Subgroups — #18087689

Order 4864.

1222 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	C₂²×#10597260	C₂	1	2	yes	maximal
2	C₂×#11926299	C₂	1	4	yes	maximal
3	C₂×C₄³×C₁₉	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₁₉⋊C₈	C₂²	1	4	yes	—
5	C₂×#10597260	—	2	4	no	—
6	C₂×#10597260	—	2	4	no	—
7	C₂×#10597260	C₄	1	8	yes	complemented
8	#11926299	C₄	1	16	yes	complemented
9	C₂²×C₄²×C₁₉	C₄	1	4	yes	—
10	C₂²×C₄²×C₁₉	C₂²	1	2	yes	—
11	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
12	C₄³×C₁₉	—	2	4	no	—
13	C₂³×C₄×C₁₉	C₂×C₄	1	2	yes	—
14	C₂³×C₄×C₁₉	C₂×C₄	1	4	yes	—
15	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
16	C₂×C₄²×C₁₉	—	2	8	no	—
17	C₂×C₄²×C₁₉	—	2	16	no	—
18	C₂×C₄²×C₁₉	—	2	4	no	—
19	C₂×C₄²×C₁₉	—	2	4	no	—
20	C₂×C₄²×C₁₉	C₂×C₄	1	8	yes	—
21	C₂×C₄²×C₁₉	C₂×C₄	1	4	yes	—
22	C₂×C₄²×C₁₉	D₄	1	2	yes	—
23	C₂×C₄²×C₁₉	D₄	1	4	yes	—
24	C₂×C₄²×C₁₉	Q₈	1	2	yes	—
25	C₂²×C₁₉⋊C₈	—	2	4	no	—
26	C₂²×C₁₉⋊C₈	—	2	4	no	—
27	C₂²×C₁₉⋊C₈	—	2	8	no	—
28	C₂²×C₁₉⋊C₈	C₂×C₄	1	16	yes	—
29	C₂²×C₁₉⋊C₈	C₂×C₄	1	8	yes	—
30	#10597260	—	2	16	no	—
31	C₂⁴×C₁₉	C₂²×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₉	—	2	4	no	—
33	C₂²×C₄×C₁₉	—	2	4	no	—
34	C₂²×C₄×C₁₉	—	2	8	no	—
35	C₂²×C₄×C₁₉	—	2	8	no	—
36	C₂²×C₄×C₁₉	C₄²	1	4	yes	—
37	C₂²×C₄×C₁₉	C₄²	1	8	yes	—
38	C₂²×C₄×C₁₉	C₂²⋊C₄	1	4	yes	—
39	C₂²×C₄×C₁₉	C₄⋊C₄	1	4	yes	—
40	C₂²×C₄×C₁₉	M₁₆	1	16	yes	—
41	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
42	C₂²×C₄×C₁₉	C₂²×C₄	1	4	yes	—
43	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
45	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
46	C₂²×C₄×C₁₉	#41	1	4	yes	—
47	C₄²×C₁₉	—	2	32	no	—
48	C₄²×C₁₉	—	2	16	no	—
49	C₄²×C₁₉	C₂²⋊C₄	1	8	yes	—
50	C₄²×C₁₉	C₄⋊C₄	1	8	yes	—
51	C₂×C₁₉⋊C₈	—	2	8	no	—
52	C₂×C₁₉⋊C₈	—	2	8	no	—
53	C₂×C₁₉⋊C₈	—	2	16	no	—
54	C₂×C₁₉⋊C₈	—	2	16	no	—
55	C₂×C₁₉⋊C₈	C₄²	1	16	yes	complemented
56	#10693	—	19	1	no	maximal, Sylow
57	C₂³×C₁₉	C₂×C₄²	1	1	yes	characteristic
58	C₂³×C₁₉	C₂×C₄²	1	2	yes	—
59	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂³×C₁₉	#117	1	2	yes	—
62	C₂³×C₁₉	C₂×M₁₆	1	8	yes	—
63	C₂×C₄×C₁₉	—	2	8	no	—
64	C₂×C₄×C₁₉	—	2	8	no	—
65	C₂×C₄×C₁₉	—	2	8	no	—
66	C₂×C₄×C₁₉	—	2	8	no	—
67	C₂×C₄×C₁₉	—	2	16	no	—
68	C₂×C₄×C₁₉	—	2	8	no	—
69	C₂×C₄×C₁₉	—	2	16	no	—
70	C₂×C₄×C₁₉	#95	1	8	yes	—
71	C₂×C₄×C₁₉	#97	1	16	yes	complemented
