Subgroups — #18073788

Order 4864.

430 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	C₄×#10597274	C₂	1	1	yes	maximal, characteristic
2	C₈×C₄⋊Dic₁₉	C₂	1	2	yes	maximal
3	C₂×#11926297	C₂	1	1	yes	maximal, characteristic
4	#14561475	C₂	1	2	yes	maximal
5	C₂×C₁₉×#322	C₂	1	1	yes	maximal, characteristic
6	C₂×#10597274	C₄	1	2	yes	—
7	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
8	C₂×C₈×Dic₁₉	—	2	2	no	—
9	C₄×C₄⋊Dic₁₉	C₂²	1	2	yes	—
10	#11926307	C₄	1	2	yes	—
11	#11926298	—	2	2	no	—
12	#11926297	C₄	1	4	yes	complemented
13	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
16	C₁₉×#322	C₄	1	4	yes	complemented
17	C₂²×C₈×C₁₉	—	2	1	no	—
18	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₉	D₄	1	2	yes	—
21	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
22	C₂×C₄⋊Dic₁₉	C₂×C₄	1	4	yes	—
23	C₂×C₄×Dic₁₉	—	2	2	no	—
24	C₂²×C₁₉⋊C₈	—	2	2	no	—
25	C₄×C₈×C₁₉	—	2	1	no	—
26	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
27	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
28	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
29	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
30	C₄×C₈×C₁₉	Q₈	1	2	yes	—
31	C₁₉⋊#95	—	2	2	no	—
32	#10597274	C₈	1	8	yes	complemented
33	C₈×Dic₁₉	—	4	4	no	—
34	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
35	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	4	yes	—
36	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
39	C₂²×Dic₁₉	—	2	4	no	—
40	C₂×C₈×C₁₉	—	2	2	no	—
41	C₂×C₈×C₁₉	—	2	2	no	—
42	C₂×C₈×C₁₉	—	2	2	no	—
43	C₂×C₈×C₁₉	—	2	2	no	—
44	C₂×C₈×C₁₉	—	4	2	no	—
45	C₂×C₈×C₁₉	C₄²	1	2	yes	—
46	C₂×C₈×C₁₉	C₄²	1	2	yes	—
47	C₂×C₈×C₁₉	C₄⋊C₄	1	4	yes	—
48	C₂×C₈×C₁₉	C₂×C₈	1	4	yes	—
49	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
50	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
51	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
52	C₄²×C₁₉	#41	1	1	yes	characteristic
53	C₄⋊Dic₁₉	—	2	4	no	—
54	C₄⋊Dic₁₉	C₂×C₈	1	8	yes	—
55	C₄×Dic₁₉	—	4	2	no	—
56	C₄×Dic₁₉	—	4	2	no	—
57	C₂×C₁₉⋊C₈	—	2	4	no	—
58	C₂×C₁₉⋊C₈	—	4	2	no	—
59	C₂×C₁₉⋊C₈	—	4	2	no	—
60	#8285	—	19	1	no	maximal, Sylow
61	C₂³×C₁₉	C₄×D₄	1	1	yes	characteristic
62	C₂×C₄×C₁₉	—	2	2	no	—
63	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
64	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
66	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
67	C₂×C₄×C₁₉	C₄×Q₈	1	1	yes	characteristic
68	C₂×C₄×C₁₉	C₄×Q₈	1	1	yes	characteristic
69	C₂×C₄×C₁₉	C₂²×C₈	1	2	yes	—
70	C₂×C₄×C₁₉	#131	1	2	yes	—
71	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
72	C₂×C₄×C₁₉	#137	1	2	yes	—
73	C₂×Dic₁₉	—	2	8	no	—
74	C₂×Dic₁₉	—	4	4	no	—
75	C₂×Dic₁₉	—	4	4	no	—
76	C₈×C₁₉	—	2	2	no	—
77	C₈×C₁₉	—	2	4	no	—
78	C₈×C₁₉	—	4	2	no	—
79	C₈×C₁₉	C₄×C₈	1	4	yes	complemented
80	C₁₉⋊C₈	—	4	4	no	—
81	C₄×#106	—	19	1	no	—
82	C₈×C₄⋊C₄	—	19	2	no	—
83	C₂×#335	—	19	1	no	—
84	C₂×#322	D₁₉	1	1	yes	characteristic
85	#1279	—	19	2	no	—
86	C₂²×C₁₉	C₄×C₄⋊C₄	1	1	yes	characteristic
87	C₂²×C₁₉	C₈×D₄	1	2	yes	—
88	C₂²×C₁₉	#440	1	2	yes	—
89	C₂²×C₁₉	#444	1	2	yes	—
90	C₄×C₁₉	—	2	1	no	—
91	C₄×C₁₉	—	2	1	no	—
92	C₄×C₁₉	C₂×C₄×C₈	1	1	yes	characteristic
93	C₄×C₁₉	#405	1	1	yes	characteristic
94	C₄×C₁₉	#428	1	2	yes	—
95	C₄×C₁₉	#441	1	1	yes	characteristic
96	C₄×C₁₉	#442	1	1	yes	characteristic
97	C₄×C₁₉	C₈×Q₈	1	2	yes	—
98	Dic₁₉	—	4	8	no	—
99	C₂×#106	—	19	2	no	—
100	C₂×C₄⋊C₈	—	19	2	no	—
101	C₂×C₄×C₈	—	38	2	no	—
102	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
103	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
