Subgroups — C₁₉×#12734 #18018693

Order 4864.

932 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₉×#135	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₉×#359	C₂	1	4	yes	maximal, complemented
3	C₂×C₁₉×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₉×#1810	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#41×C₁₉	—	2	1	no	—
6	C₂×C₁₉×#135	—	2	8	no	—
7	C₂×C₁₉×#135	—	2	1	no	—
8	C₂×C₁₉×#135	—	2	1	no	—
9	C₂×C₁₉×#135	C₄	1	2	yes	—
10	C₂×C₁₉×#135	C₂²	1	2	yes	—
11	C₂²×Q₁₆×C₁₉	C₄	1	1	yes	characteristic
12	C₂²×SD₁₆×C₁₉	—	2	1	no	—
13	C₂²×D₈×C₁₉	C₄	1	1	yes	characteristic
14	C₂³×C₈×C₁₉	C₂²	1	1	yes	characteristic
15	C₂²×C₁₆×C₁₉	C₂²	1	2	yes	complemented
16	C₂×C₁₉×#108	C₂²	1	2	yes	complemented
17	C₁₉×#407	—	2	1	no	—
18	C₁₉×#359	—	2	8	no	—
19	C₁₉×C₂²⋊C₁₆	—	2	1	no	—
20	C₁₉×C₂²⋊C₁₆	—	2	1	no	—
21	C₁₉×#341	—	2	1	no	—
22	C₂×#41×C₁₉	—	2	8	no	—
23	C₂×#41×C₁₉	—	2	2	no	—
24	C₂×#41×C₁₉	—	2	2	no	—
25	C₂×#41×C₁₉	—	4	4	no	—
26	C₂×#41×C₁₉	—	4	1	no	—
27	C₂×#41×C₁₉	—	4	1	no	—
28	C₂²×Q₈×C₁₉	—	2	1	no	—
29	C₂²×D₄×C₁₉	—	2	1	no	—
30	C₂²×D₄×C₁₉	—	4	1	no	—
31	C₂³×C₄×C₁₉	—	4	1	no	—
32	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
33	C₁₉×#135	—	2	8	no	—
34	C₁₉×#135	—	2	8	no	—
35	C₁₉×#135	—	2	4	no	—
36	C₁₉×#135	—	2	4	no	—
37	C₁₉×#135	—	2	4	no	—
38	C₁₉×#135	D₄	1	8	yes	—
39	C₂×Q₁₆×C₁₉	—	2	4	no	—
40	C₂×Q₁₆×C₁₉	—	2	1	no	—
41	C₂×Q₁₆×C₁₉	C₂×C₄	1	2	yes	—
42	C₂×SD₁₆×C₁₉	—	2	8	no	—
43	C₂×SD₁₆×C₁₉	—	2	2	no	—
44	C₂×SD₁₆×C₁₉	—	2	2	no	—
45	C₂×D₈×C₁₉	—	2	4	no	—
46	C₂×D₈×C₁₉	—	2	1	no	—
47	C₂×D₈×C₁₉	C₂×C₄	1	2	yes	—
48	C₂×M₁₆×C₁₉	—	2	2	no	—
49	C₂²×C₈×C₁₉	—	2	4	no	—
50	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
53	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
54	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
55	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
56	C₂×C₁₆×C₁₉	—	2	2	no	—
57	C₂×C₁₆×C₁₉	—	2	4	no	—
58	C₁₉×#108	—	2	4	no	—
59	C₁₉×C₂²⋊C₈	—	4	1	no	—
60	C₁₉×C₂²⋊C₈	—	4	1	no	—
61	C₂⁴×C₁₉	—	4	1	no	—
62	C₂⁴×C₁₉	SD₁₆	1	1	yes	characteristic
63	#41×C₁₉	—	2	8	no	—
64	#41×C₁₉	—	2	8	no	—
65	#41×C₁₉	—	4	8	no	—
66	#41×C₁₉	—	4	4	no	—
67	#41×C₁₉	—	4	8	no	—
68	#41×C₁₉	—	4	4	no	—
69	C₂×Q₈×C₁₉	—	2	4	no	—
70	C₂×Q₈×C₁₉	—	2	2	no	—
71	C₂×Q₈×C₁₉	—	4	2	no	—
72	C₂×Q₈×C₁₉	—	4	1	no	—
73	C₂×D₄×C₁₉	—	2	4	no	—
74	C₂×D₄×C₁₉	—	2	2	no	—
