Subgroups — #14572582

Order 2432.

356 conjugacy classes of subgroups.

The subgroup lattice has 182 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	#11926672	C₂	1	4	yes	maximal, complemented
4	C₈×Dic₃₈	C₂	1	2	yes	maximal, complemented
5	#11926572	C₂	1	2	yes	maximal
6	C₂×C₄⋊(C₁₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
7	#11926502	C₂	1	2	yes	maximal
8	C₄×M₁₆×C₁₉	C₂	1	1	yes	maximal, characteristic
9	C₂×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
10	C₂×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
11	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
12	#10597372	C₂²	1	2	yes	—
13	#10597370	C₄	1	2	yes	—
14	#10597370	C₄	1	2	yes	—
15	C₂²×C₁₉⋊C₈	C₂²	1	2	yes	—
16	#10597316	C₄	1	2	yes	—
17	#10597315	C₄	1	1	yes	characteristic
18	#10597315	C₄	1	1	yes	characteristic
19	C₄×Dic₃₈	C₂²	1	2	yes	—
20	C₄×Dic₃₈	C₂²	1	2	yes	—
21	C₁₉×#97	C₂²	1	2	yes	—
22	C₄×C₈×C₁₉	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	2	yes	—
29	C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
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	M₁₆×C₁₉	—	2	2	no	—
36	M₁₆×C₁₉	Q₈	1	4	yes	complemented
37	C₂×C₈×C₁₉	C₂³	1	2	yes	—
38	C₂×C₈×C₁₉	C₂³	1	2	yes	—
39	C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
40	C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
41	C₄²×C₁₉	C₂³	1	1	yes	characteristic
42	C₄²×C₁₉	C₂³	1	1	yes	characteristic
43	C₂²⋊Dic₁₉	C₂×C₄	1	4	yes	—
44	C₄⋊Dic₁₉	C₂×C₄	1	2	yes	—
45	C₄⋊Dic₁₉	C₂×C₄	1	2	yes	—
46	C₄⋊Dic₁₉	C₂×C₄	1	2	yes	—
47	C₄⋊Dic₁₉	C₂×C₄	1	2	yes	—
48	Dic₁₉⋊C₄	C₂×C₄	1	4	yes	—
49	Dic₁₉⋊C₄	C₂×C₄	1	4	yes	—
50	C₄×Dic₁₉	C₂³	1	4	yes	—
51	C₂×C₁₉⋊C₈	—	2	4	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	—	2	2	no	—
56	C₂×C₄×C₁₉	—	4	1	no	—
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	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×C₁₉	C₂×Q₈	1	2	yes	—
63	C₂×C₄×C₁₉	#41	1	2	yes	—
64	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
65	C₂×Dic₁₉	C₂²×C₄	1	4	yes	—
66	C₂×Dic₁₉	C₂²×C₄	1	4	yes	—
67	Dic₃₈	—	2	4	no	—
68	Dic₃₈	—	4	2	no	—
69	C₈×C₁₉	—	2	2	no	—
70	C₈×C₁₉	C₂×Q₈	1	4	yes	—
71	C₁₉⋊C₈	—	2	2	no	—
72	C₁₉⋊C₈	—	2	2	no	—
73	#2962	—	19	1	no	maximal, Sylow
74	C₂²×C₁₉	—	2	1	no	—
75	C₂²×C₁₉	—	2	1	no	—
76	C₂²×C₁₉	C₄×Q₈	1	2	yes	—
77	C₂²×C₁₉	C₂³×C₄	1	1	yes	characteristic
78	C₄×C₁₉	—	2	1	no	—
79	C₄×C₁₉	—	2	1	no	—
80	C₄×C₁₉	—	2	1	no	—
81	C₄×C₁₉	C₄×Q₈	1	2	yes	—
82	C₄×C₁₉	#131	1	2	yes	—
83	C₄×C₁₉	C₂²×Q₈	1	1	yes	characteristic
84	C₄×C₁₉	C₂×#41	1	1	yes	characteristic
85	Dic₁₉	—	2	4	no	—
86	Dic₁₉	—	2	4	no	—
87	#533	—	19	1	no	—
88	#446	—	19	4	no	—
89	#446	—	19	2	no	—
90	C₈×Q₈	—	19	2	no	—
91	#424	—	19	2	no	—
92	C₂×C₄⋊C₈	—	19	1	no	—
93	#405	—	19	2	no	—
94	C₄×M₁₆	D₁₉	1	1	yes	characteristic
95	C₂×C₁₉	—	2	2	no	—
96	C₂×C₁₉	C₂×C₄×Q₈	1	1	yes	characteristic
97	C₂×C₁₉	C₂×#131	1	1	yes	characteristic
98	C₂×C₁₉	#568	1	1	yes	characteristic
99	C₂×M₁₆	C₂×D₁₉	1	1	yes	characteristic
100	C₂×M₁₆	C₂×D₁₉	1	1	yes	characteristic
