Subgroups — C₂×C₁₉×#2921 #18066077

Order 4864.

2198 conjugacy classes of subgroups.

The subgroup lattice has 292 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	1	yes	maximal, characteristic
2	C₁₉×#2921	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₁₉×#433	C₂	1	8	yes	maximal, complemented
4	C₂×C₁₉×#423	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₉×#407	C₂	1	2	yes	maximal
6	C₂×C₄×M₁₆×C₁₉	C₂	1	2	yes	maximal, complemented
7	C₂⁴×C₄×C₁₉	C₂²	1	1	yes	characteristic
8	C₂²×M₁₆×C₁₉	C₂²	1	4	yes	—
9	C₂×C₁₉×#117	C₄	1	4	yes	—
10	C₂×C₁₉×#117	C₄	1	8	yes	—
11	C₂×C₁₉×#117	C₂²	1	4	yes	—
12	C₂×C₁₉×#117	C₂²	1	8	yes	—
13	C₂²×C₄⋊C₄×C₁₉	C₄	1	2	yes	—
14	C₂²×C₂²⋊C₄×C₁₉	C₄	1	2	yes	—
15	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
16	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
17	C₁₉×#433	C₂²	1	64	yes	complemented
18	C₁₉×#423	C₂²	1	16	yes	complemented
19	C₂×C₁₉×C₄⋊C₈	C₂²	1	8	yes	complemented
20	C₁₉×#407	C₂²	1	16	yes	—
21	C₂×C₁₉×C₂²⋊C₈	C₂²	1	8	yes	—
22	C₄×M₁₆×C₁₉	C₂²	1	16	yes	complemented
23	C₂×C₁₉×#97	C₂²	1	4	yes	complemented
24	C₂×C₄×C₈×C₁₉	C₂²	1	4	yes	complemented
25	C₂⁵×C₁₉	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₁₉	—	2	8	no	—
27	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₁₉	C₂×C₄	1	2	yes	—
29	C₂³×C₄×C₁₉	C₂×C₄	1	4	yes	—
30	C₂³×C₄×C₁₉	C₂×C₄	1	2	yes	—
31	C₂³×C₄×C₁₉	C₂×C₄	1	2	yes	—
32	C₂³×C₄×C₁₉	C₂³	1	4	yes	—
33	C₂³×C₄×C₁₉	C₂³	1	2	yes	—
34	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
35	C₂×M₁₆×C₁₉	—	2	16	no	—
36	C₂×M₁₆×C₁₉	C₂³	1	16	yes	—
37	C₂²×C₈×C₁₉	C₂³	1	8	yes	—
38	C₁₉×#117	C₂×C₄	1	16	yes	—
39	C₁₉×#117	C₂×C₄	1	32	yes	—
40	C₁₉×#117	C₂³	1	16	yes	—
41	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	8	yes	—
42	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	16	yes	—
43	C₂×C₂²⋊C₄×C₁₉	C₂×C₄	1	8	yes	—
44	C₂×C₂²⋊C₄×C₁₉	C₂×C₄	1	16	yes	—
45	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₁₉	C₂×C₄	1	4	yes	—
48	C₂×C₄²×C₁₉	C₂×C₄	1	4	yes	—
49	C₂×C₄²×C₁₉	C₂³	1	4	yes	—
50	C₂×C₄²×C₁₉	C₂³	1	4	yes	—
51	C₂×C₄²×C₁₉	C₂³	1	2	yes	—
52	C₂×C₄²×C₁₉	C₂³	1	2	yes	—
53	C₁₉×C₄⋊C₈	C₂³	1	32	yes	complemented
