Subgroups — C₁₉×#61884 #18067843

Order 4864.

1694 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#3502	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₉×#557	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#3395	C₂	1	8	yes	maximal, complemented
4	C₁₉×#3363	C₂	1	8	yes	maximal, complemented
5	C₁₉×#3209	C₂	1	6	yes	maximal
6	C₁₉×#3104	C₂	1	6	yes	maximal
7	C₁₉×#2982	C₂	1	1	yes	maximal, characteristic
8	Q₈²×C₁₉	C₂²	1	4	yes	—
9	C₁₉×#557	—	2	4	no	—
10	C₁₉×#557	—	2	4	no	—
11	C₁₉×#557	C₂²	1	8	yes	—
12	C₁₉×#557	C₂²	1	4	yes	—
13	C₁₉×#544	—	2	3	no	—
14	C₁₉×#533	C₂²	1	6	yes	—
15	C₂×C₁₉×C₄⋊Q₈	C₂²	1	3	yes	—
16	C₂×C₁₉×#125	—	2	3	no	—
17	C₁₉×#519	—	2	1	no	—
18	C₂×C₄×Q₈×C₁₉	—	2	1	no	—
19	C₂×C₄×Q₈×C₁₉	C₂²	1	3	yes	—
20	C₂×C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
21	C₁₉×#494	C₂²	1	12	yes	—
22	C₁₉×#487	C₂²	1	12	yes	—
23	C₁₉×#484	C₂²	1	4	yes	—
24	C₁₉×#477	C₂²	1	12	yes	—
25	C₁₉×#462	C₂²	1	24	yes	—
26	C₁₉×#462	C₂²	1	12	yes	—
27	C₁₉×#451	C₂²	1	12	yes	—
28	C₈×Q₈×C₁₉	C₂²	1	8	yes	—
29	C₄×Q₁₆×C₁₉	C₂²	1	24	yes	—
30	C₁₉×#431	C₂²	1	6	yes	—
31	C₂²×Q₈×C₁₉	D₄	1	1	yes	characteristic
32	C₂²×Q₈×C₁₉	C₂³	1	3	yes	—
33	C₂×Q₁₆×C₁₉	C₂³	1	24	yes	—
34	C₁₉×C₄⋊Q₈	—	2	12	no	—
35	C₁₉×C₄⋊Q₈	—	2	6	no	—
36	C₁₉×C₄⋊Q₈	C₂³	1	12	yes	—
37	C₁₉×C₄⋊Q₈	C₂³	1	6	yes	—
38	C₁₉×C₄⋊Q₈	C₂³	1	6	yes	—
39	C₁₉×#125	—	2	24	no	—
40	C₁₉×#125	—	2	12	no	—
41	C₁₉×#125	—	2	6	no	—
42	C₁₉×C₂²⋊Q₈	—	2	6	no	—
43	C₁₉×C₂²⋊Q₈	C₂³	1	4	yes	—
44	C₄×Q₈×C₁₉	—	2	8	no	—
45	C₄×Q₈×C₁₉	—	2	4	no	—
46	C₄×Q₈×C₁₉	—	2	4	no	—
47	C₄×Q₈×C₁₉	—	2	6	no	—
48	C₄×Q₈×C₁₉	—	2	6	no	—
49	C₄×Q₈×C₁₉	D₄	1	4	yes	—
50	C₄×Q₈×C₁₉	C₂³	1	12	yes	—
51	C₄×Q₈×C₁₉	C₂³	1	12	yes	—
52	C₄×Q₈×C₁₉	C₂³	1	4	yes	—
53	C₁₉×#117	—	2	6	no	—
54	C₂×C₄⋊C₄×C₁₉	—	2	3	no	—
55	C₂×C₄⋊C₄×C₁₉	—	2	3	no	—
56	C₂×C₄⋊C₄×C₁₉	—	2	3	no	—
57	C₂×C₄⋊C₄×C₁₉	D₄	1	3	yes	—
58	C₂×C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₁₉	—	2	3	no	—
60	C₂×C₄²×C₁₉	C₂³	1	3	yes	—
61	C₁₉×#107	C₂³	1	8	yes	—
62	C₁₉×C₄⋊C₈	C₂³	1	12	yes	—
63	C₁₉×#103	C₂³	1	8	yes	—
64	C₁₉×#103	C₂³	1	24	yes	—
65	C₁₉×C₂²⋊C₈	C₂³	1	4	yes	—
66	C₄×C₈×C₁₉	C₂³	1	12	yes	—
67	C₂×Q₈×C₁₉	—	2	3	no	—
