Subgroups — C₁₉×#32695 #18038654

Order 4864.

642 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#2567	C₂	1	2	yes	maximal
2	C₁₉×#2086	C₂	1	2	yes	maximal
3	C₁₉×#2057	C₂	1	4	yes	maximal
4	C₁₉×#1985	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#1944	C₂	1	2	yes	maximal
6	C₁₉×#1920	C₂	1	4	yes	maximal
7	C₂×C₁₉×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₂×C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
9	C₂×C₁₉×#107	C₂²	1	2	yes	—
10	C₂×C₁₉×#107	C₂²	1	2	yes	—
11	C₂×C₁₉×#106	C₂²	1	2	yes	—
12	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₁₉×#97	C₂²	1	1	yes	characteristic
14	C₁₉×#395	—	2	2	no	—
15	C₁₉×#395	—	2	2	no	—
16	C₁₉×#391	—	2	2	no	—
17	C₁₉×#389	C₂²	1	4	yes	—
18	C₁₉×#389	C₂²	1	4	yes	—
19	C₁₉×#384	—	2	2	no	—
20	C₄×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
21	C₁₉×#340	C₂²	1	2	yes	—
22	C₁₉×#340	C₂²	1	4	yes	—
23	C₁₉×#340	C₂²	1	4	yes	—
24	C₁₉×#336	C₂²	1	4	yes	—
25	C₂²×Q₈×C₁₉	—	2	2	no	—
26	C₂²×Q₈×C₁₉	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₉	C₂³	1	2	yes	—
28	C₂²×C₈×C₁₉	C₂³	1	2	yes	—
29	C₁₉×C₄⋊Q₈	—	4	2	no	—
30	C₁₉×C₄⋊Q₈	—	4	2	no	—
31	C₄×Q₈×C₁₉	—	4	1	no	—
32	C₄×Q₈×C₁₉	—	4	1	no	—
33	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
34	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
35	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
36	C₂×C₄⋊C₄×C₁₉	D₄	1	2	yes	—
37	C₂×C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
38	C₂×C₄⋊C₄×C₁₉	C₂³	1	4	yes	—
39	C₂×C₄⋊C₄×C₁₉	C₂³	1	4	yes	—
40	C₂×C₄²×C₁₉	—	2	2	no	—
41	C₂×C₄²×C₁₉	C₂³	1	2	yes	—
42	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
43	C₁₉×#107	—	4	2	no	—
44	C₁₉×#107	—	4	2	no	—
45	C₁₉×#106	Q₈	1	8	yes	complemented
46	C₁₉×C₄⋊C₈	—	4	1	no	—
47	C₁₉×#97	—	4	1	no	—
48	C₁₉×#95	—	2	2	no	—
49	C₁₉×#95	—	2	4	no	—
50	C₁₉×#95	—	2	4	no	—
51	C₂×Q₈×C₁₉	—	2	4	no	—
52	C₂×Q₈×C₁₉	—	4	1	no	—
53	C₂×Q₈×C₁₉	—	4	4	no	—
54	C₂×Q₈×C₁₉	—	8	4	no	—
55	C₂²×C₄×C₁₉	—	2	4	no	—
56	C₂²×C₄×C₁₉	—	2	4	no	—
57	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
59	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
62	C₂×C₈×C₁₉	—	2	2	no	—
63	C₂×C₈×C₁₉	—	2	2	no	—
64	C₂×C₈×C₁₉	—	2	2	no	—
65	C₂×C₈×C₁₉	—	2	4	no	—
66	C₂×C₈×C₁₉	C₂×Q₈	1	4	yes	—
67	C₄⋊C₄×C₁₉	—	2	4	no	—
68	C₄⋊C₄×C₁₉	—	2	4	no	—
69	C₄⋊C₄×C₁₉	—	2	4	no	—
70	C₄⋊C₄×C₁₉	—	4	2	no	—
71	C₄⋊C₄×C₁₉	—	4	1	no	—
72	C₄⋊C₄×C₁₉	—	4	4	no	—
73	C₄⋊C₄×C₁₉	—	8	4	no	—
74	C₄⋊C₄×C₁₉	—	8	4	no	—
75	C₄⋊C₄×C₁₉	C₂×Q₈	1	8	yes	—
76	C₄²×C₁₉	—	2	2	no	—
77	C₄²×C₁₉	—	2	2	no	—
78	C₄²×C₁₉	—	2	1	no	—
