Subgroups — C₁₉×#3385 #14561242

Order 2432.

340 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈²×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₉×#554	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#501	C₂	1	2	yes	maximal, complemented
4	C₁₉×#500	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₉×#479	C₂	1	2	yes	maximal, complemented
6	C₁₉×#477	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#476	C₂	1	1	yes	maximal, characteristic
8	C₁₉×#474	C₂	1	2	yes	maximal
9	C₁₉×#446	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₉×#440	C₂	1	2	yes	maximal
11	C₄×SD₁₆×C₁₉	C₂	1	1	yes	maximal, characteristic
12	C₂×SD₁₆×C₁₉	C₂²	1	1	yes	characteristic
13	C₁₉×C₄⋊Q₈	—	2	1	no	—
14	C₁₉×C₄⋊Q₈	—	2	2	no	—
15	C₁₉×C₄⋊Q₈	C₂²	1	2	yes	—
16	C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
18	C₁₉×#125	C₂²	1	2	yes	—
19	C₁₉×C₂²⋊Q₈	—	2	1	no	—
20	C₁₉×C₂²⋊Q₈	—	2	2	no	—
21	C₄×Q₈×C₁₉	—	2	1	no	—
22	C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
23	C₄×Q₈×C₁₉	C₂²	1	2	yes	—
24	C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
25	C₄×D₄×C₁₉	C₂²	1	1	yes	characteristic
26	C₄×D₄×C₁₉	C₂²	1	2	yes	—
27	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
28	C₁₉×#107	C₂²	1	2	yes	—
29	C₁₉×#107	C₂²	1	2	yes	—
30	C₁₉×#107	C₂²	1	2	yes	—
31	C₁₉×#106	C₂²	1	1	yes	characteristic
32	C₁₉×#106	C₂²	1	2	yes	—
33	C₁₉×C₄⋊C₈	C₂²	1	2	yes	—
34	C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
35	C₁₉×#103	C₂²	1	1	yes	characteristic
36	C₁₉×#103	C₂²	1	2	yes	—
37	C₁₉×#102	C₂²	1	1	yes	characteristic
38	C₁₉×#102	C₂²	1	2	yes	—
39	C₁₉×#97	C₂²	1	2	yes	—
40	C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
41	C₂×Q₈×C₁₉	—	2	1	no	—
42	C₂×Q₈×C₁₉	—	2	1	no	—
43	C₂×Q₈×C₁₉	—	2	2	no	—
44	C₂×Q₈×C₁₉	D₄	1	1	yes	characteristic
45	C₂×Q₈×C₁₉	C₂³	1	1	yes	characteristic
46	C₂×D₄×C₁₉	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₁₉	—	2	1	no	—
48	C₂²×C₄×C₁₉	—	2	2	no	—
49	SD₁₆×C₁₉	Q₈	1	4	yes	complemented
50	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
51	C₂×C₈×C₁₉	C₂³	1	2	yes	—
52	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
53	C₄⋊C₄×C₁₉	—	2	2	no	—
54	C₄⋊C₄×C₁₉	—	2	1	no	—
55	C₄⋊C₄×C₁₉	—	2	2	no	—
56	C₄⋊C₄×C₁₉	—	2	1	no	—
57	C₄⋊C₄×C₁₉	—	2	1	no	—
58	C₄⋊C₄×C₁₉	—	2	2	no	—
59	C₄⋊C₄×C₁₉	—	2	2	no	—
60	C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
61	C₄⋊C₄×C₁₉	D₄	1	2	yes	—
62	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
63	C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
64	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
65	C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
66	C₂²⋊C₄×C₁₉	—	2	1	no	—
67	C₂²⋊C₄×C₁₉	—	2	2	no	—
68	C₄²×C₁₉	—	2	2	no	—
69	C₄²×C₁₉	—	2	1	no	—
70	C₄²×C₁₉	C₂³	1	2	yes	—
71	C₄²×C₁₉	C₂³	1	1	yes	characteristic
72	C₂³×C₁₉	—	2	1	no	—
73	Q₈×C₁₉	—	2	2	no	—
74	Q₈×C₁₉	—	2	1	no	—
75	Q₈×C₁₉	—	8	2	no	—
