Subgroups — #18073790

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⋊C₄×C₁₉⋊C₈	C₂	1	2	yes	maximal
2	C₂×#11926297	C₂	1	1	yes	maximal, characteristic
3	C₂×#11926265	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)	C₂²	1	2	yes	—
7	C₂×C₄×C₁₉⋊C₈	—	2	2	no	—
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	#11926297	C₄	1	4	yes	complemented
12	#11926265	C₄	1	4	yes	complemented
13	C₂×C₁₉×#106	C₄	1	2	yes	—
14	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
16	C₁₉×#340	C₄	1	2	yes	—
17	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	4	yes	—
20	C₂×C₄²×C₁₉	—	2	2	no	—
21	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
22	C₂²×C₁₉⋊C₈	—	2	2	no	—
23	C₂²×C₁₉⋊C₈	—	2	1	no	—
24	C₂²×C₁₉⋊C₈	D₄	1	2	yes	—
25	C₁₉×#106	C₈	1	8	yes	complemented
26	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
27	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
28	C₁₉×#95	—	2	2	no	—
29	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
30	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	4	yes	—
31	C₄×C₁₉⋊C₈	—	2	1	no	—
32	C₄×C₁₉⋊C₈	—	4	4	no	—
33	C₄×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
34	C₄×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
35	C₄×C₁₉⋊C₈	Q₈	1	2	yes	—
36	C₂²×C₄×C₁₉	—	2	4	no	—
37	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
40	C₂×C₈×C₁₉	—	2	2	no	—
41	C₂×C₈×C₁₉	C₄²	1	2	yes	—
42	C₂×C₈×C₁₉	C₄²	1	2	yes	—
43	C₂×C₈×C₁₉	C₂×C₈	1	4	yes	—
44	C₄⋊C₄×C₁₉	—	2	4	no	—
45	C₄⋊C₄×C₁₉	C₂×C₈	1	8	yes	—
46	C₄²×C₁₉	—	4	2	no	—
47	C₄²×C₁₉	—	4	2	no	—
48	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
49	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
50	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
51	C₄²×C₁₉	#41	1	1	yes	characteristic
52	C₂×C₁₉⋊C₈	—	2	2	no	—
53	C₂×C₁₉⋊C₈	—	2	2	no	—
54	C₂×C₁₉⋊C₈	—	2	2	no	—
55	C₂×C₁₉⋊C₈	—	2	4	no	—
56	C₂×C₁₉⋊C₈	—	4	2	no	—
57	C₂×C₁₉⋊C₈	—	4	2	no	—
58	C₂×C₁₉⋊C₈	—	4	2	no	—
59	C₂×C₁₉⋊C₈	C₄⋊C₄	1	4	yes	—
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	—	2	8	no	—
64	C₂×C₄×C₁₉	—	4	4	no	—
65	C₂×C₄×C₁₉	—	4	4	no	—
66	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
67	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
69	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₁₉	C₄×Q₈	1	1	yes	characteristic
71	C₂×C₄×C₁₉	C₄×Q₈	1	1	yes	characteristic
72	C₂×C₄×C₁₉	C₂²×C₈	1	2	yes	—
73	C₂×C₄×C₁₉	#131	1	2	yes	—
74	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
75	C₂×C₄×C₁₉	#137	1	2	yes	—
76	C₈×C₁₉	—	2	2	no	—
77	C₈×C₁₉	C₄×C₈	1	4	yes	complemented
78	C₁₉⋊C₈	—	2	4	no	—
79	C₁₉⋊C₈	—	4	2	no	—
80	C₁₉⋊C₈	—	4	4	no	—
81	C₄×#106	D₁₉	1	1	yes	characteristic
82	C₈×C₄⋊C₄	—	19	2	no	—
83	C₂×#335	—	19	1	no	—
84	C₂×#322	—	19	1	no	—
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	—	4	8	no	—
93	C₄×C₁₉	C₂×C₄×C₈	1	1	yes	characteristic
94	C₄×C₁₉	#405	1	1	yes	characteristic
95	C₄×C₁₉	#428	1	2	yes	—
96	C₄×C₁₉	#441	1	1	yes	characteristic
97	C₄×C₁₉	#442	1	1	yes	characteristic
98	C₄×C₁₉	C₈×Q₈	1	2	yes	—
99	C₂×#106	Dic₁₉	1	2	yes	—
100	C₂×C₄⋊C₈	—	19	2	no	—
101	C₂×C₄×C₈	—	19	1	no	—
102	C₂×C₄×C₈	—	19	1	no	—
103	C₂×C₄×C₈	—	38	2	no	—
104	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
105	C₄×C₄⋊C₄	C₂×D₁₉	1	2	yes	—