54	C₁₉×C₂²⋊C₈	C₂³	1	32	yes	—
55	C₁₉×#97	C₂³	1	16	yes	complemented
56	C₄×C₈×C₁₉	C₂³	1	16	yes	complemented
57	C₂⁴×C₁₉	—	2	4	no	—
58	C₂⁴×C₁₉	—	4	4	no	—
59	C₂⁴×C₁₉	C₂²×C₄	1	2	yes	—
60	C₂⁴×C₁₉	C₂²×C₄	1	1	yes	characteristic
61	C₂⁴×C₁₉	C₂²×C₄	1	4	yes	—
62	C₂²×C₄×C₁₉	—	2	4	no	—
63	C₂²×C₄×C₁₉	—	2	16	no	—
64	C₂²×C₄×C₁₉	—	2	8	no	—
65	C₂²×C₄×C₁₉	—	2	8	no	—
66	C₂²×C₄×C₁₉	—	2	16	no	—
67	C₂²×C₄×C₁₉	—	4	4	no	—
68	C₂²×C₄×C₁₉	—	4	4	no	—
69	C₂²×C₄×C₁₉	C₂²×C₄	1	4	yes	—
70	C₂²×C₄×C₁₉	C₂²×C₄	1	4	yes	—
71	C₂²×C₄×C₁₉	C₂²×C₄	1	4	yes	—
72	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₁₉	C₂²×C₄	1	8	yes	—
74	C₂²×C₄×C₁₉	C₂²×C₄	1	4	yes	—
75	C₂²×C₄×C₁₉	C₂²×C₄	1	8	yes	—
76	C₂²×C₄×C₁₉	C₂²×C₄	1	8	yes	—
77	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₁₉	#41	1	8	yes	—
80	C₂²×C₄×C₁₉	C₂⁴	1	4	yes	—
81	C₂²×C₄×C₁₉	C₂⁴	1	2	yes	—
82	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
83	M₁₆×C₁₉	—	2	32	no	—
84	C₂×C₈×C₁₉	—	2	16	no	—
85	C₂×C₈×C₁₉	C₂⁴	1	16	yes	—
86	C₄⋊C₄×C₁₉	C₂²×C₄	1	32	yes	—
87	C₂²⋊C₄×C₁₉	C₂²×C₄	1	32	yes	—
88	C₄²×C₁₉	C₂²×C₄	1	8	yes	—
89	C₄²×C₁₉	C₂²×C₄	1	8	yes	—
90	C₄²×C₁₉	C₂²×C₄	1	4	yes	—
91	C₄²×C₁₉	C₂²×C₄	1	4	yes	—
92	C₄²×C₁₉	C₂⁴	1	4	yes	—
93	C₄²×C₁₉	C₂⁴	1	4	yes	—
94	C₂×#2921	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
95	C₂³×C₁₉	—	2	4	no	—
96	C₂³×C₁₉	—	2	8	no	—
97	C₂³×C₁₉	—	2	4	no	—
98	C₂³×C₁₉	—	2	8	no	—
99	C₂³×C₁₉	—	4	8	no	—
100	C₂³×C₁₉	—	4	16	no	—
101	C₂³×C₁₉	#117	1	4	yes	—
102	C₂³×C₁₉	C₂³×C₄	1	4	yes	—
103	C₂³×C₁₉	C₂³×C₄	1	2	yes	—
104	C₂³×C₁₉	C₂³×C₄	1	1	yes	characteristic
105	C₂×C₄×C₁₉	—	2	8	no	—
106	C₂×C₄×C₁₉	—	2	4	no	—
107	C₂×C₄×C₁₉	—	2	8	no	—
108	C₂×C₄×C₁₉	—	2	8	no	—
109	C₂×C₄×C₁₉	—	2	16	no	—
110	C₂×C₄×C₁₉	—	2	8	no	—
111	C₂×C₄×C₁₉	—	2	8	no	—
112	C₂×C₄×C₁₉	—	4	8	no	—
113	C₂×C₄×C₁₉	—	4	8	no	—
114	C₂×C₄×C₁₉	#117	1	8	yes	—
115	C₂×C₄×C₁₉	#117	1	4	yes	—
116	C₂×C₄×C₁₉	C₂³×C₄	1	8	yes	—
117	C₂×C₄×C₁₉	C₂³×C₄	1	8	yes	—
118	C₂×C₄×C₁₉	C₂³×C₄	1	4	yes	—
119	C₂×C₄×C₁₉	C₂³×C₄	1	2	yes	—
120	C₂×C₄×C₁₉	C₂³×C₄	1	1	yes	characteristic
121	C₂×C₄×C₁₉	C₂×#41	1	4	yes	—
122	C₂×C₄×C₁₉	C₂×#41	1	8	yes	—
123	C₂×C₄×C₁₉	C₂⁵	1	1	yes	characteristic
124	C₈×C₁₉	—	2	16	no	—
