Subgroups — C₁₉×#32681 #18038640

Order 4864.

914 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#2548	C₂	1	2	yes	maximal, complemented
2	C₁₉×#2054	C₂	1	2	yes	maximal, complemented
3	C₁₉×#2011	C₂	1	4	yes	maximal, complemented
4	C₁₉×#1985	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₉×#1972	C₂	1	2	yes	maximal, complemented
6	C₁₉×#1926	C₂	1	4	yes	maximal, complemented
7	C₂²×D₈×C₁₉	C₂²	1	2	yes	—
8	C₂²×D₈×C₁₉	C₂²	1	2	yes	—
9	C₂×C₁₉×#127	C₂²	1	2	yes	—
10	C₂×C₁₉×#124	—	2	2	no	—
11	C₂×C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
12	C₂×C₁₉×#106	C₂²	1	2	yes	—
13	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₁₉×#102	C₂²	1	4	yes	—
15	C₂×C₁₉×#102	C₂²	1	4	yes	—
16	C₂×C₁₉×#97	C₂²	1	1	yes	characteristic
17	C₁₉×#394	—	2	2	no	—
18	C₁₉×#394	—	2	2	no	—
19	C₁₉×#390	—	2	2	no	—
20	C₁₉×#390	C₂²	1	4	yes	—
21	C₁₉×#390	C₂²	1	4	yes	—
22	C₄×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
23	C₁₉×#340	C₂²	1	2	yes	—
24	C₁₉×#336	C₂²	1	4	yes	—
25	C₂²×Q₈×C₁₉	D₄	1	1	yes	characteristic
26	C₂²×D₄×C₁₉	—	2	2	no	—
27	C₂²×D₄×C₁₉	—	2	2	no	—
28	C₂²×D₄×C₁₉	C₂³	1	4	yes	—
29	C₂×D₈×C₁₉	—	2	4	no	—
30	C₂×D₈×C₁₉	—	4	2	no	—
31	C₂×D₈×C₁₉	—	4	2	no	—
32	C₂×D₈×C₁₉	—	4	4	no	—
33	C₂×D₈×C₁₉	—	4	2	no	—
34	C₂²×C₈×C₁₉	C₂³	1	2	yes	—
35	C₂²×C₈×C₁₉	C₂³	1	2	yes	—
36	C₁₉×#127	—	4	2	no	—
37	C₁₉×#127	—	4	2	no	—
38	C₁₉×#124	—	4	8	no	—
39	C₄×Q₈×C₁₉	—	4	1	no	—
40	C₄×Q₈×C₁₉	—	4	1	no	—
41	C₂×C₄⋊C₄×C₁₉	D₄	1	2	yes	—
42	C₂×C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
43	C₂×C₄⋊C₄×C₁₉	C₂³	1	4	yes	—
44	C₂×C₂²⋊C₄×C₁₉	—	2	4	no	—
45	C₂×C₂²⋊C₄×C₁₉	—	2	4	no	—
46	C₂×C₂²⋊C₄×C₁₉	—	2	4	no	—
47	C₂×C₂²⋊C₄×C₁₉	—	2	4	no	—
48	C₂×C₄²×C₁₉	—	2	2	no	—
49	C₂×C₄²×C₁₉	C₂³	1	2	yes	—
50	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
51	C₁₉×#106	—	4	2	no	—
52	C₁₉×C₄⋊C₈	—	4	1	no	—
53	C₁₉×#102	—	2	8	no	—
54	C₁₉×#102	—	4	4	no	—
55	C₁₉×#97	—	4	1	no	—
56	C₁₉×#95	—	2	2	no	—
57	C₂⁴×C₁₉	—	2	4	no	—
58	C₂×Q₈×C₁₉	—	2	4	no	—
59	C₂×Q₈×C₁₉	—	4	1	no	—
60	C₂×D₄×C₁₉	—	2	4	no	—
61	C₂×D₄×C₁₉	—	2	4	no	—
62	C₂×D₄×C₁₉	—	2	4	no	—
63	C₂×D₄×C₁₉	—	2	4	no	—
64	C₂×D₄×C₁₉	—	2	4	no	—
65	C₂×D₄×C₁₉	—	2	4	no	—
66	C₂×D₄×C₁₉	—	4	4	no	—
67	C₂×D₄×C₁₉	—	4	4	no	—
68	C₂×D₄×C₁₉	—	4	4	no	—
69	C₂×D₄×C₁₉	—	4	4	no	—
70	C₂×D₄×C₁₉	—	8	4	no	—
71	C₂²×C₄×C₁₉	—	2	4	no	—
72	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
