Subgroups — #14572730

Order 2432.

322 conjugacy classes of subgroups.

The subgroup lattice has 164 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11926675	C₂	1	2	yes	maximal, complemented
2	#11926675	C₂	1	4	yes	maximal, complemented
3	Q₈×C₁₉⋊C₈	C₂	1	2	yes	maximal, complemented
4	#11926605	C₂	1	4	yes	maximal
5	#11926601	C₂	1	1	yes	maximal, characteristic
6	#11926597	C₂	1	1	yes	maximal, characteristic
7	C₁₉×#533	C₂	1	1	yes	maximal, characteristic
8	C₁₉×C₄⋊Q₈	C₄	1	1	yes	characteristic
9	C₁₉×C₄⋊Q₈	C₄	1	1	yes	characteristic
10	C₁₉×#125	C₄	1	2	yes	—
11	C₁₉×C₂²⋊Q₈	C₄	1	2	yes	—
12	C₁₉×C₂²⋊Q₈	C₄	1	2	yes	—
13	C₄×Q₈×C₁₉	C₂²	1	2	yes	—
14	C₄×Q₈×C₁₉	C₂²	1	2	yes	—
15	C₁₉×#117	C₂²	1	2	yes	—
16	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
17	C₂²⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
18	C₂²⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
19	C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
20	C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
21	C₄⋊(C₁₉⋊C₈)	C₂²	1	4	yes	—
22	C₄⋊(C₁₉⋊C₈)	C₂²	1	4	yes	—
23	#10597260	C₂²	1	2	yes	—
24	#10597260	C₂²	1	4	yes	—
25	C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
26	C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
27	C₄×C₁₉⋊C₈	C₂²	1	4	yes	—
28	C₂×Q₈×C₁₉	C₂×C₄	1	2	yes	—
29	C₂×Q₈×C₁₉	C₂×C₄	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	C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
34	C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
35	C₄⋊C₄×C₁₉	C₂×C₄	1	4	yes	—
36	C₄⋊C₄×C₁₉	C₂×C₄	1	4	yes	—
37	C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
38	C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
39	C₂²⋊C₄×C₁₉	C₂×C₄	1	4	yes	—
40	C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
41	C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
42	C₄²×C₁₉	C₂³	1	1	yes	characteristic
43	C₄²×C₁₉	C₂³	1	1	yes	characteristic
44	C₄²×C₁₉	C₂³	1	4	yes	—
45	C₂×C₁₉⋊C₈	C₂³	1	2	yes	—
46	C₂×C₁₉⋊C₈	C₂³	1	2	yes	—
47	C₂×C₁₉⋊C₈	C₂³	1	4	yes	—
48	C₂³×C₁₉	C₂²×C₄	1	1	yes	characteristic
49	Q₈×C₁₉	—	2	4	no	—
50	Q₈×C₁₉	—	4	2	no	—
51	C₂×C₄×C₁₉	—	2	2	no	—
52	C₂×C₄×C₁₉	—	2	2	no	—
53	C₂×C₄×C₁₉	—	4	1	no	—
54	C₂×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
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	4	yes	—
59	C₂×C₄×C₁₉	C₂²×C₄	1	4	yes	—
60	C₂×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
62	C₁₉⋊C₈	—	2	2	no	—
63	C₁₉⋊C₈	—	2	2	no	—
64	C₁₉⋊C₈	—	2	4	no	—
65	#2985	—	19	1	no	maximal, Sylow
66	C₂²×C₁₉	—	2	1	no	—
67	C₂²×C₁₉	—	2	1	no	—
68	C₂²×C₁₉	—	2	1	no	—
69	C₂²×C₁₉	C₂³×C₄	1	1	yes	characteristic
70	C₄×C₁₉	—	2	1	no	—
71	C₄×C₁₉	—	2	1	no	—
72	C₄×C₁₉	—	2	4	no	—
73	C₄×C₁₉	—	2	1	no	—
74	C₄×C₁₉	—	2	1	no	—
75	C₄×C₁₉	—	2	4	no	—
76	C₄×C₁₉	#131	1	2	yes	—
77	C₄×C₁₉	2₋¹⁺⁴	1	2	yes	—
78	#533	D₁₉	1	1	yes	characteristic
79	#446	—	19	2	no	—
80	#446	—	19	4	no	—
81	C₈×Q₈	—	19	2	no	—
82	#433	—	19	4	no	—
83	#432	—	19	1	no	—
84	#431	—	19	1	no	—
85	C₂×C₁₉	—	4	1	no	—
