Subgroups — #18073657

Order 4864.

406 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14572569	C₂	1	1	yes	maximal, characteristic
2	#14562293	C₂	1	1	yes	maximal, characteristic
3	#14562133	C₂	1	1	yes	maximal, characteristic
4	#14561927	C₂	1	1	yes	maximal, characteristic
5	#14561538	C₂	1	1	yes	maximal, characteristic
6	#14561532	C₂	1	1	yes	maximal, characteristic
7	C₈×C₄⋊C₄×C₁₉	C₂	1	1	yes	maximal, characteristic
8	#11927101	C₂²	1	1	yes	characteristic
9	#11926674	—	2	1	no	—
10	#11926674	C₄	1	2	yes	complemented
11	C₈×Dic₃₈	—	2	1	no	—
12	C₈×Dic₃₈	C₄	1	2	yes	complemented
13	#11926597	C₂²	1	1	yes	characteristic
14	#11926597	C₂²	1	1	yes	characteristic
15	#11926570	—	2	1	no	—
16	#11926568	C₂²	1	1	yes	characteristic
17	#11926499	—	2	1	no	—
18	#11926292	C₄	1	2	yes	complemented
19	#11926289	C₄	1	2	yes	complemented
20	#11926280	C₄	1	4	yes	—
21	#11926266	—	2	1	no	—
22	#11926266	—	2	1	no	—
23	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₁₉	—	2	1	no	—
25	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
26	C₄×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
27	C₁₉×#336	—	2	1	no	—
28	C₂²×C₈×C₁₉	—	2	1	no	—
29	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
32	C₂×C₄²×C₁₉	—	2	1	no	—
33	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
34	#10597372	—	2	1	no	—
35	#10597370	—	2	1	no	—
36	#10597370	C₂×C₄	1	2	yes	—
37	C₂×#64520	—	2	1	no	—
38	#10597316	—	2	1	no	—
39	#10597315	C₂×C₄	1	1	yes	characteristic
40	#10597315	C₂×C₄	1	1	yes	characteristic
41	C₄×Dic₃₈	—	2	1	no	—
42	C₄×Dic₃₈	C₂×C₄	1	2	yes	—
43	C₁₉×C₄⋊C₈	—	2	1	no	—
44	C₁₉×C₄⋊C₈	C₂×C₄	1	2	yes	—
45	C₄×C₈×C₁₉	—	4	2	no	—
46	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
49	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
50	C₁₉×#95	C₂×C₄	1	2	yes	—
51	C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
52	C₂²⋊(C₁₉⋊C₈)	C₂×C₄	1	2	yes	—
53	#10597272	—	2	2	no	—
54	Dic₁₉⋊C₈	—	2	2	no	—
55	Dic₁₉⋊C₈	—	2	2	no	—
56	C₈×Dic₁₉	—	2	2	no	—
57	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
58	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
59	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	2	yes	—
60	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	2	yes	—
61	C₄×C₁₉⋊C₈	—	2	1	no	—
62	C₄×C₁₉⋊C₈	—	2	1	no	—
63	C₂²×C₄×C₁₉	—	2	2	no	—
64	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
67	C₂×Dic₃₈	—	2	1	no	—
68	C₂×Dic₃₈	C₄²	1	2	yes	—
69	C₂×C₈×C₁₉	—	2	1	no	—
70	C₂×C₈×C₁₉	—	2	1	no	—
71	C₂×C₈×C₁₉	—	2	1	no	—
72	C₂×C₈×C₁₉	—	2	1	no	—
73	C₂×C₈×C₁₉	—	2	2	no	—
74	C₂×C₈×C₁₉	—	4	1	no	—
75	C₂×C₈×C₁₉	—	4	1	no	—
76	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
80	C₄⋊C₄×C₁₉	—	2	4	no	—
81	C₄²×C₁₉	—	4	1	no	—
82	C₄²×C₁₉	—	4	1	no	—
83	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
84	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₁₉	#41	1	1	yes	characteristic
87	C₂²⋊Dic₁₉	—	2	2	no	—
88	C₄⋊Dic₁₉	—	2	1	no	—
89	C₄⋊Dic₁₉	—	2	1	no	—
90	C₄⋊Dic₁₉	—	2	1	no	—
91	C₄⋊Dic₁₉	C₄²	1	2	yes	—
92	Dic₁₉⋊C₄	—	2	2	no	—
93	Dic₁₉⋊C₄	—	2	2	no	—
94	C₄×Dic₁₉	—	2	2	no	—
95	#64520	—	4	1	no	—
96	#64520	—	4	1	no	—
97	C₂×C₁₉⋊C₈	—	2	2	no	—
98	C₂×C₁₉⋊C₈	—	2	2	no	—
99	C₂×C₁₉⋊C₈	—	2	2	no	—
100	#8258	—	19	1	no	maximal, Sylow