68	C₂×Q₈×C₁₉	—	2	6	no	—
69	C₂×Q₈×C₁₉	—	4	6	no	—
70	C₂×Q₈×C₁₉	Q₁₆	1	8	yes	complemented
71	C₂×Q₈×C₁₉	C₂×D₄	1	4	yes	—
72	C₂×Q₈×C₁₉	C₂⁴	1	4	yes	—
73	C₂×Q₈×C₁₉	C₂⁴	1	6	yes	—
74	C₂²×C₄×C₁₉	—	2	1	no	—
75	C₂²×C₄×C₁₉	—	2	3	no	—
76	C₂²×C₄×C₁₉	C₂×D₄	1	3	yes	—
77	C₂²×C₄×C₁₉	C₂×D₄	1	3	yes	—
78	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
79	Q₁₆×C₁₉	—	4	24	no	—
80	C₂×C₈×C₁₉	C₂⁴	1	8	yes	—
81	C₄⋊C₄×C₁₉	—	2	12	no	—
82	C₄⋊C₄×C₁₉	—	2	12	no	—
83	C₄⋊C₄×C₁₉	—	2	12	no	—
84	C₄⋊C₄×C₁₉	—	2	1	no	—
85	C₄⋊C₄×C₁₉	—	2	6	no	—
86	C₄⋊C₄×C₁₉	—	2	6	no	—
87	C₄⋊C₄×C₁₉	—	2	4	no	—
88	C₄⋊C₄×C₁₉	—	2	6	no	—
89	C₄⋊C₄×C₁₉	—	2	12	no	—
90	C₄⋊C₄×C₁₉	C₂×D₄	1	12	yes	—
91	C₄⋊C₄×C₁₉	C₂⁴	1	2	yes	—
92	C₄⋊C₄×C₁₉	C₂⁴	1	4	yes	—
93	C₂²⋊C₄×C₁₉	—	2	4	no	—
94	C₄²×C₁₉	—	2	12	no	—
95	C₄²×C₁₉	—	2	12	no	—
96	C₄²×C₁₉	C₂×D₄	1	6	yes	—
97	C₄²×C₁₉	C₂⁴	1	6	yes	—
98	#61884	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
99	C₂³×C₁₉	C₂²×D₄	1	1	yes	characteristic
100	Q₈×C₁₉	—	2	8	no	—
101	Q₈×C₁₉	—	2	12	no	—
102	Q₈×C₁₉	—	4	6	no	—
103	Q₈×C₁₉	—	8	6	no	—
104	Q₈×C₁₉	C₂×Q₁₆	1	16	yes	—
105	C₂×C₄×C₁₉	—	2	2	no	—
106	C₂×C₄×C₁₉	—	2	2	no	—
107	C₂×C₄×C₁₉	—	2	4	no	—
108	C₂×C₄×C₁₉	—	2	4	no	—
109	C₂×C₄×C₁₉	—	2	6	no	—
110	C₂×C₄×C₁₉	—	4	3	no	—
111	C₂×C₄×C₁₉	—	4	2	no	—
112	C₂×C₄×C₁₉	—	4	6	no	—
113	C₂×C₄×C₁₉	C₂×Q₁₆	1	12	yes	—
114	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₁₉	C₂²×D₄	1	6	yes	—
116	C₂×C₄×C₁₉	C₂²×D₄	1	6	yes	—
117	C₂×C₄×C₁₉	C₂⁵	1	1	yes	characteristic
118	C₈×C₁₉	—	2	8	no	—
119	#3502	C₂×C₁₉	1	1	yes	characteristic
120	C₂×#557	C₂×C₁₉	1	1	yes	characteristic
121	#3395	C₂×C₁₉	1	8	yes	complemented
122	#3363	C₂×C₁₉	1	8	yes	complemented
123	#3209	C₂×C₁₉	1	6	yes	—
124	#3104	C₂×C₁₉	1	6	yes	—
125	#2982	C₂×C₁₉	1	1	yes	characteristic
126	C₂²×C₁₉	—	2	1	no	—
127	C₂²×C₁₉	—	2	1	no	—
128	C₂²×C₁₉	C₂²×Q₁₆	1	2	yes	—
129	C₂²×C₁₉	C₂³×D₄	1	1	yes	characteristic
130	C₄×C₁₉	—	2	6	no	—
131	C₄×C₁₉	—	2	1	no	—
132	C₄×C₁₉	—	4	2	no	—
133	C₄×C₁₉	—	4	4	no	—
134	C₄×C₁₉	—	4	6	no	—
135	C₄×C₁₉	—	4	4	no	—
136	C₄×C₁₉	C₂²×Q₁₆	1	12	yes	—
137	C₄×C₁₉	#585	1	2	yes	—
138	Q₈²	C₂²×C₁₉	1	4	yes	—
139	#557	—	2	4	no	—
140	#557	—	2	4	no	—