79	C₄²×C₁₉	—	2	1	no	—
80	C₄²×C₁₉	—	4	4	no	—
81	#32695	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
82	C₂³×C₁₉	C₂²×D₄	1	1	yes	characteristic
83	Q₈×C₁₉	—	4	4	no	—
84	Q₈×C₁₉	—	8	8	no	—
85	C₂×C₄×C₁₉	—	2	2	no	—
86	C₂×C₄×C₁₉	—	2	2	no	—
87	C₂×C₄×C₁₉	—	2	1	no	—
88	C₂×C₄×C₁₉	—	2	2	no	—
89	C₂×C₄×C₁₉	—	2	1	no	—
90	C₂×C₄×C₁₉	—	2	8	no	—
91	C₂×C₄×C₁₉	—	2	2	no	—
92	C₂×C₄×C₁₉	—	2	2	no	—
93	C₂×C₄×C₁₉	—	2	4	no	—
94	C₂×C₄×C₁₉	—	4	4	no	—
95	C₂×C₄×C₁₉	—	4	4	no	—
96	C₂×C₄×C₁₉	—	4	4	no	—
97	C₂×C₄×C₁₉	—	4	4	no	—
98	C₂×C₄×C₁₉	—	4	4	no	—
99	C₂×C₄×C₁₉	#137	1	4	yes	—
100	C₂×C₄×C₁₉	C₂²×Q₈	1	2	yes	—
101	C₂×C₄×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
102	C₂×C₄×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
103	C₂×C₄×C₁₉	2₋¹⁺⁴	1	2	yes	—
104	C₈×C₁₉	—	2	4	no	—
105	C₈×C₁₉	—	4	2	no	—
106	#2567	C₂×C₁₉	1	2	yes	—
107	#2086	C₂×C₁₉	1	2	yes	—
108	#2057	C₂×C₁₉	1	4	yes	—
109	#1985	C₂×C₁₉	1	1	yes	characteristic
110	#1944	C₂×C₁₉	1	2	yes	—
111	#1920	C₂×C₁₉	1	4	yes	—
112	C₂²×C₁₉	#534	1	1	yes	characteristic
113	C₂²×C₁₉	D₄×Q₈	1	2	yes	—
114	C₂²×C₁₉	C₂×#137	1	2	yes	—
115	C₂²×C₁₉	#576	1	2	yes	—
116	C₄×C₁₉	—	2	4	no	—
117	C₄×C₁₉	—	4	8	no	—
118	C₄×C₁₉	—	4	1	no	—
119	C₄×C₁₉	—	4	1	no	—
120	C₄×C₁₉	—	4	2	no	—
121	C₄×C₁₉	—	8	4	no	—
122	C₄×C₁₉	#544	1	2	yes	—
123	C₄×C₁₉	Q₈²	1	1	yes	characteristic
124	C₄×C₁₉	#563	1	1	yes	characteristic
125	C₂×C₄⋊Q₈	C₂²×C₁₉	1	2	yes	—
126	C₂×C₄×Q₈	C₂²×C₁₉	1	1	yes	characteristic
127	C₂×#107	C₂²×C₁₉	1	2	yes	—
128	C₂×#107	C₂²×C₁₉	1	2	yes	—
129	C₂×#106	C₂²×C₁₉	1	2	yes	—
130	C₂×C₄⋊C₈	C₂²×C₁₉	1	1	yes	characteristic
131	C₂×#97	C₂²×C₁₉	1	1	yes	characteristic
132	#395	—	2	2	no	—
133	#395	—	2	2	no	—
134	#391	—	2	2	no	—
135	#389	C₂²×C₁₉	1	4	yes	—
136	#389	C₂²×C₁₉	1	4	yes	—
137	#384	—	2	2	no	—
138	C₄×C₄⋊C₄	C₂²×C₁₉	1	2	yes	—
139	#340	C₂²×C₁₉	1	4	yes	—
140	#340	C₂²×C₁₉	1	4	yes	—
141	#340	C₂²×C₁₉	1	2	yes	—
142	#336	C₂²×C₁₉	1	4	yes	—
143	C₂×C₁₉	#2734	1	1	yes	characteristic
144	C₂×C₁₉	#3195	1	1	yes	characteristic
145	C₂×C₁₉	#3199	1	1	yes	characteristic
146	C₂×C₁₉	#3385	1	4	yes	—
147	C₂²×Q₈	—	2	2	no	—
148	C₂²×Q₈	D₄×C₁₉	1	1	yes	characteristic
149	C₂²×C₈	C₂³×C₁₉	1	2	yes	—
150	C₂²×C₈	C₂³×C₁₉	1	2	yes	—
151	C₄⋊Q₈	—	4	2	no	—
152	C₄⋊Q₈	—	4	2	no	—
153	C₄×Q₈	—	4	1	no	—
154	C₄×Q₈	—	4	1	no	—
155	C₂×C₄⋊C₄	—	2	2	no	—
156	C₂×C₄⋊C₄	—	2	4	no	—
157	C₂×C₄⋊C₄	—	2	4	no	—
158	C₂×C₄⋊C₄	D₄×C₁₉	1	2	yes	—
159	C₂×C₄⋊C₄	D₄×C₁₉	1	1	yes	characteristic
160	C₂×C₄⋊C₄	C₂³×C₁₉	1	4	yes	—
161	C₂×C₄⋊C₄	C₂³×C₁₉	1	4	yes	—
162	C₂×C₄²	—	2	2	no	—
163	C₂×C₄²	C₂³×C₁₉	1	2	yes	—
