Subgroups — #18069482

Order 4864.

240 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×#11926365	C₂	1	1	yes	maximal, characteristic
2	#14561918	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1772	C₂	1	1	yes	maximal, characteristic
4	C₂×#10597269	—	2	1	no	—
5	#11926597	—	2	1	no	—
6	#11926365	—	2	1	no	—
7	#11926365	C₄	1	2	yes	complemented
8	#11926265	—	2	1	no	—
9	C₈×C₁₉⋊C₈	C₄	1	2	yes	complemented
10	C₂×C₁₉×#108	C₄	1	1	yes	characteristic
11	C₁₉×#427	C₄	1	1	yes	characteristic
12	C₁₉×#405	—	2	1	no	—
13	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
14	C₁₉×#339	—	2	1	no	—
15	C₂×M₁₆×C₁₉	—	2	1	no	—
16	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
18	C₁₉×#117	—	2	1	no	—
19	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
20	C₁₉×#108	—	2	1	no	—
21	C₁₉×#108	C₈	1	2	yes	complemented
22	C₁₉×#107	—	2	1	no	—
23	C₁₉×#106	C₈	1	2	yes	complemented
24	C₁₉×#97	—	4	1	no	—
25	C₄×C₈×C₁₉	—	4	1	no	—
26	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
29	C₄×C₈×C₁₉	Q₈	1	1	yes	characteristic
30	C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
31	#10597269	—	2	2	no	—
32	#10597269	—	4	1	no	—
33	C₂×C₁₉⋊C₁₆	—	4	1	no	—
34	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
35	C₄×C₁₉⋊C₈	—	2	2	no	—
36	C₂²×C₄×C₁₉	—	2	1	no	—
37	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
38	M₁₆×C₁₉	—	2	2	no	—
39	M₁₆×C₁₉	—	4	1	no	—
40	C₂×C₈×C₁₉	—	4	1	no	—
41	C₂×C₈×C₁₉	C₄²	1	1	yes	characteristic
42	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₁₉	C₂×C₈	1	1	yes	characteristic
46	C₂×C₈×C₁₉	C₂×C₈	1	1	yes	characteristic
47	C₂×C₈×C₁₉	M₁₆	1	1	yes	characteristic
48	C₂×C₈×C₁₉	M₁₆	1	1	yes	characteristic
49	C₂×C₈×C₁₉	D₈	1	1	yes	characteristic
50	C₂×C₈×C₁₉	SD₁₆	1	1	yes	characteristic
51	C₂×C₈×C₁₉	SD₁₆	1	1	yes	characteristic
52	C₂×C₈×C₁₉	Q₁₆	1	1	yes	characteristic
53	C₄⋊C₄×C₁₉	—	2	2	no	—
54	C₂²⋊C₄×C₁₉	—	4	1	no	—
55	C₄²×C₁₉	—	2	1	no	—
56	C₄²×C₁₉	—	4	1	no	—
57	C₄²×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
58	C₄²×C₁₉	C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₁₉⋊C₈	—	2	4	no	—
60	C₁₉⋊C₁₆	—	4	2	no	—
61	#7180	—	19	1	no	maximal, Sylow
62	C₂³×C₁₉	#104	1	1	yes	characteristic
63	C₂×C₄×C₁₉	—	2	1	no	—
64	C₂×C₄×C₁₉	—	2	1	no	—
65	C₂×C₄×C₁₉	—	4	2	no	—
66	C₂×C₄×C₁₉	—	4	2	no	—
67	C₂×C₄×C₁₉	#95	1	1	yes	characteristic
68	C₂×C₄×C₁₉	C₂²⋊C₈	1	1	yes	characteristic
69	C₂×C₄×C₁₉	C₂²⋊C₈	1	1	yes	characteristic
