Subgroups — C₁₉×#60666 #18066625

Order 4864.

1762 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#3504	C₂	1	2	yes	maximal
2	C₁₉×#3166	C₂	1	6	yes	maximal, complemented
3	C₁₉×#3164	C₂	1	6	yes	maximal, complemented
4	C₁₉×#3163	C₂	1	6	yes	maximal, complemented
5	C₁₉×#3160	C₂	1	3	yes	maximal, complemented
6	C₂×C₁₉×#499	C₂	1	3	yes	maximal, complemented
7	C₁₉×#2913	C₂	1	2	yes	maximal
8	C₁₉×#2912	C₂	1	2	yes	maximal
9	C₄×C₁₉×#131	C₂	1	1	yes	maximal, characteristic
10	C₂×C₁₉×#131	C₂²	1	1	yes	characteristic
11	C₂×C₁₉×#131	C₂²	1	1	yes	characteristic
12	C₁₉×#557	—	2	4	no	—
13	C₁₉×#554	—	2	12	no	—
14	C₁₉×#544	C₂²	1	12	yes	—
15	C₁₉×#533	C₂²	1	6	yes	—
16	C₂×C₁₉×#125	C₂²	1	6	yes	—
17	C₁₉×#520	C₂²	1	4	yes	—
18	C₄×#41×C₁₉	C₂²	1	1	yes	characteristic
19	C₁₉×#501	C₂²	1	24	yes	—
20	C₁₉×#500	C₂²	1	6	yes	—
21	C₁₉×#499	C₂²	1	3	yes	—
22	C₁₉×#499	C₂²	1	6	yes	—
23	C₁₉×#499	C₂²	1	6	yes	—
24	C₁₉×#499	C₂²	1	6	yes	—
25	C₁₉×#499	C₂²	1	1	yes	characteristic
26	C₁₉×#499	C₂²	1	6	yes	—
27	C₁₉×#498	C₂²	1	6	yes	—
28	C₁₉×#428	C₂²	1	12	yes	—
29	C₁₉×#428	C₂²	1	12	yes	—
30	C₁₉×#427	C₂²	1	6	yes	—
31	C₁₉×#427	C₂²	1	6	yes	—
32	C₂×C₁₉×#107	C₂²	1	6	yes	—
33	C₂×C₁₉×#106	C₂²	1	6	yes	—
34	C₁₉×#405	C₂²	1	3	yes	—
35	C₁₉×#405	C₂²	1	3	yes	—
36	C₄×M₁₆×C₁₉	C₂²	1	3	yes	—
37	C₂×C₄×C₈×C₁₉	C₂²	1	3	yes	—
38	C₂×#41×C₁₉	C₂³	1	1	yes	characteristic
39	C₁₉×#131	Q₈	1	8	yes	complemented
40	C₁₉×#131	Q₈	1	8	yes	complemented
41	C₂×M₁₆×C₁₉	C₂³	1	3	yes	—
42	C₂×M₁₆×C₁₉	C₂³	1	3	yes	—
43	C₂²×C₈×C₁₉	C₂³	1	3	yes	—
44	C₂²×C₈×C₁₉	C₂³	1	3	yes	—
45	C₁₉×C₄⋊Q₈	—	2	12	no	—
46	C₁₉×C₄⋊Q₈	C₂³	1	12	yes	—
47	C₁₉×#125	—	2	12	no	—
48	C₁₉×#125	—	2	12	no	—
49	C₁₉×#125	C₂³	1	6	yes	—
50	C₁₉×#125	C₂³	1	2	yes	—
51	C₁₉×#125	C₂³	1	12	yes	—
52	C₁₉×C₂²⋊Q₈	—	2	12	no	—
53	C₁₉×C₂²⋊Q₈	—	2	24	no	—
54	C₄×Q₈×C₁₉	—	2	4	no	—
55	C₄×Q₈×C₁₉	—	2	12	no	—
56	C₄×Q₈×C₁₉	D₄	1	2	yes	—
57	C₄×D₄×C₁₉	—	2	12	no	—
