Subgroups — C₁₉×#3538 #14561395

Order 2432.

1014 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#563	C₂	1	4	yes	maximal, complemented
2	C₁₉×#552	C₂	1	4	yes	maximal, complemented
3	C₁₉×#551	C₂	1	4	yes	maximal, complemented
4	D₄×Q₈×C₁₉	C₂	1	4	yes	maximal, complemented
5	C₁₉×#544	C₂	1	4	yes	maximal, complemented
6	C₁₉×#541	C₂	1	2	yes	maximal, complemented
7	C₁₉×#539	C₂	1	4	yes	maximal, complemented
8	C₁₉×#536	C₂	1	2	yes	maximal, complemented
9	C₁₉×#525	C₂	1	1	yes	maximal, characteristic
10	C₂×C₁₉×C₂²⋊Q₈	C₂	1	2	yes	maximal, complemented
11	C₂×#41×C₁₉	C₂²	1	1	yes	characteristic
12	C₂²×Q₈×C₁₉	C₂²	1	2	yes	—
13	C₂²×Q₈×C₁₉	C₂²	1	2	yes	complemented
14	C₂³×C₄×C₁₉	C₂²	1	1	yes	characteristic
15	C₁₉×C₄⋊Q₈	C₂²	1	4	yes	complemented
16	C₁₉×C₄⋊Q₈	C₂²	1	2	yes	complemented
17	C₁₉×C₄⋊Q₈	C₂²	1	8	yes	complemented
18	C₁₉×#126	C₂²	1	8	yes	—
19	C₁₉×#126	C₂²	1	8	yes	complemented
20	C₁₉×#125	C₂²	1	8	yes	complemented
21	C₁₉×#125	C₂²	1	2	yes	complemented
22	C₁₉×#125	C₂²	1	8	yes	complemented
23	C₁₉×#124	C₂²	1	4	yes	—
24	C₁₉×#123	C₂²	1	2	yes	—
25	C₁₉×#123	C₂²	1	4	yes	—
26	C₁₉×#123	C₂²	1	8	yes	—
27	C₁₉×#123	C₂²	1	4	yes	complemented
28	C₁₉×C₂²⋊Q₈	C₂²	1	4	yes	—
29	C₁₉×C₂²⋊Q₈	C₂²	1	4	yes	—
30	C₁₉×C₂²⋊Q₈	C₂²	1	4	yes	—
31	C₁₉×C₂²⋊Q₈	C₂²	1	4	yes	—
32	C₁₉×C₂²⋊Q₈	C₂²	1	2	yes	—
33	C₁₉×C₂²⋊Q₈	C₂²	1	8	yes	—
34	C₁₉×C₂²⋊Q₈	C₂²	1	4	yes	complemented
35	C₁₉×C₂²⋊Q₈	C₂²	1	8	yes	complemented
36	C₁₉×#121	C₂²	1	2	yes	—
37	C₁₉×#120	C₂²	1	2	yes	—
38	C₄×Q₈×C₁₉	C₂²	1	4	yes	complemented
39	C₄×D₄×C₁₉	C₂²	1	4	yes	—
40	C₄×D₄×C₁₉	C₂²	1	8	yes	—
41	C₁₉×#117	C₂²	1	1	yes	characteristic
42	C₁₉×#117	C₂²	1	8	yes	—
43	C₁₉×#117	C₂²	1	2	yes	complemented
44	C₂×C₄⋊C₄×C₁₉	C₂²	1	4	yes	—
45	C₂×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
46	C₂×C₂²⋊C₄×C₁₉	C₂²	1	4	yes	—
47	C₂⁴×C₁₉	C₂³	1	1	yes	characteristic
48	#41×C₁₉	—	4	2	no	—
49	C₂×Q₈×C₁₉	—	4	4	no	—
50	C₂×Q₈×C₁₉	—	4	4	no	—
51	C₂×Q₈×C₁₉	C₂³	1	4	yes	—
52	C₂×Q₈×C₁₉	C₂³	1	4	yes	—
53	C₂×Q₈×C₁₉	C₂³	1	4	yes	—
54	C₂×Q₈×C₁₉	C₂³	1	1	yes	characteristic
