Subgroups — #18081574

Order 4864.

570 conjugacy classes of subgroups.

The subgroup lattice has 218 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×#11926290	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	#11926290	—	2	8	no	—
11	C₂×C₁₉×C₄⋊Q₈	—	2	2	no	—
12	C₂×C₁₉×C₄⋊Q₈	C₂²	1	2	yes	—
13	C₁₉×#384	—	2	1	no	—
14	C₁₉×#384	—	2	1	no	—
15	C₁₉×#384	C₄	1	4	yes	—
16	C₄³×C₁₉	C₂²	1	1	yes	characteristic
17	C₂²×Q₈×C₁₉	—	2	2	no	—
18	C₁₉×C₄⋊Q₈	—	2	4	no	—
19	C₁₉×C₄⋊Q₈	—	2	4	no	—
20	C₁₉×C₄⋊Q₈	—	4	4	no	—
21	C₁₉×C₄⋊Q₈	—	4	4	no	—
22	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
23	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
24	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
25	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	4	yes	—
26	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₁₉	D₄	1	2	yes	—
29	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
31	C₂²×C₁₉⋊C₈	—	2	2	no	—
32	C₂²×C₁₉⋊C₈	—	2	2	no	—
33	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
34	C₄⋊(C₁₉⋊C₈)	—	2	4	no	—
35	C₄⋊(C₁₉⋊C₈)	—	4	4	no	—
36	C₄⋊(C₁₉⋊C₈)	D₄	1	4	yes	—
37	C₄×C₁₉⋊C₈	—	4	2	no	—
38	C₂×Q₈×C₁₉	—	4	2	no	—
39	C₂×Q₈×C₁₉	—	4	2	no	—
40	C₂×Q₈×C₁₉	—	4	4	no	—
41	C₂×Q₈×C₁₉	—	4	4	no	—
42	C₂²×C₄×C₁₉	—	2	4	no	—
43	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
44	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
48	C₄⋊C₄×C₁₉	—	2	8	no	—
49	C₄⋊C₄×C₁₉	—	4	8	no	—
50	C₄⋊C₄×C₁₉	—	4	8	no	—
51	C₄⋊C₄×C₁₉	—	4	4	no	—
52	C₄²×C₁₉	—	2	4	no	—
53	C₄²×C₁₉	—	2	2	no	—
54	C₄²×C₁₉	D₈	1	4	yes	—
55	C₄²×C₁₉	SD₁₆	1	4	yes	—
56	C₄²×C₁₉	SD₁₆	1	2	yes	—
57	C₄²×C₁₉	Q₁₆	1	2	yes	—
58	C₄²×C₁₉	C₂×D₄	1	2	yes	—
59	C₄²×C₁₉	#41	1	2	yes	—
60	C₂×C₁₉⋊C₈	—	2	4	no	—
61	C₂×C₁₉⋊C₈	—	4	2	no	—
62	C₂×C₁₉⋊C₈	—	4	2	no	—
63	C₂×C₁₉⋊C₈	—	4	2	no	—
64	C₂×C₁₉⋊C₈	—	4	4	no	—
65	#9692	—	19	1	no	maximal, Sylow
66	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	Q₈×C₁₉	—	8	4	no	—
68	Q₈×C₁₉	—	8	2	no	—
69	Q₈×C₁₉	—	8	2	no	—
70	C₂×C₄×C₁₉	—	2	2	no	—
71	C₂×C₄×C₁₉	—	2	2	no	—
72	C₂×C₄×C₁₉	—	2	2	no	—
73	C₂×C₄×C₁₉	—	4	4	no	—
74	C₂×C₄×C₁₉	—	4	8	no	—
75	C₂×C₄×C₁₉	#101	1	4	yes	—
76	C₂×C₄×C₁₉	#102	1	4	yes	—
77	C₂×C₄×C₁₉	#102	1	4	yes	—
78	C₂×C₄×C₁₉	#103	1	4	yes	—
79	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
80	C₂×C₄×C₁₉	#121	1	2	yes	—
81	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
82	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
83	C₂×C₄×C₁₉	C₂×D₈	1	2	yes	—
84	C₂×C₄×C₁₉	C₂×SD₁₆	1	2	yes	—
85	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
86	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
87	C₂×C₄×C₁₉	#137	1	2	yes	—
88	C₁₉⋊C₈	—	4	4	no	—
89	C₁₉⋊C₈	—	8	2	no	—
90	#2868	D₁₉	1	1	yes	characteristic
91	#1842	—	19	1	no	—
92	C₄×C₄⋊C₈	—	19	1	no	—
93	C₂×#332	—	19	4	no	—
94	C₂²×C₁₉	#390	1	1	yes	characteristic
95	C₂²×C₁₉	C₂×#101	1	2	yes	—
96	C₂²×C₁₉	C₂×#102	1	2	yes	—
97	C₂²×C₁₉	C₂×#103	1	1	yes	characteristic
98	C₂²×C₁₉	#419	1	1	yes	characteristic
99	C₄×C₁₉	—	2	4	no	—
100	C₄×C₁₉	—	2	2	no	—
101	C₄×C₁₉	—	8	4	no	—
102	C₄×C₁₉	#332	1	8	yes	complemented
103	C₄×C₁₉	#461	1	2	yes	—
104	C₄×C₁₉	#462	1	2	yes	—
105	C₄×C₁₉	#486	1	2	yes	—
106	C₄×C₁₉	#492	1	1	yes	characteristic
107	C₄×C₁₉	#493	1	1	yes	characteristic
108	C₂×C₄⋊Q₈	—	2	2	no	—
109	C₂×C₄⋊Q₈	C₂×D₁₉	1	2	yes	—
110	C₂×C₄⋊C₈	—	19	2	no	—
111	C₂×C₄⋊C₈	—	19	2	no	—
112	C₂×C₄⋊C₈	—	38	2	no	—
113	C₂×C₄×C₈	—	38	1	no	—