58	C₄×D₄×C₁₉	D₄	1	6	yes	—
59	C₁₉×#117	C₂³	1	12	yes	—
60	C₁₉×#117	C₂³	1	3	yes	—
61	C₂×C₄⋊C₄×C₁₉	—	2	12	no	—
62	C₂×C₄⋊C₄×C₁₉	C₂³	1	12	yes	—
63	C₂×C₄²×C₁₉	C₂³	1	3	yes	—
64	C₁₉×#107	C₂³	1	6	yes	—
65	C₁₉×#107	C₂³	1	12	yes	—
66	C₁₉×#107	C₂³	1	2	yes	—
67	C₁₉×#107	C₂³	1	12	yes	—
68	C₁₉×#106	C₂³	1	6	yes	—
69	C₁₉×#106	C₂³	1	12	yes	—
70	C₁₉×#106	C₂³	1	2	yes	—
71	C₁₉×#106	C₂³	1	12	yes	—
72	C₁₉×#97	C₂³	1	6	yes	—
73	C₄×C₈×C₁₉	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₁₉	C₂³	1	3	yes	—
75	C₄×C₈×C₁₉	C₂³	1	3	yes	—
76	C₄×C₈×C₁₉	C₂³	1	3	yes	—
77	#41×C₁₉	C₂×Q₈	1	4	yes	—
78	#41×C₁₉	C₂×Q₈	1	4	yes	—
79	C₂×Q₈×C₁₉	—	2	12	no	—
80	C₂×Q₈×C₁₉	C₂×D₄	1	1	yes	characteristic
81	C₂×D₄×C₁₉	C₂×D₄	1	3	yes	—
82	C₂²×C₄×C₁₉	—	2	12	no	—
83	C₂²×C₄×C₁₉	C₂×D₄	1	6	yes	—
84	C₂²×C₄×C₁₉	C₂⁴	1	3	yes	—
85	M₁₆×C₁₉	C₂×Q₈	1	12	yes	—
86	M₁₆×C₁₉	C₂×Q₈	1	12	yes	—
87	C₂×C₈×C₁₉	C₂×Q₈	1	12	yes	—
88	C₂×C₈×C₁₉	C₂×Q₈	1	12	yes	—
89	C₂×C₈×C₁₉	C₂⁴	1	1	yes	characteristic
90	C₂×C₈×C₁₉	C₂⁴	1	1	yes	characteristic
91	C₂×C₈×C₁₉	C₂⁴	1	3	yes	—
92	C₂×C₈×C₁₉	C₂⁴	1	3	yes	—
93	C₄⋊C₄×C₁₉	—	2	12	no	—
94	C₄⋊C₄×C₁₉	—	2	24	no	—
95	C₄⋊C₄×C₁₉	—	2	12	no	—
96	C₄⋊C₄×C₁₉	—	2	4	no	—
97	C₄⋊C₄×C₁₉	—	2	12	no	—
98	C₄⋊C₄×C₁₉	C₂×D₄	1	6	yes	—
99	C₄⋊C₄×C₁₉	C₂⁴	1	12	yes	—
100	C₄⋊C₄×C₁₉	C₂⁴	1	4	yes	—
101	C₂²⋊C₄×C₁₉	—	2	12	no	—
102	C₂²⋊C₄×C₁₉	C₂×D₄	1	6	yes	—
103	C₄²×C₁₉	—	2	12	no	—
104	C₄²×C₁₉	C₂×D₄	1	6	yes	—
105	C₄²×C₁₉	C₂⁴	1	1	yes	characteristic
106	C₄²×C₁₉	C₂⁴	1	3	yes	—
107	#60666	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
108	C₂³×C₁₉	C₂²×D₄	1	3	yes	—
109	Q₈×C₁₉	—	8	12	no	—
110	Q₈×C₁₉	C₄⋊Q₈	1	4	yes	—
111	D₄×C₁₉	C₄⋊Q₈	1	12	yes	—
112	C₂×C₄×C₁₉	—	2	4	no	—
113	C₂×C₄×C₁₉	—	2	12	no	—
114	C₂×C₄×C₁₉	—	4	6	no	—
115	C₂×C₄×C₁₉	—	8	12	no	—
116	C₂×C₄×C₁₉	C₂²×D₄	1	2	yes	—
117	C₂×C₄×C₁₉	C₂²×D₄	1	6	yes	—
118	C₂×C₄×C₁₉	C₂²×D₄	1	3	yes	—
119	C₂×C₄×C₁₉	C₂²×Q₈	1	6	yes	—
120	C₂×C₄×C₁₉	C₂²×Q₈	1	6	yes	—