75	C₂×D₄×C₁₉	—	4	2	no	—
76	C₂×D₄×C₁₉	—	4	8	no	—
77	C₂×D₄×C₁₉	—	4	2	no	—
78	C₂×D₄×C₁₉	—	4	2	no	—
79	C₂×D₄×C₁₉	—	4	1	no	—
80	C₂²×C₄×C₁₉	—	2	2	no	—
81	C₂²×C₄×C₁₉	—	2	2	no	—
82	C₂²×C₄×C₁₉	—	4	8	no	—
83	C₂²×C₄×C₁₉	—	4	1	no	—
84	C₂²×C₄×C₁₉	—	4	2	no	—
85	C₂²×C₄×C₁₉	—	4	2	no	—
86	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
87	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
88	C₂²×C₄×C₁₉	D₈	1	1	yes	characteristic
89	C₂²×C₄×C₁₉	D₈	1	1	yes	characteristic
90	C₂²×C₄×C₁₉	SD₁₆	1	1	yes	characteristic
91	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
92	Q₁₆×C₁₉	—	2	4	no	—
93	Q₁₆×C₁₉	—	2	2	no	—
94	Q₁₆×C₁₉	C₂²⋊C₄	1	4	yes	—
95	SD₁₆×C₁₉	—	2	8	no	—
96	SD₁₆×C₁₉	—	2	8	no	—
97	D₈×C₁₉	—	2	4	no	—
98	D₈×C₁₉	—	2	2	no	—
99	D₈×C₁₉	C₂²⋊C₄	1	4	yes	—
100	M₁₆×C₁₉	—	4	2	no	—
101	M₁₆×C₁₉	—	4	2	no	—
102	C₂×C₈×C₁₉	—	2	2	no	—
103	C₂×C₈×C₁₉	—	2	2	no	—
104	C₂×C₈×C₁₉	—	2	2	no	—
105	C₂×C₈×C₁₉	—	2	2	no	—
106	C₂×C₈×C₁₉	—	2	2	no	—
107	C₂×C₈×C₁₉	—	4	2	no	—
108	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₈×C₁₉	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₈×C₁₉	C₂×D₄	1	4	yes	—
112	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
113	C₁₆×C₁₉	—	4	2	no	—
114	#12734	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
115	C₂³×C₁₉	—	2	2	no	—
116	C₂³×C₁₉	—	2	2	no	—
117	C₂³×C₁₉	—	4	1	no	—
118	C₂³×C₁₉	—	4	4	no	—
119	C₂³×C₁₉	—	4	2	no	—
120	C₂³×C₁₉	—	8	2	no	—
121	C₂³×C₁₉	—	8	2	no	—
122	C₂³×C₁₉	#102	1	1	yes	characteristic
123	C₂³×C₁₉	#102	1	1	yes	characteristic
124	C₂³×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
125	Q₈×C₁₉	—	2	4	no	—
126	Q₈×C₁₉	—	4	4	no	—
127	Q₈×C₁₉	—	4	2	no	—
128	D₄×C₁₉	—	2	4	no	—
129	D₄×C₁₉	—	4	4	no	—
130	D₄×C₁₉	—	4	8	no	—
131	D₄×C₁₉	—	4	8	no	—
132	D₄×C₁₉	—	4	2	no	—
133	C₂×C₄×C₁₉	—	2	2	no	—
134	C₂×C₄×C₁₉	—	2	1	no	—
135	C₂×C₄×C₁₉	—	2	1	no	—
136	C₂×C₄×C₁₉	—	2	1	no	—
137	C₂×C₄×C₁₉	—	2	1	no	—
138	C₂×C₄×C₁₉	—	2	2	no	—
139	C₂×C₄×C₁₉	—	2	2	no	—
140	C₂×C₄×C₁₉	—	4	8	no	—
141	C₂×C₄×C₁₉	—	4	4	no	—
142	C₂×C₄×C₁₉	—	4	2	no	—
143	C₂×C₄×C₁₉	—	4	8	no	—
144	C₂×C₄×C₁₉	#102	1	1	yes	characteristic
145	C₂×C₄×C₁₉	#102	1	1	yes	characteristic
146	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
147	C₂×C₄×C₁₉	#120	1	2	yes	—
148	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
149	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
