂×C₁₉×C₄⋊Q₈	1	8	yes	—
231	C₂²	C₁₉×#544	1	4	yes	—
232	C₂²	D₄²×C₁₉	1	4	yes	—
233	C₂²	D₄×Q₈×C₁₉	1	8	yes	—
234	C₂²	Q₈²×C₁₉	1	4	yes	—
235	C₂²	C₂³×D₄×C₁₉	1	2	yes	—
236	C₂²	C₂³×Q₈×C₁₉	1	1	yes	characteristic
237	C₂²	C₂³×Q₈×C₁₉	1	2	yes	—
238	C₂²	C₂×C₁₉×H(5,𝔽₂)	1	1	yes	characteristic
239	C₂²	C₂×C₁₉×2₋¹⁺⁴	1	1	yes	characteristic
240	C₄	—	2	4	no	—
241	C₄	—	2	16	no	—
242	C₄	—	2	16	no	—
243	C₄	—	4	8	no	—
244	C₄	C₂×C₁₉×C₄⋊Q₈	1	16	yes	complemented
245	C₂	C₁₉×#2551	1	8	yes	direct product factor
246	C₂	C₂²×C₁₉×C₄⋊Q₈	1	2	yes	—
247	C₂	C₂×C₁₉×#544	1	1	yes	characteristic
248	C₂	C₂×D₄²×C₁₉	1	1	yes	characteristic
249	C₂	C₂×D₄×Q₈×C₁₉	1	2	yes	—
250	C₂	C₂×Q₈²×C₁₉	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₁₉	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₁₉
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₉×#2551	C₂	direct (N × H)
36. C₂×C₄⋊C₄×C₁₉	Q₈	semidirect
43. C₂×C₄²×C₁₉	Q₈	semidirect
75. C₄⋊C₄×C₁₉	C₂×Q₈	semidirect
77. C₄²×C₁₉	C₂×Q₈	semidirect
78. C₂×#2551	C₁₉	direct (N × H)
98. C₂×C₄×C₁₉	C₄⋊Q₈	semidirect
106. #2551	C₂×C₁₉	direct (N × H)
125. C₄×C₁₉	C₂×C₄⋊Q₈	semidirect
142. C₂×C₁₉	#2551	direct (N × H)
161. C₂×C₄⋊C₄	Q₈×C₁₉	semidirect
168. C₂×C₄²	Q₈×C₁₉	semidirect
173. C₁₉	C₂×#2551	direct (N × H)
201. C₄⋊C₄	C₂×Q₈×C₁₉	semidirect
203. C₄²	C₂×Q₈×C₁₉	semidirect
223. C₂×C₄	C₁₉×C₄⋊Q₈	semidirect
244. C₄	C₂×C₁₉×C₄⋊Q₈	semidirect
245. C₂	C₁₉×#2551	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×C₄⋊Q₈	2
C₁₉×#2551	2
C₂×C₁₉×#395	2
C₂×C₁₉×#389	2
C₂×C₁₉×#384	2
C₂×C₄×C₄⋊C₄×C₁₉	2
C₂×#2551	19

Subgroups — C2×C19×#2551 #18040883

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₉×#2551 #18040883