77	D₈×C₁₉	—	4	8	no	—
78	D₈×C₁₉	—	4	8	no	—
79	C₂×C₈×C₁₉	—	2	2	no	—
80	C₂×C₈×C₁₉	—	2	2	no	—
81	C₂×C₈×C₁₉	—	2	4	no	—
82	C₂×C₈×C₁₉	—	2	2	no	—
83	C₂×C₈×C₁₉	#41	1	4	yes	—
84	C₄⋊C₄×C₁₉	—	2	4	no	—
85	C₄⋊C₄×C₁₉	—	2	4	no	—
86	C₄⋊C₄×C₁₉	—	4	2	no	—
87	C₄⋊C₄×C₁₉	—	4	1	no	—
88	C₂²⋊C₄×C₁₉	—	4	8	no	—
89	C₂²⋊C₄×C₁₉	—	4	8	no	—
90	C₂²⋊C₄×C₁₉	—	8	4	no	—
91	C₂²⋊C₄×C₁₉	—	8	4	no	—
92	C₄²×C₁₉	—	2	2	no	—
93	C₄²×C₁₉	—	2	2	no	—
94	C₄²×C₁₉	—	2	1	no	—
95	C₄²×C₁₉	—	2	1	no	—
96	C₄²×C₁₉	—	4	4	no	—
97	#32681	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
98	C₂³×C₁₉	—	4	4	no	—
99	C₂³×C₁₉	—	4	4	no	—
100	C₂³×C₁₉	—	4	4	no	—
101	C₂³×C₁₉	—	4	4	no	—
102	C₂³×C₁₉	—	4	4	no	—
103	C₂³×C₁₉	—	4	4	no	—
104	C₂³×C₁₉	—	4	4	no	—
105	C₂³×C₁₉	C₂²×D₄	1	1	yes	characteristic
106	Q₈×C₁₉	—	4	4	no	—
107	D₄×C₁₉	—	4	4	no	—
108	D₄×C₁₉	—	4	4	no	—
109	D₄×C₁₉	—	4	4	no	—
110	D₄×C₁₉	—	4	4	no	—
111	D₄×C₁₉	—	8	8	no	—
112	D₄×C₁₉	—	8	8	no	—
113	C₂×C₄×C₁₉	—	2	2	no	—
114	C₂×C₄×C₁₉	—	2	2	no	—
115	C₂×C₄×C₁₉	—	2	1	no	—
116	C₂×C₄×C₁₉	—	2	2	no	—
117	C₂×C₄×C₁₉	—	2	1	no	—
118	C₂×C₄×C₁₉	—	2	2	no	—
119	C₂×C₄×C₁₉	—	2	2	no	—
120	C₂×C₄×C₁₉	—	2	4	no	—
121	C₂×C₄×C₁₉	—	4	4	no	—
122	C₂×C₄×C₁₉	—	4	4	no	—
123	C₂×C₄×C₁₉	—	4	4	no	—
124	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	4	yes	—
125	C₂×C₄×C₁₉	C₂×#41	1	2	yes	—
126	C₂×C₄×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
127	C₂×C₄×C₁₉	H(5,𝔽₂)	1	2	yes	—
128	C₂×C₄×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
129	C₈×C₁₉	—	2	4	no	—
130	C₈×C₁₉	—	4	2	no	—
131	#2548	C₂×C₁₉	1	2	yes	complemented
132	#2054	C₂×C₁₉	1	2	yes	complemented
133	#2011	C₂×C₁₉	1	4	yes	complemented
134	#1985	C₂×C₁₉	1	1	yes	characteristic, complemented
135	#1972	C₂×C₁₉	1	2	yes	complemented
136	#1926	C₂×C₁₉	1	4	yes	complemented
137	C₂²×C₁₉	—	8	4	no	—
138	C₂²×C₁₉	—	8	4	no	—
139	C₂²×C₁₉	—	8	4	no	—
140	C₂²×C₁₉	—	8	4	no	—
141	C₂²×C₁₉	—	8	4	no	—
142	C₂²×C₁₉	—	8	4	no	—
143	C₂²×C₁₉	—	8	4	no	—
144	C₂²×C₁₉	#534	1	1	yes	characteristic
145	C₂²×C₁₉	#550	1	2	yes	—
146	C₂²×C₁₉	C₂×AGL(1,ℤ/8)	1	2	yes	—
147	C₂²×C₁₉	#576	1	2	yes	—
148	C₄×C₁₉	—	2	4	no	—
149	C₄×C₁₉	—	4	1	no	—
150	C₄×C₁₉	—	4	1	no	—
151	C₄×C₁₉	—	4	2	no	—
152	C₄×C₁₉	—	8	4	no	—
153	C₄×C₁₉	#538	1	2	yes	—
154	C₄×C₁₉	#555	1	1	yes	characteristic
155	C₄×C₁₉	H(3,𝔽₄)	1	1	yes	characteristic