55	C₂×D₄×C₁₉	C₂³	1	4	yes	—
56	C₂×D₄×C₁₉	C₂³	1	2	yes	—
57	C₂×D₄×C₁₉	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₁₉	—	4	2	no	—
59	C₂²×C₄×C₁₉	C₂³	1	4	yes	—
60	C₂²×C₄×C₁₉	C₂³	1	2	yes	—
61	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	C₂³	1	2	yes	—
63	C₂²×C₄×C₁₉	C₂³	1	4	yes	—
64	C₂²×C₄×C₁₉	C₂³	1	2	yes	—
65	C₂²×C₄×C₁₉	C₂³	1	4	yes	—
66	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
67	C₄⋊C₄×C₁₉	C₂³	1	4	yes	—
68	C₄⋊C₄×C₁₉	C₂³	1	8	yes	—
69	C₄⋊C₄×C₁₉	C₂³	1	8	yes	—
70	C₄⋊C₄×C₁₉	C₂³	1	4	yes	—
71	C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
72	C₄⋊C₄×C₁₉	C₂³	1	8	yes	—
73	C₄⋊C₄×C₁₉	C₂³	1	8	yes	—
74	C₄⋊C₄×C₁₉	C₂³	1	8	yes	—
75	C₄⋊C₄×C₁₉	C₂³	1	4	yes	—
76	C₄⋊C₄×C₁₉	C₂³	1	8	yes	—
77	C₂²⋊C₄×C₁₉	C₂³	1	8	yes	—
78	C₂²⋊C₄×C₁₉	C₂³	1	8	yes	—
79	C₂²⋊C₄×C₁₉	C₂³	1	4	yes	—
80	C₂²⋊C₄×C₁₉	C₂³	1	2	yes	—
81	C₂²⋊C₄×C₁₉	C₂³	1	4	yes	—
82	C₂²⋊C₄×C₁₉	C₂³	1	4	yes	—
83	C₂²⋊C₄×C₁₉	C₂³	1	8	yes	—
84	C₄²×C₁₉	C₂³	1	4	yes	—
85	C₄²×C₁₉	C₂³	1	2	yes	—
86	C₄²×C₁₉	C₂³	1	8	yes	—
87	C₂³×C₁₉	—	2	4	no	—
88	C₂³×C₁₉	—	4	1	no	—
89	C₂³×C₁₉	C₂⁴	1	1	yes	characteristic
90	C₂³×C₁₉	C₂⁴	1	2	yes	—
91	C₂³×C₁₉	C₂⁴	1	2	yes	—
92	Q₈×C₁₉	—	2	8	no	—
93	Q₈×C₁₉	—	4	4	no	—
94	Q₈×C₁₉	—	4	4	no	—
95	Q₈×C₁₉	—	4	1	no	—
96	D₄×C₁₉	—	2	8	no	—
97	D₄×C₁₉	—	4	1	no	—
98	D₄×C₁₉	—	4	2	no	—
99	C₂×C₄×C₁₉	—	2	8	no	—
100	C₂×C₄×C₁₉	—	2	4	no	—
101	C₂×C₄×C₁₉	—	2	2	no	—
102	C₂×C₄×C₁₉	—	2	4	no	—
103	C₂×C₄×C₁₉	—	4	2	no	—
104	C₂×C₄×C₁₉	—	4	1	no	—
105	C₂×C₄×C₁₉	—	4	4	no	—
106	C₂×C₄×C₁₉	—	4	4	no	—
107	C₂×C₄×C₁₉	C₂⁴	1	4	yes	—
108	C₂×C₄×C₁₉	C₂⁴	1	8	yes	—
109	C₂×C₄×C₁₉	C₂⁴	1	2	yes	—
110	C₂×C₄×C₁₉	C₂⁴	1	8	yes	—
111	C₂×C₄×C₁₉	C₂⁴	1	2	yes	—
112	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
113	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
114	#3538	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
115	C₂²×C₁₉	—	2	2	no	—
116	C₂²×C₁₉	—	2	2	no	—
117	C₂²×C₁₉	—	2	2	no	—
118	C₂²×C₁₉	—	2	1	no	—
119	C₂²×C₁₉	—	2	4	no	—
120	C₂²×C₁₉	—	2	2	no	—
121	C₂²×C₁₉	—	4	4	no	—
122	C₂²×C₁₉	2₋¹⁺⁴	1	4	yes	—
123	C₂²×C₁₉	C₂⁵	1	1	yes	characteristic
