Subgroups — #14299026

Order 1920.

600 conjugacy classes of subgroups.

The subgroup lattice has 144 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄³×C₅	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₃⋊#168)	C₂	1	6	yes	maximal
3	#10600187	—	3	1	no	maximal, Hall
4	C₂×C₃×C₄²×C₅	C₄	1	3	yes	—
5	C₂×C₃×C₄²×C₅	C₄	1	1	yes	characteristic
6	C₂×C₃×C₄²×C₅	C₂²	1	3	yes	—
7	C₂²×C₃⋊#168	C₂²	1	4	yes	—
8	C₄⋊(C₃⋊#168)	—	2	12	no	—
9	#69394	—	5	1	no	maximal, Hall
10	C₄³×C₅	D₃	1	1	yes	characteristic
11	C₂×C₄⋊#168	—	3	6	no	—
12	C₂²×C₃×C₄×C₅	C₂×C₄	1	3	yes	—
13	C₂²×C₃×C₄×C₅	C₂×C₄	1	3	yes	—
14	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
15	C₃×C₄²×C₅	—	2	6	no	—
16	C₃×C₄²×C₅	C₈	1	4	yes	complemented
17	C₃×C₄²×C₅	D₄	1	6	yes	—
18	C₃×C₄²×C₅	Q₈	1	6	yes	—
19	C₂×C₃⋊#168	—	2	12	no	—
20	C₃×C₄³	D₅	1	1	yes	characteristic
21	C₂×C₄⋊(C₃⋊C₈)	—	5	6	no	—
22	C₂×C₄²×C₅	Dic₃	1	3	yes	—
23	C₂×C₄²×C₅	Dic₃	1	1	yes	characteristic
24	C₂×C₄²×C₅	C₂×D₃	1	3	yes	—
25	C₂²×#168	—	3	4	no	—
26	C₄⋊#168	—	6	12	no	—
27	#1842	—	15	1	no	Sylow
28	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₃×C₄×C₅	—	2	6	no	—
30	C₂×C₃×C₄×C₅	C₄⋊C₄	1	6	yes	—
31	C₂×C₃×C₄×C₅	C₄⋊C₄	1	6	yes	—
32	C₂×C₃×C₄×C₅	C₂×C₈	1	6	yes	—
33	C₂×C₃×C₄×C₅	M₁₆	1	6	yes	—
34	C₂×C₃×C₄×C₅	C₂×D₄	1	3	yes	—
35	C₂×C₃×C₄×C₅	C₂×Q₈	1	3	yes	—
36	C₃⋊#168	—	4	4	no	—
37	C₂×C₃×C₄²	Dic₅	1	3	yes	—
38	C₂×C₃×C₄²	Dic₅	1	1	yes	characteristic
39	C₂×C₃×C₄²	C₂×D₅	1	3	yes	—
40	C₂²×C₃⋊C₈	—	5	4	no	—
41	C₄⋊(C₃⋊C₈)	—	10	12	no	—
42	C₂²×C₄×C₅	C₂×Dic₃	1	3	yes	—
43	C₂²×C₄×C₅	C₂×Dic₃	1	3	yes	—
44	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
45	C₄²×C₅	—	2	6	no	—
46	C₄²×C₅	C₃⋊C₈	1	4	yes	complemented
47	C₄²×C₅	Dic₆	1	6	yes	—
48	C₄²×C₅	D₁₂	1	6	yes	—
49	C₂×#168	—	6	12	no	—
50	C₂×C₄⋊C₈	—	15	6	no	—
51	C₄³	D₁₅	1	1	yes	characteristic
52	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	3	yes	—
53	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
54	C₂²×C₃×C₅	C₂×M₁₆	1	3	yes	—
55	C₃×C₄×C₅	—	2	6	no	—
56	C₃×C₄×C₅	C₄⋊C₈	1	12	yes	complemented
57	C₃×C₄×C₅	#127	1	1	yes	characteristic
58	C₃×C₄×C₅	C₄⋊Q₈	1	3	yes	—
59	C₂²×C₃×C₄	C₂×Dic₅	1	3	yes	—
60	C₂²×C₃×C₄	C₂×Dic₅	1	3	yes	—
61	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
62	C₃×C₄²	—	2	6	no	—
63	C₃×C₄²	#168	1	4	yes	complemented
64	C₃×C₄²	Dic₁₀	1	6	yes	—
65	C₃×C₄²	D₂₀	1	6	yes	—
66	C₂×C₃⋊C₈	—	10	12	no	—
67	C₂³×C₅	C₂²×Dic₃	1	1	yes	characteristic
68	C₂×C₄×C₅	—	2	6	no	—
69	C₂×C₄×C₅	C₂×C₃⋊C₈	1	6	yes	—
70	C₂×C₄×C₅	#208	1	6	yes	—
71	C₂×C₄×C₅	C₄⋊Dic₃	1	6	yes	—
72	C₂×C₄×C₅	C₄⋊Dic₃	1	6	yes	—
73	C₂×C₄×C₅	C₂×Dic₆	1	3	yes	—
74	C₂×C₄×C₅	C₂×D₁₂	1	3	yes	—
75	#168	—	12	4	no	—
76	C₂²×C₈	—	15	4	no	—
77	C₂×C₄²	Dic₁₅	1	3	yes	—
78	C₂×C₄²	Dic₁₅	1	1	yes	characteristic
79	C₂×C₄²	C₂×D₁₅	1	3	yes	—
80	C₄⋊C₈	—	30	12	no	—
81	C₂×C₃×C₅	#384	1	1	yes	characteristic
