Subgroups — #18081116

Order 4864.

404 conjugacy classes of subgroups.

The subgroup lattice has 286 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×#11926296	C₂	1	2	yes	maximal
4	#14561469	C₂	1	2	yes	maximal
5	C₄×C₁₉×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
7	C₂×C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
8	C₂×C₄⋊(C₁₉⋊C₈)	C₄	1	1	yes	characteristic
9	C₂×C₄⋊(C₁₉⋊C₈)	C₄	1	1	yes	characteristic
10	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
11	C₂×C₄×C₁₉⋊C₈	C₂²	1	2	yes	—
12	#11926299	C₄	1	2	yes	—
13	#11926296	—	2	4	no	—
14	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₁₉	—	2	1	no	—
17	C₄³×C₁₉	C₂²	1	1	yes	characteristic
18	C₁₉×#336	—	2	1	no	—
19	C₂²×C₈×C₁₉	—	2	1	no	—
20	C₂²×C₈×C₁₉	—	2	1	no	—
21	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
24	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₁₉	Q₈	1	1	yes	characteristic
26	C₂×C₄²×C₁₉	Q₈	1	1	yes	characteristic
27	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
28	C₂²×C₁₉⋊C₈	—	2	2	no	—
29	C₂²×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
30	C₂²×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
31	C₁₉×C₄⋊C₈	—	2	1	no	—
32	C₁₉×C₄⋊C₈	—	2	1	no	—
33	C₁₉×C₄⋊C₈	D₄	1	2	yes	—
34	C₁₉×C₄⋊C₈	Q₈	1	2	yes	—
35	C₄×C₈×C₁₉	—	4	2	no	—
36	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
37	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
38	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
39	C₄⋊(C₁₉⋊C₈)	—	4	2	no	—
40	C₄⋊(C₁₉⋊C₈)	—	4	2	no	—
41	C₄⋊(C₁₉⋊C₈)	C₈	1	4	yes	complemented
42	C₄×C₁₉⋊C₈	—	2	2	no	—
43	C₄×C₁₉⋊C₈	—	2	2	no	—
44	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₈×C₁₉	—	2	2	no	—
52	C₂×C₈×C₁₉	—	2	2	no	—
53	C₂×C₈×C₁₉	—	4	1	no	—
54	C₂×C₈×C₁₉	—	4	1	no	—
55	C₂×C₈×C₁₉	—	4	1	no	—
56	C₂×C₈×C₁₉	—	4	1	no	—
57	C₄²×C₁₉	—	2	2	no	—
58	C₄²×C₁₉	—	2	1	no	—
59	C₄²×C₁₉	—	2	1	no	—
60	C₄²×C₁₉	—	2	1	no	—
61	C₄²×C₁₉	—	2	1	no	—
62	C₄²×C₁₉	—	4	1	no	—
63	C₄²×C₁₉	C₂×C₈	1	2	yes	—
64	C₄²×C₁₉	M₁₆	1	2	yes	—
65	C₄²×C₁₉	D₈	1	2	yes	—
66	C₄²×C₁₉	Q₁₆	1	2	yes	—
67	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
68	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
69	C₄²×C₁₉	#41	1	1	yes	characteristic
70	C₄²×C₁₉	#41	1	1	yes	characteristic
71	C₂×C₁₉⋊C₈	—	2	2	no	—
72	C₂×C₁₉⋊C₈	—	2	2	no	—
73	C₂×C₁₉⋊C₈	—	2	2	no	—
74	C₂×C₁₉⋊C₈	—	2	2	no	—
75	C₂×C₁₉⋊C₈	—	2	2	no	—
76	C₂×C₁₉⋊C₈	—	4	2	no	—
77	C₂×C₁₉⋊C₈	—	4	2	no	—
78	C₂×C₁₉⋊C₈	—	4	2	no	—
