Subgroups — #12019792

Order 1280.

198 conjugacy classes of subgroups.

The subgroup lattice has 154 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×M₆₄	C₂	1	1	yes	maximal, characteristic
2	#10602192	C₂	1	1	yes	maximal, characteristic
3	C₂×#64719	C₂	1	1	yes	maximal, characteristic
4	#10598090	C₂	1	4	yes	maximal, complemented
5	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
6	#65390	C₄	1	1	yes	characteristic
7	C₂×#3944	C₄	1	1	yes	characteristic
8	C₅×M₆₄	—	2	1	no	—
9	C₅×M₆₄	C₂²	1	2	yes	—
10	C₂×C₅×C₃₂	—	2	1	no	—
11	#64722	C₄	1	2	yes	—
12	C₁₆×Dic₅	C₂²	1	2	yes	—
13	#64719	—	2	1	no	—
14	#64719	C₂²	1	2	yes	—
15	C₂×#3926	—	2	1	no	—
16	#13630	—	5	1	no	maximal, Sylow
17	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
18	#4072	C₈	1	1	yes	characteristic
19	C₂×#681	C₈	1	1	yes	characteristic
20	C₂×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
21	C₂×C₅×C₁₆	D₄	1	1	yes	characteristic
22	C₂×C₅×C₁₆	D₄	1	1	yes	characteristic
23	C₂×C₅×C₁₆	Q₈	1	1	yes	characteristic
24	C₂×C₅×C₁₆	Q₈	1	1	yes	characteristic
25	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
26	#3947	C₈	1	2	yes	—
27	C₈×Dic₅	C₂×C₄	1	2	yes	—
28	#3944	—	2	1	no	—
29	#3944	—	2	1	no	—
30	C₂×#672	C₂×C₄	1	2	yes	—
31	C₅×C₃₂	—	2	2	no	—
32	#3926	—	2	2	no	—
33	C₂×M₆₄	—	5	1	no	—
34	C₂×M₆₄	D₅	1	1	yes	characteristic
35	#2108	—	5	1	no	—
36	#1422	—	5	4	no	—
37	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
38	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₅×C₈	C₂×C₈	1	1	yes	characteristic
41	C₂×C₅×C₈	M₁₆	1	1	yes	characteristic
42	C₂×C₅×C₈	M₁₆	1	1	yes	characteristic
43	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
44	C₂²⋊Dic₅	—	2	1	no	—
45	C₄⋊Dic₅	—	2	1	no	—
46	C₄×Dic₅	C₂×C₈	1	2	yes	—
47	#681	C₁₆	1	2	yes	—
48	#681	C₁₆	1	2	yes	—
49	C₂×#168	C₂×C₈	1	2	yes	—
50	C₅×C₁₆	C₄⋊C₄	1	1	yes	characteristic
51	C₅×C₁₆	C₄⋊C₄	1	1	yes	characteristic
52	C₅×C₁₆	C₂×D₄	1	1	yes	characteristic
53	C₅×C₁₆	C₂×Q₈	1	1	yes	characteristic
54	#672	—	2	2	no	—
55	C₂×M₃₂	—	5	1	no	—
56	C₂²×C₁₆	C₂×D₅	1	1	yes	characteristic
57	#405	—	5	1	no	—
58	M₆₄	—	2	1	no	—
59	M₆₄	—	5	2	no	—
60	M₆₄	—	10	1	no	—
61	M₆₄	C₂×D₅	1	2	yes	—
62	C₂×C₃₂	—	2	1	no	—
63	C₂×C₃₂	—	10	1	no	—
64	#346	—	5	2	no	—
65	C₄×C₁₆	—	5	2	no	—
66	C₂³×C₅	M₃₂	1	1	yes	characteristic
67	C₂×C₄×C₅	C₄⋊C₈	1	1	yes	characteristic
68	C₂×C₄×C₅	C₄⋊C₈	1	1	yes	characteristic
69	C₂×C₄×C₅	C₂×C₁₆	1	1	yes	characteristic
70	C₂×C₄×C₅	C₂×C₁₆	1	1	yes	characteristic
71	C₂×C₄×C₅	M₃₂	1	1	yes	characteristic
72	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
73	C₂×Dic₅	—	2	2	no	—
74	C₅×C₈	C₄⋊C₈	1	1	yes	characteristic
75	C₅×C₈	C₄⋊C₈	1	1	yes	characteristic
76	C₅×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₅×C₈	C₂×M₁₆	1	1	yes	characteristic
78	#168	C₂×C₁₆	1	4	yes	—
79	C₂×M₁₆	—	5	1	no	—
80	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
81	#117	—	5	1	no	—
82	M₃₂	—	10	1	no	—
83	M₃₂	—	10	1	no	—
