Subgroups — #18075793

Order 4864.

430 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14563807	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#11926297	C₂	1	1	yes	maximal, characteristic, complemented
3	#14562312	C₂	1	2	yes	maximal
4	#14561509	C₂	1	2	yes	maximal, complemented
5	C₁₉×#2931	C₂	1	1	yes	maximal, characteristic
6	C₂×#10597276	—	2	1	no	—
7	C₂×#10597276	—	2	1	no	—
8	#11926568	C₂²	1	2	yes	—
9	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
10	#11926297	—	2	1	no	—
11	#11926297	—	2	1	no	—
12	#11926290	—	2	2	no	—
13	#11926288	—	2	2	no	—
14	C₂×C₁₉×#131	—	2	1	no	—
15	C₁₉×#532	C₂²	1	1	yes	characteristic
16	C₁₉×#436	—	2	1	no	—
17	C₁₉×#436	C₄	1	2	yes	—
18	C₈×D₄×C₁₉	—	2	1	no	—
19	C₈×D₄×C₁₉	C₄	1	2	yes	—
20	C₁₉×#423	C₂²	1	1	yes	characteristic
21	C₁₉×#408	—	2	1	no	—
22	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
23	C₂×#41×C₁₉	—	2	1	no	—
24	C₁₉×#131	—	4	2	no	—
25	C₁₉×#131	—	4	2	no	—
26	C₂×M₁₆×C₁₉	—	2	1	no	—
27	C₂×M₁₆×C₁₉	—	2	2	no	—
28	C₂²×C₈×C₁₉	—	2	2	no	—
29	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
31	C₁₉×C₄⋊Q₈	D₄	1	1	yes	characteristic
32	C₁₉×#127	D₄	1	1	yes	characteristic
33	C₁₉×#124	—	2	1	no	—
34	C₁₉×#121	—	2	1	no	—
35	C₁₉×#121	Q₈	1	2	yes	—
36	C₄×D₄×C₁₉	—	2	1	no	—
37	C₄×D₄×C₁₉	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
39	C₂×#64520	—	2	2	no	—
40	C₂²×C₁₉⋊C₈	—	2	2	no	—
41	C₁₉×C₄⋊C₈	—	2	1	no	—
42	C₁₉×C₄⋊C₈	C₂×C₄	1	2	yes	—
43	C₁₉×C₂²⋊C₈	—	2	2	no	—
44	C₁₉×C₂²⋊C₈	—	2	2	no	—
45	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
48	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
49	#10597276	—	4	2	no	—
50	#10597276	—	4	2	no	—
51	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
52	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
53	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
54	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
55	#41×C₁₉	—	4	1	no	—
56	#41×C₁₉	—	4	1	no	—
57	#41×C₁₉	—	4	1	no	—
58	#41×C₁₉	—	4	1	no	—
59	C₂×Q₈×C₁₉	—	2	1	no	—
60	C₂×D₄×C₁₉	—	2	2	no	—
61	C₂×D₄×C₁₉	—	2	1	no	—
62	C₂×D₄×C₁₉	—	2	1	no	—
63	C₂×D₄×C₁₉	C₄⋊C₄	1	2	yes	—
64	C₂²×C₄×C₁₉	—	2	2	no	—
65	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
67	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
68	M₁₆×C₁₉	—	4	1	no	—
69	M₁₆×C₁₉	—	4	1	no	—
70	M₁₆×C₁₉	—	4	4	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	4	no	—
75	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
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	D₈	1	2	yes	—
80	C₂×C₈×C₁₉	SD₁₆	1	2	yes	—
81	C₄⋊C₄×C₁₉	—	2	1	no	—
82	C₄⋊C₄×C₁₉	C₄⋊C₄	1	2	yes	—
83	C₂²⋊C₄×C₁₉	—	2	2	no	—
84	C₂²⋊C₄×C₁₉	—	2	2	no	—
85	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₁₉	#41	1	1	yes	characteristic
89	#64520	—	4	2	no	—
90	#64520	—	4	2	no	—
91	C₂×C₁₉⋊C₈	—	4	2	no	—
92	C₂×C₁₉⋊C₈	—	4	2	no	—
93	C₂×C₁₉⋊C₈	—	8	2	no	—
94	#8684	—	19	1	no	maximal, Sylow
95	C₂³×C₁₉	—	2	2	no	—
96	C₂³×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
97	Q₈×C₁₉	—	4	1	no	—
98	Q₈×C₁₉	—	4	1	no	—
99	D₄×C₁₉	—	4	2	no	—
100	D₄×C₁₉	—	4	2	no	—
101	D₄×C₁₉	—	4	2	no	—
102	D₄×C₁₉	—	4	1	no	—
103	D₄×C₁₉	—	4	1	no	—
104	D₄×C₁₉	—	4	1	no	—
105	D₄×C₁₉	—	4	1	no	—
106	C₂×C₄×C₁₉	—	2	1	no	—
107	C₂×C₄×C₁₉	—	2	1	no	—
108	C₂×C₄×C₁₉	—	2	2	no	—
109	C₂×C₄×C₁₉	—	4	1	no	—
