Subgroups — C₁₉×#33382 #18039341

Order 4864.

1124 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₉×#578	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₉×#504	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₉×#2913	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₉×#2141	C₂	1	3	yes	maximal, complemented
5	C₁₉×#2139	C₂	1	3	yes	maximal, complemented
6	C₁₉×#2138	C₂	1	3	yes	maximal, complemented
7	C₂×C₁₉×#358	C₂	1	3	yes	maximal, complemented
8	C₂×C₁₉×2₋¹⁺⁴	—	2	1	no	—
9	C₁₉×#578	—	2	4	no	—
10	C₁₉×#578	C₄	1	8	yes	complemented
11	C₂×C₁₉×#137	—	2	3	no	—
12	C₂×C₁₉×#135	C₂²	1	3	yes	—
13	C₂²×Q₁₆×C₁₉	C₂²	1	3	yes	—
14	C₂×C₁₉×#131	C₂²	1	1	yes	characteristic
15	C₁₉×#520	—	2	1	no	—
16	C₁₉×#504	—	2	4	no	—
17	C₂×C₁₉×M₃₂	C₂²	1	3	yes	—
18	C₂²×C₁₆×C₁₉	C₂²	1	3	yes	—
19	C₁₉×#428	—	2	3	no	—
20	C₁₉×#427	C₂²	1	3	yes	—
21	C₂×C₁₉×#107	C₂²	1	3	yes	—
22	C₁₉×#358	C₂²	1	3	yes	—
23	C₁₉×#358	C₂²	1	6	yes	—
24	C₁₉×#358	C₂²	1	3	yes	—
25	C₁₉×#357	C₂²	1	1	yes	characteristic
26	C₁₉×#357	C₂²	1	3	yes	—
27	C₁₉×2₋¹⁺⁴	—	2	8	no	—
28	C₁₉×2₋¹⁺⁴	—	4	4	no	—
29	C₂×#41×C₁₉	—	2	3	no	—
30	C₂×#41×C₁₉	—	4	3	no	—
31	C₂×#41×C₁₉	D₄	1	1	yes	characteristic
32	C₂²×Q₈×C₁₉	—	2	3	no	—
33	C₂²×Q₈×C₁₉	—	4	1	no	—
34	C₁₉×#137	—	2	24	no	—
35	C₁₉×#135	—	2	6	no	—
36	C₁₉×#135	C₂×C₄	1	12	yes	complemented
37	C₂×Q₁₆×C₁₉	—	2	3	no	—
38	C₂×Q₁₆×C₁₉	—	2	6	no	—
39	C₂×Q₁₆×C₁₉	C₂×C₄	1	12	yes	complemented
40	C₂×Q₁₆×C₁₉	C₂³	1	3	yes	—
41	C₂×SD₁₆×C₁₉	—	2	3	no	—
42	C₂×D₈×C₁₉	C₂³	1	1	yes	characteristic
43	C₁₉×#131	C₂×C₄	1	4	yes	—
44	C₁₉×#131	D₄	1	4	yes	—
45	C₂×M₁₆×C₁₉	D₄	1	3	yes	—
46	C₂²×C₈×C₁₉	C₂³	1	3	yes	—
47	C₄×Q₈×C₁₉	—	4	1	no	—
48	C₄×D₄×C₁₉	—	4	3	no	—
49	C₁₉×#117	—	2	3	no	—
50	C₂×C₄⋊C₄×C₁₉	—	2	3	no	—
51	C₁₉×M₃₂	—	2	6	no	—
52	C₂×C₁₆×C₁₉	—	2	6	no	—
53	C₂×C₁₆×C₁₉	C₂³	1	1	yes	characteristic
54	C₂×C₁₆×C₁₉	C₂³	1	3	yes	—
55	C₁₉×#107	—	2	3	no	—
56	C₁₉×#107	C₂³	1	3	yes	—
57	C₁₉×#107	C₂³	1	1	yes	characteristic
58	C₁₉×#106	—	2	3	no	—
59	#41×C₁₉	—	2	12	no	—
60	#41×C₁₉	—	4	24	no	—
61	#41×C₁₉	—	4	6	no	—
62	#41×C₁₉	C₂²⋊C₄	1	4	yes	—
63	#41×C₁₉	SD₁₆	1	4	yes	—
64	C₂×Q₈×C₁₉	—	2	12	no	—
65	C₂×Q₈×C₁₉	—	2	3	no	—
66	C₂×Q₈×C₁₉	—	4	8	no	—
67	C₂×Q₈×C₁₉	—	4	3	no	—
68	C₂×Q₈×C₁₉	—	4	6	no	—
69	C₂×Q₈×C₁₉	D₈	1	1	yes	characteristic
70	C₂×D₄×C₁₉	—	2	1	no	—
71	C₂×D₄×C₁₉	—	4	3	no	—
72	C₂×D₄×C₁₉	D₈	1	3	yes	—