106	#340	Dic₁₉	1	2	yes	—
107	#336	—	38	2	no	—
108	#335	—	19	4	no	—
109	#322	—	19	4	no	—
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	—	19	2	no	—
115	C₂²×C₈	—	38	2	no	—
116	C₂²×C₈	—	38	1	no	—
117	C₂²×C₈	C₂×Dic₁₉	1	1	yes	characteristic
118	C₂²×C₈	C₂×Dic₁₉	1	1	yes	characteristic
119	C₂×C₄⋊C₄	C₂×Dic₁₉	1	4	yes	—
120	C₂×C₄²	—	2	2	no	—
121	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
122	#106	C₁₉⋊C₈	1	8	yes	complemented
123	C₄⋊C₈	—	19	4	no	—
124	C₄⋊C₈	—	38	2	no	—
125	C₄×C₈	—	19	2	no	—
126	C₄×C₈	—	19	2	no	—
127	C₄×C₈	—	19	2	no	—
128	C₄×C₈	—	38	1	no	—
129	C₄×C₈	—	76	4	no	—
130	C₄×C₈	C₄×D₁₉	1	2	yes	—
131	C₄×C₈	C₄×D₁₉	1	2	yes	—
132	#95	—	2	2	no	—
133	C₁₉	#8285	1	1	yes	characteristic, complemented, Sylow
134	C₂²×C₄	—	2	4	no	—
135	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
136	C₂²×C₄	#64543	1	1	yes	characteristic
137	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	19	4	no	—
140	C₂×C₈	—	38	2	no	—
141	C₂×C₈	—	38	2	no	—
142	C₂×C₈	—	38	2	no	—
143	C₂×C₈	—	38	4	no	—
144	C₂×C₈	—	76	2	no	—
145	C₂×C₈	—	76	2	no	—
146	C₂×C₈	—	76	2	no	—
147	C₂×C₈	C₂×C₁₉⋊C₈	1	4	yes	—
148	C₂×C₈	C₄×Dic₁₉	1	2	yes	—
149	C₂×C₈	C₄×Dic₁₉	1	2	yes	—
150	C₄⋊C₄	—	2	4	no	—
151	C₄⋊C₄	C₂×C₁₉⋊C₈	1	8	yes	—
152	C₄²	—	4	2	no	—
153	C₄²	—	4	2	no	—
154	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
155	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
156	C₄²	Q₈×D₁₉	1	1	yes	characteristic
157	C₄²	#64545	1	1	yes	characteristic
158	C₂³	D₄×Dic₁₉	1	1	yes	characteristic
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	8	no	—
161	C₂×C₄	—	4	4	no	—
162	C₂×C₄	—	4	4	no	—
163	C₂×C₄	#10597333	1	1	yes	characteristic
164	C₂×C₄	C₄⋊C₄×D₁₉	1	1	yes	characteristic
165	C₂×C₄	#10597338	1	1	yes	characteristic
166	C₂×C₄	#10597339	1	1	yes	characteristic
167	C₂×C₄	#10597360	1	2	yes	—
168	C₂×C₄	#10597361	1	2	yes	—
169	C₂×C₄	C₂²×C₁₉⋊C₈	1	2	yes	—
170	C₂×C₄	C₂×C₄×Dic₁₉	1	1	yes	characteristic
171	C₂×C₄	Q₈×Dic₁₉	1	1	yes	characteristic
172	C₂×C₄	#10597394	1	2	yes	—
173	C₈	—	2	2	no	—
174	C₈	—	38	4	no	—
175	C₈	—	76	2	no	—
176	C₈	—	76	4	no	—
177	C₈	C₄×C₁₉⋊C₈	1	4	yes	complemented
178	C₂²	C₄⋊C₄×Dic₁₉	1	1	yes	characteristic
179	C₂²	D₄×C₁₉⋊C₈	1	2	yes	—
180	C₂²	#11926645	1	2	yes	—
181	C₂²	#11926666	1	2	yes	—
182	C₄	—	2	1	no	—
183	C₄	—	2	1	no	—
184	C₄	—	4	8	no	—
185	C₄	C₂×C₄×C₁₉⋊C₈	1	1	yes	characteristic
186	C₄	#11926500	1	1	yes	characteristic
187	C₄	#11926583	1	2	yes	—
188	C₄	#11926650	1	1	yes	characteristic
189	C₄	#11926655	1	1	yes	characteristic
190	C₄	Q₈×C₁₉⋊C₈	1	2	yes	—
191	C₂	SD₁₆⋊(C₁₉⋊C₈)	1	4	yes	—
192	C₂	C₄⋊C₄×C₁₉⋊C₈	1	1	yes	characteristic
193	C₂	#14563521	1	1	yes	characteristic
194	C₂	#14563535	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₁₉
commutator	C₄×C₁₉	C₂×C₄×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	#8285
LCS	G ▷ 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
11. #11926297	C₄	semidirect
12. #11926265	C₄	semidirect
25. C₁₉×#106	C₈	semidirect
77. C₈×C₁₉	C₄×C₈	semidirect
122. #106	C₁₉⋊C₈	semidirect
133. C₁₉	#8285	semidirect
177. 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
C₄⋊C₄×C₁₉⋊C₈	2
C₂×#11926297	2
C₂×#11926265	2
#14561476	2
C₄×C₁₉×#106	2
#8285	19