Subgroups — #18081587

Order 4864.

882 conjugacy classes of subgroups.

The subgroup lattice has 268 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, complemented
2	C₄×C₄⋊(C₁₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#11926288	C₂	1	4	yes	maximal, complemented
4	C₁₉×#2867	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	#11926288	—	2	8	no	—
11	C₂×C₁₉×#127	—	2	1	no	—
12	C₂×C₁₉×#127	—	2	2	no	—
13	C₂×C₁₉×#127	—	2	1	no	—
14	C₂×C₁₉×#127	C₄	1	4	yes	—
15	C₂×C₁₉×#127	C₂²	1	2	yes	—
16	C₄³×C₁₉	C₂²	1	1	yes	characteristic
17	C₂²×D₄×C₁₉	—	2	4	no	—
18	C₂²×D₄×C₁₉	—	2	4	no	—
19	C₂²×D₄×C₁₉	—	2	2	no	—
20	C₂²×D₄×C₁₉	—	2	2	no	—
21	C₂²×D₄×C₁₉	C₂×C₄	1	4	yes	—
22	C₁₉×#127	—	2	4	no	—
23	C₁₉×#127	—	2	4	no	—
24	C₁₉×#127	—	4	4	no	—
25	C₁₉×#127	—	4	8	no	—
26	C₁₉×#127	—	4	2	no	—
27	C₁₉×#127	—	4	4	no	—
28	C₁₉×#127	—	4	4	no	—
29	C₁₉×#127	—	4	2	no	—
30	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₉	D₄	1	2	yes	—
33	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
35	C₂²×C₁₉⋊C₈	—	2	2	no	—
36	C₂²×C₁₉⋊C₈	—	2	2	no	—
37	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
38	C₄⋊(C₁₉⋊C₈)	—	2	4	no	—
39	C₄⋊(C₁₉⋊C₈)	—	4	4	no	—
40	C₄⋊(C₁₉⋊C₈)	D₄	1	4	yes	complemented
41	C₄×C₁₉⋊C₈	—	4	2	no	—
42	C₂⁴×C₁₉	—	2	4	no	—
43	C₂×D₄×C₁₉	—	2	8	no	—
44	C₂×D₄×C₁₉	—	4	8	no	—
45	C₂×D₄×C₁₉	—	4	8	no	—
46	C₂×D₄×C₁₉	—	4	8	no	—
47	C₂×D₄×C₁₉	—	4	8	no	—
48	C₂×D₄×C₁₉	—	4	8	no	—
49	C₂×D₄×C₁₉	—	4	8	no	—
50	C₂×D₄×C₁₉	—	4	8	no	—
51	C₂×D₄×C₁₉	—	4	4	no	—
52	C₂×D₄×C₁₉	—	4	2	no	—
53	C₂×D₄×C₁₉	—	4	2	no	—
54	C₂×D₄×C₁₉	—	4	2	no	—
55	C₂×D₄×C₁₉	—	4	4	no	—
56	C₂×D₄×C₁₉	—	4	2	no	—
57	C₂×D₄×C₁₉	—	4	4	no	—
58	C₂×D₄×C₁₉	—	4	4	no	—
59	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
60	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
61	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
64	C₄²×C₁₉	—	2	4	no	—
65	C₄²×C₁₉	—	2	2	no	—
66	C₄²×C₁₉	D₈	1	2	yes	—
67	C₄²×C₁₉	D₈	1	4	yes	—
68	C₄²×C₁₉	SD₁₆	1	4	yes	—
69	C₄²×C₁₉	SD₁₆	1	2	yes	—
70	C₄²×C₁₉	C₂×D₄	1	2	yes	—
71	C₄²×C₁₉	#41	1	2	yes	—
72	C₂×C₁₉⋊C₈	—	2	4	no	—
73	C₂×C₁₉⋊C₈	—	4	2	no	—
74	C₂×C₁₉⋊C₈	—	4	2	no	—
75	C₂×C₁₉⋊C₈	—	4	2	no	—
76	C₂×C₁₉⋊C₈	—	4	4	no	—
77	#9695	—	19	1	no	maximal, Sylow
78	C₂³×C₁₉	—	4	4	no	—
79	C₂³×C₁₉	—	4	8	no	—
80	C₂³×C₁₉	—	4	4	no	—
81	C₂³×C₁₉	—	4	8	no	—
82	C₂³×C₁₉	—	4	4	no	—
83	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	D₄×C₁₉	—	8	8	no	—
85	D₄×C₁₉	—	8	16	no	—
86	D₄×C₁₉	—	8	16	no	—
87	D₄×C₁₉	—	8	4	no	—
88	D₄×C₁₉	—	8	4	no	—
89	D₄×C₁₉	—	8	4	no	—
90	D₄×C₁₉	—	8	2	no	—
91	D₄×C₁₉	—	8	2	no	—
92	C₂×C₄×C₁₉	—	2	2	no	—
93	C₂×C₄×C₁₉	—	2	2	no	—
94	C₂×C₄×C₁₉	—	2	2	no	—
95	C₂×C₄×C₁₉	#100	1	4	yes	—
96	C₂×C₄×C₁₉	#102	1	4	yes	—
97	C₂×C₄×C₁₉	#102	1	4	yes	—
98	C₂×C₄×C₁₉	#102	1	4	yes	—
99	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
