Subgroups — #18073766

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×#10597274	C₂	1	1	yes	maximal, characteristic
2	#14563500	C₂	1	2	yes	maximal
3	C₂×C₈×C₁₉⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#14561474	C₂	1	2	yes	maximal
5	C₂×C₁₉×#334	C₂	1	1	yes	maximal, characteristic
6	C₂×#10597274	C₄	1	2	yes	—
7	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₈×Dic₁₉	—	2	2	no	—
10	C₄×C₄⋊Dic₁₉	C₂²	1	2	yes	—
11	#11926307	C₄	1	2	yes	—
12	#11926298	—	2	2	no	—
13	C₈×C₁₉⋊C₈	C₄	1	4	yes	complemented
14	C₂×C₁₉×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
16	C₁₉×#334	C₄	1	4	yes	complemented
17	C₂²×C₈×C₁₉	—	2	2	no	—
18	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
21	C₂×C₄⋊Dic₁₉	C₂×C₄	1	4	yes	—
22	C₂×C₄×Dic₁₉	—	2	2	no	—
23	C₂²×C₁₉⋊C₈	—	2	1	no	—
24	C₂²×C₁₉⋊C₈	D₄	1	1	yes	characteristic
25	C₂²×C₁₉⋊C₈	D₄	1	1	yes	characteristic
26	C₁₉×C₄⋊C₈	—	2	2	no	—
27	C₁₉×C₄⋊C₈	C₂×C₄	1	4	yes	—
28	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
29	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
30	C₁₉⋊#95	—	2	2	no	—
31	#10597274	C₈	1	8	yes	complemented
32	C₈×Dic₁₉	—	4	4	no	—
33	C₄×C₁₉⋊C₈	—	2	1	no	—
34	C₄×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
35	C₄×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
36	C₄×C₁₉⋊C₈	Q₈	1	2	yes	—
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	C₂²×Dic₁₉	—	2	4	no	—
41	C₂×C₈×C₁₉	—	2	2	no	—
42	C₂×C₈×C₁₉	—	2	4	no	—
43	C₂×C₈×C₁₉	—	4	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	C₂×C₈	1	4	yes	—
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	C₄⋊Dic₁₉	—	2	4	no	—
53	C₄⋊Dic₁₉	C₂×C₈	1	8	yes	—
54	C₄×Dic₁₉	—	4	2	no	—
55	C₄×Dic₁₉	—	4	2	no	—
56	C₂×C₁₉⋊C₈	—	2	4	no	—
57	C₂×C₁₉⋊C₈	—	2	2	no	—
58	C₂×C₁₉⋊C₈	—	2	2	no	—
59	C₂×C₁₉⋊C₈	C₄⋊C₄	1	2	yes	—
60	C₂×C₁₉⋊C₈	C₄⋊C₄	1	2	yes	—
61	C₂×C₁₉⋊C₈	SD₁₆	1	4	yes	—
62	C₈×#106	—	19	1	no	maximal, Sylow
63	C₂³×C₁₉	C₄×D₄	1	1	yes	characteristic
64	C₂×C₄×C₁₉	—	2	2	no	—
65	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
66	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
68	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₁₉	C₄×Q₈	1	1	yes	characteristic
70	C₂×C₄×C₁₉	C₄×Q₈	1	1	yes	characteristic
71	C₂×C₄×C₁₉	C₂²×C₈	1	2	yes	—
72	C₂×C₄×C₁₉	#131	1	2	yes	—
73	C₂×C₄×C₁₉	C₂×SD₁₆	1	2	yes	—
