Subgroups — #18075782

Order 4864.

430 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14572346	C₂	1	1	yes	maximal, characteristic
2	#14563806	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#11926294	C₂	1	1	yes	maximal, characteristic, complemented
4	#14562313	C₂	1	1	yes	maximal, characteristic
5	#14562310	C₂	1	1	yes	maximal, characteristic
6	#14561509	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1292	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10597353	—	2	1	no	—
9	#11927094	C₂²	1	1	yes	characteristic
10	#11926621	—	2	1	no	—
11	#11926621	C₄	1	2	yes	—
12	#11926609	—	2	1	no	—
13	#11926609	C₄	1	2	yes	—
14	C₂×#10597279	—	2	1	no	—
15	C₂×#10597279	—	2	1	no	—
16	#11926568	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
18	#11926512	—	2	1	no	—
19	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
20	#11926294	—	2	1	no	—
21	#11926294	—	2	1	no	—
22	#11926291	—	2	1	no	—
23	#11926289	—	2	1	no	—
24	#11926287	—	2	1	no	—
25	#11926287	—	2	1	no	—
26	C₁₉×#423	C₂²	1	1	yes	characteristic
27	C₁₉×#423	C₂²	1	1	yes	characteristic
28	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
29	C₂×#64541	—	2	1	no	—
30	C₂×M₁₆×C₁₉	—	2	1	no	—
31	C₂×M₁₆×C₁₉	—	2	1	no	—
32	C₂²×C₈×C₁₉	—	2	1	no	—
33	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
34	#10597376	—	2	1	no	—
35	#10597376	Q₈	1	2	yes	—
36	C₂×#64520	—	2	1	no	—
37	C₂²×C₁₉⋊C₈	—	2	1	no	—
38	C₂²×C₁₉⋊C₈	D₄	1	1	yes	characteristic
39	C₂²×C₁₉⋊C₈	D₄	1	1	yes	characteristic
40	#10597353	—	4	2	no	—
41	#10597353	—	4	2	no	—
42	C₂×#64515	—	2	2	no	—
43	C₂×C₈×D₁₉	—	2	2	no	—
44	#10597321	—	2	1	no	—
45	#10597320	D₄	1	1	yes	characteristic
46	C₄×D₇₆	—	2	1	no	—
47	C₄×D₇₆	C₂×C₄	1	2	yes	—
48	#10597315	D₄	1	1	yes	characteristic
49	C₁₉×C₄⋊C₈	—	2	1	no	—
50	C₁₉×C₄⋊C₈	—	2	1	no	—
51	C₁₉×C₄⋊C₈	—	2	1	no	—
52	C₁₉×C₄⋊C₈	—	2	1	no	—
53	C₁₉×C₄⋊C₈	—	2	1	no	—
54	C₁₉×C₄⋊C₈	C₂×C₄	1	2	yes	—
55	#10597279	—	4	2	no	—
56	#10597279	—	4	2	no	—
57	(C₂×D₁₉)⋊C₈	—	2	2	no	—
58	(C₂×D₁₉)⋊C₈	—	2	2	no	—
59	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
60	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
61	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
62	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
63	C₄×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
64	C₄×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
65	C₄×C₁₉⋊C₈	D₄	1	1	yes	characteristic
66	C₄×C₁₉⋊C₈	D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
69	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
70	C₂×C₂²⋊D₁₉	—	2	2	no	—
71	#64541	—	4	1	no	—
72	#64541	—	4	1	no	—
73	#64541	—	4	1	no	—
74	#64541	—	4	1	no	—
75	C₂×D₇₆	—	2	1	no	—
