Subgroups — #18076310

Order 4864.

454 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14563795	C₂	1	2	yes	maximal
2	#14563792	C₂	1	2	yes	maximal
3	#14563763	C₂	1	2	yes	maximal
4	C₂×C₈²×C₁₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#10597275	—	2	4	no	—
6	C₂×#10597274	—	2	4	no	—
7	#11926414	C₂²	1	4	yes	—
8	#11926307	—	2	4	no	—
9	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
12	C₈²×C₁₉	C₄	1	4	yes	complemented
13	C₂²×C₈×C₁₉	D₄	1	2	yes	—
14	C₂²×C₈×C₁₉	D₄	1	2	yes	—
15	C₂²×C₈×C₁₉	D₄	1	2	yes	—
16	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
17	C₂×C₄⋊Dic₁₉	—	2	4	no	—
18	C₂×C₄⋊Dic₁₉	—	2	4	no	—
19	C₂×C₄⋊Dic₁₉	—	2	4	no	—
20	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
21	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
22	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
23	C₄×C₈×C₁₉	Q₈	1	2	yes	—
24	C₄×C₈×C₁₉	Q₈	1	2	yes	—
25	C₄×C₈×C₁₉	Q₈	1	2	yes	—
26	#10597275	—	4	8	no	—
27	#10597274	—	4	8	no	—
28	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
29	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
30	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
31	C₂²×Dic₁₉	—	4	4	no	—
32	C₂×C₈×C₁₉	—	2	4	no	—
33	C₂×C₈×C₁₉	C₄⋊C₄	1	4	yes	—
34	C₂×C₈×C₁₉	C₄⋊C₄	1	4	yes	—
35	C₂×C₈×C₁₉	C₄⋊C₄	1	4	yes	—
36	C₂×C₈×C₁₉	D₈	1	4	yes	—
37	C₂×C₈×C₁₉	SD₁₆	1	8	yes	—
38	C₂×C₈×C₁₉	Q₁₆	1	4	yes	—
39	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
40	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
41	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
42	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
43	C₄⋊Dic₁₉	—	4	8	no	—
44	C₄⋊Dic₁₉	—	4	8	no	—
45	C₄⋊Dic₁₉	—	4	8	no	—
46	#8814	—	19	1	no	maximal, Sylow
47	C₂³×C₁₉	#127	1	1	yes	characteristic
48	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
52	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
53	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
54	C₂×C₄×C₁₉	C₂×D₈	1	2	yes	—
55	C₂×C₄×C₁₉	C₂×SD₁₆	1	4	yes	—
56	C₂×C₄×C₁₉	C₂×Q₁₆	1	2	yes	—
57	C₂×C₄×C₁₉	#135	1	4	yes	—
58	C₂×Dic₁₉	—	8	4	no	—
59	C₂×Dic₁₉	—	8	4	no	—
60	C₂×Dic₁₉	—	8	4	no	—
61	C₈×C₁₉	—	2	4	no	—
62	C₈×C₁₉	#106	1	8	yes	complemented
63	C₈×C₁₉	#107	1	8	yes	complemented
64	#1845	—	19	2	no	—
65	#1844	—	19	2	no	—
66	#1834	—	19	2	no	—
67	C₂×C₈²	D₁₉	1	1	yes	characteristic
68	C₂²×C₁₉	#384	1	1	yes	characteristic
69	C₂²×C₁₉	#492	1	2	yes	—
70	C₂²×C₁₉	#493	1	1	yes	characteristic
71	C₂²×C₁₉	#494	1	1	yes	characteristic
72	C₂²×C₁₉	#495	1	2	yes	—
73	C₄×C₁₉	C₂×#106	1	2	yes	—
74	C₄×C₁₉	C₂×#107	1	2	yes	—
75	C₄×C₁₉	#427	1	2	yes	—
76	C₄×C₁₉	#498	1	2	yes	—
77	C₄×C₁₉	#499	1	2	yes	—
78	C₄×C₁₉	#500	1	2	yes	—
79	Dic₁₉	—	16	4	no	—
80	C₂×#107	—	38	4	no	—
81	C₂×#106	—	38	4	no	—
82	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
83	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
84	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
85	#384	—	19	4	no	—
86	#340	—	38	4	no	—
87	C₈²	Dic₁₉	1	4	yes	complemented
88	C₂×C₁₉	#1706	1	2	yes	—
89	C₂×C₁₉	#1707	1	2	yes	—