124	C₄×C₁₉	—	2	1	no	—
125	C₄×C₁₉	—	2	2	no	—
126	C₄×C₁₉	—	2	2	no	—
127	C₄×C₁₉	—	2	8	no	—
128	C₄×C₁₉	—	2	8	no	—
129	C₄×C₁₉	—	2	4	no	—
130	C₄×C₁₉	H(5,𝔽₂)	1	2	yes	—
131	#563	C₂×C₁₉	1	4	yes	complemented
132	#552	C₂×C₁₉	1	4	yes	complemented
133	#551	C₂×C₁₉	1	4	yes	complemented
134	D₄×Q₈	C₂×C₁₉	1	4	yes	complemented
135	#544	C₂×C₁₉	1	4	yes	complemented
136	#541	C₂×C₁₉	1	2	yes	complemented
137	#539	C₂×C₁₉	1	4	yes	complemented
138	#536	C₂×C₁₉	1	2	yes	complemented
139	#525	C₂×C₁₉	1	1	yes	characteristic
140	C₂×C₂²⋊Q₈	C₂×C₁₉	1	2	yes	complemented
141	C₂×C₁₉	—	2	4	no	—
142	C₂×C₁₉	—	4	2	no	—
143	C₂×C₁₉	—	4	1	no	—
144	C₂×C₁₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
145	C₂×C₁₉	C₂×2₋¹⁺⁴	1	2	yes	—
146	C₂×#41	C₂²×C₁₉	1	1	yes	characteristic
147	C₂²×Q₈	C₂²×C₁₉	1	2	yes	—
148	C₂²×Q₈	C₂²×C₁₉	1	2	yes	complemented
149	C₂³×C₄	C₂²×C₁₉	1	1	yes	characteristic
150	C₄⋊Q₈	C₂²×C₁₉	1	4	yes	complemented
151	C₄⋊Q₈	C₂²×C₁₉	1	2	yes	complemented
152	C₄⋊Q₈	C₂²×C₁₉	1	8	yes	complemented
153	#126	C₂²×C₁₉	1	8	yes	—
154	#126	C₂²×C₁₉	1	8	yes	complemented
155	#125	C₂²×C₁₉	1	8	yes	complemented
156	#125	C₂²×C₁₉	1	2	yes	complemented
157	#125	C₂²×C₁₉	1	8	yes	complemented
158	#124	C₂²×C₁₉	1	4	yes	—
159	#123	C₂²×C₁₉	1	2	yes	—
160	#123	C₂²×C₁₉	1	4	yes	—
161	#123	C₂²×C₁₉	1	8	yes	—
162	#123	C₂²×C₁₉	1	4	yes	complemented
163	C₂²⋊Q₈	C₂²×C₁₉	1	4	yes	—
164	C₂²⋊Q₈	C₂²×C₁₉	1	4	yes	—
165	C₂²⋊Q₈	C₂²×C₁₉	1	4	yes	—
166	C₂²⋊Q₈	C₂²×C₁₉	1	4	yes	—
167	C₂²⋊Q₈	C₂²×C₁₉	1	2	yes	—
168	C₂²⋊Q₈	C₂²×C₁₉	1	8	yes	—
169	C₂²⋊Q₈	C₂²×C₁₉	1	4	yes	complemented
170	C₂²⋊Q₈	C₂²×C₁₉	1	8	yes	complemented
171	#121	C₂²×C₁₉	1	2	yes	—
172	#120	C₂²×C₁₉	1	2	yes	—
173	C₄×Q₈	C₂²×C₁₉	1	4	yes	complemented
174	C₄×D₄	C₂²×C₁₉	1	4	yes	—
175	C₄×D₄	C₂²×C₁₉	1	8	yes	—
176	#117	C₂²×C₁₉	1	1	yes	characteristic
177	#117	C₂²×C₁₉	1	8	yes	—
178	#117	C₂²×C₁₉	1	2	yes	complemented
179	C₂×C₄⋊C₄	C₂²×C₁₉	1	4	yes	—
180	C₂×C₄⋊C₄	C₂²×C₁₉	1	2	yes	—
181	C₂×C₂²⋊C₄	C₂²×C₁₉	1	4	yes	—
182	C₁₉	#3538	1	1	yes	characteristic, direct product factor, Sylow
