Subgroups — #18088361

Order 4864.

950 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11926410	C₂	1	2	yes	maximal
2	C₂×#11926307	C₂	1	4	yes	maximal
3	C₂²×C₄×C₈×C₁₉	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄⋊Dic₁₉	C₂²	1	4	yes	—
5	#11926410	—	2	4	no	—
6	#11926410	C₄	1	8	yes	complemented
7	C₂×C₁₉⋊#95	—	2	4	no	—
8	#11926307	—	2	8	no	—
9	C₂³×C₈×C₁₉	C₂²	1	2	yes	—
10	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₁₉	—	2	4	no	—
12	C₂×C₄×C₈×C₁₉	C₄	1	4	yes	complemented
13	C₂³×C₄×C₁₉	D₄	1	2	yes	—
14	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
15	C₂³×Dic₁₉	—	2	4	no	—
16	C₂²×C₈×C₁₉	—	2	4	no	—
17	C₂²×C₈×C₁₉	—	2	4	no	—
18	C₂²×C₈×C₁₉	C₂×C₄	1	4	yes	—
19	C₂²×C₈×C₁₉	D₄	1	4	yes	—
20	C₂²×C₈×C₁₉	Q₈	1	4	yes	—
21	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₁₉	C₂×C₄	1	4	yes	—
23	C₂×C₄²×C₁₉	D₄	1	4	yes	—
24	C₂×C₄²×C₁₉	Q₈	1	2	yes	—
25	C₂×C₄⋊Dic₁₉	—	2	8	no	—
26	C₂×C₄⋊Dic₁₉	—	2	8	no	—
27	C₂×C₄⋊Dic₁₉	—	2	4	no	—
28	C₂×C₄⋊Dic₁₉	C₂×C₄	1	8	yes	—
29	C₄×C₈×C₁₉	—	2	8	no	—
30	C₁₉⋊#95	—	2	16	no	—
31	C₂⁴×C₁₉	C₂×D₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₉	—	2	8	no	—
33	C₂²×C₄×C₁₉	C₂²⋊C₄	1	8	yes	—
34	C₂²×C₄×C₁₉	C₄⋊C₄	1	4	yes	—
35	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
36	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
38	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
39	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
40	C₂²×C₄×C₁₉	C₂×Q₈	1	2	yes	—
41	C₂²×C₄×C₁₉	#41	1	4	yes	—
42	C₂²×Dic₁₉	—	2	8	no	—
43	C₂²×Dic₁₉	—	4	4	no	—
44	C₂²×Dic₁₉	—	4	4	no	—
45	C₂²×Dic₁₉	—	4	16	no	—
46	C₂×C₈×C₁₉	—	2	4	no	—
47	C₂×C₈×C₁₉	—	2	4	no	—
48	C₂×C₈×C₁₉	—	2	8	no	—
49	C₂×C₈×C₁₉	—	2	8	no	—
50	C₂×C₈×C₁₉	C₄⋊C₄	1	8	yes	—
51	C₄²×C₁₉	C₄²	1	4	yes	—
52	C₄²×C₁₉	C₂²⋊C₄	1	4	yes	—
53	C₄²×C₁₉	C₄⋊C₄	1	8	yes	—
54	C₄⋊Dic₁₉	—	2	16	no	—
55	C₄⋊Dic₁₉	—	4	8	no	—
56	#10842	—	19	1	no	maximal, Sylow
57	C₂³×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
58	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂³×C₁₉	#124	1	2	yes	—
60	C₂³×C₁₉	#127	1	1	yes	characteristic
61	C₂³×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
62	C₂³×C₁₉	#135	1	8	yes	—
63	C₂×C₄×C₁₉	—	2	16	no	—
64	C₂×C₄×C₁₉	—	2	8	no	—
65	C₂×C₄×C₁₉	#95	1	8	yes	—
66	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
67	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
