Subgroups — #18069455

Order 4864.

282 conjugacy classes of subgroups.

The subgroup lattice has 196 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×#11926297	C₂	1	1	yes	maximal, characteristic
3	C₄×C₁₉×#107	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	#11926297	—	2	1	no	—
9	#11926297	C₄	1	2	yes	complemented
10	C₂×C₁₉×#107	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
12	C₄×C₄⋊C₄×C₁₉	—	2	1	no	—
13	C₁₉×#340	—	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	—	4	1	no	—
18	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
19	C₂²×C₁₉⋊C₈	—	4	1	no	—
20	C₁₉×#107	—	2	2	no	—
21	C₁₉×#107	C₈	1	4	yes	complemented
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	C₂×C₁₉⋊C₁₆	—	2	4	no	—
28	C₂×C₁₉⋊C₁₆	—	2	2	no	—
29	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
30	C₄⋊(C₁₉⋊C₈)	—	4	1	no	—
31	C₂²×C₄×C₁₉	—	4	2	no	—
32	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₉	D₈	1	1	yes	characteristic
34	C₂²×C₄×C₁₉	SD₁₆	1	1	yes	characteristic
35	C₂×C₈×C₁₉	—	2	2	no	—
36	C₂×C₈×C₁₉	C₄²	1	1	yes	characteristic
37	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₁₉	C₂×C₈	1	2	yes	—
41	C₂×C₈×C₁₉	M₁₆	1	2	yes	—
42	C₄⋊C₄×C₁₉	—	2	4	no	—
43	C₄⋊C₄×C₁₉	—	4	2	no	—
44	C₄²×C₁₉	—	8	1	no	—
45	C₄²×C₁₉	—	8	1	no	—
46	C₄²×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
47	C₄²×C₁₉	C₄⋊C₄	1	1	yes	characteristic
48	C₄²×C₁₉	SD₁₆	1	1	yes	characteristic
49	C₄²×C₁₉	Q₁₆	1	1	yes	characteristic
50	C₂×C₁₉⋊C₈	—	4	2	no	—
51	C₂×C₁₉⋊C₈	—	8	1	no	—
52	C₂×C₁₉⋊C₈	—	8	1	no	—
53	C₁₉⋊C₁₆	—	2	4	no	—
54	#7171	—	19	1	no	maximal, Sylow
55	C₂³×C₁₉	#102	1	1	yes	characteristic
56	C₂×C₄×C₁₉	—	2	2	no	—
57	C₂×C₄×C₁₉	—	4	4	no	—
58	C₂×C₄×C₁₉	—	8	2	no	—
59	C₂×C₄×C₁₉	—	8	2	no	—
60	C₂×C₄×C₁₉	#95	1	1	yes	characteristic
61	C₂×C₄×C₁₉	C₂²⋊C₈	1	2	yes	—
62	C₂×C₄×C₁₉	#102	1	1	yes	characteristic
63	C₂×C₄×C₁₉	#103	1	1	yes	characteristic
64	C₂×C₄×C₁₉	#103	1	1	yes	characteristic
65	C₂×C₄×C₁₉	#104	1	2	yes	—
66	C₂×C₄×C₁₉	#106	1	1	yes	characteristic
67	C₂×C₄×C₁₉	#107	1	1	yes	characteristic
68	C₂×C₄×C₁₉	D₁₆	1	1	yes	characteristic
69	C₂×C₄×C₁₉	SD₃₂	1	1	yes	characteristic
70	C₂×C₄×C₁₉	SD₃₂	1	1	yes	characteristic
71	C₂×C₄×C₁₉	Q₃₂	1	1	yes	characteristic
72	C₈×C₁₉	—	2	1	no	—
73	C₈×C₁₉	—	2	1	no	—
74	C₈×C₁₉	C₄×C₈	1	1	yes	characteristic
75	C₈×C₁₉	#97	1	1	yes	characteristic
76	C₈×C₁₉	C₄⋊C₈	1	2	yes	—
77	C₁₉⋊C₈	—	8	2	no	—
78	C₂×C₄×C₁₆	—	19	1	no	—
79	C₄×#107	D₁₉	1	1	yes	characteristic
80	C₂×#335	—	19	1	no	—
81	C₂²×C₁₉	D₄⋊C₈	1	2	yes	—
82	C₂²×C₁₉	#340	1	1	yes	characteristic
83	C₂²×C₁₉	#357	1	1	yes	characteristic
84	C₂²×C₁₉	#358	1	1	yes	characteristic
85	C₂²×C₁₉	#359	1	2	yes	—
86	C₄×C₁₉	—	2	1	no	—
87	C₄×C₁₉	—	2	1	no	—
88	C₄×C₁₉	—	8	4	no	—
89	C₄×C₁₉	Q₈⋊C₈	1	2	yes	—
90	C₄×C₁₉	#336	1	1	yes	characteristic
91	C₄×C₁₉	#339	1	1	yes	characteristic
92	C₄×C₁₉	#357	1	1	yes	characteristic
93	C₄×C₁₉	#358	1	1	yes	characteristic
94	C₄×C₁₉	#366	1	1	yes	characteristic
95	C₄×C₁₉	#367	1	1	yes	characteristic
96	C₂²×C₁₆	—	38	1	no	—
97	C₂×#107	Dic₁₉	1	2	yes	—
98	C₂×C₄⋊C₈	—	38	1	no	—
99	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
100	C₄×C₄⋊C₄	—	2	1	no	—
101	C₄×C₁₆	—	19	2	no	—
102	C₄×C₁₆	—	38	1	no	—
103	#340	—	2	1	no	—
104	#335	—	19	2	no	—
105	#335	—	38	1	no	—
106	C₂×C₁₉	#1279	1	1	yes	characteristic
107	C₂×C₁₉	#1331	1	2	yes	—
108	C₂×C₁₉	#1332	1	2	yes	—
109	C₂×C₁₉	#1380	1	1	yes	characteristic
110	C₂×C₁₉	#1382	1	1	yes	characteristic
111	C₂²×C₈	—	76	1	no	—
112	C₂²×C₈	C₂×Dic₁₉	1	1	yes	characteristic
