Subgroups — #18069094

Order 4864.

338 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊(C₁₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	#14563381	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1767	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
5	C₂×C₄×C₁₉⋊C₈	—	2	1	no	—
6	C₂×C₄×C₁₉⋊C₈	—	2	1	no	—
7	#11926299	—	2	1	no	—
8	#11926299	C₄	1	2	yes	—
9	#11926299	C₄	1	2	yes	—
10	C₂×C₁₉×C₄⋊C₈	C₄	1	2	yes	—
11	C₂×C₁₉×#97	—	2	1	no	—
12	C₄³×C₁₉	C₂²	1	1	yes	characteristic
13	C₁₉×#336	—	2	1	no	—
14	C₂²×C₈×C₁₉	—	2	2	no	—
15	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
19	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
20	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
21	C₂×C₄²×C₁₉	Q₈	1	1	yes	characteristic
22	C₂²×C₁₉⋊C₈	—	2	2	no	—
23	C₂²×C₁₉⋊C₈	—	2	2	no	—
24	C₁₉×C₄⋊C₈	—	2	2	no	—
25	C₁₉×C₄⋊C₈	C₈	1	4	yes	complemented
26	C₁₉×#97	—	4	1	no	—
27	C₁₉×#97	—	4	1	no	—
28	C₄⋊(C₁₉⋊C₈)	—	2	4	no	—
29	C₄×C₁₉⋊C₈	—	2	4	no	—
30	C₄×C₁₉⋊C₈	—	4	1	no	—
31	C₄×C₁₉⋊C₈	—	4	1	no	—
32	C₂²×C₄×C₁₉	C₄²	1	1	yes	characteristic
33	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₁₉	—	2	4	no	—
40	C₂×C₈×C₁₉	—	4	2	no	—
41	C₂×C₈×C₁₉	—	4	2	no	—
42	C₄²×C₁₉	—	2	2	no	—
43	C₄²×C₁₉	—	2	2	no	—
44	C₄²×C₁₉	—	2	2	no	—
45	C₄²×C₁₉	—	4	1	no	—
46	C₄²×C₁₉	—	4	1	no	—
47	C₄²×C₁₉	C₂×C₈	1	2	yes	—
48	C₄²×C₁₉	M₁₆	1	2	yes	—
49	C₄²×C₁₉	D₈	1	1	yes	characteristic
50	C₄²×C₁₉	SD₁₆	1	1	yes	characteristic
51	C₄²×C₁₉	SD₁₆	1	1	yes	characteristic
52	C₄²×C₁₉	Q₁₆	1	1	yes	characteristic
53	C₂×C₁₉⋊C₈	—	2	4	no	—
54	C₂×C₁₉⋊C₈	—	2	4	no	—
55	C₂×C₁₉⋊C₈	—	4	2	no	—
56	C₂×C₁₉⋊C₈	—	4	2	no	—
57	C₂×C₁₉⋊C₈	—	4	2	no	—
58	C₂×C₁₉⋊C₈	—	4	2	no	—
59	#7027	—	19	1	no	maximal, Sylow
60	C₂³×C₁₉	#95	1	1	yes	characteristic
61	C₂×C₄×C₁₉	—	2	1	no	—
62	C₂×C₄×C₁₉	—	2	1	no	—
63	C₂×C₄×C₁₉	—	2	1	no	—
64	C₂×C₄×C₁₉	—	2	1	no	—
65	C₂×C₄×C₁₉	—	2	1	no	—
66	C₂×C₄×C₁₉	—	2	1	no	—
67	C₂×C₄×C₁₉	—	2	1	no	—
68	C₂×C₄×C₁₉	—	2	1	no	—
69	C₂×C₄×C₁₉	—	2	1	no	—
70	C₂×C₄×C₁₉	—	2	1	no	—
71	C₂×C₄×C₁₉	—	2	1	no	—
72	C₂×C₄×C₁₉	—	2	1	no	—
73	C₂×C₄×C₁₉	C₄×C₈	1	1	yes	characteristic
74	C₂×C₄×C₁₉	#97	1	1	yes	characteristic
75	C₂×C₄×C₁₉	C₂²⋊C₈	1	2	yes	—
76	C₂×C₄×C₁₉	#102	1	1	yes	characteristic
77	C₂×C₄×C₁₉	#102	1	1	yes	characteristic
78	C₂×C₄×C₁₉	#103	1	1	yes	characteristic
79	C₂×C₄×C₁₉	#103	1	1	yes	characteristic
80	C₂×C₄×C₁₉	#104	1	2	yes	—
81	C₂×C₄×C₁₉	#104	1	2	yes	—
82	C₂×C₄×C₁₉	C₄⋊C₈	1	2	yes	—
83	C₂×C₄×C₁₉	#106	1	1	yes	characteristic
