Subgroups — C₁₉×#3503 #14561360

Order 2432.

1060 conjugacy classes of subgroups.

The subgroup lattice has 198 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	2	yes	maximal, complemented
2	C₁₉×#557	C₂	1	2	yes	maximal, complemented
3	C₁₉×#554	C₂	1	6	yes	maximal, complemented
4	D₄×Q₈×C₁₉	C₂	1	6	yes	maximal, complemented
5	C₁₉×#544	C₂	1	6	yes	maximal, complemented
6	C₁₉×#533	C₂	1	3	yes	maximal, complemented
7	C₂×C₁₉×C₄⋊Q₈	C₂	1	3	yes	maximal, complemented
8	C₁₉×#520	C₂	1	2	yes	maximal
9	C₄×#41×C₁₉	C₂	1	1	yes	maximal, characteristic
10	C₂×#41×C₁₉	C₂²	1	1	yes	characteristic
11	C₂²×Q₈×C₁₉	C₂²	1	6	yes	—
12	C₁₉×C₄⋊Q₈	C₂²	1	3	yes	—
13	C₁₉×C₄⋊Q₈	C₂²	1	6	yes	—
14	C₁₉×C₄⋊Q₈	C₂²	1	6	yes	—
15	C₁₉×C₄⋊Q₈	C₂²	1	6	yes	—
16	C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₁₉×C₄⋊Q₈	C₂²	1	12	yes	—
18	C₁₉×#125	C₂²	1	6	yes	—
19	C₁₉×#125	C₂²	1	12	yes	—
20	C₁₉×C₂²⋊Q₈	C₂²	1	6	yes	—
21	C₁₉×C₂²⋊Q₈	C₂²	1	6	yes	—
22	C₁₉×C₂²⋊Q₈	C₂²	1	24	yes	—
23	C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
24	C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
25	C₄×Q₈×C₁₉	C₂²	1	4	yes	—
26	C₄×Q₈×C₁₉	C₂²	1	6	yes	—
27	C₄×Q₈×C₁₉	C₂²	1	6	yes	—
28	C₄×D₄×C₁₉	C₂²	1	3	yes	—
29	C₄×D₄×C₁₉	C₂²	1	3	yes	—
30	C₄×D₄×C₁₉	C₂²	1	12	yes	—
31	C₁₉×#117	C₂²	1	3	yes	—
32	C₁₉×#117	C₂²	1	6	yes	—
33	C₂×C₄⋊C₄×C₁₉	C₂²	1	6	yes	—
34	C₂×C₄⋊C₄×C₁₉	C₂²	1	6	yes	—
35	C₂×C₄²×C₁₉	C₂²	1	3	yes	—
36	#41×C₁₉	Q₈	1	8	yes	complemented
37	C₂×Q₈×C₁₉	—	4	12	no	—
38	C₂×Q₈×C₁₉	C₂³	1	2	yes	—
39	C₂×Q₈×C₁₉	C₂³	1	6	yes	—
40	C₂×Q₈×C₁₉	C₂³	1	12	yes	—
41	C₂×Q₈×C₁₉	C₂³	1	1	yes	characteristic
42	C₂×D₄×C₁₉	C₂³	1	3	yes	—
43	C₂²×C₄×C₁₉	C₂³	1	3	yes	—
44	C₂²×C₄×C₁₉	C₂³	1	3	yes	—
45	C₂²×C₄×C₁₉	C₂³	1	12	yes	—
46	C₂²×C₄×C₁₉	C₂³	1	3	yes	—
47	C₄⋊C₄×C₁₉	C₂³	1	6	yes	—
48	C₄⋊C₄×C₁₉	C₂³	1	6	yes	—
49	C₄⋊C₄×C₁₉	C₂³	1	12	yes	—
50	C₄⋊C₄×C₁₉	C₂³	1	12	yes	—
51	C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
