Subgroups — C₁₉×#34177 #18040136

Order 4864.

2446 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	C₂²×C₁₉×#117	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2343	C₂	1	16	yes	maximal, complemented
3	C₁₉×#2315	C₂	1	2	yes	maximal
4	C₁₉×#2313	C₂	1	4	yes	maximal, complemented
5	C₁₉×#2283	C₂	1	4	yes	maximal, complemented
6	C₂×C₁₉×#381	C₂	1	2	yes	maximal
7	C₁₉×#2273	C₂	1	2	yes	maximal
8	C₂⁴×C₄×C₁₉	C₂²	1	1	yes	characteristic
9	C₂×C₁₉×#117	C₄	1	16	yes	complemented
10	C₂×C₁₉×#117	C₂²	1	8	yes	—
11	C₂²×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
12	C₂²×C₂²⋊C₄×C₁₉	C₂²	1	4	yes	—
13	C₂²×C₂²⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
14	C₂²×C₂²⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
15	C₂²×C₄²×C₁₉	C₂²	1	2	yes	—
16	C₁₉×#387	C₂²	1	32	yes	complemented
17	C₁₉×#385	C₂²	1	16	yes	—
18	C₁₉×#383	C₂²	1	8	yes	complemented
19	C₁₉×#382	C₂²	1	8	yes	complemented
20	C₁₉×#381	C₂²	1	4	yes	—
21	C₁₉×#381	C₂²	1	4	yes	—
22	C₁₉×#381	C₂²	1	8	yes	—
23	C₁₉×#380	C₂²	1	8	yes	—
24	C₄×C₄⋊C₄×C₁₉	C₂²	1	8	yes	complemented
25	C₄×C₂²⋊C₄×C₁₉	C₂²	1	8	yes	—
26	C₄×C₂²⋊C₄×C₁₉	C₂²	1	16	yes	—
27	C₁₉×#376	C₂²	1	8	yes	complemented
28	C₂×C₁₉×#95	C₂²	1	8	yes	—
29	C₂⁵×C₁₉	C₂³	1	1	yes	characteristic
30	C₂³×C₄×C₁₉	—	2	8	no	—
31	C₂³×C₄×C₁₉	C₂×C₄	1	8	yes	—
32	C₂³×C₄×C₁₉	C₂³	1	4	yes	—
33	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
34	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
35	C₂³×C₄×C₁₉	C₂³	1	8	yes	—
36	C₂³×C₄×C₁₉	C₂³	1	2	yes	—
37	C₂³×C₄×C₁₉	C₂³	1	2	yes	—
38	C₁₉×#117	C₂×C₄	1	64	yes	complemented
39	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	16	yes	complemented
40	C₂×C₄⋊C₄×C₁₉	C₂³	1	16	yes	—
41	C₂×C₄⋊C₄×C₁₉	C₂³	1	8	yes	—
42	C₂×C₂²⋊C₄×C₁₉	—	2	16	no	—
43	C₂×C₂²⋊C₄×C₁₉	C₂×C₄	1	8	yes	—
44	C₂×C₂²⋊C₄×C₁₉	C₂×C₄	1	8	yes	—
45	C₂×C₂²⋊C₄×C₁₉	C₂³	1	8	yes	—
46	C₂×C₂²⋊C₄×C₁₉	C₂³	1	8	yes	—
47	C₂×C₂²⋊C₄×C₁₉	C₂³	1	4	yes	—
48	C₂×C₂²⋊C₄×C₁₉	C₂³	1	4	yes	—
