Subgroups — C₂²×C₁₉×#462 #18066366

Order 4864.

2790 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Q₁₆×C₁₉	C₂	1	2	yes	maximal
2	C₂²×C₁₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄×Q₈×C₁₉	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₉×#462	C₂	1	24	yes	maximal, direct product factor
5	C₂²×C₁₉×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₉×#103	C₂	1	2	yes	maximal
7	C₂³×Q₈×C₁₉	C₂²	1	2	yes	—
8	C₂³×Q₈×C₁₉	C₂²	1	1	yes	characteristic
9	C₂²×Q₁₆×C₁₉	—	2	16	no	—
10	C₂²×Q₁₆×C₁₉	C₂²	1	24	yes	—
11	C₂³×C₈×C₁₉	C₂²	1	2	yes	—
12	C₂×C₁₉×C₄⋊Q₈	—	2	6	no	—
13	C₂×C₁₉×C₄⋊Q₈	C₂²	1	12	yes	—
14	C₂×C₄×Q₈×C₁₉	—	2	6	no	—
15	C₂×C₄×Q₈×C₁₉	C₂²	1	12	yes	—
16	C₂²×C₄⋊C₄×C₁₉	—	2	1	no	—
17	C₂²×C₄⋊C₄×C₁₉	—	2	1	no	—
18	C₂²×C₄⋊C₄×C₁₉	—	2	1	no	—
19	C₂²×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
20	C₂²×C₄²×C₁₉	—	2	1	no	—
21	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
22	C₁₉×#462	C₂²	1	64	yes	direct product factor
23	C₂×C₁₉×C₄⋊C₈	C₂²	1	12	yes	—
24	C₂×C₁₉×#103	C₂²	1	24	yes	—
25	C₂²×Q₈×C₁₉	—	2	3	no	—
26	C₂²×Q₈×C₁₉	—	2	6	no	—
27	C₂²×Q₈×C₁₉	—	2	4	no	—
28	C₂²×Q₈×C₁₉	—	2	8	no	—
29	C₂²×Q₈×C₁₉	—	2	8	no	—
30	C₂²×Q₈×C₁₉	D₄	1	8	yes	—
31	C₂²×Q₈×C₁₉	C₂³	1	12	yes	—
32	C₂²×Q₈×C₁₉	C₂³	1	6	yes	—
33	C₂³×C₄×C₁₉	—	2	1	no	—
34	C₂³×C₄×C₁₉	—	2	1	no	—
35	C₂³×C₄×C₁₉	—	2	2	no	—
36	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
37	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
38	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
39	C₂×Q₁₆×C₁₉	—	2	96	no	—
40	C₂×Q₁₆×C₁₉	C₂³	1	32	yes	—
41	C₂²×C₈×C₁₉	—	2	8	no	—
42	C₂²×C₈×C₁₉	C₂³	1	12	yes	—
43	C₁₉×C₄⋊Q₈	—	2	24	no	—
44	C₁₉×C₄⋊Q₈	C₂³	1	16	yes	—
45	C₄×Q₈×C₁₉	—	2	24	no	—
46	C₄×Q₈×C₁₉	C₂³	1	16	yes	—
47	C₂×C₄⋊C₄×C₁₉	—	2	12	no	—
48	C₂×C₄⋊C₄×C₁₉	—	2	12	no	—
49	C₂×C₄⋊C₄×C₁₉	—	2	12	no	—
50	C₂×C₄⋊C₄×C₁₉	—	2	3	no	—
51	C₂×C₄⋊C₄×C₁₉	C₂³	1	6	yes	—
52	C₂×C₄²×C₁₉	—	2	12	no	—
