Subgroups — #14563351

Order 2432.

424 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₁₉⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#10597372	C₂	1	1	yes	maximal, characteristic
3	#11926306	C₂	1	4	yes	maximal, complemented
4	C₂×C₁₉×#117	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₄×C₁₉	C₂²	1	1	yes	characteristic
6	C₁₉×#117	—	2	2	no	—
7	C₁₉×#117	C₄	1	4	yes	complemented
8	C₂×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
9	C₂×C₂²⋊C₄×C₁₉	—	2	1	no	—
10	C₂×C₄²×C₁₉	—	2	1	no	—
11	C₂×C₂²⋊Dic₁₉	—	2	1	no	—
12	#10597372	—	2	2	no	—
13	#10597372	C₄	1	4	yes	complemented
14	C₂×C₄⋊Dic₁₉	C₂²	1	2	yes	—
15	C₂×C₄×Dic₁₉	—	2	1	no	—
16	C₂²×C₁₉⋊C₈	—	2	2	no	—
17	C₂²×C₁₉⋊C₈	—	2	2	no	—
18	C₂²×C₁₉⋊C₈	C₄	1	4	yes	complemented
19	C₂²×C₁₉⋊C₈	C₂²	1	2	yes	—
20	C₂⁴×C₁₉	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₁₉	—	2	2	no	—
22	C₂²×C₄×C₁₉	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₁₉	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₁₉	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₁₉	D₄	1	2	yes	—
26	C₂²×C₄×C₁₉	D₄	1	2	yes	—
27	C₂²×C₄×C₁₉	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₉	Q₈	1	2	yes	—
29	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
30	C₂²×Dic₁₉	—	2	2	no	—
31	C₄⋊C₄×C₁₉	—	2	2	no	—
32	C₄⋊C₄×C₁₉	C₂×C₄	1	4	yes	—
33	C₂²⋊C₄×C₁₉	—	2	4	no	—
34	C₄²×C₁₉	—	2	4	no	—
35	C₂²⋊Dic₁₉	—	2	4	no	—
36	C₄⋊Dic₁₉	—	2	2	no	—
37	C₄⋊Dic₁₉	C₂×C₄	1	4	yes	—
38	C₄×Dic₁₉	—	2	4	no	—
39	C₂×C₁₉⋊C₈	—	2	8	no	—
40	C₂×C₁₉⋊C₈	—	2	2	no	—
41	C₂×C₁₉⋊C₈	—	2	2	no	—
42	C₂×C₁₉⋊C₈	C₂×C₄	1	4	yes	—
43	C₂³×C₁₉	—	2	4	no	—
44	C₂³×C₁₉	C₂²⋊C₄	1	2	yes	—
45	C₂³×C₁₉	C₂²⋊C₄	1	2	yes	—
46	C₂³×C₁₉	C₄⋊C₄	1	2	yes	—
47	C₂³×C₁₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₁₉	—	2	4	no	—
49	C₂×C₄×C₁₉	—	4	4	no	—
50	C₂×C₄×C₁₉	C₄²	1	4	yes	—
51	C₂×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
52	C₂×C₄×C₁₉	C₂²⋊C₄	1	4	yes	—
53	C₂×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
54	C₂×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
55	C₂×C₄×C₁₉	C₄⋊C₄	1	4	yes	—
56	C₂×C₄×C₁₉	C₄⋊C₄	1	4	yes	—
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	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
63	C₂×Dic₁₉	—	2	4	no	—
64	C₂×Dic₁₉	—	4	4	no	—
65	C₁₉⋊C₈	—	2	4	no	—
66	#1735	—	19	1	no	maximal, Sylow
67	C₂²×C₁₉	—	2	4	no	—
68	C₂²×C₁₉	—	2	4	no	—
69	C₂²×C₁₉	—	2	4	no	—
70	C₂²×C₁₉	#95	1	4	yes	—
71	C₂²×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂²×C₁₉	#135	1	4	yes	—
75	C₄×C₁₉	—	4	4	no	—
76	C₄×C₁₉	#95	1	4	yes	—
77	C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
78	C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
81	Dic₁₉	—	4	4	no	—
82	C₂³×C₈	—	19	1	no	—
83	C₂×#117	—	19	1	no	—
84	C₂×#117	D₁₉	1	1	yes	characteristic
85	#340	—	19	4	no	—
86	C₂×C₁₉	—	2	4	no	—
87	C₂×C₁₉	C₂×#95	1	1	yes	characteristic
88	C₂×C₁₉	#416	1	2	yes	—
89	C₂×C₁₉	#416	1	2	yes	—
90	C₂×C₁₉	#427	1	2	yes	—
91	C₂³×C₄	C₂×D₁₉	1	1	yes	characteristic
92	C₂²×C₈	—	19	4	no	—
93	C₂²×C₈	—	19	2	no	—
94	C₂²×C₈	—	38	2	no	—
95	C₂²×C₈	—	38	2	no	—
96	#117	—	2	2	no	—
97	#117	—	19	4	no	—
