Subgroups — C₁₉×#33371 #18039330

Order 4864.

1208 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	C₁₉×#3500	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2888	C₂	1	2	yes	maximal, complemented
3	C₁₉×#2135	C₂	1	6	yes	maximal, complemented
4	C₁₉×#2132	C₂	1	6	yes	maximal, complemented
5	C₂×C₁₉×2₋¹⁺⁴	C₄	1	1	yes	characteristic
6	C₂×C₁₉×H(5,𝔽₂)	C₂²	1	1	yes	characteristic
7	C₁₉×#568	—	2	2	no	—
8	C₁₉×#548	—	2	2	no	—
9	C₁₉×#546	—	2	6	no	—
10	C₁₉×#536	C₂²	1	3	yes	—
11	C₁₉×#535	C₄	1	3	yes	—
12	C₁₉×#532	C₂²	1	3	yes	—
13	C₂×C₁₉×#124	C₄	1	3	yes	—
14	C₄×#41×C₁₉	C₄	1	1	yes	characteristic
15	C₁₉×#413	C₂²	1	12	yes	—
16	C₂×C₁₉×#100	—	2	6	no	—
17	C₁₉×#356	C₂²	1	12	yes	complemented
18	C₁₉×#355	C₂²	1	4	yes	complemented
19	C₁₉×2₋¹⁺⁴	—	4	4	no	—
20	C₁₉×H(5,𝔽₂)	—	4	4	no	—
21	C₂×#41×C₁₉	—	2	3	no	—
22	C₂×#41×C₁₉	C₂×C₄	1	3	yes	—
23	C₂×#41×C₁₉	C₂×C₄	1	1	yes	characteristic
24	C₂×#41×C₁₉	D₄	1	2	yes	—
25	C₂×#41×C₁₉	C₂³	1	3	yes	—
26	C₂²×Q₈×C₁₉	—	2	1	no	—
27	C₂²×Q₈×C₁₉	C₂×C₄	1	3	yes	—
28	C₂²×D₄×C₁₉	C₂×C₄	1	3	yes	—
29	C₂²×D₄×C₁₉	D₄	1	6	yes	—
30	C₁₉×#131	—	4	8	no	—
31	C₂×M₁₆×C₁₉	—	2	12	no	—
32	C₁₉×C₄⋊Q₈	C₂³	1	3	yes	—
33	C₁₉×#127	C₂×C₄	1	3	yes	—
34	C₁₉×#124	—	2	6	no	—
35	C₁₉×#124	C₂×C₄	1	3	yes	—
36	C₁₉×#124	C₂×C₄	1	6	yes	—
37	C₁₉×#124	C₂³	1	1	yes	characteristic
38	C₁₉×#123	—	2	6	no	—
39	C₁₉×C₂²⋊Q₈	—	2	6	no	—
40	C₁₉×#121	—	2	6	no	—
41	C₁₉×#121	—	2	6	no	—
42	C₁₉×#120	—	2	6	no	—
43	C₄×Q₈×C₁₉	—	2	1	no	—
44	C₄×D₄×C₁₉	—	2	3	no	—
45	C₄×D₄×C₁₉	—	2	6	no	—
46	C₁₉×#117	C₂×C₄	1	3	yes	—
47	C₂×C₂²⋊C₄×C₁₉	—	2	6	no	—
48	C₂×C₄²×C₁₉	C₂×C₄	1	3	yes	—
49	C₁₉×#101	C₂³	1	8	yes	—
50	C₁₉×#100	—	2	12	no	—
51	C₂⁴×C₁₉	C₂²⋊C₄	1	6	yes	—
52	#41×C₁₉	—	2	8	no	—
53	#41×C₁₉	—	2	4	no	—
54	#41×C₁₉	—	4	6	no	—
55	#41×C₁₉	—	4	6	no	—
56	#41×C₁₉	—	4	12	no	—
57	C₂×Q₈×C₁₉	—	2	3	no	—
58	C₂×Q₈×C₁₉	—	4	6	no	—
59	C₂×Q₈×C₁₉	—	4	4	no	—
60	C₂×Q₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×Q₈×C₁₉	C₂²×C₄	1	3	yes	—
62	C₂×Q₈×C₁₉	C₂⁴	1	1	yes	characteristic
63	C₂×D₄×C₁₉	—	2	3	no	—
64	C₂×D₄×C₁₉	—	2	6	no	—
65	C₂×D₄×C₁₉	—	4	6	no	—
66	C₂×D₄×C₁₉	—	4	12	no	—
67	C₂×D₄×C₁₉	C₂²⋊C₄	1	3	yes	—
68	C₂×D₄×C₁₉	C₂²⋊C₄	1	6	yes	—