52	C₄⋊C₄×C₁₉	C₂³	1	3	yes	—
53	C₄⋊C₄×C₁₉	C₂³	1	3	yes	—
54	C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
55	C₄⋊C₄×C₁₉	C₂³	1	12	yes	—
56	C₄⋊C₄×C₁₉	C₂³	1	12	yes	—
57	C₂²⋊C₄×C₁₉	C₂³	1	12	yes	—
58	C₂²⋊C₄×C₁₉	C₂³	1	6	yes	—
59	C₄²×C₁₉	C₂³	1	1	yes	characteristic
60	C₄²×C₁₉	C₂³	1	3	yes	—
61	C₄²×C₁₉	C₂³	1	3	yes	—
62	C₄²×C₁₉	C₂³	1	3	yes	—
63	C₄²×C₁₉	C₂³	1	12	yes	—
64	C₂³×C₁₉	C₂⁴	1	3	yes	—
65	Q₈×C₁₉	—	2	12	no	—
66	Q₈×C₁₉	—	2	4	no	—
67	Q₈×C₁₉	—	4	12	no	—
68	Q₈×C₁₉	C₂×Q₈	1	4	yes	complemented
69	D₄×C₁₉	C₂×Q₈	1	12	yes	complemented
70	C₂×C₄×C₁₉	—	2	6	no	—
71	C₂×C₄×C₁₉	—	2	6	no	—
72	C₂×C₄×C₁₉	—	4	12	no	—
73	C₂×C₄×C₁₉	C₂×Q₈	1	12	yes	complemented
74	C₂×C₄×C₁₉	C₂⁴	1	4	yes	—
75	C₂×C₄×C₁₉	C₂⁴	1	3	yes	—
76	C₂×C₄×C₁₉	C₂⁴	1	3	yes	—
77	C₂×C₄×C₁₉	C₂⁴	1	12	yes	—
78	C₂×C₄×C₁₉	C₂⁴	1	3	yes	—
79	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
80	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
81	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
82	#3503	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
83	C₂²×C₁₉	—	2	3	no	—
84	C₂²×C₁₉	—	2	3	no	—
85	C₂²×C₁₉	C₂²×Q₈	1	6	yes	—
86	C₂²×C₁₉	C₂⁵	1	1	yes	characteristic
87	C₄×C₁₉	—	2	3	no	—
88	C₄×C₁₉	—	2	12	no	—
89	C₄×C₁₉	—	2	3	no	—
90	C₄×C₁₉	—	2	4	no	—
91	C₄×C₁₉	C₂²×Q₈	1	2	yes	—
92	C₄×C₁₉	C₂²×Q₈	1	6	yes	—
93	C₄×C₁₉	H(5,𝔽₂)	1	2	yes	—
94	C₄×C₁₉	2₋¹⁺⁴	1	2	yes	—
95	Q₈²	C₂×C₁₉	1	2	yes	complemented
96	#557	C₂×C₁₉	1	2	yes	complemented
97	#554	C₂×C₁₉	1	6	yes	complemented
98	D₄×Q₈	C₂×C₁₉	1	6	yes	complemented
99	#544	C₂×C₁₉	1	6	yes	complemented
100	#533	C₂×C₁₉	1	3	yes	complemented
101	C₂×C₄⋊Q₈	C₂×C₁₉	1	3	yes	complemented
102	#520	C₂×C₁₉	1	2	yes	—
103	C₄×#41	C₂×C₁₉	1	1	yes	characteristic
104	C₂×C₁₉	—	2	6	no	—
105	C₂×C₁₉	C₂³×Q₈	1	1	yes	characteristic
106	C₂×C₁₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
107	C₂×C₁₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
108	C₂×#41	C₂²×C₁₉	1	1	yes	characteristic
