Subgroups — #18075583

Order 4864.

542 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14563792	C₂	1	1	yes	maximal, characteristic, complemented
2	#14563635	C₂	1	2	yes	maximal
3	C₂×#11926295	C₂	1	1	yes	maximal, characteristic, complemented
4	#14561519	C₂	1	2	yes	maximal, complemented
5	C₂×C₈×D₄×C₁₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#10597274	—	2	1	no	—
7	C₂×#10597274	—	2	1	no	—
8	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
9	C₂×#10597289	—	2	2	no	—
10	#11926414	C₂²	1	2	yes	—
11	#11926306	—	2	2	no	—
12	#11926295	—	2	2	no	—
13	C₂³×C₈×C₁₉	—	2	1	no	—
14	C₂×C₄×D₄×C₁₉	C₂²	1	1	yes	characteristic
15	C₈×D₄×C₁₉	—	2	2	no	—
16	C₈×D₄×C₁₉	C₄	1	4	yes	complemented
17	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₁₉×C₂²⋊C₈	—	2	1	no	—
19	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
20	C₂²×D₄×C₁₉	D₄	1	1	yes	characteristic
21	C₂³×C₄×C₁₉	—	2	1	no	—
22	C₂²×C₈×C₁₉	—	2	1	no	—
23	C₂²×C₈×C₁₉	—	2	4	no	—
24	C₂²×C₈×C₁₉	—	4	2	no	—
25	C₂²×C₈×C₁₉	—	4	2	no	—
26	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
28	C₄×D₄×C₁₉	—	2	1	no	—
29	C₄×D₄×C₁₉	—	2	1	no	—
30	C₄×D₄×C₁₉	C₂×C₄	1	2	yes	—
31	C₄×D₄×C₁₉	Q₈	1	2	yes	—
32	C₂×C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
33	C₂×C₂²⋊C₄×C₁₉	—	2	1	no	—
34	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
35	C₂×C₄⋊Dic₁₉	—	2	2	no	—
36	C₂×C₄⋊Dic₁₉	—	2	2	no	—
37	C₂×C₄⋊Dic₁₉	—	2	2	no	—
38	C₂²×C₁₉⋊C₈	—	2	2	no	—
39	C₁₉×C₄⋊C₈	—	2	1	no	—
40	C₁₉×C₄⋊C₈	C₂×C₄	1	2	yes	—
41	C₁₉×C₂²⋊C₈	—	2	4	no	—
42	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
43	C₄×C₈×C₁₉	D₄	1	2	yes	—
44	#10597289	—	4	4	no	—
45	#10597274	—	4	2	no	—
46	#10597274	—	4	2	no	—
47	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
48	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
49	C₂⁴×C₁₉	—	2	1	no	—
50	C₂×D₄×C₁₉	—	2	1	no	—
51	C₂×D₄×C₁₉	—	2	2	no	—
52	C₂×D₄×C₁₉	C₄⋊C₄	1	2	yes	—
53	C₂×D₄×C₁₉	SD₁₆	1	4	yes	—
54	C₂²×C₄×C₁₉	—	2	1	no	—
55	C₂²×C₄×C₁₉	—	2	2	no	—
56	C₂²×C₄×C₁₉	—	2	2	no	—
57	C₂²×C₄×C₁₉	—	4	1	no	—
58	C₂²×C₄×C₁₉	—	4	1	no	—
59	C₂²×C₄×C₁₉	—	4	1	no	—
60	C₂²×C₄×C₁₉	—	4	1	no	—
61	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
64	C₂²×Dic₁₉	—	4	2	no	—
65	C₂×C₈×C₁₉	—	2	1	no	—
66	C₂×C₈×C₁₉	—	2	1	no	—
67	C₂×C₈×C₁₉	—	2	2	no	—
68	C₂×C₈×C₁₉	—	4	1	no	—
69	C₂×C₈×C₁₉	—	4	1	no	—
70	C₂×C₈×C₁₉	—	4	8	no	—
71	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
72	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
73	C₂×C₈×C₁₉	D₈	1	2	yes	—
74	C₂×C₈×C₁₉	SD₁₆	1	2	yes	—
75	C₄⋊C₄×C₁₉	—	2	1	no	—
76	C₄⋊C₄×C₁₉	C₄⋊C₄	1	2	yes	—