183	C₂⁴	C₂³×C₁₉	1	1	yes	characteristic
184	#41	—	4	2	no	—
185	C₂×Q₈	—	4	4	no	—
186	C₂×Q₈	—	4	4	no	—
187	C₂×Q₈	C₂³×C₁₉	1	4	yes	—
188	C₂×Q₈	C₂³×C₁₉	1	4	yes	—
189	C₂×Q₈	C₂³×C₁₉	1	4	yes	—
190	C₂×Q₈	C₂³×C₁₉	1	1	yes	characteristic
191	C₂×D₄	C₂³×C₁₉	1	4	yes	—
192	C₂×D₄	C₂³×C₁₉	1	2	yes	—
193	C₂×D₄	C₂³×C₁₉	1	1	yes	characteristic
194	C₂²×C₄	—	4	2	no	—
195	C₂²×C₄	C₂³×C₁₉	1	4	yes	—
196	C₂²×C₄	C₂³×C₁₉	1	2	yes	—
197	C₂²×C₄	C₂³×C₁₉	1	1	yes	characteristic
198	C₂²×C₄	C₂³×C₁₉	1	2	yes	—
199	C₂²×C₄	C₂³×C₁₉	1	4	yes	—
200	C₂²×C₄	C₂³×C₁₉	1	2	yes	—
201	C₂²×C₄	C₂³×C₁₉	1	4	yes	—
202	C₂²×C₄	C₂³×C₁₉	1	1	yes	characteristic
203	C₄⋊C₄	C₂³×C₁₉	1	4	yes	—
204	C₄⋊C₄	C₂³×C₁₉	1	8	yes	—
205	C₄⋊C₄	C₂³×C₁₉	1	8	yes	—
206	C₄⋊C₄	C₂³×C₁₉	1	4	yes	—
207	C₄⋊C₄	C₂³×C₁₉	1	2	yes	—
208	C₄⋊C₄	C₂³×C₁₉	1	8	yes	—
209	C₄⋊C₄	C₂³×C₁₉	1	8	yes	—
210	C₄⋊C₄	C₂³×C₁₉	1	8	yes	—
211	C₄⋊C₄	C₂³×C₁₉	1	4	yes	—
212	C₄⋊C₄	C₂³×C₁₉	1	8	yes	—
213	C₂²⋊C₄	C₂³×C₁₉	1	8	yes	—
214	C₂²⋊C₄	C₂³×C₁₉	1	8	yes	—
215	C₂²⋊C₄	C₂³×C₁₉	1	4	yes	—
216	C₂²⋊C₄	C₂³×C₁₉	1	2	yes	—
217	C₂²⋊C₄	C₂³×C₁₉	1	4	yes	—
218	C₂²⋊C₄	C₂³×C₁₉	1	4	yes	—
219	C₂²⋊C₄	C₂³×C₁₉	1	8	yes	—
220	C₄²	C₂³×C₁₉	1	4	yes	—
221	C₄²	C₂³×C₁₉	1	2	yes	—
222	C₄²	C₂³×C₁₉	1	8	yes	—
223	C₂³	—	2	4	no	—
224	C₂³	—	4	1	no	—
225	C₂³	C₂⁴×C₁₉	1	1	yes	characteristic
226	C₂³	C₂⁴×C₁₉	1	2	yes	—
227	C₂³	C₂⁴×C₁₉	1	2	yes	—
228	Q₈	—	2	8	no	—
229	Q₈	—	4	4	no	—
230	Q₈	—	4	4	no	—
231	Q₈	—	4	1	no	—
232	D₄	—	2	8	no	—
233	D₄	—	4	1	no	—
234	D₄	—	4	2	no	—
235	C₂×C₄	—	2	8	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	4	1	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	C₂⁴×C₁₉	1	4	yes	—
244	C₂×C₄	C₂⁴×C₁₉	1	8	yes	—
245	C₂×C₄	C₂⁴×C₁₉	1	2	yes	—
246	C₂×C₄	C₂⁴×C₁₉	1	8	yes	—
247	C₂×C₄	C₂⁴×C₁₉	1	2	yes	—
248	C₂×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
249	C₂×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
250	C₂²	—	2	2	no	—
251	C₂²	—	2	2	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	4	no	—
255	C₂²	—	2	2	no	—
256	C₂²	—	4	4	no	—
257	C₂²	C₁₉×2₋¹⁺⁴	1	4	yes	—
258	C₂²	C₂⁵×C₁₉	1	1	yes	characteristic