69	C₂×D₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
70	C₂×D₄×C₁₉	C₂²×C₄	1	3	yes	—
71	C₂×D₄×C₁₉	C₂×D₄	1	6	yes	—
72	C₂²×C₄×C₁₉	—	2	3	no	—
73	C₂²×C₄×C₁₉	—	2	3	no	—
74	C₂²×C₄×C₁₉	—	2	3	no	—
75	C₂²×C₄×C₁₉	C₂²×C₄	1	3	yes	—
76	C₂²×C₄×C₁₉	C₂×D₄	1	6	yes	—
77	M₁₆×C₁₉	—	2	8	no	—
78	M₁₆×C₁₉	—	4	12	no	—
79	C₂×C₈×C₁₉	—	4	12	no	—
80	C₄⋊C₄×C₁₉	—	2	3	no	—
81	C₄⋊C₄×C₁₉	—	2	6	no	—
82	C₂²⋊C₄×C₁₉	—	2	6	no	—
83	C₂²⋊C₄×C₁₉	—	2	12	no	—
84	C₂²⋊C₄×C₁₉	—	2	3	no	—
85	C₂²⋊C₄×C₁₉	—	2	2	no	—
86	C₄²×C₁₉	—	2	3	no	—
87	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₁₉	C₂²×C₄	1	3	yes	—
89	#33371	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂³×C₁₉	—	2	1	no	—
91	C₂³×C₁₉	—	4	6	no	—
92	C₂³×C₁₉	—	8	6	no	—
93	C₂³×C₁₉	C₂×C₂²⋊C₄	1	6	yes	—
94	C₂³×C₁₉	C₂×C₂²⋊C₄	1	3	yes	—
95	Q₈×C₁₉	—	4	1	no	—
96	Q₈×C₁₉	—	4	3	no	—
97	Q₈×C₁₉	—	4	6	no	—
98	Q₈×C₁₉	#99	1	4	yes	—
99	D₄×C₁₉	—	2	12	no	—
100	D₄×C₁₉	—	2	6	no	—
101	D₄×C₁₉	—	4	3	no	—
102	D₄×C₁₉	—	4	6	no	—
103	D₄×C₁₉	—	4	1	no	—
104	D₄×C₁₉	—	4	6	no	—
105	D₄×C₁₉	—	8	6	no	—
106	C₂×C₄×C₁₉	—	2	12	no	—
107	C₂×C₄×C₁₉	—	2	6	no	—
108	C₂×C₄×C₁₉	—	2	1	no	—
109	C₂×C₄×C₁₉	—	2	3	no	—
110	C₂×C₄×C₁₉	—	2	3	no	—
111	C₂×C₄×C₁₉	—	4	6	no	—
112	C₂×C₄×C₁₉	—	4	6	no	—
113	C₂×C₄×C₁₉	—	8	3	no	—
114	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	3	yes	—
115	C₂×C₄×C₁₉	C₂³×C₄	1	1	yes	characteristic
116	C₂×C₄×C₁₉	C₂²×D₄	1	2	yes	—
117	C₈×C₁₉	—	4	8	no	—
118	#3500	C₂×C₁₉	1	1	yes	characteristic
119	#2888	C₂×C₁₉	1	2	yes	complemented
120	#2135	C₂×C₁₉	1	6	yes	complemented
121	#2132	C₂×C₁₉	1	6	yes	complemented
122	C₂²×C₁₉	—	2	3	no	—
123	C₂²×C₁₉	—	2	6	no	—
124	C₂²×C₁₉	—	4	3	no	—
125	C₂²×C₁₉	—	4	6	no	—
126	C₂²×C₁₉	—	4	2	no	—
127	C₂²×C₁₉	—	4	1	no	—
128	C₂²×C₁₉	—	16	6	no	—
129	C₂²×C₁₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
130	C₄×C₁₉	—	2	2	no	—
131	C₄×C₁₉	—	2	1	no	—
132	C₄×C₁₉	—	4	1	no	—
133	C₄×C₁₉	—	4	3	no	—
134	C₄×C₁₉	—	4	3	no	—
135	C₄×C₁₉	C₂×#99	1	6	yes	—
136	C₂×2₋¹⁺⁴	C₄×C₁₉	1	1	yes	characteristic
137	C₂×H(5,𝔽₂)	C₂²×C₁₉	1	1	yes	characteristic
138	#568	—	2	2	no	—
139	#548	—	2	2	no	—
140	#546	—	2	6	no	—
141	#536	C₂²×C₁₉	1	3	yes	—
142	#535	C₄×C₁₉	1	3	yes	—