84	C₂×C₁₆	—	5	2	no	—
85	C₂×C₁₆	Dic₁₀	1	1	yes	characteristic
86	C₂×C₁₆	Dic₁₀	1	1	yes	characteristic
87	C₂×C₁₆	C₄×D₅	1	1	yes	characteristic
88	C₂×C₁₆	C₂²⋊D₅	1	1	yes	characteristic
89	C₂×C₁₆	C₂²⋊D₅	1	1	yes	characteristic
90	C₂×C₁₆	C₂²×D₅	1	1	yes	characteristic
91	#97	—	5	2	no	—
92	C₄×C₈	—	5	2	no	—
93	C₃₂	—	2	2	no	—
94	C₃₂	—	10	2	no	—
95	C₂²×C₅	—	4	1	no	—
96	C₂²×C₅	C₄⋊C₁₆	1	1	yes	characteristic
97	C₂²×C₅	C₄⋊C₁₆	1	1	yes	characteristic
98	C₂²×C₅	C₂×M₃₂	1	1	yes	characteristic
99	C₄×C₅	C₄⋊C₁₆	1	1	yes	characteristic
100	C₄×C₅	C₄⋊C₁₆	1	1	yes	characteristic
101	C₄×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
102	C₄×C₅	C₂²×C₁₆	1	1	yes	characteristic
103	Dic₅	—	4	2	no	—
104	C₂²×C₄	C₈×D₅	1	1	yes	characteristic
105	M₁₆	—	5	2	no	—
106	M₁₆	—	5	2	no	—
107	C₂×C₈	—	5	2	no	—
108	C₂×C₈	C₈×D₅	1	1	yes	characteristic
109	C₂×C₈	#676	1	1	yes	characteristic
110	C₂×C₈	#676	1	1	yes	characteristic
111	C₂×C₈	#683	1	1	yes	characteristic
112	C₂×C₈	#683	1	1	yes	characteristic
113	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
114	C₄⋊C₄	—	10	1	no	—
115	C₂²⋊C₄	—	10	1	no	—
116	C₄²	—	5	2	no	—
117	C₁₆	—	10	2	no	—
118	C₁₆	#683	1	1	yes	characteristic
119	C₁₆	#683	1	1	yes	characteristic
120	C₁₆	C₂×Dic₁₀	1	1	yes	characteristic
121	C₁₆	C₂×C₂²⋊D₅	1	1	yes	characteristic
122	C₂×C₅	—	2	1	no	—
123	C₂×C₅	—	2	1	no	—
124	C₂×C₅	—	2	1	no	—
125	C₂×C₅	C₂×C₄⋊C₁₆	1	1	yes	characteristic
126	C₂³	#3930	1	1	yes	characteristic
127	C₂×C₄	—	10	2	no	—
128	C₂×C₄	D₅×C₁₆	1	1	yes	characteristic
129	C₂×C₄	D₅×C₁₆	1	1	yes	characteristic
130	C₂×C₄	#3930	1	1	yes	characteristic
131	C₂×C₄	#3946	1	1	yes	characteristic
132	C₂×C₄	#3946	1	1	yes	characteristic
133	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
134	C₈	—	5	4	no	—
135	C₈	#3946	1	1	yes	characteristic
136	C₈	#3946	1	1	yes	characteristic
137	C₈	C₂×#676	1	1	yes	characteristic
138	C₈	C₂×#683	1	1	yes	characteristic
139	C₅	#13630	1	1	yes	characteristic, complemented, Sylow
140	C₂²	—	4	1	no	—
141	C₂²	#64721	1	1	yes	characteristic
142	C₂²	#64721	1	1	yes	characteristic
143	C₂²	C₂×#3930	1	1	yes	characteristic
144	C₄	—	20	2	no	—
145	C₄	#64721	1	1	yes	characteristic
146	C₄	#64721	1	1	yes	characteristic
147	C₄	C₂×D₅×C₁₆	1	1	yes	characteristic
148	C₄	C₂×#3946	1	1	yes	characteristic
149	C₂	—	2	1	no	—
150	C₂	—	2	1	no	—
151	C₂	—	2	1	no	—
152	C₂	C₂×#64721	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	C₂×C₂²⋊D₅
commutator	C₄×C₅	C₂²×C₁₆
Frattini	C₂×C₁₆	C₂²×D₅
Fitting	C₂×C₅×M₆₄	C₂
socle	C₂×C₅	C₂×C₄⋊C₁₆
prime radical	C₂³×C₅	M₃₂
cosocle	C₂×C₅×C₁₆	C₂³
hypercenter	C₂×M₆₄	D₅
hypocenter	C₅	#13630
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₁₆ ◁ C₂²×C₁₆ ◁ C₂×M₆₄

Complements

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

N	H ≅ G/N	type
4. #10598090	C₂	semidirect
139. C₅	#13630	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×M₆₄	2
#10602192	2
C₂×#64719	2
#10598090	2
#13630	5