Subgroups — C₁₉×#17335 #18023294

Order 4864.

814 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#2965	C₂	1	2	yes	maximal
2	C₄×C₁₉×#131	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1455	C₂	1	2	yes	maximal, complemented
4	C₁₉×#1453	C₂	1	2	yes	maximal, complemented
5	C₁₉×#1451	C₂	1	2	yes	maximal, complemented
6	C₁₉×#1450	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×M₁₆×C₁₉	C₂	1	4	yes	maximal, complemented
8	C₂×C₈²×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₁₉×#131	C₄	1	2	yes	—
10	C₄×#41×C₁₉	C₂²	1	1	yes	characteristic
11	C₁₉×#446	C₄	1	4	yes	—
12	C₁₉×#436	C₄	1	4	yes	—
13	C₁₉×#435	C₄	1	8	yes	—
14	C₁₉×#431	C₂²	1	4	yes	—
15	C₁₉×#431	C₂²	1	2	yes	—
16	C₁₉×#405	C₄	1	4	yes	—
17	C₁₉×#405	C₄	1	2	yes	—
18	C₄×M₁₆×C₁₉	C₂²	1	4	yes	—
19	C₄×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
20	C₄×M₁₆×C₁₉	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₁₉	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₁₉	C₂²	1	2	yes	—
24	C₁₉×#322	C₂²	1	4	yes	—
25	C₁₉×#322	C₂²	1	4	yes	—
26	C₁₉×#322	C₂²	1	2	yes	—
27	C₈²×C₁₉	C₂²	1	1	yes	characteristic
28	C₈²×C₁₉	C₂²	1	1	yes	characteristic
29	C₈²×C₁₉	C₂²	1	2	yes	—
30	C₈²×C₁₉	C₂²	1	2	yes	—
31	C₂×#41×C₁₉	C₂×C₄	1	1	yes	characteristic
32	C₁₉×#131	—	2	8	no	—
33	C₂×M₁₆×C₁₉	C₂×C₄	1	2	yes	—
34	C₂×M₁₆×C₁₉	C₂×C₄	1	4	yes	—
35	C₂×M₁₆×C₁₉	C₂×C₄	1	8	yes	—
36	C₂²×C₈×C₁₉	C₂×C₄	1	4	yes	—
37	C₂²×C₈×C₁₉	C₂×C₄	1	4	yes	—
38	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
39	C₄×Q₈×C₁₉	C₂×C₄	1	2	yes	—
40	C₄×D₄×C₁₉	C₂×C₄	1	4	yes	—
41	C₄×D₄×C₁₉	C₂×C₄	1	2	yes	—
42	C₁₉×#117	C₂×C₄	1	2	yes	—
43	C₁₉×#117	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₁₉	C₂³	1	2	yes	—
46	C₁₉×C₄⋊C₈	C₂×C₄	1	8	yes	—
47	C₁₉×C₄⋊C₈	C₂×C₄	1	4	yes	—
48	C₁₉×C₂²⋊C₈	C₂×C₄	1	4	yes	—
49	C₁₉×C₂²⋊C₈	C₂×C₄	1	8	yes	—
50	C₁₉×#97	C₂×C₄	1	2	yes	—
51	C₁₉×#97	C₂×C₄	1	4	yes	—
52	C₁₉×#97	C₂×C₄	1	8	yes	—
53	C₄×C₈×C₁₉	C₂×C₄	1	4	yes	—
54	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
55	C₄×C₈×C₁₉	C₂×C₄	1	4	yes	—
56	C₄×C₈×C₁₉	C₂³	1	1	yes	characteristic
57	C₄×C₈×C₁₉	C₂³	1	2	yes	—
58	C₄×C₈×C₁₉	C₂³	1	2	yes	—
59	C₄×C₈×C₁₉	C₂³	1	2	yes	—
60	C₄×C₈×C₁₉	C₂³	1	4	yes	—
61	C₄×C₈×C₁₉	C₂³	1	1	yes	characteristic
62	#41×C₁₉	—	2	4	no	—
63	C₂×Q₈×C₁₉	C₄²	1	1	yes	characteristic
64	C₂×D₄×C₁₉	C₄²	1	2	yes	—
65	C₂×D₄×C₁₉	C₄²	1	1	yes	characteristic
66	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₁₉	C₂²×C₄	1	4	yes	—
69	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
70	M₁₆×C₁₉	—	2	4	no	—
71	M₁₆×C₁₉	—	2	8	no	—
72	M₁₆×C₁₉	—	2	16	no	—
73	C₂×C₈×C₁₉	—	2	8	no	—
74	C₂×C₈×C₁₉	—	2	8	no	—
75	C₂×C₈×C₁₉	M₁₆	1	8	yes	complemented
76	C₂×C₈×C₁₉	C₂²×C₄	1	4	yes	—
77	C₂×C₈×C₁₉	C₂²×C₄	1	4	yes	—
78	C₂×C₈×C₁₉	C₂²×C₄	1	8	yes	—
