Subgroups — C₁₉×#23746 #18029705

Order 4864.

814 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#2296	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₉×#382	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#2101	C₂	1	8	yes	maximal, complemented
4	C₁₉×#1736	C₂	1	2	yes	maximal
5	C₁₉×#1736	C₂	1	2	yes	maximal
6	C₁₉×#1728	C₂	1	1	yes	maximal, characteristic
7	C₂²×M₁₆×C₁₉	C₂²	1	2	yes	—
8	C₂×C₁₉×#117	C₂²	1	1	yes	characteristic
9	C₂×C₁₉×#117	C₂²	1	1	yes	characteristic
10	C₂²×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
11	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
12	C₁₉×#389	C₂²	1	4	yes	—
13	C₁₉×#385	—	2	2	no	—
14	C₁₉×#382	—	2	2	no	—
15	C₁₉×#382	C₂²	1	4	yes	—
16	C₄×C₄⋊C₄×C₁₉	—	2	2	no	—
17	C₂×C₁₉×#95	—	2	2	no	—
18	C₁₉×#340	C₂²	1	8	yes	—
19	C₁₉×#340	C₂²	1	8	yes	—
20	C₁₉×#336	C₂²	1	4	yes	—
21	C₂³×C₄×C₁₉	—	2	2	no	—
22	C₂³×C₄×C₁₉	D₄	1	2	yes	—
23	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
24	C₂×M₁₆×C₁₉	—	2	4	no	—
25	C₂×M₁₆×C₁₉	—	2	4	no	—
26	C₂×M₁₆×C₁₉	—	2	4	no	—
27	C₂²×C₈×C₁₉	C₂³	1	4	yes	—
28	C₁₉×#117	—	2	2	no	—
29	C₁₉×#117	—	2	4	no	—
30	C₁₉×#117	Q₈	1	4	yes	—
31	C₂×C₄⋊C₄×C₁₉	—	2	8	no	—
32	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
33	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
34	C₂×C₄⋊C₄×C₁₉	C₂³	1	4	yes	—
35	C₂×C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
36	C₂×C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
37	C₂×C₂²⋊C₄×C₁₉	—	2	1	no	—
38	C₂×C₂²⋊C₄×C₁₉	—	2	1	no	—
39	C₂×C₄²×C₁₉	—	2	1	no	—
40	C₂×C₄²×C₁₉	—	2	4	no	—
41	C₂×C₄²×C₁₉	—	2	1	no	—
42	C₂×C₄²×C₁₉	D₄	1	2	yes	—
43	C₂×C₄²×C₁₉	C₂³	1	2	yes	—
44	C₁₉×#95	—	2	8	no	—
45	C₁₉×#95	—	2	4	no	—
46	C₂⁴×C₁₉	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₉	—	2	4	no	—
48	C₂²×C₄×C₁₉	—	2	2	no	—
49	C₂²×C₄×C₁₉	—	2	2	no	—
50	C₂²×C₄×C₁₉	—	2	4	no	—
51	C₂²×C₄×C₁₉	—	4	4	no	—
52	C₂²×C₄×C₁₉	—	4	8	no	—
53	C₂²×C₄×C₁₉	—	4	4	no	—
54	C₂²×C₄×C₁₉	C₂×D₄	1	4	yes	—
55	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₉	C₂×Q₈	1	2	yes	—
57	C₂²×C₄×C₁₉	#41	1	2	yes	—
58	C₂²×C₄×C₁₉	#41	1	4	yes	—
59	C₂²×C₄×C₁₉	#41	1	4	yes	—
60	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
61	M₁₆×C₁₉	—	4	8	no	—
62	C₂×C₈×C₁₉	—	2	4	no	—
63	C₂×C₈×C₁₉	—	2	4	no	—
64	C₂×C₈×C₁₉	—	2	4	no	—
65	C₄⋊C₄×C₁₉	—	2	2	no	—
66	C₄⋊C₄×C₁₉	—	2	8	no	—
67	C₄⋊C₄×C₁₉	—	2	4	no	—
68	C₄⋊C₄×C₁₉	—	4	8	no	—
69	C₄⋊C₄×C₁₉	—	4	4	no	—
70	C₄⋊C₄×C₁₉	C₂×Q₈	1	4	yes	—
71	C₂²⋊C₄×C₁₉	—	2	4	no	—
72	C₂²⋊C₄×C₁₉	—	4	2	no	—
73	C₄²×C₁₉	—	2	4	no	—
74	C₄²×C₁₉	—	2	4	no	—
75	C₄²×C₁₉	—	2	4	no	—
76	C₄²×C₁₉	—	4	2	no	—
77	#23746	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
78	C₂³×C₁₉	—	2	4	no	—
79	C₂³×C₁₉	C₂²⋊Q₈	1	2	yes	—