125	C₂²×#117	C₂×C₁₉	1	1	yes	characteristic
126	#2921	C₂×C₁₉	1	16	yes	direct product factor
127	C₂×#433	C₂×C₁₉	1	8	yes	complemented
128	C₂×#423	C₂×C₁₉	1	2	yes	complemented
129	C₂×#407	C₂×C₁₉	1	2	yes	—
130	C₂×C₄×M₁₆	C₂×C₁₉	1	2	yes	complemented
131	C₂²×C₁₉	—	2	8	no	—
132	C₂²×C₁₉	—	2	4	no	—
133	C₂²×C₁₉	—	2	4	no	—
134	C₂²×C₁₉	—	2	8	no	—
135	C₂²×C₁₉	—	4	16	no	—
136	C₂²×C₁₉	—	4	8	no	—
137	C₂²×C₁₉	C₂×#117	1	4	yes	—
138	C₂²×C₁₉	C₂×#117	1	2	yes	—
139	C₂²×C₁₉	#568	1	4	yes	—
140	C₂²×C₁₉	C₂⁴×C₄	1	1	yes	characteristic
141	C₄×C₁₉	—	2	4	no	—
142	C₄×C₁₉	—	2	8	no	—
143	C₄×C₁₉	—	2	8	no	—
144	C₄×C₁₉	C₂×#117	1	4	yes	—
145	C₄×C₁₉	C₂×#117	1	2	yes	—
146	C₄×C₁₉	C₂²×#41	1	2	yes	—
147	C₂⁴×C₄	C₂²×C₁₉	1	1	yes	characteristic
148	C₂²×M₁₆	C₂²×C₁₉	1	4	yes	—
149	C₂×#117	C₄×C₁₉	1	4	yes	—
150	C₂×#117	C₄×C₁₉	1	8	yes	—
151	C₂×#117	C₂²×C₁₉	1	4	yes	—
152	C₂×#117	C₂²×C₁₉	1	8	yes	—
153	C₂²×C₄⋊C₄	C₄×C₁₉	1	2	yes	—
154	C₂²×C₂²⋊C₄	C₄×C₁₉	1	2	yes	—
155	C₂²×C₄²	C₂²×C₁₉	1	1	yes	characteristic
156	C₂²×C₄²	C₂²×C₁₉	1	1	yes	characteristic
157	#433	C₂²×C₁₉	1	64	yes	complemented
158	#423	C₂²×C₁₉	1	16	yes	complemented
159	C₂×C₄⋊C₈	C₂²×C₁₉	1	8	yes	complemented
160	#407	C₂²×C₁₉	1	16	yes	—
161	C₂×C₂²⋊C₈	C₂²×C₁₉	1	8	yes	—
162	C₄×M₁₆	C₂²×C₁₉	1	16	yes	complemented
163	C₂×#97	C₂²×C₁₉	1	4	yes	complemented
164	C₂×C₄×C₈	C₂²×C₁₉	1	4	yes	complemented
165	C₂×C₁₉	—	2	4	no	—
166	C₂×C₁₉	—	4	4	no	—
167	C₂×C₁₉	#2921	1	4	yes	direct product factor
168	C₂×C₁₉	C₂²×#117	1	1	yes	characteristic
169	C₂×C₁₉	C₂×#568	1	2	yes	—
170	C₂⁵	C₂×C₄×C₁₉	1	1	yes	characteristic
171	C₂³×C₄	—	2	8	no	—
172	C₂³×C₄	C₂×C₄×C₁₉	1	1	yes	characteristic
173	C₂³×C₄	C₂×C₄×C₁₉	1	2	yes	—
174	C₂³×C₄	C₂×C₄×C₁₉	1	4	yes	—
175	C₂³×C₄	C₂×C₄×C₁₉	1	2	yes	—
176	C₂³×C₄	C₂×C₄×C₁₉	1	2	yes	—
177	C₂³×C₄	C₂³×C₁₉	1	2	yes	—
178	C₂³×C₄	C₂³×C₁₉	1	4	yes	—
179	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
180	C₂×M₁₆	—	2	16	no	—
181	C₂×M₁₆	C₂³×C₁₉	1	16	yes	—
182	C₂²×C₈	C₂³×C₁₉	1	8	yes	—
183	#117	C₂×C₄×C₁₉	1	16	yes	—
184	#117	C₂×C₄×C₁₉	1	32	yes	—
185	#117	C₂³×C₁₉	1	16	yes	—
186	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	8	yes	—
187	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	16	yes	—