90	C₂×C₁₉	#1844	1	1	yes	characteristic
91	C₂×C₁₉	#1845	1	1	yes	characteristic
92	C₂×C₁₉	#1846	1	1	yes	characteristic
93	C₂²×C₈	D₇₆	1	2	yes	—
94	C₂²×C₈	D₇₆	1	2	yes	—
95	C₂²×C₈	D₇₆	1	2	yes	—
96	C₂×C₄⋊C₄	—	38	4	no	—
97	C₂×C₄⋊C₄	—	38	4	no	—
98	C₂×C₄⋊C₄	—	38	4	no	—
99	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
100	#107	—	76	8	no	—
101	#106	—	76	8	no	—
102	C₄×C₈	Dic₃₈	1	2	yes	—
103	C₄×C₈	Dic₃₈	1	2	yes	—
104	C₄×C₈	Dic₃₈	1	2	yes	—
105	C₄×C₈	C₂×Dic₁₉	1	2	yes	—
106	C₄×C₈	C₂×Dic₁₉	1	2	yes	—
107	C₄×C₈	C₂×Dic₁₉	1	2	yes	—
108	C₁₉	#8814	1	1	yes	characteristic, complemented, Sylow
109	C₂²×C₄	—	76	4	no	—
110	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
111	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
112	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
113	C₂×C₈	—	2	4	no	—
114	C₂×C₈	#64516	1	8	yes	—
115	C₂×C₈	D₁₅₂	1	4	yes	—
116	C₂×C₈	Dic₇₆	1	4	yes	—
117	C₂×C₈	C₄⋊Dic₁₉	1	4	yes	—
118	C₂×C₈	C₄⋊Dic₁₉	1	4	yes	—
119	C₂×C₈	C₄⋊Dic₁₉	1	4	yes	—
120	C₄⋊C₄	—	76	8	no	—
121	C₄⋊C₄	—	76	8	no	—
122	C₄⋊C₄	—	76	8	no	—
123	C₄²	C₂×Dic₃₈	1	1	yes	characteristic
124	C₄²	C₂×Dic₃₈	1	1	yes	characteristic
125	C₄²	C₂×Dic₃₈	1	1	yes	characteristic
126	C₄²	C₂²×Dic₁₉	1	1	yes	characteristic
127	C₂³	#10597320	1	1	yes	characteristic
128	C₂×C₄	—	152	4	no	—
129	C₂×C₄	—	152	4	no	—
130	C₂×C₄	—	152	4	no	—
131	C₂×C₄	#10597315	1	1	yes	characteristic
132	C₂×C₄	#10597315	1	1	yes	characteristic
133	C₂×C₄	#10597315	1	1	yes	characteristic
134	C₂×C₄	C₂×#64516	1	4	yes	—
135	C₂×C₄	C₂×D₁₅₂	1	2	yes	—
136	C₂×C₄	#10597350	1	4	yes	—
137	C₂×C₄	C₂×Dic₇₆	1	2	yes	—
138	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
139	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
140	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
141	C₈	—	2	4	no	—
142	C₈	#10597274	1	8	yes	complemented
143	C₈	#10597275	1	8	yes	complemented
144	C₂²	#11926414	1	1	yes	characteristic
145	C₂²	#11926917	1	2	yes	—
146	C₂²	#11926921	1	1	yes	characteristic
147	C₂²	#11926924	1	1	yes	characteristic
148	C₂²	#11926928	1	2	yes	—
149	C₄	—	304	4	no	—
150	C₄	C₂×#10597274	1	2	yes	—
151	C₄	C₂×#10597275	1	2	yes	—
152	C₄	#11926582	1	2	yes	—
153	C₄	#11926942	1	2	yes	—
154	C₄	#11926946	1	2	yes	—
155	C₄	#11926949	1	2	yes	—
156	C₂	#14563227	1	2	yes	—
157	C₂	#14563232	1	2	yes	—
158	C₂	#14563792	1	1	yes	characteristic
159	C₂	#14563795	1	1	yes	characteristic
160	C₂	#14563799	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#10597320
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₂×C₈²×C₁₉	C₂
socle, prime radical	C₂³×C₁₉	#127
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂×C₈²	D₁₉
hypocenter	C₁₉	#8814
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
12. C₈²×C₁₉	C₄	semidirect
62. C₈×C₁₉	#106	semidirect
63. C₈×C₁₉	#107	semidirect
87. C₈²	Dic₁₉	semidirect
108. C₁₉	#8814	semidirect
142. C₈	#10597274	semidirect
143. C₈	#10597275	semidirect

Maximal subgroups

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

subgroup	index
#14563795	2
#14563792	2
#14563763	2
C₂×C₈²×C₁₉	2
#8814	19