Subgroups — #18076200

Order 4864.

422 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14563787	C₂	1	2	yes	maximal
2	#14561470	C₂	1	4	yes	maximal
3	C₄²×C₈×C₁₉	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
5	C₂×C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
6	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	4	yes	—
7	C₂×C₄×C₈×C₁₉	C₄	1	2	yes	—
8	C₂×C₄×C₈×C₁₉	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈×C₁₉	C₄	1	1	yes	characteristic
10	C₂×C₄×C₈×C₁₉	C₂²	1	2	yes	—
11	C₄³×C₁₉	C₂²	1	1	yes	characteristic
12	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
13	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
14	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
17	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
18	C₂×C₄²×C₁₉	Q₈	1	1	yes	characteristic
19	C₂×C₄²×C₁₉	Q₈	1	1	yes	characteristic
20	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
21	C₂²×C₁₉⋊C₈	—	2	4	no	—
22	C₄×C₈×C₁₉	—	2	4	no	—
23	C₄×C₈×C₁₉	—	2	2	no	—
24	C₄×C₈×C₁₉	—	2	4	no	—
25	C₄×C₈×C₁₉	C₈	1	4	yes	complemented
26	C₄⋊(C₁₉⋊C₈)	—	2	8	no	—
27	C₄⋊(C₁₉⋊C₈)	—	4	4	no	—
28	C₄⋊(C₁₉⋊C₈)	—	4	4	no	—
29	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
30	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₈×C₁₉	—	2	2	no	—
37	C₂×C₈×C₁₉	—	2	2	no	—
38	C₂×C₈×C₁₉	—	2	2	no	—
39	C₂×C₈×C₁₉	—	2	2	no	—
40	C₂×C₈×C₁₉	C₂×C₈	1	4	yes	—
41	C₂×C₈×C₁₉	M₁₆	1	4	yes	—
42	C₄²×C₁₉	—	2	2	no	—
43	C₄²×C₁₉	—	2	2	no	—
44	C₄²×C₁₉	—	2	2	no	—
45	C₄²×C₁₉	C₂×C₈	1	2	yes	—
46	C₄²×C₁₉	M₁₆	1	2	yes	—
47	C₄²×C₁₉	D₈	1	2	yes	—
48	C₄²×C₁₉	SD₁₆	1	4	yes	—
49	C₄²×C₁₉	Q₁₆	1	2	yes	—
50	C₄²×C₁₉	#41	1	4	yes	—
51	C₂×C₁₉⋊C₈	—	4	4	no	—
52	C₂×C₁₉⋊C₈	—	4	4	no	—
53	C₂×C₁₉⋊C₈	—	4	4	no	—
54	#8784	—	19	1	no	maximal, Sylow
55	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₁₉	—	2	1	no	—
57	C₂×C₄×C₁₉	—	2	1	no	—
58	C₂×C₄×C₁₉	—	2	1	no	—
59	C₂×C₄×C₁₉	—	2	1	no	—
60	C₂×C₄×C₁₉	—	2	1	no	—
61	C₂×C₄×C₁₉	—	2	1	no	—
62	C₂×C₄×C₁₉	C₄⋊C₈	1	2	yes	—
63	C₂×C₄×C₁₉	C₄⋊C₈	1	2	yes	—
64	C₂×C₄×C₁₉	#106	1	2	yes	—
65	C₂×C₄×C₁₉	#106	1	2	yes	—
66	C₂×C₄×C₁₉	#107	1	2	yes	—
67	C₂×C₄×C₁₉	#107	1	2	yes	—
68	C₂×C₄×C₁₉	#108	1	2	yes	—
69	C₂×C₄×C₁₉	#108	1	2	yes	—
70	C₂×C₄×C₁₉	#117	1	2	yes	—
71	C₂×C₄×C₁₉	#124	1	2	yes	—
72	C₂×C₄×C₁₉	#125	1	2	yes	—
73	C₂×C₄×C₁₉	C₂²×C₈	1	1	yes	characteristic
74	C₂×C₄×C₁₉	C₂×M₁₆	1	2	yes	—
75	C₂×C₄×C₁₉	C₂×M₁₆	1	1	yes	characteristic
