Subgroups — #18069727

Order 4864.

246 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₁₉⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	C₂×#11926295	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1846	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₁₉⋊C₁₆	—	2	1	no	—
5	C₂×C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
6	C₄⋊(C₁₉⋊C₁₆)	—	2	1	no	—
7	C₄⋊(C₁₉⋊C₁₆)	C₄	1	2	yes	complemented
8	#11926295	—	2	1	no	—
9	#11926295	C₄	1	2	yes	complemented
10	C₂×C₁₉×#107	C₄	1	2	yes	—
11	C₂×C₁₉×#106	—	2	1	no	—
12	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
13	C₁₉×#382	—	2	1	no	—
14	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
17	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
18	C₂²×C₁₉⋊C₈	—	4	1	no	—
19	C₁₉×#107	—	4	2	no	—
20	C₁₉×#106	—	4	1	no	—
21	C₁₉×#106	—	4	1	no	—
22	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
23	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
25	C₄×C₈×C₁₉	Q₈	1	1	yes	characteristic
26	C₁₉×#95	—	4	1	no	—
27	C₁₉×#95	—	4	1	no	—
28	C₂×C₁₉⋊C₁₆	—	2	2	no	—
29	C₂×C₁₉⋊C₁₆	—	4	1	no	—
30	C₂×C₁₉⋊C₁₆	—	4	1	no	—
31	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
32	C₄⋊(C₁₉⋊C₈)	—	4	1	no	—
33	C₂²×C₄×C₁₉	—	4	2	no	—
34	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₉	D₈	1	1	yes	characteristic
36	C₂²×C₄×C₁₉	SD₁₆	1	1	yes	characteristic
37	C₂×C₈×C₁₉	—	2	1	no	—
38	C₂×C₈×C₁₉	—	2	1	no	—
39	C₂×C₈×C₁₉	C₄²	1	1	yes	characteristic
40	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₁₉	D₈	1	1	yes	characteristic
44	C₂×C₈×C₁₉	SD₁₆	1	1	yes	characteristic
45	C₂×C₈×C₁₉	SD₁₆	1	1	yes	characteristic
46	C₂×C₈×C₁₉	Q₁₆	1	1	yes	characteristic
47	C₄⋊C₄×C₁₉	—	4	2	no	—
48	C₄⋊C₄×C₁₉	—	4	2	no	—
49	C₄²×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
50	C₄²×C₁₉	C₄⋊C₄	1	1	yes	characteristic
51	C₄²×C₁₉	SD₁₆	1	1	yes	characteristic
52	C₄²×C₁₉	Q₁₆	1	1	yes	characteristic
53	C₂×C₁₉⋊C₈	—	4	2	no	—
54	C₂×C₁₉⋊C₈	—	8	2	no	—
55	C₁₉⋊C₁₆	—	4	2	no	—
56	#7275	—	19	1	no	maximal, Sylow
57	C₂³×C₁₉	#102	1	1	yes	characteristic
58	C₂×C₄×C₁₉	—	2	2	no	—
59	C₂×C₄×C₁₉	—	2	2	no	—
60	C₂×C₄×C₁₉	—	8	2	no	—
61	C₂×C₄×C₁₉	—	8	2	no	—
62	C₂×C₄×C₁₉	—	8	2	no	—
63	C₂×C₄×C₁₉	#95	1	1	yes	characteristic
64	C₂×C₄×C₁₉	#100	1	1	yes	characteristic
65	C₂×C₄×C₁₉	#101	1	1	yes	characteristic
66	C₂×C₄×C₁₉	#102	1	1	yes	characteristic
67	C₂×C₄×C₁₉	#102	1	1	yes	characteristic
68	C₂×C₄×C₁₉	#103	1	1	yes	characteristic