53	C₂×C₄²×C₁₉	D₄	1	6	yes	—
54	C₂×C₄²×C₁₉	C₂³	1	6	yes	—
55	C₁₉×C₄⋊C₈	C₂³	1	16	yes	—
56	C₁₉×#103	C₂³	1	32	yes	—
57	C₂⁴×C₁₉	C₂×D₄	1	1	yes	characteristic
58	C₂×Q₈×C₁₉	—	2	24	no	—
59	C₂×Q₈×C₁₉	—	2	6	no	—
60	C₂×Q₈×C₁₉	—	2	12	no	—
61	C₂×Q₈×C₁₉	—	2	12	no	—
62	C₂×Q₈×C₁₉	—	2	24	no	—
63	C₂×Q₈×C₁₉	—	2	24	no	—
64	C₂×Q₈×C₁₉	—	4	24	no	—
65	C₂×Q₈×C₁₉	C₂×D₄	1	24	yes	—
66	C₂×Q₈×C₁₉	C₂⁴	1	8	yes	—
67	C₂×Q₈×C₁₉	C₂⁴	1	4	yes	—
68	C₂²×C₄×C₁₉	—	2	8	no	—
69	C₂²×C₄×C₁₉	—	2	4	no	—
70	C₂²×C₄×C₁₉	—	2	6	no	—
71	C₂²×C₄×C₁₉	—	2	6	no	—
72	C₂²×C₄×C₁₉	—	2	12	no	—
73	C₂²×C₄×C₁₉	—	4	8	no	—
74	C₂²×C₄×C₁₉	—	4	4	no	—
75	C₂²×C₄×C₁₉	Q₁₆	1	8	yes	complemented
76	C₂²×C₄×C₁₉	C₂×D₄	1	6	yes	—
77	C₂²×C₄×C₁₉	C₂×D₄	1	3	yes	—
78	C₂²×C₄×C₁₉	C₂×D₄	1	4	yes	—
79	C₂²×C₄×C₁₉	C₂×D₄	1	6	yes	—
80	C₂²×C₄×C₁₉	#41	1	4	yes	—
81	C₂²×C₄×C₁₉	C₂⁴	1	3	yes	—
82	Q₁₆×C₁₉	—	2	64	no	—
83	C₂×C₈×C₁₉	—	2	24	no	—
84	C₂×C₈×C₁₉	C₂⁴	1	8	yes	—
85	C₄⋊C₄×C₁₉	—	2	16	no	—
86	C₄⋊C₄×C₁₉	—	2	16	no	—
87	C₄⋊C₄×C₁₉	—	2	16	no	—
88	C₄⋊C₄×C₁₉	—	2	6	no	—
89	C₄⋊C₄×C₁₉	C₂⁴	1	4	yes	—
90	C₄²×C₁₉	—	2	16	no	—
91	C₄²×C₁₉	C₂×D₄	1	12	yes	—
92	C₄²×C₁₉	C₂⁴	1	4	yes	—
93	C₂²×#462	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
94	C₂³×C₁₉	#121	1	4	yes	—
95	C₂³×C₁₉	C₂×Q₁₆	1	4	yes	—
96	C₂³×C₁₉	#137	1	4	yes	—
97	C₂³×C₁₉	C₂²×D₄	1	3	yes	—
98	Q₈×C₁₉	—	2	24	no	—
99	Q₈×C₁₉	—	2	4	no	—
100	Q₈×C₁₉	—	2	8	no	—
101	Q₈×C₁₉	—	2	8	no	—
102	Q₈×C₁₉	—	4	24	no	—
103	Q₈×C₁₉	C₂²×D₄	1	8	yes	—
104	C₂×C₄×C₁₉	—	2	24	no	—
105	C₂×C₄×C₁₉	—	2	12	no	—
106	C₂×C₄×C₁₉	—	2	4	no	—
107	C₂×C₄×C₁₉	—	2	4	no	—
108	C₂×C₄×C₁₉	—	2	8	no	—
109	C₂×C₄×C₁₉	—	4	24	no	—
110	C₂×C₄×C₁₉	—	4	12	no	—
111	C₂×C₄×C₁₉	#121	1	12	yes	—
112	C₂×C₄×C₁₉	C₂×Q₁₆	1	24	yes	complemented
113	C₂×C₄×C₁₉	C₂²×D₄	1	4	yes	—
114	C₂×C₄×C₁₉	C₂²×D₄	1	3	yes	—
115	C₂×C₄×C₁₉	C₂²×D₄	1	6	yes	—
116	C₂×C₄×C₁₉	C₂²×D₄	1	4	yes	—
117	C₂×C₄×C₁₉	C₂×#41	1	6	yes	—
118	C₂×C₄×C₁₉	C₂⁵	1	1	yes	characteristic
