Subgroups — #18081077

Order 4864.

398 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14564354	C₂	1	2	yes	maximal
2	C₂×#11926295	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×C₁₉⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#14561476	C₂	1	2	yes	maximal
5	C₄×C₁₉×#106	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
7	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
8	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
10	#11926299	—	2	2	no	—
11	#11926295	—	2	2	no	—
12	C₈×C₁₉⋊C₈	—	2	2	no	—
13	C₂×C₁₉×#106	C₄	1	1	yes	characteristic
14	C₂×C₁₉×#106	C₄	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
17	C₁₉×#340	C₄	1	2	yes	—
18	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
21	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₁₉	—	2	2	no	—
23	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
24	C₂²×C₁₉⋊C₈	—	2	1	no	—
25	C₂²×C₁₉⋊C₈	—	2	2	no	—
26	C₂²×C₁₉⋊C₈	D₄	1	2	yes	—
27	C₁₉×#106	—	2	2	no	—
28	C₁₉×#106	C₈	1	4	yes	complemented
29	C₄×C₈×C₁₉	D₄	1	2	yes	—
30	C₄×C₈×C₁₉	Q₈	1	2	yes	—
31	C₁₉×#95	—	2	2	no	—
32	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
33	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
34	C₄⋊(C₁₉⋊C₈)	—	4	4	no	—
35	C₄×C₁₉⋊C₈	—	2	1	no	—
36	C₄×C₁₉⋊C₈	—	2	1	no	—
37	C₄×C₁₉⋊C₈	—	2	1	no	—
38	C₄×C₁₉⋊C₈	Q₈	1	2	yes	—
39	C₂²×C₄×C₁₉	—	2	2	no	—
40	C₂²×C₄×C₁₉	—	2	2	no	—
41	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
44	C₂×C₈×C₁₉	—	2	1	no	—
45	C₂×C₈×C₁₉	—	2	1	no	—
46	C₂×C₈×C₁₉	C₄⋊C₄	1	2	yes	—
47	C₂×C₈×C₁₉	C₄⋊C₄	1	2	yes	—
48	C₂×C₈×C₁₉	C₂×C₈	1	2	yes	—
49	C₂×C₈×C₁₉	M₁₆	1	2	yes	—
50	C₄⋊C₄×C₁₉	—	2	2	no	—
51	C₄⋊C₄×C₁₉	—	2	2	no	—
52	C₄⋊C₄×C₁₉	—	2	2	no	—
53	C₄⋊C₄×C₁₉	C₂×C₈	1	4	yes	—
54	C₄²×C₁₉	—	4	2	no	—
55	C₄²×C₁₉	—	4	2	no	—
56	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
57	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
58	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
59	C₄²×C₁₉	#41	1	1	yes	characteristic
60	C₂×C₁₉⋊C₈	—	2	4	no	—
61	C₂×C₁₉⋊C₈	—	2	2	no	—
62	C₂×C₁₉⋊C₈	—	2	2	no	—
63	C₂×C₁₉⋊C₈	—	2	2	no	—
64	C₂×C₁₉⋊C₈	—	4	2	no	—
65	C₂×C₁₉⋊C₈	—	4	2	no	—
66	C₂×C₁₉⋊C₈	—	4	2	no	—
67	C₂×C₁₉⋊C₈	SD₁₆	1	4	yes	—
68	#9593	—	19	1	no	maximal, Sylow
69	C₂³×C₁₉	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₁₉	—	2	2	no	—
71	C₂×C₄×C₁₉	—	2	4	no	—
72	C₂×C₄×C₁₉	—	4	2	no	—
73	C₂×C₄×C₁₉	—	4	2	no	—
74	C₂×C₄×C₁₉	—	4	2	no	—
75	C₂×C₄×C₁₉	—	4	2	no	—
76	C₂×C₄×C₁₉	—	4	2	no	—
77	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₁₉	C₄×Q₈	1	1	yes	characteristic
79	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
80	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
