Subgroups — #18075682

Order 4864.

450 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11926621	C₂	1	1	yes	maximal, characteristic
2	#14563806	C₂	1	1	yes	maximal, characteristic, complemented
3	#14562381	C₂	1	1	yes	maximal, characteristic, complemented
4	#14562061	C₂	1	1	yes	maximal, characteristic
5	#14562059	C₂	1	1	yes	maximal, characteristic
6	#14561532	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1301	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#64515	—	2	1	no	—
9	C₂×C₄×D₇₆	C₂²	1	1	yes	characteristic
10	#11926621	—	2	1	no	—
11	#11926621	—	2	1	no	—
12	#11926621	C₄	1	2	yes	—
13	#11926621	C₄	1	2	yes	—
14	#11926597	C₂²	1	1	yes	characteristic
15	C₂×#10597279	—	2	1	no	—
16	C₂×#10597279	—	2	1	no	—
17	#11926568	C₂²	1	1	yes	characteristic
18	C₂×(C₂×D₁₉)⋊C₈	—	2	1	no	—
19	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
20	#11926296	—	2	1	no	—
21	#11926294	—	2	1	no	—
22	#11926273	—	2	2	no	—
23	#11926271	—	2	2	no	—
24	C₁₉×#431	C₂²	1	1	yes	characteristic
25	C₁₉×#423	C₂²	1	1	yes	characteristic
26	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
27	C₂²×D₇₆	Q₈	1	1	yes	characteristic
28	C₂²×C₄×D₁₉	—	2	1	no	—
29	C₂×M₁₆×C₁₉	—	2	1	no	—
30	C₂²×C₈×C₁₉	—	2	1	no	—
31	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
32	C₂×(C₂×D₁₉)⋊C₄	—	2	1	no	—
33	C₂×C₄⋊Dic₁₉	Q₈	1	1	yes	characteristic
34	C₂×#64520	—	2	1	no	—
35	C₂²×C₁₉⋊C₈	D₄	1	1	yes	characteristic
36	C₂²×C₁₉⋊C₈	D₄	1	1	yes	characteristic
37	C₂×#64515	—	2	2	no	—
38	C₂×#64515	—	2	2	no	—
39	C₂×#64515	—	4	2	no	—
40	C₂×#64515	—	4	2	no	—
41	C₄×D₇₆	—	2	1	no	—
42	C₄×D₇₆	—	2	1	no	—
43	C₄×D₇₆	C₂×C₄	1	2	yes	—
44	C₄×D₇₆	D₄	1	2	yes	—
45	C₁₉×C₄⋊C₈	—	2	1	no	—
46	C₁₉×C₄⋊C₈	—	2	1	no	—
47	C₁₉×C₄⋊C₈	—	2	1	no	—
48	C₁₉×C₄⋊C₈	—	2	1	no	—
49	C₁₉×C₄⋊C₈	C₂×C₄	1	2	yes	—
50	C₁₉×C₂²⋊C₈	—	2	1	no	—
51	C₄×C₈×C₁₉	—	2	1	no	—
52	C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
53	#10597279	—	4	2	no	—
54	#10597279	—	4	2	no	—
55	(C₂×D₁₉)⋊C₈	—	2	2	no	—
56	(C₂×D₁₉)⋊C₈	—	2	2	no	—
57	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
58	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
59	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
60	C₄×C₁₉⋊C₈	—	2	1	no	—
61	C₄×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
62	C₄×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
63	C₄×C₁₉⋊C₈	D₄	1	1	yes	characteristic
64	C₄×C₁₉⋊C₈	D₄	1	1	yes	characteristic
65	C₂³×D₁₉	—	2	1	no	—
66	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