79	C₂×C₁₉⋊C₈	C₂×C₈	1	4	yes	—
80	#9601	—	19	1	no	maximal, Sylow
81	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₁₉	—	2	1	no	—
83	C₂×C₄×C₁₉	—	2	1	no	—
84	C₂×C₄×C₁₉	—	2	1	no	—
85	C₂×C₄×C₁₉	—	2	1	no	—
86	C₂×C₄×C₁₉	—	2	1	no	—
87	C₂×C₄×C₁₉	—	2	1	no	—
88	C₂×C₄×C₁₉	—	2	1	no	—
89	C₂×C₄×C₁₉	—	2	1	no	—
90	C₂×C₄×C₁₉	—	2	1	no	—
91	C₂×C₄×C₁₉	C₄⋊C₈	1	2	yes	—
92	C₂×C₄×C₁₉	C₄⋊C₈	1	2	yes	—
93	C₂×C₄×C₁₉	#107	1	2	yes	—
94	C₂×C₄×C₁₉	#107	1	2	yes	—
95	C₂×C₄×C₁₉	#108	1	2	yes	—
96	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
97	C₂×C₄×C₁₉	C₄×Q₈	1	1	yes	characteristic
98	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
99	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
100	C₂×C₄×C₁₉	#125	1	1	yes	characteristic
101	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
102	C₂×C₄×C₁₉	C₂²×C₈	1	1	yes	characteristic
103	C₂×C₄×C₁₉	C₂×M₁₆	1	1	yes	characteristic
104	C₂×C₄×C₁₉	#131	1	1	yes	characteristic
105	C₂×C₄×C₁₉	#131	1	1	yes	characteristic
106	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
107	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
108	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
109	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
110	C₈×C₁₉	—	4	2	no	—
111	C₈×C₁₉	—	4	2	no	—
112	C₁₉⋊C₈	—	2	4	no	—
113	C₁₉⋊C₈	—	4	2	no	—
114	C₁₉⋊C₈	—	8	2	no	—
115	#1842	—	19	1	no	—
116	C₄×C₄⋊C₈	—	19	1	no	—
117	C₄×C₄⋊C₈	D₁₉	1	1	yes	characteristic
118	C₂×#335	—	19	2	no	—
119	#1277	—	19	2	no	—
120	C₂²×C₁₉	#389	1	1	yes	characteristic
121	C₂²×C₁₉	C₂×C₄⋊C₈	1	1	yes	characteristic
122	C₂²×C₁₉	#424	1	1	yes	characteristic
123	C₂²×C₁₉	C₂×#107	1	1	yes	characteristic
124	C₂²×C₁₉	#428	1	1	yes	characteristic
125	C₂²×C₁₉	C₂×#108	1	1	yes	characteristic
126	C₂²×C₁₉	#430	1	1	yes	characteristic
127	C₄×C₁₉	—	2	2	no	—
128	C₄×C₁₉	—	2	2	no	—
129	C₄×C₁₉	—	2	1	no	—
130	C₄×C₁₉	—	2	1	no	—
131	C₄×C₁₉	—	4	1	no	—
132	C₄×C₁₉	#335	1	4	yes	complemented
133	C₄×C₁₉	C₈×D₄	1	1	yes	characteristic
134	C₄×C₁₉	#436	1	1	yes	characteristic
135	C₄×C₁₉	C₈×Q₈	1	1	yes	characteristic
136	C₄×C₁₉	#446	1	1	yes	characteristic
137	C₄×C₁₉	#459	1	1	yes	characteristic
138	C₄×C₁₉	#462	1	1	yes	characteristic
139	C₄×C₁₉	#474	1	1	yes	characteristic
140	C₄×C₁₉	#477	1	1	yes	characteristic
141	C₂×C₄⋊C₈	—	19	1	no	—
142	C₂×C₄⋊C₈	—	19	1	no	—
143	C₂×C₄⋊C₈	—	19	2	no	—
144	C₂×C₄⋊C₈	—	38	1	no	—
145	C₂×C₄⋊C₈	—	38	1	no	—
146	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
