Subgroups — #18079231

Order 4864.

490 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14574573	C₂	1	1	yes	maximal, characteristic
2	#14564459	C₂	1	1	yes	maximal, characteristic
3	#14562304	C₂	1	1	yes	maximal, characteristic
4	#14562199	C₂	1	1	yes	maximal, characteristic, complemented
5	#14561495	C₂	1	1	yes	maximal, characteristic, complemented
6	#14561466	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#2961	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10597320	C₂²	1	1	yes	characteristic
9	#11926918	C₄	1	2	yes	complemented
10	#11926892	C₄	1	2	yes	complemented
11	#11926744	—	2	2	no	—
12	#11926694	—	2	2	no	—
13	#11926597	C₂²	1	1	yes	characteristic
14	#11926597	C₂²	1	1	yes	characteristic
15	C₂×#10597278	—	2	1	no	—
16	C₂×#10597278	—	2	1	no	—
17	#11926410	C₂²	1	1	yes	characteristic
18	#11926287	—	2	2	no	—
19	#11926281	—	2	2	no	—
20	C₁₉×#533	C₂²	1	1	yes	characteristic
21	C₁₉×#446	—	2	2	no	—
22	C₁₉×#446	—	2	2	no	—
23	C₁₉×#424	—	2	1	no	—
24	C₁₉×#423	C₂²	1	1	yes	characteristic
25	C₁₉×#405	—	2	1	no	—
26	C₂×C₁₉×#97	C₂²	1	1	yes	characteristic
27	C₂²×D₇₆	—	2	1	no	—
28	C₂²×D₇₆	—	2	1	no	—
29	C₂²×D₇₆	—	2	1	no	—
30	C₂×M₁₆×C₁₉	—	2	1	no	—
31	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
33	C₁₉×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
34	C₁₉×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
35	C₁₉×#125	—	2	1	no	—
36	C₁₉×C₂²⋊Q₈	—	2	1	no	—
37	C₁₉×C₂²⋊Q₈	D₄	1	2	yes	—
38	C₄×Q₈×C₁₉	—	2	1	no	—
39	C₄×Q₈×C₁₉	D₄	1	2	yes	—
40	C₁₉×#117	—	2	1	no	—
41	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
42	C₂×#64527	—	2	4	no	—
43	C₂×C₄⋊Dic₁₉	—	2	1	no	—
44	#10597320	—	2	1	no	—
45	#10597320	—	2	1	no	—
46	#10597320	—	2	1	no	—
47	#10597320	C₂×C₄	1	2	yes	—
48	C₁₉×C₄⋊C₈	—	2	4	no	—
49	C₁₉×C₄⋊C₈	—	2	1	no	—
50	C₁₉×C₄⋊C₈	D₄	1	2	yes	—
51	C₁₉×#97	—	2	2	no	—
52	C₁₉×#97	—	2	1	no	—
53	C₁₉×#97	—	2	1	no	—
54	C₄×C₈×C₁₉	—	2	2	no	—
55	C₁₉⋊#95	—	2	1	no	—
56	C₁₉⋊#95	—	2	1	no	—
57	C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
58	C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
59	#10597278	—	4	2	no	—
60	#10597278	—	4	2	no	—
61	#10597266	—	2	4	no	—
62	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
63	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
64	C₄×C₁₉⋊C₈	—	2	1	no	—
65	C₄×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
66	C₂³×D₁₉	—	4	1	no	—
67	C₂×Q₈×C₁₉	—	2	1	no	—
68	C₂×Q₈×C₁₉	C₂²⋊C₄	1	2	yes	—
69	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
72	C₂²×Dic₁₉	—	4	1	no	—
73	C₂×D₇₆	—	2	2	no	—
74	C₂×D₇₆	—	2	2	no	—
75	C₂×D₇₆	—	2	2	no	—
76	C₂×D₇₆	—	4	1	no	—
77	C₂×D₇₆	—	4	1	no	—
78	C₂×D₇₆	—	4	1	no	—
79	C₂×D₇₆	—	4	1	no	—
80	C₂×D₇₆	—	4	1	no	—
81	C₂×D₇₆	—	4	1	no	—
82	C₂×D₇₆	—	4	1	no	—
83	C₂×D₇₆	—	8	2	no	—
84	C₂×D₇₆	—	8	2	no	—
85	M₁₆×C₁₉	—	4	1	no	—
86	M₁₆×C₁₉	—	4	1	no	—
