Subgroups — C₁₉×#3076 #14560933

Order 2432.

388 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×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#41×C₁₉	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#479	C₂	1	2	yes	maximal, complemented
4	C₁₉×#478	C₂	1	2	yes	maximal, complemented
5	C₁₉×#477	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₉×#476	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#475	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₉×#474	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₉×#427	C₂	1	2	yes	maximal, complemented
10	C₂×C₁₉×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₉×#416	C₂	1	2	yes	maximal
12	C₂×#41×C₁₉	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₁₉	C₂²	1	2	yes	—
14	C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₁₉×#125	C₂²	1	2	yes	—
17	C₁₉×C₂²⋊Q₈	—	2	1	no	—
18	C₁₉×C₂²⋊Q₈	—	2	1	no	—
19	C₄×Q₈×C₁₉	—	2	1	no	—
20	C₄×Q₈×C₁₉	—	2	1	no	—
21	C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
22	C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
23	C₄×D₄×C₁₉	—	2	1	no	—
24	C₄×D₄×C₁₉	—	2	1	no	—
25	C₄×D₄×C₁₉	C₂²	1	1	yes	characteristic
26	C₄×D₄×C₁₉	C₂²	1	1	yes	characteristic
27	C₁₉×#117	—	2	1	no	—
28	C₁₉×#117	C₂²	1	2	yes	—
29	C₁₉×#117	C₂²	1	1	yes	characteristic
30	C₂×C₄²×C₁₉	—	2	1	no	—
31	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
32	C₁₉×#107	C₂²	1	2	yes	—
33	C₁₉×#107	C₂²	1	2	yes	—
34	C₁₉×#106	C₂²	1	2	yes	—
35	C₁₉×#106	C₂²	1	2	yes	—
36	C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
37	C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
38	C₁₉×C₄⋊C₈	C₂²	1	2	yes	—
39	C₁₉×#103	C₂²	1	2	yes	—
40	C₁₉×#103	C₂²	1	2	yes	—
41	C₁₉×#102	C₂²	1	2	yes	—
42	C₁₉×#102	C₂²	1	2	yes	—
43	#41×C₁₉	—	2	2	no	—
44	#41×C₁₉	Q₈	1	4	yes	complemented
45	C₂×Q₈×C₁₉	—	2	1	no	—
46	C₂×Q₈×C₁₉	—	2	1	no	—
47	C₂×Q₈×C₁₉	C₂³	1	1	yes	characteristic
48	C₂×D₄×C₁₉	—	2	1	no	—
49	C₂×D₄×C₁₉	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₁₉	—	2	1	no	—
51	C₂²×C₄×C₁₉	—	2	1	no	—
52	C₂²×C₄×C₁₉	—	2	1	no	—
53	C₂²×C₄×C₁₉	D₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₉	D₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
56	C₂×C₈×C₁₉	—	2	4	no	—
57	C₂×C₈×C₁₉	C₂³	1	2	yes	—
58	C₂×C₈×C₁₉	C₂³	1	2	yes	—
59	C₄⋊C₄×C₁₉	—	2	1	no	—
60	C₄⋊C₄×C₁₉	—	2	1	no	—
61	C₄⋊C₄×C₁₉	—	2	2	no	—
62	C₄⋊C₄×C₁₉	—	2	2	no	—
63	C₄⋊C₄×C₁₉	—	2	1	no	—
64	C₄⋊C₄×C₁₉	—	2	1	no	—