147	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
148	C₂×C₄×C₈	—	2	1	no	—
149	C₂×C₄×C₈	—	19	2	no	—
150	C₄³	C₂×D₁₉	1	1	yes	characteristic
151	#336	—	2	1	no	—
152	#336	—	19	2	no	—
153	#335	—	38	4	no	—
154	C₂×C₁₉	C₂×#335	1	1	yes	characteristic
155	C₂×C₁₉	#1565	1	1	yes	characteristic
156	C₂×C₁₉	#1571	1	1	yes	characteristic
157	C₂×C₁₉	#1574	1	1	yes	characteristic
158	C₂×C₁₉	#1939	1	1	yes	characteristic
159	C₂×C₁₉	#1947	1	1	yes	characteristic
160	C₂×C₁₉	#1950	1	1	yes	characteristic
161	C₂²×C₈	—	2	1	no	—
162	C₂²×C₈	—	2	1	no	—
163	C₂²×C₈	—	19	2	no	—
164	C₂²×C₈	—	19	2	no	—
165	C₂²×C₈	—	38	2	no	—
166	C₂×C₄²	Dic₃₈	1	1	yes	characteristic
167	C₂×C₄²	Dic₃₈	1	1	yes	characteristic
168	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
169	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
170	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
171	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
172	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
173	C₄⋊C₈	—	2	1	no	—
174	C₄⋊C₈	—	2	1	no	—
175	C₄⋊C₈	—	19	4	no	—
176	C₄⋊C₈	—	38	2	no	—
177	C₄⋊C₈	—	38	2	no	—
178	C₄⋊C₈	—	38	2	no	—
179	C₄⋊C₈	—	76	2	no	—
180	C₄⋊C₈	—	76	2	no	—
181	C₄⋊C₈	Dic₃₈	1	2	yes	—
182	C₄⋊C₈	D₇₆	1	2	yes	—
183	C₄×C₈	—	4	2	no	—
184	C₄×C₈	—	38	2	no	—
185	C₄×C₈	—	38	2	no	—
186	C₁₉	#9601	1	1	yes	characteristic, complemented, Sylow
187	C₂²×C₄	Dic₁₉⋊C₄	1	1	yes	characteristic
188	C₂²×C₄	Dic₁₉⋊C₄	1	1	yes	characteristic
189	C₂²×C₄	Dic₁₉⋊C₄	1	1	yes	characteristic
190	C₂²×C₄	Dic₁₉⋊C₄	1	1	yes	characteristic
191	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
192	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
193	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	4	1	no	—
197	C₂×C₈	—	4	1	no	—
198	C₂×C₈	—	4	1	no	—
199	C₂×C₈	—	4	1	no	—
200	C₂×C₈	—	19	4	no	—
201	C₂×C₈	—	38	2	no	—
202	C₂×C₈	—	38	2	no	—
203	C₂×C₈	—	38	2	no	—
204	C₂×C₈	—	38	2	no	—
205	C₂×C₈	—	38	2	no	—
206	C₂×C₈	—	76	2	no	—
207	C₂×C₈	—	76	2	no	—
208	C₂×C₈	—	76	2	no	—
209	C₄²	—	2	1	no	—
210	C₄²	—	2	1	no	—
211	C₄²	—	2	2	no	—
212	C₄²	—	2	1	no	—
213	C₄²	—	2	1	no	—
214	C₄²	—	4	1	no	—
215	C₄²	C₈×D₁₉	1	2	yes	—
216	C₄²	#64515	1	2	yes	—
217	C₄²	#64525	1	2	yes	—
218	C₄²	#64528	1	2	yes	—
219	C₄²	C₂×Dic₃₈	1	1	yes	characteristic
220	C₄²	C₂×D₇₆	1	1	yes	characteristic
221	C₄²	#64541	1	1	yes	characteristic
222	C₄²	#64541	1	1	yes	characteristic
223	C₂³	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	#10597264	1	2	yes	—