87	C₂×C₈×C₁₉	—	2	1	no	—
88	C₂×C₈×C₁₉	—	2	1	no	—
89	C₂×C₈×C₁₉	—	2	1	no	—
90	C₂×C₈×C₁₉	—	2	2	no	—
91	C₂×C₈×C₁₉	—	2	1	no	—
92	C₂×C₈×C₁₉	—	2	1	no	—
93	C₂×C₈×C₁₉	SD₁₆	1	2	yes	complemented
94	C₄⋊C₄×C₁₉	—	2	1	no	—
95	C₄⋊C₄×C₁₉	—	2	1	no	—
96	C₄⋊C₄×C₁₉	—	2	1	no	—
97	C₄⋊C₄×C₁₉	—	2	2	no	—
98	C₄⋊C₄×C₁₉	—	2	2	no	—
99	C₄⋊C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
100	C₂²⋊C₄×C₁₉	—	2	2	no	—
101	C₄²×C₁₉	—	2	2	no	—
102	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
103	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₁₉	#41	1	1	yes	characteristic
106	#64527	—	8	4	no	—
107	C₄⋊Dic₁₉	—	4	1	no	—
108	C₄⋊Dic₁₉	—	4	1	no	—
109	C₂×C₁₉⋊C₈	—	2	2	no	—
110	C₂×C₁₉⋊C₈	—	4	1	no	—
111	#9294	—	19	1	no	maximal, Sylow
112	C₂³×C₁₉	#121	1	1	yes	characteristic
113	C₂²×D₁₉	—	4	2	no	—
114	C₂²×D₁₉	—	8	2	no	—
115	C₂²×D₁₉	—	8	2	no	—
116	C₂²×D₁₉	—	8	1	no	—
117	C₂²×D₁₉	—	8	1	no	—
118	Q₈×C₁₉	—	4	2	no	—
119	Q₈×C₁₉	—	8	1	no	—
120	C₂×C₄×C₁₉	—	2	2	no	—
121	C₂×C₄×C₁₉	—	2	2	no	—
122	C₂×C₄×C₁₉	—	4	1	no	—
123	C₂×C₄×C₁₉	—	4	1	no	—
124	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
126	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
127	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
128	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
129	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
130	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
131	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
132	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
133	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
134	C₂×Dic₁₉	—	8	1	no	—
135	C₂×Dic₁₉	—	8	1	no	—
136	C₂×Dic₁₉	—	8	1	no	—
137	D₇₆	—	4	4	no	—
138	D₇₆	—	4	2	no	—
139	D₇₆	—	4	2	no	—
140	D₇₆	—	8	2	no	—
141	D₇₆	—	8	2	no	—
142	D₇₆	—	8	2	no	—
143	D₇₆	—	8	1	no	—
144	D₇₆	—	8	1	no	—
145	C₈×C₁₉	—	2	2	no	—
146	C₈×C₁₉	—	4	1	no	—
147	C₈×C₁₉	—	4	2	no	—
148	C₁₉⋊C₈	—	4	2	no	—
149	C₁₉⋊C₈	—	8	1	no	—
150	#3208	—	19	1	no	—
151	#2961	D₁₉	1	1	yes	characteristic, complemented
152	#1961	—	19	1	no	—
153	#1545	—	19	1	no	—
154	#1518	—	19	1	no	—
155	#1286	—	19	1	no	—
156	#1275	—	19	1	no	—
157	C₂²×C₁₉	—	2	1	no	—
158	C₂²×C₁₉	—	2	1	no	—
159	C₂²×C₁₉	#386	1	1	yes	characteristic
160	C₂²×C₁₉	#460	1	1	yes	characteristic
161	C₂²×C₁₉	#464	1	1	yes	characteristic
162	C₂×D₁₉	—	8	2	no	—
163	C₂×D₁₉	—	8	2	no	—
164	C₂×D₁₉	—	8	2	no	—
165	C₂×D₁₉	—	16	2	no	—
166	C₂×D₁₉	—	16	2	no	—
167	C₄×C₁₉	—	2	1	no	—
168	C₄×C₁₉	—	2	1	no	—
169	C₄×C₁₉	—	2	1	no	—
170	C₄×C₁₉	—	2	1	no	—
171	C₄×C₁₉	—	4	2	no	—
172	C₄×C₁₉	—	4	2	no	—
173	C₄×C₁₉	C₄×SD₁₆	1	1	yes	characteristic
174	C₄×C₁₉	#440	1	1	yes	characteristic
175	C₄×C₁₉	#448	1	1	yes	characteristic
176	C₄×C₁₉	#449	1	1	yes	characteristic
177	Dic₁₉	—	16	1	no	—
178	#533	C₂×D₁₉	1	1	yes	characteristic
179	C₂×#127	—	19	1	no	—
180	#492	—	19	2	no	—
181	#486	—	19	2	no	—
182	#460	—	38	2	no	—