143	#532	C₂²×C₁₉	1	3	yes	—
144	C₂×#124	C₄×C₁₉	1	3	yes	—
145	C₄×#41	C₄×C₁₉	1	1	yes	characteristic
146	#413	C₂²×C₁₉	1	12	yes	—
147	C₂×#100	—	2	6	no	—
148	#356	C₂²×C₁₉	1	12	yes	complemented
149	#355	C₂²×C₁₉	1	4	yes	complemented
150	C₂×C₁₉	—	2	1	no	—
151	C₂×C₁₉	—	4	3	no	—
152	C₂×C₁₉	—	4	6	no	—
153	C₂×C₁₉	—	8	1	no	—
154	C₂×C₁₉	C₂²×#99	1	1	yes	characteristic
155	2₋¹⁺⁴	—	4	4	no	—
156	H(5,𝔽₂)	—	4	4	no	—
157	C₂×#41	—	2	3	no	—
158	C₂×#41	C₂×C₄×C₁₉	1	3	yes	—
159	C₂×#41	C₂×C₄×C₁₉	1	1	yes	characteristic
160	C₂×#41	D₄×C₁₉	1	2	yes	—
161	C₂×#41	C₂³×C₁₉	1	3	yes	—
162	C₂²×Q₈	—	2	1	no	—
163	C₂²×Q₈	C₂×C₄×C₁₉	1	3	yes	—
164	C₂²×D₄	C₂×C₄×C₁₉	1	3	yes	—
165	C₂²×D₄	D₄×C₁₉	1	6	yes	—
166	#131	—	4	8	no	—
167	C₂×M₁₆	—	2	12	no	—
168	C₄⋊Q₈	C₂³×C₁₉	1	3	yes	—
169	#127	C₂×C₄×C₁₉	1	3	yes	—
170	#124	—	2	6	no	—
171	#124	C₂×C₄×C₁₉	1	3	yes	—
172	#124	C₂×C₄×C₁₉	1	6	yes	—
173	#124	C₂³×C₁₉	1	1	yes	characteristic
174	#123	—	2	6	no	—
175	C₂²⋊Q₈	—	2	6	no	—
176	#121	—	2	6	no	—
177	#121	—	2	6	no	—
178	#120	—	2	6	no	—
179	C₄×Q₈	—	2	1	no	—
180	C₄×D₄	—	2	3	no	—
181	C₄×D₄	—	2	6	no	—
182	#117	C₂×C₄×C₁₉	1	3	yes	—
183	C₂×C₂²⋊C₄	—	2	6	no	—
184	C₂×C₄²	C₂×C₄×C₁₉	1	3	yes	—
185	#101	C₂³×C₁₉	1	8	yes	—
186	#100	—	2	12	no	—
187	C₁₉	#33371	1	1	yes	characteristic, direct product factor, Sylow
188	C₂⁴	C₂²⋊C₄×C₁₉	1	6	yes	—
189	#41	—	2	8	no	—
190	#41	—	2	4	no	—
191	#41	—	4	6	no	—
192	#41	—	4	6	no	—
193	#41	—	4	12	no	—
194	C₂×Q₈	—	2	3	no	—
195	C₂×Q₈	—	4	6	no	—
196	C₂×Q₈	—	4	4	no	—
197	C₂×Q₈	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
198	C₂×Q₈	C₂²×C₄×C₁₉	1	3	yes	—
199	C₂×Q₈	C₂⁴×C₁₉	1	1	yes	characteristic
200	C₂×D₄	—	2	3	no	—
201	C₂×D₄	—	2	6	no	—
202	C₂×D₄	—	4	6	no	—
203	C₂×D₄	—	4	12	no	—
204	C₂×D₄	C₂²⋊C₄×C₁₉	1	3	yes	—
205	C₂×D₄	C₂²⋊C₄×C₁₉	1	6	yes	—
206	C₂×D₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
207	C₂×D₄	C₂²×C₄×C₁₉	1	3	yes	—
208	C₂×D₄	C₂×D₄×C₁₉	1	6	yes	—
209	C₂²×C₄	—	2	3	no	—
210	C₂²×C₄	—	2	3	no	—
211	C₂²×C₄	—	2	3	no	—
212	C₂²×C₄	C₂²×C₄×C₁₉	1	3	yes	—
213	C₂²×C₄	C₂×D₄×C₁₉	1	6	yes	—
214	M₁₆	—	2	8	no	—
215	M₁₆	—	4	12	no	—
216	C₂×C₈	—	4	12	no	—
217	C₄⋊C₄	—	2	3	no	—
218	C₄⋊C₄	—	2	6	no	—