121	C₂×C₄×C₁₉	C₂⁵	1	1	yes	characteristic
122	C₈×C₁₉	C₂²×Q₈	1	2	yes	—
123	C₈×C₁₉	C₂²×Q₈	1	2	yes	—
124	C₈×C₁₉	C₂²×Q₈	1	6	yes	—
125	C₈×C₁₉	C₂²×Q₈	1	6	yes	—
126	#3504	C₂×C₁₉	1	2	yes	—
127	#3166	C₂×C₁₉	1	6	yes	complemented
128	#3164	C₂×C₁₉	1	6	yes	complemented
129	#3163	C₂×C₁₉	1	6	yes	complemented
130	#3160	C₂×C₁₉	1	3	yes	complemented
131	C₂×#499	C₂×C₁₉	1	3	yes	complemented
132	#2913	C₂×C₁₉	1	2	yes	—
133	#2912	C₂×C₁₉	1	2	yes	—
134	C₄×#131	C₂×C₁₉	1	1	yes	characteristic
135	C₂²×C₁₉	—	2	6	no	—
136	C₂²×C₁₉	C₂×C₄⋊Q₈	1	6	yes	—
137	C₂²×C₁₉	C₂³×D₄	1	1	yes	characteristic
138	C₄×C₁₉	—	2	2	no	—
139	C₄×C₁₉	—	2	6	no	—
140	C₄×C₁₉	—	4	12	no	—
141	C₄×C₁₉	—	4	4	no	—
142	C₄×C₁₉	C₂×C₄⋊Q₈	1	6	yes	—
143	C₄×C₁₉	C₂³×Q₈	1	1	yes	characteristic
144	C₄×C₁₉	C₂³×Q₈	1	1	yes	characteristic
145	C₂×#131	C₂²×C₁₉	1	1	yes	characteristic
146	C₂×#131	C₂²×C₁₉	1	1	yes	characteristic
147	#557	—	2	4	no	—
148	#554	—	2	12	no	—
149	#544	C₂²×C₁₉	1	12	yes	—
150	#533	C₂²×C₁₉	1	6	yes	—
151	C₂×#125	C₂²×C₁₉	1	6	yes	—
152	#520	C₂²×C₁₉	1	4	yes	—
153	C₄×#41	C₂²×C₁₉	1	1	yes	characteristic
154	#501	C₂²×C₁₉	1	24	yes	—
155	#500	C₂²×C₁₉	1	6	yes	—
156	#499	C₂²×C₁₉	1	3	yes	—
157	#499	C₂²×C₁₉	1	6	yes	—
158	#499	C₂²×C₁₉	1	6	yes	—
159	#499	C₂²×C₁₉	1	6	yes	—
160	#499	C₂²×C₁₉	1	1	yes	characteristic
161	#499	C₂²×C₁₉	1	6	yes	—
162	#498	C₂²×C₁₉	1	6	yes	—
163	#428	C₂²×C₁₉	1	12	yes	—
164	#428	C₂²×C₁₉	1	12	yes	—
165	#427	C₂²×C₁₉	1	6	yes	—
166	#427	C₂²×C₁₉	1	6	yes	—
167	C₂×#107	C₂²×C₁₉	1	6	yes	—
168	C₂×#106	C₂²×C₁₉	1	6	yes	—
169	#405	C₂²×C₁₉	1	3	yes	—
170	#405	C₂²×C₁₉	1	3	yes	—
171	C₄×M₁₆	C₂²×C₁₉	1	3	yes	—
172	C₂×C₄×C₈	C₂²×C₁₉	1	3	yes	—
173	C₂×C₁₉	—	2	6	no	—
174	C₂×C₁₉	C₂²×C₄⋊Q₈	1	1	yes	characteristic
175	C₂×C₁₉	#3584	1	2	yes	—
176	C₂×#41	C₂³×C₁₉	1	1	yes	characteristic
177	#131	Q₈×C₁₉	1	8	yes	complemented
178	#131	Q₈×C₁₉	1	8	yes	complemented
179	C₂×M₁₆	C₂³×C₁₉	1	3	yes	—
