Subgroups — #14572573

Order 2432.

348 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11927097	C₂	1	1	yes	maximal, characteristic
2	#11926674	C₂	1	4	yes	maximal, complemented
3	#11926672	C₂	1	4	yes	maximal, complemented
4	#11926572	C₂	1	1	yes	maximal, characteristic
5	#11926571	C₂	1	1	yes	maximal, characteristic
6	#11926567	C₂	1	1	yes	maximal, characteristic
7	#11926499	C₂	1	1	yes	maximal, characteristic
8	C₂×#10597272	C₂	1	1	yes	maximal, characteristic
9	C₁₉×#405	C₂	1	1	yes	maximal, characteristic
10	C₂×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
11	C₂²×C₈×C₁₉	C₂²	1	1	yes	characteristic
12	C₁₉×#117	C₂²	1	1	yes	characteristic
13	#10597372	C₂²	1	1	yes	characteristic
14	C₂×C₄×Dic₁₉	C₂²	1	1	yes	characteristic
15	C₂×#64520	C₂²	1	1	yes	characteristic
16	C₂²×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
17	#10597336	C₄	1	2	yes	—
18	#10597334	C₄	1	2	yes	—
19	#10597324	C₄	1	4	yes	—
20	C₄×Dic₃₈	C₂²	1	4	yes	—
21	C₁₉×#97	C₂²	1	2	yes	—
22	C₄×C₈×C₁₉	C₂²	1	2	yes	—
23	#10597272	C₂²	1	2	yes	—
24	#10597272	C₂²	1	2	yes	—
25	#10597272	C₂²	1	2	yes	—
26	Dic₁₉⋊C₈	C₂²	1	4	yes	—
27	Dic₁₉⋊C₈	C₂²	1	4	yes	—
28	C₈×Dic₁₉	C₂²	1	2	yes	—
29	C₄⋊(C₁₉⋊C₈)	C₂²	1	4	yes	—
30	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
31	C₂²×Dic₁₉	C₂×C₄	1	2	yes	—
32	C₂×Dic₃₈	C₂×C₄	1	4	yes	—
33	M₁₆×C₁₉	—	2	2	no	—
34	C₂×C₈×C₁₉	Q₈	1	4	yes	complemented
35	C₂×C₈×C₁₉	C₂³	1	2	yes	—
36	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
37	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
38	C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
39	C₂²⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
40	C₄²×C₁₉	C₂³	1	2	yes	—
41	C₂²⋊Dic₁₉	C₂×C₄	1	2	yes	—
42	C₄⋊Dic₁₉	C₂×C₄	1	4	yes	—
43	C₄⋊Dic₁₉	C₂×C₄	1	2	yes	—
44	Dic₁₉⋊C₄	C₂×C₄	1	4	yes	—
45	Dic₁₉⋊C₄	C₂×C₄	1	4	yes	—
46	C₄×Dic₁₉	C₂×C₄	1	1	yes	characteristic
47	C₄×Dic₁₉	C₂×C₄	1	1	yes	characteristic
48	C₄×Dic₁₉	C₂³	1	2	yes	—
49	C₄×Dic₁₉	C₂³	1	2	yes	—
50	#64520	—	2	2	no	—
51	C₂×C₁₉⋊C₈	—	2	2	no	—
52	C₂×C₁₉⋊C₈	C₂³	1	2	yes	—
53	C₂×C₁₉⋊C₈	C₂³	1	2	yes	—
54	C₂³×C₁₉	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₁₉	C₂²×C₄	1	4	yes	—
56	C₂×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₁₉	C₂×Q₈	1	2	yes	—
58	C₂×C₄×C₁₉	#41	1	2	yes	—
59	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
60	C₂×Dic₁₉	—	4	2	no	—
61	C₂×Dic₁₉	C₂²×C₄	1	2	yes	—
62	C₂×Dic₁₉	C₂²×C₄	1	2	yes	—
63	C₂×Dic₁₉	C₂²×C₄	1	4	yes	—
64	Dic₃₈	—	4	4	no	—
65	C₈×C₁₉	—	2	2	no	—
66	C₈×C₁₉	C₂×Q₈	1	2	yes	—
67	C₈×C₁₉	C₂×Q₈	1	2	yes	—
68	C₁₉⋊C₈	—	2	2	no	—
69	C₁₉⋊C₈	—	2	2	no	—
70	#2961	—	19	1	no	maximal, Sylow
71	C₂²×C₁₉	—	2	2	no	—
72	C₂²×C₁₉	C₄×Q₈	1	2	yes	—
73	C₂²×C₁₉	C₂³×C₄	1	1	yes	characteristic
74	C₄×C₁₉	—	2	4	no	—
75	C₄×C₁₉	C₄×Q₈	1	2	yes	—
76	C₄×C₁₉	C₂²×Q₈	1	1	yes	characteristic
77	C₄×C₁₉	C₂×#41	1	1	yes	characteristic
78	Dic₁₉	—	2	2	no	—
79	Dic₁₉	—	2	4	no	—
80	Dic₁₉	—	2	2	no	—
81	#533	—	19	1	no	—
82	#446	—	19	4	no	—
83	#446	—	19	4	no	—
84	#424	—	19	1	no	—
