Subgroups — C₂×C₁₉×#2971 #18066162

Order 4864.

2230 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#41×C₁₉	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2971	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₈×Q₈×C₁₉	C₂	1	2	yes	maximal, complemented
4	C₂×C₁₉×#436	C₂	1	6	yes	maximal, complemented
5	C₂×C₁₉×#432	C₂	1	3	yes	maximal, complemented
6	C₂×C₄×M₁₆×C₁₉	C₂	1	3	yes	maximal, complemented
7	C₂²×#41×C₁₉	C₄	1	1	yes	characteristic
8	C₂²×M₁₆×C₁₉	C₂²	1	6	yes	—
9	C₄×#41×C₁₉	C₄	1	8	yes	—
10	C₄×#41×C₁₉	C₂²	1	8	yes	—
11	C₂×C₄×Q₈×C₁₉	C₄	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₄×C₁₉	C₄	1	3	yes	—
14	C₂×C₄×D₄×C₁₉	C₂²	1	3	yes	—
15	C₂×C₁₉×#117	C₄	1	3	yes	—
16	C₂²×C₄²×C₁₉	C₂²	1	3	yes	—
17	C₈×Q₈×C₁₉	C₂²	1	16	yes	complemented
18	C₁₉×#436	C₂²	1	48	yes	complemented
19	C₁₉×#432	C₂²	1	24	yes	complemented
20	C₂×C₁₉×C₄⋊C₈	C₂²	1	6	yes	—
21	C₂×C₁₉×C₂²⋊C₈	C₂²	1	6	yes	—
22	C₄×M₁₆×C₁₉	C₂²	1	24	yes	complemented
23	C₂×C₁₉×#97	C₂²	1	3	yes	—
24	C₂×C₁₉×#97	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₁₉	C₂²	1	6	yes	—
26	C₂×#41×C₁₉	—	2	8	no	—
27	C₂×#41×C₁₉	C₂×C₄	1	8	yes	—
28	C₂²×Q₈×C₁₉	C₂×C₄	1	1	yes	characteristic
29	C₂²×D₄×C₁₉	C₂×C₄	1	3	yes	—
30	C₂³×C₄×C₁₉	C₂×C₄	1	3	yes	—
31	C₂³×C₄×C₁₉	C₂×C₄	1	3	yes	—
32	C₂³×C₄×C₁₉	C₂³	1	3	yes	—
33	C₂×M₁₆×C₁₉	—	2	24	no	—
34	C₂×M₁₆×C₁₉	C₂³	1	24	yes	—
35	C₂²×C₈×C₁₉	C₂³	1	6	yes	—
36	C₂²×C₈×C₁₉	C₂³	1	2	yes	—
37	C₄×Q₈×C₁₉	C₂×C₄	1	8	yes	—
38	C₄×Q₈×C₁₉	C₂×C₄	1	4	yes	—
39	C₄×Q₈×C₁₉	C₂³	1	4	yes	—
40	C₄×D₄×C₁₉	C₂×C₄	1	24	yes	—
41	C₄×D₄×C₁₉	C₂×C₄	1	12	yes	—
42	C₄×D₄×C₁₉	C₂³	1	12	yes	—
43	C₁₉×#117	C₂×C₄	1	24	yes	—
44	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	3	yes	—
45	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	3	yes	—
46	C₂×C₂²⋊C₄×C₁₉	C₂×C₄	1	6	yes	—
47	C₂×C₄²×C₁₉	C₂×C₄	1	3	yes	—
48	C₂×C₄²×C₁₉	C₂×C₄	1	3	yes	—
49	C₂×C₄²×C₁₉	C₂×C₄	1	12	yes	—
50	C₂×C₄²×C₁₉	C₂³	1	12	yes	—
51	C₂×C₄²×C₁₉	C₂³	1	3	yes	—
52	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
53	C₁₉×C₄⋊C₈	C₂³	1	24	yes	—
54	C₁₉×C₂²⋊C₈	C₂³	1	24	yes	—
55	C₁₉×#97	C₂³	1	4	yes	—
