Subgroups — C₁₉×#2628 #14560485

Order 2432.

488 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₁₉×#117	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#400	C₂	1	2	yes	maximal, complemented
4	C₁₉×#399	C₂	1	2	yes	maximal
5	C₁₉×#395	C₂	1	2	yes	maximal, complemented
6	C₁₉×#393	C₂	1	2	yes	maximal
7	C₁₉×#389	C₂	1	2	yes	maximal, complemented
8	C₁₉×#382	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₉×#380	C₂	1	2	yes	maximal
10	C₂²×Q₈×C₁₉	C₂²	1	2	yes	—
11	C₂³×C₄×C₁₉	C₂²	1	1	yes	characteristic
12	C₁₉×C₄⋊Q₈	—	2	4	no	—
13	C₁₉×#117	—	2	4	no	—
14	C₂×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
15	C₂×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₁₉	C₂²	1	4	yes	—
20	C₂×C₂²⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
21	C₂×C₂²⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊C₄×C₁₉	C₂²	1	4	yes	—
23	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
24	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
25	C₁₉×#95	C₂²	1	4	yes	—
26	C₁₉×#95	C₂²	1	2	yes	—
27	C₁₉×#95	C₂²	1	2	yes	—
28	C₁₉×#95	C₂²	1	4	yes	—
29	C₂⁴×C₁₉	C₂³	1	1	yes	characteristic
30	C₂×Q₈×C₁₉	—	2	4	no	—
31	C₂×Q₈×C₁₉	—	4	4	no	—
32	C₂²×C₄×C₁₉	—	2	2	no	—
33	C₂²×C₄×C₁₉	—	2	2	no	—
34	C₂²×C₄×C₁₉	Q₈	1	4	yes	complemented
35	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₁₉	C₂³	1	4	yes	—
38	C₂²×C₄×C₁₉	C₂³	1	4	yes	—
39	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
43	C₄⋊C₄×C₁₉	—	2	2	no	—
44	C₄⋊C₄×C₁₉	—	2	2	no	—
45	C₄⋊C₄×C₁₉	—	2	4	no	—
46	C₄⋊C₄×C₁₉	—	2	4	no	—
47	C₄⋊C₄×C₁₉	—	2	4	no	—
48	C₄⋊C₄×C₁₉	—	4	4	no	—
49	C₂²⋊C₄×C₁₉	—	2	2	no	—
50	C₂²⋊C₄×C₁₉	—	2	2	no	—
51	C₂²⋊C₄×C₁₉	—	4	4	no	—
52	C₄²×C₁₉	—	2	2	no	—
53	C₄²×C₁₉	D₄	1	4	yes	—
54	C₂³×C₁₉	—	2	1	no	—
55	C₂³×C₁₉	—	2	1	no	—
56	C₂³×C₁₉	—	2	2	no	—
57	C₂³×C₁₉	—	2	2	no	—
58	C₂³×C₁₉	C₂×Q₈	1	2	yes	—
59	C₂³×C₁₉	C₂⁴	1	1	yes	characteristic
60	Q₈×C₁₉	—	4	8	no	—
61	C₂×C₄×C₁₉	—	2	1	no	—
62	C₂×C₄×C₁₉	—	2	1	no	—
63	C₂×C₄×C₁₉	—	2	2	no	—
64	C₂×C₄×C₁₉	—	2	2	no	—
65	C₂×C₄×C₁₉	—	2	8	no	—
66	C₂×C₄×C₁₉	—	2	1	no	—
67	C₂×C₄×C₁₉	—	2	1	no	—
68	C₂×C₄×C₁₉	—	2	4	no	—
69	C₂×C₄×C₁₉	—	2	4	no	—
70	C₂×C₄×C₁₉	—	2	4	no	—
71	C₂×C₄×C₁₉	—	2	4	no	—
72	C₂×C₄×C₁₉	—	2	2	no	—
73	C₂×C₄×C₁₉	—	2	4	no	—
74	C₂×C₄×C₁₉	—	2	2	no	—
75	C₂×C₄×C₁₉	—	2	4	no	—
76	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
77	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