49	C₂×C₄²×C₁₉	C₂×C₄	1	16	yes	complemented
50	C₂×C₄²×C₁₉	C₂³	1	16	yes	—
51	C₂×C₄²×C₁₉	C₂³	1	4	yes	—
52	C₂×C₄²×C₁₉	C₂³	1	4	yes	—
53	C₁₉×#95	C₂³	1	16	yes	—
54	C₁₉×#95	C₂³	1	16	yes	—
55	C₁₉×#95	C₂³	1	16	yes	—
56	C₁₉×#95	C₂³	1	16	yes	—
57	C₂⁴×C₁₉	—	4	4	no	—
58	C₂⁴×C₁₉	C₂²×C₄	1	4	yes	—
59	C₂⁴×C₁₉	#41	1	8	yes	—
60	C₂⁴×C₁₉	C₂⁴	1	2	yes	—
61	C₂⁴×C₁₉	C₂⁴	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	—	2	32	no	—
63	C₂²×C₄×C₁₉	—	2	4	no	—
64	C₂²×C₄×C₁₉	—	2	4	no	—
65	C₂²×C₄×C₁₉	—	2	16	no	—
66	C₂²×C₄×C₁₉	—	2	16	no	—
67	C₂²×C₄×C₁₉	—	4	4	no	—
68	C₂²×C₄×C₁₉	—	4	4	no	—
69	C₂²×C₄×C₁₉	—	4	16	no	—
70	C₂²×C₄×C₁₉	C₂²×C₄	1	4	yes	—
71	C₂²×C₄×C₁₉	C₂²×C₄	1	4	yes	—
72	C₂²×C₄×C₁₉	C₂²×C₄	1	16	yes	—
73	C₂²×C₄×C₁₉	C₂²×C₄	1	8	yes	—
74	C₂²×C₄×C₁₉	C₂²×C₄	1	8	yes	—
75	C₂²×C₄×C₁₉	C₂⁴	1	16	yes	—
76	C₂²×C₄×C₁₉	C₂⁴	1	4	yes	—
77	C₂²×C₄×C₁₉	C₂⁴	1	2	yes	—
78	C₂²×C₄×C₁₉	C₂⁴	1	4	yes	—
79	C₂²×C₄×C₁₉	C₂⁴	1	2	yes	—
80	C₄⋊C₄×C₁₉	—	4	16	no	—
81	C₄⋊C₄×C₁₉	C₂²×C₄	1	32	yes	—
82	C₂²⋊C₄×C₁₉	—	2	16	no	—
83	C₂²⋊C₄×C₁₉	—	2	16	no	—
84	C₂²⋊C₄×C₁₉	C₂²×C₄	1	16	yes	—
85	C₂²⋊C₄×C₁₉	C₂²×C₄	1	16	yes	—
86	C₄²×C₁₉	—	4	16	no	—
87	C₄²×C₁₉	C₂²×C₄	1	16	yes	—
88	C₄²×C₁₉	C₂²×C₄	1	16	yes	—
89	#34177	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂³×C₁₉	—	2	4	no	—
91	C₂³×C₁₉	—	2	8	no	—
92	C₂³×C₁₉	—	4	8	no	—
93	C₂³×C₁₉	—	4	16	no	—
94	C₂³×C₁₉	#117	1	16	yes	—
95	C₂³×C₁₉	C₂³×C₄	1	4	yes	—
96	C₂³×C₁₉	C₂³×C₄	1	2	yes	—
97	C₂³×C₁₉	C₂×#41	1	4	yes	—
98	C₂³×C₁₉	C₂×#41	1	8	yes	—
99	C₂³×C₁₉	C₂⁵	1	1	yes	characteristic
100	C₂×C₄×C₁₉	—	2	8	no	—
101	C₂×C₄×C₁₉	—	2	8	no	—
102	C₂×C₄×C₁₉	—	2	32	no	—
103	C₂×C₄×C₁₉	—	2	4	no	—
104	C₂×C₄×C₁₉	—	2	4	no	—
105	C₂×C₄×C₁₉	—	2	8	no	—
106	C₂×C₄×C₁₉	—	2	8	no	—
107	C₂×C₄×C₁₉	—	2	16	no	—
108	C₂×C₄×C₁₉	—	2	32	no	—
109	C₂×C₄×C₁₉	—	4	8	no	—
110	C₂×C₄×C₁₉	—	4	8	no	—
111	C₂×C₄×C₁₉	—	8	16	no	—
112	C₂×C₄×C₁₉	C₂³×C₄	1	4	yes	—
113	C₂×C₄×C₁₉	C₂³×C₄	1	8	yes	—
114	C₂×C₄×C₁₉	C₂³×C₄	1	8	yes	—
115	C₂×C₄×C₁₉	C₂³×C₄	1	4	yes	—
116	C₂²×#117	C₂×C₁₉	1	1	yes	characteristic