79	C₂×C₈×C₁₉	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₁₉	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₁₉	C₂²×C₄	1	4	yes	—
82	C₄⋊C₄×C₁₉	C₄²	1	4	yes	—
83	C₄⋊C₄×C₁₉	C₄²	1	2	yes	—
84	C₂²⋊C₄×C₁₉	C₄²	1	4	yes	—
85	C₂²⋊C₄×C₁₉	C₄²	1	2	yes	—
86	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₁₉	C₂²×C₄	1	2	yes	—
88	C₄²×C₁₉	C₂²×C₄	1	2	yes	—
89	C₄²×C₁₉	C₂²×C₄	1	4	yes	—
90	C₄²×C₁₉	C₂⁴	1	1	yes	characteristic
91	#17335	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
92	C₂³×C₁₉	C₂×C₄²	1	2	yes	—
93	C₂³×C₁₉	C₂×C₄²	1	1	yes	characteristic
94	Q₈×C₁₉	—	2	2	no	—
95	D₄×C₁₉	—	2	4	no	—
96	D₄×C₁₉	—	2	2	no	—
97	C₂×C₄×C₁₉	—	2	4	no	—
98	C₂×C₄×C₁₉	—	2	4	no	—
99	C₂×C₄×C₁₉	—	2	8	no	—
100	C₂×C₄×C₁₉	C₂×C₄²	1	2	yes	—
101	C₂×C₄×C₁₉	C₂×C₄²	1	4	yes	—
102	C₂×C₄×C₁₉	C₂×C₄²	1	2	yes	—
103	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
104	C₂×C₄×C₁₉	C₂×M₁₆	1	4	yes	—
105	C₂×C₄×C₁₉	C₂³×C₄	1	1	yes	characteristic
106	C₂×C₄×C₁₉	C₂³×C₄	1	2	yes	—
107	C₈×C₁₉	—	2	4	no	—
108	C₈×C₁₉	—	2	4	no	—
109	C₈×C₁₉	—	2	8	no	—
110	C₈×C₁₉	C₂×M₁₆	1	4	yes	—
111	C₈×C₁₉	C₂×M₁₆	1	4	yes	—
112	C₈×C₁₉	#131	1	8	yes	complemented
113	#2965	C₂×C₁₉	1	2	yes	—
114	C₄×#131	C₂×C₁₉	1	1	yes	characteristic
115	#1455	C₂×C₁₉	1	2	yes	complemented
116	#1453	C₂×C₁₉	1	2	yes	complemented
117	#1451	C₂×C₁₉	1	2	yes	complemented
118	#1450	C₂×C₁₉	1	1	yes	characteristic, complemented
119	C₈×M₁₆	C₂×C₁₉	1	4	yes	complemented
120	C₂×C₈²	C₂×C₁₉	1	1	yes	characteristic, complemented
121	C₂²×C₁₉	—	2	2	no	—
122	C₂²×C₁₉	—	2	2	no	—
123	C₂²×C₁₉	—	2	4	no	—
124	C₂²×C₁₉	C₄×M₁₆	1	2	yes	—
125	C₂²×C₁₉	C₂²×C₄²	1	1	yes	characteristic
126	C₄×C₁₉	—	2	2	no	—
127	C₄×C₁₉	—	2	2	no	—
128	C₄×C₁₉	—	2	4	no	—
129	C₄×C₁₉	C₄×M₁₆	1	2	yes	—
130	C₄×C₁₉	C₂²×M₁₆	1	2	yes	—
131	C₄×C₁₉	C₂×#131	1	4	yes	—
132	C₂×#131	C₄×C₁₉	1	2	yes	—
133	C₄×#41	C₂²×C₁₉	1	1	yes	characteristic
134	#446	C₄×C₁₉	1	4	yes	—
135	#436	C₄×C₁₉	1	4	yes	—
136	#435	C₄×C₁₉	1	8	yes	—
137	#431	C₂²×C₁₉	1	4	yes	—
138	#431	C₂²×C₁₉	1	2	yes	—
139	#405	C₄×C₁₉	1	4	yes	—
140	#405	C₄×C₁₉	1	2	yes	—
141	C₄×M₁₆	C₂²×C₁₉	1	1	yes	characteristic
142	C₄×M₁₆	C₂²×C₁₉	1	2	yes	—
143	C₄×M₁₆	C₂²×C₁₉	1	4	yes	—
144	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
145	C₂×C₄×C₈	C₂²×C₁₉	1	2	yes	—
146	C₂×C₄×C₈	C₂²×C₁₉	1	2	yes	—
147	#322	C₂²×C₁₉	1	4	yes	—
148	#322	C₂²×C₁₉	1	4	yes	—
149	#322	C₂²×C₁₉	1	2	yes	—
150	C₈²	C₂²×C₁₉	1	2	yes	—
151	C₈²	C₂²×C₁₉	1	1	yes	characteristic
152	C₈²	C₂²×C₁₉	1	1	yes	characteristic
153	C₈²	C₂²×C₁₉	1	2	yes	—
154	C₂×C₁₉	—	2	2	no	—
155	C₂×C₁₉	—	4	2	no	—
156	C₂×C₁₉	C₂×C₄×M₁₆	1	1	yes	characteristic
157	C₂×C₁₉	C₄×#131	1	2	yes	—
158	C₂×#41	C₂×C₄×C₁₉	1	1	yes	characteristic
159	#131	—	2	8	no	—
160	C₂×M₁₆	C₂×C₄×C₁₉	1	2	yes	—
161	C₂×M₁₆	C₂×C₄×C₁₉	1	4	yes	—
162	C₂×M₁₆	C₂×C₄×C₁₉	1	8	yes	—
163	C₂²×C₈	C₂×C₄×C₁₉	1	4	yes	—
164	C₂²×C₈	C₂×C₄×C₁₉	1	4	yes	—