77	C₂²⋊C₄×C₁₉	—	2	2	no	—
78	C₂²⋊C₄×C₁₉	—	2	2	no	—
79	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
80	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
82	C₄²×C₁₉	#41	1	1	yes	characteristic
83	C₄⋊Dic₁₉	—	4	2	no	—
84	C₄⋊Dic₁₉	—	4	2	no	—
85	C₄⋊Dic₁₉	—	4	2	no	—
86	C₄⋊Dic₁₉	—	4	2	no	—
87	C₄⋊Dic₁₉	—	8	2	no	—
88	C₂×C₁₉⋊C₈	—	4	2	no	—
89	C₂×C₁₉⋊C₈	—	4	2	no	—
90	C₂×C₁₉⋊C₈	—	4	2	no	—
91	#8642	—	19	1	no	maximal, Sylow
92	C₂³×C₁₉	—	2	2	no	—
93	C₂³×C₁₉	—	2	4	no	—
94	C₂³×C₁₉	—	4	1	no	—
95	C₂³×C₁₉	—	4	1	no	—
96	C₂³×C₁₉	—	4	2	no	—
97	C₂³×C₁₉	#121	1	1	yes	characteristic
98	D₄×C₁₉	—	2	2	no	—
99	D₄×C₁₉	—	2	4	no	—
100	D₄×C₁₉	#106	1	4	yes	complemented
101	C₂×C₄×C₁₉	—	2	2	no	—
102	C₂×C₄×C₁₉	—	2	2	no	—
103	C₂×C₄×C₁₉	—	4	1	no	—
104	C₂×C₄×C₁₉	—	4	1	no	—
105	C₂×C₄×C₁₉	—	4	2	no	—
106	C₂×C₄×C₁₉	—	4	2	no	—
107	C₂×C₄×C₁₉	—	4	2	no	—
108	C₂×C₄×C₁₉	—	4	2	no	—
109	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
112	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
113	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
114	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
115	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
116	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
117	C₂×C₄×C₁₉	C₂×SD₁₆	1	2	yes	—
118	C₂×C₄×C₁₉	#135	1	2	yes	—
119	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
120	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
121	C₂×Dic₁₉	—	8	2	no	—
122	C₂×Dic₁₉	—	8	2	no	—
123	C₂×Dic₁₉	—	8	2	no	—
124	C₈×C₁₉	—	2	2	no	—
125	C₈×C₁₉	—	4	2	no	—
126	C₈×C₁₉	#102	1	4	yes	complemented
127	C₁₉⋊C₈	—	8	2	no	—
128	C₂×C₈×D₄	D₁₉	1	1	yes	characteristic
129	#1844	—	19	1	no	—
130	#1802	—	19	2	no	—
131	C₂×#334	—	19	1	no	—
132	#1296	—	19	2	no	—
133	C₂²×C₁₉	—	2	4	no	—
134	C₂²×C₁₉	—	4	2	no	—
135	C₂²×C₁₉	—	4	2	no	—
136	C₂²×C₁₉	—	4	4	no	—
137	C₂²×C₁₉	—	4	4	no	—
138	C₂²×C₁₉	#380	1	1	yes	characteristic
139	C₂²×C₁₉	#459	1	1	yes	characteristic
140	C₂²×C₁₉	#461	1	1	yes	characteristic
141	C₂²×C₁₉	#465	1	2	yes	—
142	C₂²×C₁₉	#471	1	2	yes	—
143	C₄×C₁₉	—	2	1	no	—
144	C₄×C₁₉	—	2	1	no	—
145	C₄×C₁₉	—	4	2	no	—
146	C₄×C₁₉	C₂×#102	1	1	yes	characteristic
147	C₄×C₁₉	#417	1	1	yes	characteristic
148	C₄×C₁₉	C₂×#106	1	1	yes	characteristic
149	C₄×C₁₉	#427	1	1	yes	characteristic
150	C₄×C₁₉	#465	1	2	yes	—
151	C₄×C₁₉	#474	1	1	yes	characteristic
152	C₄×C₁₉	#476	1	1	yes	characteristic
153	Dic₁₉	—	16	2	no	—
154	C₂³×C₈	—	2	1	no	—
155	C₂×C₄×D₄	C₂×D₁₉	1	1	yes	characteristic
156	C₈×D₄	—	2	2	no	—
157	C₈×D₄	Dic₁₉	1	4	yes	complemented
158	C₂×#106	—	38	1	no	—
159	C₂×#106	—	38	1	no	—
160	C₂×C₄⋊C₈	—	19	2	no	—