56	C₁₉×#97	C₂³	1	12	yes	—
57	C₄×C₈×C₁₉	C₂³	1	24	yes	—
58	C₂⁴×C₁₉	C₂²×C₄	1	3	yes	—
59	#41×C₁₉	—	2	32	no	—
60	C₂×Q₈×C₁₉	M₁₆	1	8	yes	complemented
61	C₂×Q₈×C₁₉	C₂²×C₄	1	4	yes	—
62	C₂×D₄×C₁₉	—	2	12	no	—
63	C₂×D₄×C₁₉	C₂²×C₄	1	12	yes	—
64	C₂²×C₄×C₁₉	—	2	12	no	—
65	C₂²×C₄×C₁₉	—	2	12	no	—
66	C₂²×C₄×C₁₉	—	4	6	no	—
67	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₉	C₂²×C₄	1	3	yes	—
70	C₂²×C₄×C₁₉	C₂²×C₄	1	6	yes	—
71	C₂²×C₄×C₁₉	C₂²×C₄	1	3	yes	—
72	C₂²×C₄×C₁₉	C₂²×C₄	1	12	yes	—
73	C₂²×C₄×C₁₉	C₂²×C₄	1	12	yes	—
74	C₂²×C₄×C₁₉	C₂²×C₄	1	3	yes	—
75	C₂²×C₄×C₁₉	C₂⁴	1	6	yes	—
76	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
77	M₁₆×C₁₉	—	2	48	no	—
78	C₂×C₈×C₁₉	—	2	12	no	—
79	C₂×C₈×C₁₉	#41	1	8	yes	complemented
80	C₂×C₈×C₁₉	C₂⁴	1	12	yes	—
81	C₂×C₈×C₁₉	C₂⁴	1	4	yes	—
82	C₄⋊C₄×C₁₉	C₂²×C₄	1	12	yes	—
83	C₄⋊C₄×C₁₉	C₂²×C₄	1	12	yes	—
84	C₂²⋊C₄×C₁₉	C₂²×C₄	1	24	yes	—
85	C₄²×C₁₉	C₂²×C₄	1	2	yes	—
86	C₄²×C₁₉	C₂²×C₄	1	12	yes	—
87	C₄²×C₁₉	C₂²×C₄	1	12	yes	—
88	C₄²×C₁₉	C₂²×C₄	1	6	yes	—
89	C₄²×C₁₉	C₂⁴	1	2	yes	—
90	C₄²×C₁₉	C₂⁴	1	6	yes	—
91	C₂×#2971	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
92	C₂³×C₁₉	—	2	6	no	—
93	C₂³×C₁₉	—	2	6	no	—
94	C₂³×C₁₉	—	2	6	no	—
95	C₂³×C₁₉	C₂³×C₄	1	6	yes	—
96	C₂³×C₁₉	C₂³×C₄	1	1	yes	characteristic
97	Q₈×C₁₉	C₂×M₁₆	1	16	yes	complemented
98	D₄×C₁₉	—	2	24	no	—
99	C₂×C₄×C₁₉	—	2	6	no	—
100	C₂×C₄×C₁₉	—	2	2	no	—
101	C₂×C₄×C₁₉	—	2	2	no	—
102	C₂×C₄×C₁₉	—	2	6	no	—
103	C₂×C₄×C₁₉	—	2	12	no	—
104	C₂×C₄×C₁₉	—	2	24	no	—
105	C₂×C₄×C₁₉	—	2	12	no	—
106	C₂×C₄×C₁₉	—	4	12	no	—
107	C₂×C₄×C₁₉	C₂×M₁₆	1	12	yes	—
108	C₂×C₄×C₁₉	C₂³×C₄	1	2	yes	—
109	C₂×C₄×C₁₉	C₂³×C₄	1	2	yes	—
110	C₂×C₄×C₁₉	C₂³×C₄	1	6	yes	—
111	C₂×C₄×C₁₉	C₂³×C₄	1	6	yes	—
112	C₂×C₄×C₁₉	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₄×C₁₉	C₂³×C₄	1	6	yes	—
114	C₂×C₄×C₁₉	C₂×#41	1	4	yes	—
115	C₂×C₄×C₁₉	C₂⁵	1	1	yes	characteristic
116	C₈×C₁₉	—	2	12	no	—
117	C₈×C₁₉	C₂×#41	1	8	yes	complemented
118	C₂×C₄×#41	C₂×C₁₉	1	1	yes	characteristic
119	#2971	C₂×C₁₉	1	16	yes	direct product factor
120	C₂×C₈×Q₈	C₂×C₁₉	1	2	yes	complemented
121	C₂×#436	C₂×C₁₉	1	6	yes	complemented