100	C₂×C₄×C₁₉	#121	1	2	yes	—
101	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
102	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
103	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
104	C₂×C₄×C₁₉	C₂×D₈	1	2	yes	—
105	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
106	C₂×C₄×C₁₉	C₂×SD₁₆	1	2	yes	—
107	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
108	C₁₉⋊C₈	—	4	4	no	—
109	C₁₉⋊C₈	—	8	2	no	—
110	#2867	D₁₉	1	1	yes	characteristic
111	#1842	—	19	1	no	—
112	C₄×C₄⋊C₈	—	19	1	no	—
113	C₂×#331	—	19	4	no	—
114	C₂²×C₁₉	—	8	8	no	—
115	C₂²×C₁₉	—	8	8	no	—
116	C₂²×C₁₉	—	8	4	no	—
117	C₂²×C₁₉	—	8	4	no	—
118	C₂²×C₁₉	—	8	4	no	—
119	C₂²×C₁₉	#390	1	1	yes	characteristic
120	C₂²×C₁₉	C₂×#100	1	2	yes	—
121	C₂²×C₁₉	C₂×#102	1	1	yes	characteristic
122	C₂²×C₁₉	C₂×#102	1	2	yes	—
123	C₂²×C₁₉	#418	1	1	yes	characteristic
124	C₄×C₁₉	—	2	4	no	—
125	C₄×C₁₉	—	2	2	no	—
126	C₄×C₁₉	#331	1	8	yes	complemented
127	C₄×C₁₉	#459	1	2	yes	—
128	C₄×C₁₉	#460	1	2	yes	—
129	C₄×C₁₉	#486	1	2	yes	—
130	C₄×C₁₉	#492	1	1	yes	characteristic
131	C₄×C₁₉	#493	1	1	yes	characteristic
132	C₂×#127	—	2	1	no	—
133	C₂×#127	—	2	1	no	—
134	C₂×#127	—	2	2	no	—
135	C₂×#127	Dic₁₉	1	4	yes	—
136	C₂×#127	C₂×D₁₉	1	2	yes	—
137	C₂×C₄⋊C₈	—	19	2	no	—
138	C₂×C₄⋊C₈	—	19	2	no	—
139	C₂×C₄⋊C₈	—	38	2	no	—
140	C₂×C₄×C₈	—	38	1	no	—
141	C₄³	C₂×D₁₉	1	1	yes	characteristic
142	#336	—	38	1	no	—
143	#331	—	38	8	no	—
144	C₂×C₁₉	—	16	4	no	—
145	C₂×C₁₉	C₂×#331	1	2	yes	—
146	C₂×C₁₉	#1545	1	2	yes	—
147	C₂×C₁₉	#1957	1	1	yes	characteristic
148	C₂×C₁₉	#1965	1	1	yes	characteristic
149	C₂×C₁₉	#1982	1	1	yes	characteristic
150	C₂²×D₄	—	2	4	no	—
151	C₂²×D₄	—	2	4	no	—
152	C₂²×D₄	—	2	2	no	—
153	C₂²×D₄	—	2	2	no	—
154	C₂²×D₄	C₂×Dic₁₉	1	4	yes	—
155	C₂²×C₈	—	38	2	no	—
156	C₂²×C₈	—	38	2	no	—
157	#127	—	2	4	no	—
158	#127	—	2	4	no	—
159	#127	—	4	8	no	—
160	#127	—	4	4	no	—
161	#127	—	4	2	no	—
162	#127	—	4	4	no	—
163	#127	—	4	4	no	—
164	#127	—	4	2	no	—
165	C₂×C₄²	C₂×Dic₁₉	1	2	yes	—
166	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
167	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
168	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
169	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
170	C₄⋊C₈	—	19	4	no	—
171	C₄⋊C₈	—	38	4	no	—
172	C₄⋊C₈	—	38	2	no	—
173	C₄⋊C₈	—	76	4	no	—
174	C₄×C₈	—	76	2	no	—
175	C₁₉	#9695	1	1	yes	characteristic, complemented, Sylow
176	C₂⁴	—	2	4	no	—
177	C₂×D₄	—	2	8	no	—
178	C₂×D₄	—	4	8	no	—
179	C₂×D₄	—	4	8	no	—
180	C₂×D₄	—	4	8	no	—
181	C₂×D₄	—	4	4	no	—
182	C₂×D₄	—	4	2	no	—
183	C₂×D₄	—	4	4	no	—
184	C₂×D₄	—	4	4	no	—
185	C₂×D₄	—	4	2	no	—
186	C₂×D₄	—	4	2	no	—
187	C₂×D₄	—	4	4	no	—
188	C₂×D₄	—	4	2	no	—
189	C₂×D₄	—	4	8	no	—
190	C₂×D₄	—	4	8	no	—
191	C₂×D₄	—	4	8	no	—
192	C₂×D₄	—	4	8	no	—