117	#2343	C₂×C₁₉	1	16	yes	complemented
118	#2315	C₂×C₁₉	1	2	yes	—
119	#2313	C₂×C₁₉	1	4	yes	complemented
120	#2283	C₂×C₁₉	1	4	yes	complemented
121	C₂×#381	C₂×C₁₉	1	2	yes	—
122	#2273	C₂×C₁₉	1	2	yes	—
123	C₂²×C₁₉	—	2	4	no	—
124	C₂²×C₁₉	—	2	8	no	—
125	C₂²×C₁₉	—	2	16	no	—
126	C₂²×C₁₉	—	2	8	no	—
127	C₂²×C₁₉	—	4	16	no	—
128	C₂²×C₁₉	C₂×#117	1	8	yes	—
129	C₂²×C₁₉	C₂×#117	1	4	yes	—
130	C₂²×C₁₉	C₂⁴×C₄	1	1	yes	characteristic
131	C₂²×C₁₉	C₂²×#41	1	2	yes	—
132	C₂²×C₁₉	#585	1	4	yes	—
133	C₄×C₁₉	—	2	4	no	—
134	C₄×C₁₉	—	2	4	no	—
135	C₄×C₁₉	—	2	8	no	—
136	C₄×C₁₉	—	2	8	no	—
137	C₄×C₁₉	—	4	16	no	—
138	C₂⁴×C₄	C₂²×C₁₉	1	1	yes	characteristic
139	C₂×#117	C₄×C₁₉	1	16	yes	complemented
140	C₂×#117	C₂²×C₁₉	1	8	yes	—
141	C₂²×C₄⋊C₄	C₂²×C₁₉	1	2	yes	—
142	C₂²×C₂²⋊C₄	C₂²×C₁₉	1	4	yes	—
143	C₂²×C₂²⋊C₄	C₂²×C₁₉	1	1	yes	characteristic
144	C₂²×C₂²⋊C₄	C₂²×C₁₉	1	1	yes	characteristic
145	C₂²×C₄²	C₂²×C₁₉	1	2	yes	—
146	#387	C₂²×C₁₉	1	32	yes	complemented
147	#385	C₂²×C₁₉	1	16	yes	—
148	#383	C₂²×C₁₉	1	8	yes	complemented
149	#382	C₂²×C₁₉	1	8	yes	complemented
150	#381	C₂²×C₁₉	1	4	yes	—
151	#381	C₂²×C₁₉	1	8	yes	—
152	#381	C₂²×C₁₉	1	4	yes	—
153	#380	C₂²×C₁₉	1	8	yes	—
154	C₄×C₄⋊C₄	C₂²×C₁₉	1	8	yes	complemented
155	C₄×C₂²⋊C₄	C₂²×C₁₉	1	8	yes	—
156	C₄×C₂²⋊C₄	C₂²×C₁₉	1	16	yes	—
157	#376	C₂²×C₁₉	1	8	yes	complemented
158	C₂×#95	C₂²×C₁₉	1	8	yes	—
159	C₂×C₁₉	—	2	4	no	—
160	C₂×C₁₉	—	2	8	no	—
161	C₂×C₁₉	C₂²×#117	1	1	yes	characteristic
162	C₂×C₁₉	#3428	1	2	yes	—
163	C₂×C₁₉	#3491	1	4	yes	—
164	C₂⁵	C₂³×C₁₉	1	1	yes	characteristic
165	C₂³×C₄	—	2	8	no	—
166	C₂³×C₄	C₂×C₄×C₁₉	1	8	yes	—
167	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
168	C₂³×C₄	C₂³×C₁₉	1	4	yes	—
169	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
170	C₂³×C₄	C₂³×C₁₉	1	8	yes	—
171	C₂³×C₄	C₂³×C₁₉	1	2	yes	—
172	C₂³×C₄	C₂³×C₁₉	1	2	yes	—
173	#117	C₂×C₄×C₁₉	1	64	yes	complemented
174	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	16	yes	complemented