165	C₂²×C₈	C₂×C₄×C₁₉	1	2	yes	—
166	C₄×Q₈	C₂×C₄×C₁₉	1	2	yes	—
167	C₄×D₄	C₂×C₄×C₁₉	1	4	yes	—
168	C₄×D₄	C₂×C₄×C₁₉	1	2	yes	—
169	#117	C₂×C₄×C₁₉	1	2	yes	—
170	#117	C₂×C₄×C₁₉	1	1	yes	characteristic
171	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
172	C₂×C₄²	C₂³×C₁₉	1	2	yes	—
173	C₄⋊C₈	C₂×C₄×C₁₉	1	8	yes	—
174	C₄⋊C₈	C₂×C₄×C₁₉	1	4	yes	—
175	C₂²⋊C₈	C₂×C₄×C₁₉	1	4	yes	—
176	C₂²⋊C₈	C₂×C₄×C₁₉	1	8	yes	—
177	#97	C₂×C₄×C₁₉	1	2	yes	—
178	#97	C₂×C₄×C₁₉	1	4	yes	—
179	#97	C₂×C₄×C₁₉	1	8	yes	—
180	C₄×C₈	C₂×C₄×C₁₉	1	4	yes	—
181	C₄×C₈	C₂×C₄×C₁₉	1	2	yes	—
182	C₄×C₈	C₂×C₄×C₁₉	1	4	yes	—
183	C₄×C₈	C₂³×C₁₉	1	1	yes	characteristic
184	C₄×C₈	C₂³×C₁₉	1	1	yes	characteristic
185	C₄×C₈	C₂³×C₁₉	1	2	yes	—
186	C₄×C₈	C₂³×C₁₉	1	2	yes	—
187	C₄×C₈	C₂³×C₁₉	1	4	yes	—
188	C₄×C₈	C₂³×C₁₉	1	2	yes	—
189	C₁₉	#17335	1	1	yes	characteristic, direct product factor, Sylow
190	#41	—	2	4	no	—
191	C₂×Q₈	C₄²×C₁₉	1	1	yes	characteristic
192	C₂×D₄	C₄²×C₁₉	1	2	yes	—
193	C₂×D₄	C₄²×C₁₉	1	1	yes	characteristic
194	C₂²×C₄	C₂²×C₄×C₁₉	1	2	yes	—
195	C₂²×C₄	C₂²×C₄×C₁₉	1	4	yes	—
196	C₂²×C₄	C₂²×C₄×C₁₉	1	2	yes	—
197	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
198	M₁₆	—	2	4	no	—
199	M₁₆	—	2	8	no	—
200	M₁₆	—	2	16	no	—
201	C₂×C₈	—	2	8	no	—
202	C₂×C₈	—	2	8	no	—
203	C₂×C₈	M₁₆×C₁₉	1	8	yes	complemented
204	C₂×C₈	C₂²×C₄×C₁₉	1	4	yes	—
205	C₂×C₈	C₂²×C₄×C₁₉	1	4	yes	—
206	C₂×C₈	C₂²×C₄×C₁₉	1	8	yes	—
207	C₂×C₈	C₂²×C₄×C₁₉	1	2	yes	—
208	C₂×C₈	C₂²×C₄×C₁₉	1	2	yes	—
209	C₂×C₈	C₂²×C₄×C₁₉	1	4	yes	—
210	C₄⋊C₄	C₄²×C₁₉	1	4	yes	—
211	C₄⋊C₄	C₄²×C₁₉	1	2	yes	—
212	C₂²⋊C₄	C₄²×C₁₉	1	4	yes	—
213	C₂²⋊C₄	C₄²×C₁₉	1	2	yes	—
214	C₄²	C₂²×C₄×C₁₉	1	1	yes	characteristic
215	C₄²	C₂²×C₄×C₁₉	1	2	yes	—
216	C₄²	C₂²×C₄×C₁₉	1	2	yes	—
217	C₄²	C₂²×C₄×C₁₉	1	4	yes	—
218	C₄²	C₂⁴×C₁₉	1	1	yes	characteristic
219	C₂³	C₂×C₄²×C₁₉	1	2	yes	—
220	C₂³	C₂×C₄²×C₁₉	1	1	yes	characteristic
221	Q₈	—	2	2	no	—
222	D₄	—	2	4	no	—
223	D₄	—	2	2	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	2	8	no	—
227	C₂×C₄	C₂×C₄²×C₁₉	1	2	yes	—
228	C₂×C₄	C₂×C₄²×C₁₉	1	4	yes	—
229	C₂×C₄	C₂×C₄²×C₁₉	1	2	yes	—
230	C₂×C₄	C₂×C₄²×C₁₉	1	1	yes	characteristic
231	C₂×C₄	C₂×M₁₆×C₁₉	1	4	yes	—
232	C₂×C₄	C₂³×C₄×C₁₉	1	1	yes	characteristic
233	C₂×C₄	C₂³×C₄×C₁₉	1	2	yes	—
234	C₈	—	2	4	no	—
235	C₈	—	2	4	no	—
236	C₈	—	2	8	no	—
237	C₈	C₂×M₁₆×C₁₉	1	4	yes	—
238	C₈	C₂×M₁₆×C₁₉	1	4	yes	—
239	C₈	C₁₉×#131	1	8	yes	complemented
240	C₂²	—	2	2	no	—
241	C₂²	—	2	2	no	—
242	C₂²	—	2	4	no	—
243	C₂²	C₄×M₁₆×C₁₉	1	2	yes	—
244	C₂²	C₂²×C₄²×C₁₉	1	1	yes	characteristic
245	C₄	—	2	2	no	—
246	C₄	—	2	2	no	—
247	C₄	—	2	4	no	—
248	C₄	C₄×M₁₆×C₁₉	1	2	yes	—
249	C₄	C₂²×M₁₆×C₁₉	1	2	yes	—
250	C₄	C₂×C₁₉×#131	1	4	yes	—
251	C₂	—	2	2	no	—
252	C₂	—	4	2	no	—
253	C₂	C₂×C₄×M₁₆×C₁₉	1	1	yes	characteristic
254	C₂	C₄×C₁₉×#131	1	2	yes	—