122	C₂×#432	C₂×C₁₉	1	3	yes	complemented
123	C₂×C₄×M₁₆	C₂×C₁₉	1	3	yes	complemented
124	C₂²×C₁₉	—	2	6	no	—
125	C₂²×C₁₉	—	2	6	no	—
126	C₂²×C₁₉	—	2	6	no	—
127	C₂²×C₁₉	—	4	12	no	—
128	C₂²×C₁₉	C₄×#41	1	2	yes	—
129	C₂²×C₁₉	C₂²×M₁₆	1	2	yes	—
130	C₂²×C₁₉	#568	1	2	yes	—
131	C₂²×C₁₉	C₂⁴×C₄	1	1	yes	characteristic
132	C₄×C₁₉	—	2	6	no	—
133	C₄×C₁₉	—	2	2	no	—
134	C₄×C₁₉	—	2	2	no	—
135	C₄×C₁₉	—	2	6	no	—
136	C₄×C₁₉	C₄×#41	1	2	yes	—
137	C₄×C₁₉	C₂²×M₁₆	1	12	yes	—
138	C₄×C₁₉	C₂²×#41	1	2	yes	—
139	C₂²×#41	C₄×C₁₉	1	1	yes	characteristic
140	C₂²×M₁₆	C₂²×C₁₉	1	6	yes	—
141	C₄×#41	C₄×C₁₉	1	8	yes	—
142	C₄×#41	C₂²×C₁₉	1	8	yes	—
143	C₂×C₄×Q₈	C₄×C₁₉	1	1	yes	characteristic
144	C₂×C₄×Q₈	C₂²×C₁₉	1	1	yes	characteristic
145	C₂×C₄×D₄	C₄×C₁₉	1	3	yes	—
146	C₂×C₄×D₄	C₂²×C₁₉	1	3	yes	—
147	C₂×#117	C₄×C₁₉	1	3	yes	—
148	C₂²×C₄²	C₂²×C₁₉	1	3	yes	—
149	C₈×Q₈	C₂²×C₁₉	1	16	yes	complemented
150	#436	C₂²×C₁₉	1	48	yes	complemented
151	#432	C₂²×C₁₉	1	24	yes	complemented
152	C₂×C₄⋊C₈	C₂²×C₁₉	1	6	yes	—
153	C₂×C₂²⋊C₈	C₂²×C₁₉	1	6	yes	—
154	C₄×M₁₆	C₂²×C₁₉	1	24	yes	complemented
155	C₂×#97	C₂²×C₁₉	1	1	yes	characteristic
156	C₂×#97	C₂²×C₁₉	1	3	yes	—
157	C₂×C₄×C₈	C₂²×C₁₉	1	6	yes	—
158	C₂×C₁₉	—	4	6	no	—
159	C₂×C₁₉	#2971	1	4	yes	direct product factor
160	C₂×C₁₉	C₂×C₄×#41	1	1	yes	characteristic
161	C₂×C₁₉	C₂³×M₁₆	1	1	yes	characteristic
162	C₂×C₁₉	C₂×#568	1	1	yes	characteristic
163	C₂×#41	—	2	8	no	—
164	C₂×#41	C₂×C₄×C₁₉	1	8	yes	—
165	C₂²×Q₈	C₂×C₄×C₁₉	1	1	yes	characteristic
166	C₂²×D₄	C₂×C₄×C₁₉	1	3	yes	—
167	C₂³×C₄	C₂×C₄×C₁₉	1	3	yes	—
168	C₂³×C₄	C₂×C₄×C₁₉	1	3	yes	—
169	C₂³×C₄	C₂³×C₁₉	1	3	yes	—
170	C₂×M₁₆	—	2	24	no	—
171	C₂×M₁₆	C₂³×C₁₉	1	24	yes	—
172	C₂²×C₈	C₂³×C₁₉	1	6	yes	—
173	C₂²×C₈	C₂³×C₁₉	1	2	yes	—
174	C₄×Q₈	C₂×C₄×C₁₉	1	8	yes	—
175	C₄×Q₈	C₂×C₄×C₁₉	1	4	yes	—
176	C₄×Q₈	C₂³×C₁₉	1	4	yes	—
177	C₄×D₄	C₂×C₄×C₁₉	1	24	yes	—
178	C₄×D₄	C₂×C₄×C₁₉	1	12	yes	—
179	C₄×D₄	C₂³×C₁₉	1	12	yes	—
180	#117	C₂×C₄×C₁₉	1	24	yes	—
181	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	3	yes	—
182	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	3	yes	—