110	C₂×C₄×C₁₉	—	4	2	no	—
111	C₂×C₄×C₁₉	—	4	2	no	—
112	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
115	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
116	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
117	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
118	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
119	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
120	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
121	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
122	C₈×C₁₉	—	2	1	no	—
123	C₈×C₁₉	—	2	1	no	—
124	C₈×C₁₉	—	4	2	no	—
125	C₈×C₁₉	#102	1	2	yes	—
126	C₈×C₁₉	#102	1	2	yes	—
127	C₁₉⋊C₈	—	8	2	no	—
128	C₁₉⋊C₈	—	8	2	no	—
129	#2931	D₁₉	1	1	yes	characteristic
130	#1848	—	19	1	no	—
131	C₂×#335	—	19	1	no	—
132	#1547	—	19	2	no	—
133	#1292	—	19	2	no	—
134	C₂²×C₁₉	—	2	1	no	—
135	C₂²×C₁₉	—	2	1	no	—
136	C₂²×C₁₉	—	4	2	no	—
137	C₂²×C₁₉	—	4	2	no	—
138	C₂²×C₁₉	—	4	2	no	—
139	C₂²×C₁₉	#380	1	1	yes	characteristic
140	C₂²×C₁₉	#474	1	1	yes	characteristic
141	C₂²×C₁₉	#476	1	1	yes	characteristic
142	C₄×C₁₉	—	2	1	no	—
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	#459	1	1	yes	characteristic
149	C₄×C₁₉	#461	1	1	yes	characteristic
150	C₄×C₁₉	#472	1	1	yes	characteristic
151	C₄×C₁₉	#473	1	1	yes	characteristic
152	C₂×#131	—	2	1	no	—
153	#532	C₂×D₁₉	1	1	yes	characteristic
154	#436	—	2	1	no	—
155	#436	Dic₁₉	1	2	yes	—
156	C₈×D₄	—	2	1	no	—
157	C₈×D₄	Dic₁₉	1	2	yes	—
158	C₂×#108	—	38	1	no	—
159	C₂×#108	—	38	1	no	—
160	#423	—	19	2	no	—
161	#423	C₂×D₁₉	1	1	yes	characteristic
162	C₂×C₄⋊C₈	—	19	2	no	—
163	#408	—	2	1	no	—
164	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
165	#335	—	38	1	no	—
166	#335	—	38	1	no	—
167	#332	—	38	2	no	—
168	#331	—	38	2	no	—
169	C₂×C₁₉	—	2	1	no	—
170	C₂×C₁₉	—	2	1	no	—
171	C₂×C₁₉	—	8	2	no	—
172	C₂×C₁₉	#1802	1	1	yes	characteristic
173	C₂×C₁₉	#1813	1	1	yes	characteristic
174	C₂×C₁₉	#1816	1	1	yes	characteristic
175	C₂×#41	—	2	1	no	—
176	#131	—	4	2	no	—
177	#131	—	4	2	no	—
178	C₂×M₁₆	—	2	1	no	—
179	C₂×M₁₆	—	2	2	no	—
180	C₂×M₁₆	—	38	2	no	—
181	C₂²×C₈	—	2	2	no	—
182	C₂²×C₈	—	38	2	no	—
183	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
184	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
185	C₄⋊Q₈	D₇₆	1	1	yes	characteristic
186	#127	D₇₆	1	1	yes	characteristic
187	#124	—	2	1	no	—
188	#121	—	2	1	no	—
189	#121	Dic₃₈	1	2	yes	—
190	C₄×D₄	—	2	1	no	—
191	C₄×D₄	C₂×Dic₁₉	1	2	yes	—
192	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
193	#108	—	76	2	no	—
194	#108	—	76	2	no	—
195	C₄⋊C₈	—	2	1	no	—
196	C₄⋊C₈	—	38	2	no	—
197	C₄⋊C₈	—	38	2	no	—
198	C₄⋊C₈	—	38	2	no	—
199	C₄⋊C₈	—	38	2	no	—
200	C₄⋊C₈	C₂×Dic₁₉	1	2	yes	—
201	C₂²⋊C₈	—	2	2	no	—
202	C₂²⋊C₈	—	2	2	no	—
203	C₄×C₈	C₂×Dic₁₉	1	1	yes	characteristic
204	C₄×C₈	C₂×Dic₁₉	1	1	yes	characteristic
205	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
206	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
207	C₁₉	#8684	1	1	yes	characteristic, complemented, Sylow
208	#41	—	4	1	no	—
209	#41	—	4	1	no	—
210	#41	—	4	1	no	—
211	#41	—	4	1	no	—
212	C₂×Q₈	—	2	1	no	—
213	C₂×D₄	—	2	2	no	—
214	C₂×D₄	—	2	1	no	—
215	C₂×D₄	—	2	1	no	—
216	C₂×D₄	C₄⋊Dic₁₉	1	2	yes	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