84	C₂×C₄×C₁₉	#107	1	1	yes	characteristic
85	C₂×C₄×C₁₉	#108	1	2	yes	—
86	C₈×C₁₉	—	4	4	no	—
87	C₁₉⋊C₈	—	4	4	no	—
88	C₁₉⋊C₈	—	4	4	no	—
89	#1767	D₁₉	1	1	yes	characteristic
90	C₄×C₄⋊C₈	—	19	1	no	—
91	#1744	—	19	1	no	—
92	C₂²×C₁₉	#336	1	1	yes	characteristic
93	C₂²×C₁₉	#339	1	1	yes	characteristic
94	C₂²×C₁₉	#339	1	1	yes	characteristic
95	C₂²×C₁₉	#340	1	1	yes	characteristic
96	C₂²×C₁₉	#341	1	1	yes	characteristic
97	C₂²×C₁₉	#344	1	1	yes	characteristic
98	C₂²×C₁₉	#344	1	1	yes	characteristic
99	C₄×C₁₉	—	2	1	no	—
100	C₄×C₁₉	—	2	1	no	—
101	C₄×C₁₉	—	2	2	no	—
102	C₄×C₁₉	—	2	1	no	—
103	C₄×C₁₉	—	2	1	no	—
104	C₄×C₁₉	—	2	2	no	—
105	C₄×C₁₉	—	2	1	no	—
106	C₄×C₁₉	—	2	1	no	—
107	C₄×C₁₉	—	2	2	no	—
108	C₄×C₁₉	D₄⋊C₈	1	2	yes	—
109	C₄×C₁₉	Q₈⋊C₈	1	2	yes	—
110	C₂×C₄⋊C₈	—	38	1	no	—
111	C₂×C₄⋊C₈	Dic₁₉	1	2	yes	—
112	C₂×#97	—	2	1	no	—
113	C₂×C₄×C₈	—	38	1	no	—
114	C₂×C₄×C₈	—	38	1	no	—
115	C₄³	C₂×D₁₉	1	1	yes	characteristic
116	#336	—	2	1	no	—
117	#336	—	19	2	no	—
118	#336	—	19	2	no	—
119	#336	—	38	1	no	—
120	C₂×C₁₉	#1278	1	1	yes	characteristic
121	C₂×C₁₉	#1279	1	1	yes	characteristic
122	C₂×C₁₉	#1281	1	1	yes	characteristic
123	C₂×C₁₉	#1300	1	1	yes	characteristic
124	C₂×C₁₉	#1301	1	1	yes	characteristic
125	C₂×C₁₉	#1302	1	1	yes	characteristic
126	C₂×C₁₉	#1303	1	1	yes	characteristic
127	C₂²×C₈	—	2	2	no	—
128	C₂²×C₈	—	38	2	no	—
129	C₂²×C₈	—	38	2	no	—
130	C₂×C₄²	Dic₃₈	1	1	yes	characteristic
131	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
132	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
133	C₂×C₄²	D₇₆	1	1	yes	characteristic
134	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
135	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
136	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
137	C₄⋊C₈	—	2	2	no	—
138	C₄⋊C₈	—	38	4	no	—
139	C₄⋊C₈	C₁₉⋊C₈	1	4	yes	complemented
140	#97	—	4	1	no	—
141	#97	—	4	1	no	—
142	C₄×C₈	—	38	4	no	—
143	C₄×C₈	—	76	1	no	—
144	C₄×C₈	—	76	1	no	—
145	C₁₉	#7027	1	1	yes	characteristic, complemented, Sylow
146	C₂²×C₄	C₄×Dic₁₉	1	1	yes	characteristic
147	C₂²×C₄	Dic₁₉⋊C₄	1	1	yes	characteristic
148	C₂²×C₄	Dic₁₉⋊C₄	1	1	yes	characteristic
149	C₂²×C₄	C₄⋊Dic₁₉	1	1	yes	characteristic
150	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
151	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
152	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
153	C₂×C₈	—	2	4	no	—
154	C₂×C₈	—	4	2	no	—
155	C₂×C₈	—	4	2	no	—
156	C₂×C₈	—	38	4	no	—
157	C₂×C₈	—	38	4	no	—
