Subgroups — #18081570

Order 4864.

570 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	1	yes	maximal, characteristic
2	C₄×C₄⋊(C₁₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂×#11926293	C₂	1	4	yes	maximal
4	C₁₉×#2868	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
6	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
8	C₂×C₄×C₁₉⋊C₈	—	2	1	no	—
9	#11926299	—	2	1	no	—
10	#11926293	—	2	8	no	—
11	C₂×C₁₉×C₄⋊Q₈	C₄	1	4	yes	—
12	C₂×C₁₉×C₄⋊Q₈	C₂²	1	2	yes	—
13	C₁₉×#384	—	2	1	no	—
14	C₁₉×#384	—	2	2	no	—
15	C₁₉×#384	—	2	1	no	—
16	C₄³×C₁₉	C₂²	1	1	yes	characteristic
17	C₂²×Q₈×C₁₉	C₂×C₄	1	4	yes	—
18	C₁₉×C₄⋊Q₈	—	2	4	no	—
19	C₁₉×C₄⋊Q₈	—	2	4	no	—
20	C₁₉×C₄⋊Q₈	—	4	8	no	—
21	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
22	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
23	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
24	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
25	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₁₉	D₄	1	2	yes	—
28	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
30	C₂²×C₁₉⋊C₈	—	2	2	no	—
31	C₂²×C₁₉⋊C₈	—	2	2	no	—
32	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
33	C₄⋊(C₁₉⋊C₈)	—	2	4	no	—
34	C₄⋊(C₁₉⋊C₈)	—	4	4	no	—
35	C₄⋊(C₁₉⋊C₈)	D₄	1	4	yes	—
36	C₄×C₁₉⋊C₈	—	4	2	no	—
37	C₂×Q₈×C₁₉	—	2	8	no	—
38	C₂×Q₈×C₁₉	—	4	8	no	—
39	C₂²×C₄×C₁₉	—	2	4	no	—
40	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
41	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
42	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
45	C₄⋊C₄×C₁₉	—	4	8	no	—
46	C₄⋊C₄×C₁₉	—	4	8	no	—
47	C₄⋊C₄×C₁₉	—	4	4	no	—
48	C₄⋊C₄×C₁₉	—	4	2	no	—
49	C₄⋊C₄×C₁₉	—	4	2	no	—
50	C₄²×C₁₉	—	2	4	no	—
51	C₄²×C₁₉	—	2	2	no	—
52	C₄²×C₁₉	SD₁₆	1	2	yes	—
53	C₄²×C₁₉	SD₁₆	1	4	yes	—
54	C₄²×C₁₉	Q₁₆	1	4	yes	—
55	C₄²×C₁₉	Q₁₆	1	2	yes	—
56	C₄²×C₁₉	C₂×D₄	1	2	yes	—
57	C₄²×C₁₉	#41	1	2	yes	—
58	C₂×C₁₉⋊C₈	—	2	4	no	—
59	C₂×C₁₉⋊C₈	—	4	2	no	—
60	C₂×C₁₉⋊C₈	—	4	2	no	—
61	C₂×C₁₉⋊C₈	—	4	2	no	—
62	C₂×C₁₉⋊C₈	—	4	4	no	—
63	#9691	—	19	1	no	maximal, Sylow
64	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	Q₈×C₁₉	—	8	8	no	—
66	C₂×C₄×C₁₉	—	2	2	no	—
67	C₂×C₄×C₁₉	—	2	2	no	—
68	C₂×C₄×C₁₉	—	2	2	no	—
69	C₂×C₄×C₁₉	—	4	4	no	—
70	C₂×C₄×C₁₉	—	4	8	no	—
71	C₂×C₄×C₁₉	#100	1	4	yes	—
72	C₂×C₄×C₁₉	#103	1	4	yes	—
73	C₂×C₄×C₁₉	#103	1	4	yes	—
74	C₂×C₄×C₁₉	#103	1	4	yes	—
75	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