76	Q₈×C₁₉	—	8	1	no	—
77	Q₈×C₁₉	—	8	1	no	—
78	Q₈×C₁₉	C₂×Q₈	1	2	yes	—
79	D₄×C₁₉	—	2	1	no	—
80	D₄×C₁₉	C₂×Q₈	1	2	yes	—
81	C₂×C₄×C₁₉	—	2	1	no	—
82	C₂×C₄×C₁₉	—	2	2	no	—
83	C₂×C₄×C₁₉	—	2	1	no	—
84	C₂×C₄×C₁₉	—	2	1	no	—
85	C₂×C₄×C₁₉	—	2	2	no	—
86	C₂×C₄×C₁₉	—	4	2	no	—
87	C₂×C₄×C₁₉	—	8	2	no	—
88	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
89	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
91	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
92	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
93	C₈×C₁₉	—	2	1	no	—
94	C₈×C₁₉	—	2	2	no	—
95	C₈×C₁₉	C₂×Q₈	1	2	yes	—
96	#3385	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
97	C₂²×C₁₉	—	2	2	no	—
98	C₂²×C₁₉	—	4	1	no	—
99	C₂²×C₁₉	—	4	1	no	—
100	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
101	C₄×C₁₉	—	2	2	no	—
102	C₄×C₁₉	—	2	2	no	—
103	C₄×C₁₉	—	2	2	no	—
104	C₄×C₁₉	—	2	1	no	—
105	C₄×C₁₉	—	4	1	no	—
106	C₄×C₁₉	—	4	2	no	—
107	C₄×C₁₉	—	4	1	no	—
108	C₄×C₁₉	—	4	2	no	—
109	C₄×C₁₉	#137	1	2	yes	—
110	C₄×C₁₉	C₂²×Q₈	1	1	yes	characteristic
111	C₄×C₁₉	2₋¹⁺⁴	1	1	yes	characteristic
112	Q₈²	C₂×C₁₉	1	1	yes	characteristic, complemented
113	#554	C₂×C₁₉	1	1	yes	characteristic
114	#501	C₂×C₁₉	1	2	yes	complemented
115	#500	C₂×C₁₉	1	1	yes	characteristic, complemented
116	#479	C₂×C₁₉	1	2	yes	complemented
117	#477	C₂×C₁₉	1	1	yes	characteristic, complemented
118	#476	C₂×C₁₉	1	1	yes	characteristic
119	#474	C₂×C₁₉	1	2	yes	—
120	#446	C₂×C₁₉	1	1	yes	characteristic, complemented
121	#440	C₂×C₁₉	1	2	yes	—
122	C₄×SD₁₆	C₂×C₁₉	1	1	yes	characteristic
123	C₂×C₁₉	—	4	2	no	—
124	C₂×C₁₉	D₄×Q₈	1	1	yes	characteristic
125	C₂×C₁₉	C₂×#137	1	1	yes	characteristic
126	C₂×C₁₉	#576	1	1	yes	characteristic
127	C₂×SD₁₆	C₂²×C₁₉	1	1	yes	characteristic
128	C₄⋊Q₈	—	2	1	no	—
129	C₄⋊Q₈	—	2	2	no	—
130	C₄⋊Q₈	C₂²×C₁₉	1	1	yes	characteristic
131	C₄⋊Q₈	C₂²×C₁₉	1	2	yes	—
132	C₄⋊Q₈	C₂²×C₁₉	1	1	yes	characteristic
133	#125	C₂²×C₁₉	1	2	yes	—
134	C₂²⋊Q₈	—	2	1	no	—
135	C₂²⋊Q₈	—	2	2	no	—
136	C₄×Q₈	—	2	1	no	—
137	C₄×Q₈	C₂²×C₁₉	1	1	yes	characteristic
138	C₄×Q₈	C₂²×C₁₉	1	2	yes	—
139	C₄×Q₈	C₂²×C₁₉	1	1	yes	characteristic
140	C₄×D₄	C₂²×C₁₉	1	1	yes	characteristic
141	C₄×D₄	C₂²×C₁₉	1	2	yes	—
142	C₂×C₄⋊C₄	—	2	1	no	—
143	#107	C₂²×C₁₉	1	2	yes	—
144	#107	C₂²×C₁₉	1	2	yes	—
145	#107	C₂²×C₁₉	1	2	yes	—
146	#106	C₂²×C₁₉	1	1	yes	characteristic
147	#106	C₂²×C₁₉	1	2	yes	—
148	C₄⋊C₈	C₂²×C₁₉	1	2	yes	—
149	C₄⋊C₈	C₂²×C₁₉	1	1	yes	characteristic
150	#103	C₂²×C₁₉	1	1	yes	characteristic
151	#103	C₂²×C₁₉	1	2	yes	—
152	#102	C₂²×C₁₉	1	1	yes	characteristic
153	#102	C₂²×C₁₉	1	2	yes	—
154	#97	C₂²×C₁₉	1	2	yes	—
155	C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
156	C₁₉	#3385	1	1	yes	characteristic, direct product factor, Sylow
157	C₂×Q₈	—	2	1	no	—
158	C₂×Q₈	—	2	1	no	—
159	C₂×Q₈	—	2	2	no	—
160	C₂×Q₈	D₄×C₁₉	1	1	yes	characteristic
161	C₂×Q₈	C₂³×C₁₉	1	1	yes	characteristic