68	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
69	C₂²×Dic₁₉	—	2	1	no	—
70	C₂×D₇₆	—	2	1	no	—
71	C₂×D₇₆	—	4	2	no	—
72	C₂×D₇₆	C₄⋊C₄	1	2	yes	—
73	C₂×C₄×D₁₉	—	2	2	no	—
74	C₂×C₄×D₁₉	—	2	2	no	—
75	C₂×C₄×D₁₉	—	4	1	no	—
76	C₂×C₄×D₁₉	—	4	1	no	—
77	C₂×C₄×D₁₉	—	4	1	no	—
78	C₂×C₄×D₁₉	—	4	1	no	—
79	M₁₆×C₁₉	—	4	1	no	—
80	M₁₆×C₁₉	—	4	1	no	—
81	C₂×C₈×C₁₉	—	2	2	no	—
82	C₂×C₈×C₁₉	—	4	1	no	—
83	C₂×C₈×C₁₉	—	4	1	no	—
84	C₂×C₈×C₁₉	—	4	1	no	—
85	C₂×C₈×C₁₉	—	4	1	no	—
86	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₁₉	#41	1	1	yes	characteristic
90	(C₂×D₁₉)⋊C₄	—	2	2	no	—
91	(C₂×D₁₉)⋊C₄	—	2	2	no	—
92	C₄⋊Dic₁₉	—	2	1	no	—
93	C₄⋊Dic₁₉	C₄⋊C₄	1	2	yes	—
94	#64520	—	4	1	no	—
95	#64520	—	4	1	no	—
96	C₂×C₁₉⋊C₈	—	2	1	no	—
97	C₂×C₁₉⋊C₈	—	2	1	no	—
98	C₂×C₁₉⋊C₈	—	4	1	no	—
99	C₂×C₁₉⋊C₈	—	4	1	no	—
100	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₁₉⋊C₈	D₈	1	2	yes	—
105	C₂×C₁₉⋊C₈	SD₁₆	1	2	yes	—
106	#64515	—	4	4	no	—
107	#64515	—	4	4	no	—
108	#8662	—	19	1	no	maximal, Sylow
109	C₂³×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
110	C₂²×D₁₉	—	2	2	no	—
111	C₂²×D₁₉	—	4	2	no	—
112	C₂²×D₁₉	—	4	2	no	—
113	C₂²×D₁₉	—	4	1	no	—
114	C₂²×D₁₉	—	4	1	no	—
115	C₂×C₄×C₁₉	—	2	2	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	C₂×Dic₁₉	—	2	2	no	—
128	C₂×Dic₁₉	—	4	1	no	—
129	C₂×Dic₁₉	—	4	1	no	—
130	D₇₆	—	2	4	no	—
131	D₇₆	—	4	2	no	—
132	C₄×D₁₉	—	4	2	no	—
133	C₄×D₁₉	—	4	2	no	—
134	C₄×D₁₉	—	4	2	no	—
135	C₄×D₁₉	—	4	2	no	—
136	C₈×C₁₉	—	4	2	no	—
137	C₈×C₁₉	—	8	1	no	—
138	C₈×C₁₉	—	8	1	no	—
139	C₁₉⋊C₈	—	2	1	no	—
140	C₁₉⋊C₈	—	2	1	no	—
141	C₁₉⋊C₈	—	8	1	no	—
142	C₁₉⋊C₈	—	8	1	no	—
143	C₁₉⋊C₈	#102	1	2	yes	—
144	C₁₉⋊C₈	#102	1	2	yes	—
145	C₂×#436	—	19	1	no	—
146	#1848	—	19	1	no	—
147	#1564	—	19	1	no	—
148	#1486	—	19	1	no	—
149	#1486	—	19	1	no	—
150	#1301	—	19	1	no	—
151	#1301	D₁₉	1	1	yes	characteristic, complemented
152	C₂²×C₁₉	—	2	1	no	—
153	C₂²×C₁₉	—	2	1	no	—
154	C₂²×C₁₉	#380	1	1	yes	characteristic
155	C₂²×C₁₉	#474	1	1	yes	characteristic
156	C₂²×C₁₉	#476	1	1	yes	characteristic
157	C₂×D₁₉	—	4	2	no	—
158	C₂×D₁₉	—	4	2	no	—
159	C₂×D₁₉	—	4	2	no	—
160	C₂×D₁₉	—	8	2	no	—
161	C₂×D₁₉	—	8	2	no	—
162	C₄×C₁₉	—	2	1	no	—
163	C₄×C₁₉	—	2	1	no	—
164	C₄×C₁₉	—	2	1	no	—
165	C₄×C₁₉	C₂×#102	1	1	yes	characteristic
166	C₄×C₁₉	#417	1	1	yes	characteristic
167	C₄×C₁₉	#430	1	2	yes	—
168	C₄×C₁₉	#459	1	1	yes	characteristic
169	C₄×C₁₉	#461	1	1	yes	characteristic
170	Dic₁₉	—	4	2	no	—
171	C₂²×M₁₆	—	38	1	no	—
172	C₂×C₄×D₄	—	19	1	no	—
173	#436	—	19	2	no	—
174	#436	—	19	2	no	—
175	#436	—	38	1	no	—