70	C₂×C₄×C₁₉	#102	1	1	yes	characteristic
71	C₂×C₄×C₁₉	#103	1	1	yes	characteristic
72	C₂×C₄×C₁₉	#104	1	1	yes	characteristic
73	C₈×C₁₉	—	4	2	no	—
74	C₈×C₁₉	C₄×C₈	1	1	yes	characteristic
75	C₈×C₁₉	#97	1	1	yes	characteristic
76	C₈×C₁₉	#102	1	1	yes	characteristic
77	C₈×C₁₉	#103	1	1	yes	characteristic
78	C₈×C₁₉	C₄⋊C₈	1	1	yes	characteristic
79	C₈×C₁₉	C₄⋊C₈	1	1	yes	characteristic
80	C₈×C₁₉	#106	1	1	yes	characteristic
81	C₈×C₁₉	#107	1	1	yes	characteristic
82	C₁₉⋊C₈	—	4	4	no	—
83	C₂×#364	—	19	1	no	—
84	#1772	D₁₉	1	1	yes	characteristic
85	#1450	—	19	1	no	—
86	C₂²×C₁₉	—	4	1	no	—
87	C₂²×C₁₉	D₄⋊C₈	1	1	yes	characteristic
88	C₂²×C₁₉	Q₈⋊C₈	1	1	yes	characteristic
89	C₂²×C₁₉	#339	1	1	yes	characteristic
90	C₄×C₁₉	—	2	2	no	—
91	C₄×C₁₉	—	4	1	no	—
92	C₄×C₁₉	—	8	2	no	—
93	C₄×C₁₉	D₄⋊C₈	1	1	yes	characteristic
94	C₄×C₁₉	Q₈⋊C₈	1	1	yes	characteristic
95	C₄×C₁₉	#336	1	1	yes	characteristic
96	C₄×C₁₉	#340	1	1	yes	characteristic
97	C₂×M₃₂	—	38	1	no	—
98	#431	—	38	1	no	—
99	C₂×#108	Dic₁₉	1	1	yes	characteristic
100	#427	Dic₁₉	1	1	yes	characteristic
101	#405	—	2	1	no	—
102	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
103	#364	—	19	2	no	—
104	#364	—	38	1	no	—
105	#339	—	2	1	no	—
106	#322	—	38	1	no	—
107	C₈²	—	19	2	no	—
108	C₂×C₁₉	—	2	1	no	—
109	C₂×C₁₉	—	2	1	no	—
110	C₂×C₁₉	—	2	1	no	—
111	C₂×C₁₉	#1279	1	1	yes	characteristic
112	C₂×M₁₆	—	2	1	no	—
113	C₂²×C₈	C₂×Dic₁₉	1	1	yes	characteristic
114	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
115	#117	—	2	1	no	—
116	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
117	M₃₂	—	38	2	no	—
118	M₃₂	—	76	1	no	—
119	C₂×C₁₆	—	76	1	no	—
120	#108	—	2	1	no	—
121	#108	C₁₉⋊C₈	1	2	yes	complemented
122	#107	—	2	1	no	—
123	#106	C₁₉⋊C₈	1	2	yes	complemented
124	C₄⋊C₈	—	38	2	no	—
125	C₂²⋊C₈	—	38	2	no	—
126	#97	—	4	1	no	—
127	C₄×C₈	—	4	1	no	—
128	C₄×C₈	—	38	2	no	—
129	C₄×C₈	Dic₃₈	1	1	yes	characteristic
130	C₄×C₈	C₄×D₁₉	1	1	yes	characteristic
131	C₄×C₈	C₄×D₁₉	1	1	yes	characteristic
132	C₄×C₈	D₇₆	1	1	yes	characteristic
133	C₁₉	#7180	1	1	yes	characteristic, complemented, Sylow
134	C₂²×C₄	—	2	1	no	—
135	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
136	M₁₆	—	2	2	no	—
137	M₁₆	—	4	1	no	—
138	C₂×C₈	—	4	1	no	—
139	C₂×C₈	—	38	4	no	—
140	C₂×C₈	C₂×C₁₉⋊C₈	1	1	yes	characteristic
141	C₂×C₈	C₂×C₁₉⋊C₈	1	1	yes	characteristic
142	C₂×C₈	#64520	1	1	yes	characteristic
143	C₂×C₈	#64520	1	1	yes	characteristic
144	C₂×C₈	C₄×Dic₁₉	1	1	yes	characteristic