219	C₂²⋊C₄	—	2	6	no	—
220	C₂²⋊C₄	—	2	12	no	—
221	C₂²⋊C₄	—	2	3	no	—
222	C₂²⋊C₄	—	2	2	no	—
223	C₄²	—	2	3	no	—
224	C₄²	C₂²×C₄×C₁₉	1	1	yes	characteristic
225	C₄²	C₂²×C₄×C₁₉	1	3	yes	—
226	C₂³	—	2	1	no	—
227	C₂³	—	4	6	no	—
228	C₂³	—	8	6	no	—
229	C₂³	C₂×C₂²⋊C₄×C₁₉	1	6	yes	—
230	C₂³	C₂×C₂²⋊C₄×C₁₉	1	3	yes	—
231	Q₈	—	4	1	no	—
232	Q₈	—	4	3	no	—
233	Q₈	—	4	6	no	—
234	Q₈	C₁₉×#99	1	4	yes	—
235	D₄	—	2	6	no	—
236	D₄	—	2	12	no	—
237	D₄	—	4	3	no	—
238	D₄	—	4	6	no	—
239	D₄	—	4	1	no	—
240	D₄	—	4	6	no	—
241	D₄	—	8	6	no	—
242	C₂×C₄	—	2	12	no	—
243	C₂×C₄	—	2	6	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	3	no	—
246	C₂×C₄	—	2	3	no	—
247	C₂×C₄	—	4	6	no	—
248	C₂×C₄	—	4	6	no	—
249	C₂×C₄	—	8	3	no	—
250	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	3	yes	—
251	C₂×C₄	C₂³×C₄×C₁₉	1	1	yes	characteristic
252	C₂×C₄	C₂²×D₄×C₁₉	1	2	yes	—
253	C₈	—	4	8	no	—
254	C₂²	—	2	3	no	—
255	C₂²	—	2	6	no	—
256	C₂²	—	4	3	no	—
257	C₂²	—	4	6	no	—
258	C₂²	—	4	2	no	—
259	C₂²	—	4	1	no	—
260	C₂²	—	16	6	no	—
261	C₂²	C₂²×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
262	C₄	—	2	2	no	—
263	C₄	—	2	1	no	—
264	C₄	—	4	1	no	—
265	C₄	—	4	3	no	—
266	C₄	—	4	3	no	—
267	C₄	C₂×C₁₉×#99	1	6	yes	—
268	C₂	—	2	1	no	—
269	C₂	—	4	3	no	—
270	C₂	—	4	6	no	—
271	C₂	—	8	1	no	—
272	C₂	C₂²×C₁₉×#99	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₁₉	C₂²×#99
commutator	C₂×C₄	C₂³×C₄×C₁₉
Frattini, cosocle	C₂×Q₈	C₂⁴×C₁₉
prime radical	C₁₉×#3500	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₉ ◁ C₂×Q₈×C₁₉ ◁ C₂×C₁₉×H(5,𝔽₂) ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₉×#2888	C₂	semidirect
3. C₁₉×#2135	C₂	semidirect
4. C₁₉×#2132	C₂	semidirect
17. C₁₉×#356	C₂²	semidirect
18. C₁₉×#355	C₂²	semidirect
89. #33371	C₁₉	direct (N × H)
119. #2888	C₂×C₁₉	semidirect
120. #2135	C₂×C₁₉	semidirect
121. #2132	C₂×C₁₉	semidirect
148. #356	C₂²×C₁₉	semidirect
149. #355	C₂²×C₁₉	semidirect
187. C₁₉	#33371	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×#3500	2
C₁₉×#2888	2
C₁₉×#2135	2
C₁₉×#2132	2
#33371	19

Subgroups — C19×#33371 #18039330

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#33371 #18039330