69	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
70	C₂×C₄×C₁₉	#117	1	2	yes	—
71	C₂×C₄×C₁₉	C₄×D₄	1	4	yes	—
72	C₂×C₄×C₁₉	C₄×Q₈	1	4	yes	—
73	C₂×C₄×C₁₉	#121	1	4	yes	—
74	C₂×C₄×C₁₉	C₂²⋊Q₈	1	4	yes	—
75	C₂×C₄×C₁₉	#125	1	2	yes	—
76	C₂×C₄×C₁₉	C₄⋊Q₈	1	2	yes	—
77	C₂×Dic₁₉	—	4	8	no	—
78	C₂×Dic₁₉	—	4	16	no	—
79	C₂×Dic₁₉	—	8	8	no	—
80	C₂×Dic₁₉	—	8	8	no	—
81	C₈×C₁₉	—	2	8	no	—
82	C₂²×C₄×C₈	D₁₉	1	1	yes	characteristic
83	C₂×#382	—	19	2	no	—
84	C₂×#340	—	19	4	no	—
85	C₂²×C₁₉	C₂×#95	1	1	yes	characteristic
86	C₂²×C₁₉	#382	1	1	yes	characteristic
87	C₂²×C₁₉	#384	1	1	yes	characteristic
88	C₂²×C₁₉	#390	1	2	yes	—
89	C₂²×C₁₉	#391	1	2	yes	—
90	C₂²×C₁₉	#416	1	8	yes	—
91	C₂²×C₁₉	#427	1	4	yes	—
92	C₂²×C₁₉	#488	1	8	yes	—
93	C₂²×C₁₉	#495	1	4	yes	—
94	C₂²×C₁₉	#499	1	4	yes	—
95	C₄×C₁₉	—	2	8	no	—
96	C₄×C₁₉	C₄×C₄⋊C₄	1	2	yes	—
97	C₄×C₁₉	#380	1	2	yes	—
98	C₄×C₁₉	#389	1	4	yes	—
99	Dic₁₉	—	8	8	no	—
100	C₂³×C₈	C₂×D₁₉	1	2	yes	—
101	C₂²×C₄⋊C₄	—	19	4	no	—
102	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
103	C₂×C₄×C₈	—	2	4	no	—
104	C₂×C₄×C₈	Dic₁₉	1	4	yes	complemented
105	#382	—	19	8	no	—
106	#382	—	38	4	no	—
107	C₂×#95	—	38	4	no	—
108	#340	—	38	8	no	—
109	C₂×C₁₉	C₄⋊#95	1	1	yes	characteristic
110	C₂×C₁₉	#1735	1	2	yes	—
111	C₂×C₁₉	#1835	1	2	yes	—
112	C₂×C₁₉	#1846	1	2	yes	—
113	C₂×C₁₉	#1958	1	4	yes	—
114	C₂×C₁₉	#1991	1	4	yes	—
115	C₂³×C₄	—	38	4	no	—
116	C₂³×C₄	D₇₆	1	2	yes	—
117	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
118	C₂²×C₈	—	2	4	no	—
119	C₂²×C₈	—	2	4	no	—
120	C₂²×C₈	Dic₃₈	1	4	yes	—
121	C₂²×C₈	D₇₆	1	4	yes	—
122	C₂²×C₈	C₂×Dic₁₉	1	4	yes	—
123	C₂×C₄⋊C₄	—	19	8	no	—
124	C₂×C₄⋊C₄	—	38	8	no	—
125	C₂×C₄⋊C₄	—	38	8	no	—
126	C₂×C₄⋊C₄	—	38	4	no	—
127	C₂×C₄²	Dic₃₈	1	2	yes	—
128	C₂×C₄²	C₄×D₁₉	1	4	yes	—
129	C₂×C₄²	C₂×Dic₁₉	1	2	yes	—
130	C₂×C₄²	C₂²⋊D₁₉	1	4	yes	—
131	C₄×C₈	—	2	8	no	—
132	#95	—	38	16	no	—
133	C₁₉	#10842	1	1	yes	characteristic, complemented, Sylow
134	C₂⁴	C₂×D₇₆	1	1	yes	characteristic
135	C₂²×C₄	—	2	8	no	—
136	C₂²×C₄	—	38	8	no	—
137	C₂²×C₄	—	76	4	no	—
138	C₂²×C₄	—	76	4	no	—
139	C₂²×C₄	—	76	16	no	—
140	C₂²×C₄	C₄⋊Dic₁₉	1	4	yes	—
141	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	8	yes	—
142	C₂²×C₄	C₂×Dic₃₈	1	2	yes	—
143	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
144	C₂²×C₄	C₂×C₄×D₁₉	1	2	yes	—
145	C₂²×C₄	C₂×D₇₆	1	2	yes	—