73	C₂²×C₄×C₁₉	—	4	3	no	—
74	C₂²×C₄×C₁₉	—	4	3	no	—
75	C₂²×C₄×C₁₉	—	4	3	no	—
76	C₂²×C₄×C₁₉	C₂×D₄	1	3	yes	—
77	Q₁₆×C₁₉	—	2	12	no	—
78	Q₁₆×C₁₉	—	2	3	no	—
79	Q₁₆×C₁₉	C₂²×C₄	1	6	yes	—
80	SD₁₆×C₁₉	—	2	12	no	—
81	D₈×C₁₉	—	2	1	no	—
82	D₈×C₁₉	C₂²×C₄	1	2	yes	—
83	M₁₆×C₁₉	C₂²⋊C₄	1	12	yes	—
84	C₂×C₈×C₁₉	C₂²×C₄	1	6	yes	—
85	C₂×C₈×C₁₉	C₂×D₄	1	3	yes	—
86	C₂×C₈×C₁₉	C₂×D₄	1	6	yes	—
87	C₂×C₈×C₁₉	C₂⁴	1	1	yes	characteristic
88	C₄⋊C₄×C₁₉	—	2	3	no	—
89	C₄⋊C₄×C₁₉	—	2	1	no	—
90	C₄⋊C₄×C₁₉	—	4	3	no	—
91	C₂²⋊C₄×C₁₉	—	4	3	no	—
92	C₄²×C₁₉	—	4	3	no	—
93	C₁₆×C₁₉	—	2	1	no	—
94	C₁₆×C₁₉	—	2	3	no	—
95	#33382	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
96	C₂³×C₁₉	—	4	1	no	—
97	C₂³×C₁₉	C₂×D₈	1	3	yes	—
98	Q₈×C₁₉	—	2	6	no	—
99	Q₈×C₁₉	—	4	3	no	—
100	Q₈×C₁₉	—	4	12	no	—
101	Q₈×C₁₉	#102	1	4	yes	—
102	D₄×C₁₉	—	2	2	no	—
103	D₄×C₁₉	—	4	12	no	—
104	D₄×C₁₉	—	4	1	no	—
105	D₄×C₁₉	#102	1	12	yes	—
106	C₂×C₄×C₁₉	—	4	3	no	—
107	C₂×C₄×C₁₉	—	4	12	no	—
108	C₂×C₄×C₁₉	—	4	12	no	—
109	C₂×C₄×C₁₉	—	4	1	no	—
110	C₂×C₄×C₁₉	—	4	3	no	—
111	C₂×C₄×C₁₉	—	16	3	no	—
112	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	6	yes	—
113	C₂×C₄×C₁₉	C₂×D₈	1	3	yes	—
114	C₂×C₄×C₁₉	C₂×SD₁₆	1	6	yes	—
115	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
116	C₈×C₁₉	C₂×C₂²⋊C₄	1	6	yes	—
117	C₈×C₁₉	C₂³×C₄	1	1	yes	characteristic
118	C₈×C₁₉	C₂²×D₄	1	1	yes	characteristic
119	C₂×#578	C₂×C₁₉	1	1	yes	characteristic
120	C₂×#504	C₂×C₁₉	1	1	yes	characteristic, complemented
121	#2913	C₂×C₁₉	1	1	yes	characteristic, complemented
122	#2141	C₂×C₁₉	1	3	yes	complemented
123	#2139	C₂×C₁₉	1	3	yes	complemented
124	#2138	C₂×C₁₉	1	3	yes	complemented
125	C₂×#358	C₂×C₁₉	1	3	yes	complemented
126	C₂²×C₁₉	—	2	3	no	—
127	C₂²×C₁₉	—	2	3	no	—
128	C₂²×C₁₉	—	4	2	no	—
129	C₂²×C₁₉	—	8	1	no	—
130	C₂²×C₁₉	—	8	1	no	—
131	C₂²×C₁₉	C₂×#102	1	6	yes	—
132	C₂²×C₁₉	C₂²×D₈	1	1	yes	characteristic
133	C₄×C₁₉	—	4	6	no	—
134	C₄×C₁₉	—	8	3	no	—
135	C₄×C₁₉	—	8	1	no	—
136	C₄×C₁₉	C₂×#102	1	6	yes	—
137	C₄×C₁₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
138	C₄×C₁₉	C₂²×SD₁₆	1	1	yes	characteristic
139	C₂×2₋¹⁺⁴	—	2	1	no	—
140	#578	—	2	4	no	—
141	#578	C₄×C₁₉	1	8	yes	complemented
142	C₂×#137	—	2	3	no	—
143	C₂×#135	C₂²×C₁₉	1	3	yes	—
144	C₂²×Q₁₆	C₂²×C₁₉	1	3	yes	—
145	C₂×#131	C₂²×C₁₉	1	1	yes	characteristic
