Subgroups — C₁₉×#34372 #18040331

Order 4864.

1910 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₉×#125	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2382	C₂	1	4	yes	maximal, complemented
3	C₁₉×#2371	C₂	1	4	yes	maximal, complemented
4	C₁₉×#2345	C₂	1	8	yes	maximal
5	C₁₉×#2336	C₂	1	8	yes	maximal, complemented
6	C₁₉×#2313	C₂	1	4	yes	maximal
7	C₁₉×#2283	C₂	1	2	yes	maximal
8	C₂×C₁₉×#125	C₄	1	16	yes	complemented
9	C₂×C₁₉×#125	C₂²	1	4	yes	—
10	C₂×C₁₉×#125	C₂²	1	2	yes	—
11	C₂×C₁₉×#125	C₂²	1	2	yes	—
12	C₂²×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
13	C₂²×C₄⋊C₄×C₁₉	C₂²	1	4	yes	—
14	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
15	C₁₉×#389	C₂²	1	16	yes	—
16	C₁₉×#387	C₂²	1	16	yes	—
17	C₁₉×#387	C₂²	1	16	yes	—
18	C₁₉×#387	C₂²	1	16	yes	—
19	C₁₉×#387	C₂²	1	16	yes	—
20	C₁₉×#383	C₂²	1	4	yes	—
21	C₁₉×#382	C₂²	1	4	yes	—
22	C₁₉×#381	C₂²	1	4	yes	—
23	C₁₉×#380	C₂²	1	4	yes	—
24	C₁₉×#380	C₂²	1	8	yes	—
25	C₄×C₄⋊C₄×C₁₉	C₂²	1	4	yes	—
26	C₄×C₄⋊C₄×C₁₉	C₂²	1	16	yes	—
27	C₄×C₂²⋊C₄×C₁₉	C₂²	1	4	yes	—
28	C₄×C₂²⋊C₄×C₁₉	C₂²	1	8	yes	—
29	C₁₉×#376	C₂²	1	4	yes	—
30	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
31	C₂³×C₄×C₁₉	C₂³	1	2	yes	—
32	C₂³×C₄×C₁₉	C₂³	1	4	yes	—
33	C₁₉×#125	C₂×C₄	1	32	yes	—
34	C₁₉×#125	C₂×C₄	1	16	yes	—
35	C₁₉×#125	C₂×C₄	1	16	yes	—
36	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	16	yes	—
37	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	32	yes	—
38	C₂×C₄⋊C₄×C₁₉	C₂³	1	8	yes	—
39	C₂×C₄⋊C₄×C₁₉	C₂³	1	16	yes	—
40	C₂×C₄⋊C₄×C₁₉	C₂³	1	8	yes	—
41	C₂×C₄⋊C₄×C₁₉	C₂³	1	4	yes	—
42	C₂×C₄⋊C₄×C₁₉	C₂³	1	4	yes	—
43	C₂×C₄⋊C₄×C₁₉	C₂³	1	8	yes	—
44	C₂×C₂²⋊C₄×C₁₉	C₂³	1	8	yes	—
45	C₂×C₄²×C₁₉	C₂×C₄	1	8	yes	—
46	C₂×C₄²×C₁₉	C₂³	1	8	yes	—
47	C₂×C₄²×C₁₉	C₂³	1	16	yes	—
48	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₁₉	C₂³	1	2	yes	—
50	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
51	C₁₉×#95	C₂³	1	8	yes	—
52	C₁₉×#95	C₂³	1	8	yes	—
53	C₁₉×#95	C₂³	1	8	yes	—
54	C₁₉×#95	C₂³	1	16	yes	—
55	C₁₉×#95	C₂³	1	16	yes	—
56	C₁₉×#95	C₂³	1	8	yes	—
57	C₂⁴×C₁₉	C₂⁴	1	1	yes	characteristic