188	C₂×C₂²⋊C₄	C₂×C₄×C₁₉	1	8	yes	—
189	C₂×C₂²⋊C₄	C₂×C₄×C₁₉	1	16	yes	—
190	C₂×C₄²	C₂×C₄×C₁₉	1	2	yes	—
191	C₂×C₄²	C₂×C₄×C₁₉	1	2	yes	—
192	C₂×C₄²	C₂×C₄×C₁₉	1	4	yes	—
193	C₂×C₄²	C₂×C₄×C₁₉	1	4	yes	—
194	C₂×C₄²	C₂³×C₁₉	1	2	yes	—
195	C₂×C₄²	C₂³×C₁₉	1	4	yes	—
196	C₂×C₄²	C₂³×C₁₉	1	2	yes	—
197	C₂×C₄²	C₂³×C₁₉	1	4	yes	—
198	C₄⋊C₈	C₂³×C₁₉	1	32	yes	complemented
199	C₂²⋊C₈	C₂³×C₁₉	1	32	yes	—
200	#97	C₂³×C₁₉	1	16	yes	complemented
201	C₄×C₈	C₂³×C₁₉	1	16	yes	complemented
202	C₁₉	C₂×#2921	1	1	yes	characteristic, direct product factor, Sylow
203	C₂⁴	—	2	4	no	—
204	C₂⁴	—	4	4	no	—
205	C₂⁴	C₂²×C₄×C₁₉	1	2	yes	—
206	C₂⁴	C₂²×C₄×C₁₉	1	1	yes	characteristic
207	C₂⁴	C₂²×C₄×C₁₉	1	4	yes	—
208	C₂²×C₄	—	2	4	no	—
209	C₂²×C₄	—	2	16	no	—
210	C₂²×C₄	—	2	8	no	—
211	C₂²×C₄	—	2	8	no	—
212	C₂²×C₄	—	2	16	no	—
213	C₂²×C₄	—	4	4	no	—
214	C₂²×C₄	—	4	4	no	—
215	C₂²×C₄	C₂²×C₄×C₁₉	1	4	yes	—
216	C₂²×C₄	C₂²×C₄×C₁₉	1	4	yes	—
217	C₂²×C₄	C₂²×C₄×C₁₉	1	4	yes	—
218	C₂²×C₄	C₂²×C₄×C₁₉	1	2	yes	—
219	C₂²×C₄	C₂²×C₄×C₁₉	1	4	yes	—
220	C₂²×C₄	C₂²×C₄×C₁₉	1	8	yes	—
221	C₂²×C₄	C₂²×C₄×C₁₉	1	8	yes	—
222	C₂²×C₄	C₂²×C₄×C₁₉	1	8	yes	—
223	C₂²×C₄	C₂²×C₄×C₁₉	1	2	yes	—
224	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
225	C₂²×C₄	#41×C₁₉	1	8	yes	—
226	C₂²×C₄	C₂⁴×C₁₉	1	4	yes	—
227	C₂²×C₄	C₂⁴×C₁₉	1	2	yes	—
228	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
229	M₁₆	—	2	32	no	—
230	C₂×C₈	—	2	16	no	—
231	C₂×C₈	C₂⁴×C₁₉	1	16	yes	—
232	C₄⋊C₄	C₂²×C₄×C₁₉	1	32	yes	—
233	C₂²⋊C₄	C₂²×C₄×C₁₉	1	32	yes	—
234	C₄²	C₂²×C₄×C₁₉	1	4	yes	—
235	C₄²	C₂²×C₄×C₁₉	1	8	yes	—
236	C₄²	C₂²×C₄×C₁₉	1	4	yes	—
237	C₄²	C₂²×C₄×C₁₉	1	8	yes	—
238	C₄²	C₂⁴×C₁₉	1	4	yes	—
239	C₄²	C₂⁴×C₁₉	1	4	yes	—
240	C₂³	—	2	4	no	—
241	C₂³	—	2	8	no	—
242	C₂³	—	2	4	no	—
243	C₂³	—	2	8	no	—
244	C₂³	—	4	8	no	—
245	C₂³	—	4	16	no	—
246	C₂³	C₁₉×#117	1	4	yes	—
247	C₂³	C₂³×C₄×C₁₉	1	4	yes	—
248	C₂³	C₂³×C₄×C₁₉	1	2	yes	—
249	C₂³	C₂³×C₄×C₁₉	1	1	yes	characteristic
250	C₂×C₄	—	2	8	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	8	no	—
253	C₂×C₄	—	2	8	no	—
254	C₂×C₄	—	2	8	no	—
255	C₂×C₄	—	2	16	no	—
256	C₂×C₄	—	2	8	no	—
257	C₂×C₄	—	4	8	no	—
258	C₂×C₄	—	4	8	no	—