146	#520	—	2	1	no	—
147	#504	—	2	4	no	—
148	C₂×M₃₂	C₂²×C₁₉	1	3	yes	—
149	C₂²×C₁₆	C₂²×C₁₉	1	3	yes	—
150	#428	—	2	3	no	—
151	#427	C₂²×C₁₉	1	3	yes	—
152	C₂×#107	C₂²×C₁₉	1	3	yes	—
153	#358	C₂²×C₁₉	1	3	yes	—
154	#358	C₂²×C₁₉	1	6	yes	—
155	#358	C₂²×C₁₉	1	3	yes	—
156	#357	C₂²×C₁₉	1	1	yes	characteristic
157	#357	C₂²×C₁₉	1	3	yes	—
158	C₂×C₁₉	—	2	6	no	—
159	C₂×C₁₉	—	8	2	no	—
160	C₂×C₁₉	C₂²×#102	1	1	yes	characteristic
161	C₂×C₁₉	#3416	1	1	yes	characteristic
162	C₂×C₁₉	#3417	1	1	yes	characteristic
163	2₋¹⁺⁴	—	2	8	no	—
164	2₋¹⁺⁴	—	4	4	no	—
165	C₂×#41	—	2	3	no	—
166	C₂×#41	—	4	3	no	—
167	C₂×#41	D₄×C₁₉	1	1	yes	characteristic
168	C₂²×Q₈	—	2	3	no	—
169	C₂²×Q₈	—	4	1	no	—
170	#137	—	2	24	no	—
171	#135	—	2	6	no	—
172	#135	C₂×C₄×C₁₉	1	12	yes	complemented
173	C₂×Q₁₆	—	2	3	no	—
174	C₂×Q₁₆	—	2	6	no	—
175	C₂×Q₁₆	C₂×C₄×C₁₉	1	12	yes	complemented
176	C₂×Q₁₆	C₂³×C₁₉	1	3	yes	—
177	C₂×SD₁₆	—	2	3	no	—
178	C₂×D₈	C₂³×C₁₉	1	1	yes	characteristic
179	#131	C₂×C₄×C₁₉	1	4	yes	—
180	#131	D₄×C₁₉	1	4	yes	—
181	C₂×M₁₆	D₄×C₁₉	1	3	yes	—
182	C₂²×C₈	C₂³×C₁₉	1	3	yes	—
183	C₄×Q₈	—	4	1	no	—
184	C₄×D₄	—	4	3	no	—
185	#117	—	2	3	no	—
186	C₂×C₄⋊C₄	—	2	3	no	—
187	M₃₂	—	2	6	no	—
188	C₂×C₁₆	—	2	6	no	—
189	C₂×C₁₆	C₂³×C₁₉	1	1	yes	characteristic
190	C₂×C₁₆	C₂³×C₁₉	1	3	yes	—
191	#107	—	2	3	no	—
192	#107	C₂³×C₁₉	1	3	yes	—
193	#107	C₂³×C₁₉	1	1	yes	characteristic
194	#106	—	2	3	no	—
195	C₁₉	#33382	1	1	yes	characteristic, direct product factor, Sylow
196	#41	—	2	12	no	—
197	#41	—	4	24	no	—
198	#41	—	4	6	no	—
199	#41	C₂²⋊C₄×C₁₉	1	4	yes	—
200	#41	SD₁₆×C₁₉	1	4	yes	—
201	C₂×Q₈	—	2	12	no	—
202	C₂×Q₈	—	2	3	no	—
203	C₂×Q₈	—	4	8	no	—
204	C₂×Q₈	—	4	3	no	—
205	C₂×Q₈	—	4	6	no	—
206	C₂×Q₈	D₈×C₁₉	1	1	yes	characteristic
207	C₂×D₄	—	2	1	no	—
208	C₂×D₄	—	4	3	no	—
209	C₂×D₄	D₈×C₁₉	1	3	yes	—
210	C₂²×C₄	—	4	3	no	—
211	C₂²×C₄	—	4	3	no	—
212	C₂²×C₄	—	4	3	no	—
213	C₂²×C₄	C₂×D₄×C₁₉	1	3	yes	—
214	Q₁₆	—	2	12	no	—
215	Q₁₆	—	2	3	no	—
216	Q₁₆	C₂²×C₄×C₁₉	1	6	yes	—
217	SD₁₆	—	2	12	no	—
218	D₈	—	2	1	no	—
219	D₈	C₂²×C₄×C₁₉	1	2	yes	—
220	M₁₆	C₂²⋊C₄×C₁₉	1	12	yes	—
221	C₂×C₈	C₂²×C₄×C₁₉	1	6	yes	—
222	C₂×C₈	C₂×D₄×C₁₉	1	3	yes	—
223	C₂×C₈	C₂×D₄×C₁₉	1	6	yes	—
224	C₂×C₈	C₂⁴×C₁₉	1	1	yes	characteristic
225	C₄⋊C₄	—	2	1	no	—
226	C₄⋊C₄	—	2	3	no	—
227	C₄⋊C₄	—	4	3	no	—
228	C₂²⋊C₄	—	4	3	no	—
229	C₄²	—	4	3	no	—
230	C₁₆	—	2	1	no	—