65	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
66	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
67	C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
68	C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
69	C₂²⋊C₄×C₁₉	—	2	1	no	—
70	C₂²⋊C₄×C₁₉	—	2	2	no	—
71	C₂²⋊C₄×C₁₉	—	2	1	no	—
72	C₂²⋊C₄×C₁₉	—	2	1	no	—
73	C₄²×C₁₉	—	2	1	no	—
74	C₄²×C₁₉	—	2	1	no	—
75	C₄²×C₁₉	—	2	1	no	—
76	C₄²×C₁₉	—	2	2	no	—
77	C₄²×C₁₉	D₄	1	1	yes	characteristic
78	C₄²×C₁₉	D₄	1	1	yes	characteristic
79	C₄²×C₁₉	C₂³	1	1	yes	characteristic
80	C₄²×C₁₉	C₂³	1	1	yes	characteristic
81	C₂³×C₁₉	—	2	1	no	—
82	C₂³×C₁₉	C₂×D₄	1	1	yes	characteristic
83	Q₈×C₁₉	—	2	1	no	—
84	Q₈×C₁₉	—	4	2	no	—
85	Q₈×C₁₉	—	4	2	no	—
86	Q₈×C₁₉	C₂×Q₈	1	2	yes	—
87	D₄×C₁₉	—	2	4	no	—
88	D₄×C₁₉	—	2	1	no	—
89	D₄×C₁₉	C₂×Q₈	1	2	yes	—
90	C₂×C₄×C₁₉	—	2	1	no	—
91	C₂×C₄×C₁₉	—	2	4	no	—
92	C₂×C₄×C₁₉	—	2	2	no	—
93	C₂×C₄×C₁₉	—	2	1	no	—
94	C₂×C₄×C₁₉	—	2	1	no	—
95	C₂×C₄×C₁₉	—	2	2	no	—
96	C₂×C₄×C₁₉	—	2	2	no	—
97	C₂×C₄×C₁₉	—	4	1	no	—
98	C₂×C₄×C₁₉	—	4	2	no	—
99	C₂×C₄×C₁₉	—	4	2	no	—
100	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×C₁₉	C₂×Q₈	1	2	yes	—
106	C₂×C₄×C₁₉	#41	1	2	yes	—
107	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
108	C₈×C₁₉	—	2	2	no	—
109	C₈×C₁₉	—	2	2	no	—
110	#3076	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
111	C₂²×C₁₉	—	2	1	no	—
112	C₂²×C₁₉	—	2	1	no	—
113	C₂²×C₁₉	—	2	2	no	—
114	C₂²×C₁₉	—	4	1	no	—
115	C₂²×C₁₉	—	4	1	no	—
116	C₂²×C₁₉	C₂²⋊Q₈	1	2	yes	—
117	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
118	C₄×C₁₉	—	2	2	no	—
119	C₄×C₁₉	—	2	1	no	—
120	C₄×C₁₉	—	2	1	no	—
121	C₄×C₁₉	—	2	1	no	—
122	C₄×C₁₉	—	4	1	no	—
123	C₄×C₁₉	—	4	2	no	—
124	C₄×C₁₉	—	4	1	no	—
125	C₄×C₁₉	—	4	2	no	—
126	C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
127	C₄×C₁₉	#135	1	2	yes	—
128	C₄×C₁₉	C₂²×Q₈	1	1	yes	characteristic
129	C₄×C₁₉	C₂×#41	1	1	yes	characteristic
130	#533	C₂×C₁₉	1	1	yes	characteristic, complemented
131	C₄×#41	C₂×C₁₉	1	1	yes	characteristic
132	#479	C₂×C₁₉	1	2	yes	complemented
133	#478	C₂×C₁₉	1	2	yes	complemented
134	#477	C₂×C₁₉	1	1	yes	characteristic, complemented
135	#476	C₂×C₁₉	1	1	yes	characteristic, complemented
136	#475	C₂×C₁₉	1	1	yes	characteristic, complemented
137	#474	C₂×C₁₉	1	1	yes	characteristic, complemented
138	#427	C₂×C₁₉	1	2	yes	complemented
139	C₂×C₄⋊C₈	C₂×C₁₉	1	1	yes	characteristic, complemented
140	#416	C₂×C₁₉	1	2	yes	—
141	C₂×C₁₉	—	2	2	no	—