180	C₂×M₁₆	C₂³×C₁₉	1	3	yes	—
181	C₂²×C₈	C₂³×C₁₉	1	3	yes	—
182	C₂²×C₈	C₂³×C₁₉	1	3	yes	—
183	C₄⋊Q₈	—	2	12	no	—
184	C₄⋊Q₈	C₂³×C₁₉	1	12	yes	—
185	#125	—	2	12	no	—
186	#125	—	2	12	no	—
187	#125	C₂³×C₁₉	1	6	yes	—
188	#125	C₂³×C₁₉	1	12	yes	—
189	#125	C₂³×C₁₉	1	2	yes	—
190	C₂²⋊Q₈	—	2	12	no	—
191	C₂²⋊Q₈	—	2	24	no	—
192	C₄×Q₈	—	2	4	no	—
193	C₄×Q₈	—	2	12	no	—
194	C₄×Q₈	D₄×C₁₉	1	2	yes	—
195	C₄×D₄	—	2	12	no	—
196	C₄×D₄	D₄×C₁₉	1	6	yes	—
197	#117	C₂³×C₁₉	1	12	yes	—
198	#117	C₂³×C₁₉	1	3	yes	—
199	C₂×C₄⋊C₄	—	2	12	no	—
200	C₂×C₄⋊C₄	C₂³×C₁₉	1	12	yes	—
201	C₂×C₄²	C₂³×C₁₉	1	3	yes	—
202	#107	C₂³×C₁₉	1	6	yes	—
203	#107	C₂³×C₁₉	1	12	yes	—
204	#107	C₂³×C₁₉	1	2	yes	—
205	#107	C₂³×C₁₉	1	12	yes	—
206	#106	C₂³×C₁₉	1	6	yes	—
207	#106	C₂³×C₁₉	1	12	yes	—
208	#106	C₂³×C₁₉	1	2	yes	—
209	#106	C₂³×C₁₉	1	12	yes	—
210	#97	C₂³×C₁₉	1	6	yes	—
211	C₄×C₈	C₂³×C₁₉	1	1	yes	characteristic
212	C₄×C₈	C₂³×C₁₉	1	3	yes	—
213	C₄×C₈	C₂³×C₁₉	1	3	yes	—
214	C₄×C₈	C₂³×C₁₉	1	3	yes	—
215	C₁₉	#60666	1	1	yes	characteristic, direct product factor, Sylow
216	#41	C₂×Q₈×C₁₉	1	4	yes	—
217	#41	C₂×Q₈×C₁₉	1	4	yes	—
218	C₂×Q₈	—	2	12	no	—
219	C₂×Q₈	C₂×D₄×C₁₉	1	1	yes	characteristic
220	C₂×D₄	C₂×D₄×C₁₉	1	3	yes	—
221	C₂²×C₄	—	2	12	no	—
222	C₂²×C₄	C₂×D₄×C₁₉	1	6	yes	—
223	C₂²×C₄	C₂⁴×C₁₉	1	3	yes	—
224	M₁₆	C₂×Q₈×C₁₉	1	12	yes	—
225	M₁₆	C₂×Q₈×C₁₉	1	12	yes	—
226	C₂×C₈	C₂×Q₈×C₁₉	1	12	yes	—
227	C₂×C₈	C₂×Q₈×C₁₉	1	12	yes	—
228	C₂×C₈	C₂⁴×C₁₉	1	1	yes	characteristic
229	C₂×C₈	C₂⁴×C₁₉	1	1	yes	characteristic
230	C₂×C₈	C₂⁴×C₁₉	1	3	yes	—
231	C₂×C₈	C₂⁴×C₁₉	1	3	yes	—
232	C₄⋊C₄	—	2	12	no	—
233	C₄⋊C₄	—	2	24	no	—
234	C₄⋊C₄	—	2	12	no	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	2	12	no	—
237	C₄⋊C₄	C₂×D₄×C₁₉	1	6	yes	—
238	C₄⋊C₄	C₂⁴×C₁₉	1	12	yes	—
239	C₄⋊C₄	C₂⁴×C₁₉	1	4	yes	—
240	C₂²⋊C₄	—	2	12	no	—
241	C₂²⋊C₄	C₂×D₄×C₁₉	1	6	yes	—
242	C₄²	—	2	12	no	—
243	C₄²	C₂×D₄×C₁₉	1	6	yes	—