175	C₂×C₄⋊C₄	C₂³×C₁₉	1	16	yes	—
176	C₂×C₄⋊C₄	C₂³×C₁₉	1	8	yes	—
177	C₂×C₂²⋊C₄	—	2	16	no	—
178	C₂×C₂²⋊C₄	C₂×C₄×C₁₉	1	8	yes	—
179	C₂×C₂²⋊C₄	C₂×C₄×C₁₉	1	8	yes	—
180	C₂×C₂²⋊C₄	C₂³×C₁₉	1	8	yes	—
181	C₂×C₂²⋊C₄	C₂³×C₁₉	1	8	yes	—
182	C₂×C₂²⋊C₄	C₂³×C₁₉	1	4	yes	—
183	C₂×C₂²⋊C₄	C₂³×C₁₉	1	4	yes	—
184	C₂×C₄²	C₂×C₄×C₁₉	1	16	yes	complemented
185	C₂×C₄²	C₂³×C₁₉	1	16	yes	—
186	C₂×C₄²	C₂³×C₁₉	1	4	yes	—
187	C₂×C₄²	C₂³×C₁₉	1	4	yes	—
188	#95	C₂³×C₁₉	1	16	yes	—
189	#95	C₂³×C₁₉	1	16	yes	—
190	#95	C₂³×C₁₉	1	16	yes	—
191	#95	C₂³×C₁₉	1	16	yes	—
192	C₁₉	#34177	1	1	yes	characteristic, direct product factor, Sylow
193	C₂⁴	—	4	4	no	—
194	C₂⁴	C₂²×C₄×C₁₉	1	4	yes	—
195	C₂⁴	#41×C₁₉	1	8	yes	—
196	C₂⁴	C₂⁴×C₁₉	1	2	yes	—
197	C₂⁴	C₂⁴×C₁₉	1	1	yes	characteristic
198	C₂²×C₄	—	2	32	no	—
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	—	2	16	no	—
202	C₂²×C₄	—	2	16	no	—
203	C₂²×C₄	—	4	4	no	—
204	C₂²×C₄	—	4	4	no	—
205	C₂²×C₄	—	4	16	no	—
206	C₂²×C₄	C₂²×C₄×C₁₉	1	4	yes	—
207	C₂²×C₄	C₂²×C₄×C₁₉	1	4	yes	—
208	C₂²×C₄	C₂²×C₄×C₁₉	1	16	yes	—
209	C₂²×C₄	C₂²×C₄×C₁₉	1	8	yes	—
210	C₂²×C₄	C₂²×C₄×C₁₉	1	8	yes	—
211	C₂²×C₄	C₂⁴×C₁₉	1	16	yes	—
212	C₂²×C₄	C₂⁴×C₁₉	1	2	yes	—
213	C₂²×C₄	C₂⁴×C₁₉	1	2	yes	—
214	C₂²×C₄	C₂⁴×C₁₉	1	4	yes	—
215	C₂²×C₄	C₂⁴×C₁₉	1	4	yes	—
216	C₄⋊C₄	—	4	16	no	—
217	C₄⋊C₄	C₂²×C₄×C₁₉	1	32	yes	—
218	C₂²⋊C₄	—	2	16	no	—
219	C₂²⋊C₄	—	2	16	no	—
220	C₂²⋊C₄	C₂²×C₄×C₁₉	1	16	yes	—
221	C₂²⋊C₄	C₂²×C₄×C₁₉	1	16	yes	—
222	C₄²	—	4	16	no	—
223	C₄²	C₂²×C₄×C₁₉	1	16	yes	—
224	C₄²	C₂²×C₄×C₁₉	1	16	yes	—
225	C₂³	—	2	4	no	—
226	C₂³	—	2	8	no	—
227	C₂³	—	4	8	no	—
228	C₂³	—	4	16	no	—
229	C₂³	C₁₉×#117	1	16	yes	—
230	C₂³	C₂³×C₄×C₁₉	1	4	yes	—
231	C₂³	C₂³×C₄×C₁₉	1	2	yes	—
232	C₂³	C₂×#41×C₁₉	1	4	yes	—
233	C₂³	C₂×#41×C₁₉	1	8	yes	—
234	C₂³	C₂⁵×C₁₉	1	1	yes	characteristic
235	C₂×C₄	—	2	8	no	—
236	C₂×C₄	—	2	8	no	—
237	C₂×C₄	—	2	32	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	2	8	no	—
241	C₂×C₄	—	2	8	no	—
242	C₂×C₄	—	2	16	no	—