142	C₂×C₁₉	—	4	2	no	—
143	C₂×C₁₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
144	C₂×C₁₉	C₂×#135	1	1	yes	characteristic
145	C₂×C₁₉	#575	1	1	yes	characteristic
146	C₂×#41	C₂²×C₁₉	1	1	yes	characteristic
147	C₂²×C₈	C₂²×C₁₉	1	2	yes	—
148	C₄⋊Q₈	C₂²×C₁₉	1	1	yes	characteristic
149	C₄⋊Q₈	C₂²×C₁₉	1	1	yes	characteristic
150	#125	C₂²×C₁₉	1	2	yes	—
151	C₂²⋊Q₈	—	2	1	no	—
152	C₂²⋊Q₈	—	2	1	no	—
153	C₄×Q₈	—	2	1	no	—
154	C₄×Q₈	—	2	1	no	—
155	C₄×Q₈	C₂²×C₁₉	1	1	yes	characteristic
156	C₄×Q₈	C₂²×C₁₉	1	1	yes	characteristic
157	C₄×D₄	—	2	1	no	—
158	C₄×D₄	—	2	1	no	—
159	C₄×D₄	C₂²×C₁₉	1	1	yes	characteristic
160	C₄×D₄	C₂²×C₁₉	1	1	yes	characteristic
161	#117	—	2	1	no	—
162	#117	C₂²×C₁₉	1	1	yes	characteristic
163	#117	C₂²×C₁₉	1	2	yes	—
164	C₂×C₄²	—	2	1	no	—
165	C₂×C₄²	C₂²×C₁₉	1	1	yes	characteristic
166	#107	C₂²×C₁₉	1	2	yes	—
167	#107	C₂²×C₁₉	1	2	yes	—
168	#106	C₂²×C₁₉	1	2	yes	—
169	#106	C₂²×C₁₉	1	2	yes	—
170	C₄⋊C₈	C₂²×C₁₉	1	1	yes	characteristic
171	C₄⋊C₈	C₂²×C₁₉	1	1	yes	characteristic
172	C₄⋊C₈	C₂²×C₁₉	1	2	yes	—
173	#103	C₂²×C₁₉	1	2	yes	—
174	#103	C₂²×C₁₉	1	2	yes	—
175	#102	C₂²×C₁₉	1	2	yes	—
176	#102	C₂²×C₁₉	1	2	yes	—
177	C₁₉	#3076	1	1	yes	characteristic, direct product factor, Sylow
178	#41	—	2	2	no	—
179	#41	Q₈×C₁₉	1	4	yes	complemented
180	C₂×Q₈	—	2	1	no	—
181	C₂×Q₈	—	2	1	no	—
182	C₂×Q₈	C₂³×C₁₉	1	1	yes	characteristic
183	C₂×D₄	—	2	1	no	—
184	C₂×D₄	C₂³×C₁₉	1	1	yes	characteristic
185	C₂²×C₄	—	2	1	no	—
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	D₄×C₁₉	1	1	yes	characteristic
189	C₂²×C₄	D₄×C₁₉	1	1	yes	characteristic
190	C₂²×C₄	C₂³×C₁₉	1	1	yes	characteristic
191	C₂×C₈	—	2	4	no	—
192	C₂×C₈	C₂³×C₁₉	1	2	yes	—
193	C₂×C₈	C₂³×C₁₉	1	2	yes	—
194	C₄⋊C₄	—	2	1	no	—
195	C₄⋊C₄	—	2	1	no	—
196	C₄⋊C₄	—	2	2	no	—
197	C₄⋊C₄	—	2	2	no	—
198	C₄⋊C₄	—	2	1	no	—
199	C₄⋊C₄	—	2	1	no	—
200	C₄⋊C₄	C₂³×C₁₉	1	2	yes	—
201	C₄⋊C₄	C₂³×C₁₉	1	2	yes	—
202	C₄⋊C₄	C₂³×C₁₉	1	1	yes	characteristic
203	C₄⋊C₄	C₂³×C₁₉	1	1	yes	characteristic
204	C₂²⋊C₄	—	2	1	no	—
205	C₂²⋊C₄	—	2	2	no	—
206	C₂²⋊C₄	—	2	1	no	—
207	C₂²⋊C₄	—	2	1	no	—
208	C₄²	—	2	1	no	—
209	C₄²	—	2	1	no	—
210	C₄²	—	2	1	no	—
211	C₄²	—	2	2	no	—
212	C₄²	D₄×C₁₉	1	1	yes	characteristic
213	C₄²	D₄×C₁₉	1	1	yes	characteristic
214	C₄²	C₂³×C₁₉	1	1	yes	characteristic
215	C₄²	C₂³×C₁₉	1	1	yes	characteristic
216	C₂³	—	2	1	no	—
217	C₂³	C₂×D₄×C₁₉	1	1	yes	characteristic
218	Q₈	—	2	1	no	—
219	Q₈	—	4	2	no	—
220	Q₈	—	4	2	no	—