259	C₄	—	2	1	no	—
260	C₄	—	2	2	no	—
261	C₄	—	2	2	no	—
262	C₄	—	2	8	no	—
263	C₄	—	2	8	no	—
264	C₄	—	2	4	no	—
265	C₄	C₁₉×H(5,𝔽₂)	1	2	yes	—
266	C₂	—	2	4	no	—
267	C₂	—	4	1	no	—
268	C₂	—	4	2	no	—
269	C₂	C₂×C₁₉×H(5,𝔽₂)	1	1	yes	characteristic
270	C₂	C₂×C₁₉×2₋¹⁺⁴	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁵×C₁₉
prime radical	C₁₉×#525	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₉×#563	C₂	semidirect
2. C₁₉×#552	C₂	semidirect
3. C₁₉×#551	C₂	semidirect
4. D₄×Q₈×C₁₉	C₂	semidirect
5. C₁₉×#544	C₂	semidirect
6. C₁₉×#541	C₂	semidirect
7. C₁₉×#539	C₂	semidirect
8. C₁₉×#536	C₂	semidirect
10. C₂×C₁₉×C₂²⋊Q₈	C₂	semidirect
13. C₂²×Q₈×C₁₉	C₂²	semidirect
15. C₁₉×C₄⋊Q₈	C₂²	semidirect
16. C₁₉×C₄⋊Q₈	C₂²	semidirect
17. C₁₉×C₄⋊Q₈	C₂²	semidirect
19. C₁₉×#126	C₂²	semidirect
20. C₁₉×#125	C₂²	semidirect
21. C₁₉×#125	C₂²	semidirect
22. C₁₉×#125	C₂²	semidirect
27. C₁₉×#123	C₂²	semidirect
34. C₁₉×C₂²⋊Q₈	C₂²	semidirect
35. C₁₉×C₂²⋊Q₈	C₂²	semidirect
38. C₄×Q₈×C₁₉	C₂²	semidirect
43. C₁₉×#117	C₂²	semidirect
114. #3538	C₁₉	direct (N × H)
131. #563	C₂×C₁₉	semidirect
132. #552	C₂×C₁₉	semidirect
133. #551	C₂×C₁₉	semidirect
134. D₄×Q₈	C₂×C₁₉	semidirect
135. #544	C₂×C₁₉	semidirect
136. #541	C₂×C₁₉	semidirect
137. #539	C₂×C₁₉	semidirect
138. #536	C₂×C₁₉	semidirect
140. C₂×C₂²⋊Q₈	C₂×C₁₉	semidirect
148. C₂²×Q₈	C₂²×C₁₉	semidirect
150. C₄⋊Q₈	C₂²×C₁₉	semidirect
151. C₄⋊Q₈	C₂²×C₁₉	semidirect
152. C₄⋊Q₈	C₂²×C₁₉	semidirect
154. #126	C₂²×C₁₉	semidirect
155. #125	C₂²×C₁₉	semidirect
156. #125	C₂²×C₁₉	semidirect
157. #125	C₂²×C₁₉	semidirect
162. #123	C₂²×C₁₉	semidirect
169. C₂²⋊Q₈	C₂²×C₁₉	semidirect
170. C₂²⋊Q₈	C₂²×C₁₉	semidirect
173. C₄×Q₈	C₂²×C₁₉	semidirect
178. #117	C₂²×C₁₉	semidirect
182. C₁₉	#3538	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×#563	2
C₁₉×#552	2
C₁₉×#551	2
D₄×Q₈×C₁₉	2
C₁₉×#544	2
C₁₉×#541	2
C₁₉×#539	2
C₁₉×#536	2
C₁₉×#525	2
C₂×C₁₉×C₂²⋊Q₈	2
#3538	19

Subgroups — C19×#3538 #14561395

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#3538 #14561395