Subgroups — C₂×#10598001 #12019790

Order 1280.

214 conjugacy classes of subgroups.

The subgroup lattice has 146 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#64719	C₂	1	2	yes	maximal
3	#10598001	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₅×C₁₆	C₄	1	1	yes	characteristic
5	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
6	C₂×C₄×C₅×C₈	C₄	1	1	yes	characteristic
7	C₄×C₅×C₁₆	C₄	1	2	yes	—
8	C₄×C₅×C₁₆	C₂²	1	2	yes	—
9	#64719	—	2	2	no	—
10	#64719	C₂²	1	4	yes	—
11	C₂×#3926	—	2	2	no	—
12	C₂×#1422	—	5	1	no	maximal, Sylow
13	C₂²×C₅×C₈	C₈	1	1	yes	characteristic
14	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄²×C₅	C₈	1	1	yes	characteristic
16	C₂×C₅×C₁₆	—	2	1	no	—
17	C₂×C₅×C₁₆	—	2	1	no	—
18	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
19	C₂×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
20	C₂×C₅×C₁₆	D₄	1	1	yes	characteristic
21	C₂×C₅×C₁₆	D₄	1	1	yes	characteristic
22	C₂×C₅×C₁₆	Q₈	1	1	yes	characteristic
23	C₂×C₅×C₁₆	Q₈	1	1	yes	characteristic
24	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
25	C₄×C₅×C₈	C₈	1	2	yes	—
26	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
27	#3926	—	2	4	no	—
28	C₂×M₆₄	—	5	2	no	—
29	C₂×C₄×C₁₆	D₅	1	1	yes	characteristic
30	#1422	—	5	4	no	—
31	C₂²×C₄×C₅	—	2	1	no	—
32	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
33	C₂×C₅×C₈	C₁₆	1	2	yes	—
34	C₂×C₅×C₈	C₁₆	1	2	yes	—
35	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₅×C₈	C₂×C₈	1	1	yes	characteristic
38	C₂×C₅×C₈	C₂×C₈	1	2	yes	—
39	C₂×C₅×C₈	M₁₆	1	1	yes	characteristic
40	C₂×C₅×C₈	M₁₆	1	1	yes	characteristic
41	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
42	C₄²×C₅	—	2	1	no	—
43	C₄²×C₅	C₂×C₈	1	2	yes	—
44	C₅×C₁₆	—	2	2	no	—
45	C₅×C₁₆	C₄⋊C₄	1	1	yes	characteristic
46	C₅×C₁₆	C₄⋊C₄	1	1	yes	characteristic
47	C₅×C₁₆	C₂×D₄	1	1	yes	characteristic
48	C₅×C₁₆	C₂×Q₈	1	1	yes	characteristic
49	C₂²×C₁₆	Dic₅	1	1	yes	characteristic
50	C₂²×C₁₆	C₂×D₅	1	1	yes	characteristic
51	C₂×C₄×C₈	Dic₅	1	1	yes	characteristic
52	M₆₄	—	5	4	no	—
53	M₆₄	—	10	2	no	—
54	C₂×C₃₂	—	10	2	no	—
55	C₄×C₁₆	Dic₅	1	2	yes	—
56	C₄×C₁₆	C₂×D₅	1	2	yes	—
57	C₂³×C₅	M₃₂	1	1	yes	characteristic
58	C₂×C₄×C₅	—	2	4	no	—
59	C₂×C₄×C₅	—	2	2	no	—
60	C₂×C₄×C₅	C₄⋊C₈	1	1	yes	characteristic
61	C₂×C₄×C₅	C₄⋊C₈	1	1	yes	characteristic
62	C₂×C₄×C₅	C₂×C₁₆	1	1	yes	characteristic
63	C₂×C₄×C₅	C₂×C₁₆	1	1	yes	characteristic
64	C₂×C₄×C₅	M₃₂	1	1	yes	characteristic
65	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
66	C₅×C₈	C₄⋊C₈	1	1	yes	characteristic
67	C₅×C₈	C₄⋊C₈	1	1	yes	characteristic
68	C₅×C₈	C₂×C₁₆	1	4	yes	—
69	C₅×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₅×C₈	C₂×M₁₆	1	1	yes	characteristic
71	C₂²×C₈	#168	1	1	yes	characteristic
72	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
73	C₂×C₄²	#168	1	1	yes	characteristic
74	C₂×C₁₆	—	2	1	no	—
75	C₂×C₁₆	—	2	1	no	—
76	C₂×C₁₆	Dic₁₀	1	1	yes	characteristic
77	C₂×C₁₆	Dic₁₀	1	1	yes	characteristic
78	C₂×C₁₆	D₂₀	1	1	yes	characteristic
79	C₂×C₁₆	D₂₀	1	1	yes	characteristic