114	#384	—	2	1	no	—
115	#384	—	2	1	no	—
116	#384	Dic₁₉	1	4	yes	—
117	C₄³	C₂×D₁₉	1	1	yes	characteristic
118	#336	—	38	1	no	—
119	#332	—	38	8	no	—
120	C₂×C₁₉	C₂×#332	1	2	yes	—
121	C₂×C₁₉	#1546	1	2	yes	—
122	C₂×C₁₉	#1957	1	1	yes	characteristic
123	C₂×C₁₉	#1966	1	1	yes	characteristic
124	C₂×C₁₉	#1981	1	1	yes	characteristic
125	C₂²×Q₈	—	2	2	no	—
126	C₂²×C₈	—	38	2	no	—
127	C₂²×C₈	—	38	2	no	—
128	C₄⋊Q₈	—	2	4	no	—
129	C₄⋊Q₈	—	2	4	no	—
130	C₄⋊Q₈	—	4	4	no	—
131	C₄⋊Q₈	—	4	4	no	—
132	C₂×C₄⋊C₄	—	2	4	no	—
133	C₂×C₄⋊C₄	—	2	4	no	—
134	C₂×C₄⋊C₄	—	2	2	no	—
135	C₂×C₄⋊C₄	C₂×Dic₁₉	1	4	yes	—
136	C₂×C₄²	C₂×Dic₁₉	1	2	yes	—
137	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
138	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
139	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
140	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
141	C₄⋊C₈	—	19	4	no	—
142	C₄⋊C₈	—	38	4	no	—
143	C₄⋊C₈	—	38	2	no	—
144	C₄⋊C₈	—	76	4	no	—
145	C₄×C₈	—	76	2	no	—
146	C₁₉	#9692	1	1	yes	characteristic, complemented, Sylow
147	C₂×Q₈	—	4	4	no	—
148	C₂×Q₈	—	4	4	no	—
149	C₂×Q₈	—	4	2	no	—
150	C₂×Q₈	—	4	2	no	—
151	C₂²×C₄	—	2	4	no	—
152	C₂²×C₄	C₂²⋊Dic₁₉	1	2	yes	—
153	C₂²×C₄	C₂²⋊Dic₁₉	1	2	yes	—
154	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
155	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
156	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
157	C₂×C₈	—	38	4	no	—
158	C₂×C₈	—	76	2	no	—
159	C₂×C₈	—	76	2	no	—
160	C₂×C₈	—	76	4	no	—
161	C₂×C₈	—	76	2	no	—
162	C₄⋊C₄	—	2	8	no	—
163	C₄⋊C₄	—	4	4	no	—
164	C₄⋊C₄	—	4	8	no	—
165	C₄⋊C₄	—	4	8	no	—
166	C₄²	—	2	2	no	—
167	C₄²	—	2	4	no	—
168	C₄²	#64525	1	4	yes	—
169	C₄²	#64526	1	4	yes	—
170	C₄²	#64527	1	2	yes	—
171	C₄²	#64528	1	2	yes	—
172	C₄²	D₄×D₁₉	1	2	yes	—
173	C₄²	#64543	1	2	yes	—
174	C₂³	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
175	Q₈	—	8	4	no	—
176	Q₈	—	8	2	no	—
177	Q₈	—	8	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	4	8	no	—
182	C₂×C₄	—	4	4	no	—
183	C₂×C₄	#10597289	1	4	yes	—
184	C₂×C₄	#10597289	1	4	yes	—
185	C₂×C₄	#10597292	1	4	yes	—
186	C₂×C₄	#10597293	1	4	yes	—
187	C₂×C₄	C₂×#64525	1	2	yes	—
188	C₂×C₄	C₂×#64526	1	2	yes	—
189	C₂×C₄	D₄×Dic₁₉	1	2	yes	—
190	C₂×C₄	#10597382	1	1	yes	characteristic
191	C₂×C₄	#10597384	1	2	yes	—
192	C₂×C₄	#10597386	1	1	yes	characteristic
193	C₂×C₄	C₂×#64527	1	1	yes	characteristic
194	C₂×C₄	#10597388	1	2	yes	—
195	C₂×C₄	C₂×#64528	1	1	yes	characteristic
196	C₈	—	76	4	no	—
197	C₈	—	152	2	no	—
198	C₂²	#11926447	1	1	yes	characteristic
199	C₂²	C₂×#10597293	1	2	yes	—
200	C₂²	C₂×#10597289	1	2	yes	—
201	C₂²	C₂×#10597292	1	1	yes	characteristic
202	C₂²	#11926553	1	1	yes	characteristic
203	C₄	—	2	4	no	—
204	C₄	—	2	2	no	—
205	C₄	—	8	4	no	—
206	C₄	#11926290	1	8	yes	complemented
207	C₄	#11926746	1	2	yes	—
208	C₄	#11926751	1	2	yes	—
209	C₄	#11926890	1	2	yes	—
210	C₄	#11926916	1	1	yes	characteristic
211	C₄	#11926920	1	1	yes	characteristic
212	C₂	C₂×#11926290	1	2	yes	—
213	C₂	#14562307	1	2	yes	—
214	C₂	#14564442	1	1	yes	characteristic
215	C₂	#14564480	1	1	yes	characteristic
216	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	#9692
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
102. C₄×C₁₉	#332	semidirect
146. C₁₉	#9692	semidirect
206. C₄	#11926290	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×#11926290	2
C₁₉×#2868	2
#9692	19