231	C₁₆	—	2	3	no	—
232	C₂³	—	4	1	no	—
233	C₂³	C₂×D₈×C₁₉	1	3	yes	—
234	Q₈	—	2	6	no	—
235	Q₈	—	4	12	no	—
236	Q₈	—	4	3	no	—
237	Q₈	C₁₉×#102	1	4	yes	—
238	D₄	—	2	2	no	—
239	D₄	—	4	12	no	—
240	D₄	—	4	1	no	—
241	D₄	C₁₉×#102	1	12	yes	—
242	C₂×C₄	—	4	12	no	—
243	C₂×C₄	—	4	3	no	—
244	C₂×C₄	—	4	12	no	—
245	C₂×C₄	—	4	1	no	—
246	C₂×C₄	—	4	3	no	—
247	C₂×C₄	—	16	3	no	—
248	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	6	yes	—
249	C₂×C₄	C₂×D₈×C₁₉	1	3	yes	—
250	C₂×C₄	C₂×SD₁₆×C₁₉	1	6	yes	—
251	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
252	C₈	C₂×C₂²⋊C₄×C₁₉	1	6	yes	—
253	C₈	C₂³×C₄×C₁₉	1	1	yes	characteristic
254	C₈	C₂²×D₄×C₁₉	1	1	yes	characteristic
255	C₂²	—	2	3	no	—
256	C₂²	—	2	3	no	—
257	C₂²	—	4	2	no	—
258	C₂²	—	8	1	no	—
259	C₂²	—	8	1	no	—
260	C₂²	C₂×C₁₉×#102	1	6	yes	—
261	C₂²	C₂²×D₈×C₁₉	1	1	yes	characteristic
262	C₄	—	4	6	no	—
263	C₄	—	8	3	no	—
264	C₄	—	8	1	no	—
265	C₄	C₂×C₁₉×#102	1	6	yes	—
266	C₄	C₂²×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
267	C₄	C₂²×SD₁₆×C₁₉	1	1	yes	characteristic
268	C₂	—	2	6	no	—
269	C₂	—	8	2	no	—
270	C₂	C₂²×C₁₉×#102	1	1	yes	characteristic
271	C₂	C₁₉×#3416	1	1	yes	characteristic
272	C₂	C₁₉×#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂²×D₈
commutator	C₈	C₂³×C₄×C₁₉
Frattini, cosocle	C₂×C₈	C₂⁴×C₁₉
prime radical	C₂×C₁₉×#578	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂×#41×C₁₉ ◁ C₂×C₁₉×#131 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₉×#504	C₂	semidirect
3. C₁₉×#2913	C₂	semidirect
4. C₁₉×#2141	C₂	semidirect
5. C₁₉×#2139	C₂	semidirect
6. C₁₉×#2138	C₂	semidirect
7. C₂×C₁₉×#358	C₂	semidirect
10. C₁₉×#578	C₄	semidirect
36. C₁₉×#135	C₂×C₄	semidirect
39. C₂×Q₁₆×C₁₉	C₂×C₄	semidirect
95. #33382	C₁₉	direct (N × H)
120. C₂×#504	C₂×C₁₉	semidirect
121. #2913	C₂×C₁₉	semidirect
122. #2141	C₂×C₁₉	semidirect
123. #2139	C₂×C₁₉	semidirect
124. #2138	C₂×C₁₉	semidirect
125. C₂×#358	C₂×C₁₉	semidirect
141. #578	C₄×C₁₉	semidirect
172. #135	C₂×C₄×C₁₉	semidirect
175. C₂×Q₁₆	C₂×C₄×C₁₉	semidirect
195. C₁₉	#33382	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₁₉×#578	2
C₂×C₁₉×#504	2
C₁₉×#2913	2
C₁₉×#2141	2
C₁₉×#2139	2
C₁₉×#2138	2
C₂×C₁₉×#358	2
#33382	19

Subgroups — C19×#33382 #18039341

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#33382 #18039341