176	#436	—	38	1	no	—
177	#431	—	19	1	no	—
178	#431	C₂×D₁₉	1	1	yes	characteristic
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	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
184	C₂×C₂²⋊C₈	—	38	1	no	—
185	C₂×C₄×C₈	—	19	1	no	—
186	#335	—	38	1	no	—
187	#334	—	38	1	no	—
188	D₄⋊C₈	—	38	2	no	—
189	D₄⋊C₈	—	38	2	no	—
190	C₂×C₁₉	—	2	1	no	—
191	C₂×C₁₉	—	2	1	no	—
192	C₂×C₁₉	#1802	1	1	yes	characteristic
193	C₂×C₁₉	#1811	1	1	yes	characteristic
194	C₂×C₁₉	#1814	1	1	yes	characteristic
195	D₁₉	—	8	2	no	—
196	C₂²×D₄	—	19	1	no	—
197	C₂³×C₄	—	38	1	no	—
198	C₂×M₁₆	—	2	1	no	—
199	C₂×M₁₆	—	38	2	no	—
200	C₂×M₁₆	—	38	2	no	—
201	C₂×M₁₆	—	38	1	no	—
202	C₂×M₁₆	—	76	2	no	—
203	C₂×M₁₆	—	76	2	no	—
204	C₂²×C₈	—	2	1	no	—
205	C₂²×C₈	—	19	1	no	—
206	C₂²×C₈	—	19	1	no	—
207	C₄×D₄	—	19	2	no	—
208	C₄×D₄	—	19	2	no	—
209	C₄×D₄	—	38	1	no	—
210	C₄×D₄	—	38	1	no	—
211	C₂×C₄⋊C₄	—	19	1	no	—
212	C₂×C₂²⋊C₄	—	38	1	no	—
213	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
214	#108	—	76	2	no	—
215	#108	—	76	2	no	—
216	C₄⋊C₈	—	2	1	no	—
217	C₄⋊C₈	—	2	1	no	—
218	C₄⋊C₈	—	2	1	no	—
219	C₄⋊C₈	—	2	1	no	—
220	C₄⋊C₈	—	38	1	no	—
221	C₄⋊C₈	—	38	1	no	—
222	C₄⋊C₈	—	38	1	no	—
223	C₄⋊C₈	C₄×D₁₉	1	2	yes	—
224	C₂²⋊C₈	—	2	1	no	—
225	C₂²⋊C₈	—	38	2	no	—
226	C₂²⋊C₈	—	38	2	no	—
227	C₂²⋊C₈	—	38	1	no	—
228	C₄×C₈	—	2	1	no	—
229	C₄×C₈	—	19	1	no	—
230	C₄×C₈	—	19	1	no	—
231	C₄×C₈	—	19	1	no	—
232	C₄×C₈	—	19	1	no	—
233	C₄×C₈	—	38	1	no	—
234	C₁₉	#8662	1	1	yes	characteristic, complemented, Sylow
235	C₂⁴	—	38	1	no	—
236	C₂×D₄	—	19	2	no	—
237	C₂×D₄	—	38	1	no	—
238	C₂×D₄	—	76	2	no	—
239	C₂²×C₄	—	38	1	no	—
240	C₂²×C₄	—	38	2	no	—
241	C₂²×C₄	—	38	2	no	—
242	C₂²×C₄	—	76	1	no	—
243	C₂²×C₄	—	76	1	no	—
244	C₂²×C₄	—	76	1	no	—
245	C₂²×C₄	—	76	1	no	—
246	C₂²×C₄	#64541	1	1	yes	characteristic
247	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
248	C₂²×C₄	Q₈×D₁₉	1	1	yes	characteristic
249	M₁₆	—	4	1	no	—
250	M₁₆	—	4	1	no	—
251	M₁₆	—	76	4	no	—
252	M₁₆	—	76	4	no	—
253	M₁₆	—	76	1	no	—
254	M₁₆	—	76	1	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	4	1	no	—
257	C₂×C₈	—	4	1	no	—
258	C₂×C₈	—	4	1	no	—
259	C₂×C₈	—	4	1	no	—
260	C₂×C₈	—	19	1	no	—
261	C₂×C₈	—	19	1	no	—
262	C₂×C₈	—	19	2	no	—
263	C₂×C₈	—	19	2	no	—
264	C₂×C₈	—	19	1	no	—
265	C₂×C₈	—	19	1	no	—
266	C₂×C₈	—	38	1	no	—
267	C₂×C₈	—	38	1	no	—
268	C₂×C₈	—	76	1	no	—
269	C₂×C₈	—	76	1	no	—
270	C₄⋊C₄	—	19	2	no	—
271	C₄⋊C₄	—	38	1	no	—
272	C₂²⋊C₄	—	38	2	no	—
273	C₂²⋊C₄	—	38	2	no	—
274	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