156	C₂²×D₈	C₂²×C₁₉	1	2	yes	—
157	C₂²×D₈	C₂²×C₁₉	1	2	yes	—
158	C₂×#127	C₂²×C₁₉	1	2	yes	—
159	C₂×#124	—	2	2	no	—
160	C₂×C₄×Q₈	C₂²×C₁₉	1	1	yes	characteristic
161	C₂×#106	C₂²×C₁₉	1	2	yes	—
162	C₂×C₄⋊C₈	C₂²×C₁₉	1	1	yes	characteristic
163	C₂×#102	C₂²×C₁₉	1	4	yes	—
164	C₂×#102	C₂²×C₁₉	1	4	yes	—
165	C₂×#97	C₂²×C₁₉	1	1	yes	characteristic
166	#394	—	2	2	no	—
167	#394	—	2	2	no	—
168	#390	—	2	2	no	—
169	#390	C₂²×C₁₉	1	4	yes	—
170	#390	C₂²×C₁₉	1	4	yes	—
171	C₄×C₄⋊C₄	C₂²×C₁₉	1	2	yes	—
172	#340	C₂²×C₁₉	1	2	yes	—
173	#336	C₂²×C₁₉	1	4	yes	—
174	C₂×C₁₉	—	16	4	no	—
175	C₂×C₁₉	#2733	1	1	yes	characteristic
176	C₂×C₁₉	#3192	1	1	yes	characteristic
177	C₂×C₁₉	#3199	1	1	yes	characteristic
178	C₂×C₁₉	#3399	1	4	yes	—
179	C₂²×Q₈	D₄×C₁₉	1	1	yes	characteristic
180	C₂²×D₄	—	2	2	no	—
181	C₂²×D₄	—	2	2	no	—
182	C₂²×D₄	C₂³×C₁₉	1	4	yes	—
183	C₂×D₈	—	2	4	no	—
184	C₂×D₈	—	4	2	no	—
185	C₂×D₈	—	4	2	no	—
186	C₂×D₈	—	4	4	no	—
187	C₂×D₈	—	4	2	no	—
188	C₂²×C₈	C₂³×C₁₉	1	2	yes	—
189	C₂²×C₈	C₂³×C₁₉	1	2	yes	—
190	#127	—	4	2	no	—
191	#127	—	4	2	no	—
192	#124	—	4	8	no	—
193	C₄×Q₈	—	4	1	no	—
194	C₄×Q₈	—	4	1	no	—
195	C₂×C₄⋊C₄	D₄×C₁₉	1	2	yes	—
196	C₂×C₄⋊C₄	D₄×C₁₉	1	1	yes	characteristic
197	C₂×C₄⋊C₄	C₂³×C₁₉	1	4	yes	—
198	C₂×C₂²⋊C₄	—	2	4	no	—
199	C₂×C₂²⋊C₄	—	2	4	no	—
200	C₂×C₂²⋊C₄	—	2	4	no	—
201	C₂×C₂²⋊C₄	—	2	4	no	—
202	C₂×C₄²	—	2	2	no	—
203	C₂×C₄²	C₂³×C₁₉	1	2	yes	—
204	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
205	#106	—	4	2	no	—
206	C₄⋊C₈	—	4	1	no	—
207	#102	—	2	8	no	—
208	#102	—	4	4	no	—
209	#97	—	4	1	no	—
210	#95	—	2	2	no	—
211	C₁₉	#32681	1	1	yes	characteristic, direct product factor, Sylow
212	C₂⁴	—	2	4	no	—
213	C₂×Q₈	—	2	4	no	—
214	C₂×Q₈	—	4	1	no	—
215	C₂×D₄	—	2	4	no	—
216	C₂×D₄	—	2	4	no	—
217	C₂×D₄	—	2	4	no	—
218	C₂×D₄	—	2	4	no	—
219	C₂×D₄	—	2	4	no	—
220	C₂×D₄	—	2	4	no	—
221	C₂×D₄	—	4	4	no	—
222	C₂×D₄	—	4	4	no	—
223	C₂×D₄	—	4	4	no	—
224	C₂×D₄	—	4	4	no	—
225	C₂×D₄	—	8	4	no	—
226	C₂²×C₄	—	2	4	no	—
227	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
228	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
229	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
230	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
231	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
232	D₈	—	4	8	no	—
233	D₈	—	4	8	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	#41×C₁₉	1	4	yes	—