74	C₂×C₄×C₁₉	#135	1	2	yes	—
75	C₂×Dic₁₉	—	2	8	no	—
76	C₂×Dic₁₉	—	4	4	no	—
77	C₂×Dic₁₉	—	4	4	no	—
78	C₈×C₁₉	—	2	2	no	—
79	C₈×C₁₉	—	4	4	no	—
80	C₈×C₁₉	C₄×C₈	1	4	yes	complemented
81	C₁₉⋊C₈	—	2	4	no	—
82	C₁₉⋊C₈	—	2	2	no	—
83	C₁₉⋊C₈	#106	1	4	yes	complemented
84	C₄×#106	—	19	1	no	—
85	C₈×C₄⋊C₄	—	19	2	no	—
86	C₂×#334	D₁₉	1	1	yes	characteristic
87	C₂×C₈²	—	19	1	no	—
88	#1279	—	19	2	no	—
89	C₂²×C₁₉	C₄×C₄⋊C₄	1	1	yes	characteristic
90	C₂²×C₁₉	C₈×D₄	1	2	yes	—
91	C₂²×C₁₉	C₄×SD₁₆	1	2	yes	—
92	C₂²×C₁₉	#443	1	2	yes	—
93	C₄×C₁₉	—	2	1	no	—
94	C₄×C₁₉	—	2	1	no	—
95	C₄×C₁₉	C₂×C₄×C₈	1	1	yes	characteristic
96	C₄×C₁₉	#405	1	1	yes	characteristic
97	C₄×C₁₉	C₂×#106	1	1	yes	characteristic
98	C₄×C₁₉	#427	1	1	yes	characteristic
99	C₄×C₁₉	C₄×SD₁₆	1	2	yes	—
100	C₄×C₁₉	C₈×Q₈	1	2	yes	—
101	Dic₁₉	—	4	8	no	—
102	C₂×#106	—	19	2	no	—
103	C₂×C₄⋊C₈	C₂×D₁₉	1	2	yes	—
104	C₂×C₄×C₈	—	19	1	no	—
105	C₂×C₄×C₈	—	19	1	no	—
106	C₂×C₄×C₈	—	38	2	no	—
107	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
108	C₄×C₄⋊C₄	—	19	2	no	—
109	#340	—	19	2	no	—
110	#336	—	38	2	no	—
111	#334	Dic₁₉	1	4	yes	complemented
112	C₈²	—	19	4	no	—
113	C₂×C₁₉	C₈×SD₁₆	1	4	yes	—
114	C₂×C₁₉	C₈×C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₁₉	#1772	1	1	yes	characteristic
116	C₂×C₁₉	C₄×#106	1	1	yes	characteristic
117	C₂²×C₈	—	2	2	no	—
118	C₂²×C₈	—	19	1	no	—
119	C₂²×C₈	—	19	1	no	—
120	C₂²×C₈	—	38	1	no	—
121	C₂²×C₈	C₄×D₁₉	1	1	yes	characteristic
122	C₂²×C₈	C₄×D₁₉	1	1	yes	characteristic
123	C₂×C₄⋊C₄	—	19	4	no	—
124	C₂×C₄²	—	38	2	no	—
125	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
126	#106	—	19	8	no	—
127	C₄⋊C₈	—	2	2	no	—
128	C₄⋊C₈	C₂×Dic₁₉	1	4	yes	—
129	C₄×C₈	—	19	2	no	—
130	C₄×C₈	—	19	2	no	—
131	C₄×C₈	—	19	2	no	—
132	C₄×C₈	—	38	1	no	—
133	C₄×C₈	—	76	4	no	—
134	C₄×C₈	C₄×D₁₉	1	2	yes	—
135	C₄×C₈	C₂×Dic₁₉	1	2	yes	—
136	#95	—	38	2	no	—
137	C₁₉	C₈×#106	1	1	yes	characteristic, complemented, Sylow
138	C₂²×C₄	—	38	4	no	—
139	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
140	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
141	C₂²×C₄	#64545	1	1	yes	characteristic
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	2	4	no	—
144	C₂×C₈	—	4	2	no	—
145	C₂×C₈	—	4	2	no	—
146	C₂×C₈	—	19	2	no	—
147	C₂×C₈	—	19	4	no	—
148	C₂×C₈	—	19	2	no	—