	subgroup	G/H
center	C₄×C₈×C₁₉	C₂³
commutator	C₂²	C₂²×C₄²×C₁₉
Frattini, cosocle	C₄²	C₂⁴×C₁₉
socle	C₂²×C₁₉	C₂²×C₄²
prime radical	C₂×#41×C₁₉	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄×C₈×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₉×#1455	C₂	semidirect
4. C₁₉×#1453	C₂	semidirect
5. C₁₉×#1451	C₂	semidirect
6. C₁₉×#1450	C₂	semidirect
7. C₈×M₁₆×C₁₉	C₂	semidirect
8. C₂×C₈²×C₁₉	C₂	semidirect
75. C₂×C₈×C₁₉	M₁₆	semidirect
91. #17335	C₁₉	direct (N × H)
112. C₈×C₁₉	#131	semidirect
115. #1455	C₂×C₁₉	semidirect
116. #1453	C₂×C₁₉	semidirect
117. #1451	C₂×C₁₉	semidirect
118. #1450	C₂×C₁₉	semidirect
119. C₈×M₁₆	C₂×C₁₉	semidirect
120. C₂×C₈²	C₂×C₁₉	semidirect
189. C₁₉	#17335	direct (N × H)
203. C₂×C₈	M₁₆×C₁₉	semidirect
239. C₈	C₁₉×#131	semidirect

Maximal subgroups

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

subgroup	index
C₁₉×#2965	2
C₄×C₁₉×#131	2
C₁₉×#1455	2
C₁₉×#1453	2
C₁₉×#1451	2
C₁₉×#1450	2
C₈×M₁₆×C₁₉	2
C₂×C₈²×C₁₉	2
#17335	19

Subgroups — C19×#17335 #18023294

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#17335 #18023294