101	C₂²×C₈	—	19	2	no	—
102	C₄⋊Q₈	—	19	1	no	—
103	C₄⋊Q₈	—	19	1	no	—
104	#125	—	19	2	no	—
105	C₂²⋊Q₈	—	19	2	no	—
106	C₂²⋊Q₈	—	19	2	no	—
107	C₄×Q₈	—	19	2	no	—
108	C₄×Q₈	—	19	2	no	—
109	#117	—	19	2	no	—
110	C₂×C₄²	C₂×D₁₉	1	1	yes	characteristic
111	C₄⋊C₈	—	19	4	no	—
112	C₄⋊C₈	—	19	4	no	—
113	C₄⋊C₈	—	19	1	no	—
114	C₄⋊C₈	—	19	1	no	—
115	C₄⋊C₈	—	19	2	no	—
116	#97	—	19	4	no	—
117	#97	C₂×D₁₉	1	2	yes	—
118	C₄×C₈	—	19	4	no	—
119	C₄×C₈	C₂×D₁₉	1	2	yes	—
120	C₁₉	#2962	1	1	yes	characteristic, complemented, Sylow
121	C₂×Q₈	—	19	2	no	—
122	C₂×Q₈	—	19	2	no	—
123	C₂²×C₄	C₄×D₁₉	1	1	yes	characteristic
124	C₂²×C₄	C₄×D₁₉	1	1	yes	characteristic
125	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
126	M₁₆	—	2	2	no	—
127	M₁₆	Dic₃₈	1	4	yes	complemented
128	C₂×C₈	—	19	2	no	—
129	C₂×C₈	—	19	2	no	—
130	C₂×C₈	—	38	4	no	—
131	C₂×C₈	C₂²×D₁₉	1	2	yes	—
132	C₂×C₈	C₂²×D₁₉	1	2	yes	—
133	C₄⋊C₄	—	19	4	no	—
134	C₄⋊C₄	—	19	4	no	—
135	C₄⋊C₄	—	19	2	no	—
136	C₄⋊C₄	—	19	2	no	—
137	C₄⋊C₄	—	19	2	no	—
138	C₄⋊C₄	—	19	2	no	—
139	C₂²⋊C₄	—	19	4	no	—
140	C₄²	—	19	4	no	—
141	C₄²	C₄×D₁₉	1	1	yes	characteristic
142	C₄²	C₄×D₁₉	1	1	yes	characteristic
143	C₄²	C₂²×D₁₉	1	1	yes	characteristic
144	C₄²	C₂²×D₁₉	1	1	yes	characteristic
145	C₂³	C₂×C₄×D₁₉	1	1	yes	characteristic
146	Q₈	—	38	4	no	—
147	Q₈	—	76	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	—	19	4	no	—
151	C₂×C₄	—	19	4	no	—
152	C₂×C₄	C₂×Dic₃₈	1	2	yes	—
153	C₂×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
154	C₂×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
155	C₂×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
156	C₂×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
157	C₂×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
158	C₂×C₄	#64541	1	2	yes	—
159	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
160	C₈	—	2	2	no	—
161	C₈	—	38	2	no	—
162	C₈	—	38	2	no	—
163	C₈	C₂×Dic₃₈	1	4	yes	—
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	C₄×Dic₃₈	1	2	yes	—
167	C₂²	C₂²×C₄×D₁₉	1	1	yes	characteristic
168	C₄	—	2	1	no	—
169	C₄	—	2	1	no	—
170	C₄	—	2	1	no	—
171	C₄	—	38	4	no	—
172	C₄	—	38	4	no	—
173	C₄	C₄×Dic₃₈	1	2	yes	—
174	C₄	#10597353	1	2	yes	—
175	C₄	C₂²×Dic₃₈	1	1	yes	characteristic
176	C₄	C₂×#64541	1	1	yes	characteristic
177	C₂	—	2	2	no	—
178	C₂	C₂×C₄×Dic₃₈	1	1	yes	characteristic
179	C₂	C₂×#10597353	1	1	yes	characteristic
180	C₂	#11927310	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₉
commutator, socle	C₂²×C₁₉	C₂³×C₄
Fitting	C₄×M₁₆×C₁₉	C₂
prime radical	C₂³×C₁₉	C₂²×C₄
cosocle	C₂×C₄×C₁₉	C₂⁴
hypercenter	C₄×M₁₆	D₁₉
hypocenter	C₁₉	#2962
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. #11926674	C₂	semidirect
3. #11926672	C₂	semidirect
4. C₈×Dic₃₈	C₂	semidirect
36. M₁₆×C₁₉	Q₈	semidirect
120. C₁₉	#2962	semidirect
127. M₁₆	Dic₃₈	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
#11926672	2
C₈×Dic₃₈	2
#11926572	2
C₂×C₄⋊(C₁₉⋊C₈)	2
#11926502	2
C₄×M₁₆×C₁₉	2
#2962	19