150	C₈×C₁₉	—	2	1	no	—
151	C₈×C₁₉	—	2	1	no	—
152	C₈×C₁₉	—	8	2	no	—
153	C₈×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
154	C₈×C₁₉	#120	1	2	yes	—
155	C₂²×#135	C₂×C₁₉	1	1	yes	characteristic
156	C₂×#359	C₂×C₁₉	1	4	yes	complemented
157	C₂×C₂²⋊C₁₆	C₂×C₁₉	1	1	yes	characteristic, complemented
158	#1810	C₂×C₁₉	1	1	yes	characteristic, complemented
159	C₂²×C₁₉	—	2	1	no	—
160	C₂²×C₁₉	—	2	1	no	—
161	C₂²×C₁₉	—	2	1	no	—
162	C₂²×C₁₉	—	2	1	no	—
163	C₂²×C₁₉	—	2	1	no	—
164	C₂²×C₁₉	—	2	1	no	—
165	C₂²×C₁₉	—	2	1	no	—
166	C₂²×C₁₉	—	4	4	no	—
167	C₂²×C₁₉	—	4	4	no	—
168	C₂²×C₁₉	—	8	8	no	—
169	C₂²×C₁₉	—	8	2	no	—
170	C₂²×C₁₉	—	8	2	no	—
171	C₂²×C₁₉	#359	1	2	yes	complemented
172	C₂²×C₁₉	C₂×#102	1	1	yes	characteristic
173	C₂²×C₁₉	#450	1	2	yes	—
174	C₄×C₁₉	—	2	1	no	—
175	C₄×C₁₉	—	2	1	no	—
176	C₄×C₁₉	—	4	4	no	—
177	C₄×C₁₉	#379	1	1	yes	characteristic
178	C₄×C₁₉	#418	1	1	yes	characteristic
179	C₄×C₁₉	#447	1	2	yes	—
180	C₂²×#41	—	2	1	no	—
181	C₂×#135	—	2	8	no	—
182	C₂×#135	—	2	1	no	—
183	C₂×#135	—	2	1	no	—
184	C₂×#135	C₄×C₁₉	1	2	yes	—
185	C₂×#135	C₂²×C₁₉	1	2	yes	—
186	C₂²×Q₁₆	C₄×C₁₉	1	1	yes	characteristic
187	C₂²×SD₁₆	—	2	1	no	—
188	C₂²×D₈	C₄×C₁₉	1	1	yes	characteristic
189	C₂³×C₈	C₂²×C₁₉	1	1	yes	characteristic
190	C₂²×C₁₆	C₂²×C₁₉	1	2	yes	complemented
191	C₂×#108	C₂²×C₁₉	1	2	yes	complemented
192	#407	—	2	1	no	—
193	#359	—	2	8	no	—
194	C₂²⋊C₁₆	—	2	1	no	—
195	C₂²⋊C₁₆	—	2	1	no	—
196	#341	—	2	1	no	—
197	C₂×C₁₉	—	2	2	no	—
198	C₂×C₁₉	—	2	2	no	—
199	C₂×C₁₉	—	4	1	no	—
200	C₂×C₁₉	—	8	4	no	—
201	C₂×C₁₉	#1786	1	1	yes	characteristic
202	C₂×C₁₉	C₂×#359	1	1	yes	characteristic
203	C₂×C₁₉	#2143	1	1	yes	characteristic
204	C₂×#41	—	2	8	no	—
205	C₂×#41	—	2	2	no	—
206	C₂×#41	—	2	2	no	—
207	C₂×#41	—	4	4	no	—
208	C₂×#41	—	4	1	no	—
209	C₂×#41	—	4	1	no	—
210	C₂²×Q₈	—	2	1	no	—
211	C₂²×D₄	—	2	1	no	—
212	C₂²×D₄	—	4	1	no	—
213	C₂³×C₄	—	4	1	no	—
214	C₂³×C₄	D₄×C₁₉	1	1	yes	characteristic
215	#135	—	2	8	no	—
216	#135	—	2	8	no	—
217	#135	—	2	4	no	—
218	#135	—	2	4	no	—
219	#135	—	2	4	no	—
220	#135	D₄×C₁₉	1	8	yes	—
221	C₂×Q₁₆	—	2	4	no	—
222	C₂×Q₁₆	—	2	1	no	—
223	C₂×Q₁₆	C₂×C₄×C₁₉	1	2	yes	—
224	C₂×SD₁₆	—	2	8	no	—
225	C₂×SD₁₆	—	2	2	no	—
226	C₂×SD₁₆	—	2	2	no	—
227	C₂×D₈	—	2	4	no	—
228	C₂×D₈	—	2	1	no	—
229	C₂×D₈	C₂×C₄×C₁₉	1	2	yes	—
230	C₂×M₁₆	—	2	2	no	—
231	C₂²×C₈	—	2	4	no	—