109	C₂²×Q₈	C₂²×C₁₉	1	6	yes	—
110	C₄⋊Q₈	C₂²×C₁₉	1	3	yes	—
111	C₄⋊Q₈	C₂²×C₁₉	1	6	yes	—
112	C₄⋊Q₈	C₂²×C₁₉	1	6	yes	—
113	C₄⋊Q₈	C₂²×C₁₉	1	6	yes	—
114	C₄⋊Q₈	C₂²×C₁₉	1	1	yes	characteristic
115	C₄⋊Q₈	C₂²×C₁₉	1	12	yes	—
116	#125	C₂²×C₁₉	1	6	yes	—
117	#125	C₂²×C₁₉	1	12	yes	—
118	C₂²⋊Q₈	C₂²×C₁₉	1	6	yes	—
119	C₂²⋊Q₈	C₂²×C₁₉	1	6	yes	—
120	C₂²⋊Q₈	C₂²×C₁₉	1	24	yes	—
121	C₄×Q₈	C₂²×C₁₉	1	1	yes	characteristic
122	C₄×Q₈	C₂²×C₁₉	1	1	yes	characteristic
123	C₄×Q₈	C₂²×C₁₉	1	4	yes	—
124	C₄×Q₈	C₂²×C₁₉	1	6	yes	—
125	C₄×Q₈	C₂²×C₁₉	1	6	yes	—
126	C₄×D₄	C₂²×C₁₉	1	3	yes	—
127	C₄×D₄	C₂²×C₁₉	1	3	yes	—
128	C₄×D₄	C₂²×C₁₉	1	12	yes	—
129	#117	C₂²×C₁₉	1	3	yes	—
130	#117	C₂²×C₁₉	1	6	yes	—
131	C₂×C₄⋊C₄	C₂²×C₁₉	1	6	yes	—
132	C₂×C₄⋊C₄	C₂²×C₁₉	1	6	yes	—
133	C₂×C₄²	C₂²×C₁₉	1	3	yes	—
134	C₁₉	#3503	1	1	yes	characteristic, direct product factor, Sylow
135	#41	Q₈×C₁₉	1	8	yes	complemented
136	C₂×Q₈	—	4	12	no	—
137	C₂×Q₈	C₂³×C₁₉	1	2	yes	—
138	C₂×Q₈	C₂³×C₁₉	1	6	yes	—
139	C₂×Q₈	C₂³×C₁₉	1	12	yes	—
140	C₂×Q₈	C₂³×C₁₉	1	1	yes	characteristic
141	C₂×D₄	C₂³×C₁₉	1	3	yes	—
142	C₂²×C₄	C₂³×C₁₉	1	3	yes	—
143	C₂²×C₄	C₂³×C₁₉	1	3	yes	—
144	C₂²×C₄	C₂³×C₁₉	1	12	yes	—
145	C₂²×C₄	C₂³×C₁₉	1	3	yes	—
146	C₄⋊C₄	C₂³×C₁₉	1	6	yes	—
147	C₄⋊C₄	C₂³×C₁₉	1	6	yes	—
148	C₄⋊C₄	C₂³×C₁₉	1	12	yes	—
149	C₄⋊C₄	C₂³×C₁₉	1	12	yes	—
150	C₄⋊C₄	C₂³×C₁₉	1	2	yes	—
151	C₄⋊C₄	C₂³×C₁₉	1	3	yes	—
152	C₄⋊C₄	C₂³×C₁₉	1	3	yes	—
153	C₄⋊C₄	C₂³×C₁₉	1	2	yes	—
154	C₄⋊C₄	C₂³×C₁₉	1	12	yes	—
155	C₄⋊C₄	C₂³×C₁₉	1	12	yes	—
156	C₂²⋊C₄	C₂³×C₁₉	1	12	yes	—
157	C₂²⋊C₄	C₂³×C₁₉	1	6	yes	—
158	C₄²	C₂³×C₁₉	1	1	yes	characteristic
159	C₄²	C₂³×C₁₉	1	3	yes	—
160	C₄²	C₂³×C₁₉	1	3	yes	—
161	C₄²	C₂³×C₁₉	1	3	yes	—
162	C₄²	C₂³×C₁₉	1	12	yes	—
163	C₂³	C₂⁴×C₁₉	1	3	yes	—
164	Q₈	—	2	12	no	—
165	Q₈	—	2	4	no	—
166	Q₈	—	4	12	no	—
167	Q₈	C₂×Q₈×C₁₉	1	4	yes	complemented
168	D₄	C₂×Q₈×C₁₉	1	12	yes	complemented
169	C₂×C₄	—	2	6	no	—
170	C₂×C₄	—	2	6	no	—