76	C₂×D₇₆	—	2	1	no	—
77	C₂×D₇₆	C₄⋊C₄	1	2	yes	—
78	C₂×C₄×D₁₉	—	2	2	no	—
79	C₂×Dic₃₈	—	2	1	no	—
80	M₁₆×C₁₉	—	4	1	no	—
81	M₁₆×C₁₉	—	4	1	no	—
82	M₁₆×C₁₉	—	4	1	no	—
83	M₁₆×C₁₉	—	4	1	no	—
84	C₂×C₈×C₁₉	—	2	2	no	—
85	C₂×C₈×C₁₉	—	4	1	no	—
86	C₂×C₈×C₁₉	—	4	1	no	—
87	C₂×C₈×C₁₉	—	8	1	no	—
88	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₁₉	#41	1	1	yes	characteristic
92	(C₂×D₁₉)⋊C₄	—	2	2	no	—
93	(C₂×D₁₉)⋊C₄	—	2	2	no	—
94	C₄⋊Dic₁₉	—	2	1	no	—
95	C₄⋊Dic₁₉	C₄⋊C₄	1	2	yes	—
96	#64520	—	4	1	no	—
97	#64520	—	4	1	no	—
98	C₂×C₁₉⋊C₈	—	2	1	no	—
99	C₂×C₁₉⋊C₈	—	2	1	no	—
100	C₂×C₁₉⋊C₈	—	4	1	no	—
101	C₂×C₁₉⋊C₈	—	4	1	no	—
102	C₂×C₁₉⋊C₈	—	8	1	no	—
103	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₁₉⋊C₈	D₈	1	2	yes	—
108	C₂×C₁₉⋊C₈	SD₁₆	1	2	yes	—
109	#64515	—	4	4	no	—
110	C₈×D₁₉	—	4	4	no	—
111	#8682	—	19	1	no	maximal, Sylow
112	C₂³×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
113	C₂²×D₁₉	—	2	2	no	—
114	C₂×C₄×C₁₉	—	2	1	no	—
115	C₂×C₄×C₁₉	—	2	1	no	—
116	C₂×C₄×C₁₉	—	4	1	no	—
117	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
120	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
121	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
122	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
123	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
124	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
125	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
126	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
127	C₂²⋊D₁₉	—	4	2	no	—
128	C₂²⋊D₁₉	—	4	2	no	—
129	C₂×Dic₁₉	—	2	2	no	—
130	D₇₆	—	4	2	no	—
131	D₇₆	—	4	1	no	—
132	D₇₆	—	4	1	no	—
133	D₇₆	—	4	1	no	—
134	D₇₆	—	4	1	no	—
135	C₄×D₁₉	—	4	2	no	—
136	C₄×D₁₉	—	4	2	no	—
137	Dic₃₈	—	4	1	no	—
138	Dic₃₈	—	4	1	no	—
139	C₈×C₁₉	—	4	2	no	—
140	C₈×C₁₉	—	8	1	no	—
141	C₈×C₁₉	—	8	1	no	—
142	C₁₉⋊C₈	—	2	1	no	—
143	C₁₉⋊C₈	—	2	1	no	—
144	C₁₉⋊C₈	—	8	1	no	—
145	C₁₉⋊C₈	—	8	1	no	—
146	C₁₉⋊C₈	#102	1	2	yes	—
147	C₁₉⋊C₈	#102	1	2	yes	—
148	#2931	—	19	1	no	—
149	#1848	—	19	1	no	—
150	C₂×#334	—	19	1	no	—
151	#1547	—	19	1	no	—
152	#1547	—	19	1	no	—
153	#1292	—	19	1	no	—
154	#1292	D₁₉	1	1	yes	characteristic, complemented
155	C₂²×C₁₉	—	2	1	no	—
156	C₂²×C₁₉	—	2	1	no	—
157	C₂²×C₁₉	#380	1	1	yes	characteristic
158	C₂²×C₁₉	#474	1	1	yes	characteristic
159	C₂²×C₁₉	#476	1	1	yes	characteristic
160	C₂×D₁₉	—	4	2	no	—
161	C₂×D₁₉	—	4	2	no	—
162	C₂×D₁₉	—	4	2	no	—
163	C₄×C₁₉	—	2	1	no	—
164	C₄×C₁₉	—	2	1	no	—
165	C₄×C₁₉	—	2	1	no	—
166	C₄×C₁₉	C₂×#102	1	1	yes	characteristic
167	C₄×C₁₉	#417	1	1	yes	characteristic
168	C₄×C₁₉	#459	1	1	yes	characteristic
169	C₄×C₁₉	#461	1	1	yes	characteristic
170	C₄×C₁₉	#471	1	1	yes	characteristic