161	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
162	C₂×#102	—	38	2	no	—
163	C₂×C₂²⋊C₈	—	2	1	no	—
164	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
165	#384	—	19	2	no	—
166	#340	—	38	2	no	—
167	#334	—	38	2	no	—
168	C₂×C₁₉	—	4	4	no	—
169	C₂×C₁₉	#1665	1	2	yes	—
170	C₂×C₁₉	#1671	1	2	yes	—
171	C₂×C₁₉	#1802	1	1	yes	characteristic
172	C₂×C₁₉	#1807	1	1	yes	characteristic
173	C₂×C₁₉	#1815	1	1	yes	characteristic
174	C₂²×D₄	D₇₆	1	1	yes	characteristic
175	C₂³×C₄	—	2	1	no	—
176	C₂²×C₈	—	2	1	no	—
177	C₂²×C₈	—	2	4	no	—
178	C₂²×C₈	—	4	2	no	—
179	C₂²×C₈	—	4	2	no	—
180	C₂²×C₈	—	38	2	no	—
181	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
182	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
183	C₄×D₄	—	2	1	no	—
184	C₄×D₄	—	2	1	no	—
185	C₄×D₄	Dic₃₈	1	2	yes	—
186	C₄×D₄	C₂×Dic₁₉	1	2	yes	—
187	C₂×C₄⋊C₄	—	38	2	no	—
188	C₂×C₄⋊C₄	—	38	2	no	—
189	C₂×C₄⋊C₄	—	38	2	no	—
190	C₂×C₄⋊C₄	D₇₆	1	1	yes	characteristic
191	C₂×C₂²⋊C₄	—	2	1	no	—
192	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
193	#106	—	76	2	no	—
194	#106	—	76	2	no	—
195	C₄⋊C₈	—	2	1	no	—
196	C₄⋊C₈	—	38	2	no	—
197	C₄⋊C₈	—	38	2	no	—
198	C₄⋊C₈	C₂×Dic₁₉	1	2	yes	—
199	#102	—	76	4	no	—
200	C₂²⋊C₈	—	2	4	no	—
201	C₄×C₈	C₂×Dic₁₉	1	2	yes	—
202	C₄×C₈	C₂²⋊D₁₉	1	2	yes	—
203	C₁₉	#8642	1	1	yes	characteristic, complemented, Sylow
204	C₂⁴	—	2	1	no	—
205	C₂×D₄	—	2	1	no	—
206	C₂×D₄	—	2	2	no	—
207	C₂×D₄	#64516	1	4	yes	—
208	C₂×D₄	C₄⋊Dic₁₉	1	2	yes	—
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	4	1	no	—
213	C₂²×C₄	—	4	1	no	—
214	C₂²×C₄	—	4	1	no	—
215	C₂²×C₄	—	4	1	no	—
216	C₂²×C₄	—	76	2	no	—
217	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
218	C₂²×C₄	#64541	1	1	yes	characteristic
219	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	4	1	no	—
224	C₂×C₈	—	4	1	no	—
225	C₂×C₈	—	4	8	no	—
226	C₂×C₈	—	76	2	no	—
227	C₂×C₈	—	76	2	no	—
228	C₂×C₈	—	76	2	no	—
229	C₂×C₈	#64525	1	2	yes	—
230	C₂×C₈	#64526	1	2	yes	—
231	C₂×C₈	C₂²⋊Dic₁₉	1	2	yes	—
232	C₂×C₈	C₂²⋊Dic₁₉	1	2	yes	—
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	—	76	2	no	—
235	C₄⋊C₄	—	76	2	no	—
236	C₄⋊C₄	—	76	2	no	—
237	C₄⋊C₄	—	76	2	no	—
238	C₄⋊C₄	—	152	2	no	—
239	C₄⋊C₄	C₄⋊Dic₁₉	1	2	yes	—
240	C₂²⋊C₄	—	2	2	no	—
241	C₂²⋊C₄	—	2	2	no	—
242	C₄²	C₂×Dic₃₈	1	1	yes	characteristic
243	C₄²	#64541	1	1	yes	characteristic
244	C₄²	C₂²×Dic₁₉	1	1	yes	characteristic
245	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
246	C₂³	—	2	2	no	—
247	C₂³	—	2	4	no	—
248	C₂³	—	4	2	no	—
249	C₂³	—	4	1	no	—
250	C₂³	—	4	1	no	—
251	C₂³	#10597376	1	1	yes	characteristic