171	C₂×C₄	—	4	12	no	—
172	C₂×C₄	C₂×Q₈×C₁₉	1	12	yes	complemented
173	C₂×C₄	C₂⁴×C₁₉	1	4	yes	—
174	C₂×C₄	C₂⁴×C₁₉	1	3	yes	—
175	C₂×C₄	C₂⁴×C₁₉	1	3	yes	—
176	C₂×C₄	C₂⁴×C₁₉	1	12	yes	—
177	C₂×C₄	C₂⁴×C₁₉	1	3	yes	—
178	C₂×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
179	C₂×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
180	C₂×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
181	C₂²	—	2	3	no	—
182	C₂²	—	2	3	no	—
183	C₂²	C₂²×Q₈×C₁₉	1	6	yes	—
184	C₂²	C₂⁵×C₁₉	1	1	yes	characteristic
185	C₄	—	2	3	no	—
186	C₄	—	2	12	no	—
187	C₄	—	2	3	no	—
188	C₄	—	2	4	no	—
189	C₄	C₂²×Q₈×C₁₉	1	2	yes	—
190	C₄	C₂²×Q₈×C₁₉	1	6	yes	—
191	C₄	C₁₉×H(5,𝔽₂)	1	2	yes	—
192	C₄	C₁₉×2₋¹⁺⁴	1	2	yes	—
193	C₂	—	2	6	no	—
194	C₂	C₂³×Q₈×C₁₉	1	1	yes	characteristic
195	C₂	C₂×C₁₉×H(5,𝔽₂)	1	1	yes	characteristic
196	C₂	C₂×C₁₉×2₋¹⁺⁴	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁵×C₁₉
prime radical	C₂×#41×C₁₉	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. Q₈²×C₁₉	C₂	semidirect
2. C₁₉×#557	C₂	semidirect
3. C₁₉×#554	C₂	semidirect
4. D₄×Q₈×C₁₉	C₂	semidirect
5. C₁₉×#544	C₂	semidirect
6. C₁₉×#533	C₂	semidirect
7. C₂×C₁₉×C₄⋊Q₈	C₂	semidirect
36. #41×C₁₉	Q₈	semidirect
68. Q₈×C₁₉	C₂×Q₈	semidirect
69. D₄×C₁₉	C₂×Q₈	semidirect
73. C₂×C₄×C₁₉	C₂×Q₈	semidirect
82. #3503	C₁₉	direct (N × H)
95. Q₈²	C₂×C₁₉	semidirect
96. #557	C₂×C₁₉	semidirect
97. #554	C₂×C₁₉	semidirect
98. D₄×Q₈	C₂×C₁₉	semidirect
99. #544	C₂×C₁₉	semidirect
100. #533	C₂×C₁₉	semidirect
101. C₂×C₄⋊Q₈	C₂×C₁₉	semidirect
134. C₁₉	#3503	direct (N × H)
135. #41	Q₈×C₁₉	semidirect
167. Q₈	C₂×Q₈×C₁₉	semidirect
168. D₄	C₂×Q₈×C₁₉	semidirect
172. C₂×C₄	C₂×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×#557	2
C₁₉×#554	2
D₄×Q₈×C₁₉	2
C₁₉×#544	2
C₁₉×#533	2
C₂×C₁₉×C₄⋊Q₈	2
C₁₉×#520	2
C₄×#41×C₁₉	2
#3503	19

Subgroups — C19×#3503 #14561360

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#3503 #14561360