145	C₂×C₈	Dic₁₉⋊C₄	1	1	yes	characteristic
146	C₂×C₈	C₄⋊Dic₁₉	1	1	yes	characteristic
147	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
148	C₂×C₈	#64525	1	1	yes	characteristic
149	C₂×C₈	#64526	1	1	yes	characteristic
150	C₂×C₈	#64527	1	1	yes	characteristic
151	C₂×C₈	#64528	1	1	yes	characteristic
152	C₄⋊C₄	—	2	2	no	—
153	C₂²⋊C₄	—	4	1	no	—
154	C₄²	—	2	1	no	—
155	C₄²	—	4	1	no	—
156	C₄²	Dic₁₉⋊C₄	1	1	yes	characteristic
157	C₄²	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
158	C₁₆	—	76	2	no	—
159	C₂³	#10597294	1	1	yes	characteristic
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	C₂²⋊(C₁₉⋊C₈)	1	1	yes	characteristic
165	C₂×C₄	C₂²⋊(C₁₉⋊C₈)	1	1	yes	characteristic
166	C₂×C₄	C₁₉⋊#95	1	1	yes	characteristic
167	C₂×C₄	#10597289	1	1	yes	characteristic
168	C₂×C₄	#10597292	1	1	yes	characteristic
169	C₂×C₄	#10597294	1	1	yes	characteristic
170	C₈	—	4	2	no	—
171	C₈	—	76	4	no	—
172	C₈	C₄×C₁₉⋊C₈	1	1	yes	characteristic
173	C₈	#10597260	1	1	yes	characteristic
174	C₈	C₄⋊(C₁₉⋊C₈)	1	1	yes	characteristic
175	C₈	C₄⋊(C₁₉⋊C₈)	1	1	yes	characteristic
176	C₈	#10597264	1	1	yes	characteristic
177	C₈	#10597265	1	1	yes	characteristic
178	C₈	#10597266	1	1	yes	characteristic
179	C₈	#10597267	1	1	yes	characteristic
180	C₂²	—	4	1	no	—
181	C₂²	D₄⋊(C₁₉⋊C₈)	1	1	yes	characteristic
182	C₂²	Q₈⋊(C₁₉⋊C₈)	1	1	yes	characteristic
183	C₂²	#11926303	1	1	yes	characteristic
184	C₄	—	2	2	no	—
185	C₄	—	4	1	no	—
186	C₄	—	8	2	no	—
187	C₄	D₄⋊(C₁₉⋊C₈)	1	1	yes	characteristic
188	C₄	Q₈⋊(C₁₉⋊C₈)	1	1	yes	characteristic
189	C₄	#11926299	1	1	yes	characteristic
190	C₄	#11926306	1	1	yes	characteristic
191	C₂	—	2	1	no	—
192	C₂	—	2	1	no	—
193	C₂	—	2	1	no	—
194	C₂	#14561476	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#10597265
commutator	C₈×C₁₉	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₁₉
Fitting	C₁₉×#1772	C₂
socle	C₂×C₁₉	#1279
prime radical	C₂³×C₁₉	#104
cosocle	C₂×C₄×C₈×C₁₉	C₂²
hypercenter	#1772	D₁₉
hypocenter	C₁₉	#7180
LCS	G ▷ C₈×C₁₉ ▷ C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ #1772

Complements

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

N	H ≅ G/N	type
7. #11926365	C₄	semidirect
9. C₈×C₁₉⋊C₈	C₄	semidirect
21. C₁₉×#108	C₈	semidirect
23. C₁₉×#106	C₈	semidirect
121. #108	C₁₉⋊C₈	semidirect
123. #106	C₁₉⋊C₈	semidirect
133. C₁₉	#7180	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926365	2
#14561918	2
C₁₉×#1772	2
#7180	19