80	C₂³×C₁₉	#123	1	2	yes	—
81	C₂³×C₁₉	#124	1	2	yes	—
82	C₂³×C₁₉	C₂²×D₄	1	1	yes	characteristic
83	C₂×C₄×C₁₉	—	2	4	no	—
84	C₂×C₄×C₁₉	—	2	4	no	—
85	C₂×C₄×C₁₉	—	2	4	no	—
86	C₂×C₄×C₁₉	—	2	4	no	—
87	C₂×C₄×C₁₉	—	4	8	no	—
88	C₂×C₄×C₁₉	—	4	4	no	—
89	C₂×C₄×C₁₉	—	4	2	no	—
90	C₂×C₄×C₁₉	—	4	4	no	—
91	C₂×C₄×C₁₉	—	4	4	no	—
92	C₂×C₄×C₁₉	—	4	8	no	—
93	C₂×C₄×C₁₉	—	8	8	no	—
94	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
95	C₂×C₄×C₁₉	#123	1	2	yes	—
96	C₂×C₄×C₁₉	#124	1	2	yes	—
97	C₂×C₄×C₁₉	#125	1	8	yes	—
98	C₂×C₄×C₁₉	#126	1	8	yes	—
99	C₂×C₄×C₁₉	C₂²×Q₈	1	1	yes	characteristic
100	C₂×C₄×C₁₉	C₂×#41	1	2	yes	—
101	C₂×C₄×C₁₉	C₂×#41	1	2	yes	—
102	C₂×C₄×C₁₉	C₂×#41	1	1	yes	characteristic
103	C₈×C₁₉	—	4	4	no	—
104	#2296	C₂×C₁₉	1	1	yes	characteristic
105	C₂×#382	C₂×C₁₉	1	1	yes	characteristic
106	#2101	C₂×C₁₉	1	8	yes	complemented
107	#1736	C₂×C₁₉	1	2	yes	—
108	#1736	C₂×C₁₉	1	2	yes	—
109	#1728	C₂×C₁₉	1	1	yes	characteristic
110	C₂²×C₁₉	—	2	4	no	—
111	C₂²×C₁₉	—	2	4	no	—
112	C₂²×C₁₉	—	2	4	no	—
113	C₂²×C₁₉	#400	1	4	yes	—
114	C₂²×C₁₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
115	C₂²×C₁₉	C₂×#123	1	1	yes	characteristic
116	C₂²×C₁₉	C₂×#124	1	1	yes	characteristic
117	C₂²×C₁₉	#575	1	4	yes	—
118	C₄×C₁₉	—	4	4	no	—
119	C₄×C₁₉	—	4	4	no	—
120	C₄×C₁₉	—	8	2	no	—
121	C₄×C₁₉	—	8	4	no	—
122	C₄×C₁₉	#400	1	4	yes	—
123	C₄×C₁₉	C₂×#125	1	2	yes	—
124	C₄×C₁₉	C₂×#126	1	2	yes	—
125	C₂²×M₁₆	C₂²×C₁₉	1	2	yes	—
126	C₂×#117	C₂²×C₁₉	1	1	yes	characteristic
127	C₂×#117	C₂²×C₁₉	1	1	yes	characteristic
128	C₂²×C₄⋊C₄	C₂²×C₁₉	1	2	yes	—
129	C₂²×C₄²	C₂²×C₁₉	1	1	yes	characteristic
130	#389	C₂²×C₁₉	1	4	yes	—
131	#385	—	2	2	no	—
132	#382	—	2	2	no	—
133	#382	C₂²×C₁₉	1	4	yes	—
134	C₄×C₄⋊C₄	—	2	2	no	—
135	C₂×#95	—	2	2	no	—
136	#340	C₂²×C₁₉	1	8	yes	—
137	#340	C₂²×C₁₉	1	8	yes	—
138	#336	C₂²×C₁₉	1	4	yes	—
139	C₂×C₁₉	—	2	4	no	—
140	C₂×C₁₉	C₂×#400	1	1	yes	characteristic
141	C₂×C₁₉	#3080	1	2	yes	—
142	C₂×C₁₉	#3093	1	2	yes	—
143	C₂×C₁₉	#3140	1	2	yes	—
144	C₂³×C₄	—	2	2	no	—
145	C₂³×C₄	D₄×C₁₉	1	2	yes	—
146	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
147	C₂×M₁₆	—	2	4	no	—
148	C₂×M₁₆	—	2	4	no	—
149	C₂×M₁₆	—	2	4	no	—
150	C₂²×C₈	C₂³×C₁₉	1	4	yes	—
151	#117	—	2	2	no	—
152	#117	—	2	4	no	—
153	#117	Q₈×C₁₉	1	4	yes	—
154	C₂×C₄⋊C₄	—	2	8	no	—
155	C₂×C₄⋊C₄	—	2	2	no	—
156	C₂×C₄⋊C₄	—	2	4	no	—
157	C₂×C₄⋊C₄	C₂³×C₁₉	1	4	yes	—
158	C₂×C₄⋊C₄	C₂³×C₁₉	1	2	yes	—
159	C₂×C₄⋊C₄	C₂³×C₁₉	1	2	yes	—
160	C₂×C₂²⋊C₄	—	2	1	no	—
161	C₂×C₂²⋊C₄	—	2	1	no	—
162	C₂×C₄²	—	2	1	no	—
163	C₂×C₄²	—	2	4	no	—
164	C₂×C₄²	—	2	1	no	—
165	C₂×C₄²	D₄×C₁₉	1	2	yes	—
166	C₂×C₄²	C₂³×C₁₉	1	2	yes	—
167	#95	—	2	8	no	—
168	#95	—	2	4	no	—