98	#117	—	38	2	no	—
99	#117	Dic₁₉	1	4	yes	complemented
100	C₂×C₄⋊C₄	—	19	2	no	—
101	C₂×C₄⋊C₄	C₂×D₁₉	1	2	yes	—
102	C₂×C₂²⋊C₄	—	2	1	no	—
103	C₂×C₂²⋊C₄	—	38	1	no	—
104	C₂×C₄²	—	2	1	no	—
105	C₂×C₄²	—	38	1	no	—
106	C₁₉	#1735	1	1	yes	characteristic, complemented, Sylow
107	C₂⁴	C₂²⋊D₁₉	1	1	yes	characteristic
108	C₂²×C₄	—	2	2	no	—
109	C₂²×C₄	—	38	2	no	—
110	C₂²×C₄	Dic₃₈	1	2	yes	—
111	C₂²×C₄	C₄×D₁₉	1	2	yes	—
112	C₂²×C₄	C₄×D₁₉	1	2	yes	—
113	C₂²×C₄	D₇₆	1	2	yes	—
114	C₂²×C₄	C₂×Dic₁₉	1	2	yes	—
115	C₂²×C₄	C₂²⋊D₁₉	1	2	yes	—
116	C₂²×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
117	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
118	C₂×C₈	—	19	4	no	—
119	C₂×C₈	—	38	8	no	—
120	C₂×C₈	—	38	2	no	—
121	C₂×C₈	—	38	2	no	—
122	C₄⋊C₄	—	2	2	no	—
123	C₄⋊C₄	—	19	4	no	—
124	C₄⋊C₄	—	38	2	no	—
125	C₄⋊C₄	C₂×Dic₁₉	1	4	yes	—
126	C₂²⋊C₄	—	2	4	no	—
127	C₂²⋊C₄	—	38	4	no	—
128	C₄²	—	2	4	no	—
129	C₄²	—	38	4	no	—
130	C₂³	—	2	4	no	—
131	C₂³	Dic₁₉⋊C₄	1	2	yes	—
132	C₂³	(C₂×D₁₉)⋊C₄	1	2	yes	—
133	C₂³	C₂²⋊Dic₁₉	1	2	yes	—
134	C₂³	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	4	4	no	—
137	C₂×C₄	—	38	4	no	—
138	C₂×C₄	—	76	4	no	—
139	C₂×C₄	C₄×Dic₁₉	1	4	yes	—
140	C₂×C₄	Dic₁₉⋊C₄	1	2	yes	—
141	C₂×C₄	Dic₁₉⋊C₄	1	4	yes	—
142	C₂×C₄	C₄⋊Dic₁₉	1	4	yes	—
143	C₂×C₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
144	C₂×C₄	(C₂×D₁₉)⋊C₄	1	4	yes	—
145	C₂×C₄	C₂²⋊Dic₁₉	1	2	yes	—
146	C₂×C₄	C₂×Dic₃₈	1	1	yes	characteristic
147	C₂×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
148	C₂×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
149	C₂×C₄	C₂×D₇₆	1	1	yes	characteristic
150	C₂×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
151	C₂×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
152	C₈	—	38	4	no	—
153	C₂²	—	2	4	no	—
154	C₂²	—	2	4	no	—
155	C₂²	—	2	4	no	—
156	C₂²	C₁₉⋊#95	1	4	yes	—
157	C₂²	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
158	C₂²	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
159	C₂²	#10597396	1	4	yes	—
160	C₂²	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
161	C₄	—	4	4	no	—
162	C₄	—	76	4	no	—
163	C₄	C₁₉⋊#95	1	4	yes	—
164	C₄	C₂×C₄×Dic₁₉	1	1	yes	characteristic
165	C₄	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
166	C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
167	C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
168	C₂	—	2	4	no	—
169	C₂	C₂×C₁₉⋊#95	1	1	yes	characteristic
170	C₂	#11926538	1	2	yes	—
171	C₂	#11926540	1	2	yes	—
172	C₂	#11926580	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²⋊D₁₉
commutator	C₄×C₁₉	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₉
Fitting	C₂×C₁₉×#117	C₂
socle	C₂³×C₁₉	C₂×D₄
prime radical	C₂⁴×C₁₉	D₄
cosocle	C₂²×C₄×C₁₉	C₂³
hypercenter	C₂×#117	D₁₉
hypocenter	C₁₉	#1735
LCS	G ▷ C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
3. #11926306	C₂	semidirect
7. C₁₉×#117	C₄	semidirect
13. #10597372	C₄	semidirect
18. C₂²×C₁₉⋊C₈	C₄	semidirect
99. #117	Dic₁₉	semidirect
106. C₁₉	#1735	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₁₉⋊C₈	2
C₂×#10597372	2
#11926306	2
C₂×C₁₉×#117	2
#1735	19