101	C₂³×C₁₉	C₄×D₄	1	1	yes	characteristic
102	C₂×C₄×C₁₉	—	2	2	no	—
103	C₂×C₄×C₁₉	—	2	4	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	C₂×C₄²	1	1	yes	characteristic
108	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
110	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
114	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
115	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
116	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
117	C₂×Dic₁₉	—	2	2	no	—
118	C₂×Dic₁₉	—	2	2	no	—
119	Dic₃₈	—	4	2	no	—
120	Dic₃₈	—	4	2	no	—
121	C₈×C₁₉	—	2	1	no	—
122	C₈×C₁₉	—	2	1	no	—
123	C₈×C₁₉	—	2	1	no	—
124	C₈×C₁₉	—	2	1	no	—
125	C₈×C₁₉	—	4	2	no	—
126	C₁₉⋊C₈	—	4	2	no	—
127	C₁₉⋊C₈	—	4	2	no	—
128	C₁₉⋊C₈	—	4	2	no	—
129	#2960	—	19	1	no	—
130	C₈×C₄⋊C₄	D₁₉	1	1	yes	characteristic
131	#1542	—	19	1	no	—
132	#1503	—	19	1	no	—
133	#1452	—	19	1	no	—
134	#1303	—	19	1	no	—
135	#1301	—	19	1	no	—
136	C₂²×C₁₉	—	2	1	no	—
137	C₂²×C₁₉	—	2	1	no	—
138	C₂²×C₁₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
139	C₂²×C₁₉	#440	1	1	yes	characteristic
140	C₂²×C₁₉	#441	1	1	yes	characteristic
141	C₄×C₁₉	—	2	1	no	—
142	C₄×C₁₉	—	2	1	no	—
143	C₄×C₁₉	—	2	1	no	—
144	C₄×C₁₉	—	4	4	no	—
145	C₄×C₁₉	#417	1	1	yes	characteristic
146	C₄×C₁₉	#419	1	1	yes	characteristic
147	C₄×C₁₉	#440	1	1	yes	characteristic
148	C₄×C₁₉	#441	1	1	yes	characteristic
149	C₄×C₁₉	#443	1	2	yes	—
150	Dic₁₉	—	4	2	no	—
151	Dic₁₉	—	4	2	no	—
152	#533	—	19	1	no	—
153	#446	—	19	2	no	—
154	#446	—	38	1	no	—
155	C₈×Q₈	—	19	2	no	—
156	C₈×Q₈	—	38	1	no	—
157	#431	—	19	1	no	—
158	#431	—	19	1	no	—
159	#424	—	38	1	no	—
160	#423	—	19	1	no	—
161	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
162	#405	—	38	1	no	—
163	C₂×C₄×C₈	—	2	1	no	—
164	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
165	C₄×C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic
166	#336	—	2	1	no	—
167	#333	—	19	2	no	—
168	#332	—	19	2	no	—
169	#328	—	19	4	no	—
170	#322	—	38	1	no	—
171	#322	—	38	1	no	—
172	C₂×C₁₉	—	2	1	no	—
173	C₂×C₁₉	—	2	1	no	—
174	C₂×C₁₉	#1757	1	1	yes	characteristic
175	C₂×C₁₉	#1763	1	1	yes	characteristic
176	C₂×C₁₉	#1764	1	1	yes	characteristic
177	C₂×M₁₆	—	38	1	no	—
178	C₂²×C₈	—	2	1	no	—
179	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
180	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
181	C₄⋊Q₈	—	19	1	no	—
182	C₄⋊Q₈	—	19	1	no	—
183	#125	—	38	1	no	—
184	C₂²⋊Q₈	—	19	2	no	—
185	C₂²⋊Q₈	—	38	1	no	—
186	C₄×Q₈	—	19	2	no	—
187	C₄×Q₈	—	38	1	no	—
188	#117	—	38	1	no	—
189	C₂×C₄⋊C₄	—	2	1	no	—
190	C₂×C₄²	—	2	1	no	—
191	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
192	C₄⋊C₈	—	2	1	no	—
193	C₄⋊C₈	—	19	2	no	—
194	C₄⋊C₈	—	19	2	no	—
195	C₄⋊C₈	—	38	2	no	—
196	C₄⋊C₈	—	38	2	no	—
197	C₄⋊C₈	—	38	1	no	—
198	C₄⋊C₈	—	38	1	no	—
199	C₄⋊C₈	C₄×D₁₉	1	2	yes	—
200	C₂²⋊C₈	—	19	2	no	—
201	C₂²⋊C₈	—	38	1	no	—
202	#97	—	38	2	no	—
203	C₄×C₈	—	4	2	no	—
204	C₄×C₈	—	38	2	no	—
205	C₄×C₈	—	38	1	no	—
206	C₄×C₈	—	38	1	no	—
207	C₄×C₈	C₄×D₁₉	1	1	yes	characteristic
208	C₄×C₈	C₄×D₁₉	1	1	yes	characteristic
209	C₄×C₈	D₇₆	1	1	yes	characteristic
210	C₄×C₈	D₇₆	1	1	yes	characteristic
211	#95	C₄×D₁₉	1	2	yes	—
212	C₁₉	#8258	1	1	yes	characteristic, complemented, Sylow