113	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
114	C₂×C₄⋊C₄	—	2	2	no	—
115	C₂×C₄²	—	4	1	no	—
116	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
117	C₂×C₁₆	—	38	4	no	—
118	C₂×C₁₆	—	38	2	no	—
119	#107	—	2	2	no	—
120	#107	C₁₉⋊C₈	1	4	yes	complemented
121	C₄⋊C₈	—	38	2	no	—
122	C₄⋊C₈	—	76	1	no	—
123	C₄×C₈	Dic₃₈	1	1	yes	characteristic
124	C₄×C₈	C₄×D₁₉	1	1	yes	characteristic
125	C₄×C₈	C₄×D₁₉	1	1	yes	characteristic
126	C₄×C₈	D₇₆	1	1	yes	characteristic
127	#95	—	4	1	no	—
128	C₁₉	#7171	1	1	yes	characteristic, complemented, Sylow
129	C₂²×C₄	—	4	2	no	—
130	C₂²×C₄	#64525	1	1	yes	characteristic
131	C₂²×C₄	#64526	1	1	yes	characteristic
132	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	76	2	no	—
135	C₂×C₈	—	152	1	no	—
136	C₂×C₈	—	152	1	no	—
137	C₂×C₈	C₂×C₁₉⋊C₈	1	2	yes	—
138	C₂×C₈	#64520	1	2	yes	—
139	C₂×C₈	C₄×Dic₁₉	1	1	yes	characteristic
140	C₂×C₈	Dic₁₉⋊C₄	1	1	yes	characteristic
141	C₂×C₈	C₄⋊Dic₁₉	1	1	yes	characteristic
142	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
143	C₄⋊C₄	—	2	4	no	—
144	C₄⋊C₄	—	4	2	no	—
145	C₄²	—	8	1	no	—
146	C₄²	—	8	1	no	—
147	C₄²	Dic₁₉⋊C₄	1	1	yes	characteristic
148	C₄²	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
149	C₄²	#64527	1	1	yes	characteristic
150	C₄²	#64528	1	1	yes	characteristic
151	C₁₆	—	38	4	no	—
152	C₂³	#10597289	1	1	yes	characteristic
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	4	4	no	—
155	C₂×C₄	—	8	2	no	—
156	C₂×C₄	—	8	2	no	—
157	C₂×C₄	#10597264	1	1	yes	characteristic
158	C₂×C₄	#10597265	1	1	yes	characteristic
159	C₂×C₄	#10597266	1	1	yes	characteristic
160	C₂×C₄	#10597267	1	1	yes	characteristic
161	C₂×C₄	#10597283	1	1	yes	characteristic
162	C₂×C₄	#10597284	1	1	yes	characteristic
163	C₂×C₄	#10597285	1	1	yes	characteristic
164	C₂×C₄	#10597286	1	1	yes	characteristic
165	C₂×C₄	C₂²⋊(C₁₉⋊C₈)	1	2	yes	—
166	C₂×C₄	C₁₉⋊#95	1	1	yes	characteristic
167	C₂×C₄	#10597292	1	1	yes	characteristic
168	C₂×C₄	#10597294	1	2	yes	—
169	C₈	—	2	1	no	—
170	C₈	—	2	1	no	—
171	C₈	—	152	2	no	—
172	C₈	C₄×C₁₉⋊C₈	1	1	yes	characteristic
173	C₈	#10597260	1	1	yes	characteristic
174	C₈	C₄⋊(C₁₉⋊C₈)	1	2	yes	—
175	C₂²	D₄⋊(C₁₉⋊C₈)	1	2	yes	—
176	C₂²	#11926306	1	1	yes	characteristic
177	C₂²	#11926345	1	1	yes	characteristic
178	C₂²	#11926348	1	1	yes	characteristic
179	C₂²	#11926351	1	2	yes	—
180	C₄	—	2	1	no	—
181	C₄	—	2	1	no	—
182	C₄	—	8	4	no	—
183	C₄	Q₈⋊(C₁₉⋊C₈)	1	2	yes	—
184	C₄	#11926299	1	1	yes	characteristic
185	C₄	#11926303	1	1	yes	characteristic
186	C₄	#11926344	1	1	yes	characteristic
187	C₄	#11926347	1	1	yes	characteristic
188	C₄	#11926369	1	1	yes	characteristic
189	C₄	#11926371	1	1	yes	characteristic
190	C₂	#14561476	1	1	yes	characteristic
191	C₂	#14561596	1	2	yes	—
192	C₂	#14561599	1	2	yes	—
193	C₂	#14561726	1	1	yes	characteristic
194	C₂	#14561730	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#64525
commutator	C₈×C₁₉	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₁₉
Fitting	C₄×C₁₉×#107	C₂
socle, prime radical	C₂³×C₁₉	#102
cosocle	C₂×C₄×C₈×C₁₉	C₂²
hypercenter	C₄×#107	D₁₉
hypocenter	C₁₉	#7171
LCS	G ▷ C₈×C₁₉ ▷ C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#107

Complements

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

N	H ≅ G/N	type
7. C₄×C₁₉⋊C₁₆	C₄	semidirect
9. #11926297	C₄	semidirect
21. C₁₉×#107	C₈	semidirect
120. #107	C₁₉⋊C₈	semidirect
128. C₁₉	#7171	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×#11926297	2
C₄×C₁₉×#107	2
#7171	19