252	D₄	—	2	2	no	—
253	D₄	—	2	4	no	—
254	D₄	#10597274	1	4	yes	complemented
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	4	1	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	152	2	no	—
264	C₂×C₄	—	152	2	no	—
265	C₂×C₄	—	152	2	no	—
266	C₂×C₄	C₂×#64516	1	2	yes	—
267	C₂×C₄	#10597350	1	2	yes	—
268	C₂×C₄	#10597370	1	1	yes	characteristic
269	C₂×C₄	#10597370	1	1	yes	characteristic
270	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
271	C₂×C₄	#10597372	1	1	yes	characteristic
272	C₂×C₄	#10597376	1	1	yes	characteristic
273	C₂×C₄	C₂×#64525	1	1	yes	characteristic
274	C₂×C₄	C₂×#64526	1	1	yes	characteristic
275	C₂×C₄	#10597395	1	1	yes	characteristic
276	C₂×C₄	#10597397	1	1	yes	characteristic
277	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
278	C₈	—	2	2	no	—
279	C₈	—	4	2	no	—
280	C₈	—	152	2	no	—
281	C₈	#10597289	1	4	yes	complemented
282	C₂²	—	2	4	no	—
283	C₂²	—	4	2	no	—
284	C₂²	—	4	2	no	—
285	C₂²	—	4	4	no	—
286	C₂²	—	4	4	no	—
287	C₂²	#11926404	1	1	yes	characteristic
288	C₂²	#11926739	1	1	yes	characteristic
289	C₂²	#11926749	1	1	yes	characteristic
290	C₂²	#11926773	1	2	yes	—
291	C₂²	#11926807	1	2	yes	—
292	C₄	—	2	1	no	—
293	C₄	—	2	1	no	—
294	C₄	—	4	2	no	—
295	C₄	—	304	2	no	—
296	C₄	C₂×#10597289	1	1	yes	characteristic
297	C₄	#11926545	1	1	yes	characteristic
298	C₄	C₂×#10597274	1	1	yes	characteristic
299	C₄	#11926582	1	1	yes	characteristic
300	C₄	#11926773	1	2	yes	—
301	C₄	#11926824	1	1	yes	characteristic
302	C₄	#11926834	1	1	yes	characteristic
303	C₂	—	4	4	no	—
304	C₂	#14562976	1	2	yes	—
305	C₂	#14563010	1	2	yes	—
306	C₂	#14563635	1	1	yes	characteristic
307	C₂	#14563660	1	1	yes	characteristic
308	C₂	#14563694	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#10597376
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₂×C₈×D₄×C₁₉	C₂
socle	C₂³×C₁₉	#121
prime radical	C₂²×D₄×C₁₉	D₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂×C₈×D₄	D₁₉
hypocenter	C₁₉	#8642
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈×D₄

Complements

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

N	H ≅ G/N	type
1. #14563792	C₂	semidirect
3. C₂×#11926295	C₂	semidirect
4. #14561519	C₂	semidirect
16. C₈×D₄×C₁₉	C₄	semidirect
100. D₄×C₁₉	#106	semidirect
126. C₈×C₁₉	#102	semidirect
157. C₈×D₄	Dic₁₉	semidirect
203. C₁₉	#8642	semidirect
254. D₄	#10597274	semidirect
281. C₈	#10597289	semidirect

Maximal subgroups

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

subgroup	index
#14563792	2
#14563635	2
C₂×#11926295	2
#14561519	2
C₂×C₈×D₄×C₁₉	2
#8642	19