141	#557	C₂²×C₁₉	1	8	yes	—
142	#557	C₂²×C₁₉	1	4	yes	—
143	#544	—	2	3	no	—
144	#533	C₂²×C₁₉	1	6	yes	—
145	C₂×C₄⋊Q₈	C₂²×C₁₉	1	3	yes	—
146	C₂×#125	—	2	3	no	—
147	#519	—	2	1	no	—
148	C₂×C₄×Q₈	—	2	1	no	—
149	C₂×C₄×Q₈	C₂²×C₁₉	1	3	yes	—
150	C₂×C₄×Q₈	C₂²×C₁₉	1	1	yes	characteristic
151	#494	C₂²×C₁₉	1	12	yes	—
152	#487	C₂²×C₁₉	1	12	yes	—
153	#484	C₂²×C₁₉	1	4	yes	—
154	#477	C₂²×C₁₉	1	12	yes	—
155	#462	C₂²×C₁₉	1	12	yes	—
156	#462	C₂²×C₁₉	1	24	yes	—
157	#451	C₂²×C₁₉	1	12	yes	—
158	C₈×Q₈	C₂²×C₁₉	1	8	yes	—
159	C₄×Q₁₆	C₂²×C₁₉	1	24	yes	—
160	#431	C₂²×C₁₉	1	6	yes	—
161	C₂×C₁₉	—	2	2	no	—
162	C₂×C₁₉	#3487	1	1	yes	characteristic
163	C₂×C₁₉	C₂³×Q₁₆	1	1	yes	characteristic
164	C₂×C₁₉	#3586	1	1	yes	characteristic
165	C₂²×Q₈	D₄×C₁₉	1	1	yes	characteristic
166	C₂²×Q₈	C₂³×C₁₉	1	3	yes	—
167	C₂×Q₁₆	C₂³×C₁₉	1	24	yes	—
168	C₄⋊Q₈	—	2	12	no	—
169	C₄⋊Q₈	—	2	6	no	—
170	C₄⋊Q₈	C₂³×C₁₉	1	12	yes	—
171	C₄⋊Q₈	C₂³×C₁₉	1	6	yes	—
172	C₄⋊Q₈	C₂³×C₁₉	1	6	yes	—
173	#125	—	2	24	no	—
174	#125	—	2	12	no	—
175	#125	—	2	6	no	—
176	C₂²⋊Q₈	—	2	6	no	—
177	C₂²⋊Q₈	C₂³×C₁₉	1	4	yes	—
178	C₄×Q₈	—	2	8	no	—
179	C₄×Q₈	—	2	4	no	—
180	C₄×Q₈	—	2	4	no	—
181	C₄×Q₈	—	2	6	no	—
182	C₄×Q₈	—	2	6	no	—
183	C₄×Q₈	D₄×C₁₉	1	4	yes	—
184	C₄×Q₈	C₂³×C₁₉	1	12	yes	—
185	C₄×Q₈	C₂³×C₁₉	1	12	yes	—
186	C₄×Q₈	C₂³×C₁₉	1	4	yes	—
187	#117	—	2	6	no	—
188	C₂×C₄⋊C₄	—	2	3	no	—
189	C₂×C₄⋊C₄	—	2	3	no	—
190	C₂×C₄⋊C₄	—	2	3	no	—
191	C₂×C₄⋊C₄	D₄×C₁₉	1	3	yes	—
192	C₂×C₄⋊C₄	C₂³×C₁₉	1	1	yes	characteristic
193	C₂×C₄²	—	2	3	no	—
194	C₂×C₄²	C₂³×C₁₉	1	3	yes	—
195	#107	C₂³×C₁₉	1	8	yes	—
196	C₄⋊C₈	C₂³×C₁₉	1	12	yes	—
197	#103	C₂³×C₁₉	1	8	yes	—
198	#103	C₂³×C₁₉	1	24	yes	—
199	C₂²⋊C₈	C₂³×C₁₉	1	4	yes	—
200	C₄×C₈	C₂³×C₁₉	1	12	yes	—
201	C₁₉	#61884	1	1	yes	characteristic, direct product factor, Sylow
202	C₂×Q₈	—	2	3	no	—
203	C₂×Q₈	—	2	6	no	—
204	C₂×Q₈	—	4	6	no	—
205	C₂×Q₈	Q₁₆×C₁₉	1	8	yes	complemented
206	C₂×Q₈	C₂×D₄×C₁₉	1	4	yes	—
207	C₂×Q₈	C₂⁴×C₁₉	1	6	yes	—
208	C₂×Q₈	C₂⁴×C₁₉	1	4	yes	—
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	—	2	3	no	—
211	C₂²×C₄	C₂×D₄×C₁₉	1	3	yes	—
212	C₂²×C₄	C₂×D₄×C₁₉	1	3	yes	—
213	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