69	C₂×C₄×C₁₉	#103	1	1	yes	characteristic
70	C₂×C₄×C₁₉	#103	1	1	yes	characteristic
71	C₂×C₄×C₁₉	#106	1	1	yes	characteristic
72	C₂×C₄×C₁₉	#107	1	1	yes	characteristic
73	C₈×C₁₉	—	2	1	no	—
74	C₈×C₁₉	—	2	1	no	—
75	C₈×C₁₉	#102	1	1	yes	characteristic
76	C₈×C₁₉	#103	1	1	yes	characteristic
77	C₈×C₁₉	#106	1	1	yes	characteristic
78	C₈×C₁₉	#107	1	1	yes	characteristic
79	C₁₉⋊C₈	—	8	2	no	—
80	C₂×C₄⋊C₁₆	—	19	1	no	—
81	#1846	D₁₉	1	1	yes	characteristic
82	C₂×#334	—	19	1	no	—
83	C₂²×C₁₉	#331	1	1	yes	characteristic
84	C₂²×C₁₉	#332	1	1	yes	characteristic
85	C₂²×C₁₉	#340	1	1	yes	characteristic
86	C₂²×C₁₉	#359	1	2	yes	—
87	C₂²×C₁₉	#360	1	2	yes	—
88	C₄×C₁₉	—	2	1	no	—
89	C₄×C₁₉	—	2	1	no	—
90	C₄×C₁₉	—	2	1	no	—
91	C₄×C₁₉	—	2	1	no	—
92	C₄×C₁₉	—	16	2	no	—
93	C₄×C₁₉	#332	1	1	yes	characteristic
94	C₄×C₁₉	#333	1	1	yes	characteristic
95	C₄×C₁₉	#340	1	1	yes	characteristic
96	C₄×C₁₉	#343	1	1	yes	characteristic
97	C₂²×C₁₆	—	38	1	no	—
98	C₂×#107	Dic₁₉	1	2	yes	—
99	C₂×#106	—	2	1	no	—
100	C₂×C₄⋊C₈	—	38	1	no	—
101	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
102	#382	—	2	1	no	—
103	C₄⋊C₁₆	—	19	2	no	—
104	C₄⋊C₁₆	—	38	1	no	—
105	#334	—	19	2	no	—
106	#334	—	38	1	no	—
107	C₂×C₁₉	#1296	1	1	yes	characteristic
108	C₂×C₁₉	#1360	1	2	yes	—
109	C₂×C₁₉	#1363	1	2	yes	—
110	C₂×C₁₉	#1382	1	1	yes	characteristic
111	C₂×C₁₉	#1386	1	1	yes	characteristic
112	C₂²×C₈	—	76	1	no	—
113	C₂²×C₈	C₂×Dic₁₉	1	1	yes	characteristic
114	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
115	C₂×C₄⋊C₄	—	2	2	no	—
116	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
117	C₂×C₁₆	—	38	2	no	—
118	C₂×C₁₆	—	76	1	no	—
119	C₂×C₁₆	—	76	1	no	—
120	#107	—	4	2	no	—
121	#106	—	4	1	no	—
122	#106	—	4	1	no	—
123	C₄⋊C₈	—	38	2	no	—
124	C₄⋊C₈	—	76	1	no	—
125	C₄×C₈	Dic₃₈	1	1	yes	characteristic
126	C₄×C₈	C₄×D₁₉	1	1	yes	characteristic
127	C₄×C₈	C₄×D₁₉	1	1	yes	characteristic
128	C₄×C₈	D₇₆	1	1	yes	characteristic
129	#95	—	4	1	no	—
130	#95	—	4	1	no	—
131	C₁₉	#7275	1	1	yes	characteristic, complemented, Sylow
132	C₂²×C₄	—	4	2	no	—
133	C₂²×C₄	#64525	1	1	yes	characteristic
134	C₂²×C₄	#64526	1	1	yes	characteristic
135	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	76	2	no	—
139	C₂×C₈	—	152	2	no	—
140	C₂×C₈	C₄×Dic₁₉	1	1	yes	characteristic
141	C₂×C₈	Dic₁₉⋊C₄	1	1	yes	characteristic
142	C₂×C₈	C₄⋊Dic₁₉	1	1	yes	characteristic
143	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
