Subgroups — #11993038

Order 1280.

462 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×C₅×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#64677	C₂	1	4	yes	maximal
3	#10600187	C₂	1	2	yes	maximal
4	C₂×C₄×C₅×C₈	C₄	1	2	yes	—
5	C₂×C₄×C₅×C₈	C₄	1	1	yes	characteristic
6	C₂×C₄×C₅×C₈	C₄	1	1	yes	characteristic
7	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
8	C₄³×C₅	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊#168	—	2	4	no	—
10	C₂×C₄⋊#168	C₂²	1	4	yes	—
11	#64677	—	2	8	no	—
12	#8805	—	5	1	no	maximal, Sylow
13	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
14	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
15	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄²×C₅	D₄	1	2	yes	—
18	C₂×C₄²×C₅	Q₈	1	2	yes	—
19	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
20	C₂²×#168	—	2	4	no	—
21	C₄×C₅×C₈	—	2	4	no	—
22	C₄×C₅×C₈	—	2	2	no	—
23	C₄×C₅×C₈	C₈	1	4	yes	complemented
24	C₄×C₅×C₈	D₄	1	4	yes	—
25	C₄×C₅×C₈	Q₈	1	4	yes	—
26	C₄⋊#168	—	2	8	no	—
27	C₄⋊#168	—	4	8	no	—
28	#1842	—	5	2	no	—
29	C₄²×C₈	D₅	1	1	yes	characteristic
30	C₂×#334	—	5	4	no	—
31	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
32	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
33	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₅×C₈	—	2	2	no	—
37	C₂×C₅×C₈	—	2	2	no	—
38	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
39	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
40	C₂×C₅×C₈	C₂×C₈	1	4	yes	—
41	C₂×C₅×C₈	M₁₆	1	4	yes	—
42	C₄²×C₅	—	2	4	no	—
43	C₄²×C₅	—	2	2	no	—
44	C₄²×C₅	C₂×C₈	1	2	yes	—
45	C₄²×C₅	M₁₆	1	2	yes	—
46	C₄²×C₅	SD₁₆	1	8	yes	—
47	C₄²×C₅	C₂×D₄	1	2	yes	—
48	C₄²×C₅	C₂×Q₈	1	2	yes	—
49	C₂×#168	—	4	4	no	—
50	C₂×#168	—	4	8	no	—
51	C₂×C₄⋊C₈	—	5	4	no	—
52	C₂×C₄⋊C₈	—	10	4	no	—
53	C₂×C₄×C₈	Dic₅	1	2	yes	—
54	C₂×C₄×C₈	Dic₅	1	1	yes	characteristic
55	C₂×C₄×C₈	Dic₅	1	1	yes	characteristic
56	C₂×C₄×C₈	C₂×D₅	1	2	yes	—
57	C₄³	C₂×D₅	1	1	yes	characteristic
58	#334	—	10	8	no	—
59	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₅	—	2	2	no	—
61	C₂×C₄×C₅	—	2	2	no	—
62	C₂×C₄×C₅	—	2	2	no	—
63	C₂×C₄×C₅	C₄⋊C₈	1	4	yes	—
64	C₂×C₄×C₅	#106	1	4	yes	—
65	C₂×C₄×C₅	#106	1	4	yes	—
66	C₂×C₄×C₅	#108	1	4	yes	—
67	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
68	C₂×C₄×C₅	#127	1	1	yes	characteristic
69	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
70	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
71	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
72	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
73	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
74	C₂×C₄×C₅	C₂×SD₁₆	1	4	yes	—
75	C₅×C₈	—	2	4	no	—
76	C₅×C₈	C₄⋊C₈	1	8	yes	complemented
77	#168	—	8	4	no	—
78	C₂²×C₈	—	10	4	no	—
79	C₂²×C₈	C₂×Dic₅	1	2	yes	—
80	C₂²×C₈	C₂×Dic₅	1	2	yes	—
81	C₂×C₄²	Dic₁₀	1	2	yes	—
82	C₂×C₄²	D₂₀	1	2	yes	—
83	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
84	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
85	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
86	C₄⋊C₈	—	10	8	no	—