214	Q₁₆	—	4	24	no	—
215	C₂×C₈	C₂⁴×C₁₉	1	8	yes	—
216	C₄⋊C₄	—	2	12	no	—
217	C₄⋊C₄	—	2	12	no	—
218	C₄⋊C₄	—	2	12	no	—
219	C₄⋊C₄	—	2	1	no	—
220	C₄⋊C₄	—	2	6	no	—
221	C₄⋊C₄	—	2	6	no	—
222	C₄⋊C₄	—	2	4	no	—
223	C₄⋊C₄	—	2	6	no	—
224	C₄⋊C₄	—	2	12	no	—
225	C₄⋊C₄	C₂×D₄×C₁₉	1	12	yes	—
226	C₄⋊C₄	C₂⁴×C₁₉	1	4	yes	—
227	C₄⋊C₄	C₂⁴×C₁₉	1	2	yes	—
228	C₂²⋊C₄	—	2	4	no	—
229	C₄²	—	2	12	no	—
230	C₄²	—	2	12	no	—
231	C₄²	C₂×D₄×C₁₉	1	6	yes	—
232	C₄²	C₂⁴×C₁₉	1	6	yes	—
233	C₂³	C₂²×D₄×C₁₉	1	1	yes	characteristic
234	Q₈	—	2	8	no	—
235	Q₈	—	2	12	no	—
236	Q₈	—	4	6	no	—
237	Q₈	—	8	6	no	—
238	Q₈	C₂×Q₁₆×C₁₉	1	16	yes	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	6	no	—
244	C₂×C₄	—	4	3	no	—
245	C₂×C₄	—	4	6	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	C₂×Q₁₆×C₁₉	1	12	yes	—
248	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
249	C₂×C₄	C₂²×D₄×C₁₉	1	6	yes	—
250	C₂×C₄	C₂²×D₄×C₁₉	1	6	yes	—
251	C₂×C₄	C₂⁵×C₁₉	1	1	yes	characteristic
252	C₈	—	2	8	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	1	no	—
255	C₂²	C₂²×Q₁₆×C₁₉	1	2	yes	—
256	C₂²	C₂³×D₄×C₁₉	1	1	yes	characteristic
257	C₄	—	2	1	no	—
258	C₄	—	2	6	no	—
259	C₄	—	4	6	no	—
260	C₄	—	4	4	no	—
261	C₄	—	4	2	no	—
262	C₄	—	4	4	no	—
263	C₄	C₂²×Q₁₆×C₁₉	1	12	yes	—
264	C₄	C₁₉×#585	1	2	yes	—
265	C₂	—	2	2	no	—
266	C₂	C₁₉×#3487	1	1	yes	characteristic
267	C₂	C₂³×Q₁₆×C₁₉	1	1	yes	characteristic
268	C₂	C₁₉×#3586	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₁₉
prime radical	C₂³×C₁₉	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂×C₄×Q₈×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₉×#3395	C₂	semidirect
4. C₁₉×#3363	C₂	semidirect
70. C₂×Q₈×C₁₉	Q₁₆	semidirect
98. #61884	C₁₉	direct (N × H)
121. #3395	C₂×C₁₉	semidirect
122. #3363	C₂×C₁₉	semidirect
201. C₁₉	#61884	direct (N × H)
205. C₂×Q₈	Q₁₆×C₁₉	semidirect

Maximal subgroups

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

subgroup	index
C₁₉×#3502	2
C₂×C₁₉×#557	2
C₁₉×#3395	2
C₁₉×#3363	2
C₁₉×#3209	2
C₁₉×#3104	2
C₁₉×#2982	2
#61884	19

Subgroups — C19×#61884 #18067843

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#61884 #18067843