72	C₂×C₄×C₁₉	C₂×C₄²	1	4	yes	—
73	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
74	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
75	C₂×C₄×C₁₉	#117	1	4	yes	—
76	C₂×C₄×C₁₉	C₄×D₄	1	4	yes	—
77	C₂×C₄×C₁₉	C₄×D₄	1	8	yes	—
78	C₂×C₄×C₁₉	C₄×Q₈	1	4	yes	—
79	C₁₉⋊C₈	—	2	16	no	—
80	C₂²×#97	—	19	2	no	—
81	C₂×C₄³	D₁₉	1	1	yes	characteristic
82	C₂×#336	—	19	4	no	—
83	C₂²×C₁₉	C₄³	1	1	yes	characteristic
84	C₂²×C₁₉	C₂×#95	1	1	yes	characteristic
85	C₂²×C₁₉	#376	1	1	yes	characteristic
86	C₂²×C₁₉	C₄×C₂²⋊C₄	1	2	yes	—
87	C₂²×C₁₉	C₄×C₄⋊C₄	1	2	yes	—
88	C₂²×C₁₉	C₂×#97	1	4	yes	—
89	C₂²×C₁₉	C₄×M₁₆	1	4	yes	—
90	C₂²×C₁₉	C₄×M₁₆	1	4	yes	—
91	C₂²×C₁₉	#407	1	4	yes	—
92	C₂²×C₁₉	#423	1	4	yes	—
93	C₂²×C₁₉	#432	1	8	yes	—
94	C₄×C₁₉	—	2	8	no	—
95	C₄×C₁₉	—	2	16	no	—
96	C₄×C₁₉	C₄×C₂²⋊C₄	1	4	yes	—
97	C₄×C₁₉	C₄×C₄⋊C₄	1	4	yes	—
98	C₂³×C₈	—	19	4	no	—
99	C₂²×C₄²	Dic₁₉	1	4	yes	—
100	C₂²×C₄²	C₂×D₁₉	1	2	yes	—
101	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
102	C₂×#97	—	19	8	no	—
103	C₂×#97	—	38	4	no	—
104	C₂×#97	—	38	4	no	—
105	C₄³	—	2	4	no	—
106	#336	—	19	16	no	—
107	C₂×C₁₉	C₄×#95	1	1	yes	characteristic
108	C₂×C₁₉	C₄×#97	1	2	yes	—
109	C₂×C₁₉	#1728	1	2	yes	—
110	C₂×C₁₉	#1745	1	2	yes	—
111	C₂×C₁₉	#1749	1	4	yes	—
112	C₂×C₁₉	#1767	1	4	yes	—
113	C₂³×C₄	C₂×Dic₁₉	1	4	yes	—
114	C₂³×C₄	C₂×Dic₁₉	1	2	yes	—
115	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
116	C₂²×C₈	—	19	16	no	—
117	C₂²×C₈	—	19	8	no	—
118	C₂²×C₈	—	38	4	no	—
119	C₂²×C₈	—	38	8	no	—
120	C₂²×C₈	—	38	4	no	—
121	C₂×C₄²	—	2	16	no	—
122	C₂×C₄²	—	2	8	no	—
123	C₂×C₄²	—	2	4	no	—
124	C₂×C₄²	—	2	4	no	—
125	C₂×C₄²	Dic₃₈	1	2	yes	—
126	C₂×C₄²	C₄×D₁₉	1	8	yes	—
127	C₂×C₄²	C₄×D₁₉	1	4	yes	—
128	C₂×C₄²	D₇₆	1	2	yes	—
129	C₂×C₄²	C₂²⋊D₁₉	1	4	yes	—
130	#97	—	38	16	no	—
131	C₁₉	#10693	1	1	yes	characteristic, complemented, Sylow
132	C₂⁴	C₂²×Dic₁₉	1	1	yes	characteristic
133	C₂²×C₄	—	2	4	no	—
134	C₂²×C₄	—	2	8	no	—
135	C₂²×C₄	—	2	8	no	—
136	C₂²×C₄	—	2	4	no	—
137	C₂²×C₄	#64520	1	16	yes	—
138	C₂²×C₄	C₄×Dic₁₉	1	8	yes	—
139	C₂²×C₄	C₄×Dic₁₉	1	4	yes	—
140	C₂²×C₄	C₄⋊Dic₁₉	1	4	yes	—
141	C₂²×C₄	C₂²⋊Dic₁₉	1	4	yes	—
142	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
143	C₂²×C₄	C₂×C₄×D₁₉	1	4	yes	—
144	C₂²×C₄	C₂×C₄×D₁₉	1	2	yes	—
145	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