162	C₂×D₄	C₂³×C₁₉	1	1	yes	characteristic
163	C₂²×C₄	—	2	1	no	—
164	C₂²×C₄	—	2	2	no	—
165	SD₁₆	Q₈×C₁₉	1	4	yes	complemented
166	C₂×C₈	C₂³×C₁₉	1	1	yes	characteristic
167	C₂×C₈	C₂³×C₁₉	1	2	yes	—
168	C₂×C₈	C₂³×C₁₉	1	1	yes	characteristic
169	C₄⋊C₄	—	2	2	no	—
170	C₄⋊C₄	—	2	1	no	—
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	2	1	no	—
173	C₄⋊C₄	—	2	1	no	—
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	—	2	2	no	—
176	C₄⋊C₄	D₄×C₁₉	1	1	yes	characteristic
177	C₄⋊C₄	D₄×C₁₉	1	2	yes	—
178	C₄⋊C₄	C₂³×C₁₉	1	1	yes	characteristic
179	C₄⋊C₄	C₂³×C₁₉	1	2	yes	—
180	C₄⋊C₄	C₂³×C₁₉	1	1	yes	characteristic
181	C₄⋊C₄	C₂³×C₁₉	1	2	yes	—
182	C₂²⋊C₄	—	2	1	no	—
183	C₂²⋊C₄	—	2	2	no	—
184	C₄²	—	2	2	no	—
185	C₄²	—	2	1	no	—
186	C₄²	C₂³×C₁₉	1	1	yes	characteristic
187	C₄²	C₂³×C₁₉	1	2	yes	—
188	C₂³	—	2	1	no	—
189	Q₈	—	2	2	no	—
190	Q₈	—	2	1	no	—
191	Q₈	—	8	2	no	—
192	Q₈	—	8	1	no	—
193	Q₈	—	8	1	no	—
194	Q₈	C₂×Q₈×C₁₉	1	2	yes	—
195	D₄	—	2	1	no	—
196	D₄	C₂×Q₈×C₁₉	1	2	yes	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	4	2	no	—
203	C₂×C₄	—	8	2	no	—
204	C₂×C₄	C₂×D₄×C₁₉	1	2	yes	—
205	C₂×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
206	C₂×C₄	C₂×D₄×C₁₉	1	2	yes	—
207	C₂×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
208	C₂×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
209	C₈	—	2	1	no	—
210	C₈	—	2	2	no	—
211	C₈	C₂×Q₈×C₁₉	1	2	yes	—
212	C₂²	—	2	2	no	—
213	C₂²	—	4	1	no	—
214	C₂²	—	4	1	no	—
215	C₂²	C₂²×D₄×C₁₉	1	1	yes	characteristic
216	C₄	—	2	2	no	—
217	C₄	—	2	2	no	—
218	C₄	—	2	2	no	—
219	C₄	—	2	1	no	—
220	C₄	—	4	1	no	—
221	C₄	—	4	2	no	—
222	C₄	—	4	1	no	—
223	C₄	—	4	2	no	—
224	C₄	C₁₉×#137	1	2	yes	—
225	C₄	C₂²×Q₈×C₁₉	1	1	yes	characteristic
226	C₄	C₁₉×2₋¹⁺⁴	1	1	yes	characteristic
227	C₂	—	4	2	no	—
228	C₂	D₄×Q₈×C₁₉	1	1	yes	characteristic
229	C₂	C₂×C₁₉×#137	1	1	yes	characteristic
230	C₂	C₁₉×#576	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₉
prime radical	C₂×D₄×C₁₉	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₄×Q₈×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. Q₈²×C₁₉	C₂	semidirect
3. C₁₉×#501	C₂	semidirect
4. C₁₉×#500	C₂	semidirect
5. C₁₉×#479	C₂	semidirect
6. C₁₉×#477	C₂	semidirect
9. C₁₉×#446	C₂	semidirect
49. SD₁₆×C₁₉	Q₈	semidirect
96. #3385	C₁₉	direct (N × H)
112. Q₈²	C₂×C₁₉	semidirect
114. #501	C₂×C₁₉	semidirect
115. #500	C₂×C₁₉	semidirect
116. #479	C₂×C₁₉	semidirect
117. #477	C₂×C₁₉	semidirect
120. #446	C₂×C₁₉	semidirect
156. C₁₉	#3385	direct (N × H)
165. SD₁₆	Q₈×C₁₉	semidirect

Maximal subgroups

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

subgroup	index
Q₈²×C₁₉	2
C₁₉×#554	2
C₁₉×#501	2
C₁₉×#500	2
C₁₉×#479	2
C₁₉×#477	2
C₁₉×#476	2
C₁₉×#474	2
C₁₉×#446	2
C₁₉×#440	2
C₄×SD₁₆×C₁₉	2
#3385	19

Subgroups — C19×#3385 #14561242

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#3385 #14561242