243	C₂×C₄	—	2	32	no	—
244	C₂×C₄	—	4	8	no	—
245	C₂×C₄	—	4	8	no	—
246	C₂×C₄	—	8	16	no	—
247	C₂×C₄	C₂³×C₄×C₁₉	1	4	yes	—
248	C₂×C₄	C₂³×C₄×C₁₉	1	8	yes	—
249	C₂×C₄	C₂³×C₄×C₁₉	1	8	yes	—
250	C₂×C₄	C₂³×C₄×C₁₉	1	4	yes	—
251	C₂²	—	2	4	no	—
252	C₂²	—	2	8	no	—
253	C₂²	—	2	16	no	—
254	C₂²	—	2	8	no	—
255	C₂²	—	4	16	no	—
256	C₂²	C₂×C₁₉×#117	1	8	yes	—
257	C₂²	C₂×C₁₉×#117	1	4	yes	—
258	C₂²	C₂⁴×C₄×C₁₉	1	1	yes	characteristic
259	C₂²	C₂²×#41×C₁₉	1	2	yes	—
260	C₂²	C₁₉×#585	1	4	yes	—
261	C₄	—	2	8	no	—
262	C₄	—	2	8	no	—
263	C₄	—	2	4	no	—
264	C₄	—	2	4	no	—
265	C₄	—	4	16	no	—
266	C₂	—	2	4	no	—
267	C₂	—	2	8	no	—
268	C₂	C₂²×C₁₉×#117	1	1	yes	characteristic
269	C₂	C₁₉×#3428	1	2	yes	—
270	C₂	C₁₉×#3491	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₉	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₁₉
Frattini, cosocle	C₂³	C₂⁵×C₁₉
prime radical	C₂⁵×C₁₉	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₉×#2343	C₂	semidirect
4. C₁₉×#2313	C₂	semidirect
5. C₁₉×#2283	C₂	semidirect
9. C₂×C₁₉×#117	C₄	semidirect
16. C₁₉×#387	C₂²	semidirect
18. C₁₉×#383	C₂²	semidirect
19. C₁₉×#382	C₂²	semidirect
24. C₄×C₄⋊C₄×C₁₉	C₂²	semidirect
27. C₁₉×#376	C₂²	semidirect
38. C₁₉×#117	C₂×C₄	semidirect
39. C₂×C₄⋊C₄×C₁₉	C₂×C₄	semidirect
49. C₂×C₄²×C₁₉	C₂×C₄	semidirect
89. #34177	C₁₉	direct (N × H)
117. #2343	C₂×C₁₉	semidirect
119. #2313	C₂×C₁₉	semidirect
120. #2283	C₂×C₁₉	semidirect
139. C₂×#117	C₄×C₁₉	semidirect
146. #387	C₂²×C₁₉	semidirect
148. #383	C₂²×C₁₉	semidirect
149. #382	C₂²×C₁₉	semidirect
154. C₄×C₄⋊C₄	C₂²×C₁₉	semidirect
157. #376	C₂²×C₁₉	semidirect
173. #117	C₂×C₄×C₁₉	semidirect
174. C₂×C₄⋊C₄	C₂×C₄×C₁₉	semidirect
184. C₂×C₄²	C₂×C₄×C₁₉	semidirect
192. C₁₉	#34177	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₁₉×#117	2
C₁₉×#2343	2
C₁₉×#2315	2
C₁₉×#2313	2
C₁₉×#2283	2
C₂×C₁₉×#381	2
C₁₉×#2273	2
#34177	19

Subgroups — C19×#34177 #18040136

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#34177 #18040136