Subgroups — #14572736

Order 2432.

330 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11927101	C₂	1	1	yes	maximal, characteristic
2	#11926674	C₂	1	2	yes	maximal, complemented
3	#11926671	C₂	1	2	yes	maximal, complemented
4	C₈×Dic₃₈	C₂	1	2	yes	maximal, complemented
5	#11926668	C₂	1	2	yes	maximal, complemented
6	#11926602	C₂	1	4	yes	maximal
7	#11926597	C₂	1	1	yes	maximal, characteristic
8	C₁₉×#431	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
10	#10597372	C₂²	1	2	yes	—
11	#10597370	C₄	1	2	yes	—
12	#10597370	C₄	1	2	yes	—
13	#10597316	C₄	1	2	yes	—
14	#10597315	C₄	1	1	yes	characteristic
15	#10597315	C₄	1	1	yes	characteristic
16	C₄×Dic₃₈	C₂²	1	2	yes	—
17	C₄×Dic₃₈	C₂²	1	2	yes	—
18	C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
19	C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₁₉×C₂²⋊C₈	C₂²	1	2	yes	—
21	C₄×C₈×C₁₉	C₂²	1	2	yes	—
22	C₂²⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
23	#10597272	C₂²	1	4	yes	—
24	Dic₁₉⋊C₈	C₂²	1	4	yes	—
25	Dic₁₉⋊C₈	C₂²	1	4	yes	—
26	C₈×Dic₁₉	C₂²	1	4	yes	—
27	C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
28	C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
29	C₄×C₁₉⋊C₈	C₂²	1	2	yes	—
30	C₂²×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
33	C₂×Dic₃₈	C₂×C₄	1	2	yes	—
34	C₂×Dic₃₈	C₂×C₄	1	2	yes	—
35	C₂×C₈×C₁₉	C₂³	1	2	yes	—
36	C₂×C₈×C₁₉	C₂³	1	2	yes	—
37	C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
38	C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
39	C₄²×C₁₉	C₂³	1	1	yes	characteristic
40	C₄²×C₁₉	C₂³	1	1	yes	characteristic
41	C₂²⋊Dic₁₉	C₂×C₄	1	4	yes	—
42	C₄⋊Dic₁₉	C₂×C₄	1	2	yes	—
43	C₄⋊Dic₁₉	C₂×C₄	1	2	yes	—
44	C₄⋊Dic₁₉	C₂×C₄	1	2	yes	—
45	C₄⋊Dic₁₉	C₂×C₄	1	2	yes	—
46	Dic₁₉⋊C₄	C₂×C₄	1	4	yes	—
47	Dic₁₉⋊C₄	C₂×C₄	1	4	yes	—
48	C₄×Dic₁₉	C₂³	1	4	yes	—
49	C₂×C₁₉⋊C₈	C₂³	1	2	yes	—
50	C₂×C₁₉⋊C₈	C₂³	1	2	yes	—
51	C₂³×C₁₉	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₁₉	—	2	2	no	—
53	C₂×C₄×C₁₉	—	2	2	no	—
54	C₂×C₄×C₁₉	—	2	1	no	—
55	C₂×C₄×C₁₉	—	2	1	no	—
56	C₂×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
62	C₂×Dic₁₉	C₂²×C₄	1	4	yes	—
63	C₂×Dic₁₉	C₂²×C₄	1	4	yes	—
64	Dic₃₈	—	2	4	no	—
65	Dic₃₈	—	2	4	no	—
66	C₈×C₁₉	—	2	2	no	—
67	C₈×C₁₉	—	2	2	no	—
68	C₁₉⋊C₈	—	2	2	no	—
69	C₁₉⋊C₈	—	2	2	no	—
70	#2986	—	19	1	no	maximal, Sylow
71	C₂²×C₁₉	—	2	1	no	—
72	C₂²×C₁₉	—	2	1	no	—
73	C₂²×C₁₉	—	2	1	no	—
74	C₂²×C₁₉	C₂³×C₄	1	1	yes	characteristic
75	C₄×C₁₉	—	2	1	no	—
76	C₄×C₁₉	—	2	1	no	—
77	C₄×C₁₉	—	2	1	no	—
78	C₄×C₁₉	#131	1	2	yes	—
79	C₄×C₁₉	#131	1	2	yes	—
80	C₄×C₁₉	2₋¹⁺⁴	1	1	yes	characteristic
81	C₄×C₁₉	2₋¹⁺⁴	1	1	yes	characteristic
82	Dic₁₉	—	2	4	no	—
83	Dic₁₉	—	2	4	no	—
84	#533	—	19	1	no	—
85	#446	—	19	2	no	—
86	#446	—	19	2	no	—
87	C₈×Q₈	—	19	2	no	—
88	C₈×Q₈	—	19	2	no	—
89	#433	—	19	4	no	—
90	#431	—	19	1	no	—
91	#431	D₁₉	1	1	yes	characteristic
92	C₂×C₁₉	—	4	1	no	—
93	C₂×C₁₉	#519	1	1	yes	characteristic
94	C₂×C₁₉	C₂×#131	1	1	yes	characteristic
95	C₂×C₁₉	C₂×#131	1	1	yes	characteristic
96	C₄⋊Q₈	—	19	1	no	—