171	C₄×C₁₉	#471	1	1	yes	characteristic
172	Dic₁₉	—	4	2	no	—
173	C₂×#131	—	38	1	no	—
174	#532	—	19	1	no	—
175	#436	—	19	2	no	—
176	#436	—	38	1	no	—
177	C₈×D₄	—	19	2	no	—
178	C₈×D₄	—	38	1	no	—
179	C₂×#108	—	38	1	no	—
180	C₂×#108	—	38	1	no	—
181	#423	—	19	1	no	—
182	#423	C₂×D₁₉	1	1	yes	characteristic
183	#423	C₂×D₁₉	1	1	yes	characteristic
184	C₂×C₄⋊C₈	—	19	1	no	—
185	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
186	#408	—	38	1	no	—
187	C₂×C₄×C₈	—	19	1	no	—
188	#334	—	38	1	no	—
189	#334	—	38	1	no	—
190	#332	—	38	1	no	—
191	#332	—	38	1	no	—
192	#331	—	38	1	no	—
193	#331	—	38	1	no	—
194	C₂×C₁₉	—	2	1	no	—
195	C₂×C₁₉	—	2	1	no	—
196	C₂×C₁₉	#1802	1	1	yes	characteristic
197	C₂×C₁₉	#1813	1	1	yes	characteristic
198	C₂×C₁₉	#1815	1	1	yes	characteristic
199	D₁₉	—	8	2	no	—
200	C₂×#41	—	38	1	no	—
201	#131	—	76	2	no	—
202	#131	—	76	2	no	—
203	C₂×M₁₆	—	2	1	no	—
204	C₂×M₁₆	—	2	1	no	—
205	C₂×M₁₆	—	38	2	no	—
206	C₂×M₁₆	—	38	1	no	—
207	C₂²×C₈	—	2	1	no	—
208	C₂²×C₈	—	19	1	no	—
209	C₂²×C₈	—	19	1	no	—
210	C₂²×C₈	—	38	2	no	—
211	C₂²×C₈	—	38	1	no	—
212	C₄⋊Q₈	—	19	1	no	—
213	#127	—	19	1	no	—
214	#124	—	38	1	no	—
215	#121	—	19	2	no	—
216	#121	—	38	1	no	—
217	C₄×D₄	—	19	2	no	—
218	C₄×D₄	—	38	1	no	—
219	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
220	#108	—	76	2	no	—
221	#108	—	76	2	no	—
222	C₄⋊C₈	—	2	1	no	—
223	C₄⋊C₈	—	2	1	no	—
224	C₄⋊C₈	—	2	1	no	—
225	C₄⋊C₈	—	2	1	no	—
226	C₄⋊C₈	—	2	1	no	—
227	C₄⋊C₈	—	38	1	no	—
228	C₄⋊C₈	—	38	1	no	—
229	C₄⋊C₈	—	38	1	no	—
230	C₄⋊C₈	—	38	1	no	—
231	C₄⋊C₈	C₄×D₁₉	1	2	yes	—
232	C₂²⋊C₈	—	38	2	no	—
233	C₂²⋊C₈	—	38	2	no	—
234	C₄×C₈	—	19	1	no	—
235	C₄×C₈	—	19	1	no	—
236	C₄×C₈	—	19	1	no	—
237	C₄×C₈	—	19	1	no	—
238	C₁₉	#8682	1	1	yes	characteristic, complemented, Sylow
239	#41	—	76	1	no	—
240	#41	—	76	1	no	—
241	#41	—	76	1	no	—
242	#41	—	76	1	no	—
243	C₂×Q₈	—	38	1	no	—
244	C₂×D₄	—	19	2	no	—
245	C₂×D₄	—	38	2	no	—
246	C₂×D₄	—	38	1	no	—
247	C₂×D₄	—	38	1	no	—
248	C₂²×C₄	—	38	2	no	—
249	C₂²×C₄	#64541	1	1	yes	characteristic
250	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
251	C₂²×C₄	Q₈×D₁₉	1	1	yes	characteristic
252	M₁₆	—	4	1	no	—
253	M₁₆	—	4	1	no	—
254	M₁₆	—	4	1	no	—
255	M₁₆	—	4	1	no	—
256	M₁₆	—	76	4	no	—
257	M₁₆	—	76	1	no	—
258	M₁₆	—	76	1	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	4	1	no	—
261	C₂×C₈	—	4	1	no	—
262	C₂×C₈	—	8	1	no	—
263	C₂×C₈	—	19	1	no	—
264	C₂×C₈	—	19	1	no	—
265	C₂×C₈	—	19	2	no	—
266	C₂×C₈	—	19	2	no	—
267	C₂×C₈	—	19	1	no	—
268	C₂×C₈	—	19	1	no	—
269	C₂×C₈	—	38	1	no	—
270	C₂×C₈	—	38	1	no	—
271	C₂×C₈	—	76	4	no	—
272	C₂×C₈	—	76	1	no	—
273	C₂×C₈	—	76	1	no	—
274	C₂×C₈	—	152	1	no	—
275	C₄⋊C₄	—	19	2	no	—