213	C₂×Q₈	—	19	2	no	—
214	C₂×Q₈	—	38	1	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
217	C₂²×C₄	#64541	1	1	yes	characteristic
218	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
219	M₁₆	—	76	1	no	—
220	M₁₆	—	76	1	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	4	1	no	—
227	C₂×C₈	—	4	1	no	—
228	C₂×C₈	—	38	2	no	—
229	C₂×C₈	—	38	2	no	—
230	C₂×C₈	—	38	2	no	—
231	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
232	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
233	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
234	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	19	2	no	—
237	C₄⋊C₄	—	38	2	no	—
238	C₄⋊C₄	—	38	2	no	—
239	C₄⋊C₄	—	38	1	no	—
240	C₄⋊C₄	—	38	1	no	—
241	C₄⋊C₄	—	38	1	no	—
242	C₂²⋊C₄	—	38	2	no	—
243	C₄²	—	4	1	no	—
244	C₄²	—	4	1	no	—
245	C₄²	—	38	2	no	—
246	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
247	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
248	C₄²	C₂×D₇₆	1	1	yes	characteristic
249	C₄²	#64541	1	1	yes	characteristic
250	C₂³	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
251	Q₈	—	76	2	no	—
252	Q₈	—	76	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	4	1	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	38	2	no	—
259	C₂×C₄	—	38	2	no	—
260	C₂×C₄	C₄²×D₁₉	1	1	yes	characteristic
261	C₂×C₄	#10597318	1	1	yes	characteristic
262	C₂×C₄	C₄×D₇₆	1	1	yes	characteristic
263	C₂×C₄	C₄×D₇₆	1	1	yes	characteristic
264	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
265	C₂×C₄	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
266	C₂×C₄	#10597378	1	1	yes	characteristic
267	C₂×C₄	#10597388	1	1	yes	characteristic
268	C₂×C₄	#10597397	1	1	yes	characteristic
269	C₂×C₄	#10597397	1	1	yes	characteristic
270	C₈	—	2	1	no	—
271	C₈	—	2	1	no	—
272	C₈	—	2	1	no	—
273	C₈	—	2	1	no	—
274	C₈	—	4	2	no	—
275	C₈	—	76	2	no	—
276	C₈	—	76	2	no	—
277	C₈	—	76	2	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	2	1	no	—
280	C₂²	C₄×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
281	C₂²	#11926644	1	1	yes	characteristic
282	C₂²	#11926649	1	1	yes	characteristic
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	4	4	no	—
287	C₄	—	76	2	no	—
288	C₄	—	76	2	no	—
289	C₄	#11926543	1	1	yes	characteristic
290	C₄	#11926552	1	1	yes	characteristic
291	C₄	#11926644	1	1	yes	characteristic
292	C₄	#11926649	1	1	yes	characteristic
293	C₄	#11926662	1	2	yes	—
294	C₂	—	2	1	no	—
295	C₂	—	2	1	no	—
296	C₂	#14563436	1	1	yes	characteristic
297	C₂	#14563468	1	1	yes	characteristic
298	C₂	#14563472	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄
commutator	C₂×C₄×C₁₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₈×C₄⋊C₄×C₁₉	C₂
socle	C₂²×C₁₉	C₄×C₂²⋊C₄
prime radical	C₂³×C₁₉	C₄×D₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₈×C₄⋊C₄	D₁₉
hypocenter	C₁₉	#8258
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
10. #11926674	C₄	semidirect
12. C₈×Dic₃₈	C₄	semidirect
18. #11926292	C₄	semidirect
19. #11926289	C₄	semidirect
212. C₁₉	#8258	semidirect

Maximal subgroups

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

subgroup	index
#14572569	2
#14562293	2
#14562133	2
#14561927	2
#14561538	2
#14561532	2
C₈×C₄⋊C₄×C₁₉	2
#8258	19