Subgroups — #18081209

Order 4864.

370 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14564398	C₂	1	2	yes	maximal
2	#14563792	C₂	1	1	yes	maximal, characteristic
3	C₂×#11926297	C₂	1	1	yes	maximal, characteristic
4	#14561519	C₂	1	2	yes	maximal
5	C₁₉×#1770	C₂	1	1	yes	maximal, characteristic
6	C₂×#10597274	—	2	1	no	—
7	C₂×#10597274	—	2	1	no	—
8	C₂×#10597265	—	2	2	no	—
9	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
10	#11926414	C₂²	1	2	yes	—
11	#11926306	—	2	2	no	—
12	#11926297	C₄	1	4	yes	complemented
13	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₁₉×#97	—	2	1	no	—
15	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
17	C₁₉×#336	—	2	1	no	—
18	C₂²×C₈×C₁₉	—	2	1	no	—
19	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₁₉	D₄	1	2	yes	—
22	C₂×C₄²×C₁₉	—	2	1	no	—
23	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
24	C₂×C₄⋊Dic₁₉	—	2	2	no	—
25	C₂×C₄⋊Dic₁₉	—	2	2	no	—
26	C₂×C₄⋊Dic₁₉	—	2	2	no	—
27	C₂²×C₁₉⋊C₈	—	2	2	no	—
28	C₁₉×C₄⋊C₈	—	2	1	no	—
29	C₁₉×C₄⋊C₈	Q₈	1	2	yes	—
30	C₁₉×#97	—	4	2	no	—
31	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₁₉	Q₈	1	2	yes	—
33	C₁₉×#95	—	2	1	no	—
34	#10597274	—	4	2	no	—
35	#10597274	—	4	2	no	—
36	#10597265	—	4	4	no	—
37	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
38	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	4	yes	—
39	C₂²×C₄×C₁₉	—	2	2	no	—
40	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
43	C₂²×Dic₁₉	—	4	2	no	—
44	C₂×C₈×C₁₉	—	2	1	no	—
45	C₂×C₈×C₁₉	—	2	1	no	—
46	C₂×C₈×C₁₉	—	2	2	no	—
47	C₂×C₈×C₁₉	—	4	1	no	—
48	C₂×C₈×C₁₉	—	4	1	no	—
49	C₂×C₈×C₁₉	C₄⋊C₄	1	2	yes	—
50	C₂×C₈×C₁₉	C₄⋊C₄	1	2	yes	—
51	C₂×C₈×C₁₉	SD₁₆	1	2	yes	—
52	C₂×C₈×C₁₉	SD₁₆	1	2	yes	—
53	C₄⋊C₄×C₁₉	—	2	2	no	—
54	C₄⋊C₄×C₁₉	—	2	2	no	—
55	C₄²×C₁₉	—	4	1	no	—
56	C₄²×C₁₉	—	4	1	no	—
57	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
58	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
59	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
60	C₄²×C₁₉	#41	1	1	yes	characteristic
61	C₄⋊Dic₁₉	—	4	2	no	—
62	C₄⋊Dic₁₉	—	4	2	no	—
63	C₄⋊Dic₁₉	—	4	2	no	—
64	C₄⋊Dic₁₉	—	4	2	no	—
65	C₄⋊Dic₁₉	—	8	2	no	—
66	C₂×C₁₉⋊C₈	—	2	4	no	—
67	C₂×C₁₉⋊C₈	—	4	2	no	—
68	C₂×C₁₉⋊C₈	—	4	2	no	—
69	#9618	—	19	1	no	maximal, Sylow
70	C₂³×C₁₉	#121	1	1	yes	characteristic
71	C₂×C₄×C₁₉	—	2	2	no	—
72	C₂×C₄×C₁₉	—	4	2	no	—
73	C₂×C₄×C₁₉	—	4	2	no	—
74	C₂×C₄×C₁₉	—	4	2	no	—
75	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
77	C₂×C₄×C₁₉	C₄×Q₈	1	1	yes	characteristic
78	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
79	C₂×C₄×C₁₉	#125	1	1	yes	characteristic
80	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
81	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