183	C₂×C₂²⋊C₄	C₂×C₄×C₁₉	1	6	yes	—
184	C₂×C₄²	C₂×C₄×C₁₉	1	3	yes	—
185	C₂×C₄²	C₂×C₄×C₁₉	1	3	yes	—
186	C₂×C₄²	C₂×C₄×C₁₉	1	12	yes	—
187	C₂×C₄²	C₂³×C₁₉	1	3	yes	—
188	C₂×C₄²	C₂³×C₁₉	1	12	yes	—
189	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
190	C₄⋊C₈	C₂³×C₁₉	1	24	yes	—
191	C₂²⋊C₈	C₂³×C₁₉	1	24	yes	—
192	#97	C₂³×C₁₉	1	4	yes	—
193	#97	C₂³×C₁₉	1	12	yes	—
194	C₄×C₈	C₂³×C₁₉	1	24	yes	—
195	C₁₉	C₂×#2971	1	1	yes	characteristic, direct product factor, Sylow
196	C₂⁴	C₂²×C₄×C₁₉	1	3	yes	—
197	#41	—	2	32	no	—
198	C₂×Q₈	M₁₆×C₁₉	1	8	yes	complemented
199	C₂×Q₈	C₂²×C₄×C₁₉	1	4	yes	—
200	C₂×D₄	—	2	12	no	—
201	C₂×D₄	C₂²×C₄×C₁₉	1	12	yes	—
202	C₂²×C₄	—	2	12	no	—
203	C₂²×C₄	—	2	12	no	—
204	C₂²×C₄	—	4	6	no	—
205	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
206	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
207	C₂²×C₄	C₂²×C₄×C₁₉	1	3	yes	—
208	C₂²×C₄	C₂²×C₄×C₁₉	1	6	yes	—
209	C₂²×C₄	C₂²×C₄×C₁₉	1	3	yes	—
210	C₂²×C₄	C₂²×C₄×C₁₉	1	12	yes	—
211	C₂²×C₄	C₂²×C₄×C₁₉	1	12	yes	—
212	C₂²×C₄	C₂²×C₄×C₁₉	1	3	yes	—
213	C₂²×C₄	C₂⁴×C₁₉	1	6	yes	—
214	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
215	M₁₆	—	2	48	no	—
216	C₂×C₈	—	2	12	no	—
217	C₂×C₈	#41×C₁₉	1	8	yes	complemented
218	C₂×C₈	C₂⁴×C₁₉	1	12	yes	—
219	C₂×C₈	C₂⁴×C₁₉	1	4	yes	—
220	C₄⋊C₄	C₂²×C₄×C₁₉	1	12	yes	—
221	C₄⋊C₄	C₂²×C₄×C₁₉	1	12	yes	—
222	C₂²⋊C₄	C₂²×C₄×C₁₉	1	24	yes	—
223	C₄²	C₂²×C₄×C₁₉	1	2	yes	—
224	C₄²	C₂²×C₄×C₁₉	1	12	yes	—
225	C₄²	C₂²×C₄×C₁₉	1	12	yes	—
226	C₄²	C₂²×C₄×C₁₉	1	6	yes	—
227	C₄²	C₂⁴×C₁₉	1	6	yes	—
228	C₄²	C₂⁴×C₁₉	1	2	yes	—
229	C₂³	—	2	6	no	—
230	C₂³	—	2	6	no	—
231	C₂³	—	2	6	no	—
232	C₂³	C₂³×C₄×C₁₉	1	6	yes	—
233	C₂³	C₂³×C₄×C₁₉	1	1	yes	characteristic
234	Q₈	C₂×M₁₆×C₁₉	1	16	yes	complemented
235	D₄	—	2	24	no	—
236	C₂×C₄	—	2	6	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	6	no	—
240	C₂×C₄	—	2	12	no	—
241	C₂×C₄	—	2	24	no	—
242	C₂×C₄	—	2	12	no	—
243	C₂×C₄	—	4	12	no	—
244	C₂×C₄	C₂×M₁₆×C₁₉	1	12	yes	—
245	C₂×C₄	C₂³×C₄×C₁₉	1	2	yes	—
246	C₂×C₄	C₂³×C₄×C₁₉	1	2	yes	—
247	C₂×C₄	C₂³×C₄×C₁₉	1	6	yes	—
248	C₂×C₄	C₂³×C₄×C₁₉	1	6	yes	—
