Subgroups — C₂×C₁₉×#2381 #18040304

Order 4864.

2326 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2381	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₁₉×#391	C₂	1	4	yes	maximal
4	C₂×C₁₉×#387	C₂	1	8	yes	maximal
5	C₂×C₁₉×#383	C₂	1	2	yes	maximal
6	C₂³×Q₈×C₁₉	C₂²	1	2	yes	—
7	C₂×C₁₉×C₄⋊Q₈	C₄	1	16	yes	complemented
8	C₂×C₁₉×C₄⋊Q₈	C₂²	1	8	yes	—
9	C₂²×C₄⋊C₄×C₁₉	C₂²	1	4	yes	—
10	C₂²×C₄⋊C₄×C₁₉	C₂²	1	4	yes	—
11	C₂²×C₄²×C₁₉	C₂²	1	4	yes	—
12	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
13	C₁₉×#391	C₂²	1	32	yes	—
14	C₁₉×#387	C₂²	1	64	yes	—
15	C₁₉×#383	C₂²	1	16	yes	—
16	C₂×C₁₉×#95	C₂²	1	8	yes	—
17	C₂×C₁₉×#95	C₂²	1	4	yes	—
18	C₂×C₁₉×#95	C₂²	1	8	yes	—
19	C₂²×Q₈×C₁₉	—	4	4	no	—
20	C₂²×Q₈×C₁₉	—	4	4	no	—
21	C₂²×Q₈×C₁₉	C₂×C₄	1	16	yes	—
22	C₂²×Q₈×C₁₉	C₂³	1	8	yes	—
23	C₂³×C₄×C₁₉	C₂³	1	8	yes	—
24	C₂³×C₄×C₁₉	C₂³	1	2	yes	—
25	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
26	C₂³×C₄×C₁₉	C₂³	1	4	yes	—
27	C₁₉×C₄⋊Q₈	C₂×C₄	1	64	yes	complemented
28	C₂×C₄⋊C₄×C₁₉	—	4	8	no	—
29	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	32	yes	—
30	C₂×C₄⋊C₄×C₁₉	C₂³	1	16	yes	—
31	C₂×C₄⋊C₄×C₁₉	C₂³	1	16	yes	—
32	C₂×C₄²×C₁₉	—	4	8	no	—
33	C₂×C₄²×C₁₉	C₂×C₄	1	8	yes	—
34	C₂×C₄²×C₁₉	C₂³	1	16	yes	—
35	C₂×C₄²×C₁₉	C₂³	1	4	yes	—
36	C₁₉×#95	C₂³	1	32	yes	—
37	C₁₉×#95	C₂³	1	16	yes	—
38	C₁₉×#95	C₂³	1	32	yes	—
39	C₂⁴×C₁₉	C₂⁴	1	1	yes	characteristic
40	C₂×Q₈×C₁₉	—	4	32	no	—
41	C₂×Q₈×C₁₉	—	4	8	no	—
42	C₂×Q₈×C₁₉	—	4	8	no	—
43	C₂×Q₈×C₁₉	C₂²×C₄	1	32	yes	—
44	C₂²×C₄×C₁₉	—	2	8	no	—
45	C₂²×C₄×C₁₉	—	2	8	no	—
46	C₂²×C₄×C₁₉	—	2	4	no	—
47	C₂²×C₄×C₁₉	—	2	8	no	—
48	C₂²×C₄×C₁₉	—	2	16	no	—
49	C₂²×C₄×C₁₉	—	2	32	no	—
50	C₂²×C₄×C₁₉	C₂²×C₄	1	8	yes	—
51	C₂²×C₄×C₁₉	C₂²×C₄	1	4	yes	—
52	C₂²×C₄×C₁₉	C₂²×C₄	1	16	yes	—
53	C₂²×C₄×C₁₉	C₂⁴	1	16	yes	—
54	C₂²×C₄×C₁₉	C₂⁴	1	8	yes	—
55	C₂²×C₄×C₁₉	C₂⁴	1	4	yes	—
56	C₂²×C₄×C₁₉	C₂⁴	1	2	yes	—
57	C₄⋊C₄×C₁₉	—	4	16	no	—
58	C₄⋊C₄×C₁₉	C₂²×C₄	1	64	yes	—
59	C₄²×C₁₉	—	4	16	no	—
60	C₄²×C₁₉	C₂²×C₄	1	16	yes	—
61	C₂×#2381	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
62	C₂³×C₁₉	C₂³×C₄	1	2	yes	—
63	C₂³×C₁₉	H(5,𝔽₂)	1	4	yes	—
64	C₂³×C₁₉	2₋¹⁺⁴	1	8	yes	—
65	C₂³×C₁₉	C₂⁵	1	1	yes	characteristic
66	Q₈×C₁₉	—	4	32	no	—