104	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
105	C₄×C₄⋊C₄	—	19	2	no	—
106	#340	—	19	2	no	—
107	#336	—	38	2	no	—
108	#335	—	19	4	no	—
109	#322	Dic₁₉	1	4	yes	complemented
110	C₂×C₁₉	SD₁₆⋊C₈	1	4	yes	—
111	C₂×C₁₉	C₈×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₁₉	#1773	1	1	yes	characteristic
113	C₂×C₁₉	#1776	1	1	yes	characteristic
114	C₂²×C₈	—	2	1	no	—
115	C₂²×C₈	—	38	2	no	—
116	C₂²×C₈	C₄×D₁₉	1	1	yes	characteristic
117	C₂²×C₈	C₄×D₁₉	1	1	yes	characteristic
118	C₂²×C₈	D₇₆	1	2	yes	—
119	C₂×C₄⋊C₄	—	19	4	no	—
120	C₂×C₄²	—	38	2	no	—
121	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
122	#106	—	19	8	no	—
123	C₄⋊C₈	—	19	4	no	—
124	C₄⋊C₈	—	38	2	no	—
125	C₄×C₈	—	2	1	no	—
126	C₄×C₈	—	76	4	no	—
127	C₄×C₈	Dic₃₈	1	2	yes	—
128	C₄×C₈	C₄×D₁₉	1	2	yes	—
129	C₄×C₈	C₂×Dic₁₉	1	2	yes	—
130	C₄×C₈	C₂×Dic₁₉	1	2	yes	—
131	C₄×C₈	C₂×Dic₁₉	1	2	yes	—
132	#95	—	38	2	no	—
133	C₁₉	#8285	1	1	yes	characteristic, complemented, Sylow
134	C₂²×C₄	—	38	4	no	—
135	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
136	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
137	C₂²×C₄	#64541	1	1	yes	characteristic
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	4	2	no	—
143	C₂×C₈	—	38	4	no	—
144	C₂×C₈	—	76	2	no	—
145	C₂×C₈	—	76	2	no	—
146	C₂×C₈	C₈×D₁₉	1	4	yes	—
147	C₂×C₈	C₄×Dic₁₉	1	2	yes	—
148	C₂×C₈	C₄×Dic₁₉	1	2	yes	—
149	C₂×C₈	C₄⋊Dic₁₉	1	4	yes	—
150	C₄⋊C₄	—	19	8	no	—
151	C₄⋊C₄	—	38	4	no	—
152	C₄²	—	76	2	no	—
153	C₄²	—	76	2	no	—
154	C₄²	C₂×Dic₃₈	1	1	yes	characteristic
155	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
156	C₄²	#64541	1	1	yes	characteristic
157	C₄²	C₂²×Dic₁₉	1	1	yes	characteristic
158	C₂³	C₄×D₇₆	1	1	yes	characteristic
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	38	8	no	—
161	C₂×C₄	—	76	4	no	—
162	C₂×C₄	—	76	4	no	—
163	C₂×C₄	C₄×Dic₃₈	1	1	yes	characteristic
164	C₂×C₄	C₄×Dic₃₈	1	1	yes	characteristic
165	C₂×C₄	C₄×D₇₆	1	1	yes	characteristic
166	C₂×C₄	C₂×C₈×D₁₉	1	2	yes	—
167	C₂×C₄	#10597347	1	2	yes	—
168	C₂×C₄	#10597354	1	2	yes	—
169	C₂×C₄	#10597355	1	2	yes	—
170	C₂×C₄	C₂×C₄×Dic₁₉	1	1	yes	characteristic
171	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
172	C₂×C₄	#10597372	1	1	yes	characteristic
173	C₈	—	2	4	no	—
174	C₈	—	2	2	no	—
175	C₈	—	4	2	no	—
176	C₈	—	76	4	no	—
177	C₈	C₈×Dic₁₉	1	4	yes	complemented
178	C₂²	C₄×C₄⋊Dic₁₉	1	1	yes	characteristic
179	C₂²	C₈×D₇₆	1	2	yes	—
180	C₂²	#11926646	1	2	yes	—
181	C₂²	#11926667	1	2	yes	—
182	C₄	—	2	1	no	—
183	C₄	—	2	1	no	—
184	C₄	—	76	8	no	—
185	C₄	C₂×C₈×Dic₁₉	1	1	yes	characteristic
186	C₄	#11926499	1	1	yes	characteristic
187	C₄	#11926586	1	2	yes	—
188	C₄	#11926652	1	1	yes	characteristic
189	C₄	#11926657	1	1	yes	characteristic
190	C₄	C₈×Dic₃₈	1	2	yes	—
191	C₂	#14562449	1	4	yes	—
192	C₂	C₈×C₄⋊Dic₁₉	1	1	yes	characteristic
193	C₂	#14563519	1	1	yes	characteristic
194	C₂	#14563532	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₇₆
commutator	C₄×C₁₉	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₂×C₁₉×#322	C₂
socle, prime radical	C₂³×C₁₉	C₄×D₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂×#322	D₁₉
hypocenter	C₁₉	#8285
LCS	G ▷ C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
12. #11926297	C₄	semidirect
16. C₁₉×#322	C₄	semidirect
32. #10597274	C₈	semidirect
79. C₈×C₁₉	C₄×C₈	semidirect
109. #322	Dic₁₉	semidirect
133. C₁₉	#8285	semidirect
177. C₈	C₈×Dic₁₉	semidirect

Maximal subgroups

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

subgroup	index
C₄×#10597274	2
C₈×C₄⋊Dic₁₉	2
C₂×#11926297	2
#14561475	2
C₂×C₁₉×#322	2
#8285	19