149	C₂×C₈	—	38	4	no	—
150	C₂×C₈	—	38	2	no	—
151	C₂×C₈	—	38	2	no	—
152	C₂×C₈	C₈×D₁₉	1	4	yes	—
153	C₂×C₈	C₄×Dic₁₉	1	2	yes	—
154	C₂×C₈	C₄×Dic₁₉	1	2	yes	—
155	C₄⋊C₄	—	19	8	no	—
156	C₄⋊C₄	—	38	4	no	—
157	C₄²	—	76	2	no	—
158	C₄²	—	76	2	no	—
159	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
160	C₄²	#64543	1	1	yes	characteristic
161	C₄²	Q₈×D₁₉	1	1	yes	characteristic
162	C₄²	C₂²×Dic₁₉	1	1	yes	characteristic
163	C₂³	#10597339	1	1	yes	characteristic
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	38	8	no	—
166	C₂×C₄	—	76	4	no	—
167	C₂×C₄	—	76	4	no	—
168	C₂×C₄	#10597333	1	1	yes	characteristic
169	C₂×C₄	C₄⋊C₄×D₁₉	1	1	yes	characteristic
170	C₂×C₄	#10597338	1	1	yes	characteristic
171	C₂×C₄	C₂×C₈×D₁₉	1	2	yes	—
172	C₂×C₄	#10597353	1	2	yes	—
173	C₂×C₄	SD₁₆×D₁₉	1	2	yes	—
174	C₂×C₄	#10597362	1	2	yes	—
175	C₂×C₄	C₂×C₄×Dic₁₉	1	1	yes	characteristic
176	C₂×C₄	D₄×Dic₁₉	1	1	yes	characteristic
177	C₂×C₄	Q₈×Dic₁₉	1	1	yes	characteristic
178	C₈	—	2	2	no	—
179	C₈	—	4	4	no	—
180	C₈	—	19	4	no	—
181	C₈	—	38	4	no	—
182	C₈	—	38	2	no	—
183	C₈	C₈×Dic₁₉	1	4	yes	complemented
184	C₂²	C₄⋊C₄×Dic₁₉	1	1	yes	characteristic
185	C₂²	#11926609	1	2	yes	—
186	C₂²	#11926631	1	2	yes	—
187	C₂²	#11926660	1	2	yes	—
188	C₄	—	2	1	no	—
189	C₄	—	2	1	no	—
190	C₄	—	76	8	no	—
191	C₄	C₂×C₈×Dic₁₉	1	1	yes	characteristic
192	C₄	#11926502	1	1	yes	characteristic
193	C₄	#106×D₁₉	1	1	yes	characteristic
194	C₄	#11926579	1	1	yes	characteristic
195	C₄	SD₁₆×Dic₁₉	1	2	yes	—
196	C₄	#11926668	1	2	yes	—
197	C₂	#14562434	1	4	yes	—
198	C₂	#14563500	1	1	yes	characteristic
199	C₂	#14563517	1	1	yes	characteristic
200	C₂	#106×Dic₁₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₁₉
commutator	C₄×C₁₉	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₂×C₁₉×#334	C₂
socle, prime radical	C₂³×C₁₉	C₄×D₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂×#334	D₁₉
hypocenter	C₁₉	C₈×#106
LCS	G ▷ C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
13. C₈×C₁₉⋊C₈	C₄	semidirect
16. C₁₉×#334	C₄	semidirect
31. #10597274	C₈	semidirect
80. C₈×C₁₉	C₄×C₈	semidirect
83. C₁₉⋊C₈	#106	semidirect
111. #334	Dic₁₉	semidirect
137. C₁₉	C₈×#106	semidirect
183. 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
#14563500	2
C₂×C₈×C₁₉⋊C₈	2
#14561474	2
C₂×C₁₉×#334	2
C₈×#106	19