81	C₂×C₄×C₁₉	#125	1	1	yes	characteristic
82	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
83	C₂×C₄×C₁₉	C₂²×C₈	1	1	yes	characteristic
84	C₂×C₄×C₁₉	C₂×M₁₆	1	1	yes	characteristic
85	C₂×C₄×C₁₉	#131	1	1	yes	characteristic
86	C₂×C₄×C₁₉	#131	1	1	yes	characteristic
87	C₂×C₄×C₁₉	C₂×SD₁₆	1	2	yes	—
88	C₂×C₄×C₁₉	#135	1	2	yes	—
89	C₈×C₁₉	—	2	2	no	—
90	C₈×C₁₉	C₄⋊C₈	1	4	yes	complemented
91	C₁₉⋊C₈	—	2	4	no	—
92	C₁₉⋊C₈	—	2	4	no	—
93	C₁₉⋊C₈	—	8	2	no	—
94	#1939	—	19	2	no	—
95	C₄×#106	D₁₉	1	1	yes	characteristic
96	C₂×#334	—	19	1	no	—
97	C₂×C₈²	—	19	1	no	—
98	#1279	—	19	2	no	—
99	C₂²×C₁₉	#389	1	1	yes	characteristic
100	C₂²×C₁₉	C₈×D₄	1	1	yes	characteristic
101	C₂²×C₁₉	#436	1	1	yes	characteristic
102	C₂²×C₁₉	C₄×SD₁₆	1	2	yes	—
103	C₂²×C₁₉	#443	1	2	yes	—
104	C₄×C₁₉	—	2	1	no	—
105	C₄×C₁₉	—	2	1	no	—
106	C₄×C₁₉	—	4	4	no	—
107	C₄×C₁₉	—	8	2	no	—
108	C₄×C₁₉	C₂×C₄⋊C₈	1	1	yes	characteristic
109	C₄×C₁₉	#424	1	1	yes	characteristic
110	C₄×C₁₉	C₈×Q₈	1	1	yes	characteristic
111	C₄×C₁₉	#446	1	1	yes	characteristic
112	C₄×C₁₉	#465	1	2	yes	—
113	C₄×C₁₉	#498	1	1	yes	characteristic
114	C₄×C₁₉	#499	1	1	yes	characteristic
115	C₂×#106	Dic₁₉	1	1	yes	characteristic
116	C₂×#106	Dic₁₉	1	1	yes	characteristic
117	C₂×C₄⋊C₈	—	19	2	no	—
118	C₂×C₄⋊C₈	—	38	2	no	—
119	C₂×C₄×C₈	—	19	1	no	—
120	C₂×C₄×C₈	—	19	1	no	—
121	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
122	C₄×C₄⋊C₄	C₂×D₁₉	1	2	yes	—
123	#340	Dic₁₉	1	2	yes	—
124	#336	—	38	2	no	—
125	#334	—	38	2	no	—
126	C₈²	—	38	2	no	—
127	C₂×C₁₉	C₈×SD₁₆	1	2	yes	—
128	C₂×C₁₉	#1590	1	2	yes	—
129	C₂×C₁₉	#1939	1	1	yes	characteristic
130	C₂×C₁₉	#1941	1	1	yes	characteristic
131	C₂×C₁₉	#1945	1	1	yes	characteristic
132	C₂²×C₈	—	19	2	no	—
133	C₂²×C₈	—	38	1	no	—
134	C₂²×C₈	—	38	2	no	—
135	C₂²×C₈	C₂×Dic₁₉	1	1	yes	characteristic
136	C₂²×C₈	C₂×Dic₁₉	1	1	yes	characteristic
137	C₂×C₄⋊C₄	C₂×Dic₁₉	1	2	yes	—
138	C₂×C₄⋊C₄	C₂×Dic₁₉	1	2	yes	—
139	C₂×C₄²	—	2	2	no	—
140	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
141	#106	—	2	2	no	—
142	#106	C₁₉⋊C₈	1	4	yes	complemented
143	C₄⋊C₈	—	38	2	no	—
144	C₄⋊C₈	—	38	2	no	—
145	C₄⋊C₈	—	76	4	no	—
146	C₄×C₈	—	19	2	no	—
147	C₄×C₈	—	38	1	no	—
148	C₄×C₈	—	38	1	no	—
149	C₄×C₈	—	38	1	no	—
150	C₄×C₈	Dic₃₈	1	2	yes	—
151	C₄×C₈	D₇₆	1	2	yes	—
152	#95	—	2	2	no	—
153	C₁₉	#9593	1	1	yes	characteristic, complemented, Sylow
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
157	C₂²×C₄	#64543	1	1	yes	characteristic
158	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
159	C₂×C₈	—	2	1	no	—
160	C₂×C₈	—	2	1	no	—
161	C₂×C₈	—	19	4	no	—
162	C₂×C₈	—	38	4	no	—
163	C₂×C₈	—	38	2	no	—
164	C₂×C₈	—	38	2	no	—
165	C₂×C₈	—	38	2	no	—