58	C₂²×C₄×C₁₉	—	2	4	no	—
59	C₂²×C₄×C₁₉	—	4	4	no	—
60	C₂²×C₄×C₁₉	—	4	8	no	—
61	C₂²×C₄×C₁₉	C₂²×C₄	1	4	yes	—
62	C₂²×C₄×C₁₉	C₂²×C₄	1	8	yes	—
63	C₂²×C₄×C₁₉	C₂²×C₄	1	16	yes	—
64	C₂²×C₄×C₁₉	C₂⁴	1	16	yes	—
65	C₂²×C₄×C₁₉	C₂⁴	1	2	yes	—
66	C₂²×C₄×C₁₉	C₂⁴	1	8	yes	—
67	C₂²×C₄×C₁₉	C₂⁴	1	2	yes	—
68	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
69	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
70	C₄⋊C₄×C₁₉	—	4	16	no	—
71	C₄⋊C₄×C₁₉	—	4	8	no	—
72	C₄⋊C₄×C₁₉	C₂²×C₄	1	32	yes	—
73	C₄⋊C₄×C₁₉	C₂²×C₄	1	16	yes	—
74	C₄⋊C₄×C₁₉	C₂²×C₄	1	16	yes	—
75	C₄⋊C₄×C₁₉	C₂²×C₄	1	32	yes	—
76	C₂²⋊C₄×C₁₉	—	2	16	no	—
77	C₄²×C₁₉	—	4	8	no	—
78	C₄²×C₁₉	—	4	16	no	—
79	C₄²×C₁₉	C₂²×C₄	1	4	yes	—
80	C₄²×C₁₉	C₂²×C₄	1	8	yes	—
81	C₄²×C₁₉	C₂²×C₄	1	4	yes	—
82	#34372	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
83	C₂³×C₁₉	—	2	4	no	—
84	C₂³×C₁₉	C₂³×C₄	1	2	yes	—
85	C₂³×C₁₉	2₋¹⁺⁴	1	4	yes	—
86	C₂³×C₁₉	C₂⁵	1	1	yes	characteristic
87	C₂×C₄×C₁₉	—	2	8	no	—
88	C₂×C₄×C₁₉	—	2	2	no	—
89	C₂×C₄×C₁₉	—	2	2	no	—
90	C₂×C₄×C₁₉	—	2	4	no	—
91	C₂×C₄×C₁₉	—	2	16	no	—
92	C₂×C₄×C₁₉	—	2	4	no	—
93	C₂×C₄×C₁₉	—	2	16	no	—
94	C₂×C₄×C₁₉	—	2	16	no	—
95	C₂×C₄×C₁₉	—	2	16	no	—
96	C₂×C₄×C₁₉	—	4	8	no	—
97	C₂×C₄×C₁₉	—	4	16	no	—
98	C₂×C₄×C₁₉	C₂³×C₄	1	2	yes	—
99	C₂×C₄×C₁₉	C₂³×C₄	1	2	yes	—
100	C₂×C₄×C₁₉	C₂³×C₄	1	4	yes	—
101	C₂×C₄×C₁₉	C₂³×C₄	1	4	yes	—
102	C₂×C₄×C₁₉	C₂³×C₄	1	16	yes	—
103	C₂²×#125	C₂×C₁₉	1	1	yes	characteristic
104	#2382	C₂×C₁₉	1	4	yes	complemented
105	#2371	C₂×C₁₉	1	4	yes	complemented
106	#2345	C₂×C₁₉	1	8	yes	—
107	#2336	C₂×C₁₉	1	8	yes	complemented
108	#2313	C₂×C₁₉	1	4	yes	—
109	#2283	C₂×C₁₉	1	2	yes	—
110	C₂²×C₁₉	—	2	4	no	—
111	C₂²×C₁₉	—	2	8	no	—
112	C₂²×C₁₉	#519	1	4	yes	—
113	C₂²×C₁₉	C₂⁴×C₄	1	1	yes	characteristic
114	C₂²×C₁₉	C₂×2₋¹⁺⁴	1	2	yes	—
115	C₂²×C₁₉	#585	1	4	yes	—
116	C₄×C₁₉	—	2	2	no	—
117	C₄×C₁₉	—	2	2	no	—
118	C₄×C₁₉	—	2	4	no	—
119	C₄×C₁₉	—	2	16	no	—
120	C₄×C₁₉	—	2	4	no	—
121	C₄×C₁₉	—	4	16	no	—
122	C₂×#125	C₄×C₁₉	1	16	yes	complemented
123	C₂×#125	C₂²×C₁₉	1	4	yes	—
124	C₂×#125	C₂²×C₁₉	1	2	yes	—
125	C₂×#125	C₂²×C₁₉	1	2	yes	—