232	C₂²×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
233	C₂²×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
234	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
235	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
236	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
237	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
238	C₂×C₁₆	—	2	2	no	—
239	C₂×C₁₆	—	2	4	no	—
240	#108	—	2	4	no	—
241	C₂²⋊C₈	—	4	1	no	—
242	C₂²⋊C₈	—	4	1	no	—
243	C₁₉	#12734	1	1	yes	characteristic, direct product factor, Sylow
244	C₂⁴	—	4	1	no	—
245	C₂⁴	SD₁₆×C₁₉	1	1	yes	characteristic
246	#41	—	2	8	no	—
247	#41	—	2	8	no	—
248	#41	—	4	8	no	—
249	#41	—	4	4	no	—
250	#41	—	4	8	no	—
251	#41	—	4	4	no	—
252	C₂×Q₈	—	2	4	no	—
253	C₂×Q₈	—	2	2	no	—
254	C₂×Q₈	—	4	2	no	—
255	C₂×Q₈	—	4	1	no	—
256	C₂×D₄	—	2	4	no	—
257	C₂×D₄	—	2	2	no	—
258	C₂×D₄	—	4	2	no	—
259	C₂×D₄	—	4	8	no	—
260	C₂×D₄	—	4	2	no	—
261	C₂×D₄	—	4	2	no	—
262	C₂×D₄	—	4	1	no	—
263	C₂²×C₄	—	2	2	no	—
264	C₂²×C₄	—	2	2	no	—
265	C₂²×C₄	—	4	8	no	—
266	C₂²×C₄	—	4	1	no	—
267	C₂²×C₄	—	4	2	no	—
268	C₂²×C₄	—	4	2	no	—
269	C₂²×C₄	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
270	C₂²×C₄	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
271	C₂²×C₄	D₈×C₁₉	1	1	yes	characteristic
272	C₂²×C₄	D₈×C₁₉	1	1	yes	characteristic
273	C₂²×C₄	SD₁₆×C₁₉	1	1	yes	characteristic
274	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
275	Q₁₆	—	2	4	no	—
276	Q₁₆	—	2	2	no	—
277	Q₁₆	C₂²⋊C₄×C₁₉	1	4	yes	—
278	SD₁₆	—	2	8	no	—
279	SD₁₆	—	2	8	no	—
280	D₈	—	2	4	no	—
281	D₈	—	2	2	no	—
282	D₈	C₂²⋊C₄×C₁₉	1	4	yes	—
283	M₁₆	—	4	2	no	—
284	M₁₆	—	4	2	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	—	2	2	no	—
288	C₂×C₈	—	2	2	no	—
289	C₂×C₈	—	2	2	no	—
290	C₂×C₈	—	4	2	no	—
291	C₂×C₈	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
292	C₂×C₈	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
293	C₂×C₈	C₂²×C₄×C₁₉	1	1	yes	characteristic
294	C₂×C₈	C₂×D₄×C₁₉	1	4	yes	—
295	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
296	C₁₆	—	4	2	no	—
297	C₂³	—	2	2	no	—
298	C₂³	—	2	2	no	—
299	C₂³	—	4	1	no	—
300	C₂³	—	4	4	no	—
301	C₂³	—	4	2	no	—
302	C₂³	—	8	2	no	—
303	C₂³	—	8	2	no	—
304	C₂³	C₁₉×#102	1	1	yes	characteristic
305	C₂³	C₁₉×#102	1	1	yes	characteristic
306	C₂³	C₂×SD₁₆×C₁₉	1	1	yes	characteristic