119	C₈×C₁₉	—	2	8	no	—
120	C₂³×Q₁₆	C₂×C₁₉	1	2	yes	—
121	C₂²×C₄⋊Q₈	C₂×C₁₉	1	1	yes	characteristic
122	C₂²×C₄×Q₈	C₂×C₁₉	1	1	yes	characteristic
123	C₂×#462	C₂×C₁₉	1	24	yes	direct product factor
124	C₂²×C₄⋊C₈	C₂×C₁₉	1	1	yes	characteristic
125	C₂²×#103	C₂×C₁₉	1	2	yes	—
126	C₂²×C₁₉	#462	1	16	yes	direct product factor
127	C₂²×C₁₉	C₂×#121	1	6	yes	—
128	C₂²×C₁₉	C₂²×Q₁₆	1	6	yes	—
129	C₂²×C₁₉	C₂×#137	1	6	yes	—
130	C₂²×C₁₉	C₂³×D₄	1	1	yes	characteristic
131	C₄×C₁₉	—	2	8	no	—
132	C₄×C₁₉	—	2	4	no	—
133	C₄×C₁₉	—	4	8	no	—
134	C₄×C₁₉	—	4	4	no	—
135	C₄×C₁₉	C₂×#121	1	6	yes	—
136	C₄×C₁₉	C₂²×Q₁₆	1	8	yes	complemented
137	C₄×C₁₉	C₂³×D₄	1	1	yes	characteristic
138	C₄×C₁₉	C₂²×#41	1	1	yes	characteristic
139	C₂³×Q₈	C₂²×C₁₉	1	2	yes	—
140	C₂³×Q₈	C₂²×C₁₉	1	1	yes	characteristic
141	C₂²×Q₁₆	—	2	16	no	—
142	C₂²×Q₁₆	C₂²×C₁₉	1	24	yes	—
143	C₂³×C₈	C₂²×C₁₉	1	2	yes	—
144	C₂×C₄⋊Q₈	—	2	6	no	—
145	C₂×C₄⋊Q₈	C₂²×C₁₉	1	12	yes	—
146	C₂×C₄×Q₈	—	2	6	no	—
147	C₂×C₄×Q₈	C₂²×C₁₉	1	12	yes	—
148	C₂²×C₄⋊C₄	—	2	1	no	—
149	C₂²×C₄⋊C₄	—	2	1	no	—
150	C₂²×C₄⋊C₄	—	2	1	no	—
151	C₂²×C₄⋊C₄	C₂²×C₁₉	1	1	yes	characteristic
152	C₂²×C₄²	—	2	1	no	—
153	C₂²×C₄²	C₂²×C₁₉	1	1	yes	characteristic
154	#462	C₂²×C₁₉	1	64	yes	direct product factor
155	C₂×C₄⋊C₈	C₂²×C₁₉	1	12	yes	—
156	C₂×#103	C₂²×C₁₉	1	24	yes	—
157	C₂×C₁₉	C₂×#462	1	12	yes	direct product factor
158	C₂×C₁₉	C₂²×#121	1	1	yes	characteristic
159	C₂×C₁₉	C₂³×Q₁₆	1	1	yes	characteristic
160	C₂×C₁₉	C₂²×#137	1	1	yes	characteristic
161	C₂²×Q₈	—	2	3	no	—
162	C₂²×Q₈	—	2	6	no	—
163	C₂²×Q₈	—	2	4	no	—
164	C₂²×Q₈	—	2	8	no	—
165	C₂²×Q₈	—	2	8	no	—
166	C₂²×Q₈	D₄×C₁₉	1	8	yes	—
167	C₂²×Q₈	C₂³×C₁₉	1	12	yes	—
168	C₂²×Q₈	C₂³×C₁₉	1	6	yes	—
169	C₂³×C₄	—	2	1	no	—
170	C₂³×C₄	—	2	2	no	—
171	C₂³×C₄	—	2	1	no	—
172	C₂³×C₄	D₄×C₁₉	1	1	yes	characteristic
173	C₂³×C₄	D₄×C₁₉	1	1	yes	characteristic
174	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
175	C₂×Q₁₆	—	2	96	no	—