97	C₄⋊Q₈	—	19	1	no	—
98	#125	—	19	2	no	—
99	C₂²⋊Q₈	—	19	2	no	—
100	C₂²⋊Q₈	—	19	2	no	—
101	C₄×Q₈	—	19	2	no	—
102	C₄×Q₈	—	19	2	no	—
103	#117	—	19	2	no	—
104	C₂×C₄²	C₂×D₁₉	1	1	yes	characteristic
105	C₄⋊C₈	—	19	4	no	—
106	C₄⋊C₈	—	19	4	no	—
107	C₄⋊C₈	—	19	1	no	—
108	C₄⋊C₈	—	19	1	no	—
109	C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
110	C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
111	C₂²⋊C₈	—	19	2	no	—
112	C₂²⋊C₈	C₂×D₁₉	1	2	yes	—
113	#97	—	19	4	no	—
114	C₄×C₈	—	19	4	no	—
115	C₄×C₈	—	19	2	no	—
116	C₄×C₈	C₂×D₁₉	1	2	yes	—
117	C₁₉	#2986	1	1	yes	characteristic, complemented, Sylow
118	C₂×Q₈	—	19	2	no	—
119	C₂×Q₈	—	19	2	no	—
120	C₂²×C₄	C₄×D₁₉	1	1	yes	characteristic
121	C₂²×C₄	C₄×D₁₉	1	1	yes	characteristic
122	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
123	C₂×C₈	—	19	2	no	—
124	C₂×C₈	—	19	2	no	—
125	C₂×C₈	C₂²×D₁₉	1	2	yes	—
126	C₂×C₈	C₂²×D₁₉	1	2	yes	—
127	C₄⋊C₄	—	19	4	no	—
128	C₄⋊C₄	—	19	4	no	—
129	C₄⋊C₄	—	19	2	no	—
130	C₄⋊C₄	—	19	2	no	—
131	C₄⋊C₄	—	19	2	no	—
132	C₄⋊C₄	—	19	2	no	—
133	C₂²⋊C₄	—	19	4	no	—
134	C₄²	—	19	4	no	—
135	C₄²	C₄×D₁₉	1	1	yes	characteristic
136	C₄²	C₄×D₁₉	1	1	yes	characteristic
137	C₄²	C₂²×D₁₉	1	1	yes	characteristic
138	C₄²	C₂²×D₁₉	1	1	yes	characteristic
139	C₂³	C₂×C₄×D₁₉	1	1	yes	characteristic
140	Q₈	—	38	4	no	—
141	Q₈	—	38	4	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	19	4	no	—
147	C₂×C₄	—	19	4	no	—
148	C₂×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
149	C₂×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
150	C₂×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
151	C₂×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
152	C₂×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
153	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
154	C₈	—	2	2	no	—
155	C₈	—	2	2	no	—
156	C₈	—	38	2	no	—
157	C₈	—	38	2	no	—
158	C₂²	—	2	1	no	—
159	C₂²	—	2	1	no	—
160	C₂²	—	2	1	no	—
161	C₂²	C₂²×C₄×D₁₉	1	1	yes	characteristic
162	C₄	—	2	1	no	—
163	C₄	—	2	1	no	—
164	C₄	—	2	1	no	—
165	C₄	—	38	4	no	—
166	C₄	—	38	4	no	—
167	C₄	#10597347	1	2	yes	—
168	C₄	#10597353	1	2	yes	—
169	C₄	#10597436	1	1	yes	characteristic
170	C₄	#10597436	1	1	yes	characteristic
171	C₂	—	4	1	no	—
172	C₂	#11927021	1	1	yes	characteristic
173	C₂	C₂×#10597347	1	1	yes	characteristic
174	C₂	C₂×#10597353	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₉
commutator, socle	C₂²×C₁₉	C₂³×C₄
Fitting	C₁₉×#431	C₂
prime radical	C₂³×C₁₉	C₂²×C₄
cosocle	C₂×C₄×C₁₉	C₂⁴
hypercenter	#431	D₁₉
hypocenter	C₁₉	#2986
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ #431

Complements

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

N	H ≅ G/N	type
2. #11926674	C₂	semidirect
3. #11926671	C₂	semidirect
4. C₈×Dic₃₈	C₂	semidirect
5. #11926668	C₂	semidirect
117. C₁₉	#2986	semidirect

Maximal subgroups

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

subgroup	index
#11927101	2
#11926674	2
#11926671	2
C₈×Dic₃₈	2
#11926668	2
#11926602	2
#11926597	2
C₁₉×#431	2
#2986	19