Subgroups — #12015726

Order 1280.

334 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#64733	C₂	1	4	yes	maximal
3	#10600585	C₂	1	1	yes	maximal, characteristic
4	#10600557	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅×M₃₂	C₂²	1	2	yes	—
6	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₂×#3951	—	2	1	no	—
8	C₂×#3951	C₂²	1	2	yes	—
9	C₂×#3949	—	2	1	no	—
10	C₂×#3949	C₂²	1	2	yes	—
11	#65224	—	2	1	no	—
12	#65213	—	2	1	no	—
13	C₅×#346	C₄	1	2	yes	complemented
14	C₄×C₅×C₁₆	C₄	1	2	yes	complemented
15	#64733	—	2	8	no	—
16	#12903	—	5	1	no	maximal, Sylow
17	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
18	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
19	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
20	C₂×C₄⋊Dic₅	—	2	2	no	—
21	C₂×C₄⋊Dic₅	—	4	2	no	—
22	C₂×#681	—	2	2	no	—
23	C₅×M₃₂	D₄	1	4	yes	—
24	C₅×M₃₂	Q₈	1	4	yes	—
25	C₂×C₅×C₁₆	C₂×C₄	1	4	yes	—
26	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
27	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
28	C₄×C₅×C₈	Q₈	1	1	yes	characteristic
29	C₄×C₅×C₈	Q₈	1	1	yes	characteristic
30	#3951	—	2	4	no	—
31	#3951	—	4	2	no	—
32	#3949	—	2	4	no	—
33	#3949	—	4	2	no	—
34	C₄⋊#168	—	4	2	no	—
35	C₂×#365	—	5	4	no	—
36	C₄×M₃₂	D₅	1	1	yes	characteristic
37	#1848	—	5	1	no	—
38	#1844	—	5	1	no	—
39	C₂²×C₄×C₅	SD₁₆	1	2	yes	—
40	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
41	C₂²×Dic₅	—	4	2	no	—
42	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₅×C₈	SD₁₆	1	2	yes	—
45	C₂×C₅×C₈	SD₁₆	1	2	yes	—
46	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
48	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
49	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
50	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
51	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
52	C₄²×C₅	SD₁₆	1	2	yes	—
53	C₄⋊Dic₅	—	4	2	no	—
54	C₄⋊Dic₅	—	4	2	no	—
55	C₄⋊Dic₅	—	8	4	no	—
56	#681	—	4	2	no	—
57	#681	—	4	2	no	—
58	C₂×#168	—	4	2	no	—
59	C₅×C₁₆	C₄⋊C₄	1	8	yes	complemented
60	C₂×M₃₂	C₂×D₅	1	2	yes	—
61	C₂×#108	—	5	2	no	—
62	C₂×#108	—	10	1	no	—
63	C₂×#106	—	5	2	no	—
64	C₂×#106	—	10	1	no	—
65	#423	—	10	1	no	—
66	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
67	#384	—	10	1	no	—
68	#365	—	10	8	no	—
69	#346	Dic₅	1	2	yes	complemented
70	C₄×C₁₆	Dic₅	1	2	yes	complemented
71	C₂³×C₅	C₂×SD₁₆	1	1	yes	characteristic
72	C₂×C₄×C₅	—	2	4	no	—
73	C₂×C₄×C₅	#106	1	2	yes	—
74	C₂×C₄×C₅	#106	1	2	yes	—
75	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₅	#127	1	1	yes	characteristic
77	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
78	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
79	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
80	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
81	C₂×Dic₅	—	8	4	no	—
82	C₂×Dic₅	—	8	2	no	—
83	C₅×C₈	#106	1	2	yes	—
84	C₅×C₈	#106	1	2	yes	—
85	C₅×C₈	C₂×C₄⋊C₄	1	2	yes	—
86	C₅×C₈	C₄⋊Q₈	1	2	yes	—
87	#168	—	8	2	no	—
88	C₂×M₁₆	—	10	2	no	—
89	C₂²×C₈	D₂₀	1	1	yes	characteristic
90	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
91	C₂×C₄⋊C₄	—	10	2	no	—
92	C₂×C₄⋊C₄	—	20	2	no	—
93	C₂×C₄²	D₂₀	1	1	yes	characteristic
94	M₃₂	Dic₁₀	1	4	yes	—
95	M₃₂	D₂₀	1	4	yes	—
96	C₂×C₁₆	C₂×Dic₅	1	4	yes	—
97	#108	—	10	4	no	—
