Subgroups — #18073736

Order 4864.

446 conjugacy classes of subgroups.

The subgroup lattice has 202 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×#11926295	C₂	1	1	yes	maximal, characteristic
4	#14561470	C₂	1	2	yes	maximal
5	C₄×C₁₉×#97	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
8	C₂×C₄×C₁₉⋊C₈	—	2	2	no	—
9	#11926299	—	2	2	no	—
10	#11926297	C₄	1	4	yes	complemented
11	#11926295	C₄	1	4	yes	complemented
12	C₂×C₁₉×#97	C₄	1	2	yes	—
13	C₂×C₁₉×#97	C₄	1	2	yes	—
14	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
16	C₄³×C₁₉	C₂²	1	1	yes	characteristic
17	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
19	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
21	C₂×C₄²×C₁₉	D₄	1	2	yes	—
22	C₂×C₄²×C₁₉	Q₈	1	2	yes	—
23	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
24	C₂²×C₁₉⋊C₈	—	2	2	no	—
25	C₂²×C₁₉⋊C₈	—	2	2	no	—
26	C₁₉×#97	—	2	4	no	—
27	C₁₉×#97	C₈	1	8	yes	complemented
28	C₄×C₈×C₁₉	—	2	1	no	—
29	C₄×C₈×C₁₉	—	2	1	no	—
30	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
31	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
32	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
33	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
34	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	4	yes	—
35	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	4	yes	—
36	C₄×C₁₉⋊C₈	—	4	4	no	—
37	C₂²×C₄×C₁₉	C₄⋊C₄	1	4	yes	—
38	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
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	C₄²	1	2	yes	—
45	C₂×C₈×C₁₉	C₄²	1	2	yes	—
46	C₂×C₈×C₁₉	C₂×C₈	1	4	yes	—
47	C₂×C₈×C₁₉	C₂×C₈	1	4	yes	—
48	C₄²×C₁₉	—	2	2	no	—
49	C₄²×C₁₉	—	2	2	no	—
50	C₄²×C₁₉	—	4	2	no	—
51	C₄²×C₁₉	C₂×C₈	1	4	yes	—
52	C₄²×C₁₉	M₁₆	1	4	yes	—
53	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
54	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
55	C₄²×C₁₉	#41	1	1	yes	characteristic
56	C₄²×C₁₉	#41	1	1	yes	characteristic
57	C₂×C₁₉⋊C₈	—	2	4	no	—
58	C₂×C₁₉⋊C₈	—	2	4	no	—
59	C₂×C₁₉⋊C₈	—	4	2	no	—
60	C₂×C₁₉⋊C₈	—	4	2	no	—
61	C₂×C₁₉⋊C₈	—	4	2	no	—
62	C₂×C₁₉⋊C₈	—	4	2	no	—
63	#8275	—	19	1	no	maximal, Sylow
64	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₁₉	—	2	4	no	—
66	C₂×C₄×C₁₉	—	2	4	no	—
67	C₂×C₄×C₁₉	—	2	2	no	—
68	C₂×C₄×C₁₉	C₄⋊C₈	1	8	yes	—
69	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
70	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
71	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
72	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
73	C₂×C₄×C₁₉	C₄×Q₈	1	1	yes	characteristic
74	C₂×C₄×C₁₉	C₄×Q₈	1	1	yes	characteristic
75	C₂×C₄×C₁₉	C₂²×C₈	1	2	yes	—
76	C₂×C₄×C₁₉	C₂×M₁₆	1	2	yes	—
77	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
78	C₂×C₄×C₁₉	#137	1	2	yes	—
79	C₈×C₁₉	—	2	2	no	—
80	C₈×C₁₉	—	2	4	no	—
81	C₈×C₁₉	C₄×C₈	1	4	yes	complemented
82	C₁₉⋊C₈	—	4	4	no	—
83	C₁₉⋊C₈	—	4	4	no	—
84	C₄×C₄⋊C₈	—	19	2	no	—
85	C₄×#97	D₁₉	1	1	yes	characteristic
86	C₂×#335	—	19	1	no	—
87	C₂×#334	—	19	1	no	—
88	#1277	—	19	2	no	—
89	C₂²×C₁₉	C₄×C₄⋊C₄	1	1	yes	characteristic
90	C₂²×C₁₉	C₂×C₄⋊C₈	1	2	yes	—
91	C₂²×C₁₉	#428	1	2	yes	—
92	C₂²×C₁₉	#430	1	2	yes	—
93	C₄×C₁₉	—	2	8	no	—
94	C₄×C₁₉	—	2	1	no	—
95	C₄×C₁₉	—	2	1	no	—
96	C₄×C₁₉	C₂×C₄×C₈	1	1	yes	characteristic
97	C₄×C₁₉	C₄×M₁₆	1	1	yes	characteristic
98	C₄×C₁₉	#431	1	2	yes	—
99	C₄×C₁₉	#440	1	2	yes	—
100	C₄×C₁₉	#441	1	1	yes	characteristic
101	C₄×C₁₉	#442	1	1	yes	characteristic
102	C₂×C₄⋊C₈	—	19	2	no	—
103	C₂×C₄⋊C₈	—	19	2	no	—
104	C₂×#97	Dic₁₉	1	2	yes	—
105	C₂×#97	Dic₁₉	1	2	yes	—
106	C₂×C₄×C₈	—	38	2	no	—