183	#450	—	38	2	no	—
184	#446	—	2	2	no	—
185	#446	—	2	2	no	—
186	#431	—	19	1	no	—
187	#431	—	19	1	no	—
188	#424	—	2	1	no	—
189	#423	C₂×D₁₉	1	1	yes	characteristic
190	C₂×#102	—	38	1	no	—
191	C₂×#102	—	38	1	no	—
192	#405	—	2	1	no	—
193	C₂×#97	C₂×D₁₉	1	1	yes	characteristic
194	#382	—	19	1	no	—
195	#331	—	38	2	no	—
196	#328	—	38	2	no	—
197	C₂×C₁₉	—	2	1	no	—
198	C₂×C₁₉	—	2	1	no	—
199	C₂×C₁₉	#1880	1	1	yes	characteristic
200	C₂×C₁₉	#1903	1	1	yes	characteristic
201	C₂×C₁₉	#1910	1	1	yes	characteristic
202	D₁₉	—	16	2	no	—
203	C₂²×D₄	—	38	1	no	—
204	C₂²×D₄	—	38	1	no	—
205	C₂²×D₄	—	38	1	no	—
206	C₂×SD₁₆	—	38	4	no	—
207	C₂×M₁₆	—	2	1	no	—
208	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
209	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
210	C₄⋊Q₈	C₄×D₁₉	1	1	yes	characteristic
211	C₄⋊Q₈	C₄×D₁₉	1	1	yes	characteristic
212	#127	—	19	2	no	—
213	#127	—	38	1	no	—
214	#127	—	38	1	no	—
215	#127	—	38	1	no	—
216	#125	—	2	1	no	—
217	C₂²⋊Q₈	—	2	1	no	—
218	C₂²⋊Q₈	D₇₆	1	2	yes	—
219	C₄×Q₈	—	2	1	no	—
220	C₄×Q₈	C₂²⋊D₁₉	1	2	yes	—
221	#117	—	2	1	no	—
222	C₂×C₄⋊C₄	—	38	1	no	—
223	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
224	C₄⋊C₈	—	2	4	no	—
225	C₄⋊C₈	—	2	1	no	—
226	C₄⋊C₈	—	38	1	no	—
227	C₄⋊C₈	—	38	1	no	—
228	C₄⋊C₈	C₂²⋊D₁₉	1	2	yes	—
229	#102	—	38	4	no	—
230	#102	—	76	2	no	—
231	#102	—	76	2	no	—
232	C₂²⋊C₈	—	38	1	no	—
233	C₂²⋊C₈	—	38	1	no	—
234	#97	—	2	2	no	—
235	#97	—	2	1	no	—
236	#97	—	2	1	no	—
237	C₄×C₈	—	2	2	no	—
238	C₄×C₈	—	19	2	no	—
239	C₄×C₈	—	38	1	no	—
240	#95	—	38	1	no	—
241	#95	—	38	1	no	—
242	C₁₉	#9294	1	1	yes	characteristic, complemented, Sylow
243	C₂⁴	—	76	1	no	—
244	C₂×Q₈	—	2	1	no	—
245	C₂×Q₈	(C₂×D₁₉)⋊C₄	1	2	yes	—
246	C₂×D₄	—	38	2	no	—
247	C₂×D₄	—	38	2	no	—
248	C₂×D₄	—	38	2	no	—
249	C₂×D₄	—	76	1	no	—
250	C₂×D₄	—	76	1	no	—
251	C₂×D₄	—	76	1	no	—
252	C₂×D₄	—	76	1	no	—
253	C₂×D₄	—	76	1	no	—
254	C₂×D₄	—	76	1	no	—
255	C₂×D₄	—	76	1	no	—
256	C₂×D₄	—	152	2	no	—
257	C₂×D₄	—	152	2	no	—
258	C₂²×C₄	—	76	1	no	—
259	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
260	C₂²×C₄	#64541	1	1	yes	characteristic
261	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
262	SD₁₆	—	152	4	no	—
263	M₁₆	—	4	1	no	—
264	M₁₆	—	4	1	no	—
265	C₂×C₈	—	2	1	no	—
266	C₂×C₈	—	2	1	no	—
267	C₂×C₈	—	2	1	no	—
268	C₂×C₈	—	2	1	no	—
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	—	2	1	no	—
271	C₂×C₈	—	38	2	no	—
272	C₂×C₈	—	76	1	no	—
273	C₂×C₈	#64527	1	2	yes	complemented
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	2	1	no	—
276	C₄⋊C₄	—	2	1	no	—
277	C₄⋊C₄	—	2	1	no	—
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	—	76	1	no	—
280	C₄⋊C₄	—	76	1	no	—
281	C₄⋊C₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
282	C₂²⋊C₄	—	2	2	no	—
283	C₄²	—	2	2	no	—
284	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