78	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
79	C₂×C₄×C₁₉	C₂×Q₈	1	2	yes	—
80	C₂×C₄×C₁₉	C₂×Q₈	1	2	yes	—
81	C₂×C₄×C₁₉	#41	1	2	yes	—
82	#2628	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
83	C₂²×C₁₉	—	2	2	no	—
84	C₂²×C₁₉	—	2	4	no	—
85	C₂²×C₁₉	—	4	1	no	—
86	C₂²×C₁₉	—	4	1	no	—
87	C₂²×C₁₉	—	4	2	no	—
88	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
89	C₂²×C₁₉	C₂²×Q₈	1	1	yes	characteristic
90	C₂²×C₁₉	C₂×#41	1	1	yes	characteristic
91	C₂²×C₁₉	H(5,𝔽₂)	1	2	yes	—
92	C₂²×C₁₉	2₋¹⁺⁴	1	2	yes	—
93	C₄×C₁₉	—	2	2	no	—
94	C₄×C₁₉	—	2	2	no	—
95	C₄×C₁₉	—	2	2	no	—
96	C₄×C₁₉	—	4	1	no	—
97	C₄×C₁₉	—	4	4	no	—
98	C₄×C₁₉	—	4	1	no	—
99	C₄×C₁₉	—	4	4	no	—
100	C₄×C₁₉	C₂²⋊Q₈	1	4	yes	—
101	C₂×C₄⋊Q₈	C₂×C₁₉	1	1	yes	characteristic, complemented
102	C₂×#117	C₂×C₁₉	1	1	yes	characteristic
103	#400	C₂×C₁₉	1	2	yes	complemented
104	#399	C₂×C₁₉	1	2	yes	—
105	#395	C₂×C₁₉	1	2	yes	complemented
106	#393	C₂×C₁₉	1	2	yes	—
107	#389	C₂×C₁₉	1	2	yes	complemented
108	#382	C₂×C₁₉	1	1	yes	characteristic, complemented
109	#380	C₂×C₁₉	1	2	yes	—
110	C₂×C₁₉	—	4	2	no	—
111	C₂×C₁₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
112	C₂×C₁₉	#535	1	1	yes	characteristic
113	C₂×C₁₉	#536	1	1	yes	characteristic
114	C₂×C₁₉	#542	1	2	yes	—
115	C₂×C₁₉	#544	1	2	yes	—
116	C₂²×Q₈	C₂²×C₁₉	1	2	yes	—
117	C₂³×C₄	C₂²×C₁₉	1	1	yes	characteristic
118	C₄⋊Q₈	—	2	4	no	—
119	#117	—	2	4	no	—
120	C₂×C₄⋊C₄	C₂²×C₁₉	1	1	yes	characteristic
121	C₂×C₄⋊C₄	C₂²×C₁₉	1	1	yes	characteristic
122	C₂×C₄⋊C₄	C₂²×C₁₉	1	2	yes	—
123	C₂×C₄⋊C₄	C₂²×C₁₉	1	2	yes	—
124	C₂×C₄⋊C₄	C₂²×C₁₉	1	2	yes	—
125	C₂×C₄⋊C₄	C₂²×C₁₉	1	4	yes	—
126	C₂×C₂²⋊C₄	C₂²×C₁₉	1	1	yes	characteristic
127	C₂×C₂²⋊C₄	C₂²×C₁₉	1	1	yes	characteristic
128	C₂×C₂²⋊C₄	C₂²×C₁₉	1	4	yes	—
129	C₂×C₄²	C₂²×C₁₉	1	1	yes	characteristic
130	C₂×C₄²	C₂²×C₁₉	1	1	yes	characteristic
131	#95	C₂²×C₁₉	1	4	yes	—
132	#95	C₂²×C₁₉	1	2	yes	—
133	#95	C₂²×C₁₉	1	2	yes	—
134	#95	C₂²×C₁₉	1	4	yes	—
135	C₁₉	#2628	1	1	yes	characteristic, direct product factor, Sylow
136	C₂⁴	C₂³×C₁₉	1	1	yes	characteristic
137	C₂×Q₈	—	2	4	no	—
138	C₂×Q₈	—	4	4	no	—
139	C₂²×C₄	—	2	2	no	—
140	C₂²×C₄	—	2	2	no	—
141	C₂²×C₄	Q₈×C₁₉	1	4	yes	complemented
142	C₂²×C₄	C₂³×C₁₉	1	1	yes	characteristic
143	C₂²×C₄	C₂³×C₁₉	1	4	yes	—
144	C₂²×C₄	C₂³×C₁₉	1	4	yes	—
145	C₂²×C₄	C₂³×C₁₉	1	1	yes	characteristic