234	C₂×C₄	#10597264	1	2	yes	—
235	C₂×C₄	Dic₁₉⋊C₈	1	2	yes	—
236	C₂×C₄	Dic₁₉⋊C₈	1	2	yes	—
237	C₂×C₄	#10597279	1	2	yes	—
238	C₂×C₄	C₄×Dic₃₈	1	1	yes	characteristic
239	C₂×C₄	#10597315	1	1	yes	characteristic
240	C₂×C₄	#10597316	1	1	yes	characteristic
241	C₂×C₄	C₄×D₇₆	1	1	yes	characteristic
242	C₂×C₄	C₂×C₈×D₁₉	1	1	yes	characteristic
243	C₂×C₄	C₂×#64515	1	1	yes	characteristic
244	C₂×C₄	#10597347	1	1	yes	characteristic
245	C₂×C₄	#10597347	1	1	yes	characteristic
246	C₂×C₄	#10597370	1	1	yes	characteristic
247	C₂×C₄	#10597376	1	1	yes	characteristic
248	C₂×C₄	C₂×#64525	1	1	yes	characteristic
249	C₂×C₄	C₂×#64528	1	1	yes	characteristic
250	C₂×C₄	#10597395	1	1	yes	characteristic
251	C₂×C₄	#10597397	1	1	yes	characteristic
252	C₈	—	4	2	no	—
253	C₈	—	4	2	no	—
254	C₈	—	38	4	no	—
255	C₈	—	76	2	no	—
256	C₈	—	152	2	no	—
257	C₂²	#11926442	1	1	yes	characteristic
258	C₂²	C₂×Dic₁₉⋊C₈	1	1	yes	characteristic
259	C₂²	#11926570	1	1	yes	characteristic
260	C₂²	C₂×#10597264	1	1	yes	characteristic
261	C₂²	#11926587	1	1	yes	characteristic
262	C₂²	C₂×#10597279	1	1	yes	characteristic
263	C₂²	#11926592	1	1	yes	characteristic
264	C₄	—	2	1	no	—
265	C₄	—	2	1	no	—
266	C₄	—	2	2	no	—
267	C₄	—	2	2	no	—
268	C₄	—	4	1	no	—
269	C₄	#11926296	1	4	yes	complemented
270	C₄	C₈×D₇₆	1	1	yes	characteristic
271	C₄	#11926623	1	1	yes	characteristic
272	C₄	C₈×Dic₃₈	1	1	yes	characteristic
273	C₄	#11926674	1	1	yes	characteristic
274	C₄	#11926739	1	1	yes	characteristic
275	C₄	#11926754	1	1	yes	characteristic
276	C₄	#11926824	1	1	yes	characteristic
277	C₄	#11926839	1	1	yes	characteristic
278	C₂	C₂×#11926296	1	1	yes	characteristic
279	C₂	#14562388	1	1	yes	characteristic
280	C₂	#14562409	1	1	yes	characteristic
281	C₂	#14562423	1	1	yes	characteristic
282	C₂	#14564353	1	1	yes	characteristic
283	C₂	#14564393	1	1	yes	characteristic
284	C₂	#14564410	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₁₉
commutator	C₂×C₄×C₁₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₄×C₁₉×C₄⋊C₈	C₂
socle, prime radical	C₂³×C₁₉	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₄×C₄⋊C₈	D₁₉
hypocenter	C₁₉	#9601
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
41. C₄⋊(C₁₉⋊C₈)	C₈	semidirect
132. C₄×C₁₉	#335	semidirect
186. C₁₉	#9601	semidirect
269. C₄	#11926296	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×#11926296	2
#14561469	2
C₄×C₁₉×C₄⋊C₈	2
#9601	19