82	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
83	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
84	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
85	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
86	C₂×C₄×C₁₉	#137	1	2	yes	—
87	C₂×Dic₁₉	—	8	2	no	—
88	C₂×Dic₁₉	—	8	2	no	—
89	C₂×Dic₁₉	—	8	2	no	—
90	C₈×C₁₉	—	2	2	no	—
91	C₈×C₁₉	—	4	2	no	—
92	C₈×C₁₉	#106	1	4	yes	complemented
93	C₁₉⋊C₈	—	4	4	no	—
94	#1948	—	19	2	no	—
95	#1844	—	19	1	no	—
96	#1770	D₁₉	1	1	yes	characteristic
97	C₂×#335	—	19	1	no	—
98	#1296	—	19	2	no	—
99	C₂²×C₁₉	#389	1	1	yes	characteristic
100	C₂²×C₁₉	#460	1	1	yes	characteristic
101	C₂²×C₁₉	#461	1	1	yes	characteristic
102	C₂²×C₁₉	#468	1	2	yes	—
103	C₂²×C₁₉	#471	1	2	yes	—
104	C₄×C₁₉	—	2	1	no	—
105	C₄×C₁₉	—	2	1	no	—
106	C₄×C₁₉	—	8	2	no	—
107	C₄×C₁₉	C₂×#106	1	1	yes	characteristic
108	C₄×C₁₉	#428	1	1	yes	characteristic
109	C₄×C₁₉	#441	1	1	yes	characteristic
110	C₄×C₁₉	#442	1	1	yes	characteristic
111	C₄×C₁₉	#475	1	1	yes	characteristic
112	C₄×C₁₉	#476	1	1	yes	characteristic
113	C₄×C₁₉	#501	1	2	yes	—
114	Dic₁₉	—	16	2	no	—
115	C₂×#106	—	38	1	no	—
116	C₂×#106	—	38	1	no	—
117	C₂×#106	—	38	2	no	—
118	C₂×C₄⋊C₈	—	19	2	no	—
119	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
120	C₂×#97	—	2	1	no	—
121	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
122	#384	—	19	2	no	—
123	C₄×C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic
124	#340	—	38	2	no	—
125	#336	—	2	1	no	—
126	#335	—	19	4	no	—
127	C₂×C₁₉	#1686	1	2	yes	—
128	C₂×C₁₉	#1691	1	2	yes	—
129	C₂×C₁₉	#1943	1	1	yes	characteristic
130	C₂×C₁₉	#1948	1	1	yes	characteristic
131	C₂×C₁₉	#1951	1	1	yes	characteristic
132	C₂²×C₈	—	2	1	no	—
133	C₂²×C₈	—	38	2	no	—
134	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
135	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
136	C₂×C₄⋊C₄	—	38	2	no	—
137	C₂×C₄⋊C₄	—	38	2	no	—
138	C₂×C₄⋊C₄	—	38	2	no	—
139	C₂×C₄⋊C₄	D₇₆	1	2	yes	—
140	C₂×C₄²	—	2	1	no	—
141	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
142	#106	—	76	4	no	—
143	#106	—	76	2	no	—
144	#106	—	76	2	no	—
145	C₄⋊C₈	—	2	1	no	—
146	C₄⋊C₈	—	19	4	no	—
147	C₄⋊C₈	—	38	2	no	—
148	C₄⋊C₈	Dic₃₈	1	2	yes	—
149	#97	—	4	2	no	—
150	C₄×C₈	Dic₃₈	1	2	yes	—
151	C₄×C₈	C₄×D₁₉	1	2	yes	—
152	#95	—	2	1	no	—
153	C₁₉	#9618	1	1	yes	characteristic, complemented, Sylow
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	—	76	2	no	—
156	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
157	C₂²×C₄	#64541	1	1	yes	characteristic
158	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
159	C₂×C₈	—	2	1	no	—
160	C₂×C₈	—	2	1	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	4	1	no	—
163	C₂×C₈	—	4	1	no	—
164	C₂×C₈	—	38	4	no	—