219	C₂²×C₄	#64541	1	1	yes	characteristic
220	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
221	M₁₆	—	4	1	no	—
222	M₁₆	—	4	1	no	—
223	M₁₆	—	4	4	no	—
224	M₁₆	—	76	2	no	—
225	M₁₆	—	76	2	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	4	4	no	—
230	C₂×C₈	—	76	2	no	—
231	C₂×C₈	—	76	2	no	—
232	C₂×C₈	—	152	2	no	—
233	C₂×C₈	#64525	1	2	yes	—
234	C₂×C₈	#64526	1	2	yes	—
235	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
236	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
237	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
238	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
239	C₄⋊C₄	—	2	1	no	—
240	C₄⋊C₄	C₄⋊Dic₁₉	1	2	yes	—
241	C₂²⋊C₄	—	2	2	no	—
242	C₂²⋊C₄	—	2	2	no	—
243	C₄²	C₂×D₇₆	1	1	yes	characteristic
244	C₄²	#64541	1	1	yes	characteristic
245	C₄²	C₂²×Dic₁₉	1	1	yes	characteristic
246	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
247	C₂³	—	2	2	no	—
248	C₂³	#10597370	1	1	yes	characteristic
249	Q₈	—	4	1	no	—
250	Q₈	—	4	1	no	—
251	D₄	—	4	2	no	—
252	D₄	—	4	2	no	—
253	D₄	—	4	2	no	—
254	D₄	—	4	1	no	—
255	D₄	—	4	1	no	—
256	D₄	—	4	1	no	—
257	D₄	—	4	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	4	1	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	#10597370	1	1	yes	characteristic
265	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
266	C₂×C₄	#10597372	1	1	yes	characteristic
267	C₂×C₄	#10597376	1	1	yes	characteristic
268	C₂×C₄	#10597376	1	1	yes	characteristic
269	C₂×C₄	C₂×#64525	1	1	yes	characteristic
270	C₂×C₄	C₂×#64526	1	1	yes	characteristic
271	C₂×C₄	#10597395	1	1	yes	characteristic
272	C₂×C₄	#10597397	1	1	yes	characteristic
273	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
274	C₈	—	2	1	no	—
275	C₈	—	2	1	no	—
276	C₈	—	4	2	no	—
277	C₈	—	152	2	no	—
278	C₈	—	152	2	no	—
279	C₈	#10597289	1	2	yes	—
280	C₈	#10597289	1	2	yes	—
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	4	2	no	—
284	C₂²	—	4	2	no	—
285	C₂²	—	4	2	no	—
286	C₂²	#11926404	1	1	yes	characteristic
287	C₂²	#11926824	1	1	yes	characteristic
288	C₂²	#11926834	1	1	yes	characteristic
289	C₄	—	2	1	no	—
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	4	2	no	—
293	C₄	C₂×#10597289	1	1	yes	characteristic
294	C₄	#11926545	1	1	yes	characteristic
295	C₄	#11926739	1	1	yes	characteristic
296	C₄	#11926749	1	1	yes	characteristic
297	C₄	#11926814	1	1	yes	characteristic
298	C₄	#11926819	1	1	yes	characteristic
299	C₂	—	2	1	no	—
300	C₂	—	2	1	no	—
301	C₂	—	8	2	no	—
302	C₂	#14563635	1	1	yes	characteristic
303	C₂	#14563685	1	1	yes	characteristic
304	C₂	#14563699	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11926404
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#2931	C₂
socle	C₂²×C₁₉	#380
prime radical	C₁₉×#532	C₂²
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#2931	D₁₉
hypocenter	C₁₉	#8684
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931

Complements

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

N	H ≅ G/N	type
1. #14563807	C₂	semidirect
2. C₂×#11926297	C₂	semidirect
4. #14561509	C₂	semidirect
207. C₁₉	#8684	semidirect

Maximal subgroups

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

subgroup	index
#14563807	2
C₂×#11926297	2
#14562312	2
#14561509	2
C₁₉×#2931	2
#8684	19