221	Q₈	C₂×Q₈×C₁₉	1	2	yes	—
222	D₄	—	2	4	no	—
223	D₄	—	2	1	no	—
224	D₄	C₂×Q₈×C₁₉	1	2	yes	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	4	1	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
236	C₂×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
237	C₂×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
238	C₂×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
239	C₂×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
240	C₂×C₄	C₂×Q₈×C₁₉	1	2	yes	—
241	C₂×C₄	#41×C₁₉	1	2	yes	—
242	C₂×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
243	C₈	—	2	2	no	—
244	C₈	—	2	2	no	—
245	C₂²	—	2	1	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	2	no	—
248	C₂²	—	4	1	no	—
249	C₂²	—	4	1	no	—
250	C₂²	C₁₉×C₂²⋊Q₈	1	2	yes	—
251	C₂²	C₂²×D₄×C₁₉	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	2	1	no	—
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	4	1	no	—
257	C₄	—	4	2	no	—
258	C₄	—	4	1	no	—
259	C₄	—	4	2	no	—
260	C₄	C₁₉×C₂²⋊Q₈	1	2	yes	—
261	C₄	C₁₉×#135	1	2	yes	—
262	C₄	C₂²×Q₈×C₁₉	1	1	yes	characteristic
263	C₄	C₂×#41×C₁₉	1	1	yes	characteristic
264	C₂	—	2	2	no	—
265	C₂	—	4	2	no	—
266	C₂	C₂×C₁₉×C₂²⋊Q₈	1	1	yes	characteristic
267	C₂	C₂×C₁₉×#135	1	1	yes	characteristic
268	C₂	C₁₉×#575	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₉	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₉
socle	C₂²×C₁₉	C₂²×D₄
prime radical	C₂×#41×C₁₉	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₉×#533	C₂	semidirect
3. C₁₉×#479	C₂	semidirect
4. C₁₉×#478	C₂	semidirect
5. C₁₉×#477	C₂	semidirect
6. C₁₉×#476	C₂	semidirect
7. C₁₉×#475	C₂	semidirect
8. C₁₉×#474	C₂	semidirect
9. C₁₉×#427	C₂	semidirect
10. C₂×C₁₉×C₄⋊C₈	C₂	semidirect
44. #41×C₁₉	Q₈	semidirect
110. #3076	C₁₉	direct (N × H)
130. #533	C₂×C₁₉	semidirect
132. #479	C₂×C₁₉	semidirect
133. #478	C₂×C₁₉	semidirect
134. #477	C₂×C₁₉	semidirect
135. #476	C₂×C₁₉	semidirect
136. #475	C₂×C₁₉	semidirect
137. #474	C₂×C₁₉	semidirect
138. #427	C₂×C₁₉	semidirect
139. C₂×C₄⋊C₈	C₂×C₁₉	semidirect
177. C₁₉	#3076	direct (N × H)
179. #41	Q₈×C₁₉	semidirect

Maximal subgroups

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

subgroup	index
C₁₉×#533	2
C₄×#41×C₁₉	2
C₁₉×#479	2
C₁₉×#478	2
C₁₉×#477	2
C₁₉×#476	2
C₁₉×#475	2
C₁₉×#474	2
C₁₉×#427	2
C₂×C₁₉×C₄⋊C₈	2
C₁₉×#416	2
#3076	19

Subgroups — C19×#3076 #14560933

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#3076 #14560933