165	C₂×C₈	—	76	2	no	—
166	C₂×C₈	—	76	2	no	—
167	C₂×C₈	Dic₁₉⋊C₄	1	2	yes	—
168	C₂×C₈	Dic₁₉⋊C₄	1	2	yes	—
169	C₂×C₈	#64526	1	2	yes	—
170	C₂×C₈	#64527	1	2	yes	—
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	76	2	no	—
174	C₄⋊C₄	—	76	2	no	—
175	C₄⋊C₄	—	76	2	no	—
176	C₄⋊C₄	—	76	2	no	—
177	C₄⋊C₄	—	152	2	no	—
178	C₄²	—	4	1	no	—
179	C₄²	—	4	1	no	—
180	C₄²	C₂×Dic₃₈	1	1	yes	characteristic
181	C₄²	C₂×Dic₃₈	1	1	yes	characteristic
182	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
183	C₄²	#64541	1	1	yes	characteristic
184	C₂³	#10597376	1	1	yes	characteristic
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	152	2	no	—
190	C₂×C₄	—	152	2	no	—
191	C₂×C₄	—	152	2	no	—
192	C₂×C₄	C₄×Dic₃₈	1	1	yes	characteristic
193	C₂×C₄	#10597315	1	1	yes	characteristic
194	C₂×C₄	#10597316	1	1	yes	characteristic
195	C₂×C₄	C₄×D₇₆	1	1	yes	characteristic
196	C₂×C₄	#10597354	1	2	yes	—
197	C₂×C₄	#10597355	1	2	yes	—
198	C₂×C₄	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
199	C₂×C₄	#10597370	1	1	yes	characteristic
200	C₂×C₄	C₂×#64526	1	1	yes	characteristic
201	C₂×C₄	C₂×#64527	1	1	yes	characteristic
202	C₂×C₄	#10597395	1	1	yes	characteristic
203	C₂×C₄	#10597397	1	1	yes	characteristic
204	C₈	—	2	2	no	—
205	C₈	—	4	2	no	—
206	C₈	—	76	4	no	—
207	C₈	#10597265	1	4	yes	complemented
208	C₂²	#11926442	1	1	yes	characteristic
209	C₂²	#11926744	1	1	yes	characteristic
210	C₂²	#11926749	1	1	yes	characteristic
211	C₂²	#11926790	1	2	yes	—
212	C₂²	#11926807	1	2	yes	—
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	—	8	2	no	—
216	C₄	—	304	2	no	—
217	C₄	C₂×#10597265	1	1	yes	characteristic
218	C₄	#11926587	1	1	yes	characteristic
219	C₄	#11926652	1	1	yes	characteristic
220	C₄	#11926657	1	1	yes	characteristic
221	C₄	#11926829	1	1	yes	characteristic
222	C₄	#11926834	1	1	yes	characteristic
223	C₄	#11926953	1	2	yes	—
224	C₂	#14563107	1	2	yes	—
225	C₂	#14563134	1	2	yes	—
226	C₂	#14564373	1	1	yes	characteristic
227	C₂	#14564398	1	1	yes	characteristic
228	C₂	#14564415	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#10597376
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1770	C₂
socle, prime radical	C₂³×C₁₉	#121
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1770	D₁₉
hypocenter	C₁₉	#9618
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1770

Complements

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

N	H ≅ G/N	type
12. #11926297	C₄	semidirect
92. C₈×C₁₉	#106	semidirect
153. C₁₉	#9618	semidirect
207. C₈	#10597265	semidirect

Maximal subgroups

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

subgroup	index
#14564398	2
#14563792	2
C₂×#11926297	2
#14561519	2
C₁₉×#1770	2
#9618	19