193	C₂²×C₄	C₂²⋊Dic₁₉	1	2	yes	—
194	C₂²×C₄	C₂²⋊Dic₁₉	1	2	yes	—
195	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
196	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
197	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
198	C₂×C₈	—	38	4	no	—
199	C₂×C₈	—	76	2	no	—
200	C₂×C₈	—	76	2	no	—
201	C₂×C₈	—	76	4	no	—
202	C₂×C₈	—	76	2	no	—
203	C₄²	—	2	2	no	—
204	C₄²	—	2	4	no	—
205	C₄²	#64525	1	2	yes	—
206	C₄²	#64525	1	4	yes	—
207	C₄²	#64526	1	4	yes	—
208	C₄²	#64526	1	2	yes	—
209	C₄²	D₄×D₁₉	1	2	yes	—
210	C₄²	#64543	1	2	yes	—
211	C₂³	—	4	8	no	—
212	C₂³	—	4	4	no	—
213	C₂³	—	4	4	no	—
214	C₂³	—	4	4	no	—
215	C₂³	—	4	8	no	—
216	C₂³	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
217	D₄	—	8	8	no	—
218	D₄	—	8	16	no	—
219	D₄	—	8	16	no	—
220	D₄	—	8	4	no	—
221	D₄	—	8	2	no	—
222	D₄	—	8	2	no	—
223	D₄	—	8	4	no	—
224	D₄	—	8	4	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	#10597289	1	4	yes	—
229	C₂×C₄	#10597289	1	4	yes	—
230	C₂×C₄	#10597289	1	4	yes	—
231	C₂×C₄	#10597290	1	4	yes	—
232	C₂×C₄	C₂×#64525	1	2	yes	—
233	C₂×C₄	C₂×#64525	1	1	yes	characteristic
234	C₂×C₄	#10597378	1	2	yes	—
235	C₂×C₄	C₂×#64526	1	2	yes	—
236	C₂×C₄	C₂×#64526	1	1	yes	characteristic
237	C₂×C₄	D₄×Dic₁₉	1	2	yes	—
238	C₂×C₄	#10597382	1	1	yes	characteristic
239	C₂×C₄	#10597384	1	2	yes	—
240	C₂×C₄	#10597386	1	1	yes	characteristic
241	C₈	—	76	4	no	—
242	C₈	—	152	2	no	—
243	C₂²	—	8	4	no	—
244	C₂²	—	8	4	no	—
245	C₂²	—	8	4	no	—
246	C₂²	—	8	8	no	—
247	C₂²	—	8	8	no	—
248	C₂²	#11926447	1	1	yes	characteristic
249	C₂²	C₂×#10597290	1	2	yes	—
250	C₂²	C₂×#10597289	1	2	yes	—
251	C₂²	C₂×#10597289	1	1	yes	characteristic
252	C₂²	#11926549	1	1	yes	characteristic
253	C₄	—	2	2	no	—
254	C₄	—	2	4	no	—
255	C₄	#11926288	1	8	yes	complemented
256	C₄	#11926736	1	2	yes	—
257	C₄	#11926741	1	2	yes	—
258	C₄	#11926890	1	2	yes	—
259	C₄	#11926916	1	1	yes	characteristic
260	C₄	#11926920	1	1	yes	characteristic
261	C₂	—	16	4	no	—
262	C₂	C₂×#11926288	1	2	yes	—
263	C₂	#14562303	1	2	yes	—
264	C₂	#14564442	1	1	yes	characteristic
265	C₂	#14564476	1	1	yes	characteristic
266	C₂	#14564563	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₁₉
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting, prime radical	C₁₉×#2867	C₂
socle	C₂³×C₁₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#2867	D₁₉
hypocenter	C₁₉	#9695
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2867

Complements

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

N	H ≅ G/N	type
1. #14563787	C₂	semidirect
2. C₄×C₄⋊(C₁₉⋊C₈)	C₂	semidirect
3. C₂×#11926288	C₂	semidirect
40. C₄⋊(C₁₉⋊C₈)	D₄	semidirect
126. C₄×C₁₉	#331	semidirect
175. C₁₉	#9695	semidirect
255. C₄	#11926288	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×#11926288	2
C₁₉×#2867	2
#9695	19