85	#424	—	19	1	no	—
86	#423	—	19	1	no	—
87	#405	—	19	1	no	—
88	#405	D₁₉	1	1	yes	characteristic
89	C₂×#97	—	19	1	no	—
90	C₂×C₁₉	—	2	2	no	—
91	C₂×C₁₉	C₂×C₄×Q₈	1	1	yes	characteristic
92	C₂×C₁₉	#568	1	2	yes	—
93	C₂×M₁₆	—	19	1	no	—
94	C₂×M₁₆	C₂×D₁₉	1	1	yes	characteristic
95	C₂²×C₈	—	19	1	no	—
96	C₂²×C₈	C₂×D₁₉	1	1	yes	characteristic
97	C₄⋊Q₈	—	19	2	no	—
98	#125	—	19	2	no	—
99	C₂²⋊Q₈	—	19	4	no	—
100	C₄×Q₈	—	19	4	no	—
101	#117	—	19	1	no	—
102	#117	C₂×D₁₉	1	1	yes	characteristic
103	C₂×C₄²	—	19	1	no	—
104	C₄⋊C₈	—	19	4	no	—
105	C₄⋊C₈	—	19	4	no	—
106	C₄⋊C₈	—	19	4	no	—
107	#97	—	19	2	no	—
108	#97	—	19	2	no	—
109	#97	—	19	2	no	—
110	#97	C₂×D₁₉	1	2	yes	—
111	C₄×C₈	—	19	2	no	—
112	C₄×C₈	C₂×D₁₉	1	2	yes	—
113	C₁₉	#2961	1	1	yes	characteristic, complemented, Sylow
114	C₂×Q₈	—	19	4	no	—
115	C₂²×C₄	—	19	2	no	—
116	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
117	M₁₆	—	2	2	no	—
118	M₁₆	—	38	2	no	—
119	C₂×C₈	—	19	2	no	—
120	C₂×C₈	—	19	2	no	—
121	C₂×C₈	—	38	2	no	—
122	C₂×C₈	Dic₃₈	1	4	yes	complemented
123	C₂×C₈	C₂²×D₁₉	1	2	yes	—
124	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
125	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
126	C₄⋊C₄	—	19	4	no	—
127	C₄⋊C₄	—	19	4	no	—
128	C₄⋊C₄	—	19	2	no	—
129	C₄⋊C₄	—	19	4	no	—
130	C₄⋊C₄	C₄×D₁₉	1	2	yes	—
131	C₂²⋊C₄	—	19	2	no	—
132	C₂²⋊C₄	C₄×D₁₉	1	2	yes	—
133	C₄²	—	19	1	no	—
134	C₄²	—	19	1	no	—
135	C₄²	—	19	2	no	—
136	C₄²	—	19	2	no	—
137	C₄²	C₂²×D₁₉	1	2	yes	—
138	C₂³	C₂×C₄×D₁₉	1	1	yes	characteristic
139	Q₈	—	76	4	no	—
140	C₂×C₄	—	19	2	no	—
141	C₂×C₄	—	19	2	no	—
142	C₂×C₄	—	19	4	no	—
143	C₂×C₄	—	76	2	no	—
144	C₂×C₄	C₂×Dic₃₈	1	2	yes	—
145	C₂×C₄	C₂×C₄×D₁₉	1	4	yes	—
146	C₂×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
147	C₂×C₄	#64541	1	2	yes	—
148	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
149	C₈	—	2	2	no	—
150	C₈	—	38	2	no	—
151	C₈	—	38	2	no	—
152	C₈	C₂×Dic₃₈	1	2	yes	—
153	C₈	C₂×Dic₃₈	1	2	yes	—
154	C₂²	—	2	2	no	—
155	C₂²	C₄×Dic₃₈	1	2	yes	—
156	C₂²	C₂²×C₄×D₁₉	1	1	yes	characteristic
157	C₄	—	2	4	no	—
158	C₄	—	38	2	no	—
159	C₄	—	38	2	no	—
160	C₄	—	38	4	no	—
161	C₄	C₄×Dic₃₈	1	2	yes	—
162	C₄	C₂²×Dic₃₈	1	1	yes	characteristic
163	C₄	C₂×#64541	1	1	yes	characteristic
164	C₂	—	2	2	no	—
165	C₂	C₂×C₄×Dic₃₈	1	1	yes	characteristic
166	C₂	#11927309	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₉
commutator, socle	C₂²×C₁₉	C₂³×C₄
Fitting	C₁₉×#405	C₂
prime radical	C₂³×C₁₉	C₂²×C₄
cosocle	C₂×C₄×C₁₉	C₂⁴
hypercenter	#405	D₁₉
hypocenter	C₁₉	#2961
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ #405

Complements

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

N	H ≅ G/N	type
2. #11926674	C₂	semidirect
3. #11926672	C₂	semidirect
34. C₂×C₈×C₁₉	Q₈	semidirect
113. C₁₉	#2961	semidirect
122. C₂×C₈	Dic₃₈	semidirect

Maximal subgroups

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

subgroup	index
#11927097	2
#11926674	2
#11926672	2
#11926572	2
#11926571	2
#11926567	2
#11926499	2
C₂×#10597272	2
C₁₉×#405	2
#2961	19