76	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
77	C₂×C₄×C₁₉	C₂×SD₁₆	1	2	yes	—
78	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
79	C₈×C₁₉	—	2	4	no	—
80	C₈×C₁₉	—	2	4	no	—
81	C₁₉⋊C₈	—	8	4	no	—
82	#1842	—	19	2	no	—
83	C₄²×C₈	D₁₉	1	1	yes	characteristic
84	#1277	—	19	4	no	—
85	C₂²×C₁₉	#382	1	1	yes	characteristic
86	C₂²×C₁₉	C₂×C₄⋊C₈	1	1	yes	characteristic
87	C₂²×C₁₉	#423	1	1	yes	characteristic
88	C₂²×C₁₉	C₂×#106	1	1	yes	characteristic
89	C₂²×C₁₉	C₂×#107	1	1	yes	characteristic
90	C₂²×C₁₉	C₂×#108	1	1	yes	characteristic
91	C₂²×C₁₉	C₂×#108	1	1	yes	characteristic
92	C₄×C₁₉	—	2	2	no	—
93	C₄×C₁₉	—	2	2	no	—
94	C₄×C₁₉	—	2	2	no	—
95	C₄×C₁₉	#334	1	4	yes	—
96	C₄×C₁₉	#335	1	4	yes	—
97	C₄×C₁₉	#431	1	2	yes	—
98	C₄×C₁₉	#432	1	2	yes	—
99	C₄×C₁₉	#486	1	2	yes	—
100	C₄×C₁₉	#487	1	2	yes	—
101	C₂×C₄⋊C₈	—	19	4	no	—
102	C₂×C₄⋊C₈	—	38	2	no	—
103	C₂×C₄⋊C₈	—	38	2	no	—
104	C₂×C₄×C₈	Dic₁₉	1	2	yes	—
105	C₂×C₄×C₈	Dic₁₉	1	1	yes	characteristic
106	C₂×C₄×C₈	Dic₁₉	1	1	yes	characteristic
107	C₂×C₄×C₈	C₂×D₁₉	1	2	yes	—
108	C₄³	C₂×D₁₉	1	1	yes	characteristic
109	C₂×C₁₉	C₂×#334	1	1	yes	characteristic
110	C₂×C₁₉	C₂×#335	1	1	yes	characteristic
111	C₂×C₁₉	#1566	1	1	yes	characteristic
112	C₂×C₁₉	#1567	1	1	yes	characteristic
113	C₂×C₁₉	#1831	1	1	yes	characteristic
114	C₂×C₁₉	#1834	1	1	yes	characteristic
115	C₂×C₁₉	#1837	1	1	yes	characteristic
116	C₂²×C₈	—	38	4	no	—
117	C₂²×C₈	C₂×Dic₁₉	1	2	yes	—
118	C₂²×C₈	C₂×Dic₁₉	1	2	yes	—
119	C₂×C₄²	Dic₃₈	1	1	yes	characteristic
120	C₂×C₄²	Dic₃₈	1	1	yes	characteristic
121	C₂×C₄²	D₇₆	1	1	yes	characteristic
122	C₂×C₄²	D₇₆	1	1	yes	characteristic
123	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
124	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
125	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
126	C₄⋊C₈	—	38	8	no	—
127	C₄⋊C₈	—	76	4	no	—
128	C₄⋊C₈	—	76	4	no	—
129	C₄×C₈	—	2	4	no	—
130	C₄×C₈	—	2	2	no	—
131	C₄×C₈	—	2	4	no	—
132	C₄×C₈	C₁₉⋊C₈	1	4	yes	complemented
133	C₁₉	#8784	1	1	yes	characteristic, complemented, Sylow
134	C₂²×C₄	C₄⋊Dic₁₉	1	1	yes	characteristic
135	C₂²×C₄	C₄⋊Dic₁₉	1	1	yes	characteristic
136	C₂²×C₄	C₄⋊Dic₁₉	1	1	yes	characteristic
137	C₂²×C₄	C₄⋊Dic₁₉	1	1	yes	characteristic
138	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
139	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
140	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	—	76	4	no	—