144	C₂×C₈	#64525	1	1	yes	characteristic
145	C₂×C₈	#64526	1	1	yes	characteristic
146	C₂×C₈	#64527	1	1	yes	characteristic
147	C₂×C₈	#64528	1	1	yes	characteristic
148	C₄⋊C₄	—	4	2	no	—
149	C₄⋊C₄	—	4	2	no	—
150	C₄²	Dic₁₉⋊C₄	1	1	yes	characteristic
151	C₄²	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
152	C₄²	#64527	1	1	yes	characteristic
153	C₄²	#64528	1	1	yes	characteristic
154	C₁₆	—	76	2	no	—
155	C₂³	#10597289	1	1	yes	characteristic
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	8	2	no	—
159	C₂×C₄	—	8	2	no	—
160	C₂×C₄	—	8	2	no	—
161	C₂×C₄	#10597264	1	1	yes	characteristic
162	C₂×C₄	#10597265	1	1	yes	characteristic
163	C₂×C₄	#10597266	1	1	yes	characteristic
164	C₂×C₄	#10597267	1	1	yes	characteristic
165	C₂×C₄	C₁₉⋊#95	1	1	yes	characteristic
166	C₂×C₄	#10597289	1	1	yes	characteristic
167	C₂×C₄	#10597290	1	1	yes	characteristic
168	C₂×C₄	#10597292	1	1	yes	characteristic
169	C₂×C₄	#10597292	1	1	yes	characteristic
170	C₂×C₄	#10597293	1	1	yes	characteristic
171	C₈	—	2	1	no	—
172	C₈	—	2	1	no	—
173	C₈	—	152	2	no	—
174	C₈	#10597264	1	1	yes	characteristic
175	C₈	#10597265	1	1	yes	characteristic
176	C₈	#10597266	1	1	yes	characteristic
177	C₈	#10597267	1	1	yes	characteristic
178	C₂²	#11926288	1	1	yes	characteristic
179	C₂²	#11926290	1	1	yes	characteristic
180	C₂²	#11926306	1	1	yes	characteristic
181	C₂²	#11926351	1	2	yes	—
182	C₂²	#11926354	1	2	yes	—
183	C₄	—	2	1	no	—
184	C₄	—	2	1	no	—
185	C₄	—	2	1	no	—
186	C₄	—	2	1	no	—
187	C₄	—	16	2	no	—
188	C₄	#11926290	1	1	yes	characteristic
189	C₄	#11926293	1	1	yes	characteristic
190	C₄	#11926306	1	1	yes	characteristic
191	C₄	#11926313	1	1	yes	characteristic
192	C₂	#14561518	1	1	yes	characteristic
193	C₂	#14561678	1	2	yes	—
194	C₂	#14561687	1	2	yes	—
195	C₂	#14561730	1	1	yes	characteristic
196	C₂	#14561740	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#10597289
commutator	C₂×C₈×C₁₉	C₄²
Frattini	C₂×C₄×C₈	C₂×D₁₉
Fitting	C₁₉×#1846	C₂
socle, prime radical	C₂³×C₁₉	#102
cosocle	C₂×C₄×C₈×C₁₉	C₂²
hypercenter	#1846	D₁₉
hypocenter	C₁₉	#7275
LCS	G ▷ C₂×C₈×C₁₉ ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1846

Complements

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

N	H ≅ G/N	type
7. C₄⋊(C₁₉⋊C₁₆)	C₄	semidirect
9. #11926295	C₄	semidirect
131. C₁₉	#7275	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₁₉⋊C₁₆)	2
C₂×#11926295	2
C₁₉×#1846	2
#7275	19