67	C₂×C₄×C₁₉	—	2	32	no	—
68	C₂×C₄×C₁₉	—	2	8	no	—
69	C₂×C₄×C₁₉	—	2	16	no	—
70	C₂×C₄×C₁₉	—	2	8	no	—
71	C₂×C₄×C₁₉	—	2	8	no	—
72	C₂×C₄×C₁₉	—	2	4	no	—
73	C₂×C₄×C₁₉	—	2	8	no	—
74	C₂×C₄×C₁₉	—	2	16	no	—
75	C₂×C₄×C₁₉	—	2	32	no	—
76	C₂×C₄×C₁₉	—	4	32	no	—
77	C₂×C₄×C₁₉	C₂³×C₄	1	8	yes	—
78	C₂×C₄×C₁₉	C₂³×C₄	1	4	yes	—
79	C₂×C₄×C₁₉	C₂³×C₄	1	16	yes	—
80	C₂²×C₄⋊Q₈	C₂×C₁₉	1	1	yes	characteristic
81	#2381	C₂×C₁₉	1	16	yes	direct product factor
82	C₂×#391	C₂×C₁₉	1	4	yes	—
83	C₂×#387	C₂×C₁₉	1	8	yes	—
84	C₂×#383	C₂×C₁₉	1	2	yes	—
85	C₂²×C₁₉	#518	1	4	yes	—
86	C₂²×C₁₉	#519	1	8	yes	—
87	C₂²×C₁₉	#563	1	16	yes	—
88	C₂²×C₁₉	C₂⁴×C₄	1	1	yes	characteristic
89	C₂²×C₁₉	C₂×H(5,𝔽₂)	1	2	yes	—
90	C₂²×C₁₉	C₂×2₋¹⁺⁴	1	4	yes	—
91	C₄×C₁₉	—	2	16	no	—
92	C₄×C₁₉	—	2	4	no	—
93	C₄×C₁₉	—	2	8	no	—
94	C₄×C₁₉	—	4	16	no	—
95	C₂³×Q₈	C₂²×C₁₉	1	2	yes	—
96	C₂×C₄⋊Q₈	C₄×C₁₉	1	16	yes	complemented
97	C₂×C₄⋊Q₈	C₂²×C₁₉	1	8	yes	—
98	C₂²×C₄⋊C₄	C₂²×C₁₉	1	4	yes	—
99	C₂²×C₄⋊C₄	C₂²×C₁₉	1	4	yes	—
100	C₂²×C₄²	C₂²×C₁₉	1	4	yes	—
101	C₂²×C₄²	C₂²×C₁₉	1	1	yes	characteristic
102	#391	C₂²×C₁₉	1	32	yes	—
103	#387	C₂²×C₁₉	1	64	yes	—
104	#383	C₂²×C₁₉	1	16	yes	—
105	C₂×#95	C₂²×C₁₉	1	8	yes	—
106	C₂×#95	C₂²×C₁₉	1	4	yes	—
107	C₂×#95	C₂²×C₁₉	1	8	yes	—
108	C₂×C₁₉	#2381	1	8	yes	direct product factor
109	C₂×C₁₉	C₂×#518	1	1	yes	characteristic
110	C₂×C₁₉	C₂×#519	1	2	yes	—
111	C₂×C₁₉	C₂×#563	1	4	yes	—
112	C₂²×Q₈	—	4	4	no	—
113	C₂²×Q₈	—	4	4	no	—
114	C₂²×Q₈	C₂×C₄×C₁₉	1	16	yes	—
115	C₂²×Q₈	C₂³×C₁₉	1	8	yes	—
116	C₂³×C₄	C₂³×C₁₉	1	8	yes	—
117	C₂³×C₄	C₂³×C₁₉	1	4	yes	—
118	C₂³×C₄	C₂³×C₁₉	1	2	yes	—
119	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
120	C₄⋊Q₈	C₂×C₄×C₁₉	1	64	yes	complemented
121	C₂×C₄⋊C₄	—	4	8	no	—
122	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	32	yes	—
123	C₂×C₄⋊C₄	C₂³×C₁₉	1	16	yes	—
124	C₂×C₄⋊C₄	C₂³×C₁₉	1	16	yes	—
125	C₂×C₄²	—	4	8	no	—
126	C₂×C₄²	C₂×C₄×C₁₉	1	8	yes	—
127	C₂×C₄²	C₂³×C₁₉	1	16	yes	—
128	C₂×C₄²	C₂³×C₁₉	1	4	yes	—
129	#95	C₂³×C₁₉	1	32	yes	—
130	#95	C₂³×C₁₉	1	16	yes	—
131	#95	C₂³×C₁₉	1	32	yes	—
132	C₁₉	C₂×#2381	1	1	yes	characteristic, direct product factor, Sylow
133	C₂⁴	C₂⁴×C₁₉	1	1	yes	characteristic
134	C₂×Q₈	—	4	32	no	—
135	C₂×Q₈	—	4	8	no	—