285	C₄²	#64541	1	1	yes	characteristic
286	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
287	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
288	C₂³	—	76	2	no	—
289	C₂³	—	152	2	no	—
290	C₂³	—	152	2	no	—
291	C₂³	—	152	1	no	—
292	C₂³	—	152	1	no	—
293	C₂³	#10597376	1	1	yes	characteristic
294	Q₈	—	4	2	no	—
295	Q₈	—	8	1	no	—
296	D₄	—	76	4	no	—
297	D₄	—	76	2	no	—
298	D₄	—	76	2	no	—
299	D₄	—	152	2	no	—
300	D₄	—	152	2	no	—
301	D₄	—	152	2	no	—
302	D₄	—	152	1	no	—
303	D₄	—	152	1	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	4	1	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	152	1	no	—
309	C₂×C₄	—	152	1	no	—
310	C₂×C₄	—	152	1	no	—
311	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
312	C₂×C₄	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
313	C₂×C₄	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
314	C₂×C₄	#10597375	1	1	yes	characteristic
315	C₂×C₄	#10597376	1	1	yes	characteristic
316	C₂×C₄	C₂×#64527	1	1	yes	characteristic
317	C₂×C₄	#10597388	1	1	yes	characteristic
318	C₂×C₄	#10597395	1	1	yes	characteristic
319	C₂×C₄	#10597396	1	1	yes	characteristic
320	C₂×C₄	#10597399	1	1	yes	characteristic
321	C₈	—	2	2	no	—
322	C₈	—	4	1	no	—
323	C₈	—	4	2	no	—
324	C₈	—	76	2	no	—
325	C₈	—	152	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	304	2	no	—
329	C₂²	—	304	2	no	—
330	C₂²	—	152	2	no	—
331	C₂²	—	152	2	no	—
332	C₂²	—	152	2	no	—
333	C₂²	#11926423	1	1	yes	characteristic
334	C₂²	#11926744	1	1	yes	characteristic
335	C₂²	#11926766	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	4	2	no	—
341	C₄	—	4	2	no	—
342	C₄	—	304	1	no	—
343	C₄	C₄×#64527	1	1	yes	characteristic
344	C₄	#11926642	1	1	yes	characteristic
345	C₄	#11926683	1	1	yes	characteristic
346	C₄	#11926689	1	1	yes	characteristic
347	C₂	—	2	1	no	—
348	C₂	—	2	1	no	—
349	C₂	—	304	2	no	—
350	C₂	#14563990	1	1	yes	characteristic
351	C₂	#14564115	1	1	yes	characteristic
352	C₂	#14564162	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11926423
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#2961	C₂
socle	C₂²×C₁₉	#386
prime radical	C₂×#10597320	C₂²
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#2961	D₁₉
hypocenter	C₁₉	#9294
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2961

Complements

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

N	H ≅ G/N	type
4. #14562199	C₂	semidirect
5. #14561495	C₂	semidirect
6. #14561466	C₂	semidirect
7. C₁₉×#2961	C₂	semidirect
9. #11926918	C₄	semidirect
10. #11926892	C₄	semidirect
93. C₂×C₈×C₁₉	SD₁₆	semidirect
151. #2961	D₁₉	semidirect
242. C₁₉	#9294	semidirect
273. C₂×C₈	#64527	semidirect

Maximal subgroups

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

subgroup	index
#14574573	2
#14564459	2
#14562304	2
#14562199	2
#14561495	2
#14561466	2
C₁₉×#2961	2
#9294	19