166	C₂×C₈	—	76	2	no	—
167	C₂×C₈	—	76	2	no	—
168	C₂×C₈	—	76	2	no	—
169	C₂×C₈	C₂×C₁₉⋊C₈	1	2	yes	—
170	C₂×C₈	#64520	1	2	yes	—
171	C₂×C₈	C₄⋊Dic₁₉	1	2	yes	—
172	C₂×C₈	C₄⋊Dic₁₉	1	2	yes	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	—	2	2	no	—
176	C₄⋊C₄	C₂×C₁₉⋊C₈	1	4	yes	—
177	C₄²	—	4	2	no	—
178	C₄²	—	4	2	no	—
179	C₄²	C₂×Dic₃₈	1	1	yes	characteristic
180	C₄²	C₂×D₇₆	1	1	yes	characteristic
181	C₄²	Q₈×D₁₉	1	1	yes	characteristic
182	C₄²	#64545	1	1	yes	characteristic
183	C₂³	D₄×Dic₁₉	1	1	yes	characteristic
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	#10597334	1	1	yes	characteristic
192	C₂×C₄	#10597336	1	1	yes	characteristic
193	C₂×C₄	#10597341	1	1	yes	characteristic
194	C₂×C₄	#10597343	1	1	yes	characteristic
195	C₂×C₄	SD₁₆×D₁₉	1	2	yes	—
196	C₂×C₄	#10597362	1	2	yes	—
197	C₂×C₄	C₂²×C₁₉⋊C₈	1	1	yes	characteristic
198	C₂×C₄	C₂×#64520	1	1	yes	characteristic
199	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
200	C₂×C₄	Q₈×Dic₁₉	1	1	yes	characteristic
201	C₂×C₄	#10597394	1	1	yes	characteristic
202	C₂×C₄	#10597394	1	1	yes	characteristic
203	C₈	—	2	2	no	—
204	C₈	—	38	4	no	—
205	C₈	—	38	4	no	—
206	C₈	—	152	2	no	—
207	C₈	C₄⋊(C₁₉⋊C₈)	1	4	yes	complemented
208	C₂²	#11926443	1	1	yes	characteristic
209	C₂²	D₄×C₁₉⋊C₈	1	1	yes	characteristic
210	C₂²	#11926622	1	1	yes	characteristic
211	C₂²	SD₁₆×Dic₁₉	1	2	yes	—
212	C₂²	#11926661	1	2	yes	—
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	—	4	4	no	—
216	C₄	—	8	2	no	—
217	C₄	C₂×C₄⋊(C₁₉⋊C₈)	1	1	yes	characteristic
218	C₄	#11926571	1	1	yes	characteristic
219	C₄	Q₈×C₁₉⋊C₈	1	1	yes	characteristic
220	C₄	#11926675	1	1	yes	characteristic
221	C₄	#11926774	1	2	yes	—
222	C₄	#11926941	1	1	yes	characteristic
223	C₄	#11926944	1	1	yes	characteristic
224	C₂	SD₁₆×C₁₉⋊C₈	1	2	yes	—
225	C₂	#14562526	1	2	yes	—
226	C₂	#14564354	1	1	yes	characteristic
227	C₂	#14564364	1	1	yes	characteristic
228	C₂	#14564384	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₁₉
commutator	C₂×C₄×C₁₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₄×C₁₉×#106	C₂
socle, prime radical	C₂³×C₁₉	C₄×D₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₄×#106	D₁₉
hypocenter	C₁₉	#9593
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×#106

Complements

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

N	H ≅ G/N	type
28. C₁₉×#106	C₈	semidirect
90. C₈×C₁₉	C₄⋊C₈	semidirect
142. #106	C₁₉⋊C₈	semidirect
153. C₁₉	#9593	semidirect
207. C₈	C₄⋊(C₁₉⋊C₈)	semidirect

Maximal subgroups

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

subgroup	index
#14564354	2
C₂×#11926295	2
C₂×C₈×C₁₉⋊C₈	2
#14561476	2
C₄×C₁₉×#106	2
#9593	19