80	C₂×C₁₆	C₂×Dic₅	1	2	yes	—
81	C₂×C₁₆	C₂×Dic₅	1	1	yes	characteristic
82	C₂×C₁₆	C₂²×D₅	1	1	yes	characteristic
83	C₄×C₈	#168	1	2	yes	—
84	C₄×C₈	C₂×Dic₅	1	2	yes	—
85	C₃₂	—	10	4	no	—
86	C₂²×C₅	—	2	2	no	—
87	C₂²×C₅	C₄⋊C₁₆	1	1	yes	characteristic
88	C₂²×C₅	C₄⋊C₁₆	1	1	yes	characteristic
89	C₂²×C₅	C₂×M₃₂	1	1	yes	characteristic
90	C₄×C₅	—	2	4	no	—
91	C₄×C₅	C₄⋊C₁₆	1	1	yes	characteristic
92	C₄×C₅	C₄⋊C₁₆	1	1	yes	characteristic
93	C₄×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
94	C₄×C₅	C₂²×C₁₆	1	1	yes	characteristic
95	C₂²×C₄	—	2	1	no	—
96	C₂²×C₄	C₂×#168	1	1	yes	characteristic
97	C₂×C₈	#672	1	2	yes	—
98	C₂×C₈	#672	1	2	yes	—
99	C₂×C₈	C₂×#168	1	1	yes	characteristic
100	C₂×C₈	C₂×#168	1	2	yes	—
101	C₂×C₈	#681	1	1	yes	characteristic
102	C₂×C₈	#681	1	1	yes	characteristic
103	C₂×C₈	C₄⋊Dic₅	1	1	yes	characteristic
104	C₂×C₈	C₄⋊Dic₅	1	1	yes	characteristic
105	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
106	C₄²	—	2	1	no	—
107	C₄²	C₂×#168	1	2	yes	—
108	C₁₆	—	2	2	no	—
109	C₁₆	C₄⋊Dic₅	1	1	yes	characteristic
110	C₁₆	C₄⋊Dic₅	1	1	yes	characteristic
111	C₁₆	C₂×Dic₁₀	1	1	yes	characteristic
112	C₁₆	C₂×D₂₀	1	1	yes	characteristic
113	C₂×C₅	—	2	1	no	—
114	C₂×C₅	—	2	1	no	—
115	C₂×C₅	#1422	1	2	yes	complemented
116	C₂×C₅	C₂×C₄⋊C₁₆	1	1	yes	characteristic
117	C₂³	#3944	1	1	yes	characteristic
118	C₂×C₄	—	2	4	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	C₄⋊#168	1	1	yes	characteristic
121	C₂×C₄	C₄⋊#168	1	1	yes	characteristic
122	C₂×C₄	C₂×#672	1	1	yes	characteristic
123	C₂×C₄	C₂×#672	1	1	yes	characteristic
124	C₂×C₄	#3944	1	1	yes	characteristic
125	C₂×C₄	C₂²×#168	1	1	yes	characteristic
126	C₈	C₄⋊#168	1	1	yes	characteristic
127	C₈	C₄⋊#168	1	1	yes	characteristic
128	C₈	C₂×#672	1	4	yes	—
129	C₈	C₂×#681	1	1	yes	characteristic
130	C₈	C₂×C₄⋊Dic₅	1	1	yes	characteristic
131	C₅	C₂×#1422	1	1	yes	characteristic, complemented, Sylow
132	C₂²	—	2	2	no	—
133	C₂²	C₄⋊#672	1	1	yes	characteristic
134	C₂²	C₄⋊#672	1	1	yes	characteristic
135	C₂²	C₂×#3944	1	1	yes	characteristic
136	C₄	—	2	4	no	—
137	C₄	C₄⋊#672	1	1	yes	characteristic
138	C₄	C₄⋊#672	1	1	yes	characteristic
139	C₄	C₂×C₄⋊#168	1	1	yes	characteristic
140	C₄	C₂²×#672	1	1	yes	characteristic
141	C₂	—	2	1	no	—
142	C₂	—	2	1	no	—
143	C₂	#10598001	1	2	yes	direct product factor
144	C₂	C₂×C₄⋊#672	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₆	D₂₀
commutator	C₄×C₅	C₂²×C₁₆
Frattini	C₂×C₁₆	C₂²×D₅
Fitting	C₂×C₄×C₅×C₁₆	C₂
socle	C₂²×C₅	C₄⋊C₁₆
prime radical	C₂³×C₅	M₃₂
cosocle	C₂×C₅×C₁₆	C₂³
hypercenter	C₂×C₄×C₁₆	D₅
hypocenter	C₅	C₂×#1422
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂²×C₁₆ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
3. #10598001	C₂	direct (N × H)
115. C₂×C₅	#1422	semidirect
131. C₅	C₂×#1422	semidirect
143. C₂	#10598001	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×C₅×C₁₆	2
C₂×#64719	2
#10598001	2
C₂×#1422	5

Subgroups — C2×#10598001 #12019790

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10598001 #12019790