146	C₂²×C₄	#64541	1	4	yes	—
147	C₂²×C₄	C₂×C₂²⋊D₁₉	1	2	yes	—
148	C₂×C₈	—	19	16	no	—
149	C₂×C₈	—	38	8	no	—
150	C₂×C₈	—	38	16	no	—
151	C₂×C₈	—	38	8	no	—
152	C₂×C₈	—	38	16	no	—
153	C₄²	—	2	32	no	—
154	C₄²	—	2	16	no	—
155	C₄²	Dic₁₉⋊C₄	1	8	yes	—
156	C₄²	(C₂×D₁₉)⋊C₄	1	8	yes	—
157	C₂³	C₂×#64520	1	8	yes	—
158	C₂³	C₂×C₄×Dic₁₉	1	1	yes	characteristic
159	C₂³	C₂×C₄×Dic₁₉	1	2	yes	—
160	C₂³	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
161	C₂³	#10597372	1	2	yes	—
162	C₂³	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
163	C₂×C₄	—	2	8	no	—
164	C₂×C₄	—	2	16	no	—
165	C₂×C₄	—	2	8	no	—
166	C₂×C₄	—	2	8	no	—
167	C₂×C₄	—	2	8	no	—
168	C₂×C₄	—	2	16	no	—
169	C₂×C₄	—	2	8	no	—
170	C₂×C₄	#10597260	1	16	yes	complemented
171	C₂×C₄	C₁₉⋊#95	1	8	yes	—
172	C₂×C₄	C₄×Dic₃₈	1	4	yes	—
173	C₂×C₄	C₄²×D₁₉	1	4	yes	—
174	C₂×C₄	#10597318	1	4	yes	—
175	C₂×C₄	C₄×D₇₆	1	4	yes	—
176	C₂×C₄	C₂×Dic₁₉⋊C₄	1	2	yes	—
177	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	2	yes	—
178	C₂×C₄	C₄×C₂²⋊D₁₉	1	8	yes	—
179	C₈	—	38	16	no	—
180	C₂²	C₄²×Dic₁₉	1	1	yes	characteristic
181	C₂²	C₂×C₁₉⋊#95	1	1	yes	characteristic
182	C₂²	#11926390	1	1	yes	characteristic
183	C₂²	C₄×C₂²⋊Dic₁₉	1	2	yes	—
184	C₂²	C₄×C₄⋊Dic₁₉	1	2	yes	—
185	C₂²	C₂×#10597260	1	4	yes	—
186	C₂²	C₄×#64520	1	4	yes	—
187	C₂²	C₄×#64520	1	4	yes	—
188	C₂²	#11926508	1	4	yes	—
189	C₂²	#11926568	1	4	yes	—
190	C₂²	#11926601	1	8	yes	—
191	C₄	—	2	8	no	—
192	C₄	—	2	16	no	—
193	C₄	C₄×(C₂×D₁₉)⋊C₄	1	4	yes	—
194	C₄	C₄×Dic₁₉⋊C₄	1	4	yes	—
195	C₂	C₄×C₁₉⋊#95	1	1	yes	characteristic
196	C₂	C₄×#10597260	1	2	yes	—
197	C₂	#14563329	1	2	yes	—
198	C₂	#14563384	1	2	yes	—
199	C₂	#14563397	1	4	yes	—
200	C₂	#14563488	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂³×C₄	C₂²×D₁₉
commutator	C₂²×C₁₉	C₄³
Fitting	C₂×C₄³×C₁₉	C₂
socle, prime radical	C₂⁴×C₁₉	C₂²×C₄
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	C₂×C₄³	D₁₉
hypocenter	C₁₉	#10693
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³×C₄ ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
7. C₂×#10597260	C₄	semidirect
8. #11926299	C₄	semidirect
55. C₂×C₁₉⋊C₈	C₄²	semidirect
71. C₂×C₄×C₁₉	#97	semidirect
131. C₁₉	#10693	semidirect
170. C₂×C₄	#10597260	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#10597260	2
C₂×#11926299	2
C₂×C₄³×C₁₉	2
#10693	19