176	C₂×Q₁₆	C₂³×C₁₉	1	32	yes	—
177	C₂²×C₈	—	2	8	no	—
178	C₂²×C₈	C₂³×C₁₉	1	12	yes	—
179	C₄⋊Q₈	—	2	24	no	—
180	C₄⋊Q₈	C₂³×C₁₉	1	16	yes	—
181	C₄×Q₈	—	2	24	no	—
182	C₄×Q₈	C₂³×C₁₉	1	16	yes	—
183	C₂×C₄⋊C₄	—	2	12	no	—
184	C₂×C₄⋊C₄	—	2	12	no	—
185	C₂×C₄⋊C₄	—	2	12	no	—
186	C₂×C₄⋊C₄	—	2	3	no	—
187	C₂×C₄⋊C₄	C₂³×C₁₉	1	6	yes	—
188	C₂×C₄²	—	2	12	no	—
189	C₂×C₄²	D₄×C₁₉	1	6	yes	—
190	C₂×C₄²	C₂³×C₁₉	1	6	yes	—
191	C₄⋊C₈	C₂³×C₁₉	1	16	yes	—
192	#103	C₂³×C₁₉	1	32	yes	—
193	C₁₉	C₂²×#462	1	1	yes	characteristic, direct product factor, Sylow
194	C₂⁴	C₂×D₄×C₁₉	1	1	yes	characteristic
195	C₂×Q₈	—	2	24	no	—
196	C₂×Q₈	—	2	6	no	—
197	C₂×Q₈	—	2	12	no	—
198	C₂×Q₈	—	2	12	no	—
199	C₂×Q₈	—	2	24	no	—
200	C₂×Q₈	—	2	24	no	—
201	C₂×Q₈	—	4	24	no	—
202	C₂×Q₈	C₂×D₄×C₁₉	1	24	yes	—
203	C₂×Q₈	C₂⁴×C₁₉	1	8	yes	—
204	C₂×Q₈	C₂⁴×C₁₉	1	4	yes	—
205	C₂²×C₄	—	2	8	no	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	6	no	—
208	C₂²×C₄	—	2	6	no	—
209	C₂²×C₄	—	2	12	no	—
210	C₂²×C₄	—	4	8	no	—
211	C₂²×C₄	—	4	4	no	—
212	C₂²×C₄	Q₁₆×C₁₉	1	8	yes	complemented
213	C₂²×C₄	C₂×D₄×C₁₉	1	6	yes	—
214	C₂²×C₄	C₂×D₄×C₁₉	1	3	yes	—
215	C₂²×C₄	C₂×D₄×C₁₉	1	4	yes	—
216	C₂²×C₄	C₂×D₄×C₁₉	1	6	yes	—
217	C₂²×C₄	#41×C₁₉	1	4	yes	—
218	C₂²×C₄	C₂⁴×C₁₉	1	3	yes	—
219	Q₁₆	—	2	64	no	—
220	C₂×C₈	—	2	24	no	—
221	C₂×C₈	C₂⁴×C₁₉	1	8	yes	—
222	C₄⋊C₄	—	2	16	no	—
223	C₄⋊C₄	—	2	16	no	—
224	C₄⋊C₄	—	2	16	no	—
225	C₄⋊C₄	—	2	6	no	—
226	C₄⋊C₄	C₂⁴×C₁₉	1	4	yes	—
227	C₄²	—	2	16	no	—
228	C₄²	C₂×D₄×C₁₉	1	12	yes	—
229	C₄²	C₂⁴×C₁₉	1	4	yes	—
230	C₂³	C₁₉×#121	1	4	yes	—
231	C₂³	C₂×Q₁₆×C₁₉	1	4	yes	—
232	C₂³	C₁₉×#137	1	4	yes	—
233	C₂³	C₂²×D₄×C₁₉	1	3	yes	—
234	Q₈	—	2	24	no	—
235	Q₈	—	2	4	no	—
236	Q₈	—	2	8	no	—
237	Q₈	—	2	8	no	—
238	Q₈	—	4	24	no	—
239	Q₈	C₂²×D₄×C₁₉	1	8	yes	—
240	C₂×C₄	—	2	24	no	—
241	C₂×C₄	—	2	12	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	2	8	no	—
245	C₂×C₄	—	4	24	no	—
246	C₂×C₄	—	4	12	no	—