259	C₂×C₄	C₁₉×#117	1	8	yes	—
260	C₂×C₄	C₁₉×#117	1	4	yes	—
261	C₂×C₄	C₂³×C₄×C₁₉	1	8	yes	—
262	C₂×C₄	C₂³×C₄×C₁₉	1	8	yes	—
263	C₂×C₄	C₂³×C₄×C₁₉	1	4	yes	—
264	C₂×C₄	C₂³×C₄×C₁₉	1	2	yes	—
265	C₂×C₄	C₂³×C₄×C₁₉	1	1	yes	characteristic
266	C₂×C₄	C₂×#41×C₁₉	1	4	yes	—
267	C₂×C₄	C₂×#41×C₁₉	1	8	yes	—
268	C₂×C₄	C₂⁵×C₁₉	1	1	yes	characteristic
269	C₈	—	2	16	no	—
270	C₂²	—	2	8	no	—
271	C₂²	—	2	4	no	—
272	C₂²	—	2	8	no	—
273	C₂²	—	2	4	no	—
274	C₂²	—	4	16	no	—
275	C₂²	—	4	8	no	—
276	C₂²	C₂×C₁₉×#117	1	4	yes	—
277	C₂²	C₂×C₁₉×#117	1	2	yes	—
278	C₂²	C₁₉×#568	1	4	yes	—
279	C₂²	C₂⁴×C₄×C₁₉	1	1	yes	characteristic
280	C₄	—	2	8	no	—
281	C₄	—	2	8	no	—
282	C₄	—	2	4	no	—
283	C₄	C₂×C₁₉×#117	1	2	yes	—
284	C₄	C₂×C₁₉×#117	1	4	yes	—
285	C₄	C₂²×#41×C₁₉	1	2	yes	—
286	C₂	—	2	4	no	—
287	C₂	—	4	4	no	—
288	C₂	C₁₉×#2921	1	4	yes	direct product factor
289	C₂	C₂²×C₁₉×#117	1	1	yes	characteristic
290	C₂	C₂×C₁₉×#568	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₁₉×#2921	C₂	direct (N × H)
3. C₂×C₁₉×#433	C₂	semidirect
4. C₂×C₁₉×#423	C₂	semidirect
6. C₂×C₄×M₁₆×C₁₉	C₂	semidirect
17. C₁₉×#433	C₂²	semidirect
18. C₁₉×#423	C₂²	semidirect
19. C₂×C₁₉×C₄⋊C₈	C₂²	semidirect
22. C₄×M₁₆×C₁₉	C₂²	semidirect
23. C₂×C₁₉×#97	C₂²	semidirect
24. C₂×C₄×C₈×C₁₉	C₂²	semidirect
53. C₁₉×C₄⋊C₈	C₂³	semidirect
55. C₁₉×#97	C₂³	semidirect
56. C₄×C₈×C₁₉	C₂³	semidirect
94. C₂×#2921	C₁₉	direct (N × H)
126. #2921	C₂×C₁₉	direct (N × H)
127. C₂×#433	C₂×C₁₉	semidirect
128. C₂×#423	C₂×C₁₉	semidirect
130. C₂×C₄×M₁₆	C₂×C₁₉	semidirect
157. #433	C₂²×C₁₉	semidirect
158. #423	C₂²×C₁₉	semidirect
159. C₂×C₄⋊C₈	C₂²×C₁₉	semidirect
162. C₄×M₁₆	C₂²×C₁₉	semidirect
163. C₂×#97	C₂²×C₁₉	semidirect
164. C₂×C₄×C₈	C₂²×C₁₉	semidirect
167. C₂×C₁₉	#2921	direct (N × H)
198. C₄⋊C₈	C₂³×C₁₉	semidirect
200. #97	C₂³×C₁₉	semidirect
201. C₄×C₈	C₂³×C₁₉	semidirect
202. C₁₉	C₂×#2921	direct (N × H)
288. C₂	C₁₉×#2921	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
C₁₉×#2921	2
C₂×C₁₉×#433	2
C₂×C₁₉×#423	2
C₂×C₁₉×#407	2
C₂×C₄×M₁₆×C₁₉	2
C₂×#2921	19

Subgroups — C2×C19×#2921 #18066077

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₉×#2921 #18066077