146	C₂²×C₄	C₂³×C₁₉	1	1	yes	characteristic
147	C₂²×C₄	C₂³×C₁₉	1	1	yes	characteristic
148	C₂²×C₄	C₂³×C₁₉	1	1	yes	characteristic
149	C₂²×C₄	C₂³×C₁₉	1	1	yes	characteristic
150	C₄⋊C₄	—	2	2	no	—
151	C₄⋊C₄	—	2	2	no	—
152	C₄⋊C₄	—	2	4	no	—
153	C₄⋊C₄	—	2	4	no	—
154	C₄⋊C₄	—	2	4	no	—
155	C₄⋊C₄	—	4	4	no	—
156	C₂²⋊C₄	—	2	2	no	—
157	C₂²⋊C₄	—	2	2	no	—
158	C₂²⋊C₄	—	4	4	no	—
159	C₄²	—	2	2	no	—
160	C₄²	D₄×C₁₉	1	4	yes	—
161	C₂³	—	2	1	no	—
162	C₂³	—	2	1	no	—
163	C₂³	—	2	2	no	—
164	C₂³	—	2	2	no	—
165	C₂³	C₂×Q₈×C₁₉	1	2	yes	—
166	C₂³	C₂⁴×C₁₉	1	1	yes	characteristic
167	Q₈	—	4	8	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	2	1	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	8	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	2	4	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	4	no	—
183	C₂×C₄	C₂×D₄×C₁₉	1	2	yes	—
184	C₂×C₄	C₂×D₄×C₁₉	1	2	yes	—
185	C₂×C₄	C₂×D₄×C₁₉	1	2	yes	—
186	C₂×C₄	C₂×Q₈×C₁₉	1	2	yes	—
187	C₂×C₄	C₂×Q₈×C₁₉	1	2	yes	—
188	C₂×C₄	#41×C₁₉	1	2	yes	—
189	C₂²	—	2	2	no	—
190	C₂²	—	2	4	no	—
191	C₂²	—	4	1	no	—
192	C₂²	—	4	1	no	—
193	C₂²	—	4	2	no	—
194	C₂²	C₂²×D₄×C₁₉	1	1	yes	characteristic
195	C₂²	C₂²×Q₈×C₁₉	1	1	yes	characteristic
196	C₂²	C₂×#41×C₁₉	1	1	yes	characteristic
197	C₂²	C₁₉×H(5,𝔽₂)	1	2	yes	—
198	C₂²	C₁₉×2₋¹⁺⁴	1	2	yes	—
199	C₄	—	2	2	no	—
200	C₄	—	2	2	no	—
201	C₄	—	2	2	no	—
202	C₄	—	4	4	no	—
203	C₄	—	4	1	no	—
204	C₄	—	4	1	no	—
205	C₄	—	4	4	no	—
206	C₄	C₁₉×C₂²⋊Q₈	1	4	yes	—
207	C₂	—	4	2	no	—
208	C₂	C₂×C₁₉×C₂²⋊Q₈	1	1	yes	characteristic
209	C₂	C₁₉×#535	1	1	yes	characteristic
210	C₂	C₁₉×#536	1	1	yes	characteristic
211	C₂	C₁₉×#542	1	2	yes	—
212	C₂	C₁₉×#544	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₉×C₄⋊Q₈	C₂	semidirect
3. C₁₉×#400	C₂	semidirect
5. C₁₉×#395	C₂	semidirect
7. C₁₉×#389	C₂	semidirect
8. C₁₉×#382	C₂	semidirect
34. C₂²×C₄×C₁₉	Q₈	semidirect
82. #2628	C₁₉	direct (N × H)
101. C₂×C₄⋊Q₈	C₂×C₁₉	semidirect
103. #400	C₂×C₁₉	semidirect
105. #395	C₂×C₁₉	semidirect
107. #389	C₂×C₁₉	semidirect
108. #382	C₂×C₁₉	semidirect
135. C₁₉	#2628	direct (N × H)
141. C₂²×C₄	Q₈×C₁₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₉×C₄⋊Q₈	2
C₂×C₁₉×#117	2
C₁₉×#400	2
C₁₉×#399	2
C₁₉×#395	2
C₁₉×#393	2
C₁₉×#389	2
C₁₉×#382	2
C₁₉×#380	2
#2628	19

Subgroups — C19×#2628 #14560485

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#2628 #14560485