146	C₂×C₈	—	76	4	no	—
147	C₂×C₈	—	76	4	no	—
148	C₂×C₈	C₂×C₁₉⋊C₈	1	4	yes	—
149	C₂×C₈	#64520	1	4	yes	—
150	C₄²	—	2	2	no	—
151	C₄²	—	2	2	no	—
152	C₄²	—	2	2	no	—
153	C₄²	#64516	1	4	yes	—
154	C₄²	D₁₅₂	1	2	yes	—
155	C₄²	Dic₇₆	1	2	yes	—
156	C₄²	C₂×C₁₉⋊C₈	1	2	yes	—
157	C₄²	#64520	1	2	yes	—
158	C₄²	#64541	1	4	yes	—
159	C₂³	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	C₄⋊(C₁₉⋊C₈)	1	2	yes	—
167	C₂×C₄	C₄⋊(C₁₉⋊C₈)	1	2	yes	—
168	C₂×C₄	#10597274	1	2	yes	—
169	C₂×C₄	#10597274	1	2	yes	—
170	C₂×C₄	#10597275	1	2	yes	—
171	C₂×C₄	#10597275	1	2	yes	—
172	C₂×C₄	#10597276	1	2	yes	—
173	C₂×C₄	#10597276	1	2	yes	—
174	C₂×C₄	#10597316	1	2	yes	—
175	C₂×C₄	#10597321	1	2	yes	—
176	C₂×C₄	C₂×#64516	1	2	yes	—
177	C₂×C₄	C₂×D₁₅₂	1	1	yes	characteristic
178	C₂×C₄	C₂×Dic₇₆	1	1	yes	characteristic
179	C₂×C₄	C₂²×C₁₉⋊C₈	1	1	yes	characteristic
180	C₂×C₄	C₂×#64520	1	2	yes	—
181	C₂×C₄	C₂×#64520	1	1	yes	characteristic
182	C₂×C₄	#10597372	1	2	yes	—
183	C₈	—	2	4	no	—
184	C₈	—	2	4	no	—
185	C₈	—	152	4	no	—
186	C₂²	#11926410	1	1	yes	characteristic
187	C₂²	C₂×C₄⋊(C₁₉⋊C₈)	1	1	yes	characteristic
188	C₂²	#11926568	1	1	yes	characteristic
189	C₂²	C₂×#10597274	1	1	yes	characteristic
190	C₂²	C₂×#10597275	1	1	yes	characteristic
191	C₂²	C₂×#10597276	1	1	yes	characteristic
192	C₂²	C₂×#10597276	1	1	yes	characteristic
193	C₄	—	2	2	no	—
194	C₄	—	2	2	no	—
195	C₄	—	2	2	no	—
196	C₄	#11926295	1	4	yes	—
197	C₄	#11926297	1	4	yes	—
198	C₄	#11926597	1	2	yes	—
199	C₄	#11926601	1	2	yes	—
200	C₄	#11926891	1	2	yes	—
201	C₄	#11926895	1	2	yes	—
202	C₂	C₂×#11926295	1	1	yes	characteristic
203	C₂	C₂×#11926297	1	1	yes	characteristic
204	C₂	#14562391	1	1	yes	characteristic
205	C₂	#14562394	1	1	yes	characteristic
206	C₂	#14563752	1	1	yes	characteristic
207	C₂	#14563763	1	1	yes	characteristic
208	C₂	#14563773	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₇₆
commutator	C₂×C₄×C₁₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₄²×C₈×C₁₉	C₂
socle, prime radical	C₂³×C₁₉	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₄²×C₈	D₁₉
hypocenter	C₁₉	#8784
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
25. C₄×C₈×C₁₉	C₈	semidirect
132. C₄×C₈	C₁₉⋊C₈	semidirect
133. C₁₉	#8784	semidirect

Maximal subgroups

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

subgroup	index
#14563787	2
#14561470	2
C₄²×C₈×C₁₉	2
#8784	19