126	C₂²×C₄⋊C₄	C₂²×C₁₉	1	2	yes	—
127	C₂²×C₄⋊C₄	C₂²×C₁₉	1	4	yes	—
128	C₂²×C₄²	C₂²×C₁₉	1	1	yes	characteristic
129	#389	C₂²×C₁₉	1	16	yes	—
130	#387	C₂²×C₁₉	1	16	yes	—
131	#387	C₂²×C₁₉	1	16	yes	—
132	#387	C₂²×C₁₉	1	16	yes	—
133	#387	C₂²×C₁₉	1	16	yes	—
134	#383	C₂²×C₁₉	1	4	yes	—
135	#382	C₂²×C₁₉	1	4	yes	—
136	#381	C₂²×C₁₉	1	4	yes	—
137	#380	C₂²×C₁₉	1	8	yes	—
138	#380	C₂²×C₁₉	1	4	yes	—
139	C₄×C₄⋊C₄	C₂²×C₁₉	1	4	yes	—
140	C₄×C₄⋊C₄	C₂²×C₁₉	1	16	yes	—
141	C₄×C₂²⋊C₄	C₂²×C₁₉	1	8	yes	—
142	C₄×C₂²⋊C₄	C₂²×C₁₉	1	4	yes	—
143	#376	C₂²×C₁₉	1	4	yes	—
144	C₂×C₁₉	—	2	4	no	—
145	C₂×C₁₉	C₂×#519	1	1	yes	characteristic
146	C₂×C₁₉	#3428	1	2	yes	—
147	C₂×C₁₉	#3553	1	4	yes	—
148	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
149	C₂³×C₄	C₂³×C₁₉	1	4	yes	—
150	C₂³×C₄	C₂³×C₁₉	1	2	yes	—
151	#125	C₂×C₄×C₁₉	1	32	yes	—
152	#125	C₂×C₄×C₁₉	1	16	yes	—
153	#125	C₂×C₄×C₁₉	1	16	yes	—
154	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	16	yes	—
155	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	32	yes	—
156	C₂×C₄⋊C₄	C₂³×C₁₉	1	8	yes	—
157	C₂×C₄⋊C₄	C₂³×C₁₉	1	16	yes	—
158	C₂×C₄⋊C₄	C₂³×C₁₉	1	8	yes	—
159	C₂×C₄⋊C₄	C₂³×C₁₉	1	4	yes	—
160	C₂×C₄⋊C₄	C₂³×C₁₉	1	4	yes	—
161	C₂×C₄⋊C₄	C₂³×C₁₉	1	8	yes	—
162	C₂×C₂²⋊C₄	C₂³×C₁₉	1	8	yes	—
163	C₂×C₄²	C₂×C₄×C₁₉	1	8	yes	—
164	C₂×C₄²	C₂³×C₁₉	1	8	yes	—
165	C₂×C₄²	C₂³×C₁₉	1	16	yes	—
166	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
167	C₂×C₄²	C₂³×C₁₉	1	2	yes	—
168	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
169	#95	C₂³×C₁₉	1	8	yes	—
170	#95	C₂³×C₁₉	1	8	yes	—
171	#95	C₂³×C₁₉	1	8	yes	—
172	#95	C₂³×C₁₉	1	16	yes	—
173	#95	C₂³×C₁₉	1	16	yes	—
174	#95	C₂³×C₁₉	1	8	yes	—
175	C₁₉	#34372	1	1	yes	characteristic, direct product factor, Sylow
176	C₂⁴	C₂⁴×C₁₉	1	1	yes	characteristic
177	C₂²×C₄	—	2	4	no	—
178	C₂²×C₄	—	4	4	no	—
179	C₂²×C₄	—	4	8	no	—
180	C₂²×C₄	C₂²×C₄×C₁₉	1	4	yes	—
181	C₂²×C₄	C₂²×C₄×C₁₉	1	8	yes	—
182	C₂²×C₄	C₂²×C₄×C₁₉	1	16	yes	—
183	C₂²×C₄	C₂⁴×C₁₉	1	16	yes	—
184	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
185	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