87	C₄⋊C₈	—	20	8	no	—
88	C₄×C₈	—	2	4	no	—
89	C₄×C₈	—	2	2	no	—
90	C₄×C₈	#168	1	4	yes	complemented
91	C₄×C₈	Dic₁₀	1	4	yes	—
92	C₄×C₈	D₂₀	1	4	yes	—
93	C₂²×C₅	#384	1	1	yes	characteristic
94	C₂²×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
95	C₂²×C₅	#423	1	1	yes	characteristic
96	C₂²×C₅	C₂×#106	1	2	yes	—
97	C₂²×C₅	C₂×#108	1	2	yes	—
98	C₄×C₅	—	2	4	no	—
99	C₄×C₅	—	2	2	no	—
100	C₄×C₅	#334	1	8	yes	complemented
101	C₄×C₅	C₂×C₄⋊C₈	1	2	yes	—
102	C₄×C₅	#423	1	2	yes	—
103	C₄×C₅	#492	1	2	yes	—
104	C₄×C₅	#498	1	2	yes	—
105	C₂²×C₄	C₄⋊Dic₅	1	2	yes	—
106	C₂²×C₄	C₄⋊Dic₅	1	2	yes	—
107	C₂²×C₄	C₂×Dic₁₀	1	1	yes	characteristic
108	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
109	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	—	20	4	no	—
113	C₂×C₈	—	20	8	no	—
114	C₂×C₈	C₂×#168	1	4	yes	—
115	C₂×C₈	#681	1	4	yes	—
116	C₂×C₈	C₄⋊Dic₅	1	4	yes	—
117	C₂×C₈	C₄⋊Dic₅	1	4	yes	—
118	C₄²	—	2	4	no	—
119	C₄²	—	2	2	no	—
120	C₄²	#677	1	8	yes	—
121	C₄²	C₂×#168	1	2	yes	—
122	C₄²	#681	1	2	yes	—
123	C₄²	C₂×Dic₁₀	1	2	yes	—
124	C₄²	C₂×D₂₀	1	2	yes	—
125	C₂×C₅	C₂×#334	1	2	yes	—
126	C₂×C₅	#1566	1	2	yes	—
127	C₂×C₅	#1842	1	1	yes	characteristic
128	C₂×C₅	#1844	1	1	yes	characteristic
129	C₂×C₅	#1848	1	1	yes	characteristic
130	C₂³	C₂×C₄⋊Dic₅	1	1	yes	characteristic
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	C₄⋊#168	1	4	yes	—
135	C₂×C₄	#3949	1	4	yes	—
136	C₂×C₄	#3949	1	4	yes	—
137	C₂×C₄	#3951	1	4	yes	—
138	C₂×C₄	#4015	1	2	yes	—
139	C₂×C₄	#4015	1	1	yes	characteristic
140	C₂×C₄	#4020	1	1	yes	characteristic
141	C₂×C₄	C₂×#677	1	4	yes	—
142	C₂×C₄	C₂²×#168	1	1	yes	characteristic
143	C₂×C₄	C₂×#681	1	2	yes	—
144	C₂×C₄	C₂×#681	1	1	yes	characteristic
145	C₂×C₄	C₂×C₄⋊Dic₅	1	2	yes	—
146	C₈	—	2	4	no	—
147	C₈	—	40	4	no	—
148	C₈	C₄⋊#168	1	8	yes	complemented
149	C₅	#8805	1	1	yes	characteristic, complemented, Sylow
150	C₂²	C₂×C₄⋊#168	1	1	yes	characteristic
151	C₂²	#65213	1	1	yes	characteristic
152	C₂²	#65224	1	1	yes	characteristic
153	C₂²	C₂×#3949	1	2	yes	—
154	C₂²	C₂×#3951	1	2	yes	—
155	C₄	—	2	4	no	—
156	C₄	—	2	2	no	—
157	C₄	#64677	1	8	yes	complemented
158	C₄	#64969	1	2	yes	—
159	C₄	#64982	1	2	yes	—
160	C₄	C₂×C₄⋊#168	1	2	yes	—
161	C₄	#65213	1	2	yes	—
162	C₂	#10600187	1	1	yes	characteristic
163	C₂	C₂×#64677	1	2	yes	—
164	C₂	#10600523	1	2	yes	—
165	C₂	#10600557	1	1	yes	characteristic
166	C₂	#10600585	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₀
commutator	C₂×C₄×C₅	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₄²×C₅×C₈	C₂
socle, prime radical	C₂³×C₅	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	C₄²×C₈	D₅
hypocenter	C₅	#8805
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
23. C₄×C₅×C₈	C₈	semidirect
76. C₅×C₈	C₄⋊C₈	semidirect
90. C₄×C₈	#168	semidirect
100. C₄×C₅	#334	semidirect
148. C₈	C₄⋊#168	semidirect
149. C₅	#8805	semidirect
157. C₄	#64677	semidirect

Maximal subgroups

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

subgroup	index
C₄²×C₅×C₈	2
C₂×#64677	2
#10600187	2
#8805	5