169	C₁₉	#23746	1	1	yes	characteristic, direct product factor, Sylow
170	C₂⁴	C₂×D₄×C₁₉	1	1	yes	characteristic
171	C₂²×C₄	—	2	4	no	—
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	—	2	4	no	—
174	C₂²×C₄	—	2	2	no	—
175	C₂²×C₄	—	4	4	no	—
176	C₂²×C₄	—	4	8	no	—
177	C₂²×C₄	—	4	4	no	—
178	C₂²×C₄	C₂×D₄×C₁₉	1	4	yes	—
179	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
180	C₂²×C₄	C₂×Q₈×C₁₉	1	2	yes	—
181	C₂²×C₄	#41×C₁₉	1	2	yes	—
182	C₂²×C₄	#41×C₁₉	1	4	yes	—
183	C₂²×C₄	#41×C₁₉	1	4	yes	—
184	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
185	M₁₆	—	4	8	no	—
186	C₂×C₈	—	2	4	no	—
187	C₂×C₈	—	2	4	no	—
188	C₂×C₈	—	2	4	no	—
189	C₄⋊C₄	—	2	2	no	—
190	C₄⋊C₄	—	2	8	no	—
191	C₄⋊C₄	—	2	4	no	—
192	C₄⋊C₄	—	4	8	no	—
193	C₄⋊C₄	—	4	4	no	—
194	C₄⋊C₄	C₂×Q₈×C₁₉	1	4	yes	—
195	C₂²⋊C₄	—	2	4	no	—
196	C₂²⋊C₄	—	4	2	no	—
197	C₄²	—	2	4	no	—
198	C₄²	—	2	4	no	—
199	C₄²	—	2	4	no	—
200	C₄²	—	4	2	no	—
201	C₂³	—	2	4	no	—
202	C₂³	C₁₉×C₂²⋊Q₈	1	2	yes	—
203	C₂³	C₁₉×#123	1	2	yes	—
204	C₂³	C₁₉×#124	1	2	yes	—
205	C₂³	C₂²×D₄×C₁₉	1	1	yes	characteristic
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	2	4	no	—
210	C₂×C₄	—	4	8	no	—
211	C₂×C₄	—	4	4	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	4	4	no	—
214	C₂×C₄	—	4	4	no	—
215	C₂×C₄	—	4	8	no	—
216	C₂×C₄	—	8	8	no	—
217	C₂×C₄	C₁₉×C₂²⋊Q₈	1	2	yes	—
218	C₂×C₄	C₁₉×#123	1	2	yes	—
219	C₂×C₄	C₁₉×#124	1	2	yes	—
220	C₂×C₄	C₁₉×#125	1	8	yes	—
221	C₂×C₄	C₁₉×#126	1	8	yes	—
222	C₂×C₄	C₂²×Q₈×C₁₉	1	1	yes	characteristic
223	C₂×C₄	C₂×#41×C₁₉	1	2	yes	—
224	C₂×C₄	C₂×#41×C₁₉	1	2	yes	—
225	C₂×C₄	C₂×#41×C₁₉	1	1	yes	characteristic
226	C₈	—	4	4	no	—
227	C₂²	—	2	4	no	—
228	C₂²	—	2	4	no	—
229	C₂²	—	2	4	no	—
230	C₂²	C₁₉×#400	1	4	yes	—
231	C₂²	C₂×C₁₉×C₂²⋊Q₈	1	1	yes	characteristic
232	C₂²	C₂×C₁₉×#123	1	1	yes	characteristic
233	C₂²	C₂×C₁₉×#124	1	1	yes	characteristic
234	C₂²	C₁₉×#575	1	4	yes	—
235	C₄	—	4	4	no	—
236	C₄	—	4	4	no	—
237	C₄	—	8	2	no	—
238	C₄	—	8	4	no	—
239	C₄	C₁₉×#400	1	4	yes	—
240	C₄	C₂×C₁₉×#125	1	2	yes	—
241	C₄	C₂×C₁₉×#126	1	2	yes	—
242	C₂	—	2	4	no	—
243	C₂	C₂×C₁₉×#400	1	1	yes	characteristic
244	C₂	C₁₉×#3080	1	2	yes	—
245	C₂	C₁₉×#3093	1	2	yes	—
246	C₂	C₁₉×#3140	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₉
prime radical	C₂⁴×C₁₉	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₉ ◁ C₂²×C₄²×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₉×#2101	C₂	semidirect
77. #23746	C₁₉	direct (N × H)
106. #2101	C₂×C₁₉	semidirect
169. C₁₉	#23746	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×#2296	2
C₂×C₁₉×#382	2
C₁₉×#2101	2
C₁₉×#1736	2
C₁₉×#1728	2
#23746	19

Subgroups — C19×#23746 #18029705

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#23746 #18029705