307	Q₈	—	2	4	no	—
308	Q₈	—	4	4	no	—
309	Q₈	—	4	2	no	—
310	D₄	—	2	4	no	—
311	D₄	—	4	4	no	—
312	D₄	—	4	8	no	—
313	D₄	—	4	8	no	—
314	D₄	—	4	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	4	8	no	—
323	C₂×C₄	—	4	4	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	4	8	no	—
326	C₂×C₄	C₁₉×#102	1	1	yes	characteristic
327	C₂×C₄	C₁₉×#102	1	1	yes	characteristic
328	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
329	C₂×C₄	C₁₉×#120	1	2	yes	—
330	C₂×C₄	C₂×D₈×C₁₉	1	1	yes	characteristic
331	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	2	yes	—
332	C₈	—	2	1	no	—
333	C₈	—	2	1	no	—
334	C₈	—	8	2	no	—
335	C₈	C₂×C₂²⋊C₄×C₁₉	1	2	yes	—
336	C₈	C₁₉×#120	1	2	yes	—
337	C₂²	—	2	1	no	—
338	C₂²	—	2	1	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	4	4	no	—
345	C₂²	—	4	4	no	—
346	C₂²	—	8	8	no	—
347	C₂²	—	8	2	no	—
348	C₂²	—	8	2	no	—
349	C₂²	C₁₉×#359	1	2	yes	complemented
350	C₂²	C₂×C₁₉×#102	1	1	yes	characteristic
351	C₂²	C₁₉×#450	1	2	yes	—
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	4	4	no	—
355	C₄	C₁₉×#379	1	1	yes	characteristic
356	C₄	C₁₉×#418	1	1	yes	characteristic
357	C₄	C₁₉×#447	1	2	yes	—
358	C₂	—	2	2	no	—
359	C₂	—	2	2	no	—
360	C₂	—	4	1	no	—
361	C₂	—	8	4	no	—
362	C₂	C₁₉×#1786	1	1	yes	characteristic
363	C₂	C₂×C₁₉×#359	1	1	yes	characteristic
364	C₂	C₁₉×#2143	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₉	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₁₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₉
socle	C₂²×C₁₉	C₂×#102
prime radical	C₂²×C₁₉×#135	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₉ ◁ C₂³×C₄×C₁₉ ◁ C₂³×C₈×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₉×#359	C₂	semidirect
3. C₂×C₁₉×C₂²⋊C₁₆	C₂	semidirect
4. C₁₉×#1810	C₂	semidirect
15. C₂²×C₁₆×C₁₉	C₂²	semidirect
16. C₂×C₁₉×#108	C₂²	semidirect
114. #12734	C₁₉	direct (N × H)
156. C₂×#359	C₂×C₁₉	semidirect
157. C₂×C₂²⋊C₁₆	C₂×C₁₉	semidirect
158. #1810	C₂×C₁₉	semidirect
171. C₂²×C₁₉	#359	semidirect
190. C₂²×C₁₆	C₂²×C₁₉	semidirect
191. C₂×#108	C₂²×C₁₉	semidirect
243. C₁₉	#12734	direct (N × H)
349. C₂²	C₁₉×#359	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₁₉×#135	2
C₂×C₁₉×#359	2
C₂×C₁₉×C₂²⋊C₁₆	2
C₁₉×#1810	2
#12734	19

Subgroups — C19×#12734 #18018693

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#12734 #18018693