244	C₄²	C₂⁴×C₁₉	1	1	yes	characteristic
245	C₄²	C₂⁴×C₁₉	1	3	yes	—
246	C₂³	C₂²×D₄×C₁₉	1	3	yes	—
247	Q₈	—	8	12	no	—
248	Q₈	C₁₉×C₄⋊Q₈	1	4	yes	—
249	D₄	C₁₉×C₄⋊Q₈	1	12	yes	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	12	no	—
252	C₂×C₄	—	4	6	no	—
253	C₂×C₄	—	8	12	no	—
254	C₂×C₄	C₂²×D₄×C₁₉	1	2	yes	—
255	C₂×C₄	C₂²×D₄×C₁₉	1	6	yes	—
256	C₂×C₄	C₂²×D₄×C₁₉	1	3	yes	—
257	C₂×C₄	C₂²×Q₈×C₁₉	1	6	yes	—
258	C₂×C₄	C₂²×Q₈×C₁₉	1	6	yes	—
259	C₂×C₄	C₂⁵×C₁₉	1	1	yes	characteristic
260	C₈	C₂²×Q₈×C₁₉	1	2	yes	—
261	C₈	C₂²×Q₈×C₁₉	1	2	yes	—
262	C₈	C₂²×Q₈×C₁₉	1	6	yes	—
263	C₈	C₂²×Q₈×C₁₉	1	6	yes	—
264	C₂²	—	2	6	no	—
265	C₂²	C₂×C₁₉×C₄⋊Q₈	1	6	yes	—
266	C₂²	C₂³×D₄×C₁₉	1	1	yes	characteristic
267	C₄	—	2	2	no	—
268	C₄	—	2	6	no	—
269	C₄	—	4	12	no	—
270	C₄	—	4	4	no	—
271	C₄	C₂×C₁₉×C₄⋊Q₈	1	6	yes	—
272	C₄	C₂³×Q₈×C₁₉	1	1	yes	characteristic
273	C₄	C₂³×Q₈×C₁₉	1	1	yes	characteristic
274	C₂	—	2	6	no	—
275	C₂	C₂²×C₁₉×C₄⋊Q₈	1	1	yes	characteristic
276	C₂	C₁₉×#3584	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₁₉
prime radical	C₂×#41×C₁₉	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₄×#41×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₉×#3166	C₂	semidirect
3. C₁₉×#3164	C₂	semidirect
4. C₁₉×#3163	C₂	semidirect
5. C₁₉×#3160	C₂	semidirect
6. C₂×C₁₉×#499	C₂	semidirect
39. C₁₉×#131	Q₈	semidirect
40. C₁₉×#131	Q₈	semidirect
107. #60666	C₁₉	direct (N × H)
127. #3166	C₂×C₁₉	semidirect
128. #3164	C₂×C₁₉	semidirect
129. #3163	C₂×C₁₉	semidirect
130. #3160	C₂×C₁₉	semidirect
131. C₂×#499	C₂×C₁₉	semidirect
177. #131	Q₈×C₁₉	semidirect
178. #131	Q₈×C₁₉	semidirect
215. C₁₉	#60666	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₉×#3504	2
C₁₉×#3166	2
C₁₉×#3164	2
C₁₉×#3163	2
C₁₉×#3160	2
C₂×C₁₉×#499	2
C₁₉×#2913	2
C₁₉×#2912	2
C₄×C₁₉×#131	2
#60666	19

Subgroups — C19×#60666 #18066625

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#60666 #18066625