276	C₄⋊C₄	—	38	1	no	—
277	C₂²⋊C₄	—	38	2	no	—
278	C₂²⋊C₄	—	38	2	no	—
279	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
280	C₄²	#64541	1	1	yes	characteristic
281	C₄²	D₄×D₁₉	1	1	yes	characteristic
282	C₄²	D₄×D₁₉	1	1	yes	characteristic
283	C₂³	—	38	2	no	—
284	C₂³	#10597342	1	1	yes	characteristic
285	Q₈	—	76	1	no	—
286	Q₈	—	76	1	no	—
287	D₄	—	76	2	no	—
288	D₄	—	76	2	no	—
289	D₄	—	76	2	no	—
290	D₄	—	76	1	no	—
291	D₄	—	76	1	no	—
292	D₄	—	76	1	no	—
293	D₄	—	76	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	38	2	no	—
298	C₂×C₄	—	76	2	no	—
299	C₂×C₄	—	76	2	no	—
300	C₂×C₄	#10597318	1	1	yes	characteristic
301	C₂×C₄	C₂²⋊C₄×D₁₉	1	1	yes	characteristic
302	C₂×C₄	#10597330	1	1	yes	characteristic
303	C₂×C₄	#10597330	1	1	yes	characteristic
304	C₂×C₄	C₄⋊C₄×D₁₉	1	1	yes	characteristic
305	C₂×C₄	#10597342	1	1	yes	characteristic
306	C₂×C₄	D₈×D₁₉	1	1	yes	characteristic
307	C₂×C₄	SD₁₆×D₁₉	1	1	yes	characteristic
308	C₂×C₄	#10597360	1	1	yes	characteristic
309	C₂×C₄	#10597364	1	1	yes	characteristic
310	C₈	—	4	2	no	—
311	C₈	—	8	1	no	—
312	C₈	—	8	1	no	—
313	C₈	—	19	2	no	—
314	C₈	—	19	2	no	—
315	C₈	—	38	1	no	—
316	C₈	—	38	1	no	—
317	C₈	—	152	1	no	—
318	C₈	—	152	1	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	76	2	no	—
322	C₂²	—	76	2	no	—
323	C₂²	—	76	2	no	—
324	C₂²	#11926403	1	1	yes	characteristic
325	C₂²	#11926823	1	1	yes	characteristic
326	C₂²	#11926833	1	1	yes	characteristic
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	76	2	no	—
331	C₄	#102×D₁₉	1	1	yes	characteristic
332	C₄	#11926544	1	1	yes	characteristic
333	C₄	#11926738	1	1	yes	characteristic
334	C₄	#11926748	1	1	yes	characteristic
335	C₄	#11926806	1	1	yes	characteristic
336	C₄	#11926810	1	1	yes	characteristic
337	C₂	—	2	1	no	—
338	C₂	—	2	1	no	—
339	C₂	—	152	2	no	—
340	C₂	#14563634	1	1	yes	characteristic
341	C₂	#14563684	1	1	yes	characteristic
342	C₂	#14563693	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11926403
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1292	C₂
socle	C₂²×C₁₉	#380
prime radical	#11927094	C₂²
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1292	D₁₉
hypocenter	C₁₉	#8682
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1292

Complements

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

N	H ≅ G/N	type
2. #14563806	C₂	semidirect
3. C₂×#11926294	C₂	semidirect
6. #14561509	C₂	semidirect
7. C₁₉×#1292	C₂	semidirect
154. #1292	D₁₉	semidirect
238. C₁₉	#8682	semidirect

Maximal subgroups

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

subgroup	index
#14572346	2
#14563806	2
C₂×#11926294	2
#14562313	2
#14562310	2
#14561509	2
C₁₉×#1292	2
#8682	19