158	C₂×C₈	—	76	2	no	—
159	C₂×C₈	—	76	2	no	—
160	C₂×C₈	—	76	2	no	—
161	C₂×C₈	—	76	2	no	—
162	C₄²	—	2	2	no	—
163	C₄²	—	2	2	no	—
164	C₄²	—	2	2	no	—
165	C₄²	—	4	1	no	—
166	C₄²	—	4	1	no	—
167	C₄²	C₂×C₁₉⋊C₈	1	2	yes	—
168	C₄²	#64520	1	2	yes	—
169	C₄²	#64525	1	1	yes	characteristic
170	C₄²	#64526	1	1	yes	characteristic
171	C₄²	#64527	1	1	yes	characteristic
172	C₄²	#64528	1	1	yes	characteristic
173	C₂³	C₁₉⋊#95	1	1	yes	characteristic
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	2	1	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	C₄×C₁₉⋊C₈	1	1	yes	characteristic
187	C₂×C₄	#10597260	1	1	yes	characteristic
188	C₂×C₄	C₄⋊(C₁₉⋊C₈)	1	2	yes	—
189	C₂×C₄	#10597264	1	1	yes	characteristic
190	C₂×C₄	#10597265	1	1	yes	characteristic
191	C₂×C₄	#10597266	1	1	yes	characteristic
192	C₂×C₄	#10597267	1	1	yes	characteristic
193	C₂×C₄	#10597279	1	2	yes	—
194	C₂×C₄	#10597282	1	2	yes	—
195	C₂×C₄	C₂²⋊(C₁₉⋊C₈)	1	2	yes	—
196	C₂×C₄	#10597289	1	1	yes	characteristic
197	C₂×C₄	#10597292	1	1	yes	characteristic
198	C₂×C₄	#10597294	1	2	yes	—
199	C₈	—	4	4	no	—
200	C₈	—	76	4	no	—
201	C₈	—	76	4	no	—
202	C₂²	#11926299	1	1	yes	characteristic
203	C₂²	#11926303	1	1	yes	characteristic
204	C₂²	#11926305	1	1	yes	characteristic
205	C₂²	#11926306	1	1	yes	characteristic
206	C₂²	#11926308	1	1	yes	characteristic
207	C₂²	#11926315	1	1	yes	characteristic
208	C₂²	#11926315	1	1	yes	characteristic
209	C₄	—	2	1	no	—
210	C₄	—	2	1	no	—
211	C₄	—	2	2	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	2	2	no	—
215	C₄	—	2	1	no	—
216	C₄	—	2	1	no	—
217	C₄	—	2	2	no	—
218	C₄	D₄⋊(C₁₉⋊C₈)	1	2	yes	—
219	C₄	Q₈⋊(C₁₉⋊C₈)	1	2	yes	—
220	C₂	#14561473	1	1	yes	characteristic
221	C₂	#14561476	1	1	yes	characteristic
222	C₂	#14561480	1	1	yes	characteristic
223	C₂	#14561527	1	1	yes	characteristic
224	C₂	#14561530	1	1	yes	characteristic
225	C₂	#14561533	1	1	yes	characteristic
226	C₂	#14561536	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²⋊Dic₁₉
commutator	C₂×C₄×C₁₉	C₄×C₈
Frattini	C₄³	C₂×D₁₉
Fitting	C₁₉×#1767	C₂
socle, prime radical	C₂³×C₁₉	#95
cosocle	C₄³×C₁₉	C₂²
hypercenter	#1767	D₁₉
hypocenter	C₁₉	#7027
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1767

Complements

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

N	H ≅ G/N	type
25. C₁₉×C₄⋊C₈	C₈	semidirect
139. C₄⋊C₈	C₁₉⋊C₈	semidirect
145. C₁₉	#7027	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊(C₁₉⋊C₈)	2
#14563381	2
C₁₉×#1767	2
#7027	19