107	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
108	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
109	C₄³	C₂×D₁₉	1	1	yes	characteristic
110	#336	—	38	2	no	—
111	#335	—	19	4	no	—
112	#334	—	19	4	no	—
113	C₂×C₁₉	#1565	1	4	yes	—
114	C₂×C₁₉	C₄×C₄⋊C₈	1	1	yes	characteristic
115	C₂×C₁₉	#1776	1	1	yes	characteristic
116	C₂×C₁₉	#1778	1	1	yes	characteristic
117	C₂²×C₈	—	38	2	no	—
118	C₂²×C₈	—	38	2	no	—
119	C₂²×C₈	C₂×Dic₁₉	1	2	yes	—
120	C₂²×C₈	C₂×Dic₁₉	1	1	yes	characteristic
121	C₂²×C₈	C₂×Dic₁₉	1	1	yes	characteristic
122	C₂×C₄²	Dic₃₈	1	2	yes	—
123	C₂×C₄²	D₇₆	1	2	yes	—
124	C₂×C₄²	C₂×Dic₁₉	1	2	yes	—
125	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
126	C₄⋊C₈	—	19	4	no	—
127	C₄⋊C₈	—	19	4	no	—
128	C₄⋊C₈	—	38	2	no	—
129	C₄⋊C₈	—	38	2	no	—
130	#97	—	2	4	no	—
131	#97	C₁₉⋊C₈	1	8	yes	complemented
132	C₄×C₈	—	2	1	no	—
133	C₄×C₈	—	2	1	no	—
134	C₄×C₈	—	76	4	no	—
135	C₄×C₈	C₄×D₁₉	1	2	yes	—
136	C₄×C₈	C₄×D₁₉	1	2	yes	—
137	C₁₉	#8275	1	1	yes	characteristic, complemented, Sylow
138	C₂²×C₄	C₄⋊Dic₁₉	1	4	yes	—
139	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
140	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
141	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	—	38	4	no	—
146	C₂×C₈	—	38	4	no	—
147	C₂×C₈	—	76	2	no	—
148	C₂×C₈	—	76	2	no	—
149	C₂×C₈	—	76	2	no	—
150	C₂×C₈	—	76	2	no	—
151	C₂×C₈	C₂×C₁₉⋊C₈	1	4	yes	—
152	C₂×C₈	C₂×C₁₉⋊C₈	1	4	yes	—
153	C₂×C₈	C₄×Dic₁₉	1	2	yes	—
154	C₂×C₈	C₄×Dic₁₉	1	2	yes	—
155	C₄²	—	2	2	no	—
156	C₄²	—	2	2	no	—
157	C₄²	—	4	2	no	—
158	C₄²	C₂×C₁₉⋊C₈	1	4	yes	—
159	C₄²	#64520	1	4	yes	—
160	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
161	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
162	C₄²	#64541	1	1	yes	characteristic
163	C₄²	#64541	1	1	yes	characteristic
164	C₂³	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	C₄⋊(C₁₉⋊C₈)	1	8	yes	—
169	C₂×C₄	C₄×Dic₃₈	1	1	yes	characteristic
170	C₂×C₄	C₄×Dic₃₈	1	1	yes	characteristic
171	C₂×C₄	C₄×D₇₆	1	1	yes	characteristic
172	C₂×C₄	C₄×D₇₆	1	1	yes	characteristic
173	C₂×C₄	#10597354	1	2	yes	—
174	C₂×C₄	#10597355	1	2	yes	—
175	C₂×C₄	C₂²×C₁₉⋊C₈	1	2	yes	—
176	C₂×C₄	C₂×#64520	1	2	yes	—
177	C₂×C₄	C₂×C₄×Dic₁₉	1	1	yes	characteristic
178	C₂×C₄	#10597372	1	1	yes	characteristic
179	C₈	—	2	4	no	—
180	C₈	—	2	2	no	—
181	C₈	—	76	4	no	—
182	C₈	—	76	4	no	—
183	C₈	C₄×C₁₉⋊C₈	1	4	yes	complemented
184	C₂²	C₄×C₄⋊Dic₁₉	1	1	yes	characteristic
185	C₂²	C₂×C₄⋊(C₁₉⋊C₈)	1	2	yes	—
186	C₂²	#11926586	1	2	yes	—
187	C₂²	#11926593	1	2	yes	—
188	C₄	—	2	8	no	—
189	C₄	—	2	1	no	—
190	C₄	—	2	1	no	—
191	C₄	C₂×C₄×C₁₉⋊C₈	1	1	yes	characteristic
192	C₄	C₄×#64520	1	1	yes	characteristic
193	C₄	#11926597	1	2	yes	—
194	C₄	#11926646	1	2	yes	—
195	C₄	#11926652	1	1	yes	characteristic
196	C₄	#11926657	1	1	yes	characteristic
197	C₂	#14562387	1	4	yes	—
198	C₂	C₄×C₄⋊(C₁₉⋊C₈)	1	1	yes	characteristic
199	C₂	#14563532	1	1	yes	characteristic
200	C₂	#14563541	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₇₆
commutator	C₄×C₁₉	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₄×C₁₉×#97	C₂
socle, prime radical	C₂³×C₁₉	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₄×#97	D₁₉
hypocenter	C₁₉	#8275
LCS	G ▷ C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
10. #11926297	C₄	semidirect
11. #11926295	C₄	semidirect
27. C₁₉×#97	C₈	semidirect
81. C₈×C₁₉	C₄×C₈	semidirect
131. #97	C₁₉⋊C₈	semidirect
137. C₁₉	#8275	semidirect
183. 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×#11926295	2
#14561470	2
C₄×C₁₉×#97	2
#8275	19