98	#108	—	20	2	no	—
99	#106	—	10	4	no	—
100	#106	—	20	2	no	—
101	C₄⋊C₈	—	20	2	no	—
102	C₄×C₈	Dic₁₀	1	1	yes	characteristic
103	C₄×C₈	Dic₁₀	1	1	yes	characteristic
104	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
105	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
106	C₂²×C₅	—	2	2	no	—
107	C₂²×C₅	C₂×#106	1	1	yes	characteristic
108	C₂²×C₅	#492	1	1	yes	characteristic
109	C₂²×C₅	#498	1	1	yes	characteristic
110	C₄×C₅	—	2	2	no	—
111	C₄×C₅	#365	1	4	yes	complemented
112	C₄×C₅	#384	1	1	yes	characteristic
113	C₄×C₅	C₂×#106	1	1	yes	characteristic
114	C₄×C₅	#492	1	1	yes	characteristic
115	C₄×C₅	#498	1	1	yes	characteristic
116	Dic₅	—	16	2	no	—
117	C₂²×C₄	—	20	2	no	—
118	C₂²×C₄	#677	1	2	yes	—
119	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
120	M₁₆	—	20	2	no	—
121	M₁₆	—	20	2	no	—
122	C₂×C₈	—	20	2	no	—
123	C₂×C₈	#677	1	2	yes	—
124	C₂×C₈	#677	1	2	yes	—
125	C₂×C₈	C₄⋊Dic₅	1	1	yes	characteristic
126	C₂×C₈	C₄⋊Dic₅	1	1	yes	characteristic
127	C₂×C₈	C₂×Dic₁₀	1	1	yes	characteristic
128	C₂×C₈	C₂×Dic₁₀	1	2	yes	—
129	C₂×C₈	C₂×D₂₀	1	2	yes	—
130	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
131	C₄⋊C₄	—	20	2	no	—
132	C₄⋊C₄	—	20	2	no	—
133	C₄⋊C₄	—	40	4	no	—
134	C₄²	#677	1	2	yes	—
135	C₄²	C₄⋊Dic₅	1	1	yes	characteristic
136	C₄²	C₄⋊Dic₅	1	1	yes	characteristic
137	C₁₆	C₄⋊Dic₅	1	8	yes	complemented
138	C₂×C₅	—	2	1	no	—
139	C₂×C₅	—	2	1	no	—
140	C₂×C₅	#1844	1	1	yes	characteristic
141	C₂×C₅	C₂×#365	1	2	yes	—
142	C₂³	C₂×#677	1	1	yes	characteristic
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	40	4	no	—
145	C₂×C₄	—	40	2	no	—
146	C₂×C₄	#3949	1	2	yes	—
147	C₂×C₄	#3949	1	2	yes	—
148	C₂×C₄	#4015	1	1	yes	characteristic
149	C₂×C₄	#4020	1	1	yes	characteristic
150	C₂×C₄	C₂×#677	1	1	yes	characteristic
151	C₂×C₄	C₂×#677	1	1	yes	characteristic
152	C₂×C₄	C₂×#677	1	1	yes	characteristic
153	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
154	C₈	—	40	2	no	—
155	C₈	#3949	1	2	yes	—
156	C₈	#3949	1	2	yes	—
157	C₈	#4015	1	2	yes	—
158	C₈	C₂×C₄⋊Dic₅	1	2	yes	—
159	C₅	#12903	1	1	yes	characteristic, complemented, Sylow
160	C₂²	—	2	2	no	—
161	C₂²	#64969	1	1	yes	characteristic
162	C₂²	#64982	1	1	yes	characteristic
163	C₂²	C₂×#3949	1	1	yes	characteristic
164	C₄	—	2	2	no	—
165	C₄	—	80	2	no	—
166	C₄	#64733	1	4	yes	complemented
167	C₄	#64969	1	1	yes	characteristic
168	C₄	#64982	1	1	yes	characteristic
169	C₄	#65224	1	1	yes	characteristic
170	C₄	C₂×#3949	1	1	yes	characteristic
171	C₂	—	2	1	no	—
172	C₂	—	2	1	no	—
173	C₂	#10600557	1	1	yes	characteristic
174	C₂	C₂×#64733	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#3949
commutator	C₂×C₅×C₈	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₄×C₅×M₃₂	C₂
socle	C₂²×C₅	C₂×#106
prime radical	C₂³×C₅	C₂×SD₁₆
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	C₄×M₃₂	D₅
hypocenter	C₅	#12903
LCS	G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
13. C₅×#346	C₄	semidirect
14. C₄×C₅×C₁₆	C₄	semidirect
59. C₅×C₁₆	C₄⋊C₄	semidirect
69. #346	Dic₅	semidirect
70. C₄×C₁₆	Dic₅	semidirect
111. C₄×C₅	#365	semidirect
137. C₁₆	C₄⋊Dic₅	semidirect
159. C₅	#12903	semidirect
166. C₄	#64733	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₅×M₃₂	2
C₂×#64733	2
#10600585	2
#10600557	2
#12903	5