Subgroups — #18073664

Order 4864.

502 conjugacy classes of subgroups.

The subgroup lattice has 176 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×#11926265	C₂	1	2	yes	maximal
3	C₁₉⋊#1273	C₂	1	2	yes	maximal
4	C₄²×C₈×C₁₉	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₁₉⋊C₈)	C₄	1	4	yes	—
6	C₂×C₄×C₁₉⋊C₈	C₂²	1	4	yes	—
7	#11926299	C₄	1	4	yes	—
8	#11926265	—	2	4	no	—
9	C₂×C₄×C₈×C₁₉	C₄	1	2	yes	—
10	C₂×C₄×C₈×C₁₉	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
13	C₄³×C₁₉	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
15	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
16	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
17	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
20	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
21	C₂²×C₁₉⋊C₈	C₂×C₄	1	8	yes	—
22	C₄×C₈×C₁₉	—	2	4	no	—
23	C₄×C₈×C₁₉	—	2	2	no	—
24	C₄×C₈×C₁₉	C₈	1	8	yes	complemented
25	C₄×C₈×C₁₉	D₄	1	2	yes	—
26	C₄×C₈×C₁₉	Q₈	1	2	yes	—
27	C₄⋊(C₁₉⋊C₈)	—	2	8	no	—
28	C₄×C₁₉⋊C₈	—	2	4	no	—
29	C₄×C₁₉⋊C₈	—	2	4	no	—
30	C₂²×C₄×C₁₉	C₄²	1	1	yes	characteristic
31	C₂²×C₄×C₁₉	C₄²	1	2	yes	—
32	C₂²×C₄×C₁₉	C₄²	1	1	yes	characteristic
33	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
34	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₈×C₁₉	—	2	2	no	—
36	C₂×C₈×C₁₉	—	2	2	no	—
37	C₂×C₈×C₁₉	C₄⋊C₄	1	4	yes	—
38	C₂×C₈×C₁₉	C₂×C₈	1	4	yes	—
39	C₂×C₈×C₁₉	C₂×C₈	1	4	yes	—
40	C₂×C₈×C₁₉	M₁₆	1	4	yes	—
41	C₄²×C₁₉	—	2	2	no	—
42	C₄²×C₁₉	—	2	4	no	—
43	C₄²×C₁₉	C₂×C₈	1	4	yes	—
44	C₄²×C₁₉	M₁₆	1	8	yes	—
45	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
46	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
47	C₄²×C₁₉	#41	1	2	yes	—
48	C₂×C₁₉⋊C₈	—	2	8	no	—
49	C₂×C₁₉⋊C₈	—	2	8	no	—
50	C₂×C₁₉⋊C₈	—	2	8	no	—
51	#8260	—	19	1	no	maximal, Sylow
52	C₂³×C₁₉	C₂×C₄²	1	1	yes	characteristic
53	C₂×C₄×C₁₉	—	2	2	no	—
54	C₂×C₄×C₁₉	—	2	2	no	—
55	C₂×C₄×C₁₉	—	2	2	no	—
56	C₂×C₄×C₁₉	C₄×C₈	1	2	yes	—
57	C₂×C₄×C₁₉	C₄×C₈	1	2	yes	—
58	C₂×C₄×C₁₉	#97	1	4	yes	—
59	C₂×C₄×C₁₉	#97	1	4	yes	—
60	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
62	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
63	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
64	C₂×C₄×C₁₉	C₂²×C₈	1	2	yes	—
65	C₂×C₄×C₁₉	C₂×M₁₆	1	4	yes	—
66	C₂×C₄×C₁₉	C₂×M₁₆	1	2	yes	—
67	C₂×C₄×C₁₉	#131	1	4	yes	—
68	C₈×C₁₉	—	2	4	no	—
69	C₈×C₁₉	C₄⋊C₈	1	8	yes	complemented
70	C₁₉⋊C₈	—	4	8	no	—
71	C₄×C₄⋊C₈	—	19	2	no	—
72	C₄²×C₈	D₁₉	1	1	yes	characteristic
73	C₂×#322	—	19	2	no	—
74	#1273	—	19	2	no	—
75	C₂²×C₁₉	C₄×C₄⋊C₄	1	1	yes	characteristic
76	C₂²×C₁₉	C₂×C₄×C₈	1	1	yes	characteristic
77	C₂²×C₁₉	C₂×#97	1	2	yes	—
78	C₂²×C₁₉	C₄×M₁₆	1	1	yes	characteristic
79	C₂²×C₁₉	#405	1	2	yes	—
80	C₄×C₁₉	—	2	2	no	—
81	C₄×C₁₉	—	2	4	no	—
82	C₄×C₁₉	#322	1	4	yes	complemented
83	C₄×C₁₉	C₂×C₄⋊C₈	1	2	yes	—
84	C₄×C₁₉	#431	1	2	yes	—
85	C₄×C₁₉	#436	1	4	yes	—
86	C₄×C₁₉	#446	1	4	yes	—
87	C₂×C₄⋊C₈	—	19	4	no	—
88	C₂×C₄×C₈	—	19	4	no	—
89	C₂×C₄×C₈	Dic₁₉	1	2	yes	—
90	C₂×C₄×C₈	Dic₁₉	1	2	yes	—
91	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
92	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
93	C₄³	C₂×D₁₉	1	1	yes	characteristic
94	#336	—	19	4	no	—