249	C₂×C₄	C₂³×C₄×C₁₉	1	1	yes	characteristic
250	C₂×C₄	C₂³×C₄×C₁₉	1	6	yes	—
251	C₂×C₄	C₂×#41×C₁₉	1	4	yes	—
252	C₂×C₄	C₂⁵×C₁₉	1	1	yes	characteristic
253	C₈	—	2	12	no	—
254	C₈	C₂×#41×C₁₉	1	8	yes	complemented
255	C₂²	—	2	6	no	—
256	C₂²	—	2	6	no	—
257	C₂²	—	2	6	no	—
258	C₂²	—	4	12	no	—
259	C₂²	C₄×#41×C₁₉	1	2	yes	—
260	C₂²	C₂²×M₁₆×C₁₉	1	2	yes	—
261	C₂²	C₁₉×#568	1	2	yes	—
262	C₂²	C₂⁴×C₄×C₁₉	1	1	yes	characteristic
263	C₄	—	2	6	no	—
264	C₄	—	2	2	no	—
265	C₄	—	2	2	no	—
266	C₄	—	2	6	no	—
267	C₄	C₄×#41×C₁₉	1	2	yes	—
268	C₄	C₂²×M₁₆×C₁₉	1	12	yes	—
269	C₄	C₂²×#41×C₁₉	1	2	yes	—
270	C₂	—	4	6	no	—
271	C₂	C₁₉×#2971	1	4	yes	direct product factor
272	C₂	C₂×C₄×#41×C₁₉	1	1	yes	characteristic
273	C₂	C₂³×M₁₆×C₁₉	1	1	yes	characteristic
274	C₂	C₂×C₁₉×#568	1	1	yes	characteristic

	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	C₂²×#41×C₁₉	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×#2971	C₂	direct (N × H)
3. C₂×C₈×Q₈×C₁₉	C₂	semidirect
4. C₂×C₁₉×#436	C₂	semidirect
5. C₂×C₁₉×#432	C₂	semidirect
6. C₂×C₄×M₁₆×C₁₉	C₂	semidirect
17. C₈×Q₈×C₁₉	C₂²	semidirect
18. C₁₉×#436	C₂²	semidirect
19. C₁₉×#432	C₂²	semidirect
22. C₄×M₁₆×C₁₉	C₂²	semidirect
60. C₂×Q₈×C₁₉	M₁₆	semidirect
79. C₂×C₈×C₁₉	#41	semidirect
91. C₂×#2971	C₁₉	direct (N × H)
97. Q₈×C₁₉	C₂×M₁₆	semidirect
117. C₈×C₁₉	C₂×#41	semidirect
119. #2971	C₂×C₁₉	direct (N × H)
120. C₂×C₈×Q₈	C₂×C₁₉	semidirect
121. C₂×#436	C₂×C₁₉	semidirect
122. C₂×#432	C₂×C₁₉	semidirect
123. C₂×C₄×M₁₆	C₂×C₁₉	semidirect
149. C₈×Q₈	C₂²×C₁₉	semidirect
150. #436	C₂²×C₁₉	semidirect
151. #432	C₂²×C₁₉	semidirect
154. C₄×M₁₆	C₂²×C₁₉	semidirect
159. C₂×C₁₉	#2971	direct (N × H)
195. C₁₉	C₂×#2971	direct (N × H)
198. C₂×Q₈	M₁₆×C₁₉	semidirect
217. C₂×C₈	#41×C₁₉	semidirect
234. Q₈	C₂×M₁₆×C₁₉	semidirect
254. C₈	C₂×#41×C₁₉	semidirect
271. C₂	C₁₉×#2971	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×#41×C₁₉	2
C₁₉×#2971	2
C₂×C₈×Q₈×C₁₉	2
C₂×C₁₉×#436	2
C₂×C₁₉×#432	2
C₂×C₄×M₁₆×C₁₉	2
C₂×#2971	19

Subgroups — C2×C19×#2971 #18066162

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₉×#2971 #18066162