164	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
165	#107	—	4	2	no	—
166	#107	—	4	2	no	—
167	#106	Q₈×C₁₉	1	8	yes	complemented
168	C₄⋊C₈	—	4	1	no	—
169	#97	—	4	1	no	—
170	#95	—	2	2	no	—
171	#95	—	2	4	no	—
172	#95	—	2	4	no	—
173	C₁₉	#32695	1	1	yes	characteristic, direct product factor, Sylow
174	C₂×Q₈	—	2	4	no	—
175	C₂×Q₈	—	4	1	no	—
176	C₂×Q₈	—	4	4	no	—
177	C₂×Q₈	—	8	4	no	—
178	C₂²×C₄	—	2	4	no	—
179	C₂²×C₄	—	2	4	no	—
180	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
181	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
182	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
183	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
184	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	2	4	no	—
189	C₂×C₈	C₂×Q₈×C₁₉	1	4	yes	—
190	C₄⋊C₄	—	2	4	no	—
191	C₄⋊C₄	—	2	4	no	—
192	C₄⋊C₄	—	2	4	no	—
193	C₄⋊C₄	—	4	2	no	—
194	C₄⋊C₄	—	4	1	no	—
195	C₄⋊C₄	—	4	4	no	—
196	C₄⋊C₄	—	8	4	no	—
197	C₄⋊C₄	—	8	4	no	—
198	C₄⋊C₄	C₂×Q₈×C₁₉	1	8	yes	—
199	C₄²	—	2	1	no	—
200	C₄²	—	2	1	no	—
201	C₄²	—	2	2	no	—
202	C₄²	—	2	2	no	—
203	C₄²	—	4	4	no	—
204	C₂³	C₂²×D₄×C₁₉	1	1	yes	characteristic
205	Q₈	—	4	4	no	—
206	Q₈	—	8	8	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	8	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	4	4	no	—
217	C₂×C₄	—	4	4	no	—
218	C₂×C₄	—	4	4	no	—
219	C₂×C₄	—	4	4	no	—
220	C₂×C₄	—	4	4	no	—
221	C₂×C₄	C₁₉×#137	1	4	yes	—
222	C₂×C₄	C₂²×Q₈×C₁₉	1	2	yes	—
223	C₂×C₄	C₁₉×H(5,𝔽₂)	1	1	yes	characteristic
224	C₂×C₄	C₁₉×H(5,𝔽₂)	1	1	yes	characteristic
225	C₂×C₄	C₁₉×2₋¹⁺⁴	1	2	yes	—
226	C₈	—	2	4	no	—
227	C₈	—	4	2	no	—
228	C₂²	C₁₉×#534	1	1	yes	characteristic
229	C₂²	D₄×Q₈×C₁₉	1	2	yes	—
230	C₂²	C₂×C₁₉×#137	1	2	yes	—
231	C₂²	C₁₉×#576	1	2	yes	—
232	C₄	—	2	4	no	—
233	C₄	—	4	1	no	—
234	C₄	—	4	2	no	—
235	C₄	—	4	1	no	—
236	C₄	—	4	8	no	—
237	C₄	—	8	4	no	—
238	C₄	C₁₉×#544	1	2	yes	—
239	C₄	Q₈²×C₁₉	1	1	yes	characteristic
240	C₄	C₁₉×#563	1	1	yes	characteristic
241	C₂	C₁₉×#2734	1	1	yes	characteristic
242	C₂	C₁₉×#3195	1	1	yes	characteristic
243	C₂	C₁₉×#3199	1	1	yes	characteristic
244	C₂	C₁₉×#3385	1	4	yes	—

	subgroup	G/H
center, socle, prime radical	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
45. C₁₉×#106	Q₈	semidirect
81. #32695	C₁₉	direct (N × H)
167. #106	Q₈×C₁₉	semidirect
173. C₁₉	#32695	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×#2567	2
C₁₉×#2086	2
C₁₉×#2057	2
C₁₉×#1985	2
C₁₉×#1944	2
C₁₉×#1920	2
#32695	19

Subgroups — C19×#32695 #18038654

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#32695 #18038654