86	C₂×C₁₉	#519	1	1	yes	characteristic
87	C₂×C₁₉	C₂×#131	1	1	yes	characteristic
88	C₂×C₁₉	#568	1	1	yes	characteristic
89	C₄⋊Q₈	Dic₁₉	1	1	yes	characteristic
90	C₄⋊Q₈	Dic₁₉	1	1	yes	characteristic
91	#125	Dic₁₉	1	2	yes	—
92	C₂²⋊Q₈	Dic₁₉	1	2	yes	—
93	C₂²⋊Q₈	Dic₁₉	1	2	yes	—
94	C₄×Q₈	C₂×D₁₉	1	2	yes	—
95	C₄×Q₈	C₂×D₁₉	1	2	yes	—
96	#117	C₂×D₁₉	1	2	yes	—
97	C₂×C₄²	C₂×D₁₉	1	1	yes	characteristic
98	C₄⋊C₈	—	19	2	no	—
99	C₄⋊C₈	—	19	2	no	—
100	C₄⋊C₈	—	19	4	no	—
101	C₄⋊C₈	—	19	4	no	—
102	C₂²⋊C₈	—	19	2	no	—
103	C₂²⋊C₈	—	19	2	no	—
104	#97	—	19	2	no	—
105	#97	—	19	4	no	—
106	C₄×C₈	—	19	1	no	—
107	C₄×C₈	—	19	1	no	—
108	C₄×C₈	—	19	4	no	—
109	C₁₉	#2985	1	1	yes	characteristic, complemented, Sylow
110	C₂×Q₈	C₂×Dic₁₉	1	2	yes	—
111	C₂×Q₈	C₂×Dic₁₉	1	2	yes	—
112	C₂²×C₄	C₂×Dic₁₉	1	1	yes	characteristic
113	C₂²×C₄	C₂×Dic₁₉	1	1	yes	characteristic
114	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
115	C₂×C₈	—	19	2	no	—
116	C₂×C₈	—	19	2	no	—
117	C₂×C₈	—	19	4	no	—
118	C₄⋊C₄	C₂×Dic₁₉	1	2	yes	—
119	C₄⋊C₄	C₂×Dic₁₉	1	2	yes	—
120	C₄⋊C₄	C₂×Dic₁₉	1	4	yes	—
121	C₄⋊C₄	C₂×Dic₁₉	1	4	yes	—
122	C₄⋊C₄	C₂×Dic₁₉	1	2	yes	—
123	C₄⋊C₄	C₂×Dic₁₉	1	2	yes	—
124	C₂²⋊C₄	C₂×Dic₁₉	1	4	yes	—
125	C₄²	C₂×Dic₁₉	1	1	yes	characteristic
126	C₄²	C₂×Dic₁₉	1	1	yes	characteristic
127	C₄²	C₂²×D₁₉	1	1	yes	characteristic
128	C₄²	C₂²×D₁₉	1	4	yes	—
129	C₄²	C₂²×D₁₉	1	1	yes	characteristic
130	C₂³	C₂²×Dic₁₉	1	1	yes	characteristic
131	Q₈	—	2	4	no	—
132	Q₈	—	4	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
137	C₂×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
138	C₂×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
139	C₂×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
140	C₂×C₄	C₂²×Dic₁₉	1	4	yes	—
141	C₂×C₄	C₂²×Dic₁₉	1	4	yes	—
142	C₂×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
143	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
144	C₈	—	38	2	no	—
145	C₈	—	38	2	no	—
146	C₈	—	38	4	no	—
147	C₂²	—	2	1	no	—
148	C₂²	—	2	1	no	—
149	C₂²	—	2	1	no	—
150	C₂²	C₂³×Dic₁₉	1	1	yes	characteristic
151	C₄	—	2	1	no	—
152	C₄	—	2	4	no	—
153	C₄	—	2	1	no	—
154	C₄	—	2	1	no	—
155	C₄	—	2	4	no	—
156	C₄	—	2	1	no	—
157	C₄	#10597394	1	2	yes	—
158	C₄	#10597433	1	2	yes	—
159	C₂	—	4	1	no	—
160	C₂	#11927022	1	1	yes	characteristic
161	C₂	C₂×#10597394	1	1	yes	characteristic
162	C₂	#11927311	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₉
commutator, socle	C₂²×C₁₉	C₂³×C₄
Fitting	C₁₉×#533	C₂
prime radical	C₂³×C₁₉	C₂²×C₄
cosocle	C₂×C₄×C₁₉	C₂⁴
hypercenter	#533	D₁₉
hypocenter	C₁₉	#2985
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ #533

Complements

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

N	H ≅ G/N	type
1. #11926675	C₂	semidirect
2. #11926675	C₂	semidirect
3. Q₈×C₁₉⋊C₈	C₂	semidirect
109. C₁₉	#2985	semidirect

Maximal subgroups

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

subgroup	index
#11926675	2
Q₈×C₁₉⋊C₈	2
#11926605	2
#11926601	2
#11926597	2
C₁₉×#533	2
#2985	19