275	C₄²	#64541	1	1	yes	characteristic
276	C₄²	D₄×D₁₉	1	1	yes	characteristic
277	C₄²	D₄×D₁₉	1	1	yes	characteristic
278	C₂³	—	38	2	no	—
279	C₂³	—	76	2	no	—
280	C₂³	—	76	2	no	—
281	C₂³	—	76	1	no	—
282	C₂³	—	76	1	no	—
283	C₂³	#10597342	1	1	yes	characteristic
284	D₄	—	38	4	no	—
285	D₄	—	76	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	4	1	no	—
288	C₂×C₄	—	38	2	no	—
289	C₂×C₄	—	76	1	no	—
290	C₂×C₄	—	76	1	no	—
291	C₂×C₄	—	76	2	no	—
292	C₂×C₄	—	76	2	no	—
293	C₂×C₄	—	76	2	no	—
294	C₂×C₄	—	76	2	no	—
295	C₂×C₄	#10597318	1	1	yes	characteristic
296	C₂×C₄	C₂²⋊C₄×D₁₉	1	1	yes	characteristic
297	C₂×C₄	#10597330	1	1	yes	characteristic
298	C₂×C₄	#10597330	1	1	yes	characteristic
299	C₂×C₄	C₄⋊C₄×D₁₉	1	1	yes	characteristic
300	C₂×C₄	#10597342	1	1	yes	characteristic
301	C₂×C₄	D₈×D₁₉	1	1	yes	characteristic
302	C₂×C₄	SD₁₆×D₁₉	1	1	yes	characteristic
303	C₂×C₄	#10597360	1	1	yes	characteristic
304	C₂×C₄	#10597364	1	1	yes	characteristic
305	C₈	—	4	2	no	—
306	C₈	—	8	1	no	—
307	C₈	—	8	1	no	—
308	C₈	—	19	2	no	—
309	C₈	—	19	2	no	—
310	C₈	—	38	1	no	—
311	C₈	—	38	1	no	—
312	C₈	—	152	1	no	—
313	C₈	—	152	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	76	2	no	—
317	C₂²	—	76	2	no	—
318	C₂²	—	76	2	no	—
319	C₂²	—	152	2	no	—
320	C₂²	—	152	2	no	—
321	C₂²	#11926403	1	1	yes	characteristic
322	C₂²	#11926823	1	1	yes	characteristic
323	C₂²	#11926833	1	1	yes	characteristic
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	76	2	no	—
328	C₄	#102×D₁₉	1	1	yes	characteristic
329	C₄	#11926544	1	1	yes	characteristic
330	C₄	#11926591	1	2	yes	—
331	C₄	#11926738	1	1	yes	characteristic
332	C₄	#11926748	1	1	yes	characteristic
333	C₂	—	2	1	no	—
334	C₂	—	2	1	no	—
335	C₂	—	152	2	no	—
336	C₂	#14563634	1	1	yes	characteristic
337	C₂	#14563676	1	1	yes	characteristic
338	C₂	#14563688	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#10597330
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1301	C₂
socle	C₂²×C₁₉	#380
prime radical	C₂²×D₇₆	Q₈
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1301	D₁₉
hypocenter	C₁₉	#8662
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1301

Complements

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

N	H ≅ G/N	type
2. #14563806	C₂	semidirect
3. #14562381	C₂	semidirect
6. #14561532	C₂	semidirect
7. C₁₉×#1301	C₂	semidirect
151. #1301	D₁₉	semidirect
234. C₁₉	#8662	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926621	2
#14563806	2
#14562381	2
#14562061	2
#14562059	2
#14561532	2
C₁₉×#1301	2
#8662	19