186	C₂²×C₄	C₂⁴×C₁₉	1	2	yes	—
187	C₂²×C₄	C₂⁴×C₁₉	1	8	yes	—
188	C₂²×C₄	C₂⁴×C₁₉	1	2	yes	—
189	C₄⋊C₄	—	4	16	no	—
190	C₄⋊C₄	—	4	8	no	—
191	C₄⋊C₄	C₂²×C₄×C₁₉	1	32	yes	—
192	C₄⋊C₄	C₂²×C₄×C₁₉	1	16	yes	—
193	C₄⋊C₄	C₂²×C₄×C₁₉	1	16	yes	—
194	C₄⋊C₄	C₂²×C₄×C₁₉	1	32	yes	—
195	C₂²⋊C₄	—	2	16	no	—
196	C₄²	—	4	8	no	—
197	C₄²	—	4	16	no	—
198	C₄²	C₂²×C₄×C₁₉	1	4	yes	—
199	C₄²	C₂²×C₄×C₁₉	1	8	yes	—
200	C₄²	C₂²×C₄×C₁₉	1	4	yes	—
201	C₂³	—	2	4	no	—
202	C₂³	C₂³×C₄×C₁₉	1	2	yes	—
203	C₂³	C₁₉×2₋¹⁺⁴	1	4	yes	—
204	C₂³	C₂⁵×C₁₉	1	1	yes	characteristic
205	C₂×C₄	—	2	8	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	2	16	no	—
210	C₂×C₄	—	2	4	no	—
211	C₂×C₄	—	2	16	no	—
212	C₂×C₄	—	2	16	no	—
213	C₂×C₄	—	2	16	no	—
214	C₂×C₄	—	4	8	no	—
215	C₂×C₄	—	4	16	no	—
216	C₂×C₄	C₂³×C₄×C₁₉	1	2	yes	—
217	C₂×C₄	C₂³×C₄×C₁₉	1	2	yes	—
218	C₂×C₄	C₂³×C₄×C₁₉	1	4	yes	—
219	C₂×C₄	C₂³×C₄×C₁₉	1	4	yes	—
220	C₂×C₄	C₂³×C₄×C₁₉	1	16	yes	—
221	C₂²	—	2	4	no	—
222	C₂²	—	2	8	no	—
223	C₂²	C₁₉×#519	1	4	yes	—
224	C₂²	C₂⁴×C₄×C₁₉	1	1	yes	characteristic
225	C₂²	C₂×C₁₉×2₋¹⁺⁴	1	2	yes	—
226	C₂²	C₁₉×#585	1	4	yes	—
227	C₄	—	2	2	no	—
228	C₄	—	2	2	no	—
229	C₄	—	2	4	no	—
230	C₄	—	2	16	no	—
231	C₄	—	2	4	no	—
232	C₄	—	4	16	no	—
233	C₂	—	2	4	no	—
234	C₂	C₂×C₁₉×#519	1	1	yes	characteristic
235	C₂	C₁₉×#3428	1	2	yes	—
236	C₂	C₁₉×#3553	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₉	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₁₉
Frattini, cosocle	C₂³	C₂⁵×C₁₉
prime radical	C₂⁴×C₁₉	C₂⁴
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₉×#2382	C₂	semidirect
3. C₁₉×#2371	C₂	semidirect
5. C₁₉×#2336	C₂	semidirect
8. C₂×C₁₉×#125	C₄	semidirect
82. #34372	C₁₉	direct (N × H)
104. #2382	C₂×C₁₉	semidirect
105. #2371	C₂×C₁₉	semidirect
107. #2336	C₂×C₁₉	semidirect
122. C₂×#125	C₄×C₁₉	semidirect
175. C₁₉	#34372	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₁₉×#125	2
C₁₉×#2382	2
C₁₉×#2371	2
C₁₉×#2345	2
C₁₉×#2336	2
C₁₉×#2313	2
C₁₉×#2283	2
#34372	19

Subgroups — C19×#34372 #18040331

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#34372 #18040331