146	C₂²×C₄	#64541	1	4	yes	—
147	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
148	C₂²×C₄	C₂×C₂²⋊D₁₉	1	2	yes	—
149	C₂×C₈	—	2	8	no	—
150	C₂×C₈	—	2	4	no	—
151	C₂×C₈	—	2	8	no	—
152	C₂×C₈	—	2	4	no	—
153	C₂×C₈	C₄⋊Dic₁₉	1	8	yes	—
154	C₄⋊C₄	—	38	16	no	—
155	C₄⋊C₄	—	76	8	no	—
156	C₄²	C₄×Dic₁₉	1	4	yes	—
157	C₄²	Dic₁₉⋊C₄	1	8	yes	—
158	C₄²	C₂²⋊Dic₁₉	1	4	yes	—
159	C₂³	#10597315	1	1	yes	characteristic
160	C₂³	#10597320	1	1	yes	characteristic
161	C₂³	#10597321	1	2	yes	—
162	C₂³	#10597350	1	8	yes	—
163	C₂³	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
164	C₂³	C₂×(C₂×D₁₉)⋊C₄	1	2	yes	—
165	C₂×C₄	—	2	16	no	—
166	C₂×C₄	—	2	8	no	—
167	C₂×C₄	—	76	16	no	—
168	C₂×C₄	—	76	8	no	—
169	C₂×C₄	—	152	8	no	—
170	C₂×C₄	—	152	8	no	—
171	C₂×C₄	C₁₉⋊#95	1	8	yes	—
172	C₂×C₄	C₄×Dic₃₈	1	4	yes	—
173	C₂×C₄	#10597315	1	2	yes	—
174	C₂×C₄	#10597316	1	2	yes	—
175	C₂×C₄	C₄×D₇₆	1	4	yes	—
176	C₂×C₄	C₂×C₄×Dic₁₉	1	1	yes	characteristic
177	C₂×C₄	C₂×Dic₁₉⋊C₄	1	2	yes	—
178	C₂×C₄	#10597370	1	4	yes	—
179	C₂×C₄	C₂×C₄⋊Dic₁₉	1	2	yes	—
180	C₂×C₄	#10597372	1	2	yes	—
181	C₂×C₄	#10597376	1	4	yes	—
182	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
183	C₈	—	2	8	no	—
184	C₂²	C₂×C₁₉⋊#95	1	1	yes	characteristic
185	C₂²	#11926410	1	1	yes	characteristic
186	C₂²	#11926414	1	1	yes	characteristic
187	C₂²	#11926446	1	2	yes	—
188	C₂²	#11926450	1	2	yes	—
189	C₂²	#11926541	1	8	yes	—
190	C₂²	#11926582	1	4	yes	—
191	C₂²	#11926899	1	8	yes	—
192	C₂²	#11926928	1	4	yes	—
193	C₂²	#11926946	1	4	yes	—
194	C₄	—	2	8	no	—
195	C₄	—	152	8	no	—
196	C₄	C₄×C₄⋊Dic₁₉	1	2	yes	—
197	C₄	#11926404	1	2	yes	—
198	C₄	#11926442	1	4	yes	—
199	C₂	#14561865	1	1	yes	characteristic
200	C₂	#14563352	1	2	yes	—
201	C₂	#14563766	1	2	yes	—
202	C₂	#14563799	1	2	yes	—
203	C₂	#14564446	1	4	yes	—
204	C₂	#14564600	1	4	yes	—

	subgroup	G/H
center	C₂⁴	C₂×D₇₆
commutator	C₂×C₄×C₁₉	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₂²×C₄×C₈×C₁₉	C₂
socle, prime radical	C₂⁴×C₁₉	C₂×D₄
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	C₂²×C₄×C₈	D₁₉
hypocenter	C₁₉	#10842
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
6. #11926410	C₄	semidirect
12. C₂×C₄×C₈×C₁₉	C₄	semidirect
104. C₂×C₄×C₈	Dic₁₉	semidirect
133. C₁₉	#10842	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926410	2
C₂×#11926307	2
C₂²×C₄×C₈×C₁₉	2
#10842	19