136	C₂×Q₈	—	4	8	no	—
137	C₂×Q₈	C₂²×C₄×C₁₉	1	32	yes	—
138	C₂²×C₄	—	2	8	no	—
139	C₂²×C₄	—	2	8	no	—
140	C₂²×C₄	—	2	4	no	—
141	C₂²×C₄	—	2	8	no	—
142	C₂²×C₄	—	2	16	no	—
143	C₂²×C₄	—	2	32	no	—
144	C₂²×C₄	C₂²×C₄×C₁₉	1	8	yes	—
145	C₂²×C₄	C₂²×C₄×C₁₉	1	4	yes	—
146	C₂²×C₄	C₂²×C₄×C₁₉	1	16	yes	—
147	C₂²×C₄	C₂⁴×C₁₉	1	16	yes	—
148	C₂²×C₄	C₂⁴×C₁₉	1	4	yes	—
149	C₂²×C₄	C₂⁴×C₁₉	1	2	yes	—
150	C₂²×C₄	C₂⁴×C₁₉	1	8	yes	—
151	C₄⋊C₄	—	4	16	no	—
152	C₄⋊C₄	C₂²×C₄×C₁₉	1	64	yes	—
153	C₄²	—	4	16	no	—
154	C₄²	C₂²×C₄×C₁₉	1	16	yes	—
155	C₂³	C₂³×C₄×C₁₉	1	2	yes	—
156	C₂³	C₁₉×H(5,𝔽₂)	1	4	yes	—
157	C₂³	C₁₉×2₋¹⁺⁴	1	8	yes	—
158	C₂³	C₂⁵×C₁₉	1	1	yes	characteristic
159	Q₈	—	4	32	no	—
160	C₂×C₄	—	2	8	no	—
161	C₂×C₄	—	2	32	no	—
162	C₂×C₄	—	2	16	no	—
163	C₂×C₄	—	2	8	no	—
164	C₂×C₄	—	2	8	no	—
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	—	2	8	no	—
167	C₂×C₄	—	2	16	no	—
168	C₂×C₄	—	2	32	no	—
169	C₂×C₄	—	4	32	no	—
170	C₂×C₄	C₂³×C₄×C₁₉	1	8	yes	—
171	C₂×C₄	C₂³×C₄×C₁₉	1	4	yes	—
172	C₂×C₄	C₂³×C₄×C₁₉	1	16	yes	—
173	C₂²	C₁₉×#518	1	4	yes	—
174	C₂²	C₁₉×#519	1	8	yes	—
175	C₂²	C₁₉×#563	1	16	yes	—
176	C₂²	C₂⁴×C₄×C₁₉	1	1	yes	characteristic
177	C₂²	C₂×C₁₉×H(5,𝔽₂)	1	2	yes	—
178	C₂²	C₂×C₁₉×2₋¹⁺⁴	1	4	yes	—
179	C₄	—	2	4	no	—
180	C₄	—	2	16	no	—
181	C₄	—	2	8	no	—
182	C₄	—	4	16	no	—
183	C₂	C₁₉×#2381	1	8	yes	direct product factor
184	C₂	C₂×C₁₉×#518	1	1	yes	characteristic
185	C₂	C₂×C₁₉×#519	1	2	yes	—
186	C₂	C₂×C₁₉×#563	1	4	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₁₉	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₁₉
Frattini, cosocle	C₂³	C₂⁵×C₁₉
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₉×#2381	C₂	direct (N × H)
7. C₂×C₁₉×C₄⋊Q₈	C₄	semidirect
27. C₁₉×C₄⋊Q₈	C₂×C₄	semidirect
61. C₂×#2381	C₁₉	direct (N × H)
81. #2381	C₂×C₁₉	direct (N × H)
96. C₂×C₄⋊Q₈	C₄×C₁₉	semidirect
108. C₂×C₁₉	#2381	direct (N × H)
120. C₄⋊Q₈	C₂×C₄×C₁₉	semidirect
132. C₁₉	C₂×#2381	direct (N × H)
183. C₂	C₁₉×#2381	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₁₉×C₄⋊Q₈	2
C₁₉×#2381	2
C₂×C₁₉×#391	2
C₂×C₁₉×#387	2
C₂×C₁₉×#383	2
C₂×#2381	19

Subgroups — C2×C19×#2381 #18040304

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₉×#2381 #18040304