95	#322	—	38	4	no	—
96	C₂×C₁₉	C₂×#322	1	1	yes	characteristic
97	C₂×C₁₉	#1450	1	1	yes	characteristic
98	C₂×C₁₉	#1451	1	2	yes	—
99	C₂×C₁₉	C₄×C₄⋊C₈	1	1	yes	characteristic
100	C₂×C₁₉	#1770	1	2	yes	—
101	C₂²×C₈	—	19	8	no	—
102	C₂²×C₈	C₂×Dic₁₉	1	2	yes	—
103	C₂²×C₈	C₂×Dic₁₉	1	2	yes	—
104	C₂×C₄²	C₄×D₁₉	1	2	yes	—
105	C₂×C₄²	C₄×D₁₉	1	2	yes	—
106	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
107	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
108	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
109	C₄⋊C₈	—	38	8	no	—
110	C₄×C₈	—	2	4	no	—
111	C₄×C₈	—	2	2	no	—
112	C₄×C₈	—	38	4	no	—
113	C₄×C₈	—	38	4	no	—
114	C₄×C₈	C₁₉⋊C₈	1	8	yes	complemented
115	C₄×C₈	Dic₃₈	1	2	yes	—
116	C₄×C₈	D₇₆	1	2	yes	—
117	C₁₉	#8260	1	1	yes	characteristic, complemented, Sylow
118	C₂²×C₄	C₄×Dic₁₉	1	1	yes	characteristic
119	C₂²×C₄	C₄×Dic₁₉	1	2	yes	—
120	C₂²×C₄	C₄×Dic₁₉	1	1	yes	characteristic
121	C₂²×C₄	C₂×C₄×D₁₉	1	2	yes	—
122	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	38	8	no	—
126	C₂×C₈	—	38	8	no	—
127	C₂×C₈	—	38	8	no	—
128	C₂×C₈	C₂×C₁₉⋊C₈	1	4	yes	—
129	C₂×C₈	C₂×C₁₉⋊C₈	1	4	yes	—
130	C₂×C₈	#64520	1	4	yes	—
131	C₂×C₈	C₄⋊Dic₁₉	1	4	yes	—
132	C₄²	—	2	4	no	—
133	C₄²	—	2	2	no	—
134	C₄²	#64515	1	8	yes	—
135	C₄²	C₂×C₁₉⋊C₈	1	4	yes	—
136	C₄²	C₂×Dic₃₈	1	1	yes	characteristic
137	C₄²	C₂×D₇₆	1	1	yes	characteristic
138	C₄²	#64541	1	2	yes	—
139	C₂³	C₂×C₄×Dic₁₉	1	1	yes	characteristic
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	C₄×C₁₉⋊C₈	1	2	yes	—
144	C₂×C₄	C₄×C₁₉⋊C₈	1	2	yes	—
145	C₂×C₄	#10597272	1	4	yes	—
146	C₂×C₄	#10597272	1	4	yes	—
147	C₂×C₄	C₄×Dic₃₈	1	2	yes	—
148	C₂×C₄	C₄×D₇₆	1	2	yes	—
149	C₂×C₄	C₂×#64515	1	4	yes	—
150	C₂×C₄	#10597347	1	4	yes	—
151	C₂×C₄	C₂²×C₁₉⋊C₈	1	2	yes	—
152	C₂×C₄	C₂×#64520	1	2	yes	—
153	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
154	C₂×C₄	#10597372	1	1	yes	characteristic
155	C₈	—	2	4	no	—
156	C₈	—	76	8	no	—
157	C₈	C₄⋊(C₁₉⋊C₈)	1	8	yes	complemented
158	C₂²	C₄×C₄⋊Dic₁₉	1	1	yes	characteristic
159	C₂²	C₂×C₄×C₁₉⋊C₈	1	1	yes	characteristic
160	C₂²	C₂×#10597272	1	2	yes	—
161	C₂²	C₄×#64520	1	1	yes	characteristic
162	C₂²	#11926499	1	2	yes	—
163	C₄	—	2	2	no	—
164	C₄	—	2	4	no	—
165	C₄	#11926265	1	4	yes	complemented
166	C₄	C₂×C₄⋊(C₁₉⋊C₈)	1	2	yes	—
167	C₄	#11926597	1	2	yes	—
168	C₄	#11926623	1	4	yes	—
169	C₄	#11926674	1	4	yes	—
170	C₂	C₂×#11926265	1	1	yes	characteristic
171	C₂	#14561918	1	1	yes	characteristic
172	C₂	#14561924	1	2	yes	—
173	C₂	C₄×C₄⋊(C₁₉⋊C₈)	1	1	yes	characteristic
174	C₂	#14563502	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₁₉
commutator	C₂²×C₁₉	C₂×C₄×C₈
Fitting	C₄²×C₈×C₁₉	C₂
socle, prime radical	C₂³×C₁₉	C₂×C₄²
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₄²×C₈	D₁₉
hypocenter	C₁₉	#8260
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄² ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
24. C₄×C₈×C₁₉	C₈	semidirect
69. C₈×C₁₉	C₄⋊C₈	semidirect
82. C₄×C₁₉	#322	semidirect
114. C₄×C₈	C₁₉⋊C₈	semidirect
117. C₁₉	#8260	semidirect
157. C₈	C₄⋊(C₁₉⋊C₈)	semidirect
165. C₄	#11926265	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×#11926265	2
C₁₉⋊#1273	2
C₄²×C₈×C₁₉	2
#8260	19