239	C₄⋊C₄	—	2	4	no	—
240	C₄⋊C₄	—	2	4	no	—
241	C₄⋊C₄	—	4	2	no	—
242	C₄⋊C₄	—	4	1	no	—
243	C₂²⋊C₄	—	4	8	no	—
244	C₂²⋊C₄	—	4	8	no	—
245	C₂²⋊C₄	—	8	4	no	—
246	C₂²⋊C₄	—	8	4	no	—
247	C₄²	—	2	1	no	—
248	C₄²	—	2	1	no	—
249	C₄²	—	2	2	no	—
250	C₄²	—	2	2	no	—
251	C₄²	—	4	4	no	—
252	C₂³	—	4	4	no	—
253	C₂³	—	4	4	no	—
254	C₂³	—	4	4	no	—
255	C₂³	—	4	4	no	—
256	C₂³	—	4	4	no	—
257	C₂³	—	4	4	no	—
258	C₂³	—	4	4	no	—
259	C₂³	C₂²×D₄×C₁₉	1	1	yes	characteristic
260	Q₈	—	4	4	no	—
261	D₄	—	4	4	no	—
262	D₄	—	4	4	no	—
263	D₄	—	4	4	no	—
264	D₄	—	4	4	no	—
265	D₄	—	8	8	no	—
266	D₄	—	8	8	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	4	yes	—
279	C₂×C₄	C₂×#41×C₁₉	1	2	yes	—
280	C₂×C₄	C₁₉×H(5,𝔽₂)	1	1	yes	characteristic
281	C₂×C₄	C₁₉×H(5,𝔽₂)	1	2	yes	—
282	C₂×C₄	C₁₉×H(5,𝔽₂)	1	1	yes	characteristic
283	C₈	—	2	4	no	—
284	C₈	—	4	2	no	—
285	C₂²	—	8	4	no	—
286	C₂²	—	8	4	no	—
287	C₂²	—	8	4	no	—
288	C₂²	—	8	4	no	—
289	C₂²	—	8	4	no	—
290	C₂²	—	8	4	no	—
291	C₂²	—	8	4	no	—
292	C₂²	C₁₉×#534	1	1	yes	characteristic
293	C₂²	C₁₉×#550	1	2	yes	—
294	C₂²	C₂×C₁₉×AGL(1,ℤ/8)	1	2	yes	—
295	C₂²	C₁₉×#576	1	2	yes	—
296	C₄	—	2	4	no	—
297	C₄	—	4	1	no	—
298	C₄	—	4	2	no	—
299	C₄	—	4	1	no	—
300	C₄	—	8	4	no	—
301	C₄	C₁₉×#538	1	2	yes	—
302	C₄	C₁₉×#555	1	1	yes	characteristic
303	C₄	C₁₉×H(3,𝔽₄)	1	1	yes	characteristic
304	C₂	—	16	4	no	—
305	C₂	C₁₉×#2733	1	1	yes	characteristic
306	C₂	C₁₉×#3192	1	1	yes	characteristic
307	C₂	C₁₉×#3199	1	1	yes	characteristic
308	C₂	C₁₉×#3399	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₉
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₉ ◁ C₂×C₄×Q₈×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₉×#2548	C₂	semidirect
2. C₁₉×#2054	C₂	semidirect
3. C₁₉×#2011	C₂	semidirect
4. C₁₉×#1985	C₂	semidirect
5. C₁₉×#1972	C₂	semidirect
6. C₁₉×#1926	C₂	semidirect
97. #32681	C₁₉	direct (N × H)
131. #2548	C₂×C₁₉	semidirect
132. #2054	C₂×C₁₉	semidirect
133. #2011	C₂×C₁₉	semidirect
134. #1985	C₂×C₁₉	semidirect
135. #1972	C₂×C₁₉	semidirect
136. #1926	C₂×C₁₉	semidirect
211. C₁₉	#32681	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₉×#2548	2
C₁₉×#2054	2
C₁₉×#2011	2
C₁₉×#1985	2
C₁₉×#1972	2
C₁₉×#1926	2
#32681	19

Subgroups — C19×#32681 #18038640

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#32681 #18038640