247	C₂×C₄	C₁₉×#121	1	12	yes	—
248	C₂×C₄	C₂×Q₁₆×C₁₉	1	24	yes	complemented
249	C₂×C₄	C₂²×D₄×C₁₉	1	4	yes	—
250	C₂×C₄	C₂²×D₄×C₁₉	1	3	yes	—
251	C₂×C₄	C₂²×D₄×C₁₉	1	6	yes	—
252	C₂×C₄	C₂²×D₄×C₁₉	1	4	yes	—
253	C₂×C₄	C₂×#41×C₁₉	1	6	yes	—
254	C₂×C₄	C₂⁵×C₁₉	1	1	yes	characteristic
255	C₈	—	2	8	no	—
256	C₂²	C₁₉×#462	1	16	yes	direct product factor
257	C₂²	C₂×C₁₉×#121	1	6	yes	—
258	C₂²	C₂²×Q₁₆×C₁₉	1	6	yes	—
259	C₂²	C₂×C₁₉×#137	1	6	yes	—
260	C₂²	C₂³×D₄×C₁₉	1	1	yes	characteristic
261	C₄	—	2	8	no	—
262	C₄	—	2	4	no	—
263	C₄	—	4	8	no	—
264	C₄	—	4	4	no	—
265	C₄	C₂×C₁₉×#121	1	6	yes	—
266	C₄	C₂²×Q₁₆×C₁₉	1	8	yes	complemented
267	C₄	C₂³×D₄×C₁₉	1	1	yes	characteristic
268	C₄	C₂²×#41×C₁₉	1	1	yes	characteristic
269	C₂	C₂×C₁₉×#462	1	12	yes	direct product factor
270	C₂	C₂²×C₁₉×#121	1	1	yes	characteristic
271	C₂	C₂³×Q₁₆×C₁₉	1	1	yes	characteristic
272	C₂	C₂²×C₁₉×#137	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₁₉	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₁₉
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₉ ◁ C₂²×C₄²×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂×C₁₉×#462	C₂	direct (N × H)
22. C₁₉×#462	C₂²	direct (N × H)
75. C₂²×C₄×C₁₉	Q₁₆	semidirect
93. C₂²×#462	C₁₉	direct (N × H)
112. C₂×C₄×C₁₉	C₂×Q₁₆	semidirect
123. C₂×#462	C₂×C₁₉	direct (N × H)
126. C₂²×C₁₉	#462	direct (N × H)
136. C₄×C₁₉	C₂²×Q₁₆	semidirect
154. #462	C₂²×C₁₉	direct (N × H)
157. C₂×C₁₉	C₂×#462	direct (N × H)
193. C₁₉	C₂²×#462	direct (N × H)
212. C₂²×C₄	Q₁₆×C₁₉	semidirect
248. C₂×C₄	C₂×Q₁₆×C₁₉	semidirect
256. C₂²	C₁₉×#462	direct (N × H)
266. C₄	C₂²×Q₁₆×C₁₉	semidirect
269. C₂	C₂×C₁₉×#462	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×Q₁₆×C₁₉	2
C₂²×C₁₉×C₄⋊Q₈	2
C₂²×C₄×Q₈×C₁₉	2
C₂×C₁₉×#462	2
C₂²×C₁₉×C₄⋊C₈	2
C₂²×C₁₉×#103	2
C₂²×#462	19

Subgroups — C22×C19×#462 #18066366

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₁₉×#462 #18066366