82	C₂×C₃×C₅	C₂×C₄⋊C₈	1	3	yes	—
83	C₂×C₃×C₅	#423	1	3	yes	—
84	C₂³×C₃	C₂²×Dic₅	1	1	yes	characteristic
85	C₂×C₃×C₄	—	2	6	no	—
86	C₂×C₃×C₄	C₂×#168	1	6	yes	—
87	C₂×C₃×C₄	#681	1	6	yes	—
88	C₂×C₃×C₄	C₄⋊Dic₅	1	6	yes	—
89	C₂×C₃×C₄	C₄⋊Dic₅	1	6	yes	—
90	C₂×C₃×C₄	C₂×Dic₁₀	1	3	yes	—
91	C₂×C₃×C₄	C₂×D₂₀	1	3	yes	—
92	C₃⋊C₈	—	20	4	no	—
93	C₂²×C₅	C₂²×C₃⋊C₈	1	1	yes	characteristic
94	C₂²×C₅	C₂×#208	1	3	yes	—
95	C₂²×C₅	C₂×C₄⋊Dic₃	1	3	yes	—
96	C₄×C₅	—	2	6	no	—
97	C₄×C₅	C₄⋊(C₃⋊C₈)	1	12	yes	complemented
98	C₄×C₅	#869	1	3	yes	—
99	C₄×C₅	#874	1	1	yes	characteristic
100	C₂²×C₄	C₂×Dic₁₅	1	3	yes	—
101	C₂²×C₄	C₂×Dic₁₅	1	3	yes	—
102	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
103	C₂×C₈	—	30	12	no	—
104	C₄²	—	2	6	no	—
105	C₄²	C₃⋊#168	1	4	yes	complemented
106	C₄²	Dic₃₀	1	6	yes	—
107	C₄²	D₆₀	1	6	yes	—
108	C₃×C₅	#1842	1	1	yes	characteristic, complemented, Hall
109	C₂²×C₃	C₂²×#168	1	1	yes	characteristic
110	C₂²×C₃	C₂×#681	1	3	yes	—
111	C₂²×C₃	C₂×C₄⋊Dic₅	1	3	yes	—
112	C₃×C₄	—	2	6	no	—
113	C₃×C₄	C₄⋊#168	1	12	yes	complemented
114	C₃×C₄	#4015	1	3	yes	—
115	C₃×C₄	#4020	1	1	yes	characteristic
116	C₂×C₅	C₂×C₄⋊(C₃⋊C₈)	1	3	yes	—
117	C₂×C₅	#4924	1	3	yes	—
118	C₂×C₅	#4935	1	1	yes	characteristic
119	C₂³	C₂²×Dic₁₅	1	1	yes	characteristic
120	C₂×C₄	—	2	6	no	—
121	C₂×C₄	C₂×C₃⋊#168	1	6	yes	—
122	C₂×C₄	#6710	1	6	yes	—
123	C₂×C₄	C₄⋊Dic₁₅	1	6	yes	—
124	C₂×C₄	C₄⋊Dic₁₅	1	6	yes	—
125	C₂×C₄	C₂×Dic₃₀	1	3	yes	—
126	C₂×C₄	C₂×D₆₀	1	3	yes	—
127	C₈	—	60	4	no	—
128	C₂×C₃	C₂×C₄⋊#168	1	3	yes	—
129	C₂×C₃	#65213	1	3	yes	—
130	C₂×C₃	#65224	1	1	yes	characteristic
131	C₅	#69394	1	1	yes	characteristic, complemented, Sylow
132	C₂²	C₂²×C₃⋊#168	1	1	yes	characteristic
133	C₂²	C₂×#6710	1	3	yes	—
134	C₂²	C₂×C₄⋊Dic₁₅	1	3	yes	—
135	C₄	—	2	6	no	—
136	C₄	C₄⋊(C₃⋊#168)	1	12	yes	complemented
137	C₄	#91758	1	3	yes	—
138	C₄	#91763	1	1	yes	characteristic
139	C₃	#10600187	1	1	yes	characteristic, complemented, Sylow
140	C₂	C₂×C₄⋊(C₃⋊#168)	1	3	yes	—
141	C₂	#11751025	1	3	yes	—
142	C₂	#11751036	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₅
commutator	C₂²×C₃×C₅	C₂²×C₈
Fitting	C₃×C₄³×C₅	C₂
socle, prime radical	C₂³×C₃×C₅	C₂²×C₄
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	C₄³	D₁₅
hypocenter	C₃×C₅	#1842
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³

Complements

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

N	H ≅ G/N	type
16. C₃×C₄²×C₅	C₈	semidirect
46. C₄²×C₅	C₃⋊C₈	semidirect
56. C₃×C₄×C₅	C₄⋊C₈	semidirect
63. C₃×C₄²	#168	semidirect
97. C₄×C₅	C₄⋊(C₃⋊C₈)	semidirect
105. C₄²	C₃⋊#168	semidirect
108. C₃×C₅	#1842	semidirect
113. C₃×C₄	C₄⋊#168	semidirect
131. C₅	#69394	semidirect
136. C₄	C₄⋊(C₃⋊#168)	semidirect
139. C₃	#10600187	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄³×C₅	2
C₂×C₄⋊(C₃⋊#168)	2
#10600187	3
#69394	5