76	C₂×C₄×C₁₉	#121	1	2	yes	—
77	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
78	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
79	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
80	C₂×C₄×C₁₉	C₂×SD₁₆	1	2	yes	—
81	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
82	C₂×C₄×C₁₉	C₂×Q₁₆	1	2	yes	—
83	C₂×C₄×C₁₉	#137	1	2	yes	—
84	C₁₉⋊C₈	—	4	4	no	—
85	C₁₉⋊C₈	—	8	2	no	—
86	#2868	D₁₉	1	1	yes	characteristic
87	#1842	—	19	1	no	—
88	C₄×C₄⋊C₈	—	19	1	no	—
89	C₂×#333	—	19	4	no	—
90	C₂²×C₁₉	#390	1	1	yes	characteristic
91	C₂²×C₁₉	C₂×#100	1	2	yes	—
92	C₂²×C₁₉	C₂×#103	1	1	yes	characteristic
93	C₂²×C₁₉	C₂×#103	1	2	yes	—
94	C₂²×C₁₉	#419	1	1	yes	characteristic
95	C₄×C₁₉	—	2	4	no	—
96	C₄×C₁₉	—	2	2	no	—
97	C₄×C₁₉	—	8	4	no	—
98	C₄×C₁₉	#333	1	8	yes	complemented
99	C₄×C₁₉	#461	1	2	yes	—
100	C₄×C₁₉	#462	1	2	yes	—
101	C₄×C₁₉	#487	1	2	yes	—
102	C₄×C₁₉	#492	1	1	yes	characteristic
103	C₄×C₁₉	#494	1	1	yes	characteristic
104	C₂×C₄⋊Q₈	Dic₁₉	1	4	yes	—
105	C₂×C₄⋊Q₈	C₂×D₁₉	1	2	yes	—
106	C₂×C₄⋊C₈	—	19	2	no	—
107	C₂×C₄⋊C₈	—	19	2	no	—
108	C₂×C₄⋊C₈	—	38	2	no	—
109	C₂×C₄×C₈	—	38	1	no	—
110	#384	—	2	1	no	—
111	#384	—	2	1	no	—
112	#384	—	2	2	no	—
113	C₄³	C₂×D₁₉	1	1	yes	characteristic
114	#336	—	38	1	no	—
115	#333	—	38	8	no	—
116	C₂×C₁₉	C₂×#333	1	2	yes	—
117	C₂×C₁₉	#1548	1	2	yes	—
118	C₂×C₁₉	#1956	1	1	yes	characteristic
119	C₂×C₁₉	#1965	1	1	yes	characteristic
120	C₂×C₁₉	#1981	1	1	yes	characteristic
121	C₂²×Q₈	C₂×Dic₁₉	1	4	yes	—
122	C₂²×C₈	—	38	2	no	—
123	C₂²×C₈	—	38	2	no	—
124	C₄⋊Q₈	—	2	4	no	—
125	C₄⋊Q₈	—	2	4	no	—
126	C₄⋊Q₈	—	4	8	no	—
127	C₂×C₄⋊C₄	—	2	4	no	—
128	C₂×C₄⋊C₄	—	2	4	no	—
129	C₂×C₄⋊C₄	—	2	2	no	—
130	C₂×C₄⋊C₄	—	2	2	no	—
131	C₂×C₄²	C₂×Dic₁₉	1	2	yes	—
132	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
133	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
134	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
135	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
136	C₄⋊C₈	—	19	4	no	—
137	C₄⋊C₈	—	38	4	no	—
138	C₄⋊C₈	—	38	2	no	—
139	C₄⋊C₈	—	76	4	no	—
140	C₄×C₈	—	76	2	no	—
141	C₁₉	#9691	1	1	yes	characteristic, complemented, Sylow
142	C₂×Q₈	—	2	8	no	—
143	C₂×Q₈	—	4	8	no	—
144	C₂²×C₄	—	2	4	no	—
145	C₂²×C₄	C₂²⋊Dic₁₉	1	2	yes	—
146	C₂²×C₄	C₂²⋊Dic₁₉	1	2	yes	—
147	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
148	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
149	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
150	C₂×C₈	—	38	4	no	—
151	C₂×C₈	—	76	2	no	—
152	C₂×C₈	—	76	2	no	—
153	C₂×C₈	—	76	2	no	—
154	C₂×C₈	—	76	4	no	—
155	C₄⋊C₄	—	4	8	no	—
156	C₄⋊C₄	—	4	2	no	—
157	C₄⋊C₄	—	4	4	no	—
158	C₄⋊C₄	—	4	2	no	—
159	C₄⋊C₄	—	4	8	no	—
160	C₄²	—	2	2	no	—
161	C₄²	—	2	4	no	—
162	C₄²	#64527	1	2	yes	—
163	C₄²	#64527	1	4	yes	—
164	C₄²	#64528	1	4	yes	—
165	C₄²	#64528	1	2	yes	—
166	C₄²	D₄×D₁₉	1	2	yes	—
167	C₄²	#64543	1	2	yes	—
168	C₂³	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
169	Q₈	—	8	8	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	4	4	no	—
174	C₂×C₄	—	4	8	no	—
175	C₂×C₄	#10597290	1	4	yes	—
176	C₂×C₄	#10597292	1	4	yes	—
177	C₂×C₄	#10597292	1	4	yes	—
178	C₂×C₄	#10597292	1	4	yes	—
179	C₂×C₄	D₄×Dic₁₉	1	2	yes	—
180	C₂×C₄	#10597382	1	1	yes	characteristic
181	C₂×C₄	#10597384	1	2	yes	—
182	C₂×C₄	#10597386	1	1	yes	characteristic
183	C₂×C₄	C₂×#64527	1	2	yes	—
184	C₂×C₄	C₂×#64527	1	1	yes	characteristic
185	C₂×C₄	#10597388	1	2	yes	—
186	C₂×C₄	C₂×#64528	1	2	yes	—
187	C₂×C₄	C₂×#64528	1	1	yes	characteristic
188	C₈	—	76	4	no	—
189	C₈	—	152	2	no	—
190	C₂²	#11926447	1	1	yes	characteristic
191	C₂²	C₂×#10597290	1	2	yes	—
192	C₂²	C₂×#10597292	1	2	yes	—
193	C₂²	C₂×#10597292	1	1	yes	characteristic
194	C₂²	#11926553	1	1	yes	characteristic
195	C₄	—	2	4	no	—
196	C₄	—	2	2	no	—
197	C₄	—	8	4	no	—
198	C₄	#11926293	1	8	yes	complemented
199	C₄	#11926746	1	2	yes	—
200	C₄	#11926751	1	2	yes	—
201	C₄	#11926894	1	2	yes	—
202	C₄	#11926918	1	1	yes	characteristic
203	C₄	#11926923	1	1	yes	characteristic
204	C₂	C₂×#11926293	1	2	yes	—
205	C₂	#14562316	1	2	yes	—
206	C₂	#14564438	1	1	yes	characteristic
207	C₂	#14564476	1	1	yes	characteristic
208	C₂	#14564559	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₁₉
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#2868	C₂
socle, prime radical	C₂³×C₁₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#2868	D₁₉
hypocenter	C₁₉	#9691
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2868

Complements

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

N	H ≅ G/N	type
98. C₄×C₁₉	#333	semidirect
141. C₁₉	#9691	semidirect
198. C₄	#11926293	semidirect

Maximal subgroups

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

subgroup	index
#14563787	2
C₄×C₄